ELASTICITY AND FRACTURE MECHANICS. Vijay G. Ukadgaonker

THEORY OF ELASTICITY AND FRACTURE MECHANICS y x Vijay G. Ukadgaonker

Theory of Elasticity and Fracture Mechanics VIJAY G. UKADGAONKER Former Professor Indian Institute of Technology Bombay Delhi-110092 2015

THEORY OF ELASTICITY AND FRACTURE MECHANICS Vijay G. Ukadgaonker 2015 by PHI Learning Private Limited, Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN-978-81-203-5141-7 The export rights of this book are vested solely with the publisher. Published by Asoke K. Ghosh, PHI Learning Private Limited, Rimjhim House, 111, Patparganj Industrial Estate, Delhi-110092 and Printed by Syndicate Binders, A-20, Hosiery Complex, Noida, Phase-II Extension, Noida-201305 (N.C.R. Delhi).

To My Teachers, Students and my Family

Contents Preface xiii 1. Analysis of Stress and Strain 1 57 1.1 Introduction 1 1.2 Importance of Theory of Elasticity Solutions 4 1.3 Analysis of Stresses 5 1.3.1 Definition of Stress 6 1.3.2 Notations 6 1.3.3 Force Equilibrium Equations in Two-dimensional Cartesian Co-ordinates 7 1.3.4 Equilibrium Equations in 3D Cartesian Co-ordinates 8 1.3.5 Equilibrium Equations in Polar Co-ordinates 8 1.3.6 Equilibrium Equations in Cylindrical Co-ordinates 9 1.3.7 Equilibrium Equations in Spherical Co-ordinates 10 1.3.8 Stress at a Point: 2D Boundary Condition 11 1.3.9 Principal Stresses and Mohr s Circles: 2D Cases 14 1.3.10 Stress Ellipsoid and Stress Director Surface 21 1.4 Analysis of Strain 22 1.4.1 Definition of Strain 22 1.4.2 Strain Displacement Relations in 2D Cartesian Co-ordinates 22 1.4.3 Strain Displacement Relations in 3D Cartesian Co-ordinates 23 1.4.4 Strain Displacement Relations in Polar Co-ordinates 24 1.4.5 Strain Displacement Relations in Cylindrical Co-ordinates 25 1.4.6 Strain Displacement Relations in Spherical Co-ordinates 27 1.4.7 Strain at a Point (2D Case) 29 1.4.8 The Rigid Body Rotation 34 1.4.9 Cubical Dilation 35 1.4.10 Plane Stress and Plane Strain 36 1.4.11 Compatibility Conditions 38 1.5 Stress Strain Relations 40 1.5.1 Properties 40 1.5.2 Definitions 40 1.5.3 Tension Test 41 1.5.4 Generalized Hooke s Law 42 1.5.5 Proof of Principal Stress Directions and Principal Strain Directions are Same 43 1.5.6 Correlation between the Elastic Constants 44 v

vi Contents 1.6 Formulation of Elastic Problems 46 1.6.1 Different Types of Fundamental Boundary Value Problems 46 1.6.2 Boundary Value Problem in Elasticity 46 1.6.3 Elastic Equations Using Index Notations 48 1.7 General Theories in Elasticity 49 1.7.1 Principle of Superposition 49 1.7.2 Theorem of Minimum Strain Energy 50 1.7.3 Principle of Virtual Work 51 1.7.4 Castigliano s Theorem 53 1.7.5 Saint Venant s Principle 55 Exercises 55 2. Finite Element Method 58 109 2.1 Introduction 58 2.2 Six Steps of FEM in the Solution of General Continuum Problems 59 2.3 Six Steps in FEM for Solid Mechanics 59 2.4 Figures for FEM 59 2.5 Element Stiffness Matrix (K) for Triangular Continuum Element 62 2.6 Assembly of Element Stiffness Matrices or Global or Overall Stiffness Matrix 66 2.7 Application 71 2.8 Basic Seven Steps in the Derivation of the Element Stiffness Characteristics 75 2.9 Theory of Elasticity Solution to the Beam Problem 75 2.10 Rectangular Element with Four Nodes 76 2.11 Rectangular Finite Element for Plate Flexure 80 2.12 Example 88 2.13 Axisymmetric Shells of Revolution 90 2.14 Example for Circular Cylindrical Thin Shell 96 2.15 Thick Shells of Revolution 101 Exercises 107 3. Two-dimensional Elasticity Problems 110 146 3.1 Introduction 110 3.2 Biharmonic Equation in 2D Problems in Rectangular Co-ordinates 110 3.3 Biharmonic Equation in Polar Co-ordinates 113 3.4 Solution of 2D Elasticity Problems in Rectangular Co-ordinates 113 3.4.1 Stress Function in the Form of a Polynomial 113 3.4.2 Bending of a Cantilever Loaded at its End 115 3.4.3 Simply Supported Beam with Uniformly Distributed Load 116 3.4.4 Solution of 2D Problem in the Form of a Fourier Series 118 3.5 2D Problems in Polar Co-ordinates 122 3.5.1 Axisymmetric Stresses 122 3.5.2 Pure Bending of Curved Bars 125

