The Monge–Ampère constraint: Matching of isometries, density and regularity, and elastic theories of shallow shells

Abstract

Please cite this article in press as: M. Lewicka et al., The Monge-Ampère constraint: Matching of isometries, density and regularity, and elastic theories of shallow shells, Ann. I. H. Poincaré -AN (2015), http://dx.Abstract. The main analytical ingredients of the first part of this paper are two independent results: a theorem on approximation of W 2,2 solutions of the Monge-Ampère equation by smooth solutions, and a theorem on the matching (in other words, continuation) of second order isometries to exact isometric embeddings of 2d surface in R 3 .In the second part, we rigorously derive the Γ-limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness h, where the depth of the shell scales like h α and the applied forces scale like h α+2 , in the limit when h → 0. We offer a full analysis of the problem in the parameter range α ∈ (1/2, 1). We also complete the analysis in some specific cases for the full range α ∈ (0, 1), applying the results of the first part of the paper.

Gaussian Curvature as an Identifier of Shell Rigidity

“…To be more precise, to our best knowledge only low energy scalings like E ∼ h α , where α ≥ 3, (which roughly speaking corresponds to bending, [15,36]), or very high energy profiles E ∼ h (which corresponds to stretching, where the limiting energy completely eliminates bending and captures only stretching of the shell) have been studied [32,33]. We also refer to the papers [15,36,24,25,35,39,40,32,33] for some results on shell deformation and theories. An interested reader can also check the work of Ciarlet for linearized rod, plate, shell and thin structures with junctions theories [1,2,3,4,5,6,7].…”

Visualization of the convex integration solutions to the Monge-Ampère equation

“…3.26 · 10 29 2.98 · 10 29 3.28 · 10 29 10 10 3.30 · 10 31 2.99 · 10 31 3.30 · 10 31 10 11 3.25 · 10 33 2.99 · 10 33 3.23 · 10 33 10 12 3.28 · 10 35 2.99 · 10 35 3.24 · 10 35 10 13 3.23 · 10 37 2.99 · 10 37 3.25 · 10 37 10 14 3.26 · 10 39 2.98 · 10 39 3.25 · 10 39 10 15 3.25 · 10 41 2.99 · 10 41 3.27 · 10 41 10 16 3.29 · 10 43 2.98 · 10 43 3.20 · 10 43 3.11 · 10 5 2.80 · 10 5 2.12 · 10 5 10 2…”

Variational Problems for Föppl--von Kármán Plates

“…Notwithstanding this difficulty, an existence theorem can be proved either assuming a sign condition on boundary forces, or an homogeneous Dirichlet condition on the transverse displacement. In the first case the work of the external forces is bounded away from zero on the kernel of the membrane energy density, thus allowing the global energy to be bounded from below; in the second one a uniqueness result in the theory of Monge-Ampère equation implies that the kernel of bending energy reduces to the null transverse displacement (see also [30], [31], [32]). These settings together with a tuning of some techniques introduced in [4] and [15] yield compactness of minimizing sequences, hence existence of minimizers via the direct method.…”

“…In Section 3 we study the limit as h → 0 of scaled Föppl-von Kármán energy F h when inplane forces in (1.11) The issues involved in the present article are closely related with a large class of instabilities, according to recent studies ( [7], [8], [9], [11], [12], [10], [17], [30], [31], [32], [41]).…”