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====== Orthotropic materials ======

===== ElastOrthoHypoMaterial =====

=== Description ===

Linear elastic orthotropic material.

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


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

=== Parameters ===

^   Name                                             ^     Metafor Code     ^
| Density                              |   ''MASS_DENSITY''       |
| Young Modulus $E_1$                |  ''YOUNG_MODULUS_1''   |
| Young Modulus $E_2$                |  ''YOUNG_MODULUS_2''   |
| Young Modulus $E_3$                |  ''YOUNG_MODULUS_3''   |
| Poisson ratio $\nu_{12}$    |  ''POISSON_RATIO_12''  |
| Poisson ratio $\nu_{13}$    |  ''POISSON_RATIO_13''  |
| Poisson ratio $\nu_{23}$    |  ''POISSON_RATIO_23''  |
| Shear modulus $G_{12}$      |  ''SHEAR_MODULUS_12''  |
| Shear modulus $G_{13}$      |  ''SHEAR_MODULUS_13''  |
| Shear modulus $G_{23}$      |  ''SHEAR_MODULUS_23''  |
| Objectivity method  \\ (Jaumann = 0, GreenNaghdi = 1)  |    ''OBJECTIVITY''       |
| Orthotropic axis                                    |    ''ORTHO_AX1_X''       |
| Orthotropic axis                                    |    ''ORTHO_AX1_Y''       |
| Orthotropic axis                                    |    ''ORTHO_AX1_Z''       |
| Orthotropic axis                                    |    ''ORTHO_AX2_X''       |
| Orthotropic axis                                    |    ''ORTHO_AX2_Y''       |
| Orthotropic axis                                    |    ''ORTHO_AX2_Z''       |

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.

===== TmElastOrthoHypoMaterial =====
:!: Metafor version >=3536
=== Description ===
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 $J$.

=== Parameters ===
^   Name                                             ^     Metafor Code     ^  Dependency ^
| Density                              |   ''MASS_DENSITY''       |  ''TO/TM''   |
| Young Modulus $E_1$                |  ''YOUNG_MODULUS_1''   |  ''TO/TM''   |
| Young Modulus $E_2$                |  ''YOUNG_MODULUS_2''   |  ''TO/TM''   |
| Young Modulus $E_3$                |  ''YOUNG_MODULUS_3''   |  ''TO/TM''   |
| Poisson ratio $\nu_{12}$    |  ''POISSON_RATIO_12''  |  ''TO/TM''   |
| Poisson ratio $\nu_{13}$    |  ''POISSON_RATIO_13''  |  ''TO/TM''   |
| Poisson ratio $\nu_{23}$    |  ''POISSON_RATIO_23''  |  ''TO/TM''   |
| Shear modulus $G_{12}$      |  ''SHEAR_MODULUS_12''  |  ''TO/TM''   |
| Shear modulus $G_{13}$      |  ''SHEAR_MODULUS_13''  |  ''TO/TM''   |
| Shear modulus $G_{23}$      |  ''SHEAR_MODULUS_23''  |  ''TO/TM''   |
| Objectivity method  \\ (Jaumann = 0, GreenNaghdi = 1)  |    ''OBJECTIVITY''    |  -   |
| Orthotropic axis                                    |    ''ORTHO_AX1_X''       |  -   |
| Orthotropic axis                                    |    ''ORTHO_AX1_Y''       |  -   |
| Orthotropic axis                                    |    ''ORTHO_AX1_Z''       |  -   |
| Orthotropic axis                                    |    ''ORTHO_AX2_X''       |  -   |
| Orthotropic axis                                    |    ''ORTHO_AX2_Y''       |  -   |
| Orthotropic axis                                    |    ''ORTHO_AX2_Z''       |  -   |
| Thermal Expansion $\alpha_1$                |  ''THERM_EXPANSION_1''   |  ''TO/TM''   |
| Thermal Expansion $\alpha_2$                |  ''THERM_EXPANSION_2''   |  ''TO/TM''   |
| Thermal Expansion $\alpha_3$                |  ''THERM_EXPANSION_3''   |  ''TO/TM''   |
| Conductivity $K_1$                |  ''CONDUCTIVITY_1''   |  ''TO/TM''   |
| Conductivity $K_2$                |  ''CONDUCTIVITY_2''   |  ''TO/TM''   |
| Conductivity $K_3$                |  ''CONDUCTIVITY_3''   |  ''TO/TM''   |
| Heat Capacity $C_p$                |  ''HEAT_CAPACITY''   |  ''TO/TM''   |
| Dissipated thermoelastic power fraction $\eta_e$               |  ''DISSIP_TE''   |  -   |
| Dissipated (visco)plastic power fraction (Taylor-Quinney factor)                |  ''DISSIP_TQ''   |  -   |

