A Treatise on the Mathematical Theory of Elasticity

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Cambridge University Press, Jan 3, 2013 - Mathematics - 662 pages
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A. E. H. Love (1863-1940) was an English mathematician and geophysicist renowned for his work on elasticity and wave propagation. Originally published in 1927, as the fourth edition of a title first published in two volumes in 1892 and 1893, this is Love's classic account of the mathematical theory of elasticity. The text provides a detailed explanation of the topic in its various aspects, revealing important relationships with general physics and applications to engineering. Also included are a historical introduction to the theory, notes section, index of authors cited and index of matters treated. This book will be of value to anyone with an interest in elasticity, physics and mathematics.
 

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Contents

HISTORICAL INTRODUCTION
1
Stokess criticism of Poissons theory The controversy concerning
24
PROBLEMS CONCERNING THE EQUILIBRIUM 0F THIN RODS
44
ART PAGE 19 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
32
57
Applications and extensions of the foregoing solution
264
The sphere with given surface displacements
265
Generalization of the foregoing solution
266
The sphere with given surface tractions
267
Plane strain in a circular cylinder
270
Applications of curvilinear coordinates
272
Symmetrical strain in a solid of revolution
274
Symmetrical strain in a cylinder
276

APPENDIX TO CHAPTER I GENERAL THEORY OF STRAIN 23 Introductory
59
Cubical dilatation
61
Reciprocal strain ellipsoid
62
Strain ellipsoid
63
Alteration of direction by the strain
64
Application to cartography
65
Finite homogeneous strain
66
Homogeneous pure strain
67
Analysis of any homogeneous strain into a pure strain and a rotation
69
Simple extension 37 Simple shear
70
Additional results relating to shea 7 1
71
Additional results relating to the composition of strains
72
Introductory
74
Surface tractions and body forces
75
Equations of motion
76
Law of equilibrium of surface tractions on small volumes
77
37
79
Transformation of stresscomponents
80
Types of stress a Purely normal stress b Simple tension or pressure 0 Shearing stress d Plane stress
81
Resolution of any stresssystem into uniform tension and shearing stress
83
The stressequations of motion and of equilibrium
84
Uniform stress and uniformly varying stress
85
Observations concerning the stressequations
86
Graphic representation of stress
88
Stressequations referred to curvilinear orthogonal coordinates
89
Special cases of stressequations referred to curvilinear orthogonal co ordinates
90
ART PAGE
92
Form of the strainenergyfunction
98
Observations concerning the stressstrain relations in isotropic solids
104
39
108
THE RELATION BETWEEN THE MATHEMATICAL
112
IEolotropy induced by permanent set
118
Recapitulation of the general theory
125
42
128
43
138
ART PAGE 97 Polar coordinates
141
Radial displacement Spherical shell under internal and external pressure Compression of a sphere by its own gravitation
142
Displacement symmetrical about an axis
143
Tube under pressure
144
Application to gun construction
145
Rotating cylinder Rotating shaft Rotating disk
146
Symmetry of structure
149
Elastic symmetry
151
46
154
Isotropic solid
155
Classification of crystals
157
Elasticity of crystals
159
Various types of symmetry
160
Material with three orthogonal planes of symmetry Moduluses
161
Extension and bending of a bar
162
Elastic constants of crystals Results of experiments
163
Curvilinear aeolotropy
164
GENERAL THEOREMS 115 The variational equation of motion
166
Applications of the variational equation
167
The general problem of equilibrium
169
Uniqueness of solution
170
Theorem of minimum energy
171
Theorem concerning the potential energy of deformation
173
Determination of average strains
174
Average strains in an isotropic solid body
175
The general problem of vibrations Uniqueness of solutio
176
Flux of energy in vibratory motion
177
Free vibrations of elastic solid bodies
178
General theorems relating to free vibrations
180
Load suddenly applied or suddenly reversed
181
Introductory
183
First type of simple solutions
185
Typical nuclei of strain
186
Local perturbations
189
Second type of simple solutions
190
Pressure at a point on a plane boundary
191
Distributed pressure
192
Pressure between two bodies in contact Geometrical preliminaries
193
48
195
Hertzs theory of impact
198
Impact of spheres
200
Effects of nuclei of strain referred to polar coordinates
201
Problems relating to the equilibrium of cones
203
CHAPTER TWODIMENSIONAL ELASTIC SYSTEMS 143 Introductory
204
Displacement corresponding with plane stress
206
Generalized plane stress
207
Introduction of nuclei of strain
208
Force operative at a point
209
Force operative at a point of a boundary
210
Case of a straight boundary 21 1
211
Typical nuclei of strain in two dimensions
213
Transformation of plane strain
214
Inversion
215
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 restin...
