Bilateral contact problem with adhesion and damage

We study a mathematical problem describing the frictionless adhesive contact between a viscoelastic material with damage and a foundation. The adhesion process is modeled by a bonding field on the contact surface. The contact is bilateral and the tangential shear due to the bonding field is included. We establish a variational formulation for the problem and prove the existence and uniqueness of the solution. The existence of a unique weak solution for the problem is established using arguments of nonlinear evolution equations with monotone operators, a classical existence and uniqueness result for parabolic inequalities, and Banach’s fixed point theorem.


Introduction
Processes of adhesion are important in industry where parts, usually non metallic, are glued together.Recently, composite materials reached prominence, since they are very strong and light, and therefore, of considerable importance in aviation, space exploration and in the automotive industry.However, composite materials may undergo delamination under stress, in which different layers debond and move relative to each other.To model the process when bonding is not permanent, and debonding may take place, we need to describe the adhesion together with the contact.A number of recent publications deal with such models, see, e.g.[4,5,7,13,14,18,19] and references therein.The main new idea in these papers is the introduction of an internal variable, the bonding field, which has values between zero and one, and which describes the fractional density of active bonds on the contact surface.Reference [11] deals with the static and quasistatic problems, and their numerical approximations.A model for the process of dynamic, frictionless, adhesive contact between a viscoelastic body and a foundation was recently considered in [13].There the contact was modeled with normal compliance and the material was assumed to be linearly viscoelastic.
The present paper represents a continuation of [13,14] and deals with a model for the dynamic, adhesive and the frictionless contact between a viscoelastic body and a foundation.The difference consits in the fact that here we assume a bilateral contact and we use a nonlinear Kelvin-Voigt viscoelastic constitutive law with growth assumptions on the viscoelastic operator, which leads to a new and nonstandard mathematical model.As in [6,11], we use the bonding field as an additional dependent variable, defined and evolving on the contact surface.Our purpose is to provide the existence of a unique weak solution to the model.
The subject of damage is extremely important in design engineering since it affects directly 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 have been investigated mathematically.General novel models for damage were derived in [8,9] from the virtual power principle.Mathematical analysis of one-dimensional problems can be found in [10].In all these papers the damage of the material is described by 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 a partial damage and the system has a reduced load carrying capacity.Contact problems with damage have been investigated in [11,17].
In this paper, the inclusion describing the evolution of damage field is where K denotes the set of admissible damage functions defined by k is a positive coefficient, ∂ϕ 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.A general viscoelastic constitutive law with damage is given by where A is the nonlinear viscosity function, G is the nonlinear elasticity function which depends on the internal state variable describing the damage of the material caused by elastics deformations, and the dot represents the time derivative, i.e., The main aim of this paper is to couple a viscoelastic problem with damage and a frictionless contact problem with adhesion.We study a dynamic problem of frictional adhesive contact.We model the material behavior with a viscoelastic constitutive law with damage and the contact with normal compliance with adhesion.We derive a variational formulation and prove the existence and uniqueness of a weak solution.
The paper is organized as follows.In Section 2 we introduce the notation and give some preliminaries.In Section 3 we present the mechanical problem, list the assumptions on the data, give the variational formulation of the problem.In Section 4 we state our main existence and uniqueness result, Theorem 3.1.The proof of the theorem is based on the theory of evolution equations with monotone operators, a fixed point argument and a classical existence and uniqueness result for parabolic inequalities.

Notation and preliminaries
In this short section, we present the notation we shall use and some preliminary material.For more details, we refer the reader to [6].
We denote by S d the space of second order symmetric tensors on R d , (d = 2, 3) while (•, •) and | • | represent the inner product and Euclidean norm on R d and S d respectively.
Let Ω ⊂ R d be a bounded domain with a regular boundary Γ and let ν denote the unit outer normal on Γ.We shall use the notations where ε : H 1 → H and Div : H 1 → H are the deformation and divergence operators, respectively, defined by Here and below, the indices i and j run from 1 to d, the summation convention over repeated indices is assumed, 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, H 1 and H 1 are real Hilbert spaces endowed with the canonical inner products given by The associated norms on the spaces H, H, H 1 and H 1 are denoted by Γ) d and let γ : H 1 → H Γ be the trace map.For every element v ∈ H 1 we also write v for the trace γv of v on Γ and we denote by v ν and v τ the normal and tangential components of v on Γ given by Similarly, for a regular (say C 1 ) tensor field σ : Ω → S d we define its normal and tangential components by We recall that the following Green's formula holds Moreover, for a real number r, we use r + to represent its positive part, that is, r + = max{0, r}.

