Combination of modified Yld2000-2d and Yld2000-2d in anisotropic pressure dependent sheet metals

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In the current research to model anisotropic asymmetric sheet metals a new non-AFR criterion is presented. In the new model, Modified Yld2000-2d proposed by Lou et al. (2013) is considered as yield function and Yld2000-2d proposed by Barlat et al. (2003) is considered as plastic potential function. To calíbrate the presented criterion, the yield function which is a pressure dependent criterion requiers ten directional yield stresses such as uniaxial tensile stresses in three directions of 0º, 45º, 90º, uniaxial compressive yield stresses in six directions of 0º, 15º, 30º, 45º, 75º, 90º from the rolling direction along with biaxial yield stress. Moreover, the plastic potential function which is a pressure indepedent criterion needs eight experimental data points such as tensile R-values in seven directions of 0º, 15º, 30º, 45º, 60º, 75º, 90º from the rolling direction and also biaxial tensile R-value. Finally with comparing the obtained results with experimental data points, it is shown that the presented non-AFR criterion predicts compressive yield stress, biaxial tensile yield stress and R-values more accurately than Modified Yld2000-2d and it would be considered as a new criterion for anisotropic asymmetric metals.

본문

# Combination of Modified Yld2000-2d and Yld2000-2d in Anisotropic Pressure Dependent Sheet Metals

Abstract
In the current research to model anisotropic asymmetric sheet
metals a new non-AFR criterion is presented. In the new model,
Modified Yld2000-2d proposed by Lou et al. (2013) is considered
as yield function and Yld2000-2d proposed by Barlat et al. (2003)
is considered as plastic potential function. To calíbrate the presented criterion, the yield function which is a pressure dependent
criterion requiers ten directional yield stresses such as uniaxial
tensile stresses in three directions of 0 _°_, 45 _°_, 90 _°_, uniaxial compressive yield stresses in six directions of 0 _°_, 15 _°_, 30 _°_, 45 _°_, 75 _°_,
90 _°_ from the rolling direction along with biaxial yield stress. Moreover, the plastic potential function which is a pressure indepedent criterion needs eight experimental data points such as tensile
_R_ -values in seven directions of 0 _°_, 15 _°_, 30 _°_, 45 _°_, 60 _°_, 75 _°_, 90 _°_ from
the rolling direction and also biaxial tensile _R_ -value. Finally with
comparing the obtained results with experimental data points, it is
shown that the presented non-AFR criterion predicts compressive
yield stress, biaxial tensile yield stress and _R_ -values more accurately than Modified Yld2000-2d and it would be considered as a
new criterion for anisotropic asymmetric metals.

Keywords
Yld2000-2d, Modified Yld2000-2d, Anisotropic asymmetric yield
functions, Non-associated flow rule, Downhill Simplex Method.

1 INTRODUCTION

Farzad Moayyedian a

Mehran Kadkhodayan b

a Ph. D. Student of Solid Mechanical
Engineering, Ferdowsi University of
Mashhad, Iran
farzad.moayyedian@gmail.com
b Professor of Solid Mechanical Engineering, Ferdowsi University of Mashhad, Iran
kadkhoda@um.ac.ir

[http://dx.doi.org/10.1590/1679-78251372](http://dx.doi.org/10.1590/1679-78251372)

Received 23.05.2014
In revised form 20.07.2014
Accepted 08.08.2014
Available online 17.08.2014

To model the behavior of anisotropic metals more precisely, their pressure dependency is considered in the recent new studies. Spitz and Richmond (1984) by the aid of experimental data of ironbased materials and aluminum showed that there is no need that the pressure dependency of yielding to be associated with irreversible plastic dilatancy. Liu et al. (1997) considered tensile and
compressive strengths far apart and presented a new criterion based on the criterion presented by
Hill and Drucker-Prager. Barlat et al. (1997) proposed a generalized yield description to account

for the behavior of the strengthened aluminum alloy sheets. It was subsequently shown that this
yield function was suitable for the description of the plastic behavior of any aluminum alloy
sheet. Barlat et al. (2003) used two linear transformations on the Cauchy stress tensor to propose
a new plane stress yield function for aluminum alloy sheets to consider the anisotropy effects.
Stoughton and Yoon (2004) proposed a non-AFR based on a pressure sensitive yield description
with isotropic hardening to account the strength differential effect (SDE) that was consistent
with the Spitzig and Richmond (1984) results. Lee et al. (2008) developed a continuum plasticity
model to consider the unusual plastic behavior of magnesium alloy in finite element analysis. A
hardening law based on two-surface model was further extended to consider the general stress–
strain response of sheet metals including the Bauschinger effect, transient behavior and unusual
asymmetry. In terms of anisotropy and asymmetry of the initial yield stress, the Drucker–Prager’s
pressure dependent yield surface was modified to include the anisotropy of magnesium alloy. Cvitanić et al. (2008) developed a finite element formulation based on non-associated plasticity. In
the constitutive formulation, isotropic hardening was assumed and an equation for the hardening
parameter consistent with the principle of plastic work equivalence was introduced. The yield
function and plastic potential function were considered as two different functions. Algorithmic
formulations of constitutive models that utilized associated or non-AFR were derived by application of implicit return mapping procedure. Aretz (2009) presented the issue of the yield function’s
convexity in the existence of hydrostatic pressure for sensitive yield conditions. Stoughton and
Yoon (2009) proposed a model based on a non-AFR and explicitly integrated it into the yield
criterion. Their results had no effects on the accuracy of the plastic strain components defined by
the gradient of a separate plastic potential function. Hu and Wang (2009) proposed a new theory
in which a corresponding constitutive model can be constructed and characterized experimentally
via two steps. The constitutive model involved two functions, yield function and plastic potential.
A relationship between two functions was suggested. Drucker–Prager yield function was employed
to express the yielding behavior of material and a differently experimental characterization of the
model was created as the corresponding plastic potential to describe the feature of plastic flow of
material. The constitutive model could also well predict stress–strain relations with different pressures loaded on the material. They showed that the feature of plastic flow was not that sensitive
to the pressure loaded on the material when the yielding stress was. Huh et al. (2010) evaluated
the accuracy of accepted anisotropic yield criteria contain Hill48, Yld89, Yld91, Yld96, Yld20002d, BBC2000 and Yld2000-18p rooted in the root-mean square error (RMSE) of the yield stresses
and the _R_ -values. They concluded that Yld2000-18 yield criterion was the most proper one to
precisely describe the yield tress and _R_ -value directionalities of sheet metals. Taherizadeh et al.
(2011) extended a generalized finite element formulation of stress integration method for nonquadratic yield and potential functions with combined non-linear hardening with non-AFR. Park
and Chung (2012) developed a novel formulation under the structure of mixed isotropic-kinematic
[hardening rule which led to the symmetric stiffness modulus for the non-AFR. Cleja-Ţigoiu and](http://www.sciencedirect.com/science/article/pii/S0749641913000156)
[Lancu (2013) introduced the rate elastic-plastic model with a plastic spin in the case of an in-](http://www.sciencedirect.com/science/article/pii/S0749641913000156)
plane rotation of the orthotropy direction and a plane stress condition, respectively. In the plane
stress case, the equation for the strain rate in the normal direction was first derived and subsequently the modified expression for the plastic multiplier associated with an in-plane rate of the

