Engineering Material 2000 Jan

Journal ofEngineering Materialsand TechnologyPublishe d Quarterl y by The America n Societ y of Mechanica l Engineers

VOLUME 122 • NUMBER 1 • JANUAR Y 2000

Technica l Papers1 A Simpl e Model for Stabl e Cycli c Stress-Strai n Relationshi p of Type 304 Stainles s Steel

Under Nonproportiona l LoadingTakamot o Itoh , Xu Chen, Toshimits u Nakagawa , and Masao Sakane

10 Representin g the Effec t of Crystallographi c Textur e on the Anisotropi c PerformanceBehavio r of Rolle d Aluminu m Plate

M. P. Mille r and N. R. Barton

18 Anisotropi c Nonlinea r Kinemati c Hardenin g Rule Parameter s From ReversedProportiona l Axial-Torsiona l Cycling

J. C. Moosbrugger

29 Uniaxia l Ratchettin g of 316FR Steel at Room Temperature—Par t I: ExperimentsM. Mizuno , Y. Mima, M. Abdel-Karim , and N. Ohno

35 Uniaxia l Ratchettin g of 316FR Steel at Room Temperature—Par t II: Constitutiv e Modelingand Simulation

N. Ohno and M. Abdel-Karim

42 Cycli c Deformatio n Behavio r and Dislocatio n Substructure s of Hexagona l Zircaloy-4Under Out-of-Phas e Loading

Xiao Lin

49 Model s for Cycli c Ratchettin g Plasticity—Integratio n and CalibrationMagnu s Ekh, Ander s Johansson , Hans Thorberntsson , and B. Lennar t Josefson

56 Model of Grain Deformatio n Metho d for Evaluatio n of Creep Lif e in In-ServiceComponents

Hyo-Ji n Kim and Jae-Ji n Jung

60 Modelin g of Irradiatio n Embrittlemen t of Reacto r Pressur e Vessel SteelsS. Murakami , A. Miyazaki , and M. Mizuno

67 High Strai n Extensio n of Open-Cel l FoamsN. J. Mill s and A. Gilchrist

74 Influenc e of Heat Treatmen t on the Mechanica l Propertie s and Damage Developmen t in aSiC/Ti-15-3 MMC

David A. Mille r and Dimitri s C. Lagoudas

80 Failur e of a Ductil e Adhesiv e Layer Constraine d by Hard AdherendsToru Ikeda, Akir a Yamashita , Deokb o Lee, and Noriyuk i Miyazaki

86 Interfacia l Stresse s and Void Nucleatio n in Discontinuousl y Reinforce d CompositesT. C. Tszeng

93 An Investigatio n of Yield Potential s In Superplasti c DeformationMarwan K. Khraisheh

98 Numerica l Analysi s of Weldin g Residua l Stres s and Its Verificatio n Usin g NeutronDiffractio n Measurement

Masahit o Mochizuki , Makot o Hayashi , and Toshi o Hattori

104 Effec t of Laser Shoc k Processin g (LSP) on the Fatigu e Resistanc e of an Aluminu m AlloyTang Yaxin , Zhang Yongkang , Zhang Hong , and Yu Chengye

108 Residua l Stres s Reductio n and Fatigu e Strengt h Improvemen t by Controllin g WeldingPass Sequences

Masahit o Mochizuki , Toshi o Hattori , and Kimiak i Nakakado

113 Consisten t and Minima l Springbac k Usin g a Steppe d Binde r Forc e Trajector y and NeuralNetwor k Control

Jian Cao, Brad Kinsey , and Sara A. Solla

This journa l is printe d on acid-fre e paper , whic h exceed s the ANSI Z39.48-1992 specificatio n for permanenc e of paper and librar y materials . jTM

85% recycle d content , includin g 10% post-consume r fibers.

Technical EditorDAVID L. MCDOWELL

GEORGE J. WENG(Past Technical Editor)

Materials DivisionAssociate Technical Editors

R. BATRA (2001)C. BRINSON (2002)

E. BUSSO (2001)K. CHAN (1999)

N. CHANDRA (2000)S. DATTA (2000)

G. JOHNSON (2001)J. W. JU (2000)S. MALL (2001)

D. MARQUIS (2001)S. MEGUID (2001)

A. M. RAJENDRA N (2001)G. RAVICHANDRA N (1999)

H. SEHITOGLU (2000)E. WARNER (2002)

H. ZBIB (2000)

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Aluminum Tubes,” ASME JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY,Vol. 105, pp. 242–249.

Taylor, G. I., 1938, “Plastic Strain in Metals,”J. Inst. Met.,Vol. 62, pp. 307–324.Wright, S. I., and Adams, B. L., 1990, “An Evaluation of the Single Orientation

Method for Texture Determination in Materials of Moderate Texture Strength,”Textures and Microstructures,Vol. 12, pp. 65–76.

Wright, S. I., and Kocks, U. F., 1996, “A Comparison of Different Texture AnalysisTechniques,”Proceedings of the Eleventh International Conference on Textures ofMaterials,Liang, Zuo, and Chu, eds., The Metallurgical Society, pp. 53–62.

Zouhal, N., Molinari, A., and Toth, L. S., 1996, “Elastic-Plastic Effects DuringCyclic Loading as Predicted by the Taylor-Lin Model of Polycrystal Viscoplasticity,”Int. J. Plas.,Vol. 12, No. 3, pp. 343–360.

(Contents continued)

119 Cold Compaction of Composite PowdersK. T. Kim, J. H. Cho, and J. S. Kim

129 On the Limit of Surface Integrity of Alumina by Ductile-Mode GrindingI. Zarudi and L. C. Zhang

135 Measurement of the Axial Residual Stresses Using the Initial Strain ApproachWeili Cheng

141 Effect of Plasma Cutting on the Fatigue Resistance of Fe510 D1 SteelM. Chiarelli, A. Lanciotti, and M. Sacchi

Announcement

146 New Reference Format for Transactions Journals

Journal of Engineering Materials and Technology JANUARY 2000, Vol. 122 / 17

Takamoto ItohAssistant Professor,

Department of Mechanical Engineering,Fukui University,

9-1, Bunkyo 3-chome, Fukui, 910-8507, Japane-mail: [email protected]

Xu ChenProfessor,

Department of Chemical Engng Machinery,Tianjin University,

Tianjin, 300072, P. R. China

Toshimitsu NakagawaGraduate Student.

Masao SakaneProfessor.

Department of Mechanical Engineering,Ritsumeikan University,

1-1-1, Noji-higashi, Kusatsu-shi Shiga,525-8577, Japan

A Simple Model for StableCyclic Stress-StrainRelationship of Type 304Stainless Steel UnderNonproportional LoadingThis paper proposes a simple two-surface model for cyclic incremental plasticity based oncombined Mroz and Ziegler kinematic hardening rules under nonproportional loading.The model has only seven material constants and a nonproportional factor whichdescribes the degree of additional hardening. Cyclic loading experiments with fourteenstrain paths were conducted using Type 304 stainless steel. The simulation has shown thatthe model was precise enough to calculate the stable cyclic stress-strain relationshipunder nonproportional loadings.

1 IntroductionMany practical components such as pressure vessels and turbine

blades receive nonproportional damage under the combination ofthermal and mechanical loadings. Nonproportional cyclic loadingcauses more damage than proportional loading and occasionallyreduces low cycle fatigue life. Type 304 stainless steel is known asa typical material which shows significant additional cyclic hard-ening under nonproportional loading (McDowell, 1983; Doong etal., 1990; Itoh et al., 1995). Low cycle fatigue lives under non-proportional loading were drastically reduced by the additionalhardening depending on strain history. The maximum reduction inlife occurred by a factor of 10 compared with the proportionalstraining (Socie, 1987; Itoh et al., 1995). On the other hand,aluminum alloys which showed a small additional hardening undernonproportional loading (Doong et al., 1990; Krempl and Lu,1983; Itoh et al., 1992a, 1992b, 1997) gave a small reduction in life(Doong et al., 1990; Itoh et al., 1995, 1997). The amount ofadditional hardening also depends on material. Thus, the reductionin low cycle fatigue life has a close connection with the additionalhardening depending on both loading history and material. Devel-opment of an accurate and convenient inelastic constitutive equa-tion has been needed for the quantitative estimates of additionalhardening.

Studies of plasticity have made great progress in these twodecades to express the cyclic deformation under complex mul-tiaxial loadings. Many inelastic constitutive equations were pro-posed for complex multiaxial loading (McDowell, 1985a, 1985b;Krempl and Lu, 1984; Benallal and Marquis, 1987; McDowell,1987; Chaboche and Nouailhas, 1989a, 1889b; Doong and Socie,1991; Ohno and Wang, 1991) using a relatively large number ofmaterial constants. The two-surface model has been recently re-ceiving attention since it provides the reasonable prediction fornonproportional cyclic stress-strain response (Krieg, 1975;Dafalias and Popov, 1976; Lamba and Sidebottom, 1978; McDow-

ell, 1985b; McDowell, 1987; Tseng and Lee, 1987; Ellyin and Xia,1989). One of the authors (Chen et al., 1996) also proposed thetwo-surface model superposing Mroz and Ziegler kinematic hard-ening rules to simulate the mechanical ratchetting and the nonpro-portional cyclic deformation of 2014 aluminum alloy. These manyinelastic constitutive models insist the validity of predicting theamount of additional hardening by comparing with a limitednumber of experimental results. However, there are many factorswhich influence the amount of additional hardening (Itoh et al.,1995, 1997), like phase shift, rotation and change of principalstrain direction, step length etc., so that the validity of constitutivemodel must be demonstrated by comparing the model analysiswith the nonproportional results including these factors. In thismeaning, there is no well demonstrated inelastic constitutivemodel for additional hardening prediction.

The objective of this paper is to present an inelastic constitutivemodel that can predict the stable cyclic nonproportional stressresponse. The inelastic constitutive model proposed by Chen et al.(1996) was improved by introducing the material parameter whichexpresses the material dependence of additional hardening and bythe nonproportional factor which expresses the severity of non-proportional loading. The inelastic constitutive model was appliedto the extensive nonproportional data under 14 types of strain pathsand the applicability of the model was discussed.

2 Experimental Procedure of Nonproportional Load-ing

The material tested was Type 304 stainless steel which receiveda solution treatment at 1373 K for one hour. Shape and dimensionsof the specimen were a thin-walled tube with 12 mm O.D., 9 mmI.D. and 4.6 mm gage length, as shown in Fig. 1. The test apparatuswas a tension-torsion servo-controlled electric-hydraulic machine.Axial and shear strains were measured on the specimen surface bya tension-torsion extensometer.

Multiaxial cyclic loading tests were carried out using 14types of proportional and nonproportional strain histories atroom temperature. Figure 2 shows the 14 strain histories, wheree and g are the axial and shear strains, respectively, and

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials Division July27, 1998; revised manuscript received February 5, 1999. Associate Technical Editor:H. Sehitoglu.

Journal of Engineering Materials and Technology JANUARY 2000, Vol. 122 / 1Copyright © 2000 by ASME

alphabets attached indicate representative points along thestrain path which will be referred in the later discussion. Case0 is a push-pull test which is the basic test for predicting thenonproportional cyclic stress and strain response. Strain pathsshown in the figure were determined so as to make clear thevarious effects in nonproportional straining. In strain paths1–13, the total axial strain range,De, had the same strainmagnitude as the total shear strain range,Dg, on Mises’ equiv-alent basis. Two total strain ranges were employed in all thetests except Case 0. They wereDe 5 0.5% and 0.8%. In Case0, the total strain range was varied from 0.5% to 1.5%. Thestrain rate was 0.1%/s on Mises’ equivalent base. A detaileddescription of the test procedure was presented in the previouspaper (Itoh et al., 1995) together with the nonproportional lowcycle fatigue lives.

3 Nonproportional Low Cycle Fatigue Strain Parame-ter

Nonproportional loading drastically reduces the low cyclefatigue life accompanied with additional hardening (McDowell,1983; Doong et al., 1990; Itoh et al., 1995, 1997). The reductionin fatigue life is closely related to the additional hardeningunder nonproportional loading, depending on the loading his-tory and material (Itoh et al., 1995, 1997). The authors proposedthe nonproportional strain parameter for predicting the lowcycle fatigue lives of Type 304 stainless steel and 6061 alumi-num alloy. The proposed parameter is expressed as (Itoh et al.,1995, 1997)

DeNP 5 ~1 1 afNP!De I (1)

whereDe I is the maximum principle strain range under nonpro-portional straining,a the material constant which discriminates thematerial dependency of additional hardening, andfNP the nonpro-portional factor which expresses the severity of nonproportionalloading.

The value ofa is defined as the ratio of stress amplitudeunder 90 degree out-of-phase loading (circular strain path in

g/=3 2 e plot) to that under proportional loading. The 90degree out-of-phase loading shows the maximum additionalhardening among all the nonproportional histories (Socie, 1987;Itoh et al., 1995). For Type 304 stainless steel, the stressamplitude under 90 degree out-of-phase loading was increasedup to 90% in comparison with the proportional loading, so thevalue of a takes 0.9.

The nonproportional factor which accounts for the severity ofnonproportional strain is calculated from strain history and isdefined as

fNP 5p

2Te ImaxE

0

T

~usin j~t!ue I~t!!dt (2)

wheree I(t) andj(t) are the absolute value of maximum principalstrain and the angle of maximum principal strain direction at time

Nomenc la tu re

E 5 Young’s modulus for elasticityG 5 shear modulusc 5 strain hardening modulusn 5 cyclic hardening exponentK 5 cyclic hardening coefficient

KNP 5 nonproportional cyclic hardeningcoefficient

Cs 5 material constanta 5 material constant for additional

hardening

fNP 5 nonproportional factore, g 5 axial and shear strains, respec-

tivelyF 5 function of yield surface

F* 5 function of limit surfaceR 5 radius of yield surface

R* 5 radius of limit surfaceXmax 5 maximum back stress

s1, s3 5 effective axial and shearstresses, respectively

e1, e3 5 effective axial and shear strains,respectively

n1, n3 5 orthogonal vectors in the stressspace

X 5 vector pointing the center ofyield surface

s 5 stress vectors* 5 stress vector on the limit surface

e 5 strain vectordeP 5 vector of plastic strain incre-

ment

Fig. 1 Shape and dimensions of the specimen tested (mm)

Fig. 2 Proportional and nonproportional strain paths

2 / Vol. 122, JANUARY 2000 Transactions of the ASME

t, respectively. The former parameter shows the strain amplitudeand the latter the principal strain direction change under nonpro-portional loading. In the equation,T and e Imax are the time for acycle and the maximum value ofe I(t) in a cycle. Detailed descrip-tion of the nonproportional low cycle fatigue parameter in Eq. (1)is omitted here and the reader is referred to the previous paper(Itoh et al., 1995) for details. The values offNP for each case arelisted in Table 1.

The reason for makingf NP an integral form is that the exper-imental results indicated that nonproportional low cycle fatiguelives are significantly influenced by the degree of principalstrain direction change and strain length after the directionchange. The value off NP takes zero under proportional strainingsuch as Case 0 and 5 in which no change of principal straindirection is occurred.f NP takes the maximum value of 1 underthe 90 degree out-of-phase straining which has the continuouschange of principal strain direction with the constant equivalentstrain corresponding toe Imax, which results in the largest addi-tional hardening and the largest reduction in the low cyclefatigue life (Socie, 1987; Itoh et al., 1995). However, when amaterial has a small value ofa, a small additional hardeningand a small reduction in fatigue life are occurred even thoughthe value off NP is large (Itoh et al., 1997). Therefore, reductionin nonproportional low cycle fatigue is connected with thedegree of additional hardening (Socie, 1987) depending on boththe loading history and materials. The material dependency ofthe additional hardening can be explained by the mechanism ofslip systems.

In nonproportional loading, the principal strain direction ischanged with proceeding cycles, so the maximum shear stressplane is changed continuously in a cycle. This causes an interac-tion between slip systems and which results in the formation ofsmall cells (Doong et al., 1990; Itoh et al., 1992a, 1992b; Kida etal., 1997) for Type 304 stainless steel. Large additional hardeningoccurred by the interaction of slip systems for that steel because oflow stacking fault energy. 6061 aluminum alloy, on the other hand,is a material of high stacking fault energy and slips of dislocationsare wavy. No large interaction occurred in 6061 aluminum alloysince dislocations change their glide planes easily following thevariation of the maximum principal strain direction (Itoh et al.,1992a, 1992b; Kida et al., 1997).

The term, (11 afNP), in Eq. (1) reflects the intensity of theinteraction between slip systems which depends on both the ma-terial and strain history (Kida et al., 1997). The equivalent strainincluding (11 afNP) in Eq. (1) gave a satisfactory correlation offatigue lives under nonproportional straining, and predicted theadditional hardening due to nonproportional loading (Itoh et al.,1995, 1997). Therefore, it is considered that the term of (11afNP) becomes a good parameter to evaluate the effect of loadinghistory and of material dependency on stress-strain relationshipunder complex nonproportional loadings. In the following, theinelastic constitutive equation which includes (11 afNP) isdeveloped.

4 Two-surface Plasticity Model Based on KinematicHardening Rule

When a thin-walled tube specimen is subjected to combinedaxial and torsional loading, the stress and strain state is expressedin terms of deviatoric vector planes. The definition of the axial-torsional subspace follows as an Ilyushin’s five-dimensional de-viatoric vector subspace and a stress vector is defined as

s 5 s1n1 1 s3n3 (3)

In this equation,s1 and s3 are the effective axial and shearstresses, respectively. Based on the Mises’ equivalent values, s1

ands3 can be given bys and=3t, wheres andt are the axial andtorsional shear stresses, respectively.n1 andn3 are the orthogonalbase vectors in the stress space.

The strain vector is defined as

e 5 e1n1 1 e3n3 (4)

Similarly, e1 is equal to the axial strain ande3 is equal to the shearstraing/=3.

Mises’ equivalent stress and strain are expressed by the follow-ing equations usings1, s3, e1, ande3.

se 5 Îs 12 1 s 3

2

ee 5 Îe 12 1 e 3

2 (5)

The total strain increment,de, is assumed to be decomposed intothe elastic and plastic strain increments,dee anddep.

de 5 de e 1 de p (6)

where the elastic strain components are given by

de 1e 5

ds1

E, de 3

e 5ds3

3G(7)

E andG are Young’s and shear moduli, respectively.Classical plasticity theory is based on the concept of the yield

surface. The movement of the yield surface under cyclic loading isdescribed by the kinematic hardening rule and the dimensionalchange of the yield surface is described by the isotropic hardeningrule. Mises’ yield surface is expressed in the stress space as

F 5 ~s 2 X ! z ~s 2 X ! 2 R2 5 0 (8)

Fig. 3 Two-surface model for superposing the two kinematic hardeningrules

Table 1 Values of the nonproportional factor


whereX is a vector indicating the center of yield surface andR isa radius of yield surface.

The limit surface,F*, in the stress space is required in thetwo-surface model which is given as,

F* 5 s* z s* 2 ~R* ! 2 5 0 (9)

where s* is a limit stress vector andR* is the radius of limitsurface equated with the radius of yield surface and the maximumback stress,Xmax.

R* 5 uXmaxu 1 R (10)

In order to describe the Bauschinger effect, a simple kinematichardening rule was proposed by Ziegler (1959) given by the nextequation:

dX 5 dmZ~s 2 X !. (11)

On the other hand, Mroz (1969) proposed a kinematic hardeningrule based on the concept of the nested yield and the loadingsurface,

dX 5 dmm~s*m 2 s!. (12)

For the two-surface model,s*m is the stress vector on the limitsurface which has the same direction as the normal vector,n, at thestressing point,s, on the yield surface, i.e.,s*m ands are locatedat similar points on the yield and limit surfaces, respectively.s*mis defined as shown in Fig. 3:

s*m 5R*

R~s 2 X !. (13)

In this study, a new superposition of the kinematic hardeningrules, which is proposed by Chen and Abel (1996), is employed. Itis expressed as

Table 2 Material constants used in the analysis

Fig. 4(a) Case 0 Fig. 4(b) Case 4

Fig. 4(c) Case 7 Fig. 4(d ) Case 10

Fig. 4 Stable cyclic stress-strain relationship for Case 0, 4, 7, and 10


dX 5 dms@~s*m 2 s! 1 Cs~s 2 X !# (14)

This equation shows that the movement of yield surface undercyclic loading complies with the direction determined by thecombination of Ziegler and Mroz hardening rules. The constant,Cs, is a function of plastic strain accumulation for transientplasticity and ratchetting. TakingCs 5 0 in Eq. (14), theproposed hardening rule corresponds with Mroz hardening ruledefined in Eq. (12). By the consistency conditiondF 5 0, dm s

is given by

dms 5~s 2 X ! z ds 2 R z dR

~s 2 X ! z ~s*m 2 s! 1 CsR2 (15)

For stable cyclic stress-strain relations, the radius of yield surfacekeeps the constant value, i.e.,dR 5 0. Therefore, the movementdirection of yield surface can be expressed as

ns 5dmnm 1 CsRnz

udmnm 1 CsRnzu

nz 5s 2 X

us 2 X u , nm 5s*m 2 s

us*m 2 su ,

dm 5 Î~s*m 2 s! z ~s*m 2 s! (16)

The plastic flow rule is given by Eq. (17) fordF 5 0 and (s 2X) z ds . 0.

de p 51

cR2 @~s 2 X ! z ds#~s 2 X ! (17)

where c denotes the strain hardening modulus.dep is collinearwith (s 2 X) for Mises’ case. For Massing type materials, theplastic modulus of the stress-plastic strain curve (Ramberg-Osgood equation) can be expressed as (Ellyin and Xia, 1989; Leeet al., 1995)

c 5 KnSse

K D ~n21!/n

. (18)

There are many ways to define the strain hardening modulus,c, under nonproportional loading (McDowell, 1985b). In themodel of this paper, the strain-hardening modulus is assumed tobe a function of the distance,d s, between the current stresspoint on the yield surface,s, and the stress point on limitsurface,s*s, i.e.,

Fig. 5 Stable cyclic stress response for Case 1–13 at De 5 0.8% Fig. 5 (Continued)


ds 5 2s z ns 1 @~s z ns!2 1 ~R* ! 2 2 usu 2# 1/2 (19)

The maximum distance is

dmax 5 2~R* 2 R!. (20)

Let D be normalized distance betweend s anddmax.

D 5dmax 2 ds

dmax(21)

The normalizedD varies between zero and one in value.n andK which characterize the stable cyclic stress-strain must

be modified to take account of the additional hardening. This studyemployed a simple idea,

KNP 5 ~1 1 afNP!K (22)

Therefore, the cyclic stable plastic modulus for nonproportionalloadings can be expressed by

c 5 ~1 1 afNP!KnS u2~R* 2 R!~D 2 1! 1 R* u~1 1 afNP!K

D ~n21!/n

. (23)

5 Analytical Results and DiscussionMaterial constants used for the analysis are listed in Table 2;

Young’s modulus is 210 GPa, shear modulus 82 GPa, radius ofthe yield surface 200 MPa, cyclic hardening coefficient 1850MPa, cyclic hardening exponent 0.29 and the material constantfor additional hardening 0.9 (Itoh et al., 1995).

The material constantCs, which is normally a function of plasticstrain accumulated for describing the transient plasticity and ratch-etting, must be determined. Under cyclic stable state, however, noor small effect ofCs was observed on the peak stress in stress-strain relationship. Since the stable stress-strain relationship wouldbe studied andCs is mainly related to the ratchetting rate (Chen etal., 1996),Cs can be taken as constant. Thus, this study takes thevalue ofCs as unity.

Figures 4(a)–(d) compare the stress-strain relationship betweenanalysis and experiment for Case 0 atDe 5 1.5% and Case 4, 7and 10 atDe 5 0.8%. In these figures, alphabets indicate therepresentative points along the strain path, Fig. 2, and the stress-strain curves, Figs. 4 and 5. In the uniaxial test of Case 0, Fig. 4(a),the proposed model predicts the stable cyclic hysteresis loop withsatisfactory accuracy. Overall fitting between the analysis andexperiment is excellent but there is a subtle difference during theunloading stage. The analytical result has a linear shape but theexperimental result has a curved shape. Since the unloading stress-

Fig. 5 (Continued) Fig. 5 (Continued)


strain relationship must be a linear shape, the difference is pre-sumably resulted from the inaccuracy in experiment.

In Case 4, Fig. 4(b), the model developed in this study predictswell the experimental stress response. Especially, the peak stressesof predicted at A;D agree well with these in experiment. Theshape of the stress-strain curves in the analysis is somewhatdifferent from that in experiment along the strain path, D-A andB-C in the axial component and that of A-B and C-D in the shearcomponent. The model estimates the larger stress amplitude alongthese strain paths.

In Case 7, Fig. 4(c), the overall fitting of the stress-strainrelationship between the prediction and experiment is good. Themodel has satisfactory accuracy for the axial component but thereexists a relatively large difference in the shear component alongthe strain paths D-E and C9-B9. The peak stresses derived by themodel precisely agree with those in experiment whereas the stress-strain curve shape differs between the model and experiment.

In Case 10, Fig. 4(d), the model estimates very close hysteresisloop to the experimental result in the axial component but itpredicts slightly larger stress response than that in experiment forthe shear component.

From the comparison of the stress response between the analysisand experiment shown in the above cases together with the otherstrain paths which were not graphically presented, the overallfitting of the stress response is good between the model andexperiment. The axial component in the analysis quite well agreedwith those in experiment but there was a small difference in the

shear component. The model predicted larger stress amplitude thanthe experiment for shear component. One of the possible causes ofthe larger prediction is resulted from the stress/strain gradient intothe tube wall of specimen and the short gage length. The gage partof the specimen has 1.5 mm in thickness and 6.4 mm in length. The

Fig. 5 (Continued)

Fig. 5 (Continued)

Fig. 5 (Continued)


thickness is slightly thicker and the gage length somewhat smallerfor the plasticity study. However, specimens were prone to buckleunder the severe nonproportional strain paths shown in Fig. 2 sothat this specimen geometry was necessary. The similar trend ofthe data fitting in the shear components was also found in theliterature (Socie, 1987).

Figure 5 compares the stress response in axial-shear stressdiagram atDe 5 0.8%. In the figures of Case 8–12, simulatedresults calculated witha 5 0 and fNP 5 0 are shown by dottedlines to be examine the effect ofa on the stress response. Incruciform strain paths, Case 1 and 2 in Fig. 5, the model predictsthe very close stress-strain shape and estimates the accurate peakstresses at the points of A;D. However, there are some pointswhere the model gives the different stress response. Cases 3 and 4have the hysteresis shape rotated about 45 degrees from that inCases 1 and 2, respectively.

Cases 5–7 have a similar path with the different number ofprincipal strain direction change steps. The model gives a goodagreement with the experimental results quantitatively at the re-spective strain path. The model follows the trend of the experi-mental results that the stress amplitude increases as the number ofsteps decreases.

For the single step and box loading of Cases 8 –13, the modelgives an accurate prediction for a simple boxy loading of Cases10 and 12. The model also gives a good prediction for a doublebox loading in Case 13. However, a small difference betweenthe prediction and experiment exists in Case 9 for axial com-ponent at C and shear component at A. This trend is also foundin Case 8 which is a similar strain path as Case 9. In the loading

path with phase shift, Case 11, the model predicts the largerstress amplitude along D-A and B-C. For the case ofa 5 0, onthe other hand, there is a large difference in stress between theanalysis and experiment shown by the dotted lines for Cases8 –12. These figures clearly show that the large contribution ofa andf NP exists in the prediction of stable cyclic stress responseunder nonproportional loading.

In conclusion, the model evaluates a relatively accurate stressresponse under nonproportional loading except a few cases. Thestress amplitude calculated by the proposed model agrees wellwith the experimental results for a wide variety of strain pathswhereas a small difference partly exists. The overall fitting isconcluded to be satisfactory.

Figure 6 shows the comparison of stress range between analysisand experiment. The axial stresses calculated by the model arewithin a factor of 10% scatter band in comparison with experi-mental data in all the tests, Fig. 6(a). The scatter of data is slightlylarger in shear component and the scatter band is a factor of 1.2,Fig. 6(b). The larger scatter in the shear component may beresulted from the geometry of the specimen.

Figure 7(a) is a replot of Fig. 6 and shows the comparison ofthe equivalent stress between analysis and experiment. Thefigure shows that the simulation estimates the equivalent peakstress within a factor of 1.2. In Fig. 7(b), where the calculationwas carried out undera 5 0 and f NP 5 0, some of the dataunder nonproportional loading are obviously underestimated by

Fig. 6(a) Axial stress range

Fig. 6(b) Shear stress range

Fig. 6 Comparison of axial and shear stress ranges between the anal-ysis and experiment

Fig. 7(a) a 5 0.9

Fig. 7(b) a, fNP 5 0

Fig. 7 Comparison of the calculated and experimental effective stressamplitudes


more than a factor of 1.2. The larger scatter of the data can beseen under the largerf NP’s tests, such as Cases 10 and 12. Themaximum scatter was almost a factor of 2 which correspondswith the value ofa for Type 304 stainless steel used in thisstudy. A contribution of (11 af NP) to the simulation is verylarge and the simple model proposed has enough precision tocalculate cyclic stable stress-strain relationship along widelyranged nonproportional straining.

6 Conclusions

1. This study proposed a simple two-surface plasticity modelwhich has only seven material constants. This model isbased on a superposed Mroz’s and Ziegler’s kinematichardening rules for describing the stress and strain relation-ship under nonproportional strain loadings.

2. The term of (11 afNP) which was proposed for thenonproportional low cycle fatigue life prediction was suc-cessfully incorporated into the plasticity model to describethe additional hardening quantitatively.a and fNP are theconstants related to the material dependency of additionalhardening and the nonproportional factor.

3. The proposed model simulated the stable cyclic stress-strainresponse under 14 types of proportional and nonpropor-tional strain histories with a satisfactory accuracy. Themodel also predicts the stress amplitude of all the testswithin a 20% scatter band.

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Applied Mechanics,Vol. 7, pp. 55–56.


M. P. Miller

N. R. Barton

Sibley School of Mechanical andAerospace Engineering,

Cornell University,Ithaca, NY 14853

Representing the Effect ofCrystallographic Texture on theAnisotropic PerformanceBehavior of Rolled AluminumPlateRolled aluminum alloys are known to be anisotropic due to their processing histories. Thispaper focuses on measuring and modeling monotonic and cyclic strength anisotropies as wellas the associated anisotropy of the elastic/elastic-plastic transition of a commercially-available rolled plate product. Monotonic tension tests were conducted on specimens in therolling plane of 25.4 mm thick AA 7075-T6 plate taken at various angles to the rollingdirection (RD). Fully-reversed tension/compression cyclic experiments were also conducted.As expected, we found significant anisotropy in the back-extrapolated yield strength. We alsofound that the character of the elastic/elastic-plastic transition (knee of the curve) to bedependent on the orientation of the loading axis. The tests performed in RD and TD(transverse direction) had relatively sharp transitions compared to the test data from otherorientations. We found the cyclic response of the material to reflect the monotonic anisotropy.The material response reached cyclic stability in 10 cycles or less with very little cyclichardening or softening observed. For this reason, we focussed our modeling effort onpredicting the monotonic response. Reckoning that the primary source of anisotropy in therolled plate is the processing-induced crystallographic texture, we employed theexperimentally-measured texture of the undeformed plate material in continuum slip poly-crystal plasticity model simulations of the monotonic experiments. Three types of simulationswere conducted, upper and lower bound analyses and a finite element calculation thatassociates an element with each crystal in the aggregate. We found that all three analysespredicted anisotropy of the back-extrapolated yield strength and post-yield behavior withvarying degrees of success in correlating the experimental data. In general, the upper andlower bound models predicted larger and smaller differences in the back-extrapolated yieldstrength, respectively, than was observed in the data. The finite element results resembledthose of the upper bound when initially cubic elements were employed. We found that byemploying an element shape that was more consistent with typical rolling microstructure, wewere able to improve the finite element prediction significantly. The anisotropy of theelastic/elastic-plastic transition predicted by each model was also different in character. Thelower bound predicted sharper transitions than the upper bound model, capturing the shapeof the knee for the RD and TD data but failing to capture the other orientations. In contrast,the upper bound model predicted relatively long transitions for all orientations. As with theupper bound, the FEM calculation predicted gentle transitions with less transition anisotropypredicted than that of the upper bound.

1 IntroductionCrystallographic texture (hereafter referred to as texture)

evolves during the processing of metals and alloys. Large inelasticdeformations associated with metal forming tend to produce char-acteristic textures, as do recrystallization processes associated withhot forming and thermal processing. Characterizing texture andlinking it to processing practices forms a main element of alloydevelopment. Often, a chief processing goal is to produce uniform,isotropic alloys. The reality, however, is that rolling producesalloys with varying degrees of heterogeneity and anisotropy. Inthin plate material primarily designed to be loaded in-plane, auniaxial yield strength that varies with the angle of the tensile axisto the rolling direction is a manifestation of the processing-induced

anisotropy. A varying material response through the thickness ofthe plate is a result of rolling-induced heterogeneities. The physicalsources of the anisotropy depend on the processing schedule andalloy. While dislocation structures can contribute to anisotropy incold-rolled materials (cf. Juul Jensen and Hansen, 1990; Hansenand Juul Jensen, 1992) as can the distribution of precipitates insome heat-treatable aluminum alloys (cf. Hosford and Zeisloft,1972; Bate et al., 1981; Barlat et al., 1996), it is well accepted thattexture is the primary source of anisotropy in rolled aluminumplate. Mechanical design with processed alloys, therefore, requiresone to link the texture to properties and performance. In thisprocess, one seeks to understand what constitutes a “good” or“bad” texture for a particular application. This is most oftenaccomplished by recharacterizing the behavior of the alloy when aprocessing change is instituted. Simulations that predict the per-formance of an alloy based on its processing-induced internalstructure, including texture, can save time and resources. While asignificant amount of research has focused on simulatingprocessing-induced texture evolution to aid alloy development,

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionFebruary 22, 1999; revised manuscript received June 15, 1999. Associate TechnicalEditor: E. Werner.

10 / Vol. 122, JANUARY 2000 Transactions of the ASMECopyright © 2000 by ASME

simulations of alloy performance under small inelastic strain con-ditions, such as those encountered during cyclic loading, thatexplicitly represent the plastic slip anisotropy associated withtexture are conspicuously absent from the literature. Employing ananalytic, anisotropic yield surface may enable one to captureorientation-dependent differences in the back-extrapolated yieldstrength, but cannot easily capture orientation-dependent differ-ences in the elastic/elastic-plastic transition.

The focus of this paper is the small strain, monotonic and cyclicresponse of a rolled aluminum alloy, AA 7075-T6. Monotonictension tests were conducted on specimens in the rolling planetaken at various angles to the rolling direction (RD). Fully-reversed, tension/compression, cyclic experiments were also per-formed using the same specimen type. The experimentally-measured texture of the undeformed material and a continuum slippolycrystal plasticity model were employed to simulate the mono-tonic experiments. Clearly, 7075-T6 is a metallurgically compli-cated alloy and texture is only one of the relevant structuralfeatures affecting its behavior. Crystallographic slip is the promi-nent inelastic deformation mechanism in the material and texture isthe major source of the experimentally-observed elastic-plasticmaterial behavior anisotropy, however, so our goal here is toinvestigate the utility of a model that explicitly represents theeffects of the orientations of the crystals in an aggregate forpredicting the anisotropic response of the alloy.

In the following section we present a brief review of the relevantliterature related to measuring and modeling the anisotropy ofrolled plate. We also explore the use of continuum slip models forsimulating small strain elastic-plastic deformations. In Section 3we present our experimental methods and results. Section 4 con-tains the model development and the simulation results. Finally,we close with discussion and conclusion sections. In this paper,bold type indicates a tensor quantity of orders one through four.Operations are as follows:

~a ^ b! ij 5 ai bj

~A z B! ij 5 Aik Bkj

A :B 5 Aij Bij

~+:A ! ij 5 + ijkl Akl

iA i 5 ÏA :A

2 Background

2.1 Aluminum Plate: Processing and Texture. Rolled AA7075 (Al Zn Mg Cu) plate is employed in a broad array ofengineering applications, including critical airframe componentsthat can experience deleterious repeated loading conditions. It iswell-established that material property anisotropies arise in thismaterial due, primarily, to the processing-induced crystallographictexture (cf. Hatch, 1984). Yield strength, fracture toughness, duc-tility, and fatigue resistance can vary significantly as the specimenloading axis is varied between the rolling direction (RD), thedirection transverse to the rolling direction in the plane of the plate(TD) and the direction normal to the plane of the plate (ND). Whilealloy development practices such as tailoring the volume fractionand distribution of strengthening precipitates can somewhat miti-gate the strength anisotropy in some alloys (cf. Hosford andZeisloft, 1972; Bate et al., 1981; Barlat et al., 1996), the factremains that designing with an alloy such as the plate used in thisstudy requires one to understand the anisotropy of its properties.

The final texture of the plate material is a product of its pro-cessing history. For heat-treatable alloys such as AA 7075, thenominal processing of a cast ingot begins with a sequence ofhot-rolling and annealing episodes. Depending on the temper, thissequence may include some early cross rolling passes where theaxis of the ingot is rotated 90°. Once the final thickness is attained

the plate is typically stretched to normalize any residual strains,then annealed and aged to bring out the precipitate structure fromwhich this material derives its strength. Certainly, small variationsin the processing schedule of the alloy can make large changes inperformance. We don’t know the history of the material investi-gated in this paper and present the above processing schedule as ageneric one for 7XXX alloys.

2.2 Models. Modeling the elastic-plastic anisotropic behav-ior of textured alloys on the macroscale often involves an analyt-ical yield surface or flow potential, which is employed to captureanisotropy of the initial or back-extrapolated yield strength (cf.Hill, 1948; Stout et al., 1983; Harvey, 1985; Barlat, 1987; Lin etal., 1993; Karafillis and Boyce, 1993). Continuum slip polycrystalmodels are alternatives. As we will show, a polycrystal model canpredict anisotropy of the elastic/elastic-plastic transition as well asinitial yield.

The philosophy of a continuum slip model is to formulate theproblem at the slip system level and to use the geometry of thecrystal class to construct a “grain.” Inelastic deformation occurs asthe result of net dislocation motion, though the formulation is stilla continuum one in that individual dislocations are not treatedexplicitly. The polycrystal response is attained by assembling anumber of grains using an intergranular constraint or mean fieldassumption (cf. Sachs, 1928; Taylor, 1938; Iwakuma and Nemat-Nasser, 1984; Asaro and Needleman, 1985; Kocks, 1987; Molinariet al., 1987; Mathur and Dawson, 1989; Lipinski and Berveiller,1989; Mathur et al., 1990; Cailletaud, 1992; Lebensohn and Tome,1994; Zouhal et al., 1996; Feyel et al., 1997; Molinari et al., 1997),or by assigning a finite element to each grain (cf. Harren andAsaro, 1989; McHugh et al., 1989; Havlicek et al., 1992; Dawsonet al., 1994; Beaudoin et al., 1995a) or subgrain (Beaudoin et al.,1995b; Mika and Dawson, 1998, 1999).

While these models have predominantly been used for simulat-ing texture evolution during large strain deformation episodes,elastic-plastic versions have been employed to simulate materialresponse during small strain cyclic deformation histories (cf. Cail-letaud, 1992; Zouhal et al., 1996; Feyel et al., 1997; Molinari et al.,1997). Rather than examining cyclic plasticity in textured alloys,however, these works have generally focused on developing slipsystem hardening formulations to model complicated multiaxialexperiments on nontextured polycrystals. It has also been estab-lished that elastic-plastic continuum slip models predict a strain-induced yield strength asymmetry or Bauschinger effect (Czyak etal., 1961; Hutchinson, 1964a, b; Zouhal et al., 1996; Barton et al.,1999) and a texture-dependent elastic/elastic-plastic transition(Barton et al., 1999). We are defining the elastic/elastic-plastictransition or knee of the curve to be the region of the stress-strainresponse where the elastic and plastic strain rates are comparablein magnitude.

Barton et al. (1999) showed that the contribution to the transi-tion behavior from the continuum slip model arises from theinteraction of the texture, the linking assumption, and the initialresidual stress state. Each crystal in the aggregate contributes to themacroscopic knee through the activation of its individual slipsystems, which depend on the slip system strength and the orien-tation of the slip systems relative to the loading axes, and themeans by which the macroscopic conditions are related to thecrystal (linking assumption). For example, in the case of the upperbound linking assumption, each crystal experiences the macro-scopic strain rate, producing a range of crystal stresses whosebreadth depends on the orientation distribution of the constituentcrystals. The duration of the transition or “knee length,” therefore,is determined by the orientation distribution function (ODF) andinitial stress state and depends on the amount of strain necessaryfor each crystal to achieve a stress state that will result in fullydeveloped plastic flow. Single crystal knee lengths are often sev-eral times longer than the strain for initial yield, as initial yieldoccurs when the first slip system becomes active and substantiallymore strain may be required to activate enough slip systems for


fully developed plasticity. Conversely, the lower bound linkingassumption assigns the macroscopic stress to each crystal. Thusthere is not a similar successive activation of slip systems inindividual crystals. An abrupt transition results as the slip systemsin the weakest crystals in the aggregate yield. A finite elementformulation that assigns each element an individual crystal orien-tation, predicts a knee duration that was closer to the upper boundin appearance, though complicated somewhat by neighborhoodinfluences. Finally, Barton et al. (1999) showed that a difference inknee length on reloading and reverse loading after prestrain arisesnaturally from the influence of the residual stress state. It is alsoworth noting (and we will show later) that the contributions to theknee from the continuum slip model dominate any slip systemhardening effects over a comparable strain range for typical hard-ening models with parameters intended to capture behavior forstrains on the order of 0.05 to unity.

3 Experimental Methods and ResultsThe aluminum alloy investigated was AA 7075-T6 25.4 mm

thick plate. We measured the texture of the plate at its centerplane,producing pole figures using X-ray diffraction. From the polefigure data, we determined the orientation distribution function(ODF) using the popLA (preferred orientation package-LosAlamos) software (Kallend et al., 1991). The ODF is shown in Fig.1 on slices taken normal to the ND direction through the Rodriguesfundamental region of orientation space. Envisioning the rotationassociated with a lattice orientation when employing an angle-axisdescription is extremely straight-forward. The rotation axis isdescribed by a vector, whose magnitude is related to the angle ofrotation. These parameterizations of orientation space do not pro-duce the highly-distorted metric tensor and singularities associatedwith Euler angles (Frank, 1988; Becker and Panchanadeeswaran,1989). A point in the fundamental region (a truncated cube forcrystals with cubic symmetry) is an orientation, which, employingthe Rodrigues parameterization, is specified relative to the RD,TD, ND frame by a rotation of angleb about an axis,n, by thevector

r 5 n tanSb

2D (1)

The texture, depicted in Fig. 1, is somewhat typical for rolled, heattreated plate of this gauge. From the ODF, we randomly selected

a sufficient number of representative orientations, which we em-ploy in the simulations described later.

We machined cylindrical specimens with a gage length of 19mm and a diameter of 6.4 mm from the centerplane of the plate atvarious angles,f, to the rolling direction. All mechanical testswere performed on a servo-hydraulic test machine operating intrue strain control employing an axial extensometer with a 12.75mm gage length. The true strain rate for all experiments was 1023

s21. We first conducted tensile tests. The data from these experi-ments are shown in Fig. 2. From Fig. 2(a), we see that beyond theelastic/elastic-plastic transition, the stress strain curves attain aroughly constant slope with the nearly identical curves from theRD and TD tests having the highest stress value at the same strainand the data from thef 5 65° experiment having the lowest value,just slightly lower than thef 5 45° data. Figure 2(b) zooms in onthe knee in the curve of the data. We see that the transitionresponse also is anisotropic, with RD and TD having abrupttransitions compared to the data from other directions. The aniso-tropic nature of the transitions is best seen in Fig. 3, which depictsthe slope of the stress-strain curves,U, as a function of accumu-lated strain. Each monotonic experiment was repeated at leastonce, with stresses at the same strain differing less than 1% fromthose depicted in the figure, in general.

We also conducted fully-reversed cyclic experiments using RD,TD, and f 5 45° specimens at strain amplitudes (e a) of 0.01,0.0125, 0.015, 0.0175, and 0.02. Figure 4 depicts the stress-strainresponse of the first 5 cycles of the RD experiment fore a 5 0.01.As can be seen, the material rapidly attained cyclic stability,exhibiting no significant difference between the magnitude of the

Fig. 2 (a) The stress strain data from the tensile tests conducted on the7075 plate. The angle, f, is the angle between the specimen axis and therolling direction. ( b) A close-up of data focusing on the elastic/elastic-plastic transition.

Fig. 1 The ODF at the 12 plane of the 7075 plate depicted on slices

through the fundamental region of orientation space parameterized withthe Rodrigues vector defined in Eq. (1). The slices were taken normal tothe plate normal direction. An ODF value of 2.42 corresponds to arandom texture.


peak tensile and compressive stress levels. The rapid attainment ofcyclic stability was characteristic of all experiments. Figure 5shows a plot of strain amplitude versus stable stress amplitude(s a), commonly referred to as the cyclic stress-strain curve(Mitchell, 1978). Again, the anisotropy of the cyclic response isevident with the 45° data falling well below the RD and TD data.In contrast to the uniaxial data, we see a slight difference in the RDand TD stress amplitudes.

4 Model Development and Simulation ResultsThe model we employ is an elastic-viscoplastic continuum slip

polycrystal model described generically in Section 2. For thedetails of the model development, the reader is referred to Marinand Dawson (1998) and Barton et al. (1999). We briefly discussthe issues most germane to the present work below.

4.1 Continuum Slip Model. The single crystal behavior ismultiplicatively decomposed into elastic and viscoplastic compo-nents with the constitutive relations formulated in an intermediateconfiguration. The shearing rates on all active slip systems com-bine to determine the inelastic rate of deformation and spin foreach crystal. A rate-dependent flow rule formulated on the slipsystem level is employed, i.e.,

g ~a! 5 go sgn~t ~a!!U t ~a!

g ~a!U1/mf

(2)

whereg o is a reference strain rate,mf is the rate sensitivity of theflow stress andg(a) is the slip system yield strength or hardness.For this work we employ a single hardness for each crystal (Millerand Dawson, 1997) with the hardness of theI th crystal evolving as

g~I ! 5 hoF1 2 Sg~I !

gsDG O

a51

n

ug ~a!u (3)

whereho andgs are material parameters.The internal variables of this model are the elastic strain tensor,

the slip system hardnesses, and the crystal orientations. The systemof equations is highly coupled and while the hardnesses evolveaccording to Eq. (2), the evolution of the elastic strains and thelattice orientations are embedded within the kinematic decompo-sition and constitutive relations (Barton et al., 1999).

We employ two methods for combining the behavior of manysingle crystals into the polycrystalline response we seek. In thefirst, we use an aggregate of crystals to model material behavior ata continuum point employing a mean field assumption to relate thecrystal quantities to their counterparts on the macroscale. A vol-ume average is used to obtain macroscale quantities. We employthe notation^[& to denote the volume average of[ over allcrystals in the aggregate so that^s& and^L & represent the volumeaverage of the Cauchy stress and velocity gradient, respectively.We conducted simulations employing either the upper bound orextended Taylor assumption,L 5 ^L &, or the lower bound assump-tion, s 5 ^s&. These are material point calculations employing nospatial discretization. The upper and lower bound simulations aredeformation-driven and are initiated by imposing a velocity gra-dient,^L &, on the aggregate of crystals. We seek to model materialbehavior during a tensile test, however, which involves a constantaxial straining rate and a stress-free condition on the lateral bound-ary of the specimen. To address this situation, we use a Newtonmethod to perturb and updateL & until the conditions on thespecified components of^s& are satisfied, effectively enabling usto conduct the material point simulation with mixed “boundaryconditions.” The other calculation employs a finite element for-mulation with a single crystal per element so that element level(macroscopic) and crystal stress and deformation quantities areidentical (Marin and Dawson, 1998). Equilibrium is satisfied in theusual finite element weak sense and compatibility is satisfiedidentically by the chosen element shape functions. The elementsare initially cubes with continuous linear velocity interpolation anddiscontinuous piecewise constant pressure interpolation. While acubic shape is clearly not representative of a typically alloy mi-crostructure, the initial focus here is on a realistic andmechanically-sound representation of the intergrain mechanicalinteractions. We will see that these interactions play a big role indetermining the character of the elastic/elastic-plastic transition ofthe aggregate. We will investigate the role of element shape in thediscussion section of the paper.

Fig. 4 The hysteresis loops for the first 5 cycles of the fully reversedcyclic experiment conducted on the 7075 plate in the rolling direction ata ea 5 0.01

Fig. 5 The stress amplitude, sa versus the strain amplitude, ea, for thecyclic experiments conducted on the 7075 plate

Fig. 3 The slopes of the stress-strain data, U, in the transition region


4.2 Simulation Results. We extracted a representative ag-gregate of crystal orientations from the ODF. For the upper andlower bound calculations we employed 1000 orientations. For thefinite element calculations we employed a 123 12 3 12 cube ofelements (1728 total), each with its own crystallographic orienta-tion as determined by the ODF. We determined the model param-eters for each linking assumption by correlating the model re-sponse to the TD tensile test data. We then ran simulations topredict the material response in the other orientations. The param-eter values we determined are given in Table 1. In the presentstudy we have focussed on the monotonic behavior, since we sawnegligible cyclic hardening or softening in the cyclic response. Inthis material, capturing the texture-dependent elastic/elastic-plastictransition is of first order importance to modeling the cyclicbehavior. We employed isotropic elasticity withE 5 69 GPa andn 5 0.3.

Figures 6, 7, and 8 depict the results for the upper bound, lowerbound, and FEM simulations, respectively. A simulation of the TDexperiment with slip system hardening neglected is also shown foreach simulation. The simulation results are discussed in the fol-lowing section.

5 DiscussionWe divide the discussion of the simulation results into two parts,

the elastic/elastic-plastic transition region of the stress strain curve(e , 1%) and the post-transition region.

5.1 Post-Transition Behavior. Each model predicts anisot-ropy of the back-extrapolated yield strength with various degreesof success in correlating the experimental data. The upper boundmodel correctly predicts the general trend but overpredicts themagnitude of the anisotropy. The RD prediction is lower than TDand thef 5 45° stress-strain curve lies well below the data. Thelower bound simulations correctly predict the coincidence of theRD and TD data, but place the stress levels of all the other tests onroughly the same level, underpredicting the magnitude of theanisotropy, in general. The finite element calculations are similarin nature to the upper bound results, however, a smaller differenceis predicted between the RD and TD stress levels.

As previously mentioned, the elements employed in the FEMsimulations were initially cubic. Increased accuracy in the FEMsimulations was attained by using elements with an initial aspectratio more representative of the as-rolled microstructure. As withmost rolled metals, we observed flat grains, elongated in the rollingdirection. We reran the FEM calculations for the TD and RD testsusing elements with an aspect ratio of RD:TD:ND5 16:4:1. Theseresults, along with the previously depicted results using cubicelements, are shown in Fig. 9. We see that the TD stress levelresponse decreases and the RD response increases, bringing thecurves closer to the coincidence we see in the data. However, asdiscussed below, the sharpness of the knee is virtually unaffectedby the aspect ratio change.

Fig. 6 The simulations of the tensile experiments using the upperbound linking assumption. Also depicted are the RD, TD, and f 5 45°data, along with a simulation of the TD experiment with the slip systemhardening neglected ( g 5 0 in Eq. (3)).

Table 1 Continuum slip plasticity model parameters

Parameter Upper bound Lower bound FEM

g o (s21) 1.0 1.0 1.0mf 0.05 0.05 0.05ho 350 1200 400gs (MPa) 480 520 600Initial g(a) (MPa) 250 345 275

Fig. 7 The simulations of the tensile experiments using the lower boundlinking assumption. Also depicted are the RD, TD, and f 5 45° data,along with the simulation of the TD experiment with slip system harden-ing neglected ( g 5 0 in Eq. (3)).

Fig. 8 The simulations of the tensile experiments using the finite ele-ment formulation. Also depicted are the RD, TD, and f 5 45° data, alongwith the simulation of the TD experiment with slip system hardeningneglected ( g 5 0 in Eq. (3)).

Fig. 9 The finite element simulations of the RD and TD experimentsemploying initial element aspect ratios of RD:TD:ND 5 1:1:1 (cube) andRD:TD:ND 5 16:4:1


5.2 Elastic/Elastic-Plastic Transition. Mechanistically, theelastic/elastic-plastic transition represents an extremely compli-cated regime. Understanding the sources of transition anisotropy isan even greater challenge. For histories involving small inelasticstrains (monotonic or cyclic), however, capturing the stress levelswithin the transition is of primary importance. Typically, thestrain-hardening model is “tuned” to capture the transition behav-ior. However, for the 7075 plate, the hardening model would needto be anisotropic. It can also be seen in Figs. 6, 7, and 8 that withthe parameters given in Table 1, the slip system hardening hasvirtually no effect on the macroscopic stress-strain curves in thetransition regime. For the continuum slip model, the character ofthe elastic/elastic-plastic transition, therefore, is governed by theactivation of the slip systems, the texture and the mean fieldassumption. The transitions predicted by each model are shown asplots of U versuse in Figs. 10, 11, and 12. The FEM predictionsshown employed cubic elements. It can seen that, while slightdifferences appear between the responses given by the upperbound and finite element models, they both predict less sharptransitions, similar to that of thef 5 45° data. The spread in thefinite element predictions is, perhaps, smaller than that of the upperbound model, especially near the end of the transition and theordering of the curves is more reminiscent of the one we see in Fig.3. However, the sharpness seen in the RD and TD data is notpredicted. The lower bound model, on the other hand, predictssharp transitions similar to the RD and TD data for all orientations.While the transition anisotropy is slight, the ordering of the curvesin the lower bound simulations is similar to that which we see inthe data.

By examining the transition anisotropy from the perspective oflower and upper bound deformation behavior, possible physicalsources can be explored. One can hypothesize that the sharp RD

and TD transition behavior is more lower bound-like while thebehavior at intermediate values off is more like that of the upperbound. As described by Kocks et al. (1998) there are plausibleupper and lower bound explanations, rooted in the intergranularconstraint condition, for yielding of a polycrystal. The lowerbound condition (each grain experiences the same stress state)results in activation of only those systems that are most favorablyoriented for slip among the grains. Since an FCC crystal will, mostlikely, have some slip systems so oriented, yielding occurs more orless simultaneously within each grain and a sharp transition resultsas the total cross section quickly yields. The upper bound conditionrequires all grains to experience the macroscopic strain rate. Theorientation of some grains will result in stresses large enough tocause extensive yielding, while others will continue to strainelastically. Eventually, all grains will yield, but the resulting tran-sition to fully developed plasticity is a gradual one. In reality, thedeformation state experienced by a grain in an aggregate liesbetween the upper and lower bound extremes. As shown in thefinite element results of Barton et al. (1999), the constraint offeredby the neighbors of an individual crystal strongly dictates themagnitudes of the resulting crystal stress components. They sawsignificant spread in crystal stress when the neighborhood waschanged. The relative strengths of surrounding crystals will changeas the loading axis is changed. It is conceivable, therefore, that themanner in which the crystals are placed in the aggregate wouldaffect the transition and its anisotropy. This may imply that thefinite element predictions would be improved by choosing crystalorientations and locations within the aggregate in such a way thatboth the ODF and the misorientation distribution function(MODF) are matched. This will necessitate measuring lattice ori-entations at individual spatial points employing such techniquessuch as the automated electron backscatter pattern (EBSP) methodin the SEM (cf. Wright and Adams, 1990; Baudin and Penelle,1993; Wright and Kocks, 1996; Miller and Turner, 1999). Sincedeformation heterogeneity plays such a key role in transitionbehavior, a refined mesh (multiple elements per crystal) may alsobe necessary. Such a study is currently being undertaken by theauthors.

Bringing emphasis on the fact that the linking assumption canhave such a profound affect on the transition prediction is, perhaps,as important as exploring the details of the individual modelpredictions. This is especially pertinent since the elastic/elastic-plastic transition is typically modeled as a strain-hardening phe-nomena, especially in macroscale models. In saturation-recoverytype hardening models, the sharpness of the knee depends on thetime constant employed in the hardening formulation. One canincrease that constant, or, as is common in macroscale cyclicplasticity models, employ additional variables in order to improvethe transition prediction. As depicted in Figs. 6, 7, and 8, however,the transition predicted by the elastic-plastic continuum slip model

Fig. 10 The slopes of the stress-strain curve, U, predicted by the upperbound model. Also shown are the RD, TD and f 5 45° data.

Fig. 11 The slopes of the stress-strain curve, U, predicted by the lowerbound model. Also shown are the RD, TD and f 5 45° data.

Fig. 12 The slopes of the stress-strain curve, U, predicted by the finiteelement model employing cubic elements. Also shown are the RD, TDand f 5 45° data.


is independent of the slip system hardening formulation. Physi-cally, it seems reasonable that the anisotropy of the transition fromelastic to elastic-plastic deformation behavior is more a product ofthe initial activation of slip systems and related intergranularconstraint effects as of stress elevation due to anisotropic strainhardening processes.

6 ConclusionsThe AA 7075 plate material investigated in this work exhibited

little cyclic hardening, which implies that an accurate representa-tion of the monotonic behavior, especially in the small strainregime, is of first order importance to cyclic model development.In our monotonic experiments, we saw anisotropy in both theback-extrapolated yield strength and the elastic/elastic-plastic tran-sition. As expected, the upper and lower bound models over- andunder-predicted the yield strength anisotropy, respectively. A finiteelement model, which associates a single orientation with eachelement, most accurately captured the yield strength anisotropy,especially when an element aspect ratio that resembled that of therolled microstructure was employed. The transition behavior andassociated anisotropy, which dictate the character of the smallstrain response, were not adequately captured by any of the mod-els. For all three mean field assumptions, prediction of the transi-tion behavior was largely independent of the slip system hardeningformulation. Larger, more detailed finite element simulations ofthe experiments may better represent the heterogeneity of slipactivation, and capture the transition anisotropy. Selecting crystalorientations for the finite element calculation that would match theMODF as well as the ODF may also be necessary. These calcu-lations are the focus of current work.

AcknowledgmentsThe authors acknowledge the Air Force Office of Sponsored

Research (grant #F49620-98-1-0401) and the National ScienceFoundation (CAREER grant CMS #9702017) for supportingthis work. In addition, NRB gratefully acknowledges the sup-port of a National Science Foundation Graduate Research Fel-lowship Award. Edward Harley, Heshan Gunawardane andTodd Turner conducted the mechanical tests and texture mea-surements and are gratefully acknowledged. Thanks also goesto Professor Paul Dawson of Cornell University for helpfuladvice and for the use of his simulation codes. Professor Ar-mand Beaudoin of the University of Illinois is acknowledgedfor many helpful discussions.

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(Contents continued)

119 Cold Compaction of Composite PowdersK. T. Kim, J. H. Cho, and J. S. Kim

129 On the Limit of Surface Integrity of Alumina by Ductile-Mode GrindingI. Zarudi and L. C. Zhang

135 Measurement of the Axial Residual Stresses Using the Initial Strain ApproachWeili Cheng

141 Effect of Plasma Cutting on the Fatigue Resistance of Fe510 D1 SteelM. Chiarelli, A. Lanciotti, and M. Sacchi

Announcement

146 New Reference Format for Transactions Journals


J. C. MoosbruggerDepartment of Mechanical and Aeronautical

Engineering,Clarkson University,

Potsdam, NY, 13699-5725

Anisotropic Nonlinear KinematicHardening Rule ParametersFrom Reversed ProportionalAxial-Torsional CyclingA procedure for determining parameters for anisotropic forms of nonlinear kinematichardening rules for cyclic plasticity or viscoplasticity models is described. An earlierreported methodology for determining parameters for isotropic forms of uncoupled,superposed Armstrong-Frederick type kinematic hardening rules is extended. For thisexercise, the anisotropy of the kinematic hardening rules is restricted to transverseisotropy or orthotropy. A limited number of parameters for such kinematic hardeningrules can be determined using reversed proportional tension-torsion cycling of thin-walled tubular specimens. This is demonstrated using tests on type 304 stainless-steelspecimens and results are compared to results based on the assumption of isotropic formsof the kinematic hardening rules.

I IntroductionSuperposition of multiple Armstrong-Frederick (A-F) type ki-

nematic hardening rules (Chaboche, Dang-Van and Cordier, 1979;Chaboche, 1993) has become a popular approach for phenomeno-logical modeling of cyclic plasticity and viscoplasticity of metals.The approach has been successfully incorporated into first-orderthermodynamics frameworks (Chaboche, 1993; McDowell, 1992)and used successfully to model responses to strain controllednonproportional loading histories (e.g., McDowell, 1985; Moos-brugger and McDowell, 1989; Moosbrugger, 1991, 1993) as wellas uniaxial cyclic loading histories. Recent modifications usingthresholding functions in the so-called “dynamic recovery term”have enabled the modeling of ratchetting responses to both uniax-ial and multi-axial nonproportional loading (Chaboche 1989;Chaboche and Nouailhas, 1989a, b; Bower, 1989; Chaboche,1991; Ohno and Wang, 1991a, b, 1993a, b; Jiang and Sehitoglu,1994a, b, 1996a, b; McDowell, 1995; Chaboche and Jung, 1997).Introduction of anisotropic forms of nonlinear kinematic hardeningrules has been reported by Sutcu and Krempl (1990), Lee andKrempl (1991), Nouailhas and Chaboche (1991), Nouailhas andCulie (1991), and Nouailhas and Freed (1992), among others.These anisotropic models have been applied primarily for thedescription of cyclic viscoplasticity of single crystals and or metal-matrix composites. Recently, Chaboche and Jung (1997) discussedsuperposed Armstrong-Frederick type kinematic hardening rules,with fixed anisotropy and with thresholding to correlate ratchettingexperiments, and showed their generalization in the framework ofquasistandard thermodynamics.

An earlier reported methodology for determining the parametersfor superposed, uncoupled A-F type kinematic hardening rulesfocused on parameters directly obtainable from hysteresis loopsand hardening modulus plots, obtained from experimental re-sponses to proportional strain controlled cycling (Moosbruggerand Morrison, 1997). This technique however, was applied only toisotropic forms of superposed A-F rules. Such forms have a limitedcapacity to correlate responses, even to proportional cyclic loadinghistories, if there is significant anisotropy present in the specimens.Of course, one alternative is to adopt anisotropic yield functions or

flow rules. However, such an approach only addresses initialyielding behavior and plastic strain rate direction in unloading-loading events.

In this paper, the methodology for obtaining the A-F rule pa-rameters is applied to determining a limited number of parametersfrom tension-torsion tests on tubular specimens, when the re-sponses exhibit distinctly anisotropic kinematic hardening behav-ior in the axial and shear components. This is seen to correspondto either orthotropic or transverse isotropic symmetries using co-ordinate axes corresponding to specimen axial, circumferential,and radial directions. Examples are given wherein the correlationof responses is shown to be significantly improved when param-eters for the axial and shear components of the A-F rules aredetermined separately, for tests on type 304 stainless steel speci-mens.

II Isotropic Framework for Determining A-F Rule Pa-rameters

The forgoing applies to conventional associated, phenomeno-logical incompressible cyclic plasticity or viscoplasticity modelswhich adopt the approximations suitable for infinitesimal defor-mations and rotations (c.f. Lubliner, 1990; Khan and Huang,1995). We concentrate on the nonlinear kinematic hardeningmodel using superposed Armstrong-Frederick (A-F) type kine-matic hardening rules originally proposed by Chaboche et al.(1979) and employed and discussed in numerous works such asthose mentioned in the introduction. The determination of param-eters for thresholding functions (e.g., Ohno and Wang, 1993a, b;McDowell, 1995; Jiang and Sehitoglu, 1996a, b; Chaboche andJung, 1997) is not discussed. Uppercase boldface symbols denotesecond rank tensors (except in Section III whereQ i and C i arefourth rank). Double bars denote the normiAi 5 [A : A] 1/2 whereA : A 5 AikAki , Aik denoting cartesian tensor components.Summation on repeated subscripts is implied unless otherwisenoted. Lower case boldface symbols denote axial-torsional sub-space (c.f. McDowell, 1985) vectors and single bars their magni-tude, e.g.,uau 5 [a z a] 1/ 2 5 [aiai ]

1/ 2, ai being vector componentsin the axial-torsional subspace. All non-bold symbols are scalarsand uxu denotes the absolute value ofx. The subscripta denotesamplitude of the subscripted quantity as inea or ueau.

The isotropic form of superposed, uncoupled A-F type kine-matic hardening rules can be written as

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionOctober 5, 1998; revised manuscript received July 14, 1999. Associate TechnicalEditor: H. M. Zbib.


A 9 5 Oi51

M

A 9i 5 Oi51

M

CiD i q (1)

with

D i 5 BiN 2 A 9i (2)

whereq 5 iEpi andN 5 (S 2 A9)/iS 2 A9i 5 Ep/iEpi; Ep isthe plastic or inelastic strain rate with cartesian componentsErs

p .The tensorD i represents the distance in the deviatoric stress plane,from the current state, to saturation of thei th backstress or kine-matic hardening subvariable whereA9i 5 A i 2 ( Arr /3)I is thedeviatoric part ofA i . The “hardening modulus” (i.e.,H 5 S :N/Ep : N) in the absence of cyclic hardening is then

H 5 Oi51

M

CiD i : N (3)

For a more convenient presentation, the axial-torsional subspaceversion of this framework (e.g., McDowell, 1985; Moosbruggerand Morrison, 1997) can be written as

a 5 Oi51

M

ai 5 Oi51

M

cid i p (4)

with

d i 5 bin 2 ai (5)

wherep 5 =2/3q, n 5 (s 2 a)/us 2 au 5 ep/p 5 ep/uepu, ep 5e1

pj 1 1 e3j 3, s 5 s1j 1 1 s3j 3, ai 5 ai1j 1 1 ai3j 3, s1 5 s, s3 5=3t, e1

p 5 e p, e3p 5 g p/=3, and j 1 and j 3 form an orthonormal

basis for the two dimensional vector subspace;s and t are theaxial and shear stress, respectively, ande p andg p are the axial andengineering shear plastic strain rates, respectively. The subspacehardening modulus (h 5 s z n/ep z n) is then

h 5 Oi51

M

cid i z n (6)

and the A-F rule parameters for the subspace areci 5 =3/ 2Ci ,bi 5 =3/ 2Bi with h 5 3H/ 2.

II Anisotropic Form of Superposed Armstrong-Frederick Type Kinematic Hardening Rules in Axial-Torsional Subspace

In a straightforward way, we can generalize Eqs. (4) and (5),writing the rate of evolution of the subspace backstress in matrixform as

F a1

a3G 5 O

i51

M F ai1

ai3G 5 O

i51

M F ci1 00 ci3

GFdi1

di3G p (7)

where

Fdi1

di3G 5 Fbi1 0

0 bi3GFn1

n3G 2 Fai1

ai3G (8)

The result is an anisotropic axial-torsional subspace version of theframework, where the axial and shear components of thei thbackstress rate are uncoupled. However, they are not uncoupledfrom the respective deformation rate components sincen1, n3 andp all depend on bothe p and g p. The isotropic form, Eqs. (4) and(5), is recovered whenci1 5 ci3 andbi1 5 bi3. With n defined asfor the isotropic case, we tacitly assume isotropy of the flow rule,though more general anisotropic forms could be employed (e.g.,Chaboche and Jung, 1997).

II.1 Parameter Determination Procedure. To determinethe parametersci1, ci3, bi1, andbi3 we follow a procedure analo-gous to that presented in (Moosbrugger and Morrison, 1997) forthe isotropic version of the A-F rules. The following definitions aremade under the assumption of pure kinematic hardening for asingle cycle response to an imposed, completely reversed cyclicloading:

h1 5a1 sgn~n1!

p5 U ds

dpU , h3 5

a3 sgn~n3!

p5 Î3U dt

dpU (9)

Distances to saturation of axial and normalized shear stress can bewritten as

d1 5 Oi51

M

di1 5 s1a 2 s1 sgn~n1! 1 DaM1,

d3 5 Oi51

M

di3 5 s3a 2 s3 sgn~n3! 1 DaM3 (10)

wheres1a 5 s a ands3a 5 =3t a denote the components (axial andnormalized shear stress amplitudes) of the subspace stress vectoramplitudesa. The algebraic sign of the subspace plastic strain ratevector components are denoted by sgn (n1) and sgn (n3).

