A FRICTIONLESS ELASTIC-VISCOPLASTIC CONTACT PROBLEM WITH NORMAL COMPLIANCE, ADHESION AND DAMAGE
LAMIA CHOUCHANE AND LYNDA SELMANI
Abstract. We study a quasistatic frictionless contact problem with nor- mal compliance, adhesion and damage for elastic-viscoplastic material.The adhesion of the contact surfaces is modeled with a surface variable, the bonding field, whose evolution is described by a first order differential equa- tion. The mechanical damage of the material, caused by excessive stess or strains, is described by a damage function whose evolution is modeled by an inclusion of parabolic type. We provide a variational formulation of the problem and prove the existence and uniqueness of a weak solution. The proofs are based on time-dependent variational equalities, classical results on elliptic and parabolic variational inequalities, differential equations and fixed point arguments.
1. Introduction
We consider a mathematical model for a quasistatic process of frictionless contact between an elastic-viscoplastic body and an obstacle, within the framework of small deformation theory. The contact is modeled with normal compliance. The effect of damage due to the mechanical stress or strain is included in the model. Such situation is common in many engeneering applications where the forces acting on the system very periodically leading to the appearence and growth of microcracks which may deteriorate the mechanism of the system. Because of the safety issue
Received by the editors: 12.06.2007.
2000Mathematics Subject Classification.74M15, 74R99, 74C10.
Key words and phrases. quasistatic process, elastic-viscoplastic material, damage, normal compliance, adhesion, weak solution, variational equality, ordinary differential equation, fixed point.
of mechanical equipments, considerable efforts were been devoted to modeling and numerically simulating damage.
Early models for mechanical damage derived from the termodyamical consid- erations appeared in [9, 10], where numerical simulations were included. Mathemat- ical analysis of one-dimensional problems can be found in [11]. In all these papers the damage of the material is described with a damage functionα, restricted to have values between zero and one. Whenα= 1 there is no damage in the material, when α= 0, the material is completely damaged, when 0< α <1 there is partial damage and the system has a reduced load carrying capacity. Quasistatic contact problems with damage have been investigated in [13, 14, 17]. In this paper, the inclusion used for the evolution of the damage field is
α.−k4α+∂ϕK(α)3Φ (σ,ε(u), α),
whereK denotes the set of admissible damage functions defined by K=
ξ∈H1(Ω)/0≤ξ≤1 a.e. in Ω ,
kis a positive coefficient,∂ϕK represents the subdifferential of the indicator function of the setKand Φ is a given constitutive function which describes the sources of the damage in the system. In the present paper we consider a rate type elastic-viscoplastic material with constitutive relation
σ. =Eε u.
+G(σ,ε(u), α),
where E is a fourth order tensor, G is a nonlinear constitutive function and α is the damage field and the adhesion between the body and the obstacle is taken into account during the conact. The adhesive contact between bodies, when a glue is added to keep surfaces from relative motion, is receiving increased attention in the mathematical literature. Analysis of models for adhesive contact can be found in [2, 3, 4, 6, 12, 15, 20]. The novelty in all the above papers is the introduction of a surface internal variable, the bonding field, denoted in the paper by β; it describes the pointwise fractional density of active bonds on the contact surface, and sometimes
referred to as theintensity of adhesion. Following [7, 8], the bonding field satisfies the restrictions 0≤β ≤1; whenβ = 1 at a point of the contact surface, the adhesion is complete and all the bonds are active, whenβ = 0 all the bonds are inactive, severed, and there is no adhesion; when 0< β <1 the adhesion is partial and only a fraction β of the bonds is active. We refer the reader to the extensive bibliography on the subject in [16,18,19].
The paper is structured as follows. In section 2 we present the notation and some preliminaries. In section 3 we present the mechanical problem, we list the assumptions and in section 4 we give and prove our main existence and uniqueness result, Theorem 4.1. The proof is based on monotone operator theory, classical results on parabolic inequalities and Banach fixed point arguments.
2. Notation and preliminaries
In this short section, we present the notation we shall use and some prelimi- nary material. For more details, we refer the reader to [5]. We denote bySdthe space of second order symmetric tensors on Rd,( d= 2,3), while (.) and |.| represent the inner product and the Euclidean norm onSd and Rd, respectively. Let Ω⊂Rd be a bounded domain with a regular boundary Γ and letν denote the unit outer normal on Γ. we shall use the notation
H =L2(Ω)d={u= (ui)/ ui∈L2(Ω)}, H={σ= (σij)/ σij =σji∈L2(Ω)},
H1={u= (ui)∈H / ε(u)∈ H}, H1={σ∈ H/ Divσ∈H},
whereε: H1→ Hand Div : H1→H are the deformation and divergence operators, respectively, defined by
ε(u) = (εij(u)), εij(u) =1
2(ui,j+uj,i), Divσ= (σij,j).
