/MAT/LAW87 (BARLAT2000)

Yield stresses can be defined either by user-defined functions (plastic strain vs. stress) or analytically by a combination of Swift-Voce model. The model is based on Barlat YLD2000 criterion. ¹

Format

Definitions

Example 1 (with Barlat parameters input I_fit=0)

In this example use Barlat parameters input (I_fit=0) and tabulated yield stress-strain curve input (Iflag=0)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  kg                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW87/1/1
Steel 
#              RHO_I
              7.8E-6                   0
#                  E                  Nu     IFlag        VP             coeff_c               exp_p              
                 210                 0.3         0         1             4.15401                3.57             
#                 A1                  A2                  A3                  A4 IFlag_fit
                 1.0                 1.0                 1.0                 1.0         0
#                 A5                  A6                  A7                  A8
                 1.0                 1.0                 1.0                 1.0
#Blank line

#              exp_a               ALPHA                NEXP                Fcut   Fsmooth     NRATE
                   2                   0                   0                             1         1
#             ASWIFT                EPSO               QVOCE                BETA                  KO
                   0                   0                   0                   0                   0
#  func_id                        YSCALE         strain rate
         4                           1.5                   1
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/4
Mat_Curev Quasi-static
#                  X                   Y
                   0                  .3
               0.007                  .5
                0.05                  .7
                 0.1                 .75
                 0.3                  .9
                   1                 1.2				 
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Example 2 (with experiment data input I_fit=1)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                   g                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW87/1/1
ALU
#              RHO_I
              2.7E-3                   0
#                  E                  Nu     IFlag        VP             coeff_c               exp_p 
               70000                 0.3         1
#              sig00               sig45               sig90                sigb     I_fit
          133.179899          133.102756          132.330693          162.330301         1
#                r00                 r45                 r90                  rb
         0.703242569         0.486264221         0.865336191         0.546807587
#empty line 

#              exp_a               ALPHA                NEXP                Fcut   Fsmooth     NRATE
                   8                0.55                0.21                             1         0
#             ASWIFT                EPSO               QVOCE                BETA                  KO
                415.             0.00220               174.7               11.19               132.4
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

1. The yield function is expressed as:(1)
   $$f=\overline{\sigma }-{\sigma }_y$$
   (2)
   $$\overline{\sigma }= \frac{1}{2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}}\left({\phi }^{\prime }\left(X^{\prime }\right)+{\phi }^{″}\left(X^{″}\right)\right){}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}$$
   (3)
   $${\phi }^{\prime }\left(X^{\prime }\right)={\left|{X^{\prime }}_1-{X^{\prime }}_2\right|}^a$$
   (4)
   $${\phi }^{″}\left(X^{″}\right)={\left|2{X^{″}}_2+{X^{″}}_1\right|}^a+{\left|2{X^{″}}_1+{X^{″}}_2\right|}^a$$
   $X\text{'}$ and $X"$ denote the principal values of the tensors $X\text{'}$ and $X"$ which are a linear transformation of the stress deviator, which leads to:(5)

   $${\phi }^{\prime }\left(X^{\prime }\right)= {\left[{\left({X^{\prime }}_{xx}-{X^{\prime }}_{yy}\right)}^2+4{\left({X^{\prime }}_{xy}\right)}^2\right]}^{\frac{a}{2}}$$

   (6)

   $$\begin{array}{l} {\phi }^{″}\left(X^{″}\right)={\left[\frac{3}{2}\left({X^{″}}_{xx}-{X^{″}}_{yy}\right)+\frac{1}{2}\sqrt{{\left({X^{″}}_{xx}-{X^{″}}_{yy}\right)}^2+4{\left({X^{″}}_{xy}\right)}^2}\right]}^a+\\ {\left[\frac{3}{2}\left({X^{″}}_{xx}-{X^{″}}_{yy}\right)-\frac{1}{2}\sqrt{{\left({X^{″}}_{xx}-{X^{″}}_{yy}\right)}^2+4{\left({X^{″}}_{xy}\right)}^2}\right]}^a\end{array}$$

   The tensors $X\text{'}$ and $X"$ are linear transformations of the stress tensor:

   $X^{\prime }=L^{\prime }\sigma and X^{″}=L^{″}\sigma$ (7)

   $$L^{\prime }=\frac{1}{3}\left[\begin{array}{ccc}2{\alpha }_1& -{\alpha }_1& 0\\ -{\alpha }_2& 2{\alpha }_2& 0\\ 0& 0& 3{\alpha }_7\end{array}\right]$$

   (8)

   $$L^{″}=\frac{1}{9}\left[\begin{array}{ccc}-2{\alpha }_3+2{\alpha }_4+8{\alpha }_5-2{\alpha }_6& {\alpha }_3-4{\alpha }_4-4{\alpha }_5+4{\alpha }_6& 0\\ 4{\alpha }_3-4{\alpha }_4-4{\alpha }_5+{\alpha }_6& -2{\alpha }_3+8{\alpha }_4+2{\alpha }_5-2{\alpha }_6& 0\\ 0& 0& 9{\alpha }_8\end{array}\right]$$
2. The yield stress could be defined either by tabulated input or using the analytic Swift-Voce model.
   • If tabulated, it is possible to add total strain rate dependency by defining a number N_rate of functions
   • The analytic Swift Voce model is expressed as:(9)
     $${\sigma }_y={\alpha }_{sv}\left[A{\left({\overline{\epsilon }}_p+{\epsilon }_0\right)}^n\right]+\left(1-{\alpha }_{sv}\right)\left[K_0+Q\left(1-\exp \left(-B{\overline{\epsilon }}_p\right)\right)\right]$$

     Where, ${\overline{\epsilon }}_p$ is the equivalent plastic strain.

3. The strain rate filtering is available to smooth strain rates when tabulated input is chosen.

   List of Animation output (in /ANIM/SHELL/USRII/JJ):

   USR 1= plastic strain

   USR 2= effective stress

   USR 3= increment of plastic strain

4. When Iflag= 1 (analytic Swift-Voce formulation is used) strain rates effect is taken into account using Cowper Symonds expression:(10)
   $${\sigma }_y={\sigma }_y\left(1+{\left(\frac{\dot{\epsilon }}{c}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}\right)$$

   If VP= 0: $\dot{\epsilon }$ is the total strain rate.

   If VP = 1: $\dot{\epsilon }$ is the plastic strain rate.

5. When Iflag= 0 (tabulated formulation) then:

   If VP= 0: ${\dot{\epsilon }}_i$ is the total strain rate.

   If VP = 1: ${\dot{\epsilon }}_i$ is the plastic strain rate.

6. If I_fit =1, the coefficients ${\alpha }_i$ will be automatically fit in the Radioss Starter. The tensile yield strengths ${\sigma }_{00},{\sigma }_{45},{\sigma }_{90}$ and Lankford ratios $r_{00},r_{45},r_{90}$ must be determined from uniaxial tension experiments along the rolling, diagonal and transverse directions at an amount of plastic work corresponding to a plastic strain equal to 0.2%. ${\sigma }_b$ and $r_b$ should be determined from biaxial test, for the same amount of plastic strain.