Wave propagation in the presence of empty cracks in an elastic medium

This paper proposes the use of a traction boundary element method (TBEM) to evaluate 3D wave propagation in unbounded elastic media containing cracks whose geometry does not change along one direction. The proposed formulation is developed in the frequency domain and handles the thin-body difficulty presented by the classical boundary element method (BEM). The empty crack may have any geometry and orientation and may even exhibit null thickness. Implementing this model yields hypersingular integrals, which are evaluated here analytically, thereby surmounting one of the drawbacks of this formulation. The TBEM formulation enables the crack to be modelled as a single line, allowing the computation of displacement jumps in the opposing sides of the crack. Furthermore, if this formulation is combined with the classical BEM formulation the displacements in the opposing sides of the crack can be computed by modelling the crack as a closed empty thin body.     

Wave slam on a sphere penetrating a free surface

Summary The problem of vertical motion of a sphere across an oscillating free surface is analysed by assuming the fluid to be inviscid and the free surface to be an equipotential surface. New analytical solutions for the added-mass coefficients of a double spherical bowl are derived. These are used in the derivation of the drag coefficient of a sphere during vertical entry and of the slamming coefficient of a fixed sphere which is exposed to wave action. An additional important parameter in hydroballistics is the wetting factor of a sphere penetrating a free surface for which a new analytic solution is also derived in this paper. A comparison between some experimental data and the analytic expressions for the slamming coefficient and the wetting factor, shows good agreement between theory and measurements.This work was sponsored by the Office of Naval Research, under contract N00014-76-C-0026, and by the National Science Foundation, under contract CME-79-21244.     

Wave propagation in and free vibrations of gradient elastic circular cylindrical shells

The governing equations of motion for a gradient elastic circular cylindrical thin shell are derived. The basic equations of dynamic equilibrium and strain-displacement relations due to Donnell are combined with the stress–strain equations of the gradient theory of elasticity involving one microstructural and one microinertial elastic constant in addition to the two classical elastic moduli. The shell governing equations of motion are first used to study the propagation of harmonic waves and then free vibrations for the particular case of a circular cylindrical shell simply supported at its two ends. The results of these analyses are compared against those of the classical case in order to assess the microstructural and microinertial effects on the dynamic behavior of the shell.     

Fatigue crack propagation for defects near a free surface

The fatigue crack growth associated with an internal flaw as it approaches a free surface was studied. Previous researchers recommended when the plastic zone at the crack tip nearest the free surface reaches one-tenth of the ligament size which exists between the free surface and the crack tip that the flaw be treated as an edge crack. The results of this study show that the assumption of an edge crack can prove to be exceptionally conservative when employed for design purposes. Underestimates of fatigue life of as much as 1000% were observed. Improved methods for predicting the fatigue growth of near surface defects are offered in the paper.     

Wave propagation in sheared rubber

Summary The speeds of propagation and polarization amplitudes are presented for finite amplitude plane shear waves propagating in rubber which is maintained in a state of static finite simple shear. The Mooney-Rivlin form of the stored-energy function is used to model the mechanical behaviour of the material. General relations are obtained between the speed of propagation of the fastest and slowest waves and the speed of propagation of the finite amplitude circularly polarized waves which may propagate along the acoustic axes. The slowness and ray surfaces are also presented.     

Wave propagation of coupled double-DWBNNTs conveying fluid-systems using different nonlocal surface piezoelasticity theories

This work is concerned with the size-dependent wave propagation of coupled double-walled boron nitride nanotubes (DWBNNTs) conveying nanoflow-systems based on Timoshenko beam theory. The two DWBNNTs are coupled by an enclosing visco-Pasternak medium. The small-scale effects are captured applying different surface piezoelasticity theories, including stress gradient, strain gradient, and strain inertia gradient. An analytical method is proposed to obtain phase velocity, cut-off, and escape frequencies of the system. Three cases of in-phase wave propagation, out-of-phase wave propagation, and wave propagation with one DWBNNT fixed are considered. Results indicate that ignoring surface and small-scale effects lead to inaccurate results.     

Wave propagation in anisotropic layered media

The determination of the natural modes of wave propagation in an anisotropic layered medium requires the solution of a transcendental eigenvalue problem that is usually approached numerically with the aid of search techniques. Such computations require great effort. The method presented in this paper provides an alternate solution to this problem in terms of a quadratic eigenvalue problem involving tridiagonal matrices, for which the eigenvalues can be found with great speed and accuracy. The technique is then illustrated by means of an example involving a cross-anisotropic Gibson solid.     

