ARL-STIRUC-R-410 eQ$~ZI41A.AP-003-961

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I.n DEPARTMENT OF. DEFENCE

DEFENCE SCIENCE AND TECHNOLOGY ORGANISATIONS,,AERONAUTICAL RESEARCH LABORATORIES

MELBOURNE, VICTORIA

STRUCTURES REPORT 410

MULTIAXIAL FATIGUE AND FRACTURE-.A LITERATURE REVIEW

ji by

P. W. BEAVER

TECWJr,,At. INFQMATION StRlVIC• ie'-' ,:I. . ,

13 toREPRODUtCi ANQ 5S.LU. THIS AtPOA?

PwLt AP O RE PUBLIC M 06

C (cOMMONWIMeITh O AUtIA, 1984

-JU LY 1914

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AR-003-961

DEPARTMENT OF DEFENCEDEFENCE SCIENCE AND TECHNOLOGY ORGANISATION

AERONAUTICAL RESEARCH LABORATORIES

STRUCTURES REPORT 410

MULTIAXIAL FATIGUE AND FRACTURE-

A LITERATURE'REVIEW

by

P. W. BEAVE•R

SUMMARY.

Problems often arise when attempting to determiae thefiatigue behaviour of stritacuralcomponents from laboratory data.. A major reason for thib lack of correlation is that mostengineering components operate in stress environments sgnificantlv more complicatedthan uniaxial tension, the stress state in which most research.studies are conducted.

A review of the literature has shown that virtually all fatigue ant fracture propertiesof metals and'components are ajfiicted bY multidirectional loading. In particular. variationsin stress state compared with uniaxial tension prodwce the following effiects:

$a) decreases in the faiigue limit by up to approximatev SO";,,-Cb) increases or decreases in low-cycle fitigue lifi by fizctors o!f tip to 20, depending

on whether the stress in the second direction is tensile; or compressive and i.sstatic or. cyclic;

') out-ovf-phase'loading also reduces the low-,ycle fatigue file by a factor of up to 4compared with in-phase loading: and

td) accelerations anti retardations o]fmatigue crack growth ra&,. by.factors of13 to 4depending on the nature of/the transverse .twess.

It is al.io evidett that multiaxial critmria used jar design ptrpvses cyn be non-con-servative, especially under out-of-phase toading, and cvn.rquettly twm lead to unsa.ie esti-,

'mates o, the f.kigue lt, ofa wtonet.

(:4j

© 'COMMONWEALTH OF AUSTRALIA 149

POSTAL ADDRESS: Director, Aeonauticai R.oamih Labomrtocs,'Box4331. P.O.. Mdboumr, Vi•otra, 3M01, Awtsralia

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CONTENTS

Page No.

I. INTRODUCTION I

2. DEFINITIONS AND NOMIENCLATURE 2'

2.1 Definitions 2

2.2 Nomenclature 3

3. MULTIAXIAL FATIGUE, TESTING TECHNIQUES 6

4.'THE EFFECTS OF MULTIAXIAL STRESS ON FATIGUE STRENGTH 9

5. MULTIAXIAL FATIGUE CRITERIA 13

6. THE EFFECTS OF BIAXIAL STRESS ON FRACTURE PROPERTIES 14

OF CRACKED COMPONENTS

6.1 Laboratory Specimens 14

6.2 Structural Components 20

7. MULTIAXIAL LOW-CYCLE FATIGUE

7.1 Cyclic Stress-Strain Data 22

7.2 Fatigue Life Relptionships 26

8. THE EFFECTS' OF BIAXIAL STRESS ON FATIGUE CRACK GROWTliRATES 28

8.1 Laboratory Specimens 28

8.2 Full-Scale Stiffened Panels Containing Cracks 33

9. THE EFFECTS OF OUT-OF-PHASE BIMAXIAL LOADING ON FATIGUE-

PROPERTIES 3t

10. THE EFFECTS OF BIAXIAL STRESS ON CRACK TIP PLASTICITY .44

11. THE EFFECTS OF BIAXIAL STRESS IN THE PRESENCE OF A NOTCH 48

12. MICROSTRUCTURAL ASPECTS OF MULTIAXiAL FATIGUJE 51

13. SUMMARY AND CONCLUSIONS 53

REFERENCES

BIBLIOGRAPHY

DISTRIBUTION LIST

DOCUMENT CONTROL DATA

4Pt0

1. INTRODUCTiGN

During the ,ast few decades a considerable amount of data concernir,• both the fatigue andfracture behaviour of commercial materials has been generatcdL Wokf'er, problems oftenarise when attempting to predict the fatigue behaviour of mechanical components and structuralelements from labcratory data and the failure of engineering structures and components byfatigue still remains as a major problem.

Several factors are responsible for the difficulties encountered in correlating laboratoryfatigilz data with the behaviour of structural components. Many engineering componentsoperate in stress environments significantly more complicated than uniaxial tension, the stressstate in which most research studies are conducted. For example, biaxial stress states exist in theleading edges 'of gas turbine blades, in pressure vessels and in the skins of aircraft wings, whilstcompopents, such as crank-shafts or propeller shafts are subjected to simultaneous; and usuallyout-of-phase, bending and torsion. In addition, most research investigations are performed usingspecimens of relatively simple geometry. In contrast, geometric discontinuities and notchesare present in engineering components, such as joints in aircraft wings or cooling holes in g.;.,turbine blades, which introduce localized regions experiencing both high strains and complexstress states. Therefore, an area of research which still requircs considerable ataention is thefatigue behaviour of ffiaterials under complex stress statLs, especially in the low-cycle, high-strainregion, and in the presence of geometric discontinuities.

At present the amount of multiaxial fatigue research of metals has been v'ery limited com-pared with that undertaken for uniaxial fatigue. This situation has arisen for the following reasons:

(1) 'There has been a lack of reliable multiaxial or biaxial testing machines as the design.development and experimental work associated with building such equipment tends tobe complicated, time consuming and expensive compared with that for uniaxial testingsystems,[ 1, 2].

(2) Intricate specimen designs are required, hence the specimens also are expensive [3].

(3)' No arcurate analytical methods for determining the stress distributions in the varioustest specimens are available [4].

(4) There is no substantial theoretical framework for analysing the data {51.

(5j The experimental work is time consuming.

(6) A large number of test resultsare required to develop and prove or disprove empiricalcriteria f31.

Momi .Multiaxial fatigue research condacted on metals has been under biaxial loading asfatigue cracks generally originate at a free surface where btaxial stress states exist. Compared wit hthat for metals, multiaxial fatigue data for libre reinforced composites are ever. more scarce.as several unusual additional problems are associated with multiaxial fatigue testing of the,.ematerials (6).

Several review papers surveying particular aspects of the multiaxial fatigue nt metals arepresently available f3, 7-141. A comprehensive review up to 1974 of the phenomenologicaIafipects of multiaxial fatigue in the low-cycle range (< 10i cycles) was undertaken by Krempl 17].

|. ,-

McDiarmid L and Parsons and Pascoe [9] have reviewed multiaxial high~cycle fatigue whilstBrown and Miller (101 have reviewed both low-cycle and high-cycle multiayial fatigue. Morerecently Brown and Miller (I 1-13] have reviewed several aspects of multiaxial fatigue, such asthe effects of stress state on the cyclic deformation behaviour and mode of crack growth in metals.together with the evolution of the various multiaxial low-cycle fatigue criteria [3]. Garud [ 14)has reviewed multiaxia, fatigue with an emphasis on the criteria for assessing the fatigue strengthof metals in multiaxial stress states at room temperature. Additional references are givern at theend of this review concerning other aspects of multiaxial fatigue, such as cumulative fatiguedamage, notch effects, cyclic deformation behaviour and creep-fatigue interactions.

To the best of the author's knowledge there is no review paper presently available regardingthe multiaxial fatigue behaviour of fibre reinforced composites, This topic will be reviewer" in alater paper.

At present there is considerabie lack of agreement in the literature regarding the apparenteffects of mul.tiaxial stress states ,on fatigue properties and especially on fatigue crack growthrates. For example. experimental results have shown that applying a tensile stress parallel to acrack and perpendicular to the principal stress can increase, decrease, or have no effect on thefatigue crack growth rate. Several factors are responsible for the inconsistency in the literatureand their effects can be sub-divided into the following groups [15]:

(1) Material effecs resulting from differences in the tpe of material or alloy tested and itsmechanical working atid heat treatment'process.

(2) Environmental effects, such as the.interaction between a growing crack and its chemicalenvironment.

-13) Geometrical effects arising from the use of a variety of test methods and/or sp.,:imenand crack geometries.

(4) Load state effects such as whether the transverse loads parallel to a crack are static orcyclic, tensile or compressive and in- or out-of-phase.

Other contributing factors, which have been mentioned previously, are (1) the lack of methodsfor accurately analysing the stress distributions in the test specimens. and (2) the absence of asatisfactory theoretical basis for interpreting the test data.

The objective of this report is to systematically review published data concerning the dfl'ectsof multiaxial stress on fatigue and fracture properties, and thereby clarify many of the conflictingresults in the literature. Emphasis is placed on the effects of biaxial loads on cyclic properties.deformation mechanisms and crack growth behaviour, as the various multiaxial fgntigu,," criteriahave been adequately reviewed elsewhere [3, 7-10. 141.

2. DEFINITIONS AND NOMENCLATIRC,

2.1. Definitions

* The stress states associated with the various biaxiai testing techniques, which %ill be-des-cribed in the next section, have been detir.d using 'a variety of parameters. For example, thebiaxial stress states in'cruciform type sp,,cimer,;, such as shown iii Figure 4la). may be ex."ressedin terms of either.

(I) the yatio of tho NOMINAL strgsses remote from the test area S,,/Sy -- A; or

(2) the ratio of the LOCAL stresses within the -test area ./u,, k.

" ." • . . - . . , . - . , , . . .. ",. .

B,,zh biaxial stress ratios have values of beteenn -I for pure shear and + I for equibiaxialtension with a value of O for uniaxial tension. Similarly. the multiaxial strain states encountered instrain controlled, low-cycle fatigue tests may be expressed in terms of the ratio 3f th, principalstrains in the test section, -.Ec. = ,. This pprameter also has Nalues between ---I (pure'shear)and + I (equibiaxial tension). However, the biaxial strain ratio has a valve of -v in uniaxialtersion, whese P is Poisson's ratio. In comparison, a parameter frequently used for combinedtension-torsion testsis the ratio of the torsional to axial strain rangey .Ay .Ac P. The ianee ofvalues for this parameter is between 0 (uniaxial tensihn) and -c (pure shear).

No single parameter, including those above, can completely specify thz- loading system asthe following factors must also be considered:

(a) The phase difference between the ti.o loading directions as the ratio of wx.a• will varywith time if a phase difference exists.

N(b) The mean loads of the cyclic loading in each of the twe directions.

'(c) Whether the loading in one of the directions is static tensile or compressive.

Therefore. two or more parameters are required to completely specify the loading system.An additional factor which must be considered in specifying the stress- state f&r biaxial

specimens cintaining cracks is the direction of the maximum principal stress with respect tothe plane of the crack.. In general, the biaxial stress ratio (Y,,!. is used to~define the stress statein cra.cked specimens. However. values of greater than I are frequently reported in the literature.By convention the loading axis perpendicular to the crack is defined as the v-direction and thatparallel to the crack as the x-direction. Consequently, v hen the biaxial stress ratio is less than I,the maximum principal stress is perpendicular to the plane of the crack. In comparison, themaximum principal stress is parallel to the'crack when the biaxial stress ratio is greater than I.Therefore, these two situations represent different ccackiloading configurations.

Many authors have used the same symbol to represent difTerent parameters. For example,somne authors hase used the y)mbol A to represent the biaxial stress ratio axioa, whereas othershave used this symbol to represent thý ratio of torsional to axial strain Ay./A. The varioussymbols used to represent such parameters in this report are as sho%% n in the following section.

Most of the biaxial fatigue test examined in this report have been used completely reversed.in-phase loading cycles. Therefore, unless stated otherwise, the cyclic loads in the x- and .-'directions can be asstimed to be in-phase and acting about a /cro mean stress.

2.2 Nomenclature

P", P. NOMINAL loads remote from the test area

Sx, SY NOMINAL stresses remote from the test area'

Ox, av LOCAL stresses within the test area

*1 -12 "principal surfa,,e strains

AP an;ial load range

A7 torsional stress range

,ACV axial local stress range

Ay torsional strain range

3

axial strain range

,A NOMINAL biaxial stress ratio (SjiSy)

k LOCAL biaxial stress ratio ..

17 ratio of torsional to axial stress range (.Ar!Aa)

ratio of torsional to axial strain range (Ay/Ai)

biaxial Strain ratio (AEi/Ati)

t fatigue strength in torsion at 107 cycl,

b fatigue strength in bending at 107cycles

St torsional stress

Sb bending stress

Smaxi Smin maximum and minimum ,'OM.INA.L streis in a cycle

R stress ratio Smin/Smr-

h, a2, O3 principal stresses (a, > a2 > a3)

ft, f2, f3 principal strains (4E > f.2 > f3)

tort ociahedral shear stress

Yoet octahedral shear strain

rMax maximurn. shear stress

Ymax maximum shear strain

fn strain normal to the plane of maximum shear

r definition of the plot of 's.• ":' ) v.,j .'t )

g, h and j life dependent constants'

K stress intensity factor

K,. critical stress intensity factor

* /the i-integral

G strain energy release rate

COCD crack opening displacem,:nt

modulus of elasticity in tension

Poisson's ratio

n cyclic hardening exponent'

k' cyclic strehgth co-efficient

A~ei2 ,elastic strain amplitude

plastic strain amplitude

A\,ri2 total strain amplitude

Nr number of'cycles to failure

•, A Manson-Coflin exponent and int.;rcept respectisel,

/, B Basquin exponent and intercept respectively

X, Y constants

0 phase angle between the torsional and axial strain

a half crack length

dafdN crack growth rate

rp plastic zone size

SKt theoretical stress concentration factor

.Kt fatigue strength reuuction factor

Subscripts

x, y cartesian co-ordinates

1, 2. 3 principal values

1, 1I, Iil mod, of crack surface displacement

• " - c , critical

oct octahedral

Conversion Units*

In. - 25.44mm

I mu -m 25.4 Am

Slbr 4-4448 N

* NOTE: All diagrams are rcrproduced from 1he original source and contain the units in use at

that time.

S'

I kip - 4"448 x 103 N

I ps.' 0,895 Nm- 2

I tonf in.- 2 ý 15'4 MNm 2

I kpsi\ in. = 1.099 MNm-

.3. MULTIAXIAL FATIGUE TESTING TECHNIQUES

A wide variety of techniques have been used tj fatigue test laboratory specimens undermultiaxial stress states. These tcchniques include:

(I) rerirsed bending of wide cantilever specimens. Varying the -,vidth-to-thickness ratioproduces biaxial stress ratios of v _> i > 0, whete P is Poisson's rUtio:

(2) anticlatic bending of square a'nd rhombic.shaped plates. The range of biaxial stressratios is 0 A > -- I depending on the ratio of the lengths of the major and minor axes.

(3) bulge testing where round or o~al-shaped plates are clamped around the periphery andsubjected to fluctuating pressure on one or both sides. An cquihiaxial stress state(A + I) is produced in the central area of the round plates %shereas the osv:l-shapedplates produce biaxial stress ratios of + I > A >.

(4) cruci/orm specimen technique Mhere in-plane orthogonal loads are applied to cross-shaped plate specimens. The complete range of biaxial stress ratios. i.e.. I. >_ A ? Ican be 6btained vith this technique: and

(5) thin-walled tubes subjected to axial tension- compression plus (a) torsion kmrding s herethe range of stress states available im limited io 0 >_ A - I. or (hj internal -eternalpressure which increases the stress states ýIsailablc to I I ,> i.

Evans 1161, Krempl [71, Lohr 1. 21 and Brown 113, ;71 hase reviewed the abose tecth iques anddiscussed the respective adviantages and disadsantages These rc :-%s sho% that the tsk met'hodsmost frequently used are:

(1) the cruciform specimen technique: and

(2) combined tension- torsion of thin-ksalled tuhular spccimens.

Examples of the types of specimens used are shown in Figures Illa and 1l0b. The 'ruciforms.pecimen technique has several advantages which make it suitable for biaxial fsttg.e studies.nomely:

(I) A full range of biaxial %tress' and str•:in ratio:s can be obtained using a singic specimengeomety.

(.2) The test section area is easily observed during a teft.

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dimensions in millimetres.

FIG. 1.' EXAMPLES OF BIAXIAL FATIGUE SPECIMENS

I.a) ACRUCIFORM SHAPED SPECIMEN 1681

*b) A THIN-WALLED TUBULAR SPECIMEN [1 2

(3) The specimen shape is'suitable for microscopy.

(4) The mean load level and amplitude on each axis can be controlled independently.

(5) It isthe most ,ersatile biaxial fatigue ,ystem available for elevated temperature work.

(6) It is-most saitable for fatigue crack gro~kth studies.

(7) Low-cycle fatigue studies are possible.

(8) The test system is suitable for component testing.

However, there are several disadvantage-, associated •ith this technique: these are:

(I) The thickness of the test section must be reduced to ensure that crack initiation andfailure occur within that region. This means that the stresses in this region may not beaccurately etimated from the load cell readings as an-unknown proportion of eachload is carried by the thick outer section [16, 17]. Local stress"s and strains must bedetermined using finite element analyses or strain gauge measurement&.

(2) The best profile for strain uniformity in the test section is a flat-bottomed profile.However. this profile is susceptible to buckling unuer cornpressive loads. This problemcan be overcome by using a continually radiused test section although tCis •kould resultin a deciease in the volume of uniform strain distributionm Therefore. the s-pecimendesign must be a compromise. preventing buckling yet producing an, adequate vblumeof uniform strain.

