Continuous orthotropic 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.

Orthotropic laws implemented in Metafor.

Damage law with delay effect for woven composites.

The strain energy density is written as: $$ \begin{eqnarray*} W_{\rm D} &=& \dfrac{1}{2}\Biggl( \dfrac{\sigma_{11}^2}{E_1\,(1-d_{11})} -2\,\dfrac{\nu_{12}}{E_1}\,\sigma_{11}\,\sigma_{22} -2\,\dfrac{\nu_{13}}{E_1}\,\sigma_{11}\,\sigma_{33} \\ && +\dfrac{\sigma_{22}^2}{E_2\,(1-d_{22})} -2\,\dfrac{\nu_{23}}{E_2}\,\sigma_{22}\,\sigma_{33} \\ &&+\dfrac{\sigma_{33}^2}{E_3} +\dfrac{\sigma_{12}^2}{G_{12}\,(1-d_{12})} +\dfrac{\sigma_{13}^2}{G_{13}\,(1-\lambda\, d_{12})} +\dfrac{\sigma_{23}^2}{G_{23}\,(1-\lambda\, d_{12})} \Biggr) \;. \end{eqnarray*} $$ Three damage variables are introduced. Delay effect is introduced with the definition of a law governing the temporal evolution of damage : $$ \dot{d}_{ij} = \frac{1}{\tau_c}\,\left( 1-e^{-a_c\,\langle d^s_{ij} - d_{ij} \rangle_+} \right) \;, $$ where $a_c$ and $\tau_c$ are delay effect parameters, $\langle x \rangle_+$ is a function equal to $x$ if $x$ is positive and 0 otherwise, and $d^s_{ij}$ is the static damage value. Along the fibers, $$ \begin{eqnarray*} d_{11}^s &=& \left\{ \begin{array}{ll} 0 & \text{ if } \left(Y_{11}<Y_{11}^{c+} \text{ and } \sigma_{11}>0\right) \text{ or } \left(Y_{11}<Y_{11}^{c-} \text{ and } \sigma_{11}<0\right) \\ 1 & \text{ otherwise } \end{array} \right. \;, \\ d_{22}^s &=& \left\{ \begin{array}{ll} 0 & \text{ if } \left(Y_{22}<Y_{22}^{c+} \text{ and } \sigma_{22}>0\right) \text{ or } \left(Y_{22}<Y_{22}^{c-} \text{ and } \sigma_{22}<0\right) \\ 1 & \text{ otherwise} \end{array} \right. \;, \end{eqnarray*} $$ where $Y_{ii}^{c+}$ and $Y_{ii}^{c-}$ are critical values of thermodynamic forces under traction and compression, respectively. Under shear, an equivalent thermodynamic force is first defined:

$$ \begin{eqnarray*} Y_{\rm eq}(t) &=& \sup_{\tau\leq t} \left( \alpha_1\,Y_{11}^+ + \alpha_2\,Y_{22}^+ + Y_{12} \right) \;, \\ Y_{ii}^+ &=& \left\{ \begin{array}{ll} Y_{ii} & \text{ if } \sigma_{ii}>0 \\ 0 & \text{ otherwise} \end{array} \right. \;, \end{eqnarray*} $$ then $$ \begin{eqnarray*} d_{12}^s = \min\left( 1, \left\langle \frac{\sqrt{Y_{\rm eq}}-\sqrt{Y_0}}{\sqrt{Y^c_{12}}-\sqrt{Y_0}} \right\rangle_+ \right) \;. \end{eqnarray*} $$