Plane Harmonic SH Waves Research Articles

Wave propagation in a three-dimensional half-space with semi-infinite irregularities

ABSTRACT Dynamic analysis of problems with complex geometries requires utilization of numerical methods. To completely capture the effects of seismic wave propagation in a system, one must consider the structure or irregularity within its encompassing half-space. Correct consideration of half-space in a numerical model is important specially when it comes to cases where the half-space contains semi-infinite irregularities. In this study, a generalized numerical methodology is presented for dynamic analysis of a half-space with semi-infinite irregularities. The methodology is first verified through comparison with analytical solution of known problems. Then the method is employed to solve the dynamic analysis of an arbitrary-shaped existing valley within a half-space under propagating plane harmonic SH wave with different angles of incidence. Results indicate complete capability of the methodology in solving wave propagation problems for complex systems without the usual restrictions of common existing methods.

Dynamic response of a functionally graded tube embedded in an elastic medium due to SH-Waves

Dynamic response of a cylindrical tube surrounded by an unbounded elastic medium due to plane harmonic SH-Waves is studied. A two-dimensional mathematical model is considered. Cylindrical coordinates are used for convenience. The surrounding medium is assumed to be homogeneous, isotropic and linear elastic. The tube is assumed to be made of linear elastic functionally graded materials (FGMs) such that shear modulus and shear wave velocity are assumed to change linearly from inner surface to outer surface. Material properties are constant along circumferential direction. It is assumed that the inner surface of the tube is traction-free and there is a welded contact between the tube and the surrounding medium. Governing equations are slightly different in the tube region and the unbounded region. Both of the governing equations are solved by applying Finite Fourier Transform in circumferential direction. The exact solution series are presented in terms of Fourier-Bessel series in the unbounded region and power series in the tube region. The presented numerical results show that when the incoming wave lengths decrease, shear stresses at the tube increase significantly. It was shown that for the shorter incoming wave lengths, tubes made of FGMs are subjected to smaller shear stresses compared to the tubes homogeneously made of outer surface material of the FG cases.

Diffraction of Plane SH Waves by a Cylindrical Cavity in an Infinite Wedge

Diffraction of plane harmonic SH waves by a cylindrical cavity in a homogeneous, isotropic and linear elastic infinite wedge is investigated. Analytical closed form solution is obtained by utilizing Wave Function Expansion Method and Image Technique. Governing equation of this two dimensional steady-state wave motion problem is the Helmholtz equation. Helmholtz equation for polar coordinates is solved by using Separation of Variables Method. Displacement fields are expressed in terms of Fourier-Bessel series in complex form. Complex coefficients of Fourier-Bessel series are to be determined from boundary conditions. Zero-stress boundary condition at cavity inner surface is satisfied directly since polar coordinate systems are used. Free stress conditions at flat surfaces of the infinite wedge are satisfied in closed form via imaging method. In this technique, imaginary cavities symmetric with respect to flat surfaces are considered. With this consideration, scattered waves from imaginary cavities would represent reflections from flat surfaces so that stress free boundary condition on flat surfaces would be automatically satisfied. This technique would generate two additional displacement fields with two additional set of unknown complex Fourier-Bessel constants. These unknowns could be determined from boundary conditions of imaginary cavities. The cylindrical scattered waves from each cavity are defined in different polar coordinate systems. Addition Theorems are used for adequate coordinate transformation. Since Image Technique is used, to maintain angular symmetry, analytical solutions are obtained for a specific wedge whose internal angle is 120 degrees. Also, direction of incident waves has to be either horizontal or on the symmetry line of the wedge and cavity center has to be on the symmetry line. In numerical results, parameters in length dimension are normalized with respect to cavity radius.

Dynamic stress concentration for multiple multilayered inclusions embedded in an elastic half-space subjected to SH-waves

The dynamic stress concentration factor (DSCF) is evaluated along the interfaces of multiple multilayered inclusions embedded in a half-space when subjected to a plane harmonic SH-wave. A weak form of Helmholtz equation is utilized to derive a non-hypersingular boundary integral equations to compute the stresses. Eliminating the need to rely on hypersingular integrals, greatly simplifies the procedure. The numerical results obtained by the proposed method, are validated against analytical solutions.Various contributing factors that can influence the DSCF are investigated, including multiple scattering, layering, stiffness of the adjacent inclusions, and impedance contrast of the layers. The DSCF is found to be highly prone to these changes, particularly with the soft materials. Therefore, accurate analysis of stresses requires models that consider multiple scattering and layering. The presented result could be used for predicting the seismic failure of pipes and underground tunnels and for estimating the stress failure in strong ground motion seismology due to subsurface irregularities.

