Elasticity: Theory and Applications

Elasticity: Theory and Applications,  now in a revised and updated second edition, has long been used as a textbook by seniors and graduate students in civil, mechanical, and biomedical engineering, since the first edition was published in 1974. The kinematics of continuous media and the analysis of stress are introduced through the concept of linear transformation of points and brought together to study in great detail the linear theory of elasticity as well as its application to a variety of practical problems. Elastic stability, the theory of thin plates, and the theory of thin shells are covered. Complex variables are introduced and used to solve two dimensional and fracture-related problems. Through theory, solved examples and problems, this authoritative book helps the student acquire the foundation needed to pursue advanced studies in all the branches of continuum mechanics. It also helps practitioners understand the source of many of the formulas they use in their designs. A solutions manual is available to instructors.

Table of Contents:
About the Author Preface Part I KINEMATICS OF CONTINUOUS MEDIA (Displacement, Deformation, Strain) Chapter 1 Introduction to the Kinematics of Continuous Media 1-1 Formulation of the Problem 1-2 Notation Chapter 2 Review of Matrix Algebra2-1 Introduction2-2 Definition of a Matrix. Special Matrices 2-3 Index Notation and Summation Convention 2-4 Equality of Matrices. Addition and Subtraction2-5 Multiplication of Matrices 2-6 Matrix Division. The Inverse Matrix Problems Chapter 3 Linear Transformation of Points3-1 Introduction 3-2 Definitions and Elementary Operations 3-3 Conjugate and principal Directions and Planes in a Linear Transformation 3-4 Orthogonal Transformations 3-5 Changes of Axes in a Linear Transformation 3-6 Characteristic Equations and Eigenvalues 3-7 Invariants of the Transformation Matrix in a Linear Transformation 3-8 Invariant Directions of a Linear Transformation 3-9 Antisymmetric Linear Transformations3-10 Symmetric Transformations. Definitions and General Theorems3-11 Principal Directions and Principal Unit Displacements of a Symmetric Transformation 3-12 Quadratic Forms 3-13 Normal and Tangential Displacements in a Symmetric Transformation. Mohr's Representation3-14 Spherical Dilatation and Deviation in a Linear Symmetric Transformation3-15 Geometrical Meaning of the aij's in a Linear Symmetric Transformation3-16 Linear Symmetric Transformation in Two Dimensions Problems Chapter 4 General Analysis of Strain in Cartesian Coordinates4-1 Introduction4-2 Changes in Length and Directions of Elements Initially Parallel to the Coordinate Axes 4-3 Components of the State of Strain at a Point 4-4 Geometrical Meaning of the Strain Components e.Strain of a Line Element4-5 Components of the State of Strain under a Change of Coordinate System4-6 Principal Axes of Strain 4-7 Volumetric Strain 4-8 Small Strain 4-9 Linear Strain 4-10 Compatibility Relations for Linear Strains Problems Chapter 5 Cartesian Tensors 5-1 Introduction5-2 Scalars and Vectors 5-3 Higher Rank Tensors 5-4 On Tensors and Matrices 5-5 The Kronecker Delta and the Alternating Symbol. Isotropic Tensors 5-6 Function of a Tensor. Invariants 5-7 Contraction 5-8 The Quotient Rule of Tensors Problems Chapter 6 Orthogonal Curvilinear Coordinates 6-1 Introduction 6-2 Curvilinear Coordinates6-3 Metric Coefficients 6-4 Gradient, Divergence, Curl, and Laplacian in Orthogonal Curvilinear Coordinates 6-5 Rate of Change of the Vectors a, and of the Unit Vectors Z, in an Orthogonal Curvilinear Coordinate System 6-6 The Strain Tensor in Orthogonal Curvilinear Coordinates 6-7 Strain-Displacement Relations in Orthogonal Curvilinear Coordinates 6-8 Components of the Rotation in Orthogonal Curvilinear Coordinates6-9 Equations of Compatibility for Linear Strains in Orthogonal Curvilinear