Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

Brought to you by | Uniwersytet Jagiellonski Authenticated Download Date | 10/29/18 8:24 AM eralizedgradientofClarke–Rockafellar,cf.[1].ThelatterwasintroducedforaclassoflocallyLipschitzfunctionsand

❆❜ai:�❛❝i:✿ We survey recent results on the mathematical modeling of nonconvex and nonsmooth contact problems arising in mechanics and engineering. The approach to such problems is based on the notions of an operator subdiffe- rential inclusion and a hemivariational inequality, and focuses on three aspects. First we report on results on the existence and uniqueness of solutions to subdifferential inclusions. Then we discuss two classes of quasi-static hemivariational ineqaulities, and finally, we present ideas leading to inequality problems with multivalued and nonmonotone boundary conditions encountered in mechanics.

▼❙❈✿ 34G20, 35A15, 35J85, 35K90, 35K85, 35L85, 35K55, 49J40, 49K40, 74Axx

❑❡②✇♦�❞a✿ Hemivariational inequality • Subdifferential • Quasi-static • Multifunction • Nonconvex • Viscoelasticity

© Versita Sp. z o.o.

✶✾✺✸

Cent. Eur. J. Math. • 10(6) • 2012 • 1953-1968 DOI: 10.2478/s11533-012- 0123-6

❈❡♥ ❛❧❊✉ ♦♣❡❛♥❏♦✉ ♥❛❧♦❢▼❛ ❤❡♠❛ ✐❝

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

StanisławMigórski^1∗

❘❡✈✐❡✇ ❆ ✐❝❧❡

1 Institute of Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6,30-348 Kraków, Poland

❘❡❝❡✐✈❡❞✶ ▼❛ ❝❤✷✵✶✷❀ ❛❝❝❡♣ ❡❞ ✷✼ ❏✉♥❡ ✷✵✶✷

1. Introduction

TonheNpoanpleinreiasraAsnhaolrytesinsed(SvNerAs2io0n11o)fhtheeldoriingiTnoarlupńa,rPtsoloafnadn,iSnevpitteedmtbaelrk7p–

re9s,e2n0t1e1d.bTyhtehegaouatlhiosrtaotrtehveieSwixtahndSyrmeppoorstiuomn thheemirveasruilattsioonfarleicneenqtuastluitdieie

ss,aonndpthroebirleampspliincavotilovinnsg. operator subdifferential inclusions in Banach spaces, quasi-static

Tgehneenraoltiizoantioonfhoefmthivearviaartiiaotniaonlainleinqeuqauliatylitwy.asThinetrhoedmuicveadriabtyioPn.aDl.iPnaenqaugailoitt yopfoourmlouslaitniotnheexepalroliyts1t9h8e0sn,octifo.n[17o,f1th8e],gaesna-

∗E-mail:stanislaw.migorski@ii.uj.edu.pl

s defined by

θ⁰(

x;v)=limsupθ(y+λv)−θ(y):

directionalderivativeθ(x;v)existsandsatisfiesθ(x;v)=θ(x;v)forallv∈X.

Le np

s e o ef

Finally,wegiveapplicationstoquasi-staticviscoelasticfrictionalcontactproblems.

(b)Tisu.s.c.xo∈neveryfinitedimensionalsubspaceofXintoX^∗endowedwiththeweaktopology;and spaceY.Als

R

o,wedenoteby2^Xt

hecollectionofallsubsetsofX.ForU⊂X,wealsowritekUkX=sup{kuk^X:u∈U}.

GivenaBanachspaceX,wedenoteitsnormbyk∗·kX.ThedualspaceisdenotedbyX^∗andh·;·iX^∗×X,denotedalsob , teu

igbe a . h - o Xnow i t a

ituaacscestsaibtilcealsovtotheineawlcoemqeurailnittehsi.sfieeld.avetriedtokeepthepresentationT con

ofthpapr so ecn o inclusionsina framework ofevolution

energyfunctionals.Thusthetheoryofhemivariationalinequalitiesprovidesmathematicalresultswhichariseandarea cal

va ti e rb olv mnt u alu io .I

inxitshteeineertialtermktendstoonzero.Wweeprotvethaetthselimiottfucnbcteihoanviisoaosfoalusteioqnueofaparaboliche mivariationalinequality.

nolnvcaonvetxnonsmocoothfuncntdioinnsanndtthceiprClarkmesInubdiffesreecnotniadlsclopsseraftiinngcluosnotnheunk nownfunction.Firstweprovethe

inebqdualiteyninwhoicuhtheunoknndownnis.Wtheeveloceityhafiteld.sTcohnetnacwtepraopplleymolueraressultinordert oprovetheuniqueweak

Tnhemgoaellionfgthisdpaaperisstoofgivoentaascutrpvheeynonmmeodeclainngboeffcoountacitnpro,b7lem,s2w0]i.thnon smoothpotentialsandonanalysisof

anrgoebstamclbe.Topmrovideaniceaxampdleelwsewhconsideercaribviscouealsaissttiactpcropbrolecmesineswohficchth efrictionalcontactismodeledwith

casethehemivariationalinequalitiescanbeformulatedasthesubstationarypointproblemsforthecorrespondingnon-d en , nsmooh

o ne gyfc T o lain fec icpro mash vai l

thniisqresuelstsinthelt.studyofoaofclassaosfdhistnory-

gdempeenntdseonthpesmeuivdaoriaotnioontoanleinoepqeuraaltities.Suchkindofproblemsarisesina

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

allowstogivevariationalformulationsofseveralmechanicalphenomenainvolvingthenonconvexandnondifferentiablepplibet o riaonaltheoryofengineringpolemsinv ingnonooonemltiv edrelatnsnthestationary ionieffeqruoadlittiiaebsleaallnoowntnoaglyivteispaonsdictnivencaonnsvweexresotorunnasoulnvetdioonraplsa.rntdiahl

elyf[5urnmsuo,l8vtedopsrooblmemsh,acnf.[9a–l12,1b5le,16s].Moeremiinforramtaiotnioanq i-s hemiariaton

in i Wh assimpleaspossibleandthismakes

turhipeleueonftsepnatrceessu.TehTehfieresptricslaasissfibnllvoowlevse.soWincalruosuiosindsewrittwhnoachlaisstsoe rsym-dfespuebnddiffeenrtetnetriamolrfsorawndhifichxewdepporinotv.idTehaennewxeisstepnecceiaalinzdela

nu erofathemat lmo ichds eq i s

ontactbetweenadeformablebodyandsuiffertial bndary c itios provt thi

b d to ahistory-dependenthemivariationalso biliyofthe rrespo gcoatroble. the

a o is,weconsidertime-dependentpossiblye nc of a wea soluti. Then s

udy th aympti r nceofsolutionswhenasmallparameter

2. Preliminaries

In this section we recall some definitions we shall use in this paper, see e.g. Zeidler [21] and Denkowski et al. [2,3].

