Experimental Observation of Elasto-Plasticity Behavior

of Type 5000 and 6000 Aluminum Alloy Sheets*1

Shohei Tamura1;*2, Satoshi Sumikawa1;*2, Takeshi Uemori2,Hiroshi Hamasaki1 and Fusahito Yoshida1

1Department of Mechanical System Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan2Faculty of Engineering, Kindai University, Higashi-Hiroshima 739-2116, Japan

Elasto-plasticity behavior of type A5052-O and AA6016-T4 aluminum alloy sheets was examined by performing several experiments ofuniaxial tension, biaxial stretching and in-plane cyclic tension-compression. Both sheets exhibit apparent r-value planar anisotropy, especiallyfor AA6016-T4 it is extremely strong, while their flow stress directionality under uniaxial tension is not so significant. Both the sheets showstrong cyclic hardening with weak Bauschinger effect. Such material behavior is well described by Yoshida-Uemori kinematic hardening modelcombined with an appropriate choice of anisotropic yield function. [doi:10.2320/matertrans.L-MZ201101]

(Received November 1, 2010; Accepted January 11, 2011; Published March 30, 2011)

Keywords: aluminum alloy sheet, anisotropy, cyclic hardening, Bauschinger effect, Yoshida-Uemori kinematic hardening model

1. Introduction

Aluminum alloy sheets are attractive lightweight materi-als, however their formability is not so good as mild steelsheets because strain localization is more likely to occur, andfurthermore, they have large springback due to their lowelastic rigidity (their Young’s modulus is about 1/3 of steel).They often show significant planar anisotropies both inr-values and flow stresses,1–3) strongly influence the sheetmetal flow in press forming. To describe such a crystallo-graphic texture-induced anisotropy, several anisotropic yieldfunctions have been proposed in the past (e.g., Hill’48,4)

Hill’90,5) Gotoh,6) Barlat et al.7–9)).The Bauschinger effect, which is a typical deformation-

induced anisotropy, extremely affects springback behavior ofsheet metals, since in many cases of sheet metal forming asheet is subjected to bending-unbending, which is a processof cyclic plasticity deformation.10) A reverse deformation at alarge strain is characterized by early re-yielding, transientBauschinger effect and permanent stress-offset (softening),and furthermore, workhardening stagnation appearing in acertain range of reverse deformation (see Hu et al.,11)

Takahashi et al.,12) Yoshida et al.10)). To describe all ofthese reverse deformation characteristics, two of the presentauthors (Yoshida & Uemori13,14)) proposed a large-straincyclic plasticity model, which are widely accepted for itshigh accuracy and flexibility in describing several types ofmaterials (e.g., see Eggertsen & Mattiason15)). Although thismodel provides a framework that can incorporate any typesof anisotropic yield function, there are still very few paperson material modeling, for aluminum sheets, which take intoaccount both the planar anisotropy and the Bauschingereffect based on experimental observations (c.f., Barlatet al.7–9) and Kuwabara18) intensively investigated theanisotropic yield loci of aluminum sheets in the framework

of the isotropic hardening model, but not the kinematichardening model).

In the first part of this paper, the elasto-plasticitydeformation characteristics of two types of aluminumsheets (A5052-O and AA6016-T4) are summarized fromnewly obtained experimental results of uniaxial tension,biaxial stretching and in-plane cyclic tension-compression.In the second part, material modeling for the planaranisotropy, as well as the Bauschinger effect and cyclichardening characteristics, is discussed in the framework ofan advanced kinematic hardening model (Yoshida-Uemorimodel,13,14) hereafter, we call it ‘Y-U model’) incorporatingwith an appropriate choice of anisotropic yield function.For such a material model, we discuss whether it is possibleto identify two sets of material parameters, one for theanisotropic yield function and the other for the kinematichardening law, separately, by using experimental data forthe yield locus and the data of cyclic plasticity, respec-tively.

2. Experimental Procedures

Test materials used in this study were type A5052-O of1.2-mm thick and AA6016-T4 of 1.0-mm thick. Uniaxialtension tests were carried out using Autograph AG-IS(SHIMADZU, load capacity is 50 kN). JIS 13A-type speci-men (parallel portion is 20-mm wide and 120-mm long) wasused in the uniaxial tension test. Stress-strain curves andr-values were determined by uniaxial tension experiments infive directions (0�, 22.5�, 45�, 67.5� and 90� from the rollingdirection).