(3) Once yielding has occurred the stress distribution in the test section cannot I- calcul-ated using resistance gauges.

(4) The load frame and actuator displacemqnts limit the range of specirien sizes.

Simi!arly. the advantages and disadvantages of the tension-torsion technique asing thin-walled specimens can he summaried as follows

A dvantages

(I) The loads in the test section can he easily determined as the applied loads are fullysupported in this region. Similarly, the strain can be easily measured using a biaxialextersometer.

(2) Accurate calculation of plastic strain is not difficult.

(3) The stresses and strains can be continuou:ily monitored thus simplifying the geInerationof hysteresis loops.

(4) High-strain low~cyclc fatigue data are readily obtainable.

(5) The system can he used at elevated temperatures.

I8

(6) This technique is suitable for understanding the fundamental behaxiour of miterials

in cmplex stress states as both strcsses and strains (elastic and plastic) are, in principle,meaiurable quantities and the ratio of shear tensile e.. can be controlled. Thislatter point is important as shear predominates in tl i ,,.i•ation and fracture pro-cesses under cyclic stresses.

Disadvantages

(I) A limited range of biaxial stress and strain ratios is available.

(2) Strain gradients through the wall of the specimen can extend the fatigue life. Thiseffect can be overcome by reducing the tube-ma'll thicknesses, but buckling ma', limitthe extent to which this can be done and thus again, the specimen design must bebased on an acceptable compromise.

Increasing the range of stress ratios possible by using pressurisation instead oftorsion introdutces additional problems. namely.

(a) Through-thickness cracks would limit pressurisation'being used for biaxial fatiguecrack growth studies.

(b) The environmental interactions of the liquid used for pressurisation maý influencethe initiation and growth of cracks.

(c) The "'hydrowedge effect" may be encountered, i.e. where high pressure' liquid enterscracks and wedges them open. This effeci provides an additional Mode I openingstress and alters the crack closure characteristics.

(d) High temperature work may not be possible because of the limited temperaturerange available with such liquids as mineral oils.

Effects (b) and (c) may be minimised by used a rubber sleeve to eliminatecontact between the oil and the specimen surface [17]. However, suitable sleevingmaterial may not be available for high temperature work.

4. THE EFFECTS OF NIULTIAXIAL STRESS ON FATIGUE ANDTENSILE STRENGTH

The effects of combined stress on high-cycle fatigue properties. st ch as fatigue strength or

fatigue limit have been studied since the turn of the century. Consequent,4y, much information

is available for laboratory specimens subjected to nominally elastic bulk stresses and w•ith fatiguelives in excess of 104 cycles. Brown and Miller 13. 101, and Garud ([14] have reviewed the mostimportant papers published on this subject. However, since infinite life is no longe a widelyused criterion for fatigue design (especially in structural components) a brief discussion onlywill be given.. Furthermore. most of the work on the effects of multiaxiality on f;itigue, strengthwas published over 20 years ago.

Most of the experimental data were obtained fr'om'combined bending arid torsion tesis.For example, Gough er al [18, 20] determined the fatigue limits at 107 cycles Jzero mean stress)for numerous ferrous alloys under alternating bending, alternating torsioii and five differentcombinations of in-phase alternating bending and torsion. Three typical sets of results are shownin Figure 2. In addition, as shown in Table 1, Forrest [21] compiled similar data from a numbetof investigators. This compilation shows that, I) the torsional fatigue strength (t) is less than thebending fatigue strength (b) and, 2) the ratio of the fatigue strength in torsion to that in bending(t/b) various markedly from one material to another. Forrest also claimed (hat the t/b ratiobears no relationship to the tensile strength of the metal.

T +

Saim-cOF Direct tr~ess due. bo bendrx" tos/m

FIG. 2. FATIGUE LIMITS OF THREE STEELS UNDER COMBINED BENDINGAND TORSION4AL ALTERNATING STRESSES f 221

'10

The large variation in the ratio of 1ib indicated that none of the traditional failure criteriacould be relied upon for predicting the fatigue strengths under other combined stress conditions,.This fact (which .',ll be discussed in the next section) was realised by Gough %ho derived twoempirical equations for this purpose, based on the experimental %alues of 1,b. Ductile metalscould be represented by an equation of the form,

TABLE I

Ratio of Fatigue Strength in Torsion (t).to that in Bending (b) for Various Materials as Compiledby Forrest [21]

Material Range of Ratio No. of Results A,ýera-e Valueit/b Considered of tib

Wrought steels 0-52-0.69 31 0.60Wrought aluminium alloys 0.43-0-74 13 0-55Wrought copper and copper alloys 0-41-0.67 7 0.56Wrought magne'sium alloys 0-49-0.60 2 0.54Titanium 0-37-0-57 3 048LCat iron 0.79-1-01 9 0-90Cast al,:,ninium and magnesium

allcys Q.71-0.91 5 0.85

Is1A2(s}) + (Sb) (I

and brittle metals by an equation of the form,

+ (N,,)2(+1~ ) + ("')(2. 1) (2)

where St and St, are the torsional and bending stresses, anJ t and b are the torsional and bendingfati'gue strengths respective;,' at lO1 cycles. Equation (1) is known as the Ellipse Quadranit relation-"ship as the data, when plotted, form part of an ellipse. Similarly, equation (2) is known a. theEllipse Arc relationship. More accurate predictions of fatigue strength under combined stressstates and for a wider range of miaterials were, possible with these equations.

The effects of both triaxial and biaxial stress states on the static' and high-cycle fatiguestrengths of a Cr-V ahd a low C-Otcel (using test specimens of the same geometry) have heenestablished in a recent paper by Habetincek [221. He found that thý ultimate tensile strengths

Sof these steels increased by up to -8% as the nsimber of loading axes increased from.one tothree, Table 2. In contrast. the fatigue strengths of the triaxial specimens were higher than inuniaxial tension by up to 48 % compared with equibiaxial or equitriaxial tension loading, Table .1.The fatigue strengths of the hitter two stress states vcrc approximately the same. Habetineckconcluded that dhe fatigue strength of.the srLecimens which he used was determined by tfiestrength of the surface layer. This was in a state of biaxial stress in both biaxially and triaxiallyloaded specimens. Similarly, other workers 123-251 have concluded that the intermediate principal

* **stress has no effect on the fatigue strengths of various metals.

-.. I

TABLE 2

The Effect of Stress State on Ultimate Tensile Strength

The values given in this table are the average of three tests [221

Material Stress State Ultimate TenilcStrengiL' '!PMal

Chromium steel (CSN 15230)(a) Heat treated and tempered uniaxial 980-6

equibiaxial 1047-3equitriaxial I I57- I

(b) Soft annealed uniaxial 64X2equibiaxial' 650-2equitriaxial. 674-6

Low carbon steel uniaxial 519-7(SCN 12010) equibiaxial 5394Normalized equitriaxial 558 .9

IAIiLE 3

The Effect of State of Stress on Fatigue limit

Tests were conduicted using:

(1) A Symmetric Cycle. Around a Zero Mean Load (R i) and

-(2) A Zero to Tension C'ycle (R 0) [22].

Fatigue L,imit> (NMPa)Material Stress State (at I07 cNclCS).

R I. R t)

Chromium steel (CSN 15230)(a) Heat treated 'and. tempered uniaxiad 75-5 144- I'

-equibiaxial 49. 0- 2cquitriaxial 42. I,5.3

(b) Sotf annealed uniaxial 70-6 137-2equibiaxial 46"o 145"3cquitriaxial 40-2 71 -•

Low catbon steel uniaxial 63.7 112"7(SCN 12010) Lquibiaxial 40.2 70.i-Normalized equitriaxial 39 -2

"*NOTE: Here. the fatigue limit if defined using the maximum stress S., and not thc stressamplitude S3.

12-

" i

5. MULTIAXIAL FATIGUE CRITERIA

The ability to predict the fatigue behaviour of structural components is an important factorin the safe design and reliable operation of many engineering systems. Since most componentsin service are subjected to complex stress states, it is necessary to consider multiaxial fatiguebehaviour when predicting the safe life or structural integrity of any component. Hokscxcr.these predictions are usually based on simple laboratory tests under uniaxial loading and util/ingmultiaxial fatigue criteria. The purpos, of these criteria is to reduce complex multiaxial stressstates to an equiva!ent uniaxial stress state.

At present more than 20 multiaxia! fatigue criteria exist [261. The theoretical -.ad experi-mental bases for many of these criteria have been reviewed recently by Broksn and Miller 1101.Krempl [71 and Garud [14). These reviews show that the early multiaxial fatigue criteria %%erebased on either static yield theories for ductile materials, or fracture criteria for brittle or notched"materials. For example, two yield theories which hae been used %kith some success for fatiguestrength correlations in the high-cycle fatigue regime are

(I) Tresca's maximum shear stress criterion.

.7. ax =constant (at a3)!i2 (3)

and (2) Von Nlises octahedral shear stress criterion, which is equixalent to a critical distor-tional strain energy criterion,

r,: constar.n . \, [(Ot -- a" + (o2 - aa)l + ("Y3--- ,i )Z1 (4)

An example of a fracture criterion is. Rankine's critical tensile stress.

at constant. (51

which was initially proposed for the fracture of brittle materials. These stress based criteria arcmore suitable for high-cycle fatigue as the stresses can be easily calculated-using elastic analyses.

* In conttrast, strain based criteria, such as the octahedral shear strain theory,

yot constant R" I C•[ . ' + (t2 . e3)" -- 4 (C3 - ei) 2 (6)

are more suitable in the low-cycle fatigue (LCF) regime, particularly since strain controlledcycling is often encountered in laboratory and 1,ractical lLCF situations.

Unfortunately, predictions using these classical theories have sometimes provmn unsatis-factory [7, 10, 12-141.' For example, Brown and Miller 1101 concluded that the octahedral shearstrain theory can be non-conservative when used to predict LCF life. Furthermore, these authorsconcluded that it is not possible, by using the classical theories, to correlate all multiaYial fatiguedata as: (I). they cannot separate out. the ý.i~o possible cracking systems, i.e. one along thesurface (Case A) and on- away from the surface (Case B) as in Figure 3, and 12) they cannotaccount-To-r tle eflects of the orientation ofthe three-dimensional strain 'ield with resptct tothe surface 111-131.

The lack of correlation between the classical theories anid expertmn: ! data has resultedin more complex theories being developed. For example. Brown and Miller 1101 proposed anew theory based on a physical interpretation of the mechanisms of 'plastic deformatiom andfatigue crack growth. They suggested that fatigue fife is controlled by the maximum. shear strain.

"",7)

4 and the strain normal to the plane of maximum shear.

( 1i'+ )

'13

Th~ese two strains govern thle direction of crack groatth, the faitigue Fife. i.e- rate (if crack groý%th,and help to differentiate bet%% cent the tso possible cracking sy stems. as described in the follo%%iinparagraph.

The theory otf Brown and Miller can be represented graphically. bv comn~oup. of constantfatigue life called -1' plots" wshich can he expressed mathemnatically as.

These I' plots consis! of t%%o separate cur~cs ýLor any g~iven faitigue life. depen~ding on Michthecr

the cracking system is Case A o r Case B as ,lhoý%n. for example. in Figure 4. Specific forms ofequation (9) were subsequently developed bN, Brosn and Miller (3. 13. 271. TlIccquafionl Msichdescribes Case A crackina is of the form

YIg I t\ lIL

where the values of th~e constants g. hi andji are lilf dependent. although useful approximationsfor i are 2 for ductile materials and I tor brittle materials' These approximation-i also correspondto the ductility effect observed in. high-Ccle faitigue byr Gough. sincec the ellipse quadrant andellipse ;trc, equations tI) and 121. mia% be transformed into straip to give the quadratic and linearforms -)f equation ( 10) respccti~ clv '1271. Vor Case B crackine. a rresca mavriumm shear srsty pe criterion vsas proposed [281

* *. consant. I

Compared \% iih the classifical theories these equations give better correlations- %%itl fiatiticu ifc'and crack grow~th rates under multiaxial conditions, and Browin 1131 advocates that theyý shouldbe used in design procedures. Ho\8evcr. thiesc nctk criteria are more comple\ -Ind req uirc addi-t,onal test information, and therefore may be more dillicult to implement. 'ur~rtnL design proec-dures are oftun based on the octahedral %hear stress or %irtiin theo~ry in %pile ofthe laick of airce-ment with -experimental results 1.29. 30[.

6. THIE EFFECTS OF BIAXIIAL STRIESS ON FRACTURE PR0VERTlFtOV CRA(Ktl)COMPONENTS

6.1 Laboratory Specimens'

Fracture mechanics Concept% can be used for predicting the fatigue lifl of craicked structuralcomponents tinder complex loading con~fletions. Consequently. seveiral theoretical and experi-mental investigations into; the effects of' biaxial stressesý on fractitre ps-operties have been~ under-taken [31-441. Hoxtever. some of tl.CSC investigations hame protluc~d contifictillg result% for thefollowing reasons

(1) An implication of linear elastic fracture mechanics (LI IkVlj is tliat fracture pxropertiesare not affected by stresse#s acting parallel,.to a'crick. For example. Uiebovwit andco-workers (31 331 havec performed linear elastic limiic elenmient analy-%es 0:1-A*%I oil

-lifte cent re-cracked specimens subjected to symmertial biaxial stresses tindeir Mode I,.ding. Their calculations shossed that thle stre%% intensity f'acto~r K_. the strain energy'

Jdease rate G and the i-inttegral are not affected by a. stress applied paraifel ito a crack.Similarly,. Miller aInd Kfouri (341 and Uiii and co-wotrkers fP5. 361 anadkscd a ccqltfc-cracked specimen in the elastic range. Ilicy found that tile stress iiitensm t'ahctor andth,: crack opening displacement WOD1) are indepiendent ot' the bixiau %t srcss ratio 4s.

14

case A4t

C vt

case9a, ,) c)

FIG.3. FATIGUE CRACK GROWTh{ SYSTEMS OBSERVED IN MULTIAXIAL STRAINFIELDS SHOWING TriE MAJOR CRACK GROWTH DIRECTIONAND CHARACTERISTIC CRACK SHAPE [131.

• -02

1.~000 case

0 0 0.2 0 0,5 06

FIG. 4. A TYPICAL r PLOT [101

15'

(2) The effects of biaxial stress on fatigue and fracture propertie-s, arc erv dependent onspecimen geometry. and the location and oricntation of the crack oith reipect to theloadings axes.

* (3) Crack tip plasticity, which-is stress state dependent, can drastically modili, the localstresses and displacements and thereby influence the critical load at ft'rcture_ T his

* effect cannot be predicted using LEFM.

EBen though some results are inconsistent. the limited experimental data 4hoi" that biaxialloads affect fracture properties. such as the critical load at fracture. fracture toughniess K. J-inte-

* gral. strain energy release rate G. the dir;:ction of crack extension and the ratte of crack arosstli.Various finite elemenit analyses hase been undertaken to determine the dependence of

* fracture properties on the biaxial stress rat'O. These analyses can he ditided into itwo Leneralgroups (I1) linear elastic FEA~s and (2) non-linear e last ic- plastic FF.A-. Liebomiti. Lee and Lt'tis13f] have shown using a linear cla~itic FEA, that thle elastic stresses art! di';-,I-"ý,i".-- in the cracktip region cannot he adequately represented lis the so-called sielrsolution". In fracturemechanics analyses these local stresses and dis.pl~acements are approximated by6 the first termof a series expansion of the ftinctions repres.rntinge them. Hfis. Subramortian and Licho~mit/(37. 381 'Showed that this approximation, wshich appears to he reasonable (in faice ;-alue. is tin-acceptable bec-ause the effects of loads applied parallel ito the crack appear entirely, in the secondierm of the series. lnL~umon of thle lon-sinL'tlar stres% terms enable-. proper A~aluation oif themaximum shear stresses. crack tip strain energ% density and crack extension direction. All: of'these parameters sho\A a biaxial load dependence t.sen in the elastic range [31. 12. 391.

Liebowitz and co-workers [31 331 also considered a noni-lincar cfastic-plastic' aal\sis of' afinite-width centre-cracked specimen subijected it) tinifornm Mode I biaitial loading.. Fihe foundthat th. strain energý release rate 6(i, the J-integral.,the stres% intensity factor h. and dthe strainintensity' factor (after fHilton and Il'utchinson 1401)1 ire all dependent oni the firaxal stress ratioA. The major part of this bliaxiai load dcpendcnc:\ comeis froni) the inelastic maiterfal hehasl iour

* at the crack tip.Mviller and Kfouri 134) and 1-1ilton ;41I) used elastic-pl, -tic anal%%c% to stdy iaxial load

effects for normal stresses beyond the smnall scale \ielduine range. Result% tit thie anal\ssN, b% Millerand KfQuri showed that changing the biaxial stress. ratio A produced diffirenat crack tip 'tressfields and crack face displacements. The equibiasial loading mode tiroduiced the greatest separ-ation stress and the smallest crack face displacement. In contrast,[ thle shear loadingj mode producedthe maximum crack face displacement and the smallest crack tio se±paration streN. Hilton shossed

*that the addition of a transiserse tensile load increased the fracture inrtiaition stress and decreasedthe plastic strain intensit.% factor. Compression loading parallel to the crack had the oppositeeffect, Furthermore, Hilton concluded. thatm the etrects oit biaiiial loads %%ere increased atl highierplastic strains. Similarly'. Smith and Pasc'oe 1421 concluded that thle increased ductilits a~s~oiaitcdwith losser strength alkly.- tends to emiphasise thle etlects of biaxial loads onl fractulre p~ropertiescompared with med ,ium and high stiength. allo)s..

