Dynamic Response of Out-Plane Line Loads by a Shallow-Embedded Circular Lining Structure

2011 â—˝  
Vol 55-57 â—˝  
pp. 1107-1110
Author(s):  
Ming Song Gao â—˝  
Zhi Gang Chen
Keyword(s):  
Dynamic Response â—˝  
Green’S Function â—˝  
Green's Function â—˝  
Stress Condition â—˝  
Ground Surface â—˝  
Arbitrary Point â—˝  
Algebraic Equations â—˝  
Dynamic Stress Concentration Factor â—˝  
Lining Structure â—˝  
Line Loads

The dynamic response problems of out-plane line loads by a shallow-embedded circular lining structure were investigated here by using the method of Green’s Function. Firstly a suitable Green’s function was constructed, which is an essential solution to the displacement field possessing a shallow-embedded circular lining structure while bearing out-plane harmonic line loads at an arbitrary point. Then we obtained a series of algebraic equations to solve this problem after constructing scattering waves that satisfied the zero-stress condition on the ground surface. Lastly, some numerical examples are given to show the effects that different parameters influence dynamic stress concentration factor (DSCF) by out-plane line source loads.

Green's Function Solution of the Semi-Cylindrical Gap with Multiple Shallow-Buried Inclusions

2011 â—˝  
Vol 261-263 â—˝  
pp. 863-867 â—˝  
Author(s):  
Hong Liang Li
Keyword(s):  
Green’S Function â—˝  
Green's Function â—˝  
Earthquake Engineering â—˝  
Displacement Field â—˝  
Line Source â—˝  
Ground Surface â—˝  
Polar Coordinate System â—˝  
Algebraic Equations â—˝  
Function Solution â—˝  
Cylindrical Inclusions

In mechanical engineering, earthquake engineering and modern municipal construction, semi-cylindrical gap and shallow-buried inclusion structure are used widely. 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 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 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 cylindrical inclusion 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 semi-cylindrical gap , the cylindrical inclusions and the location of the line source force. Based on this solution, the problem of interaction of semi-cylindrical gap , multiple cylindrical inclusions and a linear crack in semi-space can be investigated further.

Dynamic Response of Complex Structure in Half Space

2011 â—˝  
Vol 199-200 â—˝  
pp. 973-976
Author(s):  
Ming Song Gao â—˝  
Zhi Gang Chen
Keyword(s):  
Green’S Function â—˝  
Green's Function â—˝  
Complex Structure â—˝  
Surface Displacement â—˝  
Arbitrary Point â—˝  
Polar Coordinates â—˝  
Sh Wave â—˝  
Arbitrary Length â—˝  
Lining Structure â—˝  
Arbitrary Position

The problems of SH-wave scattering caused by a subsurface circular lining structure and a beeline crack with arbitrary length at an arbitrary position were studied by using the methods of Green's function, complex variables and multi-polar coordinates. A adaptive Green's function, an essential solution to the displacement field for the elastic space possessing circular lining structure while bearing out-plane harmonic lining loads at an arbitrary point, was constructed firstly, and then a crack was created using “crack-division”. Thus the expressions of displacement and stress were established while the crack and the inclusion both existed. Finally, we give some numerical examples to discuss the variety of the horizontal surface displacement in the case of different parameters.

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

2009 â—˝  
Vol 417-418 â—˝  
pp. 145-148 â—˝  
Author(s):  
Hong Liang Li â—˝  
Cun Zhang
Keyword(s):  
Green’S Function â—˝  
Green's Function â—˝  
Displacement Field â—˝  
Line Source â—˝  
Ground Surface â—˝  
Polar Coordinate System â—˝  
Algebraic Equations â—˝  
Free Condition â—˝  
Function Solution â—˝  
Cavity Structure

In mechanical engineering and modern municipal construction, shallow-buried cavity 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 cavities 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 the circle 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 circle 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. 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 cavities and the location of the line source force. Based on this solution, the problem of interaction of double circular cavities and a linear crack in semi-space can be investigated further.

The Dynamic Response of Circular Lining Structure and Multiple Cracks in Half Space under Impact Loading

2009 â—˝  
Vol 417-418 â—˝  
pp. 713-716
Author(s):  
Zai Lin Yang â—˝  
Mei Juan Xu â—˝  
Bai Tao Sun
Keyword(s):  
Dynamic Response â—˝  
Green’S Function â—˝  
Half Space â—˝  
Green's Function â—˝  
Impact Loading â—˝  
Complex Function â—˝  
Polar Coordinates â—˝  
Multiple Cracks â—˝  
Sh Wave â—˝  
Lining Structure

The dynamic response of a circular lining structure and multiple cracks in an elastic half space while bearing impact loading is studied in this paper. This problem can be seen as the problem of defending of blast. The methods of Green's function, complex function and multi-polar coordinates are used here. First, the scattering wave which satisfies the condition of stress free on the ground surface of half-space containing a shallow-embedded circular lining structure impacted by incident SH-wave is constructed based on the symmetry of SH-wave scattering and the method of multi-polar coordinates system. Then a crack or multiple cracks in any position and direction can be constructed by Green's function and means of crack-division in half space. Finally the displacement field and stress field are established in the case of coexistence of circular lining structure and cracks, and the expression of dynamic stress intensity factor (DSIF) at the tip of crack is given. The interaction of inclusion and two cracks is chosen as numerical examples finally. According to nu-merical examples, the influences of different parameters on DSIF are discussed.

