Nonlinear axial shear wave propagation in a hyperelastic incompressible solid

Summary This paper is concerned with the propagation of axial shear waves in an incompressible isotropic hyperelastic solid, whose strain energy function is expressible as a power series in (I ₁-3) and (I ₂-3) whereI ₁ andI ₂ are the first and second basic invariants of the left Cauchy-Green tensorB. Numerical solutions are presented for problems of wave propagation produced by a step function application, or a finite duration pulse, of axial shear stress at the surface of a cylindrical cavity in an unbounded medium. A modification of MacCormack's finite difference scheme [1] is proposed and is used to obtain these solutions along with a procedure for the determination of the position of the shock front for the step function application.An estimate of the breaking time of a wave, obtained from a procedure proposed by Whitham [2], is compared with the numerical results. The dissipation of mechanical energy due to shock propagation is considered.With 4 Figures     

Shock front analysis for axial shear wave propagation in a hyperelastic incompressible solid

Summary A wavefront analysis is employed to study the propagation of axial shear waves in an incompressible hyperelastic solid, whose strain energy function is expressible as a truncated power series in terms of the basic invariants of the left Cauchy-Green tensor. Waves are generated by the application of an axial shear stress at the surface of a cylindrical cavity in an unbounded medium. Depending on the nature of the boundary condition, an acceleration front or a shock front propagates from the boundary of the cavity. For an acceleration front, the coefficients in the wavefront expansion satisfy a sequence of transport equations which can be solved analytically. For a shock front, a wavefront analysis gives approximate formulas for the wave speed, shock front and intensity of the various field variables at the front. As well, our shock front analysis is used to devise a method of estimating the breaking distance of a shock front. In order to test the validity of the results of our wavefront analysis, numerical solutions are obtained for waves initiated by a step function or by a finite duration pulse at the boundary. Our numerical solutions are found by using a recently proposed relaxation scheme for systems of conservation laws.     

Finite amplitude spherically symmetric wave propagation in a compressible hyperelastic solid

Summary Some aspects of the wave propagation, resulting from the spherically symmetric expansion of a thick walled hyperelastic shell and the limiting case of expansion of a cavity in an unbounded medium, are investigated. It is assumed that the shell is isotropic and uniform in the natural reference state and its strain energy function is a particular compressible generalization of that for the neo-Hookean solid. The response of a compressible shell, due to a spatially uniform time dependent application of internal pressure, is compared with that for the neo-Hookean shell taken as a limiting case of the compressible shell. This is also done for an unbounded medium.A finite difference method which uses the relation along one of the families of characteristics is used to obtain numerical results. In order to implement this method the governing equations are expressed as a system of first order partial differential equations in conservation form.With 9 Figures     

Plane crack propagation in a hyperelastic incompressible material

We propose a crack propagation criterion for hyperelastic materials (rubber type material) within the framework of plane elasticity in finite deformation. The criterion is based on the examination of the asymptotic elastic field near the crack tip prior to propagation. According to this criterion, the propagation will take place for a critical value of the strain energy density intensity factor. The kink angle, obtained by applying the criterion of maximum opening stress, will depend on the fracture tensile stress of the actual material. We propose to use a local iterative finite element method to compute the asymptotic quantities involved in the criterion at a reasonable cost. Examples of computation for some hyperelastic laws simulating the behavior of vulcanized rubber are presented.     

The propagation and reflection of finite amplitude cylindrically symmetric shear waves in a hyperelastic incompressible solid

Summary We consider axially symmetric shear waves propagating in an incompressible hyperelastic thick-walled cylindrical shell, whose strain energy function is expressible as a truncated power series in terms of the basic strain invariants. A continuous pulse is initiated at the interior boundary of the cylinder by surface tractions of finite duration. The pulse propagates away from the interior boundary, then reflects from the outer boundary, and subsequently reflects back and forth between the two boundaries of the cylinder. We analyze shock development of the first incident and first reflected wave. The incident pulse can break before it reaches the outer boundary. Using Whitham's nonlinearization technique, we determine conditions under which the incident wave breaks and which shock waves can subsequently occur. Similar calculations are carried out for the first reflection. The formulas obtained for the incident pulse provide accurate estimates of the breaking distance and time, and the location of the shock paths, for any incident shock waves that occur. Results obtained for the reflected wave cannot be used to make similar estimates, but they do reveal that once the pulse has completely left the outer boundary, the possible shock that can occur is the same as for the incident wave. Our analysis is carried out for axial shear waves. A similar analysis can be done for torsional shear waves, but not for combined axial and torsional shear wave propagation. We illustrate the conclusions of our shock analysis with numerical solutions obtained using a relaxation scheme for systems of conservation laws. Numerical results are obtained for axial shear and for combined axial and torsional shear. These results indicate that the shock behavior indicated by our analysis of axial shear is also valid for combined axial and torsional shear wave propagation.     

Surface wave propagation in a fluid-saturated incompressible porous medium

A study of surface wave propagation in a fluid-saturated incompressible porous half-space lying under a uniform layer of liquid is presented. The dispersion relation connecting the phase velocity with wave number is derived. The variation of phase velocity and attenuation coefficients with wave number is presented graphically and discussed. As a particular case, the propagation of Rayleigh type surface waves at the free surface of an incompressible porous half-space is also deduced and discussed.     

