Matrix theory of elastic wave scattering
01 Sep 1976-Journal of the Acoustical Society of America (Acoustical Society of America)-Vol. 60, Iss: 3, pp 567-580
TL;DR: In this article, Huygen's principle is invoked to describe the scattering of waves by an obstacle of arbitrary shape immersed in an elastic medium, and conservation laws are discussed with respect to the divergence and curl of the displacement.
Abstract: Upon invoking Huygen’s principle, matrix equations are obtained describing the scattering of waves by an obstacle of arbitrary shape immersed in an elastic medium. New relations are found connecting surface tractions with the divergence and curl of the displacement, and conservation laws are discussed. When mode conversion effects are arbitrarily suppressed by resetting appropriate matrix elements to zero, the equations reduce to a simultaneous description of acoustic and electromagnetic scattering by the obstacle at hand. Unification with acoustic/electromagnetics should provide useful guidelines in elasticity. Approximate numerical equality is shown to exist between certain of the scattering coefficients for hard and soft spheres. For penetrable spheres, explicit analytical results are found for the first time.Subject Classification: [43]20.15, [43]20.30.
Citations
More filters
TL;DR: In this paper, the authors focus on variational and related methods for the overall properties of composites, such as fiber-reinforced composites or polycrystals, whose properties vary in a complicated fashion from point to point over a small, microscopic length scale, while they appear on average to be uniform.
Abstract: Publisher Summary This chapter focuses on variational and related methods for the overall properties of composites. A wide range of phenomena that are observable macroscopically are governed by partial differential equations that are linear and self-adjoint. This chapter is concerned with such phenomena for materials, such as fiber-reinforced composites or polycrystals, whose properties vary in a complicated fashion from point to point over a small, “microscopic” length scale, while they appear “on average” (that is, relative to the larger, macroscopic scale) to be uniform. This chapter treats the elastic behavior of composites, and emphasizes that a number of other properties (conductivity, viscosity of a suspension, etc.) are described by the same equations. Extensions to viscoelastic and thermoelastic behavior are presented, for both of which the variational characterization given is believed to be new. Problems, such as the resistance to flow of viscous fluid through a fixed bed of particles are mentioned, and a model problem that involves diffusion is presented in some detail. This displays the same difficulty in relation to divergence of an integral and is one problem of this type that has so far been approached variationally. Methods related to the Hashin–Shtrikman variational principle are also described in the chapter.
832 citations
TL;DR: In this article, the effects of multiple parallel planar fractures on the apparent attenuation of normally incident one-dimensional elastic waves are studied. But the authors focus on the attenuative effect of each fracture with the displacement discontinuity model, and do not consider complex interfracture multiple wave reflections with the method of characteristics.
198 citations
TL;DR: In this article, a momentum polarization was introduced to cope with density variations in a way that exactly parallels the stress polarization's correspondence with variations in moduli, and the authors derived an asymptotic formula for scattering cross-sections of penny-shaped cracks, rigid circular discs and rigid needles.
Abstract: Scattering problems in elastodynamics are formulated in terms of integral equations, whose kernels are obtained from the Green's function for a comparison body. The comparison body will usually be taken as homogeneous and elastic in applications but, at least formally, there is no bar to its being inhomogeneous, viscoelastic and non-local. The novel feature of the formulation is the introduction of a “momentum polarization” to cope with density variations in a way that exactly parallels the stress polarization's correspondence with variations in moduli. To illustrate the use of the equations, scattering by an ellipsoidal inhomogeneity in a generally anisotropic matrix is studied in the Rayleigh limit and an asymptotic formula for its scattering cross-section is given. Detailed results are presented for a spheroidal inhomogeneity in an isotropic matrix, with explicit limiting forms for the scattering cross-sections of penny-shaped cracks, rigid circular discs and rigid needles.
137 citations
TL;DR: In this paper, the T-matrix equations describing boundary value problems of potential theory and electromagnetic scattering are obtained without recourse to the Huygens principle or physically fictitious fields.
Abstract: Employing a conserved‐flux concept, the T‐matrix equations describing boundary‐value problems of potential theory and electromagnetic scattering are obtained without recourse to the Huygens principle or physically fictitious fields. For scattering by dielectric objects, tangential electric and magnetic fields on the surface are both represented in a single expansion, cutting the computation in half. In the low‐frequency limit the dynamical equations are shown to reduce to the static case, and numerical computations then indicate that in comparison with other approaches, the present method can achieve as much as an order of magnitude reduction in the number of equations and unknowns needed for a given accuracy. New exact relations are found between the electrostatic and magnetostatic problems, and analytical results are also obtained from the equations, with and without truncation.
109 citations
CIME1
TL;DR: The model validation, on the phase velocity behavior, in the case of a damage simulated by expanded polystyrene spheres in granular media shows good qualitative agreement for the changes in velocity and attenuation.
96 citations
References
More filters
9,185 citations
Book•
01 Jan 1973
TL;DR: In this article, the authors systematically present methods of solution for both steady-state and transient loading on various obstacles, and also numerical results of dynamic stress concentration on obstacles of different geometries.
Abstract: : This monograph systematically presents methods of solution for both steady-state and transient loading on various obstacles, and also numerical results of dynamic stress concentration on obstacles of different geometries. An effort is made to collect information from the open literature as well as from government agencies, industry, and individuals. (Author-PL)
949 citations