  The main result of this work is a homogenization theorem via variational
convergence for elastic materials with stiff checkerboard-type heterogeneities
under the assumptions of physical growth and non-self-interpenetration. While
the obtained energy estimates are rather standard, determining the effective
deformation behavior, or in other words, characterizing the weak Sobolev limits
of deformation maps whose gradients are locally close to rotations on the stiff
components, is the challenging part. To this end, we establish an asymptotic
rigidity result, showing that, under suitable scaling assumptions, the
attainable macroscopic deformations are affine conformal contractions. This
identifies the composite as a mechanical metamaterial with a negative Poisson's
ratio. Our proof strategy is to tackle first an idealized model with full
rigidity on the stiff tiles to acquire insight into the mechanics of the model
and then transfer the findings and methodology to the model with diverging
elastic constants. The latter requires, in particular, a new quantitative
geometric rigidity estimate for non-connected squares touching each other at
their vertices and a tailored Poincar\'e type inequality for checkerboard
structures.
