Dugdale / Neal Elements of Elasticity

The Commonwealth and International Library: Structures and Solid Body Mechanics Division
1. Auflage 2014
ISBN: 978-1-4831-9120-1
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 156 Seiten

Elements of Elasticity details the fundamental concepts in the theory of elasticity. The title emphasizes discussing the essential formulas, along with elementary matters. The text first covers stress and strain, and then proceeds to tackling the elasticity equation. Next, the selection covers plane stress and strain, along with curvilinear coordinates and polar coordinates. The next chapter deals with rotating discs and thick cylinders. Chapter 8 details strain energy in plates, while Chapter 9 discusses torsion. The last chapter covers stress propagation. The book will be of great interest to engineers, particularly those who deal with fracture mechanics.

Autoren/Hrsg.

Dugdale, D. S.

Neal, B. G.

Inhaltsverzeichnis

1;Front Cover;1
2;Elements of Elasticity;4
3;Copyright Page;5
4;Table of Contents;6
5;AUTHOR'S PREFACE;10
6;CHAPTER 1. STRAIN;14
6.1;1.1. Coordinate systems;14
6.2;1.2. Displacements;15
6.3;1.3. Strain components;17
6.4;1.4. Transformation of strains;21
6.5;1.5. Compatibility of strains;24
6.6;1.6. Displacement functions;25
6.7;1.7. Increase in area;26
6.8;Problems;28
7;CHAPTER 2. STRESS;29
7.1;2.1. Definitions;29
7.2;2.2. Stress equilibrium equations;31
7.3;2.3. Transformation of stresses;35
7.4;2.4. Graphical representation;38
7.5;2.5. Stress at boundaries;41
7.6;Problems;44
8;CHAPTER 3. ELASTICITY EQUATIONS;46
8.1;3.1. Linear medium;46
8.2;3.2. Dilatation and distortion;47
8.3;3.3. Elastic constants;49
8.4;3.4. Strain energy;51
8.5;Problems;53
9;CHAPTER 4. PLANE STRESS AND STRAIN;54
9.1;4.1. Definitions;54
9.2;4.2. Stress and displacement functions;56
9.3;4.3. Body forces and thermal strain;60
9.4;4.4. Analysis of strain gauge readings;61
9.5;4.5. Problems in rectangular coordinates;64
9.6;Problems;67
10;CHAPTER 5. CURVILINEAR COORDINATES;69
10.1;5.1. Alternative coordinates;69
10.2;5.2. Conformal transformations;72
10.3;5.3. Derivatives in curvilinear coordinates;75
10.4;5.4. Stress and strain;78
10.5;Problems;80
11;CHAPTER 6. POLAR COORDINATES;82
11.1;6.1. General equations;82
11.2;6.2. Axial symmetry;85
11.3;6.3. Harmonic solutions;86
11.4;6.4. Circular hole in a sheet;88
11.5;6.5. Stresses giving a resultant force;92
11.6;6.6. Force at apex of wedge;95
11.7;Problems;97
12;CHAPTER 7. ROTATING DISCS AND THICK CYLINDERS;101
12.1;7.1. Basic equations;101
12.2;7.2. Solid disc and disc with central hole;104
12.3;7.3. Rotating cylinder;105
12.4;7.4. Cylinders under internal and external pressure;107
12.5;7.5. Shrinkage stresses;108
12.6;7.6. Graphical construction;111
12.7;Problems;112
13;CHAPTER 8. STRAIN ENERGY IN PLATES;114
13.1;8.1. Energy balance for plane stress;114
13.2;8.2. Internal stress;116
13.3;8.3. Strain energy in flexure;118
13.4;8.4. Strain in a thin plate;120
13.5;8.5. Compatibility equation;122
13.6;8.6. Internal stress in a thin plate;124
13.7;Problems;125
14;CHAPTER 9. TORSION;126
14.1;9.1. Torsion of a uniform prism;126
14.2;9.2. Various solid sections;132
14.3;9.3. Hollow sections;136
14.4;9.4. Membrane analogy;139
14.5;Problems;142
15;CHAPTER 10. STRESS PROPAGATION;144
15.1;10.1. One-dimensional stress waves;144
15.2;10.2. Reflection from a boundary;149
15.3;10.3. Harmonic excitation;151
15.4;Problems;152
16;APPENDIX: CONDENSED SUFFIX NOTATION;153
17;INDEX;156

Leseproben

CHAPTER 1

STRAIN

Publisher Summary

This chapter discusses strain examination in coordinate systems. Rectangular or Cartesian coordinates are suitable and sufficient for expressing the general physical relationships that hold at points in the interior of a solid body. The process of solution of a problem consists of selecting mathematical expressions for the distribution of stress and strain that not only satisfy these differential equations but also give the required values at the boundaries of the body. In the typical boundary-value problem, stresses or displacements or both are specified over the surface of a body, or over part of its surface. In selecting a coordinate system for conveniently describing boundary conditions, it must be remembered that differential equations in rectangular coordinates must be recast into a new form appropriate to the new coordinates.

1.1 Coordinate systems

Neither stress nor strain can be examined in a precise way without first defining a system of coordinates. Rectangular or Cartesian coordinates are suitable and sufficient for expressing the general physical relationships that hold at points in the interior of a solid body. Alternative coordinate systems may be preferable when one has to describe a boundary of some particular shape or to specify what happens at the boundary.

