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Treatise on the Mathematical Theory 4TH Editionby Augustus E Love
Synopses & Reviews
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.
The most complete single-volume treatment of classical elasticity, this text features extensive editorial apparatus, including a historical introduction. Topics include stress, strain, bending, torsion, gravitational effects, and much more. 1927 edition.
Combining a wealth of practical applications with a thorough, rigorous discussion of fundamentals, this work is recognized as the bible on elasticity for mathematicians and physicists as well as mechanical, civil, and aeronautical engineers. Topics range from the analysis of strain and stress to the elasticity of solid bodies, including a wide range of practical material. 1927 edition.
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.
Table of Contents
Scope of History.
Enunciation of Hooke's Law.
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.
Saint-Venant's theories of torsion and flexure.
Simplifications and extensions of Saint-Venant's theories.
Jouravski's treatment of shearing stress in beams.
Kirchhoff's theory of springs.
Criticisms and applications of Kirchhoff's theory.
Vibrations of bars.
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.
CHAPTER I. ANALYSIS OF STRAIN
2 Pure shear
3 Simple shear
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
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
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
42 Traction across a plane at a point
43 Surface tractions and body forces
44 Equations of motion
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
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
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
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
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
(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
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
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
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
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
(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
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
339 Symmetrical conditions.
(a) Extensional solution
340 Tube under pressure
341 Stability of a tube under external pressure
342 Lateral forces.
(a) Extensional solution
343 General unsymmetrical conditions. Introductory.
(a) Extensional solution
(b) Approximately inextensional solutions
344 Extensional solution
345 Edge-effect. Symmetrical conditions
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
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
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