Recent Advances in Contact Mechanics: Papers Collected at the 5th Contact Mechanics International Symposium (CMIS2009), April 28-30, 2009, Chania, Greece

418 Pages·2013·18.47 MB·English

by  Jim R. Barber

Lecture Notes in Applied
and Computational Mechanics 56
SeriesEditors
Prof.Dr.-Ing.FriedrichPfeiffer
LehrstuhlBfürMechanik
TechnischeUniversitätMünchen
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85748Garching
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Prof.Dr.PeterWriggers
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Inst.BaumechanikundNumer.Mechanik
UniversitätHannover
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E-mail:[email protected]
Forfurthervolumes:
http://www.springer.com/series/4623
Recent Advances in Contact
Mechanics
Papers Collected at the 5th Contact Mechanics
International Symposium (CMIS2009),
April 28–30, 2009, Chania, Greece
Georgios E. Stavroulakis (Ed.)
ABC
Editor
Prof.Dr.Inghabil.GeorgiosE.Stavroulakis
DepartmentofProductionEngineering
andManagement
TechnicalUniversityofCrete
Chania
Greece
ISSN1613-7736 e-ISSN1860-0816
ISBN978-3-642-33967-7 e-ISBN978-3-642-33968-4
DOI10.1007/978-3-642-33968-4
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012948329
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Introduction
Contact mechanics is an active research area with deep theoretical and numerical
roots. The links between nonsmooth analysis and optimization with contact and
nonsmooth/nonconvexmechanicshavebeeninvestigatedintensivelyduringthelast
decades,especiallyinEurope.Thestudyofcomplementarityproblems,variational-,
quasivariational-andhemivariationalinequalitiesarisinginnonsmoothandcontact
mechanicsis a hottopicfor interdisciplinaryresearchand cooperation.The needs
of industryfor robust solution algorithmssuitable for large scale applicationsand
theregularupdatesofthe respectiveelementsin majorcommercialcomputational
mechanicscodes,demonstratethatthisinteractionisnotrestrictedtotheacademic
environmentandthattheinvestigationisbynomeanscompleted.
The contributions of this book have been selected from the participants of the
CMIS 2009 international conference which took place in Chania, Crete and con-
tinued a successful series of specialized contact mechanics conferences. A num-
ber of people helped me to organize the conference and eventually prepare this
book.AmongthemIwouldliketothankProfessorsMichelFre´mondandJaroslav
Haslinger.
During the preparation of the Conference, Professor J.A.C. Martins, Lisbon,
Portugal,memberoftheInternationalScientificCommittee,passedaway.Hismem-
oryandhiscontributionstothecontactmechanicscommunitywillstayalive.
Finally I would like to expressmy thanksto the editorial staff of Springer,the
group of Senior Editor Dr. Thomas Ditzinger, for their patience and support, as
wellas to theSeriesEditors,ProfessorsFriedrichPfeifferandPeter Wriggers,for
adoptingthisbookintheLectureNotesinAppliedandComputationalMechanics.
Chania,Crete,Greece GeorgiosE.Stavroulakis
Professor,TechnicalUniversityofCrete,
DepartmentofProductionEngineeringandManagement,
InstituteofComputationalMechanicsandOptimization,
(www.comeco.tuc.gr),GR-73100Chania,Greece
email:[email protected]
Privatdozent,TechnicalUniversityofBraunschweig,
DepartmentofArchitecture,CivilandEnvironmentalEngineering,
InstituteforAppliedMechanics,
(www.infam.tu-bs.de)
Germany
Contents
Chapter 1: Discrete Coulomb FrictionalSystems Subjected to
PeriodicLoading............................................... 1
JimR.Barber,YoungJuAhn
Chapter 2: Stability in Unilateral Contact Problems with Dry
Friction....................................................... 13
ElainePratt,AlainLe´ger,MichelJean
Chapter 3: On a Geometrically Exact Theory for Contact
Interactions ................................................... 31
AlexanderKonyukhov,KarlSchweizerhof
Chapter4: ApplicationsofanExistence Result forthe Coulomb
FrictionProblem............................................... 45
VincentAcary,FlorentCadoux
Chapter5:Size-ScaleEffectsontheFrictionCoefficient:FromWeak
FaultsatthePlanetaryScaletoSuperlubricityattheNanoscale ....... 67
MarcoPaggi,AlbertoCarpinteri
Chapter6:AResidualTypeErrorEstimatefortheStaticCoulomb
FrictionProblemwithUnilateralContact .......................... 85
PatrickHild,VanessaLleras
Chapter7:FrictionalContactProblemsforThinElasticStructures
andWeakSolutionsofSweepingProcess........................... 101
PatrickBallard
Chapter 8: Scalable TFETI Algorithm for Frictionless Contact
Problems:TheoryandRealWorldProblems ....................... 113
Zdeneˇk Dosta´l, Toma´sˇ Kozubek, Toma´sˇ Brzobohaty´,
