  We show that nonlinearly elastic plates of thickness $h\to 0$ with an
$\varepsilon$-periodic structure such that $\varepsilon^{-2}h\to 0$ exhibit
non-standard behaviour in the asymptotic two-dimensional reduction from
three-dimensional elasticity: in general, their effective stored-energy density
is "discontinuously anisotropic" in all directions. The proof relies on a new
result concerning an additional isometric constraint that deformation fields
must satisfy on the microscale.
