A Course in Elasticity: (29 Applied Mathematical Sciences)

This book is based on lecture notes of the late Professor de Veubeke. The subject is presented at a level suitable for graduate students in engineering, physics, or mathematics. Some exposure to linear algebra, complex analysis, variational calculus, or basic continuum mechanics would be helpful. The first third of the book contains the fundamentals of the theory of elasticity. Kinematics of continuous media, the notions of stress and equilibrium, conservation of energy, 'and the elastic constitutive law are each treated first in a nonlinear context, then specialized to the linear case. The remainder of the book is given to three classic applications of the theory, each supplemented by original re­ sults based on the use of complex variables. Each one of the three topics - Saint-Venant's theory of prismatic beams, plane deformations, and the bending of plates - is first pre­ sented and analyzed in general, then rounded out with numerous specific and sometimes novel examples. The following notational conventions are generally in force, except where noted to the contrary: lower case boldface letters denote vectors or triples of Cartesian co­ ordinates, upper case boldface letters denote 3 x 3 matrices, repeated lower case Latin subscripts are summed over (1,2,3), and non-repeated lower case Latin subscripts are assumed to range over (1,2,3).

Table of Contents:
1. Kinematics of Continuous Media.- 1.1. Material and Spatial Coordinates.- 1.2. Neighborhood Transformations.- 1.3 Composition of Changes of Configuration.- 1.4 Measure of the State of Local Deformation. Green’s and Jaumann’s Strain.- 1.5 Rigid-Body Rotations of a Neighborhood.- 1.6 The Kinematical Decomposition of the Jacobian Matrix.- 1.7 Geometric Interpretation of Infinitesimal Strains.- 1.8 The Eulerian Viewpoint in Kinematics. Almansi’s Strain.- 1.9 Eulerian Measures of Rates of Deformation and Rotation.- 1.10 Temporal, Variation of the Polar Decomposition of the Jacobian Matrix.- 2. Statics and Virtual Work.- 2.1. The Concept of Stress. True Stress.- 2.2. The Piola Stresses.- 2.3. Translational Equilibrium Equations.- 2.4. Rotational Equilibrium Equations.- 2.5. Statics and Virtual Work.- 2.6. Commutativity of the Operators ? and Di.- 2.7 Virtual Work in a Continuous Medium.- 2.8. Statics and Virtual Power for True Stresses.- 2.9. Statics and Virtual Work in Infinitesimal Changes of Configuration.- 3. Conservation of Energy.- 3.1. Constitutive Equations for Piola’s Stresses.- 3.2. The Kirchhoff-Trefftz Stresses.- 3.3 The Constitutive Equations of Geometrically Linear Elasticity.- 4. Cartesian Tensors.- 4.1. Bases and Change of Basis.- 4.2 Tensors.- 4.3 Some Special Tensors.- 4.4 The Vector Product.- 4.5. Structure of Symmetric Cartesian Tensors of Order Two. Principal Axes.- 4.6. Fundamental Invariants and the Deviator.- 4.7. Structure of Skew-Symmetric Cartesian Tensors of the Second Order.- 4.8. Matrix Representation of Tensor Operations.- 5. The Equations of Linear Elasticity.- 5.1. Compatibility of Strains in a Simply Connected Region.- 5.2. Compatibility of Strains in a Multiply Connected Region.- 5.3. Principal Elongations and FundamentalInvariants of Strain.- 5.4. Principal Stresses and Fundamental Invariants of the Stress State.- 5.5. Octahedral Stresses and Strains.- 5.6. Mohr’s Circles.- 5.7. Statics and Virtual Work.- 5.8. Taylor’s Development of the Strain Energy.- 5.9. Infinitesimal Stability.- 5.10. Hadamard’s Condition for Infinitesimal Stability.- 5.11. Isotropy and Anisotropy.- 5.12. Criteria for Elastic Limits.- 5.13. Navier’s Equations.- 5.14. The Beltrami-Michell Equations.- 6. Extension, Bending, and Torsion of Prismatic Beams.- 6.1. Green’s and Stokes’ Formulas.- 6.2. The Centroid.- 6.3. Moments of Inertia.- 6.4. The Semi-Inverse Method of Saint-Venant.- 6.5. Resultants of Stresses on a Cross Section.- 6.6. Calculation of the Transverse Displacements.- 6.7. Equations Governing the Shear Stresses.- 6.8. Calculation of the Longitudinal Displacement.- 6.9. Separation of Solutions.- 6.10. Pure Torsion.- 6.11. The Center of Torsion for a Fully Constrained Section.- 6.12. Bending without Torsion.- 6.13. The Stiffness Relation for the Twist.- 6.14. Total Energy as a Function of the Deformations of the Fibers.- 6.15. Total Energy as a Function of Generalized Forces.- 6.16. The Generalized Constitutive Equations for Bending and Torsion of Beams.- 6.17. One-Dimensional Formulation of Bending and Torsion of Beams.- 6.18. Applications.- 7. Plane Stress and Plane Strain.- 7.1. Lemmas for the Integration of Partial Differential Equations in Complex Form.- 7.2. The Structure of a Biharmonic Function.- 7.3. Structure of the Solution of the Problems of Plane Strain.- 7.4.Structure of the Solution of the Problem of Plane Stress.- 7.5. Generalized Plane Stress.- 7.6. Airy’s Stress Function.- 7.7. Complex Representation of Airy’s Function.- 7.8. Polar Coordinates.- 7.9. Applications inCartesian Coordinates.- 7.10. Applications in Polar Coordinates.- 8. Bending of Plates.- 8.1. Basic Hypotheses.- 8.2. Application of the Canonical Variational Principle.- 8.3. The Two-Dimensional Canonical Principle.- 8.4. Further Connections Between the Two- and Three-Dimensional Theories.- 8.5. Other Types of Approximations.- 8.6. Kirchhoff’s Hypothesis.- 8.7. Boundary Conditions in Kirchhoff’s Theory.- 8.8. Kirchhoff’s Variational Principle.- 8.9. Structure of the Solution of the Equations of Plates of Moderate Thickness.- 8.10. The Edge Effect.- 8.11. Torsion of a Plate.- 8.12. Saint-Venant’s Bending of a Plate.- 8.13. Particular Solutions for Transverse Load.- 8.14. Solutions in Polar Coordinates.- 8.15. Axisymmetric Bending.