Towards stochastic continuum thermodynamics.
Two lines of studies for dealing with continua with a microstructure are compared. The first one is the line developed by the author and his coworkers introducing stochastic continuum mechanics with random microstructural features on a scale smaller than the representative volume element. The other line is the mesoscopic concept introducing field quantities on an enlarged domain, a continuum mechanical frame on that enlarged domain, and an averaging procedure over the additional variables. As a common basis for connecting both approaches a mesoscale statistical volume element with random functions on it is proposed. In the case of vanishing fluctuations the classical representative volume element is recovered. For the response of the statistical volume element bonds are found via two admissible loading. Employing the paradigm of thermal conductivity the relevant Legendre transformations are presented. The latter are then generalized to the more general situation of fields governed by a quartet of Legendre transformations. The results can be summarized as follows: A formulation of continuum thermodynamics with internal variables of random media is outlined, where the intrinsic entropy production is a bilinear form of forces and fluxes. When the rates of internal variables are present in a mesoscale thermodynamical model, field variational principles of mechanics cannot be straightforwardly obtained. A method for dealing with this callenge has been outlined. It involves an ensemble averaging performed on the quartet of Legendre transformations for the random functional. This yields bounds on the response. Theoretically, as the mesoscale parameter goes to infinity, the statistical volume element tends to the representative volume element, albeit the later may not always be attainable. An example of this is the phenomenon of localization in a random microstructure, whereby the representative volume element in the conventional sense of a deterministic field theory may not exist.
scientific article; zbMATH DE number 1942718