## A Treatise on the Mathematical Theory of ElasticityA. 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 |

637 | |

638 | |

### Other editions - View all

A Treatise on the Mathematical Theory of Elasticity (Classic Reprint) Augustus Edward Hough Love No preview available - 2018 |

### Common terms and phrases

applied Article axis beam bending body forces boundary central-line centre centroid components of strain constants coordinates corresponding cos2 couple cross-section cubical curvature curve cylinder deﬁnite denote determined dilatation direction displacement ellipsoid equal equations of equilibrium expressed extension ﬁgure ﬁnd ﬁrst ﬁxed ﬂexural formulae free from traction given harmonic function homogeneous indeﬁnitely inﬁnite integral isotropic length linear elasticity linear elements load Lord Rayleigh Math method obtained origin orthogonal parallel Phil plane strain plane stress plate polar coordinates pressure principal axes prism problem Proc quadratic function quantities radius right angles rigidity rotation satisﬁed satisfy the equations shearing strain shearing stress simple shear sin2 solid harmonics solution speciﬁed sphere statically equivalent strain-components strain-energy-function stress-components stress-system surface tractions symmetry tangential traction tension theorem theory torsion transformed twist unstrained unstressed values vanish vector velocity vibrations