Nonlinear Dynamics 14: 49–56, 1997. c 1997 Kluwer Academic Publishers. Printed in the Netherlands. Generalized Method of Separation of Variables on the Nonlinear Elastic-Plastic Wave Motion YAS ¸AR PALA University of Uludaˇ g, Engineering Faculty, G¨ or¨ ukle, 16059, Bursa, Turkey (Received: 29 April 1996; accepted: 6 March 1997) Abstract. The method of separation of variables is developed in addition to the Karman method and the method of characteristics for the wave motion of uniaxial stress in rods. Rate-independent theory is considered and it is shown that the plastic wave speed is independent of the constant in the stress-strain law. Stresses are determined in explicit forms for two cases. Key words: Nonlinear, elastic, plastic, wave, motion, method of separation of variables. 1. Introduction The assumptions of linear behavior of mechanical elements, while simplifying the solution considerably, is too ideal for most materials. However, consideration of more realistic models results in nonlinear differential equations that usually do not provide any exact solution and must, therefore, be solved numerically. In addition, if the number of independent variables is greater than two, then the resulting equation becomes a nonlinear partial differential equation and unsolvable in most cases. This is perhaps the reason that the explorers have been mostly engaged with possible analytical solutions of nonlinear ordinary differential equations. One of the difficulties in the investigation of searching a possible analytical solution is to separate the nonlinear equation into two or more sub-differential equations where it is possible. These ordinary, probably nonlinear differential equations are another source of difficulty since they do not also have exact solutions in general. However, on the other hand, the more difficult problem is perhaps the justification of initial and boundary conditions into the analytical solution and this largely requires these conditions to be modified. All these may be valid for fixed boundary problems. For the moving boundary problems, one may encounter ever increasing difficulties in the analytical solution. Singularity is another source of complexity. For the most part, the solutions of nonlinear equations do not posses the functional simplicity of the solutions of linear equations. Functions with essential singularities are the rule rather than an exception. Highly restrictive conditions must be imposed if such singularities are to be excluded. Despite these obstacles, several advancements have been recorded in this field of intricate equations (see, e.g., [1, 2, 4, 5]). It is the impetus for us to investigate a possible analytical solution of the so-called generalized wave equation in one dimension for the material that does not obey Hooke’s law. The foundation of the rate-independent theory of plastic wave propagation was provided by Donnell [8] who studied the effect of a nonlinear stress-strain law on the propagation of inelastic stress in a bar. He considered a material with a bilinear stress-strain relation and predicted that two distinct wave fronts would propagate through the material. The analysis of