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ISSN 2074-1871 Уфимский математический журнал. Том 15. № 2 (2023). С. 101-119.
ANALYSIS OF A THERMO-ELASTO-VISCOPLASTIC CONTACT PROBLEM WITH WEAR AND DAMAGE
A. CHOUIA, A. AZEB AHMED AND F. YAZID
Abstract. This paper presents a quasistatic problem of a thermo-elaso-visco-plastic body in frictional contact with a moving foundation. The contact is modelled with the normal compliance condition and the associated law of dry friction. The model takes into account wear of the contact surface of the body caused by the friction and which is described by the Archard law. The mechanical damage of the material, caused by excessive stress or strain, is described by the damage function, the evolution of which is determined by a parabolic inclusion. We list the assumptions on the data and derive a variational formulation of the mechanical problem. Existence and uniqueness of the weak solution for the problem is proved using the theory of evolutionary variational inequalities, parabolic variational inequalities, first order evolution equation and Banach fixed point.
Keywords: Thermo-elasto-viscoplastic material, damage, wear, frictional contact, existence and uniqueness, fixed point arguments, weak solution.
Mathematics Subject Classification: 74H20, 74H25, 74M15, 74F05, 74R20, 49J40.
1. Introduction
Thermo-mechanical contact problems are of essential importance in analysis and design of structural elements interacting through localized or distributed contact forces and undergoing monotonic or oscillatory relative slip or sliding motion. The elevated temperature regimes are then generated either by environmental conditions or by frictional dissipation at the contact interface. The frictional sliding motion usually generates progressive wear coupled with such effects as localized plastic deformation, damage growth, oxidation and phase transformation, wear debris, etc. in the contact layer. The wear is defined as the material loss or change in surface texture occurring when two surfaces of mechanical components contact each other. As the contact process evolves, the contacting surfaces evolve too, via their wear. There exists a large engineering and mathematical literature devoted to this topic. The wear phenomenon subjects of numerous experimental and theoretical studies. We mention here the references [1], [3], [17], [18], [26], [27], [28], [30], [33], [42]. Numerical methods for wear problems with application to implanted knee joints were developed in [27]. An original analytical approach to wear was performed in [11]. General models for frictional contact with wear can be found in [33, 42] as well as in survey [41]. The mathematical analysis of various models of frictional contact with wear, including existence and uniqueness results of the weak solution, was carried out in [20], [21], [23], [25], [29]. Thermo-mechanic contact problems with the evolution of the temperature parameter with wear were treated in [5, 12, 13, 36, 37, 38, 39, 40]. The damage
A. Chouia, A. Azeb Ahmed and F. Yazid, Analysis of a thermo-elasto-viscoplastic contact
problem with wear and damage.
© A. Chouia, A. Azeb Ahmed and F. Yazid 2023.
The reported study by A. Chouia, A. Azeb Ahmed and F. Yazid was supported by the General Direction of Scientific Research and Technological Development (DGRSDT), Algeria.
Submitted April 16, 2022.
subject is extremely important in design engineering, since it directly affects the useful life of the designed structure or component. There exists a very large engineering literature on it. Models taking into account the influence of the internal damage of the material on the contact process were investigated mathematically. General novel models for damage were derived in [8], [10] from the virtual power principle. Mathematical analysis of one-dimensional damage models can be found in [7], [9]. The three-dimensional case has been investigated in [4], [17]. The damage function ( is restricted to have values between zero and one. As ( = 1, there is no damage in the material, for ( = 0 the material is completely damaged, while in the case 0 < ( < 1 there is a partial damage and the system has a reduced load carrying capacity, Quasistatic contact problems with damage were investigated in [7, 14, 16, 24],
The paper is organized as follows. In Section 2 we introduce the notation and mention some preliminary material. In Section 3 we describe the frictional contact problem, state the assumptions on the data and derive its variational formulation. Finally, in Section 4 we establish the existence of a weak solution to the model,
2. Notation and preliminaries
As it has been already mentioned in the previous section, we start by introducing the notation together with some preliminary results. For further details we refer to [6], [16], [22], Throughout the paper d G {1,2, 3} and Sd represents the space of second order svmmetrie tensors on Rd or, equivalentlv, the space of symmetric matrices of order d. The zero element of the spaces Rd and Sd will be denoted by 0. The inner product and norm on Rd and Sd are defined by
u ■ v = ulVl, ||v|| = (v ■ v)1/2 Vu = (Ui), v = (^) G Rd,
* ■ r = GijTl3, ||r|| = (r ■ r)1/2 Vct = K-), r = (ry) G
where the subscripts i, j range between 1 and d and, unless stated otherwise, the summation convention over repeated indices is adopted.
The symbol Q denotes a bounded do main of Rd with a Lipschitz continuous boundary r and r1; r2, r3 represents a partition of r into three measurable parts such that meas(r1) > 0, We use x = (xi) for a generic point in Q U T, An index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable x G Q U r, for instance, fti = df/dxi.
We use standard notation for Sobolev and Lebesgue spaces associated to Q and T, In particular, we use the spaces L2(Q)d, L2(r2)d, L2(r3)^d H 1(Q)d, endowed with their canonical inner products and associated norms. Moreover, we recall that for an element v G H 1(Q)d we sometimes write v for the trace 'jv G L2(r)d of v to r. In addition, we consider the following spaces:
H = L2(Q)d = {u = (m) : Ui G L2(Q)}, Q = {^ = ) : ^ = G L2(Q)}, H1 = {u = (m) : e(u) G Q}, Q1 = {^ G Q/Div a G H},
where e : H1 ^ Q is the deformation operator defined by
e(u) = (£ij(u)), £ij(u) = 2(ui,j + Uj,i), 1 ^ i,j ^ d, Vu G H1. (2.1) By Div : Q1 ^ H we denote the divergence operator given by the formulae
Div<j = (Diva)i = (aijj), 1 ^ i,j ^ d, Va G Q1. (2.2)
The spaces H, Q, ff^d Q1 are real Hilbert spaces endowed with the canonical inner products
(u, v)H = UiVidx Vu, v G H,
(a, t)Q = J &ijTijdx Va, r G Q, n
(u, v)Hl = (u, v)H + (e(u), e(v))Q Vu, v G Hu
(a, t)Ql = (a, t)Q + (Div a, Div t)H Va, r G Q\.