Latin American Journal of Solids and Structures 12 (2015) 92-114

deformation became available. Numerical simulations for the homogeneous deformation process
on the sheets and comparisons with experimental data made possible a selection among the plastic spins introduced in their research, aiming at obtaining a good agreement with the experiments
performed for an in-plane stress state. Lou et al. (2013) introduced a method to extend symmetric
yield functions to concern the strength differential (SD) effect in sheet metals. Adding a weighted
pressure term for anisotropic materials the SD effect was coupled with symmetric yield functions.
This method was applied to the symmetric Yld2000-2d and this yield function was modified to
describe the anisotropic and a symmetric yielding of two aluminum alloys with small and strong
[SD effects. Safaei et al. (2013) proposed a non-associated plane stress anisotropic constitutive](http://www.sciencedirect.com/science/article/pii/S0020740313001276)
model with combined isotropic-kinematic hardening. The quadratic Hill48 and non-quadratic
Yld2000-2d yield criteria were used with the non-AFR model to consider anisotropic behavior.
[Safaei et al. (2014) introduced anisotropy evolution in terms of both distortional hardening and](http://www.sciencedirect.com/science/article/pii/S0020740313001276)
variation of Lankford coefficients. A non-AFR based on Yld2000-2d anisotropic yield model was
used in which separate yield function and plastic potential were considered which could provide
[excellent accuracy and flexibility for the model. Safaei et al. (2014) showed that by scaling the](http://www.sciencedirect.com/science/article/pii/S0020740313001276)
plastic potential function, the equality of equivalent plastic strain and compliance factor can be
reserved. The effect of scaling of the non-AFR based on Barlat et al.’s (2003) anisotropic model
(called Yld2000-2d) was comprehensively studied with FE simulation of tensile loading under
uniaxial tensions along with different orientations as well as balanced biaxial stress condition. A
fully implicit return-mapping scheme was introduced for stress integration of the constitutive
model in a User-defined MATerial subroutine (UMAT). The results proved that the proposed
simplified technique was a reliable alternative for the full expression. Yoon et al. (2014) proposed
a general asymmetric yield function with dependence on the stress invariants for pressure sensitive metals. The pressure sensitivity of the proposed yield function was consistent with the experimental result of Spitzig and Richmond (1984) for steel and aluminum alloys while the asymmetry of the third invariant was preserved to model strength differential (SD) effect of pressure
insensitive materials. The proposed yield function was transformed in the space of the stress triaxaility, the von Mises stress and the normalized invariant to theoretically investigate the possible reason of the SD effect. The yield function reasonably modeled the evolution of yield surfaces
for a zirconium clock-rolled plate during in-plane and through thickness compression. The yield
function was also applied to describe the orthotropic behavior of a face-centered cubic metal of
AA2008-T4 and two hexagonal close-packed metals of high-purity a-titanium and AZ31 magnesium alloy.

In the current research the modified Yld2000-2d and Yld2000-2d are used as yield and plastic
potential functions in new criterion called Modified Yld2000-2d II. The yield and the plastic potential functions are calibrated with ten and eight experimental results, respectively. Finally it is
shown that the Modified Yld2000-2d II predicts  _C_,  _bT_, _R_  _T_ and _RbT_

more accurately than new

Modified Yld2000-2d presented by Lou et al. (2013) for anisotropic pressure dependent metals
Al2008-T4 (a BCC material) and Al2090-T3 (a FCC material).

Latin American Journal of Solids and Structures 12 (2015) 92-114

2 MODIFIED Yld2000-2d II IN PLANE STRESS PROBLEMS

In the current research a new non-AFR is introduced (Modified Yld2000-2d II) in which the Modified Yld2000-2d proposed by Lou et al. (2013) and Yld2000-2d proposed by Barlat et al. (2003)
are selected as yield and plastic potential functions, respectively. In the proposed model the pressure dependency of yield function and pressure independency of plastic potential function are
considered accurately. Moreover, the number of experimental data points for calibrating the proposed criterion are increased from ten in Modified Yld2000-2d to eighteen. In the following, calibrating of the Modified Yld2000-2d II is explained in detail.

To consider the anisotropic effects in yield stress function, two linear transformation matrices
( _Lij_  and _Lij_  ) in stress tensor are introduced. Then the first modified deviatoric stress tensor _X_ _ij_ 

can be presented by Barlat et al. (2003) as bellow:

 _X_ _xx_    _L_ 11  _L_ 12  0   _xx_ 
 _X_ _yy_    _L_ 21  _L_ 22  0  _yy_ 
 
   
 _X_ _xy_   0 0 _L_ 66    _xy_ 

_L_ 11  _L_
_L_ 21  _L_

11 12

_X_
_X_

_X_

_xy_  66   _xy_

_xx_ 

_yy_

_xx_

_x_

    

_yy_    _L_ 21  _L_ 22  0  _yy_  (1)

 

_yy_

21 22

0 0

_L_

0
0

 _xx_
 _yy_

 _xy_

_x_



in which  _ij_ is the stress tensor and _ij_  1,
_L_  can be expressed with three anisotropic parameters
#####  2 and  7 as following:

 _L_ 11    2 0 0 
   

_L_ 12   1 0 0  1 

   
  1  
 _L_ 21     0  1 0   2 
  3    
 _L_ 22    0 2 0   7 
 _L_ 66    0 0 3 

_L_
_L_
_L_
_L_
_L_

 1
 2
 7

12 1

  

21   2

22     7

 2 0 0

 1 0 0


1

 0  1 0

3 
0 2 0


 0 0 3

In this case the first modified principal deviatoric stresses _X_ 1  and _X_ 2  can be shown as:

(2)

(3)

(4)

### X 1 , X 2   1  X xx   X yy    X xx   X yy   2  4 X xy  2 

2  

###  X xx   X yy  

2
### yy    X xx   X yy    4 X xy  2 



and the second modified deviatoric stress tensor _X_ _ij_  can be introduced as:

_xx_    _L_ 11  _L_ 12 

 _X_ _xx_    _L_ 11  _L_ 12  0   _xx_ 
 _X_ _yy_    _L_ 21  _L_ 22  0  _yy_ 
 
   
 _X_ _xy_    0 0 _L_ 66    _xy_ 

_X_ _xx_    _L_ 11  _L_
_X_ _yy_    _L_ 21  _L_



_yy_

_xx_

11 12

_xx_

0
0

_yy_



  

_xy_    0 0 _L_ 66    _xy_ 



_X_ _xy_    0 0 _L_

_xy_

21 22

0 0

 _xx_
 _yy_

 _xy_

_xy_

##### in which Lij  can be shown with five anisotropic parameters of  3,  4,  5,  6 and  8 as follo
wing:

Latin American Journal of Solids and Structures 12 (2015) 92-114

 _L_ 11    2 2 8  2 0   3 
     

_L_ 12  1  4  4 4 0  4

     
  1  
 _L_ 21     4  4  4 1 0   5 
  9    
 _L_ 22    2 8 2  2 0   6 
 _L_ 66    0 0 0 0 9   8 

_L_
_L_
_L_
_L_
_L_

 2 2 8  2 0


1  4  4 4 0



1

 4  4  4 1 0

9 

 2 8 2  2 0

 0 0 0 0 9

 3
 4
 5
 6
 8

11 3

12 4

   

21   5

(5)

22     6

66     8

In this case the second modified principal deviatoric stresses _X_ 1  and _X_ 2  can be expressed as:

### X 1 , X 2   1  X xx   X yy    X xx   X yy   2  4 X xy  2 

2  

###  X xx   X yy   2

_yy_ 

2
### yy    4 X xy  2  (6)



Using these principal deviatoric stresses, the Modified Yld2000-2d can be obtained by Lou et al.
(2013) equation as:

1

_a_ _a_ _a_ _a_

_F_  _hx_  _xx_  _hy_  _yy_   _X_ 1   _X_ 2   2 _X_ 2   _X_ 1   2 _X_ 1  _X_ 2    ~~~~
 2 
 

_a_ _a_ _a_ _a_

 2 _X_   _X_   2 _X_  _X_  

~~~~ _p_
###  

1   _X_ 2  _a_  2 _X_ 2   _X_ 1  _a_  2 _X_ 1  _X_ 2 
2

_x_  _xx_  _hy_  _yy_   1 2 2 1 1 2   ~~~~ _p_

 

(7)

in which _y_
##### h x and h are the parameters for considering the pressure dependency of the anisotro
pic criterion which is an anisotropic asymmetric (pressure dependent) one. In this criterion ?= 6

 ~~~~ _p_ defines the isotropic har### for BCC and ?= 8 for FCC materials are the best choices and  

dening behavior of materials considered by Lou et al. (2013).