===== EpIsoHOrthoHypoMaterial =====

=== Description ===

Elastoplastic orthotropic material with isotropic hardening.

The elastic part follows the same relation as the [[#orthoelasthypomaterial|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 [[doc:user:elements:volumes:isohard#comp1dirplasticcriterion|criterion]] for long-fiber composites. 

=== Parameters ===

^   Name                                             ^     Metafor Code     ^
| Density                                              |   ''MASS_DENSITY''       |
| Young Modulus $E_1$                |  ''YOUNG_MODULUS_1''   |
| Young Modulus $E_2$                |  ''YOUNG_MODULUS_2''   |
| Young Modulus $E_3$                |  ''YOUNG_MODULUS_3''   |
| Poisson ratio $\nu_{12}$   |  ''POISSON_RATIO_12''  |
| Poisson ratio $\nu_{13}$    |  ''POISSON_RATIO_13''  |
| Poisson ratio $\nu_{23}$    |  ''POISSON_RATIO_23''  |
| Shear modulus $G_{12}$      |  ''SHEAR_MODULUS_12''  |
| Shear modulus $G_{13}$      |  ''SHEAR_MODULUS_13''  |
| Shear modulus $G_{23}$      |  ''SHEAR_MODULUS_23''  |
| Number of the material law which defines the yield stress $\sigma_{yield}$    |      ''YIELD_NUM''       |
| Number of the plastic criterion                     | ''PLASTICCRITERION_NUM'' |
| Objectivity method \\ (Jaumann = 0, GreenNaghdi = 1)|    ''OBJECTIVITY''       |
| Orthotropic axis                                    |    ''ORTHO_AX1_X''       |
| Orthotropic axis                                    |    ''ORTHO_AX1_Y''       |
| Orthotropic axis                                    |    ''ORTHO_AX1_Z''       |
| Orthotropic axis                                    |    ''ORTHO_AX2_X''       |
| Orthotropic axis                                    |    ''ORTHO_AX2_Y''       |
| Orthotropic axis                                    |    ''ORTHO_AX2_Z''       |

===== TmEpIsoHOrthoHypoMaterial =====
:!: Metafor version >=3536
=== Description ===
Thermomechanical elastoplastic orthotropic material with isotropic hardening. The thermal part of the law is similar to the one of the [[#orthoelasthypomaterial|linear thermoelastic orthotropic material]].

