Scaling and bounds in thermal conductivity of planar Gaussian correlated microstructures

Abstract

In this study 2d two phase microstructures closely resembling the experimentally captured micrographs of the interpenetrating phase composites are generated using a Gaussian correlation function based method. The scale dependent bounds on the effective thermal conductivity of such microstructures are then studied using Hill-Mandel boundary conditions. A scaling function is formulated to describe the transition from statistical volume element (SVE) to representative volume element (RVE), as a function of the mesoscale δ, the correlation length of the Gaussian correlation function λ, the volume fraction v, and the contrast k between the phases. The scaling function is determined through fitting the data from extensive simulations conducted over the parameter space. The scaling function shows that SVE approaches RVE as (δ/λ)−1.16. A material scaling diagram allows estimation of the RVE size, to within a chosen accuracy, of a given microstructure characterized by the correlation length of the Gaussian correlation function, contrast, and volume fraction of the phases.

Frequency-dependent scaling from mesoscale to macroscale in viscoelastic random composites

“…One extreme case δ = 1 signifies the description at the level of one grain, while the second extreme δ → ∞ is the RVE limit, and, in general, one wants to know with what precision is the RVE attained at any finite δ for specific physical and geometric parameters. Numerically generated micrograph of Gaussian correlated microstructure [33].…”

“…To better resolve the stress and strain distributions at the interface of two phases, quadratic elements are used instead of bilinear elements; the mesh density becomes more important in ensuring the accuracy of the solution obtained using FEM in the case of strong mismatch in the phases' properties [33]. For our microstructure of a random chessboard at volume fraction 0.5, a comprehensive mesh sensitivity analysis was conducted in Zhang & Ostoja-Starzewski [9], where it was found that, for the chosen constituent properties, the mesh density of 4 × 4 elements per one square of the chessboard provides a stable solution and sufficient accuracy.…”

“…without any spatial correlation between adjacent points. However, our methodology also applies to microstructures with spatial correlations such as that shown in figure 1c [33].…”

“…33) where J and K have been defined in(2.19). A contraction of C * δ and S * δ in (3.33) leads to a scaling function f…”