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

1966 ◽  
Vol 33 (1) ◽  
pp. 31-38 ◽  
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.

Derivation of Green’s function using addition theorem

2009 ◽  
Vol 36 (3) ◽  
pp. 351-363 ◽  
Author(s):  
J.T. Chen ◽  
K.H. Chou ◽  
S.K. Kao
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Addition Theorem

On the accuracy of the addition theorem for a scalar Green's function used in the FMM

2001 ◽  
Vol 31 (6) ◽  
pp. 439-442 ◽  
Author(s):  
Jian-Ying Li ◽  
Le-Wei Li ◽  
Ban-Leong Ooi ◽  
Pang-Shyan Kooi ◽  
Mook-Seng Leong
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Addition Theorem

Elastic Fields of Quantum Dots in Multilayered Semiconductors: A Novel Green’s Function Approach

2003 ◽  
Vol 70 (2) ◽  
pp. 161-168 ◽  
Author(s):  
B. Yang ◽  
E. Pan
Keyword(s):  
Quantum Dots ◽  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Point Sources ◽  
Reciprocal Theorem ◽  
Wetting Layer ◽  
Volume Integral ◽  
Elastic Fields

We present an efficient and accurate continuum-mechanics approach to predict the elastic fields in multilayered semiconductors due to buried quantum dots (QDs). Our approach is based on a novel Green’s function solution in anisotropic and linearly elastic multilayers, derived within the framework of generalized Stroh formalism and Fourier transforms, in conjunction with the Betti’s reciprocal theorem. By using this approach, the induced elastic fields due to QDs with general misfit strains are expressed as a volume integral over the QDs domains. For QDs with uniform misfit strains, the volume integral involved is reduced to a surface integral over the QDs boundaries. Further, for QDs that can be modeled as point sources, the induced elastic fields are then derived as a sum of the point-force Green’s functions. In the last case, the solution of the QD-induced elastic field is analytical, involving no numerical integration, except for the evaluation of the Green’s functions. As numerical examples, we have studied a multilayered semiconductor system of QDs made of alternating GaAs-spacer and InAs-wetting layers on a GaAs substrate, plus a freshly deposited InAs-wetting layer on the top. The effects of vertical and horizontal arrays of QDs and of thickness of the top wetting layer on the QD-induced elastic fields are examined and some new features are observed that may be of interest to the designers of semiconductor QD superlattices.

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

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

scholarly journals The Green's Function BEM for Bimaterial Solids Applied to Edge Stress Concentration Problems

2007 ◽  
Vol 1 (2) ◽  
Author(s):  
M. Denda
Keyword(s):  
Stress Concentration ◽  
Green’S Function ◽  
Green's Function ◽  
Free Edge ◽  
Fundamental Solutions ◽  
Singular Stress ◽  
Thin Coatings ◽  
Edge Stress ◽  
Line Force ◽  
Isotropic Solids

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 Somigliana’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 Green’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 Green’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.

An Arbitrary Tangential Load Underneath a Smooth Circular Punch

1990 ◽  
Vol 57 (3) ◽  
pp. 596-599 ◽  
Author(s):  
V. I. Fabrikant
Keyword(s):  
Exact Solution ◽  
Transversely Isotropic ◽  
Elastic Half Space ◽  
Reciprocal Theorem ◽  
Tangential Load ◽  
Concentrated Load ◽  
Circular Punch ◽  
Angular Displacements ◽  
The Reciprocal Theorem ◽  
First Time

The problem of a smooth circular punch penetrating a transversely isotropic elastic half space and interacting with an arbitrarily located tangential concentrated load is considered. For the first time, a closed-form exact solution is obtained for the stress distribution under the punch as well as for the linear and angular displacements of the punch. The solution is based on the results previously obtained by the author and combined with the reciprocal theorem. A numerical example is presented as an illustration.

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