Orthotropic elastoplastic model with isotropic damage - TrabBone3d ================================================================== .. _local_formulation: Local formulation ----------------- The basic equations include an additive decomposition of total strain into elastic (reversible) part and plastic (irreversible) part .. math:: :label: additiveDecomposition \veps = \veps_{\rm{e}} + \veps_{\rm{p}} the stress strain law .. math:: :label: constitutiveLaw \vsig = \left(1 - \omega\right)\bar{\vsig} = \left(1 - \omega\right)\mbf{D} : \veps_{\rm{e}} the yield function .. math:: f(\bar{\vsig}, \kappa) = \sqrt{\bar{\vsig} : \mbf{F} : \bar{\vsig}} - \sigma_Y(\kappa) loading-unloading conditions .. math:: f(\bar{\vsig}, \kappa) \leq 0 \qquad \dot{\lambda} \geq 0 \qquad \dot{\lambda} f(\bar{\vsig}, \kappa) = 0 evolution law for plastic strain .. math:: \dot{\veps}_{\rm{p}} = \dot{\lambda} \frac{\partial f}{\partial \bar{\vsig}} the incremental definition of cumulated plastic strain .. math:: \dot{\kappa} = \| \epspd \| the law governing the evolution of the damage variable .. math:: \omega(\kappa) = \omega_c (1 - \mbox{e}^{-a\kappa}) and the hardening law .. math:: \sigma_Y(\kappa) = 1 + \sigma_H (1 - \mbox{e}^{-s\kappa}) In the equations above, :math:`\bar{\vsig}` is the effective stress tensor, :math:`\mbf{D}` is the elastic stiffness tensor, :math:`f` is the yield function, :math:`\lambda` is the consistency parameter (plastic multiplier), :math:`\omega` is the damage variable, :math:`\sigma_Y` is the yield stress and :math:`s`, :math:`a`, :math:`\sigma_H` and :math:`\omega_c` are positive material parameters. Material anisotropy is characterized by the second-order positive definite fabric tensor .. math:: :label: fabric \mbf{M} = \sum_{i=1}^3 m_i (\mbf{m}_i \otimes \mbf{m}_i) normalized such that Tr\ :math:`(\mbf{M}) = 3`, :math:`m_i` are the eigenvalues and :math:`\mbf{m}_i` the eigenvectors. The eigenvectors of the fabric tensor determine the directions of material orthotropy and the components of the elastic stiffness tensor :math:`\mbf{D}` are linked to eigenvalues of the fabric tensor. In the coordinate system aligned with :math:`m_i`, :math:`i=1,2,3`, the stiffness can be presented in Voigt (engineering) notation as .. math:: \mbf{D} = \left[\begin{array}{cccccc} \frac{1}{E_1} & -\frac{\nu_{12}}{E_1} & -\frac{\nu_{13}}{E_1} & 0 & 0 & 0 \\ -\frac{\nu_{21}}{E_2} & \frac{1}{E_2} & -\frac{\nu_{23}}{E_2} & 0 & 0 & 0 \\ -\frac{\nu_{31}}{E_3} & -\frac{\nu_{32}}{E_3} & \frac{1}{E_3} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G_{23}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G_{13}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G_{12}} \end{array}\right]^{-1} where :math:`E_i = E_0 \rho^k m_i^{2l}`, :math:`G_{ij} = G_0 \rho^k m_i^l m_j^l` and :math:`\nu_{ij} = \nu_0 \frac{m_i^l}{m_j^l}`. Here, :math:`E_0`, :math:`G_0` and :math:`\nu_0` are elastic constants characterizing the compact (poreless) material, :math:`\rho` is the volume fraction of solid phase and :math:`k` and :math:`l` are dimensionless exponents. Similar relations as for the stiffness tensor are also postulated for the components of a fourth-order tensor :math:`\mbf{F}` that is used in the yield condition. The yield condition is divided into tensile and compressive parts. Tensor :math:`\mbf{F}` is different in each part of the effective stress space. This tensor is denoted :math:`\mbf{F}^{+}` in tensile part, characterized by :math:`\hat{\mbf{N}}:\bar{\vsig} \leq 0`, and :math:`\mbf{F}^{-}` in compressive part, characterized by :math:`\hat{\mbf{N}}:\bar{\vsig} \leq 0`, where .. math:: \hat{\mbf{N}} = \frac{\sum_{i=1}^{3} m_i^{-2q}}{\sqrt{\sum_{i=1}^3 