A Treatise on the Mathematical Theory of Elasticity (Google eBook)

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at the University Press, 1906 - Elasticity - 551 pages
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Contents

Resolution of any strain into dilatation and shearing strains
47
Identical relations between components of strain
49
Displacement corresponding with given strain
50
Curvilinear orthogonal coordinates
51
Components of strain referred to curvilinear orthogonal coordinates
53
Dilatation and rotation referred to curvilinear orthogonal coordinates
54
Cylindrical and polar coordinates
56
Introductory
57
Cubical dilatation
59
Reciprocal strain ellipsoid
60
Strain ellipsoid
61
Alteration of direction by the strain
62
Application to cartography
63
Finite homogeneous strain
64
Homogeneous pure strain
65
Analysis of any homogeneous strain into a pure strain and a rotation
67
Simple extension
68
Additional results relating to shear
69
Additional results relating to the composition of strains
70
ANALYSIS OF STRESS 41 Introductory
72
Surface tractions and body forces
73
Equations of motion
74
Equilibrium
75
Measure of stress
77
Transformation of stresscomponents
78
The stress quadric
79
ABT PAGE 52 Resolution of any stresssystem into uniform tension and shearing stress
81
The stressequations of motion and of equilibrium
82
Uniform stress and uniformly varying stress
84
Observations concerning the stressequations
85
Graphic representation of stress
86
Stressequations referred to curvilinear orthogonal coordinates
87
Special cases of stressequations referred to curvilinear orthogonal coordinates
89
CHAPTER III THE ELASTICITY OF SOLID BODIES to Introductory
90
Existence of the strainenergyfunction
92
Indirectness of experimental results
94
HookesLaw
95
Form of the strainenergyfuuction
96
Elastic constants
97
Methods of determining the stress in a body
98
Form of the strainenergyfunction for isotropic solids
99
Elastic constants and moduluses of isotropic solids
100
Observations concerning the stressstrain relations in isotropic solids
101
Magnitude of elastic constants and moduluses of some isotropic solids
103
Modulus of elasticity
104
Thertnoelastic equations
106
Initial stress
107
THE RELATION BETWEEN THE MATHEMATICAL THEORY OF ELASTICITY AND TECHNICAL MECHANICS 76 Limitations of the mathem...
110
Stressstrain diagrams
111
Elastic limits
113
Timeeffects Plasticity
114
Viscosity of solids
115
Eolotropy induced by permanent set
116
Hypotheses concerning the conditions of rupture
117
Scope of the mathematical theory of elasticity
119
Recapitulation of the general theory
122
Uniformly varying stress a Bar stretched by its own weight 6 Cylinder immersed in fluid c Body of any form immersed in fluid of same density d Ro...
123
ABT PAGE 87 Bar bent by couples
126
Discussion of the solution for the bending of a bar by terminal couple
127
SaintVenarits principle
129
Equations of equilibrium in terms of displacements
130
Equilibrium under surface tractions only
132
Various methods and results
133
Plane strain and plane stress
134
Bending of narrow rectangular beam by terminal load
136
Equations referred to orthogonal curvilinear coordinates
138
Radial displacement Spherical shell under internal and external pressure Compression of a sphere by its own gravitation
139
Displacement symmetrical about an axis
140
Tube under pressure
141
Application to gun construction
143
Symmetry of structure
146
Geometrical symmetry
147
Elastic symmetry
148
Isotropic solid
152
Classification of crystals
154
Elasticity of crystals
156
Various types of symmetry
157
Material with three rectangular planes of symmetry Moduluses
158
Extension and bending of a bar
159
Elastic constants of crystals Results of experiments
160
GENERAL THEOREMS 115 The variational equation of motion
163
Applications of the variational equation
164
The general problem of equilibrium 106
166
Uniqueness of solution
167
Theorem of minimum energy
168
Theorem concerning the potential energy of deformation 17i 121 The reciprocal theorem
170
Determination of average strains
171
Average strains in an isotropic solid body
172
The general problem of vibrations Uniqueness of solution
173
Flux of energy in vibratory motion
174
Free vibrations of elastic solid bodies
175
General theorems relating to free vibrations
177
Load suddenly applied or suddenly reversed
178
THE TRANSMISSION OF FORCE ABT PAGE 129 Introductory
180
First type of simple solutions
182
Typical nuclei of strain
183
Local perturbations
186
Second type of simple solutions
187
Pressure at a point on a plane boundary
188
Distributed pressure
189
Pressure between two bodies in contact Geometrical preliminaries
190
Solution of the problem of the pressure between two bodies in contact
192
Hertzs theory of impact
195
Impact of spheres
197
Effects of nuclei of strain referred to polar coordinates
198
Problems relating to the equilibrium of cones
200
TWODIMENSIONAL ELASTIC SYSTEMS 143 Introductory
201
Displacement corresponding with plane stress
203
Generalized plane stress
205
Force operative at a point
206
Force operative at a point of a boundary
207
Case of a straight boundary
208
Typical nuclei of strain in two dimensions
209
Transformation of plane strain
211
Inversion
212
Equilibrium of a circular disk under forces in its plane i Two opposed forces at points on the rim ii Any forces applied to the rim iii Heavy disk on h...
213
Examples of transformation
216
THEORY OF THE INTEGRATION OF THE EQUATIONS OF EQUILIBRIUM OF AN ISOTROPIC ELASTIC SOLID BODY 187 Nature of the pro...
217
Resum of the theory of Potential
218
Description of Bettis method of integration
220
Formula for the dilatation
221
Calculation of the dilatation from surface data
223
Formulae for the components of rotation
224
Body bounded by planeFormula for the dilatation
225
Body bounded by planeGiven surface displacements
227
Body bounded by planeGiven surface tractions
228
The sphere with given surface tractions
240
Conditions restricting the prescribed surface tractions
243
Surface tractions directed normally to the boundary
244
Solution in spherical harmonics of negative degrees
245
Sphere subjected to forces acting through its volume Particular solution
246
Sphere deformed by body force only
247
Gravitating incompressible sphere
248
