Linear viscoelasticity

stress is a linear causal functional of deformation history: in simple shear, $\sigma (t)={\int }_{-\infty }^{t}G(t-t^{\prime })\dot{\gamma }(t^{\prime })d{t}^{\prime }$. Superposition holds and complex viscoelastic modulus is amplitude-independent within the small-deformation regime.

The characteristic time for a delayed strain response to approach equilibrium under a step in stress. For a Kelvin-Voigt model with elastic modulus $E$ and viscosity $\eta$, the equation $\eta \dot{\epsilon }+E\epsilon ={\sigma }_0$ gives $\epsilon =({\sigma }_0/E)(1-e^{-t/{\tau }_r})$, where ${\tau }_r=\eta /E$. A relaxation time instead describes decay of stress after an imposed strain; the two times need not coincide in a standard linear solid model.

A four-element linear viscoelasticity model with a spring $E_0$, a Kelvin-Voigt model branch $E_1,{\eta }_1$, and a free dashpot ${\eta }_0$ in series. Adding their strains under common stress gives the displayed creep compliance, with $\tau ={\eta }_1/E_1$. There is an instantaneous elastic response, a delayed recoverable response and steady viscous flow.^[ref] Named after Johannes Martinus Burgers, it extends a Maxwell model with a delayed elastic branch, or a standard linear solid model with a free dashpot.

For a load ${\sigma }_0$ maintained for time $t_h$, the two recoverable contributions behave as in recovery after a finite-duration stress load, while the free dashpot retains ${\sigma }_0{t}_h/{\eta }_0$. HenceThis residual response models viscous flow; it does not by itself establish a plastic yield threshold or a damage mechanism in a real specimen.