inverse approach to identification of material

Inverse approach to identification of materialparameters of cyclic elasto-plasticity forcomponent layers of a bimetallic sheet

F. Yoshidaa,*, M. Urabeb, R. Hinoa,1, V.V. Toropovc

aDepartment of Mechanical System Engineering, Hiroshima University, 1-4-1, Kagamiyama,

Higashi-Hiroshima 739-8527, JapanbGraduate School of Hiroshima University, Hiroshima University, 1-4-1, Kagamiyama,

Higashi-Hiroshima 739-8527, JapancSchool of Engineering, University of Bradford, Bradford, West Yorkshire BD7 1DP, UK

Received in revised form 19 August 2002

Abstract

The present paper proposes a novel approach to the identification of the mechanical prop-

erties of individual component layers of a bimetallic sheet. In this approach, a set of materialparameters in a constitutive model of cyclic elasto-plasticity are identified for the two layers ofthe sheet simultaneously by minimizing the difference between the experimental results andthe corresponding results of numerical simulation. This method has an advantage of using the

experimental data (tensile load vs strain curve in the uniaxial tension test and the bendingmoment vs curvature diagram in the cyclic bending test) for a whole bimetallic sheet but notfor individual component layers. An optimization technique based on the iterative multipoint

approximation concept is used for the identification of the material parameters. This paperdescribes the experimentation, the fundamentals and the technique of the identification, andthe verification of this approach using two types of constitutive models (the Chaboche-Rous-

selier and the Prager models) for an aluminum clad stainless steel sheet.# 2003 Elsevier Ltd. All rights reserved.

Keywords: B. Constitutive behaviour; B. Layered materials; B. Elastic–plastic material; C. Optimization

International Journal of Plasticity 19 (2003) 2149–2170

www.elsevier.com/locate/ijplas

0749-6419/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0749-6419(03)00063-9

* Corresponding author. Tel.: +81-824-24-7541; fax: +81-824-22-7193.

E-mail address: [email protected] (F. Yoshida).1 Current address: Materials and Processing Research Center, NKK Co. Ltd., 1, Kokan-cho,

Fukuyama, Hiroshima 721-8510, Japan

http://www.elsevier.com/locate/ijplas/a4.3d

mailto:[email protected]

1. Introduction

In recent years, bimetallic sheets which consist of dissimilar metal components,such as stainless steel/aluminum, copper/steel, etc., have been widely used in manyindustrial fields because of their excellent mechanical and functional properties (e.g.,Kim and Yu, 1997; Yoshida, 1997). It is already known that the press-formability ofbimetallic sheets is quite different from that of their individual component sheets(e.g., Hawkins and Wright, 1971; Verguts and Sowerby, 1975; Semiatin and Piehler,1979b,c; Majlessi and Dadras, 1983; Yoshida et al., 1995; Yoshida and Hino, 1997;Yoshida and Urabe, 2000).The prediction of the formability of a bimetallic sheet would be possible if the

mechanical properties of its individual component metals, as well as its layer thick-ness ratio, are given (Yoshida et al., 1995; Yoshida and Hino, 1997). In order todetermine the mechanical properties, uniaxial tension tests are usually performed oneach metal layer taken from the bimetallic sheet by a mechanical or chemical pro-cessing (Semiatin and Piehler, 1979a; Yoshida and Hino, 1997). However, such ten-sion tests have the following drawbacks:

� a time-consuming process of the removal of a layer from a bimetallic sheet bya mechanical or chemical processing is necessary for the preparation of thespecimens,

� the stress–strain responses during cyclic loading cannot be obtained becauseof the buckling of the sheet in compression, although it is very important forsheet metal forming simulations to employ a constitutive model whichproperly describes the cyclic behavior of sheet metals (Yoshida and Urabe,1999; Yoshida, 2000; Wagoner et al., 2000; Chun et al., 2002a,b; Yoshida etal., 2002; Yoshida and Uemori, 2002).

If the determination of mechanical parameters of cyclic elasto-plasticity of abimetallic sheet became possible without performing such a time-consuming processof the removal of a layer, it would be a considerable technological achievement.In order to identify a set of material parameters in a constitutive model of cyclic

elasto-plasticity, which describes the complicated stress–strain responses includingthe deformation characterizations of the Bauschinger effect and cyclic strain-hard-ening, for monolithic sheet metals, the present authors (Yoshida et al., 1998) firstproposed an idea of using cyclic bending tests. In that research, material parameterswere identified by minimizing the difference between the test results and the resultsof the corresponding numerical simulation using an advanced optimization tech-nique developed by Toropov et al. (1993).Recently, similar approaches to material parameter identification of sheet metals

from bending tests were reported by Zhao et al. (2001) and Geng et al. (2002).As an extension of the cyclic bending method, the present paper proposes a novel

approach to the identification of material parameters of the individual componentlayers in a bimetallic sheet without the time-consuming process of removal of alayer. A set of material parameters in a constitutive model of cyclic elasto-plasticity

2150 F. Yoshida et al. / International Journal of Plasticity 19 (2003) 2149–2170

are identified using two different types of experimental data, namely, the tensile loadversus strain curve in the uniaxial tension test and the bending moment versus cur-vature diagram in the cyclic bending test, for a whole bimetallic sheet but not forindividual component layers. This paper describes the experimentation, the funda-mentals and the technique of the identification, and the verification of this approachusing an aluminum clad stainless steel sheet.

