Effective thermodynamic potentials and internal variables: Linear viscoelastic composites

Abstract

New theoretical relations in linear viscoelasticity are derived by combining two different points of view. On the one hand, the general thermodynamic framework makes it possible to define the energy stored and the energy dissipated in linearly viscoelastic composites. On the other hand, the correspondence principle permits to express the macroscopic strain–stress relation as ordinary differential equations for a set of effective internal variables. A finite and small number of internal variables is rigorously sufficient in several cases of interest, including in particular particulate composites. Interpreting the macroscopic response as a rheological generalized Maxwell model allows us to compute the macroscopic free and dissipated energy of the composite. This interpretation is proved to be exact in several cases of interest. Coupled with Hashin–Shtrikman estimates, these thermodynamic functions provide additional information on the statistics of the field within each individual phase of the composite.