Material properties
"Optional title"
mid, $\rho$, $E$, $\nu$, did
cid, $tsc$, $\beta$
Parameter definition
mid Unique material identification number
$\rho$ Density
$E$ Young's modulus
$\nu$ Poisson's ratio
did Damage property command ID
cid ID of a CURVE or FUNCTION defining plastic flow stress versus volumetric compression
$tsc$ Tensile stress cut-off (a positive value should be given)
$\beta$ Damping coefficient for strain rate sensitivity ($\beta$ > 0.1 is recommended)

MAT_FOAM is a simple model for crushable foams. This model is limited to isotropic behaviour under impact loading conditions (non-cyclic loading).

Stress-strain behaviour
Stress-strain behaviour

The implementation assumes a constant Young's modulus ($E$) and elastic behaviour for stress update. Trial stress is thus evaluated as:

$\sigma_{ij}^{trial}= \sigma_{ij}^n+3K( \frac{1}{3} \Delta \varepsilon_{kk} \delta_{ij})+2G( \Delta \varepsilon_{ij}- \frac{1}{3} \Delta \varepsilon_{kk} \delta_{ij})$

where $K$ is the bulk modulus and $G$ is the shear modulus.

Principal stresses $\sigma_I^{trial}$ (I=1,3) are then computed and the following criterion is checked:

$\left | \sigma_I^{trial} \right| \gt \sigma_{compaction} \Rightarrow \sigma_I^{n+1}= \sigma_{compaction} \frac{ \sigma_I^{trial}}{ \left| \sigma_I^{trial} \right|}$

The compaction curve is defined by a CURVE (compaction pressure/volumetric strain). This is done independently in each direction, implying no Poisson effect.

Principal stresses are optionally limited in tension by a tension cut-off parameter (elastic perfectly plastic behaviour).

A damping coefficient is also possibly defined in order to take into account rate sensitivity. Minimal recommended damping coefficient value is 0.1. This adds an extra damping stress as follows:

$\sigma_{ij}^{damping} = \beta \cdot \rho \cdot L^{element} \cdot c^{long} \cdot \dot{\varepsilon_ij}$

where $\beta$ is the damping coefficient, $\rho$ is the density, $L^{element}$ is the characteristic element length and $c^{long}$ is the longitudinal sound speed.