The associated norms on these spaces are denoted by || ■ \\h, || ■ ||q, || ■ Hh1 and || ■ ||qi; respectively. Let V the closed subspace of H 1(Q)d defined by
V = {v G H 1(Q)d : v = 0 onrj,
endowed with the inner product
(u, v)v = (e(u), e(v))Q Vu, v G V, (2.3)
and let H ■ || y be the associated norm:
Hv|k = H^(v)Hq Vu G V. (2.4)
Since meas(r1) > 0, Korn's inequality holds and there exists a constant Ck > 0, that depends only on Q and r1; such that
\k(v)\k ^ ck||v||hW Vv g V.
A proof of Korn's inequality may be found in ([19],p.79). It follows that || ■ ||#i(n)^d || ■ ||y are equivalent norms on V and therefore (V, || ■ ||y) is a real Hilbert space. Moreover, by the Sobolev trace Theorem, there exists a constant c0 > 0 which depends on Q, r^d r3 such that
||v||LW ^ Co||v||y Vv G V. (2.5)
Next, we introduce the closed subspace E of H 1(Q) defined by
E = V G H 1(Q) : u = 0 on r U r2},
endowed with the canonical inner product of H 1(Q). Moreover, by the Sobolev trace Theorem, there exists a constant c1 > 0 which depends on Q, r1; r2 and r3 such that
|M|L2(r3) ^ cMe VU G E. (2.6)
The following Friedriehs-Poineare inequality holds on E:
hv^hh ^ cF\M\E V^ G E. (2.7)
The space L2(Q) is identified with its dual and with a subspace of the dual E 'of E, that is, E C L2(Q) c E', and we say that the inclusions above define a Gelfand triple. The notation (■, ■,)e',e represents the duality pairing between E^d E.
For any element v G V we denote by vv and vr its normal and tangential components on the boundary r given by
Vv = v ■ V, vT = v — Vv V.
In a similar manner, we recall that the normal and tangential components of the stress field <r on the boundary r are defined by
av = (av) ■ v, <jT = <JV — avv.
Finally, it is well known that if a is a regular function, then the following Green formula holds:
J a ■ e(v) dx + J Div a ■ v dx = J av ■ v da Vv G V. (2.8)
n n r
Let T > 0. For every real Banaeh space X we use the classical notation for the spaces C(0,T; X) and C 1(0, T; X) for the space of continuous and continuously diiferentiable functions from [0, T] to X, respectively, with the norm
||f ||c (0,T ;X) = m* ||f (t)h, ||f ||ci(0,r;*) = Wb + ||f(i)||x ),
respectively. Moreover, we use the dot above to indicate the derivative with respect to the time variable.
Finally, for k G N and p G [1, ro], we use the standard notation for the Lebesgue spaces and for the Sobolev spaces Lp(0,T; X) Mid Wk,p(0,T; X), 1 ^ p ^ ro, 1 ^ k. Moreover, if X1 and X2 are real Hilbert spaces then X1 x X2 denotes the product Hilbert space endowed with the canonical inner product (■, ■ )xlXx2.
3. Mechanical and variational formulations
Let Q C Rd (d =2, 3) be a domain occupied by a thermo-viseoelastie-viseoplastie body with a Lipsehitz boundary r which is partitioned into three disjoint measurable parts r1, r^d r3, the measure of ^ is strictly positive and, in addition, r3 is a plane. Bv [0,T], T > 0, we denote the time interval of an interest. We admit an external heat source gih applied in Q x (0, T). The body is clamped on r1 x (0,T), so the displacement field vanishes there. A surface traction of density f2 act on r2 x (0,T), and a body force of densitv f0 is applied in Q x (0,T), The body is in frietional contact on r3 with a moving obstacle, the so-called foundation. In this paper we assume that the material behavior follows a thermo-viseoelastie-viseoplastie constitutive law with damage given by
t
a(t) = Ae(\i(t)) + Be(u(i)) - M(0(t)) + J Q(a(s), e(u(s)),((s))ds, (3.1)
0
e(t) - div(£V0(t)) = -M ■ vu(t) + qth(t), (3.2)
where, hereinafter <r denotes the stress tensor, u ^te displacement field, u is the
velocity, e(u) is the linearized strain tensor, B is the temperature field, M represents the thermal expansion tensor, ^d B are nonlinear operators describing the purely viscous and the elastic properties of the material, respectively, and Q is a nonlinear constitutive function which describes the viseoplastie behavior of the material, where ( is an internal variable describing the damage of the material caused by viseoplastie deformations.
The evolution of the temperature field 9 is governed by a heat equation obtained from the conservation of energy and defined by (3.2), where K = (kij) represents the thermal conductivity tensor, div(^V^) = (kijdti) and qth represents the density of volume heat sources.
The inclusion used for the evolution of the damage field is
c - AC + dVK«) 9 ^(a,s(u),e,(),
where K denotes the set of admissible damage functions defined as
K = {tf G H1 (Q) : 0 ^ •& ^ 1 a.e. in Q},
difK denotes the subdifferential of the indicator function ipK and ^ is a given constitutive function which describes the sources of the damage in the system.
