Microstructural randomness versus representative volume element in thermomechanics

Continuum thermomechanics hinges on the concept of a representative volume element (RVE), which is well defined in two situations only: (i) unit cell in a periodic microstructure, and (ii) statistically representative volume containing a very large (mathematically infinite) set of microscale elements (e.g., grains). Response of finite domains of material, however, displays statistical scatter and is dependent on the scale and boundary conditions. In order to accomplish stochastic homogenization of material response, scale-dependent hierarchies of bounds are extended to dissipative/irreversible phenomena within the framework of thermomechanics with internal variables. In particular, the free-energy function and the dissipation function become stochastic functionals whose scatter tends to decrease to zero as the material volume is increased. These functionals are linked to their duals via Legendre transforms either in the spaces of ensemble average velocities or ensemble-average dissipative forces. In the limit of infinite volumes (RVE limit (ii) above) all the functionals become deterministic, and classical Legendre transforms of deterministic thermomechanics hold. As an application, stochastic continuum damage mechanics of elastic-brittle solids is developed.