Ricerc@Sapienza

The study of thermal, mechanical and electrical properties of composite materials plays an increasingly important role in material sciences because of their wide spectrum of applica-
tions, for instance, in industrial processes, biomathematics, medical diagnosis. In this talk, we discuss some models which describe the thermal diffusivity or the electrical
conductivity in a composite medium with a nely mixed periodic structure, assuming that the microstructure of the materials under consideration is made by two different diffusive

We study a concentration and homogenization problem modelling electrical conduction in a composite material. The novelty of the problem is due to the specific scaling of the physical quantities characterizing the dielectric component of the composite. This leads to the appearance of a peculiar displacement current governed by a Laplace-Beltrami pseudo-parabolic equation. This pseudo-parabolic character is present also in the homogenized equation, which is obtained by the unfolding technique.

We prove a well-posedness result for two pseudo-parabolic problems, which can be seen as two models for the same electrical conduction phenomenon in heterogeneous media, neglecting the magnetic field. One of the problems is the concentration limit of the other one, when the thickness of the dielectric inclusions goes to zero. The concentrated problem involves a transmission condition through
interfaces, which is mediated by a suitable Laplace-Beltrami type equation.

In this paper we prove an error estimate for a model of heat conduction in composite materials having a microscopic structure arranged in a periodic array and thermally active membranes separating the heat conductive phases.

In this paper we derive a model for heat diffusion in a composite medium in which the different components are separated by thermally active interfaces. The previous result is obtained via a concentrated capacity procedure and leads to a non-stantard system of PDEs involving a Laplace-Beltrami operator acting on the interface. For such a system well-posedness is proved using contraction mapping and abstract parabolic problems theory. Finally, the exponential convergence (in time) of the solutions of our system to a steady state is proved.