Table Of ContentISDMM 2007
Defect and Material Mechanics
Proceedings of the International Symposium on
Defect and Material Mechanics (ISDMM),
held in Aussois, France, March 25–29, 2007
Edited by
CRISTIAN DASCALU
University J. Fourier, Grenoble, France
GE´ RARD A. MAUGIN
University Pierre etMarieCurie, Paris,France
and
CLAUDE STOLZ
EcolePolytechnique, Palaiseau, France
Reprinted from International Journal of Fracture
Volume 147, Nos. 1–4 (2007)
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Table of Contents
Preface 1
C. Dascalu and G.A. Maugin
Reciprocity in fracture and defect mechanics 3–11
R. Kienzler
Configurational forces and gauge conditions in electromagnetic bodies 13–19
C. Trimarco
The anti-symmetry principle for quasi-static crack propagation in Mode III 21–33
G.E. Oleaga
Configurational balance and entropy sinks 35–43
M. Epstein
Application of invariant integrals to the problems of defect identification 45–54
R.V. Goldstein, E.I. Shifrin and P.S. Shushpannikov
On application of classical Eshelby approach to calculating effective elastic moduli of 55–66
dispersed composites
K.B. Ustinov and R.V. Goldstein
Material forces in finite elasto-plasticity with continuously distributed dislocations 67–81
S. Cleja-T¸igoiu
Distributed dislocation approach for cracks in couple-stress elasticity: shear modes 83–102
P.A. Gourgiotis and H.G. Georgiadis
Bifurcation of equilibrium solutions and defects nucleation 103–107
C. Stolz
Theoretical and numerical aspects of the material and spatial settings in nonlinear 109–116
electro-elastostatics
D.K. Vu and P. Steinmann
Energy-based r-adaptivity: a solution strategy and applications to fracture mechanics 117–132
M. Scherer, R. Denzer and P. Steinmann
Variational design sensitivity analysis in the context of structural optimization and 133–155
configurational mechanics
D. Materna and F.-J. Barthold
An anisotropic elastic formulation for configurational forces in stress space 157–161
A. Gupta and X. Markenscoff
Conservation laws, duality and symmetry loss in solid mechanics 163–172
H.D. Bui
Phase field simulation of domain structures in ferroelectric materials within the context of 173–180
inhomogeneity evolution
R. Müller, D. Gross, D. Schrade and B.X. Xu
An adaptive singular finite element in nonlinear fracture mechanics 181–190
R. Denzer, M. Scherer and P. Steinmann
Moving singularities in thermoelastic solids 191–198
A. Berezovski and G.A. Maugin
Dislocation tri-material solution in the analysis of bridged crack in anisotropic bimaterial 199–217
half-space
T. Profant, O. Ševe9cek, M. Kotoul and T. Vysloužil
Study of the simple extension tear test sample for rubber with Configurational Mechanics 219–225
E. Verron
Stress-driven diffusion in a deforming and evolving elastic circular tube of single 227–234
component solid with vacancies
C.H. Wu
Mode II intersonic crack propagation in poroelastic media 235–267
E. Radi and B. Loret
Material forces for crack analysis of functionally graded materials in adaptively refined 269–283
FE-meshes
R. Mahnken
A multiscale approach to damage configurational forces 285–294
C. Dascalu and G. Bilbie
Preface
C.Dascalu · G.A.Maugin
The volume presents recent developments in the This special issue aims at bringing together recent
theoryofdefectsandthemechanicsofmaterialforces. developmentsinMaterialMechanicsandthemoreclas-
MostofthecontributionswerepresentedattheInter- sicalDefectMechanicsapproaches.Thecontributions
national Symposium on Defect and Material Forces arehighlightingrecentresearchontopicslike:fracture
(ISDMM2007), held in Aussois, France, March 25– and damage, electromagnetoelasticity, plasticity, dis-
29,2007. tributed dislocations, thermodynamics, poroelasticity,
OriginatedintheworksofEshelby,theMaterialor generalized continua, structural optimization, conser-
Configurational Mechanics experienced a remarkable vationlawsandsymmetries,multiscaleapproachesand
revivaloverthelasttwodecades.Whenthemechanics numericalsolutionstrategies.
ofcontinuaisfullyexpressedonthematerialmanifold, We expect the present volume to be a valuable
it captures the material inhomogeneities. The driving resource for researchers in the field of Mechanics of
(material)forcesoninhomogeneitiesappearnaturally DefectsinSolids.
in this framework and are requesting for constitutive We dedicate this special issue to the memory of
modelingoftheevolutionofinhomogeneitiesthrough the late Professor George Herrmann (1921–2007).
kineticlaws. G. Herrmann was a prestigious scientist, well-known
Inthisway,ageneralschemefordescribingstruc- in the international mechanics community. In the last
tural changes in continua is obtained. The Eshelbian years,hewasanactiveresearcherinthefieldofMate-
mechanicsformulationcomesupwithaunifyingtreat- rialMechanics.GeorgeHerrmannsupportedtheorga-
mentofdifferentphenomenalikefractureanddamage nizationandregisteredforattendingtheISDMM2007
evolution,phasetransitions,plasticityanddislocation Symposium in Aussois, before his sudden dead on
motion,etc. January7,2007.Noneofuswillforgethispassionfor
mechanicalsciences,hisenthusiasmandgenerosity.