2. Scheme of the material parameter identification

The procedure of the present identification problem is schematically illustrated inFig. 1. In order to determine the stress–strain response for the two individual com-ponent layers of a bimetallic sheet, at least two different types of experimental dataof the mechanical response are required. In this research, the tensile load vs axialstrain curve in the uniaxial tension test and the bending moment vs curvature dia-gram in the cyclic bending test have been used. A set of material parameters in aconstitutive model of cyclic elasto-plasticity are identified for the two layers of thesheet simultaneously by minimizing the difference between the experimental resultsand the corresponding results of numerical simulation.

2.1. Experimentation

For the experiments, a stainless steel clad aluminum sheet consisting of 1.2 mmaluminum (A1100) layer and 0.55 mm stainless-steel (type-430SS) layer, as shown inFig. 2, was employed. In the cyclic bending test, as shown in Fig. 3, one end of aspecimen was clamped and rotated by a step motor, and the other end was movingfreely in x–y directions without rotation. The above condition of the test can beregarded as uniform bending, in which the bending moment is uniformly distributedin the longitudinal direction of the sheet. The bending moment was measured by aload-cell, and the curvature of the specimen was determined from the surface strainsmeasured by strain gauges bonded on both surfaces of the specimen. The secondtype of experiment (uniaxial tension) produces the tensile load vs strain curve.In order to verify the identified material parameters, uniaxial tension tests were

also performed with the stainless steel specimen which had been taken from the cladsheet, and its stress–strain curve was obtained. The stress–strain curve of the alu-minum layer was determined using the rule of mixtures by use of the results of uni-axial tension tests both for the clad sheet and the stainless steel layer.

2.2. Constitutive models

In order to describe the stress–strain response of metals under cyclic deformation,two types of constitutive models were used, i.e., one was a model of cyclic plasticitybased on the model proposed by Chaboche and Rousselier (1983) (hereafter we callit ‘the Chaboche–Rousselier model’), and the other was the linear kinematic hard-ening model proposed by Prager (1956) (‘the Prager model’).

F. Yoshida et al. / International Journal of Plasticity 19 (2003) 2149–2170 2151

The strain rate ":is decomposed into elastic and plastic components, "

: e and ":p, as

":¼ "

: e þ ":p: ð1Þ

The yield function f and the associated flow rule are given by the equations:

f ¼1

2S� �ð Þ : S� �ð Þ �

1

3Yþ Rð Þ

2; ":p ¼

@f

@Sl:; ð2Þ

Fig. 1. Scheme of the material parameter identification for a bimetallic sheet.


e S and � denote the stress deviatior and the backstress deviatior, respectively,
wherand Y and R stand for the initial yield stress and the isotropic hardening stress,respectively. The evolution of the isotropic hardening stress is given by
R:¼ b Q� Rð Þ"�

:; "

:¼

2

3":p : "

:p

� �1=2: ð3Þ

Chaboche and Rousselier (1983) have given the expression for the backstress evolu-tion as a summation of several backstress components of the Armstrong and Frederick(1966) type (so-called ‘A-F model’). Instead of many A-F components, here we use oneof the simplest expressions of the backstress which consists of one linear kinematichardening component and one A-F component (Yoshida, 1995), �1 and �2, as

� ¼ �1 þ �2: ð4Þ

The linear kinematic hardening rule is

�:1 ¼

2

3H0"

:p: ð5Þ

The A-F kmodel is

�:2 ¼ C

2

3a":p � "�

:�2

� �: ð6Þ

Fig. 3. Schematic illustration of experimental setup for cyclic bending test.

Fig. 2. Stainless steel (type-430) clad aluminium (A1100) specimen used in the experiments.


e above constitutive model incorporates eight material parameters: two elastic
Thconstants E (Young’s modulus) and � (Poisson’s ratio); the initial yield stress Y;parameters Q and b for the isotropic hardening rule; and three parameters a, C andH0 for the kinematic hardening rules. If the linear kinematic hardening rule isexclusively considered in this constitutive model, in which the material parameter ofplasticity is H0 only, it yields the Prager kinematic hardening model. The meaningsof these material parameters in both models—how they reflect shapes of calculatedstress–strain curves under cyclic straining—are schematically shown in Fig. 4(a)–(c).
2.3. Numerical simulation and identification of material parameters

