A Course in Elasticity

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.