Similarly for plastic potential function it can be shown that _Y_ 2  (the first modified deviatoric

stress tensor) can be expressed with a linear transformation _ij_
_M_  such as:

 _Y_ _xx_    _M_ 11  _M_ 12  0   _xx_ 
 _Y_ _yy_    _M_ 21  _M_ 22  0  _yy_ 
 
   

 _Y_ _xy_    0 0 _M_ 66    _xy_ 

_Y_


 _Y_


_Y_

_M_ 11  _M_

0 0



12 

 _xx_

 _yy_

_xx_ 11 12 _xx_

11 12

  

21  _M_ 22  0  _yy_  (8)



_M_ 21  _M_



_yy_  21 22  _yy_

 

21 22

  

 _Y_ _xy_    0 0 _M_ 66   

_M_

_xy_   66   _xy_





##### where M  ij can be presented with three anisotropic parameters of  1,  2 and  7 as following:

 _M_ 11    2 0 0 
   

_M_ 12   1 0 0  1 

   
  1  
 _M_ 21     0  1 0   2 
  3    
 _M_ 22    0 2 0   7 
 _M_ 66    0 0 3 

_M_


 1 0 0


1

 0  1 0

3 
0 2 0



2 0 0

 1

 2

 7

12 1

  

21   2

(9)

_M_

22     7

0 0 3

Latin American Journal of Solids and Structures 12 (2015) 92-114

where the first modified principal deviatoric stresses _Y_ 1  and _Y_ 2  are:

### Y 1 , Y 2   1  Y xx   Y yy    Y xx   Y yy   2  4 Y xy  2 

2  

###  Y xx   Y yy  

2
### yy    Y xx   Y yy    4 Y xy  2 



(10)

and _Y_ _ij_  (the second modified deviatoric stress tensor) can be presented with another linear trans
formation _M_  _ij_ as:

 _Y_ _xx_    _M_ 11  _M_ 12  0   _xx_ 
 _Y_ _yy_    _M_ 21  _M_ 22  0  _yy_ 
 
   
 _Y_ _xy_    0 0 _M_ 66    _xy_ 

_Y_ _xx_    _M_ 11  _M_

_Y_ _yy_    _M_ 21  _M_



 _xx_

 _yy_

_xx_ 11 12 _xx_

11 12

_yy_  21 22  _yy_

 

21 22



_Y_ _xy_    0 0 _M_

0 0

_xy_   66   _xy_



##### in which M  ij are presented with five anisotropic parameters of  3,  4,  5,  6 and  8 as:

(11)

(12)

 2 2 8  2 0


1  4  4 4 0



1

 4  4  4 1 0

9 

 2 8 2  2 0

 0 0 0 0 9

 _M_ 11    2 2 8  2 0   3 
     

_M_ 12  1  4  4 4 0  4

     
  1  
 _M_ 21     4  4  4 1 0   5 
  9    
 _M_ 22    2 8 2  2 0   6 
 _M_ 66   0 0 0 0 9   8 

_M_
_M_
_M_
_M_
_M_

11 

12 

21 

22 

66   0 0 0 0 9   8 

 3
 4
 5
 6
 8

11 3

##### where the modified principal deviatoric stresses Y 1  and Y 2  are:

### Y 1 , Y 2  1  Y xx   Y yy    Y xx   Y yy   2  4 Y xy  2 

2  

###  Y xx   Y yy  

2
###    4 Y xy  2  (13)





2
### 1 , Y 2   Y xx   Y yy    Y xx   Y yy    4 Y xy  2

Finally the Yld2000-2d for anisotropic pressure independent materials can be considered as:

1

_a_ _a_ _a_ _a_
_G_  _Y_ 1  _Y_ 2   2 _Y_ 2  2 _Y_ 1   2 _Y_ 1  _Y_ 2  



 _a_  2 _Y_ 1  _Y_ 2 

1  _Y_ 2  _a_  2 _Y_ 2  _Y_ 1  _a_  2 _Y_ 1  _Y_ 2
2

2 

 

(14)

In this case ?= 6 for BCC and ?= 8 for FCC materials are the best choices as Modified
Yld2000-2d mentioned by Barlat et al. (2003).

It is noted that Eqs. (8) - (13) for plastic potential function are derived similary to analogy
in Eqs. (1) - (6) for yield function. The yield function in Eq. (7) is pressure dependent but the
plastic potential function in Eq. (14) is pressure independent. To calibrate the Yld2000-2d as
plastic function the first differentiation of this criterion with respect to  _ij_ is derived as following:

Latin American Journal of Solids and Structures 12 (2015) 92-114

#####   G  G 1  a  [  2  1   2   2  1   2  xx   1  2  2  yy  2 K 2   a  1 

 _xx_ 2 18 _K_ 2 

#####  32  M 11   M 21   M 11   M 21    M 11   M 21   xx 4 K   2  M 12   M 22   yy  3 K 1  K 2  a


#####  3  M 11   M 21   M 11   M 21    M 11   M 21   xx   ˝ M 12   M 2  2  yy  3 K 1  K 2  a
 2 4 _K_ 2 
 1  _a_
#####  G  G  [  1  2  2   2  1   2  xx   1  2  2  yy  2 K 2   a  1 
 _yy_ 2 18 _K_ 2 
#####  32  M 12   M 22   M 12   M 22    M 11   M 21   xx 4 K   2  M 12   M 22   yy  3 K 1˝  K 2˝  3 M 11   M 21   xx   M 12   M 22   yy    M 12   M 22   M 12   M 22     3 K 1  K 2 

 2 4 _K_ 2  

#####  Gxy  G 21  a   2  K 722   2 K 2   a  1   MK 66  2  2 3 K 1  K 2  a  1   MK 6  2 xy״ 6  2 3 K 1  K 2  a  1 

#####  Gxx  G 21  a  [  2  1   2   2  1   2  xx 18  K  2  1  2  2  yy  2 K 2   a  1 

#####  Gxx  G 21  a  [  2  1   2   2  1   2  xx 18  K  2  1  2  2  yy

1 

_a_

#####  2  1   2   2  1   2  xx   1  2  2  yy  2 K 2   a

_xx_ 1 2 2 _yy_  _a_

 1

#####  Gxx  G 21  a  [  2  1   2   2  1   2  xx 18  K  2  1  2  2  yy  2 K

_xx_

##### 21  a  [  2  1   2   2  1   2  xx 18  K  2  1  2  2  yy  2

##### 21  a  [  2  1   2   2  1   2  xx 18  K  2  1  2  2  yy  2 K 2   a  1

#####  M 11   M 21   M 11   M 21    M 11   M 21   xx   M 12   M 22  

##### 11   M 21   M 11   M 21    M 11   M 21   xx   M 12   M 22   yy  3 K 1  K 2  a  1

#####  32  M 11   M 21   M 11   M 21    M 11   M 21   xx 4 K   M 12   M 22   yy  3 K 1  K 2  a  1 

##### M 11   M 21   M 11   M 21    M 11   M 21   xx   M 12   M

##### 32  M 11   M 21   M 11   M 21    M 11   M 21   xx 4 K   2  M 12   M 22   yy  3

##### 11   M 21   M 11   M 21    11 21 xx  12 22

_M_ 11   _M_

#####  xx   M 12   M 22   yy

_xx_ _M_ 12 _M_ 22 _yy_   _a_

_K_ 2  12 22 _yy_  3 _K_ 1  _K_

2  _M_ 12 _M_ 22 _yy_  3 _K_ 1  _K_ 2  _a_  1

 2 11 21 11 21 4 _K_ 2  
#####  3  M 11   M 21   M 11   M 21    M 11   M 21   xx   ˝ M 12   M 2  2  yy 

 2 4 _K_ 2 

#####  M 11   M 21   M 11   M 21    M 11   M 21   xx   ˝ M 12   M 2  2 

##### 11   M 21   M 11   M 21    M 11   M 21   xx   ˝ M 12   M 2  2

#####  xx   M 12   M 2  2  yy

_xx_ 12 22 _yy_

##### M 11   M 21   M 11   M 21    M 11   M 21   xx   ˝ M 12   M

##### 32  M 11   M 21   M 11   M 21    M 11   M 21   xx 4



11   _M_ 21 

 1

##### 11   M 21   M 11   M 21    11 21 xx ˝ 12 22

11 21 _xx_ 12 2

3 _K_ 1  _K_ 2 



##### M 21   M 11   M 21    11 21 xx 4 K

˝
2

_a_ 

3 _K_ 1  _K_ 2  _a_  1]

_K_ 1  _K_

1  _K_ 2  _a_  1]



#####  Gyy  G 21  a  [  1  2  2   2  1   2  xx 18  K  2  1  2  2  yy  2 K 2   a  1 

#####  Gyy  G 21  a  [  1  2  2   2  1   2  xx 18  K  2  1  2  2  yy

1 

 _a_

#####  1  2  2   2  1   2  xx   1  2  2  yy  2 K 2   a

_xx_ 1 2 2 _yy_  _a_

#####  Gyy  G 21  a  [  1  2  2   2  1   2  xx 18  K  2  1  2  2  yy  2 K



_yy_

##### 21  a  [  1  2  2   2  1   2  xx 18  K  2  1  2  2  yy  2

##### 21  a  [  1  2  2   2  1   2  xx 18  K  2  1  2  2  yy  2 K 2   a  1

#####  M 12   M 22   M 12   M 22    M 11   M 21   xx   M 12   M 22  



#####  32  M 12   M 22   M 12   M 22    M 11   M 21   xx 4 K   2  M 12   M 22   yy  3 K 1˝  K 2˝ a  1 

##### 12   M 22   M 12   M 22    M 11   M 21   xx   M 12   M 22 

(15)

