# 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

## Stress Damage

## Tsai-Wu Yield Criteria

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

- Longitude tensile/compression (in direction 1):
(3) $${F}_{1}=-\frac{1}{{\sigma}_{1y}^{c}}+\frac{1}{{\sigma}_{1y}^{t}}$$(4) $${F}_{11}=\frac{1}{{\sigma}_{1y}^{c}{\sigma}_{1y}^{t}}$$ - Transverse tensile/compression (in direction 2):
(5) $${F}_{2}=-\frac{1}{{\sigma}_{2y}^{c}}+\frac{1}{{\sigma}_{2y}^{t}}$$(6) $${F}_{22}=\frac{1}{{\sigma}_{2y}^{c}{\sigma}_{2y}^{t}}$$ - Transverse tensile/compression (in direction 3):
(7) $${F}_{3}=-\frac{1}{{\sigma}_{3y}^{c}}+\frac{1}{{\sigma}_{3y}^{t}}$$(8) $${F}_{33}=\frac{1}{{\sigma}_{3y}^{c}{\sigma}_{3y}^{t}}$$

- Shear in plane 1-2 test:
${\sigma}_{12y}^{t}$
and
${\sigma}_{12y}^{c}$
can result from the sample tests below:
(12) $${F}_{44}=\frac{1}{{\sigma}_{12y}^{c}{\sigma}_{12y}^{t}}$$ - Shear in plane 1-3
${\sigma}_{31y}^{t}$
and
${\sigma}_{31y}^{c}$
can result from the sample tests below:
(13) $${F}_{66}=\frac{1}{{\sigma}_{31y}^{c}{\sigma}_{31y}^{t}}$$ - Shear in plane 2-3:
(14) $${F}_{55}=\frac{1}{{\sigma}_{23y}^{c}{\sigma}_{23y}^{t}}$$

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

- Plastic work
${W}_{p}$
with parameter
`B`and`n` - Strain rate
$\dot{\epsilon}$
with parameter
${\dot{\epsilon}}_{0}$
and
`c`.(15) $$F\left({W}_{p},\dot{\epsilon}\right)=\left(1+B{W}_{p}^{n}\right)\left(1+c\mathrm{ln}\frac{\stackrel{.}{\epsilon}}{{\stackrel{.}{\epsilon}}_{o}}\right)$$

- Material will be in elastic phase, if $F(\sigma )\le F\left({W}_{p},\dot{\epsilon}\right)$
- Material will be in nonlinear phase, if $F(\sigma )>F\left({W}_{p},\dot{\epsilon}\right)$

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

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

`B`,

`n`,

`c`and ${\dot{\epsilon}}_{0}$ , the yield surface is between 1 and ${f}_{\mathrm{max}}$ .