Contents vii 3.5.3 The Effect of Circular Hole on Stress Distribution in Infinite Plate (Figure 3.14) 126 3.5.4 Problems of Stress Concentration around Holes 130 3.5.5 Bending of Curved Bar by a Force Applied at its Ends 132 3.5.6 Concentrated Force at a Point on Semi-infinite Plate 136 Exercises 143 4. Complex Variable Approach 147 212 4.1 Introduction 147 4.2 Complex Functions 147 4.2.1 Representation of Biharmonic Function 147 4.2.2 Representation of Stresses 148 4.2.3 Representation of Displacements 149 4.2.4 Representation of Resultant Forces and Moments Acting on the Boundary 151 4.3 Transformation of Co-ordinates 152 4.3.1 Translation of Rectangular Co-ordinates 152 4.3.2 Rotation of Rectangular Co-ordinates 153 4.3.3 Polar Co-ordinates 155 4.4 Structure of Functions f(z) and c (z) 155 4.4.1 Arbitraryness in Choosing the Stress Functions 155 4.4.2 Finite Domain 156 4.4.3 Infinite Regions 158 4.5 Different Methods for Obtaining f(z) and c (z) 159 4.5.1 Use of Equation (4.33) 159 4.5.2 Use of Equation (4.35) 159 4.5.3 Conformal Mapping 159 4.5.4 Integro Differential Equations 160 4.5.5 Problem of Linear Relationship (Hilbert Problem) 161 4.5.6 Schwarz s Alternating Method for Multiply Connected Region 162 4.6 Application of the Theory 162 4.6.1 Circular Plate under Arbitrary Edge Thrust 162 4.6.2 Infinite Plate with a Circular Hole 165 4.6.3 Infinite Region Bounded by an Ellipse 168 4.6.4 Infinite Region with Two Circular Holes 173 4.6.5 A Novel Method of Stress Analysis of Infinite Plate with Circular Hole with Uniform Loading at Infinity 178 4.6.6 A Novel Method of Stress Analysis of an Infinite Plate with Elliptical Hole with Uniform Tensile Stress 182 4.6.7 A Novel Method of Stress Analysis of an Infinite Plate with Small Corner Radius Equilateral Triangular Hole 188 4.6.8 A Novel Method of Stress Analysis of an Infinite Plate with a Rectangular Hole of Rounded Corners under Uniform Loading at Infinity 199 Exercises 210