If we order the kinematic hardening parameters such thatci1 . ci2

. . . .. ciM then, providingai for i , M saturate prior to a reversal,Eqs. (7)–(10) dictate linearity ofh1 versusd1 andh3 versusd3 at lowh1, h3. The range of such linearity, if exhibited in the experimentalresults, is determined in much the same way as the range of stress-strain employed to estimate Young’s modulus in a uniaxial test isdetermined. QuantitiesDaM1 andDaM3 are the distances to saturationof the axial and normalized shear stress, respectively, at the tips of theaxial and normalized shear stress-strain hysteresis loops. They can bedetermined from plots ofh1 versuss1a 2 s1 sgn (n1) andh3 versuss3a 2 s3 sgn (n3) as illustrated schematically in Fig. 1.Thus,d1 andd3

can be determined directly from the responses and derivatives of theresponses to the imposed cyclic loading. If the kinematic hardeningbehavior is isotropic, theh1 versuss1a 2 s1 sgn (n1) andh3 versus

Fig. 1 Schematic illustrating expected behavior of the axial-torsionalsubspace hardening moduli for conformance to the anisotropic, super-posed A-F kinematic hardening rules: h1 versus ( s1a 2 s1 sgn ( n1)) or d1

and h3 versus ( s3a 2 s3 sgn ( n1)) or d3. The inset indicates the idealproportional loading response. Quantities taken from such plots for thedetermination of A-F rule parameters are also indicated.


s3a 2 s3 sgn (n3) plots (or h1 versusd1 and h3 versusd3) shouldcoincide. Assuming that the backstress subvariables fori , M aresaturated (ai 5 0 for i , M) at the cycle endpoints,cM1 andcM3 are theslopes of the linear region, at lowh1 andh3, of theh1 versuss1a 2 s1

sgn (n1) andh3 versuss3a 2 s3 sgn (n3) plots. The conditions

DaM1 1 aM1a 5 bM1 (11)

DaM3 1 aM3a 5 bM3 (12)

also follow, where [aM1a aM3a]T is theMth backstress amplitude.

Assumed symmetry of the hysteresis loop responses, with integra-tion of Eq. (7) for proportional cycling, leads to

2aM1a 5 ~aM1a 1 bM1!~1 2 e2cM12pa! (13)

2aM3a 5 ~aM3a 1 bM3!~1 2 e2cM32pa! (14)

wherepa 5 ueapu is the effective plastic strain amplitude. ForM $

2 other equations necessary to determine parameters using thisprocedure are

Oi51

M21

bi1 5 s1a 1 DaM1 2 us 2 aun1 2 bM1 (15)

Oi51

M21

bi3 5 s3a 1 DaM3 2 us 2 aun3 2 bM3 (16)

h1max5 Oi51

M21

ci12bi1un1u 1 cM1dM1max (17)

h3max5 Oi51

M21

ci32bi3un3u 1 cM3dM3max (18)

where

dM1max5 aM1a 1 bM1aun1u (19)

dM3max5 aM3a 1 bM3aun3u (20)

For M 5 2, the set of Eqs. (11)–(20) constitutes a linear systemsufficient to determine the parametersc11, c13, bi1 andbi3 as wellasaM1a andaM3a, oncec21 andc23 andDaM1 andDaM3 have beendetermined from experimentally derived plots such as those illus-trated in Fig. 1. The stress amplitudess1a ands3a, the plastic strainrate vector componentsn1 andn3, and the hardening moduli at theonset of plastic deformation during a reversal (h1max andh3max) areobtained from the experimental response to the proportional cy-cling.

For M $ 3, additional relationships are required which willresult in a nonlinear algebraic system of equations in the unknownparameters. An overconstrained approach would minimize an ob-jective function, such as sum of squared error between model andexperimental results. For simplicity, we will adopt here the ap-proach used for the isotropic case by Moosbrugger and Morrison(1997), and determine parameters by matching the number ofexperimental data points necessary to obtain the required addi-tional equations (to match the additional number of parameters forM . 2, i.e., 4(M 2 2) additional equations). Here, we match thestress response [s1k s3k]

T for a given reverse plastic strainpk, i.e.,

s1k 5 2aM1a sgn~n1! 1 ~aM1a sgn~n1! 1 bM1n1!~1 2 e2cM1 pk!

1 Oi51

M21

@2bi1n1 1 2bi1n1~1 2 e2ci1 pk!# 1 us 2 aun1 (21)

s3k 5 2aM3a sgn~n3! 1 ~aM3a sgn~n3! 1 bM3n3!~1 2 e2cM3 pk!

1 Oi51

M21

@2bi3n3 1 2bi3n3~1 2 e2ci3 pk!# 1 us 2 aun3 (22)

wherek 5 1 . . . 2(M 2 2) and the reverse plastic strainpk 5 uekp

2 eopu, eo

p being the subspace plastic strain vector at the reversalpoint (ea

p or 2eap depending on the sign ofn). Other possibilities

can be used such as matching hardening moduli or a combinationof stress and hardening moduli at discrete reverse plastic strainpoints. Again, this notation tacitly assumes isotropy of the flowrule. However,us 2 aun1 andus 2 aun1 in Eqs. (15), (16), (21) and(22) could be replaced with friction stresses, yield stresses orthermally activated contributions to the axial and shear compo-nents of the flow stress, which are determined from the experi-mental axial and shear responses, independently. The resultingnonlinear system of equations must be solved using a suitableprocedure such as Newton-Raphson iteration.

II.2 Demonstration of the Procedure for Axial-TorsionalCycling of Type 304 Stainless-Steel Specimens.Strain con-trolled tests were performed on thin-walled tubular specimens ofinitially annealed type 304 stainless steel. Details can be foundelsewhere (Moosbrugger, 1988). Here, we report the application ofthe described procedure on data taken from loading blocks whereinthe specimens were cycled between two symmetric strain endpoints in axial-torsional subspace (i.e., reversed proportional cy-cling) at a constant total effective strain rate ofueu 5 1023s21. Theprocedure is applied to data from two different effective total strainamplitudes:ueau 5 =e a

2 1 g a2/3 5 6.0(10)23 with e1a 5 e a 5

4.5(10)23 ande3a 5 g a/=3 5 3.9(10)23 and ueau 5 7.9(10)23

with e1a 5 6.0(10)23 and e3a 5 5.2(10)23; e a and g a are theimposed axial and engineering shear strain amplitudes, respec-tively. Experimental hardening moduli, etc. were determined usingthe same procedures reported earlier (Moosbrugger and Morrison,1997) for the isotropic case, but performed on the axial and shearcomponents of the experimental hysteresis loops separately whenusing the anisotropic model. Isotropy of the flow rule was invoked,and us 2 au 5 80 MPa was estimated from previous analysesinvolving back-extrapolation of plastic strain rate directions beforeand after nonproportional strain path direction changes (Moos-brugger, 1991). For the proportional cycling discussed here,n1 andn3 were taken as the average of these quantities computed over theindividual cycles by numerically differentiating the response ver-sus time histories.

The experimental hardening moduli plots forueau 5 6.0(10)23

are shown in Fig. 2, with the full scale shown in Fig. 2(a) and anexpanded scale at lowh1, h3 shown in Fig. 2(b). Clearly,cM1 ÞcM3 indicating the anisotropy of the kinematic hardening responsein these specimens. The straight lines shown in Fig. 2(b) are thebest fit straight lines through the data at lowh1, h3 used to estimatecM1, cM3, DaM1 and DaM3. Using the procedure, the parametersvalues determined werecM1 5 339,cM3 5 597,bM1 5 103 MPaand bM3 5 104 MPa. ForM 5 2, c11 5 7240, c13 5 15600,b11 5 120 MPa andb13 5 90.7MPa. ForM 5 3, c11 5 13200,c13 5 17700,c21 5 572, c23 5 407, b11 5 63.7 MPa, b13 580.1 MPa, b21 5 56.0 MPa andb23 5 10.7 MPa.

The correlation of the experimental results forueau 5 6.0(1023)using isotropic and anisotropic models is demonstrated in Figs.3–6. Figure 3 shows the correlation of the hysteresis loops (s1 5s versuse1 5 e ands3 5 =3t versuse3 5 g/=3) and the stresssubspace response,s1 versuss3, using the isotropic model withM 5 2. Correlation of the hardening moduli represented byh1

versusd1 and h3 versusd3 is shown in Fig. 4 for the isotropicmodel with M 5 2. Parameters could not be determined for theisotropic model withM 5 3 at this effective strain amplitude, aswas reported previously (Moosbrugger and Morrison, 1997). Cor-relations using anisotropic models with bothM 5 2 andM 5 3are shown in Figs. 5 and 6. It is interesting that parameters couldbe determined forM 5 3 when the axial and shear components of


the response were decoupled in this way (anisotropic case). Alarger “expanded range” is shown in Figs. 4(c) and 6(c) comparedto Fig. 2(b) to show the transition from linear to nonlinear behaviorof h1 versusd1 and h3 versusd3 at low h1, h3, and so that thereader can see why the ordinate and abscissa ranges were chosenin Fig. 2(b). It is seen in Figs. 4–6 that the best overall correlationof the experimental response is achieved using the anisotropicmodel with M 5 3. In comparison to the isotropic model, thecorrelation of the normalized shear stress-strain hysteresis loop (s3

versuse3) is significantly improved, as is thes1 versuss3 response.This is apparently due to the improved correlation of the hardeningmoduli at low h1, h3, since the correlation at high values of themoduli does not seem to be significantly improved over the iso-tropic model (Fig. 4), at least in terms ofh3 versusd3.

Experimental hardening moduli plots forueau 5 7.9(10)23 areshown in Fig. 7. Again it is seen that theh1 versusd1 andh3 versusd3 plots do not coincide, even at low values of the moduli, leadingto cM1 Þ cM3 and DaM1 Þ DaM3. The straight lines through thedata in Fig. 7(b) are used to determinecM1, cM3, DaM1 andDaM3.With the procedure, the parameter values determined werecM1 5253, cM3 5 413, bM1 5 73.4 MPa, andbM3 5 86.2 MPa. ForM 5 2, c11 5 4480, c13 5 4100, b11 5 131 MPa andb13 594.4 MPa. ForM 5 3, c11 5 10400,c13 5 7100, c21 5 482,c23 5 577, b11 5 69.4 MPa,b13 5 78.9 MPa,b21 5 102 MPaandb23 5 67.2 MPa.

Correlation of the experimental hysteresis loops and the stresssubspace response forueau 5 7.9(10)23, using isotropic and

anisotropic models, is summarized in Figs. 8–11. Figure 9 showsthe correlation of the hardening moduli using the isotropic modelwith M 5 3. Again, a larger “expanded range” is shown in Figs.9(c) and 11(c) compared to Fig. 4(b) to show the transition fromlinear to nonlinear behavior ofh1 versusd1 andh3 versusd3 at lowh1, h3, and so that the reader can see why the ordinate and abscissaranges were chosen in Fig. 4(b). The correlations achieved with the

Fig. 2 Experimental hardening moduli curves for a proportional cycleobtained under total strain control at e a 5 0.0045j1 1 0.0039j3 , zeaz 56.0(1023): (a) full scale and ( b) expanded scale at low values of h1 , h3

showing the linear region used to determine cM1 , cM3 , DaM1 and DaM3

Fig. 3 Comparison of experimental response and correlation using M 52 model with isotropic A-F rules ( M 5 2 ISOTROPIC MODEL) at e a 50.0045j1 1 0.0039j3 , zeaz 5 6.0(1023): (a) axial, s1 versus e1 response, ( b)normalized shear, s3 versus e3 response and ( c) stress subspace, s1

versus s3 response


anisotropic models usingM 5 2 andM 5 3 are shown in Figs.10 and 11. Again, the best correlation among these is achievedwith the anisotropic model usingM 5 3. This is true for all of theresponses shown, for this effective strain amplitude and ratio ofaxial strain amplitude to shear strain amplitude.

Limited independent investigation of the assumption of flowrule isotropy was obtained by computing friction stress(Kuhlmann-Wilsdorf and Laird, 1979) or yield stress components,s f 1 and s f 3 from the experimental axial and normalized shearstress-strain hysteresis loops. A variation of Cottrell’s method(Cottrell, 1953; Kuhlmann-Wilsdorf and Laird, 1979) was usedwith plastic strain offsets ofe1,offset

p 5 e3,offsetp 5 5(1026), 1(1025)

and 5(1025) and the procedure was performed on both branches(i.e., sgn (n1) 5 sgn (n3) 5 61) of the hysteresis loops. Specif-ically, a least squares straight line was fit to points ranging fromthe peak stress prior to a reversal to one half of the peak stressfollowing a reversal. The slope of this line was taken as the elasticmodulus for a given branch of a given hysteresis loop. The friction

stresses for different plastic strain offsets were determined byfinding the stress corresponding to a given plastic strain offsetmeasured from the least squares line, in the direction of plasticstraining of the branch, using a running average of five points todetermine when an offset equal to or greater than a specifiedplastic strain offset was achieved. One half of the absolute value of

Fig. 4 Comparison of experimental hardening moduli plots and exper-imental correlations using M 5 2 model with isotropic A-F rules ( M 5 2ISOTROPIC MODEL) at e a 5 0.0045j1 1 0.0039j3 , zeaz 5 6.0(1023): (a) fullscale and ( b) expanded scale at low values of h1 , h3

Fig. 5 Comparison of experimental response with correlations usingM 5 2 model with anisotropic A-F rules ( M 5 2 ANISOTROPIC MODEL)and M 5 3 model with anisotropic A-F rules ( M 5 3 ANISOTROPICMODEL) at e a 5 0.0045j1 1 0.0039j3 , zeaz 5 6.0(1023): (a) axial, s1 versuse1 response, ( b) normalized shear, s3 versus e3 response and ( c) stresssubspace, s1 versus s3 response


the difference between the peak stress and the correspondingplastic strain offset stress was then taken as the friction stress. Thesmoothing effect of the straight line fit, as well as the running fivepoint average used for the plastic strain offset, tended to averageout inconsistencies associated with fluctuations about theunloading/loading line due to limitations of analog-to-digital dataconversion and any noise in the data. Based on the full scale rangeof the extensometry used, the linearity of the extensometer signaland the analog-to-digital data conversion used, the minimum strainresolution was estimated to be on the order of 1026. This procedurewas performed on several hysteresis loops other than those usedfor illustration in this paper and consistent results were obtainedfor the three offsets, even for the smallest plastic strain offsetwhich is close to the estimated strain resolution limit. Table 1compares the friction stresses withus 2 aun1 and us 2 aun3 andcompares the component ratios and the plastic strain amplitude

component ratios for the two effective strain amplitudes. The totalstrain amplitude component ratios aree1a/e3a 5 1.15 ineach case.Except for the strain amplitudes, which are control parameters, theratios listed should be approximately equal for flow rule isotropy.Though the ratios deviate somewhat from isotropy in some cases,indicating flow rule anisotropy might improve correlation of thedata as well, flow rule isotropy is indicated to be a reasonableapproximation. Note the relative agreement between the computedoffset friction stresses using the lower plastic strain offsets andus2 aun1 andus 2 aun3, the latter having been estimated from strainpath direction changes (Moosbrugger, 1991).

III Relationship to Full Stress SpaceWe can generalize Eqs. (1) and (2) to an anisotropic form of the

nonlinear kinematic hardening rules by writing evolution of thei thbackstress as

A 5 Oi51

M

A i 5 Oi51

M

~Q i : E p 2 C i : A i q! (23)

where the scalar kinematic hardening rule parameters are nowfourth rank tensorsC i and Q i 5 C i : B i where [C i : B i ] rsjl 5CirsvwBivwjl (no sum on subscripti which pertains to thei th back-

Fig. 6 Comparison of experimental hardening moduli plots and exper-imental correlations using M 5 2 model with anisotropic A-F rules ( M 52 ANISOTROPIC MODEL) and M 5 3 model with anisotropic A-F rules(M 5 3 ANISOTROPIC MODEL) at e a 5 0.0045j1 1 0.0039j3 , zeaz 56.0(1023): (a) full scale and ( b) expanded scale at low values of h1 , h3

Fig. 7 Experimental hardening moduli curves for a proportional cycleobtained under total strain control at e a 5 0.0060j1 1 0.0052j3 , zeaz 57.9(1023): (a) full scale and ( b) expanded scale at low values of h1 , h3

showing the linear region used to determine cM1 , cM3 , DaM1 and DaM3


stress). Note that the backstress evolution is now written in termsof the total backstress rate, (i.e., not deviatoric). This is in principlethe form discussed, for example, by Chaboche and Jung (1997),Nouailhas and Freed (1992), Nouailhas and Chaboche (1991),Nouailhas and Culie (1991), among others.

Attaching the components of the various tensors to a particular

cartesian basis (coordinatesx1, x2, x3) we can write these equa-tions in matrix form by defining

@E p# 5 3E11

p

E22p

E33p

2E23p

2E31p

2E12p

4 , @A i # 5 3Ai11

Ai22

Ai33

Ai23

Ai31

Ai12

4 , @A# i # 5 3Ai11

Ai22

Ai33

2Ai23

2Ai31

2Ai12

4 (24)

Then

@A i # 5 @Q i #@E p# 2 @C i #@A# i #q (25)

If we suppose further that bothQ i and C i exhibit orthotropicsymmetry, and the cartesian basis corresponds to the axes oforthotropy, then we arrive at the following matrices with Voigtconstant-like entries:

Fig. 9 Comparison of experimental hardening moduli plots and exper-imental correlations using M 5 3 model with isotropic A-F rules ( M 5 2ISOTROPIC MODEL) at e a 5 0.0060j1 1 0.0052j3 , zeaz 5 7.9(1023): (a) fullscale and ( b) expanded scale at low values of h1 , h3

Fig. 8 Comparison of experimental response and correlation using M 53 model with isotropic A-F rules ( M 5 3 ISOTROPIC MODEL) at e a 50.0060j1 1 0.0052j3 , zeaz 5 7.9(1023): (a) axial, s1 versus e1 response, ( b)normalized shear, s3 versus e3 response and ( c) stress subspace, s1

versus s3 response


@Q i # 5 3Qi11 Qi12 Qi13 0 0 0Qi12 Qi22 Qi23 0 0 0Qi13 Qi23 Qi33 0 0 00 0 0 Qi44 0 00 0 0 0 Qi55 00 0 0 0 0 Qi66

4 (26)

and

@C i # 5 3Ci11 Ci12 Ci13 0 0 0Ci12 Ci22 Ci23 0 0 0Ci13 Ci23 Ci33 0 0 00 0 0 Ci44 0 00 0 0 0 Ci55 00 0 0 0 0 Ci66

4 (27)

We can then obtain the special cases of transverse isotropy andisotropy; for transverse isotropy, with the coordinatex1 normal tothe plane of isotropy,Qi22 5 Qi33, Qi13 5 Qi12, Qi55 5 Qi66 andQi44 5 (Qi22 2 Qi23)/ 2; for isotropyQi11 5 Qi22 5 Qi33 5 lQi 12mQi , Qi12 5 Qi13 5 Qi23 5 lQi with Qi44 5 Qi55 5 Qi66 5(Qi11 2 Qi12)/ 2 5 mQi (the specializations forCiJK, J, K 5 1,2 . . . 6 are analogous usingmCi and lCi). The latter special

Fig. 10 Comparison of experimental response with correlations usingM 5 2 model with anisotropic A-F rules ( M 5 2 ANISOTROPIC MODEL)and M 5 3 model with anisotropic A-F rules ( M 5 3 ANISOTROPICMODEL) at e a 5 0.0060j1 1 0.0052j3 , zeaz 5 7.9(1023): (a) axial, s1 versuse1 response, ( b) normalized shear, s3 versus e3 response and ( c) stresssubspace, s1 versus s3 response.

Fig. 11 Comparison of experimental hardening moduli plots and exper-imental correlations using M 5 2 model with anisotropic A-F rules ( M 52 ANISOTROPIC MODEL) and M 5 3 model with anisotropic A-F rules(M 5 3 ANISOTROPIC MODEL) at e a 5 0.0060j1 1 0.0052j3 , zeaz 57.9(1023): (a) full scale and ( b) expanded scale at low values of h1 , h3


(isotropy) case is equivalent to Eqs. (1) and (2) where 2mQi 5Ci Bi , 2mCi 5 Ci and Bi 5 mQi/mCi since trEp 5 Err

p 5 0.Now consider axial-torsional loading of a thin-walled tubular

specimen. We attach the axes of orthotropy to the tube axial (x1),circumferential (x2) and radial (x3) directions, respectively. Wefurther define the plastic Poisson’s ratiosn 12

p 5 2E22p /E11

p andn 13p

5 2E33p /E11

p and the ratiosj 12 5 2A922/A911 andj 13 5 2A933/A911.If the flow rule is isotropic or transverse isotropic, then for incom-pressibilityn 12

p 5 n 13p 5 j 12 5 j 13 5 1

2. For orthotropic symmetryof the kinematic hardening rules, the axial and shear componentsof the backstress evolution equations obtainable from axial-torsional tests as described earlier would then be

Ai11 5 Qi ,longE11p 2 Ci ,longAi11q (28)

Ai12 5 Qi ,shE12p 2 Ci ,shAi12q (29)

with Qi ,long 5 Qi11 2 (Qi12 1 Qi13)/ 2, Qi ,sh 5 2Qi66, Ci ,long 5 Ci11

and Ci ,sh 5 2Ci66. In deviatoric componentsA9i11 5 2Ai11/3,A9i12 5 Ai12 and, using the axial-torsional subspace vectors,ai1 5Ai11 5 3A9i11/ 2, ai3 5 =3Ai12 5 =3A9i12. With N11 5=2/3n1 5 E11

p /q andN12 5 n3/=2 5 E12p /q we find that

ci1bi1 5 Qi ,long, ci3bi3 53Qi ,sh

2(30)

and

ci1 5 Î32 Ci ,long, ci3 5 Î3

2 Ci ,sh (31)

Thus, we can determine 4M parameters, i.e.,Qi66, Ci11, Ci66, andQi ,long, i 5 1 . . . M, using the proportional axial-torsional test.There would, in the general orthotropic case, be 18M of theseparameters to determine. Though there would be less constants intotal to determine for transverse isotropic symmetry, as comparedto the orthotropic case, the two symmetries cannot be distinguishedusing the axial-torsional test alone.

VI Discussion and ConclusionsThe result, that the correlation of responses to proportional

axial-torsional straining of the tubular specimens can be improvedby adopting anisotropic forms of kinematic hardening rules, isunderstandable if the specimens possessed texture. Though a thor-ough characterization, including crystallographic texture analysis

of the specimens is not available, they were machined from 50.8mm (2 in.) diameter, extruded round bar stock before annealing. Itis likely that a duplex 100&–^111& fiber texture corresponding tothe specimen cylinder axes (extrusion direction) was retained (e.g.,Barrett and Massalski, 1980), along with grain elongation in thisdirection. Thus, a transverse isotropic symmetry about the speci-men axis (e.g., Kocks et al., 1998), in some deformation behaviors,could be expected. The fact that the yield or friction stressescorresponding to very small plastic strain offsets were, in contrast,more or less isotropic is understandable since, in spite of texture,there will always be a large number of grains oriented such thatmany slip systems are stressed at or near the maximum possiblemagnitude for a given state of stress. Upon reversal of strain ratedirection, therefore, some plastic strain rate should be detectable ator near a common uniaxial friction stress, no matter what orien-tation is tested. Kinematic hardening on the other hand is theresult, at least to some extent, of the constraint of elasticallydeforming grains on the flow of plastically deforming grains.Texture should impart an anisotropy to this phenomenon, since ahigher percentage of grains will have “harder” orientations withrespect to certain specimen directions than will have those sameorientations in an untextured specimen. In the case of the presentexperiments, an apparent^100&–^111& fiber texture resulting fromextrusion led to less “elastic, perfectly plastic-like” hysteresisloops for the axial response than for the normalized shear response.Loop shape parameters (c.f. Mughrabi, 1978; Morrison et al.,1990) were computed from the experimental responses as the ratioof the area inscribed by the stress-plastic strain hysteresis loops tothe area of the circumscribing parallelograms. These wereVH,axial 5 0.78 andVH,shear 5 0.92 for theaxial and normalizedshear stress-strain hysteresis loops, respectively atueau 56.0(1023). An elastic-perfectly-plastic response would yieldVH 51 and a purely elastic cycle would yieldVH 5 0. For ueau 57.9(1023) they wereVH,axial 5 0.79 andVH,shear 5 0.88 for theaxial and shear responses, respectively. Clearly, the normalizedshear stress-strain response appears to be less constrained thandoes the axial component. ThecM3 . cM1 result, in particular,reflects this tendency. It should be mentioned, however, that theexperiments examined were at the same axial to shear strainamplitude ratio though at two different effective total strain am-plitudes. A more complete assessment of the technique wouldrequire cycling companion specimens at different ratios, to addressvarying anglesu 5 arctan (e3a/e1a) (including in particularu 5 0,

Table 1 Experimental quantities indicating appropriateness of isotropic flow rule assumption

Effective strain amplitude,plastic strain offset, Cycle

Branchs f 1

(MPa)s f 3

(MPa)us 2 au n1

(MPa)us 2 aun3

(MPa) s f 1/s f 3 n1/n3 e1ap /e3a

p

ea 5 0.0045j 1 1 0.0039j 3

ueau 5 6.0(1023)e1,offset

p 5 e3,offsetp 5 5.0(1026)

sgn (n1) 5 sgn (n3) 5 1 54.7 58.8 60.6 51.7 0.93 1.11 1.10sgn (n1) 5 sgn (n3) 5 21 54.8 57.7 60.6 51.7 0.95 1.11 1.10

e1,offsetp 5 e3,offset

p 5 1.0(1025)sgn (n1) 5 sgn (n3) 5 1 61.8 66.1 60.6 51.7 0.93 1.11 1.10sgn (n1) 5 sgn (n3) 5 21 61.9 57.7 60.6 51.7 1.07 1.11 1.10



ea 5 0.006j 1 1 0.0052j 3

ueau 5 7.9(1023)e1,offset

p 5 e3,offsetp 5 5.0(1026)

sgn (n1) 5 sgn (n3) 5 1 57.8 49.9 59.5 53.4 1.16 1.17 1.14sgn (n1) 5 sgn (n3) 5 21 58.7 51.3 59.5 53.4 1.14 1.17 1.14






p/2 and 3p/2), at constant effective strain amplitude. If the result-ing parameters were found to be relatively independent ofu, thenthe interpretation of the kinematic hardening rule symmetrieswould be more conclusive.

By the same texture argument, cyclic loading and, in particular,nonproportional cyclic loading should have a tendency to diminishsuch anisotropy of kinematic hardening behavior. As dislocationsubstructures are developed which lead to cyclic hardening (e.g.,McDowell et al., 1988; Doong et al., 1990), those more highlydeforming grains will tend to cyclically harden somewhat morerapidly than those with “harder” orientations. Also, refinement ofstructure through sub-grain cell formation should reduce the mor-phologic anisotropy associated with grain aspect ratio. Figure 12showscM1 andcM3 versus cycle number for the proportional test atueau 5 7.9(1023). It is indicated that while both parametersdecrease as a function of cycle number, the difference betweenthem decreases very rapidly, more or less stabilizing after a fewcycles. For the same specimen, this proportional cycling wasfollowed by 25 cycles of nonproportional cycling with the samecomponent strain amplitudes but with the axial and shear strains 90degrees out of phase. This was then followed by another 25 cyclesof proportional cycling with the same component strain ampli-tudes. Figure 13 shows thecM1 and cM3 determined from theproportional cycling subsequent to the 90 degrees out of phasecycling. The ratiocM1/cM3 increased from 0.49 for the last of theinitial proportional cycles to 0.68 for the first of the proportionalcycles subsequent to the 90 degree out of phase cycling. Likewise,the ratio 2QM,long/(3QM,sh) increased from 0.67 to 1.01, for the

same cycles. Thus, the kinematic hardening behavior was signif-icantly more isotropic following the 90 degrees out of phasecycling. It appears then that processes which tend to homogenizethe deformation, such as cyclic hardening, will diminish the an-isotropy of kinematic hardening, when textures are present.

In conclusion, a procedure has been presented which can beused to determine parameters, using axial-torsional tests, for aniso-tropic, superposed Armstrong-Frederick type kinematic hardeningrules. The procedure is not sufficient to determine all of therequired constants for transverse isotropic or for orthotropic kine-matic hardening behavior, but it is a convenient means of assessingsome of the possible parameters. Correlation of responses to pro-portional axial-torsional cycling of tubular type 304 stainless steelspecimens, machined from extruded bar stock, was improvedusing the anisotropic kinematic hardening rules compared to thoseobtained assuming isotropic kinematic hardening behavior. Theaxial-torsional test, it would appear, may thus be one convenientmeans of assessing the effect of texture producing processes onsubsequent cyclic plasticity behavior. The procedure should besuitable for assessing such behaviors in other materials, providedthey exhibit anisotropic kinematic hardening behavior that is rea-sonably modeled by the anisotropic Armstrong-Frederick typerules.

AcknowledgmentsThe author gratefully acknowledges the support of the U.S.

National Science Foundation through grant no. CMS-9634707.U.S. NSF grant MSM-8601889, for support of the experiments on304 stainless steel, is also gratefully acknowledged. Helpful dis-cussions with Professor D. J. Morrison of Clarkson University arealso acknowledged.

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Fig. 12 Evolution of cM1 and cM3 with cycle number for initial 25 propor -tional cycles at e a 5 0.0060j1 1 0.0052j3 , zeaz 5 7.9(1023)

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5 0.0060j1 1 0.0052j3 , zeaz 5 7.9(1023) following 25 cycles with straincomponents 90 degrees out of phase


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M. Mizuno

Y. Mima

M. Abdel-Karim

N. Ohnoe-mail [email protected]

Department of Mechanical Engineering,Nagoya University,

Chikusa-ku, Nagoya 464-8603, Japan

Uniaxial Ratchetting of 316FRSteel at Room Temperature—Part I: ExperimentsUniaxial ratchetting characteristics of 316FR steel at room temperature are studiedexperimentally. Cyclic tension tests, in which maximum strain increases every cycle byprescribed amounts, are conducted systematically in addition to conventional monotonic,cyclic, and ratchetting tests. Thus hysteresis loop closure, cyclic hardening and visco-plasticity are discussed in the context of constitutive modeling for ratchetting. The cyclictension tests reveal that very slight opening of hysteresis loops occurs, and that neitheraccumulated plastic strain nor maximum plastic strain induces significant isotropichardening if strain range is relatively small. These findings are used to discuss theratchetting tests. It is thus shown that uniaxial ratchetting of the material at roomtemperature is brought about by slight opening of hysteresis loops as well as byviscoplasticity, and that kinematic hardening governs almost all strain hardening inuniaxial ratchetting if stress range is not large.

1 IntroductionStrain accumulation induced by cyclic loading, i.e., ratchetting,

is important in designing structural components. Ratchetting tak-ing place under uniaxial cyclic loading with nonzero mean stress isreferred to as uniaxial ratchetting, which is most fundamental andhas been studied in many works. For 304 and 316 stainless steels,uniaxial ratchetting experiments have been reported by Yoshida etal. (1988), Chaboche and Nouailhas (1989), Ruggles and Krempl(1989, 1990), Yoshida (1990), Sasaki and Ishikawa (1993), Delo-belle (1993), Delobelle et al. (1995), Haupt and Schinke (1996),and so on.

Closure of stress-strain hysteresis loops, however, has scarcelybeen studied experimentally to date. It is obvious that the closureis directly related with uniaxial ratchetting, because less ratchettingtakes place under uniaxial cyclic loading if the closure is morecomplete. It is therefore important to study the closure experimen-tally. The closure can be investigated by performing cyclic tensiontests, which are strain cycling tests with maximum strain increas-ing with every cycle (see Section 2). Such tests, which wereperformed first by Coffin (1970) to explore the effects of super-imposed cyclic and monotonic strains on the deformation andfracture of annealed Nickel A, were done recently in a joint workon the inelastic analysis of liquid-level induced thermal ratchettingof 316FR steel cylinders (Inoue and Igari, 1995; Yoshida et al.,1997).

Ohno et al. (1998) discussed the cyclic tension tests done in thejoint work mentioned above. They thus found that almost completeclosure of hysteresis loops as well as isotropic hardening depend-ing on maximum plastic strain prevailed in the tests. Kobayashi etal. (1998) then showed this finding to be essential for predictingappropriately thermal ratchetting experiments of 316FR steel cyl-inders subjected to spatial variations in temperature; they obtainedgood agreement between the experiments and the finite elementsimulations in which they implemented a constitutive model ca-pable of representing complete closure of hysteresis loops. It is,however, worth noting that the finding was deduced from cyclictension tests performed at high temperatures as high as 300 and650°C, where the so-called dynamic strain aging can be signifi-

cant. It should also be noted that extensive tests were not con-ducted (i.e., strain range was fixed to be 0.2 percent, and maximumstrain was increased by either 0.1 or 0.2 percent every cycle).

One of the structural materials developed for fast breeder reac-tors is 316FR steel, which is based on 316 stainless steel but hassuperior high temperature strength (Nakazawa et al., 1988; Taka-hashi, 1998). It has already been reported that 304 and 316 stain-less steels exhibit noticeable viscoplasticity at room temperature(e.g., Krempl, 1979; Kujawski et al., 1980; Chaboche and Rous-selier, 1983; Yoshida 1990), so that ratchetting of such austeniticstainless steels at room temperature is affected much more activelyby viscoplasticity than at high temperatures around 550°C wheredynamic strain aging tends to suppress viscoplasticity (Rugglesand Krempl, 1989; Sasaki and Ishikawa, 1993; Delobelle, 1993).This suggests that the aforementioned finding on ratchetting of316FR steel at high temperatures, i.e., almost complete closure ofhysteresis loops and isotropic hardening depending on maximumplastic strain, may not hold true at room temperature.

In the present work, therefore, the uniaxial ratchetting charac-teristics of 316FR steel at room temperature are studied withemphasis on the effects of hysteresis loop closure and viscoplas-ticity on ratchetting. Strain hardening in the presence of ratchettingis also a matter in the work. Part I of the paper describes the resultsof experiments performed to discuss the above factors. The exper-iments include cyclic tension tests, monotonic tension tests, cyclicstraining tests with zero mean strain, and uniaxial ratchetting tests.Experimental findings reported in Part I will be taken into accountin Part II, in which a new kinematic hardening model will beformulated and combined with a viscoplastic equation to simulatethe uniaxial ratchetting experiments.

2 Material and Experimental ProgramThe specimens tested had gauge sections 8 mm in diameter and

30 mm in axial length. They were machined from a 316FR steelrolled sheet 50 mm in thickness, so that they had the axis orientedin the rolling direction of the sheet. The sheet had the chemicalcomposition shown in Table 1. As seen from the table, 316FR steelis a low-carbon nitrogen-added 316 stainless steel. Each specimenwas tested in the as-received state by using a 100-kN closed-loopservohydraulic tension-compression testing machine with a digitalcontroller. A clip-on extensometer 25 mm in gauge length wasemployed to measure the axial elongation. Either the axial elon-gation or the axial load was controlled.

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionOctober 1, 1998; revised manuscript received August 6, 1999. Associate TechnicalEditor: H. Sehitoglu.


In the present work, cyclic tension tests were performed inaddition to conventional tests such as monotonic tension, cyclicstraining between fixed strain limits, and uniaxial ratchetting (Ta-ble 2). Figure 1 illustrates the strain history in the cyclic tensiontests, for which cyclic strain rangeDe and maximum strain incre-ment demax were prescribed as testing parameters. Six combina-tions ofDe anddemax were chosen in the present work to examinetheir effects systematically; i.e., for each of three strain ranges ofDe 5 0.25, 0.4 and 0.8 percent, two tests of small and largeincrements of maximum strain,demax 5 0.01 and 0.1 (or 0.05)percent, were done at a constant strain rate ofe 5 5 3 1023

percent/s. The cyclic tension tests were interrupted when maxi-mum strain reached a terminal value ofe 5 2.0 percent. Uniaxialratchetting tests, i.e., cyclic stressing tests with nonzero meanstresses, were done for four sets of stress ratioR and stress ratesby prescribing maximum stresssmax to be 280 MPa. Here and fromnow onR denotes the stress ratio of minimumsmin to maximumsmax. The ratchetting tests were continued for 100 cycles.

3 Experimental Results

3.1 Monotonic Tension Tests. Significant viscoplasticitywas observed in the monotonic tension tests performed at constantstrain rates ranging from 53 1026 to 5 3 1021 percent/s (Fig. 2).As seen from the figure, plastic flow stress became larger by aboutfive percent with the tenfold increase in strain rate. Such a signif-icant effect of strain rate can be a feature common to 304 and 316stainless steels at room temperature (e.g., Krempl, 1979; Kujawskiet al., 1980; Chaboche and Rousselier, 1983; Yoshida, 1990).

3.2 Cyclic Straining Tests. Three tests were done undercyclic straining between fixed strain limits ate 5 5 3 1023

percent/s. Weak cyclic softening and weak hardening/softeningoccurred in the tests ofDe 5 0.25 and 0.4 percent, respectively,whereas some cyclic hardening took place in the test ofDe 5 0.8percent, as shown in Fig. 3 dealing with the change in stress rangeDs as a function of accumulated inelastic strainp (5 * ude pu).Here e p denotes inelastic strain. Thus, for 316FR steel at roomtemperature, we are allowed to say that cyclic hardening dependson De, and that cyclic hardening is negligible ifDe is small.Incidentally, the symbolsE, 1 and{ in Fig. 3 stand for the cyclictension tests discussed in what follows.

3.3 Cyclic Tension Tests. The cyclic tension tests withlarge and small values of maximum strain incrementdemax gave thestress versus strain relations shown in Figs. 4(a) to (c) and Figs.5(a) to (c), respectively. The dashed lines in the figures indicate forreference the monotonic tensile curve ate 5 5 3 1023 percent/s atwhich the cyclic tension tests were performed.

Let us first discuss the cyclic tension tests shown in Figs. 4(a) to(c), in whichdemax was prescribed to be large (i.e.,demax 5 0.05 or0.1 percent). It is seen from the figures that the tests had the

following features: In the tests ofDe 5 0.25 and 0.4 percent, thehysteresis loops closed almost completely, and the tensile peakpoints came to lie on the monotonic tensile curve; moreover, stressrange varied little with the increase in the number of cycles,N,(see also Fig. 3). In the test ofDe 5 0.8 percent shown in Fig. 4(c),on the other hand, the tensile peak points had larger stresses thanin the monotonic tension test, and stress range became noticeablylarger with the increase inN. Therefore we can say that, except forthe effect of isotropic hardening, the hysteresis loops closed almostcompletely in the three tests, and that isotropic hardening wasnegligible in the tests ofDe 5 0.25 and 0.4 percent but noticeablein the test ofDe 5 0.8 percent. The same features were observedin the other cyclic tension tests shown in Figs. 5(a) to (c), in whichthe increase in maximum strain was much smaller, i.e.,demax 50.01 percent.

However, a close look at the figures discussed above revealsslightly different features in the two cases: Let us pay attention tothe cyclic tensile peak curves formed by linking the tensile peakpoints in the cyclic tension tests (Figs. 6(a) and (b)). We then seethat the cyclic tension tests ofdemax 5 0.1 or 0.05 percent gavealmost the same curves as the monotonic tension test (Fig. 6(a));on the other hand, in the cyclic tension tests ofdemax 5 0.01percent, tensile peak stress came to show smaller increase thanstress in the monotonic tension test, though tensile peak stressincreased rapidly due to cyclic hardening in about the first tencycles in the cyclic tension test ofDe 5 0.8 percent (Fig. 6(b)).Moreover, this smaller increase in tensile peak stress was found tooccur more noticeably whenDe was larger in the tests ofdemax 50.01 percent (Fig. 6(b)). It is then suggested that the hysteresis

Fig. 1 Strain history in cyclic tension tests

Fig. 2 Uniaxial tensile stress-strain curves at constant strain rates; thesolid lines indicate the simulated curves to be discussed in Section 4 ofPart II

Table 1 Chemical composition of 316FR steel (wt %)

C Si Mn P S Ni Cr Mo N

0.008 0.54 0.84 0.027 0.004 11.16 16.83 2.10 0.0754

Table 2 Loading conditions in experiments

Monotonic tension e 5 5 3 1021, 5 3 1022, 5 3 1023,5 3 1024, 5 3 1025, 5 3 1026

Cyclic straining e 5 5 3 1023 De 5 0.25, 0.4, 0.8Cyclic tensione 5 5 3 1023 De 5 0.25

De 5 0.40De 5 0.80

demax 5 0.01, 0.05demax 5 0.01, 0.10demax 5 0.01, 0.10

Ratchettingsmax 5 280 s 5 10 R 5 0.0,20.5,20.75s 5 1 R 5 20.75

stress (MPa), strain (%), time (s)


loops in the cyclic tension tests opened slightly after saturation ofcyclic hardening as a consequence of cyclic relaxation of meanstress, as will be discussed later.

3.4 Ratchetting Tests. The stress-strain curves recorded inthe uniaxial ratchetting tests are shown in Figs. 7(a) to (d). Thetensile peak strain versusN relations are plotted in Fig. 8. As seen

from the figures, ratchetting proceeded without shaking downunder the four loading conditions examined in the present work.The effect ofs or viscoplasticity was marked in the tests: Largerratchetting took place in the test ats 5 1 MPa/s than in the otherthree tests, in whichs 5 10 MPa/s. The difference in ratchettingbetween the two tests ofR 5 20.75 ats 5 1 and 10 MPa/s rapidlybecame larger withN in about the first five cycles but slowly in thesubsequent cycles (Fig. 8); in other words, the effect of viscoplas-ticity was significant especially in such early cycles. The effect ofR, on the other hand, was not so significant: The three tests ofR 50, 20.5 and20.75 ats 5 10 MPa/s had nearly the same amountof ratchetting, as was observed by Yoshida (1990) on 304 stainlesssteel at room temperature.

4 DiscussionIn this section, the features observed in the cyclic tension tests

are discussed in detail in the context of simulating the ratchettingtests. Features found in the cyclic tension tests can be summarizedas follows:

(1) Stress range changed little with the increase in the numberof cycles when strain range was small.

(2) The hysteresis loops of stress and strain closed almostcompletely except for the effect of cyclic hardening. However, inthe cyclic tension tests in which the increment of maximum strainwas small, tensile peak stress came to show smaller increase thanstress in the monotonic tension test.

Fig. 3 Variation of stress range Ds as a function of accumulated inelas-tic strain p; cyclic straining between fixed limits (—) and cyclic tension(E, 1, {)

Fig. 4 Stress versus strain relations in cyclic tension tests with largeincrement of maximum strain, demax 5 0.05 or 0.10 percent, at e 5 5 31023 percent/s

Fig. 5 Stress versus strain relations in cyclic tension tests with smallincrement of maximum strain, demax 5 0.01 percent, at e 5 5 3 1023

percent/s


Finding (1) implies that for 316FR steel at room temperatureneither accumulated plastic strain nor maximum plastic strainnoticeably induces isotropic or cyclic hardening if strain range issmall; in other words, only kinematic hardening is likely to governstrain hardening at any strain if strain range is small. Thus, we cansurmise that uniaxial ratchetting proceeds without the effect ofisotropic hardening if stress range is relatively small. To confirmthis, defining the strain rangeDe in uniaxial ratchetting as illus-trated in Fig. 9, we plot the changes inDe measured in the presentfour ratchetting tests (Fig. 10). We thus find thatDe was almostconstant and at most 0.45 percent in the four ratchetting tests,thoughDe increased very weakly with the increase inN whenR 520.75. Here we remember that no noticeable cyclic hardeningtook place in the cyclic tension tests ofDe 5 0.4 percent. Finding(1) therefore leads to the important conclusion that isotropic hard-ening can be ignored when simulating the ratchetting tests done inthe present work.

Let us take note of the cyclic tension tests of 316FR steel donepreviously at 300°C and 650°C withDe 5 0.2 percent anddemax 50.1 percent (Inoue and Igari, 1995). Ohno et al. (1998) discussedthese high temperature tests and pointed out that the isotropichardening depending on maximum plastic strain was significant at650°C, whereas negligible isotropic hardening took place at300°C. This implies that finding (1) in the present work can bevalid at least below 300°C, but must be modified at temperatureshigher than 300°C.

Now we discuss finding (2). The smaller increase in tensile peakstress than monotonic tensile stress in this finding was marked inthe cyclic tension tests ofDe 5 0.4 and 0.8 percent withdemax 50.01 percent (Fig. 6(b)). Whendemax 5 0.01 percent, large numbersof strain cycling with nonzero mean strain were imposed beforemaximum strain became equal to terminal strain,e 5 2.0 percent,to interrupt the cyclic tension tests. It is cyclic relaxation of meanstress that in general becomes noticeable with the development ofsuch strain cycling. Therefore, the smaller increase in tensile peakstress than in monotonic tensile stress can be ascribed to cyclicrelaxation of mean stress. Then, as a consequence of the cyclic

relaxation, it is concluded that the hysteresis loops in the cyclictension tests had slight opening rather than complete closure,though the opening was too slight to influence the cyclic tensiontests with large increments of maximum strain shown in Figs. 5(a)to (c).

Here we note that the smaller increase in tensile peak stressdiscussed above is not attributable to cyclic softening since nodecrease in stress range occurred before interrupting the cyclictension tests ofDe 5 0.4 and 0.8 percent withdemax 5 0.01 percent(see Fig. 3).

Slight opening of hysteresis loops allows ratchetting to proceedslowly with the increase inN. Viscoplasticity also allows ratchettingto proceed, as was discussed when comparing the ratchetting tests ats 5 1 and 10 MPa/s in the preceding section. Now of interest is whichfactor was more dominant in the present ratchetting tests. The effectof viscoplasticity can be identified with the increase in strain underunloading from maximum stress, as illustrated schematically in Fig. 9.

Fig. 6 Change of tensile peak stress in cyclic tension tests in compar-ison with stress change in monotonic tension test at e 5 5 3 1023

percent/s

Fig. 7 Stress versus strain relations in ratchetting tests


Such increase of strain under unloading was pronounced in earlycycles but declined with the increase inN (Figs. 7(a) to (d)). This isin accordance with the aforementioned experimental result that thedifference in ratchetting between the two tests ofR 5 20.75 ats 51 and 10 MPa/s increased quickly withN in early cycles but slowly insubsequent cycles (Fig. 8). Thus we can say that viscoplasticity wasa dominant factor in ratchetting, especially in early cycles in thepresent tests. Slight opening of hysteresis loops, on the other hand, canoccur in any stage of ratchetting if reverse loading takes place undercyclic stressing. It is therefore appropriate to say that slight opening ofhysteresis loops became more influential than viscoplasticity as ratch-etting proceeded, except for the test ofR 5 0, in which no reverseloading took place and only viscoplasticity was a factor in ratchetting.

5 ConclusionsIn Part I of this work, the uniaxial ratchetting characteristics of

316FR steel at room temperature were studied experimentally byperforming cyclic tension tests as well as conventional tests suchas monotonic tension, cyclic straining, and ratchetting tests. Thecyclic tension tests were done systematically by prescribing theincrease in maximum strain to be small and large for three strainranges of 0.25, 0.4 and 0.8 percent, respectively. The featuresobserved in the cyclic tension tests were then discussed in thecontext of constitutive modeling for ratchetting. The main resultsof Part I are summarized as follows:

(1) Stress range changed little with the increase in the numberof cycles when strain range was small in the cyclic tension tests,and moreover strain range was almost constant in the ratchettingtests, in which strain range was at most 0.45 percent.

(2) This suggests that, for 316FR steel at room temperature,neither accumulated inelastic strain nor maximum inelastic straininduces isotropic or cyclic hardening markedly if strain range issmall; in other words, only kinematic hardening is likely to governstrain hardening at any strain if strain range is small.

(3) Stress-strain hysteresis loops closed almost completelyexcept for the effect of isotropic hardening in the cyclic tensiontests. However, in the cyclic tension tests with small increments ofmaximum strain, tensile peak stress came to show smaller increasethan stress in the monotonic tension test.

(4) The above result implies that the hysteresis loops of stressand strain can have slight opening induced by cyclic relaxation ofmean stress. It is therefore necessary to consider slight opening ofhysteresis loops as well as viscoplasticity when simulating uniaxialratchetting of 316FR steel at room temperature.

(5) It was deduced that slight opening of hysteresis loopsbecame more influential than viscoplasticity as ratchetting pro-ceeded in the uniaxial ratchetting tests with negative stress ratio.

In Part II of the paper, a new and robust kinematic hardeningmodel capable of representing slight opening of hysteresis loopswill be developed and combined with a viscoplastic equation, andthus the effects of slight opening of hysteresis loops and visco-plasticity on the ratchetting behavior reported in Part I will bediscussed quantitatively.

AcknowledgmentThe authors are grateful to Mr. Masanaka of Nagoya University

for his technical support in performing the experiments.

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Fig. 9 Schematic illustration of a stress-strain hysteresis loop understress cycling between smax and Rsmax

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N. Ohnoe-mail [email protected]

M. Abdel-Karim


Chikusa-ku, Nagoya 464-8603, Japan

Uniaxial Ratchetting of 316FRSteel at Room Temperature—Part II: Constitutive Modelingand SimulationUniaxial ratchetting experiments of 316FR steel at room temperature reported in Part Iare simulated using a new kinematic hardening model which has two kinds of dynamicrecovery terms. The model, which features the capability of simulating slight opening ofstress-strain hysteresis loops robustly, is formulated by furnishing the Armstrong andFrederick model with the critical state of dynamic recovery introduced by Ohno and Wang(1993). The model is then combined with a viscoplastic equation, and the resultingconstitutive model is applied successfully to simulating the experiments. It is shown thatfor ratchetting under stress cycling with negative stress ratio, viscoplasticity and slightopening of hysteresis loops are effective mainly in early and subsequent cycles, respec-tively, whereas for ratchetting under zero-to-tension only viscoplasticity is effective.

1 IntroductionUniaxial ratchetting, i.e., strain accumulation under uniaxial

cyclic loading with nonzero mean stress, is a fundamental phe-nomenon in cyclic plasticity and is important in designing struc-tural components. This ratchetting can occur if materials are vis-coplastic or if stress-strain hysteresis loops have opening underuniaxial cyclic loading (Chaboche and Nouailhas, 1989). Henceviscoplasticity and hysteresis loop opening are key factors inuniaxial ratchetting, especially in the context of constitutive mod-eling.

In Part I of the present work, hysteresis loop opening of 316FRsteel at room temperature was studied experimentally by perform-ing cyclic tension tests besides conventional tests such as mono-tonic tension, cyclic straining, and ratchetting under uniaxial load-ing. It was thus found that, although hysteresis loops have onlyslight opening in the saturated state of cyclic hardening, thatopening in cooperation with viscoplasticity can result in noticeableratchetting under uniaxial cyclic stressing. This suggests the ne-cessity of a constitutive model which can simulate slight openingof hysteresis loops and viscoplasticity. It was also found thatisotropic or cyclic hardening hardly develops if strain range isrelatively small under cyclic straining, so that kinematic hardeninggoverns almost all strain hardening in ratchetting if stress range isnot large.

It is not easy to simulate ratchetting accurately, since ratchettingis strain accumulation proceeding cycle by cycle. Classical con-stitutive models thus have poor capabilities for predicting ratchet-ting (Ohno, 1990, 1998). Intensive studies in the last decade,however, have shown that although the kinematic hardening modelof Armstrong and Frederick (1966) usually overpredicts ratchet-ting, modifications of the dynamic recovery term in the model areeffective for simulating ratchetting properly (Burlet and Caille-taud, 1987; Chaboche and Nouailhas, 1989; Freed and Walker1990; Chaboche, 1991; Ohno and Wang, 1993, 1994; McDowell,1994, 1995; Jiang and Sehitoglu, 1994a, 1994b, 1996; Jiang andKurath, 1996; Wang and Barkey, 1998). Among them, Ohno andWang (1993) showed that complete or almost complete closure ofhysteresis loops is represented by introducing the critical state ofdynamic recovery, which leads to little or no uniaxial ratchetting

except for the effect of viscoplasticity. If the critical state ofdynamic recovery is assumed for each part of back stress, kine-matic hardening develops piecewise linearly or piecewise almostlinearly (Ohno and Wang, 1993), and as a consequence the mate-rial parameters concerned with kinematic hardening can be deter-mined systematically from experiments (Jiang and Sehitoglu,1996; Jiang and Kurath, 1996).

The constitutive property of complete or almost complete clo-sure of hysteresis loops stated above is, as a first approximation,just an experimental characteristic observed in the cyclic tensiontests of 316FR steel at room temperature, as was discussed in PartI. This implies that a viscoplastic constitutive model into which thecritical state of dynamic recovery of back stress is incorporatedappropriately can be successful in simulating the ratchetting testsreported in Part I. Incidentally, a rate-independent constitutivemodel expressing complete closure of hysteresis loops based onthe critical state of dynamic recovery was used to successfullysimulate cyclic tension tests and thermal ratchetting tests of 316FRsteel at high temperatures where the effect of creep was supposedto be insignificant because of dynamic strain aging (Ohno et al.,1998; Kobayashi et al., 1998).

Part II is concerned with simulation of the ratchetting experi-ments reported in Part I. A new kinematic hardening model, whichcan simulate slight opening of hysteresis loops robustly, is devel-oped by furnishing the Armstrong and Frederick model with thecritical state of dynamic recovery introduced by Ohno and Wang(1993), so that the evolution equation of kinematic hardening hastwo kinds of dynamic recovery terms. The new model is thencombined with a viscoplastic equation, and the resulting constitu-tive model is applied to simulating the experiments. Thus theeffects of viscoplasticity and slight opening of hysteresis loops onthe uniaxial ratchetting of 316FR steel at room temperature arediscussed quantitatively.

2 Kinematic Hardening ModelIn this section, a kinematic hardening model is developed to

represent slight opening of hysteresis loops robustly. Here it isassumed that straine is divided additively into elastic partee andinelastic partep, i.e.,

e 5 e e 1 e p. (1)

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionOctober 1, 1998. Associate Technical Editor: H. Sehitoglu.


It is also assumed that back stressa consists ofM parts in order toaccurately express the change ofa (Chaboche and Rousselier,1983):

a 5 Oi51

M

a i . (2)

From now on (˙) denotes the differentiation with respect to timet,(:) the inner product between second rank tensors, and^ & Ma-cauley’s bracket, i.e.,x& 5 ( x 1 uxu)/ 2.

2.1 Formulation. Now let ai be the deviatoric part ofa i . Inorder to express the evolution ofai , we consider a combination ofthe Armstrong and Frederick (1966) model and the first version ofthe Ohno and Wang (1993) model. The two models have a differ-ence with respect to the dynamic recovery ofai . In the Armstrongand Frederick model, the dynamic recovery ofai operates at alltimes in proportion toai and the accumulating rate of inelasticstrain,

p 5 ~23 e p:e p! 1/2. (3)

In the first version of the Ohno and Wang model, on the otherhand, the dynamic recovery ofai is assumed to take place only ona critical surfacef i(ai) 5 0, on which the dynamic recovery ofai

is fully activated so thatai cannot develop beyond the surfacef i 50; in other words,ai is allowed to move either inside or on thesurfacef i 5 0. The surface is defined to be a hypersphere of radiusr i in the space ofai such that

f i 5 32 ai :ai 2 r i

2 5 0. (4)

Let us assume the two kinds of dynamic recovery terms men-tioned above in order to furnish the Armstrong and Frederickmodel with the critical surface of dynamic recovery, so that theoverprediction of ratchetting by the Armstrong and Frederickmodel can be overcome. Then, with Heaviside’s step functionHand Macaulay’s bracket &, the evolution equation ofai can beexpressed as

ai 5 z i @23 r ie

p 2 m iai p 2 H~ f i !^l i&ai #, (5)

wherez i andm i are material parameters, andl i is determined tohave the following form usingf i 5 0 andf i 5 0:

l i 5 e p:ai

r i2 m i p (6)

The second and third terms in the right hand side in equation (5)express the dynamic recovery ofai based on the Armstrong andFrederick model and the Ohno and Wang model, respectively,whereas the first term is responsible for strain hardening.

If f i , 0 or l i , 0, Eq. (5) is reduced to the Armstrong andFrederick model

ai 5 z i ~23 r ie

p 2 m iai p!. (7)

On the other hand, iff i 5 0 andl i $ 0, Eq. (5) becomes identicalto the following model, i.e., the first version of the kinematichardening model formulated by Ohno and Wang (1993):

ai 5 z iF2

3r ie

p 2 H~ f i !K e p:ai

r iLaiG . (8)

Therefore, Eq. (5) changes its form from Eq. (7) to (8) as soon asai becomes located on the surfacef i 5 0. Let us emphasize that iff i 5 0 and l i $ 0, ai moves on the surfacef i 5 0; otherwiseinside the surfacef i 5 0.

It will be shown in Section 3.1 that Eq. (5) represents slightopening of hysteresis loops if 0, m i ! 1. Incidentally, Eq. (8)with H( f i) replaced by a power function (a ieq/r i)

mi was shown to

represent slight opening of hysteresis loops (Ohno and Wang,1993). Herea ieq 5 [( 3

2)ai :ai ]1/ 2. Moreover, Eq. (8) withH( f i)^ep:

ai /r i& replaced by (a ieq/r i)mip was used to simulate ratchetting in

previous works (Ohno and Wang, 1994; Chaboche, 1994; Jiangand Sehitoglu, 1996; Jiang and Kurath, 1996).

2.2 Generalization. Let us normalizeai as

a# i 5ai

r i. (9)

Then Eq. (5) is cast into an alternative form

a# i 5 z i @23 e p 2 m ia# i p 2 H~ f#i !^l# i&a# i #. (10)

wheref#i 5 (32)a# i:a# i 2 1 and˙

l# i 5 ep:a# i 2 mip. Since equation (10)satisfies the consistency conditionf#i 5 0 irrespective ofr i , thisequation can be used even whenr i changes with the development ofinelastic deformation. Equations (9) and (10) thus allow Eq. (5) to begeneralized into

ai 5 z iF2

3r ie

p 2 m iai p 2 H~ f i !^l i&aiG 1r i

r iai . (11)

Addition of the last term (r i /r i)ai to Eq. (8) was proposed byTakahashi and Tanimoto (1995).

3 Features of ModelFeatures of Eq. (5) are now explained by showing schematically

the evolution of back stress under uniaxial and multiaxial loading.An efficient method of numerical integration for Eq. (5) is alsoshown as a consequence of a multiaxial feature of the model.

3.1 Uniaxial Loading. Equation (5) has an uniaxial expres-sion

a i 5 z i @r iep 2 m ia iue pu 2 H~ f i !^l i&a i #, (12)

where f i 5 a i2 2 r i

2 and

l i 5 e p:a i

r i2 m iue pu. (13)

First let us consider monotonic tensile loading. Equation (12) isthen reduced to

a i 5 H z i ~r i 2 m ia i !e p, 0 # a i , r i ,0, a i 5 r i ,

(14)

the solution of which can be expressed as

a i 5 r iF1 2 K1 21 2 exp~2m iz ie

p!

m iLG . (15)

This tensile change ofa i , which has a corner ata i 5 r i , isillustrated in Fig. 1. Two special cases ofm i 5 0 andm i 5 1 aredealt with in the figure, too. Whenm i 5 0, Eq. (5) is reduced toEq. (8); as a result,a i changes bilinearly as follows (Ohno andWang, 1993):

a i 5 r i @1 2 ^1 2 z iep&#. (16)

This relation can be derived from Eq. (15) as the limiting case ofm i 3 0. On the other hand, whenm i 5 1, Eq. (15) becomes

a i 5 r i @1 2 exp~2z iep!#. (17)

Hence, ifm i 5 1, a i approachesr i only asymptotically; in otherwords,a i does not in effect reach the critical statef i 5 0. In thiscase, therefore, Eq. (5) takes the form of Eq. (7).

Figure 2 illustrates the monotonic tensile evolution of backstressa obtained by superposinga1 to aM. As illustrated in thefigure, if 0 # m i , 1, thea versuse p relation have corners, sinceeach a i versuse p relation has a corner under tensile loading.


Especially ifm i 5 0, thea versuse p relation is multilinear (Ohnoand Wang, 1993), as shown by the dashed line in Fig. 2. Thusz i

and r i in the case ofm i 5 0 are related with the coordinates ofcorners of the multilineara versuse p relation, a (i ) and e (i )

p , asfollows (Jiang and Sehitoglu, 1996; Jiang and Kurath, 1996):

z i 51

e ~i !p , (18)

r i 5 Fa ~i ! 2 a ~i21!

e ~i !p 2 e ~i21!

p 2a ~i11! 2 a ~i !

e ~i11!p 2 e ~i !

p G e ~i !p , (19)

wheree (0)p 5 0 anda(0) 5 0.

Now let us consider such uniaxial cyclic loading that the direc-tion of e p is changed whenever back stressa reaches either themaximum or the minimum limit prescribed,amax or amin. Thechange ofa ande p under such cyclic loading is illustrated in Figs.3(a) and (b) for m i 5 0 and 0, m i , 1, respectively. It is noticedthat Eq. (12) has two kinds of dynamic recovery terms, i.e., theterm operating always and that becoming active in the critical statef i 5 0. The former causes thea versuse p hysteresis loops to beopened, as in the Armstrong and Frederick model. The latter, onthe other hand, does not contribute to the opening, since Eq. (12)with m i 5 0, i.e., the uniaxial form of Eq. (5), describes theaversuse p hysteresis loops to be multilinear and closed (Ohno andWang, 1993). Thus, the smallerm i is set, the less opening Eq. (12)gives to the hysteresis loops, giving rise to the smaller uniaxialratchetting; especially ifm i 5 0, Eq. (12) expresses no uniaxialratchetting except for the effect of viscoplasticity. Therefore thekinematic hardening model formulated in the preceding sectioncan represent slight opening of hysteresis loops robustly.

3.2 Multiaxial Loading. Let us remember that Eq. (5) al-lows ai to move either inside or on the surfacef i 5 0, and that Eq.

(5) is reduced to Eqs. (7) and (8) whenai moves inside and on thesurfacef i 5 0, respectively. This feature of Eq. (5) causes thedirection ofai to change discontinuously as soon asai reaches thecritical surfacef i 5 0, as shown schematically in Fig. 4.

The feature above implies the following efficient method fornumerical integration of Eq. (5): First, we compute the incrementof ai based on Eq. (7),Da*i , using its incremental form

Da*i 5 z i$23 r iDe p 2 m i @ai ~t! 1 uDa*i #Dp%, (20a)

i.e.,

Fig. 1 Evolution of a i under uniaxial tensile loading

Fig. 2 Evolution of back stress a and its parts under uniaxial tensileloading ( M 5 3)

Fig. 3 Hysteresis loops of back stress a and inelastic strain ep underuniaxial cycling between amax and amin ; (a) m i 5 0, (b) 0 < m i < 1

Fig. 4 Radial return mapping for numerical integration of Eq. (5)


Da*i 5

z iF2

3r iDe p 2 m iai ~t!DpG

1 1 um iz iDp, (20b)

whereD indicates increments, and 0# u # 1. Then, noticing thatEq. (5) without the critical surfacef i 5 0 gives the above incre-ment, and defininga*i such that

a*i 5 ai ~t! 1 Da*i , (21)

we setai(t 1 Dt) 5 a*i if a*i satisfiesf i # 0; otherwise we useradial return mapping so thatai(t 1 Dt) is located on the surfacef i 5 0 (Fig. 4), i.e.,

ai ~t 1 Dt! 5 r i

a*ia*ieq

, (22)

where a*ieq 5 [( 32)a*i :a*i ]

1/ 2. The radial return mapping to thecritical surfacef i 5 0 is expected to have almost the same highaccuracy as that to the yield surface of elastic-perfectly plasticmaterials examined by Krieg and Krieg (1977), because they aresimilar to each other. Equation (5) is therefore robust also in thesense that an efficient method of numerical integration is availablefor accurately representing slight opening of hysteresis loops.

4 Viscoplastic Equation and Material ParametersThis section describes the viscoplastic equation and material

parameters used for simulating the ratchetting experiments.We assume that elastic strainee obeys Hooke’s law

e e 51 1 n

Es 2

n

E~tr s!I , (23)

whereE andn indicate elastic constants,s andI stress tensor andthe unit tensor of rank two, respectively, and tr the trace. For thechange of inelastic strainep, on the other hand, we employ aviscoplastic equation with back stress

e p 53

2f~seff!

s 2 aseff

, (24)

wheres denotes deviatoric stress, andf(s eff) represents a materialfunction of effective stressseff defined as

seff 5 @32 ~s 2 a!:~s 2 a!# 1/2. (25)

The kinematic hardening model formulated in Section 2 is used torepresent the change of back stress in Eq. (24). Isotropic hardeningon the other hand is ignored in the equation, since little isotropichardening developed in the ratchetting experiments (Part I).

Then, the uniaxial tensile form of Eq. (24),e p 5 f(s 2 a),allows viscoplastic flow stress in uniaxial tensile deformation to beexpressed approximately as

s < f 21~e! 1 a~e p!, (26)

wheref21 indicates the inverse function off, anda(e p) denotes theevolution of back stressa in tensile deformation. Now let usremember that in the ratchetting experiments the hysteresis loopsopened only slightly except for the effect of viscoplasticity (Part I),

and that the smallerm i is set the less opening of hysteresis loopsEq. (5) expresses. It is therefore appropriate to assume that

m i ! 1. (27)

Equation (5) then gives almost the same evolution of back stressunder tensile loading as the multilinear one predicted bym i 5 0(see Fig. 2). As a result,a(e p) in Eq. (5) is related with the materialparametersz i and r i through Eqs. (18) and (19).

We thus determined the material parametersz i andr i as well asthe viscoplastic functionf(s 2 a) on the basis of the uniaxialtensile experiments at constant strain rates shown in Fig. 2 of PartI. First, z i and r i were determined tentatively by approximatingmultilinearly the tensile curve ate 5 5 3 1023 percent/s and byusing Eqs. (18) and (19); the functional form off(s 2 a) wassought by analyzing the strain rate dependence of 1.0 percentoffset stress. Then, the values ofz i andr i as well as the constantsin f(s 2 a) were adjusted to best fit the tensile experiments usingEqs. (2), (5), (23), and (24). This procedure led to the viscoplasticfunction and material parameters given in Table 1 and the simu-lated curves shown by the solid lines in Fig. 2 of Part I.

The material parameterm i , which was not determined from thetensile experiments, will be used to discuss the effect of hysteresisloop opening on ratchetting in the following section. It is worth-while to point out here that the parameterm i has little influence onuniaxial tensile behavior ifm i , 0.5, asshown in a specific caseof e 5 5 3 1023 percent/s in Fig. 5.

5 Simulation of ExperimentsIn Part I we discussed four ratchetting experiments, which had

the same maximum stress ofsmax 5 280 MPa but different sets ofstress ratioR (5smin/smax) and stress rates. Figures 6(a) to (d)compare the ratchetting experiments and the simulation based onthe present constitutive model, which was computed with respectto four values of the parameterm i .

To begin with, we discuss the simulation obtained withm i 5 0.This choice ofm i renders thea versuse p hysteresis loops closed

Table 1 Viscoplastic function and material parameters

Elastic E 5 1.96 3 105, n 5 0.3Viscoplastic f(s eff) 5 2.483 10210 sinh^(seff 2 75.0)/6.20&Kinematic hardening (M 5 10) z1 5 1.003 104, r 1 5 10.0

z3 5 2.503 103, r 3 5 17.3z5 5 4.003 102, r 5 5 20.6z7 5 1.003 102, r 7 5 6.40z9 5 3.333 10, r 9 5 9.00

z2 5 5.003 103, r 2 5 14.0z4 5 1.003 103, r 4 5 22.0z6 5 1.843 102, r 6 5 14.7z8 5 5.003 10, r 8 5 10.0

z10 5 2.503 10, r 10 5 80.0

stress (MPa), strain (mm/mm), time (s).

Fig. 5 Influence of parameter m i on uniaxial tensile curve at e 5 5 3 1023

percent/s


completely (Section 3.1), bringing in only the effect of viscoplas-ticity in simulating uniaxial ratchetting. It is noticed that onlyviscoplasticity or creep can induce ratchetting especially if meanstress is high relatively to stress amplitude (Chaboche and Nouail-has, 1989). As seen from Figs. 6(a) to (d), the present model withm i 5 0 well simulates the experiment ofR 5 0 but underestimatesthe other three experiments ofR 5 20.5 and20.75. In otherwords, viscoplasticity accounts well for the increase in strain underzero-to-tension cyclic loading, but only viscoplasticity is insuffi-cient for predicting the ratchetting experiments ofR 5 20.5 and20.75. It is hysteresis loop opening that can be more or lesssubstantial in uniaxial ratchetting ofR , 0, in which reverseloading takes place. It is therefore necessary to consider hysteresisloop opening as well as viscoplasticity for simulating uniaxialratchetting of 316FR steel at room temperature.