Problem statement
The physical setting is as follows.A viscoelastic body occupies the domain Ω ⊂ R d , (d = 2, 3) with outer Lipschitz surface Γ that is divided into three disjoint measurable parts Γ 1 , Γ 2 and Γ 3 such that meas Γ 1 > 0. Let T > 0 and let [0, T] be the time interval of interest.The body is clamped on Γ 1 × (0, T) and, therefore, the displacement field vanishes there.A volume force density f 0 acts in Ω × (0, T) and surface tractions of density f 2 act on Γ 2 × (0, T).The body is in bilateral adhesive and frictionless contact with an obstacle, the so-called foundation, over the contact surface Γ 3 .Moreover, the process is dynamic, and thus the inertial terms are included is the equation of motion.We use a viscoelastic constitutive law with damage to model the material's behavior and an ordinary differential equation to describe the evolution of the bonding field.The classical formulation of the problem may be stated as follows.
Problem P ) ) ) ) ) ) Equation (3.1) represents the equation of motion in which ρ denotes the material mass density.The relation (3.2) represents the nonlinear viscoelastic constitutive law with damage introduced in Section 1, the evolution of the damage field is governed by the inclusion (3.3),where φ is the mechanical source of the damage growth, assumed to be a rather 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. Conditions (3.4) and (3.5) are the displacement and traction boundary conditions, respectively.Condition (3.6) shows that the contact is bilateral, i.e., there is no loss of the contact during the process, while condition (3.7) shows that the tangential traction depends on the intensity of adhesion and the tangential displacement.Equation (3.8) governs the evolution of the adhesion field, here H ad is a general function discussed below and R : R + → [0, L] is the truncation function defined as where L > 0 is a characteristic lenght of the bonds (see, e.g., [2]).Equation (3.9) represents a homogeneous Neumann boundary condition where ∂α ∂ν represents the normal derivative of α.
In (3.10) we consider the initial conditions where u 0 is the initial displacement, v 0 the initial velocity and α 0 the initial damage.Finally, (3.11) is the initial condition, in which β 0 denotes the initial bonding.
To obtain the variational formulation of the problem (3.1)-(3.11),we introduce subspace of H 1 defined by Since meas(Γ 1 ) > 0 Korn's inequality holds and there exists a constant C k > 0 which depends only Ω and Γ 1 such that On V we consider the inner product and the associated norms given by ) is a real Hilbert space.Moreover, by the Sobolev trace theorem there exists a constant C 0 depending only on Ω, Γ 1 and In the study of the mechanical problem (3.1)-(3.11),we make the following assumptions.The viscosity operator A : Ω × S d → S d satisfies the following assumptions.
The elasticity operator G : Ω × S d × R → S d satisfies the following assumptions.
We also suppose that the body forces and surface traction have the regularity Finally, we assume that the initial data satisfy the following conditions ) ) We define the bilinear form a : We will use a modified inner product on H = L 2 (Ω) d , given by that is, weighted with ρ, and we let ) 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 We use the notation (•, •) V ×V to represent the duality pairing between V and V.We have Keeping in mind (3.17), we observe that the integrals in (3.27) are well defined.Using standard arguments based on Green's formula (2.3) we can derive the following variational formulation of the problem P.

Problem PV
Find a displacement field u : [0, T] → V a stress field σ : [0, T] → H a damage field α : [0, T] → H 1 (Ω) and an adhesion field ) a.e.t ∈ [0, T], a.e.t ∈ [0, T], We notice that the variational problem PV is formulated in terms of the displacement, stress field, damage field and adhesion field.The existence of a unique solution of problem PV is stated and proved in the next section.
Our main result, concerning the well-posedness of the problem PV is the following.