Biaxial stretching tests were carried out using the cruci-form specimen (see Fig. 1). The specimen has two slits ineach arm, which is designed so as to release the deformationconstraint on the gauge section. The strains, "x and "y, weremeasured by strain gauge bonded on the surface of thespecimen. Initial yield locus and the subsequent equi-plasticwork loci, and furthermore, the direction of plastic strainincrement were obtained. The stresses �x and �y were

*1The Paper Contains Partial Overlap with the ICAA12 Proceedings by

USB under the Permission of the Editorial Committee.*2Graduate Student, Hiroshima University

Materials Transactions, Vol. 52, No. 5 (2011) pp. 868 to 875Special Issue on Aluminium Alloys 2010#2011 The Japan Institute of Light Metals

calculated by the following equations based on St Venant’sprinciple:

�x ¼ ðPx=A0Þ expð"xÞ; �y ¼ ðPy=A0Þ expð"yÞ; ð1Þ

where Px and Py are the applied loads, A0 is the initial cross-sectional area of the square portion (52 mm� 52 mm) of thespecimen, and "x and "y are the true strains measured bystrain gauge (YEFCA-2, Tokyo Sokki Kenkyujo Co. Ltd.).From FE simulation of biaxial stretching, it was confirmedthat the errors of thus determined stresses are within �3:1%for stress ratios of 0:5 � �y=�x � 2:0. Here, FE simulationassumed the isotropic hardening with von Mises yieldcriterion. Figures 2(a) and (b) show the contours of theequivalent stresses in a specimen under biaxial loadings ofPx : Py ¼ 1 : 1 and Px : Py ¼ 2 : 1, respectively. It is seenfrom these figures that the equivalent stresses homogene-ously distribute in enough wide area near the center of thespecimen. Note that in such a testing, the homogeneity ofstresses in whole the square region of the specimen is notnecessary. If the stress at the strain gauge area (near thecenter) is determined by eq. (1) within an acceptable smallerror, it is enough for stress determination. For example,Figs. 3(a) and (b) show FE simulation results of two stress-strain curves, one uses stresses determined by eq. (1)(solid line) and the other is the case of stresses at the centerof the specimen (dotted line), for Px : Py ¼ 1 : 1 andPx : Py ¼ 2 : 1, respectively. The two stress-strain curvesare so close to each other. Such an experiment of biaxialstretching using cruciform specimens was first performed byShiratori and Ikegami,16) and several data on yield lociobtained from the similar experiments have also beenreported by some other researchers in the last few decades(e.g., Green et al.,17) Kuwabara18)).

In-plane cyclic tension-compression tests of incrementalstep-up cyclic straining (strain history: 0, +2.5%, �2:5%,+5%,�5%, � � �) were carried out using an adhesively bondedlaminate specimen (see Fig. 4) in a servo-controlled testingmachine. Tests were carried out until the rupture or bucklingof the specimen occurs. The details of experimental tech-nique for cyclic tension-compression are described inYoshida et al.10)

3. Framework of Material Modeling

When the yield function at the initial state, f0, has a generalform:

f0 ¼ �ð�Þ � Y ¼ 0; ð2Þ

where � denotes a function of the Cauchy stress �, and Y isthe initial yield strength. For function �, we may choose anappropriate one among existing anisotropic yield functions.Assuming the kinematic hardening of the yield locus, thesubsequent yield criterion is generally written by

f ¼ �ð� � �Þ � Y ¼ 0; ð3Þ

where � stands for the backstress. The associated flow rule iswritten as

Dp ¼@ f

@�_�� ¼

@�

@�_��; ð4Þ

Fig. 1 Biaxial stretching test specimen.

(a)

(b)

Fig. 2 Contours of equivalent stresses of A5052-O. (a) Px : Py ¼ 1 : 1 and

(b) Px : Py ¼ 2 : 1.

Experimental Observation of Elasto-Plasticity Behavior of Type 5000 and 6000 Aluminum Alloy Sheets 869

where Dp denotes the plastic part of the rate of deformation.Thus the constitutive equation of plasticity is derivedby determining the evolution law of the backstress �,together with an appropriate choice of an anisotropic yieldfunction �.

