/MAT/LAW12 (3D_COMP)

The material becomes nonlinear afterwards. The Tsai-Wu criterion can be set dependent on the plastic work and strain rate in each of the orthotropic directions and in shear to model material hardening. Stress based orthotropic criterion for brittle damage and failure is available. This material is a generalization and improvement of /MAT/LAW14 (COMPSO).

Format

Definitions

Example (Carbon)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
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/MAT/LAW12/1/1
carbon
#              RHO_I
              1.5E-9
#                E11                 E22                 E33
               64000               60000                5000
#               NU12                NU23                NU31
                 .07                 .07                 .07
#                G12                 G23                 G31
                4000                2000                2000
#           sigma_t1            sigma_t2            sigma_t3               delta
                   0                   0                   0                   0
#                  B                   n                fmax               Wpref
                  50                  .5                   0                   0
#          sigma_1yt           sigma_2yt           sigma_1yc           sigma_2yc
                 600                 500                 600                 600
#         sigma_12yt          sigma_12yc          sigma_23yt          sigma_23yc
                 100                 100                  30                  30
#          sigma_3yt           sigma_3yc          sigma_13yt          sigma_13yc
                  50                  50                 100                 100
#              alpha                  Ef                   c          EPS_RATE_0       ICC
                   0                   0                   0                   0         0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

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:(1)

   $${\epsilon }_{11}=\frac{1}{E_{11}}{\sigma }_{11}-\frac{{\nu }_{21}}{E_{22}}{\sigma }_{22}-\frac{{\nu }_{31}}{E_{33}}{\sigma }_{33}$$

   (2)

   $${\epsilon }_{22}=\frac{1}{E_{22}}{\sigma }_{22}-\frac{{\nu }_{12}}{E_{11}}{\sigma }_{11}-\frac{{\nu }_{32}}{E_{33}}{\sigma }_{33}$$

   (3)

   $${\epsilon }_{33}=\frac{1}{E_{33}}{\sigma }_{33}-\frac{{\nu }_{13}}{E_{11}}{\sigma }_{11}-\frac{{\nu }_{23}}{E_{22}}{\sigma }_{22}$$

   (4)

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

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

   Strains

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

   Stresses

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

   Distortions in the corresponding material directions

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

   
   
   Figure 1.

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: (5)
     $$\begin{array}{l}\mathrm{F}\left(\sigma \right)=F_1{\sigma }_1+F_2{\sigma }_2+F_3{\sigma }_3\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}+F_{11}{\sigma }_1^2+F_{22}{\sigma }_2^2+F_{33}{\sigma }_3^2+F_{44}{\sigma }_{12}^2+F_{55}{\sigma }_{23}^2+F_{66}{\sigma }_{31}^2\\ \begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}\begin{array}{c}\end{array}+2F_{12}{\sigma }_1{\sigma }_2+2F_{23}{\sigma }_2{\sigma }_3+2F_{13}{\sigma }_1{\sigma }_3\end{array}$$
   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:(6)
     $$\mathrm{F}\left(W_p^{*},\dot{\epsilon }\right)=\left[1+B{\left(W_p^{*}\right)}^n\right]\cdot \left[1+c\cdot \ln \left(\frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)\right]$$
     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 $\delta$ . (7)

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

   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 $\alpha \cdot E_f\cdot {\epsilon }_{11}$ .