toypohl·o;g·yi.i TshehnodtaatiloitnyLp(aXir;nY)statnwdesenforXthendspXaceTofelinseyamrbbooluwndXedisopuesread tofrsrftrhoemsapaBcaenachesdpaceedXwtothaiBsawneachkLietθ:X→

bealocallyLipschitzfunction.Thegeneralizeddirectionalderivativeofθatx∈Xinthedirectionv∈X

y→x;λ↓0 λ

The generalized gradient ofθatxis the subset ofX^∗given by

∂θ(x)={ζ∈X^∗:θ⁰(x;v)≥ hζ; vi; v∈X}:

AlocallyLipschitzfunc0tionθiscalledregular(in0thesense0ofClarke,cf.[1])atx∈Xifforallv∈ Xtheone-

sided(a)tfTor:eXve→ry2^X∗.AX,ToxeirsataornoTneimspatyid,ctoonbvexpasenuddwomeaoknlytocnompiaicttssaettisifi neXs∗th;efollowingconditions:

(c) ifx→x inX,x^∗∈Txa n d limsuphx^∗;

x−xi≤ thenforeachz∈Xthereexistsx^∗(

z)∈Txsuchthat hx^∗(

n

z);x−

w

zi

ea

≤

kl

l

y

iminfhxn∗

n

;xn− zi

n

.

nn 0,

✶✾✺✹

X

X

iscontinuous [0;.T :X→X^∗is

pseudomonotoneifforeachsequence{x}⊆Xsuchthatitconvergesweakly

to

on

1]

a x h x

n

hx;xi≥c(kxk)kxkf o r every(x;x)∈Graph(T).

issurjective.

introduce the following spaces:

intoZ

ev

,b

l

y

u

kγkthenormofthetraceinL(

Z

be

;dL²(

Γ;R^s)and

byγ

m

^∗:L²( Γ;R^s)

→

no

Z^∗thead

t

j

h

ointoperatortoγ.Wealso γ0v=

0

γv

=

forallv∈V.Itiswellknownfro

it

m

y,

thetheoryofSobol

e

evspaces,cf.t[a2,3,21],th

e

a

e

t(V;H;V^∗)

and(Z;H;Z^∗) i:V→Z

lo

thee mbedding,byγ :Z→L²(

Γ

1

;

,

R

H

^s)a ndγ0:H¹( Ω;R^s)

→H^1/2(Γ;

R^s)

⊂ L²(Γ;

R^s)

t h e t raceo perators,we

inclusion of subdifferential form.

⊆ ⊆ ⊆ ⊆ arealsocontinuous,where =L(0; T;Z)and =L(0; T;V).

Wenote that in (1) the notationu(t) stands for (u )(t),i.e. u(t)=(u)(t) for allu∈ and a.e.t∈(0; T).The

S. Migórski

L

d

e

o

t

mo

L

n

:oD

to

(

n

L

e

)⊂

wit

X

hr

→

esp

X

ec

^∗tb

t

e

oD

a(l

L

in

)

e

(

a

sh

rod

r

e

tl

n

y

se

L

l

-

y

ps

d

e

e

u

fi

d

n

o

e

m

d

on

m

o

a

t

x

o

i

n

m

e

a

)

l

if

m

a

o

n

n

d

ot

o

o

n

n

l

e

y

o

if

p

(

e

a

r

)

at

a

o

n

r.

d(

Abn

)h

o

o

p

l

e

d

ra

a

t

n

o

d

rT iss a i d t o b e p s e u -

(d) i

a

f

n

{

d

xn

li

}

m

⊂

su

D

ph

(L

xn

)

∗ ;

i

x

s

n

s

i

u

≤

ch

hx

th

∗

a

;x

tix

,

n

th

→

en

x

x

w

∗

e

∈

ak

T

ly

x

i

a

n

n

X

d

,

hx

x

n∗

∈

;xn

D

i

(

→

L),hL

x

x

∗

n

;x

→

i.

Lx we ak ly inX^∗,x_n^∗∈

T xn,x_n^∗→

x^∗weaklyinX^∗

An

∗

operatorTissaidtobecoerci

∗

veifthereexistsafunctionc:R⁺→

Rw i t h c(r)→∞a s r→∞s u c h thatAsingle- valuedoperatorT:X→X^∗is

saidtobehemicontinuous,ifforallu;v;w∈Xthefunctionalt→hT(u+tv);wi

Fin

x

a

0

ll

∈

y,

X

we

an

r

dlimsuphT

fo

x

l

n

lo

;

w

xn

in

−

g

x0

s

i

ur

≤

jec

0

t

,

iv

w

it

e

y

h

re

v

s

e

ul

h

t

T

,xc

0

f.

;

[20,

−

T

x

h

i

e

≤

ore

li

m

mi

1

n

.

f

3.7

T

3

x

]

n

,;fo

n

r

−xiforallx

o

∈

pe

X

ra

.

torswhicharepseu- domonotoneweictahllretshpeectto

D(L).

Proposition2.1.

multivalued

o

If

p

X

era

i

t

s

or

a

a

r

n

e

d

fle

T

xi

:

v

X

e,

→

stri

2

c

^X

t ly ∗

\

{

co

∅

n

}

v

i

e

s

x

b

B

ou

a

n

n

d

a

e

ch

d,sc

p

o

a

e

c

r

e

ci

,

ve

L:

a

D

nd

(L

p

)

s

⊂

eud

X

om

→

ono

X

t

∗

on

i

e

s

w

a

it

l

h

in

r

e

e

a

s

r

pe

d

c

e

t

n

t

s

o

el

D

y

(

d

L

e

),

fi

t

n

h

e

e

d

nmth

a

e

xi

o

m

p

a

e

l

ra

m

to

o

r

n

L

ot

+

on

T

e

3. History-dependent subdifferentialinclusions

LVetbeΩa⊂cR^dsebdesaunbsoppaecneboofuHn1d(eΩd;Rsusb),sset≥ofR^dw=ithL2a(ΩL;iRpssc)haintzdcZon=tinHuoδu(Ωs;bRosu)nwdaitr hya∂ΩfixaedndδΓ∈⊆(1∂/Ω2;,1d).=De1n;o2t;i3n.gLbeyt

getγv γ(iv)forallv∈V.Forsimplic inwhatfollows,womitthenotionofth

mbeddingiandwewriteform o tion triples of spaces andtheem dingV⊂Zis co pact.Wede teb y ce eembeddingconstantofV

V=L²(0

;T;V); Z=L²(0

;T;Z) and Hb=L²(0

;T;H);

V

wher

Z

e0<

H

bT<

Z∗

+∞

V

.∗SincetheembeddingsV⊆Z

Z∗

⊆H2⊆Z^∗⊆

∗

V^∗are

V∗

contin

2

uous,it

∗

isknownthatthe embeddingsLetA:(0;T)×V→V^∗,

S:V→V^∗,

J:(0;T)×L²(Γ;

R^s)

→Randf:(0;T)→V^∗be

given.Weconsiderthefollowing

Problem 3.1.

