Compact solitary waves in linearly elastic chains with non-smooth on-site potential (original) (raw)
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Physical Review E, 2019
A granular chain of elastic spheres via Hertzian contact incorporates a classical nonlinear force model describing dynamical elastic solitary wave propagation. In this paper, the multiple solitary waves and their dynamic behaviors and stability in such a system are considered. An approximate KdV equation with the standard form is derived under the long-wavelength approximation and small deformation. The closed-form analytical single-and multiple-soliton solutions are obtained. The construction of the multiple-soliton solutions is analyzed by using the functional analysis. It is found that the multiple-soliton solution can be excited by the single-soliton solutions. This result is confirmed by the numerical analysis. Based on the soliton solutions of the KdV equation, the analytic solutions of multiple dark solitary waves are obtained from the original dynamic equation of the granular chain in the long-wavelength approximation. The stability of the single and multiple dark solitary wave solutions are numerically analyzed by using both split-step Fourier transform method and Runge-Kutta method. The results show that the single dark solitary wave solution is stable, and the multiple dark solitary waves are unstable.
Highly nonlinear solitary waves in chains of cylindrical particles
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We study the dynamic response of uniform granular chains composed of short cylindrical particles excited by an impulse. The particles in the chains are arranged with their axes orthogonal to the chain's axis, and the particles maintain a constant relative orientation angle. We study the formation and propagation of solitary waves in the chains varying the orientation angle (α) between particles, and show tunability of the stress transfer as a function of α. We use the general Hertzian contact theory to model the interaction between particles. We compare experimental findings with theoretical predictions based on the long wavelength approximation, and with numerical predictions based on a one-dimensional discrete particle model, and on a three-dimensional finite element approach, finding good agreement.
Solitary waves in the granular chain
Physics Reports, 2008
Solitary waves are lumps of energy. We consider the study of dynamical solitary waves, meaning cases where the energy lumps are moving, as opposed to topological solitary waves where the lumps may be static. Solitary waves have been studied in some form or the other for nearly 450 years. Subsequently, there have been many authoritative works on solitary waves. Nevertheless, some of the most recent studies reveal that these peculiar objects are far more complex than what we might have given them credit for. In this review, we introduce the physics of solitary waves in alignments of elastic beads, such as glass beads or stainless steel beads. We show that any impulse propagates as a new kind of highly interactive solitary wave through such an alignment and that the existence of these waves seems to present a need to reexamine the very definition of the concept of equilibrium. We further discuss the possibility of exploiting nonlinear properties of granular alignments to develop exciting technological applications.
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Physical Review E, 2011
We study experimentally the interaction between two solitary waves that approach one to another in a linear chain of spheres interacting via the Hertz potential. When these counter propagating waves collide, they cross each other and a phase shift respect to the noninteracting waves is introduced, as a result of the nonlinear interaction potential. This observation is well reproduced by our numerical simulations and it is shown to be independent of viscoelastic dissipation at the beads contact. In addition, when the collision of equal amplitude and synchronized counter propagating waves takes place, we observe that two secondary solitary waves emerge from the interacting region. The amplitude of secondary solitary waves is proportional to the amplitude of incident waves. However, secondary solitary waves are stronger when the collision occurs at the middle contact in chains with even number of beads. Although numerical simulations correctly predict the existence of these waves, experiments show that their respective amplitude are significantly larger than predicted. We attribute this discrepancy to the rolling friction at the beads contacts during solitary wave propagation. PACS numbers: 43.25.+y 45.70.-n