Here and below, the indices i and j run between 1 to d, the summation convention over repeated indices is used and the index that follows a comma indicates
a partial derivative with respect to the corresponding component of the independent variable. The spaces H,H, H1 and H1 are real Hilbert spaces endowed with the canonical inner products given by
(u,v)H = Z
Ω
uividx ∀u,v∈H, (σ,τ)H=
Z
Ω
σij τijdx ∀σ,τ ∈H, (u,v)H
1 = (u,v)H+ (ε(u),ε(v))H ∀u,v∈H1, (σ,τ)H
1 = (σ,τ)H+ (Div σ, Divτ)H ∀σ,τ ∈H1.
The associated norms on the spaces H,H, H1 and H1 are denoted by | |H, | |H,
| |H
1 and | |H
1, respectively. Let HΓ = H1/2(Γ)d and let γ : H1 → HΓ be the trace map. For every elementv∈H1 we also use the notationv to denote the trace γv ofvon Γ and we denote byvν andvτ thenormal andtangential components of von the boundary Γ given by
vν =v.ν, vτ =v−vνν. (2.1)
Similarly, for a regular (sayC1) tensor fieldσ : Ω →Sd, we define its normal and tangential components by
σν = (σν).ν, στ =σν−σνν, (2.2) and we recall that the following Green’s formula holds
(σ,ε(v))H+ (Divσ,v)H = Z
Γ
σν.v da ∀v∈H1. (2.3)
Finally, for any real Hilbert space X, we use the classical notation for the spaces Lp(0, T;X) and Wk,p(0, T;X), where 1 ≤ p ≤ +∞, and k ≥ 1. We denote by C(0, T;X) andC1(0, T;X) the space of continuous and continuously differentiable functions from [0, T] toX, respectively, with the norms
|f|C(0,T;X)= max
t∈[0,T] |f(t)|X,
|f|C1(0,T;X)= max
t∈[0,T] |f(t)|X+ max
t∈[0,T]
.
f(t) X
,
respectively. Moreover, we use the dot above to indicate the derivative with respect to the time variable and, for a real numberr, we user+ to present its positive part, that is r+ = max {0, r}. Finally, for the convenience of the reader, we recall the following version of the classical theorem of Cauchy-Lipschitz (see, e.g., [21, p. 60]).
Theorem 1. Assume that (X,|.|X) is a real Banach space and T > 0. LetF(t, .) : X→X be an operator defined a.e. on(0, T)satisfying the following conditions: 1-∃ LF >0 such that |F(t, x)−F(t, y)|X ≤LF |x−y|X ∀x, y∈X,a.e. t∈(0, T). 2-
∃p≥1such that t7−→F(t, x)∈Lp(0, T;X) ∀x∈X. Then for any x0∈X,there exists a unique function x∈W1,p(0, T;X)such that
x.(t) =F(t, x(t)) a.e. t∈(0, T),
x(0) =x0.
Theorem 2.1 will be used in section 4 to prove the unique solvability of the intermediate problem involving the bonding field.
Moreover, ifX1 and X2 are real Hilbert spaces, then X1×X2 denotes the product Hilbert space endowed with the canonical inner product (., .)X
1×X2. 3. Problem statement
A viscoplastic body occupies the domain Ω⊂ Rd with the boundary Γ di- vided into three disjoint measurable parts Γ1,Γ2 and Γ3 such that meas (Γ1)> 0.
The time interval of interest is [0, T] whereT >0. The body is clamped on Γ1and so the displacement field vanishes there. A volume force of densityf0acts in Ω×(0, T) and surface tractions of densityf2 act on Γ2×(0, T). We assume that the body is in adhesive frictionless contact with an obstacle, the so called foundation, over the potential contact surface Γ3. Moreover, the process is quasistatic, i.e. the inertial terms are neglected in the equation of motion. We use an elasto-viscoplastic consti- tutive law with damage to model the material’s behavior and an ordinary differential equation to describe the evolution of the bonding field. The mechanical formulation of the frictionless problem with normal compliance is as follows.
Problem P. Find a displacement field u : Ω×[0, T] → Rd, a stress field σ : Ω×[0, T] → Sd, a damage field α : Ω×[0, T] → R and a bonding field β : Γ3×[0, T]→[0,1]such that
σ. =Eε u.
+G(σ,ε(u), α), (3.1)
α.−k4α+∂ϕK(α)3Φ (σ,ε(u), α), (3.2) Divσ+f0= 0 in Ω×(0, T), (3.3)
u= 0 on Γ1×(0, T), (3.4)
σν=f2 on Γ2×(0, T), (3.5)
−σν =pν(uν)−γν β2(−R(uν))+ on Γ3×(0, T), (3.6)
στ= 0 on Γ3×(0, T), (3.7)
∂α
∂ν = 0 on Γ×(0, T), (3.8)
.