Wave propagation in elasto-plastic granular systems

Due to the nonlinear nature of the inter-particle contact, granular chains made of elastic spheres are known to transmit solitary waves under impulse loading. However, the localized contact between spherical granules leads to stress concentration, resulting in plastic behavior even for small forces. In this work, we investigate the effects of plasticity in wave propagation in elasto-plastic granular systems. In the first part of this work, a force–displacement law between contacting elastic-perfectly plastic spheres is developed using a nonlinear finite element analysis. In the second part, this force–displacement law is used to simulate wave propagation in one-dimensional granular chains. In elasto-plastic chains, energy dissipation leads to the formation and merging of wave trains, which have characteristics very different from those of elastic chains. Scaling laws for peak force at each contact point along the chain, velocity of the leading wave, local contact and total dissipation are developed.     

Wave propagation in unsaturated porous media

We present a simple macroscopical three-phase model describing wave propagation in partially saturated porous media. The model consists of a continuous non-wetting phase and a continuous wetting phase and is an extension of classical biphasic (Biot-type) models. The framework is based on kinematics, balance equations and well-known constitutive equations for single- and multi-phase continua. The final set of linearised equations gives information about the physical behaviour of three compressional waves and one shear wave. Among others, information about phase velocities, damping and displacements of the single constituents can be determined for arbitrary variations of input parameters like saturation or angular frequency (ω). The physical processes are investigated and explained by an example of Massilon sandstone filled with air and water. For the quasi-static limit case, i.e. \({\omega \mapsto 0}\) , the results of the model are identical with the phase velocity obtained with the well-known Gassmann–Wood limit. The model focuses on systems with a liquid and a gas phase. It is shown that the grain compressibility can be neglected in this case, and the amount of material parameters as well as the complexity reduces significantly compared to other three-phase approaches. This makes the well-adapted model suitable for direct application in the vadose zone or partially saturated laboratory samples with a gas phase and a liquid phase. The final model is characterised by low computational effort, general validity for application from geophysics over engineering up to biomedicine and flexibility for use of extended empirical and theoretical relationships.     

Wave propagation in initially-stressed elastic solids

Summary FollowingNovozhilov andBolotin, a linearized theory is obtained from the nonlinear theory of elasticity to investigate waves and vibrations in an elastic solid under initial stress. Wave propagation in an infinite medium is studied for three cases: (1) hydrostatic compression or tension, (2) uniaxial extension or compression, (3) uniform shear stress (for the case of plane strain). It is observed that hydrostatic pressure and uniaxial extension affect the rate of propagation. Further, uniform shear stress is found to induce coupling of dilatational and equivoluminal waves and to alter the speed of propagation. Results are discussed and compared with other theories.

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Zusammenfassung Im Anschluß anNovozhilov undBolotin wird aus der nichtlinearen Elastizitätstheorie eine linearisierte Theorie zur Untersuchung von Wellen und Schwingungen in einem vorgespannten elastischen Festkörper hergeleitet. Die Wellenausbreitung in einem unendlich ausgedehnten Medium wird für folgende drei Fälle der Vorspannung studiert: 1. Hydrostatischer Druck oder Zug; 2. Einachsiger Zug oder Druck; 3. Gleichmäßiger Schub (bei ebenem Verzerrungzustand). Es ergibt sich, daß hydrostatischer Druck und einachsiger Zug die Ausbreitungsgeschwindigkeit beeinflussen. Weiters zeigt sich, daß gleichmäßiger Schub eine Kopplung von Dilatations-und Scherwellen mit sich bringt und die Ausbreitungsgeschwindigkeit ändert. Die Ergebnisse werden diskutiert und mit anderen Theorien verglichen.

     

Wave propagation and uniqueness in magneto-elastodynamics

By using the method introduced in [1], some sufficient conditions on the acoustic tensor and on the kinetic and magnetic fields are given in order that a perturbation initially confined in a proper subset of an unbounded magnetoelastic solid, propagate with finite speed. Moreover, a strong uniqueness theorem is proved for regular solutions to the initial-boundary-value problem of magnetoelastodynamics.     

