ofMathematicalSciences,Vol.72,No.6,1994 THEASYMPTOTICSPROBLEMWITHOUT
S.A.Nazarov OFTHESOLUTIONSOFTHESIGNORINIFRICTIONORWITHSMALLFRICTION UDC517.946:539.3 Wefindthefirstfewtermsoftheasymptoticexpansionofaregularsolutionofthetwo-dimensionalSignoriniproblemwithasmallcoeO~cientoffriction.Asthefundamentalapproximationwetakethesolutionofthelimitingproblemwithoutfriction.Thissolutionisassumedtobeknown,anditisassumedthattheregionofcontactconsistsofafinitenumberofarcs,oneachofwhichoneboundaryconditionoranotherisrealized.WestudytheasymptoticsofthesolutionoftheSignoriniproblemwithoutfrictionundersmallloadvariation.Bibliography:12titles. §
1.Statementoftheproblem.
1.TheSignoriniproblem.LetftbeadomainintheplaneR2withasmoothboundary0f~representedastheunionofthreearcsSoUSpOSk.WeconsidertheSignoriniproblemwithasmallconstantcoefficientoffrictionaforahomogeneousisotropicelasticbodyinastateoftwo-dimensionalstrainundertheactionofanexternalloadpappliedtoSp.ThebodyisclampedontheportionSo,andSkisthezoneofpossiblecontactwitharigidbarrier.pletesystemofrelationssatisfiedbyaregularsolutionhastheform Lu=Oinft;u=0onSO,a(n)(u)=ponSp; (1.1) o.n(u)<0,u.<0,an~(u)un:0onSk; (1.2) la.s(u)l+~ann(u)<0,a.~(u)~s<
0,(la.~(~)l+~a~n(u))u~:0onSk, (1.3) (cf.[1]).HereListheLam~systemoperator;u=(ul,u2)isthedisplacementvector;a(u)isthestress tensor;nandsaretheunitoutwardnormalvectorandthetangentvectorto0ft;un=u•n;u~=u•s; a(n)=an;an,=n.a(n);a,~=s•a(n).Bypassagetothelimitasa--*0in(1.1)-(1.3)weobtainthe Signoriniproblemwithoutfriction(cf.[!
,2]andPar.3).Heretherelations(1.2)and(1.3)arereplacedby thefollowing: ann(u)<_O,un<_O,a~,,~(u)un=O,a,~(u)=0onSk. (1.4)
2.Theproblemofthevariationofthesolutionwhentheloadchanges.Considersolutionsu(e) andu°ofproblem(1.1)and(1.4)withright-handsidesequaltop=p0+eplandp=p0respectively.Here 1 p0,plEW27(Sp),and~>0isasmallparameter.Itisknown[2,1]thatasolutionu°CW21(f~)belongsto classW~outsideanyneighborhoodoftheendpointsofthearcsSoandSp,andconsequentlytheboundary conditions(1.5)arefulfilledalmosteverywhereonSk.Wenowintroducethesets r.={~:~eS~,u.(~)<0},r~={~:xeSk,an~(~;~)<0},r0=s~\(r~ur~) andweassumethatP0consistsofafinitenumberofinteriorpointsQ1,...,QmofthearcSk.ThenP0partitionsSkintoarcs70,71,...,7m.LetQjbeanarbitrarypointofF0.Threecasesarepossible: 1°.Oneofthearcs7j-1and7ibelongstoP,,andtheothertoP~;20."[j-l,')'jEPu;30.7j-l,TjEr~.WeshalltentativelycallthepointQjaglidingpoint,apointofdetachment,orapointofcontactrespectivelyinthesethreecases.Toconstructtheasymptoticsofu(e)ase-~0in§2werequiresolutionsui,i=0,1,ofthefollowingauxiliarylinearproblemofelasticitytheory: Lui=Oinft;ui=0onSo,a'~(ui)=pionSp; (1.5) a.~(u'•)=0onSk,u~i=0onPu,ann(U')=0onPa. (1.6) TranslatedfromProblemyMatematicheskogoAnaliza,No.12,1992,pp.82-110. 1072-3374/94/7206-3411512.50©1994PlenumPublishingCorporation 3411 Itisclearthatthevectoru°issimultaneouslyasolutionoftheSignoriniproblem(1.1),(1.4).ForthetimebeingweassumethatQj=0isaglidingpointandthedomainf~coincideswiththehalf- planeR~.={x:x2>0}nearzero.Let(r,0)bepolarcoordinates,0e(
0,Tr),andlettheray{z:0=0}bedirectedalongF,,.ordingtothegeneralresultsof[3](cf.also[4])wehavetherepresentation zti(x)=uiJ(x)-]-Ek;Jrq+½~2q(O)q-O(r3)'r---)
0, (1.7) q=0,1,
2 wherei=0,1;k~jisthecoefficientofstressintensity; u]J(;T)=e}o--~c~•lXl-~-Cij2/tXl2--(
J,~+5)(24"--~1)-1x2);V2"3"(x)-~--(x-3)(x--~1)-1(c}1--~2c~2xl)x2;(1.8)=[(2q+1)#]-1_l-1/2)cos(q+3/2)
8 +(.-q-1/2)cos(q-1/2)
0,-(q-1/2)sin(q+3/2)8+(>¢+q+1/2)sin(q-1/2)8);(1.9) cijmmadkqijarecertainconstants;andx=(3#+A)(#+A)-a,whereAand#aretheLam6constants.An importantcircumstance,ofwhichwewillmakeconstantuseinwhatfollows,isthefactthatthecoefficient k°imustvanish(cf.§
2,Par.1).
3.Theproblemwithsmallfriction.Weshallseektheasymptoticsofthesolutionoftheproblem(1.1)-(1.3)inasimplifiedformulation.Tobespecific,weshallsupposethatf~isthestrip]R×(0,12)and thatSk=R×{0},So={x:x2=12,[xll>11},andSp=JR\So.ordingto[5],forsufficientlysmallaa~dp£W?
