A General Yield Function within the Framework of Linear Transformations of Stress Tensors for the Description of

Plastic-strain-induced Anisotropy

Shun-lai Zanga ,Myoung-gyu Leeb

aSchool of Mechanical Engineering, Xi’an Jiaotong University, No.28, Xianning Road, Xi’an, Shaanxi, P.R. China

bGraduate Institute of Ferrous Technology, Pohang University of Science and Technology, San 31, Hyoja-dong, Nam-gu, Pohang, Geongbuk 790-784, Republic of Korea

Abstract. A general yield function based on linear transformations of stress tensors for orthotropic sheet metals is presented to describe plastic strain-induced anisotropy. The subsequent anisotropy is considered by introducing a set of tensorial state variables in the linear transformations of stress tensors, and defining the tensorial state variables evolved with plastic deformation. The evolution of the tensorial state variables is based on eigen decomposition. The eigenvectors are normalized and used to represent the rotation of orthotropy axes of polycrystalline materials which relates to the texture rotation. The eigenvalues are used to capture the local reorientation of the grain preferred directions. Taking the anisotropic yield function Yld2000-2d as an example, the evolution of eigenvalues and eigenvectors of linear transformation tensors is investigated for AA3003-O aluminum alloy sheet. Based on the observations, a modified Yld2000-2d yield function incorporating with isotropic hardening model was developed to predict the anisotropy and subsequent yield surfaces in uni-axial tensions to confirm the validity of current yield function.

Keywords: Anisotropic material; Yield function; Linear transformations; Plastic strain-induced anisotropy PACS: 83.60.La; 81.40.Jj

INTRODUCTION

In the last two decades, the so-called isotropic plasticity equivalent (IPE) theory generalized by Karafillis[1] has gained some popularity in developing new anisotropic yield functions. In the IPE theory, plastic anisotropy is represented by one or more linear transformation tensors on the stress tensor, further the principal values of these linear transformation tensors are substituted into an isotropic yield function to get a new anisotropic yield function.

Within the IPE framework, many anisotropic yield functions have been proposed to describe initial anisotropy of metallic sheets with phenomenological constitutive models[2-6], such as Yld2000-2d and Yld2004-13p/18p yield functions by Barlat[3,7] and Bron's anisotropic yield function[4]. The material coefficients of these anisotropic yield functions are usually constant and identified from the experimental initial tensile, shear or bi-axial yield stresses and/or r-values. However, Hu pointed out that it is difficult to get exact initial yield stress since the anisotropy feature at initial yield state is too sensitive to the definition of yield point[6]. Therefore, the plastic strain-induced anisotropy seems to be more important in relation to the constitutive models. Wagoner and Suh assumed that the coefficients characterizing anisotropy do not change with plastic deformation, while the stress exponent M associated with the shape of yield surface in most anisotropic yield functions, is a function of equivalent plastic strain[8-9]. But, since changing exponent M is not a general concept, it is difficult to apply this method to other anisotropic yield functions[6]. To obtain a general model which can cover plastic strain-induced anisotropy, a dynamic yield function was recently proposed, which assumes that the material coefficients of anisotropic yield function are not constant and identified from the current flow stresses and r-values[6]. However, the dynamic yield function is not easy to be implemented for non-quadratic yield functions because iterative solution procedures of non-linear is inevitably involved at each increment of plastic deformation. In addition, only successive strain-hardening process can be considered, which means that the plastic deformation is continuous without abrupt

The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming ProcessesAIP Conf. Proc. 1383, 63-70 (2011); doi: 10.1063/1.3623593

© 2011 American Institute of Physics 978-0-7354-0949-1/$30.00

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loading-path change and the stress path of loading-unloading-reloading is not included as well. Recently, Wang also presented a yield function to consider the plastic strain-induced anisotropy by assuming the material parameters are polynomial functions of equivalent plastic strain[10].

In this paper, a general yield function theory based on linear transformations of stress tensors is firstly presented to consider plastic strain-induced anisotropy. In order to understand the evolution of subsequent anisotropy, the eigen decompositions for the linear transformation tensors of anisotropic yield function Yld2000-2d of the AA3003-O sheet at different equivalent plastic strains are carried out. Thereafter, the evolution of eigenvalues and eigenvectors is investigated and discussed. Based on the observations, a simple evolution law was proposed and applied to predict the flow stresses and r-values in uni-axial tensions to validate the accuracy of current general yield function.

