Fracture Mechanics: With an Introduction to Micromechanics

347 Pages·2011·3.339 MB·English

by  Dietmar Gross, Thomas Seelig (auth.)

Mechanical Engineering Series
Frederick F. Ling
Editor-in-Chief
For further volumes:
http://www.springer.com/series/1161

Dietmar Gross • Thomas Seelig
Fracture Mechanics
With an Introduction to Micromechanics
Second Edition
Prof. Dr.-Ing. Dietmar Gross Prof. Dr.-Ing. Thomas Seelig
Division of Solid Mechanics Institute of Mechanics
TU Darmstadt Karlsruhe University (KIT)
Hochschulstrasse 1 Kaiserstrasse 12
D-64289 Darmstadt D-76131 Karlsruhe
Germany Germany
[email protected] [email protected]
ISSN 0941-5122 e-ISSN 2192-063X
ISBN978-3-642-19239-5 e-ISBN978-3-642-19240-1
DOI10.1007/978-3-642-19240-1
Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: 2011932493
© Springer-VerlagBerlinHeidelberg2011
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is

concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,
rep roduction on microfilm or in any other way, and storage in data banks. Duplication of this publication
or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,
1965, in its current version, and permission for use must always be obtained from Springer. Violations
are liable to prosecution under the German Copyright Law.
The use of general descriptive names, registered names, trademarks, etc. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from the relevant protective
laws and regulations and therefore free for general use.
Cover design: SPi Publisher Services
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Preface
This book has evolved from lectures on fracture mechanics and micromechanics
whichweheldforstudentsofengineeringandnaturalsciencesovertheyears.Itis
primarily meant as an aid for students learning the foundationsof these subjects.
At the same time this book may also serve as an introductioninto these fields for
researchersandpractitionersinindustryandtoprovidethetheoreticalbackground
forsolvingrespectiveproblems.
The book covers the most important areas of fracture mechanics and gives an
introductionintomicromechanics.Ourmajorconcernwasthepresentationofprin-
cipalconceptsandmethodsinaclearandsoundmannerasabasisforadeeperentry
into the matter. The presentation mainly focuseson the mechanicaldescription of
fractureprocesses;yet,materialspecificaspectsarealsodiscussed.Tokeepthetext
self-contained,continuummechanicalandphenomenologicalfoundationsarereca-
pitulatedfirst. Theyarefollowedbyabriefsurveyofclassicalfractureandfailure
hypotheses. A major part of the book is devoted to linear fracture mechanics and
elastic-plasticfracturemechanics.Furtherchaptersdealwithcreepfractureanddy-
namicfracturemechanics.Anextensivechaptertreatsfoundationsofmicromechan-
ics and homogenization.Finally, elements of damage mechanicsand probabilistic
fracture mechanics are presented. Suggestionsfor further reading are listed at the
endofeachchapter.
Thefirsteditionwas wellacceptedbythereadersmakinga neweditionneces-
sary.Wehaveusedthischancetoincorporateanumberofextensionswhichpartly
are influencedby new developmentsin the field of fracture mechanics.Discussed
are, among others, the crack initiation at notches, cohesive zone models, the peel
test, fragmentation, and strain localization due to damage and material softening.
Furthermore,followingsuggestionsfrommanystudents, supplementaryexamples
havebeenaddedasproblemsattheendofsomechapters.
The authors are indebted to all who have contributed to this book. This par-
ticularly includes those from whom we have learned or, as Roda Roda has put it
ironically:“Copyingfromfourbooksyieldsafifthprofoundbook”.Specialthanks
go to Mrs. Dipl.-Ing.H. Herbstwhohaspreparedmostof the figures.Finally,the
pleasantcooperationwiththepublisherisgratefullyacknowledged.
DarmstadtandKarlsruheinJune2011, DietmarGross
ThomasSeelig
v

