Punch Problem for Planar Anisotropic Elastic Half-Plane

2011 ◽  
Vol 27 (2) ◽  
pp. 215-226 ◽  
Author(s):  
R.-L. Lin
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Half Plane ◽  
Green's Function ◽  
Boundary Integral ◽  
Fredholm Integral Equations ◽  
Surface Traction ◽  
Rigid Punch ◽  
Anisotropic Elastic ◽  
Punch Problem

ABSTRACTThe two dimensional punch problem for planar anisotropic elastic half-plane is revisited using the Lekhnitskii's formulation with aid of the Fourier transform and boundary integral equation. Four different conditions of contact problem for the rigid punch are analyzed in this study. From the combination of surface Green's function of half-plane and Hooke's law of anisotropic material, a set of Fredholm integral equations are obtained for mixed boundary value problems. After solving the integral equation according to specified contact condition, the explicit distributions of surface traction under the punch are obtained in closed-form. From the surface traction and Green's function of anisotropic half-plane, the full-field solutions of stresses are constructed. Numerical calculations of surface traction under the rigid punch are presented base on the analysis and are discussed in detail.

Integral Equation Approach for Computing Green’s Function on Doubly Connected Regions via the Generalized Neumann Kernel

2014 ◽  
Vol 71 (1) ◽  
Author(s):  
Siti Zulaiha Aspon ◽  
Ali Hassan Mohamed Murid ◽  
Mohamed M. S. Nasser ◽  
Hamisan Rahmat
Keyword(s):  
Integral Equation ◽  
Dirichlet Problem ◽  
Linear System ◽  
Green’S Function ◽  
Green's Function ◽  
Boundary Integral ◽  
Generalized Neumann Kernel ◽  
Integral Equation Approach ◽  
Gauss Elimination ◽  
Neumann Kernel

This research is about computing the Green’s function on doubly connected regions by using the method of boundary integral equation. The method depends on solving a Dirichlet problem. The Dirichlet problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. The method for solving this integral equation is by using the Nystrӧm method with trapezoidal rule to discretize it to a linear system. The linear system is then solved by the Gauss elimination method. Mathematica plots of Green’s functions for several test regions are also presented.

scholarly journals Concentration of dynamic stresses in an elastic space with twoperiodic array of elliptical cracks

2019 ◽  
pp. 18-25
Author(s):  
Igor Zhbadynskyi
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Boundary Integral Equation ◽  
Green's Function ◽  
Scattering Problem ◽  
Mapping Method ◽  
Boundary Integral ◽  
Elastic Space ◽  
Wide Range ◽  
Elliptical Cracks

Normal incidence of the plane time-harmonic longitudinal wave on double-periodic array of coplanar elliptical cracks, which are located in 3D infinite elastic space is considered. Corresponding symmetric wave scattering problem is reduced to a boundary integral equation for the displacement jump across the crack surfaces in a unit cell by means of periodic Green’s function, which is presented in the form of Fourier integrals. A regularization technique for this Green’s function involving special lattice sums in closed forms is adopted, which allows its effective calculation in a wide range of wave numbers. The boundary integral equation is correctly solved by using the mapping method. The frequency dependencies of mode-I stress intensity factor in the vicinity of the crack front points for periodic distances in the system of elliptical cracks are revealed.

scholarly journals Integral equation with the generalized neumann kernel for computing green’s function on simply connected regions

2014 ◽  
Vol 9 (3) ◽  
Author(s):  
Ali H. M. Murid ◽  
Mohmed M. A. Alagele ◽  
Mohamed M. S. Nasser
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Gaussian Elimination ◽  
Linear Equations ◽  
Boundary Integral ◽  
System Of Linear Equations ◽  
Generalized Neumann Kernel ◽  
Simply Connected ◽  
Neumann Kernel

This research is about computing the Green’s functions on simply connected regions by using the method of boundary integral equation. The method depends on solving a Dirichlet problem using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. The numerical method for solving this integral equation is the Nystrӧm method with trapezoidal rule which leads to a system of linear equations. The linear system is then solved by the Gaussian elimination method. Mathematica plot of Green’s function for atest region is also presented.

An Integral Equation Method for Free Vibration of Circular Plate with Concentrated Mass at Arbitrary Positions

2011 ◽  
Vol 255-260 ◽  
pp. 1830-1835 ◽  
Author(s):  
Gang Cheng ◽  
Quan Cheng ◽  
Wei Dong Wang
Keyword(s):  
Integral Equation ◽  
Eigenvalue Problem ◽  
Free Vibration ◽  
Green’S Function ◽  
Circular Plate ◽  
Green's Function ◽  
Integral Equation Method ◽  
Free Vibrations ◽  
Equation Method ◽  
Concentrated Mass

The paper concerns on the free vibrations of circular plate with arbitrary number of the mounted masses at arbitrary positions by using the integral equation method. A set of complete systems of orthogonal functions, which is constructed by Bessel functions of the first kind, is used to construct the Green's function of circular plates firstly. Then the eigenvalue problem of free vibration of circular plate carrying oscillators and elastic supports at arbitrary positions is transformed into the problem of integral equation by using the superposition theorem and the physical meaning of the Green’s function. And then the eigenvalue problem of integral equation is transformed into a standard eigenvalue problem of a matrix with infinite order. Numerical examples are presented.

scholarly journals Unified double- and single-sided homogeneous Green’s function representations

2016 ◽  
Vol 472 (2190) ◽  
pp. 20160162 ◽  
Author(s):  
Kees Wapenaar ◽  
Joost van der Neut ◽  
Evert Slob
Keyword(s):  
Green’S Function ◽  
Multiple Scattering ◽  
Green's Function ◽  
Time Reversal ◽  
Wave Theory ◽  
Boundary Integral ◽  
Open Boundary ◽  
Quantum Mechanical ◽  
Holographic Imaging ◽  
Scattering Time

In wave theory, the homogeneous Green’s function consists of the impulse response to a point source, minus its time-reversal. It can be represented by a closed boundary integral. In many practical situations, the closed boundary integral needs to be approximated by an open boundary integral because the medium of interest is often accessible from one side only. The inherent approximations are acceptable as long as the effects of multiple scattering are negligible. However, in case of strongly inhomogeneous media, the effects of multiple scattering can be severe. We derive double- and single-sided homogeneous Green’s function representations. The single-sided representation applies to situations where the medium can be accessed from one side only. It correctly handles multiple scattering. It employs a focusing function instead of the backward propagating Green’s function in the classical (double-sided) representation. When reflection measurements are available at the accessible boundary of the medium, the focusing function can be retrieved from these measurements. Throughout the paper, we use a unified notation which applies to acoustic, quantum-mechanical, electromagnetic and elastodynamic waves. We foresee many interesting applications of the unified single-sided homogeneous Green’s function representation in holographic imaging and inverse scattering, time-reversed wave field propagation and interferometric Green’s function retrieval.

Theory of Electron Diffraction by Crystals

1967 ◽  
Vol 22 (4) ◽  
pp. 422-431 ◽  
Author(s):  
Kyozaburo Kambe
Keyword(s):  
Integral Equation ◽  
Electron Diffraction ◽  
General Theory ◽  
Green’S Function ◽  
Green's Function ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Two Dimensions ◽  
The Third ◽  
Third Dimension

A general theory of electron diffraction by crystals is developed. The crystals are assumed to be infinitely extended in two dimensions and finite in the third dimension. For the scattering problem by this structure two-dimensionally expanded forms of GREEN’S function and integral equation are at first derived, and combined in single three-dimensional forms. EWALD’S method is applied to sum up the series for GREEN’S function.

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