THEORY

According to the classical plasticity theory, an isotropic stress yield function can be expressed with three invariants of the stress tensor [11]. Although many sets of three invariants have been introduced, the principal values are usually adopted to represent the isotropic yield function.

� �1 2 3( ) , ,ijf F� � � � � (1)

where f is an isotropic stress yield function with respect to the Cauchy stress tensor ij� , i� are principal values of stress tensor ij� , and F is an isotropic function of its arguments, i.e.,

� � � �I II III 1 2 3, , , ,F F� � � � � � � (2) where (I,II,III) are permutations of (1,2,3). When pressure-independent plastic deformation is considered, the general formulation of the yield function can be expressed as a function of the three invariants of the deviatoric stress tensor,

� �1 2 3( ) , ,ijg s G S S S� (3)

where g is an isotropic stress yield function in the deviatoric stress space, G is an isotropic function of its arguments as well, iS represents the principal values of the deviatoric stress tensor ijs .

To formulate an anisotropic yield function, a reference frame associated with the material symmetry axes must be defined. In the isotropic plasticity equivalent approach for pressure-independent polycrystalline materials, a stress tensor ijs� is introduced as a linear transformation of deviatoric stress tensor ijs , i.e.,

s Cs CT L� = = =�� � (4) where C and L are 4th-order tensors, which contain the anisotropy coefficients expressed in the material frame, T is a 4th-order tensor as well, which transforms the Cauchy stress tensor to its deviator.

Substituting the principal values iS� of the linearly transformed stress tensor s� into Eq. (3), an anisotropic yield function � can be obtained,

� � � � � �1 2 3, ,ij ijs g s G S S S� � � � � �� (5)

Note that � is an anisotropic yield function of s , while g is still an isotropic yield function with respect to stress tensor s� .

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Considering the symmetry of the stress tensor s and the assumed symmetry of s� , C can be represented as a 6 6� matrix. The symmetry of the material and the zero trace of s further reduce the number of independent coefficients in matrix C ; for instance, 9 for an orthotropic material. Therefore, C can be written as,

12 13

21 23

31 32

44

55

66

0 0 0 00 0 0 0

0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

C CC CC C

CC

C

� �� �� �� �� �� �� �� � �

C = (6)

From a mathematical point of view, the linear transformation tensor C makes an initial isotropic yield surface

homogeneously deform to an anisotropic yield surface. In most of phenomenological constitutive models, the matrix C is assumed to be constant. Recently, the importance of the description of plastic strain-induced anisotropy has been recognized and emphasized[6,10,12]. Within the isotropy plasticity equivalent framework, the matrix C should be evolved with plastic deformation in order to consider plastic strain-induced anisotropy. Similarly as deformation kinematics, the eigen decomposition can also be applied to the matrix C ,

1�C = VAV (7) where V includes the eigenvectors representing the rotation of orthotropy axes of polycrystalline materials, and A contains the eigenvalues which can be used to capture the local reorientation of the grain preferred directions.

If we divide the plastic deformation into N steps, then the matrix C at k-step can be written as,

1k k k k

�C = V A V (8)

In Eq. (8), kA is

1

2

3

4

5

6

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

k

k

k

k k

k

k

��

��

��

� �� �� �

� � �� �� �� �� � �

A (9)

The matrix C at (k+1)-step can be written as,

11 1 1 1k k k k

�� � � �C = V A V (10)

In Eq.(9), considering the evolution of kA , its current eigenvalues can be defined as

1k k

i i i� � �� � �� (11)

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Since the local reorientation of the grain preferred directions usually influences the intensity of texture, the eigenvalues k

i� could be used to capture texture reorientation by defining proper evolution laws of i�� , i.e.,

� �, ,p pi i� � �� � � � � �� (12)

In terms of the texture rotation, if the incremental rotation between k-step and (k+1)-step is �Q , then the 1k�V

could be expressed as

1T

k k� � � �V Q V Q (13)

To consider the influence of plastic deformation on texture rotation, �Q should be related to the plastic deformation as well;

� �, ,p p�� � � � �Q Q �� (14)

Considering the physical aspects of texture evolution, and then defining proper evolution laws of Eq.(12) and (14),

the plastic strain-induced anisotropy is expected to be accurately described.