B
C.Dascalu( )
LaboratoireSolsSolidesStructures,UniversitéJoseph
Fourier,Grenoble,DomaineUniversitaire,B.P.53,
38041 Grenoblecedex9,France
e-mail:[email protected]
G.A.Maugin
UniversitéPierreetMarieCurie,InstitutJeanLeRond
d’Alembert,Case152,4placeJussieu,
75252Pariscedex05,France
e-mail:[email protected]
DefectandMaterialMechanics.C.Dascalu,G.A.Maugin&C.Stolz(eds.), 1
doi:10.1007/978-1-4020-6929-1_1,©SpringerScience+BusinessMediaB.V.2008
Reciprocity in fracture and defect mechanics
R.Kienzler
Abstract Fordefectsinsolids,whendisplacedwithin The dot marks the scalar product between the two
thematerial,reciprocityrelationshavebeenestablished vectors. In double-indexed terms, the first index indi-
recentlysimilartothetheoremsattributedtoBettiand cates the position at which the quantity is measured
Maxwell. These theorems are applied to crack- and (effect),andthesecondindexindicatesthecausedue
defect-interactionproblems. towhichthisquantityoccurs.
ToreachascalarversionofBetti’stheoremthedis-
Keywords Reciprocity·Fracture·Defect placementcomponentof u inthedirectionof F is
12 1
interaction·Materialforces introducedasuP andthecomponentofu inthedirec-
12 21
tion of F as uP (cf. Fig. 1). With the magnitude F
2 21 1
and F of F and F ,respectively,itis
2 1 2
1 Introduction
F uP = F uP. (2)
1 12 2 21
Whentreatingproblemsoflinearelasticsystems,such SinceuP isproportionalto F anduP isproportional
12 2 21
asbeams,framesortwo-andthree-dimensionalcontin-
to F influencecoefficientmaybedefinedas
1
uouselasticsolids,thereciprocitytheoremsassociated
with the names of Betti and Maxwell have proven to u1P2 =δ12F2, (3a)
bequitevaluable.Initssimplestform,Betti’stheorem uP =δ F , (3b)
21 21 1
statesthatifalinearelasticbodyissupportedproperly
and according to Marguerre (1962), Maxwell’s theo-
such that rigid body displacements are precluded and
remstates
ifanexternalforce F atpoint1whichproducesadis-
1
placement u21 at some other point 2, then a force F2 δ12 =δ21. (4)
at2wouldproduceadisplacementu at1where(cf.
12 Thereciprocityrelationsarebasedontheresultthatthe
e.g.,Marguerre1962)
energystoredinanelasticbodyafterapplicationoftwo
F1·u12 = F2·u21. (1) forcesisindependentoftheirsequenceofapplication,
andequalstheexternalworkdoneonthebody.Various
Thecontentsofthepresentpaperhasbeendevelopedtogether
applicationsofthesetheoremsaretobefoundin,e.g.,
withProf.Dr.Dr.h.c.GeorgeHerrmann,StanfordUniversity,
California,whopassedawayonJanuary7,2007. TimoshenkoandGoodier(1970),Barber(2002).
Duringtherecentdecadesanewtopichasemerged
B
R.Kienzler( )
in mechanics of elastically deformable media which
DepartmentofProductionEngineering,
is variously described as Defect Mechanics, Fracture
UniversityofBremen,Bremen,Germany
e-mail:[email protected] Mechanics,ConfigurationalMechanics,Mechanicsin
DefectandMaterialMechanics.C.Dascalu,G.A.Maugin&C.Stolz(eds.), 3
doi:10.1007/978-1-4020-6929-1_2,©SpringerScience+BusinessMediaB.V.2007
4 R.Kienzler
qqq
FFF
111 FFF
FFF 1
222
uuu1111 uuu111PPP222 2222 BBB1111 BBB1P2 2222 2
111222 uuuPPP uuu 12 BBBP BBB
222111 222111 21 21
Fig.2 Stressedelasticbodycontainingtwopointdefects
Fig.1 Elasticbodysubjectedtotwoforces
Basedonthesimilarargument(asappliedinPhysi-
calSpace)thatthechangeofenergystoredinthebody
afterapplicationoftwomaterialdisplacementsisinde-
Material Space or Eshelbian Mechanics (cf. Maugin
pendentoftheirsequenceofapplication,andequalsthe
1993;Gurtin2000;KienzlerandHerrmann2000).The
work done of the material forces in the material dis-
importance of this topic is based on the necessity of
placements, a material, Betti-like reciprocity theorem
improvedmaterialmodelingofdefectsofvarioustypes
isestablished(HerrmannandKienzler2007a)as
andscalesindeformablesolids.Andthisnecessityin
turn is the result of developing new technologies in a λ2· B21 =λ1· B12, (5)
varietyofappliedfieldssuchasdevicesinIT,aerospace
statingthattheworkinthematerialtranslationat2of
andenergysectors.
