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Critical analysis of two-dimensional heat balance equations derived for composite plates in terms of the variational principles of thermal conductivity theory. I: General two-dimensional theories. (Russian, English) Zbl 1463.74022
The authors apply variational principles to consider two approaches for deriving two-dimensional equations of stationary thermal conductivity theory of composite plates. The plate temperature is approximated by a polynomial in the transverse coordinate with unknown decomposition coefficients. In the framework of the first approach, thermal boundary conditions on the face surfaces of the plates are not taken into account. The resulting two-dimensional Euler equations and the corresponding boundary conditions at the edges are consistent from the thermophysical point of view. In this case, for an arbitrary subdomain of the plate and the entire structure as a whole, the heat balance equation is satisfied. In the framework of the second approach, the thermal boundary conditions on the face surfaces are taken into account. This causes the necessity of introducing the undetermined Lagrange multipliers and solving the variational problem for conditional extremum. It is shown that the resulting two-dimensional Euler equations and the corresponding boundary conditions at the edges are inconsistent (contradictory) from the thermophysical point of view since the heat balance equation is not satisfied for an arbitrary subdomain of the plate and the entire structure as a whole.

MSC:
74A40 Random materials and composite materials
74A15 Thermodynamics in solid mechanics
74G50 Saint-Venant’s principle
74F05 Thermal effects in solid mechanics
74K20 Plates
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