Compressible Hyperelastic Rods Research Articles

On H 2-solutions for a Camassa-Holm type equation

Abstract Camassa-Holm type equations arise as models for the unidirectional propagation of shallow water waves over a flat bottom. They also describe finite length, small amplitude radial deformation waves in cylindrical compressible hyperelastic rods. Under appropriate assumption on the initial data, on the time T T , and on the coefficients of such equation, we prove the well-posedness of the classical solutions for the Cauchy problem.

On the Modelling of Shallow-Water Waves with the Coriolis Effect

Consideration herein is a rotation-Camassa–Holm-type equation, which can be derived as an asymptotic model for the propagation of long-crested shallow-water waves in the equatorial ocean regions with the weak Coriolis effect due to the Earth’s rotation, and is also related to the compressible hyperelastic rod model in the material science. This model equation has a formal Hamiltonian structure, and its solution corresponding to physically relevant initial perturbations is more accurate on a much longer time scale. It is shown that the solutions blow up in finite time in the sense of wave breaking. A refined analysis based on the local structure of the dynamics is performed to provide the wave-breaking phenomena. The effects of the Coriolis force caused by the Earth’s rotation and nonlocal higher nonlinearities on blow-up criteria and wave-breaking phenomena are also investigated. Finally, a sufficient condition for global strong solutions to the equation in some special case is given.

On the model of the compressible hyperelastic rods and Euler equations on the circle

Considered herein is a geometric investigation on the one-parameter γ-equations modeled in the cylindrical compressible hyperelastic rods. It is shown that the family of equations can only be realized as an Euler equation on the Lie group Diff(S1) of all smooth and orientation preserving diffeomorphisms on the circle if the material parameter γ=1, which corresponds to the Camassa–Holm equation. In contrast, the Benjamin–Bona–Mahony (BBM) equation with the parameter γ=0 in this family of equations is not an Euler equation on Diff(S1) for any inertia operator.

Numerical simulation of propagation of solitary deformation waves in a compressible hyperelastic rod

AbstractIn this paper, propagation of axisymmetric deformation waves in circular cylindrical rods of compressible hyperelastic material is studied. The corresponding model equation that describes propagation of waves of moderate wave-length takes into account the coupling effect of the material nonlinearity and the geometric size of the rod. The model equation is integrated numerically under sech2-type initial conditions making use of the discrete Fourier transform based pseudospectral method. Numerical simulations are carried out over a wide range of material parameters. The analysis of the time–space behaviour of solutions demonstrates that in some domains of space of material parameters single solitary wave solutions or trains of interacting solitons can be emerged from the initial localised pulses.

Wave Breaking and Global Existence for a Generalized Two-Component Camassa-Holm System

In this paper, we study a generalized two-component Camassa–Holm system which can be derived from the theory of shallow water waves moving over a linear shear flow. This new system also generalizes a class of dispersive waves in cylindrical compressible hyperelastic rods. We show that this new system can still exhibit the wave-breaking phenomenon. We also determine the exact blow-up rate of such solutions. In addition, we establish a sufficient condition for global solutions.

H1-perturbations of Smooth Solutions for a Weakly Dissipative Hyperelastic-rod Wave Equation

We consider a weakly dissipative hyperelastic-rod wave equation (or weakly dissipative Camassa–Holm equation) describing nonlinear dispersive dissipative waves in compressible hyperelastic rods. We fix a smooth solution and establish the existence of a strongly continuous semigroup of global weak solutions for any initial perturbation from \(H^{1}(\mathbb{R}).\) In particular, the supersonic solitary shock waves [8] are included in the analysis.

Traveling waves in compressible elastic rods

We determine all weak traveling wave solutions of a model for nonlinear dispersive waves in cylindrical compressible hyperelastic rods. Besides the previously known smooth, peaked, and cusped solutions, the equation is shown to admit compactons, stumpons, and fractal-like waves.

GENERALIZED KINK WAVES IN A GENERAL COMPRESSIBLE HYPERELASTIC ROD

In this paper, we employ both qualitative analysis and numerical simulation to investigate bounded traveling waves in a general compressible hyperelastic rod. Some new bounded traveling waves are found. Their implicit expressions are obtained. Also, their planar graphs are simulated. Since they possess some properties of kink waves, we call them generalized kink waves.