Recently. Jones and Fftis 1391 undertook a general fracture miechanics airtafy.is oft a finitecentre-cracked plate. *Their calculaxions showed Ihat the fracture load %aries Esli the, Piiisosoratio Wt' and the biaxial stress ratio ( A) under both plane stress and planec straint coniditioisIFigure 5(a) and 1h). The 'frac~tgre load 4and h~ence K,~ .aluicf iiicrease with increasing! A for allvalues of Poisson~s ratio less than t~ and decrease with increasing .1 'vaiuc% -Owni x, is I.Teaterthan. ., The effect of A on fracture loud.'%%hile present. 'rather small for Poissoni\ ratiis. ictivet' li0- 3 aria 0 4. Most structural metals hiae Ploissoiis ratios within this rang-e.

-Jones and Eftis supported their analytical predictions i-ý perl'trniing hiaxial tes&I~tm materialsused extensively ini aircraft structures, 'Tests oin alumiiniu m alloy 7075 'To w. (I 3) 'specimtens.showed that the' cr~itical plane stress fractture loa~i increased 0 to 20", as the A vausincteasedfromOto 1 -8, Figures 6(al'and (b). Since the crack length and specimndimrtlnensions were identicalin all tests the critical stress .nd AK, %alucs would increase. proportionately. I Ne -dilkerence indi-cated in Figures 64,j and (b) is au~ributed to the .inisotropy' of' the minirosilrictirt' intrptduedduring the rolling proicess. i'he opposite trend was ohbse'%ed fori tes"ts in l'llxivlass (PNIMMAPwith a Poisson's ratio btxwee.cn 0-401 and 0-45, ssliere the fracture load decreased %%mtil iiie;~asimigvalue of A, Figure 7(a) and /Iii.

16

2.0-

,.5 -o 20

1.00

0.50- 7 .0 30

-05 0 05 10 1.5 20 25

LOAD BIAXIALITY

a) P2LANE STRESS

1.25-

I-c : C ý-.c

C.75 ,:- 45

025-

I 'J • ' ,I015 . To 5 • OS . 1.5 2.C 2.5

LOAD 8IAXIAUr'Y A

b) PLANE STRAIN

FIG. 5. CRITICAL STRESS VARIATION WITH BIAXIAL STRESS RATIO A ANDPOI3SON'S RATIO V (391

17 -

-, 0.30Linear regression -"

9000

8000

700010 0.4 0.8 1.2 1.6 2.o

Load Boaxiality-.

11000

10000 Linear regiession"a 0.30"

9000

8000 • ' , I ,,, I.

0 0.4 0.8 1.2 1.6 2.0Load Biaxiality .1

FIG. 6'. FRACTURE LOAD VARIATIONS WIT11 BIAXIAL STRESS RATIO I FOR ALLOY7075-T6 SHEETS 1391

a) CRACK PARALLEL TO THE ROLLING ,DIRECTION

b) CRACK PERPENDICULAR TO THE ROLLING DIRE'CTION

. . 18.

0 :0.40 - 0.45

2.00- "

Linel Regre s sionz-

S1.75- S

2 -

S1,50-O

f 1.25-'I-

-0.5 0.5 1 +0 1.5 2.0 2.5

LOAD BIAXIALITY .

.) UNIVERSITY OF WASHINGTON TESTS

a"- 0 40- 0.4S

I/4 incMh th.sk.00. 5/32 irn. ..hirk

E .01( Linear r--re1s o

"" - - -

S125

.i 4L

0to 2.0 3

i oad Biavtlity

" b) IMPERIAL COLLEGE O•F TECHINOLOGY TESTS

J .FIG. 7. FRACTURE TOUGHNESS VARIATIONS WITH BIAXIAL STRESS RATIO AFORPOLYMF.THYLMETHACRYLATE (PMMA) SHEET-1391

i I I ' I ~ ~ ~~Ii I' f I I I + I I'-+

Experimental studies by other workcrs have' indicated a %ariable intluence of biaxia~ity ,nfracture strength. Kibler and Roberts (431 found that a transierse tensile Ntre's increased thefracture toughness of Plexiglass and the alutirinium alloy 6061 by up to 25'., ccmptared withiuniaxial tension. These tests vere carried out under plane stress conditions- Kibler and Robertsexpected little or no effect for tests carried out under plane strain conditions. In centrast teesers.Radon and Culver [5] reported a slight dxcrcase in the c'itical fracture load of PM .\A. as thebiaxial load ratio increases. Tests on PMNI MA hy Ueda and co-worker-s 44.. 451 produced a similartrend when the transverse load %as parallel to the crack. 1his effect diminished a. the anglebetween , slanted crack and the transverse load increased.

The results of Jones and Eftis. Lee,,ers. Radon and Culver. and Ueda ind co-workers forPMMA are in agreement. whereas the results of Kibler and Robert< do not 6ollo.m the samretrends. This inconsistency can be explained as follows.

(1) The specimen and loading arrangements used h% Kibler and Roberts, lease some doubtas to whether the stress intensity factor remains independent of the trtns,-ersc stress[46]. The proper design of a specimen should not permit the interaction betteen thestresses in the tVo orthogonal directions [321.

(2) For an uncracked specimen of the ty.'re used b% Kibker and Roberts the stres.ses alongthe transverse v-axis %ere not constant :431.

(3) Th.: fracture tests made hb Kiiber and Robert. at different biaxial load ratios verenot conducted at the same ioadine ratec. In addition, the rate of loadine was.,t muchslower compared fith, for example. that used by Leciers. Radon and (ulvcr. There-fore. differences in results w•ould arise from time rate fclrts.associaled vvith materialssuch as PMMA 1321.

6.2 Structural Components

Information reg'ird~ng the effect of hiaxial load ratio on the fracture and tfiatigute propertiesof structural components is %ery ýcaice. Sw;ft 147, 48• has studied the effects of biaxkal-loads onthe toughness and crack growtih rates of stiffened pane.tds. such as commonly tised in aircraftand ship structures. The geometry of these panels is slhowvn in Figure 8. Tc.ts on. paiicls ,%ilthall stiffeners. intact showed that Hiaxial loading increased the critical stress intensitos factor A,..Table 4. In addition, using a semianalytical method. S(ift shoted that :

(I) The calculated stress intensity factor increased with icreatsinAn b•it'ial load ratio forpanels with a two bay craclk and the central stillkner intact. Figure Nial tthis is in agree-ment with his experimental results) : and

'21 The ouposite effect is observed when the central stillkn.r is broken. Figure, sib). Theeffct of biaxial loads was less pronounced for these panels than fior those with thecentral stiffener intact.

Maynor and co-workers (491 ,estid stecl cylindrical pressure ve.isLss with| surface cracksand compared the- critical stress, intensity ifactors KA,. with thoft olam-ned using uniaxial tensilespecimens. Their results- showed that. the average KA,. values for tlig. hiaxially strgsedd pressurevessels and the uniaxial specimens werc similar. This result'was unexpected as the avcrage Strcsat which ;he pressure vessels 'yielded was less.than that of the tensile specimnens. (Coneeqtliitly.they compa'ed their K., values for the pressure vessels with thltos obtained by lat. 150o.f,,rtensile specimens of similar strength. From this comparison it %a% concluded that the A', %ali"in biaxial tension was two-thirds the Kj,. value in uniaxial tension. Ujnfortunatcly. this compariso,,,is questionable as pressure and, geontetrical elkcets were not considered fully.

21) .

I

10. . 01 CNE"TFNRBOESTF0NE STFEE BOA

STRESS 06STRESSINTENSITY INTENSITY

K

02 OG0 0=001, 0r-00, <' -" --07" , .... ,0

101

0 4 a . 240 4 8 12 16 20 24 28HALF CRACK a.NGTH a erCmI HALF CRACK LENGTH a (Imp

FIG. 8. EFFECTS OF BIAXIAL STRESS RATIO ON STRESS INTENSITY INSTIFFENED PANELS (47]

a) STIFFENERS INTACT

b) STIFFENERS BROKEN

TABLE 4

Test Results for Lovitudinal Crack Tests on Cursed Panels of 7075-T73 AluIminium Alloy with allStiffeners Intact (481

Test Panel Pressure Hoop Axial Crack CriticalNumber: Con- at Stress Stress Lengih Stress

figuration Fast Normal Parall-I at Fast IntensityFracture to Crack to Crack F-racture Factor

(psi) (psdiI (psi) (in) (psiN in.)

one-bay 12.80 14,912 0 7.24 66460crack

2 13.36 15'622 ,2,560 8.05 72.MNX0

3 two.bay 10.80 12,600 0 12.10 65.114crack .with

4 ' centre 11.55 15,380 .28,495 283 80,573

•iframe intact

, 21

For example, tests on curved panels indicate a loss of strength of up w~ 3a(,. becau-s: of bulgingfrom internal pressure [481. Mvaxnor and co)-workers modified their equation.; 11411 calculatingKi,. for *the pressure %essels so as to allow Rir this effect. Homwexr S%.ifr [48 concluded thatthe bulging effect in pressure vessels cannot be simulated exacllý by flat panel tcune. Therefore.it is difficult to determine whether the difference het%%e.on the biaxs-aI K1, vahre of 'daN nor ind]co-workers an,' the uniaxial Ki, 4aucs of Blat is the result of stress state. or pressure aind gco-metrical effects.

IErdogan and Ratawani [51] examined thc fatigue and ,fnctture ptoperties of c~lindricalatid spherical shells containing a through craik and subJected to internal pres.Nurc or torsion.In addition,. rectangular plates and cruciforir peimn were tested. Retiuts oi' rupture testsfor the various specirrct.s. made from alumniniumn alloy 6061-T6, are shown irt Vigare 9ta and tTThese Figures show, that the nominal critical stress irtensity factor A'.,, twliesre K %.,s obtainedaisn-,g the rupture load and the initial crack "neith) necreases with increa-ang strcss ,ornponientparallel to the crack. In addition. Erdogan and R:tauani found that curvature ma,, haxe aconsiderabt, Ifect on the stress intensity iacl.-. in -hell- with cracks. litleir theorctical inal , %iNshowed that curvature increases the local stresses a the crack tip and thereb-. reduces both thecritical strain intensity factor and rupture stiegtn of the component. 'Therefoire, the diff,-rencein critical stress intensity 'values betwecen the p,-ssure %ebsels of Mavrior and :o-w~orkers andthe flat t'.iiaxta; .,-,cimens ofB~at applears to be a t.-sult of cometricat and nor. stress state effects.

7. MILLT11AXIAL LOW-0CU CI: t IGUE

Many high performance components. such as gis tu~rbine blades or riuc:ar pressure ~esselscan occasionally experience large mechanical or thermal transients during operat~on. i.e. durinu,start-up and shut-down procedures. T ese transients cats cause siienfihant C'arnaee: accumulationafter only a few hundred or a few; thowasand of these exeles. E~en under nortuai operating con-dlitions the material in the \icinity of structura; discontinuities. such a11 notcheý Or edCallexperience local cyclic plasticity. These discontinuities. "here fatigue c-aeks jstwll% nucileate.are invariably a.' t;,, free surface w hich is iii a state of biaxial strcs,. I-lcrefOre, the predlictionof lowv-cycle high-stain fatigue failure under biaxial stress conditions. is anrt. r.ortani dc*ag~n*consideration fmany structural com~ponents.

Compared with liigh-c~cle fatigue. correlation between tfli~i~ital And biaxial 'u- ,Ciefatigue is complicated by the effects of plasticity which is non-Vnear and liist~pr\ path depenident.Consequently, criteria are necessiary for both the cyclic strcss-strain behaxiouir ind ILiuti~e lift;data in low-cyclc fitiguc

7.1 Cyclic Strims-St-iia Dis.,

Limited experim ntal data are availitble oin the cyclic stress-strain behavioutr of mnaterialsunde; biaxial wtess conditions [9. 52-571. In addition, there are only a JliV publishe-d paperswhere a single test spe: men geometry has been used for the full range (,f stress states [9. 55 57!.

Parsons and Pascoe ff)] performed .train controlled tests on crucifi'. im specimens of1 twoalloy steels under states of hiaxial stress. These tests showed that the rckistance to, Cxch .e defor-mation is a functio 'n of the matetial. the applied strain and ilhc strain ratio 4 A * .1c r oi. Theannealed austenitic steel AISI 304 cyclically strain hardened at All strain amplituties. whereasthe quenched and tempered steel QT35 cyclically strain softened. kiaximum liardicniiie torAISI 304 -and maximrum softening for (ff35 occurrcel under equibiaxiial conditions (6h i I)and the minimum occurred under shear conditions 0. 1). In addition. the load range (All)applied to the specimen arms to maintain a given total strain range wais dcependent ori the strainratio. For example, the values of AP required to maintain any gowen strain rang: tat otie fifthiof the fatigue life) increased by up to a faictor of 3 as hb ichaaged from I to f. I.'

22

[ 6 061-T6 ALUM PLATE

56 44- (4 Spec• 1 2 Spec)

540 3842 - 3(?Spc

aQ

48-"x 36 -2 (

(3sec 65in Plates I 346-a 34-0

8 a Cyinders 06a 12.in. Plates 22

44- 32 o . ,axal Cross 1.3o Plates

4gI I 3C •0 005 010 0.15 020 0.25 0 0 02 03 04 05

XFIG.9. E"-ECT )F BIAXIAL STRESS RATIO ON THE RUPTURE STRENGTH Kcnof a/ 0.66mm THICK 6061-T6 ALUMNIIUM PLATES

and b/ 6061-T4 ALUAINIUM PLATE AND SHELLS IS51

23

Havard and Topper [55, 56] tested thin-wkalled tubes of normaitsed AISI MOiS mild steelunder in-phase axial ;iading and internalexternai pressure. The datz from. fultv reversed loadtests for several different stress states' were used to establish the c.vclic behaviour of the mildsteel. Each cyclic stress-strain curse could be represented by the sum of a power law term forthe plastic strain and a linear (general~ised Hooke's law) term for the elastic -.,train. The equationwhich provided the best fit for each set of data wvas of the form

_aj(l r) k.(Aa•)Act E.50 (121E 50

The %alues of the cyclic strength co-eflicient k' and the cyclic hardening exponent ni are bothdependent on the stress state. For example. Table 5 shows that the range of values of k' and n'obtained for the various biaxial stress ratios %%ere 0.083 to 0-589 andO-27t to 0-572 r_ýspecti ely.Table 5 further shows that increasing the-biaxial stresý ratio tends to increase thec~clic hardeningexponent n'. Therefore. the resistance to cyclic deformation also increases as the biaxial stre-ratio increases.

TABLE 5

Values of the Cyclic Strength Coefficient k' and Hardening Exponeu a' from Equation (12) forVarious Biaxial Stress States [551

Biaxial Stress Ratio Direction Parameters from Cyclic(circumferential to of Stress-Strain Curseaxial stress ratio) 71 Control .. .

Uniaxial (0) Axial 0-166 0.314(rolling direction)

Uniaxial (0) Axial 0-155 0.294(transverse direction)

Torsional (x). Shear 0"589 0.271 1

Biaxial ( -2.9), Circumferential 0-214 . _107Biaxial (0-50) Circumferential 0"109 0-310Biaxial (0-86) Circumferential 0-00. 0-368Biaxial ( -0.28) Axial 0.219 0-.167Biaxial (0:26) Axial 0-174 0-572

In general two criteria have been used to correlate multiaxial cycli pro'perties : ) a Trescamaximum shear stress-shear strain correlation, and (2) a Voo Miscs octahedral sheaur Aress-slearstrain correlation. Brown and Miller 571 determined the biaxial eycli" stress-strai cures f'ra I% Cr-Mo-V steel and an AISI 316 stainless steel at various temsperatures and strain-rates.In addition, these authors undertook a statistical evaluation of d. Tresca and Von- Mises corral-ations using most of the experimental data available. They foind that cyclic stress-strain cursesuider multiaxial stresses are best represented by the Tresca maximum shear 4tress-shear straincriterion, as shown for example in Figure 10.

"24

122

FIG. 10. CYCLIC STRESS-STRAIN CURVE FOR 1% Cr-Mo-V STEEL AT 20*C.I=RATIO OF TORSIONAL TO AXIAL ST•AIN [571

25

* \

• , , .

7.2 Fatigue Life Relationships

Well known relationships exist for relating low-cycle fitigue life in the uniaxia' stress stateto stable values of stress or strain amplitudes. Manson [581 and Coffin [59] used a simple powerlaw to relate fatigue life to the plastic strain range, whilst Basquin [601 used a similar law forthe stress range. These equations are

Nfl,= X (13)2

Aa Nt" = Y (14)

2

In addition, the stress amplitude in the elastic range can be expressed as,

Aor Afe2 • E (15)

2 2

Therefore, the total strain range may be represented by the sum of two power law functions offatigue life, namely

Ac A•p A~e"AC =2 + -2 = A Nt-' +8 Nr ". (16)"2 2 2

Fatigue life data under biaxial stress conditions also can be represented in terms of theseequations [55, 56, 61-631. For example. Ellison and Andrews [61J presented high-strain biaxialfatigue data for aluminium alloy. RR58 at several biaxial strain ratios and for fatigue li'es between100 and 3000 cycles. These authors used thin-walled tubes subjected to axial loading combinedwith internal/external pressure under strain control. Plots of the maximum principal plasticstrain range (Aelp) versus fatigue life (Nf) showed that the Manson-Coffin exponent increasedfrom 0.753 to 1 -436 as the biaxial strain ratio q0 increased from -I to + I. see Figure I I andTable 6: In addition, increasing the strain ratio from - I to + I decreased the fatigue life by afactor of 2 at-high plastic strain, amiplitudes (.A-j;, - 0.50 ) and by a factor of 5 at lower ampli-tudes (Aelp 0"05%).