(Sp)thereexistsasolutionueW1(~2)f3HI'~(IX)oftheproblem(1.1)-(1.3)(moreprecisely asolutionofthecorrespondinginequality--cf.§
3,Par.3).HereH=]Rx(0,l),wherelisanynumberlessthan12,andHa'~(IX)isafunctionalspacewiththenorm
1 +(1+112)1F12}(1+t~[)d~dx2, whereFuistheFourierimageofthefunctionX1~~t(Xl,X2).Inadditionsucharegularsolutionisunique andsatisfiestheestimate Ilu;(a)ll+Ilu;HI'~m)ll-3,Par.3)weestablish thatu~convergestou°asa~0,whereu°isasolutionoftheproblem(1.1),(1.4).WesetP2={xE Sk:u°(x)=O,a22(u°;x)<
O,=l=ul(x)>0},Pu=F+Ur,,andweinterpretthenotationF~,F0,andQ1,...,QminthesenseofPar.2.WenowassumethatforeachpointQjoneofthefollowingsituations holds:4°.Oneoftheintervals7j-l,7jiscontainedinP~,andtheotherinF+UF_; 5°.Oneoftheintervals7j-l,'TjiscontainedinF+andtheotherinP_. WeshalltentativelycallQjaglidingpointorapointofadhesionrespectivelyinthesetwocases.
4.Preliminarydescriptionoftheresults.In§2wefoundasymptoticrepresentationsforsolutionsu(e)exhibitedinPar.2andweobtainedanestimateoftheremainder(cf.Theorem1).Asaby-productoftheconstructionoftheasymptoticsofthesolutionwedeterminedthevariationinthestructureofthezoneofcontact.Thelatterreasoningunfortunatelyisonlyformal--itcannotbejustifiedrigorously.Neverthelessinparticularcasestheformulasobtainedareapplicableforputations.Beforegivingsuchexamplesweshalldescribe(ontheheuristiclevel)thevariationinthestructureofthezoneofcontact. ThepointsofdetachmentQjcandisappearorbereplacedbysmallregionsofdetachmentwj(~).Dependingonthedataoftheproblemthesizeofthearcswj(¢)willbeO(¢½)orO(e)(cf.(2.17)or(2.22) respectively),andinthefirstcasethearcisinthemainsymmetricwithrespecttoQjwhileinthesecond casethecenterofwj(e)canbedisplacedbyadistanceO(~).Whathasjustbeensaidremainsvalidalso forapointofcontact,butthesizesoftheregionsoftangencyareO(~Ilne[-½)andO(¢½lln¢l-½)(cf.§
2, 3412 Par.3).Thepointisthattangencygivesrisetoareactionofthesupportasymptoticallyequivalentto theactionofaforceofsizeo(¢1ln¢[-½)concentratedatthepointQj.(Wenotethattheinfluenceofthis forceisnotlocalizednearthepointofcontact,andthesolutionundergoesaglobalperturbationoforderO(e[Inel-½).)Finally,variationoftheloadleadstoadisplacementoftheglidingpointbyanamountO(e). Example1.Letthehalf-planef/=IR~_beloadedatinfinitypressivestresses~r~2=-q,a~= ai~=0,andsupposethataconcentratedforce(
0,P)isappliedatthepoint(
0,H),whereP,q>
0.The regionofpossiblecontactSkistheline{x:x2=0}.Assumingthequantityqisfixed,ietustracethesolutionoftheproblemastheparameterpq-1increases.Intheexactsolution[6,7]ofthisproblema pointofdetachmentappears--theorigin--atthevalueP_=P0=(x+t)(x+3)-~TrHq.Letusrepresent PasP=P0+eP0,Usingtheapproximateformulaforthelengthofthezoneofdetachment(of.(2.17)), wecanobtaintherelation ~(2+>c)(2~)-1d2,d=IH-
1. (1.11) Here2Iisthelengthoftheregionofcontactthatisopenedup.Intheexactsolutiontheconnectionbetweentheparametersdand¢isgivenbytheformula =(>~+3)(1+d2)~[z+3+(>c+1)d2]-1-
1. (1.12) Whenx=2.8(thePoissoncoefficientis0.3)theerrorsinformula(1.11)parisonwith(1.12)are 0.8%ford=0.2,and3.9%ford=0.5.Evenwhend=1,therelativeerroramountstoonly17%. Example2.Inthesameproblemasabove,letusconsiderthevalueoftheloadP1=2½(x+1)(x+2)-~rqHatwhichd=1intheexactsolution.IfP=P1+eP1,thentakingountoftheshiftoftheglidingpoint (of.(2.10))wededucetheapproximateformulae(Ad(s))-1~0.763.HereAd(e)istheincrementinthe parameterdandx=2.8.ordingtotheexactsolutionthisratiois0.763,0.812,and0.837ford=0.01,0.25,and0.50respectively. In§3weconstructthreetermsoftheasymptoticsofaregularsolutionoftheproblem(1.1)-(1.3)asc~~0(asthefundamentalapproximationwetakethesolutionoftheSignoriniproblemwithoutfrictionandassumethattheconditionsofPar.3hold).BothforaglidingpointandforapointofadhesionthereisanintervaloflengthO(exp(-5~-l))onwhichtheasymptoticsolutionfounddoesnotsatisfyconditions(1.2).(However,thisdoesnotpreventusfromobtainingtheestimateO(e3-~)fortheremainderduetotheexponentia!
