Elasticity with Mathematica: An introduction to continuum mechanics and linear elasticity

267 Pages·2007·2.538 MB·English

by  Andrei Constantinescu, Alexander Korsunsky

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Elasticity with MATHEMATICA(cid:1)R
Thisbookgivesanintroductiontothekeyideasandprinciplesinthetheoryofelas-
ticity with the help of symbolic computation. Differential and integral operators
on vector and tensor fields of displacements, strains, and stresses are considered
on a consistent and rigorous basis with respect to curvilinear orthogonal coordi-
nate systems. As a consequence, vector and tensor objects can be manipulated
readily, and fundamental concepts can be illustrated and problems solved with
ease.Themethodisillustratedusingavarietyofplaneandthree-dimensionalelas-
ticproblems.Generaltheorems,fundamentalsolutions,displacements,andstress
potentials are presented and discussed. The Rayleigh-Ritz method for obtaining
approximatesolutionsisintroducedforelastostaticandspectralanalysisproblems.
Thebookcontainsmorethan60exercisesandsolutionsintheformofMathemat-
icanotebooksthataccompanyeverychapter.Oncethereaderlearnsandmasters
thetechniques,theycanbeappliedtoalargerangeofpracticalandfundamental
problems.
AndreiConstantinescuiscurrentlyDirecteurdeRechercheatCNRS:theFrench
National Center for Scientific Research in the Laboratoire de Mecanique des
Solides,andAssociatedProfessoratE´colePolytechnique,Palaiseau,nearParis.He
teachescoursesoncontinuummechanics,elasticity,fatigue,andinverseproblems
atengineeringschoolsfromtheParisTechConsortium.Hisresearchisinapplied
mechanicsandcoversareasrangingfrominverseproblemsandtheidentification
ofdefectsandconstitutivelawstofatigueandlifetimepredictionofstructures.The
results have applied through collaboration and consulting for companies such as
thecarmanufacturerPeugeot-Citroen,energyprovidersE´lectricite´ deFranceand
GazdeFrance,andtheaeroenginemanufacturerMTU.
AlexanderKorsunskyiscurrentlyProfessorintheDepartmentofEngineeringSci-
ence,UniversityofOxford.HeisalsoaFellowandDeanatTrinityCollege,Oxford.
HeteachescoursesinEnglandandFranceonengineeringalloys,fracturemechan-
ics,appliedelasticity,advancedstressanalysis,andresidualstresses.Hisresearch
interestsareinthefieldofexperimentalcharacterizationandtheoreticalanalysis
ofdeformationandfractureofmetals,polymers,andconcrete,withemphasison
thermo-mechanical fatigue and damage. He is particularly interested in residual
stresseffectsandtheirmeasurementbyadvanceddiffractiontechniquesusingneu-
tronsandhigh-energyX-raysatsynchrotronsourcesandinthelaboratory.Heisa
memberoftheScienceAdvisoryCommitteeoftheEuropeanSynchrotronRadi-
ationFacilityinGrenoble,andheleadsthedevelopmentofthenewengineering
instrument(JEEP)atDiamondLightSourcenearOxford.
i
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Elasticity with
M (cid:1)
ATHEMATICA R
AN INTRODUCTION TO
CONTINUUM MECHANICS
AND LINEAR ELASTICITY
Andrei Constantinescu
CNRSandEcolePolytechnique
Alexander Korsunsky
TrinityCollege,UniversityofOxford
iii
CAMBRIDGEUNIVERSITYPRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB28RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521842013
© Andrei Constantinescu and Alexander Korsunsky 2007
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2007
ISBN-13 978-0-511-35463-2 eBook (EBL)
ISBN-10 0-511-35463-0 eBook (EBL)
ISBN-13 978-0-521-84201-3 hardback
ISBN-10 0-521-84201-8 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
PAB CUFX161-Constantinescu August13,2007 17:14
Contents
Acknowledgments pageix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1
Motivation 1
Whatwillandwillnotbefoundinthisbook 4
1 Kinematics:displacementsandstrains . . . . . . . . . . . . . . 8
outline 8
1.1. Particlemotion:trajectoriesandstreamlines 8
1.2. Strain 19
1.3. Smallstraintensor 28
1.4. Compatibilityequationsandintegrationofsmallstrains 29
summary 35
exercises 35
2 Dynamicsandstatics:stressesandequilibrium . . . . . . . . . . 41
outline 41
2.1. Forcesandmomenta 41
2.2. Virtualpowerandtheconceptofstress 42
2.3. ThestresstensoraccordingtoCauchy 46
2.4. Potentialrepresentationsofself-equilibratedstresstensors 48
summary 50
exercises 50
3 Linearelasticity . . . . . . . . . . . . . . . . . . . . . . . 56
outline 56
3.1. Linearelasticity 56
3.2. Matrixrepresentationofelasticcoefficients 58
3.3. Materialsymmetry 65
3.4. Theextensionexperiment 72
3.5. Furtherpropertiesofisotropicelasticity 75
3.6. Limitsoflinearelasticity 78
summary 80
exercises 80
v
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vi Contents
4 Generalprinciplesinproblemsofelasticity . . . . . . . . . . . 86
outline 86
4.1. Thecompleteelasticityproblem 86
4.2. Displacementformulation 88
4.3. Stressformulation 89
4.4. Example:sphericalshellunderpressure 91
4.5. Superpositionprinciple 94
4.6. Quasistaticdeformationandthevirtualworktheorem 95
4.7. Uniquenessofsolution 95
4.8. Energypotentials 96
4.9. Reciprocitytheorems 99
4.10. TheSaintVenantprinciple 101
summary 109
exercises 109
5 Stressfunctions . . . . . . . . . . . . . . . . . . . . . . . 116
outline 116
5.1. Planestress 116
5.2. AirystressfunctionoftheformA (x,y) 119
0
5.3. Airystressfunctionwithacorrectiveterm:A (x,y)−z2A (x,y) 122
0 1
5.4. Planestrain 124
5.5. AirystressfunctionoftheformA (γ,θ) 126
0
5.6. Biharmonicfunctions 126
5.7. Thedisclination,dislocations,andassociatedsolutions 130
5.8. Awedgeloadedbyaconcentratedforceappliedattheapex 133
5.9. TheKelvinproblem 137
5.10. TheWilliamseigenfunctionanalysis 139
5.11. TheKirschproblem:stressconcentrationaroundacircularhole 145
5.12. The Inglis problem: stress concentration around an elliptical
hole 147
summary 152
exercises 152
6 Displacementpotentials . . . . . . . . . . . . . . . . . . . . 157
outline 157
6.1. Papkovich–Neuberpotentials 158
6.2. Galerkinvector 182
6.3. Lovestrainfunction 183
summary 186
exercises 187
7 Energyprinciplesandvariationalformulations . . . . . . . . . . 189
outline 189
7.1. Strainenergyandcomplementaryenergy 189
7.2. Extremumtheorems 192
vi
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Contents vii
7.3. Approximatesolutionsforproblemsofelasticity 196
7.4. TheRayleigh–Ritzmethod 197
7.5. Extremalpropertiesoffreevibrations 204
summary 212
exercises 212
Appendix1.Differentialoperators 219
(cid:1)
Appendix2.MathematicaR tricks 235
Appendix3.Plottingparametricmeshes 243
Bibliography 249
Index 251
vii
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