FORMULATION OF THE HEAT CONDUCTION EQUATION FOR HETEROGENEOUS MEDIA WITH MULTIPLE SPATIAL SCALES USING REITERATED HOMOGENIZATION

Ernesto Iglesias Rodríguez¹; Manuel Ernani Cruz²; Julián Bravo-Castillero³; Raúl Guinovart-Díaz³; Reinaldo Rodríguez-Ramos³; Leslie Darien Pérez-Fernández⁴

¹ IGP Servicio Geológico de Cuba; ² UFRJ - Federal University of Rio de Janeiro, Department of Mechanical Engineering; ³ Universidad de La Habana, Departamento de Matemática; ⁴ UFPel - Federal University of Pelotas, Department of Mathematics and Statistics

doi:10.20906/CPS/COB-2015-0018

Resumo

Heterogeneous media with multiple spatial scales are finding increased importance in engineering. An example might be a large scale, otherwise homogeneous medium filled with dispersed small-scale particles that form aggregate structures at an intermediate scale. The objective in this paper is to formulate the strong-form Fourier heat conduction equation for such media using the method of reiterated homogenization. The phases are assumed to have a perfect thermal contact at the interface. The ratio of two successive length scales of the medium is a constant small parameter eps. The method is an up-scaling procedure that writes the temperature field as an asymptotic multiple-scale expansion in powers of the small parameter eps. The technique leads to two pairs of local and homogenized equations, linked by effective coefficients. In this manner the medium behavior at the smallest scales is seen to affect the macroscale behavior, which is the main interest in engineering. To facilitate the physical understanding of the formulation, an analytical solution is obtained for the heat conduction equation in a functionally graded material (FGM). The approach presented here may serve as a basis for future efforts to numerically compute effective properties of heterogeneous media with multiple spatial scales.

Palavras-chave: reiterated homogenization; heterogeneous media; heat conduction; formal asymptotic solution; functionally graded material