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 strainenergyfunction. 

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. 

SaintVenant's theories of torsion and flexure. 

Equipollent loads. 

Simplifications and extensions of SaintVenant'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 KirchhoffGehring 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 stresscomponents 
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 stresssystem into uniform tension and shearing stress 
53 Additional results 
54 The stressequations of motion and of equilibrium 
55 Uniform stress and uniformly varying stress 
56 Observations concerning the stressequations 
57 Graphic representation of stress 
58 Stressequations referred to curvilinear orthogonal coordi 
59 Special cases of stressequations referred to curvilinear orthogonal coordinates 
CHAPTER III. THE ELASTICITY OF SOLID BODIES 
60 Introductory 
61 Work and energy 
62 Existence of the strainenergyfunction 
63 Indirectness of experimental results 
64 Hooke's Law 
65 Form of the strainenergyfunction 
66 Elastic constants 
67 Methods of determining the stress in a body 
68 Form of the strainenergyfunction for isotropic solids 
69 Elastic constants and moduluses of isotropic solids 
70 Observations concerning the stressstrain 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 Thermoelastic equations 
75 Initial stress 
CHAPTER IV. THE RELATION BETWEEN THE MATHEMATICAL THEORY OF ELASTICITY AND TECHNICAL MECHANICS 
76 Limitations of the mathematical theory 
77 Stressstrain diagrams 
78 Elastic limits 
79 Timeeffects. 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 SaintVenant'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. TWODIMENSIONAL 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 boundaryconditions 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 Wavesurfaces 
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 centralline 

(b) Neutral plane 

(c) Obliquity of the strained crosssections 

(d) Deflexion 

(e) Twist 

(f) Antilclastic curvature 

(g) Distortion of the crosssections 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 