TABLE 6

Effect of Biaxial Strain Ratio 0 on the Mansdn-Coffiu Exponet and Intercept Constant forAluminium Alloy RR8 f 611]

train Ratio i IS---I . . .. ,0 + , + .1

Parameter-

X 0-753 0"761 0-938 1'218 1'436

A 0"2. 235 0.486 . .91 5"45

26

.- I. -.

• [ • ,

- 10 __0___

06 _+j

04

02

x

I Il

0040

00221

F-8~~ ~ ~ ~ __to__3__5__7__9_______34 _56_78

Similar results were obtained for a ferritic I Cr -Mo-V steel by Lolir and Ellison [1I, 2]using a modified version of' the equipment of Ellison and Andrews. In addition, a few tests atdifferent strain ratios were conducted using the aluminium alloy RR59 of Ellison and Andre~ks.Agreement between the total strain versus fatigue life curves was good. The test, on the ICr-Mo--V steel showed that the constant A in thtc Manson-Coflin equation. and both the con-stant 8 and the exponent 4 in the Basquin equation all decrease with increasingr biaxial strainlratio (b. Table 7. The exponent x in the Manson-Coffin equation was constant. Lohr and Ellisonalso observed a decrease in the fatigue lives by a faictor of 2 to 3 for both the I ",Cr-%Mo A,steel and the aluminium alloy RR58 as the biaxial strain ratio changed fromn I Lo 1. Againthis is in agreement with the trends obsersed by other workers [62. 6'31.

TABLE 7

Effect of Biaxial Strain Ratio S6 on the Nianson-C(offin and Basquim E-iplonents. and InterceptConstants for a V,, Ur-Nlo-V Steel [2)

Strain Ratio

0 1 Uniaxial 0.Paramet*er

A! anson-Co//inA 1I28 0--4, (066 1)-r8

0.9 o-9 0_ IBa~squin 0X9 ._______ ______

The fatigue life behavoiur of a ferritic steel QT35 and an austenitic %teel AIS) 304. usingflat cruciform specimens, is slightly uilfercnt frorn the ,ahose result%. Parsons and Pascoe [91 foundthat the intercept and exponent constants in the abose equations wvere different in (lhe highi strain(101' cycles) and low strain ( 10" cycles) regim~es. The change in slope of these curses occ ,rred inthe vicinity of 101 cycles. (Most of the previous authors had restricted their test% to less thi~ainapproximately 101 cycles). On subsequent examination ol' the litei~aure Parsons and Pascocconcluded that this behav iour was typical of' hardened alloys of' steel and alumintuum, andtitofcertain au~tenitic steels In geheral. increasing the ~train ratio from I to i-I decreased theintercept and exponent constants in the total strain sersus, fattigue liti: equation. (ThisequcntlN.increasing the strain ratio from I to I decrease:d the fatigue lives in the high and low-straitlregimes by factors of 10 and 20 respecti'.cl%.

8.THE EFFECTS OF BIAXIAL STRESS ON FAT11UF CRACK G~ROWFTH RAT ES

&I1 Laboratory Spaeciurn.

Thsý most popullir and widely used analytical icol for predicting the safe life oif' aircraftand many other flawed structural comnponients. based on laboratory data, is linear elastic. fracturemeclhahics. Fatigue crack growth rates tialciN can be correlated with tile cfi;:nge in thle stress-intensity factor AK. These data are usually obtained under uniasial kliadin conditions.

28

A fracture mechanics approach for the analysis of fatigue crack growth in biaxial stressfih;Jd has not.yet been successful>1 developed. Linear elastic fracture mechanics predicts thatcomponents of stress parallel to a crack should have no effect on the fatigue crack growth rate[64). In contrast, experimental results have shown that the growtth rate of fatigue cracks can beenhanced [65. 661, retarded [15. 39. 43.46. 67--72] or remain unchanged' [4. 35. 36. 73--771 bythe application of a static or cyclic tensile stress parallel to the crack. Similarly. appl% ing a staticor cyclic compressive stress in the transverse direction has been reported to increase [15. 68.69, 72], decrease [4. 74, 75, 78. 791 or hate no effect [35, 36. 76. 771 on fatigue crack cro%,thrates. Furthermore. the effects of biaxial load are enhanced by (I) cclic transverse stresses ofconstant amplitude compared with static transverse stresses. [15. 711 and. (2) cclic transxersestresses of variable amplitude compared with those of constant amplitude [35. 80].

One factor which is responsible for the inconsistency in' the literature is the method ofevaluating the stresses applied to cruciform type specimens. In most cases the practice has beento define the biaxial stress ratio k in terms of the NOMINAL stresses applied parallel (S,) andperpendicular (S.) to the. crack but remote from it [72. 811. Usualls the biaxial stress ratio

, SS•. reduces to the ratio of the loads P, P., in the loading arms of the crtociform specimens.Results from 'these tests generally indicate that biaxial stresses hate a signitfiint elffct on ftiguecrack growth rates, as shown for example in Figure 12 [681. In contrast. when tests are carriedout using the LOCAL stresses.'e.g. at the centre of an uncracked plate. the effects of the biaxialstress ratio on crack growth rates are small [4, 35 36, 74--771. (The stress distribution acrossthe test section of cruciform specimens can be evaluated using either strain gauges or finiteelement analysis. Both methods are in excellent agreement [35]). This difference betweenNOMINAL and LOCAL stress state effects can be rationalised as folloýs : The NOMINALload Py, required to maintain a constant LOCAL stress a, signiticantly increases as the LOCALbiaxial stress ratio k changes from I to i [(811. Therefore. the increase in crack growth rateassociated with the increase in P, between. sas. uniaxial tension (A 0) and equibiaxial tension(k - + i) nullifies the decrease in growth rate from the change in stress state. 'A similar counter-balancing effect occurs as the LOCAL stress state chances from unia\ial tension (A t 0) topure shear(k - I1).

Another factor may als(i be responsible fir the abo(e difference, that is. in any gi\e testthe NOMINAL and LOCAI. hiaxial stress ratios may not necessarily be the same. For example.various workers hase shown that the local biaxiil stress ratio A ,1\ a' can \,ar\ from 0.26to 0.35 for cruciform specimens loaded in uniaxWil tension. i.e. when A S, .S\ 0 [74 76].Therefore. the local stress'stateat the crack tip can be biaxial with a compressise static transsersestress although the nominal stress state may be uniaxial tension. Similarly. Kitagawa. Yuukiand Tohgo 1711 found that a static tensile load parallel to a crack produced a local transsersccompressise stress. They also found that the local stress intensity ratio. defined as RE Aglas,,.in the y'direction. varies with the biaxial load condition and crack Ingth eten if the stress ratioin the v loading arm. R, . S , ,,i. . is kept constant.

Experimental results have shown that in uniaxial tension the application of a nican stress"can increase the fatigue crack growth rates and hence shorten fatigue life [I21. ('hanging themean stress %alue' in the i dircctionalso influence, fatigue crack prowth rates in'biamial tenisioni171. 83. 841. For exampld. Iloshide.r Tanaka and Yamada 1841 found that lhecn the local stressratio RI. was 1. the fatigue crack growth rate increased as the local biaxial stress ratio 4,)decreased from I -to. 1. One the other hanid. when R 0 the growth rate was the lowest fora k value of I. Therelort. different crack.groslst rate" mayb'b produced depending on whether

,a nominal 'or local stress parameter is used to control the test.The growth of fatigue cracks in multiaxial stres, states is affected by the orientation of the

crack with respect to both thie surface of the component and the a:plicd stress fleld. In uniaxial-tension, fatigue cracks usualiv nucleate at the surface and initially grow, along planes close tothe planes of maximum shear stress, denoted as'-tage I growth by Forsyth [K5J. After the crackhas propagated through one or more grain% in depth the crack changes direction and growsalong planes perpendicular to the maximum principal stress direction, denoted as Stage II crackgrowth 1851. The stages of crack growth in muhtiaxial stress fields are the same its those justdescribed for uniaxial tension. Parsons and Pasco. ([86 also obsermed that' I'F crack groothllin biaxial strain states occurred (in or close to the planes associated with Stage I and Stage If

* cracking. However. Stage I crack growth was dominant under ihcar 1 1 I) and plane surfaice

29

o00

80-

sy .0

40-

20' 400

0a 0

2 4 6 8 t

Half crack length x 00m

FIG. 12. EFFECTS OF BIAXIAI, STRAIN RATIO ON FATIGUE CRACK P.ROPAGATI(?N RATES

FOR UNNOTCHIED CRUCIFORM SPECIMENS OF ALUMINIUM ALLOY RR38;

A = 70 MPd AND O 105 Mla f('8y ymx

strain Of =0 load ing. Similarly, .Bro%%n and M ilIler [871 found at transition from Stage Ito Stage 11crack growth for biaxial strain ratios 10') of approximately 1 ý5. At elesýated temperatures thistransition occurred fo( biaxial strain ratios (ti') of appro~ximnately 4. Brossn and Miller concludedthat the transition from Stage I to Stage If crack grosuth in ductile metals is dependent on theapplied stress or strain state, and not on the crack length.

Brown 'and Miller [101 also obser,.ed that oto tyýpes of cracks can deselop tor each stageof crack growth. Cracks can gro%% along the surfaice of a component. as occurs in ten sion-torsionitype loading. or just under but parallel to the surfa.ce as, occurs in rolling contact fotligUe. AIkernat-ivcly, cracks can grow awsay from the surfacee in the through-thickness. direction, ats occursN inthe biakial testing of pressurised tubes, plate bending (anticlastict tests, or biaxial tension testsusing itruciform specimens. Brown and Miller [101 have denoted wurfacc cracks as T% PC A cracksand t hrough-t hick ness cracks as Type B cracks. Figure 3. Type B cracking is the most di-nL'erOUscase as the crack growth rate is much faster and the flitigue strength loser than eomnlarcd ss ithType A cracks, [10, 13. 691. Consequently. thý fatigue li~es of component,. hating tspe A andType B cracks may be quite different esen though the equisalent stress;es and strains may bethe same [10. 11. 691.

The orientation of the crack plane %sith Loading direction also maiy be classified as, eithterMode 1. If or III depending on the loading configurations. Figure 13. For example. a Stage Icrack may grow under Mode I loadine in uniaxial cension. Hotkcexer. a Staae I crack uindertorsional loading conditions can gross along the surfatce under Mode 11 or in it radial directionUnder Mode III loading. In sers ice. most components. are subjected to mixeýd 1-loading conditions.For example, tubine shafts are siinnuitaiseotu., bu'uj.ctfxJ t,)i constan, torque. %,%hich pro% idesthe driving force, and an alternating bending stress. %,shich arises from thc %seieht of the rotoror any slight' imbalances [78. 791. Consequently, cracks growsing perpendicular to the surfaceeof the shaft mhay be subjected to a combiination of' Miodc I plus either Mode If or Mode Illloading.

Hourlier and Pineau [78. 791 ipsestigated the ft~igueu crack- gro%%th ratles- in four materialsunder cyclic Mode I plus steady Mode If or Mode- Ill loading conditions. They ifound thaithe simultaneous application of at cy Iclic Mode I plus a tad Mode Ill loadin& produced tssomain effects compared with Mlode I loading, namely

I) lage eceas inthefitigue crack- grossthi rates of up to Iso orders of magnitude.

(2) A significant change in the appearance of the fracture sutrf.itce.

The major effects of applying'a steady Mfode 11 to a cyclic Mode I loatding %%ere ty change thedirection and reduce the ratc of~ crack grossth compared ss ith Mode I crack gros Ili.

The special test environments somectimes used for testing hia~ial specimiens can influencefatigue crack gro%%th characteristics. For example. the fluid used in test,. oft thin-ssalled tubessubject to axial loading plus internal external pressure cain interact vsith a grossing cract. 113.16, 88]. In addition, the crack mav be subjected to the -"hvd~rossedge" effect %% here high pressuireoil enters the crack and thereby pro% ides an additional Mode I opening stress [ 13. 881.

In summarising this section. at present there are conflicting result% in the literature concerningthe effects of biaxial stresses on fatigue crack grossth charaictefristics.(ontiflicting results Iliasearisdrn between the various wsorkers ticcaiise of difllirences. associated ss.ith the maturial used.geometrical shapes of spec~imens, test control conditions, and test ensironrient lose. onemethod of overcoming these prehle~ms is to conduct a series of -changeoser tests"* as initiallyproposed by Leesers. R~adon and (ulser J70J. (hangeoser tests consist of fantigue loading aspecimen using a fixed biaxial s'tress I strain) rat io uint il Ithe crack gross th bt:has% iour bas stabiliscd.lThe biaxial stress, Istrain) ratio is thei studdenls chafteed to anmother salue and thec I~ices on thecrack growth curse monitored. This procedure can be repeated sesecral timcs 'duiring, a test..Consequently, the effects of stress state oin ft~i Igue crack grossth behaviour can be identified 'isthe one specimen, geometry ,ind environment are used for eachm stress state. In addition, thelocal stress and the stress intensity tfctor at each changreser are the same even though thecactualvalue may not be known, and therefore charges in fat igue crack grossith rates after a changeoverare legitimate and not transient 011ects.

31

Mode t Mode .r -ri

FIG. 13. BASIC MODES OF CRACK SURFACE DI PLACEMENT

32

Garrett and co-workers [I15. 72. 81] performed such changeoser using centre-cracked cruci-form specimens of mild steel. The results of some of these tests can be sutmmarised as follows

(I) An instantaneous change in stress state from uniaxial to equibiaxial tension (ao cyclic)at the same nominal applied stress normal to the crack reduces the fatigue crack grow.thrate by a factor of 2. Figure'(144a). In comparison the fatigue crack growth rate isaccelerated by a factor of 4 wlhen the stress state changes from equibiaxial tensionto uni-.xial tension. Figure 14(b). Leevers. Radon and Culser obsermed the same trendsfor .rMMA as shown in Figure 15. The changeosei from uniaxial to biaxial tension(P5x -- 2 Py and is cyclic) decreased the fatigue crack growth rate by a factor of 2 5whilst the opposite changeover increased the grow.h rate b, a factor of approximately 4.

(2) Compared with uniaxial tension, the fatigue crack growth rate is increased by a factorof 3 when a cyclic compressixe stress is applied parallel to a propagating crack. Figure

14(c).

(3) The effects of biaxial loading were the same for steel specimens in the annealed andcold worked states.

8.2 Full-Scale Stiffened Panels Containing Cracks

Many structural. elements consist of flat plates or shells reinforced with stiffeners. such asthe fuselage panels in aircraft structures. The design of ,uch structures according to damagetolerance principles requires a knowledge of fatigue crack growth characteristics. especiallyunder complex loading.

Research into the eff'-:ts of biaxial stresses on the fatigue crack growth rates of stiffenedpanels is very limited. Swift [47. 481 has examined the effects of both biaxial loads and cursaturcon the fatigue crack grewth rates in this type of structure. In this work the crack growth behasiourin a curved panel subjected to biaxial loads combined with internal pressure was compared v iththe behaviour of a flat panel in uniaxial tension with the tensile axis'parallel to the stiffener!;.The nominal axial stresses and the stress ratios (R) were the same in each case cxccpt that thecurved panel was also subjected to a cyclic internal pressure. The combined effects of cur\atureand biaxial loading caused an increase in the fatigue crack growth rates of 50 to (X)" dependingon the crack length. as shown in Figure 16.

Ansee and Morrow 1891 have investigated fatigue crack growth rates in klrge flat panelscontaining stiffeners under uniaxial and biaxial loading conditions. These panels were completelyrepresentative of the fuselage panels used in the C(Uncorde aircraft in all'respects except for curva-ture. (In contrast to Swift. these authors investigated fatigue crack growth rattqs in flat panelsonly. However. as noted bh Swift. considerable care should he exercised when' using the resultsobtained from tests on flat panels to predict the allowable stresses in curved panels). The testassembly for the stiffened panels is shown schematically in Figure 17'where the actual test panelis represented by ABCD. The cracks in each panel were symmetrical about a stiffenerwhich wasalso completely cracked. Results from these tests showed that when compared with uniaxialtension, tensile bitaxial stresses f,r c)clic) reduced the fatigue crack growth rate by a factor ofI' 5 for a nominal biaxil 'stress ratio A\ 0, 5. and by a factor of 2 2 when A I -0. Figure 18.Ansee and Morrow conchided that tensile loads parallel to a crack can hl safely neglected whendetermining the safe life of this type of structure. Itomevvr. they did not examine the effectsof compressivee ýtresses parallel to a crack.

The experimental results of Ansee and Morrow for pa.nels where both the skin and th,:central stiffener are broken are ia agreement with the theoretical analysis of Swift 1471. However.Swift predicted the reverse trend 'if the stiffetier was intact, that is hiaxial loads would increasethe fatigue crack growth rates, comparxd w ith uniaYia| loading. The finite element analysis of

Ratwani and Wilhem (90. 911 also predicted this reverse eilket,. Therelore, these theoretical

33

Annealed

10

29

.°. 8I

£4

c

23 -1:0

24-.

£3

II

10 20 30 40 so 60 70 80a N (cyrclesl) a '103.

12

Annea ed

-10.11101

0 ,9 * o

* 3 1

£2

0 to 20 30 40 so 60 T0 sob N (cycles) a10*

12

112 . . ,

*J L

9.

U I!4

""€ 0 20 30 40 so. 60 0 .

FIG. 14. BIAXIAL'"CIIANGEOVER * TESTS FOR MILD STEEr, AT CONSTANT LOAD P 1721y

a) UNIAXIAL TO EQ)UIBIAXIAL. TIENSION (0L -CYCLIC) DECREASES THEFATIGUE CRACK GROWTH RATE BY A FACTOR OF 2

b) EQUIBIAXIAL TENSION(o = CYCLIC) TO UNIAXIAL INCREASES THE FATIGUECRACK GROWTH RATE BY A FACTOR OF 4.

