Fracture Phenomena in Nature and Technology: Proceedings of the IUTAM Symposium on Fracture Phenomena in Nature and Technology held in Brescia, Italy, 1-5 July 2012 PDF
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Davide Bigoni · Angelo Carini Massimiliano Gei · Alberto Salvadori Editors Fracture Phenomena in Nature and Technology Proceedings of the IUTAM Symposium on Fracture Phenomena in Nature and Technology held in Brescia, Italy, 1–5 July 2012 Fracture Phenomena in Nature and Technology Davide Bigoni Angelo Carini • Massimiliano Gei Alberto Salvadori • Editors Fracture Phenomena in Nature and Technology Proceedings of the IUTAM Symposium on Fracture Phenomena in Nature and Technology held in Brescia, Italy, 1–5 July 2012 Reprinted from International Journal of Fracture, Volume 184, Nos. 1–2 (2013) 123 Editors Davide Bigoni Angelo Carini Massimiliano Gei Alberto Salvadori Dipartimentodi IngegneriaCivile, Dipartimentodi IngegneriaCivile, Ambientale e Meccanica Architettura, Territorio, Università diTrento Ambiente edi Matematica Trento Università diBrescia Italy Brescia Italy ISBN 978-3-319-04396-8 ISBN 978-3-319-04397-5 (eBook) DOI 10.1007/978-3-319-04397-5 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014936174 (cid:2)SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection withreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeingenteredand executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 D. Bigoni, A. Carini, M. Gei and A. Salvadori Modeling fracture by material-point erosion . . . . . . . . . . . . . . . . . . . . 3 A. Pandolfi, B. Li and M. Ortiz Crack front perturbations revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 J. R. Willis Localisation near defects and filtering offlexural waves in structured plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 S. G. Haslinger, R. C. McPhedran, N. V. Movchan and A. B. Movchan Fracture process in cortical bone: X-FEM analysis of microstructured models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 S. Li, A. Abdel-Wahab, E. Demirci and V. V. Silberschmidt Minimum theorems in 3d incremental linear elastic fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 A. Salvadori and F. Fantoni Crack patterns obtained by unidirectional drying of a colloidal suspension in a capillary tube: experiments and numerical simulations using a two-dimensional variational approach . . . . . . . . . . 75 C. Maurini, B. Bourdin, G. Gauthier and V. Lazarus Damage mechanisms in the dynamic fracture of nominally brittle polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 D. Dalmas, C. Guerra, J. Scheibert and D. Bonamy Tight sedimentary covers for CO sequestration. . . . . . . . . . . . . . . . . . 113 2 D. Leguillon, E. Karnaeva, A. Baroni and C. Putot Calibration of brittle fracture models by sharp indenters and inverse analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 V. Buljak, G. Cocchetti and G. Maier v vi Contents Statistics of ductile fracture surfaces: the effect of material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 L. Ponson, Y. Cao, E. Bouchaud, V. Tvergaard and A. Needleman Efficient pseudo-spectral solvers for the PKN model of hydrofracturing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 M. Wrobel and G. Mishuris A solution to the parameter-identification conundrum: multi-scale interaction potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 J. G. M. van Mier Remarks on application of different variables for the PKN model of hydrofracturing: various fluid-flow regimes . . . . . . . . . . . . . . 185 P. Kusmierczyk, G. Mishuris and M. Wrobel Prediction of grain boundary stress fields and microcrack initiation induced by slip band impingement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 M. Sauzay and M. O. Moussa Modeling the heterogeneous effects of retained austenite on the behavior of martensitic high strength steels. . . . . . . . . . . . . . . . 