12.1 INTRODUCTION

Soil plasticity along with all other branches of mechanics of solids requires the consideration of geometry or compatibility and of equilibrium or dynamics, and of the relation between stress and strain. Equilibrium equations are determined direct-ly by summation of forces (see [2.23]); compatibility equations ensure that strain or strain-rate components are consistent with a displacement or velocity field from which they must be derivable (see [2.24]). Compatibility and equilibrium equations are therefore independent of material properties and hence valid for elastic as well as elastic-plastic problems. The differentiating feature is the relation between stress and strain. The extreme difficulty in obtaining an exact plastic solution even with the aid of digital computers is due mainly to the fact that the stress-strain relationship in the plastic range is far more complicated than Hooke's law for linearly elastic materials.

Plastic action is load path-dependent and almost always requires step-by-step calculations that follow the history of loading. They are further complicated by the fact that the elastic plastic boundary is changing with continued loading and the stress-strain relationships for loading and unloading are different. Even without this complication, there are few solutions available for problems that consider nonlinear elasticity.

It is apparent that an exact elastic plastic solution of a practical soil mechanics problem is unlikely. Drastic simplifications and idealizations are essential for a reasonable approximate solution. The geometry or compatibility, the stress strain relations and the equations of equilibrium must all be idealized to accomplish a solution.

For example, the material may be...