AlexandrosMarkopoulos,V´ıtVondra´k
VIII Contents
Chapter9:T-FETIBasedAlgorithmfor3DContactProblemswith
OrthotropicFriction............................................ 131
JaroslavHaslinger,RadekKucˇera
Chapter10:APrimal-DualActiveSetStrategyforFiniteDeformation
DualMortarContact ........................................... 151
AlexanderPopp,MichaelW.Gee,WolfgangA.Wall
Chapter11:NumericalAnalysisof a Bone Remodelling Contact
Problem ...................................................... 173
Jose´ R.Ferna´ndez,RebecaMart´ınez
Chapter 12: Postbuckling Behaviour of a Rectangular Plate
SurroundedbyNonlinearElasticSupports ......................... 189
AlikiD.Muradova,GeorgiosE.Stavroulakis
Chapter13:ATime-SteppingSchemeforMultibodyDynamicswith
UnilateralConstraints .......................................... 205
LaetitiaPaoli
Chapter14:CoupledImplicitVariationalInequalitiesandDynamic
ContactInteractionsinViscoelasticity ............................. 221
MariusCocou
Chapter15:ComparisonsofContactForcesduringObliqueImpact:
Experimentalvs.ContinuumandFiniteElementResults ............. 239
PhilipP.Garland,RobertJ.Rogers
Chapter16:DynamicResponseofMasonryWallsConnectedwitha
ReinforcedConcreteFrame...................................... 257
MariaE.Stavroulaki,KaterinaPateraki
Chapter17:BeamsComprisingUnilateralMaterialinFrictionless
Contact:AVariationalApproachwithConstraintsinDualSpaces ..... 275
FrancoMaceri,GiuseppeVairo
Chapter 18:On the SeparationZones in Aluminium Base-Plate
Connections.NumericalSimulationandLaboratoryTesting .......... 293
Dimitrios N. Kaziolas, EvangelosEfthymiou, MichaelZygomalas,
CharalambosC.Baniotopoulos
Chapter19:CompositeSlab:AUnilateralContactProblem .......... 309
ThemistoklisS.Tsalkatidis,ArisV.Avdelas
Chapter20:TheInfluenceofSlidingFrictiononOptimalTopologies ... 327
NiclasStro¨mberg
Contents IX
Chapter 21: Derivation of the Equation of Caustics for the
ExperimentalAssessmentofDistributedContactLoadswithFriction
inTwoDimensions ............................................. 337
Vasilios Spitas, Christos Spitas, George Papadopoulos,
TheodoreCostopoulos
Chapter22:AMathematicalMethodfortheDeterminationofthe
CriticalAxialLoadsofContinuousBeamswithUnilateralConstraints
forVariousInitialGeometricImperfections ........................ 351
KonstantinosA.Tzaros,EuripidesS.Mistakidis
Chapter23:A NumericalApproachtothe Non-convexDynamic
Problemof SteelPile-SoilInteractionunder Environmentaland
Second-OrderGeometricEffects.................................. 369
AsteriosLiolios,KonstantinosLiolios,GeorgeMichaltsos
Chapter 24:Effectof DifferentTypesof MaterialHardening on
HystereticBehaviorofSphericalContactunderCombinedNormal
andTangentialLoading ......................................... 377
YuriKligerman,VadimZolotarevsky,IzhakEtsion
Chapter25:ModellingofPiezoelectricContactProblems............. 383
MikaelBarboteu,MirceaSofonea
Chapter26:A MultiResolutionStudyontheBehaviorofFractal
InterfaceswithUnilateralContactConditions ...................... 401
OlympiaK.Panagouli,EuripidesS.Mistakidis
AuthorIndex ..................................................... 419
Chapter 1
Discrete Coulomb Frictional Systems
Subjected to Periodic Loading
Jim R. Barber and Young Ju Ahn
Abstract. If elastic systems with frictional interfaces are subjected to pe-
riodic loading, the system may shake down, meaning that frictional slip is
restricted to the first few cycles, or it may settle into a steady periodic state
involving cyclic slip. Furthermore, if the system posesses a rigid-body mode,
the slip may also cause an increment of rigid-body motion to occur during
each cycle — a phenomenon known as ratcheting.
Here we investigate this behaviour for discrete systems such as finite el-
ement models, for which the contact state can be described in terms of a
finite set of nodal displacements and forces. If the system is ‘uncoupled —
i.e. if the stiffness matrix is such that the tangentialnodal displacements are
uninfluencedbythe normalnodalforces,africtionalMelan’stheoremcanbe
proved showing that shakedown will occur for all initial conditions if there
existsasafeshakedownstatefortheperiodicloadinginquestion.Forcoupled
systems, we develop an algorithm for determining the range of periodic load
amplitudes within which the long-time state might be cyclic slip or shake-
down,dependingontheinitialcondition.Theproblemisinvestigatedusinga
geometricrepresentationofthemotionofthefrictionalinequalityconstraints
in slip displacement space. Similar techniques are used to explore ratcheting
behaviour in a low-order system.