Deformation of gravitating incompressible sphere by external forces
250
Gravitating nearly spherical body
253
Tidal deformation Tidal effective rigidity of the Earth
255
Plane strain in a circular cylinder
257
Applications of curvilinear coordinates
259
Symmetrical strain in a solid of revolution
260
Symmetrical strain in a cylinder
263
VIBRATIONS OF SPHERES AND CYLINDERS 190 Introductory
265
Solution by means of spherical harmonics
266
Formation of the boundary conditions for a vibrating sphere
268
Incompressible material
271
Vibrations of the first class
272
Vibrations of the second class
273
Further investigations of the vibrations of spheres
274
Vibrations of a circular cylinder
275
Torsional vibrations
276
Transverse vibrations
278
THE PROPAGATION OF WAVES IN ELASTIC SOLID MEDIA ART PAGE 203 281 204 Waves of dilatation and waves of distortion
281
Motion of a surface of discontinuity Kinematical conditions
283
Motion of a surface of discontinuity Dynamical conditions
284
Velocity of waves in isotropic medium
285
Velocity of waves in seolotropic solid medium
286
Wavesurfaces
287
Motion determined by the characteristic equation
289
Arbitrary initial conditions
291
Motion due to body forces
293
Additional results relating to motion due to body forces
294
Waves propagated over the surface of an isotropic elastic solid body
295
TORSION 215 Stress and Strain in a twisted prism
298
The torsion problem
299
Method of solution of the torsion problem
301
Analogies with Hydrodynamics
302
Distribution of the shearing stress
304
Solution of the torsion problem for certain boundaries
305
Additional results
306
Graphic expression of the results
308
Analogy to the form of a stretched membrane loaded uniformly
310
Torsion of wolotropic prism
312
THE BENDING OF A BEAM BY TERMINAL TRANSVERSE LOAD 227 Stress in a bent beam
314
Statement of the problem
315
Necessary type of shearing stress
316
Formulae for the displacement
318
a The circle 6 Concentric circles c The ellipse d Confocal ellipses e The rectangle Additional results
322
a Curvature of the strained centralline
323
6 Neutral plane c Obliquity of the strained crosssections d Deflexion e Twist f Anticlastic curvature g Distortion of the crosssections into curved surfa...
326
Distribution of shearing stress
327
a Asymmetric loading 6 Combined strain c jEolotropic material
328
THE BENDING OF A BEAM LOADED UNIFORMLY
334
The constants of the solution
342
THE THEORY OF CONTINUOUS BEAMS
348
Single span a Terminal forces and couples 6 Uniform load Supported
357
GENERAL THEORY OF THE BENDING
365
The ordinary approximate theory
372
Rods naturally curved
379
ART PASE 260 KirchhofTs kinetic analogue
382
Extension of the theorem of the kinetic analogue to rods naturally curved
383
The problem of the elastica
384
Classification of the forms of the elastica a Inflexional elastica b Non inflexional elcustiea
386
Buckling of long thin strut under thrust
388
Computation of the strainenergy of the strut
389
Resistance to buckling
390
Elastic stability
392
Rod bent and twisted by terminal forces and couples
394
Rod bent to helical form
396
Additional results a Rod subjected to terminal couple b Straight rod with initial twist c Rod bent into circular hoop and twisted uniformly d Stability o...
398
Rod bent by forces applied along its length
402
Rod bent in one plane by uniform normal pressure
403
Stability of circular ring under normal pressure
405
VIBRATIONS OF RODS PROBLEMS OF DYNAMICAL RESISTANCE 277 Introductory
407
Extensional vibrations
408
Torsional vibrations
409
Rod fixed at one end and struck longitudinally at the other
411
Rod free at one end and struck longitudinally at the other
415
Rod loaded suddenly
416
Longitudinal impact of rods
418
Problems of dynamical resistance involving transverse vibration
420
The whirling of shafts
421
SMALL DEFORMATION OF NATURALLY CURVED RODS 287 Introductory
423
Orientation of the principal torsionflexure axes
424
Simplified formulas
426
Problems of equilibrium a Incomplete circular ring bent in its plane b Incomplete circular ring bent out of its plane
429
Vibrations of a circular ring a Flexural vibrations in the plane of the ring 6 Flexural vibrations at right angles to the plane of the ring c Torsional and ...
432
THE STRETCHING AND BENDING OF PLATES ABT PAGE 294 Specification of stress in a plate
434
Transformation of stressresultants and stresscouples
435
2 Equations of equilibrium
436
Boundary conditions
437
Relation between the flexural couples and the curvature
442
Method of determining the stress in a plate
444
Plane stress
446
Plate bent to a state of plane stress
449
Generalized plane stress
450
Plate bent to a state of generalized plane stress
452
Circular plate loaded at its centre
454
Plate bent by pressure uniform over a face
456
Plate bent by pressure varying uniformly over a face
458
Circular plate bent by uniform pressure and supported at the edge
460
Plate bent by uniform pressure and clamped at the edge
461
Plate bent by uniformly varying pressure and clamped at the edge
463
Plate bent by its own weight
464
Approximate theory of the bending of a plate by transverse forces
465
Illustrations of the approximate theory a Circular plate loaded symmetric ally 6 Application of the method of inversion c Rectangular plate supported...
470
INEXTENSIONAL DEFORMATION OF CURVED PLATES OR SHELLS 315 Introductory
471
Changes of curvature in inextensional deformation
472
Typical flexural strain
474
Method of calculating the changes of curvature
476
Inextensional deformation of a spherical shell a Formulae for the displace ment b Changes of curvature
479
Inextensional vibrations i Cylindrical shell ii Spherical shell
485
GENERAL THEORY OF THIN PLATES AND SHELLS 322 Formulae relating to the curvature of surfaces
488
Simplified formulas relating to the curvature of surfaces
490
Extension and curvature of the middle surface of a plate or shell
491
Method of calculating the extension and the changes of curvature
492
Formulae relating to small displacements
494
ABT PAGE
498
Second approximation in the case of a curved plate or shell
506
Vibrations of a thin cylindrical shell a General equations 6 Extensional
515
Vibrations of a thin spherical shell
522
Problems of stability a Buckling of a rectangular plate under thrusts in
529
Applications of the method of moving axes
536
INDEX
543

