A PROBLEM WITH WEAR INVOLVING THERMO-ELECTRO-VISCOELASTIC MATERIALS Текст научной статьи по специальности «Физика»

DOI: 10.17516/1997-1397-2022-15-2-241-254 УДК 517.9

A Problem with Wear Involving Thermo-electro-viscoelastic Materials

Aziza Bachmar* Djamel Ouchenane

Faculty of Sciences Ferhat Abbas University of Setif-1 Setif, Algeria

Received 12.09.2021, received in revised form 08.12.2021, accepted 20.01.2022

Abstract. In this paper, we consider a mathematical model of a contact problem in thermo-electro-viscoelasticity. The body is in contact with an obstacle. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. We establish a variational formulation for the model and we prove the existence of a unique weak solution to the problem. The proof is based on a classical existence and uniqueness result on parabolic inequalities, differential equations and fixed point arguments. We present a variational formulation of the problem, and we prove the existence and uniqueness of the weak solution.

Keywords: piezoelectric, temperature, thermo-electro-viscoelastic, variational inequality, wear.

Citation: A. Bachmar, D. Ouchenane, A Problem with Wear Involving Thermo-electro-viscoelastic Materials, J. Sib. Fed. Univ. Math. Phys., 2022, 15(2), 241-254. DOI: 10.17516/1997-1397-2022-15-2-241-254

1. Introduction

In the recent years, piezoelectric contact problems have been of great interest to modern engineering. General models of electroelastic characteristics of piezoelectric materials can be found in [2, 7]. The problems of piezo- viscoelastic materials have been studied with different contact conditions within linearized elasticity in [1,4] and with in nonlinear viscoelasticity in [9]. The modeling of these problems does not take into account the thermic effect. Mindlin [8] was the first to propose the thermo- piezoelectric model. The mathematical model which describes the frictional contact between a thermo-piezo- electric body and a conductive foundation is already addressed in the static case in [3]. Sofonea et al. considered in [6] the modeling of quasistatic viscoelastic problem with normal compliance friction and damage, they proved the existence and uniqueness of the weak solution,and they derived error estimates on the approximate solutions. In this paper,we consider a dynamic contact problem between a thermo-electro viscoelastic body and an electrically and thermally conductive rigid foundation which results in the wear of the contacting surface.

2. Problem statement

Problem P: Find a displacement field u : Q x [0.T] ^ rd, a stress field a : Q x [0.T] ^ Sd, the an electric potentiel field p : Q x [0.T] ^ r, the an electric displacement field D : Q x [0.T] ^ rd,

* Aziza_bechmar@yahoo.fr © Siberian Federal University. All rights reserved

a temperature field 0 : Q x [0.T] ^ r, and the wear w : r3 x [0.T] ^ r+ such that

a = A(e(u(t))) + G(e(u(t))) - £*E (y) - 0M in Q x [0.T], (2.1)

D = /3E (y) + £e (u) - (0 - 0*)p in Q x [0.T], (2.2)

pu = Div a + /o in Q x [0.T], (2.3)

div D = q0 in Q x [0.T], (2.4)

0 - div(KV0) = -M.Vu + qi in Q x [0.T], (2.5)

u = 0 on r1 x [0.T], (2.6)

av = h on r2 x [0.T], (2.7)

(2.8)

-kijirVj = ke(0 - 0r) - hT(|UT|) on r3 x [0.T], (2.11)

I av = -a |Uv | , laT| = ,

y.aT = -A (uT - v*T , A > 0, W = -kv*av, k> 0 on r3 x [0.T], y = 0 on ro x [0.T], (2.9)

Dv = q2 on rb x [0.T], (2.10)

d0 ' dv

0 = 0 in r1 U r2 x [0.T], (2.12)

u(0) = u0, v(0) = v0, 0(0) = 00, w(0) = w0 in Q. (2.13)

Where (2.1), (2.2) are represent the thermo- electro-viscoelastic constitutive law of the material in which a = (aj) denotes the stress tensor, we denote e(u) (respectively; E(y) = -Vy, A, G, £*,3, M = (mij), p = (pi) ) the linearized strain tensor (respectively; electric field, the elasticity tensor, the viscosity nonlinear tensor, the third order piezoelectric tensor and its transpose, the electric permittivity tensor thermal expansion, pyroelectric tensor),the constant 0*represents the reference temperature; (2.3) is represents the equation of motion where p represents the mass density; (2.4) is represents the equilibrium equation, we mention that Diva, divD are the divergence operators; (2.5) is represents the evolution equation of the heat field; (2.6) and (2.7) are are the displacement and traction boundary conditions; (2.8) is describes the frictional bilateral contact with wear described above on the potential contact surface r3; (2.9), (2.10) are represent the electric boundary conditions; (2.11) is pointwise heat exchange condition on the contact surface, where kij are the components of the thermal conductivity tensor, vj are the normal components of the outward unit normal v; ke is the heat exchange coefficient, 0R is the known temperature of the foundation; (2.12) represents the temperature boundary conditions. Finally, (2.13) is the initial data.