β =−h γνβ
(−R(uν))+2
−a
i
+ on Γ3×(0, T), (3.9) u(0) =u0,σ(0) =σ0, α(0) =α0 in Ω, (3.10)
β(0) =β0on Γ3. (3.11)
The relation (3.1) represents the viscoplastic constitutive law with damage, the evo- lution of the damage field is governed by the inclusion given by the relation (3.2), k is a constant, ∂ϕK denotes the subdifferential of the indicator function ϕK of K which represents the set of admissible damage functions satisfying 0≤α≤1 and Φ is a given constitutive function which describes damage sources in the system. (3.3) represents the equilibrium equation, (3.4) and (3.5) are the displacement and traction boundary conditions, respectively. (3.6) represents the normal compliance contact condition with adhesion in whichγν anda are given adhesion coefficients and Ris the truncation operator defined by
R(s) =
−L ifs≤ −L, s |s|< L, L ifs≥L.
(3.12)
HereL >0 is the caracteristic length of the bond, beyonding which it does not offer any additional traction. The introduction of R is motivated by the mathematical arguments but it is not restrictive for physical point of view, since no restriction on the size of the parameterLis made in what follows. Also,pν is a given positive function which will be decribed below. In this condition the interpenetrability between the body and the foundation is allowed, that isuνmay be positive on Γ3. The contribution of the adhesive to normal traction is represented by the term γνβ(−R(uν))+, the adhesive traction is tensile, and is proportional, with proportionality coefficientγν, to the square of the intensity of adhesion, and to the normal displacement, but as in various papers see e.g. [2, 3] and the references threin. Condition (3.7) represents the frictionless contact condition and shows that the tangential stress vanishes on the contact surface during the process. (3.8) represents a homogeneous Newmann boundary condition where ∂α∂ν represents the normal derivative ofα. Next, equation (3.9) represents the ordinary differential equation which describes the evolution of the bonding field and it was already used in [2], see also [19] for more details. Here, γν anda are given adhesion coefficients which may depend on x∈Γ3 and Ris the truncation operator given by (3.12). Notice that in this model once debonding occurs bonding connot be reestablished since, as it follows from (3.9),
.
β ≤0.In (3.10), we consider the initial conditions where u0 is the initial displacement, σ0 is the initial stress andα0 is the initial damage. Finally, (3.11) is the initial condition, in which β0 denotes the initial bonding field. LetZ denote the bonding fields set
Z=
β∈L2(Γ3) /0≤β≤1 a.e. on Γ3 ,
and for displacement field we need the closed subspace ofH1 defined by V ={v∈H1|v= 0 on Γ1}.
Since meas (Γ1) > 0, Korn’s inequality holds and there exists a constant CK > 0, that depends only on Ω and Γ1such that
|ε(v)|H≥CK|v|H
1 ∀v∈V.
OnV we consider the inner product and the associated norm given by
(u,v) = (ε(u),ε(v))H, |v|V =|ε(v)|H ∀u,v∈V.
It follows from Korn’s inequality that|.|H
1 and|.|V are equivalent norms onV and therefore (V,|.|V) is a real Hilbert space. Moreover, by the Sobolev trace theorem, there exists a constantC0, depending only on Ω, Γ1 and Γ3 such that
|v|L2(Γ3)d≤C0|v|V ∀v∈V. (3.13)
In the study of the mechanical problem (3.1)-(3.11), we make the following assump- tions. The operatorE : Ω×Sd →Sd satisfies
(a)E= (eijkh)/ eijkh∈L∞(Ω),
(b)E σ . τ=σ . A. τ ∀σ, τ∈Sd, a.e. in Ω, (c)E σ . σ≥mE |σ|2 ∀σ∈Sd, for somemE >0.
(3.14)
The operatorG: Ω×Sd×Sd×R→Sd satisfies
(a) There exists a constantLG >0 such that
|G(x,σ1, ε1, α1)− G(x,σ2,ε2, α2)| ≤LG (|σ1−σ2|+|ε1−ε2|+|α1−α2|)
∀σ1, σ2,ε1,ε2∈Sd,α1, α2∈R, a.e. x∈Ω;
(b)x7−→ G(x,σ,ε, α) is a Lebesgue measurable function on Ω
∀σ,ε∈Sd,∀α∈R; (c)x7−→ G(x,0,0,0)∈ H.
