Standard linear solid model

Example

The standard linear solid, also called the Zener model, is a spring of modulus $E_\infty > 0$ in parallel with a Maxwell material arm. The arm consists of a spring of modulus $E_1 > 0$ and a dashpot of viscosity $\eta > 0$ in series. Write $\sigma = E_\infty \varepsilon + \sigma_1$, where the arm stress satisfies $\dot\varepsilon = \dot\sigma_1 / E_1 + \sigma_1 / \eta$. Eliminating $\sigma_1$ gives $$\sigma + \frac{\eta}{E_1} \dot\sigma = E_\infty \varepsilon + \frac{\eta (E_\infty + E_1)}{E_1} \dot\varepsilon.$$ Relaxation: a strain step $\varepsilon_0$ gives $\sigma_1(t) = E_1 \varepsilon_0 e^{-t/\tau}$ with $\tau = \eta/E_1$, so the relaxation modulus is $$G(t) = E_\infty + E_1 e^{-t/\tau}.$$ Creep: a stress step $\sigma_0$ gives $\varepsilon(0^+) = \sigma_0 / (E_\infty + E_1)$, then the first-order equation $E_\infty \varepsilon + \frac{\eta (E_\infty + E_1)}{E_1} \dot\varepsilon = \sigma_0$, so $\varepsilon$ rises exponentially to $\sigma_0 / E_\infty$ with time constant $\eta (E_\infty + E_1)/(E_1 E_\infty)$.