====== 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'' |