Here let us point out that ifm i 5 0, ratchetting stops eventually

under stress cycling ofR 5 20.5 and20.75 (Figs. 6(b) to (d)).This is because Eqs. (2) and (5) withm i 5 0 make mean backstress grow monotonically as mean inelastic strain increases inratchetting. Then, as mean back stress increases, the stress versusstrain hysteresis loops become symmetrized to have the sameshapes in tension and compression sides; when completely sym-metrized, viscoplasticity induces no ratchetting.

We are now in a position to discuss the effect of hysteresis loopopening, for which the parameterm i is responsible. As seen fromFigs. 6(b) to (d), a larger value ofm i gives more significantratchetting under stress cycling ofR 5 20.5 and20.75, thoughm i has little influence under zero-to-tension cyclic loading. It isemphasized that all the experiments examined are well simulatedif m i 5 0.02.This value ofm i is so small that the hysteresis loopsof a and e p can be opened only slightly to cause very slowratchetting except for the effect of viscoplasticity. This is inaccordance with the experimental fact that strain increased only byabout 0.002 to 0.003 percent per cycle in the steady state whenR 5 20.5 and20.75 (Figs. 6(b) to (d)). We have thus ascertainedthat ratchetting under stress cycling ofR , 0 is driven by bothviscoplasticity and slight opening of hysteresis loops.

Let us compare the simulations ofm i 5 0 andm i 5 0.02shownin Figs. 6(b) to (d). They start to deviate from each other afterabout twenty and five cycles whenR 5 20.5 and20.75, respec-tively. Since only the effect of viscoplasticity is incorporated ifm i 5 0, we can conclude as follows: In the experiments ofR 520.5 and20.75, viscoplasticity was effective in ratchetting insuch early cycles, whereas slight opening of hysteresis loopsbecame influential as ratchetting proceeded, as was deduced ex-perimentally in Part I.

The stress versus strain relations given by the present constitu-tive model are shown in two typical cases of uniaxial ratchetting.Figure 7 deals with zero-to-tension cyclic loading ats 5 10MPa/s. It is seen in this case that ratchetting occurs due to visco-plasticity, which is significant near tensile peak stress. It is alsoseen that the effect of viscoplasticity decreases with the increase ininelastic strain. This is because the increase in inelastic straincauses the development of back stressa, leading to the decrease ineffective stresss 2 a near tensile peak. Here we notice again thatopening ofa versuse p hysteresis loops shows no notable effect inthis case, since no reverse loading takes place, so that the value ofm i has almost no influence on the ratchetting simulated in Fig. 7,as was seen in Fig. 6(a). Figures 8(a) and (b), on the other hand,deal with the case ofR 5 20.75 ands 5 10 MPa/s, in which thevalue of m i has great influence. Here, ifm i 5 0, ratchettingdevelops only in about the first five cycles (Fig. 8(a)), but if m i 50.02, ratchetting continues to develop properly in the subsequentcycles (Fig. 8(b)). In other words, viscoplasticity and slight open-ing of hysteresis loops are, respectively, the main driving forces inthe early and subsequent cycles, as aforementioned. Incidentally,the dashed lines in Figs. 8(a) and (b) indicate the 100th hysteresisloop in the experiment. It is seen that the model simulates fairly

Fig. 6 Increase of tensile peak strain under four stress cycling condi-tions ( smax 5 280 MPa)

Fig. 7 Simulation of uniaxial ratchetting under zero-to-tension cyclicloading ( smax 5 280 MPa, s 5 10 MPa/s)


well the shape of hysteresis loops in spite of the complete neglectof isotropic hardening.

Finally, let us show an example of the simulation of cyclictension in Figs. 9(a) to (c). The experiment for this example wasdepicted in Fig. 5(b) of Part I. It is seen that the simulation greatlydepends on the value ofm i and is close to the experiment ifm i 50.02.

6 Concluding RemarksIn Part II, a new kinematic hardening model was first developed

by furnishing the Armstrong and Frederick model with the criticalstate of dynamic recovery introduced by Ohno and Wang. This ledto a model with two kinds of dynamic recovery terms, i.e., the termoperating at all times and one that becomes active in the criticalstate. It was shown that the model is capable of representing slightopening of hysteresis loops robustly, and that it has an efficientmethod of numerical integration based on radial return mapping.The model was then combined with a viscoplastic equation withback stress, and the resulting constitutive model was appliedsuccessfully to simulating the ratchetting experiments of 316FRsteel at room temperature discussed in Part I. It was thus shownthat for ratchetting under negative stress ratio, viscoplasticity andslight opening of hysteresis loops are the main driving forces inearly and subsequent cycles, respectively, whereas for ratchettingunder zero-to-tension cyclic loading only viscoplasticity is effec-tive.

The material parameterm i was taken to be identical for all theparts of back stress in the present work. Thus, for stress cycling ofnegative stress ratio, the simulation had steady states in ratchettingsoon after saturation of the effect of viscoplasticity, whereas theexperiments exhibited slow decay in ratchetting before reachingsteady states. This slow decay may be expressed if different valuesare assigned tom1 , m2 , . . . , andmM.

Isotropic hardening, which developed slightly in the ratchettingexperiments, was ignored in the simulation accordingly. In gen-eral, however, isotropic hardening takes place and, as a result, the

parameterr i may change (McDowell, 1994, 1995; Takahashi andTanimoto, 1995; Jiang and Sehitoglu, 1996). Then Eq. (10) with(9) or Eq. (11) can be used to allow the change ofr i induced byisotropic hardening.

AcknowledgmentThe authors are grateful to Drs. M. Mizuno and M. Kobayashi of

Nagoya University for their kind discussion of the results in Part II.


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AND TECHNOLOGY, Vol. 120, No. 3, pp. 230–235.


Xiao LinState Key Laboratory for

Mechanical Behavior of Materials,Xi’an Jiaotong University,

Xi’an 710049,P.R. China

Cyclic Deformation Behaviorand Dislocation Substructuresof Hexagonal Zircaloy-4 UnderOut-of-Phase LoadingMacroscopic response and microscopic dislocation structures of Zr-4 subjected to biaxialfatigue under different phase angles of 30°, 60°, 90°, and different equivalent strainranges of 0.8%, 0.6%, 0.4% were studied. The testing results show that the delay anglebetween the stress deviators and strain increment tensors is strongly dependent on phaseangle and the equivalent strain range. When phase angle equals 60°, the delay angle hasthe minimum variation range for all specimens. The mean value of the delay angledecreases with increasing phase angle or the equivalent strain range. The variation rangeand average value of the Mises equivalent stress have the maximum in S3 with the phaseangle of 90°. They decrease as the equivalent strain range decreases. Zr-4 displays apronounced initial hardening followed by a continuous softening for all specimens duringout-of-phase cycling. The stabilized saturation stresses of Zr-4 under out-of-phase cyclingare much higher than that under uniaxial cycling. It indicates that Zr-4 displays anobvious additional hardening. As the phase angle increases, the typical dislocationstructure changes from dislocation cells to tangles. The dislocation-dislocation interac-tions increase resulting in an additional hardening. In essence, the degree of additionalhardening depends, among other factors, on the maximum shear stress ratio of resolvedshear stresses and critical resolved shear stresses (RSS/CRSS).

1 IntroductionIt has been observed in a few of experimental studies that

out-of-phase strain cycling may result in cyclic hardening beyondthe extent obtained in in-phase biaxial and uniaxial tests at thesame peak effective strain level (Benallal and Marquis, 1987;Bettge, 1995; Doong, 1990, 1991; Doquet, 1990, 1993; Feaugasand Clavel, 1997; Kida and Ohnami, 1997; Krempl, 1984, 1989;McDowell, 1988; Nishino and Ohnami, 1986). Further, the degreeof addition cyclic hardening is material dependent, and it is acomplex function of the loading history. Up to now, the degree ofout-of-phase cyclic hardening can be accurately estimated for onlya limited number of loading paths. The inability of many consti-tutive equations to effectively model the out-of-phase hardeningbehavior appears to result from the lack of knowledge on micro-scopic deformation mechanism.

A recent study by Kida and Ohnami (1997) shows that a ladderor labyrinth structure is a common structure for type 304 stainlesssteel in in-phase straining and a cell in the out-of-phase straining.Doong (1990) showed that the multislip structures (cells andsubgrains) were observed in aluminum under proportional andnonproportional loading and no additional hardening was dis-played. For copper and stainless steel, single-slip structures (planardislocations, veins and ladders) were observed after proportionalloading, whereas multislip structures (cells and labyrinths) afternonproportional loading were found. The increased cyclic harden-ing of copper and stainless steel under nonproportional loading isattributed to the change of plastic deformation mode from planar towavy slip. It was suggested by Bettge (1995) that the dislocationwalls and cells were the typical substructures for IN738LC afterproportional loading, while, it is tangled owing to the multiple slipunder nonproportional loading. So far, it is widely accepted that

the main parameter governing the degree of nonproportional hard-ening in solid solution steel is the ease of cross slip (Bettge, 1995;Doong, 1990; Kida, 1997). The materials with lower stacking faultenergies exhibit a significant addition cyclic hardening under non-proportional loading. Other materials with higher stacking faultenergies have the same amount of hardening for both proportionaland nonproportional cycling (Itoh, 1992). However, all of worksare limited on cubic metals, for example copper and stainless steeletc. Especially, little is known regarding the plastic deformationbehavior of hexagonal close-packed metals under biaxial nonpro-portional loading. The slip systems are neither so numerous nor sodispersal in hexagonal metals as in cubic metals. Some anisotropicplastic deformation behavior is, therefore, expected. The objectiveof this work is to investigate the plastic deformation behavior ofZr-4 resulting from the different out-of-phase loading paths, andprovides some microscopic structural evidence of additional hard-ening.

2 Experimental ProcedureThe material chosen for this study is a nuclear grade Zircaloy-4.

The chemical composition of this alloy is (in wt%): Sn-1.5, Fe-0.21, Cr-0.1, O-0.09, Zr-balance. The initial texture analysis showsa preferential crystallographic orientation of the prism pole (Fig.1). Thin-walled tubular specimens of outside diameter 19 mm,inside diameter 16 mm, and gage length 50 mm were machinedfrom a recrystallized bar of Zr-4. The mechanical tests werecarried out on a MTS axial-torsion servohydraulic machine. Alltests were conducted at room temperature in strain control with avon Mises equivalent strain rate of 33 1023 S21. Strain on thegage length was measured via a biaxial extensometer. Both axialand torsional strains were sinusoidal waves. The phase lag, (f),between the two sinusoidal signals was 30°, 60° and 90°, respec-tively. Recording of test results was accomplished with twoX-Yplotters.

Two groups of loading modes shown in Fig. 2 were employedas a test program. The first test program is the elliptical loading

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionNovember 3, 1998; revised manuscript received August 2, 1999. Associate TechnicalEditor: D. Marquis.


paths with the same von Mises equivalent strain range and thedifferent phase angles of 30°, 60°, 90°, so as to examine the effectof the phase angles (Fig. 2(a)). The second program examines howthe equivalent strain range affects cyclic deformation behavior incircular loading paths (Fig. 2(b)). The test program is summarizedin Table 1, where it can be seen that three phase lags and threeequivalent strains were tested.

After biaxial fatigue failure, thin foils were cut from specimensperpendicular to the axial direction by a spray machine to observe

dislocation substructures. Subsequently, these foils were mechan-ically and electrolytically thinned. Finally, the dislocation sub-structures were examined using a JEOL-200CX transmission elec-tron microscope operated at 200 kV.

3 Experimental Results

3.1 Coaxiality of the Stress Deviators and the Strain In-crement Tensors. Under combined tension-torsion of the thinwalled tubular specimens, the deviator stress,s, and strain tensors,e, can be conveniently described as two dimensional vectors in thesubspaces of the full vector spaces (Xiao, 1996)

s 5 san1 1 Î3tn3 (1)

e 5 ean1 1 ~g/Î3!n3 (2)

wheres a, t are axial and shear stresses, respectively.e a, g areaxial and shear strains. The base vectorsn1 and n3 are the unitvectors along the axese a andg/=3.

A strain increment vector can be defined as

de 5 dean1 1 d~g/Î3!n3 (3)

The accumulated strain is defined as the length of strain path,Dl , shown as follows

Dl 5 Oi51

n

@~ea,i 2 ea,i21! 2 1 ~g i 2 g i21! 2/3# 1/2 (4)

Regarding biaxial cyclic deformation, the equivalent stressrange,Ds eq, and the equivalent strain range,De eq, are defined as

Dseq 5 ~s 12 1 3t 2! 1/2 (5)

Deeq 5 ~e a2 1 g 2/3! 1/2 (6)

The coaxiality of the stress deviators and the strain incrementtensors is checked along the straining paths of combined tension-torsion specimens.

The orientation of strain increment vector with respect to theunit vectorn1 is

ude 5 tan21 ~dg/Î3de! (7)

The orientation of stress vector with respect to the unit vectorn1

is

us 5 tan21 ~Î3t/s! (8)

The coaxiality of the stress deviators and the strain incrementtensors requires thatu de 5 u s. Whereas, the stress deviator is oftendelayed to the strain increment tensor under strain controlledbiaxial test. The delay angle equals

u 5 ude 2 us (9)

u can be given by

Fig. 1 {101#0} Prism pole figure of the initial Zr-4. RD-rolling direction,TD-transverse direction.

Fig. 2 Out-of-phase loading paths with different ( a) phase angles, ( b)equivalent strain ranges

Table 1 Loading modes employed in this work

No. De a, (%) Dg, (%) De*eq, (%) f, (°)

S1 0.581 0.865 0.794 30S2 0.644 1.036 0.789 60S3 0.788 1.383 0.798 90S4 0.597 1.012 0.597 90S5 0.403 0.642 0.403 90

De*eq is the maximum of Mises equivalent stress ranges for Zr-4 underout-of-phase loading;f is the phase lag between axial and torsional strainwaves.


u 5 cos21 S s z de

usideuD (10)

Figures 3(a)–(c) illustrate the variation of the delay angles withstrain path length for Zr-4 under out-of-phase loading with thesame equivalent strain range of 0.8% and different phase angles. Itis revealed that the delay angles have the maximum variationranges in S1 with the minimum phase angle of 30° during cycling(Fig. 3(a)). It is noteworthy that the variation range of delay angleshas the minimum in S2 with a phase angle of 60° (Fig. 3(b)). Thevariation range of delay angle is larger in S3 with a phase angle of

90° than that in S2, whereas, it is smaller in S3 comparing with S1(Fig. 3(c)). The larger the phase angle, the smaller the averagevalue of delay angles.

Figure 3(c) and Figs. 4(a), (b) depict the variation of delayangles of Zr-4 during cycling with the same phase angle anddifferent equivalent strain ranges. The variation range decreases,as the equivalent strain range decreases, while the average valueincreases.

It can be concluded from the above that the variation range andaverage value of the delay angles are dependent on phase angleand equivalent strain range. When phase angle equals 60°, thedelay angle has the minimum variation ranges for all the speci-mens. The mean value of delay angle decreases with increasingphase angle or the equivalent strain range.

3.2 Macroscopic Stress Response Curves During Nonpro-portional Cycling. The effective stress response,usu, vs. strainpath length curves for Zr-4 under the same equivalent strain rangeand different phase angles are depicted in Figs. 5((a)–(c)). Theapproximate sinusoidal hardening rule is obtained during circularand elliptical loading. When phase angle equals 60°, the variationrange of effective stress has the minimum value (Fig. 5(b)).Whereas, the variation range and the average value of effectivestress in S3 with phase angle of 90° have the maximum (Fig. 5(c)).They decrease as the equivalent strain range decreases at the samecircular loading paths, as shown in Fig. 5(c) and Figs. 6(a), (b).

A plot of maximum Mises effective stress in each cycle as afunction of the logarithm of the number of cycle is shown in Fig.7. Zr-4 exhibits a pronounced initial hardening followed by acontinuous softening for all specimens. As phase angle and equiv-

Fig. 3 Delay angle, u, vs. path length increment, Dl, under the sameequivalent strain range of 0.8% and different phase angles. ( a) 30°, (b) 60°and (c) 90°

Fig. 4 Delay angle, u, vs. path length increment, Dl, under circularloading with different equivalent strain ranges. ( a) 0.597%, (b) 0.403%.


alent strain range increase, the degree of hardening and softeningis increased. It is worthy to note that no saturation stages areobtained for Zr-4 under biaxial out-of-phase cycling in comparisonwith that under uniaxial cycling. In the latter, it is known that thecyclic deformation behavior of Zr-4 is characterized with threestages of cyclic hardening, saturation and cyclic softening, and theration of three stages depends on cyclic strain range and the testingtemperature (Xiao, 1998).

3.3 Cyclic Hardening Under Out-of-Phase Loading. Typ-ical cyclic stress-strain curve of Zr-4 during biaxial tension-torsioncycling is shown in Fig. 8. For comparison, the uniaxial monotonicand cyclic stress-strain curves are also included, which are quoted

from the previous work (Xiao, 1999). They were obtained at aconstant total equivalent strain rate of 33 1023 S21. Each datumpoint under uniaxial and biaxial cycling in the figure represents anindividual specimen. Mises criterion is used in the definition of theequivalent stress and strain. There is a phase lag between the axialand shear stress loads in the case of out-of-phase straining. Theequivalent stabilized cyclic stress for specimens under nonpropor-

Fig. 5 Equivalent stress response, zsz, versus path length increment, Dl,under the same equivalent strain range of 0.8% and different phaseangles. ( a) 30°, (b) 60°, and (c) 90°.

Fig. 6 Equivalent stress response, zsz, versus path length increment, Dl,under circular loading with the different equivalent strain ranges. ( a)0.597% and (b) 0.403%.

Fig. 7 Peak stress versus cyclic number curves of Zr-4 under differentphase angles and different equivalent strain ranges


tional loading is, therefore, the maximum value of the von Misesstresses. The equivalent strain range is the von Mises equivalentstrain at the place of the corresponding maximum Mises stress,because the maximum Mises stress is not synchronous with themaximum Mises strain. The testing results show that the cyclichardening behavior is displayed for Zr-4 by the fact that the cyclicstress-strain curves lie above the monotonic one at both uniaxialand biaxial cycling. Further study shows that there is an obviousdifference in the cyclic hardening levels of Zr-4 under uniaxial andbiaxial out-of-phase cycling. An over 50 percent increase in thestabilized stress has arisen, when Zr-4 deformed under out-of-phase loading in comparison with uniaxial cycling. These resultsimply that hcp Zr-4 displays an obvious additional hardening. Theamount of hardening, which is given simply by the difference ofthe stress in the stress-strain curves, increases as the equivalentstrain range increases. The biaxial test data under circular paths atthe same phase angle of 90° and different equivalent strain rangescan be described in a power-law curve. However, the degree ofadditional hardening is decreased with decreasing the phase angleat the same equivalent strain ranges. The specimen S3 with themaximum phase angles of 90° has the highest additional harden-ing. When phase angle equals 30° and 60°, the difference ofhardening degree is small between them.

3.4 Dislocation Substructures. The dislocation substruc-tures were examined for Zr-4 after biaxial cyclic deformation.Typical dislocation structures produced by deforming Zr-4 speci-mens under various phase angles and the same equivalent strainrange are shown in Figs. 9((a)–(c)). A few of elongated dislocationcells, as shown in Fig. 9(a), were observed in S1 with the phaseangle of 30°. As the phase angle increases to 60°, cell wallsbecome thick and the amount of dislocations within cells increasesin S2 (Fig. 9(b)). When the phase angle equals 90°, no slip bandsor dislocation cells were observed, dislocations are homoge-neously distributed in the volume, and dislocation tangles arewell-developed (Fig. 9(c)). As phase angle increases at the sameequivalent strain range the dislocation structures change fromdislocation cells to dislocation tangles and the homogeneity ofdislocation distribution increases.

Figure 9(c) and Figs. 10((a), (b)) show the dislocation structuresformed in Zr-4 specimens under circular loading paths with dif-ferent equivalent strain ranges. It is known that the typical dislo-cation configuration is parallel dislocation lines resulting fromprismatic slippage for Zr-4 under uniaxial cycling at RT (Xiao,1997). Comparison of the dislocation substructures of Zr-4 formedin uniaxial cycling with biaxial cycling shows that nonproportionalloading changes slip modes of dislocation from planar to wavy

(Fig. 9(c)). With increasing strain range, multi-slip structures ap-pear to occur in circular nonproportional cycling of specimen S5and S6 (Fig. 9(c) and Fig. 10(a)). In the lower equivalent strainrange, the typical dislocation structure is some individual disloca-tion lines. On close examination, some dislocation cross-slip canbe determined in Fig. 10(b). However, cells and tangled structureshave been suggested to be impossible to occur in Zr-4 underuniaxial cycling at RT, because their formations need pyramidalslip to be activated. Pyramidal slippage is much difficult for Zr-4fatigued at RT (Xiao, 1997), and takes place only at the much highstrain range and the elevated testing temperature. The major effectof the out-of-phase loading on Zr-4 is, therefore, to decrease theminimum strain range or the testing temperature for operatingpyramidal slip and forming multislip structures such as dislocationcells and tangles.

Fig. 9 Dislocation structures of Zr-4 under the same equivalent strainrange of 0.8% and different phase angles. ( a) 30°, g 5 101#0, incidentbeam i[1#21#9], (b) 60°, g 5 101#0, incident beam i[1#21#9] and ( c) 90°, g 5101#0, incident beam i[1#21#9].

Fig. 8 Influence of the degree of nonproportionality and equivalentstrain ranges on the additional hardening. A-uniaxial, M-monotonic,C-cyclic, B-biaxial


4 DiscussionUnder out-of-phase loading, cyclic hardening is stronger than

that in proportional loading. The different cyclic hardening behav-ior can be rationalized by considering that the plane of maximumshear strain (maximum slip) changes with the direction of strain-ing. This raises a question why some materials display the signif-icant additional hardening such as austenite stainless steel (Doong,1990; Kida, 1997). Whereas, the others show a slight or noadditional hardening such as aluminum alloy (Doong, 1990;Doner, 1974) for the same cubic metals under nonproportionalcycling. The additional hardening during nonproportional loadingis, therefore, material dependent. This material dependence cannotbe explained by the increase of dislocation-dislocation interactionsdue to the rotation of maximum shear planes during nonpropor-tional cycling. Another question is that stainless steel with lowerstacking-fault energy displays planar slip. However, the deforma-tion mode changes from planar to wavy slip, when the cyclic strainrange is enough large under uniaxial cycling (Xiao, 1996). In thiscase, dislocation cells can be formed under in-phase loading,which is the same as that formed under out-of-phase loading. Thus,the additional hardening can not be expected. The slip systemsdistribute homogeneously in the three-dimension space for cubicmetals. Under out-of-phase cycling, the amount and the type of theactivated slip systems are the similar as the maximum shear strainplane rotates in the space for the cubic metals with the higherstacking fault energies. Calculations have shown that in the fullyout-of-phase case usually five of the twelve possible slip systemsof face central cubic metals in each grain will experience maxi-mum Schmid factors at some point in each cycle (McDowell,1988), i.e. five independent plastic deformation modes are pro-vided to sufficient to fulfill von Mises criterion, and the differenceof the critically resolved shear stress (CRSS) between different slipsystems is small. Thus, the additional hardening cannot occur forthese materials under out-of-phase loading. Only for cubic metalswith lower stacking-fault energies, the slippage is generally re-

stricted on the stacking fault plane. The single-slip structures suchas veins are predominant under a small-strain proportional loading.The deformation mode shows an obvious anisotropic. It will ex-perience a change of plastic deformation mode from planar slipunder proportional loading to wavy slip under out-of-phase load-ing. Multi-slip and cross-slip should result in numerousdislocation-dislocation intersections, increasing the dislocation-dislocation interactions between different slip planes and promotesignificant additional hardening.

The macroscopic deformation behavior of Zr-4 under out-of-phase loading can be elucidated by considering the microscopicdeformation mechanism. Zr-4 is a hexagonal close packed metals.It has been known that only the prismatic slip is activated for Zr-4under uniaxial cycling at RT (Pochettino, 1992; Tenckhoff, 1978;Xiao, 1997). The prismatic slip only offers two independent slipmodes resulting in a planar slip character. During out-of-phasecycling, glide on the other systems like pyramidal or basal planeswith the^c& or ^c 1 a& component dislocations is expected to playa major role as a result of the maximum shear strain plane rotating.Otherwise, in Zr-4 the operable deformation system changes fromsingle slip under in-phase cycling to multislip and cross slip duringout-of-phase cycling. The number of active slip systems increasesby reducing the mean free path of the mobile dislocations. As aresult multi-slip structures such as dislocation tangles becomedominant. The dislocation-dislocation interactions between differ-ent slip planes increase. The high additional hardening is antici-pated. On the other hand, the critical resolved shear stresses(CRSS) are different in hcp metals for various slip systems. Theslip plane listed in the order of CRSS from low to high is prismaticplane, pyramidal plane and basal plane (Pochettino, 1992; Tenck-hoff, 1978; Xiao, 1997). The CRSS of pyramidal and basal slips ishigher than that of prismatic slip. Therefore, it is believed that theapplied stress to operate the new slip systems other than prismaticslip is increased. The increasing applied stress contributes to theadditional hardening. In essence, whether additional hardeningdepends, among other factors, on ratio of the resolved shearstresses (RSS) and the CRSS. As the maximum shear stress planerotates during out-of-phase cycling, deformation systems with amaximum shear stress ratio RSS/CRSS become operative. TheCRSS is different for various slip planes of cubic metals with thelower stacking fault energy and hcp metals. In this case, anobvious additional hardening is expected. For cubic metals withthe higher stacking fault energy, the CRSS of different slip systemsis approximately constant. Little additional hardening is antici-pated. The degree of additional hardening is dependent on thevariation of the shear stress ratio RSS/CRSS. When phase angle is60°, equal resolved shear stresses for different deformation sys-tems are presented. Therefore, the degree of additional hardeningis small for Zr-4 in S2. The RSS/CRSS is increased as the phaseangles and the equivalent strain range increase, the degree ofadditional hardening is, therefore, increased.

5 Conclusions

1. The stress deviators are retarded with the strain incrementtensors for Zr-4 under out-of-phase straining. The variation rangeand average value of the delay angle are dependent on the phaselag and the equivalent strain range. The delay angles have themaximum variation ranges with the minimum phase angle of 30°.When the phase angle equals 60°, the variation of delay angles hasthe minimum. The larger the phase angle, the smaller the averagevalue of delay angles. The variation range of delay angles de-creases as the equivalent strain range decreases, while the averagevalue increases.

2. Macroscopic stress response curves show that the approx-imate sinusoidal hardening rule is obtained for Zr-4 during ellip-tical and circular cycling. The variation ranges and mean values ofeffective stresses have the maximum, when phase angle equals90°. They decrease as the equivalent strain range decreases at thesame circular loading paths.

Fig. 10 Dislocation structures of Zr-4 under circular loading with thedifferent equivalent strain ranges. ( a) 0.597%, g 5 101#0, incident beami[2#42#3] and ( b) 0.403%, g 5 101#0, incident beam i[1#21#1].


3. Zr-4 exhibits a pronounced initial hardening followed by acontinuous softening for all specimens. As the phase angle andequivalent strain range increase, the degree of hardening andsoftening increases. No saturation stage is obtained for Zr-4 underbiaxial out-of-phase cycling in comparison with uniaxial cycling.

4. An obvious additional hardening is displayed for Zr-4 underout-of-phase cycling by the fact that the cyclic stress-strain curvelies above the uniaxial cyclic one. The hardening degree increasesas the equivalent strain range and the phase angle increase.

5. The typical dislocation structures change from dislocationcells to dislocation tangles and the homogeneity of dislocationdistribution increases, as the phase angle increases. It remains thedislocation tangles under circular paths at the different equivalentstrain ranges. The smaller the equivalent strain range, the lower thedislocation density. The major effect of the out-of-phase loadingon Zr-4 is to decrease the minimum strain range or the testingtemperature for operating pyramidal slip and forming multislipstructures such as dislocation cells and tangles.

AcknowledgmentsThe financial support from the National Natural Science Foun-

dation and the Nuclear Industrial Science Foundation of China isgratefully acknowledged. I am also grateful for the financial sup-port provided by the State Key Laboratory of Structural Strengthand Vibration, Xi’an Jiaotong University.

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Magnus Ekh

Anders Johansson1

Hans Thorberntsson2

B. Lennart Josefson

Department of Solid Mechanics,Chalmers University of Technology,

SE-41296 Goteborg, Swedene-mail: [email protected]

Models for Cyclic RatchettingPlasticity—Integration andCalibrationThree well-known ratchetting models for metals with different hardening rules werecalibrated using uniaxial experimental data from Bower (1989), and implemented in theFE code ABAQUS (Hibbitt et al., 1997) to predict ratchetting results for a tension-torsionspecimen. The models were integrated numerically by the implicit Backward Euler rule,and the material parameters were calibrated via optimization for the uniaxial experimen-tal data. The algorithmic tangent stiffnesses of the models were derived to obtain efficientFE implementations. The calculated results for an FE model of the tension-torsionspecimen were compared to experimental results. The model proposed by Jiang andSehitoglu (1995) showed the best agreement both for the uniaxial and the structuralcomponent case.

1 IntroductionThe rolling contact fatigue between railway wheels and rails

may result in initiation of fatigue cracks on, or close to, the surfaceof the rail head. These surface cracks may eventually grow into therail head, and also possibly branch, which may ultimately lead tosudden breakage of the rail. This form of damage is not associatedwith material or manufacturing defects or imperfections. It israther a consequence of the increased loads (both magnitudes andintensity) in modern rail traffic. The annual cost for rail removaland renewal including preventive grinding is very high, hencethere is a strong need for methods that can predict the occurrenceof rolling contact fatigue cracks in rails. This, in turn, calls forefficient simulation tools that use constitutive relations that canaccurately model the material behavior in the rail head during thepassing of railway wheels.

For a material subjected to cyclic loading with non-zero meanload, ratchetting and/or shakedown may occur. These phenomenaare described by among others Beynon and Kapoor (1996) andChaboche and Nouailhas (1989). Ratchetting in the material occurswhen deformation continues to grow during each cycle. Thisenlarges the deformations in the material, and may lead to initia-tion of cracks. For some materials the deformation growth slowsdown, stabilizing to a cycle where the material behaves completelyelastic. This is called elastic shakedown. If the deformation growthstabilizes to a cycle where the material behaves both elasticallyand inelastically, then the shakedown is denominated plastic.

Realistic macroscopic material models for the description of theratchetting characteristics in metals can be obtained within theframework of plasticity, using the von Mises yield criterion com-bined with the Armstrong and Frederick (1966) type of nonlinearhardening rule (A-F). However, a well known fact (see McDowell,1985; Bower, 1989; Chaboche and Nouailhas, 1989) is that theA-F hardening rule, in its pure form, is often insufficient to modelcyclic material behavior. Therefore, Chaboche et al. (1979) in-cluded several kinematic hardening variables that follow the A-Fhardening rule (besides one linear kinematic hardening variable) inthe plasticity model. Other modifications of the kinematic harden-

ing rule have been suggested by, for example, Bower (1989),Chaboche (1991), Jiang and Sehitoglu (1996a, b), Ohno and Wang(1993) and McDowell (1985), to mimic the experimentally ob-served decaying ratchetting rate in engineering materials.

With increasing complexity of the phenomena modeled, thenumber of parameters and the nonlinearity in the mathematicalmodels used tend to increase. These models consist of nonlineardifferential equations, which in general must be solved numeri-cally with some incremental algorithm. The implicit BackwardEuler algorithm is favored by authors, such as Ortiz and Popov(1985), Chaboche and Cailletaud (1996), for large incrementsbecause of its stability and accuracy properties. Since the Back-ward Euler method is implicit, a nonlinear system of equationsmust be solved in each increment, which is often referred to as thelocal problem.

Calibration of the mathematical models, i.e., determination ofthe model parameters, can be done by either a mechanistic ap-proach or a simultaneous determination, as described by Mahnkenand Stein (1996). In the mechanistic approach the parameters aregiven a physical interpretation, and a sequence of ideal experi-ments are performed (see Jiang and Sehitoglu, 1996b). The exper-iments are ideal in the sense that they are designed to activate onlya few of the parameters in the model. In the simultaneous deter-mination, a sufficient number of independent experiments aredone, and all the parameters in the model are determined from aleast-squares fit.

In order to simulate the ratchetting behavior in a structuralcomponent, the material model must be implemented into anFE-code. Since the model is nonlinear, an iteration procedure onthe global (FE) level is necessary. An efficient iteration procedureis obtained by using the algorithmic tangent stiffness (ATS), whichgives the good second order local convergence properties of aNewton method (cf. Simo and Taylor, 1985).

Previous works describe the material models for ratchetting.However, it has not focused on the use in realistic FE-analyses ofstructures, hence the numerical integration technique and the cor-responding local and global problems have not been addressedsufficiently.

In this paper we investigate three different ratchetting models.We integrate these models by using the Backward-Euler technique,and show how to solve the local system of equations. Then thematerial parameters are determined by a least-squares fit to uni-axial experimental data. Furthermore, we describe a general tech-nique for deriving the ATS-tensor by using results from the localproblem. Finally, the models are implemented in the commercialFE-code ABAQUS (Hibbitt et al., 1997), and results from tension-

1 Present Address: Frontec Research Technology AB, Gårdava¨gen 1, SE-41250Goteborg, Sweden.

2 Present Address: VOLVO Information Technology AB, SE-40508 Go¨teborg,Sweden.

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionNovember 2, 1998; revised manuscript received September 9, 1999. Associate Tech-nical Editor: H. Sehitoglu.


torsion simulations of a specimen are compared to experimentaldata. This tension-torsion test is designed to resemble the loadingin a rail head during the passing of a railway wheel (cf. Bower andJohnson, 1989, 1991).

2 Description of the Models

2.1 Preliminaries. Small strains are assumed in the formu-lation of the models. Hence, the total straine can be additivelydecomposed into an elasticee and an inelastic strainep

e 5 e e 1 e p (1)

Hookes law defines the linear relation between the elastic strainand stress in the material1

s 5 E e : e e whereE e 5 2GI dev 1 Kbd ^ d (2)

whereG is the shear modulus, andKb is the bulk modulus.The stresss, the backstressX and the dragstressK define the

stress state in the material. The material is elastic whenF(s, X,K) , 0 and plastic whenF(s, X, K) 5 0, where the yieldfunction F is defined according to von Mises as

F~s, X , K! 5 Î32 utdevu 2 K 2 sY with tdev 5 sdev 2 X ,

utdevu 5 Îtdev : tdev (3)

wheresY is the initial uniaxial yield stress.The development of the plastic strainep follows the associative

flow rule

e p 5 l Î32 ndev andndev 5

def tdev

utdevu(4)

where the plastic multiplierl was introduced.The yield condition can be summarized in Kuhn-Tucker form as

l $ 0, F # 0, Fl 5 0 (5)

2.2 Kinematic Hardening Laws. The different evolutionlaws for the kinematic hardening that we examine in this paper, areall based on the nonlinear Armstrong and Frederick (1966) law

X 5 l~cÎ23 ndev 2 gX ! (6)

wherec andg are material parameters. Wheng 5 0, the kinematichardening is purely linear. On the other hand, ifg Þ 0, this law isnonlinear and gives a constant ratchetting rate for a stress con-trolled cyclic test with constant stress amplitude and nonzero meanstress (if no other type of hardening is included).

The kinematic hardening rule according to Bower in Table 1includes an additional state variableY, which gives the possibilityto model decaying and finally arresting ratchetting rate.

Finally, the kinematic hardening rule according to Jiang andSehitoglu (1995) is summarized in Table 2. This rule includesseveral backstressesX (i ), and the total backstressX is obtained

simply by adding allX (i ). The evolution law for the backstressesmakes it possible to simulate a decreasing ratchetting rate. Notethat this kinematic hardening law has been further developed andrefined, (see Jiang and Sehitoglu, 1996a, b). By introducing aso-called memory surface, the effect of a sudden change of thestress magnitude in a cyclic experiment can be accurately mod-elled.

2.3 Isotropic Hardening Law. We express the nonlinearevolution law for the isotropic hardening as

K 5 lbS1 2K

K`D (7)

whereK` is the saturated drag stress due to the isotropic harden-ing, and the parameterb governs the initial rate of the isotropichardening.

2.4 Ratchetting Models. We investigate three differentratchetting models below. In the first model, which we call A, theArmstrong-Frederick kinematic hardening law (6) and the isotro-pic hardening law (7) are combined. The isotropic hardening lawis needed in this model to simulate decreasing ratchetting rate. Thismodel is included in the commercial FE package ABAQUS (Hib-bitt et al., 1997). In both, what we have called, model B and J-Sonly kinematic hardening is considered. The evolution laws forthese two models are described in Tables 1 and 2.

3 Integration of Models

3.1 General Procedure. In this section we describe a pro-cedure to apply the implicit Backward Euler (BE) integrationtechnique, in a strain controlled fashion, to model A, B, and J-S.This step is performed in each Gaussian point in the elements of anFE mesh to determine the stress state for a given strain increment.

In order to evaluate whether the material response is elastic orplastic the Kuhn-Tucker type of conditions (5) are used. Weassume that the response is elastic whenl 5 0, and thereforeDl 5 n11l 2 nl 5 0, therefore the hardening stresses do notchange, i.e.n11K 5 nK and n11X 5 nX. This is true ifF # 0, andhence a new strain step can be taken. However, ifF . 0 theassumption does not hold, and a set of nonlinear equationsF( x) 50 must be solved. We refer to this system of equations as the localproblem. Note that the local problem for model J-S can be reducedto 7 1 2M equations, thus dim (F) 5 7 1 2M. In the case ofconstant exponentsx (i ) a reduction to 71 M equations is possible(see Chaboche and Cailletaud, 1996). The unknownsx are deter-mined by the Newton-Raphson method

x ~k11! 5 x ~k! 2 ~J! 21 n11F whereJ 5 n11F

x ~k! (8)

1 The operators “:” and “R” defines the contractiona : b 5 aij bij and the openproduct (a R b) ijkl 5 aij bkl, respectively.

Table 2 Kinematic hardening rule due to Jiang andSehitoglu (1995)

Development of the backstressesX (i ):

X ~i ! 5 Î3

2lc ~i !S r ~i !ndev 2 H uX ~i !u

r ~i ! J x ~i !

X ~i !D , i 5 1, 2, . . . ,M

Definition of the exponentsx (i ):

x ~i ! 5 x 0~i ! F 2 2 ndev :

X ~i !

uX ~i !uG , i 5 1, 2, . . . ,M

The total backstressX:

X 5 Oi51

M

X ~i !

Material parameters: r (i ), c0(i ), x 0

(i ), i 5 1, 2, . . . , M

Table 1 Kinematic hardening rule according to Bower (1989)

Development of the backstressX:

X 5 l~cÎ23 ndev 2 g1~X 2 Y !!

Development of the additional variableY:

Y 5 lg2~X 2 Y !

Material parameters: c,g 1,g 2


After the local problem has been solved, the stress and the hard-ening stresses can be updated, and a new strain step can be taken.The local problems and the stress updating are summarized inTable 3 and Table 4 for model J-S and in Appendix A for modelA and B. Note that the local problem of model A and B can bothbe reduced to 1 scalar equation, see Tables A.1 and A.3.

4 Models Calibration

4.1 Calibration Procedure. The ratchetting models are cal-ibrated against a uniaxial stress controlled experiment for a cylin-drical specimen of rail steel (BS11A). The experiment was carriedout by Bower (1989). In the experiment the stress in the steelvaries in a saw-tooth fashions (t) with a small offset valuesm,and the output is given as the strain valuese# for m time valuest i ,i 5 1, . . . , m.

The calibration is done by minimizing the objective functionf,which we define as the square root of the sum of the squareddistances between the experimental strain valuese# (t i) and thecorresponding calculated straine(t i)

f~k! 5 ÎOi51

m

~e# ~t i ! 2 e~t i , k!! 2 (9)

wherek is the vector of material parameters.We minimize the objective function by applying the Downhill

Simplex optimization algorithm (Nelder and Mead, 1965), whichis available in the program Matlab (The Math Works, 1996).

The method proposed by Jiang and Sehitoglu (1996b) to deter-mine the material parameters for model J-S is not applicable here,since the type of experimental data we use is not the same (in ourcase all parameters are determined at the same time from one setof experiment).

4.2 Calibration Results. To find the global minimum of theobjective function, different initial guesses of the material param-eters were tried. This was particularly important for model J-S(M 5 3), which we assume is because of the large number ofmaterial parameters used. When calibrating this model we foundthat no variation of the yield stress was needed, and therefore weuseds y 5 400.1 MPawhich was obtained by Bower (1989).

The results of the calibrations are summarized in Table 5, andthe experimental data (dotted lines) and the results from thecalibrated models (solid lines) are shown in Figs. 1–3. Young’smodulusE was for all the models chosen as the value from Bower(1989), i.e.,E 5 2.0982 z 1011 Pa.

Figures 1 and 2 show that neither the A model nor the B modelcan simultaneously model the nonlinear behavior of the materialand the correct ratchetting rate. However, if only one cycle is usedin the calibration a better fit is obtained to that particular cycle, butthen the ratchetting rate and the fit to the other cycles are far fromsatisfactory, which is illustrated in Fig. 4. The isotropic hardening

Table 4 Updating stresses and backstresses for model J-S

Updating of the stresses:

n11sdev 5 n11s deve 2 Î6n11ndevGDl, n11sm 5 nsm 1 3KbDem

n11X ~i ! 5 aX~i !~ nX ~i ! 1 Î3

2 c ~i !r ~i !Dl n11ndev!,n11X 5 O

i51

M

n11X ~i !

Table 3 Local problem for model J-S

Nonlinear system of equationsn11F 5 0:

n11F 5 3Î3

2 u n11tdevu 2 syn11tdev 2 Î2

3 syn11ndev

x 0~i !@2u n11X ~i !u 2 aX

~i !~ndev : nX ~i ! 1 Î32 c ~i !r ~i !Dl!# 2 n11x ~i !u n11X ~i !u, i 5 1 . . . M

aX~i !u nX ~i ! 1 Î3

2 c ~i !r ~i !Dlndevu 2 u n11X ~i !u, i 5 1 . . . M 4where

n11tdev 5 n11s deve 2 O

i51

M

aX~i !nX ~i ! 2 Dl Î3

2 ~2G 1 Oi51

M

aX~i !c ~i !r ~i !! n11ndev

n11s deve 5 nsdev 1 2GDedev, aX

~i ! 5 S 1 1 c ~i !Dl Î3

2 H u n11X ~i !ur ~i ! J n11x ~i !D 21

Unknown internal variablesx:

x 5 @Dl n11ndevu n11X ~1!u. . .u n11X ~M!u n11x ~1!. . .n11x ~M!# T

Table 5 Summary of the parameter values from the calibrations and the cycles used for the different material models

Model Parameters Optimized cycles

A c 5 6.49 GPa,g 5 0.81,b 5 10.7 MPa 15, 50,100, 200, 400 and 600K` 5 22.8 MPa, s y 5 543 MPa

B c 5 6.61 GPa,g1 5 1.28 15, 50, 100, 200, 400 and 600g2 5 0.56,s y 5 550 MPa

J-S c(1) 5 476, r (1) 5 126 MPa,x0(1) 5 0.91 1, 2, 15, 50, 100, 200, 400 and 600

c(2) 5 75.6, r (2) 5 63.4 MPa, x0(2) 5 7.8

c(3) 5 8.02, r (3) 5 81.4 MPa, x0(3) 5 3.1

s y 5 400.1 MPa(fixed)


in the A model causes the desired decreasing ratchetting rate anddecreasing width of the stress-strain cycles. In Fig. 3 model J-Sshows the best representation of both the curve shape, the width ofthe curves and the ratchetting rate (accumulated strain). We un-derline that the results from this comparison are not surprising,since the models differ much in complexity and number of modelparameters. Nevertheless this comparison motivates why a morecomplex model such as model J-S must be used. There are othercompetitive models available in literature (see Chaboche, 1991;Ohno and Wang, 1993).

To illustrate the ability to describe the correct ratchetting rate,the mean straine a for the calibrated models A and J-S are shownin Fig. 5. Model B is for clarity not included in the figure, since itsresult almost coincides with the result for model A. We define themean strain as the mean value of all the strain values in each cycle.The agreement between the experimental values (o) and the cal-ibrated models is satisfactory for all the models. Note that thisbehavior can not be simulated using Eq. (6) withg 5 0, that isusing a linear kinematic hardening rule.

5 Finite Element Implementation

5.1 Preliminaries. Following a standard finite element pro-cedure, the structure is divided into elements, each with an as-sumed displacement field. A linearization of the Principle of Vir-tual Displacements and use of the Newton-Raphson method yieldsthe global system of FE-equations to be solved iteratively for thedisplacement increment in one time (load) step (Bathe, 1996).

n11K ~i21!DU ~i ! 5 n11R 2 n11F ~i21! (10)

wheren11K (i21) is the tangent stiffness matrix,n11R is the consis-tent nodal force (which is assumed to be known beforehand) andn11F (i21) is the internal nodal force vector consistent with thepresent stress field in each element.

Information about the material behavior must be supplied in thetangent stiffness matrix and the internal nodal force vector. It shouldbe noted that to obtain the full convergence benefits of the Newton-Raphson method in the solution of the global FE-equations, an accu-rate tangent stress-strain matrix# consistent with the numerical stressintegration used in derivingF must be used. This is accomplished byemploying the algorithmic tangent stiffness.

5.2 Algorithmic Tangent Stiffness. The ATS-tensor is de-fined as:

# 5dn11s

dn11e(11)

To obtain the ATS-tensor, we first summarize the stress update forthe three ratchetting models using the following expression

n11s 5 n11smd 1 n11s deve 1 f ~x~ n11e!! (12)

Fig. 2 Cyclic stress-strain response for model B after calibration

Fig. 1 Cyclic stress-strain response for model A after calibration

Fig. 3 Cyclic stress-strain response for model J-S with M 5 3 aftercalibration

Fig. 4 Cyclic stress-strain response for model A after calibration usingmeasure points from cycle 600 only

Fig. 5 Accumulated strain versus number of cycles for model A and J-S


where the functionsf( x( n11e)) are obtained from Table 4, TableA.2, and Table A.4. Hence, the ATS-tensor becomes

# 5 E e 1fx

x

n11e(13)

where x/ n11e is derived from the local problem

F~ n11e, x~ n11e!! 5 0, ; n11e f

dF

dn11e5

F

n11e1 J

x

n11e5 0 f

x

n11e5 2J21

F

n11e(14)

5.3 FE Simulation of Tension Torsion Test. The conditionin the material of a cylindrical specimen which is subjected tocombined tension (extension) and torsion, can be assumed (Bowerand Johnson, 1989, 1991) to be similar to the condition in thematerial of rail heads subjected to combined rolling- and slidingcontact by railway wheels. To evaluate the prediction capability ofthe ratchetting models in such conditions, an FE model of acylindrical specimen was created in IDEAS (SDRC, 1994). Thegeometry of the FE model and the applied load are shown in Fig.6 and Fig. 7, respectively.

The FE model consists of two parts, one which is assumed tobehave elastically through the whole load history and anotherwhere the plastic deformation occurs. The elastic part is built up oftwo beam elements and thin shell elements (rigid interface ele-ments), which are connected to 20-nodes solid elements(C3D20R). A reduced integration technique is used to decrease theCPU-time for the analysis. The iteration technique used for findingthe global equilibrium is the Newton-Raphson method or theQuasi-Newton method (Hibbitt et al., 1997). Moreover, whensolving the system of equations in (10), a direct wavefront solverwas employed, denoted “wavefront” in Table 6 below. In somecases a direct multi wavefront sparse solver (Hibbitt et al., 1997)was used, denoted “sparse” in Table 6. For the convergencecriteria, we have chosen the default values in ABAQUS for theresiduals. However, in Table 6 the tolerance of largest residual tothe corresponding average flux norm is smaller (1028) than thedefault value (1024) (Hibbitt et al., 1997).

The load is applied to the beam node at the end of the specimen

as prescribed displacement and torque. In total, the FE-modelcontains approximately 1100 elements, 5000 nodes and 15,700degrees of freedom. Furthermore, there are 50 load cycles and ineach cycle 40 number of timesteps are taken. Also, it should benoted that geometrically linear conditions are assumed.

Figures 9–11 show the calculated displacement and rotation ofthe node where the load was applied. Only model J-S gives asatisfactory agreement between prediction and experimental datafrom Bower (1989), see Fig. 8.

Although the trend from the calculations is quite clear, there aresome limitations and uncertainties in the FE-calculations whichcould contribute to the poor correspondence between experimentsand predictions by use of model A and B. The approximation withthe elastic beam and shell elements in the FE model was created toreduce the computation time. Further, the results of the computa-tions were found to be dependent on the chosen timestep, and thepredictions must therefore be seen as approximate. It is also notedthat, for model J-S, the timestep had to be decreased during theanalysis to achieve convergence of the global iteration.

Table 6 summarizes the CPU time on an HP 735 unix workstation for the FE calculations of one load cycle. It is seen, asexpected, that use of an elastic stiffness matrix leads to excessivelylarge computational times for this case of a strongly weakening

Fig. 8 Experiment data from Bower (1989)

Fig. 6 The geometry of the FE-model of the specimen

Fig. 7 Load cycle of tension and torsion, which approximates the stresscycle of surface elements in rolling and sliding contact

Table 6 CPU-times depending on solution method(ABAQUS5.7). * 5 convergence problems, calculationstopped, ** 5 memory problems, calculation stopped

ModelIterationmethod Stiffness

CPU-time,tolerance5 1e 2 8

J-S Newton ATS 2 h 22 min (sparse)M 5 3 Newton Elastic 30 h 25 min (sparse)

QuasiNewton Elastic 2 h 23 min (wavefront)QuasiNewton ATS 1 h 17 min (wavefront)

J-SM 5 1 Newton ATS 3 h 30 min (wavefront)B Newton ATS 1 h 26 min (sparse)

Newton Elastic 93 h 11 min (sparse, **)QuasiNewton Elastic 20 h 59 min (wavefront, *)QuasiNewton ATS 32 min (wavefront)

A Newton ABAQUSinternal

39 min (sparse)

QuasiNewton ABAQUSinternal

31 min (wavefront)

Fig. 9 Result from FE-calculation with model A


material in the loading phase. For model J-S the results forM 53 are displayed. Model B and A (ABAQUS internal version) uselower computational times, primarily since a larger reduction ofthe local problem is possible and that fewer state variables need tobe stored in every Gauss point. For a given material model theQuasi-Newton method combined with the use of the ATS-tensorseems to be the best choice.

6 Concluding RemarksFrom the graphs in Section 4.2, we conclude that the capability

of model A and B to mimic the ratchetting rate and the shape of thestress strain cycles in a uniaxial experiment is not satisfactory. Thisconclusion is in agreement with other references (see McDowell,1985; Chaboche and Nouailhas, 1989) experience, at least formodel A. However, the more complex J-S model, with a largernumber of material parameters, gives satisfactory results. Thereare other sophisticated models available in literature (Chaboche,1991; Ohno and Wang, 1993), which are not evaluated in thispaper.

The material parameters in the models were determined viacalibration using the uniaxial experimental data. To determine thecorrect parameters, it is necessary to obtain enough independentdata by using measure points from several cycles, see Fig. 4. Still,for model J-S the optimal set of material parameters may bedifficult to find. This indicates that the total number of parametersshould be kept reasonably low, that is using a low value forM inTable 2. In addition, the reduced local problem arising whenapplying the fully implicit Backward Euler rule becomes moreinvolved, i.e., more unknowns and state variables are needed, formodel J-S as compared to model A and B, which increases the totalcomputational time as seen in Table 6.

Also in the FE prediction of a tension-torsion experiment on acylindrical specimen, with the calibrated material parameters,model J-S showed the best capability to mimic experimentalresults. To perform the FE calculations efficiently the algorithmictangent stiffness was derived for the models. Still there are nu-merical uncertainties in the results due to the limited computerpower, which limits the realistic number of applied timesteps.

Finally one may note two limitations of the present approachwhich may be subject to further improvement. The three-dimensional FE-simulation, which aimed at reflecting a realisticloading situation, was based on values for entering material pa-rameters determined from a simpler uniaxial simulation. This wasdue to the limited knowledge of the local behavior during thetension-torsion test, but mainly to the desire to reduce the com-plexity in the optimization process for the determination of thematerial parameters. Moreover, the present paper employs engi-neering stresses and strains only. This seems to be a reasonableapproximation based on the levels of the stresses and strains in theuniaxial and three dimensional experiments used for comparison.Formally, large displacement, or large strain approach could havebeen employed, but this was avoided to reduce the computer timeneeded in the simulations. It is believed that the qualitative resultsin the present paper will hold also when, for example, largedisplacements are used.

AcknowledgmentsWe wish to acknowledge financial support from the Brite Euram

III-project ICON (Integrated study of rolling CONtact fatigue).Valuable advice and support with the FE calculations from Dr.Hans Bjarnehed at Frontec Research and Technology (Go¨teborg,Sweden) are gratefully acknowledged.


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Fig. 10 Result from FE-calculation with model B

Fig. 11 Result from FE-calculation with model J-S


A P P E N D I X A

Local Problem and Stress Update for the Chaboche and the Bower Model

A.1 Local Problem.

Table A.1 Local problem for model A

Nonlinear equationn11F 5 0:

n11F 5 =32 u n11t devu 2 s y 2 n11K

where

u n11tdevu 5 u n11sdev 2 aXnXu 2 2

3(3G 1 caX)Dl,

n11sdeve 5 nsdev 1 2GDedev

aX 5 ~1 1 gDl! 21, n11K 51

1 1 Dlb~ nK 1 DlbK` !

Unknown internal variablex:x 5 Dl

Table A.2 Updating stresses and backstresses for model A


n11sdev 5 a1n11s dev

e 1 ~1 2 a1!aXnX ,

n11sm 5 nsm 1 3KbDem

n11X 5 a2n11s dev

e 1 ~1 2 a2!aXnX ,

n11K 51

1 1 Dlb~ nK 1 DlbK` !

where

a1 5 1 2Î6GDl

u n11s deve 2 aX

nX u , a2 5Î2

3 cDlaX

u n11s deve 2 aX

nX u

Table A.3 Local problem for model B

Nonlinear equationn11F 5 0:

n11F 5 Î32u n11tdevu 2 sy

where

u n11t devu 5 u n11sdeve 2 bX

nX 2 (1 2 bX) nYu 1 23 (3G 1 cbX)Dl

n11s deve 5 nsdev 1 2GDedev, bX 5

1 1 g2Dl

1 1 ~g1 1 g2!Dl

Unknown internal variablex:

x 5 Dl

Table A.4 Updating stresses for model B


n11sdev 5 b1n11sdev

e 1 (1 2 b1 )bXnX 1 b2

nY,

n11sm 5 nsm 1 3KbDem

n11X 5 b3n11sdev

e 1 (1 2 b3)bXnX 1 b4

nY,

n11Y 5 bY(nY 1 g 2Dl n11X)

where

b1 5 1 2Î6GDl

u n11s deve 2 bX

nX 2 ~1 2 bX! nY u , b2 5 ~1 2 b1!~1 2 bX!

b3 5Î2

3 cDlbX

u n11s deve 2 bX

nX 2 ~1 2 bX! nY u ,

b4 5 ~1 2 b3!~1 2 bX!, bY 51

1 1 g2Dl


Hyo-Jin Kim

Jae-Jin Jung

R&D Center,Korea Heavy Industries &

Construction Co., Ltd.,555 Guygok-Dong,

Changwon, 641-792, Koreae-mail: [email protected]

Model of Grain DeformationMethod for Evaluation of CreepLife in In-Service ComponentsThe creep life can be evaluated by the degree of grain deformation since the grains ofCr-Mo base material deform in the direction of stress. The grain deformation methodusing image-processing technique is developed for life assessment of in-service hightemperature components. The new assessment model for the method is presented to applyto in-service components and is verified by interrupted creep test for ex-serviced materialof 1Cr-0.5Mo steel. The proposed model, which is irrespective of stress direction, is toevaluate mean of the absolute deviation for the measured aspect ratios of the grainssections.

IntroductionLife extension of fossil fuel power plants beyond their classi-

cally defined design life is being practiced today by many utilities.The emergence of this after-market has lead to recent advances incomponent inspection, monitoring techniques, and remaining lifeassessment techniques. Failure of header and steam piping ofboiler is induced by material degradation and/or creep damage.The main damage mode in HAZ regions is grain boundary cavi-tation. It can cause the component failure if cavities coalesce incracks. In order to perform the assessment on these regions,A-parameter technique for quantifying cavitation mainly is applied(Roberts et al., 1989). In contrast, the primary damage occurring inCr-Mo base material is material softening or header-body swelling.In the material, cavities nucleate at the very last stage of creep life.Therefore, a qualitative assessment (Roberts et al., 1989) fromcoarsening of carbides or quantitative grain deformation method(Soji, 1994) is applied to remaining life assessment on the mate-rial.

Grain deformation method is applied to materials in which thedamage due to intergranular creep is predominant. The method isnondestructive technique and easy to apply to base material ofin-service high temperature component. But the method by mi-croscopy is time consuming and subjects to errors for measuringsize and deformation of grain. In this paper, image-processingtechnique is introduced to contribute to accuracy and speediness ofthe measurement. The new assessment model of the method ispresented and confirmed. For validation of the model, creep testsare performed using standard testpieces of recovered and ex-serviced materials. Replicas are taken on the specimen at intervalsof 500–1000 hours, which are repeated by the specimen rupture.From the prepared replicas, grain deformation is measured, andcreep damage and microstructures are examined.

Experimental Procedure

Specimen. The specimen is 1Cr-0.5Mo steel from a boilersuperheater header which have experienced 180,000 hours ofoperation at 538°C. To reproduce the original microstructure of thesteel, a portion of specimens is reheat-treated according to themanufacture’s initial heat treatment schedule. To evaluate themechanical properties, tensile and creep tests are performed for thereheat-treated and ex-serviced materials. The results of tensile test

are presented in Table 1 and are compared with the requirement forstrength of 1Cr-0.5Mo steel (13CrM044). It is seen that the reheat-treated material satisfies the requirement for mechanical proper-ties. The results of creep rupture test are shown in Fig. 1. In theLarson Miller Parameter ofx-axis,T andt r are operating temper-ature and rupture time, respectively. It is seen that the creepstrength for ex-serviced material reduces considerably and that thecreep strength for reheat-treated material is identical with that fornew material of NRIM (Tanaka, 1990). Therefore, it is confirmedthat the reheat-treated material is substituted for new material. Thereheat-treated material is called the recovered material in thispaper. Specimens for replication are mechanically polished.

Creep Test. Creep tests are performed using standard test-pieces of recovered and ex-serviced materials. Replicas are takenon the specimen at intervals of 500–1000 hours, which are re-peated by the specimen rupture. The replica (ASTM, 1990) isproduced by coating one side of a sheet of plastic film with asuitable solvent, such as acetone or methyle acetate, and applyingthe wetted side of the film to the prepared specimen surface. Byreplicas taken, grain deformation is measured, and creep damageand microstructures are examined.

Image-Processing TechniqueMeasuring the degree of deformation for a great deal of grains

by microscopy is time-consuming and subjects to errors. Hereimage-processing technique (Vrhovac et al., 1996; Louis et al.,1995) is introduced to contribute to accuracy and speediness forthe measurement of grains and analysis. It is assumed that thetwo-dimensional orientations represent three-dimensional grainorientations. The assumption is valid for the microstructures of acylindrical component that deforms in the direction of hoop oraxial. In this paper, the technique is applied to header and steampiping of boiler that the deformation in the direction of hoop oraxial is predominant.

Image Input System and Input. Input of metallurgical mi-crostructure images is performed by several methods, which arefrom specimen, replica, or scanner. Here images are inputted fromreplicas taken on the specimens of interrupted creep tests. Theimage-processing system consists of input apparatus such as mi-croscopy and CCD camera, main system such as A/D converterand controller, and output apparatus such as printer. The degree ofgrain deformation can differ from positions of specimen in theinterrupted creep tests. To obtain precise data, images are acquiredat constant position of specimen and input of images is performedusing the microscopy with automatic transfer equipment and thecontroller that can control image-processing program.

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials Division June8, 1998; revised manuscript received July 9, 1999. Associate Technical Editor:S. K. Datta.


Image Early Processing. The image which computer canrecognize is made from digital image obtained from CCD cameravia early processing. To measure grain deformation, grain bound-ary need to be clearly distinguished. But grain boundary is notclear by several factors such as material, material degradation,etching and replica conditions. To best image, several image-processing techniques are used. Filtering operations reduce orincrease the rate of change that occurs in the intensity transitionswithin an image. Arithmetic and logic operations upon the activeimage are performed between two image, or between an image anda constant value. Background correction is used to better distin-guish image background from image objects, making it easier toextract the objects during a counting or measurement operation.

Data Measurement. To measure grain deformation, principalstress axis is determined after early processing and grain anglesabout principal stress axis,um, are measured for grains, as shownin Fig. 2(a). The proposed model, which is irrespective of stressdirection, is to measure the ratios that are diametrical maximum tominimum dimensions for grains, as shown in Fig. 2(b). For aconsistent data measurement, a standard area with constant bound-ary is applied for input images. Also, standard deviation methodand mean method of the absolute deviation are applied for pro-cessing of measured data.

Grain Deformation MethodCreep damage occurring in base material generally is material

softening or deformation before creep cavities. A new assessment

technique, which is not a quantitative assessment method for creepcavities, need to be developed for the ductile material such as basematerial of Cr-Mo steel. In this paper, grain deformation method,which is a quantitative method for the degree of grain deformation,is introduced. The microstructure of base material for low alloyferrite Cr-Mo steel consists of ferrite plus bainite (or pearlite).Creep damage is induced during a component continues at thehigh-temperature and constant load. Grain does not deform atlow-ductile bainite and deforms toward principal stress axis athigh-ductile ferrite. As the deformation degree of grains indicatescreep life expenditure, a quantitative assessment for the degree isgrain deformation method. The method can evaluate more exactremaining life of base metal with master curve that gets from creeptests in the laboratory. The application range of the method extendsthrough development of assessment model. The model (Soji, 1994)assesses creep damage by evaluating the standard deviation fordeformation angles about principal stress axis since grains deformtoward the principal stress axis.

The quantification of grain deformation is applied using image-processing technique after replicas are taken, as shown in Fig. 3.Figure 4 shows distribution of grain deformation using image-processing technique for recovered material and material of creeplife fraction, 0.87. It is seen that distribution of deformation anglesfor grains is random in the case of recovered material and the

Table 1 Mechanical properties

PropertyMaterial

Yield strength(MPa)

Tensile strength(MPa)

Elongation(%)

Recovered 362.8 541.3 29.3Ex-serviced 273.6 492.3 33.0DIN 17175:13CrMo44 275 min. 440–590 19.0 min

Fig. 1 Creep-rupture strength of recovered and ex-serviced materials

Fig. 2 Measurement of grain deformation

Fig. 3 Measurement procedure of grain deformation using image-processing technique

Fig. 4 Distribution of grain deformation using image-processing tech-nique


ex-serviced material for the end of life has normal distributioncentering around the principal stress axis. The master curve areobtained by evaluating the standard deviation for distribution ofgrain deformation and is in the temperature range of 500°C to650°C and low stress, as shown in Fig. 5. The established modelcan subject to many errors because the principal stress axis isdetermined in a component. In this paper, the new model ispresented to apply to the creep life assessment for the componentsirrespective of principal stress axis. The proposed model is toevaluate mean of the absolute deviation for the measured ratioswhich are diametrical maximum to minimum dimensions forgrains. During interrupted creep tests are performed at 550°C and14 kg/mm2 for ex-serviced material, mean of the absolute devia-tion for the measured ratios which are diametrical maximum tominimum dimensions for grains is shown in Fig. 6. The deforma-tion distribution for an early stage shows a little uniform patternand the absolute deviation for the end of life grows large as thegrains deform largely. Therefore, it is confirmed that the proposedmodel can be applied to the creep life assessment of base materialfor in-service components.

Results and DiscussionFor recovered material, Fig. 7 shows the degree of grain defor-

mation by the established model and Fig. 8 shows the degree ofgrain deformation by the proposed model during interrupted creep

tests are performed at 538°C and 16 kg/mm2. It is shown thatstandard deviation is 46 before creep test and is 41 after creeprupture. Also, it is shown that mean of absolute deviation is 0.42before creep test and is 0.44 after creep rupture. From the results,it is shown that the deformation distribution for grains makes nodifference between an early stage and the end of creep life. It isseen that the degree of grain deformation is very small becausematerial degradation is not induced and intergranular fracture isinduced by high stress and low temperature. To obtain actualresults in the case of recovered material, It need to be lower stress,high temperature, and long-term creep test.

For ex-serviced material, Fig. 9 shows the degree of graindeformation by the established model and Fig. 10 shows the degreeof grain deformation by the proposed model during interruptedcreep tests are performed at several temperatures and stresses. Thestandard deviation is 45 before creep test. It means that thedirections of grain boundaries are disordered. Because the thick-ness of the boiler header is thick and the applied stress is verysmall, the directions of grain boundaries are disordered as thoughthe header have experienced 180,000 hours of operation. Thestandard deviation for the end of life is 20 to 30. Also, it is shownthat mean of absolute deviation is 0.42 before creep test and is 0.7to 0.9 for the end of life. The degree of grain deformation for theend of life of ex-serviced material is large because grains deformeasily by material degradation. The material degradation is due tothe increase of distance between precipitates according to coales-

Fig. 5 Relation between creep life fraction and deformation index

Fig. 6 Distribution of deformation index for creep life fraction of ex-serviced material at 550°C and 4 kg/mm 2

Fig. 7 Distribution of deformation index for creep life fraction of recov-ered material at 538°C and 16 kg/mm 2

Fig. 8 Distribution of deformation index for creep life fraction of recov-ered material at 538°C and 16 kg/mm 2


cence and coarseness of precipitate, or decrease of solute insidematrix by long term operation. From these results, it is seen thatstandard deviation and mean of absolute deviation change as thecreep life decreases. Therefore, it is confirmed that the proposedmodel of grain deformation method can be applied to the creep lifeassessment for in-service components.

The actual conditions of creep behavior for components ofpower plants are similar to these of tests at low stress and hightemperature, as shown in Figs. 9 and 10, for ex-serviced material.It is appropriate to make master curve under the conditions. On thebasis of the master curve, the introduction of image-processing issignificantly contributing to accuracy and speediness of the lifeassessment for in-service high temperature components. Exactresults can be obtained in the laboratory by the established modelbecause grain deformation by the model is measured at specimensof uniaxial creep test which principal stress axis is unidirectional.But principal stress axis must be determined in the actual compo-nent, which can subjects to errors. In this paper, the new model ispresented to apply to the creep life assessment for the componentirrespective of principal stress axis.

ConclusionsThis study is performed to characterize the correlation between

grain deformation and creep life assessment for in-service high

temperature components. The image-processing technique is in-troduced to contribute to accuracy and speediness of the assess-ment for the components such as header and steam piping of boilerthat the deformation in the direction of hoop or axial is predomi-nant. From results, the new model is presented to apply to thecreep life assessment for the components irrespective of principalstress axis and confirmed.

ReferencesRoberts, B. W., Ellis, F. V., and Henry, J. F., 1989,Remaining-Life Estimation of

Boiler Pressure Parts, Volume 4: Metallographic Models for Weld-Heat-AffectedZone,EPRI CS-5588, pp. 37–39.

Roberts, B. W., Avernue, K., and Askins, M. C., 1989,Remaining-Life Estimationof Boiler Pressure Parts, Volume 3: Base Metal Model,EPRI CS-5588, pp. 22–23.

Shoji, T., 1994,Material Degradation & Life Prediction,REALIZE INC., pp.179–186.

Tanaka, C., 1990,NRIM Creep Data Sheet,No. 35A, NRIM, pp. 5–9.ASTM Standard, 1990, “Standard Practice for Production and Evaluation of Field

Metallographic Replicas,”Annual Book,E1351-90, pp. 953–958.Vrhovac, M., Barata, J. C. R., and Rodrigues, R., 1996, “Automation of Micro-

structure Analysis by Artificial Intelligence Image Processing,”Proceedings of Fail-ures ’96,pp. 297–312.

Louis, P., and Gokhale, A. M., 1995, “Application of Image Analysis for Charac-terization of Spatial Arrangements of Features in Microstructure,”Metallurgical andMaterials Transaction,Vol. 16, pp. 1449–1456.

