Accepted Manuscript

Modeling of anisotropic hardening of sheet metals including description of theBauschinger effect

Fusahito Yoshida, Hiroshi Hamasaki, Takeshi Uemori

PII: S0749-6419(15)00033-9

DOI: 10.1016/j.ijplas.2015.02.004

Reference: INTPLA 1882

To appear in: International Journal of Plasticity

Received Date: 29 August 2014

Revised Date: 19 January 2015

Accepted Date: 3 February 2015

Please cite this article as: Yoshida, F., Hamasaki, H., Uemori, T., Modeling of anisotropic hardening ofsheet metals including description of the Bauschinger effect, International Journal of Plasticity (2015),doi: 10.1016/j.ijplas.2015.02.004.

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Modeling of anisotropic hardening of sheet metals including description of the Bauschinger effect

Fusahito Yoshidaa*, Hiroshi Hamasakib and Takeshi Uemoric*

aDepartment of Mechanical Science and Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi Hiroshima 739-8527, Japan

b Department of Mechanical Science and Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi Hiroshima 739-8527, Japan

c Department of Advanced Mechanics and Manufacturing, Okayama University, 1-1-1, Tsushima-naka, Kita-ku, Okayama 700-0082, Japan

ABSTRACT The present paper proposes a framework for constitutive modeling of plasticity to describe the evolution of anisotropy and the Bauschinger effect in sheet metals. An anisotropic yield function, which varies continuously with increasing plastic strain, is defined as an interpolation between two yield functions at two discrete levels of plastic strain. Several types of nonlinear functions of the effective plastic strain are proposed for the interpolation. A framework for a combined anisotropic-kinematic hardening model of large-strain cyclic plasticity with small elastic strain is presented, and the details of modeling which is based on the Yoshida–Uemori model (Yoshida and Uemori et al., 2002a, 2002b, 2003) are described. The shape of the yield surface and that of the bounding surface are assumed in the model to change simultaneously. The model was validated by comparing the calculated results of stress–strain responses with experimental data on r-value and stress-directionality changes in an aluminum sheet (Hu, 2007) and a stainless steel sheet (Stoughton and Yoon, 2009), as well as the variation of the yield surface of an aluminum sheet (Yanaga et al., 2014). Furthermore, anisotropic cyclic behavior was examined by performing experiments of uniaxial tension and cyclic straining in three sheet directions on a 780-MPa advanced high-strength steel sheet.

Keywords: Anisotropic hardening; B. constitutive behaviour; B. anisotropic material; B. metallic material

* Corresponding author. Tel: +81 82 424 7541; fax : +81 82 424 7541

E-mail address: fyoshida@hiroshima-u.ac.jp (F. Yoshida)

1. Introduction

The use of plasticity models that properly describe material behavior is of vital importance for accurate numerical simulation of sheet metal forming. Conventional plasticity models assume that the shape of the yield surface does not change during a plastic deformation, and consequently, r-values and flow stress directionality calculated with the models remain constant throughout the deformation. However, some metallic sheets exhibit significant changes in planar r-value anisotropy and flow stress directionality (e.g., Hu, 2007; Stoughton and Yoon, 2009; An et al., 2013; Safaei et al., 2014) and the shape of yield surface (e.g., Tozawa, 1978; Kuwabara et al., 1998; Yanaga et al., 2014; Yoon et al., 2014) as plastic strain increases.

The anisotropy of a solid is expressed by an anisotropic yield function that involves a set of material parameters, , 1, 2, ...,kC k N= . To describe the evolution of anisotropy, the material parameters should be given as functions of

the effective plastic strain ε , i.e., as ( )kC ε . The material parameters in yield functions for sheet metals are usually

determined using experimental data on uniaxial tension stresses (e.g., 0 45,σ σ ,

90σ ) and r-values (e.g., 0r ,

45r , 90r ) in

several sheet directions and some biaxial stresses (e.g., equi-balanced stress bσ ). If these stresses and r-values are

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formulated as functions of the plastic strain, i.e., 0 45 90( ), ( ), ( ), ( ),bσ ε σ ε σ ε σ ε 0 45 90( ), ( ), ( )r r rε ε ε , the

material parameters kC can be calculated from these for any effective plastic strain values.

Stoughton and Yoon (2009) employed Hill’s (1948) quadratic yield function in the framework of a non-associated flow rule to describe anisotropic hardening of several types of aluminum and stainless steel sheets. Hu (2007) used a fourth-order polynomial yield function and succeeded in reproducing both the stress and r-value anisotropies of an aluminum sheet. In their models, the material parameters of a yield function are calculated for

every time step using a given set of stress values and r-values ( 0 45 90( ), ( ), ( ), ( ),bσ ε σ ε σ ε σ ε

0 45 90( ), ( ), ( )r r rε ε ε ). However, direct calculation of material parameters for every time step of sheet metal

forming simulation is only applicable to yield functions whose material parameters are calculated explicitly as functions of

0 45,σ σ ,90σ ,

bσ ,0r ,

45r , 90r , etc.; it is not applicable to more complicated models whose material

parameters are determined using only iterative numerical approaches such as the N-R method (e.g., Barlat et al., 2003, 2005, 2007; Banabic et al. 2005; Yoshida et al. 2014).

An alternative approach is approximation equation building for material parameters, such as ( ), 1, 2, ...,kC k Nε = .

For that purpose, M sets of material parameters, ( )ikC , 1, 2, ..,i M= , determined at M discrete levels of the effective

plastic strains, , 1, 2, ..,i i Mε = , are employed (e.g., Aretz, 2008; Safaei et al., 2014; Yanaga et al., 2014). An

advantage of this approach over the aforementioned direct calculation approach is that only limited experimental data (M sets) corresponding to , 1, 2, ..,i i Mε = are needed. However, in most of cases, the approximation equations

involve many coefficients that need to be determined from analysis of experimental data. Instead of using approximation equations for material parameters, Plunkett et al. (2006) expressed the current yield function

( , )φ σ ε directly as a linear interpolation of two yield functions, ( , )i iφ σ ε and 1 1( , )i iφ σ ε+ + , within the plastic

strain range 1i iε ε ε +≤ ≤ .

