On a contact problem for elastic-viscoplastic bodies

Nonlinear

Andysir,

Theory,

Methods

& Applications,

Pergamon

Printed

Vol. 29, No. 9, pp. 1037-1050, 1997 0 1997 Elsetier Science Ltd in Great Britain. All rights reserved 0362-546X/97 $17.00+0.00

PII: SO362-546X(96)00096-X

ON A CONTACT PROBLEM FOR ELASTIC-VISCOPLASTIC BODIES MIRCEA

SOFONEA

Department of Mathematics, University of Perpignan, 52 Avenue de Villeneuve, 66 860 Perpignan, France (Received

30 May

1995; received

in revised form

5 March

1996; received

for publication

19 September

1996)

Key words andphrases: Viscoplasticity, Signorini’s problem, variational inequalities, nonlinear evolution equations, fixed point, penalized method.

1. INTRODUCTION

In this paper we propose to investigate a problem of unilateral contact between an elasticviscoplastic body and a rigid frictionless foundation. Thus, the famous Signorini’s problem in linear elasticity [l], already studied for elastic-plastic or viscoelastic bodies in [2-81, is investigated here for rate-type elastic-viscoplastic models. Everywhere in this paper we consider the case of small deformations, we denote by E = (Eij) the small strain tensor and by ct = (oij) the stress tensor; the dot above a quantity will represent the derivative with respect to the time variable of that quantity. We consider here constitutive laws of the form Ir = &i + G(a, E) (1.1) in which G and G are constitutive functions. A concrete example of an elastic-viscoplastic constitutive law of this form in the case when G does not depend on E is the Perzyna law given by i = E-lb

+ ;(o

- PKt7)

in whichp > 0 is a viscosity constant, K is a convex closed nonempty set in the space of symmetric tensors and PK represents the projection map. A relatively simple one-dimensional example of elastic-viscoplastic law of the form (1.1) in the case when a full coupling in stress and strain is involved in G is obtained by taking -k,F,@ G(a, E) =

- f(E))

if g(e) 5 o 5 f(a)

0 L w&m

if 0 > f(e)

- d

if 0 < g(e)

where k,, k, > 0 are viscosity constants and F,, F,: I?+ -+ IR are increasing

functions

with

F,(O) = F,(O) = 0 (see [9], p. 35).

Rate-type viscoplastic models of the form (1.1) are used in order to describe the behaviour of real materials like rubbers, metals, pastes, rocks and so on. Various results and mechanical interpretations concerning models of this form may be found for instance in [9] (see also the references quoted there). Existence and uniqueness results for initial and boundary value 1037

1038

M. SOFONEA

problems involving (1 .l) for different forms of G were obtained, for instance, in [IO-121 (the case when G depends only on a) and in [13, 141 (the case when a full coupling stress and strain is involved in G). In all these papers only the case of classical displacement-traction boundary conditions was considered. The aim of this paper is to investigate a quasistatic problem for the elastic-viscoplastic models (1.1) involving unilateral contact condition. So, in Section 2 the mechanical problem is stated and some notations and preliminaries are presented; in Section 3 a variational formulation of the problem is given and an existence and uniqueness result for the displacement and the stress field is obtained. Since the variational formulation proposed here is given by an evolution problem involving a variational inequality of the first kind, the ordinary differential equation arguments presented in [13,14] do not work. For this reason, a fixed point method (already used in [15] in the case of a displacement-traction problem) was used here in order to obtain the existence and the uniqueness of the solution. The continuous dependence of the solution with respect to the input data as well as a stability result are analysed in Section 4. Finally, in Section 5 we consider a penalized problem involving normal displacement conditions with friction governed by a small parameter h; for this problem we prove the existence and the uniqueness of the solution and we obtain the convergence of this solution to the solution of the contact frictionless problem when h -+ 0.

