Hyperelastic Rod Research Articles

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.

Uniform asymptotic analysis for waves in an incompressible elastic rod II. Disturbances superimposed on a pre-stressed state

This paper studies the propagation of disturbances superimposed on a pre-stressed incompressible hyperelastic thin rod. Starting from the incremental equations given by Haughton and Ogden (1979, J. Mech. Phys. Solids 27, 179-212 and 489-512), we derive a three parameter-dependent one-dimensional rod equation as the governing equation. In particular, it is found that one parameter plays a crucial role. Depending on whether it is larger or smaller than or equal to a critical value, the shear-wave velocity is larger or smaller than or equal to the bar-wave velocity. In the case that these two velocities are equal, there exist travelling-wave solutions of arbitrary form. This implies that for this particular case the initial disturbance would propagate along the rod without distortion. To see the influence of the pre-stress in detail, we further consider an initial-value problem with an initial singularity in the shear strain. The solutions are expressed in terms of integrals through the method of Fourier transform. We then conduct an asymptotic analysis for the solutions. For a material point in a neighbourhood behind the shear-wave front, the phase function of these integrals has a stationary point at infinity. Here, we use a technique of uniform asymtotic expansion to handle this case. An asymptotic expansion, correct up to order O(t-1), for the shear strain, which is uniformly valid in a neighbourhood behind the shear-wave front, is derived. For material points in other spatial domains, the method of stationary point is applicable, and asymptotic expansions (correct up to order O(t-1)) are obtained. A novelty is that we are able to deduce precise qualitative information about the waves in the far field from our analytic results. Wave profiles for two concrete examples are also provided.

Design Sensitivity for a Hyperelastic Rod in Large Displacement with Respect to Its Midcurve Shape

The paper is concerned with an optimal design problem for a hyperelastic rod. The function describing the position of a point at the line of rod cross-section centroids in its reference configuration is the variable subject to optimization. The necessary optimality condition is formulated. The continuation method combined with the gradient descent algorithm are employed to solve this problem numerically. Numerical results are provided.

Nonlinear axisymmetric waves in compressible hyperelastic rods: long finite amplitude waves

This paper investigates nonlinear axisymmetric waves in compressible hyperelastic circular cylindrical rods. We consider first a compressible Mooney-Rivlin material to obtain exact governing equations. To further study the problem, we introduce the notion of long finite amplitude waves and derive the corresponding simplified model equations, which gives the framework for studying problems like wave-interactions arising through collision or reflection. The asymptotically valid far-field equation is consequently deduced from the simplified model equations. Then, using a strained-coordinate method, we obtain the second-order solitary wave solution. The result is not only of interest itself, but also provides a suitable initial condition for wave interaction problems. Finally, the results for a general hyperelastic rod are presented.

Post-buckling behavior of a non-linearly hyperelastic thin rod with cross-section invariant under the dihedral group Dn

Post-buckling behavior of a non-linearly hyperelastic thin rod with cross-section invariant under the dihedral group Dn

Existence of solutions for a finite nonlinearly hyperelastic rod

Existence of solutions for a finite nonlinearly hyperelastic rod

An application of singularity theory to a bifurcation problem

A brief review of some results from [2] on post-buckling behavior of a nonlinearly hyperelastic thin rod with polygonal (cross-section).

Acceleration wave propagation in hyperelastic rods of variable cross-section

It is shown that when an acceleration wave propagates in hyperelastic rod with slowly varying cross-section, the transport equation for the wave intensity is a generalized Riccati equation. The three coefficients in the equation all depend on the material properties, but only the coefficient of the quadratic term is independent of the effect of area change. Three theorems are proved, based on the use of comparison equations, which establish that in general the acceleration wave intensity will become infinite (escape) after the wave has propagated only a finite distance along the rod. The existence of thresholds for the initial intensity are also established in certain cases, with their most notable property being that as the intial intensity decreases towards the threshold, so the distance the wave propagates to escape increases without bound.

Dynamical theory of hyperelastic rods

A rod is regarded as a one-dimensional mathematical model of a three-dimensional body. The exact field equations governing the motion of a hyperelastic rod are derived from the general three-dimensional theory. Then, by a suitable restriction on the number of displacement variables, a hierarchy of approximating theories is established. Because such theories are generated by a kinematic hypothesis, a precise, quantitative idea of the nature of the simplifying assumptions is furnished. An analysis of the structure of these approximating theories yields three distinct approaches by which they may be interpreted. Finally, constraints and their connection with other approximate theories are investigated. In particular, classical nonlinear theories and theories for planar motion are developed in this context of constrained theories.