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Streszczenie:
There are two main topics of this research: (i) one topic considers overall properties of a nonlinear cellular composite, treated as a model of the liver tissue, and (ii) the other topic concerns the propagation of heat in the nonlinear medium described by the homogenised coefficient of thermal conductivity.

For (i) we give a method and find the effective thermal conductivity for the model of the liver tissue, and for the point (ii) we present numerical and analytical treatment of the problem, and indicate the principal difference of heat propagation in linear and nonlinear media. In linear media, as it is well known, the range of the heat field is infinite for all times t > 0, and in nonlinear media it is finite.

Pennes’ equation, which should characterize the heat propagation in the living tissue, is in general a quasi-nonlinear partial differential equation, and consists of three terms, one of which describes Fourier’s heat diffusion with conductivity being a function of temperature T. This term is just a point of our analysis.

We show that a nonlinear character of the medium (heat conductivity dependent on the temperature) changes in qualitative manner the nature of heat transfer. It is proved that for the heat source concentrated initially (t = 0) at the space point, the range of heated region (for t > 0) is finite. The proof is analytical, and illustrated by a numerical experiment.

Słowa kluczowe:
heat transport, asymptotic homogenisation, effective heat conductivity

Streszczenie:
On the basis of the paper [1] two topics are discussed. Firstly, exact formulae for the homogenized coeffi-cients of a layered thermopiezoelectric composite are derived. Secondly, by applying the Ritz method, the local problems are solved approximately. Specific cases are also examined and illustrated.

Streszczenie:
In [9] the formulae were derived for the effective moduli of a piezoelectric composite exhibiting fine periodic structure. The static case considered was studied by the method of F-convergence, cf.[l]. Next, in [10] these results were extended by investigating the dynamic behaviour. The same formulae for the effective moduli were obtained by using the method of Bloch expansions. The aim of this contribution is to present our preliminary results concerning homogenization of the equations of the linear thermopiezoelectricity. To such a problem the method of F-convergence is not applicable. Therefore the relevant analysis was performed by using the method of two-scale asymptotic developments. Our analysis leads to interesting conclusions related to the effective moduli and to the change of the initial condition for the temperature of the homogenized body.