  We consider periodic homogenization of nonlinearly elastic composite
materials. Under suitable assumptions on the stored energy function (frame
indifference; minimality, non-degeneracy and smoothness at identity; $p\geq
d$-growth from below), and on the microgeometry of the composite (covering the
case of smooth, periodically distributed inclusions with touching boundaries),
we prove that in an open neighbourhood of the set of rotations, the multi-cell
homogenization formula of non-convex homogenization reduces to a single-cell
formula that can be represented with help of a corrector. This generalizes a
recent result of the authors by significantly relaxing the spatial regularity
assumptions on the stored energy function. As an application, we consider the
nonlinear elasticity problem for $\varepsilon$-periodic composites, and prove
that minimizers (subject to small loading and well-prepared boundary data)
satisfy a Lipschitz estimate that is uniform in $0<\varepsilon\ll1$. A key
ingredient of our analysis is a new Lipschitz estimate (under a smallness
condition) for monotone systems with spatially piecewise-constant coefficients.
The estimate only depends on the geometry of the coefficient's
discontinuity-interfaces, but not on the distance between these interfaces.
