Scaling function in conductivity of planar random checkerboards

Under investigation is the finite-size scaling of the Fourier thermal conductivity in two-phase planar random checkerboard microstructures at 50% nominal volume fraction. Examples considered include Aluminum–Copper, Constantan–Lead, Stainless Steel–Gold, Inconel X-750–Aluminum, Titanium Dioxide–Gold, Carbon Steel–Diamond, Lead–Diamond, Boron–Diamond, Molybdenum–Test, Constantan–Diamond. Mesoscale bounds are obtained using an approach consistent with the Hill–Mandel homogenization condition. Extensive numerical simulations are conducted on 10 types of microstructures with the contrast (k) ranging from 1.54 to 100. The effects of mesoscale (δ) and phases’ contrast are evaluated and generic scaling laws are established quantitatively. This is accomplished using a non-dimensional scaling function derived by contracting the mesoscale conductivity and resistivity tensors. The scaling function very closely fits all the material combinations and is given by g ( δ , k ) = 1 2 ( k - 1 / k ) 2 exp [ - 0.53 ( δ - 1 ) 0.69 ] . As a verification of our procedure, it is observed that, with increasing domain size, the mesoscale conductivity tends to the exact theoretical result for macroscopic conductivity of random checkerboards: being the geometric mean of two phases. By choosing an appropriate functional form of the scaling function, a material scaling diagram is constructed with which one can rapidly estimate the size of representative volume element for a given contrast within acceptable accuracy.