Another important issue in material modeling is describing the Bauschinger effect, especially for accurate numerical simulation of springback of high-strength steel (HSS) sheets (e.g., Yoshida and Uemori, 2003; Eggertsen and Mattiason, 2009, 2010; Wagoner et al, 2013). Although many types of kinematic hardening models have been proposed to describe the Bauschinger effect and cyclic plasticity behavior (e.g., Armstrong and Frederick, 1966; Ohno, N., 1982; Chaboche and Rousselier, 1983; Ohno and Wang, 1993; Geng and Wagoner, 2002; Yoshida et al., 2000, 2002a, 2002b, 2003, 2008; Haddadi et al. 2006; Taleb, 2013; for more details, refer to Chaboche’s review, 2008), very few models have been developed to describe stress and r-value anisotropy evolution simultaneously. So-called distortion yield function modeling is another type of formulation used to represent the Bauschinger effect and stress-strain responses under non-proportional cyclic loading (e.g., Shiratori et al, 1979; Voyadijis and Froozesh, 1990; Kurtyka and Zyczkowski, 1996; Francois, 2001; Feigenbaum and Dafalias, 2007; Barlat et al., 2011, 2013, 2014). However, to the best of the present authors’ knowledge, among these works, only Barlat et al.’s homogeneous anisotropic hardening (HAH) model (2011, 2013, 2014) reproduces the Bauschinger effect well together with the anisotropy evolution of sheet metals.

The aim of the present paper is to propose a framework for constitutive modeling of plasticity, based on the associated flow rule, to describe the evolution of anisotropy and the Bauschinger effect in sheet metals. In the

anisotropic hardening model presented in this paper, an anisotropic yield function, ( , )φ σ ε , which varies

continuously with increasing plastic strain within the range A Bε ε ε≤ ≤ , is defined as an interpolation between two

yield functions, ( )Aφ σ and ( )Bφ σ , for two discrete levels of effective plastic strains, Aε and

Bε . A full

description of anisotropy evolution requires only a limited number of yield functions, ( )iφ σ , 1, 2, ..,i M= ,

determined at a few discrete levels of plastic strains, , 1, 2, ..,i i Mε = . The proposed model was validated by

comparing the calculated stress–strain responses with experimental data on r-value and stress changes in an aluminum sheet (Hu, 2007) and a stainless sheet (Stoughton and Yoon, 2009), as well as experimental data on the variation of yield surface of an aluminum sheet (Yanaga et al., 2014). This anisotropy evolution model is able to describe the Bauschinger effect by incorporating it into the kinematic hardening model. This paper describes a combined anisotropic–kinematic hardening model constructed in the framework of the Yoshida–Uemori two-surface modeling approach (Yoshida and Uemori, 2002a, 2002b, 2003), wherein the shape of the yield surface and that of the bounding surface change simultaneously. The model was validated by comparing the numerical prediction of

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sheet-direction-dependent cyclic plasticity behavior to experimental results for a 780-MPa advanced HSS (AHSS) sheet.

2. Modeling of anisotropic hardening

2.1. Framework of modeling

In the present paper, a model is constructed within the framework of the associated flow rule of plasticity. With the assumption of small elastic and large plastic deformation, the rate of deformation D is decomposed into its

elastic and plastic parts, eD and eD , respectively, as follows: e p= +D D D . (1)

The decomposition of the continuum spin W is given as follows: p= +W WΩΩΩΩ , (2)

where pW denotes the plastic spin and ΩΩΩΩ is the spin of substructures (Dafalias, 1985). The constitutive equation of elasticity is expressed as follows:

:o

e−= + =&σ σσ σσ σσ σ C D (3)

where and o

σσσσ are the Cauchy stress and its objective rate, respectively, ΩΩΩΩ is the spin tensor; and C is the

elasticity modulus tensor. The initial yield criterion is expressed by the following equation:

( ) ( )0 0f Y Yφ σ= − = − =σ σσ σσ σσ σ , (4)

where Y is the initial yield stress and σ is the effective stress. In general, the evolution of anisotropy is expressed by

the evolutionary change of the yield function φ as a function of the effective plastic strain ε . Let us begin our

discussion with a simple case that excludes the description of the Bauschinger effect. The subsequent yield function after plastic deformation is expressed by the following equation:

( ) ( ) ( ) ( ), ( ) , ( ) , ( )F F Ff Y Hφ ε σ ε σ ε σ ε σ ε ε= − = − = − +σ σ σσ σ σσ σ σσ σ σ =0, (5)

where Fσ and FH are the flow stress and its workhardening component, respectively. Note that ( ) ( )0,0φ φ=σ σσ σσ σσ σ .

Based on the work conjugate concept, the effective plastic strain is defined as follows:

: pσ ε =& σσσσ D , dtε ε= ∫ & . (6)

The associated flow rule is written as follows

p f φλ λ∂ ∂= =∂ ∂

& &

σ σσ σσ σσ σD , (7)

where ( 0)λ ≥& is the plastic multiplier. For a homogeneous yield function of degree one, φ , the work conjugate

type definition of the effective plastic strain rate (see Eq.(6)) reduces to λ ε=& & . Thus, the elasto-plasticity constitutive equation can be written as follows:

:o

ep=σσσσ C D , (8)

if 0

: :if 0

: :

ep

H

λφ φ

λφ φ φε

= ∂ ∂ ⊗= ∂ ∂− > ∂ ∂ ∂′+ −

∂ ∂ ∂

&

&

C

C CCC

C

σ σσ σσ σσ σ

σ σσ σσ σσ σ

(9)

where

F FF

dHH H

d

σε ε

∂′ ′= = =∂

. (10)

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In the present paper, the anisotropic hardening of the yield surface is expressed as follows:

( )( , ) ( ) ( ) 1 ( ) ( )A Bφ ε µ ε φ µ ε φ= + −σ σ σσ σ σσ σ σσ σ σ for A Bε ε ε≤ ≤ (11)

Here, ( )Aφ σσσσ and ( )Bφ σσσσ are two different yield functions defined at the effective plastic strains Aε and Bε ,

respectively, and ( )µ ε an interpolation function of the effective plastic strain, where

1 ( ) ( ) ( ) 0A Bµ ε µ ε µ ε= ≥ ≥ = . (12)

Note that the types of these two yield functions, ( )Aφ σσσσ and ( )Bφ σσσσ , do not need to be the same. For example Fig.