2. PROBLEM

STATEMENT

AND

PRELIMINARIES

Let us consider an elastic-viscoplastic body whose material particles fulfil a bounded domain Sz c lRN (ZV = 1,2,3) and whose boundary I, assumed to be sufficiently smooth, is partitioned into three disjoint measurable parts I?, , I, and I,. Let meas I, > 0 and let T > 0 be a time interval. We shall assume that the displacement field vanishes on I, x (0, T), that surface tractions g act on I, x (0, T) and that body forces f act in Q x (0, T). We also suppose that the body rests on a rigid foundation S by the part I, of the boundary and that this contact is frictionless, i.e. the tangential movements are completely free. We shall finally assume the case of quasistatic processes and we shall use (1.1) as constitutive law. With these assumptions, the above mechanical problem may be modelled by the following initial and boundary value problem: 0 = &e(ti) + G(o, E(U)) in !A x (0, T) (2.1) Diva+f=O

u, I 0

Q, I 0,

in Sz x (0, T)

(2.2)

u=o

on I, x (0, T)

(2.3)

ov = g

on I, x (0, T)

(2.4)

~ri = 0,

40) = uo,

o,u, = 0

a(0) = (To

on r, x (0, T)

in CJ

(2.5) (2.6)

in which the unknowns are the displacement function II = (ui): !A x [0, T] -+ lRN and the stress function 0 = (Oij): n x [0, T] -+ S,, S, denoting the set of second order symmetric tensors on

1039

Contact problem for elastic-viscoplastic bodies

RN. In (2.1)-(2.6) E = E(U) denotes the small strain tensor, Div o represents the divergence of the tensor-valued function o, v = (vi) is the unit outward normal to Sz, QV is the stress vector, u,, (T, and oTi are given by U,

=

UiV’ ,,

TV =

a,i

a;jVjVi,

=

aij

Vj -

6, Vi

(i = l,N)

and finally u,, and crOare the initial data. We denote in the sequel by “v ” the inner product on the spaces RN and S, and by 1. ( the Euclidean norms on these spaces. The following notations are also used: H = (U = (Vi) 1Ui E L2(sZ), i = 1, N), HI

=

(U

=

(Vi)

1 Vi E H’(SZ),

X

=

(7

=

(Tij)

( Tij

=

7ji

i =

1, NJ,

E L2(n),

i, j

1, NJ,

=

32, = (T E X 1Div r E H). The spaces H, H,, X and X, are real Hilbert spaces endowed with the canonical inner products denoted by ( *, . )H, ( *, s)~~, ( *, *>x and ( *, * )X1 respectively. Let H, = [H1’2(r)]N and let y: HI + Hr be the trace map. We denote by V the closed subspace of HI given by V = (u E H, 1yu = 0 on I,). (2.7) The deformation

operator E: H, + X defined by dU)

is a linear and continuous

=

(Eij(U))v

&ijCU)

operator. Moreover,

=

+t"i,j

+

uj,i)

since meas I, > 0, Korn’s inequality

holds:

where C is a strictly positive constant which depends only on Szand I, (everywhere in this paper C will represent strictly positive generic constants which may depend on a, r, , I, G, T and do not depend on time or on input data). Let Hb = [H-1’2(r)]N be the strong dual of the space Hr and let ( *, . > denote the duality between Hb and Hr. If r E X, there exists an element yy7 E Hi such that: (~~5,

Moreover,

if

7

yu)

=

(7,

E(I&

+

is a regular (say C’) function, (Y”7,

Finally, for every real Hilbert norm on the space L”(0, T, X).

vu>

=

1 Jr

TV

(Div

7, U>H

VUEH,.

(2.9)

then - u da,

VUEH,.

(2.10)

space X we denote by I * IX the norm on X and by 1. Ia,X the

1040

M. 3. AN

EXISTENCE

SOFONEA AND

UNIQUENESS

RESULT

In this section we obtain a variational formulation for the mechanical problem (2.1)-(2.6) for which we prove an existence and uniqueness result. For this, let us suppose that f E W’,“(O, g E W’+‘(O,

T, H)

(3.1)

T, L2(l-,)N)

(3.2)

and let us denote by U,, the set of geometrically

admissible

displacement

fields defined by

U,, = (21E VI u, 5 0 on I,). Moreover,

(3.3)

for all t E [0, T] and u E Hi, let L(t, u) and Cad(tr u) be given by ut,

u) =
u>, +
(3.4)

C,,@, u) = [z E x 1(T, E(W) - &(U)>x L L(t, w - u) v w E U,,).

We have the following

(3.5)

result.

3.1. If the couple of functions (u, a) is a regular solution for the mechanical problem (2.1)-(2.6) then

THEOREM

u(t)

E

(3.6)

rr,d

(3.7)

for all t E [0, T], b(t) = &e@(t))

+ G(a(t),

e@(t)))

(3.8)

a.e. t E (0, T), o(0) = 00.

u(O) = uo, Proof.

