Nonlinear wave modulation in fluid filled distensible tubes

Summary The present work considers one dimensional wave propagation in an infinitely long, straight and homogeneous nonlinear viscoelastic or elastic tube filled with an incompressible, inviscid fluid. Using the reductive perturbation technique, and assuming the weakness of dissipative effects, the amplitude modulation of weakly nonlinear waves is examined. It is shown that the amplitude modulation of these waves is governed by a dissipative nonlinear Schrödinger equation (NLS). In the absence of dissipative effects, this equation reduces to the classical NLS equation. The examination of the coefficients of the dissipative and classical NLS equations reveals the significance of the tube wall inertia to obtain a balance between nonlinearity and dispersion. Some special solutions of the NLS equation are given and the modulational instability of the plane wave solution is discussed for various incompressible hyperelastic materials.     

Contributions of higher order terms to nonlinear waves in fluid-filled elastic tubes: strongly dispersive case

In the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thin tube and the approximate equations of an incompressible inviscid fluid, and then utilizing the modified reductive perturbation technique presented by us [15] the amplitude modulation of weakly nonlinear waves is examined. It is shown that the first order term in the perturbation expansion is governed by a nonlinear Schrödinger equation and the second order term is governed by the linearized Schrödinger equation with a nonhomogeneous term. In the longwave limit a travelling wave type of solution to these equations are also given.     

Finite amplitude transverse waves in materials with memory

We consider the propagation of finite amplitude plane transverse waves in a class of homogeneous isotropic incompressible viscoelastic solids with memory. It is assumed that the Cauchy stress may be written as the sum of an elastic part and a dissipative viscoelastic part. The elastic part is of the form of the stress corresponding to a Mooney–Rivlin material, whereas the dissipative part depends not only on current but also on previous deformations. The body is first subjected to a homogeneous static deformation. It is seen that two finite amplitude transverse plane waves may propagate in every direction in the deformed body. It is also seen that finite amplitude circularly polarized waves may propagate along either n⁺ or n⁻, where n⁺, n⁻ are the normals to the planes of the central circular section of the ellipsoid x · B⁻¹x = 1. Here B is the left Cauchy–Green strain tensor corresponding to the finite static homogeneous deformation.     

Methods for solving one-dimensional boundary-value problems in a nonlinear elastic medium

Summary One-dimensional problems connected with a mechanical exposure to the boundary of a nonlinear elastic half-space, which leads to a constantly accelerated motion of this boundary, are considered. The value , equal to the square root of the ratio between the velocity of the boundary motion and the velocity of longitudinal wave propagation in a linear elastic medium, is used as the value characterizing the intensity of this exposure. It is shown that as a result of such an exposure shock waves of small or finite amplitude may propagate in the half-space. The asymptotic matching principle and the ray method are used as methods of solution. The merits and demerits of each method are analyzed. It has been inferred that the matched asymptotic method can be applied to waves of small amplitude, and the ray method is usable when investigating the propagation of shock waves not only of small amplitude, but of finite amplitude as well if the time of a consideration of the shock process is not long. The results obtained by the two methods for shock waves of small amplitude are in close agreement. It has been demonstrated that the ray method is adaptable for solving more intricate boundary-value problems resulting in the propagation of several shock waves of finite amplitude at a time. The problem connected with the constantly accelerated motion of one of the boundaries of an initially deformed elastic layer provides an example.     

Intrinsic description of 3-dimensional shock waves in nonlinear elastic fluids

The transport equations for the amplitude of 3-dimensional shock waves in nonlinear elastic fluids are examined. It is shown that, with the exception of the term which contains the mean curvature of the shock surface, the transport equations are almost identical to the transport equations for 1-dimensional nonlinear elastic solids if we replace the stress, strain and velocity in the latter by the pressure, specific volume and normal velocity, respectively. Therefore, the results obtained for 1-dimensional shock waves regarding whether the amplitudes of jump in stress, strain and velocity grow or decay simultaneously can be applied here to the jump in pressure, specific volume and normal velocity. New compatibility equations are obtained for the velocity gradients behind the shock wave. We also obtain a universal relation between the variations of amplitudes of jump in pressure, specific volume and normal velocity.     