1 shows a case of ( )Aφ σσσσ being a Tresca-type function and ( )Bφ σσσσ being a von Mises-type function. An advantage

of this modeling framework is that, if the two yield functions ( )Aφ σσσσ and ( )Bφ σσσσ are convex, ( , )φ εσσσσ always

satisfies the convexity. The derivatives are expressed as follows:

( )

( )

( ) ( )( ) 1 ( ) ,

( )( ) ( ) .

A B

A B

φ φ φµ ε µ ε

φ µ εφ φε ε

∂ ∂ ∂= + −∂ ∂ ∂∂ ∂= −∂ ∂

σ σσ σσ σσ σσ σ σσ σ σσ σ σσ σ σ

σ σσ σσ σσ σ (13)

Fig. 1 Change of yield function from Tresca ( 1µ = ) to von Mises ( 0µ = ).

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

Tresca (µ = 1.0)Tresca-von Mises (µ = 0.3)von Mises (µ = 0)

σ y / σ

0

σx / σ

0

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Several linear and nonlinear functions can be used for the interpolation function ( )µ ε . Assuming that 0Aε = (at

initial yielding) and Bε = ∞ (at infinitely large strain) and that 0( ) ( )Aφ φ=σ σσ σσ σσ σ and ( ) ( )Bφ φ∞=σ σσ σσ σσ σ , Eq.(11)

reduces to the following

( )0( , ) ( ) ( ) 1 ( ) ( )φ ε µ ε φ µ ε φ∞= + −σ σ σσ σ σσ σ σσ σ σ , 0 ε≤ ≤ ∞ . (14)

Some examples of forms of interpolation functions are as follows:

( ) exp( )µ ε λε= − , (15)

1 2( ) exp( ) (1 )exp( )a aµ ε λ ε λ ε= − + − − , (16)

where λ , a, 1λ , and 2λ are material constants.

If we have M sets of experimental data (0 45,σ σ ,

90σ , bσ ,

0r , 45r ,

90r , etc.) for material parameter identification

corresponding to M discrete plastic strain points, 1 2 1( 0), ,..., , ,...,i i Mε ε ε ε ε+= , we can determine M sets of yield

functions 1( )φ σσσσ , 2( )φ σσσσ , …., 1( ), ( ),i iφ φ +σ σσ σσ σσ σ …, ( )Mφ σσσσ . Using an interpolation function ( )µ ε , the yield function

( , )φ εσσσσ can be defined by the following equation:

( ) 1( , ) ( ) ( ) 1 ( ) ( )i iφ ε µ ε φ µ ε φ += + −σ σ σσ σ σσ σ σσ σ σ for 1i iε ε ε +≤ ≤ . (17)

The following nonlinear equation is proposed for use as the interpolation function:

1

( ) 1ip

i

i i

ε εµ εε ε+

−= − − for 1i iε ε ε +≤ ≤ , (18)

where ( 1, 2, ..., 1)ip i M= − are material constants.

2.2. Modeling of polynomial-type yield function

Among the various types of anisotropic yield functions available, stress polynomial-type models (e.g., Hill 1948,

Gotoh 1977, Soare et al. 2008, Yoshida et al. 2013) are suitable for use in modeling anisotropy evolution. A polynomial-type yield criterion is given by the following equation:

( )( ) 0m m m mF Ff φ σ σ σ= − = − =σσσσ , (19)

where ( )( )mφ σσσσ denotes the m-th order stress polynomial-type yield function. For example, when m = 6 (Yoshida et

al. 2013) under plane stress condition,

( )( )

(6) 6 5 4 2 3 3 2 4 5 61 2 3 4 5 6 7

4 3 2 2 3 4 28 9 10 11 12

2 2 4 613 14 15 16

3 6 7 6 3

9 2 3 2

27 27 .

x x y x y x y x y x y y

x x y x y x y y xy

x x y y xy xy

C C C C C C C

C C C C C

C C C C

φ σ σ σ σ σ σ σ σ σ σ σ σ

σ σ σ σ σ σ σ σ τ

σ σ σ σ τ τ

= − + − + − +

+ − + − +

+ − + +

(20)

In the same manner as Eq.(11), when the following equation is assumed

( )( ) ( ) ( )( , ) ( ) ( ) 1 ( ) ( )m m mA Bφ ε µ ε φ µ ε φ= + −σ σ σσ σ σσ σ σσ σ σ , (21)

it reduces to an interpolation for material parameters , 1,2,...,kC k N= , as follows

( )(A) (B)( ) 1 ( )k k kC C Cµ ε µ ε= + − . (22)

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Here, (A)kC and

(B)kC are material parameters determined at the effective plastic strains, Aε and Bε , respectively.

Assuming that 0Aε = (at initial yielding) and Bε = ∞ (at infinitely large strain) and that ( ) (0)k A kC C= and

(B) ( )k kC C ∞= , Eq.(22) reduces to the following:

( )(0) ( )( ) 1 ( )k k kC C Cµ ε µ ε ∞= + − , 0 ε≤ ≤ ∞ . (23)

In discretization form:

( )(i) (i 1)( ) 1 ( )k k kC C Cµ ε µ ε += + − , 1, 2, ..., 1i M= − . (24)

2.3. Validation of the model To validate the model, calculated stress-strain responses to r-value and stress changes were compared with the corresponding experimental data for an aluminum sheet (after Hu, 2007) and a stainless steel sheet (after Stoughton and Yoon, 2009), as well as the variation of yield surface of an aluminum sheet (after Yanaga et al., 2014) . Hu (2009) presented interesting experimental data on the drastic changes in r-values (

0r , 45r ,

90r ) and flow stresses

(0 45,σ σ ,

90σ , bσ ) that occurred in a 3003-O aluminum sheet with increasing plastic strain. The flow stresses and r-

values calculated using the proposed model were compared with Hu’s experimental data. A sixth-order polynomial model, Eq. (20), was employed for the yield function. Using this yield function, flow stresses

0 45,σ σ ,90σ ,

bσ and

r-values 0r ,

45r , 90r were given as functions of material parameters 1 16~C C in explicit forms (see Yoshida et al.,

2013).

