= Solution
In <linear viscoelasticity>, <stress> is a linear functional of past <strain> increments: scaling the deformation scales the response, and responses to small deformations add. For a time-translation-invariant material, <Boltzmann superposition> writes $\sigma(t)=\int_{-\infty}^tG(t-s)\dot\epsilon(s)ds$, where $G$ is a <relaxation modulus>. The material combines reversible storage with delayed or dissipative response; the description is restricted to small deformations about the chosen state.
The <Maxwell model> places an elastic <spring> and viscous <dashpot> in series. They carry the same <shear stress> $\sigma$, and their <shear strains> add:
$$
\epsilon=\epsilon_e+\epsilon_v,\qquad
\epsilon_e=\frac\sigma{G_0},\qquad
\dot\epsilon_v=\frac\sigma\eta.
$$
Differentiating gives
$$
\boxed{\dot\epsilon=\frac{\dot\sigma}{G_0}+\frac\sigma\eta,\qquad\tau=\frac\eta{G_0}.}
$$
Here $\epsilon$ is dimensionless <shear strain>, $\sigma$ <shear stress> in pascals, $G_0$ the <spring> <shear modulus>, $\eta$ <dashpot> <viscosity> in $\mathrm{Pa\,s}$, and $\tau$ the <relaxation time>. The first term describes changing elastic <strain>; the second is viscous flow. In this model the material flows at long <times> and has zero static <shear modulus>.
Back to article page
