Magneto-electro-thermoelastic Problems: Fundamental Solutions and Green’s Function

A boundary element method (BEM) for bimaterial domains consisting of two isotropic solids bonded perfectly along the straight interface will be developed. We follow the physical interpretation of Somiglianaâ€™s identity to represent the displacement in the bimaterial domain by the continuous distributions of the line forces and dislocation dipoles over its boundary. The fundamental solutions used are the Greenâ€™s functions for the line force and the dislocation dipole that satisfy the traction and displacement continuity across the interface of two domains. There is no need to model the interface because the required continuity conditions there are automatically satisfied by the Greenâ€™s functions. The BEM will be applied to study the edge stress concentration of the bimaterial solids. We calculate the singular stress distribution at the free edge of the interface for various bimaterial configurations and loadings, in particular, for the domain consisting of thin coating over the substratum. Since the Green's function BEM does not require the boundary elements on the interface, it can handle the edge singularity on the interface accurately even for extremely thin coatings. The BEM developed here is not limited to the edge stress concentration problems and can be applied to a broad range of the bimaterial domain problems effectively.

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Equivalence between the Trefftz method and the method of fundamental solutions for Green’s function of concentric spheres using the addition theorem and image concept

Mesh Reduction Methods ◽

10.2495/be090031 ◽

2009 ◽

Cited By ~ 3

Author(s):

J. T. Chen ◽

H. C. Shieh ◽

J. J. Tsai ◽

J. W. Lee

Keyword(s):

Green’S Function ◽

Green's Function ◽

Fundamental Solutions ◽

Addition Theorem ◽

Method Of Fundamental Solutions ◽

Trefftz Method ◽

Concentric Spheres

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Exact Solutions for the Analysis of General Elastically Restrained Nonuniform Beams

Journal of Applied Mechanics ◽

10.1115/1.2899490 ◽

1992 ◽

Vol 59 (2S) ◽

pp. S205-S212 ◽

Cited By ~ 31

Author(s):

Sen Yung Lee ◽

Yee Hsiung Kuo

Keyword(s):

Exact Solutions ◽

Green’S Function ◽

Green's Function ◽

Vibrational Analysis ◽

Elastic Stability ◽

Fundamental Solutions ◽

Bending Rigidity ◽

Euler Beam ◽

Varying Coefficients ◽

Elastically Restrained

The exact solutions for the problems governed by a general self-adjoint fourth-order nonhomogeneous ordinary differential equation with arbitrarily polynomial varying coefficients and general elastic boundary conditions are derived in Green’s function form. To illustrate the analysis, the static deflection and dynamic analysis of a general eiastically end restrained Bernoulli-Euler beam with polynomial varying bending rigidity, applied axial and force, and elastic foundation modulus along the beam, subjected to an arbitrary transverse force are presented. The Green’s function is concisely expressed in terms of the four normalized fundamental solutions of the system and these fundamental solutions are given in power series forms. The characteristic equations for elastic stability and free vibrational analysis of the beam can be obtained by setting the denominator of the corresponding Green’s function equal to zero. Finally, examples are given to illustrate the accuracy and efficiency of the analysis.

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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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A Green's Function for the Domain Bounded by Nonconcentric Spheres

Journal of Applied Mechanics ◽

10.1115/1.4007071 ◽

2012 ◽

Vol 80 (1) ◽

Cited By ~ 1

Author(s):

Jeng-Tzong Chen ◽

Jia-Wei Lee ◽

Hung-Chih Shieh

Keyword(s):

Green’S Function ◽

Green's Function ◽

Fundamental Solutions ◽

Dirichlet Boundary Conditions ◽

Method Of Fundamental Solutions ◽

Circular Cylinders ◽

Image Method ◽

Eccentric Annulus ◽

Function Solution ◽

Good Agreement

The main result is the analytical derivation of Green's function for the domain bounded by nonconcentric spheres in terms of bispherical coordinates. Both surfaces, inner and outer boundaries, are specified by the Dirichlet boundary conditions. This work can be seen as an extension study for the Green's function of eccentric annulus derived by Heyda (1959, “A Green's Function Solution for the Case of Laminar Incompressible Flow Between Non-Concentric Circular Cylinders,” J. Franklin Inst., 267, pp. 25–34). To verify the solution, a semianalytical solution using the image method and a numerical solution using the method of fundamental solutions (MFS) are utilized for comparisons. Good agreement is made.

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