Hierarchy of pre-strained behaviors for thin materials

ABSTRACT:

A hierarchy of four models for thin structures was first obtained by means of a syste- matic, but formal, approach in [5]. Namely, the nonlinear membrane model, the nonlinear bending model, the slightly nonlinear von Ka`rm`an model and the linear model were reco- vered. Emphasis was put on the loading order of magnitude and on the induced magnitude of the deformations. A rigorous variational derivation of the nonlinear membrane model was then given in [7]. In particular, the degeneracy under compression was exhibited. Then, [6] provided a rigidity result that allowed to rigorously justify the other models.

Some recent papers by physicists [4] suggest models where thin bodies (leaves, gel disks) try to reach an elastic equilibrium state after some given non Euclidean metrics has been imposed and, possibly, fail when considered as 3d bodies (case when the 3d metric tensor has no realization); see also [3]. Which thin limit models should be used? It turns out, as shown in [2], [8], [9], that the driving mechanism is the structure of the 3d Riemann curvature tensor R. Depending on the vanishing of blocks of entries of R, models that share several features with the Eucidean hierarchy are derived. The method of proof requires extended quantitative rigidity estimates.

Posted on Dec 22, 2015 in Mathematical Mechanics, Seminars