Wave propagation in thin pre-stressed elastic plates

The propagation of elastic waves of small amplitude in a pre-stressed plate composed of an isotropic hyperelastic medium with a strain-energy function of general form, is investigated and the dispersion equation is derived. Approximate solutions for long waves are obtained. The implications for stability are discussed and a simple general criterion is established.     

Wave propagation in a prestressed piezoelectric half-space

Reflection of plane waves at a traction-free and electrically shorted/charge-free surface of a prestressed piezoelectric medium is studied. The reflection coefficients of qP and qSV waves are derived for electrically shorted and charge-free cases. The effect of initial stress on the reflection coefficients is discussed for a particular example of Lithium niobate.     

Wave propagation in a fluid-filled elastic tube

Summary In a recent paper [1] the present authors (T.B.M. and J.B.H.) studied dispersive wave motions in a tethered, fluid-filled elastomer tube. There the radial inertia of the fluid was taken into account by employing an approximation similar to that proposed by Love [2] for analysis of wave propagation in bars and a simple bending theory of shells was employed for the tube wall. Here, by solving the fluid equations exactly we determine conditions under which the Love approximation is valid. We then extend our previous results to include the effect of shear deformation of the tube wall and analyze this extended theory to ascertain the relative importance of including shear in fluid-filled tube models designed for biological applications.

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Wellenausbreitung in einem fluidgefüllten, elastischen Rohr

Zusammenfassung In einer vorangegangenen Arbeit [1] behandelten die beiden letztgenannten Autoren dispersive Wellenbewegungen in einem axial festgehaltenen, fluidgefüllten, elastomeren Rohr. Dort wurde die Radialträgheit des Fluids mitberücksichtigt durch Anwendung einer ähnlichen Näherung, wie sie von Love [2] für die Behandlung der Wellenausbreitung in Stäben vorgeschlagen wurde, wobei eine einfache Schalenbiegetheorie für die Rohrwand verwendet wurde. In der vorliegenden Arbeit werden durch exaktes Lösen der Gleichungen für das Fluid Bedingungen bestimmt, unter welchen die Näherung von Love gültig ist. Es werden dann die vorhergehenden Ergebnisse erweitert um Einflüsse der Schubverformung der Rohrwand mit einzuschließen und diese erweiterte Theorie wird untersucht, um die relative Bedeutung der Berücksichtigung des Schubs in fluidgefüllten Rohrmodellen, wie sie für Anwendungen in der Biologie entworfen wurden, festzustellen.

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With 9 Figures     

Wave propagation in non-homogeneous asymmetric circular plate

International Journal of Mechanics and Materials in Design - The coupling effect between the vibration of the electric components and the wave propagation is the critical factor that should be...     

Wave propagation in multilayered piezoelectric spherical plates

This paper proposes an improvement of the Legendre polynomial series method to solve the harmonic wave propagation in multilayered piezoelectric spherical plates, which are used in point-focusing transducers. The conventional Legendre polynomial method can deal with the multilayered structures only when the material properties of two adjacent layers do not change significantly and cannot obtain correctly normal stress and normal electric displacement shapes unlike the proposed improved orthogonal polynomial approach which overcomes these drawbacks. Detailed formulations are given to highlight its differences from the conventional Legendre polynomial approach. Through the comparisons of numerical results given by an exact solution (obtained from the reverberation-ray matrix method), and by the conventional polynomial approach and the improved polynomial approach, the validity of the proposed approach is illustrated. The influences of the radius-to-thickness ratio on dispersion curves, stress and electric displacement distributions are discussed. It is found that three factors determine the distribution of mechanical energy and electric energy at higher frequencies: radius-to-thickness ratio, wave speed, and position of the component material.     

Wave propagation modeling in human long bones

Summary The dynamic behavior of a dry long bone that has been modeled as a piezoelectric hollow cylinder of crystal class 6 is investigated. The solution for the wave propagation problem is expressed in terms of a potential function which satisfies an eighth-order partial differential equation, whose solutions lead to the derivation of the explicit solution of the wave equation. The mechanical boundary conditions correspond to those of stress free lateral surfaces, while the electrical boundary conditions correspond to those of short circuit. The satisfaction of the boundary conditions leads to the dispersion relation which is solved numerically. The frequencies obtained are presented as functions of various parameters and they compare well with other researchers' theoretical results.