ELASTICITYAND FRACTUREMECHANICS

ELASTICITY FRACTURE

MECHANICSAND ELASTICITYAND FRACTUREMECHANICS

THEORY OFTHEORY OF

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Vijay G. Ukadgaonker

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Theory of Elasticity and Fracture Mechanics

VIJAY G. UKADGAONKERFormer Professor

Indian Institute of Technology Bombay

Delhi-1100922015

THEORY OF ELASTICITY AND FRACTURE MECHANICSVijay 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 DefinitionofStress 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 EquilibriumEquationsinCylindricalCo-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 PrincipalStressesandMohr’sCircles:2DCases 14 1.3.10 Stress Ellipsoid and Stress–Director Surface 21 1.4 Analysis of Strain 22 1.4.1 DefinitionofStrain 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–DisplacementRelationsinSphericalCo-ordinates 27 1.4.7 StrainataPoint(2DCase) 29 1.4.8 The Rigid Body Rotation 34 1.4.9 CubicalDilation 35 1.4.10 PlaneStressandPlaneStrain 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 GeneralizedHooke’sLaw 42 1.5.5 Proof of Principal Stress Directions and Principal Strain Directions are Same 43 1.5.6 CorrelationbetweentheElasticConstants 44

v

vi Contents

1.6 FormulationofElasticProblems 46 1.6.1 DifferentTypesofFundamentalBoundaryValueProblems 46 1.6.2 BoundaryValueProbleminElasticity 46 1.6.3 ElasticEquationsUsingIndexNotations 48 1.7 GeneralTheoriesinElasticity 49 1.7.1 PrincipleofSuperposition 49 1.7.2 TheoremofMinimumStrainEnergy 50 1.7.3 PrincipleofVirtualWork 51 1.7.4 Castigliano’sTheorem 53 1.7.5 Saint–Venant’sPrinciple 55 Exercises 55

2. Finite Element Method 58–109 2.1 Introduction 58 2.2 SixStepsofFEMintheSolutionofGeneralContinuumProblems 59 2.3 SixStepsinFEMforSolidMechanics 59 2.4 FiguresforFEM 59 2.5 ElementStiffnessMatrix(K)forTriangularContinuumElement 62 2.6 AssemblyofElementStiffnessMatricesorGlobalorOverall StiffnessMatrix 66 2.7 Application 71 2.8 BasicSevenStepsintheDerivationoftheElementStiffness Characteristics 75 2.9 TheoryofElasticitySolutiontotheBeamProblem 75 2.10 RectangularElementwithFourNodes 76 2.11 RectangularFiniteElementforPlateFlexure 80 2.12 Example 88 2.13 AxisymmetricShellsofRevolution 90 2.14 ExampleforCircularCylindricalThinShell 96 2.15 ThickShellsofRevolution 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 BendingofaCantileverLoadedat itsEnd 115 3.4.3 SimplySupportedBeamwithUniformlyDistributedLoad 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 AxisymmetricStresses 122 3.5.2 PureBendingofCurvedBars 125

Contents vii

3.5.3 TheEffectofCircularHoleonStressDistributioninInfinite Plate(Figure3.14) 126 3.5.4 Problems of Stress Concentration around Holes 130 3.5.5 BendingofCurvedBarbyaForceAppliedat itsEnds 132 3.5.6 ConcentratedForceataPointonSemi-infinitePlate 136 Exercises 143

4. Complex Variable Approach 147–212 4.1 Introduction 147 4.2 ComplexFunctions 147 4.2.1 Representation of Biharmonic Function 147 4.2.2 Representation of Stresses 148 4.2.3 RepresentationofDisplacements 149 4.2.4 RepresentationofResultantForcesandMomentsActingon 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 FiniteDomain 156 4.4.3 InfiniteRegions 158 4.5 DifferentMethodsforObtainingf(z) and c(z) 159 4.5.1 UseofEquation(4.33) 159 4.5.2 UseofEquation(4.35) 159 4.5.3 ConformalMapping 159 4.5.4 Integro–DifferentialEquations 160 4.5.5 ProblemofLinearRelationship(HilbertProblem) 161 4.5.6 Schwarz’sAlternatingMethodforMultiplyConnectedRegion 162 4.6 ApplicationoftheTheory 162 4.6.1 CircularPlateunderArbitraryEdgeThrust 162 4.6.2 InfinitePlatewithaCircularHole 165 4.6.3 InfiniteRegionBoundedbyanEllipse 168 4.6.4 InfiniteRegionwithTwoCircularHoles 173 4.6.5 ANovelMethodofStressAnalysisofInfinitePlatewithCircular HolewithUniformLoadingatInfinity 178 4.6.6 ANovelMethodofStressAnalysisofanInfinitePlatewith EllipticalHolewithUniformTensileStress 182 4.6.7 ANovelMethodofStressAnalysisofanInfinitePlatewith Small Corner Radius Equilateral Triangular Hole 188 4.6.8 ANovelMethodofStressAnalysisofanInfinitePlatewitha RectangularHoleofRoundedCornersunderUniformLoading atInfinity 199 Exercises 210

