Thermodynamic Restrictions on Isothermal Linear Viscoelasticity

Abstract

In the linearised theory of isothermal viscoelasticity the stress tensor is determined by the history of the infinitesimal strain tensor through the hereditary law¹

$$ T(t) = T\left( {t,E\left( \cdot \right)} \right) = \mathcal{G}(0)E(t) + \int\limits_0^{{ + \infty }} {\dot{\mathcal{G}}(s)E\left( {t - s} \right)ds} $$

(6.1.1)

Here G(·) is the relaxation function of the material; its values are fourth order tensor. In the suffix notation (6.1.1) becomes

$$ {T_{{ij}}}(t) = \sum\limits_{{k,l}} {\left\{ {{\mathcal{G}_{{ijkl}}}(0){E_{{kl}}}(t) + \int\limits_0^{{ + \infty }} {{{\dot{\mathcal{G}}}_{{ijkl}}}(s){E_{{kl}}}\left( {t - s} \right)ds} } \right\}} $$

(6.1.2)

Because the tensors T and E are symmetric it can be assumed that G(·) satisfies the conditions

$$ {\mathcal{G}_{{ijkl}}}\left( \cdot \right) = {\mathcal{G}_{{jikl}}}\left( \cdot \right) = {\mathcal{G}_{{ijlk}}}\left( \cdot \right) $$

(6.1.3)

which imply that at most 36 of the 81 functions G(·) can be independent.