Homogenisation and spectral convergence of a periodic elastic composite with weakly compressible inclusions

Abstract

A two-phase elastic composite with weakly compressible elastic inclusions is considered. The homogenised two-scale limit problem is found, via a version of the method of two-scale convergence, and analysed. The microscopic part of the two-scale limit is found to solve a Stokes type problem and shown to have no microscopic oscillations when the composite is subjected to body forces that are microscopically irrotational. The composites spectrum is analysed and shown to converge, in an appropriate sense, to the spectrum of the two-scale limit problem. A characterisation of the two-scale limit spectrum is given in terms of the limit macroscopic and microscopic behaviours.

Keywords:

• high contrast homogenisation
• partially degenerate PDE
• eigenvalue problem
• two-scale convergence
• strong two-scale resolvent convergence

AMS Subject Classification:

• 35 Partial Differential Equations
• 35J Elliptic equations and systems
• 35P Spectral theory and eigenvalue problems
• 46 Functional analysis
• 47 Operator theory

Acknowledgments

The author would like to thank I.V. Kamotski and V.P. Smyshlyaev for their invaluable comments pertaining to this work. This work was performed at the University of Bath and was sponsored by an EPSRC PhD studentship.