Local formulation

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

$$\mbox{\boldmath$\varepsilon$} = \mbox{\boldmath$\varepsilon$} _{\rm {e}} + \mbox{\boldmath$\varepsilon$} _{\rm {p}},$$ (206)

the stress strain law

$$\mbox{\boldmath$\sigma$} = \left(1-\omega\right)\bar{\mbox{\boldmath$\sigma$}}=\left(1-\omega\right)\mbox{\boldmath$D$}:{\mbox{\boldmath$\varepsilon$}}_{\rm {e}},$$ (207)

the yield function

$$f(\bar{\mbox{\boldmath$\sigma$}},\kappa) = \sqrt{\bar{\mbox{\bold... ...gma$}} :\mbox{\boldmath$F$}:\bar{\mbox{\boldmath$\sigma$}}} - \sigma_Y(\kappa).$$ (208)

loading-unloading conditions

$$f(\bar{\mbox{\boldmath$\sigma$}},\kappa)\le 0 \qquad \dot{\lambda}\geq 0 \qquad \dot{\lambda}f(\bar{\mbox{\boldmath$\sigma$}},\kappa)=0,$$ (209)

evolution law for plastic strain

$$\dot{\mbox{\boldmath$\varepsilon$}}_{\rm {p}} = \dot{\lambda} \frac{\partial f}{\partial \bar{\mbox{\boldmath$\sigma$}}},$$ (210)

the incremental definition of cumulated plastic strain

$$\dot{\kappa} = \Vert \dot{\mbox{\boldmath$\varepsilon$}}_{\mathrm{p}}\Vert,$$ (211)

the law governing the evolution of the damage variable

$$\omega(\kappa) = \omega_c(1- ^{-a\kappa}),$$ (212)

and the hardening law

$$\sigma_Y(\kappa) = 1+\sigma_H(1- ^{-s\kappa}).$$ (213)

In the equations above, $\bar{\mbox{\boldmath $\sigma$}}$ is the effective stress tensor, $D$ 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

$$\mbox{\boldmath$M$} = \sum_{i=1}^3 m_i( \mbox{\boldmath$m$} _i \otimes \mbox{\boldmath$m$} _i),$$ (214)

normalized such that Tr $($$M$$) = 3$, $m_i$ are the eigenvalues and $m$$_i$ the eigenvectors. The eigenvectors of the fabric tensor determine the directions of material orthotropy and the components of the elastic stiffness tensor $D$ 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

$$\mbox{\boldmath$D$} =\left[\begin{array}{cccccc} \frac{1}{E_1} & -\frac{\nu_{12}}{E_1... ...1}{G_{13}} & 0\\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G_{12}} \end{array}\right]^{-1},$$ (215)

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 $F$ that is used in the yield condition. The yield condition is divided into tensile and compressive parts. Tensor $F$ is different in each part of the effective stress space. This tensor is denoted $F$$^{+}$ in tensile part, characterized by $\hat{\mbox{\boldmath $N$}}:\bar{\mbox{\boldmath $\sigma$}} \leq 0$ and $F$$^{-}$ in compressive part, characterized by $\hat{\mbox{\boldmath $N$}}:\bar{\mbox{\boldmath $\sigma$}} \leq 0$, where

$$\hat{\mbox{\boldmath$N$}} = \frac{\sum_{i=1}^{3} m_i^{-2q}}{\sqrt{\sum_{i=1}^3 m_i^{-4q}}}(\mbox{\boldmath$m$}_i \otimes \mbox{\boldmath$m$}_i)$$ (216)

$$\mbox{\boldmath$F^{\pm}$} =\left[\begin{array}{cccccc} \frac{1}{\left({\sigma_{1}^{\pm}}\ri... ...}{\tau_{13}} & 0\\ 0 & 0 & 0 & 0 & 0 & \frac{1}{\tau_{12}} \end{array}\right].$$ (217)

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

$$\frac{\chi_0^- +1}{(\sigma_0^-)^2} = \frac{\chi_0^+ +1}{(\sigma_0^+)^2}.$$ (218)

The model description and parameters are summarized in Tab. 46.

Table 46: 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 $F$
	- expp exponent $p$ in the expression for tensor $F$
Supported modes	3dMat