HISTORICAL INTRODUCTION |
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Scope of History. |
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Galileo's enquiry. |
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Enunciation of Hooke's Law. |
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Mariotte's investigations. |
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The problem of the elastica. |
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Euler's theory of the stability of struts. |
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Researches of Coulomb and Young. |
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Euler's theory of the vibrations of bars. |
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Attempted theory of the vibrations of bells and plates. |
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Value of the researches made before 1820. |
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Navier's investigation of the general equations. |
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Impulse given to the theory by Fresnel. |
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Cauchy's first memoir. |
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"Cauchy and Poisson's investigations of the general equations by means of the "molecular" hypothesis." |
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Green's introduction of the strain-energy-function. |
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Kelvin's application of the laws of Thermodynamics. |
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Stoke's criticism of Poisson's theory. |
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"The controversy concerning the number of the "elastic constants." |
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Methods of solution of the general problem of equilibrium. |
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Vibrations of solid bodies. |
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Propagation of waves. |
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Technical problems. |
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Saint-Venant's theories of torsion and flexure. |
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Equipollent loads. |
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Simplifications and extensions of Saint-Venant's theories. |
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Jouravski's treatment of shearing stress in beams. |
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Continuous beams. |
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Kirchhoff's theory of springs. |
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Criticisms and applications of Kirchhoff's theory. |
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Vibrations of bars. |
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Impact. |
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Dynamical resistance. |
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The problem of plates. |
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The Kirchhoff-Gehring theory. |
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Clebsch's modification of this theory. |
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Later researches in the theory of plates. |
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The problem of shells. |
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Elastic stability. |
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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. |
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(a) Uniform dilatation |
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(b) Simple extension |
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(c) Shearing strain |
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(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. |
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(a) Purely normal stress |
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(b) Simple tension or pressure |
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(c) Shearing stress |
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(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. |
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(a) Bar stretched by its own weight |
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(b) Cylinder immersed in fluid |
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(c) Body of any form immersed in fluid of same density |
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(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: |
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(i) the stress function |
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(ii) normal tension on a segment of a straight edge |
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(iii) force at an angle |
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(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. |
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(i) Two opposed forces at points on the rim |
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(ii) Any forces applied to the rim |
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(iii) Heavy disk resting on horizontal |
156 Examples of transformation |
APPENDIX TO CHAPTERS VIII AND IX. VOLTERRA'S THEORY OF DISLOCATIONS |
156A Introductory. |
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(a) Displacement answering to given strain |
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(b) Discontinuity at a barrier |
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(c) Hollow cylinder deformed by removal of a slice of uniform thickness |
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(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: |
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(i) Solid sphere with purely radial surface displacement |
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(ii) Solid sphere with purely radial surface traction |
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(iii) Small spherical cavity in large solid mass |
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(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: |
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(a) The circle |
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(b) Concentric circles |
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(c) The ellipse |
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(d) Confocal ellipses |
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(e) The rectangle |
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(f) Additional results |
232 Analysis of the displacement: |
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(a) Curvature of the strained central-line |
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(b) Neutral plane |
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(c) Obliquity of the strained cross-sections |
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(d) Deflexion |
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(e) Twist |
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(f) Antilclastic curvature |
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(g) Distortion of the cross-sections into curved surfaces |
233 Distribution of shearing stress |
234 Generalizations of the foregoing the |
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(a) Asymmetric loading |
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(b) Combined strain |
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(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 |