For the analysis of uniaxial tension, the isostrain condition for both the compo-nent layers of the sheet was assumed. The bending moment was calculated for agiven curvature based on the Kirchhoff–Love hypothesis with the assumption of theuniform bending under plane strain condition. The assumption has been verified bycomparing the bending moment versus curvature curve from a 3D FEM simulationof bending and the one from the plane-strain uniform bending.A set of material parameters in each of the above constitutive models were iden-

tified using two different types of experimental data simultaneously, namely, thetensile load vs strain curve (P vs ") in the uniaxial tension test and the bendingmoment vs curvature diagram (M vs �) in the cyclic bending test. The identificationwas carried out by minimizing the difference between the experimental results andthe corresponding results of numerical simulation. To make clearer our idea of usingexperimental data of uniaxial tension and cyclic bending, here we shall discuss howstrongly each material parameter reflects some specific mechanical responses in ten-sion (load vs strain curve: P vs ") and bending (bending moment vs curvature dia-gram: M vs �). Let us consider a two-ply laminate consisting of materials A and B,


Fig. 4. Schematic illustrations of cyclic stress–strain (�–") responses calculated by constitutive models:

(a) the Chaboche–Rousselier model excluding isotropic hardening; (b) the Chaboche–Rousselier model

including isotropic hardening; and (c) the Prager model, where H~ i ¼ H0i= 1þH0

i=Ei

� �, i=A or B.

whose layer thickness are tA and tB. For easier understanding, we shall begin ourdiscussion with non-isotropic hardening materials.

(i) Young’s moduli, EA and EB, can be directly determined from the linear
(elastic) parts of P–" relationship (see schematic illustration Fig. 5(a): P=K";and Fig. 5(b): P=(tAEA+tBEB)") and M–� relationship (see schematicillustrations Fig. 6(a) and (b), where elastic bending rigidity, De, can beexplicitly determined as a function of elasticity parameters: EA, EB, �A and �B,and layer thicknesses: tA and tB), since we have two linear equations fromthem for two unknown values of EA and EB.
(ii) When calculating P–" curve using the Chaboche–Rousselier model, it will
show transient workhardening (highly nonlinear P–" relationship) just after
Fig. 5. Schematic illustrations of load vs strain (P vs ") responses under uniaxial tension: (a) experimental

result; and (b) calculated result by the Chaboche–Rousselier model.


the onset of yielding, and then it approaches a linear line given by theequation [see schematic illustration in Fig. 5(b)]:

~ 0 ~ 0h i
P ¼ tA YA þ aAð Þ þ tB YB þ aBð Þ½ þ tAHA þ tBHB "� "ð Þ; ð7aÞ
0
Hi H~ i ¼1þH0
i=Ei; i ¼ A or Bð Þ: ð7bÞ

If we have an experimental P–" curve which approaches a line ofP=P*+k("�"*), as shown in Fig. 5(a), we have the following relationships:

Fig. 6. Schematic illustrations of bending moment vs curvature (M vs k) responses under cyclic bending:

(a) non-isotropic-hardening materials; and (b) isotropic-hardening materials. Elastic bending rigidity, De,

can be explicitly determined as a function of elasticity parameters: EA, EB, �A and �B, and layer thick-nesses: tA and tB. Slope D1 (asymptotic elastic-plastic rigidity) is a function of asymptotic hardening

ratios of plasticity, H0A and H0

B, together with the elasticity parameters and the layer thicknesses.


tA YA þ aAð Þ þ tB YB þ aBð Þ ¼ P; ð8aÞ

tAH~0 þ tBH~

0 ¼ k: ð8bÞ
A B
In these equations, it is impossible to split P* (and k) into two parts, one is thecontribution by material A and the other by B, such as P

A ¼ tA YA þ aAð Þ andPB ¼ tB YB þ aBð Þ kA ¼ tAH~

0A and kB ¼ tBH~

0B

� �. However, if we employ, together

with Eqs. (7) and (8), the following M–� relationship:ð
�yydA ¼ M for a given �; ð9aÞð
under the constraint of �ydA ¼ 0; ð9bÞ

thedeterminationofparameters (Y+a) andH0, for eachmaterial,will becomepossible.

(iii) Material parameters a and C, as well as the yield-strength differential between
YA andYB, directly reflect the shapes of highly nonlinear parts ofP–" curve, andalsoM–� diagram. It should be noted that the nonlinearM–� relationship is notonly due to the materials’ nonlinear stress-strain characteristics, but also to thebehavior of propagation of plastic zone from the sheet surfaces to the neutralsurface with increasing bending curvature. Therefore, to identify parameters aand C, experimental data of both P vs " andM vs � are essential.
(iv) If materials exhibit cyclic hardening characteristics under cyclic bending [see
schematic illustration Fig 6(b)], it is directly related to parameters of isotropichardening, b and Q [refer to Fig. 4(b)].
From the above discussion, it has been clarified that each material parameterstrongly reflects some specific mechanical responses in uniaxial tension and cyclicbending. Therefore, it would be possible to identify these material parameters, forindividual component layers of a bimetallic sheet, from experimental data of uni-axial tension and cyclic bending.