Findu∈V such that

A(t;u(t))+Su(t)+γ ∗∂J

(t;γu(t))3f(t) for a.e.t∈(0; T): (1)

S S S S V

symbol∂Jdenotes the Clarke subdifferential of a locally Lipschitz functionJ(t;·). We adopt the following definition.

✶✾✺✺

A functionu∈V is called asolution to Problem3.1if and only if there existsζ∈Z^∗such that for a.e.t∈(0; T),

A(t;u(t))+Su(t)+ζ(t)=f(t) and ζ(t)∈γ^∗∂J(t;γu(t)):

Definition3.2.

∗

Lemma3.3.

(III.a)J(·;u)ismeasurableon(0;T)forallu∈L²(Γ;

R^s)

(i)(III.a)–(III.d)andm1>max{c1;m2}ce2k γk²,

(III.b)J(t;·) is locally Lipschitz onL(Γ; )fora.e.t∈(0;T);

(III.c)k∂J(t;u)kL(Γ; )≤c0+c1kukL(Γ; )for allu∈L(Γ; ),a.e.t∈(0;T)withc0;c¹≥0;

todependonthecurrentvalueofthesolutionu(t).

(III.e)J⁰

2

(t;u;−u)≤d0

(1+kukL²(Γ;R^s)forallu∈L²(Γ;R^s),a.e.t∈(0;T)withd0≥0.

whereC∈L^∞0;T;(V;V^∗).Clearly,inthecaseoftheoperators(2)and(3)thecurrentvaluev(t)atthemomentt

t∈(0;T)dependonthehistoryofthesolutionuptothemomentt.Thisfeaturemakesthedifferencewithrespectto

form(2)or(3)ashistory-dependent.Weextendthisdefinitiontoalltheoperators: →^∗which satisfy conditionI I

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

In order to state a result on the solvability of Problem3.1, we need the following hypotheses.

I.

A

(I

:

.a

(0

);A

T

(

)

·

×

;v

V

)i

→

sm

V

eas

i

u

s

ra

su

b

c

le

h

o

th

n

a

(

t

0;T)forallv∈V;

(I.b)A

m

(

1

t

k

;

v

·

1

)

−

is

v2

h

k

e

V2

m

f

i

o

c

r

on

a

t

ll

in

v

u

1

o

;

u

v2

s∈an

V

dwst

i

r

t

o

h

n

m

gl

1

y

>

m

0

o

;

notonefora.e.t∈( 0;T),i.e.,hA(t;

v1)−A(t;v²);v¹−v²i^V∗×V≥

(

(

I

I

.

.

d

c)

)

k

A

A

(t

(

;

t

0

;)v)

=

kV^∗f≤

or

a

a

0

.

(

e

t

.

)+

t∈

a1

(

k

0

v

;

k

T

V

).

forallv∈V,a.e.t∈(0;T)witha0∈L²(0

;T),a0≥0anda¹>0;

II. S : V→V^∗is 0 such tha

t kSu1(t)−Su2(t)kV^∗≤LS

Z0t

ku1(s)−u2(s)kVds

for allu1; u2∈V, a.e.t∈(0; T) withLS>0.

III. J: (0;T)×L²(Γ;

R^s)

→Ris such that

2 R^s

2R^{s 2}R^s 2 R^s

(III.d)(

m

z1−

≥

z

0

2;

;

u1−u2)_L2(Γ;R^s)≥−m2ku1− u2k²_L2(Γ;R^s)forallzi∈∂J(t;ui),ui;zi∈L²(Γ;R^s),i=1;2,a.e.t∈(0;T),

We remark thatconditionIIis satisfied for the operator S : V→V^∗given by

Sv(t)=Rt;Z

0t

v(s)ds+v0! for allv∈ V; a.e.t∈(0;T);

(2)

o

w

p

h

e

e

r

r

a

e

to

R

r

:

fo

(

r

0;

a

T

.e

)

.

×

t

V

∈

→

(0;

V

T)

^∗ais

nd

su

v

c

0

h

∈

th

V

a

.

tIR

t

(

is

·;

a

v

l

)

so

is

s

m

at

e

i

a

sfi

su

e

r

d

ab

fo

l

r

eto

h

n

e

(

V

0

o

;

l

T

te

)

rr

f

a

or

o

a

p

l

e

l

ra

vt

o

∈

r

V

S:

,VR(

→

t;·

V

)∗is

gi

a

ve

L

n

ip

b

s

y

chitzcontinuous

Sv(t)=Z

0t

C(t−s)v(s)ds for allv∈ V; a.e.t∈(0;T); (3)

( L S

dependso nt heh istoryo ft hev alueso fv att hem oments0≤s≤tand,therefoSre,VwerVefert ot heo peratorso ft heainncdlu,sfioorn sth.isThreeamsoanin,wfeeatsuareyothfastucthheinsculubsdiioffnesrecnotniaslisitnscilnustihoensfaocfttthheatfothrmey(1c o)natraeinhoispteorrayt-odrespwenhdicehn,tastuabndyiffemroemnteinatl

thetime-dependentsubdifferentialinclusionsstudiedinliteratureinwhich,usually,theoperatorsinvolvedareassumed

In order to prove the existence and uniqueness for Problem3.1, we first state a result on the unique solvability of asubdifferentialinclusioninwhichthetimevariableplaystheroleofaparameter.

Assume thatIholds and f∈V^∗. If one of the following hypotheses:

(ii)IIIandm1>m²ce2k γk²

✶✾✺✻

protp;ert=iesoftheoperatorB. hypothesi establish following mhoinsotoolnleo∗peratorispseudomonotone,cf.[21,Proposition27.6].Next,wetdefinetheoperatorBd:,(h0;T)×oV→uo2^Z∗by v(∈

t;0

V

)

,=a.e.t∈(0;T).Moreover,sincetheoperatorA(t;·)satisfies(I.b)–

1

(I.c),itispseudomono

V

to

∗

n

×

e

V

fora.

1

e.tV∈(0;T).

a.e.t∈(0; T). From (I.a) and (IV.a), it is clear that F(·; v) is a measurable multifunction for allv∈V. Exploiting fundamentalsurjectivityresult,cf.e.g.[3,Theorem6.3.70],itfollowsthatF(t;·)issurjective.Thisimpliesthatfora.e.

Theorem3.4.

sou=uη^∗,whichcompletestheproof.

η(t)=Su(t) for a.e.t∈(0; T). It follows thatuis a solution to the problem (6) and, by the uniqueness ofsolutions Sincef∈V,fromtheestimate(5),weconcludethatu∈Vand(4)hold,whichcompletestheproofofthelemma.

Weshow by using theBanachcontraction principle that the operator Λ has auniquefixed pointη^∗∈^∗. Thenuη^∗ S. Migórski

is satisfied, thentheproblem

A(t;u(t))+γ^∗∂J

(t;γu(t))3f(t) for a.e.t∈(0;T)

(4) has a unique solution u∈V.

