  In the context of elasticity theory, rigidity theorems allow to derive global
properties of a deformation from local ones. This paper presents a new
asymptotic version of rigidity, applicable to elastic bodies with sufficiently
stiff components arranged into fine parallel layers. We show that strict global
constraints of anisotropic nature occur in the limit of vanishing layer
thickness, and give a characterization of the class of effective deformations.
The optimality of the scaling relation between layer thickness and stiffness is
confirmed by suitable bending constructions. Beyond its theoretical interest,
this result constitutes a key ingredient for the homogenization of variational
problems modeling high-contrast bilayered composite materials, where the common
assumption of strict inclusion of one phase in the other is clearly not
satisfied. We study a model inspired by hyperelasticity via Gamma-convergence,
for which we are able to give an explicit representation of the homogenized
limit problem. It turns out to be of integral form with its density
corresponding to a cell formula.
