/MAT/LAW52 (GURSON)

Format

Definition

Comments

1. The von Mises criteria for viscoplastic flow:
   If Iflag = 0:(1)

   $${\mathrm{\Omega }}_{vm}={\sigma }_{eq}-{\sigma }_M\sqrt{1+q_3{f}^{*2}-2q_1{f}^{*2}\cosh (\frac{3q_2{\sigma }_m}{2{\sigma }_M})}$$
   If Iflag = 1:(2)

   $${\mathrm{\Omega }}_{vm}=\frac{{\sigma }_{eq}^2}{{\sigma }_M^2}+2\text{}{q}_1{f}^{*}\cosh \left(\frac{3}{2}{q}_2\frac{{\sigma }_m}{{\sigma }_M}\right)-\left(1+q_3{f}^{\ast 2}\right)\text{}\text{}$$

   if ${\sigma }_m>0$

   (3)

   $${\mathrm{\Omega }}_{vm}=\frac{{\sigma }_{eq}^2}{{\sigma }_M^2}+2q_1{f}^{\ast }-\left(1+q_3{f}^{{\ast }^2}\right)\text{}\text{}$$

   if ${\sigma }_m\le 0$

   Where, ${\sigma }_M$ is the admissible stress, ${\sigma }_m$ is the trace[ $\sigma$ ] (hydrostatic stress), ${\sigma }_{eq}$ is the von Mises stress, and $q_1$ , $q_2$ , and $q_3$ are the material parameter for Gurson Law,

   $q_3=q_1^2$

   $f^{*}$ is the specific coalescence function.

   $f^{*}=f$ if $f\le f_c$

   $f^{*}=f_c+\frac{f_u-f_c}{f_F-f_c}(f-f_c)$ if $f>f_c$

   with

   $f_u=\frac{1}{q_1}$ corresponding to the coalescence function $f_u={\mathrm{f}}^{*}(f_F)$

2. The void volume fraction parameters must be entered so that, $f_I<f_c<f_F$ .
3. If one integration point reaches $f^{*}\ge f_F$ , the element is deleted.
4. If the I_yield flag is not activated (I_yield=0), the yield stress is computed using Cowper-Symond's law:
   (4)

   $${\sigma }_M=(A+B{\epsilon }_M{}^N)\left(1+{\left(\frac{\dot{\epsilon }}{c}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}\right)$$

   If the I_yield flag is activated (I_yield=1), the yield stress is computed directly from the Yield stress curves (Tab_ID).

5. This law is available for shell and solid elements.
6. In plot files (/TH/SHEL, /TH/SH3N and /TH/BRICK) or animation files (/ANIM), the following variables are available:
   • USR1: plastic strain ${\epsilon }_M$
   • USR2: $f^{*}$
   • USR3: admissible stress ${\sigma }_M$
   • USR4: f
   • USR5: $\text{ε}$