Fourier Variant Homogenization of the Heat Transfer Processes in Periodic Composites

Abstract

The well-known parabolic Heat Transfer Equation is a simplest recognized description of phenomena related to the heat conductivity in solids with microstructure. However, it is a tool difficult to use due to the discontinuity of coefficients appearing here. The purpose of the paper is to reformulate this equation to the form that allows to represent solutions in the form of Fourier’s expansions. This equivalent re-formulation has the form of infinite number of equations with Fourier coefficients in expansion of the temperature field as the basic unknowns. The first term in Fourier representation, being an average temperature field, should satisfy the well-known parabolic heat conduction equation with Fourier coefficients as fields controlling average temperature behavior. The proposed description takes into account changes of the composite periodicity accompanying changes in the variable perpendicular to the surfaces separating components, concerning FGM - type materials and can be treated as the asymptotic version of Heat Transfer Equation obtained as a result of a certain limit passage where the cell size remains unchanged.

Archives of Civil Engineering

“…Wierzbicki et al [12] stated as follows: "From among many known ways of the modeling of problems of the mechanics of periodic media, the tolerance modeling stands out due to relative simplicity of consideration of the scale effect. This advantage of the tolerance modeling resulted in the description and solution of many problems of the mechanics of heterogeneous media.…”