C) UNIAXIAL TO CQUIBIAXIAL TENSION/COMPRESSION (PURE SHEAR) INCREASES .

Tfir FATIGUE CRACK G, )WTH HATE BY A FACTOR OF 3

S54i

45

•

-44

~43o /LAAK-0/ 704 W•nm"'

42

4,

2 3 4 5. 6 7 e 9

N cycles

(ap

69

6K035PM'2, /% ~KO 935 Umam'"

E66

Ego

I" '

@63

#4 6e

I,

FIG. 15. BIAXIAL "CIHANGEOVERU" TESTS FOR PMMA AT COINSTANT LOAD RANGE APy

[461

a) UNIAXIAL TO BIAXIAL (P , 2;P )TENSION DECREASE THE FATIGUE

CRACK GROWTH RATE BY AX FACTORYOF 2.5

b) BIAXIAY, (Px-2.P) TENSION TO UNIAXIAL TENSION INCREASES TNEFATIGI:E CRACK GROWTH RATE BY A FACTOR OF 4

55

•3".8 r" I • ,

LONGERON 'AX AXIAL END STRESSBROKEN 15.500 PSI

3.k• 4 C05,, • / /,

TOTAL . MA" ' A- "MXPRESSUREPAEC RA C K . 93 PsI R 005LENGTH 2a

LONGERONiN. CURVED PANEL BROKEN-

CYCE' AiALSRES

.COMPARISON OF FATIGU CRACK GROWTH RATE CURVES OF M .AT ANDCURVED PANELS OF 2024-T3 SK'N MATERIA -r411

36

A B _ _

0 0 0 0 0 0

_ Ho 00 fa 0 :00 0=

c= -

0

oo

0 =0O

00

0 _

-- 2286mm

FIG.17. TEST ASSEMBLEY OF ANSEE AND MORROW [891.ABCD REPRESENTS THE ACTUAL TEST PANEL.

/1-3

31i

170- 3-3

160 ,f-.t-

E 130 7,120 f

to -

go-

dSoo

2 .6 5 £ 7 8 1 to ti 12 13 14 Is to 17Numbe" *E cycles a 1000

F IG-*10. COMPARISON BETW1ZEN THI$ FATIGUE~ CRAC -K GROWTH CURVES IN UNIAXIAL-TENSION (li-0) ANID EQUISIAXIAL, TENSION 0Aa1) 1891.

analyses suggest that neglecting the effects of tensile loads paralietl to a crat, k cou I l;c.id to i,-unsafe life estimate for panels with the stiffeners~intact and small critical crack lenoth, in t Cxin ,,

As a result of the predictions by Swift. and Ratwani and Wilhemr. lorrow f4_2j c01,1,t]iLtJa further series of tests on panels with intact stiffeners. This series -a&, c~iidtij-ctd u',ne !!ic

changeover test described in the previous section. The tests were started in htamdal tinwn .T Tthen changed to uniaxial tension after a certain period of crack grov~th. -hr Iladene c',Tt'',was then alternated between biaxial and uniaxial tension. Results from t•,e~c r:t. ,,,' ,unlike the predictions, biaxial tension reduces the fatigue crack gromth rate fth j pemic- ,,it!stiffeners intact compared with that in uniaxial tension. Figure 19. and in L".•ae- .. rt .•'K.stiffeners. Equibiaxial tension A = I reduced the fatigue crack growth rate bh a tfctcir of -in the foi'mer case and by a. factor of 2.3 in the latter case compared %tch tinamas.d tcnsi',nMorrow suggested that ciack closure and plastic zone sire effects a-- po,,rhle reason, fordifferences between his experimental results and the theoretical prediction%.. Both ,t thic c~kdt,appear to have been ignored by Swift. andRat~kani and Wilhem. Frirtherm,,re. \horrom on-eluded that a purely elastic analysis of.hiaxial loading effects in stitleIcd panel,, ,

In summary, the results of Ansee and Morrow, and .MoArrim sugge-t that thle ell.ct, of

tensile loads parallel to a crack in J, stiffened panel. with either cracked or mtact ,statfener,. t..aibe safely neglected in safe life estimates. Howexer. the effects of compre-'rc tr,,nNscr,e h,.,dNon fatigue crack growth rates in stiffened panels hate not yet been determined.. Non-ca'nscriralaeestimates of the safe life of these structures ssould occur if the effects of tran,%er,e cornprc.,ý'cstresses are similar to those observed in laboratory tests,

9. THE EFFECTS OF OUT-OF-PHASE BIAXIAL. LOADING ON FATI(; E.PROPERTIES

All of the biaxial fatigue research reoiesed so far has been conducted under proportional(in-phase) loading, that is. the ratio of the applied 4tresses or .,trains remairns constant throtugh-out the loading cycle with the peaks and the troughs ociurring at the same time. hI Nerucc.many engineering componentf aresubjected to cyclic biaxial loading, conditions %khcre thUwave' orms of mutually perpendicular stresses or strains are out-of-phase %,itfh each other. I orexample, aircraft structures are vulnerable to this typeo't loading which can reduce the f1atiguelife of the strticture [93-941.

Very little experimental work has been undertaken to establish the effects. of out-of-lphasCbiaxial loading on fatigue properties of materials and components [I I. 35.71.- 77. 95991. -Thissituation has arisen due to the difficulty in determining representatise strcsses aim- strains as themagnitude and diection of the principal stresses and strains continually change during eachcycle. Rotation of the principal stresses and strains means that the biaxial stress Istrain) ratiois not constant but also varies in a cyclic manner during each test. For example.. Lill. Dittmerand Holloway [771 round that the biaxial stress ratio in their tests could %ark by a factor of10 during a-complete cycle for out-of-phase loading of cruciform type specimens: The problemis further enhanced in low-cycle fatigue as the principal stresse and strains rotate at varingspeeds in every cycle and therefore are not coincident [951.

The rotation of the principal stress and strain directions during out-of-phasc hiaxial testingaffects the fatigue properties of materials. Foi" example, Grubisic and Simbiirger ["I6 examinedthe effects of out-of-phase loading on the hiaxial fatigue strength of' carhim steel using thin-walled -cylindrical specimens. These specimens were tested under biaxia foading conditionsby either:

1I) Axial tension plus internalpressute with a phase difference betvcen the normal stresses,a. and ',ii

(2} Tension plus torsion with out-of-plase normal and shear stresses Y,, and ri,' respect-.ively : or

(3) Combined bending and torsion.

39

..................

tip A.

605

600

~55

E

0 50-

c

16 17 1819Thousands of cycles

FIG. 19. THE EFFECTS OF *CHANGEOVER" TESTS ON THE FATIGUE CRACK GROWTHCURVES IN STIFFENED PA'NELS WITH STIFFENERS INTACT 1921

39

Results from these tests showed that a phase difference between the normal stresies. zind especiallyfor phase angles greater than 90 . decreased the fatigue strength. Similarly. the fatigue strengthwas lower for out-of-phase compared with in-phase testing for boths tension-torsion and bending-tor~ion loading.

Grubisic and Simbiirger also observed that yielding initiated at a lowser stress for in-phaseloading as a result 'of the higher peak strains encountered compared %%ith out-of-phase loading.Consequently, if the stress "amplitides wer,: the same, in-phase loading could produce plasticdeformation whereas out-of-phase loading might not..

Earlier work by Nishihara and Kassamoto [971 using combined tension and torsion showedthat the fatigue limit for ductile materials increased for out-of-phase compared i"ith in-phasecyclic loading. This result is the opposite to that obsersed b-* Grubisic and Simblirger. Little[981 re-analysed the data of Nishihara and Kawamnoto and .shom-ed that the apparent increasein fatigue strength is very misleading. Little expressed the'out-of-phase data in term-, of the trueshear stress amplitudes whereas they were originally expressed in termsK of the maximum in-phasesheir stress amplitudes. The analysis of the data by Little' showed that the fatieue limit aduallydecreases as the phase difference increases. The decrease is of the order of 25 ",~ for at sh..ar stressto normal stress ratio of 0 -5 and a phase difference of 90 . This 4;s a special case as every planein the surfacti material is a plane of maximu~m range 'of' shear stress.

Kanazawa. Miller and Browsn [951 also conducted biaxial tests using combined c~clic tensileand torsional loads with 6~atious phase differences. They found that thle f-Itigue life of, a I '

Cr-Mo--V steel is reduced by out-of-phase loading conditipns. uith phase angles (if 90 reducingthe fatigue lives by a factor of between _2 and 4. In addition. the-se authors ohser~cd that thlehighest fatigue limits "sere for in-phase loading xshich agree% %%-ill the results of' (ruhisic andSimbhirger. Usually, data from in-plias~l tests and either the Tresca or octalwedra.l shear straincriterion are used for design purposes. *Ho%%e~er. Kana/a~sa. Miller and Brom n slum~ed thatboth of these criteria are non-consersati%,e under out-ot'phase conditions, and theretfore can leadto non-conservatise estimates of' safety factors. [or esýaiple. it' the Tresca Lcriterion is used it,establish a safe life based on fatigue lire in torsion, then the sale life for a hiaxial ratio stressof 1 -5and a phase difference ol'90 ssill he m~er estimated by a ICictor of' Mt.

The cyclic strcss-strain response of' metals in hiaxial sitre-.% state'. i., alrectd by outo-oi- phase.loading. Kanazawsa. Miller and Browsn (991 showsed in other w~ork using the I "_(r %I() V steelthat out-of-phase loading produces additional wsork hardening. This %%as attributed ito theincreasing complexity of the deformation behaviour resulting from thle rotation of thle principalstress directions compared %sith in-phase loading. In addition, rotation of' thleý principal aws%distorts the hysteresis loo~ps at higher stresses \,.here gross 'plhistic deforniptioit i%ý taking, place.the hysteresis loops, as showsn in Figure 20). obtained from tension-iorsion~ tests vith phaseangles other than 0 or 180 . are difficult to analyse. The distortion of thle li~stercsiý loops ilicails

that the traditional definitions of elastic and plastic strain i-inges. used in t~atiguc fire assessmentmay be inappropriate [I111. Despite this difficulty Kana,;iwa. Miller and Brom.1n produced acyclic stress-strain curve from their results (Figure 21) by using the niamimum shear strai~n andthe corresponding shear stress during each cycle. These author% also used a correctton tactorto cornpensatti for the additional \ýork hardening produced by the rotation of the principal ases.

*The fitigue crack growth rate for metals under biaxial 4*triess cniisalois allkctvdby out-of-phase I-ading as differen't cracking systems can operate at diitk-rLn4 times in each Cceletl I]. Kitagawa. Yuuki and Tohgo 171] found, that thle fatigtue'crick growth rates in cruciformi

seiesof a structural steel increased ats the phase angle changed from 0I to ISO1 for equi-biaxial conditions, as shown in Figure 22. Similarly. Zarnrik and Frishmtith [93) ohscrsed adefinite phase effect on the crack growth direction and mode of* failure using thin walled t ubularspecimens. In contrast. Liu and IDittmer [35, 771 concluded that tile dirc-ctfon and rate ol fatiguecrack growth wsere the same for cruciform specimens under out-of-phase and in-phase loadingconditions. These conflicting, results once again highlight the problems in deiermining represeni-tative stresses and strains under out-of-phase loading conditiorts.

40

90g 9

'I 'o.L

180* /"*270- 270

90 / 90'-I I/t' --- :ogo.

270

270

FIG. 20. HYSTERESIS LOOP FOR IN- AND OUT-OF-PHASE AXIAL AND TORSIONALSTRAINS FOR A 1% Cr-MO-V STEEL AT 20*C.(a) ij 4, 0 - 01, fa 0.5%,'(b) * 4, 0 = 90*, ca - 0.5%, f99).

41

S300

~200-

2 C

420

*' 0 0 -

-: i t- '

<'1"- i '1 -

: FIG. 21. BIAXIAL CYCLIC STRESS-STRAIN CURVE FOR'OUT-OF-PIIASI& tOADING [9'91

S~42.

, . •

0 a 0O106 0

z zZZ z a zEUE 0

00 9.0 0

o 02,, o1 0 0(o. ,.'fl. u •" 6.2 0O

o3 I052C u4 I I

,5 -12"6 v wp , I I I I

Number of cycles N, %10-

FIG. 22. CRACKLENGTH VERSUS NUMBER OF LOAD CYCLES FOR VARIOUS BIAXIALFATIGUE TESTS (711

43.

10. THE. EFFECTS OF BIAXIAL STRESS ON CRACK TIP PLASTICITY

The analysis of fatigue crack grouth behas iour in biaxial stress fields usine linear elasticfracture mechanics has proven inadequate as plasticity and an isotropy effects, are not considered.In real materials fatigue crack growth rates are dependent on the amount of local plastic deform-ation at the crack tip. Consequently. fatig~ie crack growth rates can he related. to thle plasticzone size rp, or alternatively to a critical crack opening displacement COD) at the crack tip, [orexample, Hoshide, Tanaka, Yamada and Taira (4. 84, 1001 showed a unique correlation bet~keen.the rate of crack growth da/dN and the crack tip opening displacement as shown- for esaniple.in Figure 23. These authors concluded that the range of COD is a parameter which is uniquelyrelated to the fatigue crack growth rate under both elastic-plastic and general Nield conditions.In addition, they (4, 69] concluded that of' thL three fracture mnechanics- parameters e'.amined.i.e. the stress intensity factor rance AK. the etftectise srress intensits factor raneea AA,.t1 ts hlichaccounts for crack closure effects), and the ('O1) range, the latter was the best for correlatingcrack growth rate's dalldN in all stress states. Figure 23 shows that the COD range produce-,a unique relationship with dadN for all stress states compared %ssith thle relationship bet%%eenda~dNV and A~K shown in Figure 24.

The plastic deformation response of a material is dependent on the stress state f261 ia'.will be described in the last section), hence ri, and ('01) may also he expected to be stress statedependent. At present, to the best of the author's knowl[edge, no experimental dlta oin thle effectnsof biaxial loading on plastic zone size .bai~e been reported in the literature. 1-loweser. seseralelastic-plastic analyses have been unde *rtaken to establish the dependence of- both: r1, and(O1)on the biax -ial stress ratio. The elastic-plastic analyses of Miller' and Kfouri 13'41. Hilton 1411and Tanaka et al. [4) have suggested that the plastic /one size decreases as the NOMINA~Lbiaxial stress ratio %~ increases from I to -i- I as shoýs n. for example. in I-mure 25. Similrl~.0,Liu and Dittmer[35. 36] found that the plastic /one si/es for tension-compre'ssnrn ratiost A \et*are significantly larger than those for uniaxial tension (A 01 or tension-tension (.1 %elbiaxial loading conditions. The plastic /one si/e incre~as%e as, the tension-compression ratiobecomes more itegative. How4eser. these authors found that tile plastic /one %sie% for biasialratios of O-5 to I 0 are approximateiN the same and only %!ightll smaller thain those for uiiiimiialtension.

The elastic-plastic an~alyses of Adams [l011 and Licbow it. Lee and I I'lis (311 producedsimilar results to the abose. that is, the plastic zone si/e decreased its tile LOAL~ biaxial stressratio k increased from negative values, to a value of approximatlc\ 1. TFhe plastic zone si/e reacheda mihimum value at a biaxial stress ratio oif approximately I and then increascd %ser ' rapidl\as k became greater than I as shown, for example. in Figure 2f'ai. Adams also found at similardependence of COD on the biaxial stress ratio, ats shown in Figure 26b)b. ihe hiax~ial ratio atwhich rp and COD reach minimum salues. and thle percentage reduction in cach case. dependon the ratio of the applied yield stress normal ito thle crack. When thle applied yield stress ratiois 0-9 the plastic z-3ne size is decreased by up to 65',, and the crack opening displacenilent h%33",1, compared with the 'uniaxial tension salues. Adatm% concluded that thve fatigue crack groswthrate also will decrease and reach a' minimum' %alue ats the biaxial ratio increases because of thereduced values of either ,l, or COD. Hosseser. tile minima in the curses (if ri, and ('01) sersusbiaxial stress rat ir at biaxial ratios of approximately I may result fromn it chainge in the crackioading configuration., As mcntioned iný Section 2. 1. thle crack loading confip'uration for biaxialstress ratios greater. than I -is, not the same as that fo'r ratios less than 1.

Miller and Kfouri f141 performed both an elastic ;ind an elastic-plaslie analtsis to istablish'the dependence of the crack opcining displaceni~nt on the biaxial stress ratio. Results fromthe elastic analysis suggested that the crack opening displacemcrnt %as independcnst oftthe biaxialratio. However, the elastic-plastic analysis suggested that thle ('01) salue was NifinIifiantly larg'erfor pure shear (A - ) than for pure tension 4. 0). In addiftion the samue of ('Of) for puretension was slightly larger titan that for equibiaxial tension (.A +I1. The resuilts of this elastic-plastic analysis are in agreement with those of Illoshide and co-workers 18~41 jas %hi*ssn in Figure27) and those of Adams. This latter analysis, predicted that for an applied sield sitress rittio'ofbetween 0-5 and 0.6. as used by Miller and Kfouri. the (01) %.dlse would decrease %with in-creasing k until a minimum was reachcd at a valuc of( k of: approximately 1.

44 -

SM41C ao',. 153 MPa

L" d'/dN 3.96 x 10'•1•€2501

EE

z•I0- 7 _ ' . j

o 0

200 a

Z. 0 C

00 ic

5, x I0-ý 1OCOO range (250 gm behind the tip), A05250, mn

FIG. 23. RELATION BEYW;EEN CRACK GROWTH RAITE AND CRACK OPENING

DISPLACEIMENT RANGE ,J841 " .