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.     

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.     

Approximate Evaluation for Effective Elastic Moduli of Cracked Solids

A system of equations for effective elastic moduli of 2-D cracked solids is presented by combining the energy balance equation proposed by Shen and Yi (2000) with the integral equations which control the problem of an infinite solid with a finite number of cracks in a sub-region. Then, using Kachanov's method (Kachanov, 1987) for the solutions of the integral equations, 2-D effective bulk and shear moduli for solids with randomly distributed cracks are evaluated.     

A Refined Asymptotic Theory for the Nonlinear Analysis of Laminated Cylindrical Shells

Within the framework of the three-dimensional (3D) nonlinear elasticity, a refined asymptotic theory is developed for the nonlinear analysis of laminated circular cylindrical shells. In the present formulation, the basic equations including the nonlinear relations between the finite strains (Green strains) and displacements, the nonlinear equilibrium equations in terms of the Kirchhoff stress components and the generalized Hooke's law for a monoclinic elastic material are considered. After using proper nondimensionalization, asymptotic expansion, successive integration and then bringing the effects of transverse shear deformation into the leading-order level, we obtain recursive sets of the governing equations for various orders. It is shown that the von Karman-type first-order shear deformation theory (FSDT) is derived as a first-order approximation to the 3D nonlinear theory. The differential operators in the linear terms of governing equations for the leading order problem remain identical to those for the higher-order problems. The nonlinear terms related to the unknowns of the current order appear in a regular pattern and the other nonhomogeneous terms can be calculated by the lower-order solutions. It is also illustrated that the nonlinear analysis of laminated circular cylindrical shells can be made in a hierarchic and consistent way.     

Second harmonic generation of shear waves in crystals

Nonlinear self-interaction of shear waves in electro-elastic crystals is investigated based on the rotationally invariant state function. Theoretical analyses are conducted for cubic, hexagonal, and trigonal crystals. The calculations show that nonlinear self-interaction of shear waves has some characteristics distinctly different from that of longitudinal waves. First, the process of self-interaction to generate its own second harmonic wave is permitted only in some special wave propagation directions for a shear wave. Second, the geometrical nonlinearity originated from finite strain does not contribute to the second harmonic generation (SHG) of shear waves. Therefore, unlike the case of longitudinal wave, the second-order elastic constants do not involve in the nonlinear parameter of the second harmonic generation of shear waves. Third, unlike the nonlinearity parameter of the longitudinal waves, the nonlinear parameter of the shear wave exhibits strong anisotropy, which is directly related to the symmetry of the crystal. In the calculations, the electromechanical coupling nonlinearity is considered for the 6 mm and 3 m symmetry crystals. Complement to the SHG of longitudinal waves already in use, the SHG of shear waves provides more measurements for the determination of third-order elastic constants of solids. The method is applied to a Z-cut lithium niobate (LiNbO/sub 3/) crystal, and its third-order elastic constant c/sub 444/ is determined.     

Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium

This paper investigates the large amplitude vibration behavior of a shear deformable FGM cylindrical shell of finite length embedded in a large outer elastic medium and in thermal environments. The surrounding elastic medium is modeled as a Pasternak foundation. Two kinds of micromechanics models, namely, Voigt model and Mori-Tanaka model, are considered. The motion equations are based on a higher order shear deformation shell theory that includes shell-foundation interaction. The thermal effects are also included and the material properties of FGMs are assumed to be temperature-dependent. The equations of motion are solved by a two step perturbation technique to determine the nonlinear frequencies of the FGM shells. Numerical results demonstrate that in most cases the natural frequencies of the FGM shells are increased but the nonlinear to linear frequency ratios of the FGM shells are decreased with increase in foundation stiffness. The results confirm that in most cases Voigt model and Mori-Tanaka model have the same accuracy for predicting the vibration characteristics of FGM shells.     

