Heat flux averaging revisited: Virtual power equivalence and size effect

In analogy to the famous stress average rule in continuum micromechanics, it has become popular to use a heat flux averaging rule for multiscale thermodynamics: The spatial average of a microscopic heat flux field over a finite microscopic representative volume element (RVE) is set equal to a macroscopic heat flux vector. The latter is associated with a macroscopic infinitesimal volume element d V which is physically equivalent to the RVE. Here we explore the validity range of such a heat flux average rule. Therefore, we introduce microscopic virtual fields dualizing the microscopic heat balance law (i.e., the first law of thermodynamics for stationary iso-deformation conditions). The microscopic virtual fields are then “coarsened” through macroscopic virtual quantities, while requiring equivalence between the microscopic and the macroscopic formulations of the virtual powers expressing heat balance. It turns out that the heat flux average rule holds only in case of vanishing microscopic volume heat sources. Vanishing volume heat sources, together with Fourier’s heat conduction law, imply the RVE to be smaller than the ratio of the first-order to the second-order temperature gradient, which is often referred to as separation-of-scales-condition. Once significant volume heat sources occur, they do not only enter the average law for the macroscopic heat flow, but also necessitate the existence of higher-order macroscopic heat fluxes, which we call hyper-fluxes here. The latter are dual to second-order gradients of the macroscopic virtual field quantities. Associated RVEs do not fulfill the separation-of-scales-conditions, but exhibit a size which is of the order of the ratio of the first-order to the second-order temperature gradient. Hyper-fluxes provide new avenues for modeling the effects of heat sources occurring within the microstructure of a material, on the behavior of structures being built up by such materials. This is illustrated here by two examples which show the difference between concentrated and distributed microscopic heat source terms in terms of the resulting macroscopic hyper-flux quantities. The latter are deemed to be relevant for several practical cases, including wood-based energy sources and timber structures exposed to fire.