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Publisher:

Dover Publications

Publication Date:

1927

Number of Pages:

643

Format:

Paperback

Edition:

4

Price:

24.95

ISBN:

978-0486601748

Category:

Monograph

The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

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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 displac | |||||||

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 coordi | |||||||

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 | |||||||

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 | |||||||

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 s | |||||||

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 the | |||||||

(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 |

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