# 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 Figure 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 Equation 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 Figure 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, Equation 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 (Figure 4). The curves give the stress variation in function of a so-called strain ${\epsilon }_{v}$:(9) ${\epsilon }_{\nu }=1-\left(Trace\left[\epsilon \right]\right)$