3. Variational formulation and preliminaries

For a weak formulation of the problem, first we itroduce some notation. The indices i, j, k, l range from 1 to d and summation over repeated indices is implied. An index that follows a comma represents the partial derivative with respect to the corresponding component of the

dui

spatial variable, e.g: ui j = ——. We also use the following notations

dxj

H = L2(Q)d = {u = (ui)/ui e L2(Q)}, H = {a = (aij)/aij = aji e L2(Q)} , H1 = {u = (ui)/e(u) e H} = H1 (q)d, H1 = {a e H/Diva e H} . The operators of deformation e and divergence Div are defined by

e(u) = (£ij (u)), £ij (u) = 2(ui,j + uj,i), Diva = (aij,j).

The spaces H, H, H1 and H1 are real Hilbert spaces endowed with the canonical inner products given by

(u,v)h = / uividx V u,v G H, JQ

(a,T)h = / aijTij dx V a,T G H, JQ

(u,v)h1 = (u,v)h + (e(u),£(v))h Vu,v G Hi,

(a, t)hi = (a, t)h + (Diva, DivT)h, a,T G H\.

We denote by HH (respectively;! • \h, | • \Hl and | • \hi) the associated norm on the space H (respectively; H, Hi and Hi).

Let Hr = (Hi/2(r))d and y : Hi(r)d ^ Hr be the trace map. For every element v G (Hi(r))d, we also use the notation v to denote the trace map jv of v on r, and we denote by vv and vt the normal and tangential components of v on r given by

vv = v.v, vt = v — Vv v.

Similarly, for a regular (say Ci) tensor field a : Q ^ sd we define its normal and tangential components by

av = (av) .v, aT = av — avv.

We use standard notation for the lp and the Sobolev spaces associated with Q and r and, for a function ^ G H1 (Q) we still write ^ to denote it trace on r. We recall that the summation convention applies to a repeated index.

For the electric displacement field we use two Hilbert spaces

W = l2 (Q)d, W1 = {D G W, divD G l2 (Q)} .

Endowed with the inner products

(d,e)w = I DiEidx, (D,E)Wl =(D,E)W + (divD, divE)^ .

J Q

And the associated norm \.\W(respectively; \.\Wi). The electric potential field is to be found in { }

W = -¡> G H1 (Q) = 0 on ro} . Since meas (ra) > 0, the following Friedrichs-Poincare's inequality holds, thus

> CF |^|hi(Q) V^ G W, (3.1)

where cF > 0 is a constant which depends only on Q and ra. On W, we use the inner product given by

(p,^)w = (Уp, ,

and let \.\W be the associated norm. It follows from (3.1) that |.|Hi(Q) and \.\W are equivalent norms on W and therefore (W, \.\W) is a real Hilbert space.

Moreover, by the Sobolev trace Theorem, there exists a constant c0, depending only on Q, ra and r3 such that

Ml2(F3) < HW V^ G W. (3.2)

We define the space

E = {7 G H 1(Q)/y = 0 on r1 U r2}. (3.3)

We recall that when D e W1 is a sufficiently regular function,the Green's type formula holds (D, VV>)W + (div D, = j Dv.^da . (3.4)

When a is a regular function, the following Green's type formula holds

(a, e (v))h + (Diva,v)H = j av.vda Vv e H1. (3.5)

Next, we define the space

V = {u e H1/ u = 0 on r1}. Since meas (r1) > 0, the following Korn's inequality holds

|e(u)|H > ck IvIhx Vv e V, (3.6)

where ck > 0 is a constant which depends only on Q and r1. On the space V we use the inner product

(u, v)v = (e(u), e(v))H, (3.7)

let |.|v be the associated norm. It follows by (3.6) that the norms |.|Hi and |.|v are equivalent norms on V and therefore, (V, |.|v) is a real Hilbert space. Moreover, by the Sobolev trace Theorem, there exists a constant c0 depending only on the domain Q, r1 and r3 such that

|v|L2(ra)d < C0 lvlV Vv e V (3.8)