(3.9)

We have to prove (3.7); for this, let us remark that using (2.9), (2.10) and (2.2)

we have (0, E(U) - E(mc

and, using (2.3)-(2.5)

1 av-(u ,rI

= H +

r

av*(u

- u)da

vt E [o, T],

U

E uad

(3.10)

and (3.3), we obtain - u)da 2

g - (u - u) da

Vt E [o, T],

U E u,,.

(3.11)

r2

Theorem 3.1 follows.now from (3.10), (3.11), (3.4) and (3.5). Let us now remark that the converse of Theorem 3.1 also holds; indeed, if (u, a) is a regular solution of (3.6)-(3.9), after a formal computation it can be proved that (u, a) is a solution of (2.1)-(2.6). For this reason we may consider (3.6)-(3.9) as a variational formulation of the mechanical problem (2.1)-(2.6).

1041

Contact problem for elastic-viscoplastic bodies

In order to obtain assumptions:

an existence and uniqueness

E: tA x S, + S, is a symmetric (a) E,,

result,

let us consider the following

and positively definite tensor, i.e.

E L”(Q) for all i, j, k, h = 1, N

(3.12)

(b) &a * T = CJ. &r for all ~7,t E S,, a.e. in Q (c) there exists Q > 0 such that &a * o L (~lcr[~ for all cr E S, G:QxS,xS,,,+S,and (a) there exists i > 0 such that IG(x, 01, d - ‘3-c 02, @I 5 L(lo, for all c7i, 02, E,, e2 E S,, a.e. in Q 1

~21

+ I&, -

~21)

(3.13)

@I x - G(x, o, E) is a measurable function with respect to the Lebesgue measure on C& for all cr, E E S, (cl x - G(x, 0,O) E X

uoE Wad*00 E Ld(O, uo).

(3.14)

Remark 3.1. (a) Using (3.12) and (3.13) we obtain that for all cr E X and E E X the functions

x - &(X)&(X) and x H G(x, a(x), E(X)) belong to 3C hence we may consider & and G as operators with the range in X. (b) The assumption (3.14) involves a regularity condition on the initial data u. and cro as well as compatibility condition between uo, a,, fand g; as it results from the proof of Theorem 3.1, the assumption (3.14) is fulfilled if u. E H,, cro E 3C and (uo, cro) satisfy (2.2)-(2.5) for t = 0. The main result of this section is given by the following

THEOREM

problem

theorem.

3.2. Let (3.1), (3.2), (3.12)-(3.14) hold. Then there exists a unique solution (3.6)-(3.9) having the regularity u E W’Y”(O, T, II,), o E W’*“(O, T, SC,).

of the

In order to prove Theorem 3.2 we need some preliminary results. For this, let us suppose in the sequel that the assumptions of Theorem 3.2 are fulfilled and let v E L”(0, T, 32). Let also z, E W’+‘(O, T, X) be the function defined by:

t z,(t) = VW d.sJ +zo

vt E [0, T]

(3.15)

0

where zo = a, - E&(Uo).

(3.16)

1042

M.

SOFONEA

LEMMA 3.1. There exists a unique couple of functions U, E W”“(O, such that

T, H,), cq E W’*“(O,

(3.17)

u,w E ua,, a,(t) E Ld(f, &#)

= w$W)

T, X,)

(3.18)

%)(O)

(3.19)

+ z,(t)

for all t E [0, T]. Proof.

variational such that

Let t E [0, T]; using (3.1)-(3.4), (3.12), (2.8) and standard arguments of elliptic inequalities theory we obtain the existence and the uniqueness of an element u,(t) 4)w E ua, 9 (W~?#h

E(V) - ~(~Jfmc

2 L(t, u - u,(t))

L

+ (z,(t), E(V) - +4#hc vu E u,,.

(3.20)

Taking now a,(t) E X defined by (3.19), we obtain (q),

&(U) - eqfh

vv E Lg,

2 ut, u - u,(O)

(3.21)

and using (3.5) we obtain (3.18). Let us now remark that taking u = u,(t) k o, where (p E XI(Q)~, from (3.21) it follows Diva,(t)

+ f(t)

= 0

(3.22)

hence a,(t) E X,. So, we proved the existence of a couple (u,(t), a,(t)) E H, x X, such that (3.17)-(3.19) hold. Let now t,, t2 E [0, T]; using (3.20) and (3.4) we obtain lqt,)

- qhf,

5 w(tl)

- fWIH + km - mlL2(r# + Iz,(t,) - ~Jf*hc)

and, using (3.19), (3.22) and the previous inequality,

5 w(tJ

- f&h

(3.23)

it follows

+ Ig(f,) - g(fh(ry

+ Iz,w - ~JfZhc).