1/6 1/ 6 1/6

1 1 190 0 45 0 0

7 16

, 2 ,9 27 27 b

C C C

C S T U C Sσ σ σ σ σ σ

= = = + + + (25)

16 620 45 90

1 2 7 6

3 9 27, ,

2 2 12 18 2

S T U C CCr r r

C C S T U C C

− − + += = =− + + −

(26)

1 2 3 4 5 6 7

8 9 10 11 12 13 14 15

3 6 7 6 3 ,

2 3 2 , 3

S C C C C C C C

T C C C C C U C C C

= − + − + − += − + − + = − +

Figure 2 shows experimental data on the change in the flow stresses (0 45,σ σ ,

90σ ,bσ ) and the corresponding

values predicted using Eq. (23). Interpolation equation Eq. (16) was employed using material constants 0.5a = ,

1 70λ = and 2 10λ = . Material parameters of the yield function, (0)kC and ( )kC ∞ , determined at 0 andε = ∞ ,

are listed in Table 1. The calculated stresses agree fairly well with the experimental results, although the predicted

bσ is slightly lower than the experimental value at relatively small strains. The comparisons of r-values are

illustrated in Fig. 3. The predicted values of 0r ,

45r , and90r agree well with the experimental results.

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Fig. 2 Flow stresses of 3003-O aluminum sheet and values predicted using the anisotropy evolution model (see Eqs. (23) and (16)).

Fig. 3 R-values of 3003-O aluminum sheet and values predicted using the anisotropy evolution model (see Eqs. (23) and (16)).

0

20

40

60

80

100

120

140

160

0 0.05 0.1 0.15 0.2 0.25 0.3

σ0 (experiment = calculation)

σ45 (experiment)

σ45 (calculation)

σ90 (experiment)

σ90 (calculation)

σb (experiment)

σb (calculation)

σ 0, σ45

, σ 9

0, σb (M

Pa)

εp

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3

r0 (experiment)

r0 (calculation)

r45 (experiment)

r45 (calculation)

r90 (experiment)

r90 (calculation)

r 0 , r 4

5 , r 90

εp

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Table 1 Material parameters in the sixth-order polynomial-type yield function, (0)kC , k = 1~16, for 3003-O

aluminum sheet (experimental data are from Hu, 2007), determined at initial yielding ( 0ε = ) and ( )kC ∞ at

infinitely large strain (ε = ∞ ).

(0)

at 0kC

ε =

C1 C2 C3 C4 C5 C6 C7 C8 1.000 0.826 0.815 0.854 0.988 1.205 1.510 1.290

C9 C10 C11 C12 C13 C14 C15 C16 1.132 1.189 1.310 1.558 1.271 0.533 1.455 1.369

( )

atkC

ε∞

= ∞

C1 C2 C3 C4 C5 C6 C7 C8 1.000 0.767 0.742 0.660 0.705 0.827 1.378 1.043

C9 C10 C11 C12 C13 C14 C15 C16 0.795 0.830 0.822 1.194 1.017 0.401 0.950 0.866

Fig. 4 Flow stresses of 3003-O aluminum sheet and values predicted using the anisotropy evolution model (see Eqs (24) and (18)).

0

20

40

60

80

100

120

140

160

0 0.05 0.1 0.15 0.2 0.25 0.3

σ0 (experiment = calculation)

σ45 (experiment)

σ45 (calculation)

σ90 (experiment)

σ90 (calculation)

σb (experiment)

σb (calculation)

σ 0, σ45

, σ 9

0, σb (M

Pa)

εp

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Fig. 5 R-valuesof 3003-O aluminum sheet and values predicted using the anisotropy evolution model (see Eqs (24) and (18)).

Table 2 Material parameters in the sixth-order polynomial-type yield function, (i)kC , k = 1~16; i = 1~3, for 3003-O

aluminum sheet (experimental data are from Hu, 2007), determined at three levels of effective plastic strain:

0, 0.05, 0.30ε = .

(1)

at 0kC

ε =

C1 C2 C3 C4 C5 C6 C7 C8 1.000 0.826 0.815 0.854 0.988 1.205 1.510 1.290

C9 C10 C11 C12 C13 C14 C15 C16 1.132 1.189 1.310 1.558 1.271 0.533 1.455 1.369

(2)

at 0.05kC

ε =

C1 C2 C3 C4 C5 C6 C7 C8 1.000 0.789 0.696 0.623 0.670 0.927 1.386 1.119

C9 C10 C11 C12 C13 C14 C15 C16 0.919 0.908 1.037 1.368 1.063 0.136 0.944 0.901

(3)

at 0.30kC

ε =

C1 C2 C3 C4 C5 C6 C7 C8 1.000 0.770 0.737 0.656 0.701 0.839 1.378 1.053

C9 C10 C11 C12 C13 C14 C15 C16 0.807 0.835 0.839 1.199 1.028 0.404 0.986 0.871

The flow stresses and r-values calculated using Eq. (17) and the interpolation equation of Eq. (18) for M = 3

were compared to the experimental data, and the results are shown in Figs 4 and 5, respectively. Three discrete

plastic strain points, iε = 0, 0.05, and 0.3 were selected to define 1( )φ σσσσ , 2( )φ σσσσ , and 3( )φ σσσσ . The material

parameters of the yield function, (1) (2),k kC C , and (3)kC , determined at 0ε = , 0.05, and 0.3, respectively, are

listed in Table 2. The material constants used in Eq. (18) were 1 0.25p = for 0 0.05ε≤ ≤ and 2 0.60p = for

0.05 0.3ε≤ ≤ . The calculated results agree well overall with experimental results for both the stresses and r-

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3

r0 (experiment)

r0 (calculation)

r45 (experiment)

r45 (calculation)

r90 (experiment)

r90 (calculation)

r 0 , r 4

5 , r 90

εp

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values even though we used just three sets of experimental data for stresses and r-values at iε = 0, 0.05, and 0.3.

The yield surfaces at the plastic strain vlues of ε = 0, 0.05, and 0.3 are illustrated in Fig. 6.

Fig. 6 Yield surfaces at iε = 0, 0.05, and 0.3.

Plunkett et al. (2006) presented an anisotropic hardening model using a linear interpolation, i.e., corresponding to

the case of 1ip = in Eq. (18), based on Eq. (17). In their model, the flow stress, Fσ , is also expressed by linear

interpolation, together with the yield function, ( )φ σσσσ , as follows:

( )( ) ( 1)( ) 1 ( )F F i F iσ µ ε σ µ ε σ += + − , 1, 2, ..., 1i M= − , 1, 2, ...,k N= . (27)

Consequently, stress–strain curves calculated using the model are always piecewise linear, as illustrated in Fig. 7 for the case of M = 4 for a 3003-O aluminum sheet. The change in the r-values with increasing plastic strain, calculated using the model are illustrated in Fig. 8. These results show that the nonlinear interpolation model produces more realistic stress-strain responses and r-value changes (see Figs. 7 and 8) than those calculated using the linear interpolation model, although the former uses fewer sets of experimental data (M = 3) than the latter (M = 4).