Finally, for a real Banach space (X, |.|X) we use the usual notation for the space l (0.T, X) and Wk-p (0.T, X), where 1 < p < w, k =1,2,...; we also denote by C (0.T, X) and C1 (0.T, X) the spaces of continuous and continuously differentiable function on [0.T] with values in X, with the respective norms:

IxIc(0.t,x) = max lx (t)X , V ' t£[0.T] X

|x|ci(0TX) = max lx (t)|X + max lx (t)|„ .

v ' ' ' te[0.T] X te[0.T]' X

In what follows,we assume the following assumptions on the problem P. The elasticity operator A : Q x sd ^ sd

(a) 3 La > 0 such that : IA (x,e1) - A (x,e2)| < LA le1 - e2l V e1,e2 e Sd, a. e. x e Q,

(c) the mapping x ^ A (x,e) is lebesgue measurable in Q for all e e Sd, _ (d) the mapping x ^ A (x, 0) e H.

The viscosity operator G : Q x sd xsd ^ sd satisfies

'(a) 3 Lg > 0 : IG (x,e1) - G (x,e2)| < LG le1 - e1l, Ve1,e2 e Sd, p.p. x e Q,

(b) 3mG > 0 : (G (x,e1) - G (x,e2) ,e1 - e2) > m^ |e1 - e2|2 , Ve1,e2 e Sd,

(c) the mapping x ^ G (x, e) is lebesgue measurable in Q fo rall e e Sd, _ (d) the mapping x ^ G(x, 0) e H.

(3.9)

(3.10)

The thermal expansion tensor M = (mij) : Q x r ^ r, and the pyroelectric tensor p = (pi) : Q ^ rd satisfy,

J (a) mij = mji e l~(Q),

1(b) Pi e l~(Q).

(3.11)

The piezoelectric tensor £ = (eijk) : Q x Sd ^ Rd satisfies '(a) £ = (eijk) : Q x Sd ^ rd,

(b) £ (x,T) = (eijk (x) Tjk) Vt = (t^) G Sd, a.e.x G Q, (3.12)

(c) eijk = eikj G l~ (Q).

The electric permittivity tensor / = (/ij) : Q x rd ^ rd

'(a) / = (/ij) : Q x rd ^ rd,

(b) 3 (x, E) = (bij (x) Ej) VE = (Ei) G rd, a.e. x G Q

(c) bij = bji G (Q),

il Cllpn ■frto-f • 1,- - , , , ,.,. ,.,.

bij (x) EiEj

The function hT : r3 x r+ —> r+ satisfies

(3.13)

(d) 3 mp > 0 such that : bij (x) EiEj > mp \E|2 VE = (Ei) G rd, x G Q.

(a) 3 Lt > 0: h (x,ri) - hT (x,r2)\ < Lh\ri - r2\V r1,V2 G r+, p.p.x G r3,

(b) x —> hT (x, r) G l2(r3) is lebesgue measurable in r3 Vr G r+ .

(3.14)

The mass density p satisfie

p G lTO(Q) there exists p* > 0 such that p(x) > p*, a.e.x G Q. (3.15)

The body forces, surface tractions,the densities of electric charges, and the functions a and j, satisfy

'fo G l2(0.T, H), h G l2(0.T, l2(r2)d),

qo G L2(0.T, l2(Q)), q2 G L2(0.T, l2(r6)), qx G L2(0.T, l2(Q)), ke G lœ(Q, r+).

{K = (ki,j); kij = kji G l^(Q), (3 16)

Vck > 0, V(&) G rd, kij> ck a G lœ(r3), a(x) > a* > 0, a.e. on r3, j G lœ(r3), j(x) > 0, a.e. on r3.

The initial data satisfy

uo G V,0o G l2(Q), wo G l~(r3). (3.17)

We use a modified inner product on H = l2(Q)d given by

((u,v)) = (Pu,v)L2(Q)d Vu,V G H. That is, it is weighted with p. We let H be the associated norm

i

Mh = (PV,v)L2(Q)d VV G H. We use the notation (., .)V'xV to represent the duality pairing between V' and V. Then, we have

(u, v)v' xV = ((u, v)) Vu G HVv G V.

It follows from assumption (3.15) that||.||H and |.|H are equivalent norms on H, and also the inclusion mapping of (V, |.|V) into (H, ||.||H) is continuous and dense. We denote by V' the dual space of V. Identifying H with its own dual, we can write the Gelfand triple V c H = H' c V'.

