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Treatise on the Mathematical Theory 4TH Edition

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Treatise on the Mathematical Theory 4TH Edition Cover

 

Synopses & Reviews

Publisher Comments:

Combining a wealth of practical applications with a thorough, rigorous discussion of fundamentals, this work is recognized as an indispensable reference tool for mathematicians and physicists as well as mechanical, civil, and aeronautical engineers. The American Mathematical Monthly hailed it as "the standard treatise on elasticity," praising its significant content, originality of treatment, vigor of exposition, and valuable contributions to the theory.

Starting with a historical introduction, the author discusses the analysis of strain and stress, the elasticity of solid bodies, the equilibrium of isotropic elastic solids, elasticity of crystals, vibration of spheres and cylinders, propagation of waves in elastic solid media, torsion, the theory of continuous beams, the theory of plates, and other topics. A wide range of practical material includes coverage of plates, beams, shells, bending, torsion, vibrations of rods, impact, and more.

Synopsis:

The most complete single-volume treatment of classical elasticity, this text features extensive editorial apparatus, including a historical introduction. Topics include stress, strain, bending, torsion, gravitational effects, and much more. 1927 edition.

Synopsis:

Combining a wealth of practical applications with a thorough, rigorous discussion of fundamentals, this work is recognized as the bible on elasticity for mathematicians and physicists as well as mechanical, civil, and aeronautical engineers. Topics range from the analysis of strain and stress to the elasticity of solid bodies, including a wide range of practical material. 1927 edition.

Table of Contents

HISTORICAL INTRODUCTION

  Scope of History.

  Galileo's enquiry.

  Enunciation of Hooke's Law.

  Mariotte's investigations.

  The problem of the elastica.

  Euler's theory of the stability of struts.

  Researches of Coulomb and Young.

  Euler's theory of the vibrations of bars.

  Attempted theory of the vibrations of bells and plates.

  Value of the researches made before 1820.

  Navier's investigation of the general equations.

  Impulse given to the theory by Fresnel.

  Cauchy's first memoir.

  "Cauchy and Poisson's investigations of the general equations by means of the "molecular" hypothesis."

  Green's introduction of the strain-energy-function.

  Kelvin's application of the laws of Thermodynamics.

  Stoke's criticism of Poisson's theory.

  "The controversy concerning the number of the "elastic constants."

  Methods of solution of the general problem of equilibrium.

  Vibrations of solid bodies.

  Propagation of waves.

  Technical problems.

  Saint-Venant's theories of torsion and flexure.

  Equipollent loads.

  Simplifications and extensions of Saint-Venant's theories.

  Jouravski's treatment of shearing stress in beams.

  Continuous beams.

  Kirchhoff's theory of springs.

  Criticisms and applications of Kirchhoff's theory.

  Vibrations of bars.

  Impact.

  Dynamical resistance.

  The problem of plates.

  The Kirchhoff-Gehring theory.

  Clebsch's modification of this theory.

  Later researches in the theory of plates.

  The problem of shells.

  Elastic stability.

  Conclusion.

CHAPTER I. ANALYSIS OF STRAIN

1 Extension

2 Pure shear

3 Simple shear

4 Displacement

5 Displacement in simple extension and simple shear

6 Homogeneous strain

7 Relative displacement

8 Analysis of the relative displacement

9 Strain corresponding with small displacement

10 Components of strain

11 The strain quadratic

12 Transformation of the components of strain

13 Additional methods and results

14 Types of strain.

  (a) Uniform dilatation

  (b) Simple extension

  (c) Shearing strain

  (d) Plane strain

15 "Relations connecting the dilatation, the rotation and the displacement"

