= Passivity in linear viscoelasticity
{title2=$G''(\omega)\geq0\quad(\omega>0)$}
A passive <viscoelasticity> law cannot deliver net <work> in a periodic steady deformation without an external source of energy. With <strain> $\epsilon=\epsilon_0\cos\omega t$ and <stress> $\sigma=\epsilon_0(G'\cos\omega t-G''\sin\omega t)$, the cycle-averaged supplied power is $\langle\sigma\dot\epsilon\rangle=\omega\epsilon_0^2G''/2$. Thus a necessary passivity condition, using the convention $e^{i\omega t}$, is a nonnegative <loss modulus> at every positive frequency. Equivalently the <work> per cycle is $\pi\epsilon_0^2G''$. A stable differential equation for stress does not by itself ensure this energetic condition.
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