(3.24)

So, from (3.23) and (3.24) it results that t -u,(t): [0, T] + HI and t - a,(t): [O, T] + X1 are Lipschitz continuous functions, i.e. U, E WIP”(O, T, IfI), os E W’,“(O, T, X,). The unicity part in Lemma 3.1 follows from the unicity of the element u,(t), solution of (3.20).

n

Let us now remark that by (3.13) t - G(a,(t), e(u,,(t))) is a Lipschitz continuous function on [0, T] with values in X [see also Remark 3.1(a)]. This property allows us to consider the operator A: L”(0, T, X) --t W’,“(O, T, X) defined by MO

= G@,(t), M,(t)))

vrj E L-(0,

T, X),

t E [0, Tl.

(3.25)

Contact problem for elastic-viscoplastic bodies

1043

LEMMA 3.2. The operator A has a unique fixed point ft* E L”(0, T, X).

Proof. Let qr, q2 E L-(0, T, X) and t E [0, T]; for simplicity we denote: z,, = zl, z,, = z2, u VI = Ul, u,, = 4, q, = 01, Q,, = cr2. Using (3.20) and (3.12) we obtain I&M))

- ~(~2mc

and, from (3.19) and the previous inequality, Iat)

(3.26)

5 ClZl@) - z2ac

it follows

- Q2Whc 5 clz,w

(3.27)

- z2wIx.

Using now (3.25), (3.13), (3.26), (3.27) it results IAfllW

- ~tf2WI3c 5 clz,w

- Z2WlX

and, having in mind (3.15), we obtain bWt)

(3.28)

- ~tl2(Ol3c 5 C t h(s) - ~2Ckc~ 10

By recurrence, denoting by Ap the powers of the operator A, (3.28) implies IAPql(t)

- Ap~2(t)IX I C*

* IV,(~) - tlz(r)Ixdr...

d.s

0

p integrals

for all t E [0, T] and p E N. It results IA%

and, since lim(P/p!)

- Apr121m,3c 5 5

= 0, (3.29) implies

Iv, - tl21m,x

(3.29)

vp E N

that for p large enough the operator

Ap is a

contraction i: L”(0, T, X). Then there exists a unique q* E L”(0, T, X) such that Apa* = fl*. Moreover ]I* is the unique fixed point of A. Proof of Theorem 3.2. The existencepart. Let n* E L”(0, T, a,,* E WrV”(O, T, SC,) be the functions (u,, , c+) is a solution for the problem The equality (3.8) follows from (3.19), b,*(t) = &E@,*(t)) + i,*(t),

X) be the fixed point of A and let u,* E W’*“(O, T, H,), given by Lemma 3.1 for 9 = q*. We shall prove that (3.6)-(3.9). For this, we have to prove (3.8) and (3.9). (3.15) and (3.25) since i,*(t)

= q*(t) = Arl*(t)

= G(u,&),

E(u,,~O))

a.e. on (0, T). Let us now remark that from (3.14) to (3.16) we have that (uo, oo) is a solution for (3.17)-(3.19) at t = 0; since for all t E [0, T] and r7 E L-(0, T, X) the variational problem (3.17)-(3.19) has a unique solution it follows u,*(O) = uo 3 i.e. (f+, c+) satisfy (3.9).

u,*(O) = r70

1044

M. SOFONEA

The uniqueness part. In order to prove the uniqueness part, let (u,, ,a,,*) be the solution of (3.6)-(3.9) obtained above and let (u, a) be another solution of (3.6)-(3.9) having the regularity u E W”“(0, T,H,), u E I+“*” (0, T, Xi). We denote by q E L-(0, T, X) the function defined by: ~(0 = G(dO,

vt E [0, T]

dM0))

