Invariants of mesoscale thermal conductivity and resistivity tensors in random checkerboards

Abstract

Purpose – The purpose of this paper is to study the statistics of thermal conductivity and resistivity tensors in two-phase random checkerboard microstructures at finite mesoscales. Design/methodology/approach – Microstructures at finite scales are generated by randomly sampling an infinite checkerboard at 50 percent nominal fraction. Boundary conditions that stem from the Hill-Mandel homogenization condition are then applied as thermal loadings on these microstructures. Findings – It is observed that the thermal response of the sampled microstructures is in general anisotropic at finite mesoscales. Based on 1,728 boundary value problems, the statistics of the tensor invariants (trace and determinant) are obtained as a function of material contrast, mesoscale and applied boundary conditions. The histograms as well as the moments (mean, variance, skewness and kurtosis) of the invariants are computed and discussed. A simple analytical form for the variance of the trace of mesoscale conductivity tensor is proposed as a function of individual phase conductivities and the mesoscale. Originality/value – A rigorous methodology to determine the evolution of the invariants of thermal conductivity (and resistivity) tensors across a variety of length scales (microscale to macroscale) is presented. The objective is to enable setting up of constitutive equations applicable to heat conduction that are valid across all length scales.