On Boussinesq-type models for long longitudinal waves in elastic rods
journal contributionposted on 2019-03-13, 14:48 authored by F.E. Garbuzov, Karima KhusnutdinovaKarima Khusnutdinova, I.V. Semenova
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 Boussinesq-type 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.
F.E.G. and I.V.S. acknowledge the financial support from the Russian Science Foundation, Russia under the grant # 17-72-20201. K.R.K. is grateful to the UK Institute of Mathematics and its Applications (IMA) and the QJMAM Fund for Applied Mathematics for the financial support of her travel to the European Solid Mechanics Conference (ESMC2018) in Bologna, Italy in the summer of 2018 where parts of this work have been discussed and developed.
- Mathematical Sciences