CHAPTER 3 - ELASTODYNAMIC THEORY

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Budaev Bair V. and Bogy David B. 1999Rayleigh wave scattering in a wedge with mixed boundary conditionsProc. R. Soc. Lond. A.4553967–3995http://doi.org/10.1098/rspa.1999.0486SectionRestricted accessRayleigh wave scattering in a wedge with mixed boundary conditions Bair V. Budaev Bair V. Budaev Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA Google Scholar Find this author on PubMed Search for more papers by this author and David B. Bogy David B. Bogy Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA Google Scholar Find this author on PubMed Search for more papers by this author Bair V. Budaev Bair V. Budaev Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA Google Scholar Find this author on PubMed Search for more papers by this author and David B. Bogy David B. Bogy Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA Google Scholar Find this author on PubMed Search for more papers by this author Published:08 November 1999https://doi.org/10.1098/rspa.1999.0486AbstractWe consider a problem of Rayleigh wave scattering in a single elastic wedge with one face traction–free and the other face displacement–free. The problem is treated by the method previously applied to a general problem of elastic wave propagation in a composite bisolid wedge and to a problem of wave propagation in a homogeneous solid wedge with traction–free boundary conditions on both faces. The intermediate position of the configuration with the mixed boundary conditions makes this problem a convenient model to discuss, in a clear but general setting, all critical features of the developed method, including a discussion of the approach's underlying ideas and of its relationships with other approaches, such as the Wiener–Hopf method, which is widely used in the theory of diffraction. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetails This Issue08 November 1999Volume 455Issue 1991 Article InformationDOI:https://doi.org/10.1098/rspa.1999.0486Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/11/1999Published in print08/11/1999 License: Citations and impact KeywordswedgesscatteringRayleigh wavesdiffractionSommerfeld integrals

In this study, we investigated a one-dimensional diffraction problem of elastic wave propagation through gradient layer. The diffraction problem is reduced to an ordinary differential equation with the third type boundary conditions. The method of approximation of integral identities is applied to increase accuracy of the grid solution to the obtained boundary problem. The case when the elastic wave speed in the layer is constant and density changes continuously according to ρ0(1+A|sinmBx|) is considered in the paper.

In relation to the dynamic tests of materials, the approach to solve the viscoelastic wave propagations in a one dimensional viscoelastic rod was summarized. By conducting Laplace transform, the governing partial differential equations were transformed to ordinary differential equations for the image functions, which were solved analytically with suitable boundary equations. Inversely transforming these image functions gives the results of the stress, velocity, and strain in the bar. Two wave problems occurred in split Hopkinson pressure bar (SHPB) tests are analyzed: 1) the problem of evaluating the internal stress distributions in a viscoelastic specimen; and 2) the problem of stress wave propagations in a viscoelastic bar. Both problems were solved numerically by way of numerical inverse Laplace transform. For the first problem, the special case when the specimen is pure elastic was solved analytically, giving the exact solution to the problem of elastic wave propagation in a sandwich elastic media.

The problems of elastic wave propagation in the micropolar porous materials have been attempted under the interaction of micro-rotation tensor and porous distribution in this material. We obtain the existence of three coupled longitudinal and two coupled shear waves propagating with different phase speeds. Numerically, the phase speed and attenuation of five basic waves are computed. We consider the incident plane waves at the boundary of micropolar porous materials, then the dispersive nature of the incident waves is studied and the existence of critical angle for the incident shear wave is found. The amplitude and energy ratios of various reflected waves for the incident coupled longitudinal and shear waves are obtained analytically and numerically. Further, the influence of different material parameters is studied using dispersion relation and the relevant numerical values.

Dynamic stress analysis of elliptical inclusions in an inhomogeneous medium with modulus and density variations under the action of shear horizontal (SH) waves

This work is devoted to the problem of axial shear elastic wave propagation in composites reinforced with a set of unidirected cylindrical fibers. The effective field method and quasicrystalline approximation are used for the development of the dispersion equation for the wavenumber of the mean (coherent) wave field propagating in the composite. This dispersion equation serves for all frequencies of the incident field, properties and volume concentrations of the fibers. In the method, peculiarities in spatial distributions of inclusions are taken into account via a specific correlation function of the fiber set. Such a function may be constructed for random as well as for regular sets of fibers. Thus, wave propagation in random composites as well as in composites with regular fiber arrangements may be considered. In the case of a random set of fibers, different branches of the wave propagation in the composites with an epoxy matrix and glass fibers are obtained and analyzed. Such branches are constructed also for composites with regular lattices of fibers, and the position of the pass bands, where waves can propagate, and stop bands, where waves exponentially attenuate, in the frequency region are indicated. The comparison of the obtained approximate solutions with some exact solutions of the wave propagation problem for regular composites is presented.

The article considers formulation and solution of a elastic wave propagation problem in the nonlinear media with initial strains, being finite in general case. It takes into account that elastic waves propagation involves large additional strains, imposed physically on the ones, existing already. The theory of superimposed large strains is used for the problem setting. Key words: non-linear waves propagation, anisotropy, large strains, computer modeling, initial strains.