217
Examples of transformation
219
156A Introductory a Displacement answering to given strain b Discon tinuity at a barrier c Hollow cylinder deformed by removal of a slice of unifor...
221
THEORY OF THE INTEGRATION or THE EQUATIONS 0F EQUILIBRIUM OF AN ISOTROPIC ELASTIC SOLID BODY 157 Nature of the probl...
229
Résumé of the theory of Potential
230
Description of Bettis method of integration
232
Formula for the dilatation
233
Calculation of the dilatation from surface data
235
Formulae for the components of rotation
236
Calculation of the rotation from surface data
237
Body bounded by planeGiven surface displacements
239
Body bounded by planeGiven surface tractions
241
Historical Note
243
Body bounded by planeAdditional results
244
Formulae for the displacement and strain
245
Outlines of various methods of integration
247
THE EQUILIBRIUM or AN ELAsTIc SPHERE AND RELATED PROBLEMS ARR PAGE 171 Introductory
249
i Solid sphere with purely radial surface displacement ii Solid sphere with purely radial surface trac tion iii Small spherical cavity in large solid mass i...
251
Sphere subjected to body force
252
Generalization and Special Cases of the foregoing solution
254
Gravitating incompressible sphere
255
Deformation of gravitating incompressible sphere by external body force
257
Gravitating body of nearly spherical form
259
Rotating sphere under its own attraction
260
Tidal deformation Tidal effective rigidity of the Earth
261
A general solution of the equations of equilibrium
263
Introductory
278
Solution by means of spherical harmonics
279
Formation of the boundaryconditions for a vibrating sphere
281
Incompressible material
283
Frequency equations for vibrating sphere
284
Vibrations of the second class
285
Further investigations on the vibrations of spheres
286
Vibrations of a circular cylinder
287
Torsional vibrations
288
Longitudinal vibrations
289
Transverse vibrations
291
THE PROPAGATION 0F WAVES IN ELASTIC SOLID MEDIA 203 Introductory
293
Motion of a surface of discontinuity Kinematical conditions
295
Motion of a surface of discontinuity Dynamical conditions
296
Velocity of waves in isotropic medium
297
ART PAGE 208 Velocity of waves in seolotropic solid medium
298
Wavesurfaces
299
Motion determined by the characteristic equation
300
Arbitrary initial conditions
302
Motion due to body forces
304
Additional results relating to motion due to body forces
305
Waves propagated over the surface of an isotropic elastic solid bod
307
ToRsIoN 215 Stress and strain in a twisted prism
310
The torsion problem
311
Method of solution of the torsion problem
313
Analogies with Hydrodynamics
314
Distribution of the shearing stress
316
Solution of the torsion problem for certain boundaries
317
Additional results
318
Graphic expression of the results
320
Analogy to the form of a stretched membrane loaded uniformly
322
Torsion of aeolotropic prism
324
226A Bar of varying circular section
325
2261 Distribution of traction over terminal section
327
THE BENDING or A BEAM BY TERMINAL TRANsvERsE LOAD 227 Stress in bent beam
329
Necessary type of shearing stress
331
Formulas for the displacement
333
It The circle b Concentric circles c The ellipse d Confocal ellipses e The rectangle f Additional results
337
a Curvature of the strained centralline
338
b Neutral plane 0 Obliquity of the strained crosssections d De flexion e Twist f Anticlastic curvature g Distortion of the cross sections into curved surf...