3. Formulation of the material parameter identification problem

3.1. General formulation

Let us consider the material parameters in a constitutive model to be identified ascomponents of the vector x 2 RN. Then the optimization problem can be formulatedas follows (Toropov et al., 1993, 1997): Find the vector x that minimizes the objec-tive function

F xð Þ ¼XL¼1

F xð Þ

Ai 4 xi 4Bi i ¼ 1; . . . ;Nð Þ; ð10Þ


e L is the total number of individual specific response quantities (denoted by )
wherwhich can be measured in the course of experiments and then obtained as a result ofthe numerical simulation. F(x) is the dimensionless function:
F xð Þ ¼XS

s¼1

Rs � R x; �s

� �� 2( )=

XS

s¼1

Rs

� �2( )ð11Þ

which measures the deviation between the computed th individual response and theobserved one from the experiment. Here, the notations denote �: a parameter whichdefines the history of the process in the course of the experiment (e.g., the time or theloading parameter), �s ¼ 1; . . . ; L; s ¼ 1; . . . ; Sð Þ: the discrete values of � forSth data point, R

s : the value of the -th measured response quantity correspondingto the value of the experiment history parameter �s , R

x; �s� �

: the value of the sameresponse quantity obtained from the numerical simulation. : the weight coefficientwhich determines the relative contribution of information yielded by the -th set ofexperimental data, Ai, Bi: side constraints, stipulated by some additional physicalconsiderations, which define the search region in the space RN of optimizationparameters.The optimization problem (10) has the following characteristic features:

� the objective function is an implicit function of parameters x,� to calculate values of this function for the specific set of parameters x meansto use a nonlinear numerical (e.g. finite element) simulation of the processunder consideration, which may involve a large amount of computer time;

� function values present some level of numerical noise, i.e., they can only beestimated with a finite accuracy.

The direct implementation of conventional nonlinear mathematical programmingtechniques would involve a large amount of computer time, and most importantly,the convergence of the optimization cannot be guaranteed due to the presence ofnumerically induced noise in the objective function values, and even more so, itsderivatives. To solve the problem, the iterative multipoint approximation concept(Toropov et al., 1993) is used, where computationally expensive and noisy functions,F(x), (=1,. . .,L) are replaced by simplified noiseless functions obtained by theleast-squares fitting. Further details of this technique used to solve the identificationproblem can be found in references by Toropov et al. (1993, 1997) and Yoshida etal. (1998).The specific form of the multipoint approximation technique, used in this work, is

based on the algorithm of sequential quadratic programming (SQP) which builds upan approximation of the inverse of the Hessian matrix using the objective functionvalues and its derivatives. It should be noted that in the problems of parameteridentification for nonlinear constitutive models the traditional use of finite differ-ences for evaluation of derivatives would spoil the convergence of the optimizerbecause the accuracy of derivatives is severely affected by the numerical noise. As


the approximations are constructed in each iteration to evaluate the derivatives only,fairly simple approximation types can be used, e.g. quadratic polynomials withoutcross-terms:

F~k x; að Þ ¼ a0 þXNi¼1

a2 i�1ð Þþ1xi þ a2 i�1ð Þþ2x2i

� �ð12Þ

which contains 2N+1 tuning parameters to be found in each iteration by the least-squares surface fitting.

3.2. The present identification problem

The identification of the above material parameters, excluding Poisson’s ratio �,which was found to be �=0.3 for both the stainless steel and the aluminum layersfrom the conventional measurements, was performed using the tension (tensile loadvs strain :P vs ") curve (=1) and several individual bending/reversed bending(bending moment vs curvature: M vs �) diagrams (=2,. . ., L) which are regardedas individual response quantities. As for Youngs’ moduli, they could be identifieddirectly from elastic parts of P–" and M–� relationships, as already discussed inSection 2.3; however, in the present work, Young’s modulus is also treated as one ofmaterial parameters to be identified simultaneously. Here in the identificationproblem for the Chaboche–Rousselier model,

� the optimization variables x =[x1,x2,. . .,x14] are the material parameters forthe two layers: E;Y;Q; b;H0; a;Cð Þstainless steel; E;Y;Q; b;H0; a;Cð Þaluminum

� �,

� the set of values of Rs corresponds to the set of values of the tensile load

R1s ¼ P (for =1); and experimental bending moment R

s ¼ Ms (for

=2,. . ., L) both of which are found from the experiment,� the function R x; �s

� �corresponds to the calculated tensile load R1 ¼

Ð�ydA

(for =1), and bending moment R ¼ M ¼Ð�yydA (for =2, . . ., L),

� the experiment history parameter �1s is the strain "s in uniaxial tension for=1; and �s (for =2, . . ., L) is the curvature �s ,

� the index is 1 for the uniaxial tension, 2 for the first monotonic bending, 3for the subsequent reversed bending, etc. in Eqs. (10)–(12),

� under bending, a constraint condition ofÐ�ydA ¼ 0 (non axial load) is

considered.