Proof.0Wfeorprao.evi.detm∈ai(n0;sTte)p,sitoffotlhloewpsrotohfawtiAth(to;u·t)diestaciolse.rcFivirestw,istihncmeth>e o0p,eir.aet.orhAA((tt;;v·));visistrong≥lymmoknvokt2onfeoraanldlT

fowsfromthefactsthateverystronglymonotoneoperatorismonooneandeverybounde emicntinusandB(v )

γ∂J(t;γv)forallv∈V,a.e.t∈(0;T).Undereitherthe s (i) or (ii),wecan the

(

(

I

I

V

V

.

.

b

a

)

)B

kB

(·

(

;

t

v

;

)

v)

i

k

s

Z

m

^∗e≤

as

b

u

0

r

(

a

1

b

+

le

k

fo

v

r

kV

a

)

ll

fo

v

r

∈

al

V

lv

;

∈V,a.e.t∈(0;T)withb0>0;

∗

(

(

I

I

V

V

.

.

d

c)

)

f

h

o

B

r

(t

a

;

ll

v)

v

;v

∈

iV

V

∗ ×V

an

≥

d−a.e

b

.

1k

t

v

∈

kV2

(0

−

;T

b

)

2

,

k

B

vk

(t

V

;v

−

)bis

3

n

fo

o

r

ne

a

m

ll

p

v

ty

∈

,c

V

on

,

v

a

e

.

x

e

,

.

w

t

e

∈

ak

(

l

0

y

;T

co

)

m

w

p

i

a

th

ct

b

s

1

u

;

b

b

s

2

e

;

t

b3

of

≥

Z

0

;

;

(IV.e)t

v

h

n

e

→

gr

v

ap

in

hZofaB

n

(

d

t;

ζ

·)

n(

i

t

s

),

c

ζ

lo

(t

s

)

e

∈

di

Z

n

∗ ,

Z

ζ

×

n(

(

t

w

)→

−Z

ζ

^∗

(t

)

)

to

w

p

e

o

a

l

k

o

l

g

y

y

in

fo

Z

r∗a

,

.e

th

.et

n

∈

ζ(t

(0

)

;

∈

T B),(i

t

.

;

e

v

.)

.

ifζn(t)∈B(t;vn)with vn;v∈V,

Subsequently,we define themultivaluedmap F : (0; T)×V→2^V∗ F(t;v)=A(t; v)+B(t;v) for allv∈Vandby [3,Proposition6.3.66],weshowthatF(t;·)ispseudomonotoneandcoercivefora.e.t∈(0;T).Therefore,applyingthet∈(0;T)thereexistsasolutionu(

t)∈Voftheproblem(4).Furthermore,owingtothecoercivityofF(t;·),wededuce

thefollowingestimate:

ku(t)kV≤c(1+kf(t)k^V∗) fora.e.t∈(0;T)

2

wi

2

thc>0: (5) Uprsoibnlgemthe(4s)tir∗sonugnimquone.otAolnsioc,itwyeofarAe(ta;b·l)e,

(tIoII.pdr)oavnedththaetthhyepsoothluetsiiosnmo1f>themp2rcoebklγekm,

(w4)eipsraovmeenaoswurtahbaltetfhuencstoioluntioonn(t0o;tTh)e.The existence and uniqueness result for

Problem3.1reads as follows.

AssumeI,IIand f∈V^∗. If either(i)or(ii)of the hypothesis of Lemma3.3holds, then Problem3.1has a unique solution.

Proof. Weuseafixedpointargument.Letη∈V^∗.Wedenotebyuη∈Vthesolutionofthefollowingproblem:

A(t;uη(t))+γ^∗∂J(t;γuη(t))3 f(t)−η(t) for a.e.t∈(0; T):

(6)ByLemma3.3weknowthatuη∈Vexistsandisunique.Next,weconsidertheoperatorΛ:V^∗→

V^∗definedby

Λη(t)=Suη(t) for allη∈V^∗; a.e.t∈(0; T):

V

uisniaquseonluetsisonoftothPerofibxeledmpo3i.1n,twofhiΛc.hNcoanmcelulyd,esletthue∈exiVstebneceapsoalruttioofnth teotPhreoobrleemm.3T.1heanudnidqeufienneestshepaerltefmoellnotwηs∈froVm∗tbhyeto(6),weobtainu=uη.Thisimplies

Λη=Suη=Su=ηandbytheuniquenessofthefixedpointofΛwehaveη=η^∗,

Hb=L²(0

;T;H),Z^∗=L^q(0;T;Z^∗),V^∗=L^q(0;T;V^∗)

with1/p+1/q=1andW={w∈V:w⁰∈

V^∗},wherethetime withthenormkwk =kwk+kw⁰k

∗.Wehave ⊂ ⊂ ⊂b ⊂^∗⊂∗withcontinuousembeddings.Itiswell functionson[0;T]withvaluesinH),i.e.everyelementofW,afterapossiblemodificationonasetofmeasurezero,has

Z^∗⊂

V^∗,whereH^∗;Z^∗andV^∗denotedualspacestoH;ZandV,respec

∗

tively,allembeddingsarecontinuou

∗

sandV

i p

l b e yo n ad n nZndZ wl.

u(0)=u0;

inequality.

continuously,theinitialconditionu(0)hasameaninginV.

✶✾✺✼

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

4. Quasistatic hemivariationalinequalities

LetVandZbereflexive,separable Banach spaces, and letHbe a Hilbert space. Suppose thatV⊂Z⊂H≈H^∗⊂

Gsivceonmaafictxeydenmumebdedred0<inTZ.<W∞edaenndot2e≤byph<·;∞·i,twhedinutarloitducefVtheafodllVow,in gnsptahceesp:aiVri=gLb^pe(t0w;eTe;V),Za=L^p(0a;sT;eZl),derivativeisundersWtoodintVhesenseVofvector-

valWueddVistribZutionHs.ThZelattVerisaseparable,reflexiveBanachspaceknown,cf.e.g.

[3,Proposition8.4.14],thatthespaceWisembeddedcontinuouslyinC(0;T;H)(thespaceofcontinuous

auniquecontinuousrepresentativeinC(0;T;H).Moreover,sinceVisembeddedcompactlyinZ,thensoisWinZ, cCfo.n[3s,idTehretohr

eemfo8ll.o4w.1i3n]g.

evolutionary inclusion of the form:

(

A(t)u⁰(t)+Bu(t)+M^∗∂J(t;Mu(t))3f(t) fora.e.t∈(0;T);

u(0)=u0:

(7)

Weremarkthat,bythedefinitionoftheClarkesubdifferential,problem(7)isequivalenttothefollowinginequality:(A

(t)u⁰(t)+Bu(t)

−f(t);v)+

J⁰(

t;Mu(t);Mv)≥0 forallv∈V; a.e.t∈(0;T);

whereJ^0standsforthegeneralizeddirectionalderivativeofJ(t;·).Forthisreasonproblem(7)iscalledahemivariational

Definition 4.1.

