Abstract. Using the notion of Γ-convergence, we discuss the limiting behavior of the 3d nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like h β with 2 < β < 4. We establish that, for the given scaling regime, the limiting theory reduces to the linear pure bending. Two major ingredients of the proofs are: the density of smooth infinitesimal isometries in the space of W 2,2 first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces. Contents

Abstract. We study the Γ-limit of 3d nonlinear elasticity for shells of small, variable thickness, around an arbitrary smooth 2d surface. 1.

Abstract. 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 Γ-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 R 3. Contents

Abstract. 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. Contents

Abstract. We prove that the critical points of the 3d nonlinear elasticity functional on shells of small thickness h and around the mid-surface S of arbitrary geometry, converge as h → 0 to the critical points of the von Kármán functional on S, recently proposed in [8]. This result extends the statement in [16], derived for the case of plates when S ⊂ R 2. The convergence holds provided the elastic energies of the 3d deformations scale like h 4 and the external body forces scale like h 3. Contents

Abstract. We provide a derivation of the Föppl-von Kármán 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. 1.

Abstract. We provide a derivation of the Föppl-von Kármán 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. 1.

homogeneous materials Ω ⊂ R3: elastic body at rest in its reference configuration. • Study of elastic energy of the deformation u: Ω → R3 ˆ E(u) = W (∇u),

Abstract. The purpose of this note is to report on the recent development concerning the analysis and the rigorous derivation of thin film models for structures with nontrivial geometry. This includes: (i) shells with mid-surface of arbitrary curvature, and (ii) plates exhibiting residual stress at free equilibria. In the former setting, we derive a full range of models, some of them previously absent from the physics and engineering literature. The latter phenomenon has been observed in different contexts: growing leaves, torn plastic sheets and specifically engineered polymer gels. After reviewing available results, we list open problems with a promising angle of approach.

Abstract. The purpose of this paper is to report on the recent development concerning the analysis and the rigorous derivation of thin film models for structures exhibiting residual stress at free equilibria. This phenomenon has been observed in different contexts: growing leaves, torn plastic sheets and specifically engineered polymer gels. The study of wavy patterns in these contexts suggest that the sheet endeavors to reach a non-attainable equilibrium and hence assumes a non-zero stress rest configuration. 1. Elastic energy of a growing tissue and the non-Euclidean elasticity 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. Recent work has focused on some of the related questions by using variants of thin plate theory [1, 5, 4, 23]. However, the theories used are not all identical and some of them arbitrarily ignore certain terms and boundary conditions without prior justification. This suggests that it might be useful to rigorously derive an asymptotic theory for the shape of a residually strained thin lamina to clarify the role of the assumptions used while shedding light on the errors associated with the use of the approximate theory that results. Recently, such rigorous