Crushable Foam

Crushable foam models:

are used to model the enhanced ability of a foam material to deform in compression because of cell wall buckling processes (it is assumed that the resulting deformation is not recoverable instantaneously and can, thus, be idealized as being plastic for short duration events);
can be used to model the difference between a foam material's compressive strength and its much smaller tensile bearing capacity resulting from cell wall breakage in tension;
must be used with the linear elastic material model; and
are intended to simulate material response under essentially monotonic loading.

Crushable Foam Plasticity

Only linear isotropic elasticity can be used.

For the plastic part of the behavior, the yield surface is a von Mises circle in the deviatoric stress plane and an ellipse in the meridional $(p - q)$ stress plane. Two hardening models are available: the volumetric hardening model, where the point on the yield ellipse in the meridional plane that represents hydrostatic tension loading is fixed and the evolution of the yield surface is driven by the volumetric compacting plastic strain, and the isotropic hardening model, where the yield ellipse is centered at the origin in the $(p - q)$ stress plane and evolves in a geometrically self-similar manner.

The hardening curve must describe the uniaxial compression yield stress as a function of the corresponding plastic strain. In defining this dependence at finite strains, “true” (Cauchy) stress and logarithmic strain values must be given. Both models predict similar behavior for compression-dominated loading. However, for hydrostatic tension loading the volumetric hardening model assumes a perfectly plastic behavior, while the isotropic hardening model predicts the same behavior in both hydrostatic tension and hydrostatic compression.

Volumetric: Specify the volumetric hardening model.
Isotropic: Specify the isotropic hardening model.

Table 1. Hardening Type=Volumetric
Input Data	Description
Compression Yield Stress Ratio	$k = σ_{c}^{0} / p_{c}^{0}$ yield stress ratio for compression loading; $0 < k < 3$ . Enter the ratio of initial yield stress in uniaxial compression to initial yield stress in hydrostatic compression.
Hydrostatic Yield Stress Ratio	$k_{t} = p_{t} / p_{c}^{0}$ yield stress ratio for hydrostatic loading; $k_{t} \geq 0$ . Enter the ratio of yield stress in hydrostatic tension to initial yield stress in hydrostatic compression, given as a positive value.
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on one or more independent field variables. A Field column appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Table 2. Hardening Type=Isotropic
Input Data	Description
Compression Yield Stress Ratio	$k = σ_{c}^{0} / p_{c}^{0}$ yield stress ratio for compression loading; $0 < k < 3$ . Enter the ratio of initial yield stress in uniaxial compression to initial yield stress in hydrostatic compression.
Plastic Poisson's Ratio	$ν_{p}$ plastic Poisson's ratio; $- 1 < ν_{p} \leq 0.5$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on one or more independent field variables. A Field column appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Crushable Foam Hardening

The yield surface for volumetric hardening is defined as

F = \sqrt{q^{2} + α^{2} {(p - p_{0})}^{2}} - B = 0

where $p = - \frac{1}{3} t r a c e (σ)$ is the pressure stress, $q = \sqrt{\frac{3}{2} S : S}$ is the von Mises stress, $S = σ + p I$ is the deviatoric stress, $A = \frac{p_{c} + p_{t}}{2}$ is the size of the $p$ -axis of the yield ellipse, $B = α A = α \frac{p_{c} + p_{t}}{2}$ is the size of the (vertical) $q$ -axis of the yield ellipse, $α = B / A$ is the shape factor of the yield ellipse that defines the relative magnitude of the axes, $p_{0} = \frac{p_{c} - p_{t}}{2}$ is the center of the yield ellipse on the $p$ -axis, $p_{t}$ is the strength of the material in hydrostatic tension, and $p_{c} > 0$ is the yield stress in hydrostatic compression.

If volumetric hardening is chosen, the yield surface intersects the $p$ -axis at $- p_{t}$ and $p_{c}$ . We assume that $p_{t}$ remains fixed throughout any plastic deformation process. By contrast, the compressive strength, $p_{c}$ , evolves as a result of compaction (increase in density) or dilation (reduction in density) of the material. The evolution of the yield surface can be expressed through the evolution of the yield surface size on the hydrostatic stress axis, ${p_{c} + p}_{t}$ , as a function of the value of volumetric compacting plastic strain, $- ε_{v o l}^{p l}$ . Assuming that $p_{t}$ is constant and that $ε_{a x i a l}^{p l} = ε_{v o l}^{p l}$ in uniaxial compression, the compressive strength ( $p_{c}$ ) can be written as a function of uniaxial compression, $ε_{v o l}^{p l}$ :

p_{c} (ε_{v o l}^{p l}) = \frac{σ_{c} (ε_{a x i a l}^{p l}) [σ_{c} (ε_{a x i a l}^{p t}) (\frac{1}{α^{2}} + \frac{1}{9}) + \frac{p_{t}}{3}]}{p_{t} + \frac{σ_{c} (ε_{a x i a l}^{p l})}{3}} .

Thus, you provide input to the hardening law by specifying, in the usual tabular form, only the value of the yield stress in uniaxial compression as a function of the absolute value of the axial plastic strain. The table must start with a zero plastic strain (corresponding to the virgin state of the material), and the tabular entries must be given in ascending magnitude of $ε_{a x i a l}^{p l}$ . If required, the yield stress can also be a function of temperature and other predefined field variables. In this case, the model requires that the values of the strength ratios $k$ and $k_{t}$ be also specified for the same values of temperature and predefined field variables.

The yield surface for isotropic hardening is defined as

F = \sqrt{q^{2} + α^{2} p^{2}} - B = 0,

where $p = - \frac{1}{3} t r a c e (σ)$ is the pressure stress, $q = \sqrt{\frac{3}{2} S : S}$ is the von Mises stress, $S = σ + p I$ is the deviatoric stress, $A = \frac{p_{c} + p_{t}}{2}$ is the size of the $p$ -axis of the yield ellipse, $B = α p_{c} = σ_{c} \sqrt{1 + {(\frac{α}{3})}^{2}}$ is the size of the (vertical) $q$ -axis of the yield ellipse, $α$ is the shape factor of the yield ellipse that defines the relative magnitude of the axes, $p_{c}$ is the yield stress in hydrostatic compression, and $σ_{c}$ is the absolute value of the yield stress in uniaxial compression.

If isotropic hardening is chosen, a simple uniaxial compression test is sufficient to define the evolution of the yield surface. The hardening law defines the value of the yield stress in uniaxial compression as a function of the absolute value of the axial plastic strain. The piecewise linear relationship is entered in tabular form. The table must start with a zero plastic strain (corresponding to the virgin state of the materials), and the tabular entries must be given in ascending magnitude of $ε_{a x i a l}^{p l}$ . For values of plastic strain greater than the last user-specified value, the stress-strain relationship is extrapolated based on the last slope computed from the data.


Input Data	Description
Yield Stress	$σ_{c}$ , yield stress in uniaxial compression, given as a positive value.
Plastic Strain	Absolute value of the corresponding plastic strain. (The first tabular value entered must always be zero.)
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on one or more independent field variables. A Field column appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.