Closed-form solutions for the effective conductivity of two-phase periodic composites with spherical inclusions

Abstract

In this paper, we use approximate solutions of NematNasser et al. to estimate the effective conductivity of two-phase periodic composites with non-overlapping spherical inclusions. Systems with different inclusion distributions are considered: cubic lattice distributions (simple cubic, body-centred cubic and face-centred cubic) and random distributions. The effective conductivities of the former are obtained in closed form and compared with exact solutions from the fast Fourier transform-based methods. For systems containing randomly distributed spherical inclusions, the solutions are shown to be directly related to the static structure factor, and we obtain its analytical expression in the infinite-volume limit.

A numerical-analytical coupling computational method for homogenization of effective thermal conductivity of periodic composites

“…It is clear that, from (26), the effective conductivity is obtained in the same form as in the previous work [25].…”

“…Such an approximation has been shown to predict very well the overall elastic and thermal properties of two-phase composites for a large range of volume fractions of inclusions [25,29]. However, it fails at higher concentrations.…”

“…Instead of finding the full field solution of (1), the mean value of e * and the effective thermal conductivity K eff can be estimated from (7) with NIH approximation [24,25]. Such an approximation has been shown to predict very well the overall elastic and thermal properties of two-phase composites for a large range of volume fractions of inclusions [25,29].…”

“…In this paper, the remainder of the Neumann series is estimated by a combination between FFT schemes and the Nemat-Nasser-Iwakuma-Hejazi (NIH) [24,25] estimation of the effective properties. For two-phase systems with spherical inclusions, the NIH estimation in thermal problems leads to closed-form solutions which agree with numerical results for a large range of volume fractions.…”

Heat conduction and thermal conductivity of 3D cracked media

“…The analytical solution of temperature discontinuity allows estimating the effective thermal conductivity in the framework of various homogenization schemes: dilute, differential and self-consistent. The details of these techniques for thermal properties of porous geomaterial were synthesized in Do et al [32][33][34], To et al [35,36], Nguyen [37] and Chen [38]. Effective properties of micro-cracked viscoelastic materials are also successfully estimated by coupling the solution of a single crack in infinite domain and the homogenization schemes (Nguyen et al [39,40]).…”