Nonlinear torsional wave propagation modeled with Lagrangian mechanics

Abstract

Coupled nonlinear wave equations are derived for the special case of finite-amplitude torsional wave propagation in a thin rod with circular cross section. Finite-amplitude shear deformation from a torsional source produces longitudinal motion in the rod due to the normal stress effect. The derivation procedure begins by expanding the displacement field in powers of the radial coordinate, and then expressing the displacement field in terms of wave functions for the twist and longitudinal displacement by applying the stress-free nonlinear boundary conditions. The hypothesis of plane cross sections is found to be insufficient for satisfying the boundary conditions at the required order. Lagrangian mechanics are employed to obtain coupled wave equations based on the resulting expressions for the kinetic and strain energy densities. The approach thus differs from a previous derivation by Sugimoto et al. [Wave Motion 6, 247 (1984)], who did not assume the existence of a strain energy function (hypoelasticity) and employed nonlinear dynamical equations to obtain the coupled wave equations. Similarities and differences between the wave equations determined using the present approach and those obtained by Sugimoto et al. are discussed.