TU Dortmund, Raum M611

In this talk we present a quantitative geometric rigidity estimate in dimensions d = 2; 3 generalising a celebrated result by Friesecke, James and Müller to the setting of variable domains. Loosely speaking, we show that for each function y 2 H1(U;R3) and for each connected component of an open bounded set U(Rd, the L2-distance of ry from a single rotation can be controlled up to a constant by its L2-distance from the group SO(d), with the constant not depending on the precise shape of U, but only on an integral curvature functional related to @U. We further show that for linear strains the estimate can be refined, leading to a uniform control independent of the set U. The estimate can be used to establish compactness in the space of generalized special functions of bounded deformation (GSBD) for sequences of displacements related to deformations with uniformly bounded elastic energy. As an application, we rigorously derive linearized models for nonlinearly elastic materials with free surfaces by means of T-convergence. In particular, we study energies related to epitaxially strained crystalline films and to the formation of material voids inside elastically stressed solid

Leonard Kreutz

Münster