A macroscopic modelling of the heat conduction in laminates

							 A macroscopic modelling of the heat
conduction in laminates having thermal
 properties varying with temperature
Czeslaw Wozniak, Czestochowa University of Technology, Poland ;
Ewaryst Wierzbicki, Czestochowa University of Technology, Poland ;
Urszula Siedlecka, Czestochowa University of Technology, Poland


    It was shown by Artole and Duvaut, [1] that the homogenization asymptotic
technique can be succesfully applied to steady state heat conduction in micro-
periodic composities having thermal properties varying with temperature. The
first aim of this contribution is to derive a macroscopic model of heat conduction
(i.e. a model governed by PDEs with constant coefficients) which depends on
the microstructure length parameter λ and hence it is also certain boundary–
layer and initial–layer phenomena. As a tool of mathematical modelling the
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tolerance averaging technique is taken into account, cf. Wo´niak, Wierzbicki [2]
and a series of subsequent papers. Moreover, by the formal limit passage, setting
λ     0, the tolerance macroscopic model equations yield the known homogenized
results. For a sake of simplicity considerations are restricted to the two–phase
microperiodic laminates.
    The second aim of contribution is to propose a certain perturbation proce-
dure for the approximate analysis of heat conduction problems with the weak
sensibility of conductivity on temperature. At last for the steady–state problems
the results were derived from the tolerance technique.
    [1] M. Artola, G. Duvaut: Annales de la Faculte des Sciences de Toulouse,
1982.
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    [2] Cz. Wo´niak, E. Wierzbicki: Averaging technique in thermomechanics of
composite solids, Wydawnictwo Politechniki Czestochowskiej, 2000.




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