Search

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
7/2005

A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence

Gero Friesecke, Richard D. James and Stefan Müller

Abstract

We derive a hierarchy of plate models from three dimensional nonlinear elasticity by $\Gamma$-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume $\sim h^\beta$, where $h$ is the thickness of the plate.

This is in turn related to the strength of the applied force $\thicksim h^\alpha$. Membrane theory, derived earlier by Le Dret and Raoult, corresponds to $\alpha=\beta=0$, nonlinear bending theory to $\alpha = \beta =2$, Föppl von Kármán theory to $\alpha = 3$, $\beta=4$ and linearized vK theory to $\alpha > 3$. Intermediate values of $\alpha$ lead to certain theories with constraints.

A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [31] that for maps $v:(0,1)^3 \rightarrow \mathbb{R}^3$, the $L^2$ distance of $\nabla v$ from a single rotation is bounded by a multiple of the $L^2$ distance from the set $SO(3)$ of all rotations.

Received:
Jan 19, 2005
Published:
Jan 19, 2005
MSC Codes:
74K20, 49J45

Related publications

inJournal
2006 Repository Open Access
Gero Friesecke, Richard D. James and Stefan Müller

A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence

In: Archive for rational mechanics and analysis, 180 (2006) 2, pp. 183-236