(3.15) The damage function Φ : Ω×Sd×Sd×R→Rsatisfies
(a) There exists a constant L >0 such that
|Φ(x,σ1, ε1, α1)−Φ (x,σ2,ε2, α2)| ≤L(|σ1−σ2|+|ε1−ε2|+|α1−α2|)
∀σ1,σ2,ε1,ε2∈Sd,α1,α2∈R, a.e. x∈Ω;
(b)x7−→Φ (x,σ,ε, α) is a Lebesgue measurable function on Ω
∀σ,ε∈Sd, ∀α∈R; (c) x7−→Φ (x,0,0,0)∈ H.
(3.16) The normal compliance functionpν : Γ3×Rd →R+ satisfies
(a) There existsLν >0 such that
|pν(x,r1)−pν(x,r2)| ≤Lν|r1−r2| ∀r1,r2∈Rd, a.e. x∈Γ3. (b) (pν(x,r1)−pν(x,r2)).(r1−r2)≥0 ∀r1,r2∈Rd, a.e. x∈Γ3. (c) r7→pν(.,r) is Lebesgue measurable on Γ3, ∀r∈Rd.
(d) The mappingpν(.,r) = 0 for allr≤0.
(3.17)
The adhesion coefficients satisfy
γν ∈L∞(Γ3), γν ≥0,a ∈L∞(Γ3), a≥0. (3.18) We also suppose that the body forces and surface traction have the regularity
f0∈C(0, T;H), f2∈C(0, T;L2(Γ2)d). (3.19) Finally we assume that the initial data satisfy the following conditions
u0∈V, σ0∈ H1, (3.20)
α0∈K, (3.21)
β0∈Z. (3.22)
We define the bilinear forma:H1(Ω)×H1(Ω)→Rby a(ξ, ϕ) =k
Z
Ω
∇ξ. ∇ϕ dx. (3.23)
Next, we denotef : [0, T]→V the function defined by (f(t),v)V =
Z
Ω
f0(t).vdx+ Z
Γ2
f2(t).vda ∀v∈V, a.e. t∈(0, T). (3.24) The adhesion functionaljad:L∞(Γ3)×V ×V →Rdefined by
jad(β,u,v) = Z
Γ3
−γν β2(−R(uν))+vν da. (3.25) In addition to the functional (3.25), we need the normal compliance functionaljnc: V ×V →Rgiven by
jnc(u,v) = Z
Γ3
pν(uν)vν da. (3.26)
Keeping in mind (3.17)-(3.18), we observe that the integrals in (3.25) and (3.26) are well defined and we note that conditions (3.19) imply
f ∈C(0, T;V). (3.27)
Finally we assume the following condition of compatibility
(σ0,ε(v))H+jad(β0,u0,v) +jnc(u0,v) = (f(0),v)V ∀v∈V. (3.28) Using standard arguments based on green’s formula (2.3) we can derive the following variational formulation of the frictionless problem with normal compliance (3.1)-(3.11) as follows.
Problem P V. Find a displacement field u : [0, T] → V, a stress field σ : [0, T]→ Ha damage field α: [0, T]→H1(Ω)and a bonding field β : [0, T]→L2(Γ3) such that
σ. (t) =Eε u.(t)
+G(σ(t),ε(u(t)), α(t)), a.e. t∈(0, T), (3.29) α(t)∈Kfor allt∈[0, T] , (α.(t), ξ−α(t))L2(Ω)+a(α(t), ξ−α(t))
≥(Φ (σ(t),ε(u(t)), α(t)), ξ−α(t))L2(Ω) ∀ξ∈K, (3.30) (σ(t),ε(v))H+jad(β(t),u(t),v) +jnc(u(t),v)
= (f(t),v)V ∀v∈V, ∀t∈[0, T], (3.31)
.
β(t) =−h
γν β(t)
(−R(uν(t)))+2
−a
i
+ a.e. t∈(0, T), (3.32)
u(0) =u0,σ(0) =σ0, α(0) =α0, β(0) =β0. (3.33) We notice that the variational problemP V is formulated in terms of displacement, stress field, damage field and bonding field. The existence of the unique solution of problem P V is stated and proved in the next section. To this end, we consider the following remark whose estimates will be used in different places of the paper.
Remark 1. From (3.32) we obtain that β(x, t) ≤ β0(x) , since β0(x) ∈ Z then β(x, t) ≤1 for all t ≥0, a.e. on Γ3. If β(x, t0) = 0for all t = t0 it follows from (3.32)that
.
β(x, t) = 0for all t≥t0,therefore,β(x, t) = 0for allt≥t0. We conclude that 0≤β(x, t)≤1 ∀t∈[0, T],a.e. x∈Γ3.