SM41C o',••15 3 M Po

E -b

2E-

10

1 .0

o o -1. &

00 50n

250 intes ty !acto ng t, 4K, , m,G G. 23.ARELBTWEON CRACK CRACK GROTE ANTRES ANR ENITN

445

S, -- ,

I I

0 OO 50

,i, ,. . . . . I I , I I• . i

-a -a, iss 151 mpa

______!a',.666 6 MPGO.c=[-o I

N \a, -a .176.27 MPa

CRACK 'TIP

FIG. 25. EFFECT OF STRESS STATE ON CRACK TIP- PLASTIC ZONE SIZE: '341

1 46

0 a 095

0090

08 40

0 7

06 080

05 075

. 04 0.70

040

-03 065":0 3 20 0 0 2-0s, S,S,"

FIG. 26 (a) VARIATION IN PLASTIC ZONE SIZE WITH INCREASING BIAXIAL

STRESS RATIO [101]

(b) VARIATION IN CRACK OPENING DISPLACEMENT WITH INCREASINGBIAXIAL STRESS RATIO [1011

[-General yeld (Mises l

--- PZSI' % -- 0.671S• CTOD

0.o I "7,2- 7 •0 6%

.. a.

• g ~~~0 335, • '" ,.0-

, I . 1

Bioxiol stress rotto, X

FIG.' 27. EFFECT OF BIAXIAL STRZSS RATIO ON PLASTIC ZOnE SIZE AND CRACK TIP

OPENING DISPLACEMENT RELATIVE TO THE VALUES IN UNIAXIAL TENSION,

0 0. (o w UNIAXIAL YIELD STRENGTH AND 0' IS THE cTRESS It! THE

- CENTRE OF *HE CRUCIFORM SPECIMEN IN THE Y DYRECTION [841

47

L NIEL-.-

The effects of biaxial loads on COD have been established experimentally h. a rew•, %orkers,Abou-Sayed et al. [102] found that the average COD value per unit load depends almost linearlyon the biaxial stress ratio, as shown in Figure 28. The rate of decrease in COD, with increasingbiaxial ratio is greater for long cracks compared with short cracks. Hoshide. Tanaka and Yamada[84] also found that the COD valies decreased with increasing biaxial stress ratios when cyclingaround zero mean stress values (R . I ). Consequently, the fatigue crack growth rate, alsodecreased with increasing values of A, as show. n in Figure 24. However. the COD) w-as indpendentof the biaxial stress ratio when positive mean loads were applied to each cycle and the minimumloads were zero (R = 0).

Fatigue crack growth rates can be influenced by crack clo.ure effects as initially suggested, by Elber [102], even under cyclic tensile loading conditions. Elber proposed that the residualdeformation, associated with the plastic zones left in the wake of a moving fatigue crack. couldclose the crack tip and thereby reduce the effectike portion of the loading cycle. Since crackclosure is dependent on the plastic zone size. this phenomenon should also be dependent onthe biaxial stress ratio. Ogura and co-workers [104] undertook'a finite element analsi-, to predictthe fatigue crack closure behaviour for varous biaxial stress ratios. They found that the ctfecti'cfraction of the applied stress range decreased, and hence the amount of crack closure increased.as the biaxial stress ratio increased from - I to + I when cycling about zero mean stresses(R . . I). However, the biaxial stress ratio had no significant effect on crack closure %shencycling between zero and a tensile stress (R 0). The experimental results o"rHoshide. Tanakaand Yamada [841 are qualitatively in agreement with these analytical results. Those authorsfound that the effective stress intensity factor range decreased by approximately 35. 1,and conse-quently, the fatigue crack growth rate decreased by a factor of approximately 4. as V increasedfrom - I to + I when R I. The effect of ibiaxial stress ratio on crack closure %%as not signi-ficant when the sytess ratio ,•as R 0.

11. THE EFFECTS OF BIAXIAL STRESS IN THE PRUSEN(E OF A NOT(CI

The initiation and subsequent growth of fatigue cracks in engineering Component.; is usuall.yfrom geometric discontinuities such as a notches or holes. These discontinuities act as strcssand strain concentrators* and. thereby introduce localised. highly strained region,% esgleriencingcomrplex stress conditions. Consequently, the fatigue strength and fatigue life of componentscan be seriously reduced in the presence of a geometric discontinuity.

The usual method for predicting the fatigue strength or fatigue life of a notched Componentis to describe the notch severity in terms of a concentration factor. ,uch as the theoretical, telastic)stress concentration factor Kt or the fatigue strength reduction factor Kr. These correction factorscan be most useful when designing a component for an infinite life, however. they arc less usefulin the finite life regime [105]. For example, the theoretical stress concentration factor K, is,. bdefinition, independent of mean stress-Tc7.-anJ--d0cs not consider'the "size ctt'.cf' i.e.. largecompo;,nts fail sooner than 'small components. In addition. thecxperimcntall, determinedstrength reduction factors Kr are usually smaller than (and often inconsistently related* to) thecorresponding theoretical Kt values [106].' and are lifledependent.

Leis and Topper (106] suggested that three main factors are responsib.hle f'r thed,•2repanciesbetween the theoretical Kt and experimental Kr values : 1) the type of failure criteria us.d. 2) theinelastic, local stress-strain response not being aceurately accomnted fiW. and 3) the effecVts ofloca; multiaxial stresses. Newman 11071, and H1opper and Miller 1611] ha%,r shown analstic'allythat the severity of the notch effect depends on both the methiod used to represent the notchcorrection factor and on the stress state, as shown, in Figures 29 and 30. It should he noted thatNewman's results, Figure 29. were presented in terms of a stress intensity factor based on the

It should bie noted that surface roughness and metallurgial defects. such. •z. porosity,inclusions and heat affected zones also can act as stress concentrators..

48

Non - lhnew load COO curvos

*.C*p for 0 5- "ch crock

All !n.00 based ot COO - *( I P.74)

00

-. 03

I-

a002

404

010

0

FIG. 28 EFFEcr OF PIAXIAL STRESS RATIO ON CRACK OPENING DISPLACEMENT (1021

49

1.2

1.0 - -

F

.2

I0 12 1.4 2.

FIG.29. STRESS INTENSITY CORRECTION FACTORS FOR CRACK EMANATINGFROM A CIRCULAR HOLE IN AN INFINITE PLATE SUBJECTED TO BIANIAL STRESS [107]

020

Vic

0150

complex variable method of Muskhelishvili and a modified boundary collocation method-noton LEFM. In contrast. Hopper andt Miller's approach (Fig. 30) %ai to define a notch contri-bution factor as the non-dimensional ratio of th." equivalent crack length (0hich represents acrack gro%%ing in a notched field) to the crack length in an unnotched field %hcn both crackshave the same growth rates. The results oif both analyses are in agreement. For example. shearloading (A ý - I) gives the highest stress intensity correction flactors and equibiaxial loading(A = + I) gives the lowest. Therefore. the fatigue crack rate is expected to increase for a crackemanating from a notch when , -.. I and decrease when A - 1. I, %hen compared with thegrowth rates in unnotched specimens. This trend is in agreement %%ith the experimenlal resultsof Hopper and Miller. Figures 12 and 31.

Hopper and Miller also calculated the notch contribution factors from their experimentalresults. As shown in Figure 30. the closest agreement between the theoretical and the experi-mental. notch contribution factors is for equibiaxial loading (l, -.1) %hereas the greatestdifference is for the shear loading case 0 - I). Hopper and Miller suggested that the changein plastic zone size with stress state was responsible for this trend. The plastic zonesize at a cracktip ig largest under shear loading conditions. Figure 26. Therefore. the elastic anaksis used todetermine the notch contribution factor would be less accurate for this stress state comparedwith equibiaxial tension. This is in agreement with the second factor suggested by Leis and Topperto explain the discrepancies betwveen the theoretical K, and the experimental Kf %alues.

12. MICROSTRUCTURAL ASPECTS OF XUITIAXIAI. FATIGUE

The review so far has concentrated on the engineering aspects of multiaxial fatigue as mostresearch has been conducted in this area. Very few papers have been published on the micro-structural aspects of fatigue in complex stress states. For example. Habetinek and co-workers[26, 1,08] are the only researchers known to have studied the etfects of stress state on the deform-ation mechanisms and microstructure of nmetals during fatigue. They observed the dexelopmeniof the dislocation substructtre and the mode of failure- of a low-C steel under uniaxial, biaxialand treaxial stress states.

Habetinek and co-workers found that increasing the number of loading axes incrcasedthe resistance to dislocation mobility, and thereby reduced the degree of stress relaxation andretarded the development of the dislbcation substructure. Under uniaxial loading conditions acellular dislocation substructure was forming after 20",, of the fatigue failure life. How ever.under biaxial loading conditions, a similar substructure wa's not observed until 90", of thefatigue life had elapsed. No such substructure was observed in any of tht3 fatigue specimenstested under triaxial conditions. The increased resistance to didlocation motion and substructureformation which occurred as the stress state changed from uniaxial to triaxial was associatedwith a decrease in fatigue strength. In addition, the mode of failure %as dependent on the stressstate. Striations were observed on the fracture surface-under uniaxial loading conditions indi-eating a relatively continuous rhode o" crack growth. However. no striations %ere observeduinder triaxial conditions as the fracture surface consisted of smooth, brittle-type facets. A mixedmode of failure occurred under biaxial loading conditions with both small areas of striationsand brittle facets being present.

"Tanaka and co-wo-'kers [4, 75J observed a similar change in mode of failure in cruciformshaped specimens of a 0-04`%; C steel. The fracture surfaces resulting from the use of positivebiaxial stress ratios were relatively smooth, with secondary cracking and brittle-like facets beingfrequently observed. In comparison, for negative biaxial stress ratios, the fracture surfaces weremuch, roughter with secondary cracking and brittle-like facets being virtually absent. Theseauthors suggested that this change in mode of failure was a result of an increased hydrostaticstress component near the crack tip at larger biaxial stress ratios,

I. " .' S.

0

100- 0

0

20

a o 00 A +1

2 4

Hal track length x 101

FIG. 31. FATIGUE CRACK GROWTH RATES FOR NOTCHED SPECIMENS OF RR58A = 70MPa, yl.ax 105MPa 1681

52

S.. .*

The nature of the surface deformation and the direction and mode of crack croiuh af-oare dependent on stress or strain state [86. 871. Parsons and Pascoe [61 ftound that the patternof surface deformation. i.e. slip lines. %aries charaLteristically wýith the straini s•itC. In gcneral.there exists a preferred orientation for the slip lines in each strain state %%hich coincides %iththe direction of subsequent surface crack grouth. The change in the mode of crack gromkth % ithstrain state obserxed bý Brown and Miller [871 has been discussed in Section x.

Finally. the orientation of the crack path with respect to the loading axes is stres, statedependent [35. 39, 76. 109. 1101. For los biaxial stress ratios (.I < I) the crack path tends tobe stra'ght and perpendicular to the maximum principal stress ((u, ). 1-loseser. for high biaxialstress ratios (A > I) and therefore a different crack loading configuration. the crack. tends to

follow an S-shaped path centred on the initial straight notch, as shos'n for example in Figure 32.The cursature of the path increases vsith increasing biaxial stress ratio for %alues greater than I.

,13. SUINMMARY AND CONCLUSIONS

A review of the literature has sho%%n that there'are significant differences in the fatigue andfracture properties of materials and components when loadeld under uniaxial and multiaxialstress conditions. The effects of stress state on the %arious parameters used to describe theseproperties can be summarised as follows

(1) The ultimate tensile strength of steel increases by up to 18", and tile fatigue limitdecreases by up to 48,, as the stress state changes from uniaxial to biaxial to triaxial.

(2) The resistance of a metal to cyclic deformation is dependent both on the biaxial stressratio and the type of metal. Increasing the biaxial stress ratio from I (shear loading)to + I (equibiaxial loading) increases the resistance to cyclic deformation.

(3) The fatigue life of metals is dependent on the biaxial strain ratio. Increasing this ratio

from - I to + I can decrease fatigue lives in the high and lo%%-strain regimes by factors.of up to 10 and 20 respectively.

(4) Although there are conflicting results in the literature concerning the effects of biaxialstresses on fatigue crack growth rates, changeover tests hase shossn conclusiselv that

(a) An instantarneous change in the stress state from uniaxiaI-to-cqu-it+miai at thesame nominal stress applied normal to the crack decrcases the fatigue crack growthrate by a factor of 2. By. comparison. the growth rate is increased by a factor of4 when the stress state is suddenly changed from equibiaxial to-uniaxial, A similartrend is observed on, stiffened panels with the stifleners either cracked or intact.

th) An instantaneous change in stress state from uniaxial tension to pure shear, bysuddenly applying a cyclic compressive stress parallel to the crack. iacreases thefatigue crack gro.vth rate by a factor of.3.

(5) Experiments have shown that when cycling about zero mean stress the crack oleningdisplacement and the degree of crack closuie decrease with increasing biaxial stressratio. Both properties are independent of biaxial ratio when the R-value• is from zero

to a positive value.

53,

4Y

A2-1

H-- -

1=2.8

FIG. 32. CRACK TRAJECTORIES AT VARIOUS BIAXIAL STRESS RATIOS 11101

54

--* * .•L,, r *. '- -'- . .... .... . .. . . .• .- • . .. .. . . .

.-- * ..... .. . / . *..

(6) Fatigue properties' are allecte d bs out-of-phase biaxial stresses and the a"Sc~ialedrotation of the principal stresses as 1ollotss

(a) At ans gisen bi~axial ratio tItaigtuc strength is decreaised a,- the phase anele betw~eenthe normal stresses is increased.

Of) Yield is initiated at a lo'mer stress for in-phase compared wsith out-of-phaise loading.

*(c) Thecyclicst ress strain re-nonse ota material is affected by the phascangle Aiysteresisloops change shape and rotatton of thle principal .tresses produces additional'cydAic hardeniing.

(d) LCF life is reduced by uip it) a faictor of' 4 compared w~ith out-ot-phiase c~ ding -I

phatse angle of 90 eising the low'est life.

(e) Both the Tresca and octahedral shear strain criteria.. commonly used for desinnpurposes. are non-conserxati,,e under out-of-phase conditions.

(7) (a) Linear elastic fracture mechanics analyses which use -the so-called *Nnilrsolutio;'-to represent the local stresses and strains at the crack tip, predict that the xarious fractureparameters are independent of the biaxial stress ratio. liosse~r. %hen the second termof the series expansion is included. all of the fracture parameters sho. at biaxial load'depend.nce. Cexer in the elastic range.(b) Non-linear elastic-plastic analyses also show that thle fracture parameters aredependent on stress state. The malor part of this dependence comes from the inelasticmaterial response at the crack tip.

(8) The critical fracture load (a~nd critical stress intensity for fracture) are dependent on boththe biaxial stress ratio and Poisson's ratio. [racture loads increase with increasing,biaxial ratio for Poisson's ratios of less than and the rexerse occur,, wshen P isson\ratios are greater than

(9) The critical stress intensity, faictors for stillene'd panels., and cylindrical and sphericalshells containing cracks, increases swith increasin-2 biaxial stress ratio.

(10O) The stress intensity correction factors used to predict the fattigue lixes of nottihedcomponents are dependent on tlhe biaxial stress, ratio. Compared wsth uniaxial tension.a biaxial ratio of I increases the correction factors whereas a ratio of + I decreasesthe correction factors.

(11) The effects of a notch oin f~atigue criack growtih rates are greater for biakial ratios of-I and + I compared with at ratio of zerol.

.(12) The stress state affects thle deformation mechanisms and the deformed microstructure* of metals during cyclic loading. The resistance to dislocation movement and the form-

f ation of a recovered dislocation suh~siructure arc increased, and the degree of stress

relaxation is reduced, as the stress state changes from uniaxial to biaxial to triaxial.

(13) The or~ientation of the crack path and ifhe failure mode are dependent on the degree bi'I . multiaxiality.

55

REFERENCES

[I] LOHR, R. D. Biaxial high strain fatigue of.l.Cr. MoAN steel. Ph. 1). l Uni.,ersitvof Bristol. Bristol. 1977.

[2] LOHR. R. D. and ELLISON, F. G. Hia\ial high strain fatigue testingq of I r o \steel. Fatigue Engng. .tater. Strua,.. 3. 19N0. 19 37..

[3] BROWN, M. W. and MILLER. K. J. "Ib decades of progress in the asesincntof multiaxial .to,-ccle fatigue. in I oit-(. th, 144uiie at.nd Li/, Prc li, ton. AS.I %1 SI P770, C. Amzallag. B. N. Leis and P. Rabbe. lVd,,.. American Societ•. for Tetimg andMaterials. Philadelphia. 19S2, 482-499.

[4] TANAKA. K.. HOSHIDE. T.. NA..\MlI).\:. A. and TAIRA. S. Fatigue crack pro-pagation in hiaxial stress fields. Fllithu' Ln Enig. .Mlawr, Strt-t.. 2. 1979. INI 194.

[5] LEEVERS. P. S.. RADON.J. C...and CuLV.ER L. E. (rack grovath in pla,tic panel,under biaxial stress. Polrin'r. 17. 1976. 627-632..

[6] WANG. S. S.. CHIM. E. S. M. and.SO)(I[i. 1). F. Biaxial fatigue of libre-reinforcedcomposites at cryogenic temperat'ures. Part I fatigue fracture life mechanism%. J. Ing.Mater. Technol. (Trans. .4S.AE Seric-i Hl. 114, 1982. 128-136.

[7] KRE MP P, E. 'The Infltuene o/ Stair of Strv.s on I.nw-UiC-(le /1tnLue o! St t1'utt'ualMaterialv : A Literature Surie" and Inwerpreti•e Report. ASTM SIP 549, *\tcrticaiiSociety for Testing and Materials. Philadelphia. 1974.

[8] McDIARMI[. D. L. Failure crileria and cumuhttiae damagce i fatigue under multi-axial stress conditions. Ph.D. 7hews,. Cki Unisersitý. London 1972.

