Effective macroscopic description for heat conduction in periodic composites

Effects of the local resonance on the wave propagation in periodic frame structures: Generalized Newtonian mechanics

“…Similar mechanisms were also identified in double conductivity media 34 and double porosity media (e.g. Ref.…”

Effective macroscopic description for heat conduction in periodic composites

“…In (1) K e f f λ [u] is the term with the infinite order partial or ordinary differential operator K e f f λ [·] acting on the average temperature u and after limit passage with microstructural parameter λ tending to zero reduces to the form K e f f ∇u with effective conductivity matrix K e f f . From a mathematical point of view, the homogenization procedure from the heat transfer equation (2) to (1) averages the mass density , the specific heat c , conductivity matrix K and heat sources b as discontinuous coefficients in (2) as well as the temperature field ϑ converting them to their average equivalents in (1). Two different homogenization procedures can lead to different effective conductivity descriptions (1), but from a qualitative, mathematical point of view, these descriptions should be similar effective conductivity with a scale effect [1,2,3].…”

“…From a mathematical point of view, the homogenization procedure from the heat transfer equation (2) to (1) averages the mass density , the specific heat c , conductivity matrix K and heat sources b as discontinuous coefficients in (2) as well as the temperature field ϑ converting them to their average equivalents in (1). Two different homogenization procedures can lead to different effective conductivity descriptions (1), but from a qualitative, mathematical point of view, these descriptions should be similar effective conductivity with a scale effect [1,2,3]. If we deal with a periodic composites the scale effect reduces to the dependence of these descriptions on the microstructural parameter λ equal usually to the repetitive cell diameter.…”

“…Wozniak and used by his followers and collaborators, [2,4,5]. However, the most important inspiration for writing this paper is one of the first articles on homogenization of heat conduction issues written by Ariault, [1] and monograph [3]. The remaining cited references to the foundations of this theory, foe example [6,7].…”

Temperature distributions in a periodically stratified layer with a vertical dug well

“…For this reason, applications of some approximate models seem to be useful. Some effective approaches to the modeling of heat conduction problem in periodic composites are presented in papers of Aurialt [3], Bufler [4], Bufler and Meier [5]. One of the models is the homogenized model with microlocal parameters devised by Woźniak [2] and developed for microperiodic layered composites by Matysiak and Woźniak [1].…”