Transformation of the earthquake displacement field for spherical earth—Static case

1971 ◽  
Vol 61 (4) ◽  
pp. 861-874
Author(s):  
Hans R. Wason ◽  
Sarva Jit Singh
Keyword(s):  
Displacement Field ◽  
Spherical Surface ◽  
Homogeneous Medium ◽  
Addition Theorem ◽  
Polar Coordinate System ◽  
Earth Model ◽  
Static Case ◽  
Spherical Earth ◽  
Polar Coordinate ◽  
Internal Sources

abstract Explicit expressions for the static displacement field for a Volterra dislocation and a center of compression in an infinite homogeneous medium are obtained. Using an addition theorem, the field is transformed to a polar coordinate system with origin at the center of the Earth. Expressions for the discontinuity in the motion stress vector across the concentric spherical surface through the source are then obtained. These results can be used in studying the deformation of a multilayered spherical earth model induced by internal sources by the Thomson-Haskell matrix method which has so far been mostly applied to dynamic problems.

Transformation of earthquake displacement field for spherical earth

1971 ◽  
Vol 61 (2) ◽  
pp. 289-295 ◽  
Author(s):  
Hans R. Wason ◽  
Sarva Jit Singh
Keyword(s):  
Coordinate System ◽  
Displacement Field ◽  
Homogeneous Medium ◽  
Addition Theorem ◽  
Spherical Waves ◽  
The Earth ◽  
Spherical Earth ◽  
The Subject ◽  
Navier Equation

abstract An addition theorem for spherical waves is used to transform the displacement field due to an arbitrary shear dislocation in an unbounded homogeneous medium with origin at the focus to that in a coordinate system with origin at the center of the Earth. The final results are expressed in terms of the eigenvector solutions of the vector Naviér equation of dynamical elasticity. Some serious mistakes in recent papers on the subject have been pointed out.

Scattering of Anti-Plane SH-Wave by the Semi-Cylindrical Gap with Multiple Shallow-Buried Inclusions in Elastic Semi-Space

2012 ◽  
Vol 569 ◽  
pp. 78-81
Author(s):  
Hong Liang Li ◽  
Jing Guo ◽  
Li Ming Cai
Keyword(s):  
Displacement Field ◽  
Ground Surface ◽  
Analytic Solutions ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Dynamic Stress Concentration Factor ◽  
Polar Coordinate ◽  
Cylindrical Inclusions ◽  
Plane Sh Wave

Semi-cylindrical gap and Multiple circular inclusions exists widely in natural media, composite materials and modern municipal construction. The scattering field produced by semi-cylindrical gap and multiple circular inclusions determines the dynamic stress concentration factor around the gap and circular inclusions, and therefore determines whether the material is damaged or not. These problems are complicated. It is hard to obtain analytic solutions except for several simple conditions. In this paper, the solution of displacement field for elastic semi-space with semi-cylindrical gap and multiple cylindrical inclusions by anti-plane SH-wave is constructed. In complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane SH-wave and the scattering displacement field impacted by the gap and the cylindrical inclusions comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the displacement and stress condition around the edge of the gap and cylindrical inclusions. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. The total wave displacement field is the superposition of the displacement field aroused by the anti-plane SH-wave and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the cylindrical inclusions.

Interaction of Semi-Cylindrical Gap and Multiple Shallow-Buried Cavities and Inclusions

2012 ◽  
Vol 157-158 ◽  
pp. 1361-1364
Author(s):  
Hong Liang Li ◽  
Dan Sun
Keyword(s):  
Earthquake Engineering ◽  
Displacement Field ◽  
Line Source ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Displacement Condition ◽  
Polar Coordinate ◽  
Cylindrical Cavities ◽  
Inclusion Structure

In mechanical engineering, earthquake engineering and modern municipal construction, It can be found that there are shallow-buried cavity or inclusion structure everywhere. In this paper, Green's Function is studied, which is the solution of displacement field for elastic semi-space with semi-cylindrical gap and multiple shallow-buried cavities and inclusions while bearing anti-plane harmonic line source force at any point. In the complex plane, considering the symmetry of SH-wave scattering, the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field impacted by semi-cylindrical gap and multiple cylindrical cavities and inclusions are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the stress or displacement condition of the cylindrical cavities and inclusions in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. By solving these algebraic equations the value of the unknown coefficients can be obtained. So the total wave displacement field could be got. By using the expressions, an example is provided to show the effect of the change of relative location of semi-cylindrical gap, the cylindrical cavities and inclusions and the location of the line source force.

Scattering of Anti-Plane SH-Wave by Multiple Cylindrical Inclusions in Elastic Semi-Space

2012 ◽  
Vol 525-526 ◽  
pp. 305-308
Author(s):  
Hong Liang Li ◽  
Yong Yang
Keyword(s):  
Displacement Field ◽  
Ground Surface ◽  
Analytic Solutions ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Dynamic Stress Concentration Factor ◽  
Polar Coordinate ◽  
Cylindrical Inclusions ◽  
Plane Sh Wave

Multiple circular inclusions exists widely in natural media, engineering materials and modern municipal construction. The scattering field produced by multiple circular inclusions determines the dynamic stress concentration factor around the circular inclusions, and therefore determines whether the material is damaged or not. These problems are complicated, because there are many factors influenced. Researchers solved these problems by analysis and numerical methods. It is hard to obtain analytic solutions except for several simple conditions. In this paper, the solution of displacement field for elastic semi-space with multiple cylindrical inclusions by anti-plane SH-wave is constructed. In complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane SH-wave and the scattering displacement field impacted by the cylindrical inclusions comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the displacement and stress condition around the edge of cylindrical inclusions. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. The total wave displacement field is the superposition of the displacement field aroused by the anti-plane SH-wave and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the cylindrical inclusions. Based on this solution, the problem of interaction of multiple cylindrical inclusions and a linear crack in semi-space can be investigated further.

