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

Abstract

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.

Undular bores generated by fracture

“…It is worth noting that the strains encountered in our experiments are two orders of magnitude higher than the amplitudes of solitary waves in [9,10,13,54]. It is well known that small amplitude solitary waves of the Gardner equation are very close to the solitary waves of the KdV equation, while extended models are better at describing waves of moderate amplitude (see [28] and references therein).…”

“…where ν is Poisson's ratio. We assume the same scaling as in the asymptotic theory developed for waveguides with cylindrical geometry in [27,28]. Then, up to appropriate quartic terms,…”

“…Finally, following from the recent systematic asymptotic derivation of the equation of motion for a longitudinal strain wave in a nonlinear elastic rod (circular cross section) [28], we introduce the next dispersive correction and two nonlinear terms present at the same order as the cubic nonlinear term with coefficient γ [see (4)], thus the full equation is given by the following viscoelastic extended Korteweg-de Vries (veKdV) equation:…”

“…+ a 1 e ttttt + a 2 e t e tt + a 3 ee ttt = 0, (14) for some parameters a 1 , a 2 , and a 3 , which have been calculated for the rod in [28] but are currently unknown for a bar of rectangular cross section. We have also identified the effective modulus of the system with the dynamic Young's modulus E A = E .…”

“…The dominant engineering approaches have so far been based on Fourier analysis and numerical techniques, and this wave structure has not previously been identified as an undular bore. Recognition that such waves are undular bores brings a significant benefit: the Kortewegde Vries (KdV)-type models (see [23][24][25][26][27][28] and references therein) give access to additional analytical approaches. While our modeling results will be useful in the context of these classical experiments, the main focus is on bores generated by fracture.…”