Homogenization of the nonlinear bending theory for plates

Abstract

We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of $\Gamma$-convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a “cylinder” or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions of class $W^{2,2}$, using two-scale convergence. This is a non-trivial task, since one has to treat two-scale convergence in connection with a nonlinear differential constraint.