smallnessofthelength(of.Theorem1.2).)Whenthesolutionisexpandedinapowerseries,itdoesnotappeartobepossibletostudysucharegionofcontact.Thefactwearediscussingcanbemechanicallyinterpretedastheappearanceofaverynarrowzoneofadhesioninwhichtheboundaryconditionu=0holds. TheauthorisgratefultoB.A.Shoikhetforhelpfuldiscussionsandassistanceinthiswork. §
2.VariationoftheSolutionoftheSignoriniProblemWithoutFrictionwhentheLoadisVaried
1.TheasymptoticsnearaglidingpointQj.Withoutlossofgeneralityweassumethat7j-1EF~, 7jEF~,ands=(1,0)andn=(
0,-1)ataglidingpointQj.Weshallnotspecifytheindexjanyfurther. Solutionsu°andu1oftheproblems(1.5)and(1.6)havetheasymptotics u'(x)=u'(x)+k0ir1¢°
(0)+ + + -+
0, (2.1) parewith(1.7)),whereUiisavector-valuedfunctionponents U~(s,n)=b-as,Ui,(s,n)=a(3->c)(1+J~)-lTz,a,b=const, (2.2) andsandnareboundarycoordinates.Theextratermin(2.1)(parisonwith(1.7)),whichcontains•,arisesasaresultofdistortionoftheboundaryOff.Thevector-valuedfunctionqisalinearfunctionofthelogarithmandasmoothfunctionofthevariable0E[0,7r];itcanbecomputedfromthegeneralschemeof[3],butwehavenoneedofanexplicitexpressionforithere. 3413 Itfollowsfrom(2.1)and(2.2)thatwhenk0°#0theprincipalasymptoticsofun0andO'nn(UO)are determinedbythesecondtermontheright-handsideof(2.1).Itcanbeverifiedthatforeithersignofk0° oneoftheconditions(1.3)mustbeviolated.Consequentlyk°=
0.Similarlyitcanbeestablishedthatthecoefficientk°isnonpositive.Assumethat k°0, thentheseconditionsareviolatedanditisnecessarytoconstructaboundarylayer.Weintroducethe "stretched"variables Yl~--"--E--l(8--S--Eh),Y2=--c-ln, (2.4) whereSisthecoordinateofthepointQjandhisaconstanttobedetermined.Ifwepasstothevariables(2.4)andsete=0,thenthedomainftistransformedintothehalf-planeR~_.Thevector-valuedfunctionzthatursintheprincipalpartofasolutionofboundary-layertype U°(x)-~-cul(x)-{--E~z(y)~-..., (2.5) mustbesubjecttotherelations L(O/Oy)z=
0 Takingountinfinity inR~_={yEIR2:Y2>0};o12(z)=
0,O'22(Z)_<
0,Z2__~0,oftheboundarylayerwithasolutionofsmoothtypeu°+ z(y):/g0lfl2-~~1
(0)-~-]glp½~O(o)+O(fl-2/1),fl=g-lr O'22(Z)Z2euI[8,9]-~,0(
3. ----0wheny2=
0.(2.6) givesconditionsat (2.7) Wenowsolvetheequation(2.6),(2.7).Let(Ph,¢Y)bepolarcoordinateswithoriginaty=
0.Itisclear thatthevector z(y)=]¢0lPh~(I)1(~) (2.8) satisfies(2.6).Onecanverify[9]therelation z(y)=~:0lp~2(I~1
(0)--(1/2)hka0p~12°(Ù)q-O(p-½),p--+oo. (2.9) From(2.7)and(2.9)wededucethefinalequality h=-2koa(k~)-
1. (2.10) Theformulas(2.5),(2.8),and(2.10)determinethevariationofthesolutionnearQjandthevariationofthepositionoftheglidingpointitself.
2.Theasymptotics(1.6)aresmoothnearseries: nearapointofdetachmentQj.Solutionsuioftheelasticityproblems(1.5)andapointofdetachmentQjEF~,.WenowposethenormalstressesintoTaylor
3 O'nn(Ui;"s,O)=2~E(--1)mai(8--S)m-l+O(l~Sl). m---~l Usingthenotationjustintroduced,wefindthat 3ui(x)=ui(s--S,n)+EaimVm(s--S,n)+O(r4),40. m----
1 (2.11) HereUiarecertaincubicvectorpolynomialssatisfyingmainlyhomogeneousequations(1.5),(1.6)nearQj: isI][3_3n21;--n ; 2(1+~)ns . n3-3ns2 3414 Sincethevectoru°issubjectto(1.4),itfollowsthata°=a°=0anda°:>0in(2.11).Weassumethat theinequality a3°>
0 (2.12) holds.Weemphasizefurtherthatwheni=0andg>3thefollowingformulashold: u'.=o(~+1),~..(u~;~,o)=o(~),~..(u'.,~,o)=o(~). (2.13) LetaI¢
O.Thenrelations(2.13)holdwheni=q=
1.Ifinadditional>o,thenconditions(1.4) holdforthesumu°+euInearthepointQj(detachmentdoesnotur).Whenal=a~=0,formula (2.13)holds,wherei=1,q=3,andsoby(2.12)forsufficientlysmallethereisagainnodetachment.We shallassumethat aI<0oraI=0,at#
0. (2.14) Webeginbyconsideringthefirstofthesecases.Letusconstructtheboundarylayer. "stretched"coordinates
1 1 .1=-e-~(~-S),r/2=-c-~n Wepassto(2.15) andrepresentasolutionofboundary-layertypeasasum u°(~,~)+eu-l(~,~)+~~z(,)+..... (2.16) Weshallshowthatthezoneofdetachmenthasinthemaintheform {x:xecgf~,]s_SI<_¢~A,1=(-2a~)~-~(.a03")-~} (2.17) Wenowwritetheproblem(1.1),(1.4)incoordinates(2.15)andset~=
O.Wethenobtaintheresultthatinassumption(2.17),takingountofthegrowthintheboundarylayer(2.16withasolutionoftheprincipaltypeu°+eu1,thevector-vMuedfunctionzsatisfiestherelations L(o/oo)z=oin~+,a12(z)=G22(z)=0,z2>0,when1.11<
A,?
