Stability of Slender Bodies under Compression and Validity of the von Kármán Theory

Abstract

Since their formulation almost 100 years ago, the von Kármán (vK) plate equations have been frequently used both by engineers and by analysts to study thin elastic bodies, in particular their stability behaviour under applied loads. At the same time the derivation of these equations met some harsh criticism and their precise mathematical status has been unclear until very recently. Following up on a recent variational derivation of the vK theory by Friesecke, James and Müller from three-dimensional nonlinear elasticity we study the predictions and the validity of the vK equation in the presence of in-plane compressive forces. The first main result is a stability alternative: either the load leads to instability already in the nonlinear bending theory of plates (Kirchhoff–Love theory), or it leads to an instability in a geometrically linear KL theory (‘linearized instability’), or vK theory is valid. The second main result states that under suitable conditions the critical loads for nonlinear stability and linearized instability coincide. The third main result asserts this critical load also agrees with the load beyond which the infimum of the vK functional is −∞. The main ingredients are a sharp rigidity estimate for maps with low elastic energy and a study of the properties of isometric immersions from a set in \({\mathbb{R}^2}\) to \({\mathbb{R}^3}\) and their geometrically linear counterparts.