We now briefly describe the boundary conditions on the contact surface T3, based on the model derived in [34], [35]. We assume that a wear of the contact surface occurs as a result of the relative slip between the surface and the foundation and the frietional forces modify the asperities on the surface by removing small amounts of the material and possibly rearranging the remaining undulations. We introduce a wear function w : r3 x [0,T] ^ R which will
measure the wear of the contact surface T3, The wear is identified as the normal depth of the material that is lost. We use the modified version of Archard law:
w(t) = —kw ||v*||^ (t). (3.3)
where kw > 0 is a wear coefficient and v* is the tangential velocity of the foundation. For the
sake of simplicity we assume in the rest of the section that the motion of the foundation is v*
and could wear. Therefore, we assume that av (t) satisfies a normal compliance contact condition with wear, that is
—au(t) = pv(uv(t) — w(t)) on r3, (3.4)
where pv represents the normal compliance function, this condition shows that at each moment t, the reaction of the soft layer depends on the current value of the penetration represented by uv(t) — w(t). Indeed, we assume that a wear process of the foundation takes place and the debris are immediately removed from the system. Thus, the penetration becomes uv (t) — w(t) instead of uv (t) as in the case without wear.
The corresponding generalization of Coulomb law of dry friction may be stated as
||ov(t)H ^Pr(uv(t) — w(t)) on r3, (3.5)
||ov(t)H <Pr(uv(t) — w(t)) ^ ur(t) = v*, (3.6)
\\aT(t)H =pT(uv(t) — w(t)) ^ ur(t) = v* — aaT(t), a ^ 0. (3.7)
Here pT represents the tangential compliance function (called the friction bound) and ur is the tangential velocity of the body. Equations (3.6) and (3.7) can be interpreted physically in the following way. Condition (3.6) characterizes the behavior of the body in the so-called stick zone. It implies that as the tangential stress is insufficient, the boundary sticks to the foundation and moves at the same velocity as the foundation. Condition (3.7) describes the so-called slip zone, that is, as the tangential stress reaches its greatest value, the boundary does not move in tandem with the foundation. The scalar a ^ 0 is a multiplier indicating the relative direction of the slip between the body and the foundation. Finally, the associated temperature boundary condition on r3 reads as
dB
— kZ3 — v3 = ke(B — BF) — kT(|\ur — v*||) on r x (0,T), (3.8)
where Bp is the temperature of the foundation, ke is the heat exchange coefficient between the body and the obstacle, and kT : r3 x R+ ^ R+ is a given tangential function.
To simplify the notation, we do not indicate explicitly the dependence of various functions on the variables x G Q U r and t G [0,T], Then, the classical formulation of the thermo-meehanieal problem is as follows.
Problem V. Find a displacement field u : Q x [0,T] ^ Rd, a stress field a : Q x [0,T] ^ Sd, a temperature B : Q x [0,T] ^ R, a damage ( : Q x [0,T] ^ R and a wear function w :
r3 x [0,T] ^ R such, that
t
er = Ae(Ui) + Be(u)-MB + j G(v(s), e(u(s)),{ (s))ds in Q x (0,T), (3.9)
0
e - div(K-VB) = -M-Vu + qth in Q x (0,T), (3.10)
<: - AC + d^K(0 3 ^(a, s(u),B,0 in Q x (0,T), (3.11)
Diva + f0 = 0 in Q x (0,T), (3.12)
u = 0 onTi x (0,T), (3.13)
av = f2 on r2 x (0,T), 0 = 0 on r U r2 x (0,T),
- Ov
pv(uv - w), \aT || ^ Pr(uv - w),
\aT|| < pT(uv - w) \aT|| = pT(uv - w) d9
U-j
U T
v
v — aa7
on r3 x (0,T),
- kij~dxVj = ke(® - ) - kr
|Ur - V ||)
w = kwHv*Hpv(uv - w) on r3 x (0,T), dC
7^ = 0 on r x (0,T), Of
w(0) = 0 on r3, u(0) = uo, 0(0) = 0o,
a> 0, on r3 x (0,T),
C (0) = Co in a
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
Here equations (3.9) and (3.10) represent the thermo-viscoelastic-viscoplastic constitutive law with damage introduced in the first section, the evolution of the damage is governed by the inclusion of parabolic type given by relation (3.11). Equation (3.12) represents the equilibrium equations for the stress. Equalities (3.13) and (3.14) are the displacement-traction boundary conditions, respectively. Condition (3.16) describes the frictional contact with normal compliance and wear described above on the potential contact surface T3, Equation (3.18) is an ordinary differential equation which describes the evolution of the wear function. Relation
d(
(3.19) describes a homogeneous Neumann boundary condition, where — is the normal derivaos
tive of C- In equation (3.20), the identity u>(0) = 0 means that at the initial moment the body is not subject to any prior wear. Next, the functions u0, 0o and Co in (3.21) are the initial data. In the study of the mechanical problem V, we consider the following assumptions. The viscosity operator A : Q x Sd ^ Sd satisfies the following conditions:
'(a) There exists LA > 0 such that ||^.(x,e1) - ^.(x,£2)|| ^ AaH^ - £2|| for all £1, e2 G Sd,&.e, x G Q;
(b) There exists m^ > 0 such that(^.(x, e1) - ^.(x, £2)) ■ (e1 - £2) ^ m^ ||e1 - £2||2 < ,
for all £1, e2 G S , a.e. x G Q;
(c) The mapping x m ^.(x, e) is Lebesgue measurable on Q for any £ G Sd; k (d) The mapping x m ^.(x, 0) belongs to Q.
(3.22)
The elasticity operator B : Q x Sd m Sd satisfies the following conditions:
' (a) There exists a constant LB > 0 such that
||B(x, £1) - B(x, £2)|| ^ LB||£1 - £2|| for all £1, £2 G Sd, a.e. x G Q;
(b) The mapping x m B(x, e) is Lebesgue measurable on Q for any £ G Sd;
(c) The mapping x m B(x, 0) belongs to Q.
