Orthotropic elastoplastic model with isotropic damage - TrabBone3d

Local formulation

The basic equations include an additive decomposition of total strain into elastic (reversible) part and plastic (irreversible) part

(109)\[\veps = \veps_{\rm{e}} + \veps_{\rm{p}}\]

the stress strain law

(110)\[\vsig = \left(1 - \omega\right)\bar{\vsig} = \left(1 - \omega\right)\mbf{D} : \veps_{\rm{e}}\]

the yield function

\[f(\bar{\vsig}, \kappa) = \sqrt{\bar{\vsig} : \mbf{F} : \bar{\vsig}} - \sigma_Y(\kappa)\]

loading-unloading conditions

\[f(\bar{\vsig}, \kappa) \leq 0 \qquad \dot{\lambda} \geq 0 \qquad \dot{\lambda} f(\bar{\vsig}, \kappa) = 0\]

evolution law for plastic strain

\[\dot{\veps}_{\rm{p}} = \dot{\lambda} \frac{\partial f}{\partial \bar{\vsig}}\]

the incremental definition of cumulated plastic strain

\[\dot{\kappa} = \| \epspd \|\]

the law governing the evolution of the damage variable

\[\omega(\kappa) = \omega_c (1 - \mbox{e}^{-a\kappa})\]

and the hardening law

\[\sigma_Y(\kappa) = 1 + \sigma_H (1 - \mbox{e}^{-s\kappa})\]

In the equations above, \(\bar{\vsig}\) is the effective stress tensor, \(\mbf{D}\) is the elastic stiffness tensor, \(f\) is the yield function, \(\lambda\) is the consistency parameter (plastic multiplier), \(\omega\) is the damage variable, \(\sigma_Y\) is the yield stress and \(s\), \(a\), \(\sigma_H\) and \(\omega_c\) are positive material parameters.

Material anisotropy is characterized by the second-order positive definite fabric tensor

(111)\[\mbf{M} = \sum_{i=1}^3 m_i (\mbf{m}_i \otimes \mbf{m}_i)\]

normalized such that Tr\((\mbf{M}) = 3\), \(m_i\) are the eigenvalues and \(\mbf{m}_i\) the eigenvectors.

The eigenvectors of the fabric tensor determine the directions of material orthotropy and the components of the elastic stiffness tensor \(\mbf{D}\) are linked to eigenvalues of the fabric tensor. In the coordinate system aligned with \(m_i\), \(i=1,2,3\),

the stiffness can be presented in Voigt (engineering) notation as

\[\begin{split}\mbf{D} = \left[\begin{array}{cccccc} \frac{1}{E_1} & -\frac{\nu_{12}}{E_1} & -\frac{\nu_{13}}{E_1} & 0 & 0 & 0 \\ -\frac{\nu_{21}}{E_2} & \frac{1}{E_2} & -\frac{\nu_{23}}{E_2} & 0 & 0 & 0 \\ -\frac{\nu_{31}}{E_3} & -\frac{\nu_{32}}{E_3} & \frac{1}{E_3} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G_{23}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G_{13}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G_{12}} \end{array}\right]^{-1}\end{split}\]

where \(E_i = E_0 \rho^k m_i^{2l}\), \(G_{ij} = G_0 \rho^k m_i^l m_j^l\) and \(\nu_{ij} = \nu_0 \frac{m_i^l}{m_j^l}\). Here, \(E_0\), \(G_0\) and \(\nu_0\) are elastic constants characterizing the compact (poreless) material, \(\rho\) is the volume fraction of solid phase and \(k\) and \(l\) are dimensionless exponents.

Similar relations as for the stiffness tensor are also postulated for the components of a fourth-order tensor \(\mbf{F}\) that is used in the yield condition. The yield condition is divided into tensile and compressive parts. Tensor \(\mbf{F}\) is different in each part of the effective stress space. This tensor is denoted \(\mbf{F}^{+}\) in tensile part, characterized by \(\hat{\mbf{N}}:\bar{\vsig} \leq 0\), and \(\mbf{F}^{-}\) in compressive part, characterized by \(\hat{\mbf{N}}:\bar{\vsig} \leq 0\), where

\[\hat{\mbf{N}} = \frac{\sum_{i=1}^{3} m_i^{-2q}}{\sqrt{\sum_{i=1}^3 m_i^{-4q}}}(\mbf{m}_i \otimes \mbf{m}_i)\]

