Stefan Müller
Max Planck Institute for Mathematics in the Science

Plate and Membrane Theories Derived from Nonlinear Elasticity by Gamma-Convergence

ABSTRACT: A fundamental problem in elasticity is to derive theories for lower dimensional objects such as plates, shells or rods from the fully nonlinear three dimensional theory. The usual approach is to make certain assumptions on the three dimensional solutions and then to deduce a lower dimensional theory by formal or rigorous asymptotical analysis. These has lead to large variety of theories, which are sometimes not mutually compatible.

Since the early 90's a new, mathematically rigorous, approach has emerged, which is based on the variational principle and the associated notion of $\Gamma$ convergence. Le Dret and Raoult have used $\Gamma$ convergence to derive a theory for elastic membranes (these have only stretching stiffness, but no bending stiffness and cannot resist compression). In these lectures I will report on ongoing work with G. Friesecke (Munich/ Warwick) and R.D. James (Minnesota) to derive a full hierarchy of limiting theories, which are distinguished by the scaling of the elastic energy as a function of thickness. In particular I will discuss the derivation of Kirchhoff's plate theory (which captures bending) and the much debated von Kármán theory.

A key mathematical ingredient is a quantitative rigidity estimate which generalizes results of F. John for deformations with small nonlinear strain. A classical result says that any Lipschitz map from a (bounded) connected set in ${\bf R}^n$ to ${\bf R}^n$ (we are interested in $n \geq 2$) whose derivative is an element of $SO(n)$ a.e. has in fact constant derivative. The quantative rigidity estimate says that this can be extended to a linear estimate in $L^2$. More precisely for every $u \in W^{1,2}$ and every Lipschitz domain $\Omega$ we have

$$\min_{Q \in SO(n)} \int_\Omega \vert\nabla u - Q\vert^2 \leq C(\Omega) \int_\Omega \mbox{dist}^2(\nabla,SO(n)).$$ (1)

The proof of this result is surprisingly simple and will be presented in the first lecture.