Interaction of Multiple Semi-Cylindrical Gaps and a Shallow-Buried Cavity

2010 ◽  
Vol 163-167 ◽  
pp. 3910-3913
Author(s):  
Rui Zhang ◽  
Hong Liang Li
Keyword(s):  
Earthquake Engineering ◽  
Displacement Field ◽  
Cylindrical Cavity ◽  
Line Source ◽  
Ground Surface ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Free Condition ◽  
Cavity Structure ◽  
Polar Coordinate

In modern municipal construction and earthquake engineering, semi-cylindrical gap and shallow-buried cavity structure are used widely. In this paper, the solution of displacement field for elastic semi-space with multiple semi-cylindrical gaps and a shallow-buried cavity while bearing anti-plane harmonic line source force at any point is studied. In the complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field impacted by multiple semi-cylindrical gaps and a cylindrical cavity comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the stress-free condition of the gaps and the cylindrical cavity in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. The total wave displacement field is the superposition of the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of semi-cylindrical gaps , the cylindrical cavity and the location of the line source force. Based on this solution, the problem of interaction of multiple semi-cylindrical gaps , a cylindrical cavity and a linear crack in semi-space can be investigated further.

Scattering of Anti-Plane SH-Wave by Multiple Cylindrical Cavities in Elastic Semi-Space

2011 ◽  
Vol 317-319 ◽  
pp. 2497-2500
Author(s):  
Rui Zhang ◽  
Hong Liang Li
Keyword(s):  
Displacement Field ◽  
Ground Surface ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Free Condition ◽  
Scattering Field ◽  
Polar Coordinate ◽  
Cylindrical Cavities ◽  
Plane Sh Wave

In natural medium, engineering materials and structures, it can be found that there are cavities everywhere. When structure is impacted by dynamic load, the scattering field will be produced because of the cavities, and it could cause dynamic stress concentration at the edge of the cavities. In this paper, the solution of displacement field for elastic semi-space with multiple cylindrical cavities by anti-plane SH-wave is constructed. In complex plane, considering the symmetry of SH-wave scattering, the displacement field aroused by the anti-plane SH-wave and the scattering displacement field impacted by the cylindrical cavities comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the stress free condition of the cylindrical cavities in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. The total wave displacement field is the superposition of the displacement field aroused by the anti-plane SH-wave and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the cylindrical cavities. Based on this solution, the problem of interaction of multiple cylindrical cavities and a linear crack in semi-space can be investigated further.

Interaction of Multiple Cylindrical Cavities and Fixed Surface in Elastic Semi-Space

2015 ◽  
Vol 813 ◽  
pp. 147-151
Author(s):  
Hong Liang Li ◽  
Rui Zhang ◽  
Hao Zhang
Keyword(s):  
Displacement Field ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Sh Wave ◽  
Displacement Boundary Condition ◽  
Displacement Boundary ◽  
Scattering Field ◽  
Polar Coordinate ◽  
Cylindrical Cavities ◽  
Plane Sh Wave

In natural medium, engineering materials and structures, it can be found that there are cavities everywhere. Sometimes the surface of the structure is fixed, and it could be seen as a rigid line. When structure is impacted by dynamic load, the scattering field will be produced because of the cavities and the fixed surface, and it could cause dynamic stress concentration at the edge of the cavities. In this paper, the solution of displacement field for elastic semi-space with fixed surface and multiple cylindrical cavities by anti-plane SH-wave is constructed. In complex plane, considering the displacement boundary condition of the fixed surface, the displacement field aroused by the anti-plane SH-wave and the scattering displacement field impacted by the cylindrical cavities comprised of Fourier-Bessel series with undetermined coefficients are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the stress free condition of the cylindrical cavities in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. The total wave displacement field is the superposition of the displacement field aroused by the anti-plane SH-wave and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the cylindrical cavities. Based on this solution, the problem of interaction of multiple cylindrical cavities and a linear crack in semi-space with fixed surface can be investigated further.

Green's Function Solution of the Semi-Space with Double Shallow-Buried Inclusions

2011 ◽  
Vol 462-463 ◽  
pp. 518-523
Author(s):  
Guo Hui Wu ◽  
Hong Liang Li
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Displacement Field ◽  
Line Source ◽  
Ground Surface ◽  
Polar Coordinate System ◽  
Algebraic Equations ◽  
Function Solution ◽  
Polar Coordinate ◽  
Inclusion Structure

In mechanical engineering and modern municipal construction, shallow-buried inclusion structure is used widely. In this paper, Green's Function is studied, which is the solution of displacement field for elastic semi-space with double shallow-buried inclusions while bearing anti-plane harmonic line source force at any point. In complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field impacted by the circle inclusions comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the displacement and stress condition of the circle inclusions in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. Green's function, that is, the total wave displacement field is the superposition of the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the circle inclusions and the location of the line source force. Based on this solution, the problem of interaction of double circular inclusions and a linear crack in semi-space can be investigated further.

Sign in / Sign up

Export Citation Format

Share Document