On finite plane strain of an isotropic incompressible hyperelastic material

Abstract

The purpose of this paper is to derive a new formulation of the governing equations for the complex variable in the deformed configuration of an isotropic incompressible hyperelastic material, assuming that the material undergoes a constant finite axial stretch and then a finite plane strain.

As an application we consider the circular shear of a tube of such a material and, if we do not take into account torsion, it is proved that the plane deformation is given by an ordinary differential equation of second order, the integration of which requires the determination of the first derivative of the unknown function at the outer wall of the tube and the knowledge of the strain‐energy function. The axial stretch of the tube can be determined through two equations only when the strain‐energy function is specified.     

Thermoelastic wave propagation in an elastic solid containing a finite crack

Investigated in this paper is the scattering of plane harmonic thermoelastic waves around the tip of a finite crack. Integral transform techniques are used to formulate the problem and reduce it to Fredholm integral equations of the second kind. The equations are solved numerically and the singular stress field near the crack tip is determined. In particular, the variation of the stress intensity factor with the frequency of the incoming wave is exhibited graphically. The peak in the magnitude of the stress intensity factor is of paramount interest in the application of fracture mechanics to thermal stress problems.     

Comparative study of different nonconserving time integrators for wave propagation in hyperelastic waveguides

This paper addresses the formulation and numerical efficiency of various numerical models of different nonconserving time integrators for studying wave propagation in nonlinear hyperelastic waveguides. The study includes different nonlinear finite element formulations based on standard Galerkin finite element model, time domain spectral finite element model, Taylor–Galerkin finite element model, generalized Galerkin finite element model and frequency domain spectral finite element model. A comparative study on the computational efficiency of these different models is made using a hyperelastic rod model, and the optimal computational scheme is identified. The identified scheme is then used to study the propagation of transverse and longitudinal waves in a Timoshenko beam with Murnaghan material nonlinearity.     

Timoshenko-beam effects on transverse wave propagation in carbon nanotubes

This paper studies effects of rotary inertia and shear deformation on transverse wave propagation in individual carbon nanotubes (CNTs) within terahertz range. Detailed results are demonstrated for transverse wave speeds of doublewall CNTs, based on Timoshenko-beam model and Euler-beam model, respectively. The present models predict some terahertz critical frequencies at which the number of wave speeds changes. The effects of rotary inertia and shear deformation are negligible and transverse wave propagation can be described satisfactorily by the existing single-Euler-beam model only when the frequency is far below the lowest critical frequency. When the frequency is below but close to the lowest critical frequency, rotary inertia and shear deformation come to significantly affect the wave speed. Furthermore, when the frequency is higher than the lowest critical frequency, more than one wave speed exists and transverse waves of given frequency could propagate at various speeds that are considerably different than the speed predicted by the single-Euler-beam model. In particular, rotary inertia and shear deformation have a significant effect on both the wave speeds and the critical frequencies especially for CNTs of larger radii. Hence, terahertz transverse wave propagation in CNTs should be better modeled by Timoshenko-beam model, instead of Euler-beam model.     

One-dimensional wave propagation in an initially deformed incompressible medium with different behaviour in tension and compression

The propagation of 1-dimensional waves in an initially deformed incompressible medium with different moduli in tension and compression is investigated. Depending on the sign of the initial strains, various possibilities of propagation are shown to exist. The governing equations are nonlinear and a hardening or softening behaviour in shear may be present. Simple wave analytical solutions are given in a semi-infinite incompressible half-space. It is shown that in some situations, a shear pulse applied to the surface of an initially deformed half-space propagates linearly up to a specific value of the shear deformation and nonlinearly after that point.     

Axisymmetric wave propagation in a solid viscoelastic sphere

Two problems of linear wave propagation in a viscoelastic solid sphere are solved. The waves are generated by two types of impact on the surface of the sphere. The deformation has symmetry with respect to an axis through the center of the sphere. The solution is based on a superposition principle which reduces the general solution to a static elastic solution, an elastic solution of an eigenvalue problem and an integral equation of the Volterra type involving time only. The solutions are given in double infinite series involving spherical Bessel functions, Legendre polynomials and Legendre functions of the first kind and order one.     

On surface wave propagation in incompressible, transversely isotropic, pre-stressed elastic half-spaces

The problem of surface wave propagation for a transversely isotropic incompressible pre-stressed half-space is investigated in respect of a general in-plane fibre direction and a general angle of propagation. It is shown that the surface wave speed crucially depends on the normal static stress in the finite primary deformation. Some special cases leading to considerable simplification are pointed out. An important particular case of the fibres assumed almost inextensible is tackled both numerically and asymptotically to reveal an idealised fibre-reinforced material as a leading order approximation.     

