A Local Approach to the Equilibrium of Solids

Abstract

The local method assumes a given (possibly stressed and inhomogeneous) reference equilibrium state of a solid under given external loads, and analyzes its properties via the linearized equations. The term analysis of incremental elastic deformations is often employed for the method. The relevant questions include whether there exists an equilibrium state corresponding to loads close to those of the reference state, whether it is unique, etc. The existence of solutions to the equations of classical linear elasticity i.e., of the equations linearized about a stress-free state, is a particular case. No assumptions are made to exclude the nonuniqueness of solutions of the linearized equations. The unique solvability is related to the positive-definiteness of the second variation of the total stored energy on the variations compatible with the kinematical constraints. The bifurcation is likely to occur when the second variation is positive semidefinite on the class of all variations and equal to 0 on some nontrivial variation. This implies that the linearized equations lose the uniqueness (both zero variation and the just mentioned nontrivial variation solve the equations with the same data). In this situation, a further loading of the specimen can result in the occurrence of instabilities, such as barreling or buckling under compression, necking and formation of shear bands in tension or the formation of surface wrinkling. Under the uniform positivity of the second variation at the reference state, not only the linearized equations, but also the original nonlinear equations are solvable for data differing slightly from the data for the reference state. This is based on the implicit function theorem applied to the infinite dimensional spaces of displacements.