/MAT/LAW76 (SAMP)

Format

Definition

Comments

1. This material is compatible with shell, thick shell and solid elements.
2. The material damage can be modeled two ways:
   • ${\epsilon }_p^f$ (start to damage) and ${\epsilon }_p^r$ (element delete):
     $$D=\frac{{\epsilon }_p-{\epsilon }_p^f}{{\epsilon }_p^r-{\epsilon }_p^f}$$

     Where, ${\epsilon }_p$ is the current cumulated plastic strain.

   • Damage function fct_ID₁

     
     
     

     If damage function fct_ID₁ is used, then ${\epsilon }_p^f$ and ${\epsilon }_p^r$ will be ignored.

     The damage variable then influences the stiffness of the material, and thus, the stress tensor computation as:

     $$\sigma ={\sigma }_{eff}\left(1-D\right)$$
     Where,

     ${\sigma }_{eff}$

     Undamaged (effective) tensor stress

     $\sigma$

     Damaged (current) stress tensor

     $D$

     Damage variable

3. Choice of yield surface:
   $$\mathrm{f}=\{\begin{array}{c}{\sigma }_{VM}-A_0-A_1\text{}P-A_2{P}^2\text{}IQUAD=0\\ {\sigma }_{VM}^2-A_0-A\text{}\text{}\text{}\text{}{}_1\text{}P-A_2{P}^{2}IQUAD=1\end{array}$$

   Where,

   $$P=-\frac{{\sigma }_{xx}+{\sigma }_{yy}+{\sigma }_{zz}}{3}$$
   $${\sigma }_{VM}=\sqrt{\frac{3}{2}\left[{\left({\sigma }_{xx}+P\right)}^2+{\left({\sigma }_{yy}+P\right)}^2+{\left({\sigma }_{zz}+P\right)}^2+2{\sigma }_{xy}^2+2{\sigma }_{yz}^2+2{\sigma }_{xz}^2\right]}$$

   $A_0$ , $A_1$ and $A_2$ coefficients are computed from the hardening curve given for tension, compression and shear.

   For von Mises and Drucker-Prager yield surface, IQUAD=0 can be used. However, in some situations, it can be difficult for Radioss to fit the $A_0$ , $A_1$ and $A_2$ coefficients when using IQUAD=0 and an easier fit is obtained by using IQUAD=1.

4. Choice of plasticity formulation:
   • For the no associated plasticity formulation, I_form=0:

     The plastic flow rule function, $g$ , is used to describe plastic strain increment $d{\epsilon }_p=d\lambda \frac{\partial g}{\partial \sigma }$ . In this case $\frac{\partial g}{\partial \sigma }$ is not normal to the yield surface $\text{f}$ and $g$ is not associated with the yield surface $\text{f}$ .

     
     
     

     The plastic flow rule, $g$ , is given by:

     $$\mathrm{g}=\sqrt{{\sigma }_{VM}^2+\alpha P^2}$$

     Materials like soil or rock usually use the no associated plasticity formulation, I_form=0.

   • For the associated plasticity: I_form= 1, $g=\text{f}$

     In this case, the plastic strain rate is a function of the normal vector of the yield surface $\text{f}$ . Materials like metal usually use the associated plasticity formulation.

     $$d{\epsilon }_p=d\lambda \frac{\partial \mathrm{f}}{\partial \sigma }=d\lambda \frac{\partial \mathrm{g}}{\partial \sigma }$$

     
     
     

5. The convexity condition flag ICONV=1 is used to ensure stability in the material law by making sure the yield surface is convex. The yield surface may be hyperbolic for low shear yield values in tension and compression. In this case there is no unique solution and Radioss will update (increase) the shear yield stress to ensure convexity of the yield surface. Therefore, the shear yield stress may be different from input curve.
6. The tables should have a maximum dimension equal to 2. The first entry is the plastic strain and the second entry is the strain rate.
7. User variables USR2, USR3, USR4 are used to output plastic strain components in tension, compression and shear. The output is available both for shells and solids in time history and in animation file.