/MAT/LAW12 (3D_COMP)

Format

Definition

Comments

1. This material requires an orthotropic solid property (/PROP/TYPE6 (SOL_ORTH), /PROP/TYPE21 (TSH_ORTH), or /PROP/TYPE22 (TSH_COMP). It can only be used with solid elements for a 3-dimensional analysis. This law is compatible with 10-node tetrahedron and 4-node tetrahedron elements. The orthotropic material directions are set in the property entries.
2. Stress-strain relation in elastic phase.

   The stresses and strains are related as:

   $${\epsilon }_{11}=\frac{1}{E_{11}}{\sigma }_{11}-\frac{{\nu }_{21}}{E_{22}}{\sigma }_{22}-\frac{{\nu }_{31}}{E_{33}}{\sigma }_{33}$$
   $${\epsilon }_{22}=\frac{1}{E_{22}}{\sigma }_{22}-\frac{{\nu }_{12}}{E_{11}}{\sigma }_{11}-\frac{{\nu }_{32}}{E_{33}}{\sigma }_{33}$$
   $${\epsilon }_{33}=\frac{1}{E_{33}}{\sigma }_{33}-\frac{{\nu }_{13}}{E_{11}}{\sigma }_{11}-\frac{{\nu }_{23}}{E_{22}}{\sigma }_{22}$$
   $$\begin{array}{ll}{\gamma }_{12}=\frac{1}{2G_{12}}{\sigma }_{12}\hfill & \frac{{\nu }_{21}}{E_{22}}=\frac{{\nu }_{12}}{E_{11}}\hfill \\ {\gamma }_{23}=\frac{1}{2G_{23}}{\sigma }_{23}\hfill & \frac{{\nu }_{32}}{E_{33}}=\frac{{\nu }_{23}}{E_{22}}\hfill \\ {\gamma }_{31}=\frac{1}{2G_{31}}{\sigma }_{31}\hfill & \frac{{\nu }_{13}}{E_{11}}=\frac{{\nu }_{31}}{E_{33}}\hfill \end{array}$$
   Where,

   ${\epsilon }_{{}_{ij}}$

   Strains

   ${\sigma }_{{}_{ij}}$

   Stresses

   ${\gamma }_{{}_{12}}$ , ${\gamma }_{{}_{23}}$ and ${\gamma }_{{}_{31}}$

   Distortions in the corresponding material directions

   For example, for ${\gamma }_{{}_{12}}$ :

   
   
   

3. Tsai-Wu criterion:
   The material is assumed to be elastic until the Tsai-Wu criterion is fulfilled. After exceeding the Tsai-Wu criterion limit $\mathrm{F}(W_p^{*},\dot{\epsilon })$ , the material becomes nonlinear:

   • If $\mathrm{F}(\sigma )<\mathrm{F}(W_p^{*},\dot{\epsilon })$ : elastic
   • If $\mathrm{F}(\sigma )>\mathrm{F}(W_p^{*},\dot{\epsilon })$ : nonlinear
   Where,

   • Stress $\mathrm{F}\left(\sigma \right)$ in element for Tsai-Wu criterion computed as:
     $$\begin{array}{l}\mathrm{F}\left(\sigma \right)=F_1{\sigma }_1+F_2{\sigma }_2+F_3{\sigma }_3\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}+F_{11}{\sigma }_1^2+F_{22}{\sigma }_2^2+F_{33}{\sigma }_3^2+F_{44}{\sigma }_{12}^2+F_{55}{\sigma }_{23}^2+F_{66}{\sigma }_{31}^2\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}+2F_{12}{\sigma }_1{\sigma }_2+2F_{23}{\sigma }_2{\sigma }_3+2F_{13}{\sigma }_1{\sigma }_3\end{array}$$
   The coefficients of the Tsai-Wu criterion are determined from the limiting stresses when the material becomes nonlinear in directions 1, 2, 3 or 12, 23, 31 (shear) in compression or tension as:

   $F_1=-\frac{1}{{\sigma }_{1y}^c}+\frac{1}{{\sigma }_{1y}^t}$	$F_2=-\frac{1}{{\sigma }_{2y}^c}+\frac{1}{{\sigma }_{2y}^t}$	$F_3=-\frac{1}{{\sigma }_{3y}^c}+\frac{1}{{\sigma }_{3y}^t}$
   $F_{11}=\frac{1}{{\sigma }_{1y}^c{\sigma }_{1y}^t}$	$F_{22}=\frac{1}{{\sigma }_{2y}^c{\sigma }_{2y}^t}$	$F_{33}=\frac{1}{{\sigma }_{3y}^c{\sigma }_{3y}^t}$
   $F_{44}=\frac{1}{{\sigma }_{12y}^c{\sigma }_{12y}^t}$	$F_{55}=\frac{1}{{\sigma }_{23y}^c{\sigma }_{23y}^t}$	$F_{66}=\frac{1}{{\sigma }_{31y}^c{\sigma }_{31y}^t}$
   $F_{12}=-\frac{1}{2}\sqrt{\left(F_{11}{F}_{22}\right)}$	$F_{23}=-\frac{1}{2}\sqrt{\left(F_{22}{F}_{33}\right)}$	$F_{13}=-\frac{1}{2}\sqrt{\left(F_{11}{F}_{33}\right)}$

   • $\mathrm{F}\left(W_p^{*},\dot{\epsilon }\right)$ is the variable Tsai-Wu criterion limit defined:
     $$\mathrm{F}\left(W_p^{*},\dot{\epsilon }\right)=\left[1+B{\left(W_p^{*}\right)}^n\right]\cdot \left[1+c\cdot \ln \left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)\right]$$
     Where,

     $W_p{}^{ref}$

     Reference plastic work

     $W_p^{*}=\frac{W_p}{W_p^{ref}}$

     Relative plastic work

     $B$

     Plastic hardening parameter

     $n$

     Plastic hardening exponent

     ${\dot{\epsilon }}_0$

     Reference true strain rate

     $c$

     Strain rate coefficient

     $\mathrm{F}\left(W_p^{*},\dot{\epsilon }\right)$ the maximum value of the Tsai-Wu criterion limit depends on ICC:

     If ICC=1

     $f_{\max }\cdot \left(1+c\cdot \ln \left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_o}\right)\right)$

     If ICC=2

     $f_{\max }$

     Where, $f_{\max }={\left(\frac{{\sigma }_{\max }}{{\sigma }_y}\right)}^2$

4. Stress damage:

   When the limiting stress value of ${\sigma }_{ti}$ is reached in tension, the corresponding stress value is scaled as ${\sigma }_i^{\mathit{reduced}}=\left(1-D_i\right){\sigma }_{ti}$ . The value of damage $D_i$ is updated on each time step with incremental damage parameter $\delta$ .

   $$D_i={\displaystyle \sum _i{\delta }_i}$$

   After $D_i$ reaches the value of 1, the stress in corresponding direction is set to 0. The damage is irreversible, that is, if a value of $D_i$ is attained the material will not reach any lower damage value.

5. Fiber reinforcement:

   These parameters allow you to define additional fiber reinforcement in the 11 direction. Additional stress in direction 11 will be added equal to $\alpha \cdot E_f\cdot {\epsilon }_{11}$ .