All the response quantities were considered equally weighted (1=2=. . .=1) in theformulation of the objective function F(x). In addition, for the discussion on theeffect of the weighting coefficients, the identification for the case of weighting of1=1 (for uniaxial tension), 2=3=. . .=0.5 (for cyclic bending) was partly carriedout.The identification of material parameters E, Y andH0 in the Prager model was also

performed, where x ¼ x1; x2; . . . ; x6½ ¼ E;Y;H0ð Þstainless steel; E;Y;H0ð Þaluminum

� �.


4. Results of parameter identification and discussion

The material parameters were identified using two sets of experimental data of thebimetallic sheet, i.e., the tensile load vs strain curve (see Fig. 7); and the bendingmoment vs curvature (see Fig. 8). Fig. 7(a) and (b) show the comparison of the experi-mental results of the tensile load vs strain curve in uniaxial tension tests for the bime-tallic sheet and the corresponding results calculated with the constitutive models of theChaboche–Rousselier type and the Prager type, respectively, incorporating the identi-fied sets of material parameters. The results of cyclic bending and the corresponding

Fig. 7. Comparisons of experimental curves of tensile load vs strain (P vs ") in uniaxial tension and the

result of simulations with the constitutive models incorporating the sets of material parameters identified

from uniaxial tension and cyclic bending tests of the bimetallic sheet: (a) the Chaboche–Rousselier model;

and (b) the Prager model.


numerical simulations are shown in Fig. 8(a) and (b). The load vs strain curve cal-culated with the Chaboche–Rousselier model fits the experimental results well,whereas the Prager model cannot simulate the nonlinear part of the curve near theinitial yield point [see Fig. 7(b)]. The sets of material parameters identified for boththe models are listed in Tables 1 and 2. The material parameter identifications werecarried out several times by changing the initial guesses of the parameters, and theirsearch regions [i.e., side constraints Ai and Bi in Eq. (10)], and consequently, it wasfound that the differences in the obtained results between the trials were negligiblysmall. This would be an indirect verification of the uniqueness of solution for thepresent problem. As already mentioned in Sections 2.2. and 2.3., each materialparameter clearly represents some of materials’ cyclic stress–strain characteristics[see Fig. 4(a)–(c)], and it also directly reflects some of specific mechanical responses,

Fig. 8. Comparisons of experimental diagrams of bending moment vs curvature (M vs k) and the results

of simulations with the constitutive models incorporating the sets of material parameters identified from

uniaxial tension and cyclic bending tests of the bimetallic sheet: (a) the Chaboche–Rousselier model; and

(b) the Prager model.


such as the transient and asymptotic workhardening behavior in P–" curve [seeFig. 5(a) and (b)], the increase in bending-moment peaks under cyclic bending withnumber of cycles [see Fig. 6(a) and (b), in our experiment, non-cyclic-hardening wasobserved, and consequently, the optimizer gave us an answer of b and Q=0], etc.This would be a reason why we have succeeded in obtaining a unique solution.Fig. 9(a) and (b) show the stress–strain curves calculated with the models using the

identified material parameters for the stainless steel and the aluminum in the bime-tallic sheet. Both the component metals exhibit almost no cyclic strain-hardeningbecause they possess large plastic prestrain induced during the cladding by the roll-bonding process. In order to check the accuracy of this identification method, thecalculated stress–strain curves for the individual component metals were comparedwith the experimental stress–strain curves [see Fig. 10(a) and (b)]. The results cal-culated with the Chaboche–Rousselier model agree fairly well with those obtained inthe experiments. However, by the Prager model, the predicted strain-hardeningcoefficient H0 ¼ d�=d"p for the stainless steel is apparently larger than the experi-mental value. It is not attributed to the problem of the identification procedure, butto the fact that the Prager model cannot describe the nonlinear stress–strain rela-tionship near the initial yield point. Fig. 11 shows the surface strain responses (strainof the aluminum versus one of the stainless steel) during cyclic bending calculated bythe Chaboche–Rousselier model together with the corresponding experimentalresults. These simulated results agree with those obtained in experiments.There might be several reasons for a certain discrepancy between the experimental

stress—strain curves and the calculated results shown in Fig. 10(a) and (b). For thecase of the Chaboche–Rousselier model, the discrepancy is not due to the materialmodel itself, but for the case of Prager model, as already mentioned above, lessflexible bi-linear modeling strongly affects the result. Even when using exclusivelythe experimental stress–strain curve of the stainless steel for the optimization, acertain discrepancy due to bi-linear modeling in the Prager model still remains, whilegood result is obtained by the Chaboche–Rousselier model [see Fig. 12(a) and (b)].As far as the Chaboche–Rousselier model is concerned, the agreements between theexperimental data and the corresponding results of numerical simulations, both for

