Generalized Maxwell model

Example

The generalized Maxwell model, also called the Maxwell–Wiechert model, is a spring of modulus $E_\infty \ge 0$ in parallel with $n$ Maxwell material arms. Arm $i$ has a spring of modulus $E_i > 0$ and a dashpot of viscosity $\eta_i > 0$, and relaxation time $\tau_i = \eta_i / E_i$. In parallel the strain is common and the stresses add. Under a strain step $\varepsilon_0$, arm $i$ carries the stress $E_i \varepsilon_0 e^{-t/\tau_i}$ and the lone spring carries $E_\infty \varepsilon_0$. So the relaxation modulus is $$G(t) = E_\infty + \sum_{i=1}^n E_i e^{-t/\tau_i},$$ which is a Prony series when the $\tau_i$ are distinct.