Introduction

The simplicity of the collapse load theorems masks some of the more complex aspects of engineering applications involving plasticity. Solutions for fully three-dimensional elastic–plastic response will generally be difficult if not impossible to obtain in closed form. There is, however, one more class of problem for which relatively simple solutions are possible. This is the class of two-dimensional problems concerning plane plastic flow for which regions of the material are in a failure condition. The failure regions need not cover the entire body, but within the failing zone we must be assured that the yield condition is satisfied everywhere.

For these plane problems there will be three unknown components of stress: for example, σxx, σyy,σxy, where the (x, y)-plane is taken to be the plane of the problem. Within the failing region the three stresses are related by three equations: two equations of equilibrium plus the equation of the yield surface. Only first-order derivatives are involved. While this in itself does not appear overly complex, it will become apparent that considerably more simplification is possible by invoking a new coordinate system. Introducing coordinates that coincide with the potential failure surfaces, we cause the system of equations to become extremely simple. In our two-dimensional problem, the potential failure surfaces are seen simply as lines and they have come to be called slip lines.