m_i^{-4q}}}(\mbf{m}_i \otimes \mbf{m}_i) .. math:: :label: eq:Fpm \mbf{F^{\pm}}=\left[\begin{array}{cccccc} \frac{1}{\left({\sigma_{1}^{\pm}}\right)^2} & -\frac{\chi_{12}^{\pm}}{\left({\sigma_{1}^{\pm}}\right)^2}&-\frac{\chi_{13}^{\pm}}{\left({\sigma_{1}^{\pm}}\right)^2}& 0& 0&0\\ -\frac{\chi_{21}^{\pm}}{\left({\sigma_{2}^{\pm}}\right)^2} & \frac{1}{\left({\sigma_{2}^{\pm}}\right)^2}&-\frac{\chi_{23}^{\pm}}{\left({\sigma_{2}^{\pm}}\right)^2}& 0& 0&0\\ -\frac{\chi_{31}^{\pm}}{\left({\sigma_{3}^{\pm}}\right)^2} & -\frac{\chi_{32}^{\pm}}{\left({\sigma_{3}^{\pm}}\right)^2}&\frac{1}{\left({\sigma_{3}^{\pm}}\right)^2}& 0& 0&0\\ 0 & 0 & 0 & \frac{1}{\tau_{23}} & 0 & 0\\ 0 & 0 & 0 & 0 & \frac{1}{\tau_{13}} & 0\\ 0 & 0 & 0 & 0 & 0 & \frac{1}{\tau_{12}} \end{array}\right]. In the equation above :math:`\sigma_i^{\pm} = \sigma_0^{\pm}\rho^p m_i^{2q}` is uniaxial yield stress along the :math:`i`-th principal axis of orthotropy, :math:`\tau_{ij} = \tau_0 \rho^p m_i^q m_j^q` is the shear yield stress in the plane of orthotropy and :math:`\chi_{ij}^{\pm} = \chi_0^{\pm}\frac{m_i^{2q}}{m_j^{2q}}` is the so-called interaction coefficient, :math:`p` and :math:`q` are dimensionless exponents and parameters with subscript 0 are related to a fictitious material with zero porosity. The yield surface is continuously differentiable if the parameters values are constrained by the condition .. math:: :label: eq:yield_condition \frac{\chi_0^- +1}{(\sigma_0^-)^2} = \frac{\chi_0^+ +1}{(\sigma_0^+)^2}. The model description and parameters are summarized in :numref:`trabbone_table`. .. table:: Anisotropic elastoplastic model with isotropic damage - summary. :name: trabbone_table +--------------------+----------------------------------------------------------------------------------------------+ | Description | Anisotropic elastoplastic model with isotropic damage | +--------------------+----------------------------------------------------------------------------------------------+ | Record Format | :descitem:`TrabBone3d` :elemparam:`in` :elemparam:`d{rn}` :elemparam:`eps0{rn}` | | | :elemparam:`nu0{rn}` :elemparam:`mu0{rn}` :elemparam:`expk{rn}` :elemparam:`expl{rn}` | | | :elemparam:`m1{rn}` :elemparam:`m2{rn}` :elemparam:`rho{rn}` :elemparam:`sig0pos{rn}` | | | :elemparam:`sig0neg{rn}` :elemparam:`chi0pos{rn}` :elemparam:`chi0neg{rn}` | | | :elemparam:`tau0{rn}` :elemparam:`plashardfactor{rn}` :elemparam:`expplashard{rn}` | | | :elemparam:`expdam{rn}` :elemparam:`critdam{rn}` | +--------------------+----------------------------------------------------------------------------------------------+ | Parameters | - material number | | | - :param:`d` material density | | | - :param:`eps0` Young modulus (at zero porosity) | | | - :param:`nu0` Poisson ratio (at zero porosity) | | | - :param:`mu0` shear modulus of elasticity (at zero porosity) | | | - :param:`m1` first eigenvalue of the fabric tensor | | | - :param:`m2` second eigenvalue of the fabric tensor | | | - :param:`rho` volume fraction of solid phase | | | - :param:`sig0pos` yield stress in tension | | | - :param:`sig0neg` yield stress in compression | | | - :param:`tau0` yield stress in shear | | | - :param:`chi0pos` interaction coefficient in tension | | | - :param:`plashardfactor` hardening parameter | | | - :param:`expplashard` exponent in hardening law | | | - :param:`expdam` exponent in damage law | | | - :param:`critdam` critical damage | | | - :param:`expk` exponent :math:`k` in the expression for elastic stiffness | | | - :param:`expl` exponent :math:`l` in the expression for elastic stiffness | | | - :param:`expq` exponent :math:`q` in the expression for tensor \mbfF | | | - :param:`expp` exponent :math:`p` in