In this paper we revisit the derivations of model equations describing long nonlinear longitudinal bulk strain waves in elastic rods within the scope of the Murnaghan model in order to derive a Boussinesqtype model, and extend these derivations to include axially symmetric loading on the lateral boundary surface, and longitudinal pre-stretch. We systematically derive two forced Boussinesq-type models from the full equations of motion and non-zero surface boundary conditions, utilising the presence of two small parameters characterising the smallness of the wave amplitude and the long wavelength compared to the radius of the waveguide. We compare the basic dynamical properties of both models (linear dispersion curves and solitary wave solutions). We also briefly describe the laboratory experiments on generation of bulk strain solitary waves in the Ioffe Institute, and suggest that this generation process can be modelled using the derived equations.

In this paper we study nonlinear longitudinal bulk strain waves in a hyperelastic rod of circular cross section within the scope of the general weakly-nonlinear model with quadratic and cubic nonlinearities. We derive the extended Boussinesq and Korteweg -de Vries -type equations and construct a family of approximate weakly-nonlinear soliton solutions with the help of the near-identity transformations. These solutions are compared with the results of direct numerical simulations of the original fully nonlinear problem formulation.

No abstract

No abstract