We define the function f (t) e V and q : [0.T] ^ W by

(f (t),v)V = fo(t)vdx + / h(t)vdaVv e V, t e [0.T]. JQ JT2

(q(t),^)w = - qo(t)^dx + q2(t)^daV^ e W, t e [0.T].

JQ JTb

for all u, v e V, ^ e W and t e [0.T], and note that condition (3.14) imply that

f e l2(0.t,v'), q e l2(0.t,w), (3.18) We consider the wear functional j : V x V ^ r,

j(u,v) = / a \uv\ (p \vT — v*\)da (3.19) Jr3

Finally, We consider $ : V x V ^ r,

$(u,v) = a \uv\ vvda Vv e V. (3.20)

Jr3

We define for all e > 0

js(g,v) = a \gv\ (py/\vt — v*\2 + e2)daVv e V. Jrs

We define Q : [0,T] ^ E'; K : E ^ E' and R : V ^ E' by

(Q(t),p)E' xE = / ke6u(t)pds + qpdx Vp e E, (3.21)

Jr3 J Q

d f' dr dp f'

(Kt,p)e' xe = ^ kij -—-—dx + kerpds Vp e E, (3.22)

ij=1 Q j i r3

(Rv, p)E' xE = hT (\vT\)pdx — (M .Vv)pdx Vv e V,r,p e E. (3.23) Jr-3 Jq

Using the above notation and Green's formula, we derive the following variational formulation of mechanical problem P.

Problem PV : Find a displacement field u : q x [0.T] ^ V, a stress field a : q x [0.T] ^ Sd, the an electric potentiel field y : q x [0.T] ^ r, the an electric displacement field D : q x [0.T]rd, a temperature field 6: q x [0.T] ^ r, and the wear w : r3 x [0.T] ^ r+ such that

(u(t), w — u(t))v'xV + (a(t), e(w —u(t)))H +j(u, w) —j(u, u(t)) + $(u, w) — $(u, u(t)) ^

(3.24)

> (f (t),w — u(t)) Vu,w e V,

(D(t), V^)L2(Q)d + (q(t), ^)w = 0 V^ e W, (3.25)

6(t) + K6(t) = Ru(t) + Q(t) on E', (3.26)

w = —ku*av. (3.27)

4. Existence and uniqueness result

Our main result which states the unique solvability of Problem are the following. Theorem 4.1. Let the assumptions (3.9) — (3.17) hold. Then, Problem PV has a unique solution

(u,a,p,D,w) which satisfies

u G C1(0.T, H) n WL2(0.T, V) n W2-2(0.T, V'), (4.1)

a G l2(0.T, H1), Diva G l2(0.T,V'), (4.2)

p G W 1-2(0.T,W), (4.3)

D G W1'2(0.T, W1), (4.4)

e G W 1'2(0.T,E') n l2(0.T,E) n C(0.T, l2(Q)), (4.5)

w G C 1(0.T,l2(r3)). (4.6)

We conclude that under the assumptions (3.9)-(3.17), the mechanical problem (2.1)-(2.13) has a unique weak solution with the regularity (4.1)-(4.6).

The proof of this theorem will be carried out in several steps. It is based on arguments of first order evolution nonlinear inequalities, evolution equations, a parabolic variational inequality, and fixed point arguments.

First step: Let g G l2 (0.T; V) and n G l2(0.T; V') are given, we deduce a variational formulation of Problem PV.

Problem PVgn : Find a displacement field ugn : [0.T] ^ V such that

(ugV (t),w — u gn (t)) V 'X V + £(u gn (t)),W — u gn (t)) V 'x V + ^^ , W — ij, g^ ) V 'X V j w) —

—j(g, ugn(t)) > (f (t), w — ugn(t))V'XV V w G V,t G [0.T], (4.7)

ugn (0) = uo, ugn(0) = u1. We define fri(t) G V for a.e.t G [0.T] by

(fn(t),w)V'xV = (f (t) — n(t),w)V' XXV Vw G V. (4.8)

From (3.18), we deduce that

fn G l2(0.T, V'). (4.9)

Let now ugn : [0.T] ^ V be the function defined by

ugn(t)= / Vgn(s)ds + uo Vt G [0.T]. (4.10)

o

We define the operator G : V' ^ V by

(Gv,w)V' xV = (&£(v(t)),£(w))H V V,w G V. (4.11)

Lemma 4.2. For all g G l2(0.T, V) and n G L2(0.T,V'), PVgn has a unique solution with the regularity

vgn G C(0.T, H) n l2(0.T, V) and Vgn G l2(0.T, V'). (4.12)

Proof. The proof from nonlinear first order evolution inequalities, given in Refs (see [5,10]). □

In the second step, we use the displacement field ugn to consider the following variational problem.

