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Two-scale Γ-convergence of integral functionals and its application to homogenisation of nonlinear high-contrast periodic composites

Cherdantsev, Mikhail and Cherednichenko, Kirill 2012. Two-scale Γ-convergence of integral functionals and its application to homogenisation of nonlinear high-contrast periodic composites. Archive for Rational Mechanics and Analysis 204 (2) , pp. 445-478. 10.1007/s00205-011-0481-4

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Abstract

An analytical framework is developed for passing to the homogenisation limit in (not necessarily convex) variational problems for composites whose material properties oscillate with a small period ε and that exhibit high contrast of order 1/ε between the constitutive, “stress-strain”, response on different parts of the period cell. The approach of this article is based on the concept of “two-scale Γ-convergence”, which is a kind of “hybrid” of the classical Γ-convergence (De Giorgi and Franzoni in Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur (8)58:842–850, 1975) and the more recent two-scale convergence (Nguetseng in SIAM J Math Anal 20:608–623, 1989). The present study focuses on a basic high-contrast model, where “soft” inclusions are embedded in a “stiff” matrix. It is shown that the standard Γ-convergence in the L^p -space fails to yield the correct limit problem as ε tends to 0, due to the underlying lack of L^p -compactness for minimising sequences. Using an appropriate two-scale compactness statement as an alternative starting point, the two-scale Γ-limit of the original family of functionals is determined via a combination of techniques from classical homogenisation, the theory of quasiconvex functions and multiscale analysis. The related result can be thought of as a “non-classical” two-scale extension of the well-known theorem by Müller (Arch Rational Mech Anal 99:189–212, 1987).

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: Springer Verlag
ISSN: 0003-9527
Funders: EPSRC
Date of First Compliant Deposit: 30 March 2016
Last Modified: 23 Oct 2017 14:05
URI: http://orca.cf.ac.uk/id/eprint/33363

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