Whatever the shape of the body may be, stresses and strains are known to be distributed within the interior of the body in accordance with certain differential equations derived from the physical properties of the material from which the body is made. The process of solution of a problem consists of selecting mathematical expressions for the distribution of stress and strain that not only satisfy these differential equations, but also give the required values at the boundaries of the body. In the typical “boundary-value problem”, stresses or displacements or both are specified over the surface of a body, or over part of its surface. When expressions for stress and displacement have been found, and it has been shown that these are satisfactory both on the boundaries and at interior points, the problem has been solved.

In selecting a coordinate system for conveniently describing boundary conditions, it must be remembered that differential equations in rectangular coordinates must be re-cast into a new form appropriate to the new coordinates. Useful coordinate systems are necessarily orthogonal, that is, one set of coordinate lines intersect the other set at right angles at all points, but there is no further restriction on choice of coordinates. A more extended discussion of alternative coordinate systems, and the way in which differential equations are transformed, is deferred until stress and strain have been examined in terms of rectangular coordinates.

Although stress and strain are necessarily three-dimensional quantities at points in the interior of a solid body, emphasis throughout this book is placed on two-dimensional states of elastic deformation. The resulting treatment is relatively simple, and yet permits the solution of many practical problems. Strain is considered before stress so that the signs of shear stresses can be settled by referring to the signs of shear strains rather than to some arbitrary convention.

1.2 Displacements

When forces are applied to the edges of a flat sheet, the sheet may be stretched or otherwise distorted while still remaining flat. For the moment, the exact cause of the distortion is not being investigated. The problem in hand is to specify the distortion that has taken place. The result of applying loads is that all points on the sheet move in some way relative to fixed references axes and Some particular point as shown in Fig. 1.1 will move to a new position It is assumed that displacements are small in relation to the overall dimensions of the body, so that the position of a given particle at any time is described to sufficient accuracy by its coordinates measured before the distortion is imposed. More detailed restrictions on the size of displacements to ensure that strains remain infinitesimally small, will be mentioned later.

FIG. 1.1 Vector representing displacement at a point.

Displacement at a given point on a sheet may be specified by giving the length of the displacement vector together with its direction. Contours may be plotted on the sheet giving length and curves may be drawn in such a way that the tangent to the curve defines the direction of the vector. The example shown in Fig. 1.2 is for a sheet that is uniformly stretched in the -direction and uniformly compressed to the same degree in the -direction. An alternative way of specifying the displacement vector at any point is to give its components measured parallel to some particular reference axes and These components are denoted and , though occasionally the suffix notation and is preferable.

FIG. 1.2 Displacements in a plane. Full lines are contours of constant vector length , and broken lines indicate direction of vector.

Usually these components vary smoothly and continuously throughout the deformed body, but exceptions may occur. If the body initially has a rectangular shape and deforms in the central part only, while the upper and lower ends remain undeformed, as shown in Fig. 1.3a, the deformed part may be termed a kink band. It will be found later that this demands an impossible distribution of stress in an elastic body. If the upper part of the body slides over the lower part, as in Fig. 1.3b, the surface on which sliding occurs is called a dislocation. Horizontal displacements change discontinuously as this surface is crossed in a vertical direction, even though the two parts are imagined to be welded together after sliding has occurred. If it is found that vertical displacement changes discontinuously across a horizontal surface, it must be concluded that separation of the upper and lower parts has occurred, and that a gap or crack of finite width has developed.

FIG. 1.3 (a) Kink band, (b) dislocation.

1.3 Strain components

Direct and shear strains are first considered separately, and expressed in terms of displacement components. From these expressions it will be seen that values of direct and shear strains depend entirely on the coordinates chosen, so that they cannot be regarded as existing independently from each other.

Direct strain

This is defined as fractional increase in length. In the particular case of a long parallel-sided bar subjected to tensile loading, strains will be equal at all points along the bar, but in a more general kind of deformation, strain will vary from one point to another. Therefore it is necessary to consider a small element of length initially at position as shown in Fig. 1.4. After the body has been deformed, this element has moved to position The inner end of the element has moved a distance while the outer end has moved a somewhat larger distance + The final length of the element is therefore +, so the fractional increase in length is An alternative definition of strain might be obtained by expressing the increase in length as a fraction of the final length, i.e. (+). However, attention is restricted to strains that are very small in value in relation to unity. Hence the alternatively defined strains will differ from the original ones only by quantities of the second order of smallness, which may be neglected, so that the distinction between these definitions disappears. Hence, the infinitesimal strain measured in the -direction at a particular point can be written

FIG. 1.4 Direct strain.

xx=?u?x. (1.1)

(1.1)

The first suffix indicates that the displacement from which the strain is derived is measured in the -direction. The second suffix indicates that the rate of change of displacement is also measured in the -direction. The strain is measured along a line parallel to the -axis, but whether this line is taken as facing in the positive or negative direction is immaterial. This suggests that strains must transform differently from vectors for rotations of the coordinate axes. A rotation through 180° will change the sign of a vector component but will not change the sign of a strain component. The transformation properties are fundamental and will receive further attention later. It is noted that strain depends on rate of change of displacement, and not on displacement itself. If a small additional displacement 0 is given to all the particles of the body, the strains given by (1.1) will remain unchanged. Such a displacement is often called a rigid-body displacement.

In a similar way, direct strain in the -direction is given...