Bilateral Contact Problem with Adhesion between Two Bodies for Viscoelastic with Long-term Memory and Damage

Abstract We consider a quasistatic contact problem between two viscoelastic bodies with long-term memory and damage. The contact is bilateral and the tangential shear due to the bonding field is included. The adhesion of the contact surfaces is taken into account and modelled by a surface variable, the bonding field. We prove the existence of a unique weak solution to the problem. The proof is based on arguments of time-dependent variational inequalities, parabolic inequalities, differential equations and fixed point.


Introduction
Scientific research in mechanics are articulated around two main components: one devoted to the laws of behavior and other boundary conditions imposed on the body. The boundary conditions reflect the binding of the body with the outside world.
In this paper, we study a problem involving boundary conditions describing real phenomena such as bilateral contact and other very important such as the damage and the adhesion of two materials. For the constitutive law we consider an viscoelastic with long-term memory and damage given by σ = A ε(u ) + G ε(u ) + t 0 F t − s, ε(u (s)), α (s) ds, (1) where u the displacement field, σ , α the damage field and ε(u ) represent the stress and the linearized strain tensor, respectively. Here A is a given nonlinear operator, F is the relaxation operator, and G represents the elasticity operator. In (1) and everywhere in this paper the dot above a variable represents derivative with respect to the time variable t. It follows from (1) that at each time moment, the stress tensor σ (t) is split into two parts: σ (t) = σ V (t) + σ R (t), where σ V (t) = A ε(u (t)) represents the purely viscous part of the stress, and σ R (t) satisfies the rate-type elastic relation σ R (t) = G ε(u (t)) + t 0 F t − s, ε(u (s)), α (s) ds.
(2) Various results, example and mechanical interpretations in the study of elastic materials of the form (2) can be found in [1,17] and references therein.
Processes of adhesion are important in many industrial settings where parts, usually nonmetallic, are glued together. For this reason, adhesive contact between deformable bodies, when a glue is added to prevent relative motion of the surfaces, has received recently increased attention in the mathematical literature. Analysis of models for adhesive contact can be found in [2,10] and recently in the monographs [11,12]. The novelty in all these papers is the introduction of a surface internal variable, the bonding field, denoted in this paper by β. It describes the point wise fractional density of adhesion of active bonds on the contact surface, and some times it is called the intensity of adhesion. Following [6], the bonding field satisfies the restriction 0 ≤ β ≤ 1, when β = 1 at a point of the contact surface, the adhesion is complete and all the bonds are active, when β = 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. The damage is an extremely important topic in engineering, since it affects directly the useful life of the designed structure or component. There is a very large engineering literature on this topic. Models taking into account the influence of internal damage of the material on the contact process have been investigated mathematically. General models for damage were derived in [3,4] from the virtual power principle. Mathematical analysis of one-dimensional problems can be found in [5]. The three-dimensional case has been investigated in [8]. In all these papers the damage of the material is described with a damage function α , restricted to have values between zero and one. When α = 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. Contact problems with damage have been investigated in [5,14,15,16]. In this paper the inclusion used for the evolution of the damage field isα where K denotes the set of admissible damage functions defined by κ is a positive coefficient, ∂ψ K represents the subdifferential of the indicator function of the set K and φ is a given constitutive function which describes the sources of the damage in the system. In this article we consider a mathematical frictional contact problem between two viscoelastic bodies with constitutive law with long-term memory and damage. The contact is modelled with normal compliance where the adhesion of the contact surfaces is taken into account and is modelled with a surface variable, the bonding field. We derive a variational formulation of the problem and prove the existence of a unique weak solution. This article is organized as follows. In Section 2 we describe the mathematical models for the frictional contact problem between two viscoelastic bodies with long-term memory and damage. The contact is modelled with normal compliance and adhesion. In Section 3 we introduce some notation, list the assumptions on the problem's data, and derive the variational formulation of the model. In section 4 we present our main result stated in Theorem 4.1 and its proof which is based on arguments of time-dependent variational inequalities, parabolic inequalities, differential equations and fixed point.