Scattering of a plane harmonic SH wave by multiple layered inclusions

Scattering of a plane harmonic SH wave by an arbitrary number of layered inclusions in a half-space is investigated by using a direct boundary integral equation method. The inclusions of arbitrary shape and placement are embedded within an elastic half-space. The effects of multiple scattering, the geometry, and the impedance contrast of the materials for layered inclusions and pipes are considered in detail.

Scattering of a plane harmonic SH wave by a rough multilayered inclusion of arbitrary shape

Steady-state scattering of a plane harmonic SH wave by a multilayered inclusion of arbitrary shape consisting of a general number of homogeneous isotropic layers, embedded within a half space, is investigated by using a direct boundary integral equation approach. The layer interfaces are assumed to be either smooth or periodically corrugated without any sharp corners being present. The peak corrugation amplitude is arbitrarily prescribed. The corresponding integral equations are derived in terms of the unknown boundary displacements and tractions. The integral equations are discretized using linear elements leading to a system of linear algebraic equations which are solved by using the standard Gauss elimination procedure.Testing of the numerical results is performed by comparing the surface response of a cavity with those available in the literature. The tests show excellent agreement between the two results for different frequencies and angles of incidence.The numerical results include surface response of multilayered inclusions and pipes. The inclusion layers are assumed to be either softer or stiffer from the half space. The pipe layers include both soft–stiff and stiff–soft combinations of the materials.The layered inclusion results demonstrate that for the soft materials the presence of layers may significantly change the surface response when compared to the corresponding homogeneous inclusion motion. For stiff layers this effect is greatly reduced. The same phenomenon has been observed for the inclusions with periodically corrugated interfaces as well.The surface response of a layered pipe is found to strongly depend upon the nature of the incident wave and the materials of the layers.

Scattering of plane harmonic SH waves by multiple inclusions

SUMMARY Scattering of a plane harmonic SH wave by an arbitrary number of inclusions of general shape and placements within an elastic half-space is investigated by using a direct boundary integral equation approach. The integral equations have been reduced to a system of linear algebraic equations by using the linear elements. The structure of the resulting algebraic system is investigated and compared to the single-inclusion model. For convenience the multiple-inclusion models are assumed to consist of the reference and the auxiliary inclusions. The reference inclusion has the same shape, placement and the material properties as those of the corresponding single-inclusion model. The auxiliary inclusions then surround the reference inclusion assuming different patterns. Extensive testing of the proposed solution is performed. These tests include the transparency and the single-inclusion tests. In the former the material properties of the inclusions are assumed to be the same as those of the half-space and the surface response is found to be equal to the free field one. In the single-inclusion test the material properties of the auxiliary inclusions are assumed to be the same as those of the half-space while those of the reference inclusion are different. Then the resulting surface response is found to reduce to that of the single-inclusion model. For different number of multiple inclusions the surface motion atop the reference inclusion is compared with that of the corresponding single-inclusion model. In particular, a specific measure is introduced to assess accurately the importance of the multiple scattering upon the surface response. The effect of the auxiliary inclusion locations with respect to the reference inclusion upon the surface motion is investigated for two-, three- and nine-inclusion models. The results clearly show the importance of the interactions between the inclusions for a wide range of parameters present in the problem (e.g. nature of the incident wave, geometry of the model and the number of inclusions). For the problems considered here it appears that by ignoring these interactions, that is, using the single-inclusion model, will result in underestimation of the peak surface motion amplification.

Peak surface motion due to scattering of a plane harmonic SH wave by a randomly corrugated scatterer

Scattering of a plane harmonic elastic SH wave by a rough elastic inclusion embedded within an elastic half space is investigated for an anti-plane strain model by using a direct boundary integral equation method. The scatterer’s interface is assumed to be of an elliptical shape perturbed in a random fashion. The perturbations are assumed to have either uniform or normal probability distributions. The peak perturbation amplitude is assumed to be arbitrary. Surface displacement spectral amplitude is evaluated for different scatterers, impedance contrasts, angles of incidence, and range of frequencies. Corresponding contour plots are used to identify the frequencies of the peak surface spectral amplitudes for each model. The results clearly demonstrate that the presence of small roughness in the inclusion’s interface may change significantly the peak surface motion frequencies. These changes depend upon the inclusion geometry, the nature of the interface roughness, and the impedance contrast of the materials. For certain frequencies, the presence of the inclusion roughness is important for the surface response and modeling may require detailed description of the corrugated inclusion. Lack of such analysis may result in considerable error in the surface motion amplification estimates.