-200

-150

-100

-50

0

50

100

150

200

-200 -150 -100 -50 0 50 100 150 200

Experimentσ y (

MP

a)

σx (MPa)

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Fig. 7 Calculation of flow stresses for 3003-O aluminum sheet using linear interpolation equation.

Fig. 8 Calculation of r-values for 3003-O aluminum sheet using linear interpolation equation.

0

20

40

60

80

100

120

140

160

0 0.05 0.1 0.15 0.2 0.25 0.3

σ0 (experiment)

σ45

(experiment)

σ90 (experiment)

σ0 (calculation)

σ45

(calculation)

σ90 (calculation)

σ 0, σ

45, σ

90 (M

Pa)

ε p

0.4

0.5

0.6

0.7

0.8

0.9

0 0.05 0.1 0.15 0.2 0.25 0.3

r0 (experiment)

r45 (experiment)

r90 (experiment)

r0 (calculation)

r45

(calculation)

r90 (calculation)

r 0, r45, r

90

ε p

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The second example considered was the anisotropic hardening of a type 719-B stainless steel sheet (after Stoughton and Yoon, 2009). In this sheet, r-value planar anisotropy remains fixed throughout the plastic deformation. Figure 9 shows the r-values for various sheet directions, calculated by the yield function, together with the corresponding experimental data. The flow stresses,

0 45,σ σ ,90σ ,

bσ , calculated using Eq. (24) and Eq. (18),

with M = 3, are compared to the experimental data in Fig. 10. Three discrete plastic strain points, iε = 0, 0.1 and 0.5

were selected to define the sixth-order polynomial-type yield functions 1( )φ σσσσ , 2( )φ σσσσ and 3( )φ σσσσ . Material

parameters of the yield function, (1) (2),k kC C and (3)kC determined at 0ε = , 0.1 and 0.5, respectively, are listed

in Table 3. The material constants are 1 0.3p = for 0 0.1ε≤ ≤ and 2 0.5p = for 0.1 0.5ε≤ ≤ . The calculated

result agree well overall with the experimental result for the stresses. The yield surfaces at the plastic strains,ε = 0, 0.1 and 0.5, are shown in Fig. 11.

Fig. 9 R-value for type 790-B stainless steel sheet; experimental data (after Stoughton and Yoon, 2009) and predictions obtained using sixth-order polynomial-type yield function (Yoshida et al., 2013).

1

1.5

2

2.5

3

3.5

0 15 30 45 60 75 90

ExperimentCalculation

r-va

lue

r α

Tension axis angle from R.D. α (degree)

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Fig. 10 Flow stresses for type 790-B stainless steel sheet; experimental data (after Stoughton and Yoon, 2009) and predictions obtained using the proposed model (see Eqs (24) and (18)).

Fig. 11 Yield surfaces for type 790-B stainless steel sheet and experimental data (from Stoughton and Yoon, 2009)

for iε = 0, 0.1, and 0.5.

0

100

200

300

400

500

600

700

800

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

σ0 (experiment=calculation)

σ45 (experiment)

σ45 (calculation)

σ90 (experiment)

σ90 (calculation)

σb (experiment)

σb (calculation)

σ 0, σ

45, σ

90, σ

b (M

Pa)

ε p

-1000

-500

0

500

1000

-1000 -500 0 500 1000

Experiment

σ y (M

Pa)

σx (MPa)

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Table 3 Material parameters in the sixth-order polynomial-type yield function, (i)kC , k = 1~16; i = 1~3, for 790-B

stainless steel sheet (experimental data are from Stoughton and Yoon, 2009), determined at three levels of effective

plastic strain: 0, 0.10, 0.50ε = .

(1)

at 0kC

ε =

C1 C2 C3 C4 C5 C6 C7 C8 1.000 1.368 1.509 1.667 1.571 1.464 0.982 1.181

C9 C10 C11 C12 C13 C14 C15 C16 1.517 1.462 1.526 1.063 0.996 1.278 0.882 0.927

(2)

at 0.10kC

ε =

C1 C2 C3 C4 C5 C6 C7 C8 1.000 1.368 1.523 1.668 1.562 1.446 0.969 1.041

C9 C10 C11 C12 C13 C14 C15 C16 1.562 1.446 0.969 1.041 0.882 1.351 1.120 0.953

(3)

at 0.50kC

ε =

C1 C2 C3 C4 C5 C6 C7 C8 1.000 1.368 1.522 1.732 1.663 1.584 1.062 1.130

C9 C10 C11 C12 C13 C14 C15 C16 1.485 1.471 1.629 1.149 0.939 1.311 1.198 1.009

The third example considered was the variation of the yield surface of a sheet metal during plastic deformation. Yanaga et al. (2014) obtained very interesting experimental data for a 6016-T4 aluminum sheet by performing a specially designed biaxial test, in which the shape of the yield surface changed significantly with increasing plastic

strain. In the present study, the following Yld2000-2d yield function (Barlat et al., 2003), ( )φ syld2000 , was used

with Eqs (14) and (18) for 0.002 0.16ε≤ ≤ (M = 2, 1ε = 0.002 and 2ε = 0.16) to model this scenario:

( )1/

' ' " " ' ' '' '' " "1 2 1 2 1 2 2 1 1 2

1( ) ( , , , )

2

mm m m

S S S S S - S 2S + S 2S + Sφ σ = = + +

% % % % % % % % % %yld2000 yld2000s .

