Linear elastic 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.

Linear thermoelastic orthotropic material with orthotropic thermal conduction law.

Thermal conduction writes in the orthotropic frame $$\boldsymbol{q}_{cond} = \boldsymbol{K} \nabla T = \left[ \begin{array}{c c c} K_1 & 0 & 0 \\ 0 & K_2 & 0 \\ 0 & 0 & K_3 \end{array} \right] \nabla T,$$ where

Linear thermoelasticity in the orthotropic frame writes $$\boldsymbol{\sigma} = \boldsymbol{\sigma}_0 + \mathbb{H} : (\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^{th}) = \boldsymbol{\sigma}_0 + \mathbb{H} : (\boldsymbol{\varepsilon} - \boldsymbol{\alpha} \Delta T),$$ with stress tensor $\boldsymbol{\sigma}$, initial stress tensor $\boldsymbol{\sigma}_0$, Hooke's tensor $\mathbb{H}$, strain tensor (mechanical) $\boldsymbol{\varepsilon}$, and thermal strain tensor $\boldsymbol{\varepsilon}^{th}$, which is the product of the temperature variation $\Delta T$ and the thermal expansion (symmetric) tensor $\boldsymbol{\alpha}$.

Thermoelastic dissipation term $\dot{W}^{te}$ is given by the general (anisotropic) relation $$\dot{W}^{te} = -\eta_{te} \left(\sum_{i=1}^3 \sum_{j=1}^3 \mathbb{H}_{ijkl} \alpha_{kl} \right)T \frac{\dot{J}}{J},$$ with

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