(16)

#####  xx   M 12   M 22   yy

_a_  1

˝  _K_ ˝

##### M 12   M 22   M 12   M 22    M 11   M 21   xx   M 12   M

##### 32  M 12   M 22   M 12   M 22    M 11   M 21   xx 4 K   2  M 12   M 22   yy  3

##### xx   M 12   M 22   yy  ˝  ˝ a

##### 12   M 22   M 12   M 22    11 21 xx  12 22

_a_  1

_K_ 2  12 22 _yy_  3 _K_ 1˝  _K_

1 2

4 _K_ 2  12 22 _yy_  3 _K_ 1  _K_ 2  _a_  1]

_K_ 2  12 22 _yy_  3 _K_ 1  _K_

3
2

#####  M 12   M 22   M 12   M 22    M 11   M 21  

##### M 12   M 22   M 12   M 22    M 11   M 21   xx   M 12   M

##### 12   M 22   M 12   M 22    M 11   M 21   xx   M 12   M 22 

#####  M 22   M 12   M 22    M 11   M 21   xx   M 12   M 22  

##### 12   M 22   M 12   M 22    11 21 xx  12

2  _M_ 12 _M_ 22 _yy_  3 _K_ 1  _K_ 2  _a_  1

22

 _yy_

_yy_   _a_

 _xx_

_xx_

 1

_M_

##### 1  a   2  72   2 K 2   a  1   M 66  2 3 K 1  K 2  a  1   M 6  xy״ 6  2 3 K 1  K 2  a  1





_a_

#####  Gxy  G 21  a   2  K 722   2 K 2   a  1   MK 66  2  2 3 K 1  K 2  a  1   MK 6  2 xy״ 6  2 3 K 1  K 2  a  1 

##### G  G 1  a   2  72  2 K 2   a  1   M 66   2 3 K 1  K 2  a  1   M 6  6  2 3 K 1  K

##### 72   2 K 2   a  1  K 662  3 K 1  K 2  a  1  K 62 xy״ 6 3 K 1  K 2 

#####  2 K 2   a  1   M 66  2 3 K 1  K 2  a  1   M 6  6 

_a_  1 _M_ 66 _a_  1 _M_ 66 _a_

 1 _M_ 66  _a_  1 _M_ 6  6 _a_  1

6  6

6

##### 2   2 K 72   2 K 2   a  1  K 662  3 K 1  K 2  a  1  K 626 3

 _xy_ 2  _K_ 2  2 _K_ 2 

6  6

_xy_ 2  _K_ 2  2 _K_ 2  1 2 _K_ 2 _xy״_ 1 2 

_K_ 2  2 _K_ 2  1 2 _K_

where

#####  K 1    2  1   2  xx   1  2  2  yy

 6
 2
#####    2  1   2  xx   1  2  2  yy 
####  K 2    6    7
#####   M 11   M 21   xx   M 12   M 22   yy

_K_ 1 




#####  K 2    M 11   M 21   xx   M 12   M 22   yy  2 

  2 

#####  2  1   2  xx   1  2  2 

##### 2  1   2  xx   1  2

#####  1   2  xx   1  2  2 

_K_

1 

1 2 _xx_ 1 2

_xx_ 1 2 _yy_

####  2  1   2  xx   1  2  2  yy  2   7 xy 

##### 2  1   2  xx   1  2

####  1   2  xx   1  2  2  yy  2   7

2
#### 1   2  xx   1  2  2  yy    7 xy  2

_K_

2 

_xx_ 1 2 _yy_

_xy_

22 

_K_

#####  M 11   M 21   xx   M 12   M 22  

##### M 11   M 21   xx   M 12   M

11 21 _xx_ 12 22

#####   M 21   xx   M 12 

#####  xx   M 12   M 22  

_xx_ 12 22 _yy_



##### K 2    M 11   M 21   xx   M 12   M

_xx_ 12 22 _yy_

2
2

_xy_

####  M 11   M 21   xx   M 12   M 22   yy  2   M 66  xy 

#### M 11   M 21   xx   M 12   M 22   yy    M

#### 11   M 21   xx   M 12   M 22   yy  2   M 66 

#### 11 21 xx 12 22 yy    M 66 

####  xx   M 12   M 22   yy  2   M 66 

  2 

Latin American Journal of Solids and Structures 12 (2015) 92-114

3 CALIBRATION OF MODIFIED Yld2000-2d II IN PLANE STRESS PROBLEMS

In this part the yield function of Modified Yld2000-2d II which is an asymmetric function (pressure dependent criterion) is calibrated with ten stress experimental data points such as uniaxial
### tensile yield stresses  T  in 0 °, 45 °, 90 °, uniaxial compressive yield stresses  C  in 0 °, 15 °, 30 °, 45 °, 75 °, 90 ° from rolling direction along with biaxial yield stress,  bT  . On the other hand,

the plastic potential function of Modified Yld2000-2d II which is a symmetric function (pressure

 _yyp_

 _zzp_

_p_
_zz_

independent criterion) is calibrated with eight experimental data such as _R_ -values

 _R_  _T_  _d_  _yypp_ 
 
 _d_  _zz_ 

_R_  _T_  _dd_

_T_  _d_  _yyp_

in 0 _°_, 15 _°_, 30 _°_, 45 _°_, 60 _°_, 75 _°_, 90 _°_ from the rolling direction and also biaxial tensile _R_ -value

 _RbT_  _d_  _yypp_ 
 
 _d_  _xx_ 

_bT_  _dd_  _xxyypp_

_RbT_  _dd_

 _p_



. Since the yield function is pressure dependent, the experimental tests should be

employed in tensile and compressive directions. In tensile test (in ? angle from rolling direction)
the stress tensor will be:

 _xx_ 
 _yy_ 

 _xy_ 

 _xx_  _T_ _cos_ 2 
 _yy_  _T_ _sin_ 2 
 _xy_  _T_ _sin_  _cos_ 

_xx_  _T_

_yy_  _T_

_cos_



 _T_

_T_



 _T_ _sin_ 2 

_sin_  _cos_

_xy_  _T_

and similarly in compression test it is found that:

 _xx_ 
 _yy_ 

 _xy_ 

 _xx_  _Ccos_ 2 ,
 _yy_  _C_ _sin_ 2 ,
 _xy_  _C_ _sin_  _cos_  .

_xx_  _C_

_yy_  _C_

_cos_



 _C_

2 ,

 _C_ _sin_ 2 ,

_sin_  _cos_

_xy_  _C_

(17)

(18)

(19)



For biaxial test the state of stress is as below:

 _xx_ 
 _yy_ 

 _xy_ 

 _xx_  _bT_

 _yy_  _bT_

_xx_  _bT_

_yy_  _bT_



_xy_

Substituting these values in Eq. (7) the tensile and compressive yield stresses in different orientations from the rolling direction and also biaxial yield stress are obtained as following:

Latin American Journal of Solids and Structures 12 (2015) 92-114

  ~~~~ _p_
####  T    1
  _a_ _a_ _a_  _a_
 2 2 2 _K_ 1  _p_  3 _K_ 1  _p_  _K_ 2  _p_  3 _K_ 1  _p_  _K_ 1  _p_
 _hx_ _cos_  _hy_ _sin_   

 2 

  

####  C    p  1

2 2  2 _K_ 1  _p_ _a_  3 _K_ 1  _p_  _K_ 2  _p_ _a_  3 _K_ 1  _p_  _K_ 2  _p_ _a_  _a_

  _hx_ _cos_  _hy_ _sin_   
  2 

 


  ~~~~ _p_
####  bT   
 _a_
   1  2 _a_ _a_ 
  3  _L_ 11   2 _L_ 21   _L_ 12   2 _L_ 22   2 _L_ 11   _L_ 21   2 _L_ 12   _L_ 22  
 _hx_  _hy_ 
 2 

 