Global Weak Solutions to a Generalized Hyperelastic-rod Wave Equation

We consider a generalized hyperelastic-rod wave equation (or generalized Camassa--Holm equation) describing nonlinear dispersive waves in compressible hyperelastic rods. We establish existence of a strongly continuous semigroup of global solutions for any initial data from $H^1(\R)$. We also present a weak equals strong uniqueness result.

Compactons in a general compressible hyperelastic rod

Abstract There is much work in the area of solitons, but there appears very little in the way of compactons. After the approach of Adomian decomposition applied successfully (e.g. [Chaos, Solitons & Fractals 13 (2002) 321; Chaos, Solitons & Fractals 13 (2002) 1053; Appl. Math. Comput. 134 (2003) 487; Chaos, Solitons & Fractals 14 (2002) 1151]), we develop a new method to enlarge the window of compacton research, that is, the bifurcation method of traveling wave systems and numerical simulation of traveling wave equation. As above mentioned, we employ both bifurcation method and numerical simulation to investigate bounded traveling waves in a general compressible hyperelastic rod in this paper. We find some new bounded traveling waves, which are defined on finite core regions. Their planar simulative graphs and implicit expressions imply that they possess the properties of compactons.

Stability of a class of solitary waves in compressible elastic rods

We prove that the solitary waves of a model for nonlinear dispersive waves in cylindrical compressible hyperelastic rods are orbitally stable. This establishes that the shape of the wave is stable.

Uniform asymptotic analysis for transient waves in a pre-stressed compressible hyperelastic rod

This paper studies the propagation of disturbances in a pre-stressed circular rod composed of a general compressible hyperelastic material. Two coupled four-parameter dependent equations are derived as the governing equations for small axial-radial deformations superimposed on a pre-stressed rod. It is found that one parameter plays a crucial role. Depending on its value, the shear-wave velocity could be larger or smaller than or equal to the bar-wave velocity. In the case that these two velocities are equal, there exist travelling waves of arbitrary form. An initial-value problem with an initial discontinuity in the longitudinal stress is also studied. The technique of Fourier transform is used to express the solutions in terms of integrals. Then, by combining a general technique developed by us previously and the method of stationary phase, we managed to derive the asymptotic expansions for the longitudinal stress, which are uniformly valid in the whole spatial domain. The novelty is that we are able to provide completely analytical descriptions for the transient waves. The general characteristics of the disturbances is then determined. It is found that pre-stresses affect the motion greatly. In particular, the presence of pre-stresses can change the wave front from a receding type to an advancing type, or vice versa. Quantitative results for two concrete examples are also provided.

Solitary shock waves and other travelling waves in a general compressible hyperelastic rod

In the literature, it has been conjectured that solitary shock waves can arise in incompressible hyperelastic rods. Recently, it has been shown that this conjecture is true. One might guess that when compressibility is taken into account, such a wave, which is both a solitary wave and a shock wave, can still arise. One of the aims of this paper is to show the existence of this interesting type of wave in general compressible hyperelastic rods and provide an analytical description. It is difficult to directly tackle the fully nonlinear rod equations. Here, by using a non–dimensionalization process and the reductive perturbation technique, we derive a new type of nonlinear dispersive equation as the model equation. We then focus on the travelling–wave solutions of this new equation. As a result, we obtain a system of ordinary differential equations. An important feature of this system is that there is a vertical singular line in the phase plane, which leads to the appearance of shock waves. By considering the equilibrium points and their relative positions to the singular line, we are able to determine all qualitatively different phase planes. Those paths in phase planes which represent physically acceptable solutions are discussed one by one. It turns out that there is a variety of travelling waves, including solitary shock waves, solitary waves, periodic shock waves, etc. Analytical expressions for all these waves are obtained. A new phenomenon is also found: a solitary wave can suddenly change into a periodic wave (with finite period). In dynamical systems, this represents a homoclinic orbit suddenly changing into a closed orbit. To the authors9 knowledge, such a bifurcation has not been found in any other dynamical systems.