(28)

In Eq. (28), s is the stress deviator and ' '1 2,S S% % and

" "1 2,S S% % are the principal values of 's and "s , respectively,

which are defined as follows:

1

2

7

0 0

0 0

0 0

xx xx

yy yy

xy xy

s s

' = s s

s s

αα

α

′ ′ = ′

s ,

5 3 6 4

3 5 4 6

8

4 2 2 01

2 2 4 03

0 0 3

xx xx

yy yy

xy xy

s s

= s s

s s

α α α αα α α α

α

′′ − − ′′ ′′ = − − ′′

s

(29)

The Yld2000-2d parameters, 1 8~ ,mα α , for 1ε = 0.002 and 2ε = 0.16 are listed in Table 4. In Fig. 12, the yield

surfaces (equi-plastic work surfaces) for the 6016-T4 aluminum sheet under proportional loading, calculated using the proposed model (Eqs (24) and (18)), are compared with the corresponding experimental results. For the

calculation only two sets of Yld2000-2d parameters determined at 1ε = 0.002 and 2ε = 0.16, as listed in Table 4,

with an interpolation parameter, value 1 0.5p = for Eq. (18) were used. The calculated results for the yield surfaces

agree well with the experimental data. In contrast to this approach, Yanaga et al. gave the following approximation equations for

1 8~α α and m:

( )expk k k k kA B C aα ε ε= − − + , 1, 2, ..., 8k = (30)

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1 2

01 exp ( ) /

A Am

x xε−=

+ − ∆. (31)

In Eqs (30) and (31), 0, , , ,k k k kA B C a x and x∆ are coefficients (34 coefficients in total) whose values need to be

determined experimentally. Yanaga et al. demonstrated that thus formulated yield function reproduces the experimentally obtained yield surfaces very well. However, this approach to approximation equation building for material parameters might lead to a difficulty in the determination of many coefficient values. From this perspective, the present approach offers advantages in that it requires the determination of many fewer parameter values and it requires fewer sets of experimental data for parameter identification. In the case of the 6016-T4 sheet, only two

yield surface data sets at 1ε = 0.002 and 2ε = 0.16 were needed to determine the 1( )φ σσσσ and 2( )φ σσσσ parameters,

and one more data set at a plastic strain between 1ε and 2ε , e.g., at ε = 0.06 was needed to determine the

interpolation parameter, 1p . Furthermore, the proposed approach always guarantees the convexity of the yield

surface ( , )φ εσσσσ , for any type of yield function, when ( )iφ σσσσ and 1( )iφ + σσσσ are convex. In contrast, with the

approach that uses approximation equations for material parameters, convexity is theoretically guaranteed only for yield functions that satisfy the convexity for any material parameters (which is true for Yld2000-2d, for example, but not for high-order polynomial-type yield functions).

Fig. 12 Calculation of the change in yield surfaces according to the present model for a 6014-T4 aluminum sheet (experimental data are from Yanaga et al., 2014)

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400

ε = 0.002

ε = 0.02

ε = 0.04

ε = 0.06

ε = 0.10

ε = 0.16

σ y (M

Pa)

σ x (MPa)

Experiment Calculation

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Table 4 Material parameters in Yld2000-2d yield function, kα , k = 1~8, and m for 6016-T4 aluminum sheet,

determined at two levels of effective plastic strain: ε = 0.002 and 0.160 (after Yanaga et al., 2014).

and

at 0.002k mαε =

1α 2α 3α 4α 5α 6α 7α 8α m

0.907 0.993 0.922 1.045 1.032 0.988 0.830 1.429 6.7

and

at 0.160k mαε =

1α 2α 3α 4α 5α 6α 7α 8α m

0.974 1.022 0.972 1.010 1.003 0.992 1.001 1.204 38.7

3. Combined anisotropic-kinematic hardening model

3.1. Framework of modeling Kinematic hardening modeling is the most popular approach used to describe the Bauschinger effect and the

cyclic plasticity behavior of metals. The proposed anisotropic hardening model is easily combined with kinematic hardening models. In kinematic hardening modeling, the yield function f is given by the following equation:

( ) ( ) 0,

,

f Y Yφ , ε σ , ε= − − = − == −

%

%

σ α σ σ α σ σ α σ σ α σ σ σ ασ σ ασ σ ασ σ α

(32)

where αααα denotes the backstress. Based on the following definitions of the effective plastic strain and its rate

: pσ ε =& %σσσσ D , dtε ε= ∫ & , (33)

the associated flow rule is written as follows:

p f φλ λ∂ ∂= =∂ ∂

& &

% %σ σσ σσ σσ σD , (34)

where λ ε=& & . Most kinematic hardening models assume the following form of the evolution equation of the back stress:

( )o A A

Y Yε ε = − − = −

& &%α σ α σα σ α σα σ α σα σ α σx x . (35)

Here (o) denotes the objective rate. For example, in the linear kinematic hardening model:

( )o

LK LKH H

Y Yε ε′ ′

= − =& &%α σ α σα σ α σα σ α σα σ α σ , (36)

,LKA H′= = 0x .

In the Armstrong–Frederick model (1966):

( )1 12 2

o

Y Y

γ γγ ε γ ε = − − = −

& &%α σ α α σ αα σ α α σ αα σ α α σ αα σ α α σ α , (37)

1 2,A γ γ= = ααααx .

The Yoshida–Uemori kinematic hardening law has the same form (for details, see Sec. 3.2.). The constitutive equation is given in the same form as Eqs. (8) and (9), where /F FH H σ ε′ ′= = ∂ ∂ is replaced with

KH ′ as follows:

:KH H Aφ∂′ ′= = −

∂ %x

σσσσ. (38)

3.2. Anisotropic hardening based on the Yoshida-Uemori model

The Yoshida–Uemori model (2002a, 2002b, 2003) was constructed within the framework of two-surface modeling, wherein the yield surface moves kinematically within a bounding surface, as schematically illustrated in

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Fig. 13. To describe anisotropic hardening (i.e., expansion of the surface with shape change) and also kinematic hardening, the bounding surface F is expressed by the equation:

( ) ( ) 0F B Rφ ,ε= − − + =σ βσ βσ βσ β , (39)

where ββββ denotes the center of the bounding surface, and B and R are the initial size of the surface and its

workhardening component, respectively. To include the description of anisotropic hardening in the model, it is assumed that the shapes of both the yield and bounding surfaces vary simultaneously as a function of the effective plastic strain, as schematically illustrated in Fig. 14.

Fig 13 Schematic illustration of the Yoshida–Uemori two-surface model

Initial state After plastic deformation

Fig. 14 Schematic illustration of the changes of the yield surface and the bounding surface.