Afunctionu∈L^∞(0;T;V)iscalledasolutionto(7)ifandonlyifu⁰∈

Vandthereexistsη∈Z^∗such that

A(t)u⁰(

t)+

∗

Bu(t)+η(t)=f(t) fora.e.t∈(0;T);

η

u

(

(

t

0

)

)

∈

=

M

u0:

∂J(t;Mu(t)) fora.e.t∈(0;T);

Wenotethatifuisasolutionto(7),thenu∈L^∞(0;T;V)⊂V,i.e.u∈W^1;p(0;T;V).SinceW^1;p(0;T;V)⊂C(0;T;V)Inadditiontoournotation,le

tXbeaBanachspace.Weassumethefollowinghypotheses.

H(A)A∈L^∞(0;T;L(V;V^∗))isan

p

operatorsuchthatA(t)iscoercive,i.e.thereisaconstantα>0suchthatfora.e.

H(B)

t∈(0

L

;T),hA

∗

(t)v;vi≥αkvkVforallv∈V. symmetric.

H(J)

B∈

;

(V ; V) is

R

monotone (nonnegative) and

(ii)J(t;·)islocallyLipschitzonXfora.e.t∈(0;T);

✶✾✺✽

J:(0T )×X→ isafunctionsuchthat

H(M)

(i)J(·; x) is measurable on (0; T) for allx∈X;

(iii)k

L

∂J(t;x)kX^∗≤c(1+kxk²_X^/q)forallx∈X,a.e.t∈(0;T)withc>0.

M∈(Z; X).

Lemma4.4.

respectively.

S. Migórski

H0 f∈V^∗,u0∈V,u¹∈H.

H1

I

k

f

M

p

k

=

L(Z

2

;X

,)t

.

henα> 2cT ce2k

Mkmax{1;kMk},wherece>0isanembedding constantofVintoZandkMk=Thefollo

wingisthemainexistenceresultontheevolutioninclusion(7).

Under hypothesesH(A),H(B),H(J),H(M), f∈V^∗, u0∈V andH¹, the inclusion(7)admits at least one solution.

TwheecpornosoidfeorfTanheeovroelmuti4o.2nissebcoanseddoordnetrhiencslou- sciaolnleodfvtahneisfohrinmgaccelerationmethodwhichwedescribebelow.Tothisend,

( εu⁰⁰(

t)+A(t)u⁰(

√t)+0Bu(t)+M^∗∂J(t;Mu(t))3 f(t) fora.e.t∈(0;T);

(8) u(0) =u0; εu(0)=u1;

whereε>0. Forεfixed, we write for simplicityufor the solutionuεof (8).

Afunctionu∈Viscalledasolutionto(8)ifandonlyifu⁰∈

Wandthereexistsη∈Z^∗suchthat

εu⁰⁰(t)+A

∗

(t)u⁰(t)+Bu(t)+η(t)=f(t) fora.e.t∈(0;T);

η

u

(

(

t

0

)

)

∈

=

M

u0;

∂J(t;M

√u

ε

(

u

t

0

)

(

)

0)=u1:

fora.e.t∈(0;T);

SCi(m0;ilTar;lVy,)aasnbdefWore,⊂weCn(0o;teT;thHa)taifrueicsoantsinouluotuios,ntthoe(8i)n,itthiaelncuon∈diWtio¹n^;

ps(0u;

(T0;)Va)n.dSiun0c(0e)thheaveembaedmdeianngisngWi¹n^;p(V0;Ta;nVd)H⊂,First,wecommentontheexistenceresultfor(8).Thei

mportantstepistoderivethefollowinguniformestimate.

t

L

h

e

e

t

re

εe>

xi

0

sts

be

a

fi

c

x

o

e

n

d

s

.

ta

A

n

s

t

s

C

um

>

e0hy

in

p

d

o

e

th

p

e

e

s

n

e

d

s

en

H

t

(A

of

),

ε

H

s

(

u

B

c

)

h

,H

th

(

a

J

t

),H(M)an

dH⁰,andletubeasolutionto(8).Ifp>2,thenkukC(0;T;V)+ku⁰k

V+√ εku⁰k

L^∞(0;T;H)

+εku⁰⁰k

V^∗≤C{1+ku0k²V /q+

ku1k²H /q+

kfk²V /∗q

:

Moreover, this estimate still holds for p=2providedH1is satisfied.

Theorem4.2.

Definition4.3.

✶✾✺✾

From [13, Theorem 8], we obtain the following result.

s

If

o

h

lu

y

t

p

io

o

n

th

.

esesH(A),H(B),H(J),H(M),H0andH1hold,thenforeveryfixedε>0theproblem(8)admitsatleastone

The following is the existence result for the operator inclusion (7).

Theorem4.5.

Problem5.1.

interestingtoextendtheresultofTheorem4.6toaclassofproblemswithnonlinearoperatorsA(t)andB.

dynamicequation00sofmotion,representingmomentumconservation,thatgoverntheevolutionofthestateofthebody,aro ui

j r ,σi deni u ou )

appliedforces.Inmanycasesweareinterestedinsituationsinwhichthesystemconfigurationandtheexternalforcesadtaio volv

y isc thea rt s t rat a e ligb

sothattheinertial(thesecondordertimederivative)termscanbeneglected.Insuchaway,weobtainthequasistaticapro

an(equlb on fr e of n iv ,h is i ge oerto

unviquelnessofsoluatiodnstollebothindynuamsiicsataltacndmqueasisotfaticnhemivahreiantionalinequallliyty,wisleftopk ent.atItweouludesbteanlso

In this approximation, at each time instant the system is in equilibrium, and the external forces are balanced

bytheinern stsses. o s h m t clra uastic oble s irce

rap h hth

ofquasistaticcontactmodelscanbeseenfromseveralmonographsandmanypapersdedicatedtosuchphenomena,cf.DvauanLions a

o 6Sl tl[19 r rencst in we s te

is slow and the accelerations are negligible, mathematically it means that the system changes character, s

frombeingof herolicty an l i a c pe.oo

dgequasat ivari in ati e

been studied in the literature only recentlyand a recent existence result for such problemsisprovided inMigórskianO

al[13].W ntiot i b s odeedb iationalieq i swits ly nte

d

V.j:Γ×(0;T)× → is suchthat

✶✾✻✵

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

Theorem 4.6.

T

Le

h

t

eo

th

re

e

m

h

4

y

.

p

5

o

.tT

he

h

s

e

e

n

s

th

H

e

(

r

A

e

),

e

H

xis

(B

ts

),

u

H

fu

(J

l

)

fi

,

ll

H

in

(

g

M

u

),

∈

H0

L^∞

an

(0

d

;T

H

;

1

V

h

)

o

,lu

d

⁰

.

