Exact solution of the Mindlin–Herrmann model for longitudinal vibration of an isotropic rod

This paper presents a new approach to the problem of coupled longitudinal and transversal propagations of stress waves in an isotropic thick and elastic rod, based on the Mindlin–Herrmann theory. The novelty is that Hamilton’s variational principle is used not only for derivation of the governing equations and set of natural boundary conditions, but also for obtaining the exact solution in terms of Green’s functions directly from the Lagrangian. The success of this approach is based on the existence of multiple orthogonalities of the eigenfunctions. The proposed method is much easier than the standard approach of building Green’s functions. A numerical example illustrates the method of finding eigenfrequencies and eigenfunctions for isotropic Mindlin–Herrmann rod.