Non-linear elastic waves in a finite bar made of a material having non-linear stress-strain relation

Abstract

This study is devoted to the non-linear elastic wave motion in a finite bar made of a material obeying non-linear stress-strain relation. Non-linear governing partial differential equation is separated into two non-linear ordinary differential equations using a new method for the separation. Analytical solution satisfying boundary and initial conditions are constructed from the proper solutions of equations. Stresses, wave velocities and displacements in the rod are found for arbitrary conditions. Two methods are developed. While the first method brings a limitation on the solution, the second one removes this problem by selecting more practically logical conditions. The present method is also applicable to elastic-plastic wave motion in rods of engineering materials. On the contrary to Karman-Donnel elasto-plastic wave theory developed for semi infinite bars subjected to initial velocity condition only, the present method proves to be useful for rods of finite lengths subjected to both initial and boundary conditions.