Global Behavior of Buckled States of Nonlinearly Elastic Rods

Abstract

This paper furnishes a comprehensive treatment of the global qualitative behavior of buckled states of nonuniform, nonlinearly elastic rods under the action of compressive terminal thrusts. These rods can suffer not only flexure as in the classical elastica theory, but also compression and shear. The governing equations are reduced to a quasilinear ordinary functional-differential equation, which reflects the nonlinearity of the constitutive equations. It is shown that the solution branches of the nonlinear problem inherit their nodal structure from that of the problem linearized about the trivial solution. This nodal structure, which distinguishes the different branches, may be quite complicated, owing to the breadth of physical response permitted. The qualitative behavior of the solutions is particularly sensitive to the nature of the shear response, to the form of the nonhomogeneity, and to the classification of boundary conditions as statically determinate or indeterminate. The analysis is based on modifications of the theory of M. G. Crandall and P. H. Rabinowitz to accomodate the underlying physics. These modifications rely on the development of nonstandard uniqueness theorems for functional differential equations, a suitable Sturmian theory for linear functional differential equations, and effective estimates.