Explicit dispersion relation for strongly nonlinear flexural waves using the homotopy analysis method

Abstract

Using the homotopy analysis method, we present an explicit frequency-versus-wavenumber nonlinear dispersion relation for a flexural elastic beam. In our analysis, we employ the Euler–Bernoulli kinematic hypothesis and consider both a conventional transverse motion model and an inextensional planar motion model. As an example, we consider geometric nonlinearity in the form of Green–Lagrange strain. The underlying constitutive relation is formulated by linearly relating the second Piola–Kirchhoff stress to the Green–Lagrange strain, although the method is directly applicable to material nonlinearities as well. The derived analytical solution, for each model, is obtained to Mth-order accuracy and is verified by comparing with a numerical result brought about by laboriously finding the roots of the corresponding travelling-wave implicit relation governing the nonlinear elastodynamics. This is the first derivation of an explicit dispersion relation for an elastic beam undergoing strongly nonlinear finite flexural deformation. The derived relation characterizes the nature of a traveling cosine-like nonlinear wave throughout its stable pre-breaking state.