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

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 ε−1 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, E., Franzoni, T.: Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58, 842–850, 1975) and the more recent two-scale convergence (Nguetseng, G.: 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 Lp-space fails to yield the correct limit problem as ε → 0, due to the underlying lack of Lp-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. Then related result can be thought of as a “non-classical” two-scale extension of the well-known theorem by S. Müller (Arch. Rational Mech. Anal. 99, 189–212, 1987).