M. Reza Eslami Richard B. Hetnarski Jozef Ignaczak Naobumi Sumi Naotake Noda Yoshinobu Tanigawa Theory of Elasticity and Thermal Stresses Explanations, Problems and Solutions <gl Spri ringer
Contents Part I The Mathematical Theory of Elasticity 1 Mathematical Preliminaries 3 1.1 Some Formulas in Tensor Algebra 3 1.2 Alternative Definitions of a Vector and of a Tensor Using an Orthogonal Tensor 6 1.3 Further Definitions 7 1.4 Eigenproblem for a Second Order Tensor 8 1.5 A Fourth-Order Tensor C 9 1.6 Tensor Fields 9 1.7 Integral Theorems 11 1.8 Irrotational and Solenoidal Fields 11 1.9 Time-Dependent Fields 12 1.10 Problems and Solutions Related to the Mathematical Preliminaries 13 2 Fundamentals of Linear Elasticity 35 2.1 Deformation of an Elastic Body 35 2.2 Compatibility 37 2.3 Motion and Equilibrium 38 2.4 Equations of Motion 39 2.5 Constitutive Relations 40 2.6 Isotropic Elastic Body 41 2.7 The Cauchy Relations 42 2.8 Constitutive Relations for a Thermoelastic Body 42 2.9 Problems and Solutions Related to the Fundamentals of Linear Elasticity 43 3 Formulation of Problems of Elasticity 65 3.1 Boundary Value Problems of Elastostatics 65 3.2 Concept of an Elastic State 67 3.3 Concept of a Thermoelastic State 68 3.4 Formulation of Boundary Value Problems 69 xi
xii Contents 3.5 Uniqueness 70 3.6 Formulation of Problems of Nonisothermal Elastostatics 70 3.7 Initial-Boundary Value Problems of Elastodynamics 71 3.8 Concept of an Elastic Process 72 3.9 Formulation of Problems of Isothermal Elastodynamics 73 3.10 Problems and Solutions Related to the Formulation of Problems of Elasticity 74 4 Variational Formulation of Elastostatics 103 4.1 Minimum Principles 103 4.2 The Rayleigh-Ritz Method 106 4.3 Variational Principles 107 4.4 Compatibility-Related Principle 108 4.5 Problems and Solutions Related to Variational Formulation of Elastostatics 109 5 Variational Principles of Elastodynamics 127 5.1 The Hamilton-Kirchhoff Principle 127 5.2 Problems and Solutions Related to Variational Principles of Elastodynamics 129 6 Complete Solutions of Elasticity 151 6.1 Complete Solutions of Elastostatics 151 6.2 Complete Solutions of Elastodynamics 153 6.3 Complete Stress Solution of Elastodynamics 154 6.4 Problems and Solutions Related to Complete Solutions of Elasticity 155 7 Formulation of Two-Dimensional Problems 185 7.1 Two-Dimensional Problems of Isothermal Elastostatics 185 7.2 Two-Dimensional Problems of Nonisothermal Elastostatics... 187 7.3 Two-Dimensional Problems of Elastodynamics 188 7.4 Two-Dimensional Problems of Nonisothermal Elastodynamics 189 7.5 Problems and Solutions Related to Formulation of Two-Dimensional Problems 191 8 Solutions to Particular Three-Dimensional Boundary Value Problems of Elastostatics 209 8.1 Torsion of Circular Bars 209 8.2 Problems and Solutions Related to Particular Three-Dimensional Boundary Value Problems of Elastostatics Torsion Problems 212
Contents xiii 9 Solutions to Particular Two-Dimensional Boundary Value Problems of Elastostatics 219 9.1 The Two-Dimensional Version of Boussinesq-Papkovitch-Neuber Solution for a Body Under Plane Strain Conditions 219 9.2 Problems and Solutions Related to Particular Two-Dimensional Boundary Value Problems of Elastostatics 221 10 Solutions to Particular Three-Dimensional Initial-Boundary Value Problems of Elastodynamics 245 10.1 A Spherically Symmetric Green's Function of the Dynamic Theory of Thermal Stresses for an Infinite Elastic Body 245 10.2 A Spherically Symmetric Green's Function for a Parabolic Heat Conduction Equation in an Infinite Space 248 10.3 Problems and Solutions Related to Particular Three-Dimensional Initial-Boundary Value Problems of Elastodynamics 249 11 Solutions to Particular Two-Dimensional Initial-Boundary Value Problems of Elastodynamics 259 11.1 Dynamic Thermal Stresses in an Infinite Elastic Body Under Plane Strain Conditions Subject to a Time-Dependent Temperature Field 259 11.2 Dynamic Thermal Stresses in an Infinite Elastic Body Under Generalized Plane Stress Conditions Subject to a Time-Dependent Temperature Field 260 11.3 Problems and Solutions Related to Particular Two-Dimensional Initial-Boundary Value Problems of Elastodynamics 260 12 One-Dimensional Solutions of Elastodynamics 271 12.1 One-Dimensional Field Equations of Isothermal Elastodynamics 271 12.2 One-Dimensional Field Equations of Nonisothermal Elastodynamics 272 12.3 Problems and Solutions Related to One-Dimensional Initial-Boundary Value Problems of Elastodynamics 276 Part II Thermal Stresses 13 Thermal Stresses in Bars 301 13.1 Thermal Stresses in Bars 301
