A Singularly Perturbed Nonideal Transmission Problem and Application to the Effective Conductivity of a Periodic Composite

Abstract

We investigate the effective thermal conductivity of a two-phase composite with thermal resistance at the interface. The composite is obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material. The diameter of each inclusion is assumed to be proportional to a positive real parameter ǫ. Under suitable assumptions, we show that the effective conductivity can be continued real analytically in the parameter ǫ around the degenerate value ǫ = 0, in correspondence of which the inclusions collapse to points.

Effective conductivity of a singularly perturbed periodic two-phase composite with imperfect thermal contact at the two-phase interface

“…For a proof, we refer to [24]. Here, we note that if ε/ρ(ε) has a real analytic continuation around 0, then the term in the right hand side of equality (4) defines a real analytic function of the variable ε in the whole of a neighbourhood of 0.…”

“…Moreover, we give in the following Theorem 5 more information on λ eff k j [ε] for ε close to 0 by expressing Λ k j [0, r * ] by means of a certain quantity related to the solutions of a limiting transmission problem (for a proof we refer to [24]). …”

“…By means of the Implicit Function Theorem for real analytic maps in Banach spaces, we can analyse the dependence upon ε of the solutions of the system of integral equations and we can prove our main results. Further details will be presented in a forthcoming paper by the authors (see [24]). …”

Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole

“…Later on, the approach has been extended to nonlinear traction problems in elastostatics (cf., e.g., [8]), to the Stokes' flow (cf., e.g., [7]), and to the case of an infinite periodically perforated domain, also in presence of nonlinear boundary conditions (cf., e.g., [11] and [30]). Moreover, this technique has been applied to the analysis of effective properties of dilute composite materials (see [10]). The aim of the present paper is to show that such an approach can be exploited also to compute explicit power series expansions.…”

A singularly perturbed nonlinear traction problem in a periodically perforated domain: a functional analytic approach

“…For nonlinear problems in the frame of linearized elastostatics, we also mention, e.g., [24,25,26,27], and for the Stokes equation [28]. For problems for the Laplace and Poisson equations in periodically perforated domains, we mention [29,30,31,32]. We note that this paper represents the first step in the analysis of periodic boundary value problems for linearized elastostatics with this approach.…”