Mullins effect on incompressible hyperelastic cylindrical tube in finite torsion

The isotropic softening effect in non-homogeneous deformation resulting from combined effect of torsion, extension and inflation of cylindrical rubber tube is discussed. The effects of deformation induced anisotropy, permanent set and hysteresis are neglected. A general neo-Hookean parent material model is illustrated and subsequently the stress-softening effect on the same hyperelastic material is analyzed. Simple torsion of cylindrical tube with neo-Hookean material model is analyzed and the results obtained are shown in various plots. Analytical results are compared with the experimental results of Rivlin and Saunders. Universal relations are also established for incompressible, isotropic, hyperelastic material for non-homogeneous deformation with isotropic damage function in both virgin and stress-softened cases.     

The flow of an incompressible elastic-perfectly plastic solid

Summary The plane radial flow of an incompressible elastic-perfectly plastic solid through a converging channel with perfectly rough sides is considered. It is shown that the stress field is the same as for the corresponding flow of a rigid-perfectly plastic solid with the same yield stress but the velocity field is significantly different near the sides.

---

Zusammenfassung Es wird die ebene Radialströmung eines inkompressiblen elastisch-plastischen Festkörpers in einem konvergenten Kanal mit vollkommen rauben Wänden betrachtet. Das Spannungsfeld erweist sich hierbei als dasselbe wie bei der entsprechenden Strömung eines starr-plastischen Körpers mit gleicher Fließgrenze; das geschwindigkeitsfeld ist jedoch in der Nähe der Wände wesentlich verschieden.

---

---

With 1 Figure     

粘性联结的层状固体媒质中兰姆波的传播

在复合材料中 ,当有粘弹体存在时 ,粘弹体的粘性会对波的传播造成一定影响。早期的 Thomson-Haskell矩阵以面波描述板层中的场 ,而面波的幅值在各个方向都是常数 ,因此它们不能描述在一衰减媒质中反射或透射波的衰减。本文中引入体波衰减的模型 ,即 Kelvin- Voigt的粘弹性描述 [1] ,该方法导致了一个复数波数。其实部描述了波的传播 ,而虚部则描述波的衰减。1　基本理论传递矩阵可以描述弹性波在包含任意多层的层状媒质中的传播特性 [2 ] 。1.1　无限弹性固体中的波对图 1所示的计算模型 ,其运动位移方程用矢量形式可表示为 :　　 ρ 2 U …     

Finite anti-plane shearing of centrally-cracked solids of incompressible hyperelastic material

Closed-form solutions are derived for two problems of nonlinear elastic fracture mechanics. The cases considered deal with the out-of-plane deformation of a centrally-cracked cylinder of elliptic cross section involving hyperelastic materials of either Neo-Hookean or Mooney-Rivlin type. Each solid is loaded by a self-equilibrating anti-plane shear traction applied to the faces of its crack and has its remote boundary either free or fixed. It is shown that, as in the case of small strains, the equation governing the out-of-plane response decouples from those defining in-plane behavior. It is also found that a finite state of pure out-of-plane deformation can only be sustained in the presence of Poynting stresses. In particular, nonlinear solids are seen to require out-of-plane direct axial stresses to satisfy the prescribed kinematics of deformation. Additionally, the non-regular shear stresses at the crack tips retain the same power of singularity as would exist in their linear elastic counterparts. Interestingly enough, however, the direct stresses, which are regular in linear materials, exhibit a singularity of higher order than that of the shear stresses.     

A selective mass scaling method for shear wave propagation analyses in nearly incompressible materials

This paper describes a selective mass scaling method which is designed for the analysis of wave propagation problems in nearly incompressible materials. The incompressibility of materials leads to a high value of the compressional wave speed, which makes the time step extremely small in explicit time integration method. The proposed selective mass scaling method selects the eigenfrequencies related to volumetric deformation modes to decrease them, while it keeps the shear eigenmodes unchanged. This makes the time step no longer limited by the compressional wave speed but by the shear wave speed. A significant reduction of CPU time is obtained with a good accuracy for transient problems in small strains on free or largely prestressed media. Copyright © 2016 John Wiley & Sons, Ltd.     

Rayleigh wave propagation in a solid with a cold-worked surface layer

The propagation of the Rayleigh surface wave is experimentally studied along the top surface of used railroad rail under conditions where ultrasonic pulses have carrier frequencies ranging from 0.4 to 3.0 MHz and approximately 10 µs duration. The generation of the first higher (M ₂₁ or Sezawa) mode as well as the fundamental (M ₁₁) mode and their dispersion properties are observed. These phenomena are attributable to the presence of the cold-worked surface layer caused by the wheel passage. It is shown that a theoretical model of a single layer overlying a half space, whose elastic constants are determined by a destructive method, yields results which agree with the dispersion curves obtained experimentally. On the basis of this one-layered model, an inversion method to estimate the layer thickness and its elastic constants is discussed.     

A note on plane wave propagation in a linear viscoelastic solid

A new formulation of the governing equations for one spatial dimension wave propagation in a linear viscoelastic solid with more than one discrete relaxation time is proposed. The resulting system of three equations is treated as a strictly hyperbolic system of first order hyperbolic partial differential equations and the method of characteristics is adapted to obtain numerical solutions. Results are presented for a solid with two discrete relaxation times and are compared with those obtained from a predictor-corrector scheme.