16 Resolution of any strain into dilatation and shearing strains

17 Identical relations between components of strain

18 Displacement corresponding with given strain

19 Curvilinear orthogonal coordinates

20 Components of strain referred to curvilinear orthogonal coordinates

21 Dilatation and rotation referred to curvilinear orthogonal coordinates

22 Cylindrical and polar coordinates

22C Further theory of curvilinear orthogonal coordinates

APPENDIX TO CHAPTER I. GENERAL THEORY OF STRAIN

23 Introductory

24 Strain corresponding with any displacement

25 Cubical dilatation

26 Reciprocal strain ellipsoid

27 Angle between two curves altered by strain

28 Strain ellipsoid

29 Alteration of direction by the strain

30 Application to cartography

31 Conditions satisfied by the displacement

32 Finite homogeneous strain

33 Homogeneous pure strain

34 Analysis of any homogeneous strain into a pure strain and rotation

35 Rotation

36 Simple extension

37 Simple shear

38 Additional results relating to shear

39 Composition of strains

40 Additional results relating to the composition of strains

CHAPTER II. ANALYSIS OF STRESS

41 Introductory

42 Traction across a plane at a point

43 Surface tractions and body forces

44 Equations of motion

45 Equilibrium

46 Law of equilibrium of surface tractions on small volumes

47 Specification of stress at a point

48 Measure of stress

49 Transformation of stress-components

50 The stress quadratic

51 Types of stress.

  (a) Purely normal stress

  (b) Simple tension or pressure

  (c) Shearing stress

  (d) Plane stress

52 Resolution of any stress-system into uniform tension and shearing stress

53 Additional results

54 The stress-equations of motion and of equilibrium

55 Uniform stress and uniformly varying stress

56 Observations concerning the stress-equations

57 Graphic representation of stress

58 Stress-equations referred to curvilinear orthogonal coordinates

59 Special cases of stress-equations referred to curvilinear orthogonal coordinates

CHAPTER III. THE ELASTICITY OF SOLID BODIES

60 Introductory

61 Work and energy

62 Existence of the strain-energy-function

63 Indirectness of experimental results

64 Hooke's Law

65 Form of the strain-energy-function

66 Elastic constants

67 Methods of determining the stress in a body

68 Form of the strain-energy-function for isotropic solids

69 Elastic constants and moduluses of isotropic solids

70 Observations concerning the stress-strain relations in isotropic solids

71 Magnitude of elastic constants and moduluses of some isotropic solids

72 Elastic constants in general

73 Moduluses of elasticity

74 Thermo-elastic equations

75 Initial stress

CHAPTER IV. THE RELATION BETWEEN THE MATHEMATICAL THEORY OF ELASTICITY AND TECHNICAL MECHANICS

76 Limitations of the mathematical theory

77 Stress-strain diagrams

78 Elastic limits

79 Time-effects. Plasticity

79A Momentary stress

80 Viscosity of solids

81 Æolotropy induced by permanent set

82 Repeated loading

82A Elastic hysteresis

83 Hypotheses concerning the conditions of rupture

84 Scope of the mathematical theory of elasticity

CHAPTER V. THE EQUILIBRIUM OF ISOTROPIC ELASTIC SOLIDS

85 Recapitulation of the general theory

86 Uniformly varying stress.

  (a) Bar stretched by its own weight

  (b) Cylinder immersed in fluid

  (c) Body of any form immersed in fluid of same density

  (d) Round bar twisted by couples

87 Bar bent by couples

88 Discussion of the solution for the bending of a bar by terminal couple

89 Saint-Venant's principle

90 Rectangular plate bent by couples

91 Equations of equilibrium in terms of displacements

92 Relations between components of stress

93 Additional results

94 Plane strain and plane stress

95 Bending of narrow rectangular beam by terminal load

96 Equations referred to orthogonal curvilinear coordinates

97 Polar coordinates

98 Radial displacement. Spherical Shell under internal and external pressure. Compression of a sphere by its own gravitation

99 Displacement symmetrical about an axis

100 Tube under pressure

101 Application to gun construction

102 Rotating cylinder. Rotating shaft. Rotating disk

CHAPTER VI. EQUILIBRIUM OF ÆOLOTROPIC ELASTIC SOLID BODIES

103 Symmetry of structure

104 Geometrical symmetry

105 Elastic symmetry

106 Isotropic solid

107 Symmetry of crystals

108 Classification of crystals

109 Elasticity of crystals

110 Various types of symmetry

111 Material with three orthogonal planes of symmetry. Moduluses

112 Extension and bending of a bar

113 Elastic constants of crystals. Results of experiments

114 Curvilinear æolotropy

CHAPTER VII. GENERAL THEOREMS

115 The variational equation of motion

116 Applications of the variational equation

117 The general problem of equilibrium

118 Uniqueness of solution

119 Theorem minimum energy

120 Theorem of concerning the potential energy of deformation

121 The reciprocal theorem

122 Determination of average strains

123 Average strains in an isotropic solid body

124 The general problem of vibrations. Uniqueness of solution

125 Flux of energy in vibratory motion

126 Free vibrations of elastic solid bodies

127 General theorems relating to free vibrations

128 Load suddenly applied or suddenly reversed

CHAPTER VIII. THE TRANSMISSION OF FORCE

129 Introductory

130 Force operative at a point

131 First type of simple solutions

132 Typical nuclei of strain

133 Local perturbations

134 Second type of simple solutions

135 Pressure at a point on a plane boundary

136 Distributed pressure

137 Pressure between two bodies in contact. Geometrical preliminaries

138 Solution of the problem of the pressure between two bodies in contact

139 Hertz's theory of impact

140 Impact of spheres

141 Effects of nuclei of strain referred to polar coordinates

142 Problems relating to the equilibrium of cones

CHAPTER IX. TWO-DIMENSIONAL ELASTIC SYSTEMS

143 Introductory

144 Displacement corresponding with plane strain

145 Displacement corresponding with plane stress

146 Generalized plane stress

147 Introduction of nuclei of strain

148 Force operative at a point

149 Force operative at a point of a boundary

150 Case of a straight boundary

151 Additional results:

  (i) the stress function

  (ii) normal tension on a segment of a straight edge

  (iii) force at an angle

  (iv) pressure on faces of wedge

152 Typical nuclei of strain in two dimensions

153 Transformation of plane strain

154 Inversion

155 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 resting on horizontal plane

