# Metafor

ULiege - Aerospace & Mechanical Engineering

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doc:user:elements:volumes:ortho_continuousdamage

# 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