The Scattering of SH-Wave Caused by Subsurface Circular Cavities in a Layered Half-Space

2011 ◽  
Vol 488-489 ◽  
pp. 226-229
Author(s):  
Dong Ni Chen ◽  
Hui Qi ◽  
Yong Shi
Keyword(s):  
Half Space ◽  
Elastic Layer ◽  
Complex Function ◽  
Expansion Method ◽  
Elastic Half Space ◽  
Wave Functions ◽  
Large Radius ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Linear Algebraic Equations

The scattering of SH-wave caused by the subsurface circular cavities in an elastic half-space covered with an elastic layer was discussed, which was based on the complex function method ,wave functions expansion method and big circular arc postulation method in which the circular boundary of large radius was used to approximate straight boundary of surface elastic layer. By the theory of Helmholtz, the general solution of the Biot’s wave function was achieved. Utilizing the complex series expansion technology and the boundary conditions, we could transform the present problem into the problem in which we needed to solve the infinite linear algebraic equations with unknown coefficients in wave functions. Finally, the dynamic stress concentration factors around the circular cavities were discussed in numerical examples.

The Effect on Scattering of SH-Wave in a Layered Half-Space with the Subsurface Circular Cavities

2011 ◽  
Vol 194-196 ◽  
pp. 1908-1911
Author(s):  
Dong Ni Chen ◽  
Hui Qi ◽  
Yong Shi
Keyword(s):  
Half Space ◽  
Elastic Layer ◽  
Complex Function ◽  
Expansion Method ◽  
Elastic Half Space ◽  
Wave Functions ◽  
Large Radius ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Linear Algebraic Equations

The scattering of SH-wave caused by the subsurface circular cavities in an elastic half-space covered with an elastic layer was discussed, which was based on the complex function method and wave functions expansion method. The solution of scattering of SH-wave was given by using circular boundary of large radius to approximate straight boundary of surface elastic layer. According to boundary conditions, we needed to solve the infinite linear algebraic equations with unknown coefficients in wave functions. Finally, the dynamic stress concentration factors around circular cavities were discussed in numerical examples.

Dynamic Response of Shallow Buried Tunnel Subjected to SH Wave

2010 ◽  
Vol 163-167 ◽  
pp. 4265-4268
Author(s):  
Zhi Gang Chen
Keyword(s):  
Stress Concentration ◽  
Half Space ◽  
Dynamic Stress ◽  
Complex Function ◽  
Least Square ◽  
Wave Number ◽  
Dynamic Stress Concentration ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Scattering Wave

The dynamic stress concentration on quadratic and U-shaped cavities in half space, which are similar to the cross-section of the tunnels, is solved in this paper impacted by SH-wave. The analytical solution for the cavity in elastic half space is gained by the complex function method. In the complex plane, the scattering wave which satisfies the zero-stress condition at the horizontal surface can be constructed, the problem can be inverted into a set of algebraic equations to solve coefficients of the constructed scattering wave by least square method. For the earthquake-resistance researches, the numerical examples of the dynamic stress concentration around the quadratic and U-shaped cavities impacted by SH-wave are given. The influences of the dynamic stress concentration by the incident wave number and angle, the depth and shape of the cavity are discussed. It is showed that the interaction among the wave, the surface and the shallow buried tunnels should be cared in half space. In this situation, the dynamic stress concentration around the tunnel is greater obvious than the whole space.

scholarly journals Two Contact Problems for Annular Rigid Stamp on an Elastic Half Space

2018 ◽  
Vol 17 (6) ◽  
pp. 458-464
Author(s):  
S. V. Bosakov
Keyword(s):  
Functional Equation ◽  
Half Space ◽  
Bessel Functions ◽  
Infinite System ◽  
Annular Plate ◽  
Contact Problems ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

