# Elastic-Plastic Orthotropic Composite Solids

The material LAW14 (COMPSO) in Radioss allows to simulate orthotropic elasticity, Tsai-Wu plasticity with damage, brittle rupture and strain rate effects. The constitutive law applies to only one layer of lamina. Therefore, each layer needs to be modeled by a solid mesh. A layer is characterized by one direction of the fiber or material. The overall behavior is assumed to be elasto-plastic orthotropic.

Direction 1 is the fiber direction, defined with respect to the local reference frame $\stackrel{\to }{r},\stackrel{\to }{s},\stackrel{\to }{t}$ as shown in 図 1.

For the case of unidirectional orthotropy (i.e. ${E}_{33}={E}_{22}$ and ${G}_{31}={G}_{12}$) the material LAW53 in Radioss allows to simulate an orthotropic elastic-plastic behavior by using a modified Tsai-Wu criteria.

## Linear Elasticity

When the lamina has a purely linear elastic behavior, the stress calculation algorithm:
• Transform the lamina stress, ${\sigma }_{ij}\left(t\right)$, and strain rate, ${d}_{ij}$, from global reference frame to fiber reference frame.
• Compute lamina stress at time $t+\text{Δ}t$ by explicit time integration:
(1) ${\sigma }_{ij}\left(t+\text{Δ}t\right)={\sigma }_{ij}\left(t\right)+{D}_{ijkl}\text{ }{d}_{kl}\text{ }\text{Δ}t$
• Transform the lamina stress, ${\sigma }_{ij}\left(t+\text{Δ}t\right)$, back to global reference frame.
The elastic constitutive matrix $C$ of the lamina relates the non-null components of the stress tensor to those of strain tensor:(2) $\left\{\sigma \right\}=\left[D\right]\left\{\epsilon \right\}$
The inverse relation is generally developed in term of the local material axes and nine independent elastic constants:(3) $\left\{\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ {\epsilon }_{33}\\ {\gamma }_{12}\\ {\gamma }_{23}\\ {\gamma }_{31}\end{array}\right\}=\left[\begin{array}{cccccc}\frac{1}{{E}_{11}}& -\frac{{\nu }_{21}}{{E}_{22}}& -\frac{{\nu }_{31}}{{E}_{33}}& 0& 0& 0\\ & \frac{1}{{E}_{22}}& -\frac{{\nu }_{32}}{{E}_{33}}& 0& 0& 0\\ & & \frac{1}{{E}_{33}}& 0& 0& 0\\ & & & \frac{1}{2{G}_{12}}& 0& 0\\ & Symm.& & & \frac{1}{2{G}_{23}}& 0\\ & & & & & \frac{1}{2{G}_{31}}\end{array}\right]\left\{\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{12}\\ {\sigma }_{23}\\ {\sigma }_{31}\end{array}\right\}$
Where,
${E}_{ij}$
Young's modulus
${G}_{ij}$
Shear modulus
${\nu }_{ij}$
Poisson's ratios
${\gamma }_{ij}$
Strain components due to the distortion

## Orthotropic Plasticity

Lamina yield surface defined by Tsai-Wu yield criteria is used for each layer:(4) $\begin{array}{l}F=f\left({W}_{p}\right)={F}_{1}{\sigma }_{1}+{F}_{2}{\sigma }_{2}+{F}_{3}{\sigma }_{3}+{F}_{11}{\sigma }_{1}^{2}+{F}_{22}{\sigma }_{2}^{2}+{F}_{33}{\sigma }_{3}^{2}+{F}_{44}{\sigma }_{12}^{2}\\ +{F}_{55}{\sigma }_{23}^{2}+{F}_{66}{\sigma }_{31}^{2}+2{F}_{12}{\sigma }_{1}{\sigma }_{2}+2{F}_{23}{\sigma }_{2}{\sigma }_{3}+2{F}_{13}{\sigma }_{1}{\sigma }_{3}\end{array}$

with:

${F}_{i}=-\frac{1}{{\sigma }_{iy}^{c}}+\frac{1}{{\sigma }_{iy}^{t}}$ ($i$=1,2,3);

${F}_{11}=\frac{1}{{\sigma }_{1y}^{c}{\sigma }_{1y}^{t}}$; ${F}_{22}=\frac{1}{{\sigma }_{2y}^{c}{\sigma }_{2y}^{t}}$; ${F}_{33}=\frac{1}{{\sigma }_{3y}^{c}{\sigma }_{3y}^{t}}$;

${F}_{44}=\frac{1}{{\sigma }_{12y}^{c}{\sigma }_{12y}^{t}}$; ${F}_{55}=\frac{1}{{\sigma }_{23y}^{c}{\sigma }_{23y}^{t}}$; ${F}_{66}=\frac{1}{{\sigma }_{31y}^{c}{\sigma }_{31y}^{t}}$;

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

Where, ${\sigma }_{i}$ is the yield stress in direction $i$, $c$ and $t$ denote respectively for compression and tension. $f\left({W}_{p}\right)$ represents the yield envelope evolution during work hardening with respect to strain rate effects:(5) $f\left({W}_{p}\right)=\left(1+B\cdot {W}_{p}^{n}\right)\left(1+c.1\text{n}\left(\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)\right)$
Where,
${W}_{p}$
Plastic work
$B$
Hardening parameter
$n$
Hardening exponent
$c$
Strain rate coefficient
$f\left({W}_{p}\right)$ is limited by a maximum value ${f}_{\mathrm{max}}$:(6) $f\left({W}_{p}\right)\le {f}_{\mathrm{max}}={\left(\frac{{\sigma }_{\mathrm{max}}}{{\sigma }_{y}}\right)}^{2}$

If the maximum value is reached the material is failed.

In 式 5, the strain rate effects on the evolution of yield envelope. However, it is also possible to take into account the strain rate $\stackrel{˙}{\epsilon }$ effects on the maximum stress ${\sigma }_{\mathrm{max}}$ as shown in 図 3.

## Unidirectional Orthotropy

LAW 53 in Radioss provides a simple model for unidirectional orthotropic solids with plasticity. The unidirectional orthotropy condition implies:(7) $\begin{array}{l}{E}_{33}={E}_{22}\\ {G}_{31}={G}_{12}\end{array}$
The orthotropic plasticity behavior is modeled by a modified Tsai-Wu criterion (Orthotropic Plasticity, 式 4) in which:(8) ${F}_{12}=\frac{2}{{\left({\sigma }_{y}^{45c}\right)}^{2}}-\frac{1}{2}\left({F}_{11}+{F}_{22}+{F}_{44}\right)+\frac{{F}_{1}+{F}_{2}}{{\sigma }_{y}^{45c}}$
Where, ${\sigma }^{45}{}_{y}^{c}$ is yield stress in 45° unidirectional test. The yield stresses in direction 11, 22, 12, 13 and 45° are defined by independent curves obtained by unidirectional tests (図 4). The curves give the stress variation in function of a so-called strain ${\epsilon }_{v}$:(9) ${\epsilon }_{\nu }=1-\left(Trace\left[\epsilon \right]\right)$