Fig. 9 Distribution of deformation index for creep life fraction of ex-serviced material

Fig. 10 Distribution of deformation index for creep life fraction of ex-serviced material


S. MurakamiProfessor.

A. MiyazakiGraduate Student.

M. MizunoAssistant Professor.

e-mail [email protected]


Furo-cho, Chikusa-ku,Nagoya 464-8603, Japan

Modeling of IrradiationEmbrittlement of ReactorPressure Vessel SteelsA model to describe the change in the inelastic and fracture properties of reactorpressure vessel steels due to neutron irradiation in the ductile region (i.e., irradiationembrittlement) is developed. First, constitutive equations for unirradiated elastic-viscoplastic-damaged materials are developed within the framework of the irrevers-ible thermodynamics theory. To take into account the effect of hydrostatic pressure onthe nucleation and growth of microvoids, properly defined dissipation potential isused. Then, the effect of irradiation on the material behavior is incorporated into theproposed model as a function of neutron fluenceF by taking into account theinteraction between irradiation-induced defects and movable dislocations. As regardsthe damage strain threshold pD, the mechanism of void nucleation due to pile-up ofdislocations at the inclusions in the material is proposed first under unirradiatedcondition, and then the effect of irradiation on the mechanism is formulated. In orderto demonstrate the validity of this model, it is applied to the case of uniaxial tensileloading of a low alloy steel A533B cl. 1 for the pressure vessel use of light-waterreactors at 260°C. The resulting model can describe the increase in yield stress andultimate tensile strength, the decrease in total elongation and strain hardening, andthe strain rate dependence of yield stress due to neutron irradiation.

1 IntroductionThe structural materials used in nuclear reactors are subject to

neutron irradiation during their operations, and the resultingknocking out of the constituent atoms leads to the change ininternal structure of the material and thus to the change in itsmechanical properties. Embrittlement of pressure vessel steels oflight-water reactors due to the irradiation is, in particular, a crucialissue to be overcome from the viewpoint of the reliability andintegrity of the reactors (Little, 1976; Phythian and English, 1993).So far, a considerable number of papers (Wronski et al., 1965;Little, 1976; Kussmaul et al., 1990; Mansur and Farrell, 1990;Erve and Tenckhoff, 1993; Gage and Little, 1994; Little and Gage,1994; Scott, 1994) have focused on the change of solely a singlematerial property due to the irradiation, e.g., the rise of ductile-brittle transition temperatures (Phythian and English, 1993), theincrease of yield stress (Makin and Minter, 1960; Hunter andWilliams, 1971; Williams and Hunter, 1973; Steichen and Wil-liams, 1975; Phythian and English, 1993), etc. Similar studies onmaterials employed for fast reactors and fusion reactors have beenalso reported (Klueh, 1993; Klueh and Alexander, 1995, 1992;Klueh and Maziasz, 1992; Klueh and Vitek, 1991; Odette andLucas, 1992, 1991). Most of these studies deal mainly with thechange of ductile-brittle transition behavior due to the irradiation,and hence unified theory to describe the deformation and fractureof irradiated materials in the ductile region is, so far, not available.

The present paper is concerned with the mechanical modeling ofthe process of damage, embrittlement and strain hardening ofpressure vessel steels for light-water reactors under neutron irra-diation. A model is developed in the framework of continuumdamage mechanics to describe a unified process of embrittlement,damage growth and elastic-plastic deformation. First, a dissipationpotential, which can represent the effect of hydrostatic pressure ondamage evolution, is adopted on the basis of the irreversible

thermodynamics theory for constitutive equations, and the consti-tutive equations for an unirradiated material are formulated. Then,in order to extend the constitutive equations to an irradiatedmaterial, effects of irradiation-induced defects on the materialbehavior are taken into account from the viewpoint of the micro-scopic mechanisms. Finally, the resulting constitutive equationsare applied to uniaxial tensile loading of a light-water reactorpressure vessel steel, ASTM A533 grade B class 1, and the validityof the model is discussed in comparison with the correspondingexperiments.

2 Mechanisms of Irradiation EmbrittlementWhen a crystalline material is irradiated by a neutron flux, a

number of atoms are knocked out from the lattice sites to create theFrenkel defects comprised of equal numbers of interstitial atomsand vacancies. A part of the interstitial atoms and vacancies thuscreated annihilates by their recombination and by the absorptionby the sink. The point defects escaping such annihilation make upthe irradiation-induced defects or the clusters affecting the materialproperties. The irradiation-induced defects will act as an obstacleor a dislocation source for the dislocation motion on its slip planes,and bring about the irradiation hardening. This irradiation harden-ing will in turn cause the phenomena of irradiation embrittlement(Little, 1976; Phythian and English, 1993; Klueh and Alexander,1995, 1992); i.e., the reduction of fracture strain, the rise ofductile-brittle transition temperature, and the reduction of uppershelf energy in the ductile region. The rate of nucleation of Frenkeldefects, however, decreases with the increasing dose of irradiation,and the change of material properties is gradually saturated (Wron-ski et al., 1965; Little, 1976; Klueh, 1993; Oddete and Lucas,1992, 1991).

Besides the above-mentioned damage due to knocking out ofatoms, irradiation damage caused by nuclear conversion has alsobeen observed. However, regarding ASTM A533 grade B class 1steels discussed below, the effects of nuclear conversion are notsignificant, and thus we will disregard the effects of nuclearconversion in the following discussion.

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials Division April13, 1998; revised manuscript received July 9, 1999. Associate Technical Editor:S. K. Datta.


3 Modeling of Irradiation EmbrittlementPhenomena of irradiation embrittlement can be modeled as the

effects of the irradiation on the constitutive and the damage evo-lution equations of materials. If the neutron irradiation does notcause significant change in the mechanisms of deformation andfracture, these effects may be formulated by representing thematerial constants in these equations as functions of neutron flu-ence. Namely, we will first develop the constitutive and damageevolution equations for unirradiated materials. Then, by takingaccount of internal structural changes brought about by the neutronirradiation discussed above, the material functions to describe theeffects of the irradiation will be formulated.

3.1 Constitutive Equations for Unirradiated Materials.The deformation of ASTM A533 grade B class 1 steels depends onthe rate of strain. At elevated temperature around 300°C, the A533steels show ductile fracture as a result of the nucleation and growthof microvoids within the material (Shockey et al., 1980). Thedescription of such a process of deformation and fracture can befacilitated by incorporating the notion of continuum damage me-chanics (Lemaitre, 1985, 1992; Lemaitre and Chaboche, 1990;Saanouni et al., 1994) into a general thermodynamical frameworkof visco-plastic constitutive equations (Lemaitre and Chaboche,1990; Saanouni et al., 1994).

We will now assume that the material is isotropic linear elastic-viscoplastic at the initial undamaged state. Since the main purposeof the present paper is to predict a uniaxial tensile behavior, onlyan isotropic hardening variable is considered as an internal vari-able for strain hardening caused by viscoplastic deformation. Theisotropic hardening variable can represent also the effects of strainaging. However, since the experimental results necessary to dis-cuss the effects of strain aging of A533 steel are lacking, the agingeffects are not taken into account in the present paper. By suppos-ing that damage development is isotropic, a free energyC ofdamaged elastic-viscoplastic material may be formulated as fol-lows:

rC~e e, r , D! 5 12 l~1 2 D!~tr e e! 2 1 m~1 2 D! tr ~e ee e!

1 12 qQ~1 2 D!r 2, (1)

where ee, r and D represent the elastic strain tensor, isotropichardening variable and the damage variable, respectively, whileqand Q are material constants associated with strain hardening.Furthermore, the symbolr represents the density, andl andm areLame’s constants.

In order to derive inelastic constitutive equations and evolutionequations of internal state variables, the relevant dissipation po-tential should be formulated. The dissipation due to the inelasticdeformation of polycrystalline materials is caused mainly by dis-location motion caused by stress, while the dissipation due todamage evolution is primarily caused by the internal energy re-lease due to the nucleation and growth of microvoids (Lemaitreand Chaboche, 1990; Lemaitre, 1992). Thus by taking account ofthe difference in the mechanisms of these dissipation phenomena,two dissipation potentials,w*in for the inelastic deformation andw*Dfor the damage evolution, will be employed in the present paper.

According to the power function proposed by Saanouni et al.(1994), we have the dissipation potentialw*in as follows:

w*in~s, R; r , D! 5K

n 1 1K 1

K F f 11

2

R2

Q~1 2 D!

21

2q2Q~1 2 D!r 2GL n11

, (2)

where s represents stress tensor, andR is the thermodynamicconjugate force associated withr . The symbolsK and n arematerial constants to represent strain rate sensitivity, andf is ayield function given as follows:

f 5seq 2 R

~1 2 D! 1/2 2 k, (3)

Nomenc la tu re

B1, B2 5 material constantsD 5 damage variable

Dc 5 critical value of damage vari-able

E 5 Young’s modulusf 5 yield functionI 5 second-rank unit tensork 5 initial yield stress

k0 5 material constant to representthe value ofk in unirradiatedcondition

k1, k2 5 material constantskF 5 material function correspond-

ing to kK 5 material constant associated

with strain rate sensitivityK 0 5 material constant to represent

the value ofK in unirradiatedcondition

K 1, K 2 5 material constantsKF 5 material function correspond-

ing to Kl 5 distance between obstacles

l 0, l F 5 distance between obstacles inunirradiated and irradiated con-ditions

n 5 material constant associatedwith strain rate sensitivity

n9 5 number of pile-up disloca-tions

n0, nF 5 number of pile-up disloca-tions in unirradiated and irra-diated conditions

N 5 number of defects per unitvolume

p 5 magnitude of inelastic strainpD 5 threshold of inelastic strain

for damage initiationpD0, pDF 5 threshold of inelastic strain

for damage initiation in unir-radiated and irradiated condi-tions

q, Q 5 material constants associatedwith strain hardening

q0 5 material constant to representthe value ofq in unirradiatedcondition

q1, q2 5 material constantsqF 5 material function correspond-

ing to qr 5 isotropic hardening variableR 5 thermodynamic conjugate

force associated withrs, S 5 material constants associated

with damage evolutionY 5 thermodynamic conjugate

force associated withDa 5 material constant

b 5 material constant associatedwith damage evolution

g 5 material constantDs y 5 increase in yield stress

e 5 total strain tensoree 5 elastic strain tensore in 5 inelastic strain tensorh 5 material constant

l, m 5 Lame’s constantsn 5 Poisson’s ratior 5 densitys 5 stress tensor

s9 5 deviatoric stress tensors c 5 critical stress of the initiation of

debondingsEQ 5 equivalent stress

t 5 shear stress acting on slip planet0 , tF 5 plastic flow stress/shear stress

acting on slip plane in unirradi-ated and irradiated conditions

w*D 5 dissipation potential for damageevolution

w*in 5 dissipation potential for inelasticdeformation

F 5 neutron fluenceC 5 free energy of damaged elastic-

viscoplastic material


whereseq andk represent the equivalent stress and the initial yieldstress, respectively.

As regards the dissipation potentialw*D, on the other hand, thefollowing potential proposed by Lemaitre (1985) will be em-ployed:

w*D ~Y; p, D! 5S

s 1 1 S2Y

S D s11 p

~1 2 D! b , (4)

whereY is thermodynamic conjugate force associated withD, prepresents the magnitude of inelastic strain rate, andS, s andb arematerial constants with respect to damage evolution.

According to the usual procedure of irreversible thermodynam-ics (Lemaitre, 1985, 1992; Lemaitre and Chaboche, 1990; Saa-nouni et al., 1994), Eqs. (1)–(4) furnish the constitutive equationsfor unirradiated material as follows:

e 5 e e 1 e in, (5a)

e e 51

1 2 D S1 1 n

Es 2

n

Etr sID , (5b)

e in 53

2

s9

seqp, (5c)

R 5 ~1 2 D!qHQ 2R

~1 2 D! 1/2J p 2R

1 2 DD, (5d)

D 5 HS2Y

S D s p

~1 2 D! b , p $ pD ,

0, p , pD ,(5e)

p 51

~1 2 D! 1/2 K f

KL n

, (5f)

2Y 5 12 l~tr e e! 2 1 m tr ~e ee e! 1 1

2 qQr2, (5g)

wheree ande in represent the total strain tensor and inelastic straintensor, respectively, whileE and n are Young’s modulus andPoisson’s ratio. The symbolsI ands9 represent a second-rank unittensor and deviatoric stress tensor, respectively, andpD is a thresh-old of inelastic strain for the damage evolution (Lemaitre, 1985,1992). According to the notion of damage mechanics, we willfurther assume that material ruptures when the damage variableDattains to the critical valueDc (Lemaitre, 1985, 1992).

3.2 Effects of Neutron Irradiation on Material Properties.By noting that the essential mechanisms of inelastic deformationand damage do not change due to irradiation, the inelastic consti-tutive equation and the damage evolution equation of irradiatedmaterial may be given only by modifying the material constants inthe corresponding equations of the unirradiated material as mate-rial functions of neutron fluenceF. In the following discussion, wewill represent the effects of neutron irradiation on the materialconstants in view of the mechanisms of the internal structuralchanges due to irradiation.

3.2.1 Young’s Modulus.The elastic deformation is, in gen-eral, a structure-insensitive phenomena, and thus is little sensitiveto microstructural defects such as vacancies and dislocations inmaterials (Barrett et al., 1973); we will assume that Young’smodulusE is not affected by neutron irradiation.

3.2.2 Initial Yield Stress. As already described in Section 2,the irradiation hardening is caused mainly by the obstruction ofdislocation motion by irradiation-induced defects, and the sourcehardening and friction hardening have been proposed as theirmajor mechanisms (Little, 1976). The former takes place becauseirradiation-induced point defects are attracted to the vicinity ofdislocations due to the diffusion, and thus trap the dislocations.The latter, on the other hand, is a kind of dispersion strengtheningmechanisms by which the point defects or their clusters interact

with movable dislocations as dispersed obstacles. Thus mechanismof the irradiation hardening will be modeled in the present paperby noting that irradiation hardening is primarily governed by thelatter mechanism.

The increase in yield stressDs y brought about by the dispersionstrengthening is inversely proportion to the distancel betweenobstacles (Barrett et al., 1973). In view of the saturation of thedefect nucleation with the increase of fluence, the relation betweenthe number of defects per unit volumeN and the fluenceF isformulated as follows (Little, 1976):

N 5 a@1 2 exp~2gF!#, (6)

wherea and g are material constants depending on the materialand the irradiation condition. By noting dispersion strengtheningmechanism and Eq. (6), Makin and Minter (1960) proposed thefollowing relation for the increase of yield stress due to irradiation:

Dsy 5 h@1 2 exp~2gF!# 1/2, (7)

whereh and g are constants depending on the material and theirradiation conditions. If we apply Eq. (7) to the initial yield stressk, the increase ofk due to irradiation may be expressed as follows:

kF 5 k0$1 1 k1@1 2 exp~2k2F!# 1/2%, (8)

wherek0, k1 andk2 are material constants, andk0 represents thevalue ofk in the unirradiated condition.

Though the materials for experiments are different from A533steels, the increase and saturation of yield stress due to neutronirradiation in Cr-Mo steels and stainless steels have been reportedby Klueh (1993) and Odette and Lucas (1992, 1991).

3.2.3 Strain Hardening. Strain hardening of crystalline ma-terials is attributable to the increased resistances to the dislocationmotion due to the change of the dislocation configurations and theincrease in dislocation density (Barrett et al., 1973). In addition tothese mechanisms, since irradiation-induced defects act as theresistance to the dislocation motion, the hardening rateq of Eq.(5d) in irradiated materials becomes larger than that in unirradiatedmaterial. Then if we assume that the increase of dislocation densityis proportional to the number of irradiation defects on the slipplane and the effects will be saturated for large dose of irradiation,the hardening rateq may be expressed as follows:

qF 5 q0$1 1 q1@1 2 exp~2q2F!# 1/2%, (9)

whereq0, q1 andq2 are material constants, andq0 represents thevalue ofq when unirradiated.

Though the effects of neutron irradiation on strain aging can beincorporated into Eq. (9), the effects and mechanisms of aging inA533 steels have not been elucidated experimentally. Thus, theeffects of strain aging are not considered here. As regards thematerials for fast reactors and fusion reactors, i.e., mainly Cr-Mosteels, the effects of aging were reported by Klueh and Maziasz(1992) and Klueh and Vitek (1991).

As regards the magnitude of saturated hardeningQ in Eq. (5d),on the other hand, the reduction of strain hardening by neutronirradiation have been observed in the experiments of A533 steel(Hunter and Williams, 1971; Steichen and Williams, 1975). How-ever, this decrease in hardening may be caused by the reduction ofthe threshold for the damage evolution due to irradiation. Thus, wewill disregard the effect of neutron irradiation onQ.

3.2.4 Strain Rate Dependence.The reduction of strain ratesensitivity by neutron irradiation is observed in A533 steel(Steichen and Williams, 1975). Since the irradiation defects act asobstacles to the deformation, they reduce the effect of the thermalactivation on the strain rate. If the cause of the reduction in ratesensitivity may be found in irradiation defects on the slip plane,Kcan be formulated in the same way as inq:

KF 5 K0$1 2 K1@1 2 exp~2K2F!# 1/2%, (10)


whereK 0, K 1 andK 2 are material constants, andK 0 represents thevalue ofK when unirradiated.

3.2.5 Threshold for Damage Evolution.Now let us discussthe irradiation effects on the threshold inelastic strainpD of Eq.(5e) for the damage initiation. As shown in Fig. 1, the thresholdstrain for damage evolutionpD is supposed to decrease in irradi-ated materials.

A number of dimples have been observed on the rupture sur-faces of unirradiated A533 steel, and relatively large nonmetallicinclusions such as Al2O3 and MnS have been detected inside of alarge dimples. Many small dimples supposed to be generated atsubmicron particles like Fe3C have been also observed between thelarge dimples (Shockey et al., 1980). Nonmetallic inclusionsAl 2O3 , MnS, etc., in general, have weak bonding with the matrixbecause of low wetting, while particles such as carbide and nitridehave strong bonding. Accordingly, debonding of large inclusionsAl 2O3 , MnS, etc. from the matrix causes the initiation of damagein the ductile region. Such debonding is frequently caused by localstress concentration within the material (Tetelman and McEvily,1970). In the case of A533 steel, we suppose a mechanism inwhich dislocation motion is prevented by the so large inclusionsand the pill-up of the dislocations increase the local stress.

In order to evaluate the local stress at the end of the dislocationspiled up, we will employ the result of Koehler’s calculation(1952). Namely, according to the calculation, whenn9 edge dis-locations pile up on the slip plane subjected to the shear stresst,the magnitude of the tensile stress which is parallel to the slipplane at the end of the pile-up dislocations is given by:

s 5 n9t. (11)

In order to discuss the effects of irradiation on the damageinitiation, we take a representative volume element (RVE) ofunirradiated and irradiated materials at the initiation of damage asshown in Fig. 2. The figure shows the migration of a dislocationline on a slip plane. Since the rupture strain of A533 steel isrelatively large even under irradiated condition (Hunter and Wil-liams, 1971), the dislocation line goes through the inclusionsaround which dislocation loops remain as shown in the figure. Insuch case, edge dislocations with different signs will pile up onboth sides of inclusion on the same slip plane, and thus the pile-updislocations causes local tensile stress on the interface. Since theirradiation-induced defects are distributed randomly within thematerial, the effect of irradiation on the interface between thematrix and the inclusion is not significant and can be neglected.

Thus the debonding in the irradiated materials occurs when thetensile stress on the interface due to the pile-up dislocations attainsto the critical stress of the initiation of the debonding in unirradi-ated materials c. Then, the condition for the initiation of thedebonding may be expressed from Eq. (11) as follows:

sc 5 n0t0 5 nFtF , (12)

where n0, t 0, nF and tF represent the number of the pile-updislocations around the inclusion at the initiation of the debondingand the shear stress acting on the slip plane in unirradiated andirradiated materials, respectively.

In irradiated materials, the dislocation motion is hindered bymeans of the dispersion strengthening mechanism due to theirradiation-induced defects. Namely, the mean length betweenobstacles in an irradiated materiall F is smaller thanl 0 in anunirradiated material because of the existence of irradiation-induced defects as shown in Fig. 2. Accordingly, the application ofthe theory developed in 3.2.2 to the plastic flow stresst0 andtF inFig. 2 leads to the following relation:

tF 5 t0@1 1 B1$1 2 exp~2B2F!% 1/2#, (13)

where B1 and B2 are material constants. By substituting thisequation into Eq. (12), we have the following relation between thenumbers of pile-up dislocations at the inclusions before the initi-ation of debonding in unirradiated and irradiated materials:

n0 5 nF @1 1 B1$1 2 exp~2B2F!% 1/2#. (14)

Then Eq. (14) furnishes the relation between the inelastic strain ofRVE of the unirradiated and irradiated materials (Barrett et al.,1973) before the initiation of debonding:

pDF 5 pD0@1 1 B1$1 2 exp~2B2F!% 1/2# 21. (15)

This equation provides the threshold inelastic strain for the nucle-ation of voids in unirradiated and irradiated materials. The rela-tions amongt 0, tF, pD0 andpDF in the equation are depicted inFig. 1.

In the above discussion, the threshold for the initiation ofdamage in an irradiated material was specified by the magnitude ofthe inelastic strain on a single slip plane. However, in reality, thedebonding may be caused by the pile-up dislocations on a multi-tude of slip planes around inclusions (Broek, 1973). In these cases,since the dislocation motion is not affected by the hydrostaticpressure and is governed mainly by shear stress, the above discus-sion will be developed with respect to the maximum shear stress.Furthermore, since the effects of the irradiation have been incor-porated qualitatively in the above discussion, the above theory canbe applied only to the slip plane which has the largest effect on theinitiation of debonding. Therefore, the threshold inelastic strain(15) for the initiation of damage in an irradiated material may beapplied to general three dimensional states of stress.

As regards material constants besides those discussed above, theexperimental results to discuss the effects of irradiation on themare quite lacking. Thus, these material constants will be assumed tobe unaffected by irradiation.

The effects of temperature also can be taken into account easilyin the framework of the thermodynamic constitutive theory. How-ever, the main purpose of the present paper is to formulate the

Fig. 1 Schematic stress-strain curves of unirradiated and irradiatedmaterials

Fig. 2 RVE of unirradiated and irradiated materials at damage initiation


constitutive equation to describe the change in mechanical prop-erties under tensile deformation of A533 steels due to neutronirradiation by the use of continuum damage mechanics. Thus, wedon’t consider here the temperature effects to avoid complicatedformulation and discussion.

4 Application to Pressure Vessel Steel of Light-WaterReactor

Let us now apply the constitutive equations (5), (8)–(10), (15)developed in the preceding section to the light-water reactor pres-sure vessel steel A533, and the tensile behavior of unirradiated andirradiated materials under uniaxial loading will be simulated. Thevalidity of the model will be discussed in comparison with thecorresponding experiments. The experimental temperature is260°C, with the irradiation temperature 266°C. In the followingdiscussion, true stress and true strain will be employed to eliminatethe effects of the geometrical change of test specimens. However,since the details of the geometry of the specimens after the onsetof the necking are not available in the literature, we shall useherewith the average value of strain over the gauge length tocalculate the true stress and true strain.

4.1 Determination of Material Constants. We first deter-mine the material constants of Eq. (5) for unirradiated viscoplasticdeformation. These material constants were determined so that theconstitutive equations could describe the experimental results(Steiche and Williams, 1975; Mahmood et al., 1990) of the tensileproperties at different strain rates:

E 5 31.6 GPa, k 5 k0 5 293 MPa, n 5 30,

K 5 K0 5 145 MPaz s1/n, Q 5 320 MPa,

q 5 q0 5 30, S5 2.0, s 5 1.0, b 5 0.5,

pD 5 pD0 5 6.0%, Dc 5 0.45.

Then, we determine the material constants in the material func-tions and Eqs. (8)–(10), (15) for irradiated material so that theexperimental results (Hunter and Williams, 1971; Williams andHunter, 1973; Steichen and Williams, 1975) concerning thechange of the tensile properties due to irradiation can be described.

Regarding the material constants in the present model, theconstants characterizing the saturation rate of irradiation effect inEqs. (8)–(10) and (13)–(15) were assumed to be equal to oneanother; i.e.,

k2 5 K2 5 q2 5 B2. (16)

As to the material functionK that decreases with neutron irradia-tion, we assume that the strain rate dependence will disappeareventually as the fluence increases; i.e.,

K1 5 1.0. (17)

By the use of these assumptions, the material constants associ-ated with the irradiated material were determined as follows:

k1 5 1.55, K1 5 1.0, q1 5 0.5, B1 5 8.0,

k2 5 K2 5 q2 5 B2 5 9.0 3 10221 cm2/n.

4.2 Results of Calculation and Discussion. Figure 3 showsthe true stress-true strain curves for unirradiated and irradiatedmaterials calculated by Eqs. (5), (8)–(10) and (15). The figureshows qualitatively the increase in yield stress and tensile strengthtogether with the decrease in fracture strain as a typical phenom-enon of the irradiated material. The change of these properties willbe discussed quantitatively later in comparison with experimentalresults (Hunter and Williams, 1971; Williams and Hunter, 1973;Steichen and Williams, 1975).

Figure 4 shows the evolution of damage variables correspondingto the true stress-true strain curves in Fig. 3. The figure shows thetrend that increased fluence brings about earlier onset and theaccelerated rate of damage evolution.

Figures 5 to 8, furthermore, show the change in yield stress(0.2% proof stress), tensile strength, fracture strain and the amountof strain hardening due to irradiation. Figures 5, 6 and 8 show thatthe results of the present prediction are in good agreement with thecorresponding experimental data (Hunter and Williams, 1971;Williams and Hunter, 1973; Steichen and Williams, 1975); i.e.,they represent well the significant rise in the low fluence regionand the eventual saturation of the irradiation effect with increasingfluence. Regarding fracture strain in Fig. 7, though significantscatter in experimental data (Hunter and Williams, 1971; Steichenand Williams, 1975) makes the quantitative comparison difficult,they can be compared qualitatively with the experimental results.Though materials are different from A533 steels focused in thepresent paper, the qualitatively similar dependence of neutronirradiation is observed in the experiments on Cr-Mo steels byKlueh (1993) and stainless steels by Odette and Lucas (1992,1991). Thus the present model may be applicable to their experi-mental results without any essential modification.

Finally, Fig. 9 shows the dependence of yield stress (0.2% proofstress) on strain rate in unirradiated and irradiated materials.Throughout the entire range of fluence, the present model canexpress the reduction in strain-rate-dependence of yield stress with

Fig. 3 True stress-true strain curves of unirradiated and irradiatedA533B cl. 1 steel at 260°C

Fig. 4 Damage of unirradiated and irradiated A533B cl. 1 steel at 260°C


increasing fluence. The predictions particularly for the conditionsof the fluence of 4.03 1019 n/cm2 and for the unirradiatedcondition can describe well the experimental data (Steichen andWilliams, 1975), although lack of experimental data prevents usfrom satisfactory quantitative evaluation. At 1.63 1020 n/cm2 thedependence on strain rate is shown to virtually vanish.

5 Concluding RemarksIn order to evaluate the irradiation embrittlement of the pressure

vessel steel of light-water reactor in the ductile region, unifiedinelastic constitutive equations and evolution equation of damagevariable for an irradiated material were formulated from the viewpoint of continuum mechanics.

Inelastic constitutive equations and an evolution equation ofdamage variable for unirradiated material were first formulated byusing the theory proposed by Saanouni and his coworkers (1994)in the framework of thermodynamics theory for constitutive equa-tions. The resulting equations were extended to the irradiationcondition by expressing the material constants as functions ofneutron fluence. The material functions were expressed by takinginto account the interaction of dislocation motion with theirradiation-induced defects based on the mechanism of dispersionstrengthening. Finally, the constitutive equations were applied to

simulate uniaxial tensile behavior of a light-water reactor pressurevessel steel, ASTM A533B cl. 1.

The results of the present paper are summarized as follows:

Fig. 5 Predicted and experimental yield stress of unirradiated and irra-diated A533B cl. 1 steel at 260°C (Hunter and Williams, 1971; Williamsand Hunter, 1973; Steichen and Williams, 1975)

Fig. 6 Predicted and experimental ultimate tensile strength of unirradi-ated and irradiated A533B cl. 1 steel at 260°C (Hunter and Williams, 1971;Steichen and Williams, 1975)

Fig. 7 Predicted and experimental total elongation of unirradiated andirradiated A533B cl. 1 steel at 260°C (Hunter and Williams, 1971; Steichenand Williams, 1975)

Fig. 8 Predicted and experimental strain hardening of unirradiated andirradiated A533B cl. 1 steel at 260°C (Hunter and Williams, 1971; Steichenand Williams, 1975)

Fig. 9 Predicted and experimental strain rate dependence of yieldstress of unirradiated and irradiated A533B cl. 1 steel at 260°C (Steichenand Williams, 1975)


1) The present model was able to describe the true tress-truestrain curves for irradiated materials.

2) In particular, the increase in yield stress and tensile strengthas well as the reduction in fracture strain and the amount of strainhardening could be presented by the unified constitutive model.

3) Earlier onset and an accelerated rate of damage evolution inirradiated materials could be modeled in the framework of contin-uum damage mechanics.

4) Though the significant variation in the experimental data offracture strain prevented the exact comparison, qualitative agree-ment was observed in the predictions and the experiments.

As mentioned above, by introducing the notion of the continuumdamage mechanics into unified inelastic constitutive equations, thepresent model can describe the mechanical behavior (i.e., stress-strain relation of irradiated materials) up to the fracture underdifferent conditions of neutron irradiation. However, most of pre-ceding models describe only the change of specific mechanicalproperties due to the irradiation. Therefore, incorporation of thepresent constitutive model into a finite element method facilitatesthe crack analysis in nuclear structures. By using this analysis (socalled a local approach), for instance, fracture toughness of thestructural components can be predicted theoretically.

The effects of temperature are not considered in the presentpaper. However, it may be straight forward to incorporate theeffects into the present model by formulating material functions asa function of both neutron fluence and temperature. If the temper-ature effects were incorporated into the present model, DBTT shiftcould be easily predicted by analysis.

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Klueh, R. L., 1993, “Tensile Behavior of Neutron-Irradiated Martensitic Steels: AReview,” Nucl. Tech.,Vol. 102, pp. 376–385.

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Klueh, R. L., and Alexander, D. J., 1992, “Embrittlement of 9Cr-1Mo VNb and12Cr-1Mo VW Steels Irradiated in HFIR,”J. Nucl. Mater.,Vol. 187, pp. 60–69.

Klueh, R. L., and Maziasz, P. J., 1992, “Effect of Irradiation in HFIR on TensileProperties of Cr-Mo Steels,”J. Nucl. Mater.,Vol. 187, pp. 43–54.

Klueh, R. L., and Vitek, J. M., 1991, “Tensile Properties of 9Cr-1Mo VNb and12Cr-1Mo VW Steels Irradiated to 23 dpa at 390 to 550°C,”J. Nucl. Mater.,Vol. 182,pp. 230–239.

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Kussmaul, K., Fohl, J., and Weissenberg, T., 1990, “Investigation of Materialsfrom a Decommissioned Reactor Pressure Vessel—A Contribution to the Understand-ing of Irradiation Embrittlement,”ASTM STP,Vol. 1046, pp. 80–104.

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University Press, Cambridge.Little, E. A., 1976, “Neutron-Irradiation Hardening in Irons and Ferritic Steels,”

Int. Metals Rev.,Vol. 21, pp. 25–60.Little, E. A., and Gage, G., 1994, “Materials Characterization for Advanced

Pressurized Water Reactors. Part 2: Microstructure,”Nucl. Energy,Vol. 33, No. 3, pp.163–172.

Mahmood, S. T., Al-Otaibi, K. M., Jung, Y. H., and Murty, K. L., 1990, “DynamicStrain-Aging and Neutron Irradiation Effects on Mechanical and Fracture Propertiesof A533B Class 1 PV Steel and 2.25 Cr-Mo Steel,”J. Test. Eval.,Vol. 18, pp.332–337.

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N. J. Mills

A. Gilchrist

School of Metallurgy and Materials,University of Birmingham,

Edgbaston, Birmingham B15 2IT, U.K.e-mail: [email protected].

High Strain Extension ofOpen-Cell FoamsThe high strain tensile deformation of open-cell foams is analyzed, using a Kelvin foamlattice model. The stretching, bending, and twisting of elastic cell edges is analyzed, andthe deformed cell shapes predicted. The stress-strain relation and Poisson’s ratio arepredicted for strains up to 40%, for tension in the [100] and [111] directions of the BCClattice. The latter prediction is closest to stress-strain curves for polyurethane foams,especially when the cell shape anisotropy is taken into account. The change from edgebending to extension as the main deformation mechanisms, for strains exceeding 20%,increases the slope of the stress-strain curve. A comparison is made with irregular cellstructure models.

IntroductionOpen cell polyurethane (PU) foams are used in seating and

bedding. At first sight, these applications appear to involve onlycompressive strains. However, near the periphery of an indenterpressing on a foam sheet, there are high shear strains, so one of theprincipal strains can be tensile. The uniaxial tensile foam responseis considered as an initial step in dealing with biaxial tensile pluscompressive deformation.

Although the PU foams have polygonal cells of variable shapeand size, it is easier to analyze a regular structure. Lord Kelvin(1887) showed that space could be partitioned into identical tetra-kaidecahedral cells of minimal surface area. The vertices are theinterstices of a body centered cubic (BCC) lattice of atoms. Thecells have six planar faces with 4 curved sides, and eight nonplanarhexagonal surfaces. If the faces are removed, the remaining edgesform an open cell foam. Zhu et al. (1997) analyzed the high straincompression of a modified Kelvin structure with initially straightedges, called aKelvin foamby Warren and Kraynik (1997). Thestructure is almost elastically isotropic for small deformations, andits response for compression in the [111] direction is a goodapproximation to the high-strain response of PU foams.

Gent and Thomas modeled the low-strain deformation of open-cell foams (1959), emphasizing the stretching of cell edges. Theyand Lederman (1971) considered the variation of the foamYoung’s modulus with its density, and not the high strain response.Their models predict a linear variation of modulus with relativedensityR (the foam density divided by the polymer density) forfoams with lowR values. Lederman showed that the orientationdistribution of cell edges relative to the tensile stress axis affectedthe Young’s modulus. Hence PU foams with anisotropic edgeorientation distributions have anisotropic elastic properties. Ko(1965) analyzed the moduli of both face centered cubic (FCC) andclose packed hexagonal (CPH) lattices of cells, considering edgestretching and bending. The CPH model has continuous straightedges running in some directions, and both models have vertices atwhich 8 edges meet. Dement’ev and Tarakanov (1970) analyzedthe Young’s modulus of the BCC Kelvin foam in the [001]direction, assuming square cross-section edges and only consider-ing edge bending. Warren and Kraynik (1991) analyzed the elasticnonlinearity due to edge reorientation, considering both edge bend-ing and stretching, for tetrahedral units having four identical half-edges. The elastic properties of a collection of units, with randomorientations, were averaged. The bending moment is assumed to bezero at edge midpoints, due to pin-joint connection. This is a

reasonable for real foam structures, but randomly oriented tetra-hedra do not fill space, unlike the unit cells of the BCC or FCClattice models. They predicted nonlinear compressive behavior atstrains less than 15%.

Finite Element Analysis (FEA) was used to model the responseof foam seating (Pajon et al., 1996). El-Ratal and Mallick (1996)argued that the tensile response of the foams is needed for FEAmodeling, and gave such data for typical PU foams. The uniaxialtensile response considered here is a first step in the modeling thebiaxial strain response.

Modeling

Introduction. The analysis is for foams with relative density,0.05, for which the cell edge widths are much smaller than theirlengths. At relative densities.0.3 the microstructure consists ofconnected near-spherical holes (Dawson and Shortall, 1982),whereas at low densities the polygonal cell structure is more fullydeveloped. The majority of PU foam for seating and beddingapplications has a relative density less than 0.03.

Elastic anisotropy increases as a Kelvin foam deforms in tensionor compression, since the geometric nonlinearity depends on thedirection of loading. In the low strain analysis (Warren andKraynik, 1997, Zhu et al., 1997) the Young’s modulus in the [001]direction E001 is the highest, andE111 is the lowest. The cubicsymmetry means that the Young’s modulus is the same in allequivalent 100& or ^111& directions. The loading directions chosenprovide the greatest symmetry of edge deformation, simplifyingthe analysis by reducing the number of edges to be analyzed. It islikely that the [100] and [111] directions will produce the extremesof high strain behavior.

A Kelvin Foam Loaded in the [001] Direction. For concise-ness, the previous compressive analysis (Zhu et al., 1997) will bereferred to. A uniaxial tensile stresss z is applied in the [001]Miller index direction (Fig. 1). The boundary planes of the struc-tural cell (Zhu et al., 1997) in Fig. 2 remain mirror symmetryplanes at high deformations. Symmetry means that the deformationanalysis reduces to that of a single edge BC, which lies in theyzplane of the BCC lattice.Horizontaledges (those in thexy plane)lie in mirror planes, and do not bend.

The forcesP, acting on the vertices B, R, D, and G, are relatedto the applied engineering stress tensiles z by

P 5 2L 2sz (1)

using the sign convention thatP is positive for tension. In thecompressive analysis, edge axial compression was ignored com-pared with edge bending, and the results expressed as ellipticintegrals. However, to consider edge extension in the tensile mod-

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionAugust 20, 1998; revised manuscript received April 22, 1999. Associate TechnicalEditor: S. Meguid.


eling, numerical methods were used. The shear deformation of theedges was ignored in comparison with their extension, since thecell shape change at high foam extensions means that the forcevectorP acts nearly parallel with the edge BC.

If the volume of vertices is ignored, the foam relative densityRis related (Zhu et al., 1997) to the edge lengthL and cross-sectionalareaA by

R 53A

2Î2 L 2 (2)

The Plateau border edge section consists of three touching circulararcs, meeting tangentially. Its cross-sectional areaA is (=3 2p/ 2)b2, whereb is the edge breadth. Phelan et al. (1996) calcu-lated the volume of a vertex plus 4 half edges, as a function of thelengthL between vertex-centres, by minimizing the surface area.The total volumeV (per edge) was

V 5 ~Î3 2 p/2!b2~L 1 1.50b! (3)

if L . b. They did not give the distance from the vertex to theparallel sided part of the edge, or the exact vertex shape. A finiteelement calculation of the vertex deformation would show whetherour assumptions, that the edges have uniform cross-sections andthe vertices are rigid, produce significant errors. Warren et al.

(1997) calculated the second moment of areaI and the polarsecond moment of areaJ of the Plateau border section as

I 5 0.1338A2 J 5 0.0803A2 (4)

The shape of the edge BC has two-fold rotational symmetryabout its midpoint, at which the moment is zero (Zhu et al., 1997).A curvilinear coordinates, with origin O, defines the position inthe half edge OB (Fig. 2(b)), which lies in theyz plane.u is theangle between the tangent to the edge and thez axis. The momentM at a general position isP( yO 2 y) and the differential equationfor the beam curvature is

EIdu

ds5 M (5)

The tensile straine s along the edges is calculated using

es 5P cosu

EA(6)

This affects the lengthds of the 100 segments into which the halfedge is split. In an iterative solution, the momentM is calculatedfrom the original edge shape, then the curvature, calculated fromM, is used to estimate a new edge shape. The solution is partiallyrelaxed towards the new shape and the process repeated until theedge shape converges. Figure 3 shows the reduced stress (dividedby ER2) versus tensile strain for a relative density of 0.025. Thereis a steady increase in slope as the slanting edges reorient towardthe tensile axis, and the edge stretching contribution increases(Table 1). Figure 4 shows how the maximum tensile strain in theedges, at the edge midpoints, increases with the foam strain and isabout 7% when the foam strain is 40%. The relative proportion ofedge stretching to bending increases for higher relative densityfoams.

The maximum bending strain, which occurs at a cusp of thePlateau border next to a vertex (Table 1), is greater than the foammacroscopic strain. This indicates that the PU will be in the

Nomenc la tu re

A 5 edge cross-sectional areab 5 breadth of a Plateau borderC 5 constantE 5 polymer Youngs modulus

E001 5 Youngs modulus of foam loadedin the [001] direction

L 5 edge lengthI 5 second moment of area of edge

cross sectionJ 5 polar moment of area of edge

cross section

M 5 bending moment, with compo-nentsMx, My, Mz

P 5 forceR 5 foam relative density (foam

density/polymer density)s 5 position coordinate

x, y, z 5 Cartesian coordinatesa, b, g 5 angles between the edge CG

and thex, y, z axes, respec-tively

u 5 angle between the edge OD andthe z axis

f 5 twisting angle of edge CGw 5 rotation angle of vertexCe z 5 foam compressive strain

e x, e y 5 foam lateral straine s 5 edge axial strainn 5 foam Poisson’s ratio

s z 5 uniaxial tensile stress

Fig. 1 A uniaxial tensile stress sz applied in the [001] direction of aKelvin open cell foam. Chains of edges, like those in bold, take the load.

Fig. 2 (a) The structural cell for [001] direction tension of the Kelvinfoam, ( b) the effective load on the half-edge BO. Its shape is shown for20% foam strain.


nonlinear region (Zhu et al., 1997) when the foam strain is greaterthan 10%, so the calculations should really be made using anonlinear material model. The bending strains are zero at the edgemidlength where the tensile strain is largest.

As the foam strain tends to zero, the slope of the stress-straincurve becomes the Young’s modulusE001. At a strain of 0.1%, fora relative densityR 5 0.001, thesecant Young’s modulus is

E100 ;sz

ez5 1.009ER2 (7)

This is the theoretical low strain value (Zhu et al., 1997). However,if E100 is the secant modulus, the constant in Eq. (7) has increasedto 1.046 when the strain is 5% andR 5 0.03. Thehorizontal

projected lengthyL of the half edge (Fig. 2(b)) is used to calculatethe diagonal distance across the top of the deformed structural cell(Fig. 2(a)), hence the lateral tensile strain (Zhu et al., 1997).Poisson’s ratio is defined as the ratio2e y/e z of the engineeringstrains. Figure 5 shows that this increases slowly with the foamtensile strain, for strains up to 40%.

A Kelvin Foam Extended in the [111] Direction. The anal-ysis for compression (Zhu et al., 1997) used an approximation forthe edge bending curvature, hence overestimated the low strainYoung’s modulus. For edges with Plateau border cross-sections,the equivalent of Eq. (7) contained the constant 1.10 rather than the0.96 value from low strain theory. An exact expression for theedge curvature is now used, and in other parts of the analysis, someequations are simplified.

Symmetry Considerations and Boundary Conditions.A newset of axes are chosen, with thez axis parallel to the lattice [111]direction, and thex axis parallel to the projection of the unde-formed edge CG onto the plane normal toz. Edges lying initiallyin thexy plane arehorizontal.Mirror symmetry planes divide thelattice into triangular prisms, having at their centres a three-foldscrew axis parallel to thez axis. The helix of edges in each prismtransmits a forceP (Fig. 6), given (Zhu et al., 1997) by

P 54

Î3L 2sz (8)

Table 1 [100] Extension of a Kelvin foam of relative density0.025 with Plateau border edges

Foamstrain

%

Reducedstress

s

ER2Poisson’s

ratio n

Maximum edgestrain %

Reducedmoment atD

M

ER2L 3Tensile Bending

9.9 0.051 0.513 0.4 13 0.07419.8 0.292 0.570 1.3 29 0.16030.3 0.628 0.632 3.1 50 0.27440.0 1.255 0.655 6.5 74 0.411

Fig. 3 The predicted reduced stress–strain relation for the Kelvin foamloaded in the [001] and [111] directions, using Plateau border edges, plusShulmeister’s (1998) random cell prediction – – – – – – with circular sec-tion edges, all for a relative density of 0.025

Fig. 4 The maximum edge tensile strain versus the foam tensile strain,for the Kelvin foam extended in the [001] and [111] directions, with thesame parameters as Fig. 3

Fig. 5 Predicted variation of Poisson’s ratio with foam tensile strain, forthe Kelvin foam loaded in the [001] and [111] directions as in Fig. 3,compared with El-Ratal and Mallick’s Poisson’s ratio data F (1996) for acommercial PU foam (increased by 0.15 to make the comparison easier)

Fig. 6 (a) xy projection of a triangular prism showing two sloping edgesand two half horizontal edges, in both the undeformed and deformedstates, for [111] direction extension of the Kelvin foam by 20%. G liesbeneath and R above the plane of the projection ( b) a perspective view ofthe force and moments applied at G to the edge CG.


The direction of the forceP and the sign convention for the stressare opposite to that used in the compressive case. The symmetry ofthe loading direction in the lattice means that

(a) The tensile and shear forces are zero in horizontal edges.Their midpoints do not rotate about thez axis, and theydeform into circular arcs, lying in a vertical plane.

(b) The helices do not rotate relative to the prisms. The vertexC and the midpoint of CG do not rotate about thez axis,and their displacements in thexy plane are directed awayfrom the screw axis.

(c) The vertexC rotates clockwise about an axisQO by ananglew, which is positive for tension.

(d) The vertexG lies in thexz plane.

The coordinate axis origin is chosen atC (Fig. 6(a)). The anglesbetween a segment of the edge CG and thex, y, andz axes area,b, and g, respectively. A curvilinear coordinates is used forposition on CG, withs 5 0 atC ands 5 L at G. Zhu et al. (1997)gave expressions for the orientation of the undeformed edge CG,and the orientation of the edge atC in the deformed cell, when thevertexC has rotated byw.

The Equilibrium of the Moments Acting at the Vertex C.Zhuet al. (1997) analyzed the equilibrium of the moments acting atC,for compression. The vertex rotation anglew was negative, al-though this was not stated. In the tensile casew is positive. Theconstant bending momentMh in the horizontal edge BC is

Mh 5Î3 wEI

L(9)

The vector moment transmitted from the vertexG to the edge GC,has componentsMx

GMyG andMz

G. The equilibrium of the momentsacting atC gives the equation

Î3 M xG 2 M y

G 1 2P E0

L/2

cosads53wEI

L(10)

Zhu et al. (1997) used an integral for whole edge in the equivalentequation.

Torsion of the Edge CG.At a general point on the edge, thetotal momentM has components

M 5 S M xG 2 P E

0

s

cosbds,

2 M yG 1 P E

0

s

cosads, M zGD (11)

In Zhu et al. (1997) the limits of the second integral were fromsto L, and the sign of theMy

G term was positive. The moment of thecomponent parallel to the edge causes torsion. The increment oftwist df in an elementds of edge length is

df 5M z SGJ

ds (12)

whereG is the polymer shear modulus,J the edge polar secondmoment of area, andS the edge unit vector with Cartesiancomponents (cosa, cos b, cos g). Equation (12) is integratedalong the half edge, using the boundary condition thatf(0) 5w/2, giving

M xG E

0

L/2

cosads2 M yG E

0

L/2

cosbds1 M zG E

0

L/2

cosgds

2 P E0

L/2 S E0

s

cosbdsD cosads

1 P E0

L/2 S E0

s

cosadsD cosbds5 2w

2GJ (13)

Zhu et al.’s (1997) equivalent equation contains integrals for thewhole edge.

Bending of the Edge CG.The components ofM that actperpendicular toS cause edge bending. The second moment ofareaI of the Plateau border edge cross section is independent ofthe direction of bending, so the changeDS in the unit vector alongan elementds of edge is given by the vector product

DS 5M 3 S

EIds (14)

Substituting forM from Eq. (11), the components ofDS are

D cosa 5ds

EI S 2M yG cosg 2 M z

G cosb

1 P cosg E0

s

cosadsD (15)

D cosb 5ds

EI S 2M xG cosg 1 M z

G cosa

1 P cosg E0

s

cosbdsD (16)

D cosg 5ds

EI S M xG cosb 1 M y

G cosa

2 P cosa E0

s

cosads2 P cosb E0

s

cosbdsD (17)

The integral of Eq. (15) along the whole ofCG is zero sincea(0) 5 a(L). Since the integral* 0

L cosbds 5 0, the remainingterms can be analyzed to show

M yG 2 P E

0

L/2

cosads5 0 (18)

Equation (16) when integrated along the half edge provides

E0

L/2 S 2M xG cosg 1 M z

G cosa 1 P cosg E0

s

cosbdsD ds

5 2EI cosb0 (19)

The four equations (10), (13), (18), and (19) determine the un-knownsMx

G, MyG, Mz

G, andP. The tensile axial strain in the edgeis calculated from Eq. (6) with the angleg replacingu. The effectof this strain in reducing the edge moments of area is ignored.

The Stress Strain Relationship.The half edge CG is dividedinto 100 segments. For incremental rotation of the vertexC, themomentM G and forceP are found, then substituted in Eqs. (15) to(17) to estimate the new edge shape. The edge is partially relaxed


toward the new shape, and the process repeated until the change inangle is less than 1024 radian.

Figure 3 shows the variation of the reduced stresss z/ER2 withthe tensile strain. The initial slope of the graph gives the Young’smodulusE111 as 0.96ER2 for Plateau border edges, confirming thelow strain analysis. The slope of the stress-strain curve rises morerapidly at high strains than for [001] direction tension, becausesloping edges likeCG are initially within 35 deg of the tensionaxis, rather than at 45 deg. Figure 7 shows a perspective view ofa single Kelvin foam cell at a tensile strain of 25%, in which closeredges appear larger. The slanting edges align toward the tensilestress axis and bow toward the helix axis, and the triangular prismcontracts in diameter. The initially horizontal edges bend intocircular arcs of moderate curvature. Table 2 gives the dimension-less moments applied toG, the largest of which isMy. Figure 4shows that the maximum axial edge extensions, at the midpoints ofthe slanting edges, are greater for [111] than for [001] directionextension, at a particular foam strain. However, the maximumbending strain in the edge, which occurs near the vertices, is muchlarger.

Poisson’s Ratio. The edge length of the triangular prism (Fig.6) is used to calculate the lateral strain and Poisson’s ratio (Zhu etal., 1997). Figure 5 shows that the Poisson’s ratio for [111]direction extension is slightly larger than for the [100] direction,but it reaches a maximum and decreases for strains exceeding25%.

El-Ratal and Mallick (1996) found for their “commercial” PUfoam of density 17 kg m23 that the contraction in both lateral

directions was the same within experimental error. Their Poisson’sratio data in Fig. 5 is about 0.15 smaller than the predictions,showing the same increasing trend with the foam strain. Theypreferred to calculate theapparent Poisson’s ratiofrom the ratioof the true strains; these values differ from the Poisson’s ratio if thestrain are high. For high strains, a Poisson’s ratio of 0.5 does notmean that the foam volume remains constant.

Anisotropic Cell Shapes. The PU upholstery foam tested hasanisotropic cell shapes, resulting from the manufacturing process.Scanning electron microscopy showed that the cell diameter in therise direction was 30% greater than in perpendicular directions.The Kelvin foam model can be made anisotropic, with cellselongated or compressed along the [100] or [111] axes, whilemaintaining most of the structural symmetry. The angle, betweenthe sloping edges and the tensile stress axis, changes from thevalue for equiaxed cells. However, the transverse isotropy pro-duced is not exactly the same as that of a PU foam extended in theplane of the sheet.

Comparison With a Random Cell Model. Schulmeister(1998) made predictions based on regular and randomized Kelvinfoams. His predictions for the regular Kelvin foam in tension alongthe [100] direction should be the same as those reported here.However, the single reduced stress versus true strain graph ispresented on scales that preclude accurate reading of the low straindata. His predicted stress strain curve for the random model, for arelative density of 0.025 and circular cross-section edges, is almostthe same as our [111] direction prediction (Fig. 3). However,changing from Plateau border to circular edge cross-sections re-duces the second moment of area by 40%, so the agreement is notas good as it appears. He noted that the random model stress wassignificantly higher than the [100] direction Kelvin model stress atmoderate foam strains, attributing this to the chains of edgesaligned close to the stress axis.

ExperimentalA commercial upholstery PU foam, of density 26 kg m23 was

obtained in the form of a 25 mm thick sheet. This had been cutfrom a continuously foamed block, so the foam rise direction wasperpendicular to the sheet thickness. It was not possible to fullydensify this crosslinked PU, so the bulk density was assumed to be1230 kg m23, the same as a partly crosslinked PU, which had aYoung’s modulus of 50 MPa. It was not possible to evaluate thepolymer Young’s modulus. The cell diameter was approximately 1mm.

Tensile tests were carried out with an Instron, adapted to mea-sure the foam Poisson’s ratio (Mills and Gilchrist, 1997). The endsof rectangular foam specimens of length 170 mm, width 42 mmand thickness 25 mm, were bonded to flat steel plates using epoxyresin, hence the tensile stress acts perpendicular to the foam risedirection. The sample size is much greater than the cell size, soedge effects should be small. Lateral displacement transducers arespring loaded with a force of about 1 N, were fitted with 20 by 40mm PTFE plates, so that the compressive lateral stress was about1 kPa. There was low friction as the foam moved past the plates.The outputs of the load cell and displacement transducers weretaken via an analogue-to-digital converter to a computer.

Fig. 7 Predicted Kelvin foam cell shape for 20% tensile strain in the[111] direction, seen in perspective with the stress axis vertical

Table 2 [111] Deformation of a Kelvin foam of R 5 0.025 with Plateau border edges

Foamstrain

%

Reducedstress

s

ER2Poisson’s

ratio n

Maximum edge strain % Rotationw

degree

Reduced moment atG

Tensile Bending

Mx

ER2L 3

My

ER2L 3

Mz

ER2L 3

10.0 0.142 0.54 0.7 14 5.8 20.022 0.074 0.00220.5 0.561 0.62 3.3 37 11.6 20.066 0.188 20.00330.4 1.50 0.58 9.1 63 14.6 20.128 0.314 20.008


The response of the PU foam changes with the deflection cycle(Fig. 8). For the second and later cycles the response is near-constant, with a negative stress at zero strain. The change can beattributed to viscoelasticity, since the foam has been resting at zerostrain for a long time before the first cycle of extension. There issignificant hysteresis, contradicting El-Ratal and Mallick’s (1996)observation of no discernible difference between the loading andunloading response when there was a 1 minute wait at each load.The response in the absence of viscoelasticity, estimated from themean of the loading and unloading stress for a particular strain,suggests a nonlinear response. Figure 8 shows the predicted re-sponses of the Kelvin model for [111] direction extension, assum-ing a PU Young’s modulus of 100 MPa, and either equiaxed cells,or cells compressed along [111] by 30% so the slanting edges areinitially at 45 deg to the stress axis. The latter assumption gives thebetter fit to the experimental data. The prediction is shifted to thenegative stress start of the cyclic response. The assumed modulusis realistic for low density PU foams, since the use of high watercontents in the mixture causes an increase in the hard block contentand hence in the Young’s modulus of the PU.

The foam response is anisotropic. When the tensile stress is inthe plane of the sheet, Fig. 9 shows that the contraction measuredin the rise direction gives a higher Poisson’s ratio (0.8) than thecontraction in the other direction (0.5). The foam volume de-creases with the imposition of tensile strain, in agreement with theobservations of El-Ratal and Mallick on their commercial foam.The data in Fig. 9 show slight hysteresis, which may be due to thepressure and frictional shear stress from the transducers. The nearconstant slope of the data implies that Poisson’s ratio is nearlyconstant, unlike the data of El-Ratal and Mallick shown in Fig. 5.

DiscussionThe predicted response of the Kelvin foam for [111] direction

extension is closer to the measured PU foam response than is thatfor the [100] direction, the same result as for compression. How-ever, the assumed PU Young’s modulus, although reasonable,cannot be directly checked experimentally. Shulmeister’s (1998)model of a randomized Kelvin foam, if it used Plateau borderedges, would predict a 40% higher stress than for a regular Kelvinfoam, of the same density, loaded in the [111] direction to the samestrain. The regular foam model provides the quantitative linkbetween edge orientation and the stress strain curve missing from

Lederman’s (1971) paper. It is not possible to compare the pre-dictions with Warren and Kraynik (1991) tetrahedral unit modelbecause they made no tensile predictions. Kraynik et al. (1999)FEA predictions of the high strain compressive response of theKelvin open cell model, for both the [001] and [111] directions, areidentical to the compressive predictions using the method of thispaper, confirming the validity of the analyses.

The predicted Poisson’s ratio remains above 0.5, in agreementwith our experiments, and most of El-Ratal and Mallick’s data.This contrasts with compression where edge buckling is the dom-inating deformation mechanism, and Poisson’s ratio falls to lowvalues.

Since the edge tensile compliance depends inversely on the firstpower, and the edge bending compliance on the second power ofthe relative density, the low strain tensile response is dominated byedge bending for low relative density foams. The edge alignmentwith the tensile axis is almost independent of the foam density, asis the relative hardening of the stress strain curve at high strains.Only when the foam tensile strain is mainly provided by edgeextension does the stress become linearly dependent on the foamrelative density—this region is rarely met in PU foams sincefailure intercedes. This contrasts with compression, where theaxial strain in the edges can be neglected, and there is a nearplateau in stress for strains exceeding 10%. The reduced compres-sive collapse stress of about 0.1 is much less than the reducedtensile stresses shown in Fig. 3.

The response of PU foams is complicated by:

(a) Viscoelasticity. For creep loading, elastic predictions canbe converted into approximate viscoelastic solutions (Zhuand Mills, 1999), by replacing the Young’s modulus by thereciprocal of the creep complianceJ(t), except in thevicinity of the glass transition temperature. The majority ofthe hysteresis in the tensile loading and unloading responseis due to viscoelasticity.

(b) Polymer nonlinearity. The maximum polymer strain in-creases with the foam relative density, at a particular foamstrain. In commercial PU foams at strains.10%, thepolymer is in the nonlinear region, so the predicted re-sponses in Fig. 8 are overestimates.

(c) The anisotropy of the cell structure, which makes the foamresponse anisotropic.

(d) Temperature, which has a similar effect to strain rate inchanging the response.

Fig. 8 Stress strain curve for a PU upholstery foam of relative density0.021, loaded in the plane of the sheet, cycled three times to 25% strainat a nominal strain rate of 0.005 s 21. The Kelvin foam [111] predictionsare z z z z z z for equiaxed cells and – – – – – – for cells with edges initiallyat 45 deg to the stress, using E 5 100 MPa for the polyurethane.

Fig. 9 Lateral contraction in the rise —— and in-plane – – – – – – direc-tions versus tensile strain for the same experiment as Fig. 8, for cycles2 and 3


The simple shear response of open-cell foams will be influencedby the tensile principal stress, which may be greater than thecompressive principal stress. Micromechanics analysis may beable to predict whether a large compressive principal strain causessufficient edge buckling to affect the tensile response along theother principal axis. PU foam response under such biaxial strainstates, where one principal strain is tensile, needs to be investi-gated further.

AcknowledgmentThe authors acknowledge the support of ESPRC and DERA

under grant number GR/L 81307.

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Foam,”J. Mater. Sci.,Vol. 17, pp. 220–224.Dement’ev, A. G., and Tarakanov, O. G., 1970, “Model Analysis of Plastics Foams

of the Polyurethane Type,”Mekh. Polim.,Vol. 6, pp. 859–865, translated asPolymerMechanics,Vol. 6, pp. 744–749.

El-Ratal, W. H., and Mallick, P. K., 1996, “Elastic Response of Flexible Polyure-thane Foams in Uniaxial Tension,” ASME JOURNAL OF ENGINEERING MATERIALS AND

TECHNOLOGY, Vol. 118, pp. 157–161.Gent, A. N., and Thomas, A. G., 1959, “The Deformation of Foamed Elastic

Materials,”J. Appl. Polymer. Sci.,Vol. 1, pp. 107–113.Kelvin, Lord (Thompson, W.), 1887, “On the Division of Space with Minimum

Partitional Area,”Phil. Mag.,Vol. 24, pp. 503–514.

Ko, W. L., 1965, “Deformations of Foamed Elastomers,”J. Cellular Plastics,Vol.1, pp. 45–50.

Kraynik, A. M., Neilsen, M. K., Reinhelt, R. A., and Warren, W. E., “FoamMicromechanics,” NATO Adv. Sci. Inst. Series E:Appl. Sci.,Vol. 354, Foams andEmulsions,N. Rivier and J. F. Sadok, eds., pp. 187–204, Kluwer.

Lederman, J. M., 1971, “The Prediction of the Tensile Properties of FlexibleFoams,”J. Appl. Polym. Sci.,Vol. 45, pp. 693–703.

Mills, N. J., and Gilchrist, A., 1997, “The Effect of Heat Transfer and Poisson’sRatio on the Compressive Response of Closed Cell Polymer Foams,”Cell. Polym.,Vol. 16, pp. 87–119.

Pajon, M., Backacha, M. et al., 1996, “Modelling of PU Foam Behaviour—Applications in the Field of Automotive Seats,” SAE SP-1155, paper 960513.

Phelan, R., Weaire, D., Peters, E. A. J. F., and Verbist, G., 1996, “The Conductivityof a Foam,”J. Phys. Condens. Matter,Vol. 8, pp. L475–L482.

Schulmeister, V., 1998, “Modelling of the Mechanical Properties of Low DensityPolymer Foams,” Ph.D. thesis, Technical University of Delft.

Warren, W. E., and Kraynik, A. M., 1991, “The Non-Linear Elastic Behavior ofOpen-Cell Foams,” ASMEJournal Appl. Mechanics,Vol. 58, pp. 376–381.

Warren, W. E., and Kraynik, A. M., 1997, “Linear Elastic Behavior of a Low-Density Kelvin Foam with Open Cells,” ASMEJournal Appl. Mechanics,Vol. 64,pp. 787–794.

Warren, W. E., Neilsen, M. K., and Kraynik, A. M., 1997, “Torsional Rigidity ofa Plateau Border,”Mechanics Res. Comm.,Vol. 24, pp. 667–672.

Zhu, H., Knott, J. F., and Mills, N. J., 1997, “The Elastic Constants of Open CellFoams Having Tetrakaidecahedral Cells,”J. Mech. Phys. Solids,Vol. 45, pp. 319–343.

Zhu, H. X., Mills, N. J., and Knott, J. F., 1997, “Analysis of the High StrainCompression of Open-Cell Foams,”J. Mech. Phys. Solids,Vol. 45, pp. 1875–1904.

Zhu, H. X., and Mills, N. J., 1999, “Analysis of Creep in Open-Cell Foams,”J.Mech. Phys. Solids,Vol. 47, pp. 1437–1457.


David A. Miller

Dimitris C. Lagoudas

Center for Mechanics of Composites,Department of Aerospace Engineering,

Texas A&M University,College Station, TX 77843-3141

Influence of Heat Treatment onthe Mechanical Properties andDamage Development in aSiC/Ti-15-3 MMCTitanium alloys are commonly heat-treated to meet specific design requirements. In aneffort to possibly create a better composite, the influence of heat treatments on the damageevolution and strength of a SiC/Ti-15-3 metal matrix composite (MMC) was studied. Heattreatments of 450°C and 700°C for 24 hours were performed on axial and transverseunidirectional specimens. These specimens, in addition to specimens in the as-receivedcondition, were tested under nonproportional loading paths and then microstructurallyanalyzed to determine the induced damage. The axial composite with the 450°C heattreatment showed the highest elastic modulus and the lowest stiffness reduction than theother heat treatment conditions. The transverse composite in the as-received conditionshowed the highest room temperature elastic modulus and the lowest stiffness reductioncompared with other heat treatment conditions. Typical damage modes of Ti MMC’s, suchas fiber/matrix debonding and matrix microcracking, were seen in all heat treatments. Amicromechanics model based on the Mori-Tanaka averaging scheme was implemented tosimulate the effects of microcracking induced damage on composite stiffness reduction.

1 IntroductionDesigns for the next generation of supersonic and hypersonic

aircraft as well as engine components will require advanced ma-terials that can withstand elevated temperatures in structural ap-plications. Metal matrix composites (MMC’s) have recently un-dergone serious evaluation to satisfy this need because of theirhigh strength, low weight, and elevated temperature capabilities.Titanium has been the primary matrix material considered becauseof its formability, high strength-to-weight ratio, and high meltingtemperature (1668°C). Titanium also has the advantage that itsmicrostructure can be altered by alloying with metals which willstabilize thea or b phase to satisfy many different design require-ments.

Structural titanium MMC’s are often reinforced with continuoussilicon carbide (SiC) fibers fabricated through a chemical vapordeposition process. To date, the results for the SiC/Ti systems haveshown severe limitations. The composite shows improved perfor-mance in the fiber direction compared with monolithic titanium,but the transverse fiber direction properties are considerably less.Johnson (1992) has shown in an SCS-6/Ti-15-3 MMC compositewith off axis plies that fiber-matrix separation leads to a knee in thestress/strain curve well below the matrix yield level. This damagewas confirmed by a reduction in the unloading modulus and edgereplicas. Combined loading paths have been studied by Lissendenet al. (1995), in SiC/Ti tubes. Results have shown that fiber/matrixinterfacial debonding was observed in the stress-strain responseand verified through microstructural evaluations.

Mujumdar and Newaz (1992a, b, 1993) have performed roomtemperature experimental studies of the SCS-6/Ti-15-3 compositeidentifying the inelastic deformation mechanisms of plasticity anddamage. Lagoudas et al. (1995) studied the effect of surfacedamage on oxidized SiC/Ti-15-3 MMC laminates under tensileloading, and an effective stiffness reduction due to the develop-ment of cracks on the surface of the composite was evaluated.

Newaz et al. (1992) investigated the thermal cycling response ofquasi-isotropic SiC/Ti-15-3 MMC’s. Johnson et al. (1990) per-formed room temperature tests to characterize a SiC/Ti-15-3 MMCin both the as-received condition and aged at 482°C for 16 hr. Thisheat treatment was shown to increase the elastic modulus, theultimate strength and the yield stress of the matrix.

Several different temperatures have been utilized for heat treat-ments of MMC’s in previous works, (Johnson, et al., 1990; Lerchet al., 1990a) although rigorous comparisons were not madeamong different heat treatments. In fact, the motivation behind theheat treatments was not discussed, other than to stabilize themicrostructure. Wolfenden et al. (1996) utilized nondestructivetechniques to investigate the effect of heat treatments on the elasticmodulus, damping and microhardness for SiC/Ti-15-3. Lerch andSaltsman (1991), advanced Johnson’s room temperature resultswith a study of the SCS-6/Ti-15-3 MMC at 427°C. Specimenswere tested as-received and with a heat treatment of 700°C for 24hrs. However, the effect of heat treatment and resulting micro-structures on the damage development of MMC’s has not beenaddressed.

The objective for this work is to study possible improvementson a SiC/Ti-15-3 MMC through different heat treatments and toexperimentally characterize the deformation and damage mecha-nisms associated with each. The effects of heat treatment on thethermomechanical response and damage development in unidirec-tional specimens, loaded axially and transverse to the fibers, arediscussed in the second section. The third section utilizes anaveraging micromechanics technique for the correlation of crackdensity as a damage parameter with stiffness reduction and com-pares model predictions with experimental results.

2 Experimental Results on Thermomechanical Re-sponse of SiC/Ti-15-3 With Different Heat Treatments

2.1 Material Description and Specimen Preparation. Alltests in this research were performed on a composite materialmanufactured by DWA Specialty Metals using the foil-fiber-foiltechnique. The composite is a 4-ply unidirectional SiC/Ti 15-3MMC with a 32% fiber volume fraction. The matrix is reinforced

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionAugust 5, 1997; revised manuscript received May 3, 1999. Associate TechnicalEditor: S. A. Meguid.


by SiC fibers manufactured by British Petroleum. The averagefiber diameter was 100mm and has a structured carbon-titaniumdiboride coating (Allen et al., 1994). The alloy utilized in thiscomposite is a metastableb-phase alloy commonly referred to asTi 15-3, and has an actual weight percent of 15% V, 3% Al, 3% Cr,3% Sn and the balance Ti.

The composite was tested in the as-received condition and withheat treatments of 450°C for 24 hours and 700°C for 24 hours.Heat treatments were performed in a vacuum to prevent oxidationfrom degrading the material. The as-received material, see Fig. 1,contains largea-phase needles, labeled as A. Thea-particles areseen primarily along the fiber interface creating ab-depleted zonearound the fiber area. Results from an energy dispersive spectros-copy (EDS) line scan show a high carbon concentration in thearich region surrounding the fiber, possibly due to the coating.Carbon is ana-stabilizer for titanium, thus explaining thea richregion surrounding the fiber. Thea rich region surrounding thefiber seen in this composite has not been observed in other Ti-15-3systems (e.g., in the work of Lerch and Saltsman, 1991, where theSCS-6/Ti-15-3 MMC has been studied). Since the fiber coatingsdiffer for the Sigma fiber and the SCS-6 fiber, the possibility ofdifferent fiber/matrix interfaces exist.

The 450°C heat treatment produces a small finea-particleprecipitant with virtually no congregation along the grain bound-ary, as seen in Fig. 2. The distinct fiber matrix interface regionremains, but the precipitant accumulation around the fiber hasdisappeared. The aging has caused the precipitant to distributethroughout the grain structure. The grains remain the same size,but are not as apparent as before the redistribution of thea-phase.It has been shown (Lerch et al., 1990a) that this heat treatmentproduces the highest room temperature modulus for the matrixmaterial. This is to be expected since the HCPa-phase has a higherstiffness than the BCCb-phase. However, the increase ina-phase

also decreases the number of active slip planes thus reducing theductility of the material.

