Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces

Abstract

We present a quantitative geometric rigidity estimate in dimensions d = 2, 3 generalizing the celebrated result by Friesecke, James, and Müller [48] to the setting of variable domains. Loosely speaking, we show that for each y ∈ H 1 (U ; R d ) and for each connected component of an open, bounded set U ⊂ R d , the L 2 -distance of ∇y from a single rotation can be controlled up to a constant by its L 2 -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 Γ-convergence. In particular, we study energies related to epitaxially strained crystalline films and to the formation of material voids inside elastically stressed solids.

A Blake-Zisserman-Kirchhoff theory for plates with soft inclusions

“…We remark that Theorem 3.3 raises two interesting questions which are beyond the scope of this contribution: First, one might wonder if the minimal droplet assumption ( 7) in (i) and in the Compactness Theorem 4.1 can be dropped. In view of recent results in [29] it seems plausible that this condition might be further weakened. Yet, in order to understand if such a condition could be dropped completely, as in the two-dimensional seeting [58], would require an extension of the quantitative rigidity results in [30] to the three-dimensional setting which appears highly non-trivial.…”

From atomistic systems to linearized continuum models for elastic materials with voids

“…This can be interpreted as an L ∞ -diverging bound on the curvature of the boundary of the voids. In our setting, the curvature regularization term in (1.3) can be thought of as imposing an L 2 -diverging bound on the curvature: firstly, this allows the void set to concentrate at arbitrarily small scales (independently of h) and, secondly, allows the boundary of the void set to consist of a diverging (with h) number of connected components, see Example 2.5. s Our more general model comes at the expense of the necessity of more sophisticated geometric rigidity results [32] compared to [40].…”

“…We perform a careful enlargement of the voids E h according to [32,Proposition 2.8], as well as an appropriate modification of the deformations. This is the content of Propositions 3.1 and 3.2 stated at the beginning of Section 3, where we modify the deformations v h and their gradients ∇v h on a small part of the rod, such that the new deformations are actually Sobolev in a big part of the rod Ω h with a good control on their elastic energy.…”

Pressure live loads and the variational derivation of linear elasticity

“…A similar approach was then applied to different frameworks in elasticity, such as rubber-like materials [3], multiwell models [1,2,32], elasticity with residual stress [27,28], and incompressible materials [21]. Beyond elasticity we also mention the papers [9,10,25,26] for models in fracture mechanics, [12] for viscoelasticity, [23] for plasticity, and the recent contribution [11] for materials with stress driven rearrangement instabilities.…”