241 Q. Wu, P. Shanthraj and M. A. Zikry Crack nucleation from a notch in a ductile material under shear dominant loading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 A. Ghahremaninezhad and K. Ravi-Chandar DOI10.1007/978-3-319-04397-5_1 Reprinted from International Journal of Fracture, DOI10.1007/s10704-013-9896-2 Foreword D.Bigoni · A.Carini · M.Gei · A.Salvadori Publishedonline:25October2013 ©SpringerScience+BusinessMediaDordrecht2013 ThisSpecialIssueoftheInternationalJournalofFrac- in contact and friction, crystal dislocation processes, turecontainsselectedpaperspresentedattheIUTAM and atomic/electronic scale treatment of brittle crack SymposiumFracturePhenomenainNatureandTech- tips and fundamental cohesive properties. The Spe- nology that was held at the School of Engineering, cial Issue manuscripts were limited to original work University of Brescia, Italy, during the week of July andwerereviewedfollowingthestandardprocedures 1–5, 2012. The symposium focused on innovative of the Journal. The Organizing Committee is grateful contributions in fracture research, interpreted broadly to the IUTAM that fostered and supported the sym- to include new engineering and structural mechanics posium, to the financial support from the EU (PIAP- treatmentsofdamagedevelopmentandcrackgrowth, GA-2011-286110-INTERCER2), totheUniversityof large-scale failure processes as exemplified by earth- Brescia (Italy) and to the University of Trento (Italy) quake or landslide failures, ice shelf break-up, and that contributed greatly to the success of the Confer- hydraulic fracturing (natural, or for resource extrac- ence,totheMunicipalityofBresciathatwelcomedthe tion or CO sequestration), small-scale rupture phe- eventandtheparticipants. 2 nomena in materials physics including inception of GuestEditors shearbanding,voidgrowth,adhesionanddecohesion B D.Bigoni( )·A.Carini·M.Gei·A.Salvadori Trento,Italy e-mail:[email protected] 123 Reprinted from the journal 1 DOI10.1007/978-3-319-04397-5_2 Reprinted from International Journal of Fracture, DOI10.1007/s10704-012-9788-x ORIGINAL PAPER Modeling fracture by material-point erosion A.Pandolfi · B.Li · M.Ortiz Received:3August2012/Accepted:6November2012/Publishedonline:20November2012 ©SpringerScience+BusinessMediaDordrecht2012 Abstract The present work is concerned with the Keywords Meshfreeapproaches· verificationandvalidationofanimplementationofthe Materialpointerosion·Eigenfracture· eigenfractureschemeofSchmidtetal.(SIAMJMulti- Max-Entshapefunctions·BrittleFracture scaleModelSimul7:1237–1266,2009)basedonmate- rial-point erosion, which we refer to as eigenerosion. Eigenerosion is derived from the general eigenfrac- 1 Introduction ture scheme by restricting the eigendeformations in a binarysense:theycanbeeitherzero,inwhichcasethe Lagrangian meshfree methods are well-suited to a local behavior is elastic; or they can be equal to the number of areas of application, such as terminal local displacement gradient, in which case the corre- ballistics, machining, fluid-structure interaction. sponding material neighborhood is failed, or eroded. Lag-rangian meshfree methods offer significant Whencombinedwithamaterial-pointspatialdiscreti- advantagesovercompetingapproaches,suchaspurely zation,thisschemegivesrisetomaterial-pointerosion, Eulerian formulations, particle methods, purely i.e.,eachmaterialpointcanbeeitherintact,inwhich Lagrangian formulations with continuous adaptive caseitsbehavioriselastic,orbecompletelyfailed—or remeshing, arbitrary-Lagrangian–Eulerian (ALE). eroded—and has no load bearing capacity. We verify These competing approaches may suffer from a vari- theeigenerosionschemethroughconvergencestudies ety of shortcomings, for example: the introduction for mode I fracture propagation in three-dimensional of large numerical diffusion errors; large discreti- problems. By way of validation we apply the eigene- zation errors at fluid-solid interfaces; difficulties in rosionschemetothesimulationofcombinedtorsion- maintaining monotonicity, positivity and in tracking tractionexperimentsinaluminum-oxidebars. state variables; spurious modes and tensile instabili- ties;meshentanglement;theneedtoremeshorrezone B A.Pandolfi( ) arbitrary three-dimensional domains and the atten- DipartimentodiIngegneriaStrutturale,Politecnicodi dant remapping of state variables; ad-hoc transition Milano,PiazzaLeonardodaVinci32,20133Milano,Italy e-mail:pandolfi@stru.polimi.it orblendingregions;difficultiesindefiningnumerical integrationrulesandsatisfyingessentialboundarycon- B.Li·M.Ortiz ditions;unknownconvergenceandstabilityproperties; GraduateAeronauticalLaboratories,CaliforniaInstitute andothers. ofTechnology,Pasadena,CA91125,USA e-mail:[email protected] A representative example of Lagrangian mesh- free schemes is furnished by the Optimal-Transpor- M.Ortiz e-mail:[email protected] tation Meshfree (OTM) method of Li et al. (2010). 