[9] FARSONS. M. W. and PASCOE. K. J. I.o'-c~cl fatigue under biaial stresses.Proc. Invin. Itech. Engrs... .188. 1974. 61. 74.

[10] BROWN, M. W. and MILLER, K. J. A theory for fatigue failure under multiaxialstress-strain conditions. Proc. h•tit. Mlec h. Lngr.%.. 187. 1973. 65 73. Also ,ee discussion

'ibid., 1973, D229-D244.

[II] MILLER, K. J: and BROWN, M. W. Multiaxial fatigue: a rceiew. in Pro(. o/ theInt. Syrip. on Low-Ciclh, Faligutt Strength and Ehrsto-phl.stic Behaviour 14 Jhacria/.%.K.-T. Rie and E. Haiback. Eds., Stuttgart, West Gcrmany. I)DV%. 1979,415-426.

[12] MILLER, K. J.. Fatigue under multiaxial stress strain conditions. in .AIhhanial atilThl'rtnul Behaviour of' tfetalliv Vaterials. Italian Physical Soc•eL•. North |lollandPublishing Co., Amsterdam, 1982. 145--1'64.

[13] BROWN, M.'. W. Multiaxialfatigue testing and -analysis'. in' F atig t t 'h ay •npera-utsue. R. P. Skelton. Ed.. Applied Science Publishers. London. 1983, 97-133. '

S[141 GARUD, Y. S. Multiaxial fatigue a survey of the art. JA 7st. Er:aI.. 93). 19X1,165-178.

4.

4-..

[15] ANDERSON. P. R. G. and GARRETT. G. C. Fatigue crack gro%%th rate sriationsin biaxial stress fields. hit. J. Fratct., 16, 1980, RI I I-R 116.

[16] EVANS. W. J. lDe/ormati'on amd tiilure wider multia.vild *trct'.'---a .uriclr oflahoratorttechniques and e.xperimentnal data. National Gas Turbine Estahhishment C.1 P. No. 1306.Her Majesty's Stationary Office. London. 1972.

117] BROWN. M. W. Los-,cle fatigue testing under multia\ial cle\.ated temperatureconditions, in .VPL Cont. 1h'usuremnent of tlikh Temnp'raiurt, ie chtanical Propertiesof Materials. Her Maiest%*s Stationary Office. London. 191. 185 201.

[I18 GOUGH. H. J. and POLLARD. H. V. The strength of'metal, under combined alternat-ing stresses. Proc. hitny. .%teh h. tingrv.. '131. 1935. 3 -103.

[191 GOUGH. H. J. and POLLARD. H. V. Propertic. of some materials for cast crack-shafts with specia! reference to combined alternating stresses. Proc. In.stn. Auto. Engrs..31. 1937, 821-893.

[20] GCUGH. H. J., POLLARD. H.I V. and (II.-NSIIAW, W. J. Some experiment, onthe resistance of metals to fatigue under combined stress. .4cronautical Rvse.ea/ch CoutncilWetno. .Vo. 2522. His Maiest,'s Stationary Otlice, London. 1951.

[211 FORREST. P. G. Fatigue of 1Metalh. Pergamon Press, New York._ 1962. 108-113.

[221 HABETINEK. J., Fatigue strengt h and fatigue failure at complicated states of strcss.Eng. Fract. Mecih., 8. 1976. 677-689.

[23] MORRISON. T. L. M.. CROSSLAND. B. and PARRY. J. S. C. Strength of thickcylinders subjected to repeated internal pressure. Proc. hltn. Ahlech. Engrs.. 174. 1960,'95-117.

[24] BLASS. J. J. and FNI)LEY, W. N. The influence of tho intermediate principal stresson fatigue under triaxi:d stresses. Bull.l.-in. So'. Txt,. Alater. Res. Std.. 7. 1967, 254- 261.

[25] FINDLEY. W. N. and TRAC:Y. J. F. The effects of the intermediate principal stresson triaxial fatigue of 7075'-T6 aluminium alloy. J. Test. ELial., I. 1973, 432-434.

[261 GARUD. Y. S. A new approach to the evaluattion of fatigue Under multiaxial loading.J. Eng. Mater. 'echnol. (Trans. ,ISMill" .Sýeries t1), 103, 1981. 118-125.

[27] BROVN. M. W. and MILLER, K. J. High temperature loss-cycle biaxial fatigue of.two steels. Fatigue Enpgnq. Mater. Struct.. 142), 1979, 217-229.

128] BROWN. M. W. High temperature multiaxial fatigue. Ph.D. Thesis, CambridgeUniversity, 1975.

,291 GONYEA. D. C. Method of loss-cycle fatigue design indluding biaxial stress andnotch effects. in Fatiiue ai Eh7leated Tetopraturev. -ASTM STP 520. A. E. Cardon.A. J. McEvily and-C. H. Wells. Eds. American Society for Testing and Materials,Philadelphia, 1973. 679-687.

[301 FUCHS. H. 0. Fatigue research with discriminating specimens. faftigu, E Lngn. Mater.Struct.. 2, 1979. 207-215.

[311 LIEBOWITZ. H., LEE. J. D. and EFTIS. J. Biaxial load effects in fracture mechanics.in Fracture Mechanics and Technology. Vol. 11, G. C. Sih and C. L. Chow. Eds., Leyden.Sijthoff Noordhoff International Publishers, Leyden, 1977. 1021-1037 also in Engntg.

Fract. Mech. 10, 1978. 315-335.

[32] LIEBOWITZ. H., LEE. J. D. and SUBRAMONIAN. N. Theorctical amd experimentalbiaxial studies in Fracture Mechanic,. N. Perrone. H. Licbositz. D. D.Muh, dle and W.Pilkey. Eds., University Press of Virginia. Charlottsville. 1978- 593- 629.

[331 LEE. J. D. and LIEBOWITZ. H. The non-linear and biaumal etfects on encrgy releaserate. J.-integral and stress intensit, factor. EnDng. Fract. Mecih.. 9. 1977. 765- 779.

1341 MILLER. K. J. and KFOURI. A. P. An elastic-plastic finite clement analysis of cracktip field under biaxial loading conditior,. Int. J. Fract.. I0. 194. 393-404.

1351 LIU. A. F. and DITTMER. D. F. Lflect otf multiaial loading on crak gros1th.Northrop Corporation A4ircralt Group Report 'so. Nor- 78. 161- &tof. 1. 2 anad3. AFI- D L-TR-78-175-VoN. I. 2 and 3. Air Force Flight D inamics i aborato". wright-PattersonAFB. 1978.

.1361 LIU. A. F.. ALLISON. J. E. DITTMER. ). F.. and YAMANE. J. R.. Effect of biaxialstresses on crack growth, in Fracture Me -'chanics. ASTM STP 677. C. Smith Ed..American Society for Testing and Materials. Philadelphia. 1979.5--22.

(37] EFTIS. J.. SUBRAMONIAN. N. and LIEBOWITZ. H. Crack border stress anddisplacement equations revisited. Eiwtg. Fract. Mlech.. 9. 1977. 189-210.

(38] EFTIS. J.. SUBRAMONIAN. N. and LIEBOWITZ. H. Biaxial load effects on thecrack border elastic strain energy and strain energy rate. Ibid. 9(41. 1977. 753 764.

(391 JONES. D. L. and EFTIS. J. Fracture and fatigue characterization of aircraft struc-tural materials under biaxial loading. ,4irf'orce Ofice of".Viiwiific Re.warch Report N.i.AFOSR-81-O)856. George Washington University. Washington DC. 1911.

[40] HILTON, P. D. and HUT(HINSON. J. W. Plastic intensitv Gtictr% t'or cracked pl.ttc,Engng. Fract. Mech.. 3. 1971. 435- 451.

[411 HILTON. P. D. Plastic intensity factors forecracked plates,,ubhectcti tf )Hawiaf loading.Int. J. Fract., 9(2). 1973. 149-156.

[421 SMITH. E. W. and PASCOE. K. J. The behaviour of fatigue cracks ubi•ect to appliedbiaxial stress a review of experimental An idencc. Eatigu, Lmngng. Miter. "tru(t.. 6f43).1983. 201-224.

[43] KIBLER-, J. J. and ROBERTS. R. The af1"ct% oft iasi:al stresses on fatigue and fracture.

J. Engng, Ind.. Trans. .ASME. 92. 1970. 727-734.

(44] UEDA. Y., IKEDA. K.. YAO. T., AOKI. M., VOS!I.' T. and SHIlRAKLURA,.Brittle fracture initiation characteristics under hio xial loading. in Fratwre 1977. I(.'F4.Vol. 2. Waterloo, Canada. D. M. R. Taplin Ed.. I niversvdy of Waterloo, Pres,. Ontario.1977. 173-182.

[45] UEDA, Y..,IKEDA. K., YAO. T. and AOKI. 4. Characteristic' (if brittle fractureunder general combined modes -including those under biaxial tensile loads. Engrng.Frac'. Me(ih.. 18(6). 1983. 1131 -1 1S.

[46] LEEVERS. P. S., CULVER. L. E. and RADON, . C. Fatipiecrack growth in PMMAand rigid P.V.C. under biaxial stress. Ibid. 11. 1971. 487-498.

[471 SWIFT, T. The effects of fastener flexibility and .tl encr C metry on the stress intensityin stiffened cracked sheet. in Prospect's of t'ract , Mrchlueks. U.' C. Sih. H. C. VanEllst, and D. Broek, Eds., Nordhoff International Publi•hin. Leyden. 1974. 419-4%6.

S -I. _ __•

[48] SWIFT, T. The application of fracture mechanics in the d.eelopment of DC-10 fuse-lage. AGARD .4G-1 76, H. Liebo%%itz Ed.. NATO. 1974. 226-293.

[49] MAYNOR. H. W_,. WALDROP, R. S. and BUSCH, C. C. Crack tolerating abilityof high-strength biaxially stretched cylindrical pressure vessel containing a surfacecrack. Int. J. Alech. Sci.. 14, 1972. 205--213.

[50] BHAT, G. K. A study of the behaiour of small pressure tisses under biaxial stressconditions and in the preserce of surface cracks. Final Report Contracrt ao. Nonr-3764(00) IX), submitted by Melon Institute to the Navv Research Laboratory. Washing-ton DC. 1963.

[51] ERDOGAN. F. and RATAWANI. M. Vracture of cylindrical and spherical shellscontaining a crack. Nucl. Engn,. and Desig.n 20. 1972. 265-286.

[52] BENHAM. P. P. Torsional strain cycling fatigue of t:opper at Io%% endurance. J. hisf.Metalv. 91. 1962-63, 404-407.

[53] MacKENZIE. C. T., BURNS, D. J. and BENHAM, P. P. %\ comparison of the uni-axial low endurance fatigue behaviour of two steels. Proc. latnm. Mhch. L)rv., 180I.1965-66. 392-402.

[54] WHITE, D. J. and HORWOOD. G. P. Cyclic stress'-strain relations from bendingand push-pull tests. J. Strain ,,nalisis. 6. 1971. 99-107.

[55] HAVARD. D. G. and TOPPER. T. H. Biaxial fatigue at low endurancei a criterionfor mild steel. Ontario tlwdra Research Quarterl.r 234). 1971. .I-- 10.

[56] HAVARD. D. G. and TOPPER. T. H. A criterion for biaxial fatigue of mild steel atlow endurance. Proc. Is, Int. Cont. opt Structural M!echanicn in Reactor 7ichnyolohvy.Vol. 6. Berlin. 1972. 413-432.

[57] BROWN, M. W. and MILLER., K. J. Biaxial cyclic deformation behaviour of steels.Fatigue Engtig. Mater. Struct., I. 1979, 93-10(6.

[58] MANSON. J. J. Behaviour of materials under conditions of thermal stress. NACATN-2933, National Advisory Committee for Aeronautics. Cleoeland, 1954.

[59] COFFIN, L. F. Jr. A study of the effects of cvclit thermal stresses in a ductile metal.ASME Transactions, 16. 195%4. 931-950.

[60] BASQUIN. 0. H. The exponential law of endurance tests. Proc. ,S/A". It). Part If.1910. 625-630.'

[61] ELLISON. E. G. and ANDREWS. J,. M. H. Biaxial cyclic high strain fatigue ofaluminium alloy RR58. J. Strain Analyvis. 8(3), 1973. 209-219.

162] VOKOBOKI. T.. YAMANOUCHI. H. and YAMAMOTO. S. Low-cycle fatigue ofthin-walled cylindrical specimens of mild steel in uniaxial and torsional tests at constantstrain amplitude. h/t. J. Fra.t. Mech. I, 1965. 3 1.3.-

[63] LIDDLE. M. and MILLER. K. J. Multiaxial high strain fatigue. Proc. Third int.Con,:'. on Fracture, Vol.,4. Munich, 197.1. V-523'A.

[641 PARIS. P C. and SIH, G. G. Stress analysis of cracks. in Fracture Toughness Testin,and its Apoplication. ASTM STP 381. 'American Society for Testing and Materials,Philadelphia. 1965. 30-141.

[65) CHRISTENSEN, R. H. and HARMON. M. B. Limiiation of fattgac c-ck researchin the design of flight vehicles structures, in Fatigue Crack PropAgarioi. ASTM STP415. American Society for Testing and Materials. Philadelphia. 967. 5-23_

(66] JOSHI. S. R. and SHEWCHUK. J. Fatigue crack propagatitm in a biaxiat stress field.Exp. Mech.. 10. 1970. 529--533.

[671 .GU RA, K.. OHJI. K. and OH KU BO. Y. Fatigue crack gro%.th under braxial loading.Int. J. Frac'.. 10(4), 1974. 609-610.

[68) HOPPER. C. D. and MILLER. K. J. Fatigue crack propagation in htaxial str•ss fields.J. Strain .4naliysis, 12(I). 1977. 23 28.

[691 MILLER. K. J. Fatigue under complex stress. Mtaf Stiente It. 1977. 432-438.

[701 LEEVERS, P. S.. RADON. J. C. and (ULVER. L. E. Fatigue crack retardationinduced by biaxiality variations. In/. J. Fract.. 13. 1977. 874-876.

-[71) KITAGAWA. H., YUUKI. R. and TOHGO. K. A fracture mechanic-, approach tohigh-cycle fatigue crack gro%%th under in-plane biaxial loads,. Ft tigue -n. Maiter.Struct., 2. 1979. 195-206.

[72] CHARVAT, J. M. H. and GARRETT. G. G. The development of a closedloop servo-hydraulic test system for direct stress monotonic and cyclic studies under bi..xiai loading.J. Test. Eral. 8(I). 1980. 9_- 17.

[73) POOK. L. E. and'HOLMES. R. Biaxial fatigue cracK prowth tests. in Pmc. uW an /it.Conf. 'on Fatigue Testing and Design. Publ ('ily Unisersity London. Ro.al AircraftEstablishment Aeronautical Research Council. 1976. 36 I--30 -22.

[74] TAIRA. S.. TANAKA. K.. OGAWA. S. and KAN. M. Fatigue crack propagationin medium caibon steel under biaxial, stress cycling. P Jpn. (rwto .flater. Rev.21st Tokyo 1977. Soc. Mater. Sci. Kyoto. Japan, 1978. 50-55.

(751. TAIRA. S.. TANAKA. K.. KAN. M. and YAMADA. A. Effect of, non-singularstresses on fatigue crack propagation in biaxial strcis felds. Prar. Jim.. Ci'niier. 'talttrRes. 22nd, Kyoto 'Q78. Soc. Mater. Sci.'Kyoto. Japan. 1979. 13%- 137.

[76] TRUCHON. M.. AMESTOY. M. and D)ANG-VAN. K. Experimental study of fatiguecrack growth under biaxial loading., in, Au/tamees in Frarlure Rrsearch I fracture NI).ICFK. Vol. 4. D. Francois. Ed., Pergamon Press. Oxford. 1981. 1841 . 49.

[77) LIU. A. F.. DITTMEk. D. F. and HOLLOWAY. D. R. Crack growth under constaItIamplitude biaxial stresses. in JAint ('41l. on friprbi•kwia tVertruink. Socct'. forExperimental Stress Analysis. Hawahii. 1982. 900-906.

[78] HOURLIER, F. and PINEAU. A. Fatigue crack propagatlion eltawhxtur undercomplex mode loading. in .4drances in Fracture Re•swrvk (&ractt're.'•l. IC FS. Vol. 4.D. Francois, Ed.; Pergamon Press. Oxford. 1981. 18411-114g.

[79j1+ HOURLIER. F. and PINEAU. A. Propagation of falituc cractks under prtymnodalloading. Fatigue Engmg. Mater. Sutru'., 5(4), 1982. 217-302.

[801 LIU, A. F. Crack growth under variable amplitude Nitial. stres.ses. in Joint Ctoolu (lE-rperimenitl Mechanics. Society for Experimental Mechanics. Hawahii. 1982. 907 912.

[811 GARREýT, G. G.. ANDERSON, P. R.,G. and CHARVAAT. i. M. H. i'M influenceof biaxial stresses on high cycle fatigue crack propagatioam.-Lw. Mi-h.. 20. 19W0, 44-52.

[82] MADDOX. S. J. The effect of means stress on fatigue crack peopagatiorn -a literaturereview. Int. J. Frct.. 1103j, 1975. 389-40X.

[83] ZAMRIK. S. Y. and SHABARA, M. A. The effect !" ,stress ratio on fatiue craickgrowth in a biaxial stress field. J. Preys.. Vess/ T'chnol., 1977. 137-143.

[841 HOSHIDE, T., TANAKA. K. and YAMADA. A. Stress ratio effect of fatieuc crackpropagation in a hiaxial stres- field. Fati•ue Engfnv. Muter. Struct.. 40t). "198. 355 366.