(3.30)

and let z, E W”“(0, T, X) be defined by (3.15), (3.16). Since from (3.6) to (3.9) it results that (u, o) satisfy (3.17)-(3.19) and this problem has a unique solution ur E W’V”(O, T, H,), a,, E W”“(0, T, Xi), it results u = u,, CJ= ov. (3.31) Using now (3.25), (3.31) and (3.30) we get IQ = q and by the uniqueness of the fixed point of A it results: q = q*. (3.32) The uniqueness part of Theorem 3.1 is now a consequence of (3.31) and (3.32). 4. THE

CONTINUOUS

DEPENDENCE

OF

THE

SOLUTION

UPON

THE

INPUT

n DATA

In this section two solutions of the variational problem (3.6)-(3.9) for two different data are considered. An estimation of the difference of these solutions is obtained that give a continuous dependence result with respect to the input data. In particular, a finite-time stability result follows. 4.1. Let (3.12), (3.13) hold and let also (Ui, ai) be the solution of the variational problem (3.6)-(3.9) for the data fi, gi, Uoi, aoi (i = 1,2) such that (3.1), (3.2), (3.14) hold. Then there exists C > 0 which depends on Sz, I,, E, G and T such that

THEOREM

Proof. Let t E [0, T] and i E (1,2). Using (3.6), (3.7), (3.5) and (3.4) we obtain ui(t)

E r/,d 9

(oi(t), E(V) - E(Ui(t)))X L

z

(fitt)9

u

-

ui(t)>H

+

(gi(t),

Yu

-

YUi(t))L2(P2)N

vu

E

u,,

(4.2)

and, from (3.8), (3.9) it results ai

= EE(Ui(f)) + Zi(t)

(4.3)

G(oi(s), E(Ui(S)) do + ZOO

(4.4)

= OOi - G&(UOi)o

(4.5)

where Zi are defined by -1 q(t)

=

I0 ZOi

Contact problem for elastic-viscoplastic bodies

1045

Using now (4.2), (4.3), (3.12) and (2.8), after some algebra it follows l%W - ml,, 5 ahw Moreover,

+ I~,(0 - ao, -.Mfhf

+ I&(0

- &wlL2(ry

since from (4.2) it results Divai

+ fi(t)

+ Iz,w - Z2WlX).

= 0, the previous inequality

becomes

IW) - ~df)lH, + hw - %w13cl 5 C(lA(~) - JdOIH + I&W - mIL2~r~~ + MO - Z*ac).

(4.6)

Using now (4.4), (4.5), (3.12) and (3.13) we obtain

IM) - M)l,

5 Cd% - df,

+ I% - %2lx,) (4.7)

The inequality Remark

(4.1) follows now from (4.6) and (4.7).

4.2. Under the assumptions

of Theorem 4.1, if f, = f2 and g, = g, it results

1%- 4lm,H, + 13 - ~2lmJc,5 C(luo, - ~021Hl+ 1001- ~02lxJ which represents a finite time stability result for every solution of the problem (3.6)-(3.9). Some unidimensional examples can be considered in order to prove that, generally, stability does not hold (see also [14]). 5. A PENALIZED

METHOD

In this section we introduce the penalized problem of the contact problem (2.1)-(2.6) for which we obtain a variational formulation and an existence and uniqueness result. Let (uh , o,,) denote the solution of this problem (h > 0); we also obtain a convergence result of (u,, , cr,,) to the solution (u, a) of (3.6)-(3.9), when h -, 0. So, let us consider ~0: R + R the penalty function given by ifxI0

(0

(5.1) where 0 < p 5 1 and let h > 0. We consider the following b = &E(C) + G(a, E(U))

Diva+f=O

mixed problem:

in Sz x (0, T) in n x (0, T)

(5.2)

(5.3)

u=o

on I, x (0, T)

(5.4)

CTV= g

on I, x (0, T)

(5.5)

1046

M. SOFONEA

a, =

0

if u, I 0,

--u l P h ”

if u, > 0,

U,i

=

(5.6)

in Q

a(0) = uo

u(O) = uo,

on r, x (0, T)

0

(5.7)

in which the unknowns are the displacement function U: 52 x [0, T] + IRN and the stress function 6: !A x [0, T] -+ S,. Let us remark that problem (5.2)-(5.7) is similar to problem (2.1)-(2.6) except the fact that the contact frictionless condition (2.5) on I, x (0, T) was replaced by (5.6); from the mechanical point of view, (5.6) represents a normal displacement condition with friction in which l/h may be interpreted as a normal friction coefficient; the case h = 0 corresponds to an infinite normal friction coefficient, i.e. to the inpenetrability condition between Q and S (formally, (5.6) becomes (2.5) when h + 0). For more details concerning the formulation of frictional contact laws see, for example, [16, 171. In the study of the penalized problem (5.2)-(5.7) we suppose that (3.1), (3.2) hold; for all t E [0, T] and u E HI let Cid(f, V) denote the set given by Cid (t, U) =

(t, e(w) - E(u))x + $ j(w) - k j(u) 1 L(t, w - u) V w E I/ 1

(5.8)

where V and L are defined by (2.7), (3.4) and j: H, + IR is given by

j(u) = We have the following

cp(u,) da. r3

result.

