Quantitative rigidity results, besides from their inherent geometric interest, have played a prominent role in the mathematical study of models related to elasticity\plasticity. For instance, the celebrated rigidity estimate of Friesecke, James, and Müller has been widely used in problems related to linearization, discrete-to-continuum or dimension-reduction issues for functionals within the framework on nonlinear elasticity. In this talk, I will present a generalization (in the physically relevant dimensions d=2,3) of this result to the setting of variable domains, where the geometry of the domain comes into play, in terms of a suitable integral curvature functional of its boundary. The estimate can be used to establish compactness in the space of generalized special functions of bounded deformation 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 Gamma-convergence. In particular, we study energies related to epitaxially strained crystalline films and to the formation of material voids inside elastically stressed solids. This is joint work with Manuel Friedrich and Leonard Kreutz