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A Treatise on the Mathematical Theory of Elasticity

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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 Aeolotropy 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 AeOLOTROPIC 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 aeolotropy 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 Resume 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 Formulae for the components of rotation 163 Calculation of the rotation from surface data 164 Body bounded by plane?Formulae 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 Formulae 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 aeolotropic 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 aeolotropic 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 Formulae 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) Aeolotropic 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 Formulae 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 formulae 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

About the Author

In addition to his work on elasticity, Augustus Edward Hough Love (1863-1940) studied wave propagation and was awarded the prestigious Adams Prize in 1911 for his development of a mathematical model of surface waves known as Love waves.

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