Table 1

Identified material parameters in the Chaboche–Rousselier model for stainless steel clad aluminum sheet

E (GPa)
Y0 (MPa) H01 (MPa) a (MPa) C Q (MPa) b
Aluminum
71.2 56.2 707 29.2 1210 0 0
Stainless steel
204.3 308.1 996 277.3 935 0 0
Table 2

Identified material parameters in the Prager model for stainless steel clad aluminum sheet

E (GPa)
Y0 (MPa) H01 (MPa)
Aluminum
56.7 82.6 789
Stainless steel
234.1 538.1 7532

the uniaxial tension (Fig. 7) and the cyclic bending (Fig. 8) of the bimetallic sheet,are rather good. Hence, it would be concluded that the assumption of equallyweighted (1=2=. . .=1) response quantities of uniaxial tension (=1) and eachbending/unbending (=2, 3, . . .) is acceptable, and small change of the weightingparameters might have minor effect. For the case of the Prager model, on the con-trary, the values of weighting coefficients may have some effect on the results. For

Fig. 9. Cyclic stress–strain responses, for the stainless steel and aluminium layers in the bimetallic sheet,

calculated with the constitutive models incorporating the sets of material parameters identified from uni-

axial tension and cyclic bending tests of the bimetallic sheet: (a) the Chaboche–Rousselier model; and

(b) the Prager model.


example, when using the smaller values of weighting coefficients for cyclic bending,slightly smaller value of strain-hardening coefficient H0 was obtained [e.g., for thecase of 1=1 (for uniaxial tension) and 2=3=. . .=0.5 (for cyclic bending),H0=6743, while for the equally weighted case H0=7532]. Even so, the main cause ofthe discrepancy in this model was not the bad choice of the weighting parameters,but less flexible model itself. The residual stress which had been induced in eachmetal during cladding by roll-bonding process would also affect the results, since theeffect of the residual stress on P–" curve would not just the same as that on M–�diagram, however, its details were not examined in the present work. Another pos-sible reason for the discrepancy would be the non-J2 effect (or anistropy) of the

Fig. 10. Comparisons of the stress–strain curves in uniaxial tension and the calculated results with the

constitutive models for the stainless steel (SS) and aluminum (Al) layers in the bimetallic sheet: (a) the

Chaboche–Rousselier model; and (b) the Prager model. The sets of material parameters were identified

using the experimental data of uniaxial tension and cyclic bending.


materials. In the present work, the yield function was assumed to be J2 (von Mises)type. However, it is well known that some metals are not J2 type, e.g., aluminumand its alloys are rather Tresca-material (e.g., Shiratori et al., 1976a,b; Kanetake etal., 1981), and in some cases texture-induced anisotropy would also exist. Since theplane-strain flow stress, which appears under sheet bending, is directly related to thetypes of yield function employed in the calculation, the choice of yield function isvery important for the material parameter identification using sheet bendingexperiments. Generally speaking, to obtain the proper material parameters of plas-ticity, the experimental data of enough large strain are essential (at least, a strainvalue where the transient workhardening [or transient Bauschinger effect] finishes isnecessary, and ‘‘how large?’’ is also dependent on applications, e.g., for ordinal sheetmetal forming the required strain would be level of more than 0.2, but for structuralelement applications it would not be so large). Furthermore, if the componentmaterials have significant cyclic hardening nature, data of enough numbers ofbending cycles, until the cyclic hardening stabilization takes place, will be necessary.In order to illustrate the convergence of the optimization process, the change of

the square root of the objective function F(x) as a function of the number of itera-tions is shown in Fig. 13, in both cases of using the Chaboche–Rousselier and thePrager models. From this figure, it is found that the objective function approaches toa small asymptotic value after some 10 iterations in both the cases. The CPU time forthe identification with the Chaboche–Rousselier model was about 26 min by EWSSUN Ultra 1 (140 MHz) (SPEC fp95: 7.9), and 9 min in the case of using the Pragermodel. Such a great difference in CPU time depending on the type of constitutivemodels is attributed to the difference of the numbers of material parameters to beidentified (14 parameters for the Chaboche–Rousselier model and six parameters for

Fig. 11. Comparison of experimental surface strains [strain of the aluminum (Al) versus that of the

stainless steel (SS)] during cyclic bending and the calculated result with the Chaboche–Rousselier model

incorporating the sets of material parameters identified from uniaxial tension and cyclic bending tests of

the bimetallic sheet.