Scattering of a plane harmonic SH wave by a completely embedded corrugated scatterer

AbstractAnti‐plane‐strain model for steady‐state scattering of elastic waves by a rough inclusion or a cavity embedded in a half space is considered by using a direct boundary integral equation method. The roughness of the scatterer is assumed to be periodic with arbitrary amplitude and period.Detailed testing of the numerical results is presented. The motion along the half‐space surface is evaluated for different corrugations, frequencies and impedance contrast of the materials. The importance of the scatterer roughness upon the displacement field is clearly demonstrated. It was shown that larger corrugation amplitudes, shorter corrugation periods and higher frequencies may produce significant change in the displacement field when compared with the corresponding smooth scatterer result. This effect strongly depends upon the impedance contrast of the materials. Copyright © 2008 John Wiley & Sons, Ltd.

Scattering of Waves by a Sedimentary Basin with a Corrugated Interface

Antiplane strain and plane strain models for scattering of harmonic waves by a sedimentary basin with corrugated interface is investigated using an indirect boundary integral equation method. The corrugated interface is generated by adding to a smooth semielliptical shape a sinusoidal variation of prescribed amplitude and period or a random perturbation. Surface response is simulated for incident plane harmonic SH, P, SV , and Rayleigh waves for different frequencies, depth of the reference basin, amplitude, and period of corrugation. The results show that the presence of the corrugation may result in a significant reduction of peak surface ground motion atop the basin when compared with the corresponding smooth basin response. This reduction depends on polarization and frequency of the incident wave, the fundamental shape of the basin, and the nature of the corrugation. Especially pronounced reduction of the corrugated basin response is observed for incident SV waves. In addition, it is shown that the presence of the corrugation will affect the fundamental frequency of the basin as well when compared with the corresponding smooth basin case. Therefore, the results presented clearly demonstrate that neglecting the interface irregularities in modeling the basin response may result in a significant overestimate of the surface ground motion.

On diffraction of normally incident SH-waves by a line crack in micropolar elastic medium

The problem of diffraction of waves due to plane harmonic SH-waves incident normally on a line crack situated in an infinite micropolar elastic medium has been considered. The solution of the problem is obtained for both low and high frequencies for small coupling parameter. The stress-intensity factors in micropolar elastic medium have been derived. The stress-intensity factor for such problem in an elastic medium can be deduced from results obtained in this paper. It is also found that the effect of micropolarity in the propagation of waves is more significant in high frequencies than low frequencies.

Plane SH-waves at a corrugated interface between two dissimilar perfectly conducting self-reinforced elastic half-spaces

In this paper, we have attempted a problem of reflection and refraction of plane harmonic SH-wave at a corrugated interface between two different perfectly conducting self-reinforced elastic half-spaces. Rayleigh's method is employed to find out the expressions of reflection and refraction coefficients for first- and second-order approximation of the corrugation. The expressions of these coefficients show that they depend on the properties of half-spaces, angle of incidence, frequency of the incident wave and are strongly influenced by the corrugation of the interface. Numerical computations are performed for a particular model having special type of interface and the variation of these coefficients are depicted graphically against the angles of incidence, frequency parameter, corrugation parameter at different values of reinforcement parameters. Results of some earlier works are reduced as a particular case of this formulation. Copyright © 2005 John Wiley & Sons, Ltd.

A note on a plate having a circular cavity excited by plane harmonic SH waves

A solution for the plate having a circular cavity subject to plane harmonic SH waves, horizontally polarized shear waves, is presented in this paper. The method of solution involves series expansion of incident and reflected SH waves from the cavity and two free surfaces in terms of cylindrical wave functions and the image method, respectively. The image method proposed in this article is used to satisfy the boundary condition of the traction free surfaces on the plate. Then, the boundary condition of the cavity is applied to the solution having unknown coefficients. In order to simplify the problem, only the lowest mode is taken into account in the plate. The results obtained in this study provide important information about the behaviour of the plate near the discontinuity. Numerical results show that the distance between the upper surface and the nearest cavity boundary (at θ=0, r=a ) has primary effect. If the radius of the cavity is relatively small compared with the thickness of the plate, the solutions approach the full space's solutions for the cavity, well-known in literature. Also, those solutions for the plate correspond to the solutions of the plate without cavity. Dynamic stress concentration factors around the cavity and displacements in the cross-section of the plate are obtained near the cavity for various wave numbers. It is concluded that the presented solutions are partially analytical solutions, so they can be used for construction and verification of approximate numerical techniques such as the boundary element method (BEM), finite element method (FEM), finite difference method (FDM), etc.