the expression for tensor \mbfF | +--------------------+----------------------------------------------------------------------------------------------+ | Supported modes | 3dMat | +--------------------+----------------------------------------------------------------------------------------------+ Nonlocal formulation - TrabBoneNL3d ----------------------------------- The model is regularized by the over-nonlocal formulation with damage driven by a combination of local and nonlocal cumulated plastic strain .. math:: :label: eq:overKappa \hat{\kappa} = (1-m)\kappa + m\bar{\kappa}, where :math:`\bar{\kappa}` is the nonlocal contribution to the plastic strain. The nonlocal contribution is given by .. math:: :label: eq:nonlocal_contribution \bar{\kappa} = \int_{\Omega} \frac{1}{|x-y|} \kappa(y) \, dV(y), where :math:`\Omega` is the domain of the material and :math:`|x-y|` is the distance between points :math:`x` and :math:`y`. The over-nonlocal parameter :math:`m` controls the degree of nonlocality. The model description and parameters are summarized in :numref:`trabboneNl_table`. .. table:: Nonlocal formulation of anisotropic elastoplastic model with isotropic damage -- summary. :name: trabboneNl_table +--------------------+----------------------------------------------------------------------------------------------+ | Description | Nonlocal anisotropic elastoplastic model with isotropic damage | +--------------------+----------------------------------------------------------------------------------------------+ | Record Format | :descitem:`TrabBoneNL3d` :elemparam:`in` :elemparam:`d{rn}` :elemparam:`eps0{rn}` | | | :elemparam:`nu0{rn}` :elemparam:`mu0{rn}` :elemparam:`expk{rn}` :elemparam:`expl{rn}` | | | :elemparam:`m1{rn}` :elemparam:`m2{rn}` :elemparam:`rho{rn}` :elemparam:`sig0pos{rn}` | | | :elemparam:`sig0neg{rn}` :elemparam:`chi0pos{rn}` :elemparam:`chi0neg{rn}` | | | :elemparam:`tau0{rn}` :elemparam:`plashardfactor{rn}` :elemparam:`expplashard{rn}` | | | :elemparam:`expdam{rn}` :elemparam:`critdam{rn}` :elemparam:`m{rn}` :elemparam:`R{rn}` | +--------------------+----------------------------------------------------------------------------------------------+ | Parameters | - material number | | | - :param:`d` material density | | | - :param:`eps0` Young modulus (at zero porosity) | | | - :param:`nu0` Poisson ratio (at zero porosity) | | | - :param:`mu0` shear modulus (at zero porosity) | | | - :param:`m1` first eigenvalue of the fabric tensor | | | - :param:`m2` second eigenvalue of the fabric tensor | | | - :param:`rho` volume fraction of the solid phase | | | - :param:`sig0pos` yield stress in tension | | | - :param:`tau0` yield stress in shear | | | - :param:`chi0pos` interaction coefficient in tension | | | - :param:`chi0neg` interaction coefficient in compression | | | - :param:`plashardfactor` hardening parameter | | | - :param:`expplashard` exponent in the hardening law | | | - :param:`expdam` exponent in the damage law | | | - :param:`critdam` critical damage | | | - :param:`expk` exponent :math:`k` in the expression for elastic stiffness | | | - :param:`expl` exponent :math:`l` in the expression for elastic stiffness | | | - :param:`expq` exponent :math:`q` in the expression for tensor \mbfF | | | - :param:`expp` exponent :math:`p` in the expression for tensor \mbfF | | | - :param:`m` over-nonlocal parameter | | | - :param:`R` nonlocal interaction radius | +--------------------+----------------------------------------------------------------------------------------------+ | Supported modes | 3dMat | +--------------------+----------------------------------------------------------------------------------------------+