the Prager model). About 90% of the CPU time was consumed by the calculation ofbending moment.Furthermore, instead of using the set of experimental data of tensile load vs strain

in uniaxial tension and bending moment vs curvature in cyclic bending, as discussedabove, we may use the other mechanical responses, e.g., the bending moment vssurface strains in cyclic bending for both the stainless and aluminum layers. Fig. 14shows the calculated stress–strain curves for the stainless steel and the aluminum bythe constitutive model which incorporates the identified material parameters using

Fig. 12. Comparisons of the stress–strain curve in uniaxial tension and the calculated results with the

constitutive models for the stainless steel layer in the bimetallic sheet: (a) the Chaboche–Rousselier model;

and (b) the Prager model. The sets of material parameters were identified using exclusively the experi-

mental data of uniaxial tension of the stainless steel (E=204.0 GPa, Y=358.1 MPa, a =206.3 MPa,

C=1525, H01 ¼ 30:0 MPa for the Chaboche–Rosselier model; and E=204.2 GPa, Y=528.7 MPa, H0

1 ¼

5999 MPa for the Prager model, cf. Tables 1 and 2).


only the surface strain data, together with the experimental stress–strain curves. InFig. 14, a certain discrepancy between the simulated stress–strain curves and theexperimental results is found, whereas the simulated results for the bending momentand curvature agree well with the experimental results (see Fig. 15). One of the rea-sons for the discrepancy is that the strain in the cyclic bending is not large enoughfor the determination of plastic properties. Especially for the bimetallic sheet, strainin the stainless steel layer is much smaller than in the aluminum layer because of theshift of the neutral surface from the mid-plane of the sheet (Verguts and Sowerby,1975; Majlessi and Dadras, 1983). If the curvature in the cyclic bending test had

Fig. 13. Change of the square root of the objective function F(x) with increasing number of iteration.

Fig. 14. Comparison of the stress–strain curves and the simulated results with the Chaboche–Rousselier

model incorporating the identified set of material parameters for the stainless steel (SS) and aluminum

(Al) layers in the bimetallic sheet. The material parameter identification is based on the experimental data

of surface strains.


been large enough, the results of the identification would have been better. However,in practice, it is not so easy to give large surface strains by bending for thin sheetmetals, therefore, the use of experimental data of both uniaxial tension and cyclicbending is recommended for material parameter identification for bimetallic sheets.

5. Concluding remarks

A method for the identification of mechanical properties of the individual com-ponent layers in a bimetallic sheet by a mixed experimental–numerical approach hasbeen presented. As an example, for a stainless steel clad aluminum sheet, a set ofmaterial parameters in each of two constitutive models of cyclic elasto-plasticity: anonlinear kinematic/isotropic hardening model (the Chaboche–Rousselier model)and a linear kinematic hardening model (the Prager model), have been successfullyidentified by using the experimental data of the uniaxial tension and cyclic bending.It should be noted that an advanced optimization technique based on the iterativemultipoint approximation concept allows us to solve the present identificationproblem within an acceptable calculation time. By this new approach to materialparameter identification, the determination of mechanical properties of a bimetallicsheet has become possible without performing a time-consuming process of theremoval of a layer from the sheet for the preparation of the specimens. It is a con-siderable technological achievement. Moreover, it should be emphasized that thisapproach allows us to identify material parameters not only for the monotonicdeformation but also for the cyclic behavior characterized by the Bauschinger effectand the cyclic hardening nature of materials.

Fig. 15. Comparison of experimental diagram of bending moment versus curvature and the simulated

result with the constitutive model incorporating the identified set of material parameters. The material

parameter identification is based on the experimental data of surface strains.


Acknowledgements

The present work has been done within a Royal Society joint project ‘Optimiza-tion and Inverse Problems of Large Deformation Plasticity’ between University ofBradford and Hiroshima University, as well as a Monbukagakusho (The Ministryof Education, Culture, Sports, Science and Technology, Japan) project, grant-in aidfor scientific research No. 12555029.

References

Armstrong, P.J., Frederick, C.O., 1966. A mathematical representation of the multiaxial Bauschinger

effect (GEGB report RD/B/N731). Berkeley Nuclear Laboratories.

Chaboche, J.L., Rousselier, G., 1983. On the plastic and viscoplastic constitutive equations, part I and II.

ASME J. Pressure Vessel Technol. 105, 153–164.

Chun, B.K., Jinn, J.T., Lee, J.K., 2002a. Modeling the Bauschinger effect for sheet metals, part I: theory.

Int. J. Plasticity 18, 571–595.

Chun, B.K., Jinn, J.T., Lee, J.K., 2002b. Modeling the Bauschinger effect for sheet metals, part II:

applications. Int. J. Plasticity 18, 597–616.

Geng, L., Shen, Y., Wagoner, R.H., 2002. Anisotropic hardening equations derived from reverse-bending

testing. Int. J. Plasticity 18, 743–767.

Hawkins, R., Wright, J.C., 1971. Mechanical properties and press-formability of copper/mild steel sand-

wich sheet materials. J. Inst. Metal. 99, 357–371.