EVOLUTION OF LINEAR TRANSFORMATION TENSOR

In this section, taking the anisotropic yield function Yld2000-2d as an example, the evolution of eigenvectors and eigenvalues of linear transformation tensors is investigated. For this yield function, two linear transformation tensors are used to represent the anisotropy in plane stress condition, and total eight anisotropy coefficients, 1 8� �� , are usually adopted to transform the Cauchy stress to stress tensor ijs� , as shown in Eq. (15)[11].

1

2

7

0 00 00 0

��

�

� �� � �� � �

C = and 5 3 6 4

3 5 4 6

8

4 2 2 01 2 2 4 03

0 0 3

� � � �� � � �

�

� � � ��� � �� �� � �

C = (15)

where these eight anisotropy coefficients are usually calculated from the measured experimental data, 0� , 45� , 90� ,

b� , 0r , 45r , 90r and br . Here �� and r� are yield stresses and r-values along different angles to rolling direction, b� and br are bi-axial yield stress and r-value, respectively. To describe plastic strain-induced anisotropy, the

subsequent yield surfaces should evolve with plastic deformation. It implies that these experimental data at the same subsequent hardening state should be determined to re-calculate the anisotropy coefficients. Regarding the experimental data, there are two methods to determine the experimental data in relation to a subsequent yield state. One method assumes that the anisotropy of yield function is a function of plastic work, which indicates that the experimental data at the same plastic work should be used to determine the subsequent yield surface. The other is based on the so-called principle of equivalent strain-hardening work, which can well explain the hardening behavior of perfect plastic body in which no further hardening occurs we may still have the response of the increment of plastic strain[6]. Here, the first order plastic work principle is used that can be defined as

p pd d dij ijw � �� � ��� � (16)

In this work, the aluminum alloy 3003-O in literature [6] was used as experimental data, and the Eqs. (17) - (19) were adopted to fit all experimental stress-strain curves and r-values as suggested by Hu [6].

� � � �,n ne p ek k� �

� � � � � � �� � � � �� � � (17)

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� � � �,b bn ne p eb b b b b b bk k� � � � �� � � (18)

11 1 11 1 1w w

w w w

n n nn n n n n n

w wr n k n k n k� � �

� � � � � �� � � � � � � � � �� � �

�� � �� �� �� �� �� �

(19)

where k� , n� , wk � , wn � are the constants of material parameters of tensile tests at angle � to rolling direction, bk ,

bn material parameters of equibiaxial tension, and e�� , e

b� are the limit elastic strains of corresponding experiments. All the material parameters are listed in Table 1. Base on the above plastic work principle, the material properties of aluminum 3003-O at different hardening states are calculated as listed in Table 2.

TABLE 1. Material parameters of aluminum 3003-O[6]. Directions: Rolling Transverse Diagonal Equibiaxial

Regarding the strain: 0p� 0

pw� 90

p� 90p

w� 45p� 45

pw� p

b� k values 195.0 240.0 183.0 240.0 187.0 232.0 171.3 n values 0.214 0.217 0.215 0.227 0.222 0.229 0.17

TABLE 2. Material properties of aluminum 3003-O at different hardening states.

Directions: Plastic strain:

Rolling (MPa)

Transverse (MPa)

Diagonal (MPa)

Equibiaxial (MPa) 0r 90r 45r

Yield stress (0.2%) 51.58 48.1 47.1 49.98 0.704 0.662 0.842 0.0077 72.32 68.26 67.71 75.56 0.677 0.576 0.769 0.0120 78.23 73.91 73.49 81.04 0.672 0.559 0.754 0.0159 82.42 77.91 77.58 84.81 0.668 0.548 0.745 0.0195 85.77 81.10 80.85 87.76 0.665 0.54 0.737 0.0402 99.03 93.73 93.80 99.09 0.655 0.511 0.712 0.0740 112.33 106.38 106.83 110.07 0.646 0.488 0.690 0.1087 121.75 115.34 116.09 117.67 0.641 0.473 0.677 0.1446 129.3 122.53 123.52 123.68 0.637 0.463 0.667 0.1818 135.71 128.63 129.84 128.73 0.633 0.455 0.660 0.2203 141.34 133.99 135.39 133.13 0.631 0.449 0.654

As shown in Table 2, four stresses and three r-values are available for AA3003-O aluminum alloy sheet. With

these data, only seven anisotropy coefficients can be identified. Therefore, 3 6� �� is assumed for the anisotropic yield function Yld2000-2d.