156 Examples of transformation

APPENDIX TO CHAPTERS VIII AND IX. VOLTERRA'S THEORY OF DISLOCATIONS

156A Introductory.

  (a) Displacement answering to given strain

  (b) Discontinuity at a barrier

  (c) Hollow cylinder deformed by removal of a slice of uniform thickness

  (d) Hollow cylinder with radial fissure

CHAPTER X. THEORY OF THE INTEGRATION OF THE EQUATIONS OF EQUILIBRIUM OF AN ISOTROPIC ELASTIC SOLID BODY

157 Nature of the problem

158 Résumé of the theory of Potential

159 Description of Betti's method of integration

160 Formula for the dilatation

161 Calculation of the dilatation from surface data

162 Formulæ for the components of rotation

163 Calculation of the rotation from surface data

164 Body bounded by plane?Formulæ for the dilatation

165 Body bounded by plane?Given surface displacements

166 Body bounded by plane?Given surface tractions

167 Historical Note

168 Body bounded by plane?Additional results

169 Formulæ for the displacement and strain

170 Outlines of various methods of integration

CHAPTER XI. THE EQUILIBRIUM OF AN ELASTIC SPHERE AND RELATED PROBLEMS

171 Introductory

172 Special solutions in terms of spherical harmonics

173 Applications of the special solutions:

  (i) Solid sphere with purely radial surface displacement

  (ii) Solid sphere with purely radial surface traction

  (iii) Small spherical cavity in large solid mass

  (iv) Twisted sphere

174 Sphere subjected to body force

175 Generalization and Special Cases of the foregoing solution

176 Gravitating incompressible sphere

177 Deformation of gravitating incompressible sphere by external body force

178 Gravitating body of nearly spherical form

179 Rotating sphere under its own attraction

180 Tidal deformation. Tidal effective rigidity of the Earth

181 A general solution of the equations of equilibrium

182 Applications and extension of the foregoing solution

183 The sphere with given surface displacements

184 Generalization of the foregoing solution

185 The sphere with give surface tractions

186 Plane strain in a circular cylinder

187 Applications of curvilinear coordinates

188 Symmetrical strain in a solid of revolution

189 Symmetrical strain in a cylinder

CHAPTER XII. VIBRATIONS OF SPHERES AND CYLINDERS

190 Introductory

191 Solution by means of spherical harmonics

192 Formation of the boundary-conditions for a vibrating sphere

193 Incompressible material

194 Frequency equations for vibrating sphere

195 Vibrations of the first class

196 Vibrations of the second class

197 Further investigations on the vibrations of spheres

198 Radial vibrations of a hollow sphere

199 Vibrations of a circular cylinder

200 Torsional vibrations

201 Longitudinal vibrations

202 Transverse vibrations

CHAPTER XIII. THE PROPAGATION OF WAVES IN ELASTIC SOLID MEDIA

203 Introductory

204 Waves of dilatation and waves of distortion

205 Motion of a surface of discontinuity. Kinematical conditions

206 Motion of a surface of discontinuity. Dynamical conditions

207 Velocity of waves in isotropic medium

208 Velocity of waves in æolotropic medium

209 Wave-surfaces

210 Motion determined by the characteristic equation

211 Arbitrary initial conditions

212 Motion due to body forces

213 Additional results relating to motion due to body forces

214 Waves propagated over the surface of an isotropic elastic solid body

CHAPTER XIV. TORSION

215 Stress and strain in a twisted prism

216 The torsion problem

217 Method of solution of the torsion problem

218 Analogies with Hydrodynamics

219 Distribution of the shearing stress

220 Strength to resist torsion

221 Solution of the torsion problem for certain boundaries

222 Additional results

223 Graphic expression of the results

224 Analogy to the form of a stretched membrane loaded uniformly

225 Twisting couple

226 Torsion of æolotropic prism

226A Bar of varying circular section

226B Distribution of traction over terminal section

CHAPTER XV. THE BENDING OF A BEAM BY TERMINAL TRANSVERSE LOAD

227 Stress in bent beam

228 Statement of the problem

229 Necessary type of shearing stress

230 Formulæ for the displacement

231 Solution of the problem of flexure for certain boundaries:

  (a) The circle

  (b) Concentric circles

  (c) The ellipse

  (d) Confocal ellipses

  (e) The rectangle

  (f) Additional results

232 Analysis of the displacement:

  (a) Curvature of the strained central-line

  (b) Neutral plane

  (c) Obliquity of the strained cross-sections

