Continuous isotropic damage

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

Laws implemented in Metafor

Lemaitre & Chaboche damage model [1,2]. $$ \dot D = \left(\dfrac{\bar \sigma^2 R_\nu}{2ES\left(1-D\right)^2}\right)^s \dot \varepsilon^{pl} \mbox{, if } \varepsilon^{pl} > \varepsilon^{pl}_D \mbox{, and } \eta > \eta_D $$

where the triaxiliaty function is defined as:

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

where $ p $ is the pressure, $ \bar \sigma $ is Von Mises stress and $\eta$ is the stress triaxiality ratio.

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

$$ \dot d =f(d, \rho_0)kS_v(d)\dot r $$

where

$S_v(d)$ is the surface per unit volume available for remodeling (polynomial of degree 5 in $d$)

and where

$$ \begin{align*} \dot r &= \ cf_1(d, \rho_0)g_f&\text{ if }g_f>0 \\ \dot r &= -cf_1(d, \rho_0)g_r&\text{ if }g_r>0 \end{align*} $$ with $$ \begin{eqnarray*} g_f &=& N^{1/4}u(\sigma)-(1+\omega)\psi \\ g_r &=& \dfrac{1}{N^{1/4}u(\sigma)}-\dfrac{1}{(1-\omega)\psi} \end{eqnarray*} $$ $ f, f_1 $ are functions in the damage variable, $ u $ is a measure of the elastic strain energy.

This law is defined for the remodeling of the alveolar bone. Damage evolution also depends on pressure.

$$ \dot d =f(d, \rho_0)kS_v(d)\dot r $$

where

$S_v(d)$ is the surface per unit volume available for remodeling (polynomial of degree 5 in $d$)

$$ \begin{align*} \dot r &= cf_1(d, \rho_0)g_f &\text{ if }g_f>0 \text{ and } p>0 \\ \dot r &= -cf_1(d, \rho_0)g_f &\text{ if }g_f>0 \text{ and } p<0 \\ \dot r &= -cf_1(d, \rho_0)g_r &\text{ if }g_r>0 \end{align*} $$ with $$ \begin{eqnarray*} g_f &=& N^{1/4}u(\sigma)-\psi \\ g_r &=& \dfrac{1}{N^{1/4}u(\sigma)}-\dfrac{1}{\psi} \end{eqnarray*} $$ $ f, f_1 $ are functions in the damage variable, $ u $ is a measure of the elastic strain energy.

Same law than the previous one, except that remodeling constants are different in formation and resorption.

$$ \begin{align*} \dot r &= c_ff_1(d, \rho_0)g_f\;\;&\text{ if }g_f>0\;\;\text{ and } p>0 \\ \dot r &= -c_rf_1(d, \rho_0)g_f\;\;&\text{ if }g_f>0\;\;\text{ and } p<0 \\ \dot r &= -c_rf_1(d, \rho_0)g_r\;\;&\text{ if }g_r>0 \end{align*} $$ with <note> some definitions are lacking </note> ... === Parameters === ^ Name ^ Metafor Code ^ | Coefficient $ N $ | BONE_REMOD_N |

$$ \dot D = D_C\dfrac{\dot \varepsilon^{pl}}{\varepsilon^{pl}_f-\varepsilon^{pl}_D} \mbox{ if } \varepsilon^{pl} > \varepsilon^{pl}_D $$

where the plastic strain at failure is defined as:

$$ \varepsilon^{pl}_f = \left(D_1 + D_2 \exp\left(D_3\dfrac{p}{\bar\sigma}\right)\right) \left(1+\ln\dfrac{\dot \varepsilon^{pl}}{\dot \varepsilon^{pl}_0}\right)^{D_4} \left(1-D_5\dfrac{T-T_{room}}{T_{melt}-T_{room}}\right) $$

where $p$ is the pressure and $ \bar \sigma $ the Von Mises stress.

Damage evolution law following Geers's models. Several laws actually exist, all of the same author, which is why they are gathered in a same class. If the full Geers's model, damage is integrated globally on the structure, and not locally at each integration point. I can give references if needed. All models are based on a characteristic quantity, $\kappa$.

Power law. $\kappa$ is the equivalent plastic strain $\varepsilon^{pl}$:

$$ D = 1 - \left(\dfrac{\kappa_i}{\kappa}\right)^{n_1} \left(\dfrac{\kappa-\kappa_i}{\kappa_c-\kappa_i}\right)^{n_2} \mbox{ if } \kappa_i\leq\kappa\leq\kappa_c $$

Exponential law. $\kappa$ is the equivalent plastic strain $\bar\varepsilon^{pl}$

$$ D = 1 - \exp\left(-\beta\left(\kappa-\kappa_i\right)\right) $$

Hyperbolic tangent. $\kappa$ is the equivalent plastic strain $\varepsilon^{pl}$

$$ D = \dfrac{1}{2\tanh\left(3\right)} \left(\tanh\left(6\dfrac{\kappa-\kappa_i}{\kappa_c-\kappa_i}-3\right)+\tanh\left(3\right)\right) $$

Law linear. $\kappa$ is a function of the stress triaxiality and the equivalent plastic strain $\varepsilon^{pl}$

$$ \dot{\kappa} = C\left<1+A\dfrac{p}{\bar\sigma}\right> \left(\varepsilon^{pl}\right)^B \dot\varepsilon^{pl} $$ where $p$ is the pressure, and $ \overline{\sigma} $ the Von Mises stress. $\langle .\rangle$ are Macaulay symbols( $\langle \alpha\rangle = \alpha $ if $ \alpha \ge 0 $ and $ 0 $ otherwise)

$$ \dot D = \dfrac{\dot\kappa}{\kappa_c-\kappa_i} $$

[1] Lemaitre J. A continuous damage mechanics model for ductile fracture. Journal of Engineering Materials and Technology 1985;107:9–83.

[2] Chaboche JL. Description thermodynamique et phénoménologique de la viscoélasticité cyclique avec endommagement. PhD Thesis, Université Pierre et Marie Curie, Paris VI, 1978.

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