General solution of plane problem of piezoelectric media expressed by ?harmonic functions?

1997 ◽  
Vol 18 (8) ◽  
pp. 757-764 ◽  
Author(s):  
Ding Haojiang ◽  
Wang Guoqing ◽  
Chen Weiqiu
Keyword(s):  
General Solution ◽  
Plane Problem ◽  
Harmonic Functions ◽  
Piezoelectric Media

The Uniform Flexure of an Incomplete Tore

1949 ◽  
Vol 2 (4) ◽  
pp. 469
Author(s):  
W Freiberger ◽  
RCT Smith
Keyword(s):  
General Solution ◽  
Cross Section ◽  
Harmonic Functions ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Derivatives Of ◽  
Special Case

In this paper we discuss the flexure of an incomplete tore in the plane of its circular centre-line. We reduce the problem to the determination of two harmonic functions, subject to boundary conditions on the surface of the tore which involve the first two derivatives of the functions. We point out the relation of this solution to the general solution of three-dimensional elasticity problems. The special case of a narrow rectangular cross-section is solved exactly in Appendix II.

V.—On Whittaker's Solution of Laplace's Equation

1944 ◽  
Vol 62 (1) ◽  
pp. 31-36
Author(s):  
E. T. Copson
Keyword(s):  
Harmonic Function ◽  
General Solution ◽  
Arbitrary Function ◽  
Harmonic Functions ◽  
Singular Points ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Image Position

In 1902, Professor E. T. Whittaker gave a general solution of Laplace's equation in the formwhere f is an arbitrary function of the two variables. It appears that this is not the most general solution, since there are harmonic functions, such as r−1Q0(cos θ), which cannot be expressed in this form near the origin. The difficulty is naturally connected with the location of the singular points of the harmonic function. It seems therefore to be worth while considering afresh the conditions under which Whittaker's solution is valid.

scholarly journals Stress Analysis of Rotary Vibration of Rigid Friction Pile and Stress General Solution of Central Symmetry Plane Elastic Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Fu-yao Zhao ◽  
Er-xiang Song ◽  
Jun Yang
Keyword(s):  
General Solution ◽  
Stress Analysis ◽  
Plane Problem ◽  
Symmetry Plane ◽  
Angular Frequency ◽  
Central Symmetry ◽  
Friction Pile ◽  
Soil Interface ◽  
Rotary Vibration ◽  
Plane Elastic Problem

The rotary vibration of rigid friction pile can be seen approximately as a central symmetry plane problem in elasticity. The stress general solution of central symmetry plane problem in elasticity can be constructed by technique such as the Laurent expansion of the volume force. This solution has some decoupling, generalized, and convergent properties, and it can be used in stress analysis of the rotary vibration of pile. The analysis results show that the maximum value of displacement will not occur at the edge of the pile and the assumption that pile cross section remains unchanged is no longer applicable, if the value of one dimensionless quantity, reflecting the angular frequency of the rotation, radius, and material properties of the pile, is larger than 1.84. Once the rotary vibration of rigid friction pile happens, the pile may lose its bearing capacity under the comprehensive effect of normal and shear stress of the pile-soil interface and it will be very difficult to recover.

Notes on problems in hexagonal aeolotropic materials

1951 ◽  
Vol 47 (2) ◽  
pp. 401-409 ◽  
Author(s):  
R. T. Shield
Keyword(s):  
General Solution ◽  
Stress Function ◽  
Harmonic Functions ◽  
Present Note ◽  
Three Dimensional ◽  
Stress Distributions ◽  
Axially Symmetric ◽  
Isotropic Materials ◽  
Stress Functions ◽  
Single Stress

Three-dimensional stress distributions in hexagonal aeolotropic materials have recently been considered by Elliott(1, 2), who obtained a general solution of the elastic equations of equilibrium in terms of two ‘harmonic’ functions, or, in the case of axially symmetric stress distributions, in terms of a single stress function. These stress functions are analogous to the stress functions employed to define stress systems in isotropic materials, and in the present note further problems in hexagonal aeolotropic media are solved, the method in each case being similar to that used for the corresponding problem in isotropic materials. Because of this similarity detailed explanations are unnecessary and only the essential steps in the working are given below.

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