123 Reprinted from the journal 3 A.Pandolfietal. In the quasistatic setting of interest here, the OTM elasticenergy.Inaddition,intheeigenfracturescheme method combines: (i) Maximum-entropy (max-ent) thefractureenergyissettobeproportionaltothevol- meshfree interpolation Arroyo and Ortiz (2006) from ume of the θ-neighborhood of the support of the eig- a nodal-point set; and (ii) material-point sampling endeformationfield,suitablyscaledby1/θ.Theopti- (cf.,e.g.,Sulskyetal.1994)inordertotrackthelocal malcrack setisobtained by minimizingthe resulting stateofmaterialpoints,carryoutcomplexconstitutive energyfunctionalwithrespecttoboththedisplacement updates and perform spatial integrals. Max-ent inter- andtheeigendeformationfields,subjecttoirreversibil- polationArroyoandOrtiz(2006)offerstheadvantage ity constraints. We note that other two-field approxi- of being meshfree and entirely defined—essentially mation schemes for brittle fracture, most notably the explicitly—by the current nodal-set positions, thus Ambrosio-Tortorellischeme(AmbrosioandTortorelli effectivelysidesteppingtheneedforcontinuousreme- 1992;BraidesandDefranceschi1998),havebeenpro- shing in simulations of unconstrained flows. In addi- posed in the past and used as a basis for numerical tion,max-entinterpolationsatisfiesaKronecker-delta approximations(BourdinandChambolle2000;Bour- propertyattheboundary,whichgreatlyfacilitatesthe dinetal.2000;Bourdin2007),buttheuseofeigende- enforcementofessentialboundaryconditions,andhas formations to describe brittle fracture in a variational good accuracy convergence and monotonicity condi- frameworkdoesnotappeartohavebeenpursuedprior tions. Because of interpolatory nature, OTM is free toSchmidtetal.(2009).Wealsonotethatotherdam- fromthetensilenumericalinstabilitiesthatplaguepar- ageregularizationsofbrittlefracture(BraidesandDal ticlemethods.Indynamicproblems,theOTMmethod Maso 1997; Braides 2002; Braides and Defranceschi additionally draws on optimaltransportation concept, 1998;Negri2005)havebeenproposedinthepastand such as the Wasserstein distance between successive showntobeconvergent. massdensities,inordertodiscretizetheactionintegral Inthepresentwork,wespecificallyconsideramesh- in time. The optimal-transportation approach to time free approach based on maximum-entropy (max-ent) discretization leads to geometrically-exact updates of interpolation(ArroyoandOrtiz2006)combinedwith the local volumes and mass densities, and exact con- material-point sampling and integration (Li et al. servationpropertiesincludingsymplecticity,linearand 2010). We confine our attention throughout to quasi- angularmomentum. static problems. Extensions of the max-ent meshfree Many of the applications where Lagrangian mesh- approachtodynamics,basedonoptimal-transportation freeschemes,suchastheOTMmethod,areattractive theory, may be found in Li et al. (2010). In the max- involve material failure and fracture of some kind. ent/material-pointschemeconsideredhere,thespatial However, there is limited experience at present con- discretizationisbasedontwosetsofpoints:thenodal cerning the simulation of fracture and fragmentation points and the material points. Specifically, the nodal processeswithintheframeworkofmeshfreeinterpola- points carry the information concerning the displace- tionschemes.Notableexceptionsarethecontributions ments, whereas the material points carry the material in the meshfree Galerkin approximation (Belytschko state, including eigendeformations. The link between et al. 1993; Lu et al. 1995; Belytschko et al. 1996), materialpointsandnodesisestablishedthroughmax- and in the smooth particle hydrodynamics method entinterpolation.Themax-entshapefunctionsexhibit (RabczukandEibl2003;Rabczuketal.2004;Karekal rapid decay and their support can be restricted to a etal.2011).Inthispaperweassesstheperformanceof finite range, with the result that every material point arecentlyproposedapproachtofracture,termedeigen- is connected to a limited number of nodes within its fracture(Schmidtetal.2009),withinameshfreeframe- immediateenvironment. work.Theeigenfractureschemeresortstotheclassical When combined with the max-ent/material-point device of eigendeformations (Mura 1987; Colonnetti scheme, eigenfracture may be implemented as mate- 1917)inordertoaccountformaterialfracture.Tothis rial-point erosion, i. e., the material-points can be end, the energy functional depends on two fields: the either intact, in which case their behavior is elastic, displacementfieldu andaneigendeformationfieldε∗ orbecompletelyfailed—oreroded—andhavenoload that describes such cracks as may be present in the bearing capacity. The implementation of the method, body. Specifically, eigendeformations allow the dis- included the all-important θ-neighborhood