Thermodynamic potentials for viscoelastic composites

Explicit expressions for the free-energy and dissipation densities of viscoelastic composites at fixed temperature are proposed. The composites are comprised of an arbitrary number of distinct constituents exhibiting linear Maxwellian rheologies and distributed randomly at a length scale that is much smaller than that over which applied loads vary significantly. Central to their derivation is the recognition that any viscous deformation field can be additively decomposed into an irrotational field and a solenoidal field in such a way that variational approximations available for elastic potentials become applicative to viscoelastic potentials. The thermodynamic potentials conform to a generalized standard model with a finite number of effective internal variables with explicit physical meaning. Specific approximations of the Hashin–Shtrikman and the Self-Consistent types are worked out in detail. Under particular circumstances, these approximations may turn out exact. Macroscopic stress–strain relations and intraphase statistics of the stress field up to second order are also provided.