====== 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.


===== WovenCompositeDamage =====
=== Description ===

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 
$$
\begin{eqnarray*}
  d_{12}^s = \min\left(
    1,
    \left\langle
      \frac{\sqrt{Y_{\rm eq}}-\sqrt{Y_0}}{\sqrt{Y^c_{12}}-\sqrt{Y_0}}
    \right\rangle_+
  \right) \;.
\end{eqnarray*}
$$

=== Parameters ===

^                       Name                            ^      Metafor Code       ^
|  $Y_{11}^{c+}$  |  WOVEN_YCP11      |
|  $Y_{11}^{c-}$  |  WOVEN_YCM11      |
|  $Y_{22}^{c+}$  |  WOVEN_YCP22      |
|  $Y_{22}^{c-}$  |  WOVEN_YCM22      |
|  $Y_0$  |  WOVEN_Y0         |
|  $Y^c_{12}$  |  WOVEN_Y12C       |
|  $\lambda$  |  WOVEN_LAMBDA     |
|  $\alpha_1$  |  WOVEN_ALPHA1     |
|  $\alpha_2$  |  WOVEN_ALPHA2     |
|  $a_c$  |  TIME_DELAY_AC    |
|  $\tau_c$  |  TIME_DELAY_TAUC  |