Towards stochastic continuum thermodynamics

We explore the relation of a line of studies over the last dozen years on mesoscale material response below the Representative Volume Element with the line of studies on mesoscopic continuum physics. The objective is the development of stochastic continuum thermodynamics able to grasp random microstructural features absent in deterministic continuum theories. We first compare two lines of studies - those dealing with the response on a scale smaller than the Representative Volume Element (RVE) vis-à-vis those on mesoscopic continuum physics - and observe the dilemma of choice of the random field approximation. As a common basis for connecting both approaches, we propose a mesoscale Statistical Volume Element (SVE) and consider bounding its response via two admissible loadings. Employing the paradigm of thermal conductivity, we also present the relevant Legendre transformations. The latter are then generalized to the more general situation of fields governed by a quartet of Legendre transformations.