Elementary Theory of One-Dimensional Stress Waves in Bars

This chapter discusses the primary one-dimensional theory of stress waves in bars. It begins by discussing the two coordinates that exist in continuum mechanics. These coordinates are used to study the movement of the medium. These are material coordinate system and spatial coordinate system. The authors use the material coordinate system (Lagrange system) to study the longitudinal motion of a bar with uniform cross section. Characteristic lines and the compatibility relationships along the characteristic lines are discussed with explanatory equations. In this discussion, the directional derivative method is used to deduce the equations of characteristics. Further, it explains the longitudinal stress waves propagating in a semi-infinite long bar. The chapter presents an insight into strong discontinuity, weak discontinuity, shock waves, and continuous waves. Dispersion effects induced by the transverse inertia are explained in one section. Finally, the last section concludes the chapter by describing the propagation of a simple kind of transverse wave, namely the elastic torsion wave in cylindrical bars. The torsion wave theory based on the invariable plane cross-section assumption leads solutions that are identical with Pochhammer's elastodynamic exact solution.