xiv Contents 13.2 Problems and Solutions Related to Thermal Stresses in Bars 303 14 Thermal Stresses in Beams 317 14.1 Thermal Stresses in Beams 317 14.1.1 Thermal Stresses in Beams 317 14.1.2 Thermal Stresses in Composite Beams 320 14.1.3 Thermal Deflection in Beams 322 14.1.4 Curved Beams 323 14.2 Problems and Solutions Related to Thermal Stresses in Beams 325 15 Heat Conduction 353 15.1 Heat Conduction Equation 353 15.2 One-Dimensional Heat Conduction Problems 355 15.3 Problems and Solutions Related to Heat Conduction 359 16 Basic Equations of Thermoelasticity 391 16.1 Governing Equations of Thermoelasticity 391 16.1.1 Stress and Strain in a Cartesian Coordinate System 391 16.1.2 Navier's Equations, Compatibility Equations and Boundary Conditions 392 16.1.3 General Solution of Navier's Equations 395 16.1.4 Thermal Stresses in a Cylindrical Coordinate System 396 16.1.5 Thermal Stresses in a Spherical Coordinate System 399 16.2 Problems and Solutions Related to Basic Equations of Thermoelasticity 403 17 Plane Thermoelastic Problems 423 17.1 Plane Strain and Plane Stress 423 17.2 Problems and Solutions Related to Plane Thermoelastic Problems 429 18 Thermal Stresses in Circular Cylinders 445 18.1 One-Dimensional Problems 445 18.2 Plane Problems 447 18.3 Two-Dimensional Axisymmetric Problems (r, z) 448 18.4 Three-Dimensional Problems 450 18.5 Problems and Solutions Related to Thermal Stresses in Circular Cylinders 453
Contents xv 19 Thermal Stresses in Spherical Bodies 475 19.1 One-Dimensional Problems in Spherical Bodies 475 19.2 Two-Dimensional Axisymmetric Problems 477 19.3 Problems and Solutions Related to Thermal Stresses in Spherical Bodies 480 20 Thermal Stresses in Plates 513 20.1 Basic Equations for a Rectangular Plate 513 20.2 Basic Equations for a Circular Plate 518 20.3 Problems and Solutions Related to Thermal Stresses in Plates 520 21 Thermally Induced Instability 535 21.1 Instability of Beam-Column 535 21.2 Instability of Plate 536 21.3 Problems and Solutions Related to Thermally Induced Instability 539 Part III Thermal Stresses Advanced Theory and Applications 22 Heat Conduction 573 22.1 Problems in Rectangular Cartesian Coordinates 573 22.1.1 Steady Two-Dimensional Problems: Separation of Variables 573 22.1.2 Fourier Series 575 22.1.3 Double Fourier Series 577 22.1.4 Bessel Functions and Fourier-Bessel Series 579 22.1.5 Nonhomogeneous Differential Equations and Boundary Conditions 587 22.2 Problems and Solutions Related to Heat Conduction 588 22.2.1 Lumped Formulation 596 22.3 Problems in Cylindrical Coordinates 600 22.4 Problems in Spherical Coordinates 604 22.4.1 Steady-State Two- and Three-Dimensional Problems 604 23 Thermal Stresses in Beams 629 23.1 Thermal Stresses in Beams 629 23.2 Deflection Equation of Beams 633 23.3 Boundary Conditions 635 23.4 Problems and Solutions of Beams 636 24 Thick Cylinders and Spheres 653 24.1 Problems and Solutions of Cylinders and Spheres 653
xvi Contents 25 Piping Systems 677 25.1 Thermal Expansion of Piping Systems 677 25.1.1 Definition of the Elastic Center 678 25.2 Piping Systems in Two-Dimensions 682 25.3 Piping Systems in Three-Dimensions 686 25.4 Problems and Solutions of Piping Systems 689 26 Coupled Thermoelasticity 701 26.1 Governing Equations, Classical Theory 701 26.2 Problems and Solutions of Coupled Thermoelasticity 703 Part IV Numerical Methods 27 The Method of Characteristics 715 27.1 Basic Equations for Plane Thermoelastic Waves 715 27.2 Characteristics and Characteristic Equations 716 27.3 Derivation of Difference Equations and Numerical Procedure 718 27.4 Finite-Difference Solution for Temperature 719 27.5 Problems and Solutions Related to The Method of Characteristics 720 28 Finite Element of Coupled Thermoelasticity 727 28.1 Galerkin Finite Element 727 28.2 One-Dimensional Problem 733 28.3 Problems and Solutions Related to Coupled Thermoelasticity 736 29 Boundary Element, Coupled Thermoelasticity 755 29.1 Governing Equations 755 29.2 Boundary Integral Equation 757 29.3 Fundamental Solution 758 29.4 Numerical Formulation 761 29.5 Problems and Solutions Related to Coupled Thermoelasticity 763 Errata to Three Books 777 Bibliography 781 Index 783