# Boltzmann Viscoelastic Model (LAW34)

This law valid for solid elements can be used for viscoelastic materials like polymers, elastomers, glass and fluids.

Elastic bulk behavior is assumed. Air pressure may be taken into account for closed cell foams: (1)
$P=-K{\epsilon }_{kk}+{P}_{air}$
with: (2)
${P}_{air}=-\frac{{P}_{0}\gamma }{1+\gamma -\Phi }\text{ };\text{ }\gamma =\frac{V}{{V}_{0}}-1+{\gamma }_{0}$
and: (3)
${\epsilon }_{kk}=\mathrm{ln}\left(\frac{V}{{V}_{0}}\right)$
Where,
$\gamma$
Volumetric strain
$\text{Φ}$
Porosity
${P}_{0}$
Initial air pressure
${\gamma }_{0}$
Initial volumetric strain
$K$
Bulk modulus
For deviatoric behavior, the generalized Maxwell model is used. The shear relaxation moduli in Viscous Materials, 式 19 is then defined as:(4)
$\Psi \left(t\right)={G}_{l}+{G}_{s}{e}^{-\beta t}$
(5)
${G}_{s}={G}_{0}-{G}_{l}$
Where,
${G}_{0}$
Short time shear modulus
${G}_{l}$
Long time shear modulus
$\beta$
Decay constant, defined as the inverse of relaxation time ${\tau }_{s}$ :(6)

$\beta =\frac{1}{{\tau }_{s}}$ ; with ${\tau }_{s}=\frac{{\eta }_{s}}{{G}_{s}}$

The coefficients ${\eta }_{s}$ , ${G}_{s}$ and ${G}_{l}$ are defined for the generalized Maxwell model, as shown in 図 1.

From 式 4, the value of $\beta$ governs the transition from the initial modulus ${G}_{0}$ to the final modulus ${G}_{l}$ . For $t$ =0, you obtain $\Psi \left(t\right)\to {G}_{0}$ and when $t\to \infty$ , then $\Psi \left(t\right)\to {G}_{l}$ . For a linear response, put ${G}_{0}={G}_{l}$ .