Second step: We use the displacement field ugn to consider the following variational problem. Problem Pegn : Find Qgn € E such that

Qgn (t) + K0gV (t) = RUgn (t) + Q(t) on E . (4.13)

Lemma 4.3. Under the assumptions (3.9)-(3.17), the problem Pegn has a unique solution

Qgn € w 1-2(0.t,e') n l2(0.t,e) n c(o.t,l2(q)).

Proof. Since we have the Gelfand triple E c l2(Q) c E . We use a classical result on first order evolution equations given in [11] to prove the unique solvability of (4.13). Now, we have Qo € l2(Q). The operator K is a linear and continuous, so a(r,p) = (Kr,p)E'XE is bilinear, continuous and coercive, we use the continuity of a(.,.) and from (3.16), we deduce that

a(T,p) = (kt,p)e'xe < |k|l~(fi)dxd \vt\e \vp\e + \ke\l~ (r3) \T ll2(r3) hl2(r3) < C \T \E he .

We have

d f dr dr f

a{r,T ) = (Kr,r )E' xE = ^ kij dX.dXdx + ker2ds.

ij=1 ^ j i r3

By (3.16) there exists a constants C > 0 such that

(KT,T)e' xe > C \TIE .

We have Qo € l2(Q). Let

F(t) € E' : (F(t),T)e'xe = Rgn(t) + Q(t),T) Vt € E.

Under the assumptions(3.14), (3.16) we have

, t , T , T

/ \RU\E' dt < / \Q(t)\E' dt < / |F\E' dt < x>.

Jo Jo Jo

We find

F € L2(0.T, E'). By a classical result on first order evolution equations

3 !Qgn € W 1-2(0.T,E') n L2(0.T,E) n C(0.T,l2(q)).

□

Third step: We use the displacement field ugn and the temperature field 6gv to consider the following variational problem.

Problem PVgfv : Find an electric potential field tpgn : q x [0.T] ^ W such that

(ßV^gV (t), - (&(Ugv (t)), V^L2(Q)d - ((9gv (t) - 9*gv (t))pi, V^ L2 (n)d = ( 14)

= (q(t)^)w ^ e W,t e [0.T]. .

We have the following result for PVgn

Lemma 4.4. There exists a unique solution pgn G W L2(0.T, W) satisfies (4.14), moreover if Piand p2 are two solutions to (4.14). Then, there exists a constants c> 0 sach that

|P1(t) — P2(t)|w < c (Ju1(t) — u2(t)|V + |e1(t) — e2(t)|L2(n^ Vt G [0.T]. (4.15)

Proof. Let t G [0.T], we use the Riesz-frechet representation theorem to define the operator Agn : W ^ W by

(Agn (t)p,^)w = (t), V^)L2(Q)d - (&(UgV (t)), V^)L2(Q)d -

- ((0gn(t) - V*gv(t))pi, V^)L2(Q)d V t G [0.T] .

(4.16)

For all p, M G W. Let p1, p2 G W, then assumptions (3.11)—(3.13) imply

(Agn(t)p1 — Agn(t)p1 ,p1 — p2)w > mp |p1 — p2 |W . (4.17)

In other hand, from (3.11)—(3.13), it results

(Agn (t)p1 — Agn (t)p1 ,'ip)w ^ Cp — P2 |w M tw,

where cp is a positive constant which depends on ¡3. Thus

Agn(t)P1 — Agn(t)P2|w ^ CP |P1 — P2|w . (4.18)

Inequalities (4.17) and (4.18) show that the operator Agn(t) is a strongly monotone, Lipschits continuous operator on W and, therefore, there exists a unique element Pgn (t) G W such that

Agn Pgn (t) = q (t). (4.19)

We combine (4.16) and (4.17) and find that Pgn (t) G W is the unique solution of the nonlinear variational equation (4.14).

