The Linearized Theory of Elasticity, 2002

This book is derived from notes used in teaching a first-year graduate-level course in elasticity in the Department of Mechanical Engineering at the University of Pittsburgh. This is a modern treatment of the linearized theory of elasticity, which is presented as a specialization of the general theory of continuum mechanics. It includes a comprehensive introduction to tensor analysis, a rigorous development of the governing field equations with an emphasis on recognizing the assumptions and approximations in­ herent in the linearized theory, specification of boundary conditions, and a survey of solution methods for important classes of problems. Two- and three-dimensional problems, torsion of noncircular cylinders, variational methods, and complex variable methods are covered. This book is intended as the text for a first-year graduate course in me­ chanical or civil engineering. Sufficient depth is provided such that the text can be used without a prerequisite course in continuum mechanics, and the material is presented in such a way as to prepare students for subsequent courses in nonlinear elasticity, inelasticity, and fracture mechanics. Alter­ natively, for a course that is preceded by a course in continuum mechanics, there is enough additional content for a full semester of linearized elasticity.

1 Review of Mechanics of Materials.- 1.1 Forces and Stress.- 1.2 Stress and Strain.- 1.3 Torsion of Circular Cylinders.- 1.4 Bending of Prismatic Beams.- Problems.- 2 Mathematical Preliminaries.- 2.1 Scalars and Vectors.- 2.2 Indicial Notation.- 2.3 Tensors.- 2.4 Tensor Calculus.- 2.5 Cylindrical and Spherical Coordinates.- Problems.- 3 Kinematics.- 3.1 Configurations.- 3.2 Strain Tensors: Referential Formulation.- 3.3 Strain Tensors: Spatial Formulation.- 3.4 Kinematic Linearization.- 3.5 Cylindrical and Spherical Coordinates.- Problems.- 4 Forces and Stress.- 4.1 Stress Tensors: Referential Formulation.- 4.2 Stress Tensors: Spatial Formulation.- 4.3 Kinematic Linearization.- 4.4 Cylindrical and Spherical Coordinates.- Problems.- 5 Constitutive Equations.- 5.1 Elasticity.- 5.2 Constitutive Linearization.- 5.3 Material Symmetry.- 5.4 Isotropic Materials.- 5.5 Cylindrical and Spherical Coordinates.- Problems.- 6 Linearized Elasticity Problems.- 6.1 Field Equations.- 6.2 Boundary Conditions.- 6.3 Useful Consequences of Linearity.- 6.4 Solution Methods.- Problems.- 7 Two-Dimensional Problems.- 7.1 Antiplane Strain.- 7.2 Plane Strain.- 7.3 Plane Stress.- 7.4 Airy Stress Function.- Problems.- 8 Torsion of Noncircular Cylinders.- 8.1 Warping Function.- 8.2 Prandtl Stress Function.- Problems.- 9 Three-Dimensional Problems.- 9.1 Field Theory Results.- 9.2 Potentials in Elasticity.- 9.3 Dislocation Surface.- 9.4 Eshelby’s Inclusion Problems.- Problems.- 10 Variational Methods.- 10.1 Calculus of Variations.- 10.2 Energy Theorems in Elasticity.- 10.3 Approximate Solutions.- Problems.- 11 Complex Variable Methods.- 11.1 Functions of a Complex Variable.- 11.2 Antiplane Strain.- 11.3 Plane Strain/Stress.- Problems.- Appendix: General Curvilinear Coordinates.- A.l General VectorBases.- A.1.1 Covariant and Contravariant Components.- A.1.2 Reciprocal Bases.- A.l.3 Higher-Order Tensors.- A.2 Curvilinear Coordinates.- A.2.1 Cartesian Coordinates.- A.2.2 Cylindrical Coordinates.- A.2.3 Spherical Coordinates.- A.2.4 Metric Tensor in a Natural Vector Basis.- A.2.5 Transformation Rule for Change of Coordinates.- A.3 Tensor Calculus.- A.3.l Gradient.- References.

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