]2~---
O, axz(z)=zz=0,azz(z)<_0when1.xl->A,r/2=
O, zk(rl)=d°Vd(rI)+aIV:(rl)+o
(1),Irll~oo,h=1,
2. (2.1s)(2.19) Thesolutionoftheproblem(2.18),(2.19)(withanarbitraryA>0andwithoutthecondi*ionsz2>0anda22(z)_<0)canberepresentedasthesumofthevectorpolynomialY=a°V3+a~V1andthesolutionoftheproblemofacrackwithanormalloadappliedtoitsedges q(rh)=-a22(Y;~1,0)=2#(a~r]~q-a~),irhl<
A. (2.20) Suchasolutionisdeterminedordingtoformulas(2.8)and(2.7)of[
6,Chapt.II]fromthefollowinganalyticfunctionwithargumentr]=rh+ir/2: Zl(rl)=#{a°(q2-(1/2)rl(2r/2-A2)(r/2-A2)-½-a~(1-r/(r/2-A2)-½)}. (2.21) Therequirementthatthecoefficientsofstressintensityvanishattheendsofthecrack(cf.§
1,Par.2:§
2,Par.1)leadstothevalueofAindicatedin(2.17).Bydirectverificationweprovethatalltheinequalitiesof(2.18)hold.Thustheboundarylayerisnowconstructed. Wenowreturntothesecondcasegivenin(2.14).Weestablishthatthezoneofdetachmenthasmainly(ase---*0)theform {x:xeOFt,Is-S-~hl_1,Ala~l(2a°)-l}. (2.22) 3415 Inordancewith(2.22)weintroducethestretchedvariables(2.4),andweshallseekasolutionofboundarylayertypeintheformofasum g%,n)+~v-l(~,~)+~3z(y)+.... (2.2a) Proceedingasbefore,wefindthatthevectorzsatisfiesrelations(2.18).Thegrowthconditionsmeanthat zk(y)=a3oVi(ayl-h,y~)-a2V1i(y2,-h,y~)+O
(1),lyl--~oo,k=1,
2. (2.24) Thesolutionoftheproblem(2.18),(2.24)foranarbitraryAwithouttheinequalitiesz2_>0anda22_<0canberepresentedasthesumofthevectoryh(y)=a33(Yl-h,Y2)-a21V2(yl-h,y2),andthesolutionoftheproblemofacrackwithanormalload q(Yl)=--a22(Yh;Yl,0)----2~[a°(yl--h)2_1_al(yl_h)] (2.25) actingonitsedges.Ifweseth=a21(2a°)-½,theexpression(2.25)assumestheform(2.20),wherea]=a°h2-alh=(-a~)2(4a°)-
1.Consequentlyitremainsonlytorepeattheprecedingreasoningconnectedwiththefunction(2.21).
3.TheasymptoticsnearapointofcontactQj.LetusassumethatonthesetP0thereisonlyonepointofcontactQj.SolutionsuioftheproblemsofthetheoryofelasticityaresmoothinaneighborhoodofQj.Expandingthetraceson0f~ofthenormaldisplacementsinTaylorseries 8u~(s,0)=~-~(-1)mai(s-S) rn----
1 "n-1+O(Is-S[3), i.e.,introducingthenotationami,weobtaintherepresentations
3 ~'(~)=U'(~-S,n)+~~'~Wm-I(~-
S,~)+O(~), rll--~
1 ~~
0. (2.26) HereUiarequadraticpolynomialsparewith(2.11)); w°(,,,)= [°1;]wl(~,n)= [:1; w2(~,n)= [_~
2 + (~ 25?
2_3)(~ + ]1)-1n~ . Sinceu°satisfies(1.4),itfollowsthata°=a°=0anda°_>0in(2.26).Inaddition,wheni=0and q=2,relations(2.13)hold.Assumethatinequality(2.12)holds.Ifal>0,thencondition(1.4)holdsfor thesumu°+eu1nearthepointQj(tangencydoesnotur).Weassumethat alo, (2.27) andwepassto"stretched"coordinates(2.15).Asbefore,asolutionzofboundary-layertypemustbesubjecttotherelations L(0/0~)z=0inR~,a12(z)=a22(z)=0,z2_>0when[qll_>A,r]2=
0,~12(z)=0,z2=
0,~r22(z)_>0when[~]11--(A,7]2=
0. (2.28) Relations(2.26)and(2.27)implytheexpansionszk(r/)--a30W[:2(q)+a]W°(71)+o
(1),M--+oo,k=1,
2. (2.29) 3416 However,onecanverifythatthereisnosolutionoftheproblem(2.28),(2.29).Therefore,following[10,8,11],wealtertheformoftheexteriorandinteriorasymptoticexpansions: u,,~u°+e(u1+TG(n))when[x-Qjl>-el~4,u(x)~.,U°(s,n)+eU~(s,n)+ez(q)when[x-Qjl-<~/4. (2.30)(2.31) HereTisaconstanttobedetermined,G(n)isasolutionoftheproblemwithaconcentratedunitnormal forceappliedatthepointQanddirectedtotheinterioroff~(inotherwordsG(n)=-Gn(Qj),whereGis theGreen'smatrixoftheproblem(1.5),(1.6)withasingularityatthepointQi)"UsingthenotationF(z) forsolutionsoftheFlamantproblem(cf.,forexample,[
7,§10.9]),wewritetheasymptoticrepresentation nearthepointQj: G('~)(z)=F(x-Qj)+g+O(lx-Qj].inIx-Qj[-1). (2.32) Byvirtueof(2.26)and(2.32)thegrowthconditionsontheexpansions(2.30)and(2.3!
