Local formulation

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

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

the stress strain law

$\displaystyle \mbox{\boldmath$\sigma$}$ (207)

the yield function


loading-unloading conditions


evolution law for plastic strain


the incremental definition of cumulated plastic strain

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

the law governing the evolution of the damage variable

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

and the hardening law

e (213)

In the equations above, $ \bar{\mbox{\boldmath $\sigma$}}$ is the effective stress tensor, $ D$ is the elastic stiffness tensor,  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$, and $ \omega_c$ are positive material parameters. Material anisotropy is characterized by the second-order positive definite fabric tensor


normalized such that Tr $ M$, are the eigenvalues and $ m$ 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 , $ i = 1, 2, 3$, the stiffness can be presented in Voigt (engineering) notation as

$\displaystyle \mbox{\boldmath$D$}$ (215)

where , and . Here, , and are elastic constants characterizing the compact (poreless) material, 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 that is used in the yield condition. The yield condition is divided into tensile and compressive parts. Tensor is different in each part of the effective stress space. This tensor is denoted in tensile part, characterized by and in compressive part, characterized by , where



In the equation above is uniaxial yield stress along the $ i$-th principal axis of orthotropy, is the shear yield stress in the plane of orthotropy and is the so-called interaction coefficient, $ p$ and 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


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 in the expression for tensor
  - expp exponent $ p$ in the expression for tensor
Supported modes 3dMat

Borek Patzak