Papers by Keyword: Green's Function

Abstract: The Green quasifunction method (GQM) is applied to solve the bending problem of clamped orthotropic thin plates with trapezoidal boundary shape on Winkler foundation. Firstly the governing differential equation of the problem is reduced to the boundary value problem of the biharmonic operator, and then it is reduced to the Fredholm integral equation of the second kind by Green’s formula. A Green quasifunction is established by using the fundamental solution and the boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The comparison with ANSYS finite element solution shows good agreement. The proposed method is a novel and effective mathematical one.

Abstract: A new method-mixed boundary grid method (FBGM) is proposed to analyze the free vibration of cantilever plates with variable thickness. The fundamental differential equations of the plate are established for the bending problem. By transforming these equations into integral equations in a small area and using the trapezoidal rule of the approximate numerical integration, the solution of these equations is obtained and chosen as Green function to obtain the characteristic equation of the free vibration. The accuracy of the numerical results for the natural frequency parameter calculated by the proposed method is investigated.

Abstract: The Green quasifunction method(GQM) is employed to solve the bending problem of clamped orthotropic thin plates with trapezoidal boundary shape. Firstly the governing differential equation of the problem is reduced to the boundary value problem of the biharmonic operator, and then it is reduced to the Fredholm integral equation of the second kind by Green’s formula. A Green quasifunction is established by using the fundamental solution and the boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The irregularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. A numerical example demonstrates the feasibility and efficiency of the proposed method, and it is a novel mathematical method.

Abstract: A new soil model named as 3d combined-layer soil model is firstly proposed. According to the concept of image, the Green function of this soil structure is derived using complex image method. Based on it, the surface potential distribution (SPD) of different soil structure configuration is calculated, moreover, the influence of soil parameters on SPD is analyzed. It can be seen from the results that the variations of the soil parameters such as soil thickness and resistivity can bring different influence to the SPD of the soil, and it is related not only to the distances, but also the angle for the grounding electrode. Therefore, the 3DSM may be more accurate for the calculation of SPD.

Abstract: 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.

Abstract: This paper concerns the determination of qualitative properties of linear vibrational systems, in particular for a single branch structure consisting of a pinned beam-rod system. First, we establish the characteristic equations satisfied by the Green’s function for this structure. The Green’s functions corresponding to support conditions where the left end of the beam was pinned-end are deduced by adopting the direct integral method. Using the theory of oscillation kernels established by Gantmakher and Krein, oscillatory properties of the Green's function for the beam-rod system are proved. Furthermore, four oscillation properties associated with frequencies and mode functions for the system are given.

Abstract: 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.

Abstract: 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.

Abstract: Annular plates are commonly found in the fields of engineering. The present study is concerned with the integral equation method for the free vibrations of annular plates with elastic supports. A set of complete systems of orthogonal functions, which is constructed by Bessel functions of the first and the second kind is used to construct the Green's function of annular plates. The eigenvalue problem of free vibration of annular plates with Elastic Supports is transformed into the eigenvalue problem of integral equation. And then, the problem of integral equation is transformed into a standard eigenvalue problem of a matrix with infinite order. Numerical example shows the significant advantages of the present method.

Abstract: 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.