)leadtotherelations zk(~)=a°3Uk(~)+alU°(~)+T(Fk(~)-~-gk--(4#Tr)-l(1-~:.¢)lnc½)-f-o(
J.),[r][~a<),]¢=1,
2.(2.33) Theproblem(2.28),(2.33)canbesolvedbytheKolosov-Muskhelishvilimethodplexpotentials(el.,forexample,[6,7])inthecasewhen 2aI+a°A2=T{(4~r#)-l(2+(1+x)ln(Ac½/2)-g2}. (2.34) Asbefore,theinequalitiesoftheproblem(2.28)arefulfilledifthestressescanbearrangedsoasto bebounded,i.e.,thecoefficientofstressintensityiszero.Hence,takingintoounttheformulasfrom[
6, Ch.II],wededucethat T=4zr#(1+x)-la°A2, (2.35) Consequentlythezoneofcontactisasfollows: {x:xE0ft,Is-S]_¢)-1(1-27r#g2)-l}. (2.37) ForsufficientlysmalleEq.(2.37)hasauniquesmallpositivesolution.Forthissolutionthefollowingformulaholds: A=2[a°(al)-llne]-½+o([ln¢[-½),e~
0. Wenowconsiderthecase a°>0,al=0,a~-~
0. (2.38) ThestretchedcoordinatesfortheCorrespondingboundarylayeraregivenbyEqs.(2A),andthezoneofcontactisdefinedbytheformula {x:xe0ft,Is-S-ehl_eA}. (2.39) Theexteriorexpansionhastheform(2.30),andtheinteriorhastheform u(x)~,,U°(s,n)+e2z(y)when]x-Qjt(1), lyl---*ec,k=t,
2. (2.41) 3417 Whenh=-a21(2ao3)-1,wefindthat aa0U~2(yl-h,y2)+a21Uk1(
Y,--h,y2)=a°U~(y)-(al)2(4a°)-lU°(y). Forthatreasontherelation(2.41)canbeobtainedfromrelation(2.33)byreplacingthequantitya~by-(a~)2(4a°)-1ande½byE.Consequentlyformulas(2.34)and(2.35)holdwiththesesubstitutions,andEq.(2.37)fordeterminingAcanberewrittenasfollows: -(al)2=2(a°)2A2{ln(~A/2)+2(1+x)-l(1-27r#g2)-I). (2.42)
4.Analgorithmforconstructingtheasymptotics.Asformula(2.30)shows,apointofcontact introducesaperturbationoforderO(~llnsl-½)intothestress-strainstateatagreatdistancefromthepointofcontact.SincethepurposeistoconstructtheasymptoticsuptoorderO(~2),itisnecessarytotakeountoftheinfluenceofpointsofcontactonaglidingpointorapointofdetachment,andalsotheinteractionofpointsofcontact(ifthereareseveralofthem).WeemphasizethattheconstructionsoftheinternalexpansionsfoundinPar.1.2arelocalincharacter. WedenotebyJthesetofindicesjbywhichthepointsofcontactarelabeled.Weendowthequantities a~j,Tj,Aj,andhjthaturinformulas(2.26),(2.30),(2.36),and(2.39)withanindexjeJ.SupposeinadditionthatG(n'J)arethesolutionsoftheproblemsmentionedinPar.3withforcesconcentratedatthepointsQi"Thevectorginthecorrespondingexpansion(2.32)willbedenotedgJJ,andinadditionthe representations G(.,J)(z)=giq+o(ix_Qql),x--.Qq, arevalidnearthepointsQq,whereqEJ,q¢j. TakingountofthemutualinfluenceofpointsofcontactchangessomeoftheformulasofPar.3that connecttheunknownquantitiesTj,Aj,andhi.Wenowgiveequationsthatareobtainedusingthesame reasoningasbeforefromthegrowthconditionsandtherequirementthatthestressesintheboundary-layer problembebounded.ForanyjEJthequantitiesTjaregivenbyEq.(2.35)inwhicha°andAarereplacedbya°jandAj.Ifinequalities(2.27)holdforconstantsa]janda311,wehave,inanalogywith(2.37) 2a~j_.~a°jA~.{ln(Aje½/2)--1~t_2(1-~-x)-1}_47r#(1+x)-IEa3oq.,.,2tq_gj2q• qEJ Thezoneofcontactwasdefinedbytheformula(2.36),whereS=Sj,A=Aj,andhj=
O.Andifrelations(2.38)holdforconstantsa°3j,a~j,a~j,theninanalogywith(2.42) (2.43) -(a~j)2(2a°j)=a°jA~{ln(Aj~/2) -1+ 2(1+x)-1}--4r#(l+x)-
1,) a30qA.~2qq2jq, hj=
1 0-
1 --a2j(2a3j)• qEJ (2.44) Thezoneofcontacthastheform(2.39),whereS,h,andAarereplacedbySj,hj,andAj. Thesystemofnonlinearequations(2.43),(2.44)(theindexjrangesoverthewholesetJ)containsthelargeparameter]lncI.ItsprincipalpartisalinearsystemwithrespecttoA2withadiagonalnegative- definitematrix.Byvirtueof(2.27)and(2.38),theleft-handsidesof(2.43)and(2.44)containnegative numbers.ordingtoBanach'sprincipleforsmall~thesystem(2.43),(2.44)hasauniquesmallpositive solution{Aj:jEJ}.Itdeterminestheleadingtermsintheasymptoticsofthesolutionoftheoriginal problem(1.1),(1.4)everywhereexceptinsmallneighborhoodsofpointsofcontactandpointsofdetachment.Theboundarylayerscorrespondingtothesepointsareconstructedordingtotheschemedescribedin Paragraphs1and2.Whendoingthisitisnecessarytotakeountoftheextraterm~-~jAqG("'q),which, inparticular,leadstoreplacingthequantitiesai1andk~intheformulas(2.14)(2.17),(2.22),and(2.10) bythesums ai1q-ETqaiGlqandklq-ETqkq
J J 3418 respectively.Hereailadandk~arecoefficientsinexpansionsofthetype(2.11),(2.1)forthevector-valued functionsG(n'q).