In the sequel we consider thatCis a generic positive constant which depends on Ω,Γ1,Γ3, γν, Land may change from place to place. First, we remark thatjadand jncare linear with respect to the last argument and therefore
jad(β,u,−v) =−jad(β,u,v), jnc(u,−v) =−jnc(u,v). (3.34) Next, using (3.25) as well as the properties of the operatorR, (3.12), we find jad(β1,u1,v)−jad(β2,u2,v) =
Z
Γ3
γν β21[(−R(u2ν))+−(−R(u1ν))+]vν da
+ Z
Γ3
γν (β22−β21) (−R(u2ν))+ vν da≤C Z
Γ3
|β1−β2| |v| da,
and from (3.13) we obtain
jad(β1,u1,v)−jad(β2,u2,v)≤c |β1−β2|L2(Γ3) |v|V . (3.35) Now, we use (3.26) to see that
|jnc(u1,v)−jnc(u2,v)| ≤ Z
Γ3
|pν(u1ν)−pν(u2ν)| |vν| da,
and therefore (3.17) (a) and (3.13) imply
|jnc(u1,v)−jnc(u2,v)| ≤C|u1−u2|V |v|V . (3.36)
We use again (3.26) to see that
jnc(u1,u2−u1) +jnc(u2,u1−u2) = Z
Γ3
(pν(u1ν)−pν(u2ν)) (u2ν−u1ν) da,
and therefore (3.17) (b) implies
jnc(u1,u2−u1) +jnc(u2,u1−u2)≤0. (3.37) The inequalities (3.35)-(3.37) combined with equalities (3.34) will be used in various places in the rest of the paper.
4. Well posedness of the problem
The main result in this section is the following existence and uniqueness result.
Theorem 2. Assume that (3.14)-(3.22) and (3.28) hold. Then, problem P V has a unique solution {u,σ, β, α} which satisfies
u∈C(0, T;V), σ∈C(0, T;H1), β∈W1,∞ 0, T;L2(Γ3)
, α∈W1,2 0, T;L2(Ω)
∩L2 0, T;H1(Ω)
. (4.1)
A quadruplet (u,σ, β, α) which satisfies (3.29)-(3.33) is called a weak solution to the compliance contact problem P. We conclude that, under the stated assump- tions, problem (3.1)-(3.11) has a unique weak solution satisfying (4.1). We turn now to the proof of Theorem 4.1 which is carried out in several steps. To this end, we assume in the following that (3.14)-(3.22) and (3.28) hold. Below,Cdenotes a generic positive constant which may depend on Ω,Γ1,Γ3,E, γν, LandT but does not depend ont nor of the rest of input data, and whose value may change from place to place.
Moreover, for the sake of simplicity, we supress, in what follows, the explicit depen- dence of various functions onx∈Ω∪Γ. The proof of Theorem 4.1 will be carried out in several steps. In the first step we solve the differential equation in (3.32) for the
adhesion field, whereuis given, and study the continuous dependence of the adhesion solution with respect tou.
Lemma 3. For every u∈C(0, T;V), there exists a unique solution βu∈W1,∞ 0, T;L2(Γ3)
satisfying
.
βu(t) =−h
γν βu(t)
(−R(uν(t)))+2
−a
i
+ a.e. t∈(0, T), βu(0) =β0.
Moreover, βu(t)∈Z for t ∈[0, T], a.e. on Γ3, and there exists a constant C >0, such that, for all u1,u2∈C(0, T;V),
βu1(t)−βu2(t)
2
L2(Γ3)≤C Z t
0
|u1(s)−u2(s)|2V ds ∀t∈[0, T].
Proof. Consider the mappingF: [0, T]×L2(Γ3)→L2(Γ3) defined by F(t, β) =−h
γνβ(t)
(−R(uν))+2
−ai
+
,
∀t ∈ [0, T] and β ∈ L2(Γ3). It follows from the properties of the truncation op- erator R that F is Lipschitz continuous with respect to the second argument, uni- formly in time. Moreover, for anyβ ∈L2(Γ3), the mapping t7−→F(t, β) belongs to L∞ 0, T, L2(Γ3)
. Thus, the existence and the uniqueness of the solutionβufollows from the classical theorem of Cauchy-Lipschitz given in Theorem 2.1. Notice also that the argument used in Remark 3.1 shows that 0≤βu(t) ≤1 for all t ∈[0, T], a.e. on Γ3.Therefore, from the definition of the setZ, we find thatβu(t)∈Z for all t∈[0, T], which concludes the proof of the Lemma. Now letu1,u2∈C(0, T;V) and lett∈[0, T].We have, fori= 1,2,
βui(t) =β0− Z t
0
h
γν βui(t)
(−R(uiν(t)))+2
−a
i
+ ds, and then
βu1(t)−βu2(t) L2(Γ
3)
≤C Z t
0
βu
1(s)
(−R(u1ν(s)))+2
−βu
2(s)
(−R(u2ν(s)))+2 L2(Γ
3)
ds.