  

 ~~~~ _p_
####  T  

1

 ~~~~ _p_
#### T  

~~~~ _p_
####  





 _a_ _a_ _a_  _a_

2 2 _K_ 1  _p_  3 _K_ 1  _p_  _K_ 2  _p_  3 _K_ 1  _p_  _K_ 1  _p_

 _hy_ _sin_   
 2 
 

 _a_ _a_ _a_  _a_

2 2 2 _K_ 1  _p_  3 _K_ 1  _p_  _K_ 2  _p_  3 _K_ 1  _p_  _K_ 1  _p_

_x_ _cos_  _hy_ _sin_   

 



2 2 2 _K_ 1  _p_  3 _K_ 1  _p_  _K_ 2  _p_  3 _K_ 1  _p_  _K_

_hx_ _cos_  _hy_ _sin_  



2 2 1 _p_ 1 _p_ 2 _p_ 1 _p_ 1

 _h_ _sin_  

2  _hy_ _sin_ 2 

2 _K_ 1  _p_  3 _K_ 1  _p_  _K_ 2  _p_  3

 ~~~~ _p_
####  C  

1

 ~~~~ _p_
#### C  

~~~~ _p_
####  





2 2  2 _K_ 1  _p_ _a_  3 _K_ 1  _p_  _K_ 2  _p_ _a_  3 _K_ 1  _p_  _K_ 2  _p_ _a_ 

 _hx_ _cos_  _hy_ _sin_   
 2 
 

2 2  2 _K_ 1  _p_ _a_  3 _K_ 1  _p_  _K_ 2  _p_ _a_  3 _K_ 1  _p_  _K_ 2  _p_ _a_  _a_

_x_ _cos_  _hy_ _sin_   

 

2 2  2 _K_ 1  _p_  3 _K_ 1  _p_  _K_ 2  _p_  3 _K_ 1  _p_  _K_

_hx_ _cos_  _hy_ _sin_  



2 2 1 _p_ 1 _p_ 2 _p_ 1 _p_ 2

 _h_ _sin_  

 _hy_ _sin_ 2 

2 _K_ 1  _p_  3 _K_ 1  _p_  _K_ 2  _p_  3

 ~~~~ _p_
####  bT  

 ~~~~ _p_
#### bT  

1



~~~~ _p_
####  

(20)

(21)

_a_

  1  3  2 _a_  _L_ 11   2 _L_ 21   _L_ 12   2 _L_ 22  _a_  2 _L_ 11   _L_ 21   2 _L_ 12 

 _hy_ 

 1  2

_a_

 1 3 2  _L_ 11   2 _L_ 21   _L_ 12   2 _L_ 22  _a_  2 _L_ 11   _L_

_hx_  _hy_ 

3 2  _L_ 11   2 _L_ 21   _L_ 12   2 _L_ 22  _a_  2 _L_ 11   _L_ 21   2

1 2  _L_ 11   2 _L_ 21   _L_ 12   2 _L_ 22  _a_  2 _L_ 11   _L_ 21 

_L_ 12   _L_ 22  _a_  _a_

12 22

_x_ _y_

in which,

####  2  1  2  cos 2   1  2  2 

####  K 1  p  2  1  2  cos 2   1  2  2  sin 2 

 6
####    2  1  2  cos 2   1  2  2  sin 2  2  K 2  p  L  11   L 21   cos 2   L 612   L 22   sin 2   

_K_ 1  _p_



####  1  2  cos 2   1  2  2  sin 2 

#### 1  p  2  1  2  cos 2   1  2  2  sin 2

#### K 1  p  2  1  2  cos 2   1  2  2  sin

#### 2  1  2  cos 2   1  2

_p_

2
#### 2  p   2  1  2  cos 2   1  2  2  sin 2    7 sin  cos  2

#### K 2  p   2  1  2  cos 2   1  2  2  sin 2    7 sin  cos

#### 2  1  2  cos 2   1  2

_p_

2
####  2  1  2  cos 2   1  2  2  sin 2    7 sin  cos 

2
####  1  2  cos 2   1  2  2  sin 2    7 sin  cos 

#### K 1  p  L 11   L 21   cos 2   L 12   L 22   sin










#### 1  p  L 11   L 21   cos 2   L 12   L 22   sin 2

####  L 11   L 21   cos 2   L 12   L 22  

####   L 12   L 22   sin 2 

_p_

2
#### 2  p    L 11   L 21   cos 2   L 12   L 22   sin 2    L 66  sin  cos  2

2
####  K 2  p    L 11   L 21   cos 2   L 12   L 22   sin 2    L 66 

  2 

#### K 2  p    L 11   L 21   cos 2   L 12   L 22   sin 2    L 66  sin  cos

_p_

2
####  L 11   L 21   cos 2   L 12   L 22   sin 2    L 66  sin  cos 

2
####   L 12   L 22   sin 2    L 66  sin  cos 

Using Eqs. (20) and (21) along with stress components in Eqs. (17) to (19), the Modified
Yld2000-2d as a yield function can be calibrated.

In the current research the non-AFR is employed and the Yld2000-2d as a pressure independent criterion is used for plastic potential function. Using non-AFR rule it is found that:

Latin American Journal of Solids and Structures 12 (2015) 92-114


 _d_  _xxp_  _d_ 
 ~~~~ 

 _d_  _yyp_  _d_ ~~~~ 


 _d_  _xyp_  _d_ 
 ~~~~ 

_d_  _xxp_  _d_  _G_

~~~~ 

 _xxp_  _d_ _G_
~~~~ 

_p_
_xx_

_xx_

_d_  _yyp_  _d_  _G_

~~~~ 

(22)