(3.24)
The viscoplastic operator Q : Q x Sd x Sd x R m Sd satisfies the following conditions: ' (a) There exists a constant Lg > 0 such that ||£(x, 01, £1, Cl) — Q(x, 02, £2,C2)H ^ LQ(H^1 — + \ki — ^H + ICl — Cai) for all 0 1, 02, e1, £2 G Sd, C1, C2 G R, a.e. x G Q. (b) The mapping x m Q(x, a, £,() is Lebesgue measurable on Q, for all a, e G Sd and < G R. ^ (c) The mapping x m Q (x, 0, 0, 0) belongs to Q.
The damage source function ^ : Q x Sd x Sd x R x R m R satisfies the following conditions:
(a) There exists a constant L^ > 0 such that |^(x, 0-1, £1,81,(1) — ^(x, 02, £2,02,(2)1
^ L4, (H^1 — + He1 — £2H + I#1 — 021 + IC1 — C2I)
< for all 01,02, £1, £2 G Sd, 01,02, (1,(2 G R, a.e. x G Q; (3.25)
(b) The mapping x m ^(x, a, £,0,() is Lebesgue measurable on Q, for all a, £ G Sd and 0,( G R;
(c) The mapping x m ^(x, 0, 0, 0, 0) belongs to L2(Q).
The normal compliance function pv : r3 x R m R+ satisfies the following conditions: (a) There exists Lv > 0 such that Ipv(x,u1) — pv(x,u2)I ^ Lv |w1 — u2I for all u1, u2 G R, a.e. x G r3; ^ (b) The mapping x m pv(x, u) is measurable on r3 for all u G R; (3.26)
(c) The mapping x m pv(x, 0) blongs to L2(r3). The tangential compliance function pT : r3 x R m R+ satisfies the following conditions: (a) There exists LT > 0 such that Ipr(x,u1) — pT(x,u2)I ^ LT Iu1 — u2I for all u1, u2 G R, a.e. x G r3; * (b) The mapping x m pT(x, u) is measurable on r3 for all u G R; (3.27)
(c) The mapping x m pT(x, 0) blongs to L2(r3). The tangential function kT : r3 x R+ m R+ satisfies the following conditions:
(a) There exists MT > 0 such that \kT(x, r1) — kT(x, r2)I ^ MT Ir1 — r2I for all n, f'2 G R+ , a.e. x G r3; (3,28)
(b) The mapping x m kT (x, 0) G L2(r3) is measurable on r3, V r G R+. A particular example of a tangential function kT is given by
kT(x,r) = p(x)r, x G r3,
where p G L^(r3; R+) represents some rate coefficient for the gradient of the temperature.
The thermal expansion tensor M = (mij) : Q x Rd m Rd, the heat sources density qth and the thermal conductivity tensor K : Q x Rd m Rd satisfy the following conditions:
M = (mZ3), mZ3 = m3Z G L™(Q), qth G C([0,T]; L2(Q)), (3.29)
£ = (h3) = (kji) G L™(Q), (3.30)
3ck > 0, such that kij&& ^ ck&& V£ G Rd. (3.31)
The densities of forces are assumed to have the following regularity:
fo e C([0,T]; L2(Q)d), f2 e C([0,T]; L2(r2)d). (3.32)
We assume that the initial conditions satisfies the following condition:
uo e V, do e E, Co e K, dp e L2(0,T; L2(r3)), ke e R+) (3.33) Finally, we define mappings
a : H1(Q) x H1(Q) ^ R, P : V x V x L2(r3) ^ R, f : V x V ^ R, 5 : [0,T] ^ R, and the functions 2 : E ^ E^d R : V ^ E', respectively, by
a«,£) = y VC -V^, (3.34) n
P(w, u, v) = Jpv(uv — w)vvda + JpT(uv — w)||vr — v*||da, (3.35) rs rs
f (u, v) = J f0.vdx + J f2 ■ v da, (3.36)
n r2
{S(t),^)E'xE = J qth(t)ydx + J ke0F(t)ydx, (3.37)
n rs
{Zt,V)E'xE = J kj + /ker^da, (3.38)
n J 1 rs
{Rv,^ xE = —J (M.Vv)^dx + j kT (||vr — v* ||)^da. (3.39)
n rs
We end this section with the remark that Problem V represents the classical formulation of the frictional problem. In general, this problem has no classical solution, which has all the necessary classical derivatives. For this reason, as usual in the analysis of frictional contact problems, there is a need to associate to Problem V a new problem, a so-called variational formulation. Using standard arguments we obtain the variational formulation of the thermo-mechanical problem (3,9)-(3,21),
Problem Vv. Find a displacement field u : [0,T] ^ V, a stress field a : [0,T] ^ Q1; a temperature field 9 : [0,T] ^ E, a damage field ( : [0,T] ^ H1 (Q) and a wear function w : [0,T] ^ L2^) such that for all t e [0,T]
t
a(t) = Ae(ui(t)) + £e(u(i)) — M6(t) + j Q(a(s), e(u(s)),((s))ds, (3.40)
o
(a(t), e(v) — e(u(i)))0 + P(w(t), u(t), v) — P(w(t), u(t), u(t))
> (f (t), v — u(i))v Vv e V, j
m,i)—c (t))i2(n)+a(c (t),# — c (t))
> №(r(t),e(u(t)),9(t),((t)),# — C(t))L2(n), ((t) e K, W e K, 9(t) + Z e(t) = Ru(t) + S (t) in E', (3.43)
w(t) = kw Hv*(t)Hpu (u„ (t) — w(t)), (3.44)
u(0) = uo, 0(0) = 0o, C (0) = Co, ^(0) = wo. (3.45)
We note that variational Problem Vy is formulated in terms of the displacement field, stress field, temperature field, damage field and wear function. The functions u, 9, (and w satisfying (3,40)-(3,45) are called weak solution to contact problem V. The existence of the unique solution to Problem Vy is stated and proved in the next section.
