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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.

Table of contents provided by Syndetics

• 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

Author notes provided by Syndetics