Using the definition of the truncation operator R given by (3.12) and considering βu1 =βu1−βu2+βu2 we find
βu1(t)−βu2(t) L2(Γ
3)
≤C
Z t
0
βu
1(s)−βu
2(s) L2(Γ
3) ds+ Z t
0
|u1(s)−u2(s)|L2(Γ3)d ds
. Applying Gronwall’s inequality, it follows that
βu1(t)−βu2(t)
2
L2(Γ3)≤C Z t
0
|u1(s)−u2(s)|2L2(Γ3)d ds,
and using (3.13) we obtain the second part of Lemma 4.2.
Now we consider the following viscoplastic problem and we prove an existence and uniqueness result for (3.29), (3.31) and (3.33) with the corresponding initial condition.
Problem QV. Find a displacement field u : [0, T] → V, a damage field α: [0, T]→H1(Ω) and a stress field σ: [0, T]→ Hsatisfying (3.29) and
(σ(t),ε(v))H+jad(βu(t),u(t),v) +jnc(u(t),v)
= (f(t),v)V ∀v∈V, ∀t∈[0, T], (4.2) u(0) =u0,σ(0) =σ0, α(0) =α0. (4.3) Let (η,ω)∈C(0, T;H ×L2(Ω)) and let Zη(t) =Rt
0
η(s) ds+σ0− Eε(u0), then
Zη∈C1(0, T;H), and consider the following variational problem.
Problem QVη. Find a displacement field uη : [0, T] → V and a stress field ση: [0, T]→ Hsuch that
ση(t) =Eε(uη(t)) +Zη(t), ∀t∈[0, T], (4.4) (ση(t),ε(v))H+jad(βuη(t),uη(t),v) +jnc(uη(t),v)
= (f(t),v)V ∀v∈V, ∀t∈[0, T], (4.5) uη(0) =u0,ση(0) =σ0. (4.6) To solve problem QVη we consider θ ∈ C(0, T;V) and we construct the following intermediate problem.
Problem QVηθ.Find a displacement field uηθ : [0, T]→V and a stress field σηθ: [0, T]→ Hsuch that
σηθ(t) =Eε(uηθ(t)) +Zη(t), (4.7) (σηθ(t),ε(v))H+ (θ(t),v)V = (f(t),v)V ∀v∈V, ∀t∈[0, T], (4.8) uηθ(0) =u0,σηθ(0) =σ0. (4.9) Lemma 4. There exists a unique solution (uηθ,σηθ)of the problem QVηθ which sat- isfies uηθ∈C(0, T;V),σηθ∈C(0, T;H1).
Proof. We define the operatorA:V →V by
(A u,v)V = (Eε(u),ε(v))H, ∀u,v∈V. (4.10) Using (3.14), it follows that A is a strongly monotone Lipschitz operator, thus Ais invertible andA−1:V →V is also a strongly monotone Lipschitz operator. It follows that there exists a unique functionuηθ which satisfies
uηθ ∈C(0, T;V), (4.11)
A uηθ(t) =hηθ(t), (4.12)
wherehηθ∈C(0, T;V) is such that
(hηθ(t),v)V = (f(t),v)V −(Zη(t),ε(v))H−(θ(t),v)V ∀v∈V, ∀t∈[0, T]. (4.13) It follows from, (4.12) that uηθ ∈ C(0, T;V). Considerσηθ defined in (4.7), since, Zη∈C1(0, T;H),uηθ ∈C(0, T;V) we deduce thatσηθ∈C(0, T;H). Since Divσηθ =
−f0 ∈C(0, T;H), we further haveσηθ ∈C(0, T;H1). This concludes the existence part of Lemma 4.3. The uniqueness of the solution follows from the unique solvability of the time-dependent equation (4.12). Finally (uηθ,σηθ) is the unique solution of problemQVηθ obtained in Lemma 4.3, which concludes the proof.
Let Λθ(t) denote the element ofV defined by
(Λθ(t),v)V =jad(βuηθ(t),uηθ(t),v)+jnc(uηθ(t),v) ∀v∈V, ∀t∈[0, T]. (4.14) We have the following result.
Lemma 5. For each θ ∈ C(0, T;V) the function Λθ : [0, T] → V belongs to C(0, T;V). Moreover, there exists a unique element θ∗ ∈ C(0, T;V) such that Λθ∗=θ∗.