Fig. 1 shows the evolution of eigenvalues of linear transformation tensors with plastic deformation. It can be seen from Fig. 1(a), the eigenvalues of linear transformation tensor �C gradually saturate to the constants after large plastic deformation. However, for tensor ��C , the obvious abrupt changes exist for all eigenvalues when plastic deformation is small. It might result from that the deformation at early stage is quite unstable due to the fact that the early plastic range may not be completely plastic, or may be elastic-plastic transient regions.

Fig. 2 shows the evolution of eigenvectors of linear transformation tensor ��C with plastic deformation. Here, the evolution of eigenvectors is respectively represented by the angles between current eigenvector and the initial one as shown in Fig. 2(a), and the angles between current eigenvector and previous one as shown in Fig. 2(b). Supposed

1kn , 2

kn , and 3kn are the current eigenvectors of linear transformation tensor ��C , 1

1k�n , 21k�n , and 3

1k�n are the previous eigenvectors, 1

0n , 20n , and 3

0n are initial one, the angles used here are respectively defined as

0

0

arcosi ik

i i ik

�� ��� ��� ��� �

n nn n

and 1

1

d arcosi ik k

i i ik k

� �

�

� ��� ��� ��� �

n nn n

(20)

Fig. 2(a) indicates that the eigenvector evolved with plastic deformation. When the traditional initial yield stresses

and r-values, i.e., assuming the equivalent plastic strain is zero, are selected to determine the initial eigenvectors, the

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subsequent eigenvectors have abrupt changes. However, when the eigenvectors which correspond to equivalent plastic strain of 0.0077 and 0.0120 are used to represent the initial state, the evolution of subsequent eigenvectors is continuous and almost linear. Fig. 2(b) shows that the eigenvectors have a rapid change rate at small plastic deformation, and after that the rates of i� to equivalent plastic strain are almost constant. It implies that the evolution of directions of eigenvectors has a linear relation to equivalent plastic strain.

(a) Linear transformation tensor �C (b) Linear transformation tensor ��C

FIGURE 1. Evolution of eigenvalues of linear transformation tensors with plastic deformation.

(a) Angles between current eigenvectors and initial one (b) Angles between current eigenvectors and previous one

FIGURE 2. Evolution of eigenvectors of linear transformation tensor ��C with plastic deformation.

PREDICTION OF THE ANISOTROPY AND SUBSEQUENT YIELD SURFACES

As shown in Fig. 1, the eigenvalues of linear transformation tensors tend to saturate towards constants after large deformation. Two possible reasons result in such behavior. One is that only monotonic loading paths are considered in the present work. In such cases, the evolution of anisotropy and hardening behavior is continuous, and thus no abrupt changes exist. It implies that the evolution of the eigenvalues should be continuous too. The other reason is that, for most polycrystalline metallic materials, the hardening rate decreases with plastic deformation. Therefore, it can be inferred that the change rate of anisotropy and hardening behavior should decrease with plastic deformation as well. In this work, for simplicity the eigenvectors of linear transformation tensor ��C are assumed to be fixed during plastic deformation as a first attempt and Eq. (21) is used to describe the evolution of eigenvalues.

� �0 1p

ibi i iQ e �� � �� � � (21)

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where iQ and ib are material parameters, p� equivalent plastic strain, the plus-minus denotes that the eigenvalues increase or decrease. This sign might be related to the plastic strain rate when strain path changes exist. In our case, for simplicity, the signs are directly defined based on the evolution of eigenvalues as shown in Fig. 1.

The important part of the current general yield function is to define the initial eigenvalues and eigenvectors of linear transformation tensors. As mentioned above, the definitions of initial yield stresses and r-values might be very difficult and inaccurate with traditional methods. To overcome it, the early stable anisotropy at small plastic deformation should be chosen to represent the initial anisotropy. In this work, the experimental data from equivalent plastic strain of 0.0077 was used to fit the initial eigenvalues and eigenvectors.