5.1. If the couple of functions (u, a) is a regular solution of the mechanical (5.2)-(5.7) then u(t) E v

THEOREM

a(t)E CLu, WN

problem (5.10) (5.11)

for all t E [0, T], c+(t) = E&i(t)) a.e.

+ G(a(t), c(u(t)))

(5.12)

a(0) = a,.

(5.13)

t E (0, T), 40) = uo,

Proof. The proof of Theorem 5.1 is similar to the proof of Theorem

r

av*(v

- u)dar

g*(v-u)da+i r2 (~(4 da r3

3.1, replacing (3.11) by

COW da r3

vt E [O, T], u E v.

Contact problem for elastic-viscoplastic bodies

1047

The previous theorem allows us to consider (5.10)-(5.13) as a variational formulation of the mechanical problem (5.2)-(5.7). In the study of (5.10)-(5.13) let us consider the following assumption: (5.14) uo E v, 00 E CL (0, uoh As in Section 3, we remark that (5.14) (which represents a regularity condition on the initial data u. , o. as well as a compatibility condition between u o, oo, f and g) is fulfilled if u. E Hi, o. E X and if (uo, ao) satisfy (5.3)-(5.6) for t = 0. THEOREM 5.2. Let (3.1), (3.2), (3.12), (3.13) and (5.14) hold. Then, there exists aunique solution of the problem (5.10)-(5.13) having the regularity u E W’*“(O, T, H,), r~ E W”“(O, T, X,). Proof. The proof of Theorem 5.2 is similar to the proof of Theorem obtained in three steps, as follows: (i) for all q E L-(0, T, X) there exists a unique solution

u/x,,E W’*“(O, T, HI), for the variational

ohg E W’+‘(O,

3.2 and it can be

T, X,)

problem (5.15)

41s(t) E if

(5.16)

~hJf) E C”,d (t, ktlW) fJh,W = w4Jf))

(5.17)

+ z,(t)

for all t E [0, T]; (ii) the operator A,,: L”(0, T, X) + WrP”(O, T, X) defined by Ldf)

= W/JO,

&,,,WN

vq E L”(0, T, X),

has a unique fixed point $ E L”(0, T, 32); (iii) the couple u = uhYh*, 0 = ahVh’ is the unique (5.10)-(5.13). n

solution

t E [O, T]

of the variational

(5.18) problem

The solution of the variational problem (5.10)-(5.13) depends on h > 0; the behaviour of this solution when h + 0 is given by the following theorem. THEOREM 5.3. Let (3.1), (3.2), (3.12)-(3.14)

solution of the problem t E [0, T] we have IW)

(5.10)-(5.13) - WIH,

and (5.14) hold. For all h > 0 let (u,,, oh) be the and let (u, a) be the solution of (3.6)-(3.9). Then, for all

+ 0, la&)

- a(t)lq

+ 0

when h * 0.

(5.19)

In order to prove Theorem 5.3 we need some preliminary results. For this, let us suppose in the sequel that the assumptions of Theorem 5.3 are fulfilled. We start with an estimation of the difference between the solutions of the variational problem (5.15)-(5.17) constructed with two different functions q.

1048

M. SOFONEA

LEMMA 5.1. Let vi E L-(0, T, X), i = 1, 2 and let h > 0. Then, denoting by (Ui, ai) the solution of (5.15)-(5.17) for 17 = rli, there exists C > 0 such that

IUlW- UdOlH, + IN) - adthc,5 c f 1171(s) - ?&c d.s I

vi E [0, T].