Creep and recovery of nonlinear viscoelastic materials of the differential type

In this work, the creep and recovery properties of rubberlike viscoelastic materials in simple shear are studied by two special constitutive equations for isotropic, nonlinear incompressible viscoelastic material of the differential type. The creep and recovery processes are of significant importance to both the mechanics analysis and engineering applications. The constitutive equations introduced in this work generalize the Voigt-Kelvin solid and the 3-parameter model of classical linear viscoelasticity. They describe the uncoupled non-Newtonian viscous and nonlinear elastic response of an isotropic, incompressible material. The creep and recovery processes are treated for simple shear deformation superimposed on a longitudinal static stretch. Closed form solutions are provided and both processes are described effectively by the exponential function.     

Flexural- and capillary-gravity waves due to fundamental singularities in an inviscid fluid of finite depth

Wave motion due to line, point and ring sources submerged in an inviscid fluid are analytically investigated. The initially quiescent fluid of finite depth, covered by a thin elastic plate or by an inertial surface with the capillary effect, is assumed to be incompressible and homogenous. The strengths of the sources are time-dependent. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The perturbed flow is decomposed into the regular and the singular components. An image system is introduced for the singular part to meet the boundary condition at the flat bottom. The solutions in integral form for the velocity potentials and the surface deflexions due to various singularities are obtained by means of a joint Laplace-Fourier transform. To analyze the dynamic characteristics of the flexural- and capillary-gravity waves due to unsteady disturbances, the asymptotic representations of the wave motion are explicitly derived for large time with a fixed distance-to-time ratio by virtue of the Stokes and Scorer methods of stationary phase. It is found that the generated waves consist of three wave systems, namely the steady-state gravity waves, the transient gravity waves and the transient flexural/capillary waves. The transient wave system observed depends on the moving speed of the observer in relation to the minimal and maximal group velocities. There exists a minimal depth of fluid for the possibility of the propagation of capillary-gravity waves on an inertial surface. Furthermore, the results for the pure gravity and capillary-gravity waves in a clean surface can also be recovered as the flexural and inertial parameters tend to zero.     

On free oscillations of elastic incompressible bodies in finite shear

Large amplitude free oscillations of thick-walled elastic, incompressible bodies are studied resulting from three types of shearing deformations. The material strain-energy function is expressed in a general form of finite series expansion so that the governing equations of motion in each case reduce to a system of non-linear second order partial differential equations. The degree of non-linearity is shown to be dependent on the number of terms retained in the power series. Approximate results for the non-linear systems are gained by a regular perturbation scheme. Illustrative problems using a strain-energy function for some rubber-like materials are also included.     

Modulation of nonlinear waves in a viscous fluid contained in a tapered elastic tube

In the present work, treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube and the blood as a Newtonian fluid, we have studied the amplitude modulation of nonlinear waves in such a fluid-filled elastic tube, by use of the reductive perturbation method. The governing evolution equation is obtained as the dissipative nonlinear Schrödinger equation with variable coefficients. It is shown that this type of equations admit solitary wave solutions with variable wave amplitude and speed. It is observed that, the wave speed increases with distance for tubes of descending radius while it decreases for tubes of ascending radius. The dissipative effects cause a decay in wave amplitude and wave speed.     

An extension of Key's principle to nonlinear elasticity

A variational principle for finite isothermal deformations of anisotropic compressible and nearly incompressible hyperelastic materials is presented. It is equivalent to the nonlinear elastic field (Lagrangian) equations expressed in terms of the displacement field and a scalar function associated with the hydrostatic mean stress. The formulation for incompressible materials is recovered from the compressible one simply as a limit. The principle is particularly useful in the development of finite element analysis of nearly incompressible and of incompressible materials and is general in the sense that it uses a general form of constitutive equation. It can be considered as an extension of Key's principle to nonlinear elasticity. Various numerical implementations are used to illustrate the efficiency of the proposed formulation and to show the convergence behaviour for different types of elements. These numerical tests suggest that the formulation gives results which change smoothly as the material varies from compressible to incompressible.     

