Plastic Buckling

Abstract

In a typical boundary value problem involving prescribed nominal traction rates on a part S _F of the boundary surface, and prescribed velocities on the remainder S _v , more than one mode of deformation may be possible when the applied load reaches a critical value. The lack of uniqueness of the deformation mode under given boundary conditions is commonly referred to as bifurcation, the current shape and mechanical state of the body being supposed to be given or previously determined. For a linear solid, in which the strain rate is a unique linear function of the stress rate during both loading and unloading, a bifurcation mode corresponds to an eigensolution of the field equations and represents a mode quasi-statically possible under constant loads on S _F and rigid constraints on S _v . In dealing with the conventional elastic/plastic solid, which is bilinear in the sense that the strain rate is related to the stress rate by separate linear functions for loading and unloading, it is convenient to introduce a linear comparison solid with identical boundary conditions (Section 1.5). While bifurcation in the linearized solid can occur under any given traction rates on S _F and velocities on S _v when the load becomes critical, bifurcation in the actual elastic/plastic solid would occur only under those traction rates for which there is no instantaneous unloading of the material that is currently plastic. The incremental theory of plasticity will be almost exclusively used in this chapter for the estimation of the critical load.