(5.20)

0

Proof. Let t E [0, T] and let Z,i be defined by (3.15), (3.16) for 9 = vi, i = 1,2. Using (5.15)-(5.17) and (5.8) we obtain

(GE(Ui(f)),

dU)

2

L(t,

U

-

-

E(Ui(f)))X

+

(Zi(t),

E(U)

E(Ui(f)))$C

+

k.j(U)

-

ij(Ui(t))

1,2.

Vuc~,i=

Ui(t))

-

(5.21)

Using now (5.21), (3.12) and (2.8) we obtain IUlW - U~(f)lfq 5 Clz,(t)

(5.22)

- Z*(f)lm

Using again (5.17) and (5.21) it results Div o,(t) = Diva2(t) = -f(t) (5.22), it follows Ial(t) - a,(t)l,, 5 Clz,(t) - z&c.

hence, from (3.12) and (5.23)

5.2. Let q E L”(0, T, X) and let z, E W’v”(O, T, X) be the function defined by (3.15), (3.16). Let also (u,, , cr,,) be the solution of the problem (3.17)-(3.19) and, for all h > 0, let of (5.15)-(5.17). Then, for all t E [0, T] we have @hq 3 ohha)be the solution LEMMA

bh,#)

Proof.

-

U,,(f)iN,

-+

bh,(t)

Let t E [0, T]. Using (5.15)-(5.17) Uh,#)

-

~,(t)lJC,

-+

when h -+ 0.

0

(5.24)

and (5.8) we obtain

E v,

(&&&)),

c(U)

+

Taking

O,

; j(u)

-

-

‘$U,Q(t)))~

+

; j@,,,(t))

1

L(t,

k,(t),

E(u)

u -

-

&&)))X

VUEV.

(5.25)

&@h,#))kC

(5.26)

Uh,#))

u = 0 in (5.25) and using (3.12) and (2.8) we have cluh,(t)l&

+

;

.i(&&))

5

L(t,

Uh,&f))

+

k,(t),

and, after some algebra, we obtain that (uhn(t))h is a bounded sequence in Hr. Therefore there exists an element u’,(t) E H, and a subsequence (uh’,,(t))h’ C (uhv(t))h such that Uhf,,@)

- ii,(t)

in H,,

when h’ + 0.

(5.27)

1049

Contact problem for elastic-viscoplastic bodies

Moreover, since (z+J~))~, is a bounded sequence in H,, from (5.26) we obtain that there exists C > 0 such that j(u,,,(t)) I Ch’ vh’ > 0. (5.28) Using now the lower semicontinuity (5.9), (5.1) and (3.3) it follows

ofj, from (5.27) and (5.28) it resultsj(u’,(t))

= 0 and by (5.29)

~,V) E uaci. Using again (5.25), (5.27) and standard lower semicontinuity ~wqt)),

E(V) - E(fi,(W)X

1 L(t, u - c,(t))

arguments we obtain

+
vu E Uad.

From (5.29) and (5.30) we find that i?,(t) is a solution of (3.20) and from the uniqueness of the solution for this variational inequality we obtain c’,(t) = u,(t). Since u,(t) is the unique weak limit of any subsequence of (~,,~(f))~, we deduce that the whole sequence (~~~(0)~ is weakly convergent in H, to u,(t): 4lJf)

when h + 0.

- u,(t)

(5.31)

In order to obtain the strong convergence let us remark that from (3.12) and (2.8) it follows cI%#

- ~,Wl~,

5 (w%#)),

&4n#))

- (W~JO), and, putting

a4lJfN

Ebl~UN

- E@,Whc

5 ut, h&) + cqt),

Proof and 5.2 operator q = q*,

(5.32)

- GAfhc

u = u,(t) in (5.25), we obtain (Wb&)),

Lemma

- ~(~,Whc

- qt)) ~(~,(O) - 4&rJfh

*

(5.33)

5.2 follows now from (5.32), (5.33) and (5.31). of Theorem 5.3. Let h > 0 and t E [0, T]; as it results from the proof of Theorems 3.2 we have u = z+, CJ= g,,+, u,, = u,,,,;, a,, = a,,,,; where f~* is the fixed point of the A,, defined by (5.18). So, denoting by (uh9*, oh+) the solution of (5.15)-(5.17) for it results

IMf) - WIH, + l%(t) - ~Wlx, 5 l&,;(t) - 4q.(fhz, + h,;(t) - ~ht1401X, + I%pw - ~,*Wlff, + IGpw - ~,*Wlx, * Using now Lemma

(5.34)