Problem Statement
We describe the model for the process, we present its variational formulation. The physical setting is as follows. Let us consider two viscoelastic bodies with long-term memory occupying two bounded domains Ω 1 , Ω 2 of the space R d (d = 2, 3). For each domain Ω , the boundary Γ is assumed to be Lipschitz continuous, and is partitioned into three disjoint measurable parts Γ 1 , Γ 2 and Γ 3 , on one hand, such that meas Γ 1 > 0. Let T > 0 and let [0, T ] be the time interval of interest. The body Ω is subjected to f 0 forces. The bodies are assumed to be clamped on Γ 1 × (0, T ). The surface tractions f 2 act on Γ 2 × (0, T ). The two bodies can enter in contact along the common part Γ 1 3 = Γ 2 3 = Γ 3 . The bodies are in adhesive contact with an obstacle, over the contact surface Γ 3 . With the assumption above, the classical formulation of the contact problem with adhesion and damage between two viscoelastics bodies with long-term memory is following.

Problem P
Find the displacement field u = (u 1 , u 2 ) such that u : Ω × [0, T ] → R d , a stress field σ = (σ 1 , σ 2 ) such that σ : Ω × [0, T ] → S d , a damage field α = (α 1 , α 2 ) such that α : Ω × [0, T ] → R and a bonding field β : Div ∂α ∂ν First, equations (5) represent the viscoelastic constitutive law with long term-memory and damage, the evolution of the damage is governed by the inclusion of parabolic type given by the relation (6) where φ is the mechanical source of the damage growth, assumed to be a rad her general function of the strains and damage itself, and ∂ϕ K is the subdifferential of the indicator function of the admissible damage functions set K. Equations (7) are the equilibrium equations for the stress, in which "Div" denote the divergence operator for tensor. (8) and (9) are the displacement and traction boundary conditions, respectively. Condition (10) shows that the contact is bilateral, i.e. there is no separation between the bodies during the process, while condition (11) shows that the resistance to tangential motion is generated by the glue, in comparison to which the frictional traction can be neglected. Moreover, the tangential traction depends only on the bonding field and on the relative tangential displacement. Next, equation (12) governs the evolution of the adhesion field is assumed to depend generally on β and |u|, can be found in [19,21,22,23]. We do not impose sign restrictions on the process, and thus, cycles of debonding and rebonding may take place, as a result of imposed periodic forces. In addition, we include here the possibility that, as the cycle of bonding and debonding go on, there is a decrease in the bonding effectiveness. Therefore, the process is also assumed to depend on the time history of the bonding, which we denote by The whole process is assumed to be governed by the differential equatioṅ Here, H ad is a general function discussed below, which vanishes when its first argument vanishes.
An example of such that a function is where γν is the bonding energy constant and β + is the maximal tensile normal traction that the adhesive can provide. We note that in this case only debonding is allowed. Another example, in which H ad depends on all three variables is Here the magnitude of the displacement r causes the debonding, while naturally there is a tendency to rebond. However, the bonding cannot exceed β = 1, and, moreover, the rebonding becomes weaker as the process goes on, which is represented by the factor 1 + ζ 2 β in the denominator. Equation (13) represents a homogeneous Newmann boundary condition where ∂ α ∂ ν represents the normal derivative of α. In (14) we consider the initial conditions where u 0 is the initial displacement, and α 0 the initial damage. Finally, (15) is the initial condition, in which β 0 denotes the initial bonding.