The paper presents solutions of two contact problems for the annular plate die on an elastic half-space under the action of axisymmetrically applied force and moment. Such problems usually arise in the calculation of rigid foundations with the sole of the annular shape in chimneys, cooling towers, water towers and other high-rise buildings on the wind load and the load from its own weight. Both problems are formulated in the form of triple integral equations, which are reduced to one integral equation by the method of substitution. In the case of the axisymmetric problem, the kernel of the integral equation depends on the product of three Bessel functions. Using the formula to represent two Bessel functions in the form of a double row on the works of hypergeometric functions Bessel function, the problem reduces to a functional equation that connects the movement of the stamp with the unknown coefficients of the distribution of contact stresses. The resulting functional equation is reduced to an infinite system of linear algebraic equations, which is solved by truncation. Under the action of a moment on the annular plate  die, the distribution of contact stresses is searched as a series by the products of the Legendre attached functions with a weight corresponding to the features in the contact stresses at the die edges. Using the spectral G. Ya. Popov ratio for the ring plate, the problem is again reduced to an infinite system of linear algebraic equations, which is also solved by the truncation method. Two examples of calculations for an annular plate die on an elastic half-space on the action of axisymmetrically applied force and moment are given. A comparison of the results of calculations on the proposed approach with the results for the round stamp and for the annular  stamp with the solutions of other authors is made.

The Scattering Field of SH-Wave in Half-Space With a Semi-Cylindrical Hill and a Horizontal Circular Tunnel

2005 ◽  
Author(s):  
Guoqing Wang ◽  
Liming Dai ◽  
Diankui Liu
Keyword(s):  
Wave Function ◽  
Half Space ◽  
Stress Condition ◽  
Complex Function ◽  
Sh Wave ◽  
Coordinate Method ◽  
Linear Algebraic Equations ◽  
Scattering Field ◽  
Moving Coordinate ◽  
The Hill

The scattering field of SH-wave in a half-space with a semi-cylindrical hill and a subsurface horizontal hole is studied in the present research by utilizing a complex function and the moving-coordinate method. Based on the concept of ‘conjunction,’ the domain considered is divided into two subdomains. The first subdomain is a cylindrical one which includes the surface of the hill, while the rest is the second subdomain. In the cylindrical subdomain, a standing wave function is constructed which automatically satisfies the zero-stress condition at the hill surface and arbitrary-stress condition at the other part of the circular subdomain. For the second subdomain, which contains a semi-cylindrical canyon and a subsurface hole, a scattering wave function is assumed, which satisfies the zero-stress condition on the horizontal surface. By employing the moving-coordinate method, the solutions of the mathematical model established for the SH-wave can be obtained with the satisfaction of the continuous conditions of stress and displacement across the junction interface together with the zero-stress condition at the surface of the tunnel. The solutions such obtained consist of a series of infinite linear algebraic equations, which can be solved numerically with consideration of the first finite terms corresponding to the frequencies of the wave. For demonstrating the application of the model developed, the displacements of the horizontal and semi-cylindrical hill surfaces are quantified with different properties of wave and geometry parameters.

The Dynamic Response of Elastic Half Space Including a Semi-Lining Hill

2013 ◽  
Vol 753-755 ◽  
pp. 1712-1718
Author(s):  
Jing Fu Nan
Keyword(s):  
Dynamic Response ◽  
Half Space ◽  
Line Source ◽  
Complex Function ◽  
Arbitrary Point ◽  
Elastic Half Space ◽  
Displacement Function ◽  
Algebraic Equations ◽  
Out Of Plane ◽  
Horizontal Interface

The dynamic response of elastic half space including semi-cylindrical lining hill while bearing out-of-plane harmonic line source loading on horizontal interface is investigated using the method of complex function and Greens function. the displacement function of incidence wave is given while a out-of-plane harmonic line source is loaded on arbitrary point of horizontal interface;and the solution domain is divided into two domains, an elastic half space with the semi-circular canyon and a cylindrical lining; the scattering wave of semi-cylindrical canyon and the standing wave of cylindrical lining are constructed. Finally, it conjoins the two domains, and a series of infinite algebraic equations can be obtained to settle this problem. In the end, the numerical expressions of the ground motion in the horizontal surface are discussed.

Several Elliptical Punches on an Elastic Half Space

1986 ◽  
Vol 53 (2) ◽  
pp. 390-394 ◽  
Author(s):  
V. I. Fabrikant
Keyword(s):  
Half Space ◽  
General Theorem ◽  
Elastic Half Space ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Generalized Displacements

A general theorem is established which relates the resulting forces, acting on a set of arbitrary punches, with their generalized displacements through a system of linear algebraic equations. The theorem is applied to the case of arbitrarily located elliptical punches. Several specific examples are considered.

