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Viscoelasticity

This chapter describes the viscoelastic constitutive law adopted by FrontISTR. For details on selection and input specifications, see Material Data in the Functions section.

Phenomenon (Viscoelastic Response)

TBD (to be written in the next phase).

Generalized Maxwell Model

FrontISTR adopts a generalized Maxwell model. Let the deviatoric strain be \(e\) and, on branch \(m\), let the deviatoric viscous strain be \(q^{(m)}\); define the deviatoric elastic strain \(h^{(m)}\) as

\[ h^{(m)} \equiv e - q^{(m)} \]

Then the constitutive equation becomes a function of \(e\) and \(h^{(m)}\) as follows.

\[ \sigma(t) = K \, \operatorname{tr}\varepsilon \, I + 2 G_0 (\mu_\infty e(t) + \mu h) \]

where

\[ \mu h = \sum_{m=1}^{M} \mu_m h^{(m)}(t), \qquad \mu_\infty + \sum_{m=1}^{M} \mu_m = 1 \]

Also, \(h^{(m)}\) is obtained from

\[ \dot{h}^{(m)}(t) + \frac{1}{\lambda_m} h^{(m)}(t) = \dot{e}(t) \]

where \(\lambda_m\) is the relaxation time.

Prony Series

The relaxation modulus \(G\) is represented by the following Prony series.

\[ G(t) = G_0 \left[ \mu_\infty + \sum_{m=1}^{M} \mu_m \exp\!\left( \frac{-t}{\lambda_m} \right) \right] \]