Orthotropic materials

Linear orthotropic material.

The strain-stress relation in the orthotropic frame is written as:

$$ \left[ \begin{array}{c} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ \varepsilon_{23} \\ \varepsilon_{31} \\ \varepsilon_{12} \end{array} \right] = \left[ \begin{array}{cccccc} \frac{1}{E_{1}} & -\frac{\nu_{12}}{E_{1}} & -\frac{\nu_{13}}{E_{1}} & 0 & 0 & 0 \\ -\frac{\nu_{12}}{E_{1}} & \frac{1}{E_{2}} & -\frac{\nu_{23}}{E_{2}} & 0 & 0 & 0 \\ -\frac{\nu_{13}}{E_{1}} & -\frac{\nu_{23}}{E_{2}} & \frac{1}{E_{3}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2\,G_{23}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2\,G_{13}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2\,G_{12}} \end{array} \right] \left[ \begin{array}{c} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{array} \right] $$

Only the first two orthotropic axes are computed using ORTHO_AX{1,2}_{X,Y,Z}, the third one being computed as the cross product of the first two.

Elastoplastic orthotropic material with isotropic hardening.

The elastic part follows the same relation as the linear orthotropic material.

As in the isotropic case, the yield stress verifies the constraint:

$$ f=\overline{\sigma}-\sigma_{yield}=0 $$

where $\overline{\sigma}$ is an equivalent stress, specific to orthotropic materials. See for example the criterion for long-fiber composites.

Elastoplastic orthotropic material with isotropic hardening and damage.

The elastoplastic part has the same characteristics as the elastoplastic orthotropic material

The damage part consists in a material softening governed by one or several damage variables $d_{ij}$, whose value is included between 0 and 1. Typically, a modulus equal to $E_i$ before damage becomes $(1-d_i)\,E_i$ once damage appears, but not always. The way damage is induced depends on the law defined by the parameter DAMAGE_NUM. See for example the basic laws