Proof. Letθ ∈C(0, T;V) and lett1, t2 ∈ [0, T]. Using (3.35), (3.36) and (4.14) we obtain
|Λθ(t1)−Λθ(t2)|V ≤C
βuηθ(t1)−βuηθ(t2) L2(Γ
3)+|uηθ(t1)−uηθ(t2)|V
. (4.15) By Lemma 4.3,uηθ ∈C(0, T;V) and, by Lemma 4.2,βu
ηθ ∈W1,∞ 0, T;L2(Γ3) , then we deduce from inequality (4.15) that Λθ ∈ C(0, T;V). Let now θ1,θ2 ∈ C(0, T;V) and denote uηθi = ui and βu
ηθi = βu
i for i = 1,2. Using again the relations (3.35), (3.36) and (4.14) we find
|Λθ1(t)−Λθ2(t)|2V ≤C
βu1(t)−βu2(t)
2
L2(Γ3)+|u1(t)−u2(t)|2V
. (4.16) Then by Lemma 4.2, we have
βu1(t)−βu2(t)
2
L2(Γ3)≤C Z t
0
|u1(s)−u2(s)|2L2(Γ3) ds,
and by (3.13) we get
βu1(t)−βu2(t)
2
L2(Γ3)≤C Z t
0
|u1(s)−u2(s)|2V ds.
Use the previous inequality in (4.16) to obtain
|Λθ1(t)−Λθ2(t)|2V ≤C
|u1(t)−u2(t)|2V + Z t
0
|u1(s)−u2(s)|2V ds
. (4.17) Moreover, from (4.8) it follows that
(Eε(u1)− Eε(u2),ε(u1)−ε(u2))H+ (θ1−θ2,u1−u2)V = 0 on (0, T). (4.18)
Hence
|u1(t)−u2(t)|V ≤C |θ1(t)−θ2(t)|V ∀t∈[0, T]. (4.19) Now from the inequalities (4.17) and (4.19) we have
|Λθ1(t)−Λθ2(t)|2V ≤C
|θ1(t)−θ2(t)|2V + Z t
0
|θ1(s)−θ2(s)|2V ds
∀t∈[0, T]. Applying Gronwall’s inequality we obtain
|Λθ1(t)−Λθ2(t)|2V ≤C Z t
0
|θ1(s)−θ2(s)|2V ds∀t∈[0, T].
Reiterating this inequalityntimes yields
|Λnθ1−Λnθ2|2C(0,T;V)≤(CT)n
n! |θ1−θ2|2C(0,T;V),
which implies that for n sufficiently large a power Λn of Λ is a contraction in the Hilbert space C(0, T;V). Then, there exists a unique θ∗ ∈ C(0, T;V) such that Λnθ∗=θ∗ andθ∗is also the unique fixed point of Λ.
Lemma6. There exists a unique solution of problem QVη satisfying uη∈C(0, T;V), ση∈C(0, T;H1).
Proof. Letθ∗∈C(0, T;V) be the fixed point of Λ, Lemma 4.3 implies that (uηθ∗,σηθ∗)∈ C(0, T;V)×C(0, T;H1) is the unique solution of QVηθ for θ = θ∗. since Λθ∗ =θ∗ and from the relations (4.14), (4.7), (4.8) and (4.9), we obtain that (uη,ση) = (uηθ∗,σηθ∗) is the unique solution ofQVη. The uniqueness of the solution is a consequence of the uniqueness of the fixed point of the operator Λ given in (4.14).
Now for (η,ω)∈C(0, T;H ×L2(Ω)), we suppose that the assumptions of The- orem 4.1 hold and we consider the following intermediate problem for the damage field.
ProbemP Vω. Find a a damage field αω: [0, T]→H1(Ω) such that αω(t)∈ K, for allt∈[0, T] and
(α.ω(t), ξ−αω(t))L2(Ω)+a(αω(t), ξ−αω(t))
≥(ω(t), ξ−αω(t))L2(Ω) ∀ξ∈K, a.e. t∈(0, T) (4.20)
αω(0) =α0 (4.21)
Lemma 7. Problem P Vω has a unique soltion αω such that
αω∈W1,2 0, T;L2(Ω)
∩L2 0, T;H1(Ω)
. (4.22)
Proof. We use (3.21), (3.23) and a classical existence and uniqueness result on parabolic inequalities (see for instance [1 p. 124]).
As a consequence of the problemsQVη andP Vω, we may define the operator L:C(0, T;H ×L2(Ω))→C(0, T;H ×L2(Ω)) by
L(η,ω) = (G(ση, ε(uη), αω),Φ(ση, ε(uη), αω)), (4.23) for all (η,ω)∈C(0, T;H ×L2(Ω)). Then we have.
Lemma 8. The operator L has a unique fixed point
(η∗,ω∗)∈C(0, T;H ×L2(Ω)).