∈

F

W

or

,es

v

u

e

c

r

h

y

t

ε

ha

fi

t

x

t

e

h

d

e

,

f

l

o

e

l

t

lo

u

w

ε

in

b

g

e

c

a

on

s

v

o

e

l

r

u

g

t

e

io

n

n

ce

t

s

o

h

(

o

8

l

)

d

g

:iven

by

√u^ε→uweakly*inL^∞(0;T;V); u⁰ε→u⁰weakly inV;

εu⁰_ε→0weakly*inL^∞(0;T;H);

εu⁰_ε⁰→

0weaklyinV^∗;asε→0.Moreover,thelimitfunctionuisasolutiontotheproblem(7).

Thestudyofinclusionsoftype(7)ismotivatedbyseveralcontactproblemsofsolidmechanics.Itiswellknownthatthee f the form−σi ;j=fi,whe euis a displacement s the stress tensor andfis the s ty (per nit v l me of n r ct ns e e slowl in time n u h a way tha t ccele a ion in the sys em are her small nd n g i le, p xim tio i i rium equati s) o th

equations motio ,−Dσ=fw ere Div the d ver nce p

ar.talreRigroumateaiat etmentofq isatprmsent.Theidgrowtoft ee ory

u t d [4], Han nd Sof nea [ ], hil or ea. ]ande f e e here. When as ume

thath procesay p b peto el ipt c or a parb ol i t y T urknowle

isti c hem ational equ

lieshavsLeipdsecrahcithzimoppoerrtaatnotrsenwmeercehaconnnshigdaetrgqeudqaisnais[t6a,t1c9ip].roWoledemalslso mpcooinlttaoctuytptvhaartomtheenare.suFnlitnaoufaTlihteeoerermemh4a.r6trcoahnngbtehmaopqpolicoanbioleantoof

5. History-dependent hemivariationalinequality

IansstohcisiasteedctiwointhwPeropbrolevimde3.a1.reWseultadoonpetxtihsetennocteatainodnoufniSqeucetnioenss3o.f

Tahesoplruotbiolenmtounadcelrascsonosfidheermaitvioanriarteiaodnsalaisnefoqluloawlisti.esFindu∈V such that hA(t;u(t));viV^∗×V+hSu(t);viV^∗×V+Z

Γ

j⁰(

t;γu(t);γv)dΓ≥ hf(t);viV^∗×V

(9)forallv∈Vanda.e.t∈(0;T).

Ihnyptohtehesstiusd.yofthehemivariationalinequality(9),inadditiontothepreviousassumptions,weneedthefollowing R^s R

(V.a)j(·;·; ξ) is measurable on Γ×(0; T) for allξ∈R^sandj(·;·;0)∈L¹(Γ×(0; T));

(V.b)j(x;t;·)islocallyLipschitzonR^sf o r

a.e.(x;t)∈Γ×(0;T);

(V.b)j(x;t;·)islocallyLipschitzonR^sf o r

a.e.(x;t)∈Γ×(0;T);

T

(V.c)k∂j(x;t;ξ)k ≤c0+c1kξk fora.e.(x;t)∈Γ×(0;T),allξ∈ withc0;c¹≥0;

(V.e)j⁰(

2

x;t;ξ;−ξ)≤d0(1+kξk^Rs)fora.e.(x;t)∈Γ×(0;T),allξ∈R^swith d0≥0.

e

withanobstacleonΓC,theso-calledfoundation.Thecontactisfrictionalandismodeledwithsubdifferentialiboundary conditions.

oftheform(9)inwhichtheunknowniseitherthevelocityorthedisplacementfield.InbothcasestheabstractresultsofS i 4n o

p i v h o

Inthissectionweillustratetheuseoftheseresultsinthestudyoftwocontactproblems.T phscl i g o te r o cprole s slo isel

whitehsuyrfaicae∂sΩettwnhichfishpafirtsittiocnnedtainttothbreemdiisjaoinftomlewass.urAabvlecpoaratsstΓicb;oΓdyaoΩ

ncdcuΓpiessuachdothmaatinm(Γo)fIR,.dW=e2a;r3e,D N C D>

thneΓaNccaenledravtoiounmoefftohrecessyosftedmeisityf0actinaΩnd.,Wtheeraesforem,eththeptrohcesfsorcesandtract ionschange,tshloewlyintimiesothatnegligible isquasistatic.Moreover body is ncontact

u·v=uivi; kvkd=(v·v) for allu=(ui); v=(vi)∈

;

✶✾✻✶

S. Migórski

R^s R^s R^s

(V.d)m(ζ1−

≥

ζ

0

2

;

)·(ξ1−ξ2 )≥−m2kξ1−ξ2k²R^sforall ζi;ξi∈R^s,ζi∈∂j(x;t;ξi),i=1;2,a.e .(x;t)∈Γ×(0;T)with

Nexoistetenthcaetainndthuenicqounednietsiosnre(Vsu.dlt)ftohrethdeothdeemniovatersiatthioenainlnienreqpuraodliutyct(9in).Rsh.eFdreo tmailTshoenorietsmp3ro.4o,fwceandbedeufcoeuntdheinfo[l1l4o]w.ing

Theorem 5.2.

AssumethatIandIIholdandf

√∈V^∗.Ifoneofthefollowinghypotheses:

(

(

ii

i

)

)V(V.

a

a

n

)–

d

(V

m

.d

1

)

>

an

m

d

2

m

c2

1

kγ

>

k2

max{3c1;m²}c^e2k γk²,

is satisfied, then Problem5.1has a solution u∈V. If, in addition, the regularity condition:

eitherj(x;t;·)o r −j(x;t;·)is regular onR^sf o r a . e .

(x;t)∈Γ×(0;T) holds, then the solution of Problem5.1is unique.

6. Quasi-static frictional contactproblems

Alargenumberofquasistaticcontactproblemswithelasticorviscoelasticmaterialsleadstoahemivariationalinequalityectons3,a d5wrkandcanbeusedtor ovidetheunqueweaksolabilityoftecrrespondingcontactproblems.

d

oiTnt>ere0s.teTdheinbltohdeyeivsoclulatimopnepdrooncnsesΓsDoafntdhesomtehcehadnisicpalalcsseutmateentoffiaethltdeevbaondiyshi enstthheerbeo.uSnudrefdacientterarvcatiloonfstoimfdee[n0s;0Tty],fwNhearcet

tWheeduissepltahceemnoetnattivoenctνor=,th(νei)stfroerssthteenosuotrw,aarnddutnhietlnionremarailzeadt∂stΩra.inWteendseonro,trees bpyecutiv=el(yu.i)W,σedr=ec(aσlilj)t,haantdthεe(uc)om=p(oεnije(nut)s)

osyfmthmeetlriniceaterinzseodrssotrnaiRn^dteonr,soerquεi(vua)leanrtely,gtivheenspbayceε ijo(fus)y=mm(ueti;rjic+muajt;ir)i/c2e.sWofeorddeenrodte.TbyS

thes p a cinenoefrsperocodnudctsoradnedr

thecorrespondingnormsonR^dandS^daregivenby he canonical

R 1/2 R^d

o:τ=σijτij; kτkS^d= (τ:τ)^1/2 forall σ=(σij); τ=(τij)∈S^d;

rWesitphecthtievseelyp.reliminaries,theclassicalformulationofthequasistaticcontactproblemweconsiderinthissectionisth e

following.

