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Macroscopic thermal properties of quasi-linear cellular medium on example of the liver tissue

Gambin B., Kruglenko E., Gałka A. A., Wojnar R.

Computer Assisted Methods in Engineering and Science

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2015

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Vol. 22, no. 4

329--346

EN

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.

2

Stepped basic function in the asymptotic homogenization of an elliptic system

Mitkowski P. J., Mitkowski W.

Automatyka / Automatics

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2012

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Vol. 16, no. 1

45-57

EN

The problem of asymptotic homogenization for one-dimensional stationary elliptic system was studied. The value of effective conductivity coefficient of homogeneous elliptic macroscopic system was determined for the basic cell approximated by a "stepped" function. The conditions to obtain given conductivity coefficient, in the considered homogenization problem, were shown. Results of numerical calculations were provided to illustrate discussed issues.

PL

W pracy rozważa się problem asymptotycznej homogenizacji dla jednowymiarowego stacjonarnego systemu eliptycznego. Dokonano aproksymacji "schodkowej" funkcji bazowej i wyznaczono przybliżoną wartość efektywnego współczynnika przewodności eliptycznego jednorodnego układu makroskopowego. Przedstawiono również jak w omawianym procesie homogenizacji uzyskać z góry zadany efektywny współczynnik przewodności. Rozważania zostały zilustrowane wynikami obliczeń numerycznych.

3

Analytical expressions of effective constants for a piezoelectric composite reinforced with square cross-section fibres

Otero J., Bravo-Castillero J., Guinovart-Diaz R., Rodriguez-Ramos R., Maugin G. A.

Archives of Mechanics

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2003

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Vol. 55, nr 4

357-371

EN

The purpose of this paper is to present analytical expresiion of the effective elastic, piezoelectric and dielectric constants of reinforced piezoelectric composite materials with unidirectional fibers periodically distributed in a square matrix, as obtained by means of the "double asymptotic homogenization" method. The cross-section of the fibers is square. Each periodic cell of the medium is a binary piezoelectric composite wherein both phases are homogeneous piezoelectric materials with transversely isotropic properties. Comparison between the derived theoretical predictions of characteristic parameters and the existing experimental results shows a rather good agreement. The results obtained in the present paper were verified by means of the universal relations of Schulgasser. Numerical computation of the effective properties can be realized without difficulties.