thechangeofthematerialforceduetoamaterialtrans-
In approaching configurational mechanics one can
lationat1equalstheworkinthematerialtranslationat
saythatamaterialforceisassociatedwithadefectin
1ofthechangeofthematerialforceduetoamaterial
astressedelasticbody,becauseifthisdefectweredis-
translationat2.
placedwithinthebody,thetotalenergyofthesystem
NotethedifferencebetweenPhysicalandMaterial
wouldbechangedandthenegativeratioofthischange
Space: in Physical Space, the applied physical forces
andthedisplacement,inthelimitasthedisplacement
arethecausesof(thechangeof)physicaldisplacements
ismadetoapproachzero,isthematerialforceonthe
(effects) whereas in Material Space the material dis-
defect.
placementcause(changesof)materialforces(effects).
Let us consider a linearly elastic body of arbitrary
Inanalogy,ascalarversionofEq.5isreachedbyintro-
geometryproperlysupportedandsubjectedtosurface ducingthecomponentof B inthedirectionofλ as
21 2
tractionsand/orbodyforces.Andletusassumethatthe BP,thecomponentofB inthedirectionofλ asBP
21 12 1 12
bodycontainsanarbitrarynumberandtypeofdistrib- (cf.Fig.2)andthemagnitudesofλ andλ asλ and
1 2 1
utedorlocalized(point)defects,suchasdislocations, λ ,respectively,andiffollows
2
cracks,inclusions,cavities,etc.Letusfocusattention
on two localized defects placed at positions 1 and 2 λ2B2P1 =λ1B1P2. (6)
and let the associated material forces be the vectors
Since linearity is implied, material influence coeffi-
B and B respectively. Now, defect 1 will be dis-
10 20 cientsmaybedefinedas
placed by an amount λ relatively to the material in
1
BP =β λ , (7a)
whichthedefectispositioned(materialdisplacement). 21 21 1
In turn the material force at 1 will be changed by an
BP =β λ , (7b)
amountB andthematerialforce2willbechangedby 12 12 2
11
B21.Thematerialdisplacementisassumedtobesmall and a material, Maxwell-like reciprocity theorem is
inthesensethatlinearityisimpliedbetweenmaterial obtained(HerrmannandKienzler2007a)
displacements and material forces, e.g., B ∝ λ . If
21 1 β =β . (8)
defect2wouldbemateriallydisplacedbyλ themate- 21 12
2
rialforceat1and2wouldbechangedby B and B , It states that the change of the material force at 2 in
12 22
respectively. thedirectionofthematerialdisplacementλ duetoa
2
Reciprocityinfractureanddefectmechanics 5
Fig.3 Crackconfiguration bb
withinabarintension
aa
aa 22
11
NN NN
aa
11 aa
22
bb
unit-materialtranslationat1equalsthechangeofthe (a)
materialforceat1inthedirectionofλ duetoaunit-
1
materialtranslationat2.
Theimplicationsofanon-linearformulationofthe
reciprocityrelation(5)havebeendealtwithinKienzler
andHerrmann(2007).Byconsideringastressedelastic
bodysubjectedsequentiallytoamaterialdisplacement
aa
of a defect and the application of a physical force a
novel type of coupling of Physical and Material
Mechanicsbymeansofareciprocitytheoremhasbeen aa aa
establishedinHerrmannandKienzler(2007b).
It is the intention of the present contribution to (b)
exploresomeapplicationsofthetwotheorems(5)and
(8)infractureanddefectmechanics.
