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 $${\varepsilon^{-1}}$$ 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 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 Lp-space fails to yield the correct limit problem as $${\varepsilon \to 0,}$$ 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. 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).