Three-dimensional stress distributions in hexagonal aeolotropic crystals

1948 ◽  
Vol 44 (4) ◽  
pp. 522-533 ◽  
Author(s):  
H. A. Elliott ◽  
N. F. Mott
Keyword(s):  
Harmonic Functions ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Quadratic Equation ◽  
Stress Distributions ◽  
Third Order ◽  
Isotropic Solid ◽  
Single Stress ◽  
Image Position ◽  
Special Case

The conditions for equilibrium in an elastically stressed hexagonal aeolotropic medium (transversely isotropic) are formulated, and solutions are found in terms of two ‘harmonic’ functions ø1, ø2, which are solutions ofν1, ν2 being the roots of a certain quadratic equation.It is also shown that in the case of axially symmetrical stress systems the solution may be expressed in terms of the third-order differential coefficients of a single stress function Φ.The solutions for an isotropic medium may be deduced as a special case.The problems of nuclei of strain in such a hexagonal solid are solved, and the results for zinc and magnesium contrasted with those for an isotropic solid.

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.

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.

Properties of Harmonic Functions of Three Real Variables Given by Bergman-Whittaker Operators

1963 ◽  
Vol 15 ◽  
pp. 157-168 ◽  
Author(s):  
Josephine Mitchell
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Rectifiable Curve ◽  
Image Position

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and setThe Bergman-Whittaker operator defined by

A Point Heat Source on the Surface of a Semi-Infinite Transversely Isotropic Piezothermoelastic Material

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
Peng-Fei Hou ◽  
Wei Luo ◽  
Andrew Y. T. Leung
Keyword(s):  
Heat Source ◽  
Cadmium Selenide ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Numerical Results ◽  
Point Heat Source ◽  
Elementary Functions ◽  
General Solutions ◽  
Coupled Field

We use the compact harmonic general solutions of transversely isotropic piezothermoelastic materials to construct the three-dimensional Green’s function of a steady point heat source on the surface of a semi-infinite transversely isotropic piezothermoelastic material by four newly introduced harmonic functions. All components of the coupled field are expressed in terms of elementary functions and are convenient to use. Numerical results for cadmium selenide are given graphically by contours.

Fatigue Failure Criteria for Unidirectional Fiber Composites

1981 ◽  
Vol 48 (4) ◽  
pp. 846-852 ◽  
Author(s):  
Z. Hashin
Keyword(s):  
Fatigue Failure ◽  
Failure Modes ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Failure Criteria ◽  
Fiber Composites ◽  
Cyclic Stress ◽  
Unidirectional Fiber ◽  
Single Stress ◽  
Material Information

Three-dimensional fatigue failure criteria for unidirectional fiber composites under states of cyclic stress are established in terms of quadratic stress polynomials which are expressed in terms of the transversely isotropic invariants of the cyclic stress. Two distinct fatigue failure modes, fiber mode, and matrix mode, are modeled separately. Material information needed for the failure criteria are the S-N curves for single stress components. A preliminary approach to incorporate scatter into the failure criteria is presented.

Approximate Elasticity Solution for Orthotopic Cylinder Under Hydrostatic Pressure and Band Loads

1970 ◽  
Vol 37 (1) ◽  
pp. 101-108 ◽  
Author(s):  
A. P. Misovec ◽  
J. Kempner
Keyword(s):  
Approximate Solution ◽  
Shell Theory ◽  
Good Accuracy ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
External Pressure ◽  
Theory Of Elasticity ◽  
Numerical Comparison ◽  
Axial Loads ◽  
Special Case

An approximate solution to the Navier equations of the three-dimensional theory of elasticity for an axisymmetric orthotopic circular cylinder subjected to internal and external pressure, axial loads, and closely spaced periodic radial loads is developed. Numerical comparison with the exact solution for the special case of a transversely isotropic cylinder subjected to periodic band loads shows that very good accuracy is obtainable. When the results of the approximate solution are compared with previously obtained results of a Flu¨gge-type shell solution of a ring-reinforced orthotropic cylinder, it is found that the shell theory gives fairly accurate representations of the deformations and stresses except in the neighborhood of discontinuous loads. The addition of transverse shear deformations does not improve the accuracy of the shell solution.

The Stress Field Caused by a Circular Cylindrical Inclusion in a Transversely Isotropic Elastic Solid

2003 ◽  
Vol 70 (6) ◽  
pp. 825-831 ◽  
Author(s):  
H. Hasegawa ◽  
M. Kisaki
Keyword(s):  
Stress Field ◽  
Strain Energy ◽  
Elastic Solid ◽  
Transversely Isotropic ◽  
Cylindrical Inclusion ◽  
Stress Distributions ◽  
Axisymmetric Stress ◽  
Displacement Fields ◽  
Isotropic Solid ◽  
Stress And Displacement Fields

Exact solutions are presented in closed form for the axisymmetric stress and displacement fields caused by a circular solid cylindrical inclusion with uniform eigenstrain in a transversely isotropic elastic solid. This is an extension of a previous paper for an isotropic elastic solid to a transversely isotropic solid. The strain energy is also shown. The method of Green’s functions is used. The numerical results for stress distributions are compared with those for an isotropic elastic solid.

scholarly journals 4. Note on Professor Tait's “Quaternion Path” to Determinants of the Third Order

1869 ◽  
Vol 6 ◽  
pp. 121-125
Author(s):  
Hugh Martin
Keyword(s):  
Third Order ◽  
The Third ◽  
Image Position ◽  
Special Case

I have read with much interest Professor Tait's “Note on Determinants of the Third Order” in the Proceedings of this Session (pp. 59–61), and admire the method of discovering new properties of Determinants. I am not sure, however, that the properties, when discovered, are more difficult of proof by Determinant methods, and I venture to submit the following as simple and elementary:—The first property, namely,is true under greater generality, and the Determinant proof is the same as for the special case.

scholarly journals On the Solutions of Three-Dimensional Rational Difference Equation Systems

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
H. S. Alayachi ◽  
A. Q. Khan ◽  
M. S. M. Noorani
Keyword(s):  
Difference Equation ◽  
Mathematical Program ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Boundedness Of Solutions ◽  
Real Numbers ◽  
Third Order ◽  
Numerical Examples ◽  
Rational Difference Equation ◽  
Special Case

In this paper, we are interested in a technique for solving some nonlinear rational systems of difference equations of third order, in three-dimensional case as a special case of the following system: x n + 1 = y n z n − 1 / y n ± x n − 2 , y n + 1 = z n x n − 1 / z n ± y n − 2 ,  and  z n + 1 = x n y n − 1 / x n ± z n − 2 with initial conditions x − 2 , x − 1 , x 0 , y − 2 , y − 1 , y 0 , z − 2 , z − 1 ,  and  z 0 are nonzero real numbers. Moreover, we study some behavior of the systems such as the boundedness of solutions for such systems. Finally, we present some numerical examples by giving some numerical values for the initial values of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program MATLAB to confirm the obtained results.

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