1.1 Introduction
Many engineering systems comprise one or more contacting elastic bodies
in nominally static contact. Examples include bolted joints between ma-
chine components and the centrifugally loaded contact between aero engine
Jim R. Barber · YoungJu Ahn
Department of Mechanical Engineering, University of Michigan, AnnArbor,
MI48109-2125, USA
e-mail: {jbarber,yjahn}@umich.edu
G.E.Stavroulakis(Ed.):RecentAdvances inContactMechanics,LNACM56,pp.1–11.
springerlink.com (cid:2)c Springer-VerlagBerlinHeidelberg2013
2 J.R. Barber and Y.J. Ahn
turbine blades and the blade disk. These systems are typically subjected to
mechanical vibrations, which can cause the contact tractions to exceed the
limiting frictionconditionatpartofthe interface,leadingto a state ofcyclic
microslip.Thisinturnresultsinenergydissipationwhichaffectsthedynam-
icsofthe systemandmayalsoleadto theinitiationoffrettingfatigue cracks
emanating from the microslip region.
The Coulomb friction law is still arguably the best simple approximation
to the observed behaviour of unlubricated contacts and it introduces a his-
tory dependence into the problem. In particular, the steady cyclic state will
generally differ from that during the first cycle of loading. We would like
to be able to solve for this steady state directly, and hence determine the
location and magnitude of damage due to fretting fatigue and/or estimate
the energy loss so as to define an equivalent (frequency-dependent) damping
element.However,thesteadycyclicstateisofteninherentlynon-unique,with
the state achieved depending on the initial condition or the initial transient
period of loading.
1.2 Shakedown and Melan’s Theorem
If the time-independent component in the compressive normal tractions is
sufficiently large,the systemmayshake down, meaningthat the steadystate
is one in which all points on the interface remain in a state of stick after an
initial transient that may involve microslip.
Shakedown is a well known phenomenon in the analogous process of elas-
tic/plastic deformation, where it can be predicted using Melan’s theorem [9]
whichbroadlyspeakingstatesthatifthesystemcanshakedown,itwilldoso
regardlessof initial conditions. For frictional systems, anequivalent theorem
might be stated as “If a set of time-independent tangential displacements at
the interface can be identified such that the corresponding residual stresses
when superposed on the time-varying stresses due to the applied loads cause
the interface tractions to satisfy the conditions for frictional stick throughout
the contact area at all times, then the system will eventually shake down to a
stateinvolvingnoslip,thoughnotnecessarilytothestatesoidentified.”Tran-
sient studies of cyclic frictional systems seem to confirm the validity of this
theorem [7], but the proof of Melan’s theorem depends on the associativity
of the plastic flow rule, whereas the Coulomb friction law is non-associative.
The theorem has recently been proved in both discrete [8] and continuum
[4] formulations, but only for the restricted class of systems in which there
is no coupling between normal tractions and tangential displacements. This
class includes the much studied case ofthe contactof two similar elastic half
planes, and more generally, any system that is symmetric about the contact
plane.
1 Discrete Coulomb Frictional Systems Subjected to Periodic Loading 3
The discrete theorem is established by defining a non-negative norm
1
A= (v˜−v)T κ(v˜−v) , (1.1)
2
where v is a vector of instantaneous nodal slip displacements, v˜ is a ‘safe’
shakedown vector and κ is the reduced stiffness matrix. The norm A is a
measure of the deviation of the instantaneous deviation of the system from
the shakedown state and the theorem is established by demonstrating that
for all permissible slip motions, the time derivative A˙ < 0 and hence the
shakedown state is approachedmonotonically.
1.3 Coupled Systems
That the normal and tangential elastic problems be uncoupled is both a
necessary and sufficient condition for Melan’s theorem to apply, except for
certain very special low order discrete systems [8]. For coupled systems, it is
alwayspossibletoconstructcounterexamplestothetheorem—i.e.periodic
loading scenarios for which the long term state of the system may be either
shakedown or cyclic slip depending only on the initial conditions.
To explore this phenomenon, we consider the behaviour of a two-
dimensionalN-nodediscretesystemsubjectedtoexternalloadingoftheform
F(t)=F +λF (t), (1.2)
0 1
where F is a time-invariant mean load, F (t) is a periodic load, t is time
0 1
and λ is a scalar loading factor.
Thediscretedescriptionoftheelasticsystemcanbecondensedsoastoin-
cludeonlythecontactdegreesoffreedom,givingasystemoflinearequations
q = qw+A v +B w
j j ji i ij i
p = pw+B v +C w , (1.3)
j j ji i ji i
where v ,w are respectively the tangential and normal nodal displacements,
i i
q ,p arethetangentialandnormal(compressive)nodalforces,qw,pw arethe
i i j j
nodal reactions that would be generated by the external forces F if all the
nodal displacements were constrained to be zero and A,B,C are partitions
of the reduced stiffness matrix κ. We note that with this terminology, the
coupling between tangential displacements and normal reactions is defined
by the matrix B and hence the condition for Melan’s theorem to apply is
B=0.
We define the Coulomb friction law for node i by the relations
w ≥ 0; p ≥0 (1.4)
i i
w >0 ⇒p =q =0 (1.5)
i i i