Common terms and phrases

Popular passages

Page 2 - ... the power of any spring is in the same proportion with the tension thereof: that is, if one power stretch or bend it one space, two will bend it two, and three will bend it three, and so forward.
Page 15 - An Essay on the application of Mathematical Analysis to the Theories of Electricity and Magnetism...
Page 297 - It is not improbable that the surface waves here investigated play an important part in earthquakes, and in the collision of elastic solids. Diverging in two dimensions only, they must acquire at a great distance from the source a continually increasing preponderance.
Page 149 - ... x cos 0 + y sin 0 y
Page 4 - The modulus of the elasticity of any substance is a column of the same substance, capable of producing a pressure on its base which is to the weight causing a certain degree of compression, as the length of the substance is to the diminution of its length.
Page 188 - ... surface of the body, and the positive direction of the axis of z to- be that which goes into the interior of the body. The local effect of force applied at the origin being very great, we suppose the origin to be excluded by a hemispherical surface. The displacement expressed by (15) could be maintained in the body by tractions over the plane boundary, which are expressed by the equations ' AX Y = ^ Ay Z -0 ' *° 5' ^~U> and by tractions over the hemispherical boundary, which are expressed by...
Page 4 - This introduction of a definite physical concept, associated with the coefficient of elasticity which descends, as it were from a clear sky, on the reader of mathematical memoirs, marks an epoch in the history of the science.
Page 11 - One of the advantages of this method, of great importance, is, that we are necessarily led by the mere process of the calculation, and with little care on our part, to all the equations and conditions which are requisite and sufficient for the complete solution of any problem to which it may be applied.
Page 170 - ... the equilibrium state) acting through the displacements from the unstressed state to the state of equilibrium.
Page 42 - The strain quadric has the property that the reciprocal of the square of its central radius vector in any direction is proportional to the extension of a line in that direction. If the...

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