Maxwell material

Example

The Maxwell material is the model of a linear spring of modulus $E > 0$ in series with a linear dashpot of viscosity $\eta > 0$. In series the stress $\sigma$ is the same in both elements and the strains add, so with $\sigma = E \varepsilon_1$ and $\sigma = \eta \dot\varepsilon_2$ one gets $$\dot\varepsilon = \frac{\dot\sigma}{E} + \frac{\sigma}{\eta}.$$ Let $\tau = \eta / E$ be the relaxation time. Relaxation: a strain step $\varepsilon_0$ at time $0$ gives $\sigma(0^+) = E \varepsilon_0$, and then $\dot\varepsilon = 0$ gives $\dot\sigma = -\sigma / \tau$, so the relaxation modulus is $G(t) = E e^{-t/\tau}$. Creep: a stress step $\sigma_0$ gives $\varepsilon(t) = \sigma_0 / E + \sigma_0 t / \eta$, so the creep compliance is $J(t) = 1/E + t/\eta$. The stress depends on the past strain rates and fades with time, so the Maxwell material is viscoelastic.