viii Contents 5. Interaction Effect on Stresses of Two Unequal Holes in Infinite Plate 213 271 5.1 Schwarz s Alternating Method 213 5.2 Convergence of Schwarz s Alternating Method 215 5.3 Infinite Plate with Two Unequal Circular Holes Subjected to Uniform Pressures 216 5.3.1 Results 217 5.4 Infinite Plate with Two Unequal Circular Holes Subjected to Uniform Shearing Stresses 219 5.4.1 First Approximation 220 5.4.2 Second Approximation 220 5.4.3 Third Approximation 221 5.4.4 Numerical Results 222 5.5 Infinite Plate with Two Unequal Circular Holes Subjected to Uniform Tension at Infinity along the Line of Symmetry 225 5.5.1 First Approximation 226 5.5.2 Second Approximation 226 5.5.3 Third Approximation 227 5.5.4 Stress Field 227 5.5.5 Numerical Results 230 5.6 Two Unequal Circular Holes Subjected to Uniform Tension at Infinity Perpendicular to the Line of Symmetry 232 5.6.1 First Approximation 233 5.6.2 Second Approximation 233 5.6.3 Third Approximation 235 5.6.4 Stress Field 235 5.6.5 Numerical Results 237 5.7 Plate with two Unequal Circular Holes Subjected to Uniform in Plane Shearing Stress at Infinity 240 5.8 Two Unequal Arbitrarily Oriented Elliptical Holes or Cracks 245 5.8.1 First Approximation 246 5.8.2 Second Approximation 247 5.8.3 Uniform Shear at Infinity 251 5.8.4 Stress Field 253 5.9 Two Unequal Collinear Elliptical Holes 260 5.10 Stress Analysis of Door and Window of a Passenger Aircraft 262 Exercises 271 6. Anisotropic Elasticity 272 405 6.1 Basic Cases of Elastic Symmetry 272 6.2 Plane of Elastic Symmetry 273 6.3 Three Planes of Elastic Symmetry (Orthotropic Body) 274 6.4 Axis of Rotational Symmetry or Transversely Isotropic Body 276 6.5 Isotropic Body 277 6.6 Curvilinear Anisotropy 278 6.7 Cylindrical Anisotropy 278

Contents ix 6.8 Spherical Anisotropy 280 6.9 Some Anisotropic Elastic Constants for Engineering Materials 280 6.10 Fundamental Equations of the Theory of Anisotropic Elasticity 284 6.11 Complex Representation of the Stress Functions 285 6.12 Boundary Conditions 287 6.13 Representation of Resultant Forces and Moments 288 6.14 Expressions for the Functions f(z 1 ) and y(z 2 ) for Multiply-connected Region 290 6.15 First Fundamental Problem for Infinite Anisotropic Plate with Elliptical Hole 292 6.15.1 An Anisotropic Plate with Elliptical Hole with Uniform Tensile Stresses at Infinity at an Angle 294 6.15.2 Uniform Tension p in x-direction 297 6.15.3 Crack of Length 2a 303 6.15.4 Edge of Elliptical Hole Subjected to Uniform Tangential Stress T 305 6.15.5 Elliptical Hole Subjected to Uniform Pressure p 305 6.15.6 A Novel Method of Stress Analysis of an Infinite Anisotropic Plate with Elliptical Hole or Crack with Uniform Tensile Stress 306 6.15.7 Stress Distribution around Triangular Holes in Anisotropic Plates 312 6.15.8 A General Solution for Stress Resultants and Moments around Holes in Unsymmetric Laminates 327 6.15.9 A General Solution for Moments Around Holes in Symmetric Laminates 344 6.15.10 A General Solution for Stresses Around Holes in Symmetric Laminates under Inplane Loading 365 6.15.11 Stress Analysis for an Orthotropic Plate with an Irregular Shaped Hole for Different In-plane Loading Conditions Part 1 385 Exercises 405 7. Introduction to Fracture Mechanics 406 416 7.1 Importance of Fracture Mechanics 406 7.2 Modes of Fracture 406 7.3 The Griffith Criterion: As Surface Energy 407 7.4 A Crack Structure 408 7.5 The Stresses at the Crack Tip and Irwin s Stress Intensity Factor 410 7.6 Relationship betwee K I and G I 413 7.7 Crack Propagation and Paris Law 413 7.8 Fracture Control Plans 415 Exercises 416 8. Stress Analysis of Fracture 417 505 8.1 Westergaard s Stress Function 417 8.2 Stresses in a Cracked Body 418 8.3 Mode I Crack Problem with Biaxial Stress and Using Wertergaard s Stress Function 420 8.4 Crack in Finite Width Plate 423

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