The 700°C heat treatment, Fig. 3, results in largea-phaseneedles dispersed throughout theb-phase grains, while the grainsize remains unchanged. Theb-depleted region around the fibers isstill present, as well as the grain boundary phases. It has beenshown that this heat treatment slightly reduces the stiffness of thetitanium from the as-received condition (Lerch et al., 1990a).

2.2 Thermomechanical Testing Description. All testswere performed on an MTS servo-hydraulic load frame under loadcontrol at a rate of 10.4 MPa/sec. Elevated temperature tests wereperformed using a three-zone clamshell resistance furnace. Strainmeasurements for all tests were made with a one-inch gage hightemperature extensometer.

In order to study the evolution of inelastic deformation through-out the entire loading path, cyclic tension tests with an increasingstress amplitude were performed at room temperature and at 427°C(800°F). Measurement of the unloading elastic modulus for eachcycle quantified the damage imparted into the specimen at theapplied stress level. Cyclic loading paths were used to study thedevelopment of inelastic deformation in the composite and mon-itor the degradation of the initial elastic modulus. The loading pathconsisted of unloading to 10% of the maximum applied stress ineach cycle with the maximum stress incrementally increasing ineach successive cycle to a value approaching the failure stress ofthe composite.

A total of six unidirectional composite specimens loaded in thefiber direction (axial specimens) were investigated in this studyallowing for one cyclic test for each heat treatment and testingtemperature. Also investigated were nine specimens loaded trans-verse to the fiber direction (transverse specimens). The transversetests consisted of one monotonic test to failure for each heattreatment at room temperature and one cyclic test for each heattreatment and testing temperature. Due to the small number ofspecimens tested, this study only qualitatively addresses the influ-ence of heat treatment on the material parameters (e.g., elasticmodulus) of the composite. Since limited material did not allow formultiple tests of the same heat treatment, possible variations of thematerial parameters due to material heterogeneity cannot be ad-dressed in this study. However, all specimens were taken from thesame composite plate, thus, the manufacturing process was iden-tical for each specimen. Although the present experiments cannotidentify material parameters without a large margin of error, thedegradation of material properties reported here should be indic-ative of damage induced changes in similarly produced Ti-15-3systems.

Microstructural evaluations were performed after each test todetermine the state of damage in the composite. The specimenswere carefully sectioned, polished and etched for a detailed mi-crostructural evaluation with light microscopy and scanning elec-tron microscopy.

Fig. 1 Microphotograph of a tested as-received specimen polished andetched to reveal the microstructure

Fig. 2 Microphotograph of a tested 450°C/24 hour heat treatment spec-imen polished and etched to reveal the microstructure

Fig. 3 Microphotograph of a tested 700°C/24 hour heat treatment spec-imen polished and etched to reveal the microstructure


2.3 Axial Tension Tests. Three cyclic tests, one for eachheat treatment, were conducted at room temperature and at 427°Cfor axial specimens. For brevity, only the results of the unloadingelastic modulus as a function of the applied stress are shown in Fig.4. As seen in Fig. 4, the 450°C specimen has the highest elasticunloading modulus for both room temperature and elevated tem-perature tests. This was expected since the 450°C heat treatmentcreates the highest room temperature modulus and yield point inthe matrix due to the high concentration of HCPa-phase presentthroughout the matrix. The 700°C specimen tested at elevatedtemperature has the lowest initial elastic modulus of the threetreatments, although the as-received specimens have similar val-ues. These specimens have similar microstructures with the pri-mary difference being the size of the precipitant as seen in Figs. 1and 3.

A reduction in elastic modulus is seen throughout the entire testfor each specimen as the applied stress increases, indicating thatdamage is present in all specimens. However, the 450°C specimentested at room temperature settles to a constant unloading modulusabove 1500 MPa. The inelastic deformation in the axial directionhas been shown to be primarily dominated by matrix plasticity,rather than damage (Majumdar and Newaz, 1992a), although theresults seen in Fig. 4 show that damage is present. Microstructuralevaluation of the specimens reveal both fiber cracks and matrixcracks in room and elevated temperature tests, however, a uniformspacing of cracks was not found. Matrix cracks were seen to travelacross fibers, without causing fiber failure, although, on failedspecimens the fiber/matrix interface was debonded. From the cat-astrophic failure seen and heard when the composite failed, it canbe assumed that the fibers fail almost simultaneously. Regardlessof the heat treatment and testing temperature, all specimens failedin the range of the fiber strain limit of 0.9–1.0%.

2.4 Transverse Tension Tests. Three cyclic tests, one foreach heat treatment condition, were conducted at room tempera-ture and at 427°C. Also, a monotonic tension test to failure wasperformed at room temperature for each heat treatment to identifythe regions of inelastic deformation to be studied in the cyclic tests.For brevity, only the results of the unloading elastic modulus arepresented in Fig. 5. The as-received specimen tested at roomtemperature and the 700°C specimens tested at both room andelevated temperature showed inelastic deformation with no per-manent strain, thus, the inelastic deformation is attributed only todamage. The 450°C heat treatment specimen tested at room tem-perature and the as-received specimen tested at elevated tempera-ture showed inelastic deformation with a permanent strain. How-ever, these tests both contained a knee, or stiffening, in theunloading curve, most likely due to the closing of the cracks alongthe debonded fiber/matrix interface. For both of these tests, thecomposite would unload to the origin without showing a perma-

nent offset if the knee were not present. The 450°C heat treatmenttested at elevated temperature showed inelastic deformation with apermanent strain, however, the knee is not observed in the unload-ing cycles. This implies that the permanent strain for this specimenis due to matrix plasticity, in contrast to all other transverse testsmentioned here.

The as-received specimen at room temperature, as seen in Fig.5, showed the highest initial elastic modulus, but damage reducedthe modulus as higher load levels were reached. The 700°C spec-imen had the highest elevated temperature modulus, but alsoshowed the greatest loss of stiffness, while the as-received speci-men at elevated temperature and the 700°C specimen at roomtemperature showed a slow rate of damage growth with the appliedload. This point could be explained by the accumulation of micro-cracks reaching a saturation point at which the modulus no longerdecreases. The 700°C and 450°C specimens tested at elevatedtemperature and the as-received and 450°C specimen tested atroom temperature show a continuous drop in the unloading mod-ulus, while the remaining tests settle to a nearly constant value.

The initial elastic modulus for the transverse tensile tests closelymatch published results for similar systems, however, the inelasticdeformation described above occurs at stress levels well below theelastic limit seen in similar systems (about 275 MPa for roomtemperature SCS-6/Ti-15-3, Mujumdar and Newaz, 1992a; Lerchand Saltsman, 1991). The inelastic deformation is facilitated by themanufacturing flaws seen in the material, resulting in failure for asreceived, 450°C and 700°C specimens at 242 MPa, 274 MPa and244 MPa respectively at room temperature. Observations of sucha low failure stress similar to those presented here have not beenreported in literature, but microstructural evaluations suggest a

Fig. 6 Example of representative tested unetched transverse speci-mens showing cracks emanating from manufacturing flaw and propagat-ing toward fiber

Fig. 4 Unloading axial elastic modulus versus applied stress for axialspecimens

Fig. 5 Unloading transverse elastic modulus versus applied stress fortransverse specimens


correlation between the low failure stress and the pre-existingmanufacturing defects in the specimens tested in this work.

Microstructural evaluation shows that damage for all of thespecimens primarily occurs in the form of matrix cracking ema-nating from the fiber/matrix interface in the direction of the load-ing. Figs. 1, 2, 3, and 6 show examples of crack development in thetransverse specimens. Figure 6 shows crack initiation from man-ufacturing flaws, while Figs. 1, 2, and 3 show the crack develop-ment along grain boundaries revealed through etching. Grainboundaries are prone to crack development in these specimens dueto the brittlea-phase accumulation. From the etched microstruc-tures seen in Figs. 1, 2, and 3 the highest level ofa-phase grain

boundary accumulation is seen in the 700°C heat treatment spec-imens and the lowest accumulation is seen in the 450°C heattreatment specimens. The premature failure of the specimens andlack of large permanent deformation, leads to the conclusion thatcrack development along grain boundaries, initiated by pre-existing manufacturing flaws, led to damage development andultimate failure of the specimens at stress levels well below thecomposite elastic limit.

3 Thermomechanical Modeling of Damage in SiC/Ti-15-3 MMC

3.1 Model Development. In this section an attempt will bemade to correlate the experimental observations with theoreticalmodeling. The actual deformation mechanisms are complex andinvolve damage evolving during the loading. An approximateaveraging micromechanics method will be used to model theinfluence of the loading direction and material properties on themechanical response of the composite. The model will be based onthe extension of the Mori-Tanaka micromechanics averagingmethod (Mori and Tanaka, 1973) to include damage effects.

Averaging methods have been developed which model damagein fibrous composites. In the present study, an attempt will bemade to incorporate evolving damage in an incremental formula-tion. For the present implementation of the incremental formula-tion, the increment of total strain,De# t, for each increment ofapplied overall stress,Ds# , is expressed as the sum of the elasticincrement, and strain increment due to damage, i.e.,

De# t 5 @M d|s# 1Ds# 2 M d|s# #s# 1 M d|s# 1Ds# Ds# , (1)

where the first term calculates the strain increment introduced bydamage through the change in the elastic damage complianceM d,and the second term is the elastic increment using the current valueof the compliance.

The Mori-Tanaka method (Weng, 1984; Benveniste, 1987) uti-lizes the Eshelby equivalence tensor,S, to calculate the elasticstress concentration factor,Ba

e, which relates the overall appliedstress,s# , to the stress in thea phase by

sa 5 B aes# , a 5 f, m (2)

where a indicates the fiber or matrix phase. For a two-phasecomposite, a computationally convenient stress concentration fac-tor based on the Mori-Tanaka approximation is given by (Gavazziand Lagoudas, 1990)

B fe 5 @I 1 cmL m

e ~I 2 S!~M me 2 M m

e !# 21 (3)

whereL ae andM a

e are the elastic stiffness and compliance tensorsrespectively,ca is the volume fraction for each phase andI isidentity matrix. The effective elastic compliance for the MMC canthen be found using the following relation (Hill, 1965),

M e 5 M m 1 cf ~M f 2 M m!B fe. (4)

Substituting the elastic modulus for each heat treatment into theabove equations yields an evaluation of effective elastic compli-ance for each heat treatment of the composite.

As the loading is increased, matrix cracks develop in the uni-

Table 1 Material properties for each heat treatment and testing temperature utilized in modeling (Lerch et al., 1990a; Johnsonet al., 1990; Wolfenden et al., 1996)

Matrix Properties Fiber Properties

Room temperature Elevated temperature, 427°C Room temperature and elevated temperature

As-Rcvd 450°C 700°C As-Rcvd 450°C 700°C

E, GPa 92.4 104.8 90.0 78.0 83.0 77.5 395.1n .32 .32 .32 .32 .32 .32 .19

Fig. 7 A comparison between the simulated stress/strain response andexperimental measurements for ( a) 700°C specimen tested at elevatedtemperature and ( b) 450°C specimen tested at room temperature


directional composite laminate and the above evaluation of theelastic compliance must be modified to account for the developingdamage. The effect of cracks on the overall compliance of thecomposite is modeled using the methods derived by Laws andDvorak (1987), and Dvorak et al. (1985) which relate the com-posite compliance to the density of cracks in the composite.Motivated by experimental observations, two different types ofcracks are modeled, slit-cracks in the transverse specimens andpenny-shaped matrix cracks in axial specimens.

The transverse crack density,b tr , a function of the overallapplied stress, is defined as the number of cracks of width,a, in asquare cross-section of material with sides of lengtha. The dam-aged composite compliance,M d, is calculated as a function of thecrack densityb tr . The Mori-Tanaka method is used to first gen-erate the undamaged effective composite compliance, which is

then revised as a function of the crack density (Dvorak et al.,1985).

In the axial specimens, cracks are modeled as a collection ofpenny-shaped matrix cracks transverse to the fibers. The crackdensity,b a, is a function of applied load and is defined as theaverage number of cracks of specific diameter,d, in a cube withsides of length,d. The effective compliance of the composite iscalculated in two steps. First, the matrix is regarded as a volume ofhomogeneous material with similar cracks of densityb a, for whichan effective compliance,M m

d , is calculated as a function of thecrack density (Dvorak et al., 1985). The second step utilizes theMori-Tanaka method to generate an effective compliance of thedamaged composite,M d, using the properties of the fiber and thedamaged matrix,M m

d .

3.2 Implementation and Comparison With Experiments.Implementation of the averaging micromechanics approach topredict the mechanical response for damage requires knowledge ofthe evolution of the crack density,b, as a function of applied load.A damage growth model relating the crack density to the appliedload is not utilized in this paper due to the inherent complexitiesthose models contain. In the absence of a damage growth criterion,an experimental measurement will be utilized to determine thecontribution of damage to the total inelastic deformation during agiven loading path. Specifically, the damage will be determinedfrom the experimentally measured components of the elastic un-loading stiffness,L d(3, 3), shown in Fig. 4 andL d(2, 2) shown inFig. 5, whereL d 5 [M d] 21.

Using the experimental data from Figs. 4 and 5, the overallstress vs. strain response can be simulated accounting for damageevolution utilizing the method described above. Simulations weremade for all specimens, however, for brevity only two transversespecimens are presented. A comparison between the simulatedstress/strain response and experimental measurements are shownin Fig. 7 for the 700°C specimen tested at elevated temperature andthe 450°C specimen tested at room temperature. In the simulationof the 700°C specimen at elevated temperature, Fig. 7(a), theinelastic deformation is only due to damage, as evidenced by thelack of permanent deformation. For the 450°C specimen tested atroom temperature, the simulation shows that the knee in theunloading curve is the cause of the permanent deformation, sug-gested by the simulation matching the initial unloading modulusabove the knee and fully unloading with no permanent deforma-tion, Fig. 7(b).

A prediction of crack density as a function of applied load canbe determined by solving for the transverse crack density,b tr , andthe axial crack density,b a, that yields the experimentally mea-sured component of the unloading elastic stiffness. Since uniaxialstress states are investigated, only the components of compliancein the loading direction will be calculated. The model prediction ofthe crack density as a function of the applied stress is shown in Fig.8 using the material properties listed in Table 1. To verify themodel predictions for the resulting crack density as a function ofapplied load, post-test microstructural evaluations were made. Theaverage crack densities measured from the specimens are com-pared to the predicted crack densities at the maximum appliedstress shown in Fig. 8. Table 2 contains the measured and pre-dicted values of the crack densities for each heat treatment and testtemperature for the transverse composite.

Table 2 Comparison of experimental and predicted crack densities,b tr, after final loading for transverse specimens

Room temperature Elevated temperature, 427°C

As-Rcvd 450°C 700°C As-Rcvd 450°C 700°C

Experiment .33 .35 .22 .23 .39 .28

Model Prediction .50 .69 .38 .35 .76 .92

Fig. 8 Prediction of crack density as a function of applied stress for ( a)axial crack density, ba and (b) transverse crack density, btr


In transverse specimens, the experimental crack density wasmeasured from a representative microphotograph of each sample,and calculated as the number of cracks of average measured crackwidth, a, divided by the number of squares with sides of length,a,in a square area encompassing the entire cross-section of thespecimen. The crack width was taken to be the extent of the matrixcrack often developing adjacent to a fiber, and perpendicular to theloading direction, as seen in Fig. 6. Note that the matrix cracks,which propagate along the grain boundary in the direction of theload, are not considered in the measured transverse crack density.The predicted crack densities for the transverse specimens, shownin Table 2, are higher than the experimentally measured valuesespecially for the elevated temperature tests. The pre-existingmanufacturing defects in the composite may cause a larger exper-imentally measured stiffness reduction, thus leading to an over-prediction of the crack density. In addition, matrix cracks bridgingthe fibers in the loading direction, not accounted in the measuredcrack densities, may play a role in the higher stiffness reductionleading to an overprediction of crack density. When the transversematrix crack is coupled with the cracks developing along grainboundaries, the result is a higher stiffness reduction for the ele-vated temperature tests. Caution should be finally drawn to the factthat the model assumes non-intersecting matrix slit cracks, whilethe observed damage indicates crack coalescence in many of theexamined specimens.

4 ConclusionsA SiC/Ti-15-3 MMC was studied to determine the effects of

different heat treatments on the damage evolution at both roomtemperature and at 427°C. The thermomechanical study revealedthat the heat treatments could affect the overall composite com-pliance and damage accumulation. Mechanical tests show that the450°C heat treatment creates a microstructure with a consistentlyhigh elastic modulus and high damage tolerance for all loadingconditions and temperatures.

Microstructural evaluation identified the primary damage modesfor both the transverse and axial specimens. The axial specimensshowed evidence of cracks developing perpendicular to the load-ing direction starting from the fiber/matrix interface. The trans-verse specimens showed cracks emanating from areas of poorconsolidation resulting in cracks propagating in the loading direc-tion along grain boundaries. A model based on the Mori-Tanakamethod was developed and implemented which incorporates dam-age in an incremental formulation. The developed method success-fully simulates the stress vs. strain response of the compositeaccounting for damage. A prediction of the crack density wasmade as a function of overall applied load. It was shown that thepredicted crack density at the final load level overpredicts thecrack densities measured from a post-test microstructural analysisfor the transverse specimens, most likely due to the pre-existingdamage along the fiber/matrix interface.

5 AcknowledgmentsThe authors acknowledge the support of AFOSR grant No.

F49620-94-1-0341.

ReferencesAllen, D. H., Eggleston, M. R., and Hurtado, L. D., 1994, “Recent Research on

Damage Development in SiC/Ti Continuous Fiber Metal Matrix Composites,”Frac-ture of Composites,E. A. Armanios, ed.,Key Engineering Materialsseries, TransTech Publications.

Benveniste, Y., 1987, “A New Approach to the Application of the Mori-Tanaka’sTheory in Composite Material,”Mech. Materials,Vol. 6, pp. 147–157.

Dvorak, G. J., Laws, N., and Hejazi, M., 1985, “Analysis of Progressive MatrixCracking in Composite Laminates I. Thermoelastic Properties of a Ply with Cracks,”J. Composite Mat.,Vol. 19, pp. 216–234.

Gavazzi, A. C., and Lagoudas, D. C., 1990, “Incremental Elastoplastic Behavior ofMetal Matrix Composites Based on Averaging Schemes,”Proceedings IUTAM Sym-posium on Inelastic Deformation of Composite Materials,G. J. Dvorak, ed., Springer-Verlag, pp. 465–485.

Hill, R., 1965, “Theory of Mechanical Properties of Fibre-Strengthened Materials:III. Self-Consistent Model,”J. Mechanics and Physics of Solids,Vol. 13, pp. 189–198.

Johnson, W. S., 1992, NASA Technical Memorandum 107597, Lewis ResearchCenter.

Johnson, W. S., S. J. Lubowinski, and A. L. Highsmith, 1990, “MechanicalCharacterization of Unnotched SCS-6/Ti-15-3 Metal Matrix Composites at RoomTemperature,”Thermal and Mechanical Behavior of Metal Matrix and CeramicMatrix Composites,ASTM STP 1080, J. M. Kennedy et al., eds., ASTM, Philadel-phia, PA, pp. 193–218.

Lagoudas, D. C., X. Ma, D. A. Miller, and D. H. Allen, 1995, “Modeling ofOxidation in Metal Matrix Composites,”Int. J. Engng. Sci.,Vol. 33, No. 15, pp.2327–2343.

Laws, N., and Dvorak, G. J., 1987, “The Effect of Fiber Breaks and AlignedPenny-Shaped Cracks on the Stiffness and Energy Release Rates in Uni-directionalComposites,”Int. J. Solids and Structures,Vol. 23, No. 9, pp. 1269–1283.

Lerch, B. A., Gabb, T. P., and MacKay, R. A., 1990a, “Heat Treatment Study ofthe SiC/Ti-15-3 Composite System,” NASA TP 2970.

Lerch, B. A., and Saltsman, J. F., 1991, “Tensile Deformation Damage in SiCReinforced Ti-15V-3Cr-3Al-3Sn,” NASA Technical Memorandum 103620, LewisResearch Center.

Lissenden, Herakovich, C. T., and Pindera, M. J., 1995, “Response of SiC/Ti underCombined Loading part I: Theory and Experiment for Imperfect Bonding,”Journalof Composite Materials,Vol. 29, No. 2, pp. 130–155.

Majumdar, B. S., and G. M. Newaz, 1992a, 1992b, 1993, NASA Contractor Report189095, 189096, 191181, Lewis Research Center.

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Newaz, G. M., B. S. Majumdar, and F. W. Brust, 1992, “Thermal CyclingResponse of Quasi-Isotropic Metal Matrix Composites,” ASME JOURNAL OF ENGI-NEERING MATERIALS AND TECHNOLOGY, Vol. 114, pp. 156–161.

Okada, M., and Banergee, D., 1984, “Tensile Properties of Ti-15V-3Al-3Cr-3SnAlloy,” Titanium Science and Technology: Proceedings of the Fifth InternationalConference on Titanium,Munich, Germany, Luetjering ed., Deutsch Gesellschaft furMetallkunde, Pub., pp. 1835–1842.

Sun, C. T., J. L. Chen, G. T. Sha, and W. E. Koop, 1990, “Mechanical Charac-terization of SCS-6/Ti-6-4 Metal Matrix Composite,”J. Composite Mat.,Vol. 24, pp.1029–1059.

Weng, G. J., 1984, “Some Elastic Properties of Reinforced Solids, with SpecialReference to Isotropic ones Containing Spherical Inclusions,”Int. J. Ingrg. Sci.,Vol.22, pp. 845–856.

Wolfenden, A., K. D. Hall, and B. A. Lerch, 1996, “The effect of heat treatment onYoung’s modulus, damping, and microhardness of SiC/Ti-15-3,”J. of MaterialsScience,Vol. 31, pp. 1489–1493.


Toru IkedaChemical Engineering Group,

Department of Materials Process Engineering,Graduate School of Engineering,

Kyushu University,6-10-1, Hakozaki, Higashi-ku,

Fukuoka, 812-8581, Japan

Akira YamashitaBoiler Plant Division,

Kawasahi Heavy Industry Ltd.,2-11-1, Minamisuna, Koto-ku,

Tokyo 136-0076, Japan

Deokbo Lee

Noriyuki Miyazaki

Chemical Engineering Group,Department of Materials Process Engineering,

Graduate School of Engineering,Kyushu University,

Fukuoka, 812-8581, Japan

Failure of a Ductile AdhesiveLayer Constrained by HardAdherendsThe evaluation of a fracture from a thin layer constrained by a hard material is importantin relation to the structural integrity of adhesive joints and composite materials. It hasbeen reported that the fracture toughness of a crack in a ductile adhesive joint dependson the bond thickness, but the mechanism has not yet been elucidated clearly. In this study,the J-integral and the near-tip stress of a crack in an adhesive joint are investigated. Itis determined that a decrease of the bond thickness increases the stress ahead of a cracktip, which results in the decrease of fracture toughness.

1 IntroductionThe estimation of fracture behavior of a crack in an adhesive

joint or a thin ductile layer constrained by hard adherends isimportant from the viewpoint of the structural integrity of com-posite materials and adhesive structures. Bond thickness is one ofthe important design parameters for adhesive structures because ofthe dependence of the fracture toughness on the bond thickness.Several experimental studies of the effect of bond thickness on thefracture toughness have been performed. For example, Gardon(1963) determined that the peeling force, which is proportional tothe fracture energy, decreases with the decrease of bond thickness.Mostovoy and Ripling (1971) reported the same phenomenon in atapered double cantilever beam adhesive joint specimen (TDCB)with an epoxy adhesive. Bascom et al. (1975, 1976) investigatedthe TDCB for a rubber toughened epoxy adhesive and found thatthe fracture energy is maximized when the bond thickness is equalto the diameter of the plastic zone size 2r p for plane strain.r p is thefamous first order estimate of the plastic zone size proposed byIrwin,

5 r p 51

2p

EaG

s 02 Plane Stress

r p 51

6p

EaG

s 02~1 2 n 2!

Plane Strain, (1)

whereG is the energy release rate of a crack in an adhesive layer,Ea Young’s modulus,n a Poisson’s ratio ands0 the yield stress, allof them of the adhesive. Kinloch and Shaw (1981) performed thesame experiment as Bascom et al., and verified the same depen-dence of the fracture energy on the bond thickness, as shown inFig. 1. They considered that the fracture energy of the adhesivejoint is attributed to the volume of a plastic zone, as also illustratedin Fig. 1. Their explanation is as follows. At first, the volume of theplastic zone and the fracture energyGIc are the same as those of a

crack in a bulk adhesive when the bond thickness is much largerthan 2r p(t c @ 2r p). Then the plastic zone develops andGIc

increases with the bond thickness approaching to 2r p due to theconstraint of the adherend (t b . 2r p). The plastic zone reaches themaximum volume andGIc gives a maximum when the bondthickness is equal to 2r p (tm 5 2r p). For t a , 2r p, the decreaseof the bond thickness causes restriction in the development of theplastic zone, andGIc decreases. This qualitative explanation isplausible. It is, however, questionable that the fracture toughnessdirectly relates to the volume of the plastic deformation zonearound a crack tip because the plastic deformation zone is notequal to the process zone. Furthermore, the variation of the plasticdeformation zone with the bond thickness has not been verified byany numerical analysis or any experimental observations.

The effect of confinement by adherends on the fracture tough-ness should be argued in relation to a stress field around a crack tip.Varias et al. (1991) investigated stress fields around a crack tip inmetal foil constrained by ceramic adherends. They determined theincrease of the near tip stress with the decrease of the foil thick-ness. Hsia et al. (1994) studied the effect of confinement byceramic adherends on the dislocation around a crack tip in ex-tremely thin metal foil. Tvergaard and Hutchinson (1996) simu-lated the crack extension of a crack in constrained metal foil.

However, the target of these researches is a crack in metal foilbetween ceramic adherends. The difference in Young’s modulibetween ceramics and metal is about 5 times, which is muchsmaller than that between metal and plastic adhesives. The plasticstrain around a crack tip is dominant around a crack tip withinmetal foil; however, the elastic strain still in the vicinity of a crackin a plastic adhesive can not be ignored. The process zone arounda crack tip in a plastic adhesive is much larger than that in themetal form. Therefore, it is necessary to investigate the stress fieldand a plastic zone around a crack tip within an adhesive layerconfined by hard adherends. We investigated the elastic-plasticstress field around a crack tip within a constrained adhesive layer,using a combination of the finite element method (FEM) and theboundary element method (BEM) (Ikeda et al., 1995). Daghyani etal. (1995) conducted fracture tests of compact tension (CT) adhe-sive joints with several bond thicknesses and observed a transition

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionAugust 5, 1998; revised manuscript received June 22, 1999. Associate TechnicalEditor: G. Ravichandran.


from cohesive fracture in thick bonds to interface fracture in thinbonds. They also performed a large deformation elastic-plasticanalysis for investigating stress near the crack tip. In this study, weexamined the bond thickness effect on theJ-integral of the ECPand the TDCB first. Then we performed the detailed large defor-mation elastic-plastic analyses of stresses ahead of crack tips inadhesive joints.

2 Bond Thickness Effect on theJ-IntegralMany experiments (Mostovoy and Ripling, 1971; Bascom et al.,

1975; Bascom and Cottington, 1976; Kinloch and Shaw, 1981)used approximate equations obtained by the beam theory to cal-culate the fracture toughness, in which adhesive layers are ignored.However, it is possible that the elastic deformation of an adhesiveregion affects the estimation of the fracture toughness due to thesmall elastic constant. The plastic deformation of the adhesive maynot be ignored. We compared the energy release rate obtained bythe approximate equation, that of the linear elastic analysis and theJ-integral obtained by the elastic-plastic analysis. Both the elasticand the elastic-plastic analyses are assumed to be plane strain. Theenergy release rate obtained by the approximate equation and thatby the numerical analysis are denoted byGapp. andG, respectively,hereafter distinguished each from the other.

2.1 Method for J-integral Analysis. TheJ-integrals of sev-eral bond thicknesses were obtained for two types of specimens,the edge cracked plate adhesive joint specimen (ECP) and theTDCB. The combination method of the FEM and the BEM wasused for the analyses (Ikeda et al., 1998). The adherend andadhesive regions were assumed to be a linear elastic material andan elastic-plastic material and modeled by the BEM and the FEM,respectively. Both regions are connected along the interface usingthe conjugate gradient method (Mori, 1986). The algorithm of thiscombination method is based on the traditional incremental straintheory. This combination method has the advantage of saving CPUtime and memory storage size over the finite element method. The

contour integral method was utilized to obtain theJ-integral. Thenumerical integration of the contour integral was performed accu-rately by the adaptive automatic integration technique (Miyazaki etal., 1993). In this method, aJ-integral path can be located both inthe FEM and the BEM regions, and we can obtain the accurateJ-integrals using integral paths located far from a crack tip. Theelastic energy release rateG was also obtained in the same way byassuming both adhesive and adherend regions to be linear elasticmaterials.

2.2 J-Integral of the ECP. The geometry of the ECP underuniform tension is illustrated in Fig. 2. The adherend and adhesivewere assumed to be mild steel and a rubber toughened epoxy,respectively. Combined meshes comprising about 2000 eight-nodeisoparametric finite elements and about 180 quadratic boundaryelements were used to model a symmetrical half of the ECP. Theadherend region was modeled by the BEM as an elastic material,and the adhesive region was modeled by the FEM as an elastic-plastic material, whose stress-strain relation is given by theRamberg-Osgood equation as follows.

e

e05

s

s01 aS s

s0D n

, (2)

where e and s are strain and stress, respectively,E Young’smodulus,s0 the yield stress,a andn the constants. We separated(2) into two parts, the elastic part and the elastic-plastic part, forthe convenience of the numerical analysis.

5e

e05

s

s0. . . ~s # s0!

e

e05

s

s01 aS s

s0D n

. . . ~s . s0! ~e0 5 s0/E!. (3)

We can avoid calculating plastic strain at integral points in finiteelements where the stress is lower than the yield stress at eachincremental step. Both strain-stress relationships obtained by (2)and (3) are almost identical. Although this separation causes ajump of strain frome0 to (11 a)e0 at s beings0 , the effect of thisjump on the stress distribution in the plastic deformation zone issmall. Material constants of each material are shown in Table 1.These material constants are the same as those of Kinloch’s andShaw’s experiment (1981) except for the parametersa andn forthe Ramberg-Osgood equation. The parametersa and n wereassumed to be the same as another rubber toughened epoxy whosestress-strain curve we have measured using strain gauges and atesting machine. TheJ-integral analyses were performed with

Fig. 1 Concept of the bond thickness effect on the fracture toughnessof a crack in an adhesive layer

Fig. 2 Two types of adhesive specimens


several adhesive thicknesses. The elastic energy release rateG wasobtained by assuming that both mild steel and epoxy resin arelinear elastic materials. The energy release rate of the ECP withoutthe adhesive region for plane strain was approximated by (4) and(5) (Murakami et al., 1987).

HKI 5 s fÎpa F~j !, j 5 a/W~j # 0.6, 60.5%!F~j ! 5 1.122 0.231j 1 10.55j 2 2 21.72j 3

1 30.39j 4(4)

Gadherends7K I

2~1 2 n s2!

Es5

$s fÎpa F~j !% 2~1 2 n s2!

Es(5)

To roughly estimate the contribution of an adhesive region to theenergy release rate, the steel adherends are assumed to be a rigidbody in comparison with the adhesive because of the large differ-ence in elastic constant. If we ignore the rotation of the adherends,the adhesive layer can be modeled into a gripped layer betweenrigid adherends under uniform displacement. The energy releaserate of a crack in an adhesive region, subjected to the uniformdisplacement in plane strain shown in Fig. 3, is approximated bythe following equation.

Gadhesive7Ea~1 2 na!

~1 1 na!~1 2 2na!z

n 02

H(6)

whereEa andn a are Young’s modulus and Poisson’s ratio of theadhesive, respectively. Then0 is the uniform displacement of therigid body, theH is half of the bond thicknesst. If we assumed then0 to be equal to the average displacement of the ligament part ofthe ECP,n0 can be expected to be

n0 7HWs f ~1 2 n a

2!

~W 2 a!Ea(7)

The total energy release rate of the ECP can be obtained bycombining (5) and (6),

Gapp.7 Gadherends1 Gadhesive7$s fÎpa F~j !% 2~1 2 n s

2!

Es

1~1 2 n a

2!~1 2 na! 2W2s f2

2~1 2 2na!~W 2 a! 2Eat. (8)

(8) obviously shows a proportional relationship between the en-ergy release rate and the bond thickness.

Figure 4 shows theJ-integral and theG of the ECP with thebond thickness. TheJ-integral obtained by the elastic-plastic anal-ysis corresponds well to theG obtained by the liner elastic anal-ysis, but they are not constant and increase with the bond thick-ness. Thus, in this case, the effect of plasticity in the adhesive can

be ignored, but the effect of bond thickness cannot. The reason forthe energy release rate’s dependence on the bond thickness is thatthe total compliance is affected by that of the adhesive layer. Theapproximated energy release rateGapp. for the ECP calculated by(8) is also shown in Fig. 4. (8) approximates the energy release rateof the ECP relatively well, but it is still smaller than the energyrelease rate calculated by numerical analyses. The two interfacesbetween an adhesive layer and adherends of an actual ECP are notparallel because of the rotation of the adherends, which is apossible reason for the lower evaluation of the energy release rateobtained by (8).

2.3 J-Integral of the TDCB. The next example is theTDCB, shown in Fig. 2. The geometry of the TDCB is the sameas that of Kinloch’s and Shaw’s experiment (1981). All of thematerial constants used in this analysis are the same as those ofthe ECP, as shown in Table 1. TheJ-integral and the elasticenergy release rateG were obtained in the same way as theECP. Combined meshes comprising about 2500 eight-nodeisoparametric finite elements and about 250 quadratic boundaryelements were used to model a symmetrical half of the TDCB.The elastic energy release rate of the TDCB for plane strain,ignoring the adhesive region, can be approximated by the beamtheory as

Gapp.54P2m~1 2 ns!

EsB2

m 5 3a2/h3 11

h~mm21! (9)

whereP is a load,Es Young’s modulus of the adherend,B a widthof the specimen,h a height of the beam,a a crack length, andma constant (Kinloch and Shaw, 1981). TheJ-integral, the elasticenergy release ratesG andGapp., are shown in Fig. 4 as a functionof the bond thicknesst. The J-integral and the elastic energyrelease rateG are almost equivalent. Furthermore, they are inde-pendent of the bond thickness and agree well withGapp. over theactual bond thickness. This tendency is very different from that ofthe ECP. The effect of the adhesive region is negligible due to thesmall amount of strain energy stored in the adhesive region com-pared with that stored in the beams; thus, theJ-integral of TDCBcan be approximated by (9).

Figure 5 shows a comparison of the ratio of stored strain energywithin an adhesive layer in the TDCB and in the ECP. The smallratio of the energy release rate is stored in the adhesive layer in theTDCB, whereas the relatively large ratio of it is stored in theadhesive layer in the ECP, which depends on the bond thickness.The strain energy released from the adhesive layer at a crackextension is very limited for the TDCB, but considerable for theECP.

Table 1 Material constants of adhesive joint

Fig. 3 A crack in an adhesive region under uniform displacement

Fig. 4 J-integral, G and Gapp. of the ECP and the TDCB as a function ofbond thickness


3 Near Crack Tip Stress Analysis of a Crack in anAdhesive Joint

3.1 Analysis of a Crack in Homogeneous Material. Whenthe material property is characterized by the Ramberg-Osgoodequation shown in (1), the stress field in the close vicinity of acrack tip in a plastic zone can be approximated by the HRR fieldas follows (Rice, 1968; Hutchinson, 1968; Rice and Rosengren,1968):

1

s0S s rr s ru

s ru suuD 5 S J

ae0s0 I nrD1/~n11!S s rr s ru

s ru suuD (10)

wherer and u are polar coordinates centered at a crack tip,I n afunction of n, s rr , s uu ands ru functions ofn andu. The near tipstress field in a plastic zone of a metal can be approximated wellby the HRR field. It is, however, not known if the HRR field existsaround a crack tip in a plastic material such as epoxy. The stressaround a crack tip in a CT specimen made of a bulk rubbertoughened epoxy, shown in Table 1, was analyzed. The finitedeformation elastic-plastic analysis by the updated Lagrange algo-rithm was performed by the MARC program, which is a commer-cial software of the finite element method. About 1800 four-nodeisoparametric finite elements were used for modeling a symmet-rical half of the CT specimen. The crack is modeled as a slit,whose width is 2mm for the finite deformation analysis. The finiteelement mesh around each crack tip is common to the CT, theECP, and the TDCB mentioned in the following sections.

The normalized hoop stress along thex-axis near a crack tipobtained from the analysis atJ 5 1500 N/m iscompared with theHRR field and the K-field in Fig. 6. It can be seen that the

analytical results are considerably below the HRR field. The HRRfield is much higher than the stress field of the bulk-CT specimenin the region of 10–100mm distance from a crack tip. The stressahead of a crack tip in the bulk-CT specimen corresponds with theK-field rather than the HRR-field. The HRR field in the sufficientlydeveloped elastic-plastic region must be lower than the K-field.The HRR field is derived by assuming that the 1st term of the rightside of (1) can be neglected well within the developed plasticdeformation zone (Hutchinson, 1968; Rice and Rosengren, 1968).However, the elastic part of the strain cannot be ignored in theplastic zone of a rubber toughened epoxy because of the highelastic strain, and the development of the plastic zone is poor incomparison with that in metal. If we assume the small strain theoryeven in the extreme vicinity of a crack tip, the HRR field may beobserved within a region less than 5mm distant from the crack tip.However, this region has already been covered by the large defor-mation zone, and the adaptability of the homogeneous mechanicsis also doubtful because the diameter of the rubber particle is morethan 1mm. We judge the absence of the HRR field around a cracktip in a rubber toughened epoxy from this indirect evidence.

3.2 Analysis of a Crack in Adhesive Joints. The stressesnear a crack tip of the ECP and the TDCB with several bondthicknesses were also analyzed in the same way as the bulkadhesive CT specimen. The geometries of specimens and thematerial properties are shown in Fig. 2 and Table 1, as well as inthe analyses in the previous chapter. Symmetrical halves of bothspecimens were modeled by about 3000 four-node isoparametric

Fig. 7 An example of the finite element meshes. TDCB specimen with anadhesive layer whose thickness is 0.2 mm.

Fig. 5 Ratio of the strain energy stored in adhesive ( Wadhesive ) to thatstored in whole specimen ( Wwhole ) with bond thickness

Fig. 6 Comparison among the hoop stress distributions near a crack tipof a CT specimen, the HRR-field and the K-field at u 5 0


finite elements. An example of the FEM mesh of the TDCB isshown in Fig. 7. The distributions of the normalized hoop stressess yy/s 0 and triaxial stressessH/s 0 (sH 5 (s xx 1 s yy 1 s zz)/3)near a crack tip of both specimens onx-axis atJ 5 1500 N/m areshown in Figs. 8 and 9. Thes yy/s 0 and sH/s 0 of the bulk CTspecimen are also plotted in these figures.

The hoop stress and triaxial stress distributions near a crack tipare considered to be a function only of theJ-integral and the bondthickness because these stress distributions both for the ECP andthe TDCB with the same bond thickness correspond with eachother. When the bond thickness is thicker than 2.8 mm, the stressdistribution near a crack tip is identical to that of a crack in thebulk CT specimen. In the case of the bond thickness being thinnerthan 0.5 mm, the distributions of both the hoop stress and thetriaxial stress near a crack tip are higher than those of the bulk CTspecimen in the region more than 10mm distant from a crack tip.The length of a failure region observed around a crack tip in therubber modified epoxy is usually from 100mm to 1 mm (Lee et al.,1998). The stress around the failure region is expected to havedirect influence on the fracture toughness. Figure 10 shows thes yy

andsH normalized by those of the bulk CT specimen at a 50mmdistance from each crack tip as the function of the bond thicknesst. Both thes yy andsH increase sharply in the bond thickness lessthan 0.5 mm. One of the major mechanisms of the failure of rubbertoughened epoxy is cavitations of rubber particles by the triaxialstress (Chen and Jan, 1992; Lee et al., 1998). It is expected that thehigher triaxial stress around a crack tip in a thinner bond regionaccelerates the failure process, which results in the decrease of thefracture toughness. A similar mechanism of the decrease of thefracture toughness was reported by Varias et al. (1991) for metalfoil constrained by hard adherends. They expected cavitation andvoid growth in the thin ductile metal layer between hard ceramicadherends.

The influence of bond thickness on the crack tip opening dis-placement (CTOD) was investigated for the ECP and the TDCB.

Deformed crack faces around crack tips in several bond thick-nesses of the ECP and the TDCB atJ 5 1500 N/m areshown inFig. 11. As can be seen from this figure, the CTOD of the ECPalmost corresponds to that of a crack in the homogeneous adhe-sive. The CTOD of the TDCB is not shown in this paper becauseit is almost identical with the case of the ECP. The strip yieldmodel is one of the estimation models of the CTOD for small scaleyielding conditions (Anderson, 1995). This model provides theCTOD for plane strain,

CTOD5G

2s0. (11)

In this case, one half of the CTOD obtained by (11) is about 5.5mm, which corresponds well with one half of the CTOD obtainedby the finite element analysis, as shown in Fig. 11. The CTOD ofthe finite element analysis was determined by the displacement atthe intersection of a 90 degree vertex with the crack flanks. Thisresult takes issue with other researchers. The CTOD obtained byVarias et al. (1991) for a crack in metal foil bonded between twoceramic substrates depends on metal foil thickness. We obtained adifferent result from in our previous paper using coarser mesh thanthat of our current paper, in which the CTOD decreases with thedecrease of bond thickness (Ikeda et al., 1995). Daghyani, Lin andMai also showed the decrease of the CTOD with the decrease ofthe bond thickness (1995). However, the bond thickness depen-dence of the CTOD was almost removed by using finer meshes inour current study. The numerical analysis of the CTOD of a crackin an adhesive layer between hard adherends is very sensitive tothe mesh size. We need more detailed research to make a finaldecision.

The distributions of the plastic deformation zone around crack tipsof several bond thicknesses of the TDCB atJ 5 1500 N/m areillustrated in Fig. 12. The areas of plastic zones around the crack tip

Fig. 8 Distributions of hoop stress near a crack tip on the x-axis of theECP and the TDCB for several bond thicknesses at J 5 1500 N/m

Fig. 9 Distributions of triaxial stress near a crack tip along the x-axis ofthe ECP and the TDCB for several bond thicknesses at J 5 1500 N/m

Fig. 10 Variation of the hoop stress and the triaxial stress near a cracktip along the x-axis at r 5 50 mm as a function of bond thickness

Fig. 11 Variation of the crack tip opening displacement in the severalbond thicknesses of the ECP at J 5 1500 N/m


in both the ECP and the TDCB are plotted with the bond thickness inFig. 13. The area of the plastic zone increases continuously with adecrease in the bond thickness. There is obviously no direct correla-tion between the volume of the plastic zone and the fracture energybecause the increase of the volume of the plastic zone is not related tothe decrease of the fracture toughness, but to the increase. The fracturetoughness of adhesive joints can not be attributed to the volume of theplastic zone around a crack tip.

4 Concluding RemarksFirst, we investigated the influence of the bond region on the

J-integral of two different types of cracked adhesive joints, theECP and the TDCB. TheJ-integral of the ECP increases propor-tionally with the bond thickness, but that of the TDCB is indepen-dent of the bond thickness and corresponds with the approximate

equation which derives from the beam theory. The reason why theJ-integral of the TDCB is independent of the bond thickness is thatalmost all of the strain energy is not stored in the adhesive regionbut in the adherends.

Then, we analyzed precisely the stress fields near a crack tip ofboth the ECP and the TDCB. The following are the results ob-tained from the analyses. The volume of the plastic deformationzone around a crack tip constrained by hard adherends can not berelated to the fracture toughness directly. The stress distributionsnear a crack tip of the ECP and the TDCB agree well with eachother in the case of the same bond thickness. Therefore, the stressdistribution near a crack tip is expected not to depend on the shapeof the specimens, but only on theJ-integral and the bond thick-ness. The CTOD of a crack in an adhesive joint keeps a uniquerelationship with theJ-integral.

The distribution of the triaxial stress near a crack tip increases withthe decrease of the bond thickness in the region more than 10mmdistant from a crack tip. It is expected that higher triaxial stress aheadof a crack tip in a thinner adhesive layer than 0.5 mm accelerates thefailure process and causes smaller fracture toughness. However, weneed other reasons for the drastic decrease of the fracture toughness ina very thin adhesive layer (,0.5 mm). For example, the transitionfrom the cohesive fracture to the interface fracture at a thin adhesivelayer is one of the possible reasons.

ReferencesAnderson, T. L., 1995,Fracture Mechanics,CRC Press, pp. 117–181.Bascom, W. D., Cottington, R. L., Jones, R. L., and Peyser, P., 1975, “The Fracture

of Epoxy- and Elastomer-Modified Epoxy Polymers in Bulk and as Adhesives,”Journal of Applied Polymer Science,Vol. 19, pp. 2545–2562.

Bascom, W. D., and Cottington, R. L., 1976, “Effect of Temperature on theAdhesive Fracture Behavior of an Elastomer-Epoxy Resin,”Journal of Adhesion,Vol. 7, pp. 333–346.

Chen, T. K., and Jan, Y. H., 1992, “Fracture Mechanism of Toughened EpoxyResin with Bimodal Rubber-particle Size Distribution,”Journal of Materials Science,Vol. 27, pp. 111–121.

Daghyani, H. R., Ye, L., and Mai, Y. W., 1995, “Mode I Fracture Behaviour ofAdhesive Joints. Part I. Relationship between Fracture Energy and Bond Thickness,”Journal of Adhesion,Vol. 53, pp. 149–162.

Daghyani, H. R., Ye, L., and Mai, Y. W., 1995, “Mode I Fracture Behaviour ofAdhesive Joints. Part II. Stress Analysis and Constraint Parameters,”Journal ofAdhesion,Vol. 53, pp. 163–172.

Gardon, J. L., 1963, “Peel Adhesion. I. Some Phenomenological Aspects of theTest,” Journal of Applied Polymer Science,Vol. 7, pp. 625–641.

Hsia, K. J., Suo, Z., and Yang, W., 1994, “Cleavage duo to Dislocation Confine-ment in Layered Materials,”Journal of the Mechanics and Physics of Solids,Vol. 42,No. 6, pp. 877–896.

Hutchinson, J. W., 1968, “Singular Behavior at the End of a Tensile Crack in aHardening Material,”Journal of the Mechanics and Physics of Solids,Vol. 16, pp. 13–31.

Ikeda, T., Miyazaki, N., Yamashita, A., and Munakata, T., 1995, “Elastic-PlasticAnalysis of Crack in Ductile Adhesive Joint,”Composites for the Pressure VesselIndustry ASME PVP-302,pp. 155–162.

Ikeda, T., Yamashita, A., and Miyazaki, N., 1998, “Elastic-Plastic Analysis ofCrack in Adhesive Joint by Combination of Boundary Element and Finite ElementMethod,” Computational Mechanics,Vol. 21, No. 6, pp. 533–539.

Kinloch, A. J., and Shaw, S. J., 1981, “The Fracture Resistance of a ToughenedEpoxy Adhesive,”Journal of Adhesion,Vol. 12, pp. 59–77.

Lee, D., Ikeda, T., Todo, M., Miyazaki, N., and Takahashi, K., 1999, “TheMechanism of Damage around Crack Tip in Rubber-Modified Epoxy Resin,”Trans-actions of JSME,Series A, Vol. 65, No. 631, pp. 439–446 (in Japanese).

Miyazaki, N., Ikeda, T., Soda, T., and Munakata, T., 1993, “Stress Intensity FactorAnalysis of Interface Crack Using Boundary Element Method; Application ofContour-Integral Method,”Engineering Fracture Mechanics,Vol. 45, No. 5, pp.599–610.

Mori, M., 1986, “FORT77; Numerical Programming,” Iwanami, pp. 90–114 (inJapanese).

Mostovoy, S., and Ripling, E. J., 1971, “Effect of Joint Geometry on the Toughnessof Epoxy Adhesives,”Journal of Applied Polymer Science,Vol. 15, pp. 661–673.

Murakami, Y. (Editor in Chief), 1987,Stress Intensity Factors Handbook 1,Pergamon Press, p. 9.

Rice, J. R., 1968, “A Path Independent Integral and the Approximate Analysis ofStrain Concentration by Notches and Cracks,” ASMEJournal of Applied Mechanics,Vol. 35, pp. 379–386.

Rice, J. R., and Rosengren, G. F., 1968, “Plane Strain Deformation Near a CrackTip in a Power-Law Hardening Material,”Journal of the Mechanics and Physics ofSolids,Vol. 16, pp. 1–12.

Tvergaard, V., and Hutchinson, J., 1996, “On The Toughness of Ductile AdhesiveJoints,”Journal of the Mechanics and Physics of Solids,Vol. 44, No. 5, pp. 789–800.

Varias, A. G., Suo, Z., and Shih, C. F., 1991, “Ductile Failure of a ConstrainedMetal Foil,” Journal of the Mechanics and Physics of Solids,Vol. 39, No. 7, pp.963–986.

Fig. 12 Distributions of the plastic zone around crack tips in severalbond thickness regions of the TDCB at J 5 1500 N/m (sM: von Mises’sequivalent stress)

Fig. 13 Variation of the area of plastic deformation zone with the bondthickness


T. C. Tszeng1

Department of Mechanical, Materials andAerospace Engineering,

Illinois Institute of Technology,Chicago, IL 60616

Interfacial Stresses and VoidNucleation in DiscontinuouslyReinforced CompositesThis paper presents the theoretical predictions of the stress state at the inclusion-matrixinterface in discontinuous metal matrix composites by the generalized inclusion method.In the author’s previous works, this method had been extended to the elastoplasticdeformation in the matrix material. The present analysis of the ellipsoidal inclusionproblem indicates that the regions at the pole and the equator of the particle/matrixinterface essentially remain elastic regardless of the level of deformation, although thesize of the elastic region keeps decreasing as deformation becomes larger. It was alsofound that, when the composite is undergoing a relatively large plastic deformation(strain), the maximum interfacial normal stress is approximately linearly dependent uponthe von Mises stress and the hydrostatic stress. Based on the stress criterion for voidnucleation, the author determined the void nucleation loci and nucleation strain for acomposite subjected to an axisymmetric macroscopic stress state. The influences ofinterfacial bonding strength, inclusion shape, and volume fraction on the occurrence ofvoid nucleation have been determined. The interfacial bonding strength in a SiC-aluminum system was re-evaluated by using existing experimental evidence.

IntroductionStress state at the inclusion/matrix interface relative to the

bonding strength plays a very important role in determining themechanical strength of discontinuously (e.g., whisker, short fiber,or particulate) reinforced metal matrix composites. As far as duc-tile fracture is concerned, a large number of studies have beenconducted which focused on the mechanism of void nucleation inparticle-strengthened alloys (Fisher and Gurland, 1981; Brown andStobbs 1971; Goods and Brown, 1979; Argon et al., 1975; Hunt etal., 1989; Embury, 1985; LeRoy et al., 1981; Needleman, 1987;Tszeng, 1993; Tvergaard, 1990). The previous studies have led tothe conclusion that void nucleation at the interface between theinclusion and the matrix is one of the controlling phenomena inductile fracture in particle-strengthened alloys, especially in caseswhere the volume fraction of the second phase particles is high.There are two prevailing criteria for void nucleation, namely, theenergy criterion and the stress criterion (see, for example, Fisherand Gurland, 1981). Typical SiCp-reinforced composites usuallycontain SiC particles of a size on the order of 10mm (Weng, et al.,1992; Lloyd, 1989). According to most of the previous studies (seeTszeng, 1993 for a brief review), the energy criterion for voidnucleation on a particle of this size is automatically satisfied. Thatis, the occurrence of void nucleation is solely governed by thestress criterion. This criterion states that voids will nucleate at theinterface if the tensile normal stresss n at the inclusion-matrixinterface exceeds the bonding strengths I of the interface.

While the present paper is mainly focused on interfacial failure,one has to notice that other forms of failure may actually prevail ina process, including particle fracture (Barnes, et al., 1995), voidformation in the matrix (Christman and Suresh, 1988), shearfailure in the matrix (Vasudevan et al., 1989). In addition, theeffects of temperature are very often significant in determining theform of failure (Barnes et al., 1995). With this reservation, this

study only intends to provide an investigation on the influence ofexternal/macroscopic load on the interfacial stress.

Among the theoretical studies of inclusion-matrix interfacialstress surrounding the particle, Argon et al. (1975) and Thomsonand Hancock (1984) used a continuum mechanics based finiteelement method (FEM) while Brown and Stobbs (1971) and Le-Roy et al. (1981) employed the theory of dislocations. The ap-proach based on FEM certainly gives a picture of the detailedstress state in the vicinity of interface, but the choice of unit cell forcomputation and the associated boundary conditions becomesmore difficult with increasing volume fraction of second phaseparticles. This is the case associated with discontinuously rein-forced metal matrix composites, which is the material consideredin the present study.

Any theoretical prediction of void nucleation from the inclusion/matrix interface also requires a good estimate of the bondingstrengths I at the interface. To the author’s knowledge, the onlydata of the bonding strength in a SiC-Al system was reported byFlom and Arsenault (1986), who proposed a lower bound of 1690MPa for the 6061 aluminum alloy reinforced by 1 vol% SiCp.

The objective of this paper is to provide a theoretical view of thestress state at the interface between inclusion and matrix, and torelate these theoretical predictions to the occurrence of void nu-cleation. The level of bonding strength of a SiC-aluminum alloysystem will be re-evaluated. The SiC/2124 aluminum alloy com-posite will be used as the model material and the influences ofhydrostatic pressure will be examined.

TheoryThe reinforcements are represented by hard, elastic ellipsoidal

inclusions of an aspect ratioa embedded in a matrix materialwhich can deform elastoplastically. In a typical short fiber-reinforced composite, the orientation and aspect ratio varies fromfiber to fiber. The effects of variable aspect ratio have beenexamined elsewhere (Tszeng, 1994b), and the present study as-sumed that all the fibers have the same aspect ratio. The analysismethod is an extension of the Eshelby inclusion method (Eshelby,1957) combined with the Mori-Tanaka mean field theory (1973).The full formulation can be found in Tszeng (1994a, b). In usingthis line of approach, one has to know that the legitimacy of the

1 Formerly affiliated with Department of Mechanical Engineering, University ofNebraska-Lincoln.

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials Division June30, 1997; revised manuscript received June 21, 1999. Associate Technical Editor:J. W. Ju.


method becomes suspect when the material is deformed at arelatively large plastic strain, or in a system comprising highvolume fraction of reinforcement (Christensen, 1990). In addition,the use of the secant method in the generalized inclusion methodfor finite plasticity also invites concerns. To work within thelimitations, we will use the model only in a class of proportional,monotonic loading problems.

In the following, we provide a brief account of the formulationfor calculating interfacial stress by the generalized inclusionmethod (Tszeng, 1994a, b). The stress in the considered reinforce-ment,sF, is calculated by (bold-faced variables denote tensors):

s F 5 L F~eA 1 eC 1 e9! Inhomogeneous composite (1)

5 L M~eA 1 eC 1 e9 2 eT!

Reference homogeneous composite (2)

whereeA is the strain in homogeneous, unreinforced matrix metal,eC is the constrained strain,e9 is the additional term due to theinteraction between inclusions,eT is the incompatible transforma-tion strain, andL F and L M are the stiffness tensors of the rein-forcement and matrix material, respectively. In fact,L M is thesecant stiffness tensor of the matrix material atsM andeM when itgoes into plastic deformation, i.e.,sM 5 L MeM. The secantYoung’s modulus and the corresponding Poisson’s ratio for plasticincompressibility are, respectively,

1

ES 51

EM 1e# p

sY(3)

n S 51

22 S1

22 n MD ES

EM (4)

where EM and nM are, respectively, the Young’s modulus andPoisson’s ratio of the matrix material, ande# p andsY are, respec-tively, the effective plastic strain and the flow stress of the matrixmaterial. In general, the flow stresssY is dependent on the effec-tive plastic strain,e# p. The von Mises yield criterion is used for thematrix, and the components of the plastic strain in the matrix aredetermined by the flow rule

deP 53

2

de# p

sYs (5)

wheres is the deviatoric stress of the average matrix stresssM.The reinforcement stress in Eqs. (1) or (2) is the stress in a single

particle of particular orientation in the matrix. For a compositecomprising reinforcement of different orientation, each individualreinforcement would experience a different stress state dependingon its individual orientation. A volume average procedure isneeded for composites having reinforcements with different ori-entations. The formulation given by Tszeng (1994a) shows theaverage transformation strainêT& is obtained by solving the fol-lowing equation:

@~1 2 f !L M~^S& 2 I ! 2 L F~~1 2 f !^S& 1 f I !#êT&

5 ~L F 2 L M!~I 1 L# !eA. (6)

The volume average is defined in Tszeng (1994a). In Eq. (6),f isthe volume fraction of the reinforcement,S is the fourth-orderEshelby tensor which depends only on the inclusion geometry(aspect ratio) and the Poisson’s ratio of the matrix,I is thefourth-order identity tensor andL# 5 (L M)21(L R 2 L M), L R is thesecant stiffness tensor of the matrix material atsA and eA, i.e.,sA 5 L ReA. After the average transformation strainêT& is ob-tained, the transformation strain,eT, associated with the consideredinclusion can be obtained (Tszeng, 1994a).

The stress state at a point immediately outside the inclusionsurface is obtained by modifying the original formulation ofEshelby (1957). At point P in Fig. 1, the unit normaln is

n 5 ~cosf cosc, sin f cosc, sin c!. (7)

The anglec is related tou by

tan c 51

a2 tan u(8)

wherea is the aspect ratio. The strain at the interface is:

e~Int! 5 eC~Int! 1 eM, (9)

whereeM 5 êT& 1 e9, is the average matrix strain. The hydro-static and deviatoric part of the interfacial constrained straineC(Int)

are respectively given by, in indicial form (Eshelby, 1957):

eoC~Int! 5 eo

C 21

3

1 1 n S

1 2 n S eoT 2

1 2 2n S

1 2 n S 9eijTni nj , (10)

and

9eilC~Int! 5 9eil

C 11

1 2 n S 9ejkT nj nkni nl 2 9eik

T nknl 2 9elkT nkni

11 2 2n S

3~1 2 n S!9ejk

T nj nkd il 21

3

1 1 n S

1 2 n S eoTSni nl 2

1

3d ilD . (11)

whereeT is the stress free transformation strain associated with theconsidered inclusion. In these expressions, a variable primed at theleft top corner represents the deviatoric part, and a subscript “o”represents the hydrostatic part. Note thatnS in above equations isthe secant Poisson’s ratio of the matrix material defined in Eq. (4).The corresponding interfacial stresss(Int) is obtained through sim-ple calculation, i.e.,s(Int) 5 L (Int)e(Int), where L (Int) is the secantmodulus ate(Int). Since a material point at the “interface” is actuallylocated immediately outside the inclusion surface, the mechanicalproperties of the interface are assumed to be the same as those ofmatrix material. For convenience, the interfacial stress is trans-formed to the local coordinate which has the 3-axis coincident withthe surface outer normaln (Fig. 1).

The present paper is focused on the occurrence of void nucle-ation based on the stress criterion (Tszeng, 1993), no void growthor coalescence will be discussed. More precisely, we are investi-gating the limiting conditions for void nucleation, instead of thepost-nucleation behavior. The latter subject has been studied byseveral investigators in the past (e.g., Needleman, 1987).

Fig. 1 The coordinate system and conventions for ellipsoidal inclusion.At a surface point P which makes angles ( u, f) with axes 3 and 1, theouter normal from the surface has an angle c with the horizontal plane.


Results and DiscussionIn the following, the mechanical properties are based on SiC

reinforced 2124 aluminum alloys which were recently studied byseveral investigators (Christman et al., 1989; Tvergaard, 1990;Tszeng, 1994a, b). This composite system is chosen simply as amodel material, the general observations to be derived from thetheoretical calculations are rather generic and therefore should beapplicable to other composite systems as well. The Young’s mod-ulus and Poisson’s ratio areEM 5 60 GPa andnM 5 0.3 for thematrix andEF 5 342.6 GPa andnF 5 0.21 for theinclusion. Theflow stress,sY, of the matrix material is given by the modifiedLudwik relation:sY 5 s o 1 h(e# p) n , with s o 5 300 MPa,h 5442.1MPa, andn 5 0.478,which is very close to what was usedby Tvergaard (1990). The macroscopic stress is axisymmetric withs 11

A 5 s 22A 5 rs 33

A , wherer is the proportional factor.

Interfacial Stresses on Spherical Particles. To elucidate thecalculations, we first consider a simple composite comprisingellipsoidal reinforcement of an aspect ratioa 5 1 (i.e., sphericalparticulate), volume fractionf 5 0.1, andloaded by a uniaxialstresss 33

A 5 1 MPa in the principal (long) direction (i.e., the3-axis). The distribution of interfacial stress is plotted as a functionof the angleu from the pole (see Fig. 1) in Fig. 2. The stressesshown in Fig. 2 actually represent the specific stress (stress dividedby the macroscopic stresss 33

A ) in the elastic range of the compositedeformation. As shown in Fig. 2, the normal stress (along direction39 in Fig. 1) shows a maximum tension,s nmax, at the pole (u 5 0).Figure 2 also shows the the von Mises effective stress(5=1

2((s 11 2 s 22)2 1 (s 22 2 s 33)

2 1 (s 11 2 s 33)2) 1/ 2, where

s11, s22, s33 are the principal stress in the local coordinate system).In contrast, the von Mises effective stress (normalized to theoverall uniaxial stress) has a peak value of about 1.5 atu 5 45 degfrom the pole. For a rigid, spherical inclusion in an elastic matrix,Goodier (1933) found that the maximum tensile stress duringuniaxial loading iss nmax 5 2s 33

A . By increasing the Young’smodulus of the particle material and decreasing the particle vol-ume fraction, we found essentially the same results as that ofGoodier’s, see Table 1.

In the present analysis, there is no separate yield criterion for theinterface, yielding is allowed to occur only in the matrix which, bydefinition, includes the interface. It is of interest to examine thepossibility of yielding at the interface. We apply the von Misesyield criterion on the data presented in Fig. 2. According to Fig. 2,the specific von Mises effective stress has a peak value of about 1.5at u 5 45 deg from the pole. Since the initial yield stress is 300MPa, yielding at the interface first occurs at a macroscopic stressof about 200 MPa (5300 MPa/1.5) in the region aroundu 5 45deg fora 5 1, and spreads along the interface as the macroscopicstress becomes higher. The macroscopic stress iss 33

A 5 324 MPa

when initial yielding occurs globally in the matrix as based on theaverage matrix stresssM. At that moment, the interfacial vonMises effective stress atu ; 0 deg actually just exceeds the yieldstress, whereas the interface away from the pole (e.g.,u . 70 deg)remains elastic. As the macroscopic stress continues increasing,Fig. 3 illustrates the stress distributions at different levels ofdeformation. Several interesting phenomena can be seen in Fig. 3.First, the peak tensile normal stress always appears at the pole, andincreases with the deformation. In this respect, Argon et al. (1975)determined that the normal stress in the polar region seemed tohave a flat distribution in the case of a cylindrical inclusion,whereas both Orr and Brown (1974) (on cylindrical inclusion) andThomson and Hancock (1984) (on spherical inclusion) indicatedthat the maximum normal stress moves away from the pole as thedeformation becomes larger. Although the inclusions studied bythe works of Orr and Brown (1974) and Thomson and Hancock(1984) had a very low volume fraction (,1%), we have foundessentially the same pattern of distribution as of Figs. 2 and 3 forcomposites with different level of particle volume fraction. Thus,difference in volume fraction is not the cause of the discrepancy.Further study is needed to clarify the cause of the discrepancy.

Let the maximum normal stress be expressed in the form afterThomson and Hancock (1984):

snmax 5 Cs# 1 sm (12)

whereC is the concentration factor, ands# andsm are the macro-scopic von Mises effective stress and hydrostatic stress (5(sm1 1sm2 1 sm1)/3), respectively. Note that the original form of Eq.(12) proposed by Argon et al. (1975) actually assumedC 5 1. Aplot of the concentration constantC versus the effective plasticstrain is given in Fig. 4 (the curve withr 5 0). The concentrationfactor is increasing with deformation, as observed by Thomson andHancock (1984). However, qualitative differences do exist be-tween our calculation ofC and that of Thomson and Hancock.First, the value ofC becomes more level at a larger compositestrain, compared with that of Thomson and Hancock which in-creases almost linearly with the strain. Second, our calculations

Fig. 2 The distribution of interfacial specific stresses (interfacial stressdivided by s33

A ) as a function of the angle from the pole in a compositedeformed elastically during uniaxial loading. The aspect ratio of theinclusion a 5 1 and 5, volume fraction f 5 0.1.

Table 1 Maximum interfacial tensile stress as a function ofvolume fraction and ratio of Young’s moduli

Volumefraction

Ratio of Young’s moduli (inclusion to matrix)

1 10 100

0.1 1.559 1.759 1.7820.01 1.653 1.906 1.9350.001 1.663 1.922 1.9520.0001 1.664 1.923 1.954

Fig. 3 The distribution of normal stress and von Mises stress at theparticle-matrix interface in a composite deformed plastically during uni-axial loading at a range of overall uniaxial stress s33

A . The aspect ratio ofthe inclusion a 5 1, volume fraction f 5 0.1.


(not shown) indicate that, in the elastic range, the value ofCremains constant whereas that of Thomson and Hancock shows arapid decrease followed by a sudden increase in the transition fromelastic to plastic deformation.

It is desirable to know whether Eq. (12) holds for a more generalstate of macroscopic stress. In this respect, we consider an axi-symmetric, proportional macroscopic stress withs11 5 s22 5 rs33,wherer remains constant during the loading history. By changingthe value ofr, the role of hydrostatic stress can be examined. Thecase withr 5 0 corresponds to the uniaxial loading (s11 5 s22 50) and a more positiver corresponds to a more tensile hydrostaticstress. For the considered macroscopic stress with21 # r # 1, theinterfacial normal stress always has a peak at the pole of thespherical particles. The maximum interfacial normal stress at arange of effective plastic strain is calculated, and the correspond-ing concentration factorC is plotted in Fig. 4. The concentrationfactor, C, increases very rapidly at small plastic strains, andbecomes more level at larger deformation. Note that the curves inFig. 4 do not collapse into one single curve, the value ofC istherefore dependent on the type of macroscopic stress state. It is,however, noticed that the values ofC for differentr’s seem to bevery close to each other at a larger strain. This means that thesimple form of Eq. (12) is applicable at a relatively large plasticstrain for a more general state of macroscopic stress.

There are two other ways to look at the empirical Eq. (12). First,at a specified macroscopic von Mises effective stress, Eq. (12)implies that the maximum interfacial normal stress is linearlydependent on the hydrostatic stress with a unity proportionalityconstant. The relation according to our calculations is plotted inFig. 5 for three levels of von Mises effective stress. Indeed, a linearrelationship does exist between the maximum interfacial normalstress and the hydrostatic stress. The slopes of these curves arevery close to unity, although it decreases slightly at a higher Misesstress. Second, at a specified macroscopic hydrostatic stress, Eq.(12) implies that the peak stress is linearly dependent on the vonMises effective stress with a proportionality constantC. Therelation according to our calculations is plotted in Fig. 6 for threelevels of hydrostatic stress. Again, the relationship is approxi-mately linear, although the proportionality constantC becomesslightly lower at a higher hydrostatic stress. According to theabove elaboration, it seems that the linearity as represented by Eq.(12) does exist when either the Mises stress or the hydrostaticstress hold constant.