Variational formulation and the main result
In this section, we list the assumptions on the data and derive a variational formulation for the contact problem. To this end, we need to introduce some notation and preliminary material. Here and below, S d represent the space of secondorder symmetric tensors on R d . We recall that the inner products and the corresponding norms on S d and R d are given by Here and below, the indices i and j run between 1 and d and the summation convention over repeated indices is adopted. Now, to proceed with the variational formulation, we need the following function spaces: The spaces H , H 1 , H and H 1 are real Hilbert spaces endowed with the canonical inner products given by and the associated norms · H , · H 1 , · H , and · H 1 respectively. Here and below we use the notation When σ is a regular function, the following Green's type formula holds, To obtain the variational formulation of the problem (5)-(15), we introduce for the bonding field the set and for the displacement field we need the closed subspace of H 1 defined by Since meas Γ 1 > 0, the following Korn's inequality holds: where the constant c K denotes a positive constant which may depends only on Ω , Γ 1 (see [11]). Over the space V we consider the inner product given by and let · V be the associated norm. It follows from Korn's inequality (20) that the norms · H 1 and · V are equivalent on V . Then (V , · V ) is a real Hilbert space. Moreover, by the Sobolev trace theorem and (21), there exists a constant c 0 > 0, depending only on Ω , We define the set V of admissible displacement fields by We also introduce the spaces E 0 = L 2 (Ω ), E 1 = H 1 (Ω ). To simplify notation, we define the product spaces The spaces V and E 1 , and are real Hilbert spaces endowed with the canonical inner products denoted by (·, ·) V and (·, ·) E1 . The associate norms will be denoted by · V and · E1 respectively.
64 Bilateral Contact Problem with Adhesion between Two Bodies for Viscoelastic with Long-term Memory and Damage Finally, for any real Hilbert space X, we use the classical notation for the spaces L p (0, T ; X), W k,p (0, T ; X), where 1 ≤ p ≤ ∞, k ≥ 1. We denote by C(0, T ; X) and C 1 (0, T ; X) the space of continuous and continuously differentiable functions from [0, T ] to X, respectively, with the norms respectively. Moreover, we use the dot above to indicate the derivative with respect to the time variable and, for areal number r, we use r + to represent its positive part, that is r + = max{0, r}. For the convenience of the reader, we recall the following version of the classical theorem of Cauchy-Lipschitz (see, [16, p.48]).
Theorem 3.1 Assume that (X, · X ) is a real Banach space and T > 0. Let F (t, ·) : X → X be an operator defined a.e. on (0, T ) satisfying the following conditions: There exists a constant L F > 0 such that Then for any x 0 ∈ X, there exists a unique function x ∈ W 1,p (0, T ; X) such thaṫ a.e. t ∈ (0, T ), This theorem 4.1 will be used in section 4 to prove the unique solvability of the intermediate problem involving the bonding field.
In the study of the Problem P, we consider the following assumptions: The viscosity function A : Ω × S d → S d satisfies: The elasticity operator G : Ω × S d → S d satisfies: The relaxation function F : The damage source function φ : Ω × S d × ×R → R satisfies: The tangential contact function pτ : (d) pτ (x, β, r).ν(x) = 0 ∀r ∈ R d such that r.ν(x) = 0, a.e.x ∈ Γ 3 .

(27)
The adhesion function H ad : Γ 3 ×R×[−L, L] → R satisfies: We also suppose that the body forces and surface tractions satisfy and, finally, the initial data satisfies Finally we assume that the initial data satisfy the following conditions We define the bilinear forme a : The microcrack diffusion coefficient verifies We define the map a.e.t ∈ (0; T ). We note that, using (29) we obtain the following regularity Using on formula Green's and (7), we deduce that for = 1, 2 we have a.e. t ∈ (0; T ).

Problem PV
Find a displacement field u = (u 1 , u 2 ) : [0, T ] → V , a a damage field α = (α 1 , α 2 ) : [0, T ] → E 1 , and a bonding field We notice that the variational problem PV is formulated in terms of displacement field, damage field and bonding field. The existence of the unique solution of problem PV is stated and proved in the next section. To this end, we consider the following remark which is used in different places of the paper.

