Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-15T02:11:57.672Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-local homogenized limits for composite media with highly anisotropic periodic fibres

Published online by Cambridge University Press:  12 July 2007

K. D. Cherednichenko ,
V. P. Smyshlyaev  and
V. V. Zhikov
K. D. Cherednichenko
Affiliation:
St John's College, Oxford OX1 3JP, UK (cheredni@maths.ox.ac.uk)
V. P. Smyshlyaev
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK (vps@maths.bath.ac.uk)
V. V. Zhikov
Affiliation:
Department of Mathematics, Vladimir Pedagogical University, Vladimir 600024, Russia (zhikov@vgpu.vladimir.ru)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a homogenization problem for highly anisotropic conducting fibres embedded into an isotropic matrix. For a ‘double porosity’-type scaling in the expression of high contrast between the conductivity along the fibres and the conductivities in the transverse directions, we prove the homogenization theorem and derive two-scale homogenized equations using a version of the method of two-scale convergence, supplemented in the case when the spectral parameter λ = 0 by a newly derived variant of high-contrast Poincaré-type inequality. Further elimination of the 'rapid' component from the two-scale limit equations results in a non-local (convolution-type integro-differential) equation for the slowly varying part in the matrix, with the non-local kernel explicitly related to the Green function on the fibre. The regularity of the solution to the non-local homogenized equation is proved.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 136 , Issue 1 , February 2006 , pp. 87 - 114
Copyright
Copyright © Royal Society of Edinburgh 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)