Proof. Let (η1,ω1), (η2,ω2)∈C(0, T;H ×L2(Ω)), lett∈[0, T] and use the notationuηi=ui,σηi=σi,Zηi=Zi andαωi =αi fori= 1,2. Taking into account the relations (3.15), (3.16) and (4.23), we deduce that
|L(η1,ω1)− L(η2,ω2)|H×L2(Ω)
≤C
|u1(t)−u2(t)|V +|α1(t)−α2(t)|L2(Ω)+|σ1(t)−σ2(t)|H
. (4.24) Using (4.5) we obtain
(Eε(u1)− Eε(u2), ε(u1)−ε(u2))H=jad(βu2,u2,u1−u2)−jad(βu1,u1,u1−u2) +jnc(u2,u1−u2)−jnc(u1,u1−u2) + (Z2−Z1, ε(u1)−ε(u2))H a.e. t∈(0, T).
(4.25) Keeping in mind (3.35), (3.37) and (3.14) we find
|u1(t)−u2(t)|V ≤C
βu1(t)−βu2(t) L2(Γ
3)+|Z1(t)−Z2(t)|H
, (4.26)
and
|u1(t)−u2(t)|2V ≤C βu
1(t)−βu
2(t)
2
L2(Γ3)+|Z1(t)−Z2(t)|2H . By Lemma 4.2, we obtain
|u1(t)−u2(t)|2V ≤C
|Z1(t)−Z2(t)|2H+ Z t
0
|u1(s)−u2(s)|2V ds
≤C(
Z t
0
|η1(s)−η2(s)|2H ds+ Z t
0
|u1(s)−u2(s)|2V ds). (4.27) Applying Gronwall’s inequality yields
|u1(t)−u2(t)|2V ≤C Z t
0
|η1(s)−η2(s)|2H ds, (4.28) which implies
|u1(t)−u2(t)|V ≤C Z t
0
|η1(s)−η2(s)|H ds. (4.29) Moreover, by (4.4) we find
|σ1(t)−σ2(t)|H ≤C(|u1(t)−u2(t)|V +|Z1(t)−Z2(t)|H). Substituting (4.29) in the previous inequality we obtain
|σ1(t)−σ2(t)|H ≤C Z t
0
|η1(s)−η2(s)|H ds. (4.30) From (4.20) we deduce that
α.1, α2−α1
L2(Ω)+a(α1, α2−α1)
≥(ω1, α2−α1)L2(Ω) a.e. t∈(0, T), and
α.2, α1−α2
L2(Ω)+a(α2, α1−α2)
≥(ω2, α1−α2)L2(Ω) a.e. t∈(0, T). Adding the previous inequalities we obtain
α.1−α.2, α1−α2
L2(Ω)+a(α1−α2, α1−α2)
≤ |ω1−ω2|L2(Ω)|α1−α2|L2(Ω) a.e. t∈(0, T).
Integrating the previous inequality on [0, t], after some manipulations we obtain 1
2|α1(t)−α2(t)|2L2(Ω)≤C Z t
0
|ω1(s)−ω2(s)|L2(Ω)|α1(s)−α2(s)|L2(Ω) ds
+C Z t
0
|α1(s)−α2(s)|2L2(Ω) ds.
Applying Gronwall’s inequality to the previous inequality yields
|α1(t)−α2(t)|L2(Ω)≤C Z t
0
|ω1(s)−ω2(s)|L2(Ω) ds. (4.31)
Substituting (4.29), (4.30) and (4.31) in (4.24), we obtain
|L(η1,ω1)− L(η2,ω2)|H×L2(Ω)
≤C Z t
0
|(η1,ω1) (s)−(η2,ω2) (s)|H×L2(Ω) ds. (4.32)
Lemma 4.7 is a consequence of the result (4.32) and Banach’s fixed point Theorem.
Now, we have all ingredients to solveQV.
Lemma 9. There exists a unique solution (u,σ, α) of problem P V satisfying u ∈ C(0, T;V),σ∈C(0, T;H1),α∈W1,2 0, T;L2(Ω)
∩L2 0, T;H1(Ω) .
Proof. Let (η∗,ω∗) ∈ L2(0, T;H ×L2(Ω)) be the fixed point of L given by (4.24), by Lemma 4.5, we deduce that (uη,ση) = (uηθ∗,σηθ∗) ∈ C(0, T;V)× C(0, T;H1) is the unique solution of QVη. SinceL(η∗,ω∗) = (η∗,ω∗), from the rela- tions (4.4), (4.5), (4.6) and Lemma 4.6 we obtain that (u,σ, α) = (uη∗θ∗,ση∗θ∗, αω∗) is the unique solution ofQV. The regularity of the solution follows from Lemma 4.6.
The uniqueness of the solution results from the uniqueness of the fixed point of the
operatorL.
Theorem 4.1 is now a consequence of Lemma 4.2 and Lemma 4.8.
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Department of Mathematics, University of Setif,
19000 Setif, Algeria
E-mail address: l [email protected]
Department of Mathematics, University of Setif,
19000 Setif, Algeria
E-mail address: [email protected]