Coordinates Problems Part II THEORY OF STRESSChapter 7 Analysis of Stress7-1 Introduction 7-2 Stress on a Plane at a Point. Notation and Sign Convention 7-3 State of Stress at a Point. The Stress Tensor7-4 Equations of Equilibrium. Symmetry of the Stress Tensor. Boundary Conditions7-5 Application of the Properties of Linear Symmetric Transformations to the Analysis of Stress 7-6 Stress Quadric 7-7 Further Graphical Representations of the State of Stress at a Point. Stress Ellipsoid. Stress Director Surface 7-8 The Octahedral Normal and Octahedral Shearing Stresses 7-9 The Haigh-Westergaard Stress Space 7-10 Components of the State of Stress at a Point in a Change of Coordinates 7-11 Stress Analysis in Two Dimensions 7-12 Equations of Equilibrium in Orthogonal Curvilinear Coordinates Problems Part III THE THEORY OF ELASTICITY - APPLICATIONS TO ENGINEERING PROBLEMS Chapter 8 Elastic Stress-Strain Relations and Formulation of Elasticity Problems 8-1 Introduction 8-2 Work, Energy, and the Existence of a Strain Energy Function 8-3 The Generalized Hooke's Law 8-4 Elastic Symmetry 8-5 Elastic Stress-Strain Relations for Isotropic Media 8-6 Thermoelastic Stress-Strain Relations for Isotropic Media 8-7 Strain Energy Density8-8 Formulation of Elasticity Problems. Boundary-Value Problems of Elasticity 8-9 Elasticity Equations in Terms of Displacements8-10 Elasticity Equations in Terms of Stresses 8-11 The Principle of Superposition 8-12 Existence and Uniqueness of the Solution of an Elasticity Problem 8-13 Saint-Venant's Principle 8-14 One Dimensional Elasticity 8-15 Plane Elasticity8-16 State of Plane Strain 8-17 State of Plane Stress 8-18 State of Generalized Plane Stress 8-19 State of Generalized Plane Strain 8-20 Solution of Elasticity Problems ProblemsChapter 9 Solution of Elasticity Problems by Potentials9-1 Introduction9-2 Some Results of Field Theory9-3 The Homogeneous Equations of Elasticity and the Search for Particular Solutions 9-4 Scalar and Vector Potentials. Lame's Strain Potential9-5 The Galerkin Vector. Love's Strain Function. Kelvin's and Cerruti's Problems 9-6 The Neuber-Papkovich Representation. Boussinesq's Problem9-7 Summary of Displacement Functions 9-8 Stress Functions 9-9 Airy's Stress Function for Plane Strain Problems 9-10 Airy's Stress Function for Plane Stress Problems 9-11 Forms of Airy's Stress Function Problems Chapter 10 The Torsion Problem10-1 Introduction 10-2 Torsion of Circular Prismatic Bars 10-3 Torsion of Non-Circular Prismatic Bars 10-4 Torsion of an Elliptic Bar 10-5 Prandtl's Stress Function 10-6 Two Simple Solutions Using Prandtl's Stress Function 10-7 Torsion of Rectangular Bars 10-8 Prandtl's Membrane Analogy 10-9 Application of the Membrane Analogy to Solid Sections 10-10 Application of the Membrane Analogy to Thin Tubular Members 10-11 Application of the Membrane Analogy to Multicellular Thin Sections10-12 Torsion of Circular Shafts of Varying Cross Section10-13 Torsion of Thin-Walled Members of Open Section in which some Cross Section is Prevented from WarpingA-10-1 The Green-Riemann FormulaProblems Chapter 11 Thick Cylinders, Disks, and Spheres 11-1 Introduction 11-2 Hollow Cylinder with Internal and External Pressures and Free Ends11-3 Hollow Cylinder with Internal and External Pressures and Fixed Ends11-4 Hollow Sphere Subjected to Internal and External Pressures 11-5 Rotating Disks of Uniform Thickness 11-6 Rotating Long Circular Cylinder 11-7 Disks of Variable Thickness 11-8 Thermal Stresses in Thin Disks 11-9 Thermal Stresses in Long Circular Cylinders 11-10 Thermal Stresses in Spheres Problems Chapter 12 Straight Simple Beams12-1 Introduction 12-2 The Elementary Theory of Beams 12-3 Pure Bending of Prismatical Bars 12-4 Bending of a Narrow Rectangular Cantilever by an End Load 12-5 Bending of a Narrow Rectangular Beam by a Uniform Load 12-6 Cantilever Prismatic Bar of Irregular