 _yyp_  _d_ _G_
~~~~ 

_p_
_yy_

_yy_

_d_  _xyp_  _d_  _G_

~~~~ 

 _xyp_  _d_
~~~~ 

_p_
_xy_

_xy_

With considering the new presented plastic potential function in Eq. (14) which is pressure independent the assumption of incompressibility can be stated as bellow:

_d_  _zzp_  _d_  _xxp_  _d_  _yyp_ (23)

Moreover, from the definition of _R_ -values it can be shown that:

 _p_  _G_ _sin_ 2   _G_ _cos_ 2  
 _R_  _T_  _dd_  _yyzzp_   _xx_  _Gyy_  _G_ 


 _xx_  _yy_

 _G_

 _RbT_  _dd_  _xxyypp_   _Gyy_
  _xx_

_R_  _T_  _dd_  _yyzzpp_   _Gxx_ _sin_ 2   _GGyy_ _cos_ 2  _G_   _Gxy_ _sin_  _cos_



 _yyp_  _Gxx_ _sin_ 2   _Gyy_ _cos_ 2   _Gxy_ _sin_  _cos_ 

_T_  _d_  _yyp_   _Gxx_ _sin_ 2   _Gyy_ _cos_ 2   _Gxy_



 _zzp_

_p_
_zz_

 _xx_  _yy_

_xx_ _yy_

 _G_

_RbT_  _dd_  _xxyypp_   _Gyy_

 _yyp_ 

_bT_  _dd_  _xxyypp_   _Gyy_





_xx_

(24)

obtained from Eqs. (15) and (16)

Using the stress components in Eqs. (17) to (19) and also

 _G_
 _ij_

the Yld2000-2d as a plastic potential function can be calibrated.

4 PARAMETER EVALUATION AND ROOT MEAN SQUARE ERRORS
OF THE YIELD STRESSES AND R-VALUES
#### Ten material constants denoted as  i  i  1,8 , h x and hy for the yield function and also eight parameters of  i  i  1,8  for the plastic potential function can be computed from eighteen expe
rimental data points such as as  _T_ 0,  _T_ 45,  90 _T_,  _bT_,  0 _C_,  15 _C_,  45 _C_,  60 _C_,  75 _C_,  90 _C_, _R_ 0 _T_, _R_ 15 _T_,

_R_ 30 _T_, _R_ 45 _T_, _R_ 60 _T_, _R_ 75 _T_, _R_ 90 _T_ and _RbT_ for Modified Yld2000-2d II. These experimental data points
##### are utilized to set up two error functions, i.e. the first for the yield function  E 1   and the second
#### for the plastic function  E 2  of Modified Yld2000-2d II as following:

Latin American Journal of Solids and Structures 12 (2015) 92-114

###  T 45   T 45 

###  90 T   90 T 

2 2 2 2

_T_    _T_    _T_    _T_

0 30 45 90
### 1    T 0  predexp ..  1     30 T  predexp ..  1     T 45  predexp ..  1     90 T  predexp ..

_E_

2 2 2 2
###   T 0  pred .  1    30 T  pred .  1    T 45  pred .  1    90 T  pred .  1     T 0  exp .    30 T  exp .    T 45  exp .    90 T  exp . 

###  T 0 

###  T 0    30 T    T 45    90 T

###  T 0 

###  T 0     30 T     T 45     T

_T_    _T_    _T_    _T_

30 45 90

_pred_ .     _pred_ .     _pred_ .     _pred_ .

1    _pred_ .  1    _pred_ .  1    _pred_ .  1 

###  30 T   30 T 

30 45 90

_exp_ .   _exp_ .   _exp_ .   _exp_

###  bT   bT   90 C   90 C 

###  0 C   0 C 

###  15 C   15 C 

###  60 C   60 C 

2 2 2

###  45 C    60 C

_C_    _C_

2 2 2
###  bT  pred .  1    0 C  pred .  1    15 C  pred .  1     bT  exp .    0 C  exp .    15 C  exp . 

2 2
###  45 C  pred .  1    60 C  pred .  1     45 C  exp .    60 C  exp . 

###  45 C   45 C 

2 2

(25)

(26)

###  bT    0 C    15 C

### bT  pred .  1    0 C  pred .  1    15 C  pred bT  exp .    0 C  exp .    15 C  exp .

0 15
.     _pred_ .     _pred_ .

1    _pred_ .  1    _pred_ .  1

###  45 C     60 C

45 60

_pred_ .     _pred_ .

_C_    _C_

_pred_ .     _pred_

1    _pred_ .  1

###  bT     0 C     C

0 15
.   _exp_ .   _exp_ .

45 60

_exp_ .   _exp_ .

2
###  90 C  pred .  1     90 C  exp . 

 90 _C_

_pred_

_C_

_pred_ .

1   0

###  90 C

_exp_ .

_C_

_exp_ .

and

###  R 0 T 

###  R 30 T   R 30 T 

###  R 45 T   R 45 T 

###   R 0 T  pred .  1    R 15 T  pred .  1    R 30 T  pred .  1    R 45 T  pred .  1    R 0 T  exp .    R 15 T  exp .    R 30 T  exp .    R 45 T  exp . 

_T_   _R_ _T_   _R_ _T_   _R_ _T_

### E 2   R 0 T  pred .  1    R 15 T  pred .  1    R 30 T  pred .  1    R

0 15 30 45
### 2    R 0 T  predexp ..  1     R 15 T  predexp ..  1     R 30 T  predexp ..  1     R 45 T  predexp ..

15 30 45

_pred_ .     _pred_ .     _pred_ .     _pred_

###  R 0 T 

1    _pred_ .  1    _pred_ .  1    _pred_ .  1

###  R 15 T   R 15 T 

_T_   _R_ _T_   _R_ _T_   _R_ _T_

### R 0 T     R 15 T     R 30 T     R

15 30 45

_exp_ .   _exp_ .   _exp_ .   _exp_ .

###  R 60 T 

###  R 90 T   R 90 T 

_T_
### b  pred

2
###   R 60 T  pred .  1    R 75 T  pred .  1    R 90 T  pred .  1     R 60 T  exp .    R 75 T  exp .    R 90 T  exp . 

###  RbT  pred .  1    RbT  exp . 

###  RbT   RbT 

_T_   _R_ _T_   _R_ _T_

### R 60 T    R 75 T    R 90 T    R

.

60 75 90

_pred_ .     _pred_ .     _pred_ .

75 90

_pred_ .     _pred_ .     _pred_

### RbT  exp .

###  R 60 T 

1    _pred_ .  1    _pred_ .  1

###  R 75 T   R 75 T 

_T_   _R_ _T_   _R_ _T_

### R 60 T     R 75 T     R

60 75 90

_exp_ .   _exp_ .   _exp_ .

75 90

_exp_ .   _exp_ .   _exp_ .

These error functions can be minimized by the Downhill Simplex method to identify the material
parameters which were more explained by Hu et al. (2010) and Barlat et al. (2013).
## The root-mean square errors (RMSEs) of the tensile  E  T  and compressive  E  C  yield stresses, the tensile biaxial stress  E  Tb , the tensile R -values  E RT  and the tensile biaxial R -value
###  E  Tb  can be computed as follow:

###  T 0  exp .   T 0 

###  15 T  exp .   15 T 

###  15 T 

###  30 T  exp .   30 T 

###  30 T 

 _T_ _T_ 2 _T_ _T_ 2 _T_ _T_ 2  2
###   0  exp .   0  pred .   15  exp .   15  pred .   30  exp .   30  pred .  

        

 _T_ _T_ _T_ 
###    0  exp .    15  exp .    30  exp .  

 2 2 2 
###   T 45  exp .   T 45  pred .    T 60  exp .   T 60  pred .    T 75  exp .   T 75  pred .        T 45  exp .    T 60  exp .    T 75  exp .     90 T  exp .   90 T  pred .  2     90 T  exp .  

 

2 2 2

2 2 2
###  T 0  exp .   T 0  pred .    15 T  exp .   15 T  pred .    30 T  exp .   30 T  pred .     T 0  exp .    15 T  exp .    30 T  exp . 

1
2

###  T 0    T 0    15 T    15 T    30 T    30 T

_T_   _T_    _T_   _T_    _T_   _T_

0 0 15 15 30 30

_exp_ . _pred_ .    _exp_ . _pred_ .    _exp_ . _pred_ .

0 15 15 30 30

_exp_ . _pred_ .    _exp_ . _pred_ .    _exp_ . _pred_

###  T 0 

###  T 0     15 T     T

_T_    _T_    _T_

0 15 30

_exp_ .   _exp_ .   _exp_ .

15 30

_exp_ .   _exp_ .   _exp_

###  T 60  exp .   T 60 

###  T 60 

_pred_ .     100

_T_  
### 

.

2 2

###  T 75  exp .   T 75 

2 2
###  1   T 45  exp .   T 45  pred .    T 60  exp .   T 60  pred .    T 75  exp .   7    T 45  exp .    T 60  exp .    T 75 

1
7

###  T 45  exp .   T 45 

###  T 45    T 45    T 60    T 60    T 75    T

### T 45  exp .   T 45  pred .    T 60  exp .   T 60  pred .    T 75  exp .   T 75

###  T 45 

###  T 45     T

45 60

_exp_ .   _exp_ .

_T_ 1 45 _exp_ . 45 _pred_ . 60 _exp_ . 60 _pred_ . 75 _exp_ . 75 _pred_

        

_T_ _exp_ . _pred_ . _exp_ .
### E   7    T 45  exp .     T 60  exp .

(27)

_exp_ .

_T_    _T_

 _T_

_exp_ .

###  90 T  exp .   90 T 

###  90 T    T

_T_   _T_

90 90

_exp_ . _pred_ .

_exp_ . _pred_

###  90 T 

 _T_

_exp_

_T_

_exp_ .

Latin American Journal of Solids and Structures 12 (2015) 92-114

and,

####  0 C  exp .   0 C 

####  15 C  exp .   15 C 

####  15 C 

####  30 C  exp .   30 C 

####  30 C 

2 2 2

2 2 2
####  0 C  exp .   0 C  pred .    15 C  exp .   15 C  pred .    30 C  exp .   30 C  pred .     0 C  exp .    15 C  exp .    30 C  exp . 

1
2

####  0 C    0 C    15 C    15 C    30 C    30 C

_C_   _C_    _C_   _C_    _C_   _C_

0 0 15 15 30 30

_exp_ . _pred_ .    _exp_ . _pred_ .    _exp_ . _pred_ .

0 15 15 30 30

_exp_ . _pred_ .    _exp_ . _pred_ .    _exp_ . _pred_ .