following.

more details andmechanicalinterpretation.T A B

perators A and B with respect to the time variable allows to model situations when the properties of the material

od i

evo

no

N

tatio

0

n

;

vaa

n

ndv

C

f

=

ort h

C

e

×

n

(

o

0

rm

T

a

)

la

F

n

o

d

r

tan

l

g

v

entia

H

l

dwe

o

s

fv

ll

on∂

o

Ω

te

g

b

iven

v

b

t

y

he

vt

r

=

ace

v·fν

v

an

o

d

n

v

Γa

=

nd

vw−vν.We

rall at

ν τ

components ν τ ν

istheequilibriumequation,whereDivrepresentsthedivergenceoperator,i.e.Divσ=(σij;j).Conditions(10c)and(10d)a et r o

o y n and

νandτforσandu⁰ⁱndicatenormalandtangentialcomponentsoftensorsandvectors.Thesymbol∂jdenotestheClarkesbdi tlof

i heata o psofici l e u re

functionu0denotestheinitialdisplacementfield.

boundaryconditionsoftheform(10e),(10f)withthefunctionsjνandjτsatisfyingassumptionsVIIIandIXbelowcanbefundi 25].

e, e tuelv mark emp i clet v t o

withnonmonotonenormaldampedresponse,associatedtoanonmonotonefrictionlaw,toTresca’sfrictionlawortoapwer- awftion rr

ls l re l thecr d qu i be

(VI.a)A(·;·;ε)ismeasurableonQforallε∈S^d;

(VII.a)B(·;·;ε)ismeasurableonQforallε∈S^d; (VI.e)A(x;t;0)=0fora.e.(x;t)∈Q.

(VI.b) (x;t;·)iscontinuouson fora.e.(x;t)∈Q;

(VI.c)( (x;t;ε1)−(x;t;ε2)):(ε1− ε²)≥mkε1−ε²k^dfor allε¹;ε2∈ , a.e.(x; t)∈Qwithm>0;

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

Problem 6.1.

Find the displacement fieldu: Ω×[0; T]→R^dand the stress fieldσ: Ω×[0; T]→S^dsuch that, for allt∈(0; T),

σ(t)=A(t;ε(u⁰(

t)))+B(t;ε(u(t))) inΩ; (10a)

Divσ(t)+f

u

0((

t

t)

)

=

=

0

0

io n n

Ω

Γ

;

D;

((1100bc))

−σν(t)∈

σ(t

∂

)

j

ν

ν(t

=

;u

f

⁰

N

(t

(t

))

) o

o

n

_{n Γ}

Γ

N

;

;

(

(

1

1

0

0

d

e

)

)

C

−στ(t)∈∂j

u

τ(

(

t

0

;

)

u⁰(t

u

)) o

in

nΩΓ; ((1100gf))

=

τ

0 :

C

WepresentashortdescriptionoftheequationsandconditionsinProblem6.1andwereferthereaderto[6,15,19]for

thheeveiqscuoastiitoyno(1p0eara)troerpraensdenetslatshtiecivtiyscoopeelarasttiocr,corenssptietcuttiivveelyla.wTihne wehxipchlicitadnedpendaernecegivoefnthneonvliisnceoasritoypearnadtoersla,sctailclietdyepend on the temperature, which plays the role of a parameter, .e. its lutionin time is prescribed.Equality(10b)re the displacem n and t acti n b undar co ditions, respectively, (10g) is the initial condition in which the

Conditions(10e)and(10f)representthefrictionalcontactconditionsinwhichjνandjτaregivenfunctions.Thesubscripts uffereniajw th respect totlsv riable.

C ncrete exam le f rtona mod ls which lead tos bdiffe ntial o n [1,1H re we r stric o rs es to re that these xa les n ud he iscous contac and the cntact

IΓnot×he(lsTtu)dyricodfΣth.eOcuonΓtaecstupt;robbel.eomw(a10aal)v–

a(1id0∈gf)orw1e(Ωu;sRoer)setsapnodnatridingndoetnaatisoinstaQytic=frΩcti×on(0a;lTco)o,ntΣaDct=proΓDl×m(s0.;Te) u,sΣeNth=e

σec=σthνthσeνn.ormaxlt,awndetianntrgoednutciaeltchoemsppoanceenstsVoftheHstressfieldbσyoVnt=he{vbo=un(dva)rya Hre1(dΩe;fiRnedd):bvyσν0=(aσ.eν.)·νΓand}

an

τ

dH=−L²(Ω

ν

;S^d

N

).

e

and ,

defined ⁱ∈ = onD

We assume that the viscosity operator A and the elasticity operator B satisfy VI. A:Q×S^d→S^dis suchthat

A S^d

A A A 2

S S^d A

(VI.d)kA(x; t;ε)kS^d≤a0(x; t)+a1kεkS^dfor allε∈S^d, a.e. (x; t)∈Qwitha0∈L²(Q),a0≥0 anda1>0;

VII. B:Q×S^d→S^dis suchthat

ν

✶✾✻✷

(VII.b)kB(x; t;ε1)−B(x; t;ε2)kS^d≤LBkε1−ε2kS^dfor allε1;ε2∈S^d, a.e. (x; t)∈QwithLB>0;

(VII.c)B(·;·;0)∈L²( Q;S^d).

(VII.c)B(·;·;0)∈L²( Q;S^d).

IX.jτ:ΣC× R^d→ is suchthat VIII.jν:ΣC× → is suchthat

(IX.b)jτ(x;t;·)islocallyLipschitzonR^df o r

a.e.(x;t)∈ΣC; (VIII.b)jν(x;t;·)islocallyLipschitzonRfora.e.(x;t)∈Σ^C;

(VIII.d)(ζ1−ζ²)(r¹−r²)≥−m^ν|r1−r²|²f o r a l l

ζi∈∂j^ν(x;t;rⁱ),ri∈R,i=1;2, a.e.(x;t)∈Σ^Cwithm^ν≥ 0;

(IX.e)j_τ⁰(x;t;ξ;−ξ)≤dτ(1+kξkR^d)forallξ∈R^d,a.e.(x;t)∈ΣCw i t h dτ≥0.

(IX.c)k∂jτ(x;t;ξ)k ≤c0τ+ c1τkξk for allξ∈ ,a.e.(x;t)∈ΣCwithc0τ;c1τ≥ 0;

whichimplythat Z Z

0 0 0

0

✶✾✻✸

S. Migórski

The contact potentialsjνandjτsatisfy the following hypotheses.

(VIII.a)jν(·;·; r) is measurable on ΣCfor allr∈Randjν(·;·;0)∈L¹(ΣC);(VIII.c)|∂jν(x;

t; r)| ≤c0ν+c1ν|r|for allr∈R, a.e. (x; t)∈ΣCwithc0ν; c1ν≥0;(VIII.e)jν0( x;t;r;

−r)≤dν(1+|r|)forallr∈R,a.e.(x;t)∈Σ^Cwithd^ν≥0.