Cross Section Subjected to a Transverse End Force 12-7 Shear Center Problems Chapter 13 Curved Beams 13-1 Introduction 13-2 The Simplified Theory of Curved Beams 13-3 Pure Bending of Circular Arc Beams 13-4 Circular Arc Cantilever Beam Bent by a Force at the End Problems Chapter 14 The Semi-Infinite Elastic Medium and Related Problems14-1 Introduction 14-2 Uniform Pressure Distributed over a Circular Area on the Surface of a Semi-Infinite Solid 14-3 Uniform Pressure Distributed over a Rectangular Area 14-4 Rigid Die in the Form of a Circular Cylinder14-5 Vertical Line Load on a Semi-Infinite Elastic Medium 14-6 Vertical Line Load on a Semi-Infinite Elastic Plate 14-7 Tangential Line Load at the Surface of a Semi-Infinite Elastic Medium 14-8 Tangential Line Load on a Semi-Infinite Elastic Plate 14-9 Uniformly Distributed Vertical Pressure on Part of the Boundary of a Semi-Infinite Elastic Medium 14-10 Uniformly Distributed Vertical Pressure on Part of the Boundary of a Semi-Infinite Elastic Plate14-11 Rigid Strip at the Surface of a Semi- Infinite Elastic Medium 14-12 Rigid Die at the Surface of a Semi-Infinite Elastic Plate14-13 Radial Stresses in Wedges 14-14 M. Levy's Problems of the Triangular and Rectangular Retaining Walls Chapter 15 Energy Principles and Introduction To Variational Methods 15-1 Introduction 15-2 Work, Strain and Complementary Energies. Clapeyron's Law 15-3 Principle of Virtual Work 15-4 Variational Problems and Euler's Equations 15-5 The Reciprocal Laws of Betti and Maxwell 15-6 Principle of Minimum Potential Energy 15-7 Castigliano's First Theorem 15-8 Principle of Virtual Complementary Work 15-9 Principle of Minimum Complementary Energy 15-10 Castigliano's Second Theorem 15-11 Theorem of Least Work 15-12 Summary of Energy Theorems 15-13 Working Form of the Strain Energy for Linearly Elastic Slender Members 15-14 Strain Energy of a Linearly Elastic Slender Member in Terms of the Unit Displacements of the Centroid G and of the Unit Rotations 15-15 A Working Form of the Principles of Virtual Work and of Virtual Complementary Work for a Linearly Elastic Slender Member 15-16 Examples of Application of the Theorems of Virtual Work and Virtual Complementary Work 15-17 Examples of Application of Castigliano's First and Second Theorems 15-18 Examples of Application of the Principles of Minimum Potential Energy and Minimum Complementary Energy 15-19 Example of Application of the Theorem of Least Work15-20 The Rayleigh-Ritz MethodProblems Chapter 16 Elastic Stability: Columns and Beam-Columns 16-1 Introduction 16-2 Differential Equations of Columns and Beam-Columns 16-3 Simple Columns 16-4 Energy Solution of the Buckling Problem 16-5 Examples of Calculation of Buckling Loads by the Energy Method16-6 Combined Compression and Bending 16-7 Lateral Buckling of Thin Rectangular Beams Problems Chapter 17 Bending of Thin Flat Plates 17-1 Introduction and Basic Assumptions. Strains and Stresses 17-2 Geometry of Surfaces with Small Curvatures 17-3 Stress Resultants and Stress Couples 17-4 Equations of Equilibrium of Laterally Loaded Thin Plates 17-5 Boundary Conditions 17-6 Some Simple Solutions of Lagrange's Equation17-7 Simply Supported Rectangular Plate. Navier's Solution 17-8 Elliptic Plate with Clamped Edges under Uniform Load 17-9 Bending of Circular Plates 17-10 Strain Energy and Potential Energy of a Thin Plate in Bending 17-11 Application of the Principle of Minimum Potential Energy to Simply Supported Rectangular Plates Problems Chapter 18 Introduction to the Theory of Thin Shells 18-1 Introduction 18-2 Space Curves 18-3 Elements of the Theory of Surfaces 1) Gaussian surface coordinates. First fundamental form. 2) Second fundamental form. 3) Curvature of a normal section. Meunier's theorem. 4) Principal directions and lines of curvature. 5) Principal curvatures, first and second curvatures. 6) Euler's theorem. 