Some results on finite amplitude elastic waves

Summary In this paper some special types of finite amplitude wave motions are considered, for which kinematical non-linearities do not arise in the equations of motion of an elastic solid. Consequently, only constitutive non-linearities occur and, for special classes of materials, solutions may be read off from corresponding solutions in the linear theory. These include SH-waves¹ and Love waves in layered or inhomogeneous media. Finite amplitude transverse circularly-polarized harmonic progressive waves are shown to propagate in any compressible or incompressible isotropic elastic material. Some effects of homogeneous pre-stress are also investigated.

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Zusammenfassung Es werden Spezialfälle von Wellenbewegungen mit endlicher Amplitude betrachtet, für die keine kinematischen Nichtlinearitäten in den Bewegungsgleichungen des elastischen Festkörpers auftreten. Es kommen also nur Material-Nichtlinearitäten vor. Für gewisse Klassen von Materialien können die Lösungen aus den entsprechenden der linearen Theorie gewonnen werden. Hierher gehören horizontal polarisierte Scherwellen und Lovesche Wellen in geschichteten oder inhomogenen Medien. Es wird gezeigt, daß sich harmonische, zirkularpolarisierte Transversalwellen in jedem kompressiblen oder inkompressiblen elastischen Material fortpflanzen. Einige Effekte homogener Vorspannung werden ebenfalls untersucht.

     

Radial propagation of rotary shear waves in a finitely deformed elastic solid

A study is made of the radial propagation of rotary shear waves in an incompressible elastic solid under finite radial deformations. Basic equations are derived on the basis of Biot's mechanics of incremental deformations, and analysis is made by specializing the initial deformations to two cases: (a) an infinite solid with a cylindrical bore is inflated by all-around tension, and (b) a cuboid is rounded into a ring and its ends are bonded to each other. The influence of the inhomogeneities of such deformations upon the laws of shear wave propagation is presented in the form of curves.     

Evolution equation for nonlinear Scholte waves

The evolution equations for nonlinear Scholte waves (finite amplitude elastic waves propagating along liquid/solid interface), which account for the second order nonlinearity of a liquid, are derived for the first time. For mathematical simplicity the nonlinearity of the solid, which influence is expected to be weak in the case of weak localization of the Scholte wave, is not taken into consideration. The analysis of these equations demonstrates that the nonlinear processes contributing to the evolution of the Scholte wave can be divided into two groups. The first group includes nonlinear processes leading to wave spectrum broadening which are common to bulk pressure waves in liquids and gases. The second group includes the nonlinear processes which are active only in the frequency down-conversion (leading to wave spectrum conservation or narrowing), which are specific to the confined nature of the interface wave. It is demonstrated that the nonlinear parameters, which characterize the efficiency of various nonlinear processes in the interface wave, strongly depend on the relative properties of the contacting liquid and solid (or, in other words, on the deviation of the Scholte wave velocity from the velocities of sound in liquid and in solid). In particular, the sign of the nonlinear parameter responsible for the second harmonic generation can differ from the sign of the nonlinear acoustic parameter of the liquid. It is also verified that there are particular liquid/solid combinations where the nonlinear processes, which are inactive in the frequency up-conversion, dominate in the evolution of the Scholte wave. In this case distortionless propagation of the finite amplitude harmonic interface wave is possible. The proposed theory should find applications in nonlinear acoustics, geophysics, and nondestructive testing.     

Stabilized finite elements for time-harmonic waves in incompressible and nearly incompressible elastic solids

The propagation of waves in elastic solids at or near the incompressible limit is of interest in many current and emerging applications. Standard low-order Galerkin finite element discretization struggles with both incompressibility and wave dispersion. Galerkin least squares stabilization is known to improve computational performance of each of these ingredients separately. A novel approach of combined pressure-curl stabilization is presented, facilitating the use of continuous, equal-order interpolation of displacements and pressure. The pressure stabilization parameter is determined by stability considerations, while the curl stabilization parameter is determined by dispersion considerations. The proposed pressure-curl–stabilized scheme provides stable and accurate results on a variety of numerical tests for incompressible and nearly incompressible elastic waves computed with linear elements.