Viscoelasticity¶
This chapter describes FrontISTR's viscoelastic constitutive law. Selection and input details are in the Material Data chapter.
Generalized Maxwell Model¶
FrontISTR adopts a generalized Maxwell model. Letting \(e\) be the deviatoric strain and \(q^{(m)}\) the deviatoric viscous strain on branch \(m\), define the deviatoric elastic strain \(h^{(m)}\) as
\[
h^{(m)} \equiv e - q^{(m)}.
\]
The constitutive equation is then written in terms of \(e\) and \(h^{(m)}\):
\[
\sigma(t) = K \, \operatorname{tr}\varepsilon \, I + 2 G_0 (\mu_\infty e(t) + \mu h),
\]
where
\[
\mu h = \sum_{m=1}^{M} \mu_m h^{(m)}(t), \qquad \mu_\infty + \sum_{m=1}^{M} \mu_m = 1,
\]
and \(h^{(m)}\) satisfies
\[
\dot{h}^{(m)}(t) + \frac{1}{\lambda_m} h^{(m)}(t) = \dot{e}(t),
\]
with \(\lambda_m\) the relaxation time.
Prony Series¶
The relaxation modulus \(G\) is represented by a Prony series:
\[
G(t) = G_0 \left[ \mu_\infty + \sum_{m=1}^{M} \mu_m \exp\!\left( \frac{-t}{\lambda_m} \right) \right].
\]