Kelvin–Voigt material

Example

The Kelvin–Voigt material is the model of a linear spring of modulus $E > 0$ in parallel with a linear dashpot of viscosity $\eta > 0$. In parallel the strain $\varepsilon$ is the same in both elements and the stresses add, so $$\sigma = E \varepsilon + \eta \dot\varepsilon.$$ Let $\tau = \eta / E$ be the retardation time. Creep: a stress step $\sigma_0$ at time $0$ with $\varepsilon(0) = 0$ gives the linear equation $\tau \dot\varepsilon + \varepsilon = \sigma_0 / E$, whose solution is $$\varepsilon(t) = \frac{\sigma_0}{E} \left(1 - e^{-t/\tau}\right),$$ so the creep compliance is $J(t) = (1 - e^{-t/\tau})/E$. The strain approaches the elastic value $\sigma_0/E$ with a delay, which is the rate dependence of a viscoelastic material.