Equivalence between the Trefftz method and the method of fundamental solution for the annular Green's function using the addition theorem and image concept

In this paper, the method of Green’s function is used to investigate the problem of dynamic stress concentration of circular lining and interior linear crack impacted by incident SH-wave. The train of thought for this problem is that: Firstly, a Green’s function is constructed for the problem, which is a fundamental solution of displacement field for an elastic space possessing a circular lining while bearing out-of-plane harmonic line source force at any point in the lining. In terms of the solution of SH-wave’s scattering by an elastic space with a circular lining, anti-plane stresses which are the same in quantity but opposite in direction to those mentioned before, are loaded at the region where the crack existent actually, we called this process “crack-division”. Finally, the expressions of the displacement and stress are given when the lining and the crack exist at the same time. Then, by using the expressions, some example is provided to show the effect of crack on the dynamic stress concentration around circular lining.

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Covariance Operator = Green’s Function = Resolvent Kernel = Euclidean Propagator = Fundamental Solution

Quantum Physics ◽

10.1007/978-1-4684-0121-9_7 ◽

1981 ◽

pp. 123-143 ◽

Cited By ~ 1

Author(s):

James Glimm ◽

Arthur Jaffe

Keyword(s):

Fundamental Solution ◽

Green’S Function ◽

Green's Function ◽

Covariance Operator ◽

Resolvent Kernel

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On the condition number of integral equations in linear elasticity using the modified Green's function

The ANZIAM Journal ◽

10.1017/s1446181100008129 ◽

2003 ◽

Vol 44 (3) ◽

pp. 431-446

Author(s):

E. Argyropoulos ◽

D. Gintides ◽

K. Kiriaki

Keyword(s):

Fundamental Solution ◽

Green’S Function ◽

Optimal Condition ◽

Integral Equations ◽

Green's Function ◽

Condition Number ◽

Linear Elasticity ◽

Boundary Integral Equations ◽

Boundary Integral ◽

Function Technique

AbstractIn this work the modified Green's function technique for an exterior Dirichlet and Neumann problem in linear elasticity is investigated. We introduce a modification of the fundamental solution in order to remove the lack of uniqueness for the solution of the boundary integral equations describing the problems, and to simultaneously minimise their condition number. In view of this procedure the cases of the sphere and perturbations of the sphere are examined. Numerical results that demonstrate the effect of increasing the number of coefficients in the modification on the optimal condition number are also presented.

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On Fundamental Solutions and Green’s Functions in the Theory of Elastic Plates

Journal of Applied Mechanics ◽

10.1115/1.3625022 ◽

1966 ◽

Vol 33 (1) ◽

pp. 31-38 ◽

Cited By ~ 7

Author(s):

A. Kalnins

Keyword(s):

Green’S Function ◽

Green's Function ◽

Bessel Functions ◽

Arbitrary Point ◽

Fundamental Solutions ◽

Addition Theorem ◽

Reciprocal Theorem ◽

Elastic Plates ◽

Concentrated Load ◽

The Reciprocal Theorem

This paper is concerned with fundamental solutions of static and dynamic linear inextensional theories of thin elastic plates. It is shown that the appropriate conditions which a fundamental singularity must satisfy at the pole follow from the requirement that the reciprocal theorem is satisfied everywhere in the region occupied by the plate. Furthermore, dynamic Green’s function for a plate bounded by two concentric circular boundaries is derived by means of the addition theorem of Bessel functions. The derived Green’s function represents the response of the plate to a harmonically oscillating normal concentrated load situated at an arbitrary point on the plate.

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Green's functions and Riemann's method

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500008981 ◽

1965 ◽

Vol 14 (4) ◽

pp. 293-302 ◽

Cited By ~ 3

Author(s):

A. G. Mackie

Keyword(s):

Fundamental Solution ◽

Green’S Function ◽

Boundary Value Problems ◽

Green's Function ◽

Elliptic Equations ◽

Green's Functions ◽

Green’S Functions ◽

Hyperbolic Equations ◽

Boundary Value ◽

Auxiliary Function

Methods for solving boundary value problems in linear, second order, partial differential equations in two variables tend to be somewhat rigidly partitioned in some of the standard text-books. Problems for elliptic equations are sometimes solved by finding the fundamental solution which is defined as a solution with a given singularity at a certain point. Another approach is by way of Green's functions which are usually defined as solutions of the original homogeneous equations now made inhomogeneous by the introduction of adelta function on the right hand side. The Green's function coincides with the fundamental solution for elliptic equations but exhibits a totally different type of singularity for parabolic or hyperbolic equations. Boundary value problems for hyperbolic equations can often by solved by Riemann's method which depends on the existence of an auxiliary function called the Riemann or sometimes the Riemann-Green function. The main object of this paper is to show the close relationship between Riemann's method and the method of Green's functions. This not only serves to unify different methods of solution of boundary value problems but also provides an additional method of determining Riemann functions for given hyperbolic equations. Before establishing these relationships we shall survey the general approach to boundary value problems through the use of the Green's function.

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