(112)\[\begin{split}\mbf{F^{\pm}}=\left[\begin{array}{cccccc} \frac{1}{\left({\sigma_{1}^{\pm}}\right)^2} & -\frac{\chi_{12}^{\pm}}{\left({\sigma_{1}^{\pm}}\right)^2}&-\frac{\chi_{13}^{\pm}}{\left({\sigma_{1}^{\pm}}\right)^2}& 0& 0&0\\ -\frac{\chi_{21}^{\pm}}{\left({\sigma_{2}^{\pm}}\right)^2} & \frac{1}{\left({\sigma_{2}^{\pm}}\right)^2}&-\frac{\chi_{23}^{\pm}}{\left({\sigma_{2}^{\pm}}\right)^2}& 0& 0&0\\ -\frac{\chi_{31}^{\pm}}{\left({\sigma_{3}^{\pm}}\right)^2} & -\frac{\chi_{32}^{\pm}}{\left({\sigma_{3}^{\pm}}\right)^2}&\frac{1}{\left({\sigma_{3}^{\pm}}\right)^2}& 0& 0&0\\ 0 & 0 & 0 & \frac{1}{\tau_{23}} & 0 & 0\\ 0 & 0 & 0 & 0 & \frac{1}{\tau_{13}} & 0\\ 0 & 0 & 0 & 0 & 0 & \frac{1}{\tau_{12}} \end{array}\right].\end{split}\]

In the equation above \(\sigma_i^{\pm} = \sigma_0^{\pm}\rho^p m_i^{2q}\) is uniaxial yield stress along the \(i\)-th principal axis of orthotropy, \(\tau_{ij} = \tau_0 \rho^p m_i^q m_j^q\) is the shear yield stress in the plane of orthotropy and \(\chi_{ij}^{\pm} = \chi_0^{\pm}\frac{m_i^{2q}}{m_j^{2q}}\) is the so-called interaction coefficient, \(p\) and \(q\) 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

(113)\[\frac{\chi_0^- +1}{(\sigma_0^-)^2} = \frac{\chi_0^+ +1}{(\sigma_0^+)^2}.\]

The model description and parameters are summarized in Table 52.

Table 52 Anisotropic elastoplastic model with isotropic damage - summary.

Description	Anisotropic elastoplastic model with isotropic damage
Record Format	TrabBone3d in d (rn) eps0 (rn) nu0 (rn) mu0 (rn) expk (rn) expl (rn) m1 (rn) m2 (rn) rho (rn) sig0pos (rn) sig0neg (rn) chi0pos (rn) chi0neg (rn) tau0 (rn) plashardfactor (rn) expplashard (rn) expdam (rn) critdam (rn)
Parameters	material number d material density eps0 Young modulus (at zero porosity) nu0 Poisson ratio (at zero porosity) mu0 shear modulus of elasticity (at zero porosity) m1 first eigenvalue of the fabric tensor m2 second eigenvalue of the fabric tensor rho volume fraction of solid phase sig0pos yield stress in tension sig0neg yield stress in compression tau0 yield stress in shear chi0pos interaction coefficient in tension plashardfactor hardening parameter expplashard exponent in hardening law expdam exponent in damage law critdam critical damage expk exponent \(k\) in the expression for elastic stiffness expl exponent \(l\) in the expression for elastic stiffness expq exponent \(q\) in the expression for tensor mbfF expp exponent \(p\) 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

(114)\[\hat{\kappa} = (1-m)\kappa + m\bar{\kappa},\]

where \(\bar{\kappa}\) is the nonlocal contribution to the plastic strain. The nonlocal contribution is given by

(115)\[\bar{\kappa} = \int_{\Omega} \frac{1}{|x-y|} \kappa(y) \, dV(y),\]

where \(\Omega\) is the domain of the material and \(|x-y|\) is the distance between points \(x\) and \(y\). The over-nonlocal parameter \(m\) controls the degree of nonlocality.

The model description and parameters are summarized in Table 53.

Table 53 Nonlocal formulation of anisotropic elastoplastic model with isotropic damage – summary.

Description	Nonlocal anisotropic elastoplastic model with isotropic damage
Record Format	TrabBoneNL3d in d (rn) eps0 (rn) nu0 (rn) mu0 (rn) expk (rn) expl (rn) m1 (rn) m2 (rn) rho (rn) sig0pos (rn) sig0neg (rn) chi0pos (rn) chi0neg (rn) tau0 (rn) plashardfactor (rn) expplashard (rn) expdam (rn) critdam (rn) m (rn) R (rn)
Parameters	material number d material density eps0 Young modulus (at zero porosity) nu0 Poisson ratio (at zero porosity) mu0 shear modulus (at zero porosity) m1 first eigenvalue of the fabric tensor m2 second eigenvalue of the fabric tensor rho volume fraction of the solid phase sig0pos yield stress in tension tau0 yield stress in shear chi0pos interaction coefficient in tension chi0neg interaction coefficient in compression plashardfactor hardening parameter expplashard exponent in the hardening law expdam exponent in the damage law critdam critical damage expk exponent \(k\) in the expression for elastic stiffness expl exponent \(l\) in the expression for elastic stiffness expq exponent \(q\) in the expression for tensor mbfF expp exponent \(p\) in the expression for tensor mbfF m over-nonlocal parameter R nonlocal interaction radius
Supported modes	3dMat