Relaxation modulus

Remarks

Example: for the law $\dot\varepsilon = \dot\sigma / E + \sigma / \eta$ of a spring and a dashpot in series, the step gives $\sigma(0^+) = E \varepsilon_0$ and then $\dot\sigma = -\sigma E / \eta$, so $G(t) = E e^{-t/\tau}$ with $\tau = \eta / E$. Non-example: for a nonlinear law such as $\sigma = E \varepsilon + c \varepsilon^3$ with $c \neq 0$, the ratio $\sigma(t)/\varepsilon_0$ depends on $\varepsilon_0$, so no relaxation modulus exists. The creep compliance $J(t)$ is defined in the same way from a stress step $\sigma(t) = \sigma_0 H(t)$, by $\varepsilon(t) = \sigma_0 J(t)$. The two functions are linked through the Laplace transform by $\hat G(s) \hat J(s) = 1/s^2$, or equivalently $\int_0^t G(t - u) J(u) \, du = t$ for $t > 0$.