# LAW12 and LAW14

Describes orthotropic solid material which use the Tsai-Wu formulation. The materials are 3D orthotropic-elastic, before the Tsai-Wu criterion is reached. LAW12 is a generalization and improvement of LAW14.

## Elastic Phase

Both material laws require Young's modulus, shear modulus and Poisson ratio (9 parameters) to describe the material orthotropic in elastic phase.
$\left[\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ {\gamma }_{12}\\ {\gamma }_{23}\\ {\gamma }_{31}\end{array}\right]=\left[\begin{array}{cccccc}\frac{1}{{E}_{11}}& -\frac{{\nu }_{12}}{{E}_{11}}& -\frac{{\nu }_{31}}{{E}_{33}}& 0& 0& 0\\ & \frac{1}{{E}_{22}}& -\frac{{\nu }_{23}}{{E}_{22}}& 0& 0& 0\\ & & \frac{1}{{E}_{33}}& 0& 0& 0\\ & & & \frac{1}{2{G}_{12}}& 0& 0\\ & symm.& & & \frac{1}{2{G}_{23}}& 0\\ & & & & & \frac{1}{2{G}_{31}}\end{array}\right]\cdot \left[\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{12}\\ {\sigma }_{23}\\ {\sigma }_{31}\end{array}\right]$

## Stress Damage

Stress limits (in tensile/compression) are requested for damage. These stress limits could be observed from a tensile test in 3 related directions.
Once stress limit is reached, then damage to material begins (stress reduced with damage parameter $\delta$ ). If Damage (${\text{D}}_{\text{i}}={\text{D}}_{\text{i}}+\text{δ}$ ) reaches D=1, then stress is reduced to 0.

## Tsai-Wu Yield Criteria

In LAW12 (3D_COMP), the Tsai-Wu yield criteria is:

$\begin{array}{l}F\left(\sigma \right)={F}_{1}{\sigma }_{1}+{F}_{2}{\sigma }_{2}+{F}_{3}{\sigma }_{3}+{F}_{11}{\sigma }_{1}^{2}+{F}_{22}{\sigma }_{2}^{2}+{F}_{33}{\sigma }_{3}^{2}+{F}_{44}{\sigma }_{12}^{2}+{F}_{55}{\sigma }_{23}^{2}\\ +{F}_{66}{\sigma }_{31}^{2}+2{F}_{12}{\sigma }_{1}{\sigma }_{2}+2{F}_{23}{\sigma }_{2}{\sigma }_{3}+2{F}_{13}{\sigma }_{1}{\sigma }_{3}\end{array}$

The 12 coefficients of the Tsai-Wu criterion could be determined using the yield stress from the following tests:

Tensile/Compression Tests
• Longitude tensile/compression (in direction 1):
${F}_{1}=-\frac{1}{{\sigma }_{1y}^{c}}+\frac{1}{{\sigma }_{1y}^{t}}$
${F}_{11}=\frac{1}{{\sigma }_{1y}^{c}{\sigma }_{1y}^{t}}$
• Transverse tensile/compression (in direction 2):
${F}_{2}=-\frac{1}{{\sigma }_{2y}^{c}}+\frac{1}{{\sigma }_{2y}^{t}}$
${F}_{22}=\frac{1}{{\sigma }_{2y}^{c}{\sigma }_{2y}^{t}}$
• Transverse tensile/compression (in direction 3):
${F}_{3}=-\frac{1}{{\sigma }_{3y}^{c}}+\frac{1}{{\sigma }_{3y}^{t}}$
${F}_{33}=\frac{1}{{\sigma }_{3y}^{c}{\sigma }_{3y}^{t}}$

Then the interaction coefficients can be calculated as:

${F}_{12}=-\frac{1}{2}\sqrt{\left({F}_{11}{F}_{22}\right)}$
${F}_{23}=-\frac{1}{2}\sqrt{\left({F}_{22}{F}_{33}\right)}$
${F}_{13}=-\frac{1}{2}\sqrt{\left({F}_{11}{F}_{33}\right)}$

Shear Tests
• Shear in plane 1-2 test: ${\sigma }_{12y}^{t}$ and ${\sigma }_{12y}^{c}$ can result from the sample tests below:
${F}_{44}=\frac{1}{{\sigma }_{12y}^{c}{\sigma }_{12y}^{t}}$
• Shear in plane 1-3 ${\sigma }_{31y}^{t}$ and ${\sigma }_{31y}^{c}$ can result from the sample tests below:
${F}_{66}=\frac{1}{{\sigma }_{31y}^{c}{\sigma }_{31y}^{t}}$
• Shear in plane 2-3:
${F}_{55}=\frac{1}{{\sigma }_{23y}^{c}{\sigma }_{23y}^{t}}$
The parameters shown below in LAW12 and LAW14 are requested to calculate the Tsai-Wu criteria:

The yield surface for Tsai-Wu is $F\left(\sigma \right)=1$ . As long as $\left(F\left(\sigma \right)\le 1\right)$ , the material is in the elastic phase. Once $\left(F\left(\sigma \right)>1\right)$ , the yield surface is exceeded and the material is in nonlinear phase.

In these two material laws, the following factors could also be considered for the yield surface.
• Plastic work ${W}_{p}$ with parameter B and n
• Strain rate $\stackrel{˙}{\epsilon }$ with parameter ${\stackrel{˙}{\epsilon }}_{0}$ and c.
$F\left({W}_{p},\stackrel{˙}{\epsilon }\right)=\left(1+B{W}_{p}^{n}\right)\left(1+c\mathrm{ln}\frac{\stackrel{.}{\epsilon }}{{\stackrel{.}{\epsilon }}_{o}}\right)$
Then the yield surface will be $F\left(\sigma \right)=F\left({W}_{p},\stackrel{˙}{\epsilon }\right)$ .
• Material will be in elastic phase, if $F\left(\sigma \right)\le F\left({W}_{p},\stackrel{˙}{\epsilon }\right)$
• Material will be in nonlinear phase, if $F\left(\sigma \right)>F\left({W}_{p},\stackrel{˙}{\epsilon }\right)$

This yield surface $F\left({W}_{p},\stackrel{˙}{\epsilon }\right)$ will be limited with ${f}_{\mathrm{max}}$ ( $F\left({W}_{p},\stackrel{˙}{\epsilon }\right)\le {f}_{\mathrm{max}}$ ), where ${f}_{\mathrm{max}}$ is the maximum value of the Tsai-Wu criterion limit.

${f}_{\mathrm{max}}={\left(\frac{{\sigma }_{\mathrm{max}}}{{\sigma }_{y}}\right)}^{2}$

Depending on parameter B, n, c and ${\stackrel{˙}{\epsilon }}_{0}$ , the yield surface is between 1 and ${f}_{\mathrm{max}}$ .