Spherical Elastic Waves Research Articles

Three-dimensional internal failure propagation by spherical elastic waves in fused silica glass plate subjected to hypervelocity impact of debris

Three-dimensional internal failure propagation by spherical elastic waves in fused silica glass plate subjected to hypervelocity impact of debris

The Study of Characteristics of Sphere-Radial Phononic Crystals

By deducing the spherical elastic wave equation in theory, the concept of sphere-radial phononic crystal is proposed, and then the equations to determine the acoustic band structures is deduced. A numerical example is given for steel/nitrile rubber phononic crystal. The numerical simulation results suggest that the band gaps of sphere-radial phononic crystals do exist, which have better attenuation characteristics and practical application performance than the one-dimensional phononic crystals.

Features of blast-induced vibration source and identification of geostructures

An analysis of the behavior of spherical elastic wave radiation from an impulsively loaded cavity embedded in an infinite space has been carried out. It is confirmed that the propagating waveform is dominated by the natural vibration mode of the cavity and the magnitude of the dominant response depends only on the impulse of the time function of the load, and not the profile of the time function. This finding supports the fact that the dynamic signals recoded away from the blasting explosive contain mainly the characteristics of the natural vibrations of the geological structures near the cavity created by a blast. Based on this finding, two promising methods are proposed for the identification of geo-structures and parameters.

Scattering of spherical elastic waves from a small-volume spherical inclusion

SUMMARY A theory is presented for the scattering of spherical elastic waves by a small, spherical inclusion. The domain of validity of the treatment is ka′« 1, where k is the wavenumber and a′ is the radius of the spherical inclusion. The treatment is an extension of that of Gritto, Korneev & Johnson (1995), who derived exact expressions for the scattered wavefield from a plane P wave incident on a spherical inclusion in the low-frequency limit. Since neither treatment assumes small perturbations, they are also valid for an arbitrary material property contrast between the spherical inclusion and the background medium. For spherical elastic wavefields propagating through an infinite, homogeneous medium and interacting with a single small, spherical inclusion, the scattered wavefield is obtained directly as a sum of spheroidal motions of degrees 0, 1 and 2 in scatterer-centred coordinates, with nine independent amplitude coefficients which depend on the contrast in material properties and are proportional to the volume of the inclusion. If the inclusion is embedded in a stratified and bounded medium, then these scattering coefficients provide a valid local description of the scattered wavefield in the vicinity of the inclusion. Exploiting this fact, source equivalents in the frequency domain (six moment tensor elements and three single forces) are derived from the scattering coefficients. To first order in material property contrasts, equivalence is established between these source equivalents and the coupled mode theory developed for surface waves and long-period body waves using perturbation theory.

Interaction of two spherical elastic waves in a nonlinear five-constant medium

The interaction of two spherical longitudinal waves produced by noncoincident sources with the generation of a transverse difference-frequency wave is studied theoretically. The equation of motion from five-constant elasticity theory to the second order in the displacement is used as a mathematical model. The general solution for the difference-frequency field outside the wave interaction region is obtained in the form of a scattering integral. An analytical solution for plane intersecting beams is derived as a limit case of small interaction volumes. For the general case, the solution is analyzed numerically, and the effect of the sphericity of interacting waves is compared with the results of plane-wave resonance interaction. In the limit of large interaction volumes, backscattering at the difference frequency is formed. This effect suggests the use of this type of nonlinear interaction for remote sensing of the nonlinear properties of the Earth’s crust.

Numerical Model of the Spherical Elastic Wave Propagation in a Nonlinear Medium

There are new experimental results showing that nonlinear effects are significant in seismic wave propagation through the upper part of the geological medium [1]. Nevertheless, no adequate models exist in seismology to describe theoretically such events. I describe here an attempt to derive a wave equation for the spherical nonlinear elastic wave and to solve it numerically.

Spherical elastic waves

Spherical elastic waves

High Frequency Cylindrical and Spherical Elastic Waves in a Heterogenous Half-Space

A high-frequency asymptotic representation of the displacement field in an inhomogeneous elastic half-space is secured for periodic normal line or point loading. It is assumed that the constitutive moduli of the medium depend solely, and analytically, on distance from the solid’s plane boundary. Our technique involves a transformation of the linearized elastodynamic equations in order to recast these as a first order system of ordinary differential equations, the utilization of an asymptotic theory of Birkhoff and inversion by means of the method of stationary phase and the geometrical theory of diffraction. If the solid is homogeneous, the leading term of our asymptotic expansion in inverse powers of frequency is identically equal to the known displacement vector. In a half-space whose wavespeeds decrease monotonically, the formation of shadow zones and the functional dependence of the field on the material parameters are discussed.

Diffraction of a spherical elastic wave by a wedge: PMM vol. 40, no. 5, 1976, pp. 898–908

A three-dimensional nonstationary problem of spherical elastic wave diffraction by a smooth solid wedge with arbitrary apex angle is considered. An exact solution in the form of a sum of two terms, the known acoustic solution and an additional part describing the influence of elasticity, and caused by the appearance of additional longitudinal and transverse diffraction waves, is obtained by the method of integral transforms with extraction of the singularities in the neighborhood of an edge. This latter term essentially distinguishes the elastic from the acoustic solution. The particular case of an incident wave with a jump in the stresses at the front is investigated in detail. The corresponding acoustic problem has been examined in [1–4], where the solution in elementary functions was first obtained in [2]. Only the solution for the plane wave diffraction problem [5] is known for a wedge in the elastic case. Solutions found earlier for plane diffraction problems by a smooth wedge [6] and a smooth half-plane [7], which agree with the solution of the corresponding acoustic problems, are not true because of neglecting the condition at the edge, which indeed resulted in nonintegrable stresses in the neighborhood of the edge.

