Table Of ContentDavide 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
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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
