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:

• 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 $D$:

$$\begin{align} \dot{D}&=\dfrac{D_{N}}{s_N\sqrt{2\pi}}\exp\left[-\dfrac{1}{2}\left[\dfrac{\bar{\varepsilon}^{vp}-\varepsilon_N}{s_N}\right]^2\right]\dot{\bar{\varepsilon}}^{vp}+\left[1-D\right]\text{tr}\left(\mathbf{D^{irr}}\right) &\text{ si }D<D_{crit} \notag \\ &=\dfrac{D_{ult}-D_{crit}}{\Delta \varepsilon}\dot{\bar{\varepsilon}}^{vp}& \text{ si } D>D_{crit} \notag \end{align}$$

• 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:

$$\eta = \dfrac{3}{2} \dfrac{m+1}{m} \sinh\left(2\dfrac{2-m}{2+m}\dfrac{p}{\bar{\sigma}}\right)$$

• 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:

$$\eta = \dfrac{3}{2} \dfrac{m+1}{m} \sinh\left(2\dfrac{2-m}{2+m}\dfrac{|p|}{\alpha_\eta\sigma_{yield}}\right)$$