As for the kinematic hardening law, here we use Y-Umodel.13,14) This model is constructed in the framework oftwo-surface modeling, wherein the yield surface moveskinematically within the bounding surface, as schematicallyillustrated in Fig. 5. The bounding surface F is expressed bythe equation

F ¼ �ð�;�Þ � ðBþ RÞ ¼ 0; ð5Þ

where � denotes the center of the bounding surface, and B

and R are its initial size and isotropic hardening (IH)component. One of the strong originalities of this model is aproposition of a kinematic hardening law that accuratelyexpresses the Bauschinger effect of materials. The relativekinematic motion of the yield surface with respect to thebounding surface is expressed by

�� ¼ �� �; ð6Þ

For the evolution of ��, the following equation is assumed:

��� ¼ C

a

Y

� �ð� � �Þ �

ffiffiffiffiffiffia

����

r��

� �_pp; ð7Þ

_pp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

3Dp : Dp

r; ���� ¼ �ð��Þ; a ¼ Bþ R� Y ; ð8Þ

where _pp is the equivalent plastic strain rate, defined as thesecond invariant of Dp, and C and a are material parametersthat control the rate of the kinematic hardening. For theisotropic hardening of the bounding surface, the followingevolution equation is assumed:

_RR ¼ mðRsat � RÞ _pp; ð9Þ

where Rsat is the saturated value of the isotropic hardeningstress R at infinitely large strain, and m is a materialparameter that controls the rate of isotropic hardening. Forthe kinematic hardening of the bounding surface, thefollowing evolution equation is assumed:

�� ¼ mb

Y

� �ð� � �Þ � �

� �_pp; ð10Þ

where b denotes a material parameter. Here, parameter m isassumed to be the same as in the evolution equation of theisotropic hardening stress. To describe the phenomenon ofworkhardening stagnation appearing in a reverse deforma-tion, the model of non-IH hardening surface is introduced(refer to Yoshida & Uemori13,14)).

Besides the kinematic hardening model, the followingmodel of plastic-strain dependent Young’s modulus (Yoshidaet al.10)) was used in the calculation:

E ¼ E0 � ðE0 � EaÞf1� expð�� �""pÞg; ð11Þ

where E0 and Ea stands for Young’s modulus for virgin andinfinitely large pre-strained materials, respectively, and � is amaterial constant.

As for anisotropic yield functions, here we examine thefollowings:

(a)

(b)

Fig. 3 Stress-equivalent plastic strain curves of A5052-O. (a) Px : Py ¼1 : 1; and (b) Px : Py ¼ 2 : 1.

Fig. 4 In-plane cyclic tension-compression test specimen.

Fig. 5 Schematic illustration of Yoshdia-Uemori model.

870 S. Tamura, S. Sumikawa, T. Uemori, H. Hamasaki and F. Yoshida

- Hill’48 quadratic function:4)

f ¼1

2fF�2

yy þ G�2xx þ Hð�xx � �yyÞ2 þ 2N�2xyg

�Gþ H

2���2 ¼ 0;

ð12Þ

where F, G, H and N are anisotropic material parameters.Gotoh’s bi-quadratic function:6)

f ¼ A1�4xx þ A2�

3xx�yy þ A3�

2xx�

2yy þ A4�xx�

3yy

þ A5�4yy þ ðA6�

2xx þ A7�xx�yy þ A8�

2yy�

2xy

þ A9�4xy � A1 ���4 ¼ 0; ð13Þ

where A1A9 are anisotropic material parameters.- Barlat Yld2000-2d model:8)

� ¼ �0 þ �00 ¼ 2 ���a; �0 ¼ jX01 � X02ja;

�00 ¼ j2X002 þ X001 ja þ j2X001 þ X002 j

að14Þ

where X01, X02, X001 and X002 denote principal values of X0 ¼L0 : � and X00 ¼ L00 : �, respectively. L0 and L00 are trans-formation matrix that contain eight anisotropic materialparameters. The exponent a is usually chosen as either 6 or 8.

Most of previous papers on material modeling foraluminum sheets focus either on the description of theBauschinger effect (e.g., Boger et al.19)) or on the anisotropicyield function (e.g., Barlat et al.;7–9) Kuwabara et al.;20) Nakaet al.21)), in contrast, the present paper discusses both thematerial characteristics in the framework of cyclic plasticitymodeling with the anisotropic yield function.

4. Experimental Results and Discussion

4.1 Deformation behavior under uniaxial tensionStress-strain curves of A5052-O and AA6016-T4 under

uniaxial tension are illustrated in Figs. 6(a) and (b), respec-tively, for the specimens cut from the sheets in threedirections � ¼ 0�, 45� and 90� with respect to their rollingdirection. Type A5052-O sheet exhibits apparent yieldplateau and serrations in the stress-strain curves, which iscaused by well-known Portevin-Le Chatelier (PLC) effect.From these figures it is found that the flow stress direction-ality is not very significant for both sheets. The mechanicalproperties of the sheets in terms of yield strengths, ��(defined as stresses at 3% or 4% plastic strain), and r-values,r� at � ¼ 0�, 22.5�, 45�, 67.5� and 90�, are summarized inTable 1. In addition, biaxial yield strength, �bi, determinedfrom biaxial stretching experiment is also indicated. Thesesheets exhibit apparent r-value planar anisotropy, especiallyfor AA6016-T4 it is extremely strong.