[85] FORSYTH. P. J. E. A tmet •tage process of fatigue crack grossih. Proc. Cratk Pro-pagation Srnip.. Cranfield. H\.SO. London. 196i. 76--94.

,[861 PARSONS. M. W. and PASCOF. K. J. Obsersations of surface dcformation.:crackinitiation and crack growth in Io\%-c.cle fatigue under hiaxial stre,. Mater. Sci, ng..22. 1976. 31-50.

[87] BROWN. M. W. and MILLER. K. J. Initiation and growth of cracks in hiaial fati-nte.Fatigue Engng. Mater. Struct.. I. 1979. 231-246.

188] BROWN, M. W. Lov cycle fatigue testing under multiamial stresses at clevatcdliempeia-tures. in Mtfeasurenwnt of Iii,'h , lemperature t lechlrnical Propertie• ol .Alotcrftal'. M. S.Loveday. M. F. Day and B. F. Dyson, Eds.. Her Majesty's Stationary Office, London.1982, 185'-203.

[89] ANSTEE, R. F. W. and MORROW. S. Wi. The effects'of bitaxial loading on thc pro-pagation of cracks in integrally stiffened panels. Proc. iO1h .S~imp. I.C.,-l.F.. A Nlaenhaut.Ed.. Aeronautics Administration, Belgium. 1979. 3-9 1.3.9 40.

[90] RATWANI. M.. M. and WILHEIM. I). P. l)esclopment and e.zluation of methodsof plane stress fracture anlalysis-- application of the residual strength prediction techniqueto complex aircraft structure%. Alir Force Flight DVnvtuipy I.Laoratort Report .IFf.DL-TR-73-42, Part Ill. Wright Patterson AFB. 1975.

[911 RATWANI. M. M. and WILHEM. D. P. Influence of biaxial loading on anal)sisof cracked stiffened panels. Enving. Fract. Mech.. II. 1979. 585 593.

[92] MORROW. S. M. Crack propagation in a stiffened panel .%ith either an intact orbroken stiffener under uni~axial or biaxial loading. R,4IE-T,%f-S7'RUCi994, HMSO.London. 198 I.

?931 ZAMRIK. S. Y. and FRISHMUTH. R. E. The effects of out-of-phase biasial straincyclic on low-cyclc fatigue. Ep. Mlech.. 13. 1973. 204- 209,

f94] rOOR. P. M.' A unified engineering approach to the prediction of multiaxialfatiguefracture of aircraft structure. Eng. Fract. Melv:.; 7. 1975, 731 74.1.

[951 KANAZAWA. K.. MfLLER. K. J. and BROWN. NJ. W. Low-c.cle fatigue under -

oui-of-phase loading conditions. J Engng.. ItteYr. 7iechnol. (Tranv ,4SMi Series ti).99. 1977. 222-228.

1961 GRUBISIC. V. and SIMBUJRGER..A. Fatigue under combined out-of-phase multi-axial stresses. in Fatigue Tv.stiig andIDc.-imn, 'V.l. 2. S; F. E. Int. Conf. City University,London. 1976, 27. 27'8.

1971 NISHIHAR'\..T. and KAWAMOTO. M. The strength of metals under combined

alternating bending and torsion with a phase difference. Meroirs. College oLngim'eringKyoto Imperial Unirer.irv. II. 1945. 85--112.

lem

[98] LITTLE. R. E. A note on the shear stress criterion for fatigue failure under combinedstress. Aeronaut. Q., 20. 1969. 57-60.

[991 KANAZAWA, K., MILLER, K. J. and BROWN, M. W. Cyclic deformation of I'%,Cr-Mo-V steel under out.of-phase loads. Fatigue Engng. Mater. Struewr. 2. 1979.217-228.

[1001 TAIRA. S., TANAKA, K. and OGAWA. S. Fatigue crack propagation law underelasticiplastic and general yield condition. J. Mater. Sci. Japan; 26, 1977. 93-98-

(101) ADA MS, N. J. I. Some comments on the effect of biaxial stress on fatigue crackgrowthand fracture. Engng. Fraci. Mech.. 5, 1973. 983-991.

(102) ABOU-SAYED, i. S., BROEK, D.. FORTE, T. P. and STONESIFER, R. B. Planestress fracture under biaxial loading, in. Advances in Fracture Research (Fracture "81)ICF5. Vol. 4, D. Fracois, Ed., Pergamon Press, Oxford. 1981, 1708-1713.

(103) ELBER, W. The significance of fatigue crack closure, in Damage Tolerance in AircraftStructures, ASTM STP 486, American Society for Testing and Materials. Phi!adephia.1971, 230-242.

[104) OGURA, K., OHJI. K. and HONDA, K. Influence of mechanical factors on the fatiguecrack closure, in Fracture 1977, ICF4, Vol. 2. D. M. R. Taplin, Ed.. University ofWaterloo Pres, Ontario, 1977, 1035-1047.

[1051 SMITH, R. A. and MILLER, K. .t. Fatigue cracks at notches. Int. J. Meck. Sci.. 19,1977, 11-22.

[1061 LEIS, B. N. and TOPPER. T. H. Long life strength reduction due to local biaxialstate of stress. J. Engng. Mater. Technol. (Trans. ASME Series H). 99(3). 1977. 215-221.

[107] NEWMAN, J. C Jr. An improved method of collocation for the stress analysis ofcracked plates with various shaped boundaries. NASA Report TN D-6376. NASA,Washington DC. 1971.

[1081 KOUTNIK. M., HOLUB. J.,-and HABETINEK. J. Some microstructural aspects ofmultiaxial loading in low-C steel. Practical Metallography 12. 1975. 33-41.

(1091 RADON, J. C., LEEVERS, P. S. and.CULVER, L. E. Fracture toughness of PMMAunder biaxial stress. in Fracture 1977, ICF4, Vol. 3, Waterloo. Canada. D. M. R. Taplin,Ed., University of Waterloo Press. Ontario, 1977. I 1 3 71 1 18.

[1101 LEEVERS,'P. S.. RADON. 3. C. and CULVER, L. E. Frac-ture trajectories in abiaxially stressed plate. J. Mech. Phys. Solids, 24. 1976, 381-395.

BIBLIOGRAPHY

Multiaxial Fatigue Testing

ANDREWS. J. M. H. and ELLISON. E. G. A testing rig for cycling at high biaxial strains.J. Strain Analysis. 843). 1973. I6<--175.

BROWN. M. W. and MILLER. K. J. A biaxial fatigue machine for elesated temperature testing.J. Test. Eral., 9(4). !981. 202-208.

PARSONS. M. W. and PASCOE. K. J. Deselopment of a biaxial fatigue testing rig. J. StrainAnalysis, 10(l). 1975. 1 9.

PENN. R. W., FONG, J. T. and KEARSLEY. E. A. Experience in data acquisition and reduc-tion for a biaxial mechanical testing program. in Use of Computers in the Fatýgue Laboratory.ASTM STP 613, American Society for Testing and Materials. Philadelphia. 1976, 78-93.

WILSON. I. H. and WHITE. D. J. Cruciform .specimens for bwxial fatigue tests: and investig-ation using finite element -anal)sis and photoclastic coating techniques. J. Strain .4,na/lsis.6. 1971. 27-37.

Multiaxial Fatigue Criteria

LEIS, B. N. and LAFLEN. J. H. Problems in damage analy~is under non-proportional cycling."J. Engng. Mater. Tech. ITrans. ASME Series I1). 102. 1980. 127-134.

McDIARMID. 1). L. Cumulatie damage in fatigue under m'ultiaxial stress conditions. Proc.Insta. Mech. Eng.. 188. 40,74. 1974. 423-430 and D101-D103.

McrOWELL. D. L., SOCIE. D. F. and LAMBA. H. S.- Multiaxial nonproportional cyclicdeformation, in Low-Crele iatique anl I.ite Prediction. ASTM STP 770. C., Amraliag.B. N Leis and P. Rabbe. Eds.. American Society of Testing and Materials. Philadelph ia.1982. 500-518.

MOWBRAY, D•.F. A hydrostatic strevs-scnsitise relalionship for fatigue under biaxial strc,%conditions. J Test. Eral.. 8g1). 1980. 3-8.

SINES. G. and ONGI. G. Fatigue crit.ria under combincd stresses or starins. J. Eng. Mat r.Tech. (Trans. ASME Series I1). 103. 1981, 82-90.

Multiaxial Fracture

DALLY, T. W. and SANFORD. R. J. (•lassilication of stress intensity factbw% from isomhromatcfringe patterns. Exp. ffech.. 18. 1978. 441-448.

N

MILLER, K. J. and KFOURI. A. P. A comparison ofr lastic-plastic fracture parameters inbiaxial stress states, in Elastic-Plastic Fracture, ASTM STP 668& J- D. Landes, J. A. Begleyand G. A. Clarke, Eds., American Society for Testing and Materials, Philadelphia. 1979,214-228.

SMITH. S. H. Analysis of energy quantities for fracture under biaxial stresses, in Prospectsof Fracture Mechanicr. G. C. Sih, H. C. Van Elst and D. Broek. E4s. Noordhoff Inter-national Publishing, Leyden, i974. 367-388.

Multiaxial Low-Cycle Fatigue

BLASS, J. J. and ZAMRIK. S. Y. Multiaxial low-cycle fatigue of type 304 stainless steel. Proc.Symposium on Creep-Fatigue Interactions, ASME, MPL New York. 1976. 129-159.

ELLYIN, F. znd VALERE, B. High strain multiaxial fatigue. J. Engng. Mater. Tech. (Trans.ASME Series H), 104, 1982, 165-173.

ELLYIN, F., LAFEBVRE. D. F. and NEALE, K. W. High strain biaxial fatigue of 2024-T351 aluminium under combined axial stress and torsion, in Advances in Material Technologyin the A"nericas-1980, Vol. 2. Materials. Processing and Performance Materials Division,ASME, 1980. 17-21.

KONTER, A. W. A. and JANSSEN, G. T. Effect of biaxial loading and geometry on predictionof low cycle fatigue. Proc. 5th /mt. Conf. on Structural Mechanics in Reactor Technology,Berlin, Division L. 1979. paper 13/I, 1-10.

LEFEBVRE, D.. NEALE. K. W. and ELLYIN. F. A criterion for low cycle fatigue i'ailureunder biaxial states of slress. J. Eng. Mater. 7ech. ITrans. ASME. Series H ). 103. 1981, I -6.

LOHR, R. D. and ELLISON, E. G. A simple theory for low-cycle multiaxial fatigue. Fatigue8ngng. Mater. Struct. 3. 1980, 1-17.

MOGUEROU. A.. VASSAL. R., VESSIERG, G. and BAHUAUD. J. Low-cycle fatigue underbiaxial strain, in Low-Cycle Fatigue and Life Prediction, ASTM STP 770. C. Amzallag.B. N. Leis and'P. Rabbe. Eds., American Society for Testing and Materials., Philadelphia.1982, 519-546.

OHJI, K., OGURA, K., HARADA, S. and SENGAý H. Biaiial loo-cnl fatigue of anisotropicrolled steel'plates. Bull, JSME, 17(1031., 1974, 32-40.

OHJI, K., OGURA, K. and HARADA, S. Observation of low-cycle fatigue.crack initiationand propagation in aniwotropic. rolled steel under biaxial stressing. Ibid.. 18, 1975. 17-24.

Multlaxial Fatigue Crack Growth

BECK, E. J. Fatigue flaw growth behaviour in stiffened and unmiffened panels loaded in-biaxialtension. NASA-CR-128W04, Martin Marietta Corp. Den)er. Colorado. 1973..166p,

MILLER. K. J. Multiaxial fatigue. in Conference on Fall~e Teoing an d Design. Society ofEnvironmental Engineers, London. Vol. I. 1976. Paper N4 13.

SMITH. E. W. and PASCOE. K. J. Fatigue crack initiation and growth in H. Y. 100 steelsubie,', d to in-plane biaxial loading. Camnbridtge Uni. Engng. Dept. Tech. Rep. CUED/C-MAt.4 R. 95, Cambridge. England, 1982.

ZAMRIK. S. Y. and SHABARA, M. A. The application of fracture mechanics analysis to crackgrowth rate in a biaxial stress field, in The IV1Inter-American Conf. on Materials TechnologyVenezuela. 1975, 472-477.

Out-o/-Phase Multia vial Loading

McDIARMID, D. L. Fatigue behaviour under out-of-phase bending arid torsion. The Aero-nautical Journal of RAeS, 85. 1981. I 18-122.

McDIARMID, D. L. Fatigue under out-of-phase biaxial stresses at different frequencies. Paper

presented at the ASTM International Symposium on Multiaxial Fatigue, San, FranciscoDecember 1982. To be published in ASTM STP.

Notch Effects

GRANT, A. F. Jr. A general stress intensity factor solution for through cracked fastener holes.Int. J. Fract., 11(2), 1975, 283-294.

WALKER, E. K. Multiaxial stress-strain approximaition for notch fatigue behaviour: J. Te.l't.Eval.. 5(2), 1977, 106-113.

Multiaxial Creep-Fatigue Interactions

FONG, J. T. and PENN, -R. W. Multiaxial fatigue and creep at elevated tem;'.ratures andmathematical modelling, in Frarture Mechanics and Technology. Vol. I. 1977, 145-159.

HENDERSON, J. and DYSON', B. F. Multiaxial creep testing. in Measureiment offtth Tempera-ture Mechanical Properties of MAaterials, M. S. Loveday, M. F. Day and B, F. Dyson, Eds..Her Majesty's Stationary Office, London, 1982, 171-184.

LOBITZ, D. W. and NICKEL. R. E. Mulliaxial creep-fatigue damage. Nue.. Engng. Design,51(1), 1978, 61-67.

KELLY, D. A. Problems in creep testing under biaxial stress systems. A. qf Strain Anah'sis,11(1), 1976, 1-6.

MANSON, S. S.. and HALFORD, G. R. Treatment of multiaxial 1Lreep-fatigue by strain rangepartitioning, in 1976 ASM E-MPCSymposiumon Creep-Fatigue lieraation, MPC.3, American.Society of Mechanical Engineers, New York, 1976, 299"-322.

w.. ) -:

MILLER, K. J. and PASCOE, K. J. Some recent high temperature biaxial low-cycle fatigueresults. Conf. Mechanics and Physics of Fracture, Inst. of Physics, Cambridge, London,1975, Paper 29.

SAKANE, 'M. and OHNAMI, M. Effect of multiaxiality ,f stress on metallic creep-fatigueinteraction at elevated temperatures. 3rd International Conference on Pressure Vessel Tech-nology, Tokyo, Japan, American Society of Mech. Eng., 1977, 779-786.

ZAMRIK, S. Y. The application of strain-range partitioning method to multiaxial creep-fatigue interactions. AGARD Conf.• Proc. No. 243, 1978, 16-1-16-26.

ZAMRIK, S. Y. and BILIR, 0. G. Analysis of biaxial fatigue damage at elevated temperatures.Exp. Mech., 16, 1976, 373-379.

ZAMRIF', S. Y. and BILIR, 0. G. The application of strain range partitioning method totorsional creep-fatigue interaction. J. Eng. Mater. Tech. (Trans. ASME, Series H), 98,1976, 244-248.

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DOCUMENT CONTROL DATA

I. a. AR No. I. b. Establishment No. 2. Document Date 3. Task No.AR-003-961 ARL-STRUC-R-410 July, 1984 DST 82/008

4. Title 5. Security 6. No. PagesMULTIAXIAL FATIGUE AND FRACTURE- a. document 55A LITERATURE REVIEW Unclassified

b. title c. abstract 7. No. Refs'j. U. 150

8. Author(s) 9. Downgrading lnsz.uctionsP. W. BEAVER

1S. Coporate Author and Addre .' II. Authority (as appropriate)Aeronautical Research Laboratories, a. Sponsor c. Downgrading

P.O. Box 4331, MELBOURNE. VIC. 3001 b. Security d. Approval

12. Secondary Distribution (of this docuawni)Approved for Public Release

Overseas enquirers outside stated limitations should be referred through ASDIS. Defence Information ServicesBranch, Department of Defence, Campbell Park. CANBERRA. ACT, 2601.

13. a. This document may be ANNQUNCED in catalopes and awareness sesvices available to...No limitations

i13. b. Citation for other purposes (i.e. casual eanemicement) may be (select) unrestricted (or) as for 13 a.

14. Duc-tors 15. COSATI Group'Multiaxial stress 11130

Low cycle fatigue.,Fracture properties.,

Fatigue tests"Crack propagationsMetal fatigue.

16. AbstractA review of the literature has shown that virtually alfatigae and(fracture properties of metalsand components are affected by mu,'tiaxial loadin g. In particular. rarittions in stress .tatecompared with uniaxial tension produre thefollo*,af effctis:

Aa) Decreases in the./atigue limit bh up to approximatelv 50%P.

jb) Increases or decreases in low-cycle fatigur life by J'aiowr of up to 20,. depending onwhether the stress in the second direwtim is tensile or compressiev, and is statik or

cyclic.

(c) Out.of-phase loading also reduces te kw.cyde fatigs lif by a fimor of 4 t ompar'dwith in-phase loading, l

i -i • •- -i

This page is to be used to record information which is required by the Establishment for its own use butwhich will not be added to the DISTIS data bIse unless specifically requested.

16. Abstract (Contd)(d) Accelerations or retardations oj fatigue crack growth rates by factors of 3 to 4 depend-

ing on the nature of the transverse stress.

4t is also evident that multiaxial criteria tied for design purposes can be non-c, -. serrative.especially under out-of-phase loading, and conequently can lead to unsafe estimates of thefatigue life of a component.

17. ImprintAeronautical Research Laboratories, Melbourne

18. Document Series and Number 19; Cost Code 20. Type of Report and Period Covered.Struc'ures Report 410 25 7021

21. Computer Programs Used

22. Establishment File Ref(s)

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