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The kinematic hardening of the yield surface describes the transient Bauschinger deformation, which is

characterized by early re-yielding and subsequent rapid change in workhardening rate. The relative kinematic motion of the yield surface with respect to the bounding surface is expressed by the following equation:

* = −α α βα α βα α βα α β . (40)

The evolution of *αααα is given by the following equation:

( )* * ** *

a a Ca aC C

Y Yε ε

α α = − − = −

& &%οοοοα σ α α σ αα σ α α σ αα σ α α σ αα σ α α σ α , (41)

* *( )α φ= αααα ,a B R Y= + − . (42)

An Armstrong–Frederick-type evolution equation is used to express the kinematic hardening of ββββ .

( )o b kb

k kY Y

ε ε = − − =

& &%β σ α β σ − ββ σ α β σ − ββ σ α β σ − ββ σ α β σ − β . (43)

Thus, in Eq. (35),

* *

,a a

A Ca kb C C kα α

= + = −

α − βα − βα − βα − βx , (44)

and in Eq. (9),

* *

:K

a aH H Ca kb C C k

φα α

∂ ′ ′= = + − − ∂ %α − βα − βα − βα − β

σσσσ. (45)

With respect to the expansion of the bounding surface, i.e., the evolution of R, in the first version of the Yoshida–Uemori model (2002b, 2003), the following equation based on the Voce hardening law was proposed:

1 exp( )Voce satR R R kε= = − − , (46)

written as

( )Voce sat VoceR R k R R ε= = −& & & ,. (47)

However, it is not necessary to use the Voce-type formulation. For example, based on the Swift law:

0 0( )n nSwiftR R K ε ε ε= = + − , (48)

the following evolution equation can be obtained

( )( 1)/1/0

n nn nSwift SwiftR R nK R Kε ε

−= = +& & & . (49)

Furthermore, a combination of the above two hardening laws, Eqs (48) and (49), is also possible and can be expressed as follows:

( )1 , 0 1Swift VoceR R Rω ω ω= + − ≤ ≤& & & , (50)

where ω is a weighting coefficient. This model has high flexibility in describing various levels of workhardening at large strain levels.

One of the features of the Yoshida–Uemori model is that it is able to describe the workhardening stagnation that appears in a reverse stress-strain curve for a certain range of reverse deformation (see Hasegawa and Yakou, 1975; Christodoulou et al., 1986). This phenomenon is closely related to the strain-range and mean-strain dependency of cyclic hardening. Specifically, the larger the cyclic strain range is, the larger the saturated stress amplitudes are. This dependency is expressed by the stagnation of the expansion of the bounding surface for a certain range of reverse

deformation. The states of hardening ( 0R>& ) and non-hardening ( 0R=& ) of the bounding surface are determined

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for a so-called non-IH (isotropic hardening) surface,gσ , defined in the stress space as follows and schematically

illustrated in Figs 15(a) and (b):

( ), 0g r rσ φ= − =σ −σ −σ −σ − q , (51)

where q and r denote the center and size of the non-IH surface, respectively. It is assumed that the center of the

bounding surface q exists either on or inside of the surfacegσ . The expansion of the bounding surface takes place

only when the center point of the bounding surface, q, lies on the surface gσ (see Fig. 15(b)), i.e.,

0R>& when

( ) ( ), , 0g r r rσ φ= − =β − β −β − β −β − β −β − β −q q and ( ),: 0

og rσΓ∂

= >∂β −β −β −β −

ββββββββ

q (52a)

0R=& otherwise. (52b)

In an analysis of some experimental data, the plastic strain region of workhardening stagnation was found to

increase with the accumulated plastic strain. To describe this phenomenon, it was assumed that the surface gσ

moves kinematically as it expands. The governing equations of the kinematic motion and expansion of the surface are given by Eqs (53) and (54), respectively.

( )(1 )o h

r

Γ−= −q qββββ , (53)

r hΓ=& . (54)

Here, h is a parameter that controls the strength of the workhardening stagnation characteristic. A larger value of h corresponds to a larger strain region within which workhardening stagnation occurs, and as a result, a larger value of

h leads to weaker cyclic hardening of a material. We may assume that the shape of the surface gσ , is fixed 0φ φ= ,

or even φ = von Mises type, throughout the deformation, because the shape of gσ has not been measured

experimentally yet, and its effect on the stress-strain calculation would be rather minor.

In the proposed model, the size of the yield surface is held constant. However, if we carefully observe the stress-strain response during unloading after plastic deformation, we find that the stress–strain curve is no longer linear but rather is slightly curved due to very early re-yielding and the Bauschinger effect. To describe this phenomenon, in the model, the following equation for plastic-strain-dependent Young’s modulus is introduced:

( ) 1 exp( )o o aE E E E pξ= − − − − , (55)

where oE and

aE are Young’s modulus for virgin and infinitely large pre-strained materials, respectively, and ξ is a

material constant.

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Fig. 15 Schematic illustration of the non-IH surface gσ defined in the stress space, when expansion of the bounding

surface (a) stops, and (b) takes place.

3.3. Validation of the model

To validate the model, cyclic stress-strain responses for a 780R AHSS sheet were examined. Cyclic strain tests and uniaxial tension tests were conducted using specimens cut from the sheet in three directions: 0o (rolling direction), 45o and 90o. The flow stresses and r-values of the three specimens were significantly different, depending on the sheet direction. The uniaxial tension test results showed that the stress directionality, measured

with respect to normalized stresses, 45 0/σ σ and 90 0/σ σ , varies with increasing plastic strain, whereas r-values

do not change much throughout the deformation range ( 0 0.46 0.04r = ± , 45 1.12 0.10r = ± , 90 0.62 0.06r = ± ).

A sixth-order polynomial yield function (Yoshida et al., 2013, see Eq. (20)) was employed for the calculations. For the evolution law for material parameters, , 1,2,...,16kC k = , Eq. (23), with the interpolation equation Eq. (15), was

employed, with 50.0λ = . The material parameters of the yield function, (0)kC and ( )kC ∞ , are listed in Table 5.