4. An existence and uniqueness result
Theorem 4.1. Let assumptions (3.22)-(3.33) hold. Then there exists a unique solution (u, a, 0, (, w) to problem Vy. Moreover, this solution possesses the following properties:
u G C1 (0,T; V), (4.1)
0 G C([0, T]; Q1), (4.2)
0 G C([0, T]; L2(Q)) n L2(0,T; E) n W 1'2(0,T; E'), (4.3)
C G W 1'2(0, T; L2(Q)) n L2(0, T; H 1(Q)), (4.4)
w G C 1([0,T]; L2(r3)). (4.5)
The proof of theorem 4.1 consists in several steps, which we shall provide in what follows. Throughout this section the assumptions of Theorem 4.1 are supposed to hold true. By C we denote generic positive constants which may depend on Q, r2, r3, B, M, pv, pT and T but are independent of t and the rest of input data, and whose value may change from place to place.
First, let (A, 'q, G C([0, T]; L2(r3) x Q x Q) be given and consider the following variational problem.
Problem Find a displacement field uXv^ : [0,T] m V and a stress field : [0,T] m Q1 such that for all t G [0,T]
(t) = Ae(u Xvii (t)) + Be(uXvi (t)) + v(t) + £(t), (4.6)
(t), e(v) — (t)))Q + P(X(t), uwVt(t), v)
— P(X(t), uXvi(t), uXvi(t)) ^ (f (t), v — uXvi(t))v Vv G V, ( ' j
uA^ (0) = uo. (4.8)
For Problem we have the following result.
Lemma 4.1. There exists a unique solution (uA^, ) to Problem Vxv£ and it has its regularity expressed in (4.1)-(4.2).
Proof. We use the Riesz representation theorem to define the operators A : V m V, B : V m V and the function F : [0,T] m V by
(Au, v)v = (Ae(u), e(v))Q, (4.9)
(Bu, v)v = (Be(u), e(v))Q, (4.10)
(F(t), v)v = (f(t), v)v — (V(t)+ ((t), e(v))Q, (4.11)
for all u, v G V and t G [0,T], Assumptions (3.22)(b), (3.22)(c) and (3.22) imply that the operator A is a strongly monotone and Lipschitz continuous operator on V and B is a Lipsehitz continuous operator on V. For u G V and A G L2(r3^, the toctional P(X, u, ■) is convex and lower semieontinuous on V. We use (2.5), (3.26) and (3.27) to find
P(X, u1, v2) — P(X, u1, v1) + P(X, u2, v1) — P(X, u2, v2) ^ c^(Lu + LT)|u1 — u2Hy 11v 1 — v2Hy,
(4.12)
for all ui, u2, vi, v2 G V. Moreover, using (3,32) it is easy to see that the function f defined by (3,36) satisfies f G C([0,T]; V) and, keeping in mind that (iq, £) G C([0, T"]; Q x Q), we deduce from (4,11) that F G C([0,T]; V). It follows from a class of abstract evolutionary variational inequalities (see for example [15]) that there exists a unique function U\v£ G C([0,T]; V).
We use relation (4,6), assumptions (3,22), (3,23) and the regularity of the functions ^d £ with the properties of the deformation tensor e to obtain that G C([0,T]; Q).
We choose a function v = vXv£(t) ± z in (4.7), where z G £>(n)d is arbitrary and we use definitions (3,35) and (3,36) to obtain
Div *Xvt (t) + f o(i) = 0, (4.13)
for all t G [0,T], The regularity of the functions f 0, the relation (4.13) and since G
C([0, T]; Q)show that G C([0, T]; Qi). The prooO is complete. □
In the second step, let fa G C([0,T]; P2(^)) be given and consider the following variational problem for the damage field.
Problem V^. Find a damage field : [0,T] ^ H 1(Q) such that
(Ut),$ - Ut))L2{n) + a(Mt),# - CW)
^ (fa(t),§ - UVhm C(t) G K, W G X, a.e. t G [0,T],
(4.14)
C(0) = Co. (4.15)
Lemma 4.2. Problem V^ has a unique solution which satisfies the regularity expressed in (4.4). Moreover, if Q is the solution of Problem V^. corresponding to fa G C([0,T]; L2(Q)), i = 1,2, then there exists C > 0 such, that
t
IKiW - <2(t)\\hm ^ c J Iifai(s) - toWWhmds vt g [0,t]. (4.16)
0
Proof To solve "Pj, we use a classical existence and uniqueness result on parabolic variational inequalities (see, for instance, [2]). By (4.14) we get:
(Ci - C2, Ci - C2)L2(Q) + a(Ci - C2, Ci - C2) ^ - fai - fa2)L2(n) v G [0, T].
Integrating the above inequality with respect to time and using the initial conditions Ci(0) = C2(0) = C0 and the inequalitv a((i - C2, Ci - C2) ^ 0, we find
t
1 IKiW - C2(t)W^(n) (fai(s) - fa2(s),(i(s) - C2(s))l2(H) ds vt G [0,T],
0
which implies that
t t
IICi(i) - C2(i)W^2(n) ^J \fa(s) - fa(s)\lHn)ds + J IICi(s) - C2(s)W^2(n) ds.
00
This inequality combined with Gronwall inequality leads to (4.16) which completes the proof.
□
At the third step, we define the operator £ : C([0,T]; Q) ^ C([0,T]; Q) by
t
£G = Q(^, e(uXvi),Q where £(t) =i G(s) ds VG G C([0,T]; Q), (4.17)
and is the stress field
= M(u) + Be(uXvt) + 'q + i. (4.18)
We observe that since ('q, £) G C([0,T]; Q x Q), it is straightforward to see that G
C([0,T]; Q). We hence obtain the following result.
Lemma 4.3. The operator £ has a unique fixed point G* G C([0,T]; Q).