####  0 C 

####  0 C     15 C     30 C

_C_    _C_    _C_

0 15 30

_exp_ .   _exp_ .   _exp_ .

15 30

_exp_ .   _exp_ .   _exp_ .

####  60 C  exp .   60 C 

####  60 C 

. _pred_ .     100

_C_  
#### 

2 2

####  75 C  exp .   75 C 

2 2
####  1   45 C  exp .   45 C  pred .    60 C  exp .   60 C  pred .    75 C  exp .   7    45 C  exp .    60 C  exp .    75 C 

1
7

 _C_ _C_ 2 _C_ _C_ 2 _C_ _C_ 2  2
####   0  exp .   0  pred .   15  exp .   15  pred .   30  exp .   30  pred .  

        

 _C_ _C_ _C_ 
####    0  exp .    15  exp .    30  exp .  

 2 2 2 
####   45 C  exp .   45 C  pred .    60 C  exp .   60 C  pred .    75 C  exp .   75 C  pred .        45 C  exp .    60 C  exp .    75 C  exp .     90 C  exp .   90 C  pred .  2     90 C  exp .  

 

####  45 C  exp .   45 C 

####  45 C    45 C    60 C    60 C    75 C    75 C

#### 45 C  exp .   45 C  pred .    60 C  exp .   60 C  pred .    75 C  exp .   75 C

####  45 C 

####  45 C     60 C

45 60

_exp_ .   _exp_ .

_C_  1   45 _exp_ . 45 _pred_ .    60 _exp_ . 60 _pred_ .    75 _exp_ . 75 _pred_

.

#### E  C  7   exp  . 45 C  exp . pred .    exp  . 60 C  exp .

(28)

_exp_ .

_C_    _C_

 _C_

_exp_ .

####  90 C  exp .   90 C 

####  90 C    90 C

_C_   _C_

90 90

_exp_ . _pred_ .

_exp_ . _pred_

####  90 C 

####  90 C

_exp_ .

_C_

_exp_ .

and,

###  bT  exp .   bT 

###   bT  exp .   bT  pred .  100 (29)

###  bT    bT

### Tb   bT  exp .   bT  pred

_b_

. _pred_ .

100

### E  Tb  exp  . bT  exp .

###  bT 

 _bT_

####  R 0 T  exp .   R 0 T 

####  R 15 T  exp .   R 15 T 

####  R 15 T 

####  R 30 T  exp .   R 30 T 

####  R 30 T 

2 2 2

2 2 2
####  R 0 T  exp .   R 0 T  pred .    R 15 T  exp .   R 15 T  pred .    R 30 T  exp .   R 30 T  pred .     R 0 T  exp .    R 15 T  exp .    R 30 T  exp . 

1
2

_T_  _R_ _T_   _R_ _T_  _R_ _T_   _R_ _T_  _R_ _T_

#### R 0 T    R 0 T    R 15 T    R 15 T    R 30 T    R

0 0 15 15 30 30

_exp_ . _pred_ .    _exp_ . _pred_ .    _exp_ . _pred_ .

0 15 15 30 30

_exp_ . _pred_ .    _exp_ . _pred_ .    _exp_ . _pred_ .

####  R 0 T 

_T_   _R_ _T_   _R_ _T_

#### R 0 T     R 15 T     R

0 _exp_ .   15 _exp_ .   30 _exp_ .

_exp_ .   15 _exp_ .   30 _exp_ .

####  R 60 T  exp .   R 60 T 

####  R 60 T 

_pred_ .     100 .

_T_  
#### 

2 2

####  R 75 T  exp .   R 75 T 

2 2
####  1   R 45 T  exp .   R 45 T  pred .    R 60 T  exp .   R 60 T  pred .    R 75 T  exp .   7    R 45 T  exp .    R 60 T  exp .    R 75 T 

#### E RT  1   R 45 T  exp .   R 45 T  pred .    R 60 T  exp .   R 60 T  pred .    R 75 T  exp .   R

1
7

 _T_ _T_ 2 _T_ _T_ 2 _T_ _T_ 2  2
####   R 0  exp .   R 0  pred .   R 15  exp .   R 15  pred .   R 30  exp .   R 30  pred .  

        

 _T_ _T_ _T_ 
####    R 0  exp .    R 15  exp .    R 30  exp .  

 2 2 2 
####   R 45 T  exp .   R 45 T  pred .    R 60 T  exp .   R 60 T  pred .    R 75 T  exp .   R 75 T  pred .        R 45 T  exp .    R 60 T  exp .    R 75 T  exp .     R 90 T  exp .   R 90 T  pred .  2     R 90 T  exp .  

 

####  R 45 T  exp .   R 45 T 

#### 45 T  exp .   R 45 T  pred .    R 60 T  exp .   R 60 T  pred .    R 75 T  exp .   R 75 T

####  R 45 T 

#### R 45 T     R

45 60

_exp_ .   _exp_ .

_exp_ .   _exp_

#### RT  17   45 exp  R . T  45 pred .    60 exp  R . T  60 pred .    75 exp  R . T  75 pred

.

(31)

_exp_ .

_R_

_exp_ .

####  R 90 T  exp .   R 90 T 

_T_  _R_ _T_

#### R 90 T    R

90 90

_exp_ . _pred_ .

####  R 90 T 

_R_

_T_

90 _exp_ .

_exp_

and,

###  RbT  exp .   RbT 

###  RbT  exp .   RbT  pred .

  100 (32)

### RTb   RbT  exp  R . bT  expRbT .  pred

### E RTb   RbT  exp .   Rb

_b_

. _pred_ .

100

###  RbT 

_R_

The tensile and compressive yield stresses along with the tensile _R_ -values determined from experiments required for the above error functions as they are shown in Tables (1) to (3) for Al2008T4 and Al2090-T3.

Latin American Journal of Solids and Structures 12 (2015) 92-114

_T_ _T_ _T_ _T_ _T_ _T_ _T_ _T_
M aterial  0  15  30  45  60  75  90  _b_

Al 2008-T4 211.67 211.33 208.5 200.03 197.3 194.3 191.56 185.0

Al 2090-T3 279.62 269.72 255 226.77 227.5 247.2 254.45 289.4

Table 1: Experimental results for Al 2008-T4 and Al 2090-T3 in tension presented by Lou et al. (2013).

_C_ _C_ _C_ _C_ _C_ _C_ _C_
M aterial  0  15  30  45  60  75  90

Al 2008-T4 213.79 219.15 227.55 230.25 222.75 220.65 214.64

Al 2090-T3 248.02 260.75 255 237.75 245.75 263.75 266.48

Table 2: Experimental results for Al 2008-T4 and Al 2090-T3 in compression presented by Lou et al. (2013).

_T_ _T_ _T_ _T_ _T_ _T_ _T_ _T_
M aterial _R_ 0 _R_ 15 _R_ 30 _R_ 45 _R_ 60 _R_ 75 _R_ 90 _Rb_

Al 2008-T4 0.87 0.814 0.634 0.5 0.508 0.506 0.53 1.000

Al 2090-T3 0.21 0.33 0.69 1.58 1.05 0.55 0.69 0.670

Table 3: Experimental results for Al 2008-T4 and Al 2090-T3 for _R_ -value in tension presented by Lou et al.

(2013).

5 RESULTS AND DISCUSSIONS

The Modified Yld2000-2d II constructed by Modified Yld2000-2d employed as yield and Yld20002d used as plastic potential functions. These criteria are compared with experimental results in
Figs. (1) to (6) for Al 2008-T4 (as a BCC material) and Al 2090-T3 (as a FCC material). The
mechanical properties of these materials in different angles from the rolling direction are available
##### in Tables (1) to (3). In Tables (4) and (5) parameters  i ( i  1 8), h x and hy along with  i ( i  1 8) are computed with minimizing the error functions E 1 and E 2 in Eqs. (26) and (27)

using Downhill Simplex Method for Al 2008-T4 and Al 2090-T3.

##### M aterial  1  2  3  4  5  6  7  8 h x hy

Al 2008-T4 -0.0904 1.8021 1.8149 0.8574 -0.8184 0.0519 0.6291 1.6161 0.0173 0.0732

Al 2090-T3 1.1685 1.1393 0.5730 0.1067 0.6042 1.9914 -0.9788 1.8900 -0.0504 0.0385

Table 4: Material parameters in yield function in Modified Yld2000-2d of Al 2008-T4 and Al 2090-T3.

Latin American Journal of Solids and Structures 12 (2015) 92-114

##### M aterial  1  2  3  4  5  6  7  8

Al 2008-T4 0.0189 -0.0001 0.0047 0.0095 0.0079 0.0078 0.0076 0.0132

Al 2090-T3 0.1442 0.0755 0.0354 0.1284 0.1552 0.2693 0.1156 -0.0000

Table 5: Material parameters in plastic potential function in Yld2000-2d of Al 2008-T4 and Al 2090-T3.

Figures (1) and (2) show the  _xx_  _yy_ for yield surfaces using Eq. (7) and coefficients given in

Table (4) and plastic potential surfaces using Eq. (14) and coefficients given in Table (5) for Al
2008-T4 and Al 2090-T3. In these figures the differences between the yield and plastic potential
surfaces for both materials can be observed because of non-AFR assumption. For both materials
the yield surfaces of Modified Yld2000-2d and the Modified Yld2000-2d II are fitted to experimental data with acceptable accurately. For Al 2008-T4, both criteria are close to each other but
their difference in the third quadrant is more than other parts.

Figures (3) and (4) show tensile and compressive yield stresses in different angles from the rolling direction for Al 2008-T4 and Al 2090-T3 and the obtained results are compared with experimental results. For Al 2090-T3, these criteria differ essentially and since the available experimental data are limited, it cannot be precisely said which one is more accurate.

