Hyper Visco-elastic Law for Foams (LAW62)

Experimental tests on foam specimens working in compression illustrate that the material behavior is highly nonlinear. The general behavior can be subdivided into three parts related to particular deformation modes of material cells. When the strain is small, the cells working in compression deform in membrane without causing buckling in its lateral thin-walls. In the second step, the lateral thin-walls of the cells buckle while the material undergoes large deformation. Finally, in the last step the cells are completely collapsed and the contact between the lateral thin-walled cells increases the global stiffness of the material.

As the viscous behavior of foams is demonstrated by various tests, it is worthwhile to elaborate a material law including the viscous and hyper elasticity effects. This is developed in ¹ where a decoupling between viscous and elastic parts is introduced by using finite transformations. Only the deviatoric part of the stress tensor is concerned by viscous effects.

Material LAW62 corresponds to a hyper-elastic solid material using the Ogden formulation for rubber material. The strain energy functional ² is given by:(1)

W (C) = \sum_{i = 1}^{N} \frac{2 μ_{i}}{α_{i}^{2}} (λ_{1}^{α_{i}} + λ_{2}^{α_{i}} + λ_{3}^{α_{i}} - 3 + \frac{1}{β} (J^{- α_{i} β} - 1))

Where,

C

is the right Cauchy Green Tensor,

C = F^{t} F

with

F

the deformation gradient matrix,

λ_{i}

are the eigenvalues of

F

J = \det F

β = \frac{ν}{(1 - 2 ν)}

ν \neq 0

and

ν \neq \frac{1}{2}

Note: For rubber materials which are almost incompressible, the bulk modulus is very large compared to the shear modulus.

The ground shear modulus is given by:(2)

μ = \sum_{i = 1}^{N} μ_{i}

W

can be written as:(3)

W (C) = \bar{W} (\bar{C}) + U (J)

Where,

$\bar{C} = {\bar{F}}^{t} \bar{F}$
$\bar{F} = J^{- 1 / 3} F$
$\bar{C}$: Deviatoric part of the right Cauchy Green Tensor
$U$ and $\bar{W}$: Volumetric and deviatoric parts of the stored energy functions and $S_{0}$ the second Piola-Kirchhoff stress tensor given by:

(4)

S_{0} = \frac{\partial W}{\partial E} = 2 \frac{\partial W}{\partial C} = 2 \frac{\partial \bar{W}}{\partial C} + 2 \frac{\partial U}{\partial C} = S_{0}^{d e v} + S_{0}^{v o l}

With $E = \frac{1}{2} (C - I)$

The Green-Lagrange strain tensor:

$S_{0}^{d e v} = 2 \frac{\partial \bar{W}}{\partial C}$ and $S_{0}^{v o l} = 2 \frac{\partial U}{\partial C}$ are the deviatoric and volumetric parts of the second Piola-Kirchhoff stress tensor $S_{0}$ .

Rate effects are modeled through visco-elasticity using a convolution integral using Prony series. This corresponds to an extension of small strain theory or finite deformation to large strain. The rate effect is applied only to the deviatoric stress. The deviatoric stress is computed as:(5)

S^{d e v} (t) = γ_{\infty} S_{0}^{d e v} (t) - J^{- 2 / 3} D E V [\sum_{i = 1}^{M_{i}} Q_{i} (t)]

Where,

Q_{i}

is the internal variable given by the following rate equations:(6)

{\dot{Q}}_{i} (t) + \frac{1}{τ_{i}} Q_{i} (t) = \frac{γ_{i}}{τ_{i}} D E V [2 \frac{\partial \bar{W}}{\partial \bar{C}} (t)]

$\lim Q_{i} (t) = 0$ , $t \to - \infty$

$γ_{i} \in [0, 1]$ , $τ_{i} > 0$

Q_{i}

is given by the following convolution integral:(7)

Q_{i} (t) = \frac{γ_{i}}{τ_{i}} \int_{- \infty}^{t} \exp [- (t - s) / τ_{i}] \frac{d [D E V {2 \partial_{\bar{C}} {\bar{W}}^{0} [\bar{C} (s)]}]}{d s} d s

Where,

$γ_{\infty} = G_{\infty} / G_{0}$
$1 = γ_{\infty} + \sum_{i = 1}^{M_{i}} γ_{i}$
$γ_{i} = G_{i} / G_{0}$
$G_{0} = G_{\infty} + \sum_{i = 1}^{M_{i}} G_{i}$
$d e v (•) = • - \frac{1}{3} (• : C) C^{- 1}$

Where, $G_{0}$ is the initial shear modulus; $G_{0}$ should be exactly the same as the ground shear modulus $μ$ . $G_{\infty}$ is the long-term shear modulus that can be obtained from long-term material testing. $τ_{i}$ are the relaxation times.

The relation between the second Piola-Kirchhoff stress tensor

S = S^{d e v} + S_{0}^{v o l}

and Cauchy stress tensor

σ

is:(8)

σ = \frac{1}{\det F} F S F^{t}

¹ Simo J.C., “On a fully three-dimensional finite strain viscoelastic damage model: Formulation and Computational Aspects”, Computer Methods in Applied Mechanics and Engineering, Vol. 60, pp. 153-173, 1987.

² Ogden R.W., “Nonlinear Elastic Deformations”, Ellis Horwood, 1984.