For the other material parameters of anisotropy-evolved yield function Yld2000-2d, inverse identification is carried out by optimization using uni-axial tensile tests with a developed Matlab toolbox SMAT. A cost function is defined in the least square sense and minimized, starting from an initial guess of parameters. Finally, the identified material parameters are summarized in Table 3. The initial eigenvectors are shown in Eq. (22).

TABLE 3. Material parameters of anisotropy-evolved yield function Yld2000-2d for aluminum 3003-O. Transformation tensors: 0

1� 1Q 1b 02� 2Q 2b 0

3� 3Q 3b �C 0.9033 0.0507 17.435 1.0351 0.1005 17.006 1.0204 0.0268 16.796 ��C 0.9524 0.0 0.0 1.1922 0.0 0.0 1.2522 0.0 0.0

0

0.8402 0.6546 00.5423 0.7560 0

0 0 1.0

� � �� �� �� � �

V = (22)

Fig. 3 is the simulated flow stresses and r-values compared with experimental data. It is found that the stress-

strain and r-value curves predicted by the current yield function are in good agreement with the experimental curves in all three directions, especially, for the evolution of r-values. For the case of constant material coefficients of yield function, r-values are usually kept when isotropic hardening model is adopted. However, the current method gives a good correlations with simplified evolution equation, Eq. (21).

(a) Cauchy stress (b) R-value

FIGURE 3. Predicted Cauchy stresses and r-values in uni-axial tensions.

When a hardening state is selected, its corresponding yield surface could be predicted by the current yield function. Choosing the hardening states presented in Table 2, the initial and subsequent yield surfaces are plotted as shown in Fig. 4 when the shear stress is zero. For comparisons, the yield surfaces calculated by the constant and variable material coefficients of yield function are expressed in the same graph, marked with ‘anisotropy-unevolved’ and ‘anisotropy-evolved’, respectively. The results show that the subsequent yield surfaces predicted by the two methods are rather different, especially, for the bi-axial tensile stress states.

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FIGURE 4. Initial and subsequent yield surfaces with anisotropy-unevolved and anisotropy-evolved yield functions.

CONCLUSIONS

A general yield function based on linear transformations of stress tensors for orthotropic sheet metals is presented to model plastic strain-induced anisotropy. The subsequent anisotropy is considered by defining proper evolution laws for the eigenvalues and eigenvectors of linear transformation tensors. Taking the anisotropic yield function Yld2000-2d and AA3003-O aluminum alloy sheet, the evolution of eigenvalues and eigenvectors of linear transformation tensors is investigated. To confirm the validity of the present method, the flow stresses and r-values of uni-axial tensions are predicted by a modified Yld2000-2d yield function which can cover plastic strain-induced anisotropy under successively strain-hardening process. The results show that the present model can accurately describe the anisotropy and hardening behavior, especially, the evolution of r-values. As a future work, the model will be applied to a real sheet metal forming process to verify the importance of the plastic induced anisotropy.

ACKNOWLEDGMENTS

Shun-lai Zang would like acknowledge financial support by the Specialized Research Fund for the Doctoral Program of Higher Education (No.200806981025), and by the National Natural Science Foundation of China (No.11002105) and by the Opening Project of Key Laboratory of Testing Technology for Manufacturing Process (Southwest University of Science and Technology), Ministry of Education (No.10ZXZK03).

REFERENCES

1. A.P. Karafillis and M.C. Boyce, J. Mech. Phys. Solids 41, 1859-1886 (1993). 2. D. Banabic et al, Int. J. Plast. 21, 493-512 (2005). 3. F. Barlat et al, Int. J. Plast. 19, 1297-1319 (2003). 4. F. Bron and J. Besson, Int. J. Plast. 20, 937-963 (2004). 5. O. Cazacu et al, Int. J. Plast. 22, 1171-1194 (2006). 6. W. Hu et al, Int. J. Plast. 23, 620-639 (2007). 7. F. Barlat et al, Int. J. Plast. 21, 1009-1039 (2005). 8. R.H. Wagoner, Metallurgical and Materials Transactions A 11, 165-175 (1980). 9. Y.S. Suh et al, Int. J. Plast. 12, 417-438 (1996). 10. H.B. Wang et al, Comput. Mater. Sci. 47, 12-22 (2009). 11. F. Barlat et al, Int. J. Plast. 23, 876-896 (2007). 12. J.H. Hahm and K.H. Kim, Int. J. Plast. 24, 1097-1127 (2008).

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