Orthotropic materials

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.

Metafor version >=3536

Linear thermoelastic orthotropic material with orthotropic thermal conduction law.

Thermal conduction writes in the orthotropic frame $$ \boldsymbol{K}~\nabla T = \left[ \begin{array}{c c c} K_1 & 0 & 0 \\ 0 & K_2 & 0 \\ 0 & 0 & K_3 \end{array} \right] \nabla T, $$ where $\boldsymbol{K}$ is the orthotropic conduction matrix (in material axes) and $\nabla T$ is the temperature gradient.

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 fraction of heat dissipated thermoelastic energy $\eta_{te}$ and determinant of the Jacobian matrix $J$.

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.

Metafor version >=3536

Thermomechanical elastoplastic orthotropic material with isotropic hardening. The thermal part of the law is similar to the one of the linear thermoelastic orthotropic material.

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