340
Distribution of shearing stress
341
a Asymmetric loading 6 Com bined strain 0 IEolotropic material
343
Analogy to the form of a stretched membrane under varying pressure
345
a A method of determining the shearing stress in the case of rectangular sections b Extension of this method to curved boundaries 0 Form of boundar...
346
ART PAGE
349
The constants of the solution
356
Extension of the theory of the bending of beams
365
Graphic method of solution of the problem of continuous beams
375
Introductory
381
The ordinary approximate theory
388
258A Small displacement
395
Height consistent with stability
425
ART PAGE 294 Specification of stress in a plate
455
Transformation of stressresultants and stresscouples
456
Equations of equilibrium
457
Boundaryconditions
458
THEORY OF MODERATELY THICK PLATES 299 Method of determining the stress in a plate
465
Plane stress
467
Plate bent to a state of plane stress
470
Generalized plane stress
471
Plate bent to a state of generalized plane stress
473
Circular plate loaded at its centre
475
Plate bent by pressure uniform over a face
477
Plate bent by pressure varying uniformly over a face
479
Circular plate bent by a uniform pressure and supported at the edge
481
Plate bent by uniform pressure and clamped at the edge
482
Additional deflexion due to the mode of fixing the edge
484
Plate bent by uniformly varying pressure and clamped at the edge
485
Plate bent by its own weight
486
Note on the theory of moderately thick plates
487
Illustrations of the approximate theory a Circular plate loaded sym
489
metrically 6 Application of the method of inversion c Rectangular plate i Variable pressure Two parallel edges supported ii Uniform pressure suppor...
497
INEXTENSIONAL DEFORMATION or CURVED PLATES OR SHELLS 315 Introductory
499
Typical flexural strain
502
Method of calculating the changes of curvature
503
Inextensional deformation of a cylindrical shell a Formulae for the dis
505
placement b Changes of curvature
506
Inextensional deformation of a spherical shell a Formulae for the dis placement 6 Changes of curvature
507
Inextensional vibrations i Cylindrical shell ii Spherical shell
512
ART PAGE 322 Formulae relating to the curvature of surfaces
515
Simplified formulae relating to the curvature of surfaces
517
Method of calculating the extension and the changes of curvature
519
Formulae relating to small displacements
520
Nature of the strain in a bent plate or shell
524
Specification of stress in a bent plate or shell
527
Approximate formulae for the strain the stressresultants and the stress couples
528
Second approximation in the case of a curved plate or shell _ _ _
532
Equations of equilibrium
534
Boundaryconditions
536
332A Buckling of a rectangular plate under edge thrust
537
Theory of the vibrations of thin shells
538
Vibrations of a thin cylindrical shell a General equations b Ex tensional vibrations c Inextensional vibrations d Inexactness of the inextensional displa...
547
Vibrations of a thin spherical shell
549
CHAPTER XXIVA EQUILIBRIUM 0F THIN PLATES AND SHELLS 3350 Large deformations of plates and shells
553
Large thin plate subjected to pressure
557
335i Long strip Supported edges
559
Long strip Clamped edges
563
EQUILIBRIUM OF THIN SHELLS 336 Small displacement
564
The middle surface a surface of revolution
565
Torsion
567
CYLINDRICAL SHELL 339 Symmetrical conditions a Extensional solution b Edgeeffect
568
Tube under pressure
570
Stability of a tube under external pressure
571
Lateral forces a Extensional solution b Edgeeffect
575
General unsymmetrical conditions Introductory a Extensional solution b Approximately inextensional solution 0 Edgeeffect
579
SPHERICAL SHELL 344 Extensional solution
583
Edgeeffect Symmetrical conditions
586
ART PAGE
590
Extensional solution Unsymmetrical conditions
601
NOTES
614
Applications of the method of moving axes
628
Matters treated
637
293
638
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