====== 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 some definitions are lacking ... === 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]