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====== 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

===== LemaitreChabocheContinuousDamage =====

=== Description ===

Lemaitre & Chaboche damage model [[doc:user:elements:volumes:continuousdamage#References|[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.

=== Parameters ===

^   Name                                            ^     Metafor Code     ^  Dependency ^ 
| Young Modulus $ E $                                |      ''LEMAITRE_YOUNG''      |     ''TO/TM''      |
| Poisson Ratio $\nu$                                |       ''LEMAITRE_NU''        |     ''TO/TM''      |
| Exponent $ s $                                     |     ''LEMAITRE_SMALL_S''     |     ''TO/TM''      |
| Coefficient $ S $                                  |      ''LEMAITRE_BIG_S''      |     ''TO/TM''      |
| Plastic strain threshold $ \varepsilon^{pl}_D $    |  ''LEMAITRE_EPL_THRESHOLD''  |     ''TO/TM''      |
| Triaxiality threshold $ \eta_D $    |  ''LEMAITRE_TRIAX_THRESHOLD''  |     ''TO/TM''      |
===== BoneRemodContinousDamage =====

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.


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

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.

=== Parameters ===

^   Name                                            ^     Metafor Code     ^
| 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 $                                  |  ''BONE_REMOD_C''                  |
| Density of undamaged material$ \rho_0 [kg/m^3] $        |  ''BONE_REMOD_MASS_DENSITY''       |


==== AlvBoneRemodContinousDamage ====

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


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

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.

=== Parameters ===
^   Name                                            ^     Metafor Code     ^ 
| Coefficient $ N $                                       |        ''BONE_REMOD_N''            |   
| Percentage of available surface $ k $                   |       ''BONE_REMOD_K''            |   
| Reference elastic strain energy $ \psi $                |     ''BONE_REMOD_PSI''          |  
| Remodeling speed $ c $                                  |  ''BONE_REMOD_C''            |
| Density of undamaged material $\rho_0 [\mbox{kg}/\mbox{m}^3] $  |  ''BONE_REMOD_MASS_DENSITY'' |


==== AlvBoneRemodContinousDamage2constant ====

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

=== Description ===

$$
\begin{align*}
\dot r &= c_ff_1(d, \rho_0)g_f\;\;&\text{ if }g_f>0\;\;\text{ and } p>0 \\
\dot r &= -c_rf_1(d, \rho_0)g_f\;\;&\text{ if }g_f>0\;\;\text{ and } p<0 \\
\dot r &= -c_rf_1(d, \rho_0)g_r\;\;&\text{ if  }g_r>0 
\end{align*}
$$

with

<note>
some definitions are lacking
</note>
...

=== Parameters ===
^   Name                                            ^     Metafor Code     ^ 
| Coefficient $ N $                                       |        ''BONE_REMOD_N''            |   
| Percentage of available surface $ k $                   |       ''BONE_REMOD_K''            |   
| Reference elastic strain energy $ \psi $                |     ''BONE_REMOD_PSI''          |   
| Remodeling speed $ c_f $                                |  ''BONE_REMOD_CF''            |
| Remodeling speed $ c_r $                                |  ''BONE_REMOD_CR''            |
| Density of undamaged material $\rho_0 [\mbox{kg}/\mbox{m}^3] $  |  ''BONE_REMOD_MASS_DENSITY'' |
 
===== LangsethContinousDamage =====

=== Description ===

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

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.

=== Parameters ===

^   Name                                            ^     Metafor Code     ^  Dependency ^ 
| $ D_1 $                             |        ''LANGSETH_D1''       |     ''TO/TM''      |
| $ D_2 $                             |       ''LANGSETH_D2''        |     ''TO/TM''      |
| $ D_3 $                             |      ''LANGSETH_D3''         |     ''TO/TM''      |
| $ D_4 $                             |      ''LANGSETH_D4''         |     ''TO/TM''      |
| $ D_5 $                             |      ''LANGSETH_D5''         |     ''TO/TM''      |
| Damage $ D_C $                      |      ''LANGSETH_DC''         |     ''TO/TM''      |
| $ \dot \varepsilon^{pl}_0 $         |      ''LANGSETH_EPSP0''      |     ''TO/TM''      |
| Room temperature $ T_{room} $       |      ''LANGSETH_ROOM''       |         -          |
| Melting temperature $ T_{melt} $    |      ''LANGSETH_TMELT''      |         -          |
| Plastic strain threshold $ \varepsilon^{pl}_D $ |  ''LANGSETH_EPL_THRESHOLD''  |    -    |


===== GeersContinuousDamage =====

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. [[mail:ppjeunechamps@ulg.ac.be|I]] can give references if needed. All models are based on a characteristic quantity, $\kappa$.

=== Parameters common to all models ===

^   Name                                 ^     Metafor Code     ^  Dependency ^ 
| Initiation value $ \kappa_i $          |  ''GEERS_KAPPA_I''  |     ''TO/TM''      |
| Critical value   $ \kappa_c $          |  ''GEERS_KAPPA_C''  |     ''TO/TM''      |


==== PowGeersContinuousDamage ====

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

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

^   Name   ^     Metafor Code     ^  Dependency ^ 
| $ n_1 $  |  ''GEERS_N1''   |     ''TO/TM''      |
| $ n_2 $  |  ''GEERS_N2''   |     ''TO/TM''      |


==== ExpGeersContinuousDamage ====

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

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

^   Name     ^     Metafor Code     ^  Dependency ^ 
| $ \beta $  |  ''GEERS_BETA''  |     ''TO/TM''      |



==== TanhGeersContinuousDamage ====

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

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

^   Name                                            ^     Metafor Code     ^  Dependency ^ 
| Initiation value $ \kappa_i $  |  ''GEERS_KAPPA_I''  |     ''TO/TM''      |
| Critical value $ \kappa_c $          |  ''GEERS_KAPPA_C''  |     ''TO/TM''      |





==== LinGeersContinuousDamage ====

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)

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

^   Name                                            ^     Metafor Code     ^  Dependency ^ 
| $ A $  |   ''GEERS_A''   |     ''TO/TM''      |
| $ B $  |   ''GEERS_B''   |     ''TO/TM''      |
| $ C $  |   ''GEERS_C''   |     ''TO/TM''      |

===== References =====

[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.

[3]

[4]