Abstract
We consider a model of a composite material with “inextensible membrane type” interface conditions. An analytic solution of a stationary heat conduction problem in an unbounded doubly periodic two-dimensional composite whose matrix and inclusions consist of isotropic temperature-dependent materials is given. In the case where the conductive properties of the inclusions are proportional to those of the matrix, the problem is transformed into a fully linear boundary value problem for doubly periodic analytic functions. The solution makes it possible to calculate the average properties over the unit cell and discuss the effective conductivity of the composite. We present numerical examples to indicate some peculiarities of the solution.
Highlights
The importance and applications of composite materials are increasing very fast in the last decades
In a composite material we typically have several reinforcements such as particles, flakes and fibers which are embedded in a matrix of metals, polymers or ceramics
It is relevant to consider the boundary conditions as transmission conditions in some bounded manifold in the interior of the domain
Summary
The importance and applications of composite materials are increasing very fast in the last decades. It is relevant to consider the boundary conditions as transmission conditions in some bounded manifold in the interior of the domain This allows the proposal materials to have thermal and electrical conduction of very different strengths. An extensive and complete overview of the employed methods can be found in the fundamental work [22] Bearing all this in mind, in the present paper we are proposing a model of a composite material which is determined by some general boundary conditions (and so incorporating different scenarios). The spaces where we will consider the problem are convenient to prove the existence and uniqueness of a corresponding solution (upon some conditions) This will allow us to understand the general properties of such solution and to interpret their most useful properties as it concerns the mechanical and physical behaviour. We will be concerned with the description of the effective conductivity tensor of a steady-state heat conduction problem in 2D unbounded doubly periodic composite materials with temperature dependent conductivities
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