====== Damage ====== The ''Damage'' class manages all damage evolution laws. When defining a new law, the following must be defined: * The stress associated to damage which is taken into account in the plastic criterion $ \sigma_{damage} $ * The evolution of the damage variable $ D $ * The softening of the elastic limit $\omega $ Laws in Metafor: ===== GursonTvergaardDamage ===== === Description === * Stress associated to damage, $ \sigma_{damage}$: $$ \sigma_{damage}=K\left[\dfrac{\dot{\bar{\varepsilon}}^{vp}\bar{\sigma}\left[1-D\right]}{\bar{\sigma}+p\dfrac{\partial f}{\partial p}}\right]^{m}\left[\bar{\varepsilon}^{vp}\right]^{n} $$ * Softening of elastic limit $ \omega $: $$ \omega \left(D,p,\sigma_{yield}\right) = \sqrt{1-2\dfrac{D}{D_{ult}}\cosh\left(\dfrac{3\alpha p}{\sigma_{yield}}\right)+\left[\dfrac{D}{D_{ult}}\right]^2} $$ * Evolution of the damage variable $ D $: \begin{align} \dot{D}&=\dfrac{D_{N}}{s_N\sqrt{2\pi}}\exp\left[-\dfrac{1}{2}\left[\dfrac{\bar{\varepsilon}^{vp}-\varepsilon_N}{s_N}\right]^2\right]\dot{\bar{\varepsilon}}^{vp}+\left[1-D\right]\text{tr}\left(\mathbf{D^{irr}}\right) &\text{ si }DD_{crit} \notag \end{align} === Parameters === ^ Name ^ Metafor Code ^ Dependency ^ | Viscosity ($ K $) | ''GURSON_K'' | - | | Sensitivity to strain rate ($ m $) | ''GURSON_M'' | - | | Hardening of viscous terms ($ n $) | ''GURSON_N'' | - | | Damage value at failure ($D_{ult}$) | ''GURSON_D_ULT'' | - | | Damage value at coalescence ($D_{crit}$) | ''GURSON_DCRIT'' | - | | Parameter of nucleation law ($\alpha$) | ''GURSON_ALPHA'' | - | | Maximal number of nucleated microvoids ($D_{N}$) | ''GURSON_D_N'' | - | | Variance of the nucleation distribution function ($s_N$) | ''GURSON_S_N'' | - | | Average strain at nucleation ($\varepsilon_N$) | ''GURSON_EPS_N'' | - | | Coalescence parameter $\Delta\varepsilon$ | ''GURSON_DELTA_EPS'' | - | ===== KhaleelDamage ===== === Description === * Stress associated to damage, $ \sigma_{damage}$: $$ \sigma_{damage}=K\left(1-\sqrt{D}\right)\left[\dfrac{\dot{\bar{\varepsilon}}^{vp}\bar{\sigma}\left[1-D\right]}{\bar{\sigma}+p\dfrac{\partial f}{\partial p}}\right]^{m}\left[\bar{\varepsilon}^{vp}\right]^{n} $$ * Softening of elastic limit $ \omega $: $$ \omega \left(D,p,\sigma_{yield}\right) = \left(1-\sqrt{D}\right) - \dfrac{\alpha_\omega 3p}{\sigma_{yield}} $$ * Evolution of the damage variable $ D $: $$ \begin{align} \dot{D}&=B\sigma_{vp}^v \left(\bar{\varepsilon}^{vp}\right)^b \dot{\bar{\varepsilon}}^{vp}+\left[1-D\right]E_v\eta\left(p\right)\text{tr}\left(\mathbf{D^{irr}}\right)&\text{ si }DD_{crit} \notag \end{align} $$ where $\eta\left(p\right)$ is defined as: $$ \eta = \dfrac{3}{2} \dfrac{m+1}{m} \sinh\left(2\dfrac{2-m}{2+m}\dfrac{p}{\bar{\sigma}}\right) $$ === Parameters === ^ Name ^ Metafor Code ^ Dependency ^ | Viscosity ($ K $) | ''KHALEEL_K'' | - | | Sensitivity to strain rate ($ m $) | ''KHALEEL_M'' | - | | Hardening of viscous terms ($ n $) | ''KHALEEL_N'' | - | | Damage value at failure ($D_{ult}$) | ''KHALEEL_D_ULT'' | - | | Damage value at coalescence ($D_{crit}$) | ''KHALEEL_DCRIT'' | - | | Sensitivity to pressure ($\alpha_\omega$) | ''KHALEEL_ALPHA'' | - | | First cavity nucleation parameter ($ B $) | ''KHALEEL_BIGB'' | - | | Second cavity nucleation parameter ($ b $) | ''KHALEEL_SMALLB'' | - | | Cavity growth parameter ($ E_v $) | ''KHALEEL_EV'' | - | | Cavity coalescence parameter ($ F $) | ''KHALEEL_FACT_EV'' | - | ===== AdamKhaleelDamage ===== === Description === * Stress associated to damage, $ \sigma_{damage}$: $$ \sigma_{damage}=K\left(1-\sqrt{D}\right)\left[\dfrac{\dot{\bar{\varepsilon}}^{vp}\bar{\sigma}\left[1-D\right]}{\bar{\sigma}+p\dfrac{\partial f}{\partial p}}\right]^{m}\left[\bar{\varepsilon}^{vp}\right]^{n} $$ * Softening of elastic limit $ \omega $: $$ \begin{align*} \omega \left(D,p,\sigma_{yield}\right) &= 1-\sqrt{D}\left(1+\dfrac{\alpha_\omega 3|p|}{\sigma_{yield}}\right) &\text{ si } |p| > \dfrac{p_{lim}}{PLIM} \\ &= \sqrt{\dfrac{3}{2}}\dfrac{\zeta+\sqrt{\beta^2-p^2}}{\sigma_{yield}} &\text{ si } |p| < \dfrac{p_{lim}}{PLIM} \end{align*} $$ where $$ \begin{eqnarray*} &p_{lim} &= \dfrac{1-\sqrt{D}}{\sqrt{D}} \dfrac{\sigma_{yield}}{3\alpha_\omega} \\ &\zeta &= \sqrt{\dfrac{2}{3}} \left(1 - \left(1+\dfrac{3\alpha p_{lim}}{\sigma_{yield}PLIM}\right) \sqrt{D}\right) \sigma_{yield} - \sqrt{\dfrac{3}{2}} \dfrac{p_{lim}}{3\alpha\sqrt{D}PLIM} \\ &\beta &= \sqrt{ \left(\dfrac{p_{lim}}{PLIM}\right)^2 + \dfrac{3}{2}\left(\dfrac{p_{lim}}{3\alpha\sqrt{D}PLIM}\right)^2} \end{eqnarray*} $$ * Evolution of the damage variable $ D $: $$\begin{align} \dot{D}&=B\sigma_{vp}^v \left(\bar{\varepsilon}^{vp}\right)^b \dot{\bar{\varepsilon}}^{vp}+\left[1-D\right]E_v\eta\left(p\right)\text{tr}\left(\mathbf{D^{irr}}\right)&\text{ si }DD_{crit} \notag \end{align} $$ where $\eta\left(p\right)$ is defined as: $$ \eta = \dfrac{3}{2} \dfrac{m+1}{m} \sinh\left(2\dfrac{2-m}{2+m}\dfrac{|p|}{\alpha_\eta\sigma_{yield}}\right) $$ === Parameters === ^ Name ^ Metafor Code ^ Dependency ^ | Viscosity ($ K $) | ''ADAM_K'' | - | | Sensitivity to strain rate ($ m $) | ''ADAM_M'' | - | | Hardening of viscous terms ($ n $) | ''ADAM_N'' | - | | Damage value at failure ($D_{ult}$) | ''ADAM_D_ULT'' | - | | Damage value at coalescence ($D_{crit}$) | ''ADAM_DCRIT'' | - | | Cavity growth parameter ($ E_v $) | ''ADAM_EV'' | - | | Cavity coalescence parameter ($ F $) | ''ADAM_FACT_EV'' | - | | Sensitivity to pressure ($\alpha_\omega$) | ''ADAM_ALPHA'' | - | | First cavity nucleation parameter ($ B $) | ''ADAM_BIGB'' | - | | Second cavity nucleation parameter ($ b $) | ''ADAM_SMALLB'' | - | | Sensitivity of cavity growth to pressure ($ \alpha_\eta $) | ''ADAM_ALPHA_ETA'' | - | | Parameter smoothing the viscoplastic criterion ($ PLIM $) | ''ADAM_PLIM'' | - |