Proof. The continuity of £G is a straightforward implication of the continuity of and and (3.24) . Moreover, let £2 G C([0,T]; Q) and G1, G2 G C([0,T]; Q) be their corresponding integrals in time and let G C([0,T]; L2(Q)). For the sake of simplicity, we
use the notation = u^ u= u^x^ = ff^d = Q for i = 1, 2. Given t G [0,T] by (3.24) we find that
||£G1(i) — £G2(t)HQ ^ C(\l*1(t) — V2(t)\\Q + IMi) — u2(t)Hv + IIC1W — (2(t)Wmn)). (4.19) It follows from (4.17) and (4.18) that
t
W°1(t) — (t)H2Q ^ ^ IMi) — u 2 (t*) || y + ||u1(i) — u2(t)H^ + J\G1(s) — G^ft d8y (4.20)
o
Moreover by (4.6) and (4.7) we obtain
(^e(u 1 (t)) — As(u2(t)), e(u 1(t)) — e(ii2)(t))Q
+ (Be(u1(t)) — Be(u2(t)), e(ii 1(t)) — e(ii2(t)))Q ^P(X, u1(t), u2(t)) — P(X, u1(t), u 1(i)) + P(X, u2(t), u 1(i)) — P(X, u2(t), u2(t)) — (%1(t) — %2(t), (e(u 1(t)) — e(u2(t)))Q. Also it follows from (3.22)(b), (3.23)(a) and (4.12) that
|\u 1(t) — u2(t)\\v ^ C(|u1(t) — u2(t)Hv + ||^1(t) — miiQ).
Using this result, the inequality 2ab ^ a2 + b2 and (4.17), we obtain
t
||u 1(t) — u2(t*)|| y ^ C(Hu1(t) — u2(t)H2v + j ||G1(s) — G2(s)H2Qds).
o
Since u1 (0) = u2 (0) = u0, we have
t
IMi) — u2(t)H2v ^ cj Hu 1(S) — u2(5)Hyds.
o
It follows from (4.23), (4.24) and (4.16) that
|\u 1(t) — u2(t)Hy + ||C1(t) — C2(i)H|2(n) ^ c( j I\u 1(S) — u2(5)||y ds
o
t
+ i ||^1(s) — ^(s) H|2(Q) ds (4.25)
(4.21)
(4.22)
(4.23)
(4.24)
+ J ||Gi(s) - G2(s)\\2q ds 0
)
t
Bv using the Gronwall Lemma, we find that
t
|\u 1(t) — u 2(t)H2v + HC1W — <2(t)Wh(n) ^ cj ||G1 (8) — G2(s~) H q ds. (4.26)
o
This inequality combined with (4.20), (4.24), (4.26), (4.19) lead us to the following estimate:
t
\\£G1(t) — £G2(t)H2Q ^ cj ||G1 (s) — G2(s)H2q ds. (4.27)
o
Reiterating inequality (4.27) n times, we get:
\\£nG1 — £nG2HC([o,T];Q) ^ , — G2\\c(\O,t];q). (4.28)
Therefore, for n large enough, £n is a contractive operator on the Banach C([0,T]; Q) and we conclude that there exists a unique G* G C([0,T]; Q) such that £G* = G*. The proof is complete. □
At the forth step, let (A, 'q) G C([0,T]; L2(r3) x Q) be given and we consider the following auxiliary problem.
Problem Vxv- Find a displacement field uA^ : [0,T] ^ V, a stress field aXv : [0,T] ^ Q1 and a temperature field 9Xv : [0,T] ^ E such, that
t
aXv(t) = M(uXv(t)) + Be(uXv(t)) + v(t) + j Q(vXv(s), e(uXv(s)),^(s)) ds, (4.29)
o
(aXv(t), e(v) — e(uXv(t)))Q + P(X(t), uXv(t), v)
— P(X(t), uXv (t), u Xr) (t)) > (f(t), v — u Xr) (t))v Vv G V, j
9Xn (t) + Z dXv (t) = 1luXn (t) + S (t) in E', (4.31)
uA^ (0) = uo, eXv (0) = do. (4.32)
Lemma 4.4. There exists a unique solution (uA^, cXn,9Xv} to the auxiliary problem V^ obeying regularity (4.1)-(4.3). Moreover, if (u», <Ji,9i} and (i represents the solutions of Problems VXVi and Vq, respectively, for ('qi,pi) G C([0,T]; Q x L2(Q)), i = 1, 2, then there exists C > 0 such, that
t
2
IMi) — °2(t)II2Q Hu 1(t) — Mm'v + j Hu 1(8) — u2(8)\\2vds
o t
+ KW — V2(t)II2Q + j IIC1 (s) — <2(s)\lh(n)
(4.33)
o
t
Hu 1(t) — u2(t)H2v ^ W^t) — V2(t)II2Q + j ll<1(s) — <2(s)llhm d*), (4.34)
o
t
№1 (t) — 02(t)Hh(n) ^ c\ Hu 1(S) — u2(5)||y ds. (4.35)
Proof. Let G* e C([0,T]; Q) be the fixed point of the operator C defined by (4,17) and uX^ = u\v£*, a\v = be the solution to problem obtained in Lemma 4,1 for £ = £* =
t
f G*(s)ds. Equation CG* = G* combined with (4.17) shows that uX^, <JXr, satisfies (4,29) 0
and (4,30), Then conditions (4,32), regularities (4,1) and (4,2) follow from Lemma 4,1, Using now the displacement field uX^ obtained in lemma 4,4, we obtain (4,31) and we know that the inclusion mapping of (E, || ■ ||E) into (L2(Q), || ■ ||l2(q)) is continuous and dense, we can write the Gelfand evolution triple
E C L2(H) c E'.
