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.

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.

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.

Three-Dimensional Stress Singularities at Conical Notches and Inclusions in Transversely Isotropic Materials

1986 ◽  
Vol 53 (1) ◽  
pp. 89-96 ◽  
Author(s):  
Nihal Somaratna ◽  
T. C. T. Ting
Keyword(s):  
General Solution ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Solution Procedure ◽  
Legendre Functions ◽  
Spherical Coordinates ◽  
Stress Singularities ◽  
Isotropic Materials ◽  
Transversely Isotropic Materials

This study examines analytically the possible existence of stress singularities of the form σ = ρδf(θ,φ) at the apex of axisymmetric conical boundaries in transversely isotropic materials. (ρ, θ, φ) refer to spherical coordinates with the origin at the apex. The problems of one conical boundary and of two conical boundaries with a common apex are considered. The boundaries are either rigidly clamped or traction free. Separation of variables enables the general solution to be expressed in terms of Legendre functions of the first and second kind. Imposition of boundary conditions leads to an eigenequation that would determine possible values of δ. The degenerate case that arises when the eigenvalues of the elasiticity constants are identical is also discussed. Isotropic materials constitute only a particular case in this class of degenerate materials and previously reported eigenequations corresponding to isotropic materials are shown to be recoverable from the present results. Numerical results corresponding to a few selected cases are also presented to illustrate the solution procedure.

The effect of a hole on certain stress distributions in aeolotropic and isotropic plates

1944 ◽  
Vol 40 (2) ◽  
pp. 172-188 ◽  
Author(s):  
S. Holgate
Keyword(s):  
Circular Hole ◽  
Infinite Plate ◽  
Complex Stress ◽  
Stress Distributions ◽  
Finite Form ◽  
Isotropic Materials ◽  
Isotropic Plates ◽  
General Manner ◽  
Alternative Method ◽  
Stress Functions

1. The problems of stress distributions in an infinite plate containing a circular hole were solved in a general manner by Bickley(1) for isotropic materials. An alternative method, using complex stress functions, was later given by Green(5) and extended so as to apply to aeolotropic materials. In the present paper Green's method is employed to determine the stress distributions that arise when the stresses are produced by isolated forces acting at points on, or near to, the edge of the circular hole, and though some of the solutions are cumbersome they are all obtained in finite form.

scholarly journals IX. Plane stress and plane strain in bipolar co-ordinates

1921 ◽  
Vol 221 (582-593) ◽  
pp. 265-293 ◽  
Keyword(s):  
Stress Function ◽  
Three Dimensional ◽  
Linear Partial Differential Equation ◽  
Elastic Solid ◽  
Great Difficulty ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Applied Forces ◽  
Special Solutions ◽  
Single Stress

The problem of the equilibrium of an elastic solid under given applied forces is one of great difficulty and one which has attracted the attention of most of the great applied Mathematicians since the time of Euler. Unlike the kindred problems of hydrodynamics and electrostatics, it seems to be a branch of mathematical physics in which knowledge comes by the patient accumulation of special solutions rather than by the establishment of great general propositions. Nevertheless, the many and varied applications of this subject to practical affairs make it very desirable that these special solutions should be investigated, not only because of their intrinsic importance but also for the light which they often throw on the general problem. One of the most powerful methods of the mathematical physicist in the face of recalcitrant differential equations is to simplify his problem by reducing it to two dimensions. This simplification can only imperfectly be reproduced in the Nature of our three-dimensional world, but, in default of more general methods, it provides an invaluable weapon. It was shown by Airy that in the two-dimensional case the, stresses may be derived by partial differentiations from a single stress function, and it was shown later that, in the absence of body forces, this stress function satisfies the linear partial differential equation of the fourth order ∇ 4 X = 0, where ∇ 4 = ∇ 2 . ∇ 2 , and ∇ 2 is the two-dimensional Laplacian ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 .

scholarly journals FIRST PRINCIPLES DERIVATION OF A STRESS FUNCTION FOR AXIALLY SYMMETRIC ELASTICITY PROBLEMS, AND APPLICATION TO BOUSSINESQ PROBLEM

2017 ◽  
Vol 36 (3) ◽  
pp. 767-772
Author(s):  
CC Ike
Keyword(s):  
First Principles ◽  
Stress Function ◽  
Three Dimensional ◽  
Homogeneous Medium ◽  
Solution Process ◽  
Point Load ◽  
Axially Symmetric ◽  
Boussinesq Problem ◽  
Linear Elastic ◽  
Elasticity Problems

In this work, a stress function is derived from first principles to describe the behaviour of three dimensional axially symmetric elasticity problems involving linear elastic, isotropic homogeneous materials. In the process, the fifteen governing partial differential equations of linear isotropic elasticity were reduced to the solution of the biharmonic problem involving the stress function. thus simplifying the solution process. The stress function derived was found to be identical with the Love stress function. The stress function was then applied to solve the axially symmetric problem of finding the stress fields, strain fields and displacement fields in the semi-infinite linear elastic, isotropic homogeneous medium subject to a point load P acting at the origin of coordinates also called the Boussinesq problem. The results obtained in this study for the stresses and displacements were exactly identical with those from literature, as obtained by Boussinesq.http://dx.doi.org/10.4314/njt.v36i3.15

scholarly journals Displacements and Stress Functions of Straight Dislocations and Line Forces in Anisotropic Elasticity: A New Derivation and Its Relation to the Integral Formalism