A study on the scattering field of SH-waves in a topographical area consisting of a semi-cylindrical hill and a subsurface horizontal hole

2007 ◽  
Vol 221 (3) ◽  
pp. 451-465 ◽  
Author(s):  
G Wang ◽  
L Dai ◽  
D Liu
Keyword(s):  
Stress Condition ◽  
Numerical Solutions ◽  
Complex Function ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Sh Waves ◽  
Linear Algebraic Equations ◽  
Scattering Field ◽  
Moving Coordinate ◽  
Infinite Linear

This research intends to investigate the scattering field of SH-wave in a half-space containing a semicylindrical hill and a subsurface horizontal hole. A mathematical model is established in a two-dimensional plane on the basis of the characteristics of SH-waves, the ‘division-conjunction’ concept, the complex function, and moving-coordinate methods. The whole domain considered is divided into two subdomains, and the wave expressions are assumed in each subdomain. In the cylindrical subdomain, the wave function is constructed with the satisfaction of the zero-stress condition on the hill's surface automatically. In the other subdomain, the solution of the scattering waves is postulated under the stress-free condition on the horizontal surface. The analytical solutions of themodel established are obtained through a series of infinite linear algebraic equations, under the conditions that both the stress and displacement across the conjunction interface of the two subdomains are continuous. The numerical solutions are developed by truncating the infinite linear algebraic equations. The numerical simulations are performed for quantifying the displacements of the horizontal and semicylindrical hill surfaces subjected to incident SH waves, and the numerical results are verified with a comparison to the existing results of a case without subsurface hole.

Interaction of SH-Wave on Multiple Linear Cracks near Shallow-Embedded Structure

2011 ◽  
Vol 488-489 ◽  
pp. 440-443
Author(s):  
Zai Lin Yang ◽  
Hua Nan Xu ◽  
Mei Juan Xu ◽  
Bai Tao Sun
Keyword(s):  
Green’S Function ◽  
Half Space ◽  
Green's Function ◽  
Complex Function ◽  
Surface Displacement ◽  
Elastic Half Space ◽  
Polar Coordinates ◽  
Multiple Cracks ◽  
Sh Wave ◽  
Lining Structure

The surface displacement of a circular lining structure and multiple cracks in an elastic half space by incident SH-wave is studied in this paper based on the methods of Green's function, complex function and multi-polar coordinates. Firstly, we construct a suitable Green’s function which indicates a fundamental solution to the displacement field for an elastic half space possessing a circular lining structure and cracks while bearing out-plane harmonic line loads at arbitrary point. Then using the method of crack-division, a crack is created. Thus expressions of displacement and stress field are established at the existence of the structure and the cracks. Finally, the interaction of inclusion and two cracks is chosen as numerical examples and the influences of different parameters on the surface displacement are discussed.

Dynamics of a Mode III Crack Impacted by Elastic Wave in Half Space with a Removable Rigid Cylindrical Inclusion

2006 ◽  
Vol 324-325 ◽  
pp. 679-682 ◽  
Author(s):  
Zai Lin Yang ◽  
Dian Kui Liu ◽  
Xiao Lang Lv
Keyword(s):  
Green’S Function ◽  
Half Space ◽  
Green's Function ◽  
Complex Function ◽  
Elastic Half Space ◽  
Cylindrical Inclusion ◽  
Sh Wave ◽  
Mode Iii ◽  
Scattering Wave ◽  
Moving Coordinate

Scattering of SH wave by a crack is studied in elastic half space with a removable rigid cylindrical inclusion by Green’s function, complex function and moving coordinate method. In half space, firstly the scattering wave function of removable rigid cylindrical inclusion is constructed; next a suitable Green’s function is solved for present problem, then using crack-division to make a crack. Thus the solution of problem can be obtained. Numerical examples are provided and discussed.

SH-Wave Scattering From the Interface Defect

2017 ◽  
Vol 5 (1) ◽  
pp. 45-50
Author(s):  
Myron Voytko ◽  
◽  
Yaroslav Kulynych ◽  
Dozyslav Kuryliak
Keyword(s):  
Half Space ◽  
Wave Diffraction ◽  
Scattered Field ◽  
Wave Scattering ◽  
Elastic Layer ◽  
Analytical Form ◽  
Structure Parameters ◽  
Sh Wave ◽  
Interface Defect ◽  
Hopf Method

The problem of the elastic SH-wave diffraction from the semi-infinite interface defect in the rigid junction of the elastic layer and the half-space is solved. The defect is modeled by the impedance surface. The solution is obtained by the Wiener- Hopf method. The dependences of the scattered field on the structure parameters are presented in analytical form. Verifica¬tion of the obtained solution is presented.

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