(IX.a)jτ(·;·;ξ) is measurable on ΣCfor allξ∈R^dandjτ(·;·;0)∈L¹(ΣC);

R^d R^d R^d

(IX.d)(ζ₁−ζ²)·(ξ¹−ξ²)≥−m^τkξ₁−ξ²k^R2dforallζi∈∂j^τ(x;t;ξi),ξ_i∈R^d,i=1;2,a.e.(x;t)∈ΣCwithm^τ≥0;

The volume force and traction densities satisfy

f0∈ L²(

0;T;L²(Ω;

R^d); fN∈ L²( 0;T;L²(Γ

N;R^d) (11)

and, finally, the initial displacement issuchthat

u0 V: ∈

(12)w

W

h

e

ic

t

h

ur

s

n

o

n

lv

o

e

w

(

1to0ath

)–

e

(1

va

0g

ri

)

a

.

ti

L

o

e

n

t

a

v

lf

∈

orm

V

u

.

la

T

t

h

i

e

o

n

n

,

o

u

f

s

P

in

r

g

ob

(

l

1

e

0

m

b)

6

,

.1

w

.

e

S

h

u

a

p

v

p

e

osethat(u;σ)isacoupleofsuffic ientlysmoothfunctions(σ(t);ε(v))H=(f0(t);v)L²(Ω;R^d)+ Z

∂Ω

σ(t)ν·vdΓ for a.e.t∈(0;T): (13) WetakeintoacZcounttheboundarZyconditions(10cZ)and(10d)toseethat

∂Ω

σ(t)ν·vdΓ=Γ

N

fN(t)·vdΓ+Γ

C

⁽σν(t)v^ν+στ(t)·vτd Γ for a.e.t∈(0;T): (14) On the other hand, from the definition of the Clarke subdifferential, (10e) and (10f), we have

−σν(t)v^ν≤j^ν0(

t;u⁰ν(t);v^ν); −στ(t)·vτ≤j^τ0(

t;u⁰τ(t);v^τ) onΣC;

ΓC

⁽σν(t)v^ν+ στ(t)·vτd Γ≥ − Γ

C

⁽jν(t;u^ν(t);v^ν)+jτ(t;u^τ(t);v^τ)d Γ

(15)fora.e.t∈(0;T).Considerthefunctionf:(0;T)→V^∗givenby

hf(t);viV∗×V=(f0(t);v)L²(Ω;R^d)+(fN(t);v)L²(Γ_N;R^d) (16)

for allv∈Vand a.e.t∈(0; T). We com

Z

bine (13)–(16) to obtain

C

^{(0 0} 0 0

(σ(t);ε(v))H+

Γ

jν(t; uν(t);vν)+jτ(t;uτ(t);vτ)dΓ≥ hf(t);viV^∗×V

R

H

!!

!

(u;σ)iscalledaweaksolutionofthefrictionalcontactproblem(10a)–(10g).Weconclude,underthehypothesesof

Wenowpasstothesecondproblemofthissection.Weconsiderthequasistaticviscoelasticcontactwithnonmonotonen a

lae ion. t fist ,e ume tatt o fo sa acetractionschange

slowlyintimesothattheaccelerationinthesystemisnegligible.Weshowthatthequasistaticmodelcanbeformulateda t

dp e e riati inulit frm and e a su fTo 3.4is applicablein

the displacement field, byσ:Q→

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

and, using the constitutive law (10a), it follows (A(t;ε(u⁰(

t)));ε(v)_H+

(B(t;ε(u(t)));ε(v)_H+

ZΓC

⁽j_ν⁰(

t;u⁰_ν(t);v^ν)+jτ0(

t;u⁰τ(t);v^τ)d

Γ≥ hf(t);viV^∗×V (17)

ffoolrloawllsvth∈atVanda.e.t∈ (0;T).Letw=Zut^0denotethevelocityfi el d. Then,b yusi ngthe initial con di ti on (10g),i t Therefore,(17)and(18)leadtotheu(t)=

0

w(s)ds+u0 forall t∈[0;T]:

)–(10g),intermsofvelocity.

(18)

Problem6.2.

following variational formulation of problem (10a

Find a velocity fieldw∈V such that

(A(t;ε(w(t)));ε(v)_H+Bt;ε

Z0t

w(s)ds+u0 ;ε(v) +Z

ΓC

⁽j_ν⁰(

t;wν(t);v^ν)+jτ0(

t;wτ(t);vτ)d

Γ≥hf(t);viV^∗×V

for allv∈Vand a.e.t∈(0; T).

oTbhteaihnetmhievafroilalotiwoninaglrineseuqlutaolintyPirnobPlermobl6e.2m.6 .2 isof th e form of t he inequality in Probl em5 . 1.FromTheorem5 .2,w e

As

(i

s

)

um

(

VeIItI.h

a

a

)–

t((

V

V

I

I

I

.

I

a.d),

),

V

(

I

I

I

X

,.(

a

1

)

1

–

)

(

IaXn.dd

)

(

a

1

n

2

d

)h

m

o

1

ld

>

.I

m

fao

x

n

{

e

√o

3

f(t

c

h

1

e

ν+

fol

c

l

1

o

τ

w

);

i

m

ng

ν;

h

m

y

τ

p

}

o

c

th

e2

e

k

s

γ

e

k

s

2

:

,

(ii)VIII,IXandm1>max{m^ν;mτ}ce2k γk²

ssatisfied,thenProblem6.2hasatleastone solution. If, in addition,

either jν(x; t;·); jτ(x;t;·)are regularor −jν(x;t;·);− j^τ(x;t;·)areregularfora.e.(x;t)∈Σ^C; (19)

then the solution of Problem6.2is unique.

Letwbeasolution of Problem6.2and denotebyuandσthe functionsdefinedby (18)and (10a). Then, thecoupleTheorem6.3,thatthefrictionalcontactproblem(10a)–(10g)hasatleastoneweaksolutionwiththefollowingregularity:

u∈W^1;2(0

;T;V)and σ∈ L²(0

;T;H) withD i v σ∈L²(0

;T;V^∗): If,inaddition,theregularitycondition(19)holds,thentheweaksolutionofProblem6.1isunique.orm l comp i nc

andfrcti As in he r problemw a ss h he vlumerce ndsurf

thsisacaimsee.Feoretnhdemntechhamniivcaalforomdnuallatioenqoafthyeopfrothceessowe(u7s)ethetnhotaatbidosntrincttrore ducletdoabohvee.reWmesetQ=Ω×(0;T).

Aεg(auin)=we(∂duen+ot∂eub)y/2ut:heQst→rRtensor.WeassumealinearviscoelasticSthewstirtehssthetecnosnosrtiatunt divebylaεw(uo)ft=he{Kεeijl(vui)n}–,

V

i

o

j

igttypej

i ij

ain Theorem6.3.

i

✶✾✻✹

σij

=aijkl εkl (u⁰)+bijkl εk l

(u)

model inQ;