Longitudinal bulk strain solitons in a hyperelastic rod with quadratic and cubic nonlinearities

TL;DR

This paper derives and analyzes weakly-nonlinear soliton solutions in a hyperelastic rod with quadratic and cubic nonlinearities, validated by numerical simulations showing stable propagation and novel 'table-top' solitons.

Contribution

It introduces extended Boussinesq and KdV-type equations for hyperelastic rods and constructs approximate soliton solutions validated by numerical simulations.

Findings

Excellent agreement between analytical solutions and simulations for small amplitude waves.

Stable propagation of 'table-top' solitons observed.

Method extends applicability beyond small amplitude regime.

Abstract

We study long nonlinear longitudinal bulk strain waves in a hyperelastic rod of circular cross section within the scope of the general weakly-nonlinear elasticity leading to a model with quadratic and cubic nonlinearities. We systematically derive the extended Boussinesq and Korteweg - de Vries - type equations and construct a family of approximate weakly-nonlinear soliton solutions with the help of near-identity transformations. These solutions are compared with the results of direct numerical simulations of the original nonlinear problem formulation, showing excellent agreement within the range of their asymptotic validity (waves of small amplitude) and extending their relevance beyond it (to the waves of moderate amplitude) as a very good initial guess. In particular, we were able to observe a stably propagating "table-top" soliton..