In Figs. 7(a) and (b), the calculated results of flow stressdirectionality (flow stress �� normalized by �0), for A5052-Oand AA6016-T4, respectively, by several yield functions arecompared with the corresponding experimental data. Here,for Hill’48’s yield criterion, completely different two setsof anisotropic parameters can be determined depending onthe selection of mechanical properties used for parameteridentification, where r-values (r0, r45 and r90) are used forHill’48-r and yield stresses (�0, �45 and �90) for Hill’48-�.For Yld2000-2d’s yield function, exponent of a ¼ 6 was usedin the simulation. Among these, Yld2000-2d (a ¼ 6) gives

the best fit to the experimental results, and the results byGotoh’s yield function are also reasonably well. The Gotoh’scalculation of �� at � ¼ 67:5� on AA6016-T4 shows a bitlarge error, since in this material parameter identification �0,�22:5, �45, �90, �bi and r0, r22:5, r45, r90 are solely employed asGotoh proposed in his paper, but not r67:5 and �67:5. Instead ofusing r22:5, if we used �0, �22:5, �45, �67:5, �90, �bi, as Hu22)

proposed, the results would become better. Hill’48-� modelpredicts the flow stress directionalities on both the sheets,since it employs �0, �45, �90 for material parameteridentification. In contrast, the predictions by Hill’48-r modelare very poor.

In Figs. 8(a) and (b), the calculated results of r-valueplanar anisotropies, for A5052-O and AA6016-T4, respec-tively, by several yield functions are compared with the

(a)

(b)

Fig. 6 Stress-strain curves under uniaxial tension of aluminum sheets in

three directions of 0�, 45� and 90� with respect to their rolling direction.

(a) A5052-O; and (b) AA6016-T4.

Table 1 Mechanical properties for A5052-O and AA6016-T4.

MaterialAngle from rolling direction

0� 22.5� 45� 67.5� 90� biaxial

Stress/MPaA5052-O�1 166 169 166 163 163 166

AA6016-T4�2 195 197 187 191 186 195

r-valueA5052-O�3 0.72 0.60 0.51 0.56 0.59 —

AA6016-T4�3 0.76 0.40 0.26 0.33 0.61 —

Superscripts �1, �2 and �3 indicate plastic strains of 0.04, 0.03 and 0.1,

respectively, where stresses and/or r-values were obtained.

Experimental Observation of Elasto-Plasticity Behavior of Type 5000 and 6000 Aluminum Alloy Sheets 871

corresponding experimental data. The predictions byYld2000-2d (a ¼ 6), Gotoh and Hill’48-r are in goodagreement with the experimental results, since all thesemodels employ at least three r-values (r0, r45, r90) formaterial parameter identification. In contrast, Hill’48-�model fails in the prediction.

In order to investigate the dependency of plastic strain ofYoung’s modulus, incremental tension-unloading experi-ments were performed, and Young’s moduli of pre-strainedsheets were obtained from unloading stress-strain slopes.Figure 9 shows the relationship between Young’s moduli andaccumulated plastic strain on A5052-O and AA6016-T4. Theresult of A5052-O showed that Young’s modulus is almostkept constant independently on the plastic strain, while thatfor AA6016-T4 decreases slightly with increasing plasticstrain, which can be expressed by eq. (10).

4.2 Equi-plastic work loci under biaxial stretchingFigures 10(a) and (b) show the comparison of normalized

equi-plastic work loci (defined as stresses at 3% or 4% plasticstrain) with the corresponding predictions by several yieldfunctions. Note that the equi-plastic work loci are identicalwith the yield loci only when the isotropic hardening of theyield loci is assumed.

From the experimental results, it is found that the shape ofequi-plastic work locus of A5052-O is almost symmetric with

respect to equi-biaxial stress axis (�x ¼ �y). Contrary to this,the yield locus of AA6016-T4 is not symmetry, and itexpands toward plane strain side (�x : �y ¼ 2 : 1). The yieldloci calculated by Gotoh’s function agrees well with theexperimental results of both A5052-O and AA6016-T4. Thepredictions of the loci by Yld2000-2d are mostly in goodagreement with the experimental results of both sheets, whilea certain discrepancy is found in the yield stress of AA6016-T4 at �x : �y ¼ 2 : 1. Since von Mises’s yield function

(a)

(b)

Fig. 7 Comparison of normalized flow stresses under tensile directions

with the corresponding predictions by various yield functions for (a)

A5052-O; and (b) AA6016-T4.