Predominant motion of the Los Angeles sedimentary basin

Predominant motion of the Los Angeles sedimentary basin is investigated theoretically and experimentally. Theoretical investigation of the predominant periods of the basin is done through use of an indirect boundary integral equation method. Sedimentary basis is modeled as a single dipping layer embedded within a baserock half-space. The basin is subjected to vertically incident plane harmonic SH, P, and SV waves. For each of the incident waves the fundamental period of the predominant motion of the basin is calculated numerically. In experimental part of investigation long period microtremor recordings were made along two characteristics sections at eleven observation sites of the basin. From these measurements the fundamental period of the predominant motion of the basin is identified. Comparison of the experimental and theoretical periods of predominant motion show that the predominant motion of the basin is associated with corresponding SV -wave resonance.

Scattering of plane SH-waves by underground cavities

A simplified method is presented to treat the complicated problem of scattering of plane harmonic SH-waves by cavities of any shape, either in full or half space. The essence of the method is the discretization of the boundary of the cavity into a series of straight strips. By doing this only plane waves will be encountered in the analysis. An approximate method to include the effects of diffraction due to grazing waves is also incorporated. Results compare reasonably well with existing closed-form solutions. The angle of incidence of the waves is seen to be a dominant parameter in the analysis.

Scattering of plane harmonic SH, SV, P and Rayleigh waves by non-axisymmetric three-dimensional canyons: a wave function expansion approach : Eshraghi, H; Dravinski, M Earthq Engng Struct DynamV18, N7, Oct 1989, P983–998

Scattering of plane harmonic SH, SV, P and Rayleigh waves by non-axisymmetric three-dimensional canyons: a wave function expansion approach : Eshraghi, H; Dravinski, M Earthq Engng Struct DynamV18, N7, Oct 1989, P983–998

Scattering of elastic waves by nonaxisymmetric three‐dimensional dipping layer

AbstractUsing a boundary method, we investigated the scattering of elastic plane harmonic SH, SV, P, and Rayleigh waves by three‐dimensional nonaxisymmetric dipping layers embedded in an elastic half‐space. The valley was subjected to incident Rayleigh wave and oblique incident SH, SV, and P waves. The method utilized spherical wave functions to express the unknown scattered field. These functions satisfy the equation of motion and radiation conditions at infinity but they do not satisfy the stress‐free boundary conditions at the surface of the half‐space. The boundary and continuity conditions are imposed locally in the least‐square‐sense at several points on the layer interface and on the surface of the half‐space. A comparative study was done to examine the validity and limitations of the two‐dimensional approximations (antiplane and plane strain models) of three‐dimensional models. It is demonstrated that the two‐dimensional approximations may be inadequate to represent actual displacement field for three‐dimensional irregularities.

Scattering of plane harmonic SH, SV, P and rayleigh waves by non‐axisymmetric three‐dimensional canyons: A wave function expansion approach

AbstractScattering of elastic waves by three‐dimensional canyons embedded within an elastic half‐space is investigated by using a wave function expansion technique. The geometry of the canyon is assumed to be non‐axisymmetric. The canyon is subjected to incident plane Rayleigh waves and oblique incident SH, SV and P waves. The unknown scattered wavefield is expressed in terms of spherical wave functions which satisfy the equations of motion and radiation conditions at infinity, but they do not satisfy stress‐free boundary conditions at the half‐space surface. The boundary conditions are imposed locally in the least‐squares sense at several points on the surface of the canyon and the half‐space.Through a comparative study the validity and limitations of two‐dimensional approximations (antiplane strain and plane strain models) have been examined. It is shown that scattering of waves by three‐dimensional canyons may cause substantial change in the surface displacement patterns in comparison to the two‐dimensional models. These results emphasize the need for three‐dimensional modelling of realistic problems of interest in strong ground motion seismology and earthquake engineering.

Scattering of plane harmonic SH wave by dipping layers of arbitrary shape : Dravinski, M Bull Seismol Soc AmV73, N5, Oct 1983, P1303–1319

Scattering of plane harmonic SH wave by dipping layers of arbitrary shape : Dravinski, M Bull Seismol Soc AmV73, N5, Oct 1983, P1303–1319

Scattering of plane harmonic SH wave by dipping layers or arbitrary shape

The antiplane strain model for scattering of plane SH waves by dipping layers of arbitrary shape is investigated by using the indirect boundary integral method. The dipping layers are of finite length perfectly bonded together. The material of the layers is assumed to be homogeneous, linearly elastic, and isotropic. Presented numerical results incorporate variation in the number of layers, the angle of incidence, the frequency of the incident wave, the material properties, and the shape of the layers.