====== 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

===== AnisoDamageDummy =====

A dummy testing all possible variations of the damage variable.

===== LemaitreChabocheContinuousAnisoDamage =====

Anisotropic extension of [[doc:user:elements:volumes:continuousdamage|Lemaitre isotropic damage law]]

=== Description  ===

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.

=== Parameters === 
^          Name                                      ^  Metafor Code  ^ Dependency         ^
| Young Modulus $ E $                                |        ''LEMAITRE_E''        |     ''TM''      |
| Poisson ratio $\nu$                                |       ''LEMAITRE_NU''        |     ''TM''      |
| Exponent $ s $                                     |     ''LEMAITRE_SMALL_S''     |     ''TM''      |
| Coefficient $ S $                                  |      ''LEMAITRE_BIG_S''      |     ''TM''      |
| Plastic strain threshold $ \varepsilon^{pl}_D $    |  ''LEMAITRE_EPL_THRESHOLD''  |     ''TM''      |

===== BoneRemodContinuousAnisoDamage =====

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.

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

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. - [[http://orbi.ulg.ac.be/handle/2268/126082|cfr p131-132 my thesis]]

=== Parameters === 
^          Name                                        ^  Metafor Code  ^ Dependency         ^   
| Coefficient $ N $                                        |        ''BONE_REMOD_N''            |   
| Percentage of available surface $ k $                    |       ''BONE_REMOD_K''            |   
| Reference elastic strain energy $ \psi $                 |     ''BONE_REMOD_PSI''          |  
| Half width of the dead zone   $ \omega $                 |      ''BONE_REMOD_OMEGA''        |     
| Remodeling speed $ c_f $                                 |  ''BONE_REMOD_CF''            |
| Remodeling speed $ c_r $                                 |  ''BONE_REMOD_CR''            |
| Density of undamaged material $ \rho_0 $                 |  ''BONE_REMOD_MASS_DENSITY'' |
| "weight" of anisotropy, $ \eta $                         |     ''BONE_REMOD_ETA''                     |



==== AlvBoneRemodContinuousAnisoDamage ====
This law is defined for the remodeling of the alveolar bone. Damage evolution also depends on pressure.
cfr [[http://orbi.ulg.ac.be/handle/2268/126082 |p140-142 of my thesis]]