5.Foundationoftheasymptotlcs.TheSignoriniproblem(1.1),(1.4)admitsaformulationasavariationalinequality E(u(e),v U(C))>//.s, (v-u(e))ds~, (2.45) forallveW~+(w;So,Sk)={veW~(a):v=0onSo,vn_<0onSk}[1,2].HereE(u,u)isthestrainenergyfunctional. TodetermineanapproximatesolutionVfromtheformalasymptoticsjustfoundweuseamodificationofthemethodofgrowthexpansions.Wedescribethisprocedureusingtheexampleofaglidingpoint.ForotherpointsofF0theschemeofconstructionremainsthesame,buttheformulasaremorecumbersome.Let{eC°°(R)and{(t)=1forItI<
1,{(t)=0forltl>1,0_<{_<
1.Nearaglidingpoint,inordancewith(2.1),(2.5),(2.8),and(2.10),weset V(x)=(1-¢(r/cd)){u°(x)+cu'(x)}+¢(r/D){U°(x)+¢U'(x)+c~z(y)}-(1-¢(r/~d))¢(r/D){U°(x)+~Ul(x)+k°r~l(O)+koCr~1d2°(O)} (2.46) ThefirsttermV(1)ontheright-handsideofEq.(2.46)istheexternalsolutionsmoothednearasingularity;thesecondtermV(2)istheboundarylayeractinginsomeneighborhoodofthepointQj,andthesubtractedtermV(a)eliminatesoneoftheleadingtermsthatarecountedtwice(inVO)andV
(2))intheasymptoticsofuiasr--+0andintheasymptoticsofzasp--+oo(thesetermsareequalsincethegrowthconditionshold). ordingtothedefinitionandpropertiesofthesetF0theinequalityu,0~+eu~_<0isviolatedonSkonlyinsmallneighborhoodsofthepointsQ1,...,Qm.Keepinginmind(2.1),(2.2),(1.9),(2.6),and(2.8)onecanverifythatduetothechoiceofthequantitiesdandDin(2.45)theinequalityV,,_<0holdsinaneighborhoodofaglidingpointQj;thesameassertionistrueforotherpointsonP0.ThevectorVleavesdiscrepanciesconcentratednearthesingularpointsQ1,...,Qmintherelations(1.1)and(1.4),wherepisreplacedbyp~.Sincewehavetakenountoftheleadingtermintheboundarylayer,wehavetherelations LV=-Finf~,a(n)(V)-pl+p2p2_<0,p]=0onSk, m IF(x)l_S,, 3419 whereR(x)=p(x)(1+Ilnp(x)l)andp(x)=dist{x,r0}.Itremainsonlytonotethatthislastintegralispositiveandmakeuseofrelations(2.48).Asaresult weobtainthefollowingtheorem. Theorem1.Fortheasymptoticapproximation(2.46)justconstructedtheestimateIlu(e)-V;W~(f~ll3.TheAsymptoticsoftheSolutionoftheSignoriniProblemWithaSmallCoefficientofFriction
1.Thefirsttermsoftheasymptotics.AtagreatdistancefromI'0werepresentthesolutionusoftheproblem(1.1)-(1.3)asasumu°+(~v1.Replacingusbyu°+avIin(1.1)-(1.3),weobtainthefollowingequationsforvI(withk=1): Lvk:0in•,vk:0onSo,o'(n)(vk):0onSp,O"12(?
k))=a22(Vk)=0onr~, (3.1) vI=0,o12(v1)=To'22(u0)onr-i-. (3.2) SinceSo¢~,thereexistsauniquesolutionV1oftheproblem(3.1),(3.2).By[3],wehavethefollowingexpansions vl(x):ulJ(x)~-]~ljr3l~.°(Oj)-~-rff(]e~jf~.l(oj):yf=]¢,°jZl(lnrj,Oj))-~-0(r32.),rj----+
0, (3.3) vl(x)=uli(x)-]-TiriT(lnri,Oi)-~-O(r~llnril2),ri~
0. (3.4) Heretheindicesjandicorrespondtoglidingpointsandpointsofadhesion,andthesign5=ischosenordingtocondition4°of§1; ulq(x)=(c01~-C1q1Xl2r-(24-3)(xJr1.')~--1cq11x2) (el.(1.8));cq]andklqarecertainconstants;Ti=a22(u°;Qi); (2~(t,O),E~(i,O))=(1-x)(1+x)-lTr-l(t-2/3)(ff1
(0),~0
(0))+(6#(1+,))-1(2r)-½{((2x+3)sin5/20+(2x+3)sin1/20,(2x+3)(cos5/20-cos1/28))+(27r)-1(1-x)[8((3-2x)sin1/28-sin5/20,(3+2x)cos1/20-cos5/28)+2(cos5/20-cos1/20,sin1/20-sin5/20)]}.(3.5) (Tr(t,0),To(t,O))=(Tr#)-'[(0-7r/2)(sin20,cos20-1)+t((1->~)(1+,7,f)-1-cos20,sin20)].(3.6) By(3.3)-(3.6)thestressesa(v1)havesingularitiesatthepointsQI,.-.,Qm-Inparticularifklj7£
O, then,paring(1.7)and(3.3)andtakingountof(1.9)withq=0,1,wefindthatthesumu°+av-1 doesnotsatisfyrelations(1.2).Asin§2,weconstructaboundarylayerneartheglidingpoints.Wenow passto"stretched"coordinates yl=o:--l(xl--X31-ozhj(oL)),y2----o!
-lx2 (3.7) parewith(2.4)),whereahj(a)isthedisplacement(tobedetermined)oftheglidingpointQjwith coordinatex~.(Hereandbelow,whenwerefertotheformulasof§2,wereplace¢bya.)WeseektheprincipaltermsoftheboundarylayernearQqintheform(2.5),wherez-zlj.Thevector-valuedfunctionzljisasolutionoftheproblem(2.6),(2.7)andisdefinedbyEq.(2.8).Makingtheexpansion(2.9)moreprecise(cf.[9]),wehave z!
J(y)=kl0jP2z[ol(o)-hj(a)(2p)-lO°(O)+(16~)-1(1"-~x)(hj(ol)D-1)2~(O)+0(p-3/2),D-+O(
D,(3.8) where =(1+x)-l(8?
