BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

Elasticity theory is the central model of solid mechanics. Properly formulated, it gives rise to formidable nonlinear problems whose understanding is in many cases beyond the reach of present-day mathematics. Nevertheless, the last quartercentury has seen substantial advances in this understanding, due largely to the development and application of new methods of nonlinear analysis. An important part in this development has been played by Stuart Antman’s pioneering studies of the existence and bifurcation of solutions for various rod and shell problems. His treatment of the theory in the monograph under review is thus of particular interest. Perhaps the most famous nonlinear problem of elasticity is that of the buckling of a rod, and it well illustrates many of the difficulties of problem formulation and analysis typically encountered in the theory. Suppose we are given an initially straight cylindrical rod, which we try to compress by applying opposing forces to its two ends. If the rod is sufficiently thin and we push the ends together hard enough, the rod does not remain straight, but instead buckles into a curved configuration. This will happen however perfectly the rod is made and however careful we are to prevent asymmetries either in the composition of the rod or in the application of the forces. (Of course gravity will produce such an asymmetry, but this can be avoided partially by orienting the rod vertically, or almost completely by performing the experiment in a spacecraft.) Other similar buckling behaviour occurs for thin curved sheets under applied forces (sometimes accompanied by associated noises, as in the mistreatment of plastic coffee cups). How can we model buckling of a rod mathematically? Suppose the rod has length L. We can identify the material points of the rod by their positions x in the open subset Ω = (0, L)×D of R, where the cross-section D is a bounded domain in R. We call this undeformed configuration of the rod its reference configuration. We can thus describe any other configuration of the rod by the mapping y : Ω → R which takes each material point x to its deformed position y(x). If the rod is made from a material such as steel, wood or rubber, we can hope to model it as elastic, that is, as a material for which the (Piola-Kirchhoff) stress tensor S(x) at the point x ∈ Ω in the configuration y depends only on the deformation gradient Dy(x), which in rectangular Cartesian coordinates can be identified with the 3 × 3 matrix of partial derivatives ∂yi ∂xj (x). We write this dependence as S = σ(Dy). If Σ is a smooth oriented surface passing through the point x ∈ Ω and having unit normal N(x) there, then S(x)N(x) gives the contact force per unit undeformed area acting at y(x) across the deformed surface y(Σ). Of course buckling is a dynamic phenomenon, so that we are interested in motions of the rod described by one-parameter families of configurations x 7→ y(x, t) where the parameter t is the time. However, we shall first consider a static theory in which y = y(x) is the only unknown. The governing partial differential equations