The Transition Matrix Formalism for the Scattering of an Alluvium on an Elastic Half-Space

2007 ◽  
Author(s):  
Wen-I Liao ◽  
Tsung-Jen Teng ◽  
Shiang-Jung Wang
Keyword(s):  
Half Space ◽  
Transition Matrix ◽  
Plane Waves ◽  
Three Dimensional ◽  
Scattering Matrix ◽  
Elastic Half Space ◽  
Basis Functions ◽  
Matrix Formalism ◽  
T Matrix ◽  
Singular Wave

This paper develops the transition matrix formalism for scattering from an three-dimensional alluvium on an elastic half-space. Betti’s third identity is employed to establish orthogonality conditions among basis functions that are Lamb’s singular wave functions. The total displacements and associated tractions exterior and interior to the surface are expanded in a Rayleigh series. The boundary conditions are applied and the T-matrix is derived. A linear transformation is utilized to construct a set of orthogonal basis functions. The transformed T-matrix is related to the scattering matrix and it is shown that the scattering matrix is symmetric and unitary and that the T-matrix is symmetric. Typical numerical results obtained by incident plane waves for verification are presented.

Guided Surface Waves on an Elastic Half Space

1971 ◽  
Vol 38 (4) ◽  
pp. 899-905 ◽  
Author(s):  
L. B. Freund
Keyword(s):  
Wave Propagation ◽  
Surface Wave ◽  
Surface Waves ◽  
Half Space ◽  
Body Wave ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
Normal Displacement ◽  
Rigid Barrier ◽  
Wave Disturbances

Three-dimensional wave propagation in an elastic half space is considered. The half space is traction free on half its boundary, while the remaining part of the boundary is free of shear traction and is constrained against normal displacement by a smooth, rigid barrier. A time-harmonic surface wave, traveling on the traction free part of the surface, is obliquely incident on the edge of the barrier. The amplitude and the phase of the resulting reflected surface wave are determined by means of Laplace transform methods and the Wiener-Hopf technique. Wave propagation in an elastic half space in contact with two rigid, smooth barriers is then considered. The barriers are arranged so that a strip on the surface of uniform width is traction free, which forms a wave guide for surface waves. Results of the surface wave reflection problem are then used to geometrically construct dispersion relations for the propagation of unattenuated guided surface waves in the guiding structure. The rate of decay of body wave disturbances, localized near the edges of the guide, is discussed.

Contact Stress Analysis of a Layered Transversely Isotropic Half-Space

1992 ◽  
Vol 114 (2) ◽  
pp. 253-261 ◽  
Author(s):  
C. H. Kuo ◽  
L. M. Keer
Keyword(s):  
Half Space ◽  
Mixed Boundary ◽  
Contact Region ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Hankel Transform ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Stress Components ◽  
Contact Failure

The three-dimensional problem of contact between a spherical indenter and a multi-layered structure bonded to an elastic half-space is investigated. The layers and half-space are assumed to be composed of transversely isotropic materials. By the use of Hankel transforms, the mixed boundary value problem is reduced to an integral equation, which is solved numerically to determine the contact stresses and contact region. The interior displacement and stress fields in both the layer and half-space can be calculated from the inverse Hankel transform used with the solved contact stresses prescribed over the contact region. The stress components, which may be related to the contact failure of coatings, are discussed for various coating thicknesses.

Transient Thermoelastic Stress Fields in a Half-Space

2002 ◽  
Vol 125 (1) ◽  
pp. 33-43 ◽  
Author(s):  
Shuangbiao Liu ◽  
Qian Wang
Keyword(s):  
Stress Field ◽  
Half Space ◽  
Frictional Heating ◽  
Three Dimensional ◽  
Thermoelastic Stress ◽  
Parabolic Type ◽  
Elastic Half Space ◽  
Transform Method ◽  
Rough Surface Contact ◽  
Thermoelastic Stress Field

Computing the thermoelastic stress field of a material subjected to frictional heating is essential for component failure prevention and life prediction. However, the analysis for three-dimensional thermoelastic stress field for tribological problems is not well developed. Furthermore, the pressure distribution due to rough surface contact is irregular; hence the frictional heating can hardly be described by an analytical expression. This paper presents a novel set of frequency-domain expressions (frequency response functions) of the thermoelastic stress field of a uniformly moving three-dimensional elastic half-space subjected to arbitrary transient frictional heating, where the velocity of the half-space, its magnitude and direction, can be an arbitrary function of time. General formulas are expressed in the form of time integrals, and important expressions for constant velocities are given for the transient-instantaneous, transient-continuous, and steady-state cases. The thermoelastic stress field inside a translating half-space with constant velocities are illustrated and discussed by using the discrete convolution and fast Fourier transform method when a parabolic type or an irregularly distributed heat source is applied.