=== Parameters ===
^   Name                                             ^     Metafor Code     ^  Dependency ^
| Density                              |   ''MASS_DENSITY''       |  ''TO/TM''   |
| Young Modulus $E_1$                |  ''YOUNG_MODULUS_1''   |  ''TO/TM''   |
| Young Modulus $E_2$                |  ''YOUNG_MODULUS_2''   |  ''TO/TM''   |
| Young Modulus $E_3$                |  ''YOUNG_MODULUS_3''   |  ''TO/TM''   |
| Poisson ratio $\nu_{12}$    |  ''POISSON_RATIO_12''  |  ''TO/TM''   |
| Poisson ratio $\nu_{13}$    |  ''POISSON_RATIO_13''  |  ''TO/TM''   |
| Poisson ratio $\nu_{23}$    |  ''POISSON_RATIO_23''  |  ''TO/TM''   |
| Shear modulus $G_{12}$      |  ''SHEAR_MODULUS_12''  |  ''TO/TM''   |
| Shear modulus $G_{13}$      |  ''SHEAR_MODULUS_13''  |  ''TO/TM''   |
| Shear modulus $G_{23}$      |  ''SHEAR_MODULUS_23''  |  ''TO/TM''   |
| Number of the material law which defines the yield stress $\sigma_{yield}$    |      ''YIELD_NUM''       |  -   |
| Number of the plastic criterion                     | ''PLASTICCRITERION_NUM'' |  -   |
| Objectivity method  \\ (Jaumann = 0, GreenNaghdi = 1)  |    ''OBJECTIVITY''    |  -   |
| Orthotropic axis                                    |    ''ORTHO_AX1_X''       |  -   |
| Orthotropic axis                                    |    ''ORTHO_AX1_Y''       |  -   |
| Orthotropic axis                                    |    ''ORTHO_AX1_Z''       |  -   |
| Orthotropic axis                                    |    ''ORTHO_AX2_X''       |  -   |
| Orthotropic axis                                    |    ''ORTHO_AX2_Y''       |  -   |
| Orthotropic axis                                    |    ''ORTHO_AX2_Z''       |  -   |
| Thermal Expansion $\alpha_1$                |  ''THERM_EXPANSION_1''   |  ''TO/TM''   |
| Thermal Expansion $\alpha_2$                |  ''THERM_EXPANSION_2''   |  ''TO/TM''   |
| Thermal Expansion $\alpha_3$                |  ''THERM_EXPANSION_3''   |  ''TO/TM''   |
| Conductivity $K_1$                |  ''CONDUCTIVITY_1''   |  ''TO/TM''   |
| Conductivity $K_2$                |  ''CONDUCTIVITY_2''   |  ''TO/TM''   |
| Conductivity $K_3$                |  ''CONDUCTIVITY_3''   |  ''TO/TM''   |
| Heat Capacity $C_p$                |  ''HEAT_CAPACITY''   |  ''TO/TM''   |
| Dissipated thermoelastic power fraction $\eta_e$               |  ''DISSIP_TE''   |  -   |
| Dissipated (visco)plastic power fraction (Taylor-Quinney factor)                |  ''DISSIP_TQ''   |  -   |
===== DamageEpIsoHOrthoHypoMaterial =====

=== Description ===

Elastoplastic orthotropic material with isotropic hardening and damage.

The elastoplastic part has the same characteristics as the [[#episohorthohypomaterial|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 [[doc:user:elements:volumes:ortho_continuousdamage|basic laws]]

=== Parameters ===

^   Name                                             ^     Metafor Code     ^
| Density                                              |   ''MASS_DENSITY''       |
| Young Modulus $E_1$                |  ''YOUNG_MODULUS_1''   |
| Young Modulus $E_2$                |  ''YOUNG_MODULUS_2''   |
| Young Modulus $E_3$                |  ''YOUNG_MODULUS_3''   |
| Poisson ratio $\nu_{12}$    |  ''POISSON_RATIO_12''  |
| Poisson ratio $\nu_{13}$    |  ''POISSON_RATIO_13''  |
| Poisson ratio $\nu_{23}$    |  ''POISSON_RATIO_23''  |
| Shear modulus $G_{12}$      |  ''SHEAR_MODULUS_12''  |
| Shear modulus $G_{13}$      |  ''SHEAR_MODULUS_13''  |
| Shear modulus $G_{23}$      |  ''SHEAR_MODULUS_23''  |
| Number of the material law which defines the yield stress $\sigma_{yield}$    |      ''YIELD_NUM''       |
| Number of the plastic criterion                     |  ''PLASTICCRITERION_NUM''  |
| Number of the damage law                            |  ''DAMAGE_NUM''          |
| Maximal value of damage variables (failure)         |  ''DAMAGE_MAX''          |
| Objectivity method \\ (Jaumann = 0, GreenNaghdi = 1)|    ''OBJECTIVITY''       |
| Orthotropic axis                                    |    ''ORTHO_AX1_X''       |
| Orthotropic axis                                    |    ''ORTHO_AX1_Y''       |
| Orthotropic axis                                    |    ''ORTHO_AX1_Z''       |
| Orthotropic axis                                    |    ''ORTHO_AX2_X''       |
| Orthotropic axis                                    |    ''ORTHO_AX2_Y''       |
| Orthotropic axis                                    |    ''ORTHO_AX2_Z''       |