A Treatise on the Mathematical Theory of Elasticity

Augustus Edward Hough Love

at the University Press, 1906 - Elasticity - 551 pages

What people are saying

User Review -

Page 466 is essentially illegible. So the first sentence of the .pdf file: "This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project
to make the world’s books discoverable online" does not seem entirely true.
The word "carefully" seems to be an overstatement. 

User Review -

Page 20 is partly blank.

Contents

• HISTORICAL INTRODUCTION: 1
• PROBLEMS CONCERNING THE EQUILIBRIUM OF THIN RODS: 3
• Scope of History Galileos enquiry Enunciation of Hookes Law Mariottes: 7
• ANALYSIS OF STRAIN: 32
• Components of strain: 40
• ART PAGE 13 Additional methods and results: 43
• Types of strain a Uniform extension b Simple extension e Shearing strain d Plane strain: 45
• Relations connecting the dilatation the rotation and the displacement: 46
• 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
• Historical Note: 230
• Body bounded by planeAdditional results: 231
• Formulae for the displacement and strain: 232
• Outlines of various methods of integration: 234
• ART PAGE: 235
• THE EQUILIBRIUM OF AN ELASTIC SPHERE AND RELATED PROBLEMS 171 Introductory: 236
• The sphere with given surface displacements: 238
• Generalization of the foregoing solution i Integration by means of poly nomials ii Body force required to maintain a state of strain with zero surface ...: 239
• 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

Bibliographic information

• Title: A Treatise on the Mathematical Theory of Elasticity
• Author: Augustus Edward Hough Love
• Edition: 2
• Publisher: at the University Press, 1906
• Original from: the New York Public Library
• Digitized: Apr 28, 2006
• Length: 551 pages