It is observed that the Modified Yld2000-2d II is more successful than the Modified Yld20002d in predicting compressive yield stresses compared with experimental results. However, in predicting tensile yield stresses, the Modified Yld2000-2d is still more accurate. Figures (5) and (6)
show the _R_ -values in different angles from the rolling directions for Al 2008-T4 and Al 2090-T3.
It is seen that the Modified Yld2000-2d II can predict the experimental results more precisely
than Modified Yld2000-2d especially for Al 2090-T3.

(1-a)

Latin American Journal of Solids and Structures 12 (2015) 92-114

(1-b)

Figure 1: (a) Yield function of Modified Yld 2000-2d and Modified Yld 2000-2d II in compared
with experimental results, (b) plastic potential surface of Modified Yld 2000-2d II for Al 2008-T4.

(2-a)

Latin American Journal of Solids and Structures 12 (2015) 92-114

(2-b)

Figure 2: (a) Yield function of Modified Yld 2000-2d and Modified Yld 2000-2d II in compared with

experimental results, (b) plastic potential surfaces of Modified Yld 2000-2d II for Al 2090-T3.

(3-a)

Latin American Journal of Solids and Structures 12 (2015) 92-114

(3-b)

Figure 3: Comparison of the yield stress directionality for Al 2008-T4:
(a) uniaxial tensile yield stress; and (b) uniaxial compressive yield stress.

(4-a)

Latin American Journal of Solids and Structures 12 (2015) 92-114

(4-b)

Figure 4: Comparison of the yield stress directionality for Al 2090-T3:
(a) uniaxial tensile yield stress; and (b) uniaxial compressive yield stress.

Figure 5: Comparison of the _R_ -value directionality for Al 2008-T4.

Latin American Journal of Solids and Structures 12 (2015) 92-114

Figure 6: Comparison of the _R_ -value directionality for Al 2090-T3.

As it is observed, to calibrate the yield function in compressive yield stresses the experimental
### data points are increased from two  0 C,  90 C  in Modified Yld2000-2d to seven in Modified Yld2000-2d II  0 C,  15 C,  45 C,  60 C,  75 C,  90 C  and consequently Modified Yld2000-2d II can predict

the compressive yield stresses more accurately, Figs. (3-b) and (4-b). To calibrate the yield fun### ction for tensile yield stresses, however, three experimental data points  T 0,  T 45,  90 T  are selected

for both criteria and it is seen that the Modified Yld2000-2d is closer to experimental data than
Modified Yld2000-2d II, Figs. (3-a) and (4-a). To calibrate the yield function for biaxial yield
### stress  bT  one data point is used for both criteria and they could predict the experimental data

point reasonably, Figs. (1-a) and (2-a). The relative errors for these cases are computed in Tables
(6) and (7).

In order to calibrate the plastic potential function in Modified Yld2000-2d II seven experimen### tal data points are used  R 0 T, R 15 T, R 45 T, R 60 T, R 75 T, R 90 T  which are much more than three data points in yield function of Modified Yld2000-2d  R 0 T, R 45 T, R 90 T  and consequently the obtained results

are more accurate than those of Modified Yld2000-2d, Figs. (5) and (6). For _RbT_, one point is

selected for both criteria and they predict the experimental data point properly. The relative
error for this case computed in Tables (6) and (7).

The main point of the current study is using separate yield and plastic potential functions for
anisotropic pressure dependent materials, i.e. Modified Yld2000-2d (a pressure dependent criterion) for yield function and Yld2000-2d (a pressure independent criterion) for plastic potential.
Using these functions increase the number of required experimental data points to calibrate the
criterion from 10 for Modified Yld2000-2d and 8 for Yld2000-2d to 18 for Modified Yld2000-2d II.

Latin American Journal of Solids and Structures 12 (2015) 92-114

In Modified Yld2000-2d and Yld2000-2d the experimental data points employed for calibration
are a combination of yield stresses and R-values in different angles from rolling direction, while
for the Modified Yld2000-2d II they are yield stresses and _R_ -values for the yield and plastic potential functions, respectively. In the following the relative errors are computed in Tables (6) and
(7) for Al 2008-T3 and Al2008-T4 from Eqs. (28) to (32) and compared with experimental results.

_E_ _RTb_

Criterion

_E_  _T_ _E_  _C_ _E_  _Tb_ _E_ _RT_

Modified Yld2000-2d 0.2704 1.5915 6.0536e-007 3.985 0.0187

Modified Yld2000-2d II 0.5838 0.4617 1.8179e-009 1.8179e-009 0.00076

Table 6: The obtained computation errors for Al 2008-T4 compared with experimental results (in percentage).

_E_ _RTb_

Criterion

_E_  _T_ _E_  _C_ _E_  _Tb_ _E_ _RT_

Modified Yld2000-2d 0.7350 2.4651 2.3763e-010 12.889 0.0017

Modified Yld2000-2d II 1.2207 0.6645 8.2811e-010 0.754 0.000098

Table 7: The obtained computation errors for Al 2008-T4 compared with experimental results (in percentage).

Tables (6) and (7) show that Modified Yld2000-2d II predicts  _C_,  _bT_, _R_  _T_ and _R_  _T_ more accu
rately than Modified Yld2000-2d for Al 2008-T4 (a BCC material) and Al2090-T3 (a FCC material).

In the following to check the proposed criterion for strong strength differential effect, AZ31
which is a HCP material at 3% plastic strain is choosen for study. Its experimental data points is
utilized from Yoon et al. (2014). Figure (7) show the yield function for AZ31 in  _xx_  _yy_ and as

it seen the Modified Yld2000-2d predicts experimental results with good accuracy. In Figure (8)
tensile and compressive yield stresses are compared with experimental results and as it obsereved
the Maodified Yld2000-2d II is succesful to predicti a the mechanical behavior of a HCP material
like FCC and BCC maerials.

Latin American Journal of Solids and Structures 12 (2015) 92-114

Figure 7: The Modified Yld2000-2d II and experimental data points for AZ31.

(8-a)

Latin American Journal of Solids and Structures 12 (2015) 92-114

(8-b)

Figure 8: Comparison of the Modified Yld2000-2d II with experimental results for AZ31:

(a) uniaxial tensile yield stress, (b) uniaxial compressive yield stress.

In Table 9, the obtained errors of Modified Yld2000-2d II in predicting tensile yield stress, compressive and biaxial yield stresses in compared with experimental results are shown and as it is
seen the proposed criterion is also successful for a HCP material.

Criterion

_E_  _T_ _E_  _C_ _E_  _Tb_

Modified Yld2000-2d II 0.1980 0.3896 1.9173e-010

Table 9: The obtained computation errors for AZ31 compared with experimental results (in percentage).

6  CONCLUSIONS

A non-AFR criterion with considering Modified Yld2000-2d as yield function and Yld2000-2d as
plastic potential function called Modified Yld2000-2d II was presented. The yield and plastic potential functions were calibrated with ten and eight experimental data points, respectively. To
verify the Modified Yld 2000-2d II in compared with experimental results, three anisoropic materials were selected contain Al 2008-T4 (a BCC material) and Al 2090-T3 (a FCC material) for
slight strength differential effect and AZ31 (a HCP material) for strong strength differential effect. The obtained results showed that the Modified Yld2000-2d II (non-AFR) predicted experimental results with good accuracy and better than Modified Yld2000-2d (AFR) proposed by Lou
et al. (2013) for these materials.

Latin American Journal of Solids and Structures 12 (2015) 92-114

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