Contents
Introduction ....................................................... 1
1 Elementsofsolidmechanics..................................... 5
1.1 Stress .................................................... 5
1.1.1 Stressvector ........................................ 5
1.1.2 Stresstensor ........................................ 7
1.1.3 Equilibriumconditions ............................... 10
1.2 Deformationandstrain...................................... 11
1.2.1 Straintensor ........................................ 11
1.2.2 Strainrate .......................................... 13
1.3 Constitutivelaws........................................... 14
1.3.1 Elasticity ........................................... 15
1.3.2 Viscoelasticity....................................... 19
1.3.3 Plasticity ........................................... 22
1.4 Energyprinciples........................................... 27
1.4.1 Energybalance ...................................... 27
1.4.2 Principleofvirtualwork .............................. 28
1.4.3 TheoremsofClapeyronandBetti....................... 30
1.5 Planeproblems ............................................ 31
1.5.1 Planestress,planestrain,longitudinalshear .............. 31
1.5.2 Linearelasticity,complexmethod ...................... 33
1.5.3 Perfectlyplasticmaterial,slip-linefields................. 35
1.6 Furtherreading ............................................ 38
2 Classicalfractureandfailurehypotheses.......................... 39
2.1 Basicconcepts............................................. 39
2.2 Failurehypotheses.......................................... 41
2.2.1 Principalstresshypothesis............................. 41
2.2.2 Principalstrainhypothesis............................. 42
2.2.3 Strainenergyhypothesis .............................. 42
2.2.4 Coulomb-Mohrhypothesis ............................ 43
vii
viii Contents
2.2.5 Drucker-Pragerhypothesis ............................ 46
2.2.6 Johnson-Cookcriterion ............................... 47
2.3 Deformationbehaviorduringfailure........................... 48
2.4 Problems ................................................. 49
2.5 Furtherreading ............................................ 49
3 Microandmacrophenomenaoffracture ......................... 51
3.1 Microscopicaspects ........................................ 51
3.1.1 Surfaceenergy,theoreticalstrength ..................... 51
3.1.2 Microstructureanddefects ............................ 53
3.1.3 Crackformation ..................................... 56
3.1.4 Percolation ......................................... 57
3.2 Macroscopicaspects ........................................ 59
3.2.1 Crackgrowth........................................ 59
3.2.2 Typesoffracture..................................... 60
3.3 Furtherreading ............................................ 61
4 Linearfracturemechanics ...................................... 63
4.1 Generalremarks ........................................... 63
4.2 Crack-tipfield ............................................. 64
4.2.1 Two-dimensionalcrack-tipfields ....................... 64
4.2.2 Mode-Icrack-tipfield ................................ 70
4.2.3 Three-dimensionalcrack-tipfield....................... 71
4.3 K-concept ................................................ 72
4.4 K-factors ................................................. 73
4.4.1 Examples........................................... 74
4.4.2 Integralequationformulation .......................... 80
4.4.3 Methodofweightfunctions ........................... 82
4.4.4 Crackinteraction .................................... 85
4.4.5 Stressintensityfactorsandstressconcentrationfactors..... 90
4.5 FracturetoughnessK ..................................... 92
Ic
4.6 Energybalance ............................................ 94
4.6.1 Energyreleaseduringcrackpropagation................. 94
4.6.2 Energyreleaserate................................... 96
4.6.3 Compliance,energyreleaserate,andK-factors........... 99
4.6.4 Energybalance,Griffith’sfracturecriterion ..............100
4.6.5 J−integral..........................................106
4.7 Small-scaleyielding ........................................113
4.7.1 Plasticzonesize,Irwin’scracklengthcorrection..........113
4.7.2 Qualitativeremarksontheplasticzone ..................115
4.8 Stablecrackgrowth.........................................116
4.9 Mixed-modeloading........................................119
4.10 Crackinitiationatcavitiesandnotches.........................124
4.11 Fatiguecrackgrowth........................................126
4.12 Interfacecracks ............................................128
Contents ix
4.13 Piezoelectricmaterials ......................................136
4.13.1 Basicprinciples .....................................136
4.13.2 Thecrackinaferroelectricmaterial.....................138
4.14 Problems .................................................140
4.15 Furtherreading ............................................143
5 Elastic-plasticfracturemechanics................................145
5.1 Introduction ...............................................145
5.2 Dugdalemodel.............................................146
5.3 Cohesivezonemodels.......................................150
5.4 Crack-tipfield .............................................153
5.4.1 Perfectlyplasticmaterial..............................153
5.4.2 Totalstraintheory,HRR−field.........................159
5.5 Fracturecriterion...........................................164
5.6 DeterminationofJ .........................................166
5.7 DeterminationofJ ........................................167
c
5.8 Crackpropagation..........................................171
5.8.1 J–controlledcrackgrowth ............................171
5.8.2 Stablecrackgrowth ..................................173
5.8.3 Steady-statecrackgrowth .............................175
5.9 Essentialworkoffracture....................................181
5.10 Problems .................................................183
5.11 Furtherreading ............................................184
6 Creepfracture.................................................185
6.1 Introduction ...............................................185
6.2 Fractureoflinearviscoelasticmaterials ........................186
6.2.1 Crack-tipfield,elastic-viscoelasticanalogy ..............186
6.2.2 Fractureconcept.....................................188
6.2.3 Crackpropagation ...................................190
6.3 Creepfractureofnonlinearmaterials ..........................194
6.3.1 Secondarycreep,constitutivelaw.......................194
6.3.2 Stationarycrack,crack-tipfield,loadingparameters .......195
6.3.3 Creepcrackgrowth ..................................199
6.4 Furtherreading ............................................204
7 Dynamicfracturemechanics ....................................207
7.1 Introduction ...............................................207
7.2 Somefoundationsofelastodynamics ..........................208
7.3 Dynamicloadingofastationarycrack .........................209
7.3.1 Crack-tipfield,K-concept .............................209
7.3.2 Energyreleaserate,energeticfracturecriterion ...........210
7.3.3 Examples...........................................211
7.4 Crackpropagation..........................................214
7.4.1 Crack-tipfield.......................................214