On the scaling from statistical to representative volume element in thermoelasticity of random materials

Under consideration is the finite-size scaling of effective thermoelastic properties of random microstructures from a Statistical Volume Element (SVE) to a Representative Volume Element (RVE), without invoking any periodic structure assumptions, but only assuming the microstructure's statistics to be spatially homogenous and ergodic. The SVE is set up on a mesoscale, i,e. any scale finite relative to the microstructural lenght scale. The Hill condition generalized to thermoelasticity dictates uniform Neumann and Dirichlet boundary conditioned, which, with the help of two variational principles, lead to scale dependant hierarchies of mesoscale bounds on effective (RVE level) properties: thermal expansion and stress coefficients, effective stiffness, and specific heats. Due to the presence of a non-quadratic term in the energy formulas, the mesoscale bounds to the thermal expansion and more complicated than those for the stiffness tensor and the heat capacity. To quantitatively asses the scaling trend towards the RVE, the hierarchies are computed for a planar matrix-inclusion composite, with inclusions (of circular disk shape) located at points of planar, hard core Poisson point field. Overall, while the RVE is attained exactly on scales infinitely large relative to the microscale, depending on the microstructural parameters, the random fluctuations in the SVE response may become very weak on scales an order of magnitude larger than the microscale, thus already approximating the RVE.