The operator 2 is linear and coercive. Using Friedriehs-Poineare inequality, we have
(ZT,T)exe' > C||r||E. (4.36)
By (3.30) and (3.38) for all T,U e E, we have
d
(ZT,U)e' xe ^ Y1 ^hA^mhl hi^U^ ||L2(n) + ke^T ||L2(r3)|^|L2(r3). ij=1
Using (2.6), we find
(ZT,U)e'xe < c||r||e|M|e. (4.37)
On the other hand, from the definitions of R, S and the regularity of iiXv we deduce that
<pXv = mixv + S e L2(0,T; E'). (4.38)
Since 90 e E, from inequalities (4.36), (4.37) and regularity (4.38), it follows that the operator 2 is hemieontinuous and monotone. Then by using classical arguments of functional analysis concerning parabolic equations (see, for instance, [31]), we prove easily the existence and uniqueness of 9Xr) satisfying
'dXv e C([0,T]; L2(^)) n L2(0,T; E) U W 1,2(0,T; E'),
eXv (t) + Z eXv (t) = pXv (i) in E', (4.39)
eXv (0) = 00.
Consider now (iq1,^1), (lH2,fa2) e C([0,T]; Q x £2(^)) and for i = 1, 2, we denote
u XVi = ui, &XVi = 0'г, = Ow^i , C^i = Ci. Using (4.29) and (3.24), we have
lki(*) - ^ q llui(t) - mt)\\2v + l|ui(t) - u2(t)||f
+ kw - ^wllq + i lm*) - ^2(s)||q ds
(4.40)
+ J ||ui (s) - u2(s)||£ dS + j ||Ci(s) - C2(s)||2(n) d^j . 00 Using a Gronwall argument and (4.24) in the above inequality, we deduce (4.33).
We use (4.29), (4.30), (4.33) and arguments similar to those used in the proof of (4.23) to obtain
t
||u 1 - u 2 || V ^ ^ jjui(t) - u2(t)H2v +J ||ui(s) - u2(s)||£ ds
+ ||ui(s) - ds + (s) - V2(s)\\qds
0 0 t
+ IICl(s) - <2(8
0
This inequality combined with (4,24), (4,16) and Gronwall inequality leads us to (4,34), Now for G C([0,T]; V) we have for t G [0,T]-.
(di(t) - d2(t),6i(t) - d2(t)}E'XE + (z6i(t) - zd2(t),di(t) - d2(t)}E'XE
(4,41)
= (nil l(t) -nil 2(t),di(t) - 92 (t))E' xE.
Then by integrating over (0,t), we get (4,35) by using (3,39), (3,29), (3,30) and (3,31), The proof is complete, □
Finally, as a consequence of these results and by using the properties of the operator M, and the function for t G [0,T] we consider the element
nx(V,p)(t) = (nPx(v)(t), n\(V,p)(t)) G Q x L2(Q), (4.42)
defined by the identities
n°x(v)(t) = -M(9Xr, (t)), (4.43)
n\(V,p)(t)= 0(vXn(t), uXv(t),ex>ri(t),(,(t)). (4.44)
We have the following result.
Lemma 4.5. Forfar) G C ([0,T]; Q x L2(tt)) the function nx(^, p) : [0,T] ^ Q x L2(tt) is continuous and there is a unique element (/q*,p*) e C([0,T]; Q x L2(Q)) such, that nx(/q*,p*) =
(v*,P* )•
Proof Let G C ([0,T]; Q x L2(Q)) and ti,t2 G [0,T]. Using (4.43) and (3.29), we have
\n°x(v)(ti) - n°x(v)(t2)\Q ^ i\M(eXv(ti)) -M(eXv(t2))\\Q
^ Lm\\9\v (ti) - Q^V (¿2)||L2(П),
where Lm = sup \\mi,j||i2(Q^ Then due to the regularity of 9Xr) stated in Lemma 4.3 we deduce
from (4.45) that n0x(^) G C([0,T]; Q). By similar arguments, from (4.44), (2.4) and (3.25) it follows that
\nlx(V,p)(ti) - nlx(V,p)(t2)I\LHn) ^C01(ti) - UA^(t2)\\V
n 4.46)
+ (ti) - 0Xv (t2)\\v + \\Uti) - Ut2)lL2(n))
Therefore, n\(<n,ii) G C([0,T]; L2(Q)) and nx(^,p) G C([0,T]; Q x L2(Q)). Let now (tqi,pi), (*q2,p2) G C([0,T]; Q x L2(Q)). We use the notation
Uwn. = Ui, uXn. = ui, eXn. = Oi, = (i for i = 1, 2.
We use similar arguments as in the proof of relations (4.45) and (4.46) to find that
\nx(Vi,pi)(t) - n^^mwlxmn) (||ui(i) - U2(t)l2v
+ Iim - 02(t)II2L2{Q] + ||Ci(i) - <2(t)\\hm), '
t
t
for all t e [0,T]. We employ (4.24), (4.16), (4.34) and (4.35) to obtain
||nA(Wl)(i) - nx(V2,»2)(t)\\2QxL2(n) ^ cj ||(wi)(s) - (V2,Ms)\\2QxL2{n)ds Vi e [0,T].
0
Reiterating this inequality m times, we get:
C mr^m
||nm(wi) — nm(^2,^2)|c ([0,T];QxL2(Q)) C([0,T ]; QxL2(Q)).
Thus, for m sufficiently large, nm is a contraction on the Banaeh space C([0,T]; Q), and so nA has a unique fixed point. The proof is complete. □
Let A e C([0,T]; L2(r3)). At the fifth step, we consider the following variational problem. Problem Vv. Find a displacement field u A : Q x [0, T] ^ Rd, a stress field a x : Q x [0, T] ^ Sd, a temperature field 6A : Q x [0,T] ^ R and a damage field (A : Q x [0,T] ^ R such, that
a x(t) = Ae(Ux(t)) + B(e(ux(t))) - M0x(t) + [ Q(a a(s), e(ua(s)),(a(s)) ds, (4.48)
(4.49)
(4.50)
(a x(t), e(v - u x(t))) q + P(X(t), ux(t), v) - P(X(t), ux(t), ux(t))
^ (f (t), v - ux(t))v Vv e V,
(Ut),# - Ca(^))l2(q) + a(C(t),# - Cx(t))
> (^(°A, e(uA(t)),ex(t),(w(t)),ti - Cx(i))L2(n) Cx(t) e K VV e X, ^x(i) + 2 0x(t) = Rux(i) + S(t) in E', (4.51)
ux (0) = uo, 0a(0) = do, Ca(0) = Co. (4.52)
Lemma 4.6. Problem Vv has a unique solution (u A,&A,9A,(A) satisfying (4,l)-(4,5).