5.1 we obtain

b+;(f) - %Iq*wlH1+ I%&) - %J*wlE, 5 c r bl,*<@- tl*wlx~ c0 and, since $ = A,,?$ = G(ah, E(u~)), q* = Aq* = G(o, E(U)), by (3.13) it results

(5.35)

1050

M. SOFONEA

Let us now consider E > 0; using Lemma 5.2 we obtain that there exists 6 > 0 which depends on e, t and q* such that for all 0 < h < 6 we have Ihs*(f)

- ~e*uhf,

+ Iah,*w - ~rl*wlxl

5 E*

(5.36)

So, if 0 < h < 6, from (5.34) to (5.36) it results b/l(f) - wlff, + MO t SC hw (.r 0 Using now a Gronwall-type inequality,

- 4x,

ml,,

+ b&) - WIHJ ds + E. >

(5.37)

(5.37) implies

IW) - WIH, + hw - mix, 5 cc for all h such that 0 < h < S, which proves (5.19).

n

Remark 5. I. The mechanical interpretation of the previous convergence result is the following: at each time moment t E [0, T], the solution (u(t), a(t)) of the rigid contact frictionless problem (2.1)-(2.6) may be obtained as the limit of the solution (u,,(f), a,,(t)) of the friction contact problem (5.2)-(5.7) when the normal friction coefficient l/h tends to infinity. REFERENCES 1. Fichera, G., Boundary value problem of elasticity with unilateral constraints. In Encyclopedia ofPhysics, Vol. VI a/2, ed. S. Flugge. Springer-Verlag, Berlin, 1972. 2. Haslinger, J. and HlavaEek, I., Contact between elastic bodies-I. Continuous problem. Applik. Math., 1980, 25, 324347. 3. Haslinger, J. and HlavaEek, I., Contact between elastic perfectly plastic bodies. Applik. Math., 1982, 27, 27-45. 4. HlavaEek, I. and NeEas, J., Mathematical Theory of Elastic and Elastoplar;tic Bodies: An Introduction. Elsevier, Amsterdam, 1981. 5. HlavaEek, I. and NeEas, J., Solution of Signorini’s contact problem in the deformation theory of plasticity by secant modules method. Applik. Mat., 1983, 28, 199-214. 6. Burguera, M., Analisis numeric0 de una clase de problemas de contact0 en plasticidad perfecta. Thesis, University of Santiago de Compostela, 1991. 7. JaruSek, J., Solvability of the variational inequality for a drum with a memory vibrating in the presence of an obstacle. Bollettino U.M.I., 1994, 8-A, 113-122. 8. JaruSek, J., Dynamic contact problems for viscoelastic bodies. Proceedings of the 2nd Contact Mechanics International Symposium, Carry-Le-Rouet, ed. M. Raous, M. Jean and J. J. Moreau. Plenum Press, New York, 1995. 9. Cristescu, N. and Suliciu, I., Viscoplasticity. Martius Nijhoff, Editura Tehnica, Bucarest, 1982. 10. Duvaut, G. and Lions, J. L., Les incfquations en m&unique et en physique. Dunod, Paris, 1972. 11. Suquet, P., Evolution problems for a class of dissipative materials. Quart. Appl. Maths., 1981, 391-414. 12. Suquet, P., Sur les equations de la plasticite: existence et rtgularitt des solutions. Journal de MPcanique, 1981, 20(l), 3-39. 13. Ionescu, I. R. and Sofonea, M., Quasistatic processes for elastic-viscoplastic materials. Quart. Appl. Maths., 1988, 2, 229-243. 14. Ionescu, I. R. and Sofonea, M., Functional and Numerical Methods in Viscoplasticity. Oxford University Press, 1993. 15. Djabi, S. and Sofonea, M., A fixed point method in quasistatic rate-type viscoplasticity. Appl. Math. and Comp. Sci., 1993, 3(2), 269-279. 16. Jean, M. and Moreau, J. J., Uniterality and dry friction in the dynamics of rigid body collections. In Proceedings of Contact Mechanics international Symposium, ed. A. Curnier. Presses Polytechniques et Universitaires Romandes, Lausanne, 1992, pp. 31-48. 17. Jean, M., Frictional contact in collections of rigid of deformable bodies: numerical simulation of geometrical motions. In Mechanics of Geomaterial Interfaces, ed. A. P. S. Salvadurai. Elsevier Science, 1994.