An existence and uniqueness result
Now, we propose our existence and uniqueness result.
The proof of Theorem 4.1 is carried out in several steps that we prove in what follows, everywhere in this section we suppose that assumptions of Theorem 4.1 hold, and we consider that C is a generic positive constant which depends on Ω , Γ 1 , Γ 2 , Γ 3 , pτ , A , G , F , φ , κ , and L. Let a η = (η 1 , η 2 ) ∈ C(0, T ; V ) be given. In the first step we consider the following variational problem.
We have the following result for the problem PV u η . Lemma 4.2 There exists a unique solution to problem P Vη which satisfies the regularity (45).
Proof. We use the Riesz representation theorem to define the operators A : V → V , such that It follows from (51) and (23)(a) that which shows that A : V → V is Lipschitz continuous. Now, by (51) and (23) (b), we find i.e., that A : V → V is a strongly monotone operator on V . therefore A is invertible and its inverse A −1 is also strongly monotone operator Lipschitz continuous on V . Moreover using Riesz Representation Theorem we may define an element fη ∈ (0, T ; V ) by It follows now from classical result that there exists a unique function vη ∈ (0, T ; V ) which satisfies Avη(t) = fη(t). (54) It follows from (51)-(55) that uη is a solution of the variational problem P Vη and it satisfies the regularity expressed in (45). This concludes the existence part of lemma1. The uniqueness of the solution follows from the uniqueness of the solution of the problem (54).
In the second step, we use the displacement field uη obtained in Lemma 4.2 and we consider the following initial-value problem.
Problem PV β η Find the adhesion βη : We have the following result.
The property β ηζ (t) ∈ Z so we can consider the operator Λη : for which we prove that a single fixed point.
Applying the Gronwall lemma we find and by integrating this inequality in Γ 3 was From (61) and (62) we find Reiterating this inequality n times it comes It concludes that n sufficiently large, an iterated Λ n η of Λη is a contraction in the Banach space L ∞ (0, T ; L ∞ (Γ 3 )). Then, there exists a unique ζ * η :∈ L ∞ (0, T ; L ∞ (Γ 3 )) such that Λ n η ζ * η = ζ * η and moreover ζ * η is also the unique fixed point Λη. Let βη = β ηζ * η the solution of (58), (59) for ζ = ζ * η . Using (61) and the relation (34) we obtain ∀t ∈ [0, T ] and keeping in mind (58)-(60) it follows that βη is a solution to the problem P β V and satisfies (47) and βη ∈ Z. Which concludes the existence part of Lemma 4.3. The uniqueness follows from the uniqueness of the fixed point of the operator Λη given by (61).
In the third step we let θ ∈ C(0, T ; E 0 ) be given and consider the following variational problem for the damage.
Theorem 4.4 Let X ⊂ Y = Y ⊂ X be a Gelfand triple. Let F be a nonempty, closed, and convex set of X. Assume that a(·, ·) : X × X → R is a continuous and symmetric bilinear form such that for some constants α > 0 and c 0 , Then, for every u 0 ∈ F and f ∈ L 2 (0, T ; Y ), there exists a unique function u ∈ H 1 (0, T ; Y ) ∩ L 2 (0, T ; X) such that We prove next the unique solvability of Problem PV ζ θ . Lemma 4.5 There exists a unique solution ζ θ of Problem PV ζ θ and it satisfies Proof. The inclusion mapping of (E 1 , · E1 ) into (E 0 , · E0 ) is continuous and its range is dense. We denote by E 1 the dual space of E 1 and, identifying the dual of E 0 with itself, we can write the Gelfand triple We use the notation (·, ·) E 1 ×E1 to represent the duality pairing between E and E 1 . We have and we note that K is a closed convex set in E 1 . Then, using (33), (34) and the fact that α 0 ∈ K in (32), it is easy to see that Lemma 4.5 is a straight consequence of Theorem 4.4.
Finally as a consequence of these results and using the properties of the operator G , the operator F , the functional j and the functional φ , for t ∈ [0, T ], we consider the element defined by the equations 68 Bilateral Contact Problem with Adhesion between Two Bodies for Viscoelastic with Long-term Memory and Damage Here, for every (η, θ) ∈ C(0, T ; V × E 0 ), uη, βη and α θ represent the displacement field, bonding field and the damage field obtained in Lemmas 4.2, 4.3 and 4.5 respectively, We have the following result.
Proof of Uniqueness. The uniqueness of the solution is a consequence of the uniqueness of the fixed point of the operator Λ defined by (66)-(68) and the unique solvability of the Problems PV u η , PV β η , and PV α η .