Continuous orthotropic damage

The ContinousAnisoDamage class manages the continuous orthotropic damage evolution laws. When defining a new law, the evolution of the damage variable $\delta H$ must be defined, and so must be its derivatives with respect to pressure, plastic strain and damage.

Laws implemented in Metafor

A dummy testing all possible variations of the damage variable.

Anisotropic extension of Lemaitre isotropic damage law

The damage tensor is denoted $D$

$$ \dot D = \left(\dfrac{\tilde\sigma_{eq}^2 R_\nu}{2ES}\right)^s |D^{pl}| \mbox{ if } \varepsilon^{pl} > \varepsilon^{pl}_D $$

where $|D^{pl}|$ is a tensor with the same eigenvectors as $D^{pl}$, and eigenvalues equal to the absolute value of $D^{pl}$ eigenvalues. The triaxiality function is defined as :

$$ R_\nu = \dfrac{2}{3}\left(1+\nu\right) + 3\left(1-2\nu\right) \left(\dfrac{p}{\sigma_{eq}}\right)^2 $$

where $ p $ is the pressure and $ \sigma_{eq} $ Von Mises stress.

This law is used for bone remodeling (extracted from Doblaré's law, used only in elasticity). Damage variation depends mostly on damage, surface available for remodeling and a “remodelling rate” function, which itself depends on stress state.

$$ \dot H =f(H, \rho_0)kS_v(d_h)\dot r $$

where

$S_v(d_h)$ is the surface per unit volume available for remodeling (polynomial of degree 5 in $d$), and $d_h$ is the average damage ($d_h = d_{ii}/3$)

\begin{align*} \dot r &= c_f(H, \rho_0)g_f\;\;&\text{ if }g_f>0 \\ \dot r &= -c_r(H, \rho_0)g_r\;\;&\text{ if }g_r>0 \end{align*}

with

\begin{align*} g_f &= N^{1/4}u(\sigma)-(1+\omega)\psi\\ g_r &= -N^{1/4}u(\sigma)+(1-\omega)\psi \end{align*}

$ u $ is a measure of the elastic strain energy. - cfr p131-132 my thesis

This law is defined for the remodeling of the alveolar bone. Damage evolution also depends on pressure. cfr p140-142 of my thesis