(a)

(b)

Fig. 8 Comparison of r-values under tensile directions with the corre-

sponding predictions by various yield functions for (a) A5052-O; and

(b) AA6016-T4.

Fig. 9 Relationships between Young’s modulus and plastic strain.

872 S. Tamura, S. Sumikawa, T. Uemori, H. Hamasaki and F. Yoshida

assumes isotropy of materials, the discrepancies betweenexperimental results and the predictions of r-values and yieldstresses are very large especially for AA6016-T4. As alreadymentioned, Hill’48-� hardly predicts r-values for both sheets(see Table 1) although the calculated results of the equi-plastic work loci agree fairly well with the experimentalresults. On the other hand, Hill’48-r completely fails insimulating the loci.

Furthermore, the validation of normality rule was alsocarried out. Figures 11(a) and (b) show the comparison ofdirection of plastic strain incremental vectors betweenexperimental results and the corresponding calculated onesusing various yield functions. In the experiments, thedirections of plastic strain rate were kept constant duringplastic deformation. Gotoh’s and Yld2000-2d’s yield func-tions predict well both equi-plastic work loci and plasticstrain vector.

4.3 Cyclic tension-compressionFigures 12(a) and (b) show the stress-strain curves

obtained from in-plane cyclic tension-compression test anduniaxial tension test. In these figures, the calculated results bythe IH (isotropic hardening) model and Y-U model are

illustrated. From these, significant cyclic workhardening andweak Bauschinger effect (compared to high strength steelsheets10)) are found for both materials. Y-U model success-fully describes such characteristics of cyclic stress-strainresponses. Note that works on cyclic plasticity modeling foraluminum sheets based on the experimental data, which areof importance for numerical simulation of multi-stage sheetmetal forming, are still very limited, although there are somediscussions on the Bauschinger effect in reverse deforma-tions (e.g., Lee et al.23)).

5. On Modeling of Cyclic Plasticity for AnisotropicMaterials

Figures 13(a) and (b) show the comparisons of thecalculated results by two models of the IH and Y-U, forequi-plastic work loci and the directions of plastic strainincrements, respectively, under proportional (radial) loadingson A5052-O. Here Gotoh’s yield function is used in bothmodels. From these, it is found that the calculated results byIH and Y-U models are almost the same.

Note that it does not mean that IH model is sufficient forthe description of cyclic plasticity behavior (see Fig. 12:cyclic stress-strain response calculated by IH model). Foraccurate simulation of the Bauschinger effect and cyclichardening characteristics, the use of a non-linear kinematichardening model is essential. An important conclusion

(a)

(b)

Fig. 10 Comparisons of normalized equi-plastic work loci with the

corresponding predictions by various yield functions for (a) A5052-O;

and (b) AA6016-T4.

(a)

(b)

Fig. 11 Comparisons of direction of incremental plastic strain vector with

the corresponding predictions by various yield functions for (a) A5052-O;

and (b) AA6016-T4.

Experimental Observation of Elasto-Plasticity Behavior of Type 5000 and 6000 Aluminum Alloy Sheets 873

derived from the results shown in Figs. 13(a) and (b) is thatwe can identify two sets of material parameters, one for theanisotropic yield function and the other for the kinematichardening law, separately, by using experimental data for theyield locus and the data of stress-strain responses of uniaxialcyclic plasticity, respectively.

6. Conclusions

Elasto-plasticity behavior of type A5052-O and AA6016-T4 aluminum alloy sheets was examined by various experi-ments. From the comparison of the experimental observa-tions of the stress-strain responses with the correspondingsimulations using several models of anisotropic plasticity,appropriate material modeling was discussed. The highlightsof the present findings are summarized as follows:(1) Both sheets exhibit apparent r-value planar anisotropy,

especially for AA6016-T4 it is extremely strong, whiletheir flow stress directionality under uniaxial tension isnot so significant.

(2) Both the sheets show strong cyclic hardening withcomparatively weak Bauschinger effect.

(3) Such material behavior is described accurately by Y-Ukinematic hardening model with an appropriate choiceof anisotropic yield function. As for anisotropic yieldfunctions, either Gotoh’s bi-quadratic function orBarlat’s Yld2000-2d is a good choice for thesealuminum sheets.

(4) One of the advantages of the present framework ofmaterial modeling is that two sets of material param-eters, one for the anisotropic yield function and theother for the kinematic hardening law can be identifiedseparately. For the yield function parameters we solelyuse experimental data of the yield loci (or equi-plasticwork loci) under proportional (monotonic) loading, andfor the parameters of kinematic hardening law only dataof stress-strain responses of uniaxial cyclic plasticityare necessary.

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