The Yoshida–Uemori model parameters are listed in Table 6. It should be emphasized that, in the framework of this anisotropic-kinematic hardening model, a set of kinematic hardening parameters are determined from uniaxial experiments (uniaxial tension and cyclic plasticity) only in the rolling (0o) direction and remain fixed throughout the plastic deformation. The calculated stress-strain curves in the three sheet directions in uniaxial tension were compared to the corresponding experimental results, as shown in Figs 16(a) and (b). The calculated results agree well with the experimental results for plastic-strain-dependent stress directionality. The experimental and calculated results for stress-strain responses under cyclic straining are illustrated in Figs 17(a) and (b), respectively. The calculated results reflect the sheet-direction-dependent cyclic hardening characteristic well.

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Table 5 Material parameters in the sixth-order polynomial-type yield function, (0)kC , k = 1~16, for 780R AHSS

sheet, determined at initial yielding ( 0ε = ) and ( )kC ∞ at infinitely large strain (ε = ∞ ).

(0)

at 0kC

ε =

C1 C2 C3 C4 C5 C6 C7 C8 1.0000 0.6315 0.3813 0.2635 0.3632 0.4843 0.6357 1.0678

C9 C10 C11 C12 C13 C14 C15 C16 0.6219 0.3354 0.4604 0.7029 1.2282 0.7522 0.8913 1.0707

( )

atkC

ε∞

= ∞

C1 C2 C3 C4 C5 C6 C7 C8 1.0000 0.6306 0.4069 0.2608 0.3798 0.6406 .8381 1.1349

C9 C10 C11 C12 C13 C14 C15 C16 0.6651 0.3976 0.5905 0.8057 1.2639 0.4497 0.9255 1.2436

Table 6 Yoshida–Uemori parameters

(a) plasticity parameters

(MPa)Y (MPa)B (MPa)b (MPa)satR 1C 2C k h

550 750 100 140 800 250 12.0 0.2

* 1C C= for 0 0.005ε≤ ≤ ; 2C C= for 0.005ε > .

(b) parameters for plastic-strain-dependent Young’s modulus

0 (GPa)E (GPa)aE ξ

200.0 160.0 120.0

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Fig. 16 Stress-strain curves for 780R AHSS under uniaxial tension in three sheet directions.

0

200

400

600

800

1000

0 0.02 0.04 0.06 0.08 0.1

0o experiment calculation

45o experiment calculation

90o experiment calculation

Tru

e st

ress

(M

Pa)

True strain

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(a) Experiment

(b) Simulation

Fig. 17 Stress-strain curves for 780R AHSS under cyclic loading in three sheet directions.

-1200

-800

-400

0

400

800

1200

-0.1 -0.05 0 0.05 0.1

0o

45o

90o

Tru

e st

ress

(M

Pa)

True strain

-1200

-800

-400

0

400

800

1200

-0.1 -0.05 0 0.05 0.1

0o

45o

90o

Tru

e s

tres

s (M

Pa)

True strain

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4. Concluding remarks

The present paper proposes a framework for constitutive modeling of plasticity to describe the evolution of anisotropy and the Bauschinger effect in sheet metals. The highlights are summarized as follows. - An anisotropic yield function, which varies continuously with plastic strain, is defined by a nonlinear

interpolation function of the effective plastic strain using a limited number of yield functions determined at a few discrete points of plastic strain (see Eqs (11) and (14)). In this modeling framework, it is possible to use any type of yield function; the convexity of the yield surface is always guaranteed.

- When using a polynomial type of yield function, this model reduces to the interpolation of its material parameters (see Eqs (22) -(24)).

- This approach, which requires only one interpolation equation, offers a great advantage over other approaches in that it involves fewer material parameters; some other approaches use many approximation equations for each material parameter.

- Several types of nonlinear equations (see Eqs (15), (16) and (18)) that involve very few material parameters can be used as interpolation functions.

- The description of the Bauschinger effect and the evolution of anisotropy is made possible by incorporating the

proposed anisotropic hardening model in a kinematic hardening model. A set of kinematic hardening parameters can be identified from experiments independent of anisotropic hardening parameters, and their values remain fixed throughout the plastic deformation (see Appendix: scheme of anisotropic-kinematic hardening model).

- The proposed anisotropic hardening model was validated by comparing calculated results of stress–strain

responses, r-value variations, and changes in the shape the yield surface, with experimental data for several types of aluminum and stainless steel sheets. For cyclic plasticity, details of the modeling approach, which is based on the Yoshida–Uemori model, were described. The model was validated by comparison with the results of uniaxial tension and cyclic tension–compression tests for several sheet directions for an AHSS sheet.

Acknowledgement The authors would like to thank Professor T. Kuwabara of Tokyo University of Agriculture and Technology for allowing us to use his published data on the evolutionary change of the yield surfaces on 6016-T4 sheet. We are grateful to Mr. H. Hasegawa of Hiroshima University for his help with experiment on an AHSS sheet. Appendix. Scheme of material parameter identification in the anisotropic-kinematic hardening model

In this framework of anisotropic-kinematic hardening modeling based on the Yoshida-Uemori model, parameters in an anisotropic yield function and in the kinematic hardening law are determined independently, as follows: (a) Anisotropic parameters

The stress-strain responses under monotonic proportional loadings predicted by the Yoshida-Uemori model, using an anisotropic yield function, are almost the same as those calculated by the isotropic hardening model (refer to Yoshida and Uemori, 2003). Thus the anisotropic parameters in a yield function can be identified in just the same way as that in the isotropic hardening modeling, where we use experimental data of flow stresses (e.g.,

0 45,σ σ ,90σ ) and r-values (e.g.,

0r , 45r ,

90r ) in several sheet directions and some biaxial stresses (e.g.,

equi-balanced stress bσ ). Note that, to describe the anisotropy evolution, only a few sets of these experimental

data at discrete levels of plastic strains, , 1, 2, ..,i i Mε = are needed, e.g., for 3003-O aluminum in Section 2.3

(case of M = 3), 45 90, , bσ σ σ and

0r , 45r ,

90r data at ε = 0 (initial yielding), 0.05 and 0.30 were used together with the

continuous stress-strain data of 0 0

pσ ε− .

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(b) The Yoshida-Uemori model parameters

This model contains 7-8 plasticity (kinematic hardening) parameters (e.g., see Table 6(a)) and 3 elasticity parameters (e.g., see Table 6(b)). The plasticity parameters are identified from uniaxial tension and cyclic straining experiments,

0 0σ ε− , in rolling direction (refer to Yoshida and Uemori, 2002b). The elasticity

parameters are determined from the sequential tension-unloading experiment (refer to Yoshida et al., 2002a).

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