We show that Pgn G W 12(0.T, W). To this end, let t1,t2 G [0.T] and, for the sake of simplicity, we write Pgn (ti) = Pi, ugn (ti) = ui, egn(ti) = e», q (ti) = qi, for i = 1.2. From (4.14), (3.11)-(3.13) it results

mP |P1 — P2|w ^

w (4.20)

< c(|u1 — u2|v |P1 — P2|w + |e1 — e2 |L2(Q) |P1 — P2|w + |q1 — q2|w |P1 — P2|w) We find

|P1 (t) — P2 (t)|w <

/ \ (4.21)

< C (Ju1 (t) — u2 (t)|v + |e1 (t) — e2 (t)|L2(fi) + |q1 (t) — q2 (t)|^ Vt G [0.T] .

We also note that assumption (3.16), combined with definition imply that q G W12 (0.T, W). Since ugn G C1 (0.T, V), egn G C1 (0.T, E), inequality (4.21) implies that Pgn G W12 (0.T, W).

Let: n1 ,n2 G C (0.T,V'), g1,g2 G C (0.T,V) and let Pgn (ti) = Pi, ugn (ti) = uh we use (4.20) and arguments similar to those used in the proof of (4.15) to obtain

mP P — P2w < c(|u1 — u2|V + |e1 — e2|L2(fi)). For all t G [0.T]. This inequality leads to (4.15) which concludes the proof. □

Consider the operator

A : L2(0.T,V x V') ^ l2 (0.T, V x V')

a(g,n) = (ai(g), a2W), Vg G L2(0.T, V), VV G L2(0.T,V'),

ai(g) = vav,

(4 22)

(a2(V),w)v'Xv = (A(e(u(t))) - dM - CE(v),e(w))h + $(g,w), ;

\A(g2,m)-A(gi,V2)\l2(o .T ;VxV')

= \(ai(g2), a2(m))-(ai(gi), a2(m))\l2{o .T ;VxV') = = \Ai(g2) - Ai(gi)\l2(0.T ;V XV') + \A2(V2) - A2 (Vi)\l2(0.T ;V XV') .

We have the following result.

Lemma 4.5. The mapping a : L2(0.T, VxV') ^ L2(0.T,VxV') has a unique element (g*,n*) G L2(0.T,V x V'), such that

a(g*,n*) = (g*,V*). (4.23)

Proof. Let (gji/qi) G L2(0.T,V x V'). We use the notation (ui,^i). For (g,n) = (gi,ni), i = 1.2. Let t G [0.T]. We have

ai(g) = van. (4.24)

So

\gi(t) - g2(t)\V < \vi(t) - V2(t)\V . (4.25)

It follows that

(vVi(t) - V>2(t), vi(t) - V2(t)) + (Ge(vi(t)) - &e(v2(t)), e(vi(t)) - e(v2(t)))+

(4.26)

+ (ni(t)-m(t),vi(t)-v2(t))+ j(gi,vi(t)) - j(gi,v2(t))- j(g2,vi(t))+ j(g2,v2(t)) < 0 From the definition of the functional j given by (3.19), and using (3.8), (3.16) we have

j(g2,v2(t)) - j(g2,vi(t)) - j(gi,v2(t))+ j(gi,vi(t)) < C \gi - g2\V \vi - v2\v (4.27)

Integrating the (4.26) inequality with respect to time, using the initial conditions v2(0) = = vi(0) = v0, using (3.8), (3.10), (4.26) using Cauchy-Schwartz's inequality and the inequality

C mm 1 2ab ^ -a2 +—— b2 et 2ab ^ -a2 + m& b2, by Gronwall's inequality we find

m& C m&

\vi(t) - v2(t)\V < C^ Jo \gi(s) - g2(s)\V ds + ^ \ni(s) - m(s)\V' (4.28)

\gi - g2\V < C^I \gi(s) - g2(s)\V ds + Jo \m(s) - m(s)\V' ds). (4.29)

So

And, we have

(A2(n),w)vxV = (a(e(u(t))) - dM - eE(v),e(w)) + $(g,w). (4.30)

V / h

From the definition of the functional $ given by (3.20), and using (3.8), (3.16) we have

4>(:gi,v2(t)) - $(gi,vi(t)) - $(g2,v2(t)) + $(g2,vi(t)) < C\gi - g2\v \vi - v2\v . (4.31) So

\ni(t) - V2(t)\V' < C (\ui(t) - u2 (t) \V + /0 \ui(s) - u2(s)\V ds+

+ \vi(t) - M(t)\Wv + \°i(t) - °2(t)\l2{n) + \gi(t) - g2(t)\V).

