# 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.

For the case of unidirectional orthotropy (that is, ${E}_{33}={E}_{22}$ 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

- 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
$t+\text{\Delta}t$
by explicit time integration: $${\sigma}_{ij}\left(t+\text{\Delta}t\right)={\sigma}_{ij}\left(t\right)+{D}_{ijkl}\text{\hspace{0.05em}}{d}_{kl}\text{\hspace{0.05em}}\text{\Delta}t$$
- Transform the lamina stress, ${\sigma}_{ij}\left(t+\text{\Delta}t\right)$ , 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:

The inverse relation is generally developed in term of the local material axes and nine independent elastic constants:

- ${E}_{ij}$
- 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:

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:

- ${W}_{p}$
- Plastic work
- $B$
- Hardening parameter
- $n$
- Hardening exponent
- $c$
- Strain rate coefficient

$f\left({W}_{p}\right)$ is limited by a maximum value ${f}_{\mathrm{max}}$ :

If the maximum value is reached the material is failed.

(a) Strain rate effect on ${\sigma}_{\mathrm{max}}$ | (b) No strain rate effect on ${\sigma}_{\mathrm{max}}$ |
---|---|

$\sigma ={\sigma}_{y}\left(1+c.1\text{n}\left(\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\right)\right)$ ${\sigma}_{\mathrm{max}}={\sigma}_{\mathrm{max}}^{0}\left(1+c.1\text{n}\left(\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\right)\right)$ ${f}_{\mathrm{max}}={\left(\frac{{\sigma}_{\mathrm{max}}}{{\sigma}_{y}}\right)}^{2}$ |
$\sigma ={\sigma}_{y}\left(1+c.1\text{n}\left(\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\right)\right)$ ${\sigma}_{\mathrm{max}}={\sigma}_{\mathrm{max}}^{0}$ |

## Unidirectional Orthotropy

LAW 53 in Radioss provides a simple model for unidirectional orthotropic solids with plasticity. The unidirectional orthotropy condition implies:

The orthotropic plasticity behavior is modeled by a modified Tsai-Wu criterion (Orthotropic Plasticity, Equation 4) in which: