/MAT/LAW87 (BARLAT2000)

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.
   • Iflag=0: Tabulated.
     • It is possible to add total strain rate dependency by defining a number N_rate of functions.
   • Iflag=1: 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,

     ${\overline{\epsilon }}_p$

     Equivalent plastic strain.

   • Iflag=2: Hansel hardening model is considered.(10)
     $${\sigma }_y=\left\{B_{HS}-\left(B_{HS}-A_{HS}\right)e^{\left(-m{\left[{\overline{\epsilon }}^p+{\epsilon }_0\right]}^n\right)}\right\}\left(K_1-K_{2}T\right)+\text{Δ}{H}_{\gamma \alpha }{V}_m$$
     Temperature is updated in the law when adiabatic conditions:(11)

     $$\text{Δ}T=\eta \frac{\overline{\sigma }d{\overline{\epsilon }}_p}{\rho C_p}$$
     The martensite rate equation is computed as follows:(12)

     $$\frac{\partial V_m}{\partial \epsilon }=\left\{\begin{array}{cc}0& if\epsilon <{\epsilon }_0\\ \frac{B}{A}\cdot e^{\left(\frac{Q}{T}\right)}\cdot {\left(\frac{1-V_m}{V_m}\right)}^{\left(\frac{B+1}{B}\right)}\cdot \frac{{\left(V_m\right)}^p}{2}\cdot \left(1-\tanh \left[C+DT\right]\right)& f\epsilon \ge {\epsilon }_0\end{array}\right.$$
3. If Chard>0, a kinematic hardening model of Chaboche Rousselier is used:
   • The back stress is calculated as:(13)
     $$a={\displaystyle \sum _{i=1}^4{a}_i}$$
     With,(14)
     $$a_i=A_i{C}_{i}d{\epsilon }_p-C_i{a}_i\text{Δ}{\overline{\epsilon }}_p$$
   • The yield stress is computed as follows when combined isotropic kinematic hardening is chosen:(15)
     $${\sigma }_y=(1-Chard).{\sigma }_{iso\_hard}+Chard.{\sigma }_{kin\_hard}$$
4. 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
5. When Iflag=1 (analytic Swift-Voce formulation is used) strain rates effect is taken into account using Cowper-Symonds expression:(16)
   $${\sigma }_y={\sigma }_y\left(1+{\left(\frac{\dot{\epsilon }}{c}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}\right)$$

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

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

   If c=0 or p=0, the strain rate effects are not taken into account.

6. 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.

7. Strain rate filtering:

   If VP=0 (dependency on strain rate), the default value of F_cut = 10KHz.

   If VP=1 (dependency on plastic strain rate), F_smooth and F_cut are ignored.

8. 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.