r)-½(3COS}8--(2X+1)cos50,2(x--1)sin--3sin}0). 3420 Supposethatforallglidingpointsthecoefficientk~°j(whichisnonpositive,cf.§2ofPar.1)issubject toinequality(2.3).Then,replacinghi(a)byh°=hj(O)in(3.8)paring(2.7)and(3.8),wefindthathi(O)isdeterminedordingto(2.10). LetusnowconsiderapointofadhesionQi.Thefollowingrepresentationforthestressescr22isaconsequenceof(3.4)and(3.6): - 1)(lnr+2xT--g3-)f+ --'
O. (3.9) Thereforeforsufficientlysmallrithestressa22(u°+oLvl;Xl,0)ispositiveandhencethesumu°+avIdoesnotsatisfythefirstconditionof(1.2).Howeverthezoneinwhichthisconditionisviolatedisexponentially small(sincea]inri[exp(-ba-1),cf.(3.9)),andcanbeneglected.Ifc0i1#0in (3.4),thenthepointofadhesionundergoesaperturbationthatcanbestudiedasaglidingpointisstudied. Forbrevityweshallassumethatvl(Qi)=v~(Qi)=
0. 2.Thesecondtermsintheasymptofics(foraglidingpoint).Wefindthesmallest-ordertermsoftheinteriorandexteriorexpansions: ,/An(X)e,~U0(X)+oLvl(x)+OI2V2(X)-2V•.•' It°~(X),~u°J(x)AwoLUlJ(x)2Vol~zlj(yj)+oL,2u2J(x).2Vo~bz2j(yJ).Ac.... (3.1o) Substituting(3.10)into(1.1)-(1.3)andseparatingthecoefficientsofa2,weobtainEq.(3.1)forv2,k=2,andtheboundaryconditions v2=0,a,2(v2)=TG22(V')onr± (3.11) inthezoneofcontact.Keepinginmind(3.8)andtakingountofthegrowthconditions,wearriveatthe relations vZ(x)=-(16#)-1(1+x)ku0(hj0)2rj--!
2~
(0)+O
(1),rj--~
O. (3.12) Thereforethesolutionoftheproblem(3.1),(3.11),(3.12)mustbesoughtintheclassofvector-valuedfunctionswithinfiniteelasticenergy.Suchasolutionexistsbyvirtueof[3]and[12],isunique,andadmitsarepresentation v2(x)=V(x)_(16#)-1(1+~)E'/~0[~0~2r(j)(x), (3.13) inwhichVisasolutionwithfiniteenergy,thesummation~'extendsovertheglidingpoints,#(J)isa solutionofthehomogeneousproblem(1.5),(1.6)thatisboundedoutsideanyneighborhoodofQjandhas1 singularityrj~tg(0j)(cf.[12]and[
4,§3.8]).Thefollowingexpansion,whichimproves(3.12),isvalid: v2(x)=--(16#)-1(1q-X)k0lj(h°)2rj_1.2ff2(Oj)q-U2d.d-r/1[k2j¢°(Oj)q:k~jZ2(lnrj,Oj)]+O(rj),rj--+
O.(3.14) HereU2j=(c~,0)isacertainconstantdependinginparticularonh~(cf.(3.13)); (Z2(t,0),Z~(t,0))=(1-x)(1+x)-lzr-l(t-2)((
I,°
(0),~5°
(0))+[#(1+,)]-I(87r)-½{((2x+1)sin3/20-(2x-1)sin1/20,(2x+1)(cos3/20-cos1/20))--(27r)-1(1--.)[O(sin3/20+(2•-1)sinl/20,cos3/20+(2x+1)cosl/20)2(cos3/20+cosl/20),sin3/20+sinU20)]}. AssumethatatallglidingpointsQj uO1~(Qj.J~>0oru°(Qi)<
O. (3.15) 3421 FordefinitenesssupposethatlocallythesetFaislocatedtotheleftofthepointQj.Thentakingountof(2.6)and(3.8)wefindthatthevectorz2'jmustsatisfytheproblem Z(O/Oy)2'J=0inR+2,ak2(z2'j)=O,k=l,2wheny~"=0,y~<
0, Z2,j~-
O,O'12(Z2,j)=::Fo'22(zl'j)wheny3=0,y~>
0. (3.16) Thesign4-in(3.16)correspondstothesignin(3.15).Sinceintransitiontothecoordinates(3.7)the expressioninrjacquiresadependenceonInc~,werepresentthequantityhj(o~)as hj(v~),,~h~+oth}(lna)+.... (3.17) Takingountofthenecessityofthegrowthintheexpansions(3.10),ordingto(1.7),(3.3),and(3.14),weconcludethat z2,J(yj)=k°jp~jO2(Oj)+p~(Ic~jOl(Oj)+Tlc°jZa(lnpj+Ince,0j))"JCtO'~[(]g02j-kljh0j(l1no~)/2)@°(Oj):FkljZ2(lnpj+lna,Oj)]+o(p-j~), Wefindasolutionoftheproblem(3.16),(3.18)asasum pi--,oo. (3.18) =k2jPhjO(¢flj)+fl2j(bj(lnol)21(j)Tk°jZl(lnphj,j)), (3.19) wherebj(lna)isanunknownquantityand(Phi,¢2j)arepolarcoordinatesanalogousto(2.8).(Forthe 1 reasonsmentionedin§
2,Par.1thetermO(P~hj)iseliminatedfrom(3.19).WealsocallattentiontothepresenceofInphiin(3.19).Thecorrespondingtermviolatescondition(1.2)inanexponentiallysmallzone,which,asshowninPar.1,isinessential.)Thefollowingrelationsholdaspj~oo: s ~
3 a1 3--'2½
1 3 _
3 1
1 1 fl~Jff21(~J)=fl]ol(oj)--"ZhJ(°l)PJ~2°(OJ)q-O(fl;7)' (3.20) Zq(lnpj+lna,Oj)==.q(lnpj,Oj)+lna(1x)(1+x)-lTr-ie22-q(Oj),q=1,
2, (3.21) p~~l(lnphj,~j)=p~Zl(lnpj,Oj)q-fl~jhj(ce)((1-x)(1+x)-lTr-lff2°(Oj)-Z2(lnpj,Oj)//2)-1-O(pjlnpj).hj~3.22) Wenowsubstitute(3.20)and(3.22)into(3.19),replacehj(a)byh°,paretheresultwiththe expansion(3.18).TakingountofEq.(3.21),weobtainthefollowingformulasforthequantitiesbj(lna) and bj(lnoz)=k~j+3hjok2oj/2:q:(1-x)(1+x)-lk°jlno~, (3.23) h}(lna)=2(kl)j°-a{koj2+hO(bj(lna)_3hOk~ff2)q:(l_x)(l+x)-lTr-l(k~jlna_hjokloj)}.(3.24) Relations(3.23)and(3.19)definetheboundarylayer,and(2.10),(3.24)determinetheasymptoticsof thedeviationoftheglidingpointQj.Thusthesecondtermsoftheexpansions(3.10)and(3.17)havebeen computed.
