Table of Contents
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 $:
\begin{align} \dot{D}&=\dfrac{D_{N}}{s_N\sqrt{2\pi}}\exp\left[-\dfrac{1}{2}\left[\dfrac{\bar{\varepsilon}^{vp}-\varepsilon_N}{s_N}\right]^2\right]\dot{\bar{\varepsilon}}^{vp}+\left[1-D\right]\text{tr}\left(\mathbf{D^{irr}}\right) &\text{ si }D<D_{crit} \notag \\ &=\dfrac{D_{ult}-D_{crit}}{\Delta \varepsilon}\dot{\bar{\varepsilon}}^{vp}& \text{ si } D>D_{crit} \notag \end{align}
Parameters
| Name | Metafor Code | Dependency |
|---|---|---|
| Viscosity ($ 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 $:
$$ \begin{align} \dot{D}&=B\sigma_{vp}^v \left(\bar{\varepsilon}^{vp}\right)^b \dot{\bar{\varepsilon}}^{vp}+\left[1-D\right]E_v\eta\left(p\right)\text{tr}\left(\mathbf{D^{irr}}\right)&\text{ si }D<D_{crit} \notag \\ &=B\sigma_{vp}^v \left(\bar{\varepsilon}^{vp}\right)^b \dot{\bar{\varepsilon}}^{vp}+\left[1-D\right]F E_v\eta\left(p\right)\text{tr}\left(\mathbf{D^{irr}}\right)&\text{ si }D>D_{crit} \notag \end{align} $$ 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 $:
$$\begin{align} \dot{D}&=B\sigma_{vp}^v \left(\bar{\varepsilon}^{vp}\right)^b \dot{\bar{\varepsilon}}^{vp}+\left[1-D\right]E_v\eta\left(p\right)\text{tr}\left(\mathbf{D^{irr}}\right)&\text{ si }D<D_{crit} \notag \\ &=B\sigma_{vp}^v \left(\bar{\varepsilon}^{vp}\right)^b \dot{\bar{\varepsilon}}^{vp}+\left[1-D\right]F E_v\eta\left(p\right)\text{tr}\left(\mathbf{D^{irr}}\right)&\text{ si }D>D_{crit} \notag \end{align} $$ 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 | - |