According to Fig. 3, the von Mises effective stress at theinterface is increasing with the overall stress. However, distinctlydifferent phenomenon occurs at the pole and the equator. Asdiscussed earlier, the von Mises stress at the interface between

angles 0 and 70 deg from the pole already exceeds the initial yieldstress of the matrix material (300 MPa) when the composite startsto yield globally. In the transition from elastic to plastic deforma-tion, the von Mises stress at the pole decreases and becomes lowerthan the matrix’s yield stress. To further illustrate the stress statein the polar region, the von Mises stress at the pole is plotted as afunction of the macroscopic effective stress, shown in Fig. 7. Asindicated, the von Mises stress at the pole does exceed the yieldingstress in the transition from elastic to plastic deformation, followedby an asymptotic decrease. The interface in the polar regiontherefore experiences unloading after the early, very low level ofplastic deformation. To facilitate discussion, let us use two anglesu1 andu2 in the way that the plastic zone at the interface is definedby u1 # u # u2. That is, the region between 0# u # u1 is in elasticdeformation. The angles are plotted as functions of the effectiveplastic strain in Fig. 8. The angleu1 is equal to 0 deg at the elasticlimit (meaning the pole is in plastic deformation) and increasesvery rapidly right after the composite goes into plastic deforma-tion. After reaching a peak, the angleu1 then monotonicallydecreases as the deformation becomes even larger. The FEMcalculations of Thomson and Hancock (1984) also indicated thesimilar tendency of a very low level of plastic deformation at thepole (see Figs. 9 and 14 of Thomson and Hancock, 1984). Aspointed out by Thomson and Hancock (1984), the observed yield-ing at the polar element in their study may be a consequence of the

Fig. 4 The concentration factor C as a function of the effective strain forcomposites comprising spherical particles of volume fraction 0.1. s11

A 5s22

A 5 rs33A

Fig. 5 The relationship between the maximum interfacial normal stressand the hydrostatic stress at several different levels of von Mises stress(labeled by the corresponding curves). All stresses are in MPa. Thevolume fraction of spherical particle is 0.1.

Fig. 6 The relationship between the maximum interfacial normal stressand the von Mises stress at several different levels of hydrostatic stress(labeled by the corresponding curves). All stresses are in MPa. Thevolume fraction of spherical particle is 0.1.


finite dimensions of the finite element mesh, and therefore may notrepresent the actual feature.

Similar phenomena also appear at the equator, where the vonMises stress reaches its peak of 101 MPa (,s o 5 300 MPa) at themacroscopic elastic limit, and decreases monotonically after thecomposite yields globally (see Fig. 7). The plastic zone at theinterface spreads toward the equator at larger deformation, asindicated by the increasingu2 in Fig. 8. However, the material inthe region of the equator would not go into plastic deformation,regardless of the level of macroscopic deformation.

Based on the above observations, we see that the plastic zonefirst appears atu ; 45 deg and spreads in both directions along theinterface; the pole (u 5 0 deg) and the equator (u 5 90 deg)essentially remain elastic throughout the deformation. This con-clusion is generally in accord with the FEM calculations of Thom-son and Hancock (1984) (see Fig. 7 of that paper).

According to Table 1, a lower particle volume fraction leads toa higher maximum interfacial normal stress,s nmax at the samemacroscopic stress. However, the same statement does not standwhen the maximum interfacial normal stress is plotted as a func-tion of the macroscopic strain. This point will be re-examined laterin this paper.

Interfacial Bonding Strength. As mentioned earlier, the onlydata of the bonding strength in an SiC-Al composite system wasreported by Flom and Arsenault (1986), who suggested a lowerbound of 1690 MPa for the 6061 aluminum alloy reinforced by 1vol% SiCp. In determining the actual maximum interfacial normalstress, Flom and Arsenault used the original form of Eq. (12)proposed by Argon et al. (1975), i.e.,C 5 1, for the axisymmetrictensile specimen with a semi-circular, circumferential groove. Dueto the lack of exact material properties of the aluminum alloymatrix considered by Flom and Arsenault (1986), we are not ableto use the present method for an estimate of the bonding strength.

Further, as shown in the Appendix, the bonding strength of 1690MPa is not reproducible based on the data provided by Flom andArsenault (1986).

A recent work of Weng et al. (1992) on the microfracture ofSiCp reinforced 6061-T6 aluminum alloy provides other possibledata of the bonding strength of the SiC-aluminum system. For thestandard tensile specimen of composite comprising 10 vol% SiCparticles of mean diameter 10mm, matrix debonding starts appear-ing at a uniaxial stress of 253 MPa. By using Eq. (12) withC 51 and assuming the effective stress in the matrix when the particlesare absent is 253 MPa, one obtained the bonding strength as 337MPa. Since, in general, the value ofC in Eq. (12) is greater than1, the actual interfacial normal stress, and therefore the bondingstrength, should be higher than 337 MPa. By using the presentmethod with matrix properties:EM 5 70 GPa,nM 5 0.33, s o 5220 MPa,h 5 190 MPa, andn 5 0.56, themaximum interfacialnormal stresss nmax at s 33

A 5 253 MPa was found to be 416 MPaand the corresponding macroscopic strain is 0.00619 in the loadingdirection. Thus, the bonding strength of the considered compositesystem is about 416 MPa.

According to these data for the bonding strength of SiC-aluminum alloy system, the value of 1690 MPa suggested by Flomand Arsenault (1986) seems to be too high. A plausible bondingstrength should range between 400 MPa and 900 MPa (see theAppendix). However, it is understood that the bonding strength fora given material system depends very much upon the processinghistory. Further, void nucleation itself is a stochastic process(Thomson and Hancock, 1984). Consequently, a statistical ap-proach may be needed for a better description of the bondingstrength. Similar studies on the particle cracking have been con-ducted in the past, e.g., Mochida, Taya and Lloyd (1991). It will bea sizable effort to determine the statistical nature of the interfacialbonding strength and to include this description in the theoreticalmodel. That effort is beyond the scope of the present paper.Instead, two different levels of bonding strength will be used toexplore their influences on the void nucleation, as will be discussedin the next section.

Void Nucleation. In the following calculations regarding voidnucleation for the considered SiC-2124 aluminum alloy matrixcomposite, two levels of interfacial bonding strengths I are to beused, namely, 500 and 1000 MPa. A bonding strength of 1000MPa may not be achievable by the current technology, as will beseen later. The macroscopic stress is axisymmetric withs 11

A 5 s 22A

5 rs 33A . For a void to nucleate from the particle/matrix interface,

the maximum interfacial normal stresss nmax which appears at thepole has to exceed the bonding strengths I of the interface. Thenucleation loci for both levels of bonding strength are calculatedand shown in Fig. 9 after LeRoy et al. (1981). The same figure alsoshows the loci corresponding to the initial yielding and the hydro-static stress state, respectively. The bonding strength does have animportant influence on the occurrence of void nucleation. With thehigher bonding strength of 1000 MPa, the composite shows veryhigh resistance to void nucleation. Note that void nucleation in thecomposite with a lower bonding strength occurs approximately atthe elastic limit during uniaxial loading (s 11

A 5 s 22A 5 0).

The nucleation strains in square root are plotted as a function ofthe hydrostatic stresssm in Fig. 10.A higher hydrostatic (tensile)stress corresponds to a smaller nucleation strain. This figure is inagreement with the widely accepted observation that a compres-sive hydrostatic stress can inhibit or delay the damage formation/accumulation. Again, the important influence of bonding strengthis obvious. According to the present calculation, for a compositecomprising spherical particles of volume fraction 10% and abonding strength of 1000 MPa, the nucleation strain in uniaxialloading is as high as 0.169. Obviously, this strain is even largerthan most of theelongation strain at rupture reported in theliterature. That is, a bonding strength of 1000 MPa may be unre-alistically high from the standpoint of current technology.

Fig. 7 The changing of interfacial von Mises at the pole and at theequator as functions of the effective plastic strain

Fig. 8 The angles ( u1 , u2) defining the plastic zone at the interface arechanging as the deformation proceeds


Effects of Aspect Ratio and Volume Fraction. The curves ofspecific interfacial stress for a composite comprising ellipsoidalinclusions (fibers) of an aspect ratio of 5 are also shown in Fig. 2.Obviously, at the same macroscopic stress, inclusions of a higheraspect ratio experience a high peak stress at the inclusion-matrixinterface. Thus, particulate-reinforced composites are expected topossess a better ductility compared with the whisker-reinforcedcomposites, which is actually what is generally observed (Rackand Ratnaparkhi, 1988). To further explore this point, the macro-scopic uniaxial stress (i.e.,r 5 0) needed to bring the maximuminterfacial normal stress to the bonding strength is plotted as afunction of the fiber aspect ratio in Fig. 11. In these calculations,the bonding strength is taken to be 1000 MPa. The same figure alsoshows the effects of reinforcement volume fraction. As indicatedin this figure, the nucleation strain decreases very rapidly with theaspect ratio. At an aspect ratio of 5, void nucleation actually takesplace in the elastic range. It is also noticed that the nucleationstrain approaches an asymptotic value at a large aspect ratio.

In Fig. 11, the volume fraction has a more important role in thecomposite comprising inclusions of a smaller aspect ratio. In thisrespect, a larger volume fraction leads to a lower nucleation strain.This is in agreement with the general observation that the ductilityand fracture toughness of discontinuously reinforced metals isusually inferior to that of unreinforced matrix metals, and that ahigher volume fraction results in a smaller elongation strain (duc-tility).

According to the curve corresponding to an aspect ratio of 5 inFig. 2, initial yielding at the interface may have started at thelocation of peak Mises stress at as 33

A as low as 110 MPa, which is

about one-third of the initial yield stress of the matrix aluminumalloy. To ensure the composite is free from localized plasticdeformation, the macroscopic stress has to be below the elasticlimit which is about 110 MPa in the considered case. As mentionedearlier, the corresponding limit stress in composites comprisingspherical particulate is 200 MPa. It is generally accepted that localconcentrations of plastic deformations should be avoided for abetter fatigue resistance (Dieter, 1976). Hence, a higher elasticlimit corresponds to better fatigue resistance, especially in thecases involving a high level of stress. Since it has been shown thatthe elastic limit is lower for composites comprising fibers of alarger aspect ratio, one may expect composites comprising fibersof a higher aspect ratio should possess a lower fatigue resistance.This remark is actually contradictory to what is generally observed(Rack and Ratnaparkhi, 1988). This is a topic subjected to furtherstudy.

ConclusionSeveral concluding remarks can be derived from our calcula-

tions and discussion for composites subjected to an axisymmetricmacroscopic stress state.

1. When the composite is undergoing a relatively large plasticdeformation (strain), the maximum interfacial normal stress isapproximately linearly dependent upon the von Mises stress andthe hydrostatic stress (Eq. (12)). That is, the general form of Eq.(12) is valid only at a relatively large plastic strain. However, thelinearity does exist when either the Mises stress or the hydrostaticstress remains constant in a loading process.

2. The regions at the pole and the equator of the particle/matrix interface essentially remain elastic regardless of the level ofdeformation, although the size of the elastic region keeps decreas-ing as deformation becomes larger.

3. Interfacial bonding strength is the most influential singlefactor as far as void nucleation is concerned. A plausible value ofbonding strength in SiC-aluminum systems is in the range between400 MPa and 900 MPa.

4. A higher level of reinforcement volume fraction leads to alower nucleation strain. Its effects becomes diminishing in com-posites comprising inclusions of a larger aspect ratio.

AcknowledgmentThe author would like to thank the reviewers for their elaborated

comments on the original manuscript.

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Ductile Fracture,”Metallurgical Transactions,Vol. 6A, pp. 825–837.

Fig. 9 A diagram showing the comparison of the calculated nucleationloci for composites comprising spherical particles of volume fraction f 50.1 subjected to axisymmetric deformation s11

A 5 s22A 5 rs33

A . Two levelsof interfacial bonding strength are used.

Fig. 10 A plot of the square root of nucleation strain ( =eN) as a functionof macroscopic mean stress for composites comprising spherical parti-cles of volume fraction f 5 0.1 subjected to axisymmetric deformation s11

A

5 s22A 5 rs33

A . Two levels of interfacial bonding strength are used.

Fig. 11 A plot of the nucleation strain ( =eN) for composites comprisingellipsoidal inclusions of a range of aspect ratio and volume fractionduring uniaxial loading. The interfacial bonding strength is assumed tobe 1000 MPa.


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A P P E N D I X

Bonding Strength According to the Experiments of Flomand Arsenault (1986)

According to Flom and Arsenault (1986), the hydrostatic stressreaches its maximum value at the bottom of the semi-circulargroove, which is given by

sm 5c

Î1 2 ~a/a# ! 2 sY (A1)

wheresY is the flow stress,c 5 0.48 anda# 5 2.3 mm, and 2ais the current diameter of the ligament. However, the works ofHancock and MacKenzie (1976) and Hancock and Brown (1983)indicated that the hydrostatic stress in the transverse cross sectionactually has a minimum value at the bottom of the groove andreaches maximum at the center. According to Flom and Arsenault(1986), particle debonding could barely be seen at the bottom ofthe circumferential groove at an uniaxial load of 750 kgf on thespecimen. The corresponding flow stresssY 5 750 3 9.81/pa2

MPa, wherea is not available from the paper of Flom andArsenault (1986). Since the actual dimensions of the specimen arenot available from the paper, a reverse procedure is to be used toassess those missing data. In fact, by using the original formula fora# andc (Eqs. (3) and (4) of Flom and Arsenault, 1986) and usinga ranging from 1.85 through 2.25 (initial diameter at the groovedarea), we were not able to obtain thec 5 0.48 anda# 5 2.3 mmgiven by Flom and Arsenault (1986). According to our calculation,(c, a# ) 5 (0.4826, 2.3) ata 5 2.2, and (c, a# ) 5 (0.48, 1.97)at a 5 1.97. Thecorresponding maximum interfacial normalstresss nmax 5 1285 and 1692 MPa, respectively. Also, the corre-sponding flow stresssY 5 484 and 670 MPa, respectively. Now,according to Hancock and MacKenzie (1976), the hydrostaticstress at the bottom of the groove is1

3sY. Using Eq. (12) withC 51, we obtained the lower bound of the bonding strength as 645 and893 MPa, respectively, corresponding to the flow stress of 484 and670 MPa. Thus, a plausible bonding strength should range between400 MPa and 900 MPa.


Marwan K. KhraishehDepartment of Mechanical Engineering,King Fahd University of Petroleum and

Minerals (KFUPM),Dhahran 31261, Saudi Arabia

e-mail [email protected]

An Investigation of YieldPotentials In SuperplasticDeformationRecent results (Khraisheh et al., 1995 and 1997) have indicated that superplastic mate-rials exhibit a strong degree of anisotropy and that the plastic flow cannot be describedby the isotropic von Mises flow rules. In this study, the yield potential for the model Pb-Snsuperplastic alloy is constructed experimentally for different effective strain rates usingcombined tension/torsion tests. A generalized anisotropic “dynamic” yield function is alsoproposed to represent the experimentally constructed yield potentials. The anisotropicfunction is not only capable of describing the initial anisotropic state of the yield potential,it can also describe its evolution through the evolution of unit vectors defining thedirection of anisotropy. The anisotropic yield function includes a set of material constantswhich determine the degree of deviation of the yield potential from the isotropic von Misesyield surface. It is shown that the anisotropic yield function successfully represents theexperimental yield potentials, especially in the superplastic region.

I IntroductionThere has been a significant amount of research done in the area

of superplasticity in recent years due to the potential benefits in thearea of metal forming. Many parts used in the aerospace industryare currently manufactured from Ti and Al superplastic alloys, andrecently, superplastic forming has been considered for use in theautomotive industry due to the many advantages over conventionalforming techniques (Hamilton, 1985; Ghosh, 1985; and Smith,1998). Greater design flexibility, lower die cost, and the ability toform complex shapes from hard materials are some of benefits thatsuperplastic forming offers. However, for the superplastic formingprocess to become more acceptable as an efficient metal formingtechnique suitable for high production rate, there is a need fornumerical tools that can predict the deformation behavior of su-perplastic materials under multiaxial loading conditions, as en-countered in actual forming operations. In order to achieve that,accurate constitutive relations for superplastic deformation mustbe developed. One important aspect in constructing the constitu-tive relations is the selection of appropriate yield function, in thecase of rate-independent materials, or equivalently the yield po-tential (dynamic yield function) in the case of rate-dependentmaterials (e.g., superplastic materials).

Most of the work that has been directed toward studying themechanics of superplastic materials is based on uniaxial loadingconditions and assumes isotropic behavior (Holt, 1970; Hamiltonet al., 1991; Dutta and Mukherjee, 1992; and Carrino and Guiliano,1997). Moreover, the von Mises yield criterion is usually em-ployed to extend the uniaxial models to multiaxial loading condi-tions through definitions of effective stress and strain. Until re-cently (Khraisheh et al., 1995 and Khraisheh et al., 1997), therewas no or little information reported on anisotropy in superplasticdeformation, which led to the use of the isotropic von Mises flowrules. The recent results indicate that the Pb-Sn superplastic alloyexhibits a strong degree of deformation-induced anisotropy. Theseresults included induced axial stresses in torsion and distortionaleffects in tension/torsion tests. The source of the deformationinduced anisotropy is believed to be the severe grain shape distor-tion and grain rotation. For more details on the mechanism(s) of

deformation induced anisotropy, see Khraisheh et al. (1995) andMayo and Nix (1989). The results suggest that in order to accu-rately model the deformation behavior of superplastic materialsunder general loading conditions, an appropriate anisotropic dy-namic yield function must be used in representing the yield po-tential.

For rate-independent materials subjected to multiaxial loading,anisotropic yield surfaces are used to describe anisotropic effectsresulting from texture development. There is a considerableamount of work done in this area, for example, Hill (1950), Barlat(1987), Barlat and Fricke (1988), and Barlat and Lian (1989). Forrate-dependent materials, however, the anisotropic dynamic yieldsurface (yield potential) is the equivalent of the anisotropic yieldsurface. Very little is known about the yield potentials in super-plastic materials. In this paper, the yield potential for the Pb-Snsuperplastic alloy is constructed experimentally using combinedtension/torsion tests at different effective strain rates. In addition,a general anisotropic dynamic yield function is used to accuratelyrepresent the experimentally constructed yield potentials.

II ExperimentalCombined tension/torsion tests were conducted on a modified

screw-type Instron machine capable of controlling the axial andradial motions and especially equipped with data acquisition sys-tem. The tests were conducted on the Sn-38.1% Pb superplasticalloy, for which superplastic forming is possible at room temper-ature. The superplastic alloy was prepared in air by melting com-mercially pure Pb and Sn and casting into a mold. The casting wasthen upset in one direction and rolled to a final thickness resultingin a thickness strain of22.71. The resulting microstructure wasexamined by scanning electron microscope and can be character-ized by equiaxed grains with an average grain size of;5.8 mm.Cylindrical bar test specimens with gage length of 2.54 cm and adiameter of 0.46 cm were then machined from the rolled sheet. Formore details on the experimental set-up, material preparation, andthe mechanical properties of the alloy used here, see Khraisheh etal. (1995), and Khraisheh et al. (1997).

For rate-dependent materials and using the general associatedflow rule, it is shown that for every effective strain rate,e# , there isone value for the dynamic yield function (Dafalias, 1990 andKhraisheh et al., 1997). Hence the yield potential in the stressspace can be constructed from experiments by maintaining theeffective strain rate constant for various loading paths. Therefore,

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionNovember 15, 1998; revised manuscript received May 14, 1999. Associate TechnicalEditor: N. Chandra.


combined tension/torsion tests at constant effective strain rateswere conducted to construct the yield potential. The effectivestrain rate for simple tension, pure torsion, and combined tension/torsion cases using von Mises definition are given by:

Simple Tension

e# 5 e (1)

Pure Torsion

e# 5g

Î3(2)

Combined Tension-Torsion

e# 5 Îe 2 1g 2

3(3)

where e is the axial strain rate andg is the radial strain rate.Proportional loading tests withe 5 kg were conducted, wherek isthe proportionality constant. For a specific effective strain rate,different combinations of the axial and radial strain rates wherechosen to obtain different points in thes-t plane. The axial strainrate can be expressed in terms of the effective strain rate andk as:

e 5 e# Î 3k2

1 1 3k2 (4)

Five data points were used to construct the yield surfaces. Eachpoint corresponds to a different value ofk. One point correspondsto a k value of` (simple tension), another point corresponds to ak value of 0 (pure torsion), and three points corresponds to finitevalues ofk (combined tension/torsion).

An important issue in constructing the yield surface is how todefine the yield point. For most metals, the initial yield point isclear and the proportional limit or the 0.2% offset methods areusually used to determine the yield point. For most superplasticmaterials, however, there is no well defined yield point and thedefinition of the yield point (or flow stress value) is controversial.The problem arises because plastic deformation for superplasticmaterials at the superplastic temperature takes place almost in-stantly and the value of the flow stress changes significantly at thevery early stages of deformation, where the yield point is usuallydefined (e.g., 0.2% strain). For the Pb-Sn superplastic alloy usedhere, this problem was solved by examining the stress/straincurves. The flow stress is almost constant and no significanthardening is observed. Therefore, the steady or the “saturated”value of the flow stress (axial and shear) was used in constructingthe yield surface. Hence the yield surface presented here representsthe collection of points in the stress space, at which superplasticflow occurs. As an illustration, the axial and shear stress compo-nents for a combined tension/torsion test at an effective strain rateof 6.5 3 1024 s21 andk 5 0.45 areshown in Figs. 1and 2. It isclear that the axial and shear components saturate approximately tovalues of 16 MPa and 7.75 MPa, respectively. These values areused to locate one point on the yield potential (Fig. 5).

Fig. 1 The axial stress component in a combined tension/torsion test.Effective strain rate 6.5 3 1024 s21, k 5 0.45.

Fig. 2 The shear stress component in a combined tension/torsion test.Effective strain rate 6.5 3 1024 s21, k 5 0.45.

Table 1 Results of experiments performed to construct the yield potential for an effective strain rate of 33 1024 s21

Prop. constant (k 5 e/g)Axial strain rate,

e (s21)Radial strain rate,

g (s21)Saturated axial

stress, MPaSaturated shear

stress, MPa

0 (pure torsion) 0 5.203 1024 2* (induced) 7.650.25 1.193 1024 4.773 1024 8.25 5.700.6 2.163 1024 3.603 1024 12.50 4.501.4 2.773 1024 1.983 1024 15.5 2.0` (pure tension) 3.003 1024 0 20.6 0

* Induced axial stress during fixed-end torsion tests, for more details see Khraisheh et al. (1995).

Table 2 Results of experiments performed to construct the yield potential for an effective strain rate of 6.53 1024 s21





stress, MPa

0 (pure torsion) 0 1.133 1023 1.75* (induced) 11.000.15 1.603 1024 1.093 1023 8.75 9.500.45 3.93 1024 9.03 1024 16.00 7.751.1 5.73 1024 5.43 1024 21.25 4.00` (pure tension) 6.53 1024 0 23.50 0



The experimental data used in constructing the yield potentialsfor effective strain rates of 33 1024, 6.53 1024, and 13 1023 s21

are summarized in Tables 1, 2, and 3, respectively. The corre-sponding yield potentials are plotted in Figs. 3, 4, and 5.

Fore superplastic materials which exhibit significant hardening(e.g., Al7474 superplastic alloys), a consistent meaningful proce-dure need to be employed to determine the yield point. A possibleway is to use the value of the stress at zero plastic strain. However,

this technique involves graphical extrapolation and could presenterror and scatter in the results.

III Yield Potential RepresentationThe selection of the appropriate form for the dynamic yield

function is very important in constitutive modeling. First, theisotropic von Mises yield function is considered and comparedwith the experimental data. It has the form:

f 5 Î3J2

J2 5 12 S.S (5)

wheref is the yield function andJ2 is the second invariant of thedeviatoric stress tensor,S. For combined tension/torsion condition,the yield function is reduced to:

f 5 Îs 2 1 3t 2 (6)

s is the axial stress andt is the shear stress. Equation (6) ismodified such that the yield function reduces to the tensile yieldstress in simple tension (f 5 s y) leading to:

s y2 5 s 2 1 3t 2 (7)

Equation (7) represents the von Mises yield surface and is plottedagainst the experimental data in Figs. 3, 4, and 5. The values forsy

were taken from Tables 1, 2, and 3 for simple tension case (k 5 `).It is clear from the figures that the experimental data cannot be

represented by the isotropic von Mises yield function. This distor-tion of the yield potential can be attributed to the deformation-induced anisotropy that was observed for this alloy (Khraisheh etal., 1995). Therefore, a general anisotropic dynamic yield function,Fig. 3 The yield potential for the Pb-Sn superplastic alloy. Effective

strain rate 3 3 1024 s21. Experimental data versus von Mises and theanisotropic functions.

Table 3 Results of experiments performed to construct the yield potential for an effective strain rate of 13 1023 s21





stress, MPa

0 (pure torsion) 0 1.733 1023 1.25* (induced) 14.20.2 3.253 1024 1.6253 1023 12.5 11.40.5 6.253 1024 1.33 1023 20.0 7.751.1 8.93 1024 8.03 1024 25.0 4.5` (pure tension) 1.03 1023 0 27.8 0


Fig. 4 The yield potential for the Pb-Sn superplastic alloy. Effectivestrain rate 6.5 3 1024 s21. Experimental data versus von Mises and theanisotropic functions.

Fig. 5 The yield potential for the Pb-Sn superplastic alloy. Effectivestrain rate 1 3 1023 s21. Experimental data versus von Mises and theanisotropic functions.


in which anisotropy can evolve is now considered. The yieldfunction which is defined in reference to the axisxi (i 5 1, 2, 3)has the following form (Hill, 1950; Dafalias, 1990; and Khraishehand Zbib, 1997):

f 2 5 32 S.S1 c1~M.S! 2 1 c2~N1.S! 2 1 c3~N2.S! 2

N1 5 a1 # a1 N2 5 a2 # a2

M 5 12 @a1 # a2 1 a2 # a1# (8)

wherea1, a2, and a3 are orthonormal vectors (a3 5 a1 3 a2)along the axis of anisotropyx9i (i 5 1, 2, 3), c1, c2, andc3 arematerial parameters, andV is the diadic product. Forc1 5 c2 5c3 5 0, the anisotropic yield function reduces to the isotropic vonMises yield function (equation 5). The directions ofa1 anda2 aredefined by the angle betweenx1 anda1 measured positive coun-terclockwise (f), as shown in Fig. 6. The direction tensorsN1, N2,andM can then be expressed in terms off as:

N1 5 F cos2 f cosf sin f 0cosf sin f sin2 f 0

0 0 0G

N2 5 F sin2 f 2cosf sin f 02cosf sin f cos2 f 0

0 0 0G

M 5 12 F 22 cosf sin f cos2 f 2 sin2 f 0

cos2 f 2 sin2 f 2 cosf sin f 00 0 0

G (9)

For combined tension/torsion case, Eq. (8) reduces to:

f 2 5 AIs2 1 AIIst 1 AIII t

2

AI 5 1 1c1

9cos2 f sin2 f 1

c2

9~2 cos2 f 2 sin2 f! 2

1c3

9~2 sin2 f 2 cos2 f! 2

AII 5 2 23 cosf sin f$c1~cos2 f 2 sin2 f! 2 2c2~2 cos2 f

2 sin2 f! 1 2c3~2 sin2 f 2 cos2 f!%

AIII 5 3 1 c1~cos2 f 2 sin2 f! 2 1 4c2 cos2 f sin2 f

1 4c3 cos2 f sin2 f (10)

The values of the material parameters (c1, c2, andc3) and theanglef determine the degree of deviation of the anisotropic yieldsurface from the isotropic von Mises yield surface. Equation (10)can be written in a nondimensional form such as:

1 5 AISs

f D2

1 AIISs

f DS t

fD 1 AIIIS t

fD2

(11)

The nondimensional yield surface is plotted in Figs. 7, 8, 9, and10 for different values of the anglef, and the material parametersc1, c2, andc3, respectively. The figures clearly show how the yieldsurface is affected by each parameter. In general, the anglef

Fig. 6 Schematic illustration of the axis of anisotropy and the angle f

Fig. 7 The effect of the angle f on the anisotropic yield surface fortension-torsion case (equation 11). c1 5 1, c2 5 5, and c3 5 4. The dashedline represents von Mises surface ( c1 5 c2 5 c3 5 0).

Fig. 8 The effect of c1 on the anisotropic yield surface for tension-torsion case (Eq. (11)). f 5 30°, c2 5 5, and c3 5 4. The dashed linerepresents von Mises surface ( c1 5 c2 5 c3 5 0).



determines the deviation of orientation of the anisotropic yieldsurface from the von Mises isotropic one. The material parametersc1, c2, and c3 determine the magnitude or the intensity of thedeviation.

In order to compare the anisotropic yield surface with the experi-mental data in Figs. 3, 4, and 5, the conditionf 5 sy is imposed, wheresy is the yield stress in simple tension. This leads to:

f 2 51

AI@AIs

2 1 AIIst 1 AIII t2# 5 s y

2 (12)

Equation (12) is plotted against the experimental data and vonMises in Figs. 3, 4, and 5.

IV Discussion and ConclusionsIt is clear from Figs. 3, 4, and 5 that the yield surface (or flow

curve) of the Pb-Sn superplastic alloy is anisotropic and cannot berepresented by the isotropic Mises surface. This is especially truefor the low strain rate (33 1024 s21) which lies in the superplasticregion (region II of the logs 2 log e curve). As the strain rateincreases, we move away from the superplastic region, and vonMises surface presents better representation of the experimentaldata. Hence, when modeling the deformation behavior of super-plastic materials in the superplastic region, it is inappropriate toassume isotropic behavior and use the von Mises stress/strainequivalents. The departure from isotropy explains why von Mises-based models failed to predict the deformation behavior of thePb-Sn superplastic alloy under biaxial loading in bulge forming(Khraisheh and Zbib, 1999).

The anisotropic yield function proposed in this study is not onlycapable of describing the initial anisotropic nature of the yieldsurface, it can also describe its evolution through the evolution ofthe unit vectors,a1 and a2 (i.e., the evolution of the anglef),which define the direction of the axis of anisotropy. The materialdependent parameters,c1, c2, andc3 have significant effects on theshape of the yield surface. They define the intensity of the depar-ture from the isotropic yield surface.

After rigorous fitting trials to the experimental data shown inFigs. 3, 4, and 5, the values (c1 5 1, c2 5 5, c3 5 4, andf 530°) present the best combination of the parameters which resultedin a very good fit to the experimental data for the three differentstrain rates. Of course, a more accurate treatment can result if thematerial parameters can be expressed in terms of the strain rate,however, for simplicity, constant values are used here. The aniso-tropic yield function for these values are compared against theexperimental data and Von Mises in Figs. 3, 4, and 5. It is evidentthat this anisotropic yield function represents the experimental datavery well, especially in the superplastic region (e 5 3 3 1024 s21).

It is worth mentioning that only experimental data in the firstquadrant is available, and these values of the parameters representthe best fit in the first quadrant only. However, if results of otherloading directions are available (i.e., data in other quadrants), theproposed yield function (with the range of parameters that itincludes) can easily fit almost any experimental data.

In a previous study (Khraisheh et al., 1997), the isotropic vonMises yield surface was used in constructing a constitutive modelfor superplastic materials. Although the model was able to capturea number of phenomena, including uniaxial behavior and stressrelaxation, however, it failed to predict the actual trend of thedeformation-induced anisotropy. Under preparation is a study inwhich the anisotropic yield potential proposed here is employed inthe constitutive model developed earlier to predict thedeformation-induced anisotropy. The anglef, which indicates thedeviation of the orientation of the anisotropic function, will play asignificant role in predicting the induced anisotropy and its evo-lution.

In summary, the present study clearly showed that superplasticdeformation is anisotropic, as the experimentally constructed yieldpotentials deviate considerably from the isotropic von Mises yieldsurface. The proposed anisotropic dynamic yield function is capa-ble of representing the experimental results and also in capturingthe evolution of the induced-anisotropy.

AcknowledgmentThe author wishes to acknowledge the continuous collaboration

and fruitful discussions with Professor Hussein Zbib of Washing-ton State University which motivated this research. The support ofKFUPM is also acknowledged.

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ing Limits of Sheet Metals,”Material Science and Engineering,Vol. 91, pp. 55–72.Barlat, F., and Fricke, W. G., Jr., 1988, “Prediction of Yield Surfaces, Forming

Limits and Necking Directions for Textured FCC Sheets,”Proceedings of the EighthInternational Conference on Textures of Materials,J. S. Kallend and G. Gottstein,eds., The Metallurgical Society, pp. 1043–1050.

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Hamilton, C. H., Zbib, H. M., Johnson, C. H., and Richter, S. K., 1991, “DynamicGrain Coarsening and its Effects on Flow Localization in Superplastic Deformation,”Proceedings of the Second International SAMPE Symposium,Chipa, Japan, pp.272–279.

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Khraisheh, M. K., Bayoumi, A. E., Hamilton, C. H., Zbib, H. M., and Zhang, K.,1995, “Experimental Observations of Induced Anisotropy During the Torsion ofSuperplastic Pb-Sn Eutectic Alloy,”Scripta Metallurgica et Materialia,Vol. 32, No.7, pp. 955–959.

Khraisheh, M. K., Zbib, H. M., Hamilton, C. H., and Bayoumi, A. E., 1997,“Constitutive Modeling of Superplastic Deformation. Part I: Theory and Experi-ments,”International Journal of Plasticity,Vol. 13, No. 1–2, pp. 143–164.

Khraisheh, M. K., and Zbib, H. M., 1997, “Constitutive Modeling of AnisotropicSuperplastic Deformation,”Physics and Mechanics of Finite Plasticity and Visco-plastic Deformation,A. S. Khan, ed., NEAT Press, Fulton MD, pp. 45–46.

Khraisheh, M. K., and Zbib, H. M., 1999, “Optimum Forming Loading Paths forPb-Sn Superplastic Sheet Materials,” ASME JOURNAL OF ENGINEERING MATERIALS AND

TECHNOLOGY, Vol. 121, No. 3, pp. 341–345.Mayo, M. J., and Nix, W. D., 1989, “Direct Observation of Superplastic Flow

Mechanisms in Torsion,”Acta Metallurgica,Vol. 37, No. 4, pp. 1121–1143.Smith, M. T., 1998, “Research Toward the Increased Use of Superplastic Forming

in Lightweight Structures,”Proceedings of the Symposium on Superplasticity andSuperplastic Forming,35th Annual Meeting of the Society of Engineering Science,M. Khaleel, ed., Pacific Northwest National Laboratory, PNNL-SA-30406.



Masahito MochizukiDepartment of Manufacturing Science,

Osaka University,Suita, Osaka 565-0871,

Japan(Formerly, Hitachi, Ltd.)

e-mail: [email protected]

Makoto Hayashi

Toshio Hattori

Mechanical Engineering Research Laboratory,Hitachi, Ltd.,

Tsuchiura, Ibaraki 300-0013,Japan

Numerical Analysis of WeldingResidual Stress and ItsVerification Using NeutronDiffraction MeasurementDirect measurements and computed distributions of through-thickness residual stress in apipe butt-welded joint and a pipe socket-welded joint are compared. The analyticalevaluation methods used were inherent strain analysis and thermal elastic-plastic anal-ysis. The experimental methods were neutron diffraction for the internal residual stress,and X-ray diffraction and strain-gauge measurement for the surface stress. The residualstress distributions determined using these methods agreed well with each other, both forinternal stress and surface stress. The characteristics of the evaluation methods and thesuitability of these methods for each particular welded object to be evaluated arediscussed.

IntroductionWelding residual stress can eventually result in brittle fracture,

fatigue failure, and stress corrosion cracking in welded structures.Therefore, when a welded structure needs to be strength-designedor crack propagation needs to be evaluated, it is important to knowthe distribution of residual stress in detail including through-thickness distributions (Almen and Black, 1963; Todoroki andKobayashi, 1988). A method that can be used to directly measureresidual stress in the welded structure should be established be-cause each structure naturally contains dispersion of welding re-sidual stress. There are many difficulties in the stress measurementbecause the various conditions of the measurement are limited bythe environment of the welded structure or by the specification ofthe measurement method. Therefore, we should also establish ananalytical method for welding residual stress at the same time asthat we are developing a measurement method.

We have developed analytical evaluation methods for weldingresidual stress by using inherent strain analysis (1995; 1997) andthermal elastic-plastic analysis (1993; 1996). The analytical resultswere verified by comparing them with those obtained from otherevaluation techniques. That is, for the stress on the surface ofwelded structures, calculated residual stress were compared withstrain-gauge measurements and they showed good agreement. Wealso compared the through-thickness residual stress from the in-herent strain analysis and the thermal elastic-plastic analysis. Andit was shown that both stress distributions agree well. However,through-thickness residual stress obtained by the analytical methodhas hardly been directly compared with measured values forwelded structures.

Neutron diffraction is a common method for measuring internalstress of a structure (McEwen et al., 1983; Allen et al., 1985). Theprinciple of neutron diffraction is the same as that of X-raydiffraction. Residual stress at the structure surface can be measuredby using X-ray diffraction because the penetration depth of X-raysis extremely shallow (about 20mm from the surface). On the otherhand, the penetration depth of neutrons is deep, about 50 mm fromthe surface even in case of steel, so it can be used to measure thethrough-thickness residual stress in steel welded structures.Through-thickness residual stress in various kinds of welded joints

has been measured using neutron diffraction (Hutchings, 1990;Wikander et al., 1994; Hayashi and Ishiwata, 1995). However, theverification of measured values by neutron diffraction are seldomdone for welded structures. In particular, the measured values forthrough-thickness welding residual stress by neutron diffractionhave not been validated by comparing them with other evaluationresults. Furthermore, it is commonly considered to be difficult forneutron diffraction to measure residual stress at a stress-concentration point because the minimum volume to be measureddepends on the width of neutron beam.

In this paper, we compare direct measurements and computeddistributions of through-thickness residual stress in welded joints.Residual stresses are evaluated in a pipe butt-welded joint and apipe socket-welded joint. Inherent strain analysis and thermalelastic-plastic analysis are adopted as the analytical evaluationtechnique and neutron diffraction is used as the measurementtechnique for evaluating through-thickness residual stress. Stressmeasurement method using a strain gauge and X-ray diffractionmethod for measuring residual stress on the structure surface weresimultaneously used for comparing analyzed residual stress distri-butions on the structure surface. It is shown that the most suitablemethod for evaluating residual stress can be selected by consider-ing each characteristics of the evaluation technique and the con-dition of the welded object.

Residual Stress In Pipe Butt-Welded Joint

The Object for Residual Stress Evaluation. The geometry anddimensions of a 4-in.-diameter pipe butt-welded joint, which is madeof carbon steel (JIS STPT410), used to evaluate residual stress areshown in Fig. 1. The angle of the V-shaped groove is 60 degrees, andthe groove root width is 3 mm. There are four welding layers and fourwelding passes. The heat input is 1.2 kJ/mm for each welding pass.Strain-relief treatment was done before welding in order to removeresidual stress due to machining.

Evaluating Residual Stress

Inherent Strain Analysis. When stress is induced in a bodywithout the action of external load, the stress is called residualstress which is self-equilibrating, and the source of residual stressshould exist in the body. The source of residual stress is incom-patible strain which is called inherent strain. In the case of weldedjoints, the source of residual stress is plastic strain includingdislocations. The plastic strain is regarded as inherent strain.

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionOctober 7, 1998; revised manuscript received May 11, 1999. Associate TechnicalEditor: N. Chandra.


The following elastic response equations are generally derivedas the relationships between the vectors of inherent strain {e*},elastic strain {e e}, and residual stress {s} which occur at anylocation in an elastic body owing to the inherent strain (Ueda andFukuda, 1989).

$s% 5 @D#$e e%5 @D#@H* #$e* %J . (1)

Here the matrix [H*] is the elastic strain-inherent strain matrix andthe matrix [D] is the elastic stress-strain matrix. The matrix [H*]links the overall fields of inherent strain to the overall fields ofelastic strain, and depends on the domain occupied by the body andits boundary conditions. And the componentH*ij of [H*] can bederived as thei th component of the elastic response strains subjectto the j th component of the inherent strains being unit.

The distribution zone and the magnitude of inherent strain donot change as long as no new inherent strains are added by plasticdeformation resulting from mechanical cutting, external loads, andso on. Therefore, even though the geometry of an object is changeddue to cutting, the inherent strain can be estimated from theobserved value of elastic strain as follows (Mochizuki et al., 1995).

First, to measure the elastic strain induced in a structural mem-ber, as many changes of strain as possible need to be observed bycutting or the like. Since various kinds of errors in measurementmay be induced in the observed strains, the measurement equationis as follows.

$me e% 2 @H* #$e* % 5 $r % (2)

where {e*} is the most probable value for inherent strain and {r }is the residual.

The most probable value {e*} for the inherent strain can bedetermined from a condition that minimizes the sum of squares ofthe residual {r } as in the following equation.

$e* % 5 ~@H* # T@H* #! 21@H* # T$me e% (3)

The most probable value {e*} for welding residual stress at anylocation can be determined by substituting the most probable value{ e*} obtained for the inherent strain {e*} in Eq. (1).

The procedure for evaluating residual stress in the pipe butt-welded joint is shown in Fig. 2. Strain gauges are bonded on thesurface of the welded joint. Cutting the welded joint using anabrasive cut-off machine and bonding the strain gauges to a sectionafter cutting are repeated one after the other. Inherent strain iscalculated from measured elastic strain {me e} by using Eq. (3).The detailed procedure is as follows.

After the strain gauges are bonded on the surface of the weldedjoint, the T-specimen and theLi-specimens (i 5 1 to n) are cutfrom the original pipe butt-welded joint. The inherent strain in theoriginal welded joint is the same in theT-specimen and theLi-specimens. This is because inherent strain does not change evenif the specimen is cut. Residual stress remaining in eachLi-specimen results only from by circumferential inherent straine*u.The inherent strains in the other directions (radial and axial) do notcontribute to residual stress in theLi-specimens. The shear com-

ponents of inherent strains can be ignored because they are muchsmaller. Residual stress remaining in theT-specimen also occursby axial inherent straine*z and radial inherent straine*r .

Each released strain is measured after cutting around the straingauges on the specimens to form dice-like pieces. By substitutingthese released strains into Eq. (3), circumferential inherent straine*u can be calculated from the measured data of theLi-specimens.And axial inherent straine*z and radial inherent straine*r can becalculated from the measured data of theT-specimen. The originalinherent strain distribution of the whole welded joint can beconstructed by gathering the calculated inherent strain from eachsmall specimen. Three-dimensional residual stress includingthrough-thickness stress can be calculated by using the completeinherent strain in the pipe butt-welded joint. This inherent strainanalysis was done by the elastic finite element method.

Thermal Elastic-Plastic Analysis.Thermal elastic-plastic be-havior during each welding step caused by temperature changewas simulated by thermal elastic-plastic analysis. The detailedflow is described as follows (Mochizuki et al., 1993; Mochizuki etal., 1995).

The material properties at each temperature from room temper-ature to melting point must be known for the materials used in theanalysis. The temperature distribution at each time due to the heatinput of welding was computed by transient heat conductionanalysis during welding procedure. Transient stress during thewelding steps was successively calculated from thermal loadsobtained by heat conduction analysis, which was done by thermalelastic-plastic analysis with the nonlinearity of materials. Residualstress can be determined as the stress when the material is cooleddown to room temperature after welding. The residual stress in thepipe butt-welded joint was computed by thermal elastic-plasticanalysis using an axi-symmetric model.

Neutron Diffraction Measurement.Neutron diffraction is usedfor measuring the spacingd between the atomic planes of a crystal

Fig. 2 Sequences of measuring released strains for calculating inherentstrains in a pipe butt-welded joint

Fig. 1 Configuration of a carbon-steel pipe butt-welded joint


lattice (Allen et al., 1985). A neutron beam of known wavelengthl is diffracted by a scattering angle 2u from its incident directionaccording to Bragg’s law.

2d z sin u 5 nl (4)

Elastic straine is determined by comparing the measured valueof d to the value measured in a suitable stress-free referenced0 byusing the differentiated equation (4).

e 5Dd

d05

d 2 d0

d05 2cot u z Du. (5)

Measurements of the change of a spacing between the atomicplanes of a crystal lattice,Dd 5 d 2 d0, or a scattering angleDucan be used to calculate elastic straine. To calculatee, it isnecessary to measured0 or u0 under stress-free conditions. Ac-cordingly, the reference scattering angleu0 must be measured aftera part of the specimen of the measured object is cut out and treatedby stress-relief annealing. A characteristic of neutron diffraction isthat the measurement ofu0 is essential, whereas it is not necessaryfor X-ray diffraction.

Strain is measured in a direction that bisects the incident anddiffracted neutron beams. To obtain values for strain in the prin-cipal directions of the object to be measured (that is, axial straine z,radial straine r , and circumferential straineu), the object must bereoriented so that a bisector is located in the required direction, asFig. 3 shows. The incident and diffracted beams are shaped by slitsin neutron-absorbing cadmium masks.

Measurements are made within the region of intersection ofthese two beams. With three measurements of strain at eachlocation, the principal residual stresses, axial stresss z, radial stresss r , and circumferential stresssu, can be calculated through ageneralized Hooke’s law.

sz 5EN

1 1 n H ez 1n

1 2 2n~ez 1 e r 1 eu !J (6)

s r 5EN

1 1 n H e r 1n

1 2 2n~ez 1 e r 1 eu !J (7)

su 5EN

1 1 n H eu 1n

1 2 2n~ez 1 e r 1 eu !J (8)

Here EN is Young’s modulus for neutron diffraction andn isPoisson’s ratio.

The diffraction plane dependence due to elastic anisotropy ofthe material affect elastic constants for neutron or X-ray diffrac-tion. Thus, the Young’s modulus for neutron diffractionEN isdifferent from the mechanical Young’s modulusE which is typi-cally used. The appropriate elastic constraints for neutron diffrac-tion areEN 5 243 GPa andn 5 0.28 for elastic strain determinedfrom shifts in the (211) diffraction peak in a pipe butt-welded jointmade of ferritic steel (Hayashi et al., 1995a), and internal residualstress was measured. The neutron diffractometer at Chalk RiverNuclear Laboratories of Atomic Energy of Canada Limited wasused in the measurement (Hayashi and Ishiwata, 1995).

X-Ray Diffraction Measurement.The principle of X-ray dif-fraction measurement is the same as that of neutron diffraction.However, only residual stress near the surface can be measuredbecause the penetration depth of X-rays is extremely shallow(about 20mm). The stress component of the normal direction onthe surface can therefore be regarded as 0, and it is not necessaryto measure the scattering angle. Residual stress in the pipe butt-welded joint was measured by a sin2 w 2 6 points method (Societyof Material Science, Japan Ed., 1981). Cr-Kb X-rays were used forthe measurement.

Strain-Gauge Measurement.Measuring stress by strain gaugecan obtain the residual stress on the surface of the structure to beevaluated (Lu Ed., 1996). Once small pieces are cut off from the

object, residual stress in the small pieces is physically released.Residual stress is converted into the released elastic strain which isobtained as the difference between before and after cutting. Thedetailed procedure for measuring the residual stress on the surfaceof the pipe butt-welded joint is as follows.

First, strain gauges were bonded on the inside and outsidesurface of the welded joint. The periphery of the bonded straingauges in the welded joint was cut into small cubes with about10-mm sides. Residual stress in the small pieces was released bycutting the welded joint, and axial released straine z and circum-ferential released straineu were measured. Axial residual stresss z

and circumferential residual stresssu can be obtained from thefollowing equations using measured strains.

sz 5 2E

1 2 n 2 ~ez 1 neu ! (9)

su 5 2E

1 2 n 2 ~eu 1 nez! (10)

HereE is Young’s modulus andn is Poisson’s ratio for the weldedjoint.

Comparison of Residual Stress Distributions. The weldingresidual stress in a carbon-steel pipe butt-welded joint was evalu-ated using each method. Through-thickness residual stress wascompared with the results obtained from inherent strain analysis,

Fig. 3 Scattering geometries for measurement of three-components oflattice strain in a pipe butt-welded joint


thermal elastic-plastic analysis, and neutron diffraction measure-ment. Residual stress on the surface was evaluated by inherentstrain analysis, thermal elastic-plastic analysis, X-ray diffractionmeasurement for the outer surface, and strain-gauge measurement.

Axial residual stresses along the weld-center line through thethickness direction are compared in Fig. 4. The horizontal axis isthe radial distance from the inner surface to the outer surface.These stress distributions by inherent strain analysis, thermalelastic-plastic analysis, and neutron diffraction measurement agreewell with each other. Axial residual stresses along the heat-affected zone are compared in Fig. 5. Residual stress from all threemethods agree except the measurements near the inner surface byneutron diffraction. These measurement values are a little lowerthan the analytical results.

Circumferential residual stresses along the weld-center line andalong the heat-affected zone through the thickness direction areshown in Figs. 6 and 7, respectively. Both figures indicate that theresults for the three evaluation methods completely agree well. Itis thought that these results verify the validity of stress distributionby comparing each other.

Axial residual stress on the outer surface of the pipe butt-weldedjoint is shown in Fig. 8. The stress distributions by inherent strain

analysis, thermal elastic-plastic analysis, X-ray diffraction mea-surement, and strain-gauge measurement are compared in thefigure. Analytical results by thermal elastic-plastic analysis arelittle higher than those of others, but the other residual stresseswholly agree well. And thermal elastic-plastic analysis givesslightly different results in this figure.

Axial residual stresses on the pipe inner surface are compared inFig. 9. Residual stresses by using each evaluation method agreewell, except the results of thermal elastic-plastic analysis are alittle different to those of others.

Residual Stress in Pipe Socket-Welded JointThrough-thickness residual stress in a pipe socket-welded joint was

also evaluated as another example for the comparison of internalresidual stress. The configuration for the pipe socket-welded joint isshown in Fig. 10. Two small-diameter pipes, which are made ofcarbon steel (JIS STPT410), are fillet-welded with a socket, which ismade of carbon steel (JIS S25C). The number of welding layers aretwo and the number of passes are three. The heat input is 1.4 kJ/mmfor each welding pass. Strain-relief treatment was done before weld-ing in order to remove residual stress by machining.

Through-thickness residual stresses in the welding deposit weremeasured by using neutron diffraction. The measuring positions for

Fig. 4 Comparison of axial residual stress across the weld metal centerin a pipe butt-welded joint

Fig. 5 Comparison of axial residual stress across the heat-affectedzone in a pipe butt-welded joint

Fig. 6 Comparison of circumferential residual stress across the weldmetal center in a pipe butt-welded joint

Fig. 7 Comparison of circumferential residual stress across the heat-affected zone in a pipe butt-welded joint


the neutron diffraction are shown in Fig. 11 (Hayashi et al., 1995b).The sampling volume was about 1 mm3 1 mm 3 2 mm with thelong dimension tangent to the curvature of the pipe for the axial andradial strains, and 1 mm3 1 mm3 4 mm for circumferential strain.Residual stress in the socket-welded joint was also calculated by usinginherent strain analysis and thermal elastic-plastic analysis. The min-imum mesh size for the thermal elastic-plastic analysis was nearly thesame as the measured area of the neutron beam.

Axial and circumferential residual stresses in the weld-metal ofthe pipe socket-welded joint are compared in Figs. 12 and 13,

respectively. Both distributions are agreed with each other in eachfigure, except near the welding root, in which the stress is con-centrated. It is surely shown to be difficult for neutron diffractionto measure residual stress at the stress-concentration point, becausethe minimum range to be measured depends on the area of theneutron beam and because a value of the residual stress by neutrondiffraction is obtained as the average in the measuring volume.

Fig. 8 Comparison of axial residual stress on the outer surface in a pipebutt-welded joint

Fig. 9 Comparison of axial residual stress on the inner surface in a pipebutt-welded joint

Fig. 10 Configuration of a small-diameter pipe socket-welded joint

Fig. 11 Measuring points of residual stress in a pipe socket-weldedjoint by neutron diffraction

Fig. 12 Comparison of axial residual stress in the deposit in a pipesocket-welded joint

Fig. 13 Comparison of circumferential residual stress in the deposit ina pipe socket-welded joint


Discussion for Evaluation MethodsWe confirmed that residual stress evaluated by using each

method agreed very well for two pipe welded joints. This meansthat each method can precisely evaluate welding residual stress forboth internal and surface stresses. Therefore, the characteristics ofeach evaluation method are examined as follows by consideringthe conditions under which each evaluation method was applied toan actual welded structure.

Characteristics of Each Evaluation Method

Inherent Strain Analysis. This method can be used to simplycalculate residual stress by elastic analysis. It can easily be appliedto a lot of welded structures with various configurations, once thedatabase for inherent strain distribution is constructed (Mochizukiet al., 1997). On the other hand, the evaluations for structuralshape, welding method, or welding conditions which do not con-form to a database are difficult. Even if these are done, it isnecessary to assume the inherent strain distribution to be used inthe evaluation. Furthermore, transient thermal stress or stress his-tory in the welding process cannot be calculated by using inherentstrain analysis.

Thermal Elastic-Plastic Analysis.Welding residual stress canbe accurately calculated by simulating the change of mechanicalbehavior caused by thermal elastic-plastic phenomenon during thewelding process. In thermal elastic-plastic analysis, the time spentis extremely long. Also, the material properties of the object fromroom temperature to melting point should be known. However,thermal elastic-plastic analysis can be used to calculate residualstress with some degree of precision, even material property can bedefined precisely, and even the welded objects can be treated bythe two-dimensional models. In other words, thermal elastic-plastic analysis becomes a powerful technique when the analysiscondition is appropriately known. It is also suitable for a weldedstructure with special shape, in which the application of inherentstrain analysis is difficult, and for materials that are difficult toinvestigate, such as irradiated material.

Neutron Diffraction Measurement.This technique can be useddirectly to obtain internal residual stress by nondestructive mea-surement. However, the objects which can be measured are limiteddepending on the condition of the neutron source device. Thespacing between the atomic planes of the crystal lattice or scatter-ing angle under stress-free conditions must be measured as areference. There are many difficulties in direct application to anactual structure. But neutron diffraction becomes very effectivewhen the residual stress in a small test piece is to be measured orwhen the verification using a mock-up specimen is necessary.

X-Ray Diffraction Measurement.Nondestructive measurementon the structural surface is possible using X-ray diffraction. However,the measured value depends on the surface conditions of the object tobe measured (Park et al., 1994). Therefore, scrupulous attention todetail is necessary in measurement. The measurement becomes dif-ficult so that the diffraction profile is hard to obtain at the place wherethe crystalline structure of the welded metal is large, as in somematerials. It can be used to measure an actual structure because thebody for the X-ray equipment can be miniaturized.

Strain-Gauge Measurement.Method of stress measurementusing the strain-gauge can obtain average stress in small piecesthat have been cut into cubes. It is very difficult to apply to actualstructures because strain-gauge measurement is a destructivemethod. However, it is the most standard technique for the residualstress measurement in a welded specimen. Through-thickness re-sidual stress can be evaluated when the inherent strain method isused in combination. Measuring a steep grade in a minute domainsuch as a thin film is difficult with the strain-gauge method.

Most Suitable Method. In conclusion, the most suitablemethod of evaluating residual stress in welded structures should be

selected depending on the purpose of measurement by considering theevaluating location and the operating conditions of the welded objectto be evaluated. Using several kinds of evaluations together is alsoeffective to increase the reliability of residual stress evaluation.

SummaryThrough-thickness residual stress in a pipe butt-welded joint and

a pipe socket-welded joint was compared. The analytical evalua-tion methods were inherent strain analysis and thermal elastic-plastic analysis and the experimental methods were neutron dif-fraction for the internal stress. And X-ray diffraction and strain-gauge measurement for the surface residual stress were also used.The residual stress distributions determined using these methodsagreed well with each other, both for internal stress and surfacestress. The residual stress distributions obtained by each evaluationmethod were shown to agree mutually. The characteristics of theevaluation methods were summarized and it was found that themost suitable method under any particular situation can be selectedby considering the evaluated location and the operating conditionsof the welded object to be evaluated.

ReferencesAllen, A. J., Hutchings, M. T., Windsor, C. G., and Andreani, C., 1985, “Neutron

Diffraction Methods for the Study of Residual Stress Fields,”Advances in Physics,Vol. 34, pp. 445–473.

Almen, J. O., and Black, P. H., 1963,Residual Stresses and Fatigue in Metals,McGraw-Hill, New York.

Hayashi, M., Ishiwata, M., Minakawa, N., Funahashi, S., and Root, J. H., 1995a,“Diffraction Plane Dependence of Elastic Constants in Ferritic Steel in NeutronDiffraction Stress Measurement,”Journal of the Society of Materials Science,Japan(in Japanese), Vol. 44, pp. 1115–1120.

Hayashi, M., Ishiwata, M., Minakawa, N., Funahashi, S., and Root, J. H., 1995b,“Residual Stress Measurement of Socket Welded Joint by Neutron Diffraction,”Journal of the Society of Materials Science,Japan (in Japanese), Vol. 44, pp.1464–1469.

Hayashi, M., and Ishiwata, M., 1995, “Residual Stress Measurement of ButtWelded Carbon Steel Pipe by Neutron Diffraction,”Proceedings of the 13th Inter-national Conference on NDE in the Nuclear and Pressure Vessel Industries,Kyoto,Japan, pp. 22–25.

Hutchings, M. T., 1990, “Neutron Diffraction Measurement of Residual StressFields, The Answer to the Engineers Prayer,”Journal of Nondestructive Testing andEvaluation,Vol. 5, pp. 395–403.

Klassen, R. J., Root, J. H., and Holden, T. M., 1991, “Neutron DiffractionMeasurements of Strain in a CANDU-3 Rolled Joint Hub,” AECL Report, ANDI-50.

Lu, J., Edited, 1996,Handbook of Measurement of Residual Stresses,Society forExperimental Mechanics, Inc., Fairmont Press, Inc., Lilburrn.

McEwen, S. R., Holden, T. M., Hobson, R. R., and Cracknell, G., 1983, “ResidualStrains in Rolled Joints,”Transactions of the 9th International Conference onStructural Mechanics in Reactor Technology,Chicago, Div. G, pp. 183–188.

Mochizuki, M., Enomoto, K., Okamoto, N., Saito, H., and Hayashi, E., 1993,“Welding Residual Stresses at the Intersection of a Small Diameter Pipe Penetratinga Thick Plate,”Nuclear Engineering and Design,Vol. 144, No. 3, pp. 439–447.

Mochizuki, M., Saito, N., Enomoto, K., Sakata, S., and Saito, H., 1995, “A Studyon Residual Stress of Butt-Welded Plate Joint Using Inherent Strain Analysis,”Transactions of the 13th International Conference on Structural Mechanics in Reac-tor Technology,Porto Alegre, Brazil, Vol. 2, Div. F, pp. 243–248.

Mochizuki, M., Hayashi, M., and Hattori, T., 1996, “Effect of Welding Sequenceon Residual Stress in Multi-Pass Butt-Welded Pipe Joints,”Transactions of the JapanSociety of Mechanical Engineers(in Japanese), Vol. 62A, No. 604, pp. 2719–2725.

Mochizuki, M., Hayashi, M., Nakagawa, M., Tada, N., and Shimizu, S., 1997, “ASimplified Analysis of Residual Stress at Welded Joints Between Plate and Penetrat-ing Pipe,”JSME International Journal(Series A), Vol. 40, No. 1, pp. 8–14.

Park, W., Kobayashi, H., Nakamura, H., and Koide, T., 1994, “Measurement andEvaluation of Welding Residual Stresses for Austenitic Stainless Steel by X-rayDiffraction Method,” Proceedings of the 1994 Annual Meeting of JSME/MMD(inJapanese), Vol. B, No. 940-37, pp. 70–71.

The Society of Materials Science, Japan, Edited, 1981,Standard for X-ray StressMeasurement,Yokendo Ltd. (in Japanese).

Todoroki, A., and Kobayashi, H., 1988, “Prediction of Fatigue Crack Growth Ratein Residual Stress Field (Application of Superposition Technique),”Transactions ofthe Japan Society of Mechanical Engineers(in Japanese), Vol. 54A, No. 497, pp.30–37.

Ueda, Y., and Fukuda, K., 1989, “New Measuring Method of Three-DimensionalResidual Stresses in Long Welded Joints Using Inherent Strain as Parameters—LzMethod,” ASME JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY, Vol. 111, pp.1–8.

Wikander, L., Karlsson, L., Nasstrom, M., and Webster, P., 1994, “Finite ElementSimulation and Measurement of Welding Residual Stresses,”Modeling and Simula-tion in Materials Science and Engineering,Vol. 2, pp. 845–864.


Tang Yaxin

Zhang Yongkang

Zhang Hong

Yu Chengye

Department of Mechanical Engineering,Nanjing University of Aeronautics &

Astronautics,Nanjing, 210016, P.R. China

Effect of Laser ShockProcessing (LSP) on theFatigue Resistance of anAluminum AlloyThe laser shock induced stress wave is described and measured with a polyvinylidenefluoride (PVDF) transducer. A principle for selecting laser parameters is proposed. Asmall sized laser with a high power is used for Laser Shock Processing (LSP). The fatiguelife of the aluminum alloy 2024T62 is greatly improved after LSP. With 95% confidence,the mean fatigue life of LSP specimens is 4.5–9.8 times that of unshocked ones. The fatigueand fracture resistance mechanisms of LSP such as the variation of the surface hardness,the microstructure and the fracture section of specimens before and after LSP areanalyzed.

1 Introduction/BackgroundLaser Shock Processing (LSP) uses an intense laser beam with

a high power density (GW/cm2) and a short duration (tens of ns)to irradiate the surface of metals. It forms a stress wave (shockwave) which is transmitted into the metal. If the peak pressure ofthe stress wave is larger than the Hugoniot Elastic Limit (HEL) ofthe metal, plastic deformation is produced and a strengthened layeris formed on the surface of the metal, and the mechanical proper-ties of the metal are improved. Compared with other laser treat-ment methods, the duration of laser shock processing is very short,so LSP can mitigate harmful heat effects in the metal. LSP haswide applications in many manufacturing industries, such as en-gines, tankers, ships as well as automobiles [1–4].

LSP can be very useful in fatigue and fracture resistance areasof aircraft structures, in which the structural material is expected tobe subjected to a large number of fatigue cycles. In some detailedstructures such as small holes, small slots, it is very difficult forconventional strengthening technology such as the shot peeningand the cold expansion to be used due to limited accessibility. ButLSP can easily reach these areas by using optical methods. LSPcan also provide aircraft structures with high-amplitude residualcompressive stresses and an excellent original fatigue qualities ofstructural details to lengthen the useful life of airplanes. Therefore,LSP technology is of great value for increasing fatigue and fractureresistance of aircraft structures in the development of modernairplanes [5].

2 Experimental Procedure

2.1 Calculation of the Laser Shock Induced Stress Wave.A schematic of the LSP process is shown in Fig. 1. When thesurface of a metal is irradiated with a high power laser beam, thesurface temperature of the metal rapidly rises to its vaporizationtemperature. For example, if the input power density is 1 GW/cm2,the vaporization temperature of the metal will be reached in lessthan 1 ns. The irradiated zone is vaporized violently, which pro-duces an expanding plasma and causes a stress wave in the metal.

According to the theory of explosive gas dynamics [6, 7], thefollowing can be used for estimating the shock pressureP:

P 5 S k 1 1

2k D 2k/k21

z@2~k2! 2 1!# 2/3

k 1 1z r 1/3 z ~AI0! 2/3 (1)

where

k — ratio of specific heatr — metal densityA — ratio of laser absorptionI 0 — laser power density

The ratio of laser absorptionA was measured as 0.8. Assumingthe metal steam is an ideal single-atom gas wherek ' 1.67,formula (1) can be reduced as follow:

r < 0.266r 1/3 z I 02/3 (2)

2.2 Measurement of the LSP Stress Wave Using the PVDFTransducer. The polyvinylidene fluoride (PVDF) film has beenwidely used to measure shock loads: plate impacts, explosions, etc.These films have a good reproducibility for measuring a widerange of pressures up to 20 GPa [8]. In order to measure lasershock pressures, a special PVDF transducer (shown in Fig. 2) hasbeen developed. The transducer consists of a 50mm thick PVDFfilm and two electrodes of 100mm thick aluminum foils. Aresistor,R is connected with the ends of both electrodes. When thePVDF film is shocked by a pulsed laser, it produces an electriccurrent through the circuit, which is represented by the voltageacross the resistor,R. The output voltage of this resistor is fed toan oscilloscope (25 MHz, dual channels). When the laser powerdensity is 5.73 108 W/cm2 (wavelength 1.06mm, pulse duration50 ns), the stress wave induced by laser shock in the PVDF film isshown in Fig. 3, and the measured peak pressure is 0.3 GPa. As the

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials Division May21, 1998; revised manuscript received March 10, 1999. Associate Technical Editor:K. S. Chan.

Fig. 1 Schematic of the LSP process


impedance mismatch between the Al foil (ZAl 5 1.33 107 Kg/m2 zs) and the PVDF (ZPVDF 5 0.35 3 107 Kg/m2 z s) is taken intoaccount, the peak pressure applied on the backside of Al foil canbe determined. In this case, the peak pressure is 0.8 GPa. Whileusing formula (2), the calculated value of peak pressure under thesame parameters is 1.05 GPa, which is close to the experimentalresult. Formula (2) can be accepted to estimate the peak pressureof LSP.

2.3 Fatigue Specimen. The fatigue specimen is of the dualdog-bone type geometry and made of the aluminum alloy2024T62. Two fatigue holes with diameter 2 mm are bored with aprecise jig boring machine, which are in both the LSP area and theunshocked area, respectively, as shown in Fig. 4. It is ensured thatthe qualities of the two holes are similar. The roughness is less than0.1 mm on the inside surface of the two holes.

2.4 Laser and Laser Parameters Selection. A Q-switchedNd: glass laser with wavelength of 1.06mm and pulse duration of30 ns was used. The laser has three amplifier stages, namely, thepre-set amplifier, the triple amplifier and the post-set amplifier.The modulation crystal is a deuterated potassium dihydrogen phos-phate (KD* P) crystal. A storage oscilloscope (7834Tek) with aresolution of 1 ns is used to monitor the wave shape of the laser atregular intervals. Eight percent of the laser beam energy was cut bya plate and transferred directly to an energy meter to achieveon-line control.

Eleven specimens were LSP shocked successively on both sides.The shocked surface was coated with a black paint and coveredwith a overlay of K9 optical glass. The selected laser parametersare shown in Table 1. Using formula (2), the evaluated value ofpeak pressure for the stress wave is about 1.2–2.2 GPa, which islarger than the Hugoniot Elastic Limit of 2024-T62 of 0.53 GPa. Itis obvious that the strengthened zone is formed to different extents.

2.5 Fatigue Test. All of the fatigue specimens were tested tofailure at a maximum load of 4.2 kN, and a stress ratio ofR 5 0.1under constant-amplitude load control with precision65%, usinga test frequency 13 Hz, i.e., low frequency fatigue failure.

In order to identify the mechanism of LSP, the surface hardness,the microstructure, and the residual stress of the LSP specimenwere measured and were compared with those of the unshockedmaterial. The fatigue fractures of shocked and unshocked speci-mens were also analyzed.

3 Results and Discussion

3.1 Results of Fatigue Test. The fatigue test results areshown in Table 2. Considering the specimen number 8 seemsunusual, its result is left out. So the fatigue life of 2024-T62increases by 2.2–8.7 times, with an average increase of 5.2 times.Based on the consideration of 95% confidence, the mean fatiguelife of LSP specimens is 4.5–9.8 times that of the unshocked ones.

3.2 Hardness. The microhardness distribution of the LSParea is measured with a Model 71 micro-hardness meter under aload of 50 g and a pressure period of 15 s. The result is shown inFig. 5. It is shown that the hardness of the unshocked area is 72–93Hv and the hardness of the LSP area is 98–137 Hv. There are largehardness variations across the LSP area. In general, the hardness ofthe center is higher than that of fringe of the LSP area. Theincrease of hardness means that it is difficult to produce the grainboundary slip and form fatigue slip-bands, i.e., the fatigue lifeincreases.

3.3 Microstructure. The microstructure of speciments be-fore and after LSP was examined by using a transmission electronmicroscope (TEM). The TEM photos are shown in Fig. 6. It isshown that the dislocation density of the aluminum alloy increasesdistinctly after LSP, which is thought to contribute to the improve-ment in the fatigue life.

3.4 Residual Stress. The residual stress on the surface of theLSP specimen was measured using X-ray stress meter with a spotof 2.5 3 3.5 mm, and the average residual stress of LSP area is230.1 MPa (acting as a compressive stress), while in the unshockarea the residual stress is approximately zero. The existence of a

Table 1 LSP parameters of 2024T62

Specimenno. Energy (J)

Pulseduration

(ns)

Spotdiameter

(mm)

Powerdensity

(GW/cm2)

Peakpressure

evaluation(GPa)

1 14.10, 15.10 30 7 1.22, 1.31 1.75, 1.842 18.07, 16.72 30 7 1.57, 1.45 2.08, 1.973 16.44, 19.03 30 7 1.42, 1.65 1.94, 2.154 16.85, 16.04 30 7 1.46, 1.39 1.98, 1.915 16.31, 15.63 30 7 1.41, 1.35 1.93, 1.886 16.44, 16.44 30 7 1.42, 1.42 1.94, 1.947 17.40, 16.99 30 7 1.51, 1.47 2.02, 1.998 18.35, 19.16 30 7 1.59, 1.66 2.09, 2.159 20.20, 8.10 30 7 1.75, 0.70 2.23, 1.21

10 14.0, 18.10 30 7 1.21, 1.57 1.74, 2.0811 19.71, 19.30 30 7 1.71, 1.67 2.20, 2.00

Fig. 3 Stress wave on the front face of PVDF film

Fig. 4 Bi-detail fatigue specimen of 2024T62 AL alloy. All dimensionsare in millimeters.