111
222
2 Interactingcracks
aa aa
11 22
Interactingcrackshavebeentheobjectofresearchover
severalyears(cf.,e.g.,Erdogan1962;Panasyuketal.
aa bb aa
1977;Gross1982).Thematerialforcesatcracktipsare 11 22
usually calculated from the path-independent J inte-
gral (Rice 1968). Reciprocity relations are, therefore, Fig.4 Singlecrack(a),twoneighbouredcracks(b)
concernedwiththechangeofthe J integralduetothe
translation of some other defect, e.g., the change of
andloadedbyanaxialforceintensionasdepictedin
lengthofacrack2intheneighbourhoodoftheorigi-
Fig.3.
nal crack 1. Usually, the solution of crack-interaction
If a single crack is considered first, the key idea is
problemsinvolveeithersomeadvancesanalyticaltools
thattheenergy-releaserateisproportionaltothelength
or an extended numerical investigation. Based on the
oftheshadedstripinFig.4a,i.e.,
√ √
strength-of-materials theories, a simple first estimate
G = J ∝2a 2; J =C2a 2. (9)
forinteractingedgecracksinanelasticbarunderten-
sionhavebeengivenrecentlybyRohdeandKienzler TheconstantC,say,wouldbeproportionaltothesquare
(2005)andRohdeetal.(2005),andthereciprocityrela- of the applied load, inverse proportional to Young’s
tionsareappliedwithinthissimplifiedproblemsetting. modulus,andwouldcombinesomeinformationabout
InRohdeandKienzler(2005)andRohdeetal.(2005) thegeometryofthebar.
bars are investigated with a set of 2×2 edge cracks Inthepresenceofasecondcrack,cf.Fig.4b,both
symmetricallypositionedwithrespecttoitslengthaxis stripscannotdevelopcompletelyduetoshieldingsuch
6 R.Kienzler
that the energy-release rate of crack 1 is proportional (a)
tothelengthofthedarklyshadedareaontheleftand
theenergyreleaserateofcrack2isproportionaltothe 11
lightlyshadedareaontheright,i.e.,
√
1
G = J = B = 2C(3a −a +b), (10a)
1 1 10 1 2
2 111 222
√
1
G = J = B = 2C(3a −a +b), (10b)
2 2 20 2 1
2
where a and a are the crack lengths of crack 1 and
1 2 (b)
crack 2, respectively, and b is the distance of both
cracks.Eq.10isvalidaslongas
22
|a −a |<b<a +a . (11)
1 2 1 2
It is observed, of course, that the crack configuration 111 222
represents a mixed-mode problem and the vector J
doesnotpointinthedirectionofapotentialself-similar
crack extension. It is sufficient, however, to consider Fig.5 Changeofenergy-releaseratesduetocrackextensionλ1
only the component of J in the direction of (cid:4)a1 or ofcrack1(a)andλ2ofcrack2(b)
(cid:4)a ,correspondingto BP or BP.Detailsoftheanal-
2 12 21
ysisandthevalidationoftheresultsaregiveninRohde
andKienzler(2005)andRohdeetal.(2005). reciprocity relations in Material Space. In words: the
With the simple relations (10) at hand it is easy to changeofthe J integralatcrack1duetoachangeof
check the reciprocity theorems. A crack extension of thecracklengthofcrack2equalsthechangeofthe J
crack1,(cid:4)a ,correspondstoamaterialtranslationλ . integralatcrack2duetothechangeofthecracklength
1 1
Duetoλ ,thematerialforces B and B changeto ofcrack1.Thisrelationmayalsobeverifiedwithina
1 10 20
morerigorousapproacheitheranalyticallyornumeri-
B11 = B1λ1 −B10 cally and may be used to establish influence surfaces
√
1
= 2C[3(a +λ )−a +b−3a +a −b] for defects within the stress state of a crack tip. This
1 1 2 1 2
2
problemwillbefurthertreatedinthenextsection.
√
3
=+ 2Cλ , (12a)
1
2
B21 = B2λ1 −B10
√
1
= 2C[3a2−(a1+λ1)+b−3a2+a1−b] 3 Interactionofanedgedislocationwithacircular
2
√ hole
1
=− 2Cλ ,
1
2
=β λ . (12b) Asafurtherexamplefortheapplicationofthematerial
21 1
reciprocityrelationsletusconsiderthefollowingdefect
Ifcrack2isextendedbyanamountλ2thechangeofthe configuration.Withinaninfinitelyextendedplaneelas-
materialforcesatcrack1and2iscalculatedsimilarly tic sheet an edge dislocation is situated in the origin
as of a plane Cartesian coordinate system (x ,x ) with
1 2
√
B =−1 2Cλ =β λ , (12c) Burgersvectorpointing(withoutlossofgenerality)in
12 2 2 12 2 x -direction.Atx = ξ (i = 1,2)acircularholewith
√ 2 i i
B =+3 2Cλ . (12d) radiusr0iscentred,or,inpolarcoordinates,atdistance
22 2 2 d > r (ε = d/r > 1) under the angle ϕ measured
0 0
The changes of the various energy-release rates have fromthex -axisasdepictedinFig.6.
1
beenmadegraphicinFig.5 Duetoamaterialtranslationξ →ξ +λ thetotal
i i i
Substitution of (12b) and (12c) into relations (5) energy(cid:8)ofthesystemchangesandtheenergy-release
or (6) and (8) confirms and illustrates the validity of rateisdefinedas
