Scaling laws for non-Euclidean plates and the W2,2W^{2,2} isometric immersions of Riemannian metrics

This paper concerns the elastic structures which exhibit non-zero strain at free equilibria. Many growing tissues (leaves, flowers or marine invertebrates) attain complicated configurations during their free growth. Our study departs from the 3d incompatible elasticity theory, conjectured to explain the mechanism for the spontaneous formation of non-Euclidean metrics. Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its $\Gamma$-convergence under the proper scaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a $W^{2,2}$ isometric immersion of a given 2d metric into $\mathbb R^3$.Comment: 18 pages, 1 figur

The infinite hierarchy of elastic shell models: some recent results and a conjecture

We summarize some recent results of the authors and their collaborators, regarding the derivation of thin elastic shell models (for shells with mid-surface of arbitrary geometry) from the variational theory of 3d nonlinear elasticity. We also formulate a conjecture on the form and validity of infinitely many limiting 2d models, each corresponding to its proper scaling range of the body forces in terms of the shell thickness.Comment: 11 pages, 1 figur

Rigidity and regularity of co-dimension one Sobolev isometric immersions

We prove the developability and $C^{1,1/2}$ regularity of $W^{2,2}$ isometric immersions of $n$-dimensional domains into $R^{n+1}$. As a conclusion we show that any such Sobolev isometry can be approximated by smooth isometries in the $W^{2,2}$ strong norm, provided the domain is $C^1$ and convex. Both results fail to be true if the Sobolev regularity is weaker than $W^{2,2}$.Comment: 43 pages, 15 figure

The von Karman equations for plates with residual strain

We provide a derivation of the Foppl-von Karman equations for the shape of and stresses in an elastic plate with residual strains. These might arise from a range of causes: inhomogeneous growth, plastic deformation, swelling or shrinkage driven by solvent absorption. Our analysis gives rigorous bounds on the convergence of the three dimensional equations of elasticity to the low-dimensional description embodied in the plate-like description of laminae and thus justifies a recent formulation of the problem to the shape of growing leaves. It also formalizes a procedure that can be used to derive other low-dimensional descriptions of active materials.Comment: 26 page