On The Scattering of Spherical Elastic Waves by a Spherical Cavity

The distribution of scattered energy due to the interaction of spherical elastic dilatational waves with a spherical cavity is found to differ from that due to plane waves for long wavelengths even when the radius of curvature is much larger than the cavity radius.

Comment on "A Direct Numerical Analysis Method for Cylindrical and Spherical Elastic Waves"

Comment on "A Direct Numerical Analysis Method for Cylindrical and Spherical Elastic Waves"

Natural vibrations of finitely deformable structures.

1 Mehta, P. K. and Davids, N., A direct numerical analysis method for cylindrical and spherical elastic waves, AIAA J. 4, 112-117 (1966). 2 Mehta, P. K., Discrete analysis methods in mechanics of deformable solids, thesis in Engineering Mechanics, The Pennsylvania State Univ. (June 1966). 3 Davids, N. and Mehta, P. K., analysis methods for stress waves and spherical cavities, College of Engineering, The Pennsylvania State Univ. Engineering Research Bulletin B-92 (May 1965). 4 Mehta, P. K. and Davids, N., Elastic stress waves in multilayered cylindrical and spherical shells by a direct unified computational analysis, The India Society of Theoretical and Applied Mechanics, 10th Annual Congress, Madras, India (December 20-24, 1965). 5 Davids, N., Mehta, P. K., and Johnson, O. T., Spherical elastoplastic waves in materials, ASME Winter Annual Meeting (American Society of Mechanical Engineers, New York, 1965), pp. 125-137. 6 Davids, N., Minnich, H. R., and Sliney, J., A penetration method for determining impact yield strength, Proceedings of VII Hypervelocity Symposium on Impact (U. S. Army, Air Force, and Navy, Tampa, Florida, 1964), Vol. 3, p. 261. 7 Davids, N. and Berger, R. L., A computer analysis method for thermal diffusion in biochemical systems, Commun. Assoc. Computer Machinery 7, 547-551 (1964). 8 Hopkins, H. G., Dynamic expansion of spherical cavities in metals, Progress in Solid Mechanics (North Holland Publishing Co., Amsterdam, 1960), Vol. 1, pp. 83-164. 9 Prescot, J., Applied Elasticity (Dover Publications Inc., New York, 1961) S726, p. 330. 10 Olsson, H. F., Acoustical Engineering (D. Van Nostrand Company Inc., New York, 1957), pp. 5-7. 11 Prager, W. and Hodge, P. G., Theory of Perfectly Plastic Solids (John Wiley & Sons Inc., New York, 1961), pp. 100-101.

Optical Lever Observation of Hypervelocity Impact Shock Waves

An optical lever system was used to observe the impact shock wave arrival and free-surface motion at the rear surface of a 2.54-cm-thick 2024-T4 aluminum target plate. The projectile was a 0.636-cm-diam. steel ball and struck the target at a velocity of 5.28±0.11 mm/μsec. The shock-wave data, in the range 30–1 kbar, extend data by Fowles to lower pressures. The measured elastic shock velocity of 6.23±3% agrees with ultrasonic values and with elastic shock values by other investigators. Elastic shock amplitudes do not maintain a constant value as in one-dimensional experiments, but decay at a rate faster than predicted for spherical elastic waves.

A Unified Approach to Cylindrical and Spherical Elastic Waves by Method of Characteristics

A set of generalized equations is presented which governs the propagation of plane, cylindrical, and spherical dilatation waves in elastic media. The corresponding characteristic equations are then derived, including the propagation of abrupt changes (discontinuous wave fronts). Procedures of numerical integration along the characteristic directions are established and carried out for several examples on a digital computer. The solutions of four of the specific examples calculated show excellent agreement with existing solutions by other methods.

A direct numerical analysis method for cylindrical and spherical elastic waves.

This paper describes a general approach for determining axisymmetric and spherical symmetric elastic stresses and deformations in cylindrical and spherical shells subjected to dynamic internal and external pressures by a direct numerical approach referred to as discontinuous-step analysis. The pressures are any arbitrary functions of time p = p(t) and q = q(t). This analysis is based on the direct use of the governing physical laws and initial and boundary conditions to effect changes in the system by finite steps. A single general solution is obtained covering the cases of traveling wave solutions of cylindrical (plane stress and plane strain cases) and spherical elastic stress waves. Reflection of stress waves from both the inner and the outer boundaries, for specified boundary conditions, is automatically generated. Validity of the method is checked by comparing its numerical results with those obtained from an existing mathematical solution of spherical elastic waves in an infinite medium. The agreement between them is very good. Illustrative new numerical results are given for cylindrical elastic stress wave generated by a dynamic internal pressure pulse of constant magnitude pQ. Nomenclatu re t = time p(t) = internal pressure function q(t) = external pressure function rt d, z = radial, tangential, axial coordinates o- = stresses e = strains u = particle displacement v — particle velocity E = elastic modulus v = Poisson's ratio p = mass density F = yield stress in uniaxial tension a = cavity (cylindrical or spherical) radius 6 = external radius of cylinder or sphere I = length of cylinder A = inner surface area of hemispherical or semicylindrical element m = mass of element