  (d) Deflexion

  (e) Twist

  (f) Antilclastic curvature

  (g) Distortion of the cross-sections into curved surfaces

233 Distribution of shearing stress

234 Generalizations of the foregoing theory:

  (a) Asymmetric loading

  (b) Combined strain

  (c) Æolotropic material

234C Analogy to the form of a stretched membrane under varying pressure

235 Criticisate or shell

325 Method of calculating the extension and the changes of curvature

326 Formulæ relating to small displacements

327 Nature of the strain in a bent plate or shell

328 Specification of stress in a bent plate of shell

329 "Approximate formulæ for the strain, the stress-resultants, and the stress-couples"

330 Second approximation in the case of a curved plate or shell

331 Equations of equilibrium

332 Boundary-conditions

332A Buckling of a rectangular plate under edge thrust

333 Theory of the vibrations of thin shells

334 Vibrations of a thin cylindrical shell.

  (a) General equations

  (b) Extensional vibrations

  (c) Inextensional vibrations

  (d) Inexactness of the inextensional displacement

  (e) Nature of the correction to be applied to the inextensional displacement

335 Vibrations of a thin spherical shell

CHAPTER XXIVA. EQUILIBRIUM OF THIN PLATES AND SHELLS

335C Large deformations of plates and shells

335D Plate bent to cylindrical form

335E Large thin plate subjected to pressure

335F Long strip. Supported edges

335G Long strip. Clamped edges

EQUILIBRIUM OF THIN SHELLS

336 Small displacement

337 The middle surface a surface of revolution

338 Torsion

CYLINDRICAL SHELL

339 Symmetrical conditions.

  (a) Extensional solution

  (b) Edge-effect

340 Tube under pressure

341 Stability of a tube under external pressure

342 Lateral forces.

  (a) Extensional solution

  (b) Edge-effect

343 General unsymmetrical conditions. Introductory.

  (a) Extensional solution

  (b) Approximately inextensional solutions

  (c) Edge-effect

SPHERICAL SHELL

344 Extensional solution

345 Edge-effect. Symmetrical conditions

CONICAL SHELL

346 Extensional solution. Symmetrical conditions

347 Edge-effect. Symmetrical conditions

348 Extensional solution. Lateral forces

349 Edge-effect. Lateral forces. Introductory.

  (a) Integrals of the equations of equilibrium

  (b) Introduction of the displacement

  (c) Formation of two linear differential equations

  (d) Method of solution of the equations

350 Extensional solution. Unsymmetrical conditions

351 Approximately inextensional solution

352 Edge-effect. Unsymmetrical conditions?Introductory.

  (a) Formation of the equations

  (b) Preparation for solution

  (c) Solution of the equations

NOTES

A. Terminology and Notation

B. The notation of stress. Definition of stress in a system of particles. Lattice of simple point-elements (Cauchy's theory). Lattice of multiple point-elements

C. Applications of the method of moving axes

INDEX

  Authors cited

  Matters treated

Product Details

ISBN:
9780486601748
Author:
Love, Augustus E.
Publisher:
Dover Publications
Author:
Love, Augustus E.
Author:
Love, A. E. H.
Author:
Engineering
Author:
Love, Augustus Edward Hough
Location:
New York :
Subject:
General
Subject:
Mechanics
Subject:
Elasticity
Subject:
Mechanics - General
Subject:
Engineering - Civil
Subject:
Physics-Classical Mechanics
Edition Number:
4
Edition Description:
Trade Paper
Series:
Dover Books on Engineering
Publication Date:
20110631
Binding:
TRADE PAPER
Language:
English
Pages:
672
Dimensions:
9.25 x 6.13 in 1.93 lb

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Treatise on the Mathematical Theory 4TH Edition New Trade Paper
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Product details 672 pages Dover Publications - English 9780486601748 Reviews:
"Synopsis" by ,
The most complete single-volume treatment of classical elasticity, this text features extensive editorial apparatus, including a historical introduction. Topics include stress, strain, bending, torsion, gravitational effects, and much more. 1927 edition.
"Synopsis" by , Combining a wealth of practical applications with a thorough, rigorous discussion of fundamentals, this work is recognized as the bible on elasticity for mathematicians and physicists as well as mechanical, civil, and aeronautical engineers. Topics range from the analysis of strain and stress to the elasticity of solid bodies, including a wide range of practical material. 1927 edition.
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