Proof Let (vA,fa) e C([0, T]; Q x L2(Q)) the fixed point of nx defined by (4.42)-(4.44) and u A = u Avx, dA = 0Avx, (a = (Vx be the solutions to problems Vvv and VV obtained in Lemmata 4.4 and 4.2 for = (,qA,faA). Let
a x(t) = Ae(ux(t))+ B(e(ux(t))) - MQx(t) + J Q(v a(s), s(ua(s)),Ca) ds.
0
Equation nA(^ A) = 'qA and n A(^ A,faA) = combined with (4.43) and (4.44) shows that (u A, uA,dA,(A) satisfies (4.48)-(4.51). Then (4.52) and the regularities (4,l)-(4,4) follow from Lemmata 4.2, 4.4 and assumptions on A, B, Q and M which concludes the existence part of the Lemma 4.6.
The uniqueness part of Lemma 4.6 is a consequence of the uniqueness of the fixed point of the operator nA defined by (4.42)-(4.44) and the unique solvabilitv of problems "P^ and VVX-The proof is complete. □
Now we consider an operator $ : C([0,T]; L2(r3)) ^ C([0,T]; L2(r3)) defined by
$A = kw ||v*|| ^pv(uv - X)ds Vt e [0,T]. (4.53)
0
Lemma 4.7. The operator $ has a unique fixed point \* e C([0,T]; L2(r3)).
Proof. Let Ai, A2 G C([0, T]; L2(r3)) and denote by ui; i = 1, 2, the solution of (4,49) in problem for A = Xi, that is, u = u,^ and we also let Ui = uFurthermore, in what follows by C we denote various positive constants which may depend on kw and v*. Taking into consideration (4,53), (2,5) and (3,26), we deduce that
I I ^Xi(t) - $A2 (t) 11 ¿2(p3) ^C ( I I Ui(s) - U2(s) 11 I dS
0
t
(4.54)
+ J I I Ai(s) - X2(^ 11 ^2(r3) ds). 0
We use similar arguments that those used in the proof of the relation (4.23) to find that t t
J I I U i(S) - U 2(s) 11 ^ ds ^ C (/I I Ui(s) - U2(s) 11 ^ ds
00
t
+ j I I Ai(s) - A2(s) 11 ^ ds).
0
Since ui(0) = u2(0) = u0, by using (4.55) we obtain
t
I I Ui(i) - U2 (t) 11 \v ^ cj I I U i (8) - 112(8) 11 2V ds
0
(4.55)
(4.56)
^ Cj || Ui (S) - U2(S) || I ds + CJ || Ai(s) - A2^) || ^2(F3) ds. 00 Applying Gronwall inequality, we deduce
t
| I Ui (i) - U2(t) 11 *y ^ cj I I Ai(s) - A2(s) 11 |2 (P3 )d8. (4.57)
0
It follows from this inequality that
t t s
J I I Ui(s) - U2(s) 11 *v ds ^ G^ J I I Ai(r) - X2(r) 11 ^ drds. 0 0 0
Since s ^ t,
t s t t
jJ I I Ai(r) - X2(r) 11 ^ drds ^ C^ J I I Ai (r) - X2(r) 11 ^ drds
0 0 0 0
t t
= C J I I Ai(r) - X2(r) 11 h^drj ds. 00
Then
t t
i I I Ui(s) - U2(s) 11 *v ds ^ C i I I Ai(s) - A2(s) 11 ^2(P3) ds. (4.58)
t
t
t
Combining now (4,54) with (4,58), we obtain
t
||$Ai(i) - $A2(i)^2(r3) ^ cj ||Ai(s) - A2(S)^2(r3) ds. (4.59)
0
Reiterating this inequality n times, we arrive at
Cn
H$nAi - $"'A2|c([0,T];L2(r3)) ^ n! Pi - ^2|C([0,T];L2(rs)).
Thus, for n sufficiently large, $n is a contraction on C([0, T"]; L2(r3)), and so $ has a unique fixed point A* in this Banaeh space. The proof is complete, □
We now have all ingredients needed to provide the proof of Theorem 4,1,
Proof. Let A* be the fixed point of the operator £ given by (4,53), With (4,48)-(4,53) it is easy to verify that (u A*, u \*, 0X*, (\*, A*) is the unique solution to problem Vv possessing regularities (4.1)-(4.5). □
Acknowledgements
The authors would like to thank the anonymous referees for very helpful suggestions and comments which lead to the improvement of this paper. This research work is supported by the General Direction of Scientific Research and Technological Development (DGRSDT), Algeria,
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Abdallah Chouia, Faculty of Exact Sciences,
Laboratory of Operator Theory and PDE: Foundations and Applications University of El Oued, El Oued 39000, Algeria E-mail: chouia-abdallah@univ-eloued.dz
Abdelaziz Azeb Ahmed, Faculty of Exact Sciences,
Laboratory of Operator Theory and PDE: Foundations and Applications University of El Oued, El Oued 39000, Algeria E-mail: aziz-azebahmed@univ-eloued. dz
Fares Yazid, Faculty of Sciences,
Laboratory of pure and applied mathematics University of Laghouat, Laghouat 03000, Algeria E-mail: f.yazid@lagh-univ.dz