Kanetake, K., Tozawa, Y., Kato, T., Aiba, S., 1981. Effect of texture on deformation behavior of alumi-

num sheets. J. Jpn Inst. Light Metals 31, 307–312.

Kim, J.-K., Yu, T.-X., 1997. Forming and failure behaviour of coated, laminated and sandwiched sheet

metals: a review. Journal of Materials Processing Technology 63, 33–42.

Majlessi, S.A., Dadras, P., 1983. Pure plastic bending of sheet laminates under plane strain condition. Int.

J. Mech. Sci. 25, 1–14.

Prager, W., 1956. A new method of analyzing stresses and strains in work-hardening plastic solids. J. Appl.

Mech. 23, 493–496.

Semiatin, N.L., Piehler, H.R., 1979a. Deformation of sandwich sheet materials in uniaxial tension. Metal.

Trans. 10A, 85–96.

Semiatin, N.L., Piehler, H.R., 1979b. Formability of sandwich sheet materials in plane strain compression

and rolling. Metal. Trans. 10A, 97–107.

Semiatin, N.L., Piehler, H.R., 1979c. Forming limits of sandwich sheet materials. Metal. Trans. 10A,

1107–1118.

Shiratori, E., Ikegami, K., Yoshida, F., 1976a. The subsequent yield surfaces after proportional preload-

ing of the Tresca-type material. Bulletin of the JSME 19, 1122–1128.

Shiratori, E., Ikegami, K., Yoshida, F., Kaneko, K., Koike, S., 1976b. The subsequent yield surfaces after

preloading under combined axial load and torsion. Bulletin of the JSME 19-134, 877–883.

Toropov, V.V., Filatov, A.A., Polynkin, A.A., 1993. Multiparameter stractural optimization using FE

and multipoint explicit approximates. Structral Optimization 6, 7–14.

Toropov, V.V., Yoshida, F., van der Giessen, E., 1997. Material parameter identification for large plas-

ticity models. In: Sol, H., Oomens, C.W.J. (Eds.), Proceedings of the EUROMECH Colloqium on

Material Identification Using Mixed Numerical Experimental Methods. Kluwer Academic, pp. 81–92.

Verguts, H., Sowerby, R., 1975. The pure plastic bending of laminated sheet metals. Int. J. Mech. Sci. 17,

31–51.

Wagoner, R.H., Geng, L., Balakrishnan, V., 2000. Role of hardening law in springback. In: Khan, A.S.,

Zhan, H., Yuen Y. (Eds.), Proc. 8th Int. Symp. on Plasticity and Its Current Applications. NEAT

Press, pp. 609–611.

Yoshida, F., Hino, R., Okada, T., 1995. Deformation and fracture of stainless-steel/aluminium sheet


metal laminates in stretch bending/unbending. In: Tanimura, S., Khan, A.S. (Eds.), Proceedings of the

5th International Symposium on Plasticity and Its Current Applications. Gordon and Breach, PP. 869–

872.

Yoshida, F., 1995. Ratchetting behaviour of 304 stainless steel at 650 C under multiaxially strain-con-

trolled and uniaxially/multiaxially stress-controlled conditions. European J. Mechanics, A/Solids 14,

97–117.

Yoshida, F., 1997.Deformation and fracture of sheet metal laminates in plastic forming. In: Hui,D. (Ed.),

Proceedings of 4th International Conference on Composite Engineering, pp. 61–64.

Yoshida, F., Hino, R., 1997. Forming limit of stainless steel-clad aluminium sheets under plane stress

condition. Journal of Materials Processing Technology 63, 66–71.

Yoshida, F., Urabe, M., Toropov, V.V., 1998. Identification of material parameters in constitutive model

for sheet metals from cyclic bending tests. Int. J. Mech. Sci. 40, 237–249.

Yoshida, F., Urabe, M., 1999. Computer-aided process design for the tension levelling of metallic strips.

Journal of Materials Processing Technology 89/90, 218–223.

Yoshida, F., Urabe, M., 2000. Computer-aided process design for the tension levelling of clad sheet metal.

Key Engineering Materials 177-180, 503–508.

Yoshida, F., 2000. A constitutive model of cyclic plasticity. Int. J. Plasticity 16, 359–380.

Yoshida, F., Uemori, T., Fujiwara, K., 2002. Elastic-plastic behavior of steel sheets under in-plane cyclic

tension-compression at large strain. Int. J. Plasticity 18, 633–659.

Yoshida, F., Uemori, T., 2002. A model of large-strain cyclic plasticity describing the Bauschinger effect

and workhardening stagnation. Int. J. Plasticity 18, 661–686.

Zhao, K.M., Lee, J.K., 2001. Generation of cyclic stress-strain curves for sheet metals. ASME, J. Eng.

Mater. Technol. 123, 391–397.


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