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1721
Author(s):  
Markus Lazar
Keyword(s):  
Plane Strain ◽  
Stress Function ◽  
Function Fields ◽  
Three Dimensional ◽  
Anisotropic Elasticity ◽  
Green Tensor ◽  
Two Dimensional ◽  
Integral Formalism ◽  
Stress Functions ◽  
Generalized Plane Strain

The displacement and stress function fields of straight dislocations and lines forces are derived based on three-dimensional anisotropic incompatible elasticity. Using the two-dimensional anisotropic Green tensor of generalized plane strain, a Burgers-like formula for straight dislocations and body forces is derived and its relation to the solution of the displacement and stress function fields in the integral formalism is given. Moreover, the stress functions of a point force are calculated and the relation to the potential of a Dirac string is pointed out.

Discontinuous Maxwell–Rankine stress functions for space frames

2018 ◽  
Vol 33 (1) ◽  
pp. 35-47
Author(s):  
Allan McRobie ◽  
Chris Williams
Keyword(s):  
Stress Function ◽  
Three Dimensional ◽  
Shear Forces ◽  
Space Frames ◽  
Stress Functions

This article shows how bending and torsional moments in three-dimensional frames can be represented via a discontinuous Maxwell–Rankine stress function. The associated Rankine reciprocal contains polygonal faces whose areas represent forces. These faces are orthogonal to the member forces (which may include shear forces) and need not be orthogonal to the beams.

scholarly journals Analytical graphic statics

2021 ◽  
pp. 095605992110016
Author(s):  
Tamás Baranyai
Keyword(s):  
Stress Function ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Linear Functionals ◽  
The Past ◽  
Force Diagram ◽  
Graphic Statics ◽  
Reciprocal Diagrams ◽  
Force Systems ◽  
Stress Functions

Graphic statics is undergoing a renaissance, with computerized visual representation becoming both easier and more spectacular as time passes. While methods of the past are revived, little emphasis has been placed on studying the mathematics behind these methods. Due to the considerable advances of our mathematical understanding since the birth of graphic statics, we can learn a lot by examining these old methods from a more modern viewpoint. As such, this work shows the mathematical fabric joining different aspects of graphic statics, like dualities, reciprocal diagrams, and discontinuous stress functions. This is done by introducing a new, three dimensional force diagram (containing the old two dimensional force diagram) depicting the three dimensional equilibrium of planar force systems. A corresponding three dimensional “form diagram” (dual diagram) is introduced, in which forces are treated as linear functionals (dual vectors). It is shown that the polyhedral stress function introduced by Maxwell is in fact a linear combination of these functionals; and the projective dualities connecting these three dimensional diagrams are also explained.

scholarly journals Comparison of Tooth- and Bone-Borne Appliances on the Stress Distributions and Displacement Patterns in the Facial Skeleton in Surgically Assisted Rapid Maxillary Expansion—A Finite Element Analysis (FEA) Study

Materials ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1152
Author(s):  
Rafał Nowak ◽  
Anna Olejnik ◽  
Hanna Gerber ◽  
Roman Frątczak ◽  
Ewa Zawiślak
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Three Dimensional ◽  
Element Model ◽  
Rapid Maxillary Expansion ◽  
Stress Distributions ◽  
Maxillary Expansion ◽  
Facial Skeleton ◽  
Element Analysis ◽  
Maxillary Constriction

The aim of this study was to compare the reduced stresses according to Huber’s hypothesis and the displacement pattern in the region of the facial skeleton using a tooth- or bone-borne appliance in surgically assisted rapid maxillary expansion (SARME). In the current literature, the lack of updated reports about biomechanical effects in bone-borne appliances used in SARME is noticeable. Finite element analysis (FEA) was used for this study. Six facial skeleton models were created, five with various variants of osteotomy and one without osteotomy. Two different appliances for maxillary expansion were used for each model. The three-dimensional (3D) model of the facial skeleton was created on the basis of spiral computed tomography (CT) scans of a 32-year-old patient with maxillary constriction. The finite element model was built using ANSYS 15.0 software, in which the computations were carried out. Stress distributions and displacement values along the 3D axes were found for each osteotomy variant with the expansion of the tooth- and the bone-borne devices at a level of 0.5 mm. The investigation showed that in the case of a full osteotomy of the maxilla, as described by Bell and Epker in 1976, the method of fixing the appliance for maxillary expansion had no impact on the distribution of the reduced stresses according to Huber’s hypothesis in the facial skeleton. In the case of the bone-borne appliance, the load on the teeth, which may lead to periodontal and orthodontic complications, was eliminated. In the case of a full osteotomy of the maxilla, displacements in the buccolingual direction for all the variables of the bone-borne appliance were slightly bigger than for the tooth-borne appliance.

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