A Boundary Value Problem in the Theory of Plastic Wave Propagation* By

In the analysis of boundary value problems in the theory of plasticity, the general situation arises that different partial differential equations are to be satisfied depending on whether the material is in the plastic or elastic state. The criterion determining the state at any material point depends on the dependent variables and their derivatives with respect to time. Thus the regions of application of the different differential equations must be determined from the boundary and initial conditions as the solution is developed. The theory of the propagation of plastic waves in one dimension is a case in which the solution, including the determination of the unknown plastic-elastic boundaries, can be treated. An example is presented in this paper which illustrates the many types of boundary determination conditions which must be used. The method is based on the numerical integration along the characteristics of the hyperbolic equations arising, one linear and one quasi-linear. The development is possible since forward integration along characteristics enables the unknown boundaries to be determined independently of the subsequent solution. This situation is contrasted with other problems in the theory of plasticity. The complexity of the procedure indicates the difficulty to be anticipated with analytical treatment of such problems, and with the numerical treatment of problems involving more extensive plastic flow.