The analysis of elastic wave propagation is a critical problem in both science and engineering, with applications in structural health monitoring and seismic wave analysis. However, the efficient and accurate numerical simulation of large-scale three-dimensional structures has posed significant challenges to traditional methods, which often struggle with high computational costs and limitations. This paper presents a novel two-and-a-half-dimensional generalized finite difference method (2.5D GFDM) for efficient simulation of elastic wave propagation in longitudinally invariant structures. The proposed scheme integrates GFDM with 2.5D technology, reducing 3D problems to a series of 2D problems in the wavenumber domain via Fourier transforms. Subsequently, the solutions to the original 3D problems can be recovered by performing inverse Fourier transforms on the solutions obtained from the 2D problems. The 2.5D GFDM avoids the inherent challenge of mesh generation in traditional methods like FEM and FVM, offering a meshless solution for complex 3D problems. By employing sparse coefficient matrices, it offers significantly improved computational efficiency. The new approach achieves significant computational advantages while maintaining high accuracy, as validated through three representative examples, making it a promising tool for solving large-scale elastic wave propagation problems in longitudinally invariant structures.

A three-dimensional FEM (finite element method) for solving elastic or viscoelastic wave propagation problems are tested. Viscoelastic wave equations are introduced by three relaxation mechanisms. Second-order polynomial interpolation function and perfectly matched layer (PML) absorbing boundary condition are composed to this FEM framework. Also, it performs effective CPU time reduction by parallel computing. The accuracy of the FEM is evaluated by three tests. FEM is compared to the analytical solution for Lamb’s problem and the semi-analytical solutions for layered elastic or viscoelastic media. Especially, FEM and the analytical solution for Lamb’s problem showed almost perfect convergence. Therefore, we conclude that FEM produces very exact waveform. Moreover, this conclusion can support the accuracy of FEM for viscoelastic wave equations.

We show that the pseudospin symmetry that Akito Arima discovered many years ago (with collaborators) is a symmetry of the the Dirac Hamiltonian for which the sum of the scalar and vector potentials are a constant. In this paper we discuss some of the implications of this relativistic symmetry and the experimental data that support these predictions. In his original paper Akito also discussed pseudo-U(3) symmetry. We show that pseudo-U(3) symmetry is a symmetry of the Dirac Hamiltonian for which the sum of harmonic oscillator vector and scalar potentials are equal to a constant, and we give the generators of pseudo-U(3) symmetry. Going beyond the mean field we summarize new results on non relativistic shell model Hamiltonians that have pseudospin symmetry and pseudo-orbital angular momentum symmetry as a dynamical symmetries.

This work is devoted to determine appropriate domain and range of the forward map from the coefficients to the solutions of the multi-dimensional wave equation for which the forward map attains certain regularity properties. The first result concerns an explicit upper bound of its linearization or formal derivative and the properties of the coefficients on which the bound depends. In view of results for the smooth coefficient case, the estimate is optimal. We then present recent results on continuity and differentiability of the forward map as well as continuity of the linearized forward map. Information concerning regularity properties of the forward map is indispensable in the study of the inverse problem via smooth optimization methods. The usefulness of our results and some directions for future research are also discussed.

The problems of propagation and diffraction of non-stationary waves in acoustic and elastic media are of great theoretical and practical importance in such fields of science and engineering as aircraft manufacturing, shipbuilding, seismic exploration of mineral resources, seismic resistance of structures, and many others.This paper considers the problem of non-stationary wave propagation from a spherical cavity located near a stationary rigid sphere in an acoustic space.Formulas are derived for the particle velocity vector and pressure in the medium.Numerical experiments are presented, the results of which are presented in graphical form.The results of this paper can be used in shipbuilding, aircraft manufacturing, by design organizations during the construction of structures, and in the design of underwater reservoirs.

Dirac equation with scalar and vector quadratic potentials and Coulomb-like tensor potential

The problem of propagation of Love waves in elastic waveguides loaded on the surface by a viscous (Newtonian) liquid is important in many applications such as geophysics, seismology, investigation of the physical properties of liquids. Love wave energy is concentrated near the waveguide surface, so that Love waves are especially suited to study the material properties of surface layers. In this work, the direct problem and the inverse problem of the Love wave propagation in a layered elastic waveguides loaded with a viscous liquid have been formulated and solved. The inverse problem relies on the determination of the material parameters (e.g., the unknown value of liquid viscosity) from measurements of the dispersion curves of Love waves.

Electromagnetism is a simple example of a gauge theory where the underlying potentials -- the vector and scalar potentials -- are defined only up to a gauge choice. The vector potential generates magnetic fields through its spatial variation and electric fields through its time-dependence. We experimentally produce a synthetic gauge field that emerges only at low energy in a rubidium Bose-Einstein condensate: the neutral atoms behave as charged particles do in the presence of a homogeneous effective vector potential. We have generated a synthetic electric field through the time dependence of an effective vector potential, a physical consequence even though the vector potential is spatially uniform.