3.Justificationoftheasymptotics.TheSignoriniproblem(1.1)-(1.3)admitsaformulationasaninequality l-I,ll)dXl>/p.(,-u)dXlVveWg+(a;S0,Sk).(3.25) Sk Sp 3422 Inthedefinition(3.7)ofthe"stretched"coordinatesyJwereplacethequantityhi(a)bythesumh~+ah}(lna),andwedefineanapproximatesolutionoftheproblem(1.1)-(1.3)bytheformulas v(~,x)=z(o~,z)V(~)(o~,z)+EI~(rilD)(1~(o~-*ri))V(OJ(a,yj) -~'¢(rUD)(:l-¢(a-lr~))V(~)J(a,y~)+~¢(a-trj)V(~)(a,64j)(3.26) parewith(2.46)).HereV(*)andV(Ojarethetermsfoundintheexteriorandinteriorexpansions(theright-handsidesofrelations(3.10));V(°')jistheasymptoticrepresentationofthesolutionintheintermediatezone(itisobtainedifeachtermofthesumu°+avI+a2v2isreplacedbytheasymptotictermsexhibitedin(1.7),(3.3),and(3.14));Z(¢,x)isaquantitythatequalsunityoutsidesmallneighborhoodsofthepointsQ1,...,Qm,whileinneighborhoodsofglidingpointsandpointsofadhesionsitequals1-¢(a-lrj)and1-¢(a-trj)respectively;tisanynumberfromtheset[3,c~)(theexponenttischosensothattheerrorsthatarisefrommultiplyingthelower-ordertermsoftheasymptoticsby1-~(a-trj)aresmall); denotessummationwithrespecttoj=1,...,rnand~'denotessummationoverjusttheindicesthat correspondtoglidingpoints.ByconstructionthevectorVbelongstoW2~+(fi;So,Sk)forsufficientlysmalla,(i.e.,itissubjectto therelations(1.2)).ItalsosatisfiestheequationsLu=-Fin~,u=0onSo,a(n)(u)=ponSpparewith(1.1),andtheinequalities 022(V)_<71,0"12(V)sgnYl>~'/'2,l~,z(v)l+~z2(v)_<~-~. (3.27) Takingountoftheboundary-valueproblemsfromwhichthetermsoftheouterandinnerexpansionsaredetermined,weverifybyputationthatforanypositive5thefollowingestimateshold: IF(o~,x)l_-~~C(2"j)(1+~--1r'~--1~,(3.28) icr22(V)V2dxl<_c6a6-~.sk (3.29) Using(3.27),(3.29),andtheBettiformulas,weobtain E(V,u-V)-fF.(u-Y)dx-fp:(u-V)dzl=-fal~(V)(ul-Yl)+~2(Y)(u2-Vd2xl)>_-c,~6-' 12 Sp S~ -/(it1,",us,+'~3i"i~11+'~'2i"iVxi)dxl+
0,I0"22(V)(iux'[Vll)dxl.(3.30) eJ S~ Sk Addinginequality(3.25)withv=Vandinequality(3.30),wehave E(V-u,V-u)_<.,<~°-'f(i~,i"lu.i+l~,l"iV,i+I~,!
"i-,i)d-,÷jiFi•~,-Vld. Sk I2 +'~/1o~(~)-<~(V)ltu~-V~!
dz,. Sk Asin§
2,Par.5,applyingKorn'sinequalityandvariantsoftheHardyinequality I1(1-P"G-lrj)-l(1q-Ilnrjl)-lu;LZ(f~)ll--<¢llu;W~(~)ll~,H(1"-t'-oz--l?
"j)--½(1+]lnrji-½u;L2(Sk)li2,Par.5).Forthatreasontheyareomittedatthispoint.Wemerelypointoutthatthe"loss"ofthefactora2in(3.34)paredwith(3.33))cameaboutasaresultoftheimpossibilityofapplyingtheHardyinequality(3.32). Theorem2.Thereexistsa(regular)solutionofinequality(3.25)(ortheproblem(1.1)-(1.3))suchthatfortheasymptoticapproximation(3.26)justconstructedtheestimateIlu-V;W~(~)I[1.G.Duvaut,InequalitiesinMechanicsandPhysics,Springer,NewYork(1976).2.G.Fichera,ExistenceTheoremsintheTheoryofElasticity[Russiantranslation],Moscow(1974).3.V.A.Kondrat'ev,"Boundary-valueproblemsforellipticequationsindomainswithconicorcusp points,"TrudyMosk.Mat.Obshch.,16,209-292(1967).4.V.Z.PartonandP.I.Perlin,MethodsoftheMathematicalTheoryofElasticity[inRussian],Moscow (1981).5.N.Glava~ek,
J.Haslinger,
I.NeSas,andJ.Lovi~ek,SolutionofVariationalInequalitiesinMechanics [Russiantranslation],Moscow(1986).6.L.I.Sedov,ACourseinContinuumMechanics,Wolters-Noordhoff,Groningen(1971).7.Yu.N.R~botnov,MechanicsofaDeformableSolidBody[inRussian],Moscow(19SS).8.A.M.II'in,MatchingofAsymptoticExpansionsofSolutionsofBoundary-ValueProblems,American MathematicalSociety,Providence(1992).9.S.A.Nazarov,"Localstabilityandinstabilityofnormaltearcracks,"Izv.Akad.NaukSSSR,Mekh. Tver.Tela,No.3,124-129(1988).10.A.M.I'lin,"Aboundary-valueproblemforasecond-orderellipticequationinadomainwithacrack," Mat.Sb.,103,No.2,265-284(1977).11.V.G.Maz'ya,
S.A.Nazarov,andB.A.Plamenevskii,AsymptoticsofSolutionsofEllipticBoundary- valueProblemswithSingularPerturbationsoftheDomain[inRussian],Tbilisi(1981).12.V.G.Maz'yaandB.A.Plamenevskii,"Onthecoefficientsintheasymptoticsofthesolutionsofelliptic boundary-valueproblemsinadomainwithconicpoints,"Math.Nachr.,76,29-60(1977). 3424