The Linearized Theory of Elasticity

by William S. Slaughter

Overview

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.

Product Details

ISBN-13:	9780817641177
Publisher:	Birkhäuser Boston
Publication date:	12/14/2001
Edition description:	2002
Pages:	543
Product dimensions:	6.10(w) x 9.25(h) x 0.05(d)

Table of Contents

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.

Editorial Reviews

"There is a good balance between theory and practical applications…[the] approach acknowledges the basic concepts of continuum mechanics without burdening the presentation with excessive generalities. The assumptions required to obtain linear results from nonlinear results are clearly described. This enables students to clearly understand the limitations of linear results…the book includes a good range of discussion and examples…to motivate and complement the theory…[it] is written in a clear style…[and] can be recommended as a good example of a modern textbook in this field." —Applied Mechanics Review

"This very accessible book will be of interest in teaching or learning linear elasticity." —Zentralblatt Math

"The book presents classical parts of the linearized theory of elasticity in a selfcontained way that seems to be a fine compromise between the necessity of a deep mathematical insight and the accessibility of exposition. The author points out that the book is intended as a text for a first-year graduate course in mechanical or civil engineering. ...Many figures and solved examples contribute to the clarity of exposition. Moreover, each chapter finishes with a subsection of unresolved problems, hints being often given. The material in the book is well organized, presented in a lucid way, and can reach a fairly broad audience spanning from advanced undergraduate students to graduate students. Professionals and researchers may enjoy this book for its clarity and instructive examples, as well as a refreshing reminder of the classical results of the linearized theory of elasticity." —Applications of Mathematics

"This book is a modern treatment of the linearized theory of elasticity, presented as a specialization of the general theory of continuum mechanics. It is derived from notes used by the author in teaching a first-yeargraduate-level course in elasticity…. Presented [are] various results connected to functions of a complex variable, strain, plane strain/stress, etc.… Each chapter ends with a useful list of problems.

The book is clearly written, with rigorous presentation, in a pleasant and accessible style. This new text is an excellent resource devoted to introduce the students in mechanical or civil engineering to the linearized theory of elasticity. It is warmly recommended to all researchers in the field." —Revue D’Analyse Numérique et de Théorie de L’Approximation

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