viii Contents

5. InteractionEffectonStressesofTwoUnequalHolesinInfinitePlate 213–271 5.1 Schwarz’sAlternatingMethod 213 5.2 ConvergenceofSchwarz’sAlternatingMethod 215 5.3 InfinitePlatewithTwoUnequalCircularHolesSubjectedtoUniform Pressures 216 5.3.1 Results 217 5.4 InfinitePlatewithTwoUnequalCircularHolesSubjectedto UniformShearingStresses 219 5.4.1 FirstApproximation 220 5.4.2 SecondApproximation 220 5.4.3 ThirdApproximation 221 5.4.4 Numerical Results 222 5.5 InfinitePlatewithTwoUnequalCircularHolesSubjectedtoUniform TensionatInfinityalongtheLineofSymmetry 225 5.5.1 FirstApproximation 226 5.5.2 SecondApproximation 226 5.5.3 ThirdApproximation 227 5.5.4 Stress Field 227 5.5.5 Numerical Results 230 5.6 TwoUnequalCircularHolesSubjectedtoUniformTensionatInfinity PerpendiculartotheLineofSymmetry 232 5.6.1 FirstApproximation 233 5.6.2 SecondApproximation 233 5.6.3 ThirdApproximation 235 5.6.4 StressField 235 5.6.5 NumericalResults 237 5.7 PlatewithtwoUnequalCircularHolesSubjectedtoUniforminPlane ShearingStressatInfinity 240 5.8 TwoUnequalArbitrarilyOrientedEllipticalHolesorCracks 245 5.8.1 FirstApproximation 246 5.8.2 SecondApproximation 247 5.8.3 UniformShearatInfinity 251 5.8.4 Stress Field 253 5.9 TwoUnequalCollinearEllipticalHoles 260 5.10 StressAnalysisofDoorandWindowofaPassengerAircraft 262 Exercises 271

6. Anisotropic Elasticity 272–405 6.1 BasicCasesofElasticSymmetry 272 6.2 PlaneofElasticSymmetry 273 6.3 ThreePlanesofElasticSymmetry(OrthotropicBody) 274 6.4 AxisofRotationalSymmetryorTransverselyIsotropicBody 276 6.5 IsotropicBody 277 6.6 CurvilinearAnisotropy 278 6.7 CylindricalAnisotropy 278

Contents ix

6.8 SphericalAnisotropy 280 6.9 SomeAnisotropicElasticConstantsforEngineeringMaterials 280 6.10 FundamentalEquationsoftheTheoryofAnisotropicElasticity 284 6.11 ComplexRepresentationoftheStressFunctions 285 6.12 BoundaryConditions 287 6.13 RepresentationofResultantForcesandMoments 288 6.14 ExpressionsfortheFunctionsf(z1) and y(z2) for

Multiply-connectedRegion 290 6.15 FirstFundamentalProblemforInfiniteAnisotropicPlatewithEllipticalHole 292 6.15.1 AnAnisotropicPlatewithEllipticalHolewithUniformTensile StressesatInfinityatanAngle 294 6.15.2 UniformTensionp in x-direction 297 6.15.3 CrackofLength2a 303 6.15.4 EdgeofEllipticalHoleSubjectedtoUniformTangentialStressT 305 6.15.5 EllipticalHoleSubjectedtoUniformPressurep 305 6.15.6 ANovelMethodofStressAnalysisofanInfiniteAnisotropic PlatewithEllipticalHoleorCrackwithUniformTensileStress 306 6.15.7 StressDistributionaroundTriangularHolesinAnisotropicPlates 312 6.15.8 AGeneralSolutionforStressResultantsandMoments aroundHolesinUnsymmetricLaminates 327 6.15.9 AGeneralSolutionforMomentsAroundHolesinSymmetric Laminates 344 6.15.10 AGeneralSolutionforStressesAroundHolesinSymmetric LaminatesunderInplaneLoading 365 6.15.11 StressAnalysisforanOrthotropicPlatewithanIrregular ShapedHoleforDifferentIn-planeLoadingConditions—Part1 385 Exercises 405

7. Introduction to Fracture Mechanics 406–416 7.1 ImportanceofFractureMechanics 406 7.2 ModesofFracture 406 7.3 TheGriffithCriterion:AsSurfaceEnergy 407 7.4 A Crack Structure 408 7.5 TheStressesat theCrackTipandIrwin’sStressIntensityFactor 410 7.6 RelationshipbetweeKI and GI 413 7.7 CrackPropagationandParisLaw 413 7.8 Fracture Control Plans 415 Exercises 416

8. Stress Analysis of Fracture 417–505 8.1 Westergaard’sStressFunction 417 8.2 Stresses in a Cracked Body 418 8.3 ModeICrackProblemwithBiaxialStressandUsingWertergaard’s Stress Function 420 8.4 CrackinFiniteWidthPlate 423

Theory Of Elasticity And FractureMechanics

Publisher : PHI Learning ISBN : 9788120351417Author : UKADGAONKER,VIJAY G.

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