# Visco-elasto Materials for Foams (LAW33)

This material law can be used to model low density closed cell polyurethane foams, impactors, impact limiters. It can only be used with solid elements.

- The components of the stress tensor are uncoupled until full volumetric compaction is achieved (Poisson's ratio = 0.0).
- The material is isotropic.
- The effect of the enclosed air is considered via a separate Pressure vs Volumetric
Strain relation:
(1) $${P}_{air}=-\frac{{P}_{0}\cdot \gamma}{1+\gamma -\Phi}$$with:(2) $$\gamma =\frac{V}{{V}_{0}}-1+{\gamma}_{0}$$Where,- $\gamma $
- Volumetric strain
- $\text{\Phi}$
- Porosity
- ${P}_{0}$
- Initial air pressure
- ${\gamma}_{0}$
- Initial volumetric strain

- The structural stresses
$\sigma $
follow the Maxwell-Kelvin-Voight viscoelastic model (Generalized Kelvin-Voigt Model (LAW35), Equation 12 before the limiting yield curve is
reached):
(3) $${\sigma}_{ij}^{}\left(t+\text{\Delta}t\right)={\sigma}_{ij}^{}\left(t\right)+\left[E{\dot{\epsilon}}_{ij}-\left(\frac{E+{E}_{t}}{\eta}{\sigma}_{ij}^{s}\left(t\right)\right)+\frac{E\cdot {E}_{t}}{\eta}{\epsilon}_{ij}\right]\cdot \text{\Delta}t$$ - The Young's modulus used in the calculation is: $E=\mathrm{max}(E,{E}_{1}\dot{\epsilon}+{E}_{2})$
- Yield is defined by a user-defined curve vs volumetric strain, $\gamma $ , or $\sigma =A+B\left(1+C\gamma \right)$
- Yield is applied to the principal structural stresses.
- Unloading follows Young's modulus, which results in viscous unloading.
- The full stress tensor is obtained by adding air pressure to the structual stresses:
(4) $${\sigma}^{total}{}_{ij}\left(t\right)={\sigma}_{ij}^{}\left(t\right)-{P}_{air}{\delta}_{ij}$$