Fig. 2 Sketch of PVDF transducer


residual compressive stress state in the LSP area is very useful toresist the initiation and propagation of the fatigue crack.

3.5 SEM Analyses of Fatigue Fractures. The fatigue frac-tures were analyzed by a scanning electron microscope (SEM).There is no obvious fatigue origin on the fracture surface andsometimes there are multiple fatigue origins after LSP. In general,the fatigue crack is initiated in the center of the inside surface ofthe hole and sometimes in the subsurface of the LSP area (asshown in Fig. 7(a)). But there is a clear fatigue origin in thefracture of the unshocked specimen and the crack is initiatedgenerally in a square corner of the hole (as shown in Fig. 7(b)).The fatigue fracture of the LSP’d specimen is smoother than thatof the unshocked specimen, and does not exhibit fatigue striations.In contrast, the fatigue fracture of unshocked specimen is ratherrough and has clear fatigue striations.

LSP results in surface hardening, an increase in dislocation

density, and an increase in the compressive residual stress in thesurface layer. Among these factors, the compressive residual stressis the major factor; the surface hardening and dislocation densityincreases are relevant to the creation of this compressive residualstress.

4 Conclusions

(1) LSP with a high power density (GW/cm2) and a shortduration (tens of ns) can induce a stress wave in metals, which canbe estimated from the theory of explosive gas dynamics andmeasured with a PVDF transducer. When the peak pressure of thestress wave is larger than the Hugoniot Elastic Limit of thematerial, the metal can be fatigue strengthened.

(2) The fatigue life of 2024T62 aluminum alloy is improvedgreatly after LSP. With 95% confidence, the mean fatigue life ofLSP specimens is 4.5–9.8 times than that of unshocked specimens.

(3) LSP results in surface hardening, increased dislocationdensity and a compressive residual stress in the strengthened layer.The compressive residual stress is the major factor for improvingthe fatigue life of the aluminum alloy.

AcknowledgmentThe authors wish to thank the reviewers for their kindness in

reviewing and editing the paper.

References1 Vaccari, J. A., “Laser shocking extends fatigue life,”American Machinist,

1992, 7, pp. 62–64.2 Peyre, P., Merrien, P., and Lieurade, H. P., et al., “Renforcementd’alliages-

d’aluminum moul’esparondes dechocaser,”Mat’ eriaux & techniques,1993, Vol.6–7, pp. 7–12.

Fig. 5 Hardness distribution in the LSP area of 2024T62

Fig. 6 The microstructure comparision of 2024T62 aluminum alloy be-fore and after LSP. ( a) Microscopic photo of 2024T62 before LSP 3 20 K;(b) Microscopic photo of 2024T62 after LSP 3 20 K.

Fig. 7 The fatigue fracture of 2024T62. ( a) Fracture of LSP specimen 3200; (b) fracture of unshocked specimen 3 200

Table 2 Results of 2024T62 fatigue test

Specimenno.

Max. load(kN)

Min. load(kN)

Stress radio(R)

Testfrequency

Cycles ofun-shockedspecimens

Cycles of LSPspecimens

Increase(%)

1 4.2 0.42 0.1 13 96 580 356 930 269.62 4.2 0.42 0.1 13 77 710 755 870 872.73 4.2 0.42 0.1 13 73 850 297 750 303.24 4.2 0.42 0.1 13 88 630 719 640 711.95 4.2 0.42 0.1 13 93 580 .523 080 .4596 4.2 0.42 0.1 13 73 700 567 610 670.27 4.2 0.42 0.1 13 98 000 414 270 322.78 4.2 0.42 0.1 13 95 340 2 355 900 2 3719 4.2 0.42 0.1 13 81 480 259 360 218.3

10 4.2 0.42 0.1 13 107 320 799 840 645.311 4.2 0.42 0.1 13 96 440 823 210 753.6Average 690.8


3 Grevey, D., Maiffredy, L., and Vannes, A. B., “Laser shock on a TRIP alloy:mechanical and metalurgical consequences,”Journal of Materials Science,1992, Vol.27, pp. 2110–2116.

4 Puig, T., Decamps, B., and Bourda, C., et al., “Deformation of ag/g9waspaloy after laser shock,”Material Science and Engineering,1992, Vol. A154, pp.183–191.

5 Zhang Hong, Tang Yaxin, and Yu Chengye, et al., “Effects of laser shockprocessing on the fatigue life of fastener holes,”Chinese Journal of Lasers,Vol. A23,No. 12, pp. 1112–1116.

6 Zeldovich, Y. B., and Raizer, Y. P.,Physics of shock wave and hightemperature hydrodynamics of explosive gas,New York, Academic Press, 1966, pp.154–167.

7 Henrych, J.,The dynamics of explosion and its use,Elsevier, NY, 1979, pp.63–72.

8 Romain, J. P., Bauer, F., and Zanouri, D., “Measurement of laserinduced shock pressures using PVDF gauges,”High Pressure Science and Tech-nology, 1993, S. C. Schmidf, ed., American Institute of Physics, 1994, pp.1915–1919.


Masahito Mochizuki1

Department of Manufacturing Science,Osaka University,

Suita, Osaka 565-0871,Japan


Toshio HattoriHitachi, Ltd.,


Kimiaki NakakadoHitachi Construction Machinery Co., Ltd.,


Residual Stress Reduction andFatigue Strength Improvementby Controlling Welding PassSequencesThe effects of residual stress on fatigue strength at a weld toe in a multi-pass fillet weldjoint were evaluated. The residual stresses in the weld joints were varied by controllingthe sequence of welding passes. The residual stress at the weld toe was 80 MPa in thespecimen whose last welding pass was on the main plate side, but it was 170 MPa in thespecimen whose last pass was on the attachment side. The fatigue strength was nearly thesame at high stress amplitude for both specimens, but the fatigue strength of the specimenwhose last weld pass on the main plate was higher than that of the other specimen at lowstress amplitude. This difference is due to the magnitude of the initial residual stress andthe relaxation of the residual stress under fatigue cycling. The effects of the residual stresswere shown in a modified Goodman diagram, in which residual stress is treated as a meanstress.

IntroductionIn designing a welded structure, it is important to know the

relation between fatigue strength of welds and factors which affectthe fatigue strength (e.g., Almen and Black, 1963). The factors thataffect the fatigue strength are residual stress, stress concentration,mechanical properties of the material, and macro/micro-structure.Residual stress is one of the most important factors, and it is wellknown that residual stress is more effective on high-cycle fatiguethan the other factors.

There are two methods for evaluating fatigue strength underresidual stress fields. They are fracture mechanics and fatiguestrength curve. In the method of fracture mechanics, which usesthe principle of elastic stress superposition, stress intensity factordue to initial residual stress is analyzed and the effects of residualstress are evaluated as the change of stress ratio (Buchalet andBamford, 1976; Todoroki and Kobayashi, 1988). There are twosubsidiary methods that uses fracture mechanics to evaluate fa-tigue strength: one uses the redistribution of residual stress causedby crack propagation (Glinka, 1979; Nelson, 1982) and the otheruses effective stress intensity factor range due to crack closure(Parker, 1982; Beghini et al., 1994). On the other hand, a methodusing a fatigue strength curve is generally used by treating residualstress as a mean stress (Maddox, 1991; Yamashita et al., 1996).The effects of residual stress on fatigue strength are evaluated byusing a fatigue limit diagram like a modified Goodman diagram.

Most of these evaluations were conducted by using a specimenwhich was made specifically for the evaluation. That is, residualstress distribution or crack propagation path are intentionally de-termined according to the evaluation conditions. It is very impor-tant for engineers and researchers to obtain a fundamental princi-ple regarding the effects of residual stress on fatigue strength fromthese evaluation results. On the other hand, the evaluations of therelation between residual stress and fatigue strength in an actualweld joint were difficult especially at the weld toe or the weld rootwhere fatigue failure easily occurs (e.g., Hudak et al., 1985;Yamashita et al., 1996). One of the reasons is considered that the

precision of residual stress analysis and measurement at the weldtoe or the weld root in an actual weld joint had not been enough toaccurately evaluate the effect on the fatigue strength (Mochizuki etal., 1995). Furthermore, it is difficult to vary residual stress by onlychanging the welding process without treatment after welding suchas stress-relief heat treatment or shot peening (Mochizuki et al.,1996).

However, it is possible to vary residual stress value by changingwelding pass sequences at a multi-pass fillet weld joint. In thiscase, the other factors which affect fatigue strength, such asconfiguration or mechanical properties, can be kept the same. Thismethod does not need any treatment after welding for varyingresidual stress. Therefore, fatigue strength can be evaluated for twodifferent residual stress revel. In this paper, the effects of theresidual stress on fatigue strength at a weld toe in a multi-pass filletweld joint were clarified experimentally and theoretically. Theapplicability of improving fatigue strength by optimizing weldingpass sequence was studied. A method using a fatigue strengthcurve was applied to these evaluations because this method is easyto apply to the strength design of welded structures.

Multi-Pass Fillet Weld JointA weld joint was fabricated for evaluating residual stress and

fatigue strength. Two attachments were fillet-welded on both sidesof a main plate as shown in Fig. 1. External loading during thefatigue test was added along the main plate. The base material wasa carbon steel (JIS SS400), and the welding material is nearly thesame. Chemical compositions of the materials, their mechanicalproperties, and the welding conditions are listed in Tables 1, 2, and3, respectively. Two types of weld joints were fabricated bychanging welding pass sequences, as shown in Fig. 2. One was aweld joint whose final welding pass in each quadrant was depos-ited at the attachment side, and the other was a joint whose finalpass was deposited on the main plate side. The weld joints weremade to have the same shape in the weld toe which is subjected tostress concentration.

Stress concentration factora was calculated using an elasticfinite-element analysis. The shape of the weld toe after weldingwas traced and the finite elements were divided. The stress con-centration factor of the weld joint whose final welding pass wasdeposited at the main plate side wasa 5 3.76 to 4.12, and thefactor of the joint whose final pass was at the attachment side was

1 Formerly, Hitachi, Ltd., Ibaraki, Japan.Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-

ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionSeptember 17, 1998; revised manuscript received June 28, 1999. Associate TechnicalEditor: K. S. Chan.


a 5 3.68 to 4.10. In the multi-pass fillet weld joint, the effect ofthe shape of the weld toe on fatigue strength can be negligible.Residual stress and fatigue strength of a similar weld joint heat-treated after welding (two hours at 625°C) were also measured.

Relation Between Welding Sequence and ResidualStress

Method of Residual Stress Analysis and Measurement.Re-sidual stresses in the two multi-pass fillet weld joints were mea-sured by using strain-gauges (Lu, 1996) and X-ray diffraction (TheSociety of Materials Science, Japan, 1981). First, residual stresseson the center of the weld toe and the base metal were non-destructively measured by X-ray diffraction. Irradiation range ofX-rays was 5-mm width for the welding longitudinal direction and1.5-mm length for the transverse direction (Park et al., 1994).Next, strain gauges were pasted on the specimen surface in thesame position as that for X-ray diffraction. Residual stress was

calculated from the released elastic strains measured when theweld joint was cut into small pieces from around the strain gauges.

Welding residual stresses were also calculated by thermalelastic-plastic analysis using the finite element method for the twowelding pass sequences in a multi-pass fillet weld joint. Thetemperature distribution at each time resulting from the heat inputof welding was computed by transient heat conduction analysisduring the welding procedure. Transient stress during a weldingstep was successively calculated from thermal loads obtained byheat conduction analysis. This calculation involved thermalelastic-plastic analysis with nonlinearity of materials. Residualstress can be determined as the stress when the material is cooleddown to room temperature after welding. The details of the ana-lytical method and material properties are described in the previ-ous papers by the authors (1993, 1995).

Results of Residual Stress Analysis and Measurement.Transverse residual stressessRy, which is perpendicular fromlongitudinal welding direction, are compared in Figs. 3 and 4 fortwo kinds of specimens, respectively. These stresses were obtainedby measurements and numerical analysis. The measurement is theaverage stress of the four measured values from each quadrant.Measurements and analytical distributions of residual stress agreewell for two multi-pass fillet weld joints. Therefore, the result fromthermal elastic-plastic analysis was used as a standard for residualstress used for evaluating fatigue strength.

Transverse residual stressessRy in the weld toe on the mainplate are 170 MPa, for the weld joint whose final welding pass wasdeposited on the attachment side, and 80 MPa for the joint whosefinal pass was on the main plate side. Transverse residual stresssRy

Fig. 1 Configurations of a multi-pass fillet weld joint (Units: mm)

Table 1 Chemical composition of the materials used

Table 2 Mechanical properties of the materials used

Table 3 Welding conditions of a multi-pass fillet weld joint

Fig. 2 Sequences of welding pass in multi-pass fillet weld joints

Fig. 3 Comparison of residual stress in a multi-pass fillet weld joint (lastpass on the attachment)


on the weld toe in a multi-pass fillet weld joint can be changed bycontrolling the laminating location and sequence of the weldingpass.

The measured value of transverse residual stress at the weld toewas25 MPa in a fillet weld joint, which was heat-treated for stressrelief after welding, and in this paper it was treated as zero MPa.

Production of Initial Residual Stress Caused by Laminatingof Welding Pass. The generation of residual stress on the weldtoe of the main plate side is shown in Figs. 5 and 6 for the twotypes of weld specimens, respectively. These histories of residualstress at the end of each welding pass were obtained from thermalelastic-plastic analysis. Numbers on the horizontal axis show thewelding pass sequence. Each transverse residual stresssRy on theweld toe of the main plate is considered to generate by thefollowing mechanism.

In the weld joint whose final welding pass was deposited on theattachment side,sRy slightly increases from the first to the thirdwelding passes. Then, highersRy is added by the fourth weldingpass because the deposit on the fourth welding pass shrinks itselfand the weld toe of the main plate is pulled by the fourth pass. Onthe other hand, in the weld joint whose final pass was on the mainplate side,sRy also increases from the first to the third welding

passes. Then, residual stress present up to the third welding pass iscompletely released because the weld toe is heated over the melt-ing temperature. Residual stress due to the fourth welding pass isregenerated after the stress relaxation. As a result,sRy on the weldtoe of the main plate becomes smaller than that of the attachment.Maximum residual stress of welding longitudinal directionsRx isgenerated during the last welding pass because longitudinal resid-ual stress is caused by the shrinkage itself. Therefore,sRx on theweld toe of the main plate is larger when the final pass wasdeposited on the attachment side than when the final pass wasdeposited on the main plate side.

Residual stresses in two kinds of multi-pass fillet weld jointsbecome different as a results of the different production of residualstress. And the sequence of welding passes affects residual stressin a multi-pass weld joint.

Effect of Residual Stress on Fatigue Strength

Method of Fatigue Test. The fatigue test was done by apply-ing a tensile loading to the main plate with a stress ratioR 5 0 fortwo multi-pass fillet weld joints. The maximum force of the testmachine was63 MN and the frequency was 8 to 9.5 Hz.

Results of Fatigue Test. The fatigue strength curves in twomulti-pass fillet weld joint are compared in Fig. 7. The vertical axisshows the nominal stress rangeDs y along the loading directionand the horizontal axis shows the number of cycles to failureNf.It was confirmed from the observation during the fatigue test thatthe initial surface crack nucleated at the center of the weld toe andit propagated as a semi-elliptical crack.

The fatigue strength of the two kinds of multi-pass fillet weldjoints are nearly the same in the high stress range. A clear differ-ence appears around the fatigue cycle of 13 105 cycles. Theseresults show that high cycle fatigue strength can be improved byvarying the sequence of welding passes.

The results of a fatigue test using the fillet weld joints after heattreatment for stress relief are also shown in Fig. 7. The fatiguestrength at 13 107 cycles for the weld joint in which residualstress can be negligible is higher than the joint in which residualstress exists. That is, the fatigue strength of the fillet weld jointwithout residual stress is higher than that of the joint with residualstress.

Relation Between Residual-Stress Relaxation Behavior andFatigue Strength. The effect of the residual stress relaxation onthe fatigue strength in a multi-pass fillet weld joint is discussed.The change of residual stress in the weld joint was measured byX-ray diffraction for every fatigue cycles of 10n (n 5 0, 1,

Fig. 4 Comparison of residual stress in a multi-pass fillet weld joint (lastpass on the main plate)

Fig. 5 History of residual stress at the weld toe in a multi-pass fillet weldjoints by multipass welding (last pass on the attachment)

Fig. 6 History of residual stress at the weld toe in a multi-pass fillet weldjoint by multipass welding (last pass on the main plate)


2, . . . , 7). The relaxation behavior of transverse residual stresssRy at the weld toe of the main plate was evaluated.

Stress relaxation behaviors ofsRy at the weld toe of the mainplate are shown in Figs. 8 and 9 for the two weld specimens,respectively. These figures show that the weld joint whose lastwelding pass was deposited on the attachment side and the jointwhose last pass was on the main plate side exhibit similar tenden-cies for stress relaxation. Initial transverse residual stresssRy istensile, and gradually decreases when the stress range is large.Residual stress becomes zero near the 13 104 cycles. On the otherhand, residual stress hardly changes when the stress range is small.In addition, residual stress rapidly decreases after crack initiationand propagation at high fatigue cycles.

The presence of plastic deformation in the weld toe is discussednext. The maximum transverse stress at the weld toes y z max isassumed to be elastically equal to the sum ofsRy and a z Ds y,wheresRy is the transverse residual stress at the weld toe of themain plate,a is stress concentration factor, andDs y is nominalstress range. The yield stress of the material ands y z max are com-pared in the following. Plastic deformation at the weld toe isgreatly generated when the stress range is large and residual stressis gradually released. On the other hand, plastic deformation doesnot occur in the small stress range in which residual stress is hardlyrelaxed. It is thought that the relaxation behavior of residual stressis determined by the presence of plastic deformation in the weldtoe during fatigue cycling.

From these results of residual stress relaxation, it can be con-cluded that at the high stress range, the magnitude of residual stressdoes not affect the fatigue strength because residual stress isrelaxed at the low fatigue cycles. At the low stress range, fatiguestrength depends on the initial residual stress because residualstress hardly changes by fatigue loading. The results of the fatiguetest support these conclusions on residual stress relaxation duringfatigue cycling.

DiscussionFigure 10 shows a fatigue limit diagram, which relates mean

stress and stress amplitude, obtained from three types of multi-passfillet weld joints. Residual stress was treated as a part of meanstress and was added to the stress by external loading. The fatiguestrength at 13 107 cycles was used for the fatigue limit. A straightline in the figure represents the modified Goodman diagram. Theeffects of residual stress on fatigue strength in a fillet-weld jointare shown by the modified Goodman diagram when residual stressis treated as mean stress.

The fatigue limit diagram for a simple rectangular plate made ofthe same material is also shown in Fig. 10. The fatigue limit wasobtained from the fatigue strength at 13 107 cycles when themean stress is zero. From this figure, fatigue strength reductionfactor b 5 3.53 is calculated. Factorb is smaller than stressconcentration factora.

It is commonly considered that the effect of residual stress onfatigue strength can be evaluated by treating the residual stress asa mean stress. From the results of this paper, the commonly held

Fig. 7 Relation between nominal stress range and fatigue life in multi-pass fillet weld joints

Fig. 8 Residual stress relaxation by stress cycles at the weld toe in amulti-pass fillet weld joint (last pass on the attachment)

Fig. 9 Residual stress relaxation by stress cycles at the weld toe in amulti-pass fillet weld joint (last pass on the main plate)


view can be validated even when the object to be evaluated iscomplicated such as a multi-pass fillet weld joint.

The method for improving fatigue strength was developed byoptimizing the sequence of welding passes and by reducing resid-ual stress at the critical position. Rapid evaluation of weldingresidual stress will help expand the application of the proposedmethod.

SummaryThe effects of residual stress on fatigue strength at a weld toe in

a multi-pass fillet weld joint were evaluated. The residual stressesin the specimen were varied by controlling the sequence of weld-ing passes. Residual stress calculated by thermal elastic-plasticanalysis agreed well with the measurements obtained by straingauge and X-ray diffraction methods. The residual stress at theweld toe was 80 MPa in the specimen whose last welding pass wason the main plate, but it was 170 MPa in the specimen whose lastpass was on the attachment. The fatigue strength of the specimens

was nearly the same at high stress amplitude, but the fatiguestrength of the specimen whose last welding pass on the main platewas higher than that of the other specimen at low stress amplitude.This difference is due to the magnitude of the initial residual stressand the relaxation of the residual stress under the fatigue cycling.The effects of the residual stress were shown in the modifiedGoodman diagram, in which residual stress is treated as a meanstress. The method for improving fatigue strength was proposed byoptimizing the sequence of welding passes and reducing residualstress.

ReferencesAlmen, J. O., and Black, P. H., 1963,Residual Stresses and Fatigue in Metals,

McGraw-Hill, NY.Beghini, M., Bertini, L., and Vitale, E., 1994, “Fatigue Crack Growth in Residual

Stress Fields, Experimental Results and Modelling,”Fatigue and Fracture for Engi-neering Materials and Structure,Vol. 17, pp. 1433–1444.

Buchalet, C. B., and Bamford, W. H., 1976, “Stress Intensity Factor Solutions forContinuous Surface Flaws in Reactor Pressure Vessels,”Mechanics of Crack Growth,ASTM STP, Vol. 590, pp. 385–402.

Hudak, Jr. S. J., Burnside, O. H., and Chan, K. S., 1985, “Analysis of CorrosionFatigue Crack Growth in Welded Tubular Joints,” ASMEJournal of Energy Re-sources Technology,Vol. 107, pp. 212–219.

Glinka, G., 1979, “Effect of Residual Stress on Fatigue Crack Growth in SteelWeldments under Constant and Variable Amplitude Loads,”Fracture Mechanics,ASTM STP, Vol. 677, pp. 198–214.

Lu, J., Edited, 1996,Handbook of Measurement of Residual Stresses,The FairmontPress, Inc.

Maddox, S. J., 1991,Fatigue Strength of Weld Structures,Abington Publishing,Cambridge, pp. 152–154.

Mochizuki, M., Enomoto, K., Okamoto, N., Saito, H., and Hayashi, E., 1993,“Welding Residual Stresses at the Intersection of a Small Diameter Pipe Penetratinga Thick Plate,”Nuclear Engineering and Design,Vol. 144, No. 3, pp. 439–447.

Mochizuki, M., Saito, N., Enomoto, K., Sakata, S., and Saito, H., 1995, “A Studyon Residual Stress of Butt-Welded Plate Joint Using Inherent Strain Analysis,”Transactions of the 13th International Conference on Structural Mechanics in Reac-tor Technology,Porto Alegre, Brazil, Vol. 2, Div. F, pp. 243–248.

Mochizuki, M., Hayashi, M., and Hattori, T., 1996, “Effect of Welding Sequenceon Residual Stress in Multi-Pass Butt-Welded Pipe Joints,”Transactions of the JapanSociety of Mechanical Engineers(in Japanese), Vol. 62A, No. 604, pp. 2719–2725.

Nelson, D. V., 1982, “Effects of Residual Stress on Fatigue Crack Propagation,”Residual Stress Effects in Fatigue,ASTM STP, Vol. 776, pp. 172–194.

Park, W., Kobayashi, H., Nakamura, H., and Koide, T., 1994, “Measurement andEvaluation of Welding Residual Stresses for Austenitic Stainless Steel by X-rayDiffraction Method,” Proceedings of the Annual Meeting of JSME/MMD(in Japa-nese), Vol. B, No. 940-37, pp. 70–71.

Parker, A. P., 1982, “Stress Intensity Factors, Crack Profiles, and Fatigue CrackGrowth Rates in Residual Stress Fields,”Residual Stress Effects in Fatigue,ASTMSTP, Vol. 776, pp. 13–31.

The Society of Materials Science, Japan, Edited., 1981,Standard Method of X-rayStress Measurement(in Japanese), Yokendo Ltd.

Todoroki, A., and Kobayashi, H., 1988, “Prediction of Fatigue Crack Growth Ratein Residual Stress Field (Application of Superposition Technique),”Transactions ofthe Japan Society of Mechanical Engineers(in Japanese), Vol. 54A, No. 497, pp.30–37.

Yamashita, T., Hattori, T., Iida, K., Nomoto, T., and Sato, M., 1997, “Effects ofResidual Stress on Fatigue Strength of Small Diameter Welded Pipe Joint,” ASMEJournal of Pressure Vessel Technology,Vol. 119, pp. 428–434.

Fig. 10 Effect of residual stress on fatigue strength in multi-pass filletweld joints using a modified Goodman diagram


Jian Cao

Brad Kinsey

Department of Mechanical Engineering,Northwestern University,

Evanston, IL 60208

Sara A. SollaDepartment of Physics and Astronomy,

Northwestern University,Evanston, IL 60208;

Department of Physiology,Northwestern University Medical School,

Chicago, IL 60611

Consistent and MinimalSpringback Using a SteppedBinder Force Trajectory andNeural Network ControlOne of the greatest challenges of manufacturing sheet metal parts is to obtain consistentpart dimensions. Springback, the elastic material recovery when the tooling is removed,is the major cause of variations and inconsistencies in the final part geometry. Obtaininga consistent and desirable amount of springback is extremely difficult due to the nonlineareffects and interactions between process and material parameters. In this paper, theexceptional ability of a neural network along with a stepped binder force trajectory tocontrol springback angle and maximum principal strain in a simulated channel formingprocess is demonstrated. When faced with even large variations in material properties,sheet thickness, and friction condition, our control system produces a robust final partshape.

IntroductionObtaining consistent and accurate part dimensions is crucial

in today’s competitive manufacturing industry. Inconsistenciesin part dimensions slow new product launches, increasechangeover times, create difficulties in downstream processes,require extra quality assurance efforts, and decrease customersatisfaction and loyalty for the final product. In the sheet metalforming process, a major factor preventing accurate final partdimensions is springback in the material. Springback is thegeometric difference between the part in its fully loaded con-dition, i.e., conforming to the tooling geometry, and when thepart is in its unloaded, free state. For a complicated 3-D part,undesirable twist is another form of springback. The unevendistribution of stress through the sheet thickness direction andacross the stamping in the loaded condition relaxes duringunloading, thus producing springback. Factors that affect theamount of springback include variations in both process andmaterial parameters, such as friction condition, tooling geom-etry, material properties, sheet thickness, and die temperature.Because controlling all of these variables in the manufacturingprocess is nearly impossible, springback, in turn, cannot bereadily controlled. Adding to the difficulty is the fact thatspringback is a highly nonlinear effect; therefore, simulationsand correcting methods are complicated. There has been atremendous amount of research interest related to springback inrecent years as is evident in proceedings of Society of Auto-motive Engineers, NUMIFORM, and NUMISHEET confer-ences.

Springback can be reduced through modifications to the formingprocess. Several researchers have proposed to use a stepped binderforce trajectory to accomplish this objective (Ayres, 1984; Hishidaand Wagoner, 1993; Sunseri et al., 1996). A stepped binder forcetrajectory is an instantaneous jump from a low binder force (LBF)value to a high binder force (HBF) at a specified percentage of thetotal punch displacement (PD%) (see Fig. 1). Sunseri et al. (1996)investigated springback in the Aluminum channel forming processshown in Fig. 2. Their work was conducted through experimentsand simulations at specific values for process and material param-

eters. In a production environment, however, the amount of spring-back will deviate from the desired level due to variations in theprocess. Therefore, a control system that accommodates variationsin process parameters is required.

In recent years, many research groups have investigated the useof artificial neural networks to control sheet metal forming pro-cesses. Metal forming is an ideal candidate for neural networkcontrol due to the nonlinear effects and interactions of the processparameters. Cho et al. (1997) used a neural network to predict theforce in cold rolling, and Di and Thomson (1997) predicted thewrinkling limit in square metal sheets under diagonal tension.Among others (Elkins, 1994 and Yang et al., 1992), Forcellese etal. (1997) used a neural network to control springback in a 60 degaluminum V-punch air bending process. Their system was trainedusing experimentally obtained examples consisting of five param-eters from the punch force trajectory, an off-line measurement ofsheet thickness, and the target bend angle as the inputs into theneural network and the punch displacement as the output. Inanother research project, Ruffini and Cao (1998) proposed to usea neural network to control springback angle in a channel formingprocess with punch force trajectory as the sole source for identi-fying the process variations and adjusting the HBF used in Sunseriet al. (1996). Preliminary results showed this approach to bepromising.

In this paper, the springback of an aluminum channel is con-trolled via a stepped binder force trajectory and neural networkcontrol. The neural network determines the HBF and PD% of thestepped binder force trajectory. Punch force trajectory was iden-tified as the key parameter that reflects variations in materialproperties, sheet thickness, and friction coefficient. Therefore, fourpolynomial coefficients from curve fitting the punch force trajec-tory were used as inputs into the neural network. Figure 3 showsa flowchart of our proposed control system for this application.Despite large variations in material properties, sheet thickness (t),and friction coefficient (m), the springback angle (u) was main-tained between 0.2 and 0.6 deg and the maximum strain (e) waslimited to between 8% and 10%. Finally, a comparison with theclosed-loop control method proposed by Sunseri et al. (1996) isincluded to show the benefits of our control method. While onlynumerical simulation results are presented here, the control systemwill be physically implementation in the future to verify improve-ment claims.

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials Division June20, 1998; revised manuscript received June 14, 1999. Associate Technical Editor:N. Chandra.


Channel Forming ProcessA simple geometry to investigate springback is a channel form-

ing process. Therefore, the same aluminum channel forming pro-cess used by Sunseri et al. (1996) is employed here (see Fig. 2).First, the effect of constant binder force (CBF) on springback wasevaluated. As the CBF was increased while all other process andmaterial parameters were held constant in our simulations, thespringback angle,u, was reduced as shown graphically in Fig. 4and physically in Fig. 5. However, the increased binder forcecaused a subsequent increase in the maximum strain1 in the ma-terial, solid line in Fig. 4, to levels that exceed the maximumstretchability of aluminum (Graf and Hosford, 1993). By utilizinga stepped binder force trajectory, moderate maximum strain levelsin the sidewall were obtained while reducing springback in theprocess as demonstrated in the table of Fig. 1.

To produce a robust process when faced with deviations in thefriction coefficient, Sunseri et al. (1996) implemented closed-loopvariable binder force control to follow the punch force trajectoryobtained from the stepped binder force case with nominal processconditions. This control method was able to produce consistentspringback levels when the friction coefficient was varied from avalue of 0.1, for the nominal case, to 0.25. However, whether thismethodology could withstand variations of other parameters suchas material properties and sheet thickness was not determined.

Finite Element Model. A commercial Finite Element Analy-sis package (ABAQUS, 1997) was used for our numerical simu-lations of the channel forming process. Since the problem is closeto a plane-strain condition and is symmetric, only one sixteenth ofthe width and half of the length of the entire blank (220 mm3 46mm) was modelled. The binder, the die, and the punch weremodelled as three separate rigid surfaces. Each surface was mod-elled using four-node interface elements (ABAQUS type IRS4),and a Coulomb friction law was assumed. Our blank mesh had anuneven distribution of 40 four-node, reduced integration shellelements (ABAQUS type S4R) with a more dense concentration ofelements where the blank contacted the punch and binder cornerradii. Boundary conditions were specified to create a plane-straincondition. The material was modelled to be isotropic, elasto-plastic

following the von Mises yield criterion and isotropic strain hard-ening. The elastic properties were the Young’s modulus,E, of 70GPa and Poisson’s ratio,n, of 0.3. The plastic behavior of the sheetmaterial was modelled using a power law relation (s 5 Ke n). Ournominal material, denoted Material 1, had a material strengthcoefficient,K, of 528 MPa and a strain hardening exponent,n, of0.265.

Proposed Control SystemIn a channel forming process, springback in general is extremely

sensitive to variations in material and forming parameters. In thiswork, we develop a methodology for controlling springback whileproducing an acceptable amount of maximum strain in the materialthrough a combination of a stepped binder force trajectory andneural network control. In a stepped binder force trajectory, twocritical values need to be determined, the magnitude of the HBFand the PD% of the total punch displacement. These two param-eters are the outputs from our neural network. From our previousresearch experience in sheet metal forming process control(Ruffini and Cao, 1998; Kinsey and Cao, 1997; and Sunseri et al.,1996), the punch force trajectory was selected as the parameter thatprovides information about the current process. Therefore, poly-nomial coefficients from curve fitting the punch force trajectoryare used as inputs into the neural network.

A flowchart of our proposed control system is shown in Fig. 3.The forming process would proceed normally at a constant initialbinder force, 16 kN, to a depth of 10 mm. While the punchdisplacement continues, the polynomial coefficients from curvefitting of the punch force trajectory are calculated and fed into aneural network. The outputs from the neural network, the HBF andPD% for the stepped binder force trajectory, would be obtained intime to make the appropriate HBF adjustment at the specifiedpunch location, 19 mm multiplied by the PD%.

The value of 10 mm, where the coefficients from the punchforce trajectory are calculated, was chosen for two reasons. At thisdistance, the punch force trajectory is well defined and enough datapoints are available for accurate curve fitting to occur. Figure 6shows the effect variations int and m have on the punch forcetrajectory from 0 to 10 mm. Secondly, the distance of 10 mmallows adequate CPU time for calculations to be conducted. As-suming a punch speed of 50 mm/sec and setting the minimumPD% as 57.5% of the total punch displacement, 10.925 mm,approximately 18.5 ms are available to compute inputs into theneural network and predict the HBF and PD% of the steppedbinder force trajectory. Curve fitting and neural network programson our Pentium II, 233 MHz computer required approximately 10

1 In this work, maximum strain refers to the maximum principal logarithmic strainin the channel.

Nomenc la tu re

CBF 5 constant binder forceLBF 5 low binder forceHBF 5 high binder forcePD% 5 punch displacement percentage

t 5 sheet thicknessm 5 friction coefficientu 5 springback angle

e 5 maximum principal strainu 5 displacement of node in finite ele-

ment modelf 5 angular rotation of node in finite

element modelE 5 Young’s modulus

n 5 Poisson’s ratios 5 true stressK 5 material strength coefficientn 5 strain hardening exponentki 5 integral gainkp 5 proportional gain

Fig. 1 Stepped binder force trajectory

Fig. 2 Channel forming geometry


ms to compute values. How close to a true step function isproduced will depend on the speed of the binder force varyingmechanism.

The punch force trajectory was broken up into three regions,Region A, a transition region, and Region B (see Fig. 6). Thegroups of lines in Region A correspond to varying values of sheetthickness, and a second-degree polynomial was used to fit the datain this region. Despite the fact that it was a second degree poly-nomial, only two input coefficients were used to characterize thisregion since the punch force trajectory nearly passed through theorigin. In Region B, the dissimilar slopes are mainly caused bychanges in the friction coefficient. A linear interpolation providedtwo additional inputs for the network. Thus, a total of four poly-nomial coefficients were used as inputs into our neural network.

Note that the punch force trajectories are obtained from numer-ical simulations; therefore, they are smooth and have consistentpatterns due to variations int andm. In an actual forming process,noise in the data acquisition equipment would produce disparitiesin these curves. By using polynomial coefficients from curvefitting the punch force trajectory, the inputs into the neural networkare in essence filtered to account for these practical considerations.

To implement this control system in an actual forming opera-tion, training data would need to be produced. This informationcould be derived from actual production of stampings during dietry out by varying process parameters. For instance, to account forbatch to batch variation in the material, material could be obtainedand formed which represents the full range of potential sheetthicknesses and material properties. Also, the lubrication statecould be varied. Using various combinations of these processparameters, the HBF and PD% values, which produce the desiredamount of springback, could be determined. In addition, numericalsimulations, calibrated to experimental results to assure accuracy,could be used to expeditiously increase the amount of trainingdata. Once the neural network is well trained with the entire rangeof potential process parameter values, actual values of the materialproperties, sheet thickness, and friction state are not required sincethe inputs into the neural network are polynomial coefficients frompunch force trajectory curve fitting.

Any increasing binder force trajectory could be implemented toreduce springback. However, the simplicity of the stepped binderforce trajectory requires only two input parameters, HBF andPD%. In addition, existing presses in industry have the ability toproduce a stepped binder force trajectory. Therefore, a steppedbinder force trajectory is an ideal choice.

Neural Networks. Artificial neural networks have been stud-ied for many years in the hope of mimicking the human brain’sability to solve problems that are ambiguous and require a largeamount of processing. Human brains accomplish this data process-ing by utilizing massive parallelism, with millions of neuronsworking together to solve complicated problems. Similarly, artifi-cial neural network models consist of many computational ele-ments, called “neurons” to correspond to their biological counter-parts, operating in parallel and connected by links with variableweights. These weights are adapted during the training process,most commonly through the backpropagation algorithm (Rumel-hart and McClelland, 1986), by presenting the neural network withexamples of input-output pairs exhibiting the relationship thenetwork is attempting to learn. The goal is for the neural networkto generalize, or extract, the pattern given in the input-outputexamples. Further details on neural networks, in general, can befound in Widrow and Lehr (1990). For our particular application,the structure of the neural network was determined to be four inputparameters, five hidden neurons, and two outputs. A sigmoidalactivation function was used for the hidden neurons while theoutputs utilized a linear one. A more detailed discussion on howthis optimal structure was determined is given in Kinsey (1998).

To initially determine the feasibility of using a neural network tocontrol and minimize springback, the ability of the neural networkto handle large variations in thet and m was investigated. Sheetthickness values from 0.8 to 1.4 mm (in increments of 0.1 mmbetween 0.8 and 1.2 mm) and friction coefficient levels from 0.04to 0.20 (in increments of 0.01 between 0.04 and 0.12 and after-wards in increments of 0.02) were considered. These variations int andm values are obviously larger than what would be seen in anactual mass production forming process but are used here for

Fig. 4 Springback angle and maximum strain versus constant binderforce

Fig. 5 Final part shape with increasing constant binder force

Fig. 6 Punch force trajectory for 0 to 10 mm of punch travel

Fig. 3 Flowchart of proposed neural network control system


demonstration purposes to show the capability of neural networksto control even considerable changes in variables. Training datawas generated through trial and error simulations for 104 combi-nations of these two process parameters to determine the HBF andPD% values which produced a springback angle,u, in the range of0.4 to 0.5 deg and a maximum strain,e, in the range of 8% to 10%.This extremely narrow springback range was intentionally chosenwith the realization that the predicted values for the HBF and PD%from the trained network may not result in the springback amountin the same tight range. A given HBF and PD% combination wasnot necessarily the only one that would have provided the valuesfor u ande in the specified ranges. However, by choosing theu ande ranges intentionally small, we guaranteed that the network wouldreceive training data that was within a narrow window of possibleHBF and PD% values. These 104 simulation runs provided theinput-output pairs for the training data with four curve fittingpolynomial coefficients from the punch force trajectories servingas the inputs and the HBF and PD% representing the desiredoutputs.

ResultsOnce the neural network was well trained, fourt andm combi-

nations left out of the training set as well as four additionalt andm combinations not included in the previous range oft and mvalues were “fedforward” in the network to predict HBF and PD%values for the stepped binder force trajectory following the proce-dure outlined in Fig. 3 and the previous section. The resultingspringback angle,u, and maximum strain,e, from the process werethen calculated. Table 1 shows the excellent results that wereobtained for these eight cases. All of the springback angles andmaximum strain values are within a range of 0.3 to 0.6 deg and 8%to 10%, respectively.

The ability of the neural network to provide HBF and PD%values to use when faced with large variations int andm demon-strates the potential of neural networks in this application. How-ever, there are other parameters that vary during the process, whichhave a similar effect on final part shape. Material properties, forinstance, have been shown to cause severe dimensional variationsin sheet metal stampings (Kinsey and Cao, 1997). When theoriginal trained network for was used to predict HBF and PD%values with different material properties, unacceptable levels ofspringback and maximum strain were produced. This is not sur-prising since the network was not trained to accommodate discrep-ancies in material properties. Therefore, additional training dataincorporating material variations was required. Deviations on thetrue stress-strain curve for the nominal material, Material 1, werecreated by varying the strength coefficient,K, by 610% (Materials2 and 3) and620% (Materials 6 and 7) and the strain hardeningcomponent,n, by approximately616% (Materials 4 and 5).

Seven combinations oft and m (t 5 0.9/m 5 0.04, t 51.1/m 5 0.06, t 5 1.4/m 5 0.08, t 5 1.0/m 5 0.10, t 50.8/m 5 0.12, t 5 1.0/m 5 0.16, andt 5 1.2/m 5 0.20) werechosen from which training examples with the new materials werecreated. Again through trial and error, 36 training examples with agiven material,t, andm combination were produced that had thesame ranges foru ande as were previously used, 0.4 to 0.5 deg and8% to 10% respectively. One of the six new materials at each of

thet andm combinations, was left out of the training set to be usedas a check to see if the neural network was able to predict accuratevalues of HBF and PD%. After these additional training exampleswith variations in material properties were added to the originaltraining set, the network was retrained with the same neuralnetwork structure used previously.

Table 2 shows the results when the cases from the six combi-nations that were left out of the training set and four new materialthickness,t, and friction coefficient,m, combinations were fedfor-ward in the newly trained network. Once again, the neural networkwas able to provide values of HBF and PD% that producedacceptable levels ofu, 0.2 to 0.6 deg, ande, 8% to 10%. However,these simulation outputs were not always within the narrow spring-back angle range of 0.3 to 0.6 deg as was obtained in thet andmresults. This indicates that the neural network has more difficultygeneralizing over variations in material. As previously stated, thenarrow range onu during training was intentionally created toallow for reasonable discrepancies in the feedforward process.Furthermore, additional training sets or a more clever neural net-work structure, such as giving “hints” to the network about thematerial properties, could improve the ability of the neural networkto handle variations in material properties.

Comparison of Results to Closed-Loop ControlThe control system proposed by Sunseri et al. (1996) utilized

closed-loop control of the binder force to follow the punch forcetrajectory in order to control springback in the same channelforming process. Regarding the friction coefficient as possibly themost significant process parameter, the robustness of their controlsystem was tested against variations in the friction coefficient, andexcellent results were obtained. However, variations in the mate-rial properties and sheet thickness were not investigated in theirwork. Therefore, further closed-loop control simulations withthese variations were conducted here in order to form a compari-son with the neural network control system.

First, the punch force trajectory from our nominal case, Material1, t of 1.0 mm, andm of 0.10, was created. A Proportional plusIntegral (PI) controller was used to adjust the binder force to allowfor following of the nominal punch force trajectory, as was done inSunseri et al. (1996). The maximum punch displacement betweenadjustments of the binder force was 0.02 mm. Listed in Table 3 arethe proportional gains,kp, and the integral gains,ki , used by thecontroller for the three cases investigated. Figure 7 shows howwell the punch force trajectory for these three cases (ki 5 0.8 forMaterial 4) tracked the nominal punch force trajectory. Figure 8shows how the binder force varied over the punch displacement to

Table 1 Results of variation in thickness and friction coefficient Table 2 Results of variation in material properties, thickness, and fric-tion coefficient

Table 3 Comparison of results from neural network and closed-loopcontrol systems


allow for following the nominal punch force trajectory (ki 5 0.8for Material 4). Note that the binder force trajectories level off atthe end of the forming process similar to a stepped binder forcetrajectory.

The u ande values from these closed-loop control experimentsare also shown in Table 3 along with the results from the neuralnetwork control system. This table clearly indicates that the spring-back angle for the neural network control system was considerablyless and closer to the original range of 0.4 to 0.5 deg for all threecases. Even for the case where the nominalt andm values, 1.0 mmand 0.10, respectively, were used with Material 5, the neuralnetwork control system outperformed the closed-loop control sys-tem.

Though a neural network system requires more up front workproducing a sufficient number of examples to train the network, thebenefits for controlling springback and maximum strain are unde-niable. Also, there are additional benefits to the neural networkcontrol system over this closed-loop control strategy for this ap-plication. Note that the value ofki had to be adjusted for theMaterial 4 case in Table 3, compared to the cases for Materials 3and 5, in order for the punch force trajectory to be followedaccurately. That is, as the material and process parameters change,the gains necessary to follow a punch force trajectory closely mayalso change. Figure 7 also shows what would have been the actualpunch force trajectory for the Material 4 case if theki from thecases for Materials 3 and 5, 2.0, was used. Therefore, this closed-loop control system with pre-selected fixed gains is not robustwhen faced with large variations in process and material parame-ters. Furthermore, presses currently found in industry are capableof readily producing a stepped binder force trajectory, as is used inthe neural network control system, while following a continuouslyvarying punch force trajectory would require a more vigorouscontrol system.

ConclusionsIn this paper, a neural network system, along with a stepped

binder force trajectory, was proposed to control springback andmaximum strain in a simulated Aluminum channel forming pro-cess. A neural network was chosen due to its ability to handle thehighly non-linear coupled effects that are found in metal-formingprocesses when variations in the material and process parametersoccur. The punch force trajectory was identified as the processparameter that provides the greatest insight into deviations ofvarious material and process variables. Therefore, polynomial co-efficients from curve fitting of the punch force trajectory were usedas inputs into the neural network. The results show that the neuralnetwork was successful at providing the high binder force (HBF)

and punch location (PD%) values for the stepped binder forcetrajectory which produced acceptable values of springback (u), 0.2to 0.6 deg, and maximum strain (e), 8% to 10%, in the finalproduct when faced with variations in the material strength (K) of620%, the strain hardening exponent (n) of 616%, the sheetthickness (t) of 625%, and the friction coefficient (m) of 665%.Also when compared with the closed-loop control strategy pro-posed by Sunseri et al. (1996) for the same process, the neuralnetwork system was shown to be superior at minimizing spring-back in the presence of variations in material properties, as well asbeing a more robust system to implement if adequate training datais available.

While this work was conducted using simulations, the method-ology developed could be easily extended to an actual formingprocess or experiments, which will be conduct in the future tovalidate our claims. The only hardware requirements would be aCPU, the ability to measure punch force trajectory, and the abilityto vary the binder force once during the process cycle. With thisneural network control system in place, a sheet metal formingprocess would be robust to inevitable variations in material andprocess parameters; thus, creating consistent final part dimensionsthat are critical to downstream processes and customer satisfaction.

AcknowledgmentThis research was funded in part by NSF grants #CMS-9622271

and #DMI-9703249.

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Hornik, K., Stinchcombe, M., and White, H., 1989, “Multilayer FeedforwardNetworks are Universal Approximators,”Neural Networks,Vol. 2, pp. 359–66.

Fig. 7 The target and actual punch force trajectories in closed-loopcontrol

Fig. 8 Resulting binder force trajectories for closed-loop control follow-ing punch force trajectory


Kinsey, B., 1998, “Process Control in Sheet Metal Forming,” Master’s thesis,Department of Mechanical Engineering, Northwestern University.

Kinsey, B., and Cao, J., 1997, “An Experimental Study to Determine the Feasibility ofImplementing Process Control to Reduce Part Variation in a Stamping Plant,”Sheet MetalStamping: Development Applications,SAE Paper 970713, SP-1221, pp. 107–12.

Ruffini, R., and Cao, J., 1998, “Using Neural Network for Springback Minimiza-tion in a Channel Forming Process,”Developments in Sheet Metal Stamping,SAEPaper 98M-154, SP-1322, pp. 77–85.

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Sunseri, M., Cao, J., Karafillis, A. P., and Boyce, M. C., 1996, “Accommodationof Springback Error in Channel Forming Using Active Binder Force Control: Nu-merical Simulations and Results,” ASME JOURNAL OF ENGINEERING MATERIALS AND

TECHNOLOGY, Vol. 118, No. 3, pp. 426–35.Widrow, B. and Lher, M. A., 1990, “30 Years of Adaptive Neural Networks:

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K. T. KimProfessor.

Assoc. Mem. ASMEe-mail: [email protected]

J. H. Cho

J. S. Kim

Department of Mechanical Engineering,Pohang University of Science and Technology,

Pohang 790-784, Korea

Cold Compaction of CompositePowdersDensification behavior of composite powders was investigated under cold compaction.Experimental data were obtained for mixed copper and tungsten powders with variousvolume fractions of tungsten powder under cold isostatic pressing and die compaction. Amodel was also proposed for densification of mixed—soft and hard—metal powders undercold compaction. Theoretical predictions from the proposed model and models in theliterature were compared with experimental data. The agreements between experimentaldata and theoretical predictions from the proposed model are very good for compositepowders at initial stage under cold isostatic pressing. Theoretical predictions, however,underestimate experimental data under cold die compaction.

IntroductionP/M parts are typically produced in a sequence of cold compac-

tion, sintering, and finishing. Nonuniform density distribution ofpowder compacts under cold compaction has a great influence onthe subsequent process such as sintering, thus it is important topredict densification and density distribution of metal powderunder cold compaction. So far, the research on powder compactionhas been mainly focused on densification of pure powders (Gur-son, 1977; Shima and Oyane, 1976; Arzt, 1982; Helle et al., 1985;Fleck et al., 1992; Fleck, 1995; Larsson et al., 1996; Fleck et al.,1996). Recently, it has been extended to mixed powders (Lange etal., 1991; Besson and Evans, 1992; Gurson and McCabe, 1992;Turner and Ashby, 1996; Jefferson et al., 1994; Besson, 1995),since P/M has been considered as a major route for processingcomposite materials.

Lange et al. (1991) investigated densification behavior of mixedaluminum and steel powders under cold compaction. Besson andEvans (1992) studied densification of copper powder reinforcedwith ceramic powder such as SiC and alumina. Gurson and Mc-Cabe (1992) examined the yield function for mixed metal powdersby using data from triaxial compression test. By considering var-ious amounts, sizes and shapes of rigid inclusions, Turner andAshby (1996) investigated densification behavior of mixed pow-ders under cold isostatic pressing. Jefferson et al. (1994) andBesson (1995) theoretically studied densification behavior of pow-ders surrounding an isolated rigid inclusion.

Recently, Bouvard (1993), Zavaliangos and Wen (1997) andStoråkers et al. (1997) proposed theoretical models for densifica-tion of mixed powders. Assuming that the matrix powder withpower-law creep response has a spherical shape with the same sizeas the rigid inclusion, Bouvard (1993) proposed a model fordensification of mixed powders under hydrostatic compression. Byemploying the effect of the mixing quality on densification, Zaval-iangos and Wen (1997) proposed a model for densification ofmixed powders with perfectly plastic behaviors under cold iso-static pressing. Storåkers et al. (1997) obtained densification mod-els for mixed spherical powders in different sizes with the samehardening parameters and creep exponents under cold isostaticpressing and die compaction. However, theoretical predictionsfrom these models did not agree well with experimental data forcomposite powders under cold compaction.

The present paper reports on densification behavior of mixed—soft and hard—powders during cold compaction. Experimentaldata were obtained for mixed—copper and tungsten—powders

under cold isostatic pressing and die compaction. Based on themodels in the literature, densification equations were proposed forcomposite powders under cold isostatic pressing and die compac-tion. In the model, soft and hard powders have a spherical shapewith the same size and different hardening parameters. Theoreticalpredictions from the proposed model and models in the literaturewere compared with experimental data for mixed—copper andtungsten—powders under cold compaction.

Experiments

Materials and Specimens. Gas atomized copper powder (Py-ron, USA) with particle sizes between 150–200mm and hydrogenreduced tungsten powder (Korea tungsten, Korea) with particlesizes between 150–200mm were used. Figure 1 shows that theshape of copper powder is spherical and that of tungsten powder ispolygonal.

The plastic flow stress for the matrix material of copper powderwas obtained from solid copper produced in this work underuniaxial compression test. That of tungsten powder was obtainedfrom experimental data by Yiu and Wang (1979).

Solid copper was produced by hot isostatic pressing as follows:A hot isostatic press (HIP System 30T, Kobe steel, Japan) wasused to produce dense copper specimens. A copper tube was usedas a container of copper powder during hot isostatic pressing. Thecontainer is 23 mm in inner diameter and 1 mm in thickness. Thepowder in the container was degassed for 5 h at400°C and vacuumsealed. The sealed sample was hot isostatically pressed for 1 hunder 50 MPa at 800°C. After hot isostatic pressing, the containerwas removed by machining. A specimen was machined into acylindrical shape with 12 mm in height and 10 mm in diameter.Then, the specimen was annealed for 1 h at600°C in a hydrogenatmosphere. The final relative density of the specimen was nearlyr . 99%.

Uniaxial stress-strain response of the matrix material of copperpowder was obtained from uniaxial compression of solid copper atroom temperature by using an MTS servohydraulic testing ma-chine. To reduce the friction between the specimen and compres-sion platens, teflon sheets were inserted. The flow stress of thematrix material was obtained by measuring the height and diam-eter of the specimen by interrupting the test. Teflon sheets werechanged at each interruption.

Cold Isostatic Pressing. Copper powder was deoxidized for45 min at 300°C in a hydrogen atmosphere. Mixtures of copperand tungsten powders with various volume fractions of tungstenpowder were produced by using a gravity mixer for 1 h to obtainuniform mixing. Composite powders were poured in a rubbermold, then sealed with a latex envelope. The sealed sample wascompacted in a cold isostatic press (ABB Autoclave Engineers,

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials Division April13, 1998; revised manuscript received July 16, 1999. Associate Technical Editor:H. M. Zbib.


USA) under hydrostatic pressures of 150–350 MPa. The compos-ite powder compact is about 13 mm in height and 10 mm indiameter. Relative density of the compact was measured byArchimedes’ method with paraffin treatment.

Cold Die Compaction. Densification behaviors of compositepowders with various volume fractions of tungsten powder wereinvestigated under cold die compaction. Composite powders of10 g was poured in a closed die and compacted under pressure of150–650 MPa. To reduce the friction between the powder and diewall, zinc stearate was sprayed on the die wall. Relative density ofthe compact was measured by Archimedes’ method with paraffintreatment after the compact was ejected from the die.

AnalysisThis paper reports on densification behavior of composite pow-

ders under cold compaction. The densification equations were

obtained for mixture of soft and hard powders under cold isostaticpressing and cold die pressing. For simplicity, powders are as-sumed to have the same radiusR with power-law hardeningparametersns for soft powder andnH for hard powder. Also, bothpowders are assumed to have a spherical shape. Thus, strainhardening responses of the matrix material for soft and hardpowders can be written, respectively,

emS5 SsmS

soSD nS

, emH 5 SsmH

soHD nH

(1)

where sm, em and s o, respectively, are uniaxial stress, plasticstrain and a constant for the matrix material. The subscriptsS andH represent the soft and hard powder, respectively.

Cold Isostatic Pressing. The hydrostatic pressureP for den-sification of composite powders under cold isostatic pressing canbe written (Bouvard, 1993; Zavaliangos and Wen, 1997)

P 51

3V Ocontact

F ~c!d ~c! (2)

whereF (c) are the interparticle contact forces,d(c) are the center tocenter distances of the particles adjacent to each contact and thesuperscriptc denotes the contact type.

Assuming that the powder compact with relative densityDconsists ofN particles, the volume of the powder compact can bewritten

V 5 S4

3pR3D N

D. (3)

For composite powders with the volume fraction of soft powderf S and that of hard powderfH, Eq. (2) becomes

P 5D

4pR3 F fSZSSFSS

dSS

21 fSZSHFSH

dSH

2

1 fH ZHSFHS

dHS

21 fH ZHH FHH

dHH

2 G (4)

whereZ denotes the number of contacts per particle.Zab (a, b 5S, H) are the average coordination numbers ofb-powder contact-ing with ana-powder.Fab anddab are the contact forces and thecenter to center distances ofb-powder contacting with ana-powder, respectively.

For a single type powder compact, Arzt (1982) suggested thatthe number of particle centersG within radiusr around a referenceparticle with radiusR can be expressed by

G~r ! 5 Zo 1 CS r

2R2 1D ~r $ 2R! (5)

where the initial coordination numberZo 5 7.3 and the constantC 5 15.5 forrandom dense packing. Densification is modelled bythe concentric growth of the spherical particles. When the spheresgrow to the radiusR9, some of them overlap. The number of theseoverlap can be written from Eq. (5). Thus, if we take the originalradiusR to be 1,Z(D) can be written

Z~D! 5 G@2R9~D!# 5 Zo 1 C~R9 2 1! ~r , R9! (6)

Applying Eq. (6) for densification of composite powders, thecoordination numbersZab for various types of contact may bewritten by a relationship betweenR9 and dab (Bouvard, 1993;Zavaliangos and Wen, 1997). Thus,

Zab 5 fbFZo 1 CS 2R

dab

2 1DG ~a, b 5 S, H!. (7)

When the same kind of spherical powders are in contact, therelationship between the contact force and the center to center

Fig. 1 Scanning electron micrographs of ( a) Cu powder and ( b) Wpowder


distance of the particles can be obtained as follows: when bothpowders have the same radius with work-hardening response,Matthews (1980) obtained the relationship between the contactforce F and the circle radiusa of the contact area. Thus,

F 56nso

2n 1 1 S 8a

9pRD1/n

~pa2! (8)

where s o and n denote the work-hardening coefficient and thereciprocal of the usual work-hardening index. Matthews (1980)also obtained a relationship between the circle radiusa of thecontact area and the center to center distanced when sphericalpowders with the same radius are in contact. Thus,

d 5 2R F1 2 S 2n

2n 1 1D2~n11! a2

R2G (9)

Hence, the relationship between the contact forceF and the centerto center distanced can be written from Eqs. (8) and (9). Thus,

F 56np 121/n

2n 1 1 S8

9D1/n S1 2

d

2RD~2n11!/2n

3 S 2n

2n 1 1D~12n!~2n11!/n

soR2 (10)

When spherical powders of different materials, for instance, softand hard particles are in contact, we assume that only the softparticle deforms. Considering the contact problem of two sphereswith different diameters, for instance,D 1 5 aD 2 with a positiveconstanta, the relationship between the contact forceF and thecircle radiusa of the contact area can be written from Storåkers(1997). Thus,

F

pa2soS6R

a D 1/n

5 3~1 1 a! 1/n (11)

where s o and n are material constants of the soft particle. Therelationship between the center to center distanced and the circleradiusa of the contact area can be obtained from the combinedboundary value problem and the result by Biwa and Storåkers(1995) for deformation of a specimen in hardness test. Thus,

d 5 2R 2~1 1 a!a2

2Rj(12)

wherej can be written as a function of hardening parametern. Forinstance,j 5 1.08 whenn 5 4.115 forcopper powder. Hence,whena 5 1, i.e., two spheres have the equal size, the relationshipbetween the contact forceF and the center to center distanced canbe written from Eqs. (11) and (12). Thus,

F 5 6p S Î2

3 D 1/n

j ~2n11!/2n soR2 S1 2d

2RD~2n11!/2n

(13)

Bouvard (1993) and Storåkers et al. (1997) assumed that thecontact forces are the same regardless of various types of contact.Thus,

FHH 5 FSH 5 FHS 5 FSS (14)

Zavaliangos and Wen (1997), however, obtained the differentcontact forces depending on the types of contact. Thus,

FSH 5 FHS < ~1 1 0.6fH !FSS,

FHH < SRfH~2 2 exp~1 2 SR!! 12fH FSS (15)

whereSR(5s oH/s oS) is the ratio of yield strengths of the soft andhard powders. By substituting Eqs. (10) and (13) into Eq. (14) or(15), dSH(5dHS) anddHH can be written in terms ofdSS as shown

in Eqs. (A7)–(A8) when the contact forces are the same and in Eqs.(A9)–(A10) when they are different.

Hence, ZSH(5ZHS), ZHH, FSH(5FHS) and FHH can also bewritten in terms ofdSS. Thus,

Zab 5 Zab ~dSS! ~a, b 5 S, H! (16)

Fab 5 Fab ~dSS! ~a, b 5 S, H!. (17)

By using the balance of energy, a relationship for the pressureand densification rate can be written (Bouvard, 1993; Zavaliangosand Wen, 1997)

PD

D5 2

1

V Ocontact

F ~c!d ~c!. (18)

By substituting Eq. (4) andfHZHS 5 f SZSH from Eq. (7) into Eq.(18), we can write

dD

D5 23

L1

M1ddSS (19)

where

L1 5 fSSZSSFSS1 2ZSHFSH

ddSH

ddSSD 1 fH ZHH FHH

ddHH

ddSS(20)

M1 5 fS~ZSSFSSdSS1 2ZSHFSHdSH! 1 fH ZHH FHH dHH (21)

By substituting Eqs. (16) and (17) into Eqs. (19) and (4), we canwrite

dD 5 23DL1~dSS!

M1~dSS!ddSS (22)

P 5D

4pR3 L2~dSS! (23)

where

L2~dSS! 5 fSZSSFSS

dSS

21 fSZSH~dSS!FSH~dSS!dSH~dSS!

1 fH ZHH ~dSS!FHH ~dSS!dHH ~dSS!

2. (24)

Hence, the variation of relative density with pressure for compositepowders under cold isostatic pressing can be obtained from Eqs.(22) and (23).

Cold Die Compaction. When powder aggregates of uniformsphere are subjected to macroscopic stressesS ik so that powderswith their centers at a distancer (5rn) from the center of areference powder are deformed by the stretch ratiol(5d/ 2R),Fleck et al. (1997) showed that macroscopic stressesS ik are relatedto the contact forceF given by

S ik 5 2RZo

VA EA

FIlni nkdA

21

2VA EA

lHE2R

2R/l SFrdZ

drdrDJni nkdA (25)

whereA is the surface area of reference powder andFI is the initialcontact force.V and n are the aggregate volume and the unitvector, respectively.

For composite powders, we substitute the number ofb-powderstouching ana-powder initially fa fbZo, and the number of thoseduring densificationfaZab into Zo andZ in Eq. (25), respectively.Thus, Eq. (25) can be rewritten


S ik 5 2RZo

VA EA

$ Oa,b5S,H

fa fb FIablab%ni nkdA

21

VA EAH O

a,b5S,H

falab E2R

2R/lab HFab rdZab

drdrJJ

3 ni nkdA ~a, b 5 S, H! (26)

whereFIab is the initial contact force betweena- andb-powders.Fleck et al. (1997) also showed that the relationship betweenFIab

andFab can be written

Fab 5 S1 2 lab r /2R

1 2 labD ~2na11!/2na

FIab ~a, b 5 S, H! (27)

whereFIab can be obtained from Eqs. (10) and (13). Thus,

FIaa 56na

2na 1 1 S 2na

2na 1 1D~12na !~2na11!/na

p 12~1/na !

3 S8

9D1/na

soa R2~1 2 laa ! ~2na11!/2na ~a 5 S, H! (28)

FISH 5 6pS Î2

3 D 1/nS

j ~2nS11!/2nS soSR2~1 2 lSH! ~2nS11!/2nS (29)

We now assume that the average stretch ratiol av of compositepowders can be written as the sum of the stretch ratio for each typeof contact multiplied by the volume fractions for the correspondingtype of contact. Thus,

lav 5 f S2lSS1 2fS fHlSH 1 f H

2 lHH . (30)

During closed die compaction, the compaction ratioL(5h/ho)can be written as the ratio of compacted heighth to initial heightho of a powder compact. Assuming no friction exists between thepowder and die, the relationship between the compaction ratioLand the average stretch ratiol av can be written

l av22 5 1 1 ~L 22 2 1! cos2 f (31)

wheref is the angle between a given orientationn and thez-axis,i.e., the direction of die compaction (see Fig. 1 in Storåkers et al.,1997).

By substituting Eqs. (27)–(31) into Eq. (26), macroscopicstressesS ik can be written at a given compaction ratioL. ThepressureS33 under die compaction can be written as Eq. (B9) inAppendix B. Thus,

S33 5 2RZo

VA E0

p/2

cos2 fpR2 sin fdf@G1~f!#

21

2VA E0

p/2

cos2 fpR2 sin fdf@G2~f!# (32)

whereG1(f) andG2(f) are written in Eq. (B5) and (B6), respec-tively. The relationship between the compaction ratio and relativedensity for composite powders with initial relative densityDo

during die compaction can be written

D 5Do

L. (33)

Hence, the variation of relative density with pressureP(5S 33) forcomposite powders under die compaction can be obtained fromEqs. (32) and (33).

Results and DiscussionFigures 2 and 3, respectively, show uniaxial stress-plastic strain

responses of solid copper and solid tungsten under uniaxial com-pression. The circular points represent experimental data. Thesolid curves are obtained from Ludwik’s equation to representexperimental data. Thus,

emS5 S smS

480D4.115

, emH 5 S smH

2900D3.435

(34)

where subscriptsS andH denote the soft powder (copper powder)and the hard powder (tungsten powder), respectively. The yieldstrength and Young’s modulus of copper powder are 149 MPa and110.3 GPa and those of tungsten powder are 700 MPa and 344.8GPa, respectively.

Cold Isostatic Pressing. Figure 4 shows optical micrographsof composite powder compacts with 40 vol% W in Cu compactedat various hydrostatic pressures during cold isostatic pressing. Thedark areas are pores in the compact. As pressure increases, copperpowders deform markedly, but tungsten powders do not deform asmuch. From deformation between tungsten powders, it is observedthat hard powders also deform by contact and do not behave asrigid inclusions. From contacts between copper and tungsten pow-ders, it is also observed that deformation in copper powder is verylarge, but that in tungsten powder is negligibly small. Hence, theassumption we made, the soft particle deforms only when soft andhard particles are in contact, seems to be plausible.

Figure 5 shows optical micrographs of composite powder

Fig. 2 Uniaxial stress-plastic strain relation for dense copper

Fig. 3 Uniaxial stress-plastic strain relation for dense tungsten


compacts with various volume fractions of tungsten powderunder hydrostatic pressure of 250 MPa during cold isostaticpressing. Figure 5 show that more pores are observed in com-

posite powder compacts as the volume fraction of tungstenpowder increases.

Figure 6 shows comparisons between experimental data andtheoretical predictions for the variation of relative density with

Fig. 4 Optical micrographs of 40 vol% of W in Cu composite powdercompacts at various hydrostatic pressures

Fig. 5 Optical micrographs of composite powder compacts with variousvol% of W in Cu under cold isostatic pressing at 250 MPa


pressure of composite powder compacts with various vol% of Wunder cold isostatic pressing. The data points represent experimen-tal data. The dashed and solid curves were obtained by using Eqs.

(22) and (23) with Eq. (14) for the same contact forces and withEq. (15) for different contact forces, respectively. Figure 6 showsthat the dashed curves with the same contact forces lie higher thanthe solid curves with different contact forces. The calculatedresults from Eqs. (22) and (23) show that relative densities increasealmost linearly as pressure increases, not as observed in experi-mental data. Densification behavior at final stage may not beproperly modelled without including the interference between dif-ferent types of contacts at final stage. However, theoretical pre-dictions from Eqs. (22) and (23) with Eq. (15) for different contactforces show reasonably good agreements with experimental data atinitial stage of compaction.

Figures 7 show comparisons between experimental data andtheoretical predictions from the models by Bouvard (1993), Zaval-iangos and Wen (1997), Storåkers et al. (1997) and the presentmodel with Eq. (15) for the variations of relative density withpressure of composite powders with various vol% of W. In themodel by Bouvard (1997), we assumed that the soft powderbehaves as a work-hardening material in stead of a power-lawcreep material. The calculated results from the model by Bouvard(1997) underestimate experimental data of composite powders,since the model assumed that the hard powder behaves as a rigidinclusion. The model by Zavaliangos and Wen (1997) assumedthat the particles behave as perfectly-plastic materials. Thus, thecalculated results from the model by Zavaliangos and Wen (1997)overestimate experimental data. The model by Storåkers et al.(1997) assumed that the same work-hardening parameter for two

Fig. 7(a) Fig. 7(b)

Fig. 7(c) Fig. 7(d)

Fig. 7 Comparisons between experimental data and theoretical predictions from various models for the variation of relative density withhydrostatic pressure of ( a) 20 (b) 40 (c) 60 (d ) 80 vol% of W in Cu under cold isostatic pressing

Fig. 6 Comparisons between experimental data and theoretical predic-tions from the proposed model for the variation of relative density withhydrostatic pressure of composite powders with various vol% of W in Cuunder cold isostatic pressing


different types of particles. From Eq. (34), we obtained the similarstress-strain relations for copper and tungsten powders with thesame work-hardening parameter. Thus,

emS5 S smS

490D3.75

, emH 5 S smH

2680D3.75

(35)

Fig. 8 Optical micrographs of 40 vol% of W in Cu composite powdercompacts at various pressures

Fig. 9 Optical micrographs of composite powder compacts with variousvol% of W in Cu under cold die compaction at 300 MPa


The calculated results from the model by Storåkers et al. (1997)also overestimate experimental data.