7) Rate of change of the vectors a, and the corresponding unit vectors along the parametric lines. 8) The Gauss-Codaizi conditions. 9) Application to surfaces of revolution. 10) Important remarks. 18-4 Basic Assumptions and Reference System of Coordinates 18-5 Strain-Displacement Relations 18-6 Stress Resultants and Stress Couples 18-7 Equations of Equilibrium of Loaded Thin Shells 18-8 Boundary Conditions 18-9 Membrane Theory of Shells 18-10 Membrane Shells of Revolution 18-11 Membrane Theory of Cylindrical Shells 18-12 General Theory of Circular Cylindrical Shells 18-13 Circular Cylindrical Shell Loaded Symmetrically with Respect to its Axis Problems Chapter 19 Solutions of Elasticity Problems by Means of Complex Variables19-1 Introduction 19-2 Complex Variables and Complex Functions: A Short Review 19-3 Line Integrals of Complex Functions. Cauchy's Integral Theorem 19-4 Cauchy's Integral Formula 19-5 Taylor Series 19-6 Laurent Series, Residues and Cauchy 's Residue Theorem 19-7 Singular Points of an Analytic Function 19-8 Evaluation of Residues 19-9 Conformal Representation or Conformal Mapping19-10 Examples of Mapping by Elementary Functions 19-11 The Theorem of Harnack and the Formulas of Schwarz and Poisson 19-12 Torsion of Prismatic Bars Using Complex Variables 19-13 Torsion of Prismatic Bars with Various Shapes 19-14 The Plane Stress and Strain Problems and the Solution to the Biharmonic Equation 1) Displacements and stresses. 2) Boundary conditions. 3) The structure of the functions +(i)and ~(7.) in simply connected regions. 4) The structure of the functions +(z) and ~(z)in finite, multiply connected regions. 5) The structure of the functions +(z) and ~(z)in infinite, multiply connected regions. 6) The first and second boundary value problems in plane elasticity. 7) Displacements and stresses in curvilinear coordinates. 8) Conformal mapping for plane problems. 9) Solution by means of power series for simply connected regions. 10) Mapping of infinite regions. 19-15 Solutions Using Timoshenko's Equations. Westergaard's Stress Function 19-16 Simple Examples Using Complex Potentials 19-17 The Infinite Plate with a Circular Hole 19-18 The Infinite Plate under the Action of a Concentrated Force and Moment19-19 The Infinite Plate with an Elliptic Hole Subjected to a Tensile Stress Normal to the Major Principal Axis of the Ellipse 19-20 Infinite Plate with an Elliptic Hole Subjected to a Uniform all around Tension S 19-21 Conformal Mapping Applied to the Problem of the Elliptic Hole 19-22 Infinite Plate with an Elliptic Hole Subjected to a Uniform Pressure P 19-23 Application to Fracture MechanicsReferencesAppendicesAddendum: Comments and Detailed Explanations. Additional Solved Examples. Additional Problems. Index

About the Author :
Adel S. Saada (Ing., E.C.P., Ph.D., Princeton University) is presently Professor of Civil Engineering at Case Western Reserve University, Cleveland, Ohio. Dr. Saada received his Ingénieur des Arts et Manufactures degree from École Centrale des Arts et Manufactures de Paris, France, and the equivalent of a Master of Science degree from the University of Grenoble, France. Before coming to Princeton University the author was a practicing structural engineer in France. Dr. Saada's teaching activities are in two major areas: the first is that of the mechanics of solids and in particular elasticity; the second is that of mechanics applied to soils and foundations. His research activities are primarily in the area of stress-strain relations and failure of transversely isotropic geomaterials. Much of his research work has been supported by personal grants from the National Science Foundation, the U.S. Army, and the U.S. Air Force. Dr. Saada is a member of several professional societies, a consulting engineer, and the author of numerous papers published in both national and international journals.