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Carnegie Mellon Center for Nonlinear Analysis


Scaling Laws for Non-Euclidean Plate Models Derived from Incompatible Elasticity

Marta Lewicka
University of Minnesota
School of Mathematics

Abstract: Various structures in nature exhibit residual stress at free equilibria. In other words, the existence of intrinsic constraints does not allow the ideal configuration to be realized. This phenomenon occurs in particular in growing tissues (as leaves or flowers), where it is postulated that the cell division provides a mechanism for the formation of a 'target metric'. As a result, the tissue tries to adapt its shape to the metric; on the other hand geometric constraints usually preclude the existence of a 3d configuration acquiring the target.

In this talk, we shall give an overview of recent results studying such phenomena from a variational point of view. We first introduce the 3d incompatible elastic energy, measuring the pointwise distance between the deformation gradient and the set of orientation preserving realizations of a given target metric with non-zero Riemann curvature. This latter condition guarantees that the infimum of the energy (in the absence of boundary conditions or body forces) is positive. In this setting, we derive energy scaling laws for thin non-Euclidean plates, and the corresponding limiting theories as vanishing thickness Gamma-limits. We obtain the Kirchhoff-like theory when the 3d prescribed metric does not depend on the thickness. The von Karman-type theories arise when the 3d metric is a perturbation of the Euclidean one. As a corollary, we also obtain new conditions for existence of isometric immersions of 2d Riemannian metrics into 3d.