Cold Die Compaction. Figures 8 show optical micrographs ofcomposite powder compacts with 40 vol% of W in Cu compactedat various pressures during cold die compaction. It is observed thatcomposite powder compacted by cold die compaction is lesshomogeneous than that by cold isostatic pressing because of de-viatoric stress. The inhomogeneity during die compaction inducesthe enlarged contact area of the powders. Thus, this leads theproposed model to underestimate the experimental data.

Figures 9 show optical micrographs of composite powder com-pacts with various vol% of W in Cu during cold die compactionunder pressure of 300 MPa. Figure 10 shows comparisons betweenexperimental data and theoretical predictions for the variations ofrelative density with pressure of composite powders with variousvol% of W in Cu under cold die compaction. The dashed and solidcurves were obtained from Eqs. (32) and (33) with Eq. (14) for thesame contact forces and with Eq. (15) for different contact forces,respectively. Theoretical predictions from Eqs. (32) and (33) withEq. (14) for the same contact forces shows reasonably goodagreements with experimental data under cold die compaction,although we assumed that only the soft material will deform whenthe different particles are in contact.

Figure 11 shows comparisons between experimental data andtheoretical predictions from the model by Storåkers et al. (1997)and the present model for the variations of relative density withpressure of composite powders with various vol% of W in Cuunder cold die compaction. The dashed curves were obtained fromthe model by Storåkers et al. (1997). The solid curves wereobtained from Eqs. (32) and (33) with Eq. (14) for the samecontact forces and the dash-dotted curves from Eqs. (32) and (33)with Eq. (15) for different contact forces. Theoretical predictionsfor the same contact forces from Eqs. (32) and (33) with Eq. (14)show reasonably good agreements with experimental data.

ConclusionsDensification behaviors of composite powders during cold com-

paction were investigated. Experimental data were obtained for mixedcopper and tungsten powders under cold isostatic pressing and diecompaction. Densification equations were also proposed for compos-ite powders under cold isostatic pressing and die compaction.

Experimental data were compared with theoretical predictionsfrom the proposed model and models in the literature. At initialstage of compaction, theoretical predictions from the assumptionof the different contact forces agreed well with experimental data

for mixed copper and tungsten powders under cold isostatic press-ing. Theoretical results from the assumption of the same contactforces for various types of contact are higher than that fromdifferent contact forces under cold isostatic pressing. Theoreticalpredictions from models in the literature, however, did not agreewell with experimental data under cold isostatic pressing. In diecompaction, theoretical predictions from the assumption for thesame contact forces show reasonably good agreements with ex-perimental data.

AcknowledgmentsThis work was financially supported by a grant from the Korean

Science and Engineering Foundation (KOSEF) under grant no.971-1007-042-2. We are grateful for this support.

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Fig. 10 Comparisons between experimental data and theoretical predic-tions from the present model for the variations of relative density withpressure of composite powders with various vol% of W in Cu under colddie compaction

Fig. 11 Comparisons between experimental data and theoretical predic-tions from the present model and the model by Storåkers et al. (1997) forthe variations of relative density with pressure of composite powderswith various vol% of W in Cu under cold die compaction


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in Powder Metallurgy and Particulate Materials,18. T. M. Cadle, K. S. Narasimhan,eds., MPIF, Princeton, NJ, pp. 137–148.

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A P P E N D I X AWhen the contact forces are assumed to be the same or different

for various types of contact, the center to center distances of theparticles adjacent to each contact can be obtained as follows:

From Eqs. (10) and (13), the contact forcesFSS, FSH andFHH,respectively, can be written

FSS~dSS! 5 KSSS1 2dSS

2RD~2nS11!/2nS

(A1)

FSH~dSH! 5 KSHS1 2dSH

2RD~2nS11!/2nS

(A2)

FHH ~dHH ! 5 KHHS1 2dHH

2RD~2nH11!/2nH

(A3)

whereKSS, KSH andKHH are

KSS56nS

2nS 1 1 S 2nS

2nS 1 1D~12nS!~2nS11!/nS

3 p 12~1/nS! S8

9D1/nS

soSR2 (A4)

KSH 5 6pS Î2

3 D 1/nS

j ~2nS11!/2nS soSR2 (A5)

KHH 56nH

2nH 1 1 S 2nH

2nH 1 1D~12nH !~2nH11!/nH

3 p 12~1/nH ! S8

9D1/nH

soHR2 (A6)

In Eqs. (A1)–(A6),nS and nH are hardening parameters for softand hard particles, respectively.

When the contact forces are assumed to be the same regardlessvarious types of contact, the center to center distances for eachtype of contact can be written in terms ofdSS from Eq. (14) andEqs. (A1)–(A6). Thus,

dSH~5dHS! 5 2RS1 2 S KSS

KSHD 2nS/~2nS11! S1 2

dSS

2RDD (A7)

dHH 5 2RS1 2 S KSS

KHHD 2nH /~2nH11! S1 2

dSS

2RD~2nS11!nH /nS~2nH11!D

(A8)

When the contact forces are assumed to be different for differenttypes of contact, the center to center distances for each type of

contact can be written in terms ofdSS from Eq. (15) and Eqs.(A1)–(A6). Thus,

dSH~5dHS!

5 2RS1 2 S ~1 1 0.6fH !KSS

KSHD 2nS/~2nS11! S1 2

dSS

2RDD (A9)

dHH 5 2RS1 2 SSRfH~2 2 exp~1 2 SR!! 12fH

KSS

KHHD 2nH /~2nH11!

3 S1 2dSS

2RD~2nS11!nH /nS~2nH11!D (A10)

A P P E N D I X B

For densification of composite powders under cold die compac-tion, the compacting pressureS33 can be obtained as follows:

When i 5 3, k 5 3, Eq. (26) becomes

S33 5 2RZo

VA EA

n3n3dA@ f S2FISSlSS1 2fS fH FISHlSH

1 f H2 FIHHlHH # 2

1

2VA EA

n3n3dA

3 FfSlSS E2R

2R/lSS

FSSrdZSS

drdr 1 fSlSH E

2R

2R/lSH

FSHrdZSH

drdr

1 fHlHS E2R

2R/lHS

FHSrdZHS

drdr

1 fHlHH E2R

2R/lSH

FHH rdZHH

drdrG . (B1)

The relationship betweenlab anddab can be written

lab 5dab

2R~a, b 5 S, H!. (B2)

When the contact forces are assumed to be the same or differentfor various types of contact, Eq. (B2) can be written

lab 5 lab ~lSS! ~a, b 5 S, H! (B3)

by using Eqs. (A7) and (A8) for the same contact forces and Eqs.(A9) and (A10) for different contact forces.

By substituting Eqs. (B3) and (27) into (B1) withdA 5 2pR sinfdf, Eq. (B1) can be written

S33 5 2RZo

VA E0

p/2

cos2 fpR2 sin fdf@G1~lSS!#

21

2VA E0

p/2

cos2 fpR2 sin fdf@G2~lSS!# (B4)

whereG1(lSS) andG2(lSS), respectively, are

G1~lSS! 5 f S2FISSlSS1 2fS fH FISHlSH 1 f H

2 FIHHlHH , (B5)


G2~lSS! 5 CH f S2FISS

1 2 lSS

lSSS nS

4nS 1 12 ~1 2 lSS!

nS

6nS 1 1D1 2fS fH FISH

1 2 lSH

lSHS nS

4nS 1 12 ~1 2 lSH!

nS

6nS 1 1D1 f H

2 FIHH

1 2 lHH

lHHS nH

4nH 1 12 ~1 2 lHH !

nH

6nH 1 1DJ ~B6!

From Eqs. (30) and (B3), the average stretchl av for compositepowders can be written

lav 5 lav ~lSS!. (B7)

By substituting Eq. (31) into Eq. (B7), we may write

l av22~lSS! 5 1 1 ~L 22 2 1! cos2 f (B8)

Now, by substituting Eq. (B8) into Eqs. (B5) and (B6), Eq. (B4)can be rewritten

S33 5 2RZo

VA E0

p/2

cos2 fpR2 sin fdf@G1~f!#

21

2VA E0

p/2

cos2 fpR2 sin fdf@G2~f!# (B9)

Hence, the compacting pressureS33 can be obtained for a givencompaction ratioL by integrating Eq. (B9).


I. Zarudi

L. C. ZhangMem. ASME

Department of Mechanical andMechatronic Engineering,The University of Sydney,

NSW 2006, Australiae-mail: [email protected]

On the Limit of SurfaceIntegrity of Alumina byDuctile-Mode GrindingThis paper investigates both experimentally and theoretically the subsurface damage inalumina by ductile-mode grinding. It found that the distribution of the fractured area ona ground mirror surface, with the Rms roughness in the range from 30 nm to 90 nm,depends on not only the grinding conditions but also the pores in the bulk material.Surface pit formation is the result of interaction of abrasive grains of the grinding wheelwith pores. Thus the surface quality achievable by ductile-mode grinding is limited by theinitial microstructure of a material. The investigation shows that median and radialcracks do not appear and hence are not the cause of fracture as usually thought.

1 IntroductionHigh surface integrity of hard, brittle materials with perfect

mirror-like and fracture-free surfaces of monocrystalline materialscan be achieved by ultra-precision grinding (Zarudi and Zhang,1997a, Suzuki, 1997). It has been shown theoretically and exper-imentally that the mechanism of material removal can be purelyductile when five independent slip and twin systems are activated(Zarudi and Zhang, 1997b). However, in the case of grindingpolycrystals a certain amount of fractured surface areas alwaysappear causing the degradation of its surface integrity (Zarudi andZhang, 1997b, Komanduri, 1996). This leads to controversialopinions on the nature of these fractured surface areas. For exam-ple, one has considered that the cause of such surface fracturecould be the result of dislodgments of grains (Komanduri, 1996),or of the median and radial cracks developed in the subsurface(Bifano et al., 1991). Recently, Tele (Tele, 1995) admitted thatmanufacturing defects such as pores or voids could also contributeto the surface integrity in machining.

The present paper aims to investigate the mechanisms of surfaceintegrity in grinding alumina components.

2 ExperimentPolycrystalline alumina of 99.99% purity produced (Kyocera,

Japan) with grain size of 1mm and 25mm were ground with anultra-precision grinder, a modified Minini Junior 90 CF CNCM286 (measured loop stiffness5 80 N/mm). The grinding param-eters used are listed in Tables 1 to 3. The horizontal and verticalgrinding forces,Ft andFn, were measured with a three-axis Kistlerdynamometer (Type 9257B). At least five samples were groundunder the same grinding conditions.

The properties of a ground surface were explored by means ofHigh Resolution Scanning Electron Microscope (HRSEM) JSM-6000F and Atomic Force Microscope (AFM). The structure ofpores in alumina after sintering and the subsurface structure ofground specimens were studied by means of Transmission Elec-tron Microscope (TEM) EM 430. Plan view samples were used forexaminations of the alumina structure in the initial state. Fivesamples were checked for every type of alumina. Twenty areaswere randomly recorded for every sample and the surface coveredby pores was quantified by means of the image analysis system“MD301.” Parameters of pores were determined by the methoddescribed by Glasbey (1994).

To prepare a cross-sectional view sample a ground specimenwas first sectioned into thin slices, of 1 mm thick, perpendicular tothe ground surface (Fig. 1(a)). After that two slices were gluedtogether in the manner shown in Fig. 1(b) and thinned downmechanically and then by ion milling to a thickness of 40 nmsuitable for TEM study (more details can be found in Zarudi et al.,1996). Five cross-sectional view samples were examined for everygrinding regime. The density of dislocations was determined byintersection method (Hirsh, 1971). At least 20 fields were checkedfor every grinding regime. To determine the surface area coveredby pits, AFM records were quantified by means of “MD301.” Atleast 10 records were examined for every ground regime.

3 Results and Discussion

3.1 Material Porosity Before Grinding. Pores were de-tected in both types of alumina after sintering. It showed that poresin the 25mm-grained alumina with an average diameter of 2mmcovered 5–8%62% of the whole ground surface and those in the 1mm-grained alumina featuring an average diameter of 0.3mmcovered 1–2%60.5% of the ground surface.

3.2 Surface Topography After Grinding. Typical topog-raphy of ground surfaces is shown in Fig. 2. Mirror surfaces weregenerated under all the table speeds listed in Table 1. Groovescould be clearly observed under AFM Fig. 2(a, c). However, somepits were also seen (Fig. 2(b, e)). The values of Rms roughnessvstable speed for both types of alumina are shown in Fig. 3. It is clearthat the Rms roughness increases with the increase of table speed.For the 1mm-grained alumina the Rms roughness grows from 30nm to 50 nm when the table speed changes from 0.02 m/min to 1m/min. For the 25mm-grained alumina, it increased from 33 nm to88 nm and thus the effect was more pronounced. Clearly, thesurface pits must have affected the measured surface roughnessvalues, because in the parts without pits Rms roughness was onlyfrom 15 nm to 20 nm.

The surface areas covered by pits (SACP) are shown in Fig. 4.It is obvious that the SACP decreases significantly when the tablespeed of grinding decreases. In other words, SACP becomessmaller if the nominal chip thickness in grinding is smaller. Fur-thermore, the percentage of SACP on the ground surface of the 1mm-grained alumina is always much less than that on the 25mm-grained alumina. In the range of table speed from 0.2 m/minto 0.5 m/min, the rate of increment of SACP on the 25mm-grainedalumina is quite high.

To study the nature of fractured surfaces of SACP we usedHRSEM. The detailed topography of a typical pit can be seen inFig. 5. In the central part of the pit no fractured facets could be

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionOctober 4, 1997; revised manuscript received April 27, 1999. Associate TechnicalEditor: S. A. Meguid.


seen and its inner structure could be attributed to the cross-sectionof a pore. However, the edges of the pit show some fracturedfragments. Under higher magnifications (Fig. 6), cracks were eas-ily located in the vicinity of the pore edges (Fig. 6(b)). Thetopography of pits suggests that their formation not by graindislodgment but by the interaction of abrasives of the grindingwheel with pores inside materials.

3.3 Subsurface Structure. Before ultra-precision grinding,the damaged subsurface layer induced by specimen preparation

(conventional grinding) was determined by means of TEM oncross-section view samples. These initially damaged zones werefirst removed completely during the subsequent ultra-precisiongrinding. Thus in all the cases below, the characterisation of thesubsurface structure were purely caused by the ultra-precisiongrinding.

Figure 7 shows two distinguished regions in the subsurface ofboth types of alumina. The first region, which is immediately

Table 1 Grinding conditions

Table 2 Wheel dressing conditions

Table 3 Wheel truing conditions

Fig. 1(a)

Fig. 2(a)

Fig. 1(b)

Fig. 1 Preparation of cross-section view samples for TEM investiga-tions: ( a) cutting into thin slices; ( b) gluing two slices together

Fig. 2(b)

Fig. 2(c)

Fig. 2(d)

Fig. 2 Topography of surfaces after ductile-regime grinding: ( a) and (b)the 1 mm-grained alumina; ( c) and (d) the 25 mm-grained alumina; ( a) and(c) 3-d image of the surface; ( b) and (d) general view


under the ground surface, is characterised by an extremely highdensity of dislocations. Individual dislocations can be hardlyrecognised in this region. The dislocation density exceeds 1012

cm22. In the 25mm-grained alumina, the depth of the region isfrom 0.2mm to 0.4mm, depending on the table speed used. In the1 mm-grained alumina it varies from 0.1 to 0.3mm. The density ofdislocations in the second region was 108 to 109 cm22 (Fig. 7). Thisregion has a thickness of 0.2mm to 0.5mm in the 25mm-grainedalumina and of 0.1mm to 0.2mm in the 1mm-grained alumina. Nomicrocracks were found in the subsurface areas without poresalthough 10 areas were examined under each group of grindingconditions. This proved to great extent that in the areas withoutpores, material removal was under a real ductile-mode.

The crack-free behavior in the first region with high dislocationdensity can be explained by the initiation of sufficient number oftwin and slip systems (Chin, 1975), because in this region grindingactivated at least five independent such systems (Zarudi andZnang, 1997). In the second zone, however, less than five inde-pendent slip and twin systems were activated and thus microcrack-ing should be highly possible there. However, this is inconsistentwith the above experimental observations. Such phenomenon maybe elucidated by the possibility of microcracking due to the pile-upof dislocations on an active slip plane in the second zone. Thethreshold of an effective resolve shear stress (t s) for cracking atthe leading edge of the pile-up can be determined by (Hagan,1979):

t s2 5

3p

8 F gm

~1 2 n!LG (1)

whereg is the fracture surface energy,m is shear modulus,n isPoisson’s ratio, andL is the length of pile-up. If the pile-up length

Fig. 3 Effect of table speed on Rms roughness of alumina with differentgrain sizes

Fig. 4 Effect of table speed on surface area covered by pits

Fig. 5(a)

Fig. 6(a)

Fig. 5(b)

Fig. 5 Topography of pits: ( a) the 1 mm-grained alumina, ( b) the 25mm-grained alumina

Fig. 6(b)

Fig. 6 Subsurface structure in the vicinity of a pit: ( a) general view; ( b)a detailed view of the pore edge with microcracks


decreases,t s increases according to the above formula, and thismay eliminate crack initiation.

Based on this and considering that the length of the pile-ups in thespecimens (i.e., penetration depth of dislocations in the second areawas in the range from 150 to 500 nm, we can conclude that thepile-ups are not sufficient for microcracking. In addition, it is worthnoting that the interaction of different slip systems was a rare event inthe second region because of the extremely low density of disloca-tions. Hence, no microcracks can be created via interaction.

From the above discussion, it is clear that the initiation ofdislocations in subsurface does not create sufficient stress concen-trators for microcracking. In addition, importantly, no median orradial cracks were found, although five cross-sectional view sam-ples were examined under each grinding condition and twentysubsurface areas. These indicate that the existing explanationsabout the formation of fractured surfaces are not suitable forunderstanding the SACP in ultra-precision grinding of alumina.Our following consideration, though incomplete and very approx-imate, provides additional information about the fracture aroundpore edges and the variation of SACP on the ground surfaces ofdifferent types of alumina.

3.4 Grinding Materials Containing Pores

3.4.1 Assumptions. For simplicity, we assume that forces ingrinding are distributed equally among active abrasive grains in thegrinding zone and that all grains have an equal diameter of 1mm forthe grinding wheel used in the present study. Experimentally deter-mined forces listed in Table 4 can then be used in calculations. Usingthe method shown by Zhang and Howes (1995) the number of activegrains per micron in our experiment is found to be 2.5.

In grinding a material containing pores, as shown in Fig. 8, weencounter two cases, an abrasive grain interacting with a closed

pore in the subsurface or with an open pore on the surface. Thedetails of the abrasive-pore interaction can be very complex.Nonuniform loading of bending, shear, and dynamic impact allcontribute to the local fracture. However, to facilitate the analysis,consider the bending-induced fracture, which, we think, is one ofthe major causes of fracture around a pore. Following the approachused for impact loads (Galiev, 1996) we assume further that thestress state in the material induced by abrasive impact is similar tothat under an equivalent static load and that the shape of a pore isaxisymmetrical about the normal of the ground surface.

3.4.2 Closed Pore in the 1mm-Grained Alumina. In thiscase, the layer of alumina on the top of a pore is modeled by aclamped circular plate with a variable thickness. The cross-sectionof such a plate model is shown in Fig. 9(a). At the peripheral part,the plate has a linearly varying thickness. The extension of line ABintersects with the top surface at centreO. The ratiob/a was takenas 0.8.

As the average radius of pores was only 0.3mm in the 1mm-grained alumina, interacting stress from single grit is roughlyapproximated by a uniform pressure over the whole plate with adiameter of 0.3mm (equal to the average diameter of pores). Themaximum bending moment of the plate is therefore (Timoshenkoand Woinowsky-Krieger, 1976):

Mmax 5 M0 2P

8p1 g1P, (2)

where

M0 5P~1 1 n!a2

16, (3)

Fig. 7(a)

Fig. 8 Ductile-mode grinding of alumina containing pores

Fig. 7(b)

Fig. 7 Subsurface structure of alumina after ultra-precision grinding: ( a)the 1 mm-grained alumina, ( b) the 25 mm-grained alumina

Table 4 Measured grinding forces


P is the total load applied on a pore through an active abrasivegrain,y is Poisson’s ratio, anda is the radius of the plate. In Eq.(2) g1 is a function ofb/a and can be taken as 0.05 forb/a 5 0.8(Timoshenko and Woinowsky-Krieger, 1976).

3.4.3 A Closed Pore in the 25mm-Grained Alumina. As theaverage radius of pores in the 25mm-grained alumina is 2mm, thediameter of the plate model is also taken to be 2mm. Theinteracting stress in this case distributes only partially on the platebounded by a circle of radius equal to 1mm, the average diameterof abrasive grains, as shown in Fig. 9(b).

Under this loading condition, the maximum bending moment inthe plate is determined by (Timoshenko and Woinowsky-Krieger,1976):

Mmax 5 M0 2P

4p S1 2c2

2a2D 1 g1P, (4)

where

M0 5P

4p F ~1 1 n! loga

c1 1 2

~1 2 n!c2

4a2 G . (5)

3.4.3 An Open Pore. When a pore opens partially to thegrinding surface, the problem is considered as a plate with a centralhole subjected to an interacting stress from an abrasive grain, asshown in Fig. 9(c). The maximum bending moment in this case is(Timoshenko and Woinowsky-Krieger, 1976):

Mmax 5P

4pF ~1 1 n!a2

d2 1 1 2 nG3 F ~1 2 n!Sa2

d2 2 1D 1 2~1 1 n!a2

d2 loga

dG , (6)

whered is radius of the hole.On the other hand, the stress intensity factor of a plate with

voids can be expressed as (Rooke and Cartwrite, 1976):

K1 56M

~h 2 g! 3/2 f~g/h!, (7)

whereM is the bending moment,h is the thickness of the plate inthe central part, andg is the length of a possible imperfections.Some imperfection always exists in ceramics, which can be grainboundaries, faults of structure etc. To be conservative, we took thesmallest imperfection size of 15 nm by assuming that the materialis almost perfect.

The stress intensity factors obtained above can then be used toexamine the fracture behavior of alumina around pore edges.Assume that a fracture will take place when the stress intensityfactor exceeds the corresponding fracture toughness, which is 0.7MPa3 m1/2 for the 25mm-grained alumina and 2 MPa3 m1/2 forthe 1mm-grained alumina (Xu et al., 1996).

Table 5 shows the critical thickness of the layer (hc) above apore, at which fracture occurs. For instance, in grinding the 25mm-grained alumina for a closed pore,hc is 200 nm under the tablespeeds of 0.02 m/min and 0.1 m/min. Thus microcracking will takeplace if the thickness of the layer above a pore is less than 200 nm.Generally,hc is smaller under lower table speeds, indicating thatfracture initiation is harder. This means that under a lower tablespeed, SACP will be smaller and the overall surface roughnessmay also be smaller.

In the case with the 1mm-grained alumina,hc is considerablysmaller and is in the range from 20 nm to 50 nm when table speedvaries. This is consistent with the fact that the microfracturingeffect of pores in the 1mm-grained alumina is much lower thanthat with the 25mm-grained alumina as presented by Fig. 4.

When a pore is open, edge fracture always exists since the edgethickness of the open pore is normally variable and is often lessthanhc. It is necessary to emphasize that the present model onlyinvolves bending effect. In a real case, imperfections in materialand effect of shear and dynamic impact will make fracture easier(hc larger). This will in turn intensify the process of the micro-crack initiation at a pore edge.

In short, the pore effect on microcracking makes SACP largerand limits the surface integrity level achievable in the ultra-precision grinding of polyscrystalline alumina containingpores. Microcracking around pore edges always occurs, al-though a pure ductile-mode of material removal can be

Fig. 9(a)

Fig. 9(b)

Fig. 9(c)

Fig. 9 Model for interaction between an abrasive grain and materialcontaining pores: ( a) and (b) closed pores, ( c) an open pore; ( a) the 1mm-grained alumina, ( b) the 25 mm-grained alumina

Table 5 Pore effect on microcracking in grinding


achieved by activating sufficient slip and twin systems in theareas without pores.

4 Conclusions

1. The nature of SACP on ground surfaces is related to poresin the bulk material.

2. The subsurface structure of alumina after ultra-precisiongrinding consists of a layer with a high density of disloca-tions (more than 1012 cm22) and another with lower densityof dislocations (about 109 cm22).

3. A real ductile-mode of material removal is possible dueto existence of more than five independent slip and twinsystems in the first layer. However, the absence of mi-crocracks in the second layer is due to the small length ofpile-ups.

4. No radial and lateral cracks were found in the subsurfacearea of the alumina after the ultra-precision grinding. Thissuggests that the formation of SACP is not related to thesecracks. The bending models of open and closed pores,though rough, showed acceptable explanation to the varia-tion of SACP.

5. The existence of pores in alumina has an effect on the levelof the surface integrity achievable by ultra-precision grind-ing. This seems to indicate that the ultimate surface integ-rity is determined not only by the depth of cut as usuallythought, but also to certain extent, by the percentage andsize of pores.

AcknowledgmentsThe authors wish to thank the Australian Research Council

(ARC) for continuing support of this project. We thank also theElectron Microscope Unit for use of its facilities.

ReferencesBifano, T. G., Dow, T. A., and Scattergood, R. O., 1991, “Ductile-Regime

Grinding: A New Technology for Machining Brittle Materials,” ASME JOURNAL

OF ENGINEERING MATERIALS AND TECHNOLOGY, Vol. 113, pp. 184 –189.Chin, G. Y., 1975, “Slip and Twinning Systems in Ceramic Crystals,”Deformation

of Ceramic Materials,R. C. Brand and R. E. Tressler, eds., pp. 25–44.Galiev, S. U., 1996, “Experimental Observations and Discussion of Counterintui-

tive Behaviour of Plates and Shallow Shells Subjected to Blast Loading,”Int. J.Impact Eng.,Vol. 18, pp. 783–802.

Glasbey, C. A., 1995,Image Analysis for the Biological Science,Chichester, NewYork.

Hagan, T., 1979, “Micromechanics of Crack Nucleating During Indentation,”J.Mat. Sci.,Vol. 14, pp. 2975–2980.

Hirsh, P. B., 1971,Electron Microscopy of Thin Crystals,Butterworth, London.Komanduri, R., 1996, “On Material Removal Mechanisms in Finishing of Ad-

vanced Ceramics and Glasses,”Cirp Annals,Vol. 45, pp. 09–514.Rooke, D. P., and Cartwrite, D. J., 1976,Compendium of Stress-Intensity Factors,

Her Majesty’s Stationery Office, London.Suzuki, H., Wajima, N., Zahmaty, M. S., Kuriyagava, T., and Syoji, K., 1997,

“Precision Grinding of a Spherical Surface Accuracy Improving by On-MachineMeasurement,”Advances in Abrasive Technology,L. C. Zhang and N. Yasunaga,eds., World Scientific, Singapore, pp. 116–121.

Tele, R., 1995, “Ceramography of High Performance Ceramics. Part IX: Pores andChips,” Prakt. Metallogr,Vol. 32, pp. 440–466.

Timoshenko, S. P., and Woinowsky-Krieger, S., 1976,Theory of Plates and Shells,McGraw-Hill, Singapore.

Zarudi, I., and Zhang, L. C., 1997a, “Subsurface Structure Change of Silicon AfterUltra-Precision Grinding,”Advances in Abrasive Technology,L. C. Zhang and N.Yasunaga, eds., Vol. 33, World Scientific, Singapore, pp. 33–38.

Zarudi, I., and Zhang, L. C., 1997b, “Surface and Subsurface Structure ofAlumina After Ductile Mode Grinding,”Proc. of 12-th ASPE Annual Meeting,pp.267–270.

Zarudi, I., 1996, “Plastic Zone Formation in Alumina Associated with Ultra-Precision Grinding,” Proc. of the 11th ASPE, 610–613.

Zarudi, I., Zhang, L. C., and Mai, Y.-W., 1996, “Subsurface Damage in AluminaInduced by Single-Point Scratching,”J. of Mater. Sci.,Vol. 31, pp. 905–914.

Zhang, Bi, and Howes, T. D., 1995, “Subsurface Evaluation of Ground Ceramics,”Annals of CIRP,Vol. 44, pp. 263–266.

Xu, H. H. K., Wei, L., and Jahanmir, S., 1996, “Influence of Grain Size on theGrinding Response of Alumina,”J. Am. Ceram. Soc.,Vol. 79, pp. 1307–1313.


Weili ChengBerkeley Engineering And Research, Inc.,

896 Seville Place,Fremont, CA 94539


Measurement of the AxialResidual Stresses Using theInitial Strain ApproachTraditionally, the unknown variables to be estimated in residual stress measurement arebased on the approximation of the stresses. In this paper an alternative approach ispresented, in which the initial strain is used as the unknown variable to be estimated inthe measurement. A useful feature of the approach is that it does not require themeasurement to be made on the original part if the stress is uniform in the axial direction.Instead, the original residual stress can be computed once the initial strains are obtainedfrom a specimen removed from the original part. This approach is incorporated into tworecently developed methods: the crack compliance method and the single-slice method formeasurement of the axial stress in plane strain. Experimental validation shows that theapproach based on initial strains leads to a result that agrees very well with the analyticalprediction.

IntroductionResidual stresses in many long structural parts can be idealized

as distributed uniformly in the axial direction with little out-of-plane shear stresses. Examples are rods, beams, and long butt orfillet welds between two plates. In many cases the residual stressesarise under conditions which are not easily simulated by numericalcomputation. Figure 1(A) shows schematically a section of rod andbeam while Fig. 1(B) shows a section of fillet weld between twoplates. In these examples the constraint is often such that residualstress in thez-direction would be expected to be the most severe.Therefore, an essential aspect in assessing the integrity of theseparts is to measure the distribution of axial (z-axis) stress.

To obtain the residual stresses, the unknown quantities to beestimated usually correspond to the stresses or average forcesreleased by removing or separating material. Thus, it is commonlybelieved that the residual stresses are “lost” if any deformation dueto partial relieve of the stresses, such as fracture of a part, is notsufficiently recorded. This is true for most methods based on anapproximation of stresses. One exception is the single-slicemethod (Cheng and Finnie, 1998) developed for axisymmetricbodies for which the original residual stress may be estimated froma section removed from the original body. However, for an arbi-trary prismatic body the analysis based on an approximation of thestress becomes much more complicated and it is not easy to find aseries of functions that always satisfy the equilibrium condition.

To overcome these limitations, we turn to an alternative ap-proach based on the approximation of the incompatible initialstrain field from which the residual stresses may be obtained. Therelation between residual stresses and initial strains has beenaddressed in many classic textbooks on elasticity (Fung, 1965;Timoshenko and Goodier, 1970). However, its use for residualstress measurement is relatively new. Ueda and his colleagues firstproposed their “inherent strain” method in the mid-70’s. Instead ofsolving the stresses alone, the inherent strain method aims atestablishing the incompatible initial strain field that causes theresidual stresses. This involves measuring the deformation in dif-ferent directions due to cutting and/or sectioning the body intostrips and many small segments (Ueda et al., 1986; Ueda andFukuda, 1989; Ueda et al., 1996). With elaborate efforts in both

experiment and computation, they demonstrated that initial strainscan be successfully used to estimate residual stresses.

In the present approach initial strains are introduced only forgenerating a series of stress distributions on the plane of measure-ment that are used to approximate the axial stresses released priorto and/or during measurement. For this reason it is informative tomention some distinctive features of the approach.

First, a residual axial stress field is uniquely determined by agiven incompatible initial strain field but the reverse is not true.This implies that there exists more than one incompatible initialstrain field that lead to the same axial residual stress distribution onthe plane of measurement.

Second, if the residual stress is altered by cuts made on planeson which the out-of-plane shear stresses are small enough to beneglected before cutting, the original residual axial stress may beestimated using the initial strain field measured from a section ofthe part as long as little permanent deformation occurs duringcutting. Thus, the initial strain field estimated from a fractured partmay be used to compute the original stress in the part beforefracture if the original axial stress was uniform over a shortdistance in the direction normal to the plane of fracture.

Third, for residual stresses produced by local incompatibledeformation, such as due to welding, it is only necessary to definethe initial strains in a subregion containing the weld (Ueda et al.,1996). This feature makes the approximation of the initial strainsdue to welding easier than the approximation of the stress distri-bution over the entire region of the part. It also implies that theoriginal residual stress distribution can be estimated only if thespecimen contains the entire initial strain field. For a welded part,for instance, it means that the cross-section of the specimen shouldbe large enough to contain the entire cross-section of the weld.

It is worth of noting that the first feature mentioned above isunique to the present approach and, as demonstrated later, it makesthe construction of the initial strain distributions for numericalcomputation relatively easier than the “inherent strain” approach.

In this paper an analysis based on the approximation of initialstrains is incorporated into the crack compliance method (Chengand Finnie, 1994; Prime, 1999) and the single-slice method (Chengand Finnie, 1998). A computational procedure is then presented toobtain the change of strains due to stresses released for each giveninitial strain function. After the initial strain field is determined bya least squares fit over the experimentally measured strains, theresidual axial stress can be computed.

To demonstrate the use of initial strains for residual stressmeasurement, an experiment using the crack compliance method

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials Division April24, 1998; revised manuscript received July 16, 1999. Associate Technical Editor:S. K. Datta.


was carried out on a beam which had been cut apart in a previoustest. The original stress in the beam before it was cut apart wasthen estimated using the initial strain approach. It is shown that theresult agrees very closely with the analytical solution.

Initial Strain Approach for the Crack ComplianceMethod: Axial Stress in a Beam

The crack compliance method has been used to measure residualstresses through-the-thickness in parts of different configurations(Cheng and Finnie, 1994; Prime, 1999). In this method the residualstress distribution to be measured is approximated by a series ofcontinuous or piecewise functions withn unknown coefficients.From the linear superposition shown in Fig. 2, the deformation dueto introducing a progressively increasing depth of cut is the sameas that produced by applying the stress being released on the facesof the cut with the sign reversed. This allows the deformation orthe “compliance function” due to each individual function in theapproximate stress distribution to be obtained. Since the numbermof measurements recorded during cutting is usually much largerthan the numbern of unknown coefficients, a least squares fit iscommonly used to determine the unknowns.

To demonstrate how the initial strain approach works for thecrack compliance method, we consider the residual stress producedby a beam subjected to pure bending beyond its elastic limit asshown in Fig. 3. Since the residual stress is uniform along thelength between the two inner support pins, it is sufficient to assumethat the initial strain field is uniform along the entire length of thebeam. Also, since there exist more than one initial strain field thatproduce the same axial stress in the beam, we choose the one thatcorresponds to the deformation caused by a temperature variatione( y), which, when expressed in terms of a series of LegendrepolynomialsLi( y) (Johnson, 1982), leads to

e~y! 5 ex~y! 5 ey~y! 5 ez~y! 5 Oi52

n

Ai Li ~y!

for 21 # y # 1 (1)

in which y is the normalized distance measured from the neutralaxis. It is seen that the uniform and linear terms are omitted fromEq. (1) because they do not produce any residual stress for a beamfree of any constraint. The stress in the axial (z) direction canreadily be obtained using the analytical solution available forthermal stresses in a long strip (Timoshenko and Goodier, 1970).That is,

sz~y! 5 2Ee~y! 1E

2 E21

1

e~y!dy 1 3Ey

2 E21

1

e~y!ydy (2)

in which E is the elastic modulus. Since the integrals in Eq. (2)vanish for all Legendre polynomials withi $ 2, we arrive at avery simple solution for the stress,

sz~y! 5 2Ee~y! 5 2E Oi52

n

Ai Li ~y! for 21 # y # 1 (3)

Now we introduce a cut of increasing depth to the beam. Thedeformation due to release of the stress on the plane of cut can beobtained as shown in Fig. 4(A). Thus, without specifying explicitlythe stress distribution on the faces of cut, the deformation due toreleasing the stress by cutting can be determined from the differ-ence between the bodies with and without the cut. This analysis

Fig. 1(A)

Fig. 1(B)

Fig. 1 Illustrations of ( A) a beam and a rod and ( B) a long butt weldbetween two plates with residual axial stresses distributed uniformlyalong the length except near the ends

Fig. 2 Linear superposition used for crack compliance method basedon an approximation of the stress

Fig. 3 A beam subjected to four-points bending to produce a uniformaxial stress distribution in the midsection between the two inner supportpins


can be generalized to other situations, as shown in Fig. 4(B), inwhich the shape of the beam is altered by material removal withoutany plastic deformation. Although the stresses in latter cases aredifferent from the original ones, the deformation due to introduc-tion of a cut of increasing depth is caused by the same initial strainor axial stress. Therefore, the original stress can still be computeddirectly from Eq. (3) once the initial strain field is estimated. Onthe other hand, if the existing stresses on the plane of cut are to bemeasured, they need to be computed using the estimated initialstrains for the exact geometry of the part shown in Fig. 4(B)without the cut.

Denoting the strains produced by the initial strains with andwithout the cut bye c ande o respectively, the change in strain dueto cutting is then given by

e~aj ! 5 ec~aj ! 2 eo 5 Oi52

n

Ai @eci ~aj ! 2 e io# 5 O

i52

n

Aie ie~aj ! (4)

whereaj is thej th depth of cut ande ie is the strain produced by the

i th order functionLi( y) in Eq. (1). For a numberm of depths Eq.(4) can be written in a matrix form

@e e# z A 5 e (5)

where

@e e# 5 3e 21

e : e i1e : e n1

e

: : :

e 2je : e ij

e : e nje

: : :

e 2me : e im

e : e nme4 ,

A 5 3A2

:

Ai

:

An

4 and e 5 3e1

:

e j

:

em

4 (6)

For m . n a least squares fit can be used to obtain the unknowncoefficient vectorA from the measured strain vectore. This leadsto

A 5 $@e e# t@e e#% 21@e e# te (7)

where a superscriptt denotes a transposed matrix. Substituting thecoefficientsAi computed from Eq. (7) into Eq. (3), the originalresidual stress distribution can be obtained.

Initial Strains Approach for the Single-Slice Method:Axial Stress in a Rod

The single-slice method, presented recently by Cheng andFinnie (1998), is primarily for measuring the axial stress dis-tribution in the mid-section of a part. Experimentally, it in-volves measuring the strain due to removing a slice from themid-section. First, a complete cut is made to separate the part inthe mid-section. On one of two sections strain gages are in-stalled on the surface exposed by the first cut and a second cutis made to remove a slice containing the strain gages while thechange of strains due to removing the slice is recorded. Theaxial stress variation over the cross-section can then be esti-mated using the strain data.

Because the analysis developed by Cheng and Finnie (1998) foraxisymmetric problems is based on the approximation of the stress,a rigorous solution is given here to show that the initial strainapproach provides an improved alternative. Consider a rod in planestrain with an axisymmetric residual stress field, shown in Fig.5(A). As mentioned earlier, the choice of the initial strains is notunique. To illustrate this point, an initial strain field which isuniform in thez-direction is chosen as

ez~r ! 5 e~r !

er ~r ! 5 eu ~r ! 5 be~r ! (8)

where r is the normalized radial distance andb is an arbitraryconstant. Obviously, a variety of initial strain fields may be pro-duced by using different values ofb. In this case the analyticalsolution for the corresponding stress distribution in thez-directionmay be obtained as

sz~r ! 5E~bm 1 1!

1 2 m 2 S 2 E0

1

erdr 2 eD (9)

whereE is the elastic modulus andm the Poisson’s ratio. It isseen from Eq. (9) that any initial strains defined by Eq. (8) leadto similar distributions of the axial stress even though theyproduce different stresses in ther andu-directions. Thus, theyall can be used to estimate the same residual axial stressbecause the deformation measured by cutting off the slice isonly dependent on the release of the axial stress. Particularly,

Fig. 5 Schematics of ( A) a long cylinder with an axisymmetric residualstress is separated at the midsection z 5 0; (B) a slice cut out along planez 5 a while hoop and/or radial strains are recorded on plane z 5 0; (C)linear superposition for the initial strain approach

Fig. 4 Schematics of ( A) crack compliance method for a beam sub-jected to initial strains with and without a cut of increasing depth; ( B)after undergoing cutting or material removal the same beam is used tomeasure the initial strains by introducing a cut of increasing depth


the axial stress can always be generated by specifying an initialstrain in the axial direction.

For axisymmetric problems, a complete polynomial series con-sists of only even orders 2i and the initial straine may beexpressed as

e~r ! 5 Oi51

n

Ai r2i (10)

in which Ai is the amplitude coefficient for the 2i th order term.From Eq. (9) the residual axial stress produced by the initial straingiven in Eq. (8) becomes

sz~r ! 5E~bm 1 1!

1 2 m 2 Oi51

n

AiS 1

i 1 12 r 2iD (11)

It can readily be shown that equilibrium condition is alwayssatisfied by Eq. (11). Since Eq. (10) is defined by a complete evenorder polynomial series, the corresponding axial stress distributionis also a complete even order polynomial series. Therefore, theaxial stress distribution obtained earlier by Cheng and Finnie(1998) can be constructed exactly by the polynomials given by Eq.(11).

Now the strain or deformation produced by each term of thestress given in Eq. (11) can be computed using each term of theinitial strains given in Eq. (10). Settingb 5 1, the assumed initialstrain field becomes one produced by a temperature distribution,which can be handled directly by most finite element programs.Settingb 5 0, on the other hand, the only non-zero initial strain isin the axial direction, which can be simulated by specifying atemperature distribution while setting thermal expansion coeffi-cients in ther andu directions to zero.

We now separate the rod shown in Fig. 5(A) by a complete cutin the mid-section. For elastic deformation the initial strain givenby Eq. (10) remains unchanged and the new stress state for eitherhalf of the body can be determined from the same initial strainfield. We now introduce another complete cut on planez 5 a toremove a slice of thicknessa as shown in Fig. 5(B). The change ofstrain on the surface of the slice exposed by the first cut is due torelease of the stresses on the planez 5 a. If the approach basedon the approximation of the stress were used (Cheng and Finnie,1998), the normal and shear stress distributions on the planez 5a before the second cut would have to be computed first. Then,with a reversed sign, they would be used as the loading conditionson the face exposed by the second cut shown in Fig. 5(B) tocompute the deformation of the slice due to the second cut. For theapproach based on initial strains, however, we do not need tocompute the normal and shear stresses existing on planez 5 a, nordo we need to specify any additional loading conditions other thanthe initial strains in the slice. A simple procedure based on linearsuperposition shown in Fig. 5(C) may be used to obtain thedeformation due to cutting out the slice. It is seen that the defor-mation produced by loading on a slice of thicknessa without anyinitial strain, shown as (III), is equal to the difference in thedeformation obtained for the slice, shown as (II), subjected to onlyinitial strain and the deformation for one half of the rod, shown as(I), subjected to only the same initial strain. Thus, the change ofstrain on the face of the slice,z 5 0, due to cutting off the slicecan be obtained for each function given in Eq. (10) withoutknowing the stresses on planez 5 a. This leads to a morestraightforward implementation than that based on the approxima-tion of the stress.

Using the linear superposition approach shown in Fig. 5(C), thechange in strain, say in theu-direction, due to cutting off the slicemay be expressed as

eu ~r j ! 5 e uc~r j ! 2 e u

o~r j ! 5 Oi52

n

Ai @e uic ~r j ! 2 e ui

o ~r j !#

5 Oi52

n

Aie uie ~r j ! (12)

wherer j is the radial location of thej th strain gage ande uie is the

strain produced by thei th order functionr 2i in Eq. (10). For anumberm of strain gages Eq. (12) can be written in a matrix form

@e ue# z A 5 eu (13)

where matrix [eue], column vectorsA andeu have the same forms

as the counterparts in Eq. (6). Similarly, form . n, the unknowncoefficient vectorA can be obtained by a least squares fit as givenby Eq. (7).

Experimental ValidationThe residual stress produced by four-point bending, as illus-

trated in Fig. 3, has been used to validate the crack compliancemethod in the past (Prime, 1991). This is because the residualstress can be predicted accurately using the stress-strain relationobtained from the same beam (Mayville and Finnie, 1982). Also,it provides an ideal benchmark for checking the capability of themethod for measuring a rapidly varying residual stress fieldthrough the thickness since residual stresses with different magni-tude and gradient can be generated by controlling the extent of theplastic deformation. As demonstrated by Finnie and Cheng (1996),a ninth order residual stress distribution was measured successfullyby the crack compliance method using the compliance functionscomputed either analytically or numerically by finite elementmethod. In this paper we present a more severe test. A residualstress distribution was generated in another beam, which had asteep gradient near the mid-plane ofy/t 5 0.5. Both beams weremade of stainless steel withE 5 196 GPs or 28.53 106 psi.Because of different yield stresses in compression and tension, theresidual stress distributions produced by bending is not exactlyanti-symmetric and the peak tensile stress is always a few percenthigher than the peak compressive stress. As a comparison, the twostress distributions estimated analytically from two bending testsdenoted as A and B are shown in Fig. 6. The experimental resultsgiven by Finnie and Cheng (1996) using crack compliance methodfor beam A are also reproduced in the figure. It is seen that a ninthorder stress distribution is sufficient to represent the entire stressdistribution for beam A. For beam B, however, the stress gradient

Fig. 6 Two different residual stresses (data lines) produced by four-points bending used for validation of the crack compliance method.Dotted line—estimated using the compliance functions computed byFEM.


changes much more abruptly near the peak stress and definitelyrequires a higher-order approximation.

The beam (B) was subsequently separated by electric dischargewire machining (EDWM) near the mid-section. Obviously, thestress had been partially released within a region about one thick-ness from the end exposed by cutting.

In the present study one section of the beam was used tomeasure the original residual stress by introducing a cut of pro-gressively increasing depth on a plane about one half (51%)thickness away from the end, as shown in Fig. 7, while the changein strain was recorded on the back face directly opposite the cut.First, the residual stress on the plane of cut (about half thickness

away from the end) was estimated based on approximation of thestress using Legendre polynomial series of orders from 10 to 19.Convergent results were obtained when the order of approximationwas larger than 14. Figure 8 shows the estimated stress distributionrepresented by a seventeenth order polynomial as a dashed line.The estimated peak stress is seen to be about 27% lower than theanalytically predicted original residual stress. Then, the approachpresented in this paper was used to estimate the initial strain filedfrom which the original residual stress distribution was obtainedusing Eq. (3). In this case initial strains were also represented byLegendre polynomial series of orders 10 to 19. Figure 9 shows thestress distributions estimated by polynomial series of orders 15 to19. It is seen that the results stay remarkably stable fory/t . 0.2and converge consistently fory/t , 0.2 with t being the thicknessof the beam. The average of the last three orders (17, 18, and 19)is used as the final result and also shown in Fig. 8 as a solid line.Now the measured stress agrees closely with the predicted residualstress. It is worthy of noting that the measured peak tensile stressis also found to be a few percent higher than the peak compressivestress, which is consistent with the analytically predicted stresses.

In both cases, the crack compliance functions were computed byFEM. In the first case, a fine mesh was required only in the regionnear the cut and a coarser mesh was used in the region away fromthe cut. In the second case, the same fine mesh was used for theentire section of the beam to maintain a uniformly distributedinitial strain field along the length of the beam, which required aconsiderably larger amount of computation. Fortunately, the tre-mendous increase in computing power in the last few years hasmade the computations feasible for most two-dimensional prob-lems. In fact, all of the computation required for the presentmeasurement was carried out on a personal computer.

DiscussionAn approach using initial strains for measurement of residual

axial stresses has been presented. Analytical solutions for the crack

Fig. 7 Configuration of the test for measurement of the initial strains ina section of the beam using the crack compliance method

Fig. 8 Residual stress distributions measured on the plane of cut(dashed line) and in the original beam (solid line) compared with thatpredicted from the bending test

Fig. 9 Results of convergence study using Legendre polynomials se-ries of orders 15–19 to approximate the initial strain field in a beam dueto bending beyond elastic limit


compliance method and the single slice method were obtained forbeams and rods. Experimental validation was carried out by mea-suring the original residual stresses in a beam subjected to four-points bending and subsequently separated in the midsection. Theresidual stress distribution to be measured had a gradient that isconsiderably steeper than any of the residual stresses estimatedpreviously by a single continuous function. With the strains re-corded when a cut of progressively increasing depth was intro-duced on a plane about half thickness away from the end, thepresent approach successfully estimated the original stress distri-bution which agrees well with the analytically predicted stressdistribution.

The steep stress gradient was approximated by a single contin-uous function using Legendre polynomials of different orders.Convergent results were obtained when the order was larger than14. As shown in Fig. 9 the estimated stress distribution staysremarkably stable for normalized distancey/t . 0.2 and con-verges consistently fory/t , 0.2. Thestudy of convergence alsodemonstrates that the approach based on initial strains has virtuallyidentical capability as that based on stresses in representing rapidlyvarying residual stresses for the crack compliance method.

The initial approach is also shown to provide a simplifiedanalysis for the single slice method for the measurement of axi-symmetric residual axial stresses, such as the water-quenched roddescribed by Cheng and Finnie (1998). Using the same experi-mental data, the axial stress distribution in the water-quenched rodwas also obtained by the present approach and was found to agreeclosely with that obtained earlier by the stress-based approach.

In summary, the paper shows that the original residual stress canbe measured even if the stress has been partially released bycutting as long as the permanent deformation introduced by cuttingis negligible and the original stress was uniform over a shortdistance along the length. This is true for both the crack compli-ance method and the single slice method.

AcknowledgmentThis work was supported by the Electric Power Research Insti-

tute. The author is grateful to the Project Manager, Dr. Raj Patha-

nia, for his assistance. The author also thanks Professor EmeritusIain Finnie of University of California, Berkeley, for his supportand the many stimulating discussions that we shared for more thana decade. The cutting of the specimen was performed at PrecisionTechnologies, Livermore, CA and the assistance of Mr. MarkBernhard and Mr. Erbin Burgonio with the experiment is greatlyappreciated.

ReferencesCheng, W., and Finnie, I., 1994, “An Overview of the Crack Compliance Method

for Residual Stress Measurement,” SEM,Proceedings of the Fourth InternationalConference on Residual Stresses,pp. 449–458.

Cheng, W., and Finnie, I., 1998, “The Single-Slice Method for Measurement ofAxisymmetric Residual Stresses in Solid Rods or Hollow Cylinders in the Region ofPlane Strain,” ASME JOURNAL OF ENGINEERINGMATERIALS AND TECHNOLOGY, Vol. 120,pp. 170–176.

Finnie, I., and Cheng, W., 1996, “Residual Stress Measurement by the Introductionof Slots or Cracks,”Localized Damage IV,Nisitani et al., eds. Computation Mechan-ics Publications, pp. 37–51.

Fung, Y. C., 1965,Foundations of Solid Mechanics,Prentice-Hall, EnglewoodCliffs, NJ.

Prime, M. B., 1991, “Experimental Verification of the Crack Compliance Method,”M.S. project report, University of California, Berkeley.

Prime, M. B., 1999, “Residual Stress Measurement by Successive Extension of aSlot: The Crack Compliance Method,”Appl. Mech. Rev.,Vol. 52, pp. 75–96.

Timoshenko, S. P., and Goodier, J. N., 1970,Theory of Elasticity,3rd edition,McGraw-Hill, New York.

Johnson, D., and Johnson, J., 1982,Mathematical Methods in Engineering andPhysics,Prentice-Hall, Englewood Cliffs, NJ.

Mayville, R., and Finnie, I., 1982, “Uniaxial Stress-Strain Curves from a BendingTest,” Experimental Mechanics,Vol. 22, p. 197.

Ueda, Y., Fukuda, K., and Kim, Y. C., 1986, “New Measuring Method of Axi-symmetric Three-Dimensional Residual Stresses Using Inherent Strains as Parame-ters,” ASME JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY, Vol. 108, pp.328–334.

Ueda, Y., and Fukuda, K., 1989, “New Measuring Method of Three-DimensionalResidual Stresses in Long Welded Joints Using Inherent Strains as Parameters—Lz

Method,” ASME JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY, Vol. 111, pp.1–8.

Ueda, Y., Murakawa, H., and Ma, N. X., 1996, “Measuring Method for ResidualStresses in Explosively Clad Plates and a Method of Residual Stress Reduction,”ASME JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY, Vol. 118, pp. 576–582.


M. ChiarelliAssistant Professor.

A. LanciottiAssociate Professor.

Department of Aerospace Engineering,University of Pisa,

Via Diotisalvi 2,56126 Pisa, Italy

M. SacchiEngineer,

Breda Railway Constructions,Via Ciliegiole 110b,51100 Pistoia, Italy

Effect of Plasma Cutting on theFatigue Resistance of Fe510 D1SteelThe paper describes the results of a research programme, carried out at the Departmentof Aerospace Engineering of the University of Pisa, for the assessment of the influence ofplasma cutting on the physical and mechanical properties of Fe510 D1, a low carbon steelwidely used in carpentry. The activity started by observing that several industries reworkplasma cut edges, particularly in the case of fatigue structures, in spite of the good qualityof the plasma cut edges in a fully automatic process. Obviously, reworking is veryexpensive and time-consuming. Comparative fatigue tests demonstrated that the fatigueresistance of plasma cut specimens in Fe510 steel was fully comparable to that of milledspecimens, as the consequence of the beneficial residual stresses which formed in theplasma cut edges.

1 IntroductionA research programme has been conducted at the Department of

Aerospace Engineering of the University of Pisa concerning theeffect of plasma cutting on the fatigue properties of Fe510 D1, asteel typically used for the construction of railway vehicles and incarpentry, thanks to its excellent quality in welding.

Different options exist to profile a sheet or a plate; laser, plasma,oxy-fuel, water-jet and mechanical profiling are those most fre-quently used. Limiting our attention to railway constructions andrailway trucks in particular, they are typically welded structuresbuilt by starting from plates with a thickness in the range of 6 to12 mm. Plasma cutting in this case is cheaper and faster than laseror water-jet cutting, and it provides better edge finish than oxy-fuel. A fundamental requirements in railway constructions is thefatigue resistance of structures. In the case of free edges (edgeswhich are not subsequently welded), in spite of the good quality ofthe plasma cut edges in a fully automatic process, reworking bygrinding or mechanical profiling is often performed. The heataffected zone, the residual stresses, the possible defects and thedecarburized area are eliminated in this way, but obviously re-working is very expensive and time-consuming. In the case ofedges which are subsequently incorporated in a welded joint thisproblem is not likely to exist, as these effects are to be eliminatedduring the subsequent fusion. Notwithstanding this, in many casesmilling is used to prepare the chamfers which are necessary toweld thick plates. Reworking of free and occasionally of weldededges, too, is not an uncommon practice; “where doubts exist” it isperformed in the case of welded bridges (Harris, 1996), eventhough these structures are less critical from the fatigue point ofview.

An exhaustive review of the effects of gas cutting on the fatigueresistance of mild and high tensile structural steels was carried outby Gurney (1979). The main conclusion drawn is that the qualityof the cutting greatly influences the fatigue resistance of steels,which can be as high as that of the parent metal up to one half ofthis value.

In this context, it was interesting to quantify the fatigue resis-tance of plates cut by means of the equipment now on the market,particularly since in recent years the technology of plasma cuttinghas been greatly improved by the introduction of High Definition

Plasma Arc Cutting (HDPAC). This technique uses a small orifice,0.4 4 0.7 mm, instead of 34 4.5 mm for normal plasma cutting.This allows us to obtain close tolerances and square edges, com-parable to the results that can be obtained by laser cutting, but ata significantly lower cost (Kirkpatrick, 1995; Harris and Lowery,1996; Hoult et al., 1995). Therefore, it is possible that in the futurea system of this type will be accepted as a finishing process forfatigue loaded structures.

2 Experimental

2.1 Base Material. Fe510 D1 in standard plate supply has aferritic structure; the chemical composition of this material isgiven in Table 1, together with its mechanical properties (yieldstress,s y, ultimate stress,s u, and elongation to failure A), mea-sured in the longitudinal (L) and in the transversal (T) directions.

Dogbone specinens, 40 mm wide, were machined from plateswith thickness of 8 mm; plates of this thickness are typically usedin the construction of railway trucks. The external surfaces of thespecimens were not machined, so as to maintain, as in real con-structions, the “as-received” condition of the plates. Specimenswere saw cut and milled in the longitudinal direction of the plates;a notch was introduced in some of them, a central hole, 4 mm inthe diameter, to investigate the notch sensitivity of the material.

2.2 Plasma Cut Specimens. A group of specimens wasobtained by cutting them with a numerically controlled plasmacutting machine. The torch was water cooled and had a nozzle withan outlet diameter of 2.5 mm; the plasma gas was oxygen, 0.05m3/s, at a pressure of 10.5 bar. Water was injected in the arc, 36cc/s; the injection of water dilutes the plasma jet and improves thequality of the cut edges. A current setting of 400 amps at 150 voltswas used. The distance between the torch and the plate was 3 mm;the cutting speed was 92 mm/s. The plasma cut specimens werealso obtained in the longitudinal direction of the plates. The plasmacut surfaces did not look as regular as the milled surfaces. Theplasma cut edges were not straight and the width of the plate on thereverse side was about 0.8 mm smaller than that on the torch side,Fig. 1. In addition, the width of the specimens ranged from 37.3 to40.05 mm, while the nominal dimension was 40 mm. Thesedifferences are generally meaningless in large structures, but canbe important in small structures, so that it can be concluded thatclose tolerances cannot be obtained by standard plasma cutting.Besides, small scratches were present on the cut surfaces. Theloads to be applied in the tests on plasma cut specimens wereevaluated by taking into account their actual dimensions.

Contributed by the Materials Division for publication in the JOURNAL OF ENGINEER-ING MATERIALS AND TECHNOLOGY. Manuscript received by the Materials DivisionDecember 11, 1998; revised manuscript received April 6, 1999. Associate TechnicalEditor: A. E. Werner.


Other important aspects concerning plasma arc cutting are localstructural modification of the material and the presence of residualstresses. The base material has a ferritic structure; the rapid coolingafter plasma cutting transforms the ferritic structure into fineperlite or martensite, like a thermal process of quenching. Figure 2shows a transverse section of a plasma cut specimen (detail A inFig. 1); the heat affected zone, which is about 0.5 mm wide on theplasma torch side and about 0.2 mm wide on the rear side, can beclearly seen. It was composed of martensite (see the detail in Fig.2), a harder structure than the basic ferrite. The hardness in the heataffected zone was about 500 Vickers, more than twice the hardnessof the core material, Fig. 3.

As far as residual stresses are concerned, gas cutting alwaysproduces thermal residual stresses: during the cutting process, thearea around the heat source is violently heated and locally fused.The hot material tends to expand but the cooler surrounding areapartially prevents this expansion, producing a plastic deformationof the hot area, also as a consequence of the low yield stress of thehot material. So during the cooling phase, tensile residual stressesare generated in the plastically compressed area. These stresses arebalanced by compressive residual stresses in the inner zones of theplates. At the same time, as a function of the cooling rate, micro-structural transformations can take place in the material; theyproduce changes in the volume and, as a consequence, residualstresses. In particular, if the cooling rate is sufficiently high, thetransformation from ferrite to martensite takes place and thisproduces an increase in volume with consequent compressiveresidual stresses, which add to the thermal tensile residual stresses.Measurements were carried out to establish the final distribution ofthe residual stresses. A plasma cut edge was instrumented withstrain gauges and dissected to allow the relaxation of the internalstresses. A chain of nine strain gauges, located at 1 mm from eachother, was used for this measurement, together with a single straingauge placed on the plasma cut surface. Figure 4 shows theposition of the strain gauges together with the dissection scheme.

The results obtained are shown in Fig. 5. A very thin layer ofmaterial near the plasma cut edge, about 0.2 mm thick, had beensubject to a compressive residual stress due to the micro-structuraltransformation from ferrite to martensite. The maximum compres-sive stress at the external surface was about 150 MPa. This stresswas superimposed on the thermal residual stress field, whichinvolved the deeper layers of the specimen. The presence of acompressive residual stress at the external surface partly explains

Table 1(b) Mechanical properties of Fe510 D1

Fibres s y (MPa) s u (MPa) A (%)

L 455 620 31.0T 405 575 30.5

Table 1(a) Chemical composition of Fe510 D1

C: 0.193%Mn: 1.520%P: 0.015%S: 0.0004%Si: 0.360%Al: 0.026%

Fig. 1 Aspect of the edges after plasma cutting

Fig. 2 Heat affected zone after plasma cutting

Fig. 3 Micro-hardness of the plasma cut heat affect zone


the good fatigue behavior of plasma cut specimens, as will beshown in the following. Note that the presence of a surfacecompressive stress, instead of a thermal tensile residual stress, isnot a general result. It depends on the material, its thickness and onthe cutting conditions (gas, velocity, diameter of the orifice and soon). By changing these parameters a different stress field can beproduced.

2.3 Test Equipment and Programme. Fatigue tests werecarried out at room temperature by means of a servo-hydraulicfatigue machine with a maximum loading capacity of 200 KN.Sinusoidal loading was applied. The load frequency ranged from10 to 18 Hertz as a function of the load applied; the ratio betweenthe minimum and the maximum stresses in the load cycle cyclewasR 5 0. Plasma cut specimens were also tested atR 5 0.5.

Additional tests were also carried out to investigate the notchsensitivity of the material, the effect of thermal relaxation onplasma cut specimens, and the effect of the micro-scratches ob-served on the plasma cut surfaces. For this purpose, artificialmicro-notches were introduced in some specimens.

3 Results and DiscussionFigure 6 shows the results concerning the milled specimens (the

arrow on the right of a symbol indicates a fatigue test suspendedwithout evidence of fatigue damage). The fatigue limit of un-notched specimens was about 370 MPa and shows a good resultcompared with the yield stress of the material,s y 5 455 MPa.

The fatigue limit decreased to 205 MPa in the case of thenotched specimens (central hole, 4 mm in diameter). By takinginto account the value of the stress concentration factor of thesespecimens,Kt 5 2.73 (Peterson, 1974), the notch sensitivityfactor wasq 5 0.46. This value is in agreement with certainexperimental results available in the literature (Juvinall, 1967),which allow the estimation of the notch sensitivity factor of a steelon the basis of the radius of the notch and the yield stress of thematerial.

The results of milled and of the plasma cut specimens arecompared in Fig. 7; as can be observed, they were fully compa-rable. Therefore, it can be concluded that plasma cutting in thismaterial, thickness 8 mm, is not detrimental from the fatigueresistance point of view. This observation is confirmed by the factthat fatigue crack nucleation took place both in the plasma cutedges and in the base material.

Prokopenko (1991) carried out tests on the fatigue behavior ofplasma cut edges. In this case, an increase of 60% in hardness andcomparable fatigue behavior was observed in plasma cut edgeswhen compared with the results of the initial nonhardened edges ofa structural steel.

The experimental activity was continued in order to use the

Fig. 4 Strain gauge map for residual strain evaluation in a plasma cutedge

Fig. 5 Distribution of the longitudinal residual stress in a plasma cutedge

Fig. 6 Results of the fatigue tests carried out on notched and un-notched specimens. Edge preparation: milling.

Fig. 7 Results of the fatigue tests carried out on milled and plasma cutspecimens


results obtained from the practical point of view. Two aspects wereexamined in some detail: the first was related to the possiblerelaxation of the compressive residual stress, the second to thepossible brittle behavior of the martensite layer. With regard to thefirst aspect, complex welded structures were thermal relaxed toeliminate the residual stresses introduced by the welding process;it was important to verify that the fatigue resistance of a plasma cutfree edge included in a welded structure was preserved after thethermal relaxation process. This operation was carried out inaccordance with the following procedure:

● maximum starting temperature: 200°C;● heating gradient: 60°C/hr4 150°C/hr;● reaching maximum temperature of 590°C6 15°C and main-

tenance for 1 hr;● cooling gradient: 120°C/hr;● exit temperature:,200°C;● cooling in calm air.

Observations using an optical microscope showed that the sur-face layer was apparently unchanged after thermal relaxation. Theproperties of the martensite layer were certainly changed after thehigh temperature process, but there were no effects on the fatigueresistance, so that, the fatigue results obtained from this group ofspecimens were fully comparable with the previous results.

A second group of specimens was used for the analysis of thenotch sensitivity of the plasma cut edges. Small scratches arealways present on gas surfaces; the aim of these tests was toinvestigate the effect of a bigger scratch produced during gascutting or accidentally introduced in the structure. Such damagesare not important in ductile materials, but it was important toinvestigate their effects in the brittle martensite layer. Small sharpnotches were introduced in the specimens by means of a smallabrasive disk, thickness 0.25 mm. The notch consisted in a cutalong the whole thickness of the specimens, Fig. 8. Differentdepths of the notch, “a” in Fig. 8, from 0.1 to 0.5 mm measured onthe plasma torch side, were tested. The results show that a notchwith a depth of 0.1 to 0.3 mm did not affect fatigue resistance; thiswas confirmed by the fact that the fatigue nucleation took place insome specimens in a different location from the starting notch. Allthe specimens cracked at the micro-notch location when its depthwas increased to 0.5 mm. The results obtained in this case areshown in Fig. 9. The micro-notch decreased the fatigue stress byabout 5%, a relatively small amount. The conclusion of these testswas that small scratches in a plasma cut edge are not detrimental.

Some tests on plasma cut specimens were carried out at a loadratio R 5 0.5. This value is representative of several railwayapplications in which the fatigue stresses superimposed the impor-tant static stress produced by the weight of the vehicle. Figure 10shows the results of these tests, compared with the results relevantto R 5 0. The scatter in the results was very high in this case andseveral run-outs were also recorded at the higher stress levels. Onthe other hand, by increasing the stress ratio, the maximum stressin the fatigue cycle is almost the yield stress of the material, so anincrease in the scatter is not surprising.

4 ConclusionsA research programme has been conducted at the Department of

Aerospace Engineering of the University of Pisa concerning theeffect of plasma cutting on the fatigue properties of Fe510 D1.

Plasma cutting is widely used in industry but in spite of the goodquality obtained by automatic processes, plasma cut edges incritical fatigue structures are often reworked by grinding or me-chanical profiling.

Comparative fatigue tests were carried out on dogbone speci-mens obtained from plate by milling or plasma cutting. Obviously,the plasma cut surfaces did not look as regular as the milledsurfaces; the edges were not straight and close tolerances aredifficult to obtained. In addition, small scratches were present onthe cut surfaces.

Fig. 8 Specimen with micro-notch

Fig. 9 Comparison between the fatigue results of notched and un-notched plasma cut specimens

Fig. 10 Comparison between the results of the fatigue tests carried outon plasma cut specimens under different stress ratios


The optical microscope showed that the heat-affected zone of aplasma cut edge was composed of martensite, a harder structurethan the basic ferrite. The hardness of this area was more thantwice the hardness of the parent material.

A compressive residual stress of about 150 MPa was measuredat the external surface of a plasma cut edge. This stress is super-imposed on the thermal residual stress field, which involved thedeeper layers of the specimen.

The results of fatigue tests carried out on milled and plasma cutspecimens were fully comparable. So it can be concluded thatplasma cutting in this material, thickness 8 mm, is not detrimentalfrom the fatigue resistance point of view. It is important to notethat the extrapolation of the results obtained to other materials, todifferent thickness or different cutting parameters, is not necessar-ily possible. Therefore, additional tests must be carried out whenone or more parameters are changed.

Additional fatigue tests confirmed the good fatigue resistance ofplasma cut specimens after thermal relaxation even in the presence

of small artificial defects in the hard layer produced by plasmacutting.

ReferencesGurney, T. R., 1979,Fatigue of Welded Structures,Second Edition, Cambridge

University Press.Harris, I. D., 1996, “Plasma Arc Cutting of Bridge Steels,,”Proceedings of the

International Conference on Welded Sturctures in Particular Welded Bridges,Budap-est 2–3 September, pp. 363–364.

Harris, I. D., and Lowery, J., 1996, “High Tolerance Plasma Arc Cutting,”Weldingin the World,Vol. 37, No. 6, pp. 283–287.

Hoult, A. P., Pashby, I. R., and Chan, K., 1995, “Fine Plasma Cutting of AdvancedAerospace Materials,”Journal of Materials Processing Technology,Vol. 48, Jan., pp.825–831.

Juvinall, R. C., 1967,Stress, Strain, and Strength,McGraw-Hill, New York.Kirkpatrick, I., 1995, “High Definition Plasma—An Alternative to Laser Technol-

ogy,” Sheet Metal Industries,Vol. 72, pp. 18–19, Sept.Peterson, R. E., 1974,Stress Concentration Factors,John Wiley, New York.Prokopenko, A. V., et al., 1991, “Effect of Plasma Cutting on the Fatigue and

Cracking Resistance of Steel,”Strength of Materials(English translation of ProblemyProchnosti), Vol. 22, No. 12; Aug., pp. 1738–1740.


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