construc- placement field to develop jumps at no cost in local tion, is exceedingly simple and applies to general 112233 4 Reprinted from the journal Modelingfracturebymaterial-pointerosion situations, possibly involving complex three-dimen- 2 Variationalformulationoffracturemechanics sionalfracturepatternssuchasbranchingandfragmen- tation. The accuracy and convergence of the eigene- In this section we succinctly summarize the formula- rosion approach is comparable—at a much reduced tion of fracture mechanics that we take as the basis implementationcostandcomplexity—tothatofother for subsequent developments. We specifically follow numericalfractureschemes.Wenotethatelementero- Larsenetal.(2009)andPandolfiandOrtiz2012,which sionhasbeenextensivelyusedtosimulatefractureina maybeconsultedforadditionalmathematicaldetail. numberofareasofapplication,includingterminalbal- We consider an elastic body occupying a domain listics (Johnson and Stryk 1987; Belytschko and Lin φ ⊂ Rn, n ≥ 2. The boundary ∂φ of the body con- 1987; Ortiz and Giannakopoulos 1990; Johnson and sists of an exterior boundary ω, corresponding to the Stryk1990;WhirleyandHallquist1991;Borviketal. boundary of the uncracked body, and a collection of 2008). However, some of these methods fail to con- cracks jointly defining a crack set C. In addition, ω vergeorconvergetothewronglimit(Negri2003).By is partitioned into a displacement boundary ω and a 1 contrast,theeigenfractureschemeisknowntoproperly traction boundary ω . The body undergoes deforma- 2 convergetoGriffithfracture(Griffith1920)inthelimit tions under the action of body forces, displacements ofvanishinglysmallmeshsizes(Schmidtetal.2009). prescribedoverω andtractionsappliedoverω ,sub- 1 2 Inparticular,thelocal-neighborhoodaveragingofthe jecttoacontactconditiononC.Undertheseconditions, energywhichunderliesthecalculationoftheeffective thepotentialenergyofthebodyis energy-release has the effect of eliminating spurious ⎧(cid:6) (cid:6) mesh-dependencies. ⎪⎨ W(x,u,∇u)dx+ V(x,u)dHn−1,if[[u]]·ν≥0, Webaseourassessmentofthemethodonselected E(u,C,t)= ⎪⎩φ ω2 verification and validation test cases. We verify the +∞, otherwise, approach by means of convergence studies for mode (1) Ifracturepropagationinthree-dimensionalplates.We where,hereandsubsequently,Hdisthed-dimensional additionallypresentavalidationofthemethodthrough Hausdorffmeasure,1νisaunitnormaltoC,[[u]]isthe simulationsofcombined traction-torsionexperiments displacementjump, on aluminum oxide bars (Suresh and Tschegg 1987). Wefindthattheeigenerosionschemeindeedresultsin [[u]]·ν ≥0 (2) convergentapproximations,bothasregardscrackpaths defines the contact constraint, W is the elastic strain as well as the attendant deformation fields and struc- energy density of the body—possibly including dis- turalresponse.Wealsofindthattheschemeenablesthe tributed body forces— and V is the potential of the simulationofexceedinglycomplexthree-dimensional applied tractions. The explicit dependence of E on t fracturepatterns.Therangeandversatilityaffordedby in (1) is meant to reflect the time dependence of the theapproachisallthemoreremarkablegiventhesim- forcing,namely,theappliedforcesandprescribeddis- plicityofitsimplementation. placements. Suppose now that the applied loads and The paper is organized as follows. In Sect. 2, prescribeddisplacementsareincrementedoverthetime we begin by formulating the problem to be approx- interval[t,t +πt]andthat,inresponsetothisincre- imated, namely, the problem of quasi-static crack- mental loading, the crack set extends from C(t) to growth in an otherwise linear-elastic solid. We C(t +πt).Owingtotheirreversibilityoffracturewe continue with brief review to the max-ent/material- mustnecessarilyhavethat point approach in Sect. 3. Then we recall briefly the eigenerosionmethodinSect.4.InSect.5.1weverify C(t)⊂C(t +πt), (3) theapproachbymeansofconvergencestudiesformode i.e.,thecracksetmustbemonotonicallyincreasingin I fracture propagation in three-dimensional plates. In time.Lettheelasticenergyincrementrecordedduring Sect.5.2wepresentavalidationofthemethodthrough thetimeincrementbeπE.Then,aclassicalcalculation simulationsofcombined traction-torsionexperiments on aluminum oxide bars, taken from Suresh and 1 cf.,e.g.,DalMasoandToader(2002);onsmoothcurves,dH1 Tschegg (1987). We conclude with some comments istheelementoflength;onsmoothsurfaces,dH2istheelement ontheactualresultsandpossibleextensionsinSect.6. ofarea. 123 Reprinted from the journal 5
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