Reflection Coefficient for Plane Waves in a Fluid Incident on a Layered Elastic Half-Space

1983 ◽  
Vol 50 (2) ◽  
pp. 405-414 ◽  
Author(s):  
D. B. Bogy ◽  
S. M. Gracewski
Keyword(s):  
Reflection Coefficient ◽  
Half Space ◽  
Plane Waves ◽  
Elastic Half Space ◽  
Wave Number ◽  
Solid Boundary ◽  
Inviscid Fluid ◽  
Interface Waves ◽  
Plate Models ◽  
Layered Solid

The reflection coefficient is derived for an isotropic, homogeneous elastic layer of arbitrary thickness that is perfectly bonded to such an elastic half-space of a different material for the case when plane waves are incident from an inviscid fluid onto the layered solid. The derived function is studied analytically by considering several limiting cases of geometry and materials to recover previously known results. Approximate reflection coefficents are then derived using various plate models for the layer to obtain simpler expressions that are useful for small values of σd, where σ is the wave number and d is the layer thickness. Numerical results based on all the models for the propagation of interface waves localized near the fluid-solid boundary are obtained and compared. These results are also compared with some previously published experimental measurements.

Viscous-Elastic Half-Space Vibration Stimulated by High-Speed Moving Loads of Different Speeds

2012 ◽  
Vol 518-523 ◽  
pp. 3874-3877
Author(s):  
Tao Qian ◽  
Xiao Ping Shui ◽  
Yong Fa Zhang ◽  
Yong Gang Guo ◽  
Meng Ma
Keyword(s):  
Half Space ◽  
Coordinate Transformation ◽  
High Speed ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
Moving Loads ◽  
Cylindrical Coordinates ◽  
Relative Coordinate ◽  
The Laplace Transform ◽  
Practical Security

A rule of response of an infinite viscous-elastic half-space stimulated by the moving loads of different speeds is outlined in this paper. In order to obtain a three-dimensional analytical solution of the Viscous-elastic half-space with the moving loads of different speeds, the Laplace transform and relative coordinate transformation in cylindrical coordinates are used. Then, the IFFT and relative coordinate transformation are used to solve two-dimensional infinite integration which can greatly improve the operational efficiency. The rules of responses of different velocities from the results by using the principle of dynamics and energy dissipation are also analyzed and induced in this paper, and obtain the incentives of displacement distortion by the super-Rayleigh wave velocity at surface. The results could be referred in improving the practical security in the project.

THREE-DIMENSIONAL SURFACE CRACK ANALYSIS OF ELASTIC HALF-SPACE UNDER ROLLING CONTACT LOAD

2009 ◽  
Vol 06 (02) ◽  
pp. 317-332 ◽  
Author(s):  
MENG-CHENG CHEN ◽  
HUI-QIN YU
Keyword(s):  
Numerical Solution ◽  
Integral Equations ◽  
Half Space ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
Rolling Contact ◽  
Contact Load ◽  
Hypersingular Integral ◽  
Space Body ◽  
Planar Crack

In this work a three-dimensional planar crack on the surface of elastic half-space was analyzed under rolling contact load. The stresses interior to an elastic half-space body under rolling contact load and those produced by an infinitesimal displacement jump loop for the elastic half-space body were used to reduce the planar crack problem to the solution of a system of two-dimensional hypersingular integral equations with unknown displacement jump. The ideas of finite element discretization were employed to construct numerical solution schemes for solving the integral equations. An appropriate treatment of the associated hypersingular integral in the numerical solution to the integral equations was proposed in Hadamard's finite-part integral sense. The numerical results showed that the present procedure yields solutions with high accuracies. The stress intensity factors near the crack front edge under rolling contact load were indicated in graphical form with varying the crack shape, the radius of rolling contact zone and the friction coefficients, respectively. In addition, the influence of the lubricant infiltrating the crack surfaces on the crack propagation was also discussed in the paper.

scholarly journals Plane waves in thermoelasticity with one relaxation time

2001 ◽  
Vol 26 (4) ◽  
pp. 225-232
Author(s):  
Jun Wang ◽  
Wen Dong Chang
Keyword(s):  
Relaxation Time ◽  
General Solution ◽  
Laplace Transform ◽  
Half Space ◽  
Plane Waves ◽  
Elastic Half Space ◽  
Second Sound ◽  
Boundary Temperature ◽  
Explicit Expressions

We apply the thermoelastic equations with one relaxation time developed by Lord and Shulman (1967) to solve some elastic half-space problems. Laplace transform is used to find the general solution. Problems concerning the ramp-type increase in boundary temperature and stress are studied in detail. Explicit expressions for temperature and stress are obtained for small values of time, where second sound phenomena are of relevance. Numerical values of stress and temperature are calculated and displayed graphically.

Exact Transient Response of an Elastic Half Space Loaded Over a Rectangular Region of Its Surface

1969 ◽  
Vol 36 (3) ◽  
pp. 516-522 ◽  
Author(s):  
F. R. Norwood
Keyword(s):  
Half Space ◽  
Real Axis ◽  
Transient Response ◽  
Plane Waves ◽  
Elastic Half Space ◽  
Impulsive Loading ◽  
Rectangular Region ◽  
The Real ◽  
Algebraic Expressions ◽  
Straight Lines

The response of an elastic half space to a normal impulsive loading over one half and also over one quarter of its bounding surface is considered. By a simple superposition the solution is obtained for a half space loaded on a finite rectangular region. In each case the solution was found to be a superposition of plane waves directly under the load, plus waves emanating from bounding straight lines and the corners of the loaded region. The solution was found by Cagniard’s technique and by extending the real transformation of de Hoop to double Fourier integrals with singularities on the real axis of the transform variables. Velocities in the interior of the half space are given for arbitrary values of Poisson’s ratio in terms of single integrals and algebraic expressions.

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