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

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

scholarly journals Tectonomagnetic modeling based on the piezomagnetism: a review

10.4401/ag-3602 â—˝  
2001 â—˝  
Vol 44 (2) â—˝  
Author(s):  
Y. Sasai
Keyword(s):  
Green’S Function â—˝  
Green's Function â—˝  
Function Method â—˝  
Ground Surface â—˝  
3D Models â—˝  
Analytic Solutions â—˝  
Surface Integral â—˝  
Green’S Function Method â—˝  
Magnetic Poles â—˝  
2D And 3D

Development of tectonomagnetic modeling on the basis of the piezomagnetic effect is reviewed for the period since the early 1990's. First, the basic theory is briefly summarized, in which the representation theorem or the surface integral representation for the piezomagnetic potential and the Green's function method are presented. In the 1990's, several field observations in earthquakes and volcanoes were interpreted with the aid of analytic solutions based on the Green's function method. A general formula was developed for an inclined rectangular fault with strike-slip, dip-slip and tensile faulting. The surface integral method has been applied to 2D and 3D models, as well as to fault models in the inhomogeneously magnetized crust. When the magnetic field is measured within a bore hole, the effect of magnetic poles around the hole should be taken into account. As a result, tectonomagnetic signals are much enhanced in a bore hole compared with on the ground surface. Finally, piezomagnetic field changes associated with the Parkfield fault model are introduced and the new aspect of the model is discussed.

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

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

The Discrete Green's Function for Convective Heat Transfer—Part 2: Semi-Analytical Estimates of Boundary Layer Discrete Green's Function  

10.1115/1.4047516 â—˝  
2020 â—˝  
Vol 142 (10) â—˝  
Author(s):  
John K. Eaton â—˝  
Pedro M. Milani
Keyword(s):  
Boundary Layer â—˝  
Green’S Function â—˝  
Boundary Layers â—˝  
Green's Function â—˝  
Similarity Solutions â—˝  
Element Formulation â—˝  
Algebraic Equations â—˝  
Stagnation Line â—˝  
Thermal Boundary Conditions â—˝  
Discrete Green’S Function

Abstract This is the second paper in a set that defines the discrete Green's function (DGF). This paper focuses first on the turbulent boundary layer and presents two different methods to estimate the DGF. The long-element formulation defines the DGF with just two simple algebraic equations, but it is not quantitatively accurate for short element lengths. A short element correction is derived, but must be recalculated for each selection of flow parameters and element lengths. A similarity solution is derived that allows accurate estimates of the DGF diagonal elements for laminar boundary layers and for turbulent boundary layers discretized with short element lengths. To illustrate other methods to derive DGFs in more complex flows, a low-resolution DGF for laminar stagnation line boundary layers is determined using the skin-friction formulation combined with similarity solutions for two different thermal boundary conditions. Stagnation line flow is shown to be highly sensitive to the thermal boundary condition, and this can be analyzed effectively using the DGF.

The Green’S Function Solution of Right-Angle Plane Including a Semi-Cylindrical Canyon

2013 â—˝  
Vol 275-277 â—˝  
pp. 830-835
Author(s):  
Chun Xiang Zhao â—˝  
Hui Qi
Keyword(s):  
Free Boundary â—˝  
Green’S Function â—˝  
Wave Field â—˝  
Green's Function â—˝  
Complex Function â—˝  
Algebraic Equations â—˝  
Function Solution â—˝  
Out Of Plane â—˝  
Stress Boundary Conditions â—˝  
The Right

The Green’s function of a right-angle plane including semi-cylindrical canyon while bearing out-of-plane harmonic line source load on horizontal interface have been considered using the methods of complex function and image. Firstly, the wave field of right-angle plane was imaged half space, the scattering wave field, which satisfies the free stress boundary conditions of the right-angle plane on the vertical interface could be constructed. Secondly, a series of infinite algebraic equations be obtained to settle this problem by considering the stress free boundary condition of semi-cylindrical canyon. Finally, some examples for ground motion of a right-angle plane were given and discussed. Numerical results show that displacement of the horizontal surface is influenced by right-angle free boundary.

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