7.1 Introduction

In previous sections we have considered cases when the boundary and initial conditions conform to the assumption that only a single component of stress and a single component of velocity occur. Sometimes, in these cases, a constitutive equation is not used at all and the procedure is somehow similar to that used in some static problems, i.e., the equation of motion and continuity condition are only combined with a yield condition but not a constitutive equation. Using this procedure a sufficient number of equations is obtained for the number of unknown quantities required. Sometimes else, a constitutive equation is used, but this is in fact a one-dimensional stress-strain relation, and therefore as a rule, the methods used do not greatly differ from those used for the study of the propagation of longitudinal waves in thin bars. In all the cases examined the propagation of a single type of wave has been considered if the equation of motion is of the hyperbolic type. Otherwise, propagation is assumed to occur by diffusion according to an equation of parabolic type.

First let as present shortly the static, plane stress, problem (Sokolovski [1969]) when at the orifice one is given a pressure. For a Mises type of yield condition

the problem is hyperbolic close to the orifice, but elliptic at farther distances Fig. 7.1.1. The characteristics are logarithmic spiral lines. The figure corresponds to p= when b ?5.13 a. On this circle the problem is parabolic.