(4.32)

By (4.26), using the inequality 2ab ^ -a2 +--Gb2 and 2ab ^ -a2 +--Gb2, we find

mg 2C m@ 2

1 2 r 2 i r 2

- |vi(t) - V2(t)\V + mW |vi(s) - V2(s)\V ds < - |ni(s) - %(s)|V' ds+

2 V Jo V m< Jo V ft ft

+ m< |vi(s) - V2(s)|V ds + C x - |gi(s) - g2(s)|V ds+ (433)

4 Jo m< Jo

+ C x 4— .0 |vi(s) - V2(s)|V ds-

So

J |vi(s) - V2(s)|V ds < c( J |ni(s) - n2(s)|V' ds + J |gi(s) - g2(s)|V d^j- (4.34) By (3.26), we find

fa (t) - è2 (i), et (i) - e2 (i)) + (Kè ) - kè ), et (t) - e2 (t)E xB =

V /Ex E

= (rv ) - R(v2 ),è, (t) - è2 (t))

(4.35)

2\°)je-xE ■

We integrate (4.35) over [0.T] we use the initial conditions 0\ (0) = 02 (0) = 90, and we use the coercivety of K and the Lipschitz continuity of R to deduce that

1 |èi (t) - è2 (t)^) + C J |èi (s) - è2 (s)^) ds ^ < — (I |vi (s) - V2 (s)|v |èi (s) - è2 (s)|L2(n) ds) .

using the inequality 2ab < 2 a2 + 2b2, we find

1 |èi (t) - è2 (t)^) + — J |èi (s) - è2 (s)^) ds < C ft

< 4 Jo Vi (s) - V2 (s)|v ds + C |èi (s) - è2 (s)|L2(Q) ds-

Also

|èi (t) - è2 (t)|L2(n) < c J |vi (s) - V2 (s)|V ds. (4.36)

By (4.34) , we find

|èi (t) - è2 (t)^) < — ( fQ |ni (s) - m (s)|V' ds + £ g (s) - g2 (s)|V ds) . (4.37)

Also

And

2 t 2

Ui(t) - U2(t)|V + / |ui(s) - U2(s)|V ds < jo

< C^ J |vi(s) - V2(s)|V + J |ui(s) - U2(s)|V^ ds. |ui(t) - U2(t)|V > 0.

n|ui(r) - «2(^)|V drds > 0.

(4.38)

So

\ui — U2 \v + \ui — uify ds < J0

< C / (\vi(s) — V2(s)\V + \ui(s) — U2(s)\V)ds + / / \ui — U2 \V drds,

Jo J0J0

\ui — U2 \V + \ui — U2 \V ds < Jo

f * f S

< C / (\vi(s) — V2(s)\V + \ui(s) — U2(s)\V + / \ui(r) — U2(r)\V dr)ds

00

by Gronwall's inequality, and using (4.34) we have

\ui — u2\y + J \ui — u2\V ds < C^ J \ni(s) — m(s)\V' ds + J \gi(s) — 92(s)\y ds) (4.39) and using (4.29) and (4.32) we find

f *

\a(gi,ni) — A(92,V2)\l2(0.T;VxY') < C jo \(9i,Vi) — (92,V2)\2y XV ' ds■ (4.40)

Thus, for m sufficiently large, am is a contraction on l2(0.T, Vx V') and so a has a unique fixed point in this Banach space. □

We consider the operator L : C(0.T, l2(^)) ^ C(0.T, l2(rs)),

Lw(t) = —kv*[ av (s)ds yt G [0.T]■ (4.41)

0

Lemma 4.6. The operator L : C(0.T, l2(^)) ^ C(0.T, l2(^)) has a unique element w* G C(0.T, l2(r3)); such that

Lw* = w*.

Proof. Using (4.41), we have

\Lwi(t) — Lw2(t)\12{r3) < ku* j \ai(s) — a2(s)\H ds. (4.42)

From (2.1),we have

\L^i(t) - Lu2(t)\w(ra) < C j (\ui(t) - U2(t)\2v + j \ui(s) - U2(s)\2v ds+ (4

+ \vi(s) - V2(s)\W + \ei(s) - 62(s)\l2{n))dt. By (4.15) and (4.36), we find f *

|Ui - u2\v + I \U1 - u2 \v

ds + \<pi (t) - V2(t)\t2(Ci) + \6l(t) - 62(t)\l2{Q) < t

^ \vi(s) - v2(s)\V ds.

