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Improved variational bounds for conductive periodic composites with 3D microstructures and nonuniform thermal resistance. (English) Zbl 1331.49015

Summary: Improved variational bounds for the effective conductivity of a matrix-inclusion conductive periodic composite are obtained. The studied composite is macroscopically anisotropic with nonuniform interfacial thermal resistance between isotropic phases. The homogenization theory is applied to a three-dimensional heat conduction problem which is stated in terms of nondimensional parameters. The Biot number is explicitly given in the variational formulation of the local problems and in the related minimization problems. The approach is based on the Lipton-Vernescu variational principles which allow to derive narrower bounds by incorporating more detailed morphological information. The bounds depend on the concentration and the conductivity of each phase, the periodic distribution and the shape of the inclusions, the Biot number and the nonuniform interfacial resistance.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49M30 Other numerical methods in calculus of variations (MSC2010)
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
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