Jo

So

43)

(4.44)

2 t 2 2 2 - ^^ + / |ui - ^^ ds + <Wi(t) - ¥2 WW + |èi(t) - è2(t)|£2(Q) <

< C( / |vi(s) - V2(s)|V ds + K(t) - ^2 (t) |L2(r3

So, we have

/t

|ui - u2 V ds + <Wi(t) - f2(t)|w + |èi(t) - è2(t)|£2(Q) <

< C |Wi(t) - ^2 (t) |l2 (r3) .

(4.45)

By (4.43), we find

\-L^x(t) - LW2(t)|L2(p3) < C/ Ms) - W2(s)|L2(p3) ds.

J 0

Thus, for m sufficiently large, Lm is a contraction on C(0.T, L2(r3)) and so L has a unique fixed point in this Banach space. □

Now, we have all the ingredients to prove Theorem 4.1. Existence

Let (g*,n*) G l2(0.T, V x V') be the fixed point of A defined by (4.22), let w* G C(0.T, l2(r3)) be the fixed point of Lw* defined by (4.41), and let (u,e,P) = (ug*n* ,eg*n* ,Pg*n*) be the solutions of Problems PVg*n*, Pggn and PVg>n*. It results from (4.7), (4.13) and (4.16) that (ug*n*,eg*n*, Pg*n*) is the solutions of Problems PV. Properties (4.1)-(4.6) follow from Lemmas 4.2, 4.3 and 4.4 .

Uniqueness

The uniqueness of the solution is a consequence of the uniqueness of the fixed point of the operators A, L defined by (4.22), (4.41), and the unique solvability of the Problem PVgn , Pggn and PVgn which completes the proof.

References

[1] L.E.Anderson, Aquasistatic frictional problem with normal compliance, Nonlinear Analysis: Theory, Methods, Applications, 16(1991), no. 4, 347-370.

[2] R.C.Batra, J.S.Yang, Saint-Venant's principle in linear piezoelectricity, Journal of Elasticity, 38(1995), no. 2, 209-218.

[3] H.Benaissa, E.-H.Essou, R.Fakhar, Existence results for unilateral contact problem with friction of thermo- electro-elasticity, Applied Mathematics and Mechanics, 36(2015), no. 7, 911-926. DOI: 10.1007/s10483-015-1957-9

[4] M.Cocu, E.Pratt, M.Raous, Formulation and ap- proximation of quasistatic frictional contact, International Journal of Engineering Science, 34(1996), no. 7, 783-798.

DOI: 10.1016/0020-7225(95)00121-2

[5] G.Duvaut, J.L.Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1988.

[6] W.Han, M.Shillor, M.Sofonea, Variational and nu- merical analysis of a quasistatic viscoelas-tic problem with normal compliance, friction and damage, Journal of Computational and Applied Mathematics, 137(2001), no. 2, 377-398. DOI: 10.1016/S0377-0427(00)00707-X

[7] F.Maceri, P.Bisegna, The unilateral frictionless contact of a piezoelectric body with a rigid support, Mathematical and Computer Modelling, 28(1998), no. 4-8, 19-28.

[8] R.D.Mindlin, Elasticity, piezoelectricity and crystal lattice dynamics, Journal of Elasticity, 2(1972), no. 4, 217-282.

[9] M.Rochdi, M.Shillor, M.Sofonea, Aquasistaticcontact problem with directional friction and damped response, Applicable Analysis, 68(1998), no. 3-4, 409-422.

DOI: 10.1080/00036819808840639

[10] M.Selmani, L.Selmani, Frictional contact problem for elastic-viscoplastic materials with thermal effect, Berlin Helberg, 2013.

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Проблема износа термоэлектровязкоупругих материалов

Азиза Бахмар Джамель Ошенан

Факультет наук

Университет Ферхата Аббаса де Сетифа (UFAS)

Сетиф, Алжир

Аннотация. В данной работе рассматривается математическая модель контактной задачи тер-моэлектровязкоупругости. Тело соприкасается с препятствием. Контакт фрикционный и двусторонний с подвижным жестким основанием, что приводит к износу контактирующей поверхности. Устанавливается вариационная формулировка модели и доказывается существование единственного слабого решения задачи. Доказательство основано на классическом факте существования и единственности параболических неравенств, дифференциальных уравнений и аргументов с фиксированной точкой. Приводится вариационная постановка задачи, доказывается существование и единственность слабого решения.

Ключевые слова: пьезоэлектрик, температура, термоэлектровязкоупругость, вариационное неравенство, износ.