.36)
A quadruple {u, σ, α, β} which satisfies (3.28)-(3.32) is called a weak solution to the Problem P. We conclude that under the stated assumptions, problem (3.1)-(3.11)has a unique solution satisfying (3.33)-(3.36).The proof of Theorem (3.1) will be carried out in several steps and is based on the theory evolution equations with monotone operators, a fixed point argument and a classical existence and uniqueness result for parabolic inequalities.To this end, we assume in the following that (3.14)-(3.23)hold.Below, C denotes a generic positive constant which may depend on Ω, Γ 1 , Γ 2 , Γ 3 , A, G, p τ , L and T but does not depend on t nor on the rest of the input data, and whose value may change from place to place.Moreover, for the sake of simplicity, we suppress, in what follows, the explicit dependence of various functions on x ∈ Ω ∪ Γ.
The proof of Theorem 3.1 will be provided in the next section.

Existence and uniqueness result
Let η ∈ L 2 (0, T; V ) be given.In the first step we consider the following variational problem.

Problem P η V
Find a displacement field u η : [0, T] → V such that To solve Problem P η V , we apply an abstract existence and uniqueness result which we recall for the convenience of the reader.Let V and H denote real Hilbert spaces such that V is dense in H and the inclusion map is continuous, H is identified with its dual and with a subspace of V , i.e., V ⊂ H ⊂ V we say that these inclusions define a Gelfand triple.The notations | • | V , | • | V and (•, •) V ×V represent the norms on V and on V and the duality pairing between them, respectively.The following abstract result may be found in [20, page 48].Theorem 4.1.Let V, H be as above, and let A : V → V be a hemicontinuous and monotone operator which satisfies ) for some constants ω > 0, C > 0 and λ ∈ R.Then, given u 0 ∈ H and f ∈ L 2 (0, T; V ) there exists a unique u which satisfies We apply this theorem to problem P η V .Lemma 4.2.There exists a unique solution to problem P η V possessing the regularity condition expressed in (3.33).
It now follows from Theorem 3.1 that there exists a unique function v η which satisfies ) ) and it has the regularity expressed in (3.33).This concludes the proof of the existence part of Lemma 4.2.The uniqueness of the solution to problem (4.9)-(4.10),guaranteed by Theorem 4.1.
In the second step, we use the displacement field u η obtained in Lemma 4.2 and consider the following initial-value problem.
A. Aissaoui and N. Hemici (4.12) We have the following result.

Lemma 4.3.
There exits a unique solution β η of problem P β V and it satisfies Proof.For the sake of simplicity we suppress the dependence of various functions on x ∈ Γ 3 .Notice that the equalities and inequalities below are valid a.e.x ∈ Γ 3 .We consider the mapping It easy to check that F is Lipschitz continuous with respect to the second variable, uniformly in time; also, for all Thus, the existence of a unique function β η which satisfies(4.12)-(4.13)follows from a version of the Cauchy-Lipschitz theorem.
We now study the dependence of the solution of problem P β V with respect to η. Lemma 4.4.Let η i ∈ L 2 (0, T; V ) and let β η i , i = 1, 2, denote the solution of problem P β V , then Proof.Let t ∈ [0, T] The equalities and inequalities below are valid a.e.x ∈ Γ 3 and, as usual, we do not depict the dependence on x explicitly.Using (4.12) and (4.13), we can write where β i = β η i and u i = u η i .Using now the adhesion rate function H ad and the definition (3.18)(a) of the truncation R, we obtain Bilateral contact problem with adhesion and damage

11
We apply now Gronwall's inequality to deduce that Integrating the last inequality over Γ 3 we find and from (3.13), we obtain In the third step, let θ ∈ L 2 (0, T; L 2 (Ω)) be given and consider the following variational problem for the damage field.
We apply this theorem to problem P θ V .
Lemma 4.6.Problem R θ V has a unique solution α θ such that The inclusion of (H ) is continuous and its range is dense.We denote by (H 1 (Ω)) the dual space of H 1 (Ω) and, identifying the dual of L 2 (Ω) with itself, we can write the Gelfand triple We use the notation (α, ζ) (H 1 (Ω)) ×H 1 (Ω) for the duality pairing between (H 1 (Ω)) and H 1 (Ω).We have and we note that K is closed convex set in H 1 (Ω).Then, using the definition (3.24) of the bilinear form a and the fact that α 0 ∈ K in (3.22), it is easy to see that Lemma 4.6 is a straightforward consequence of Theorem 4.5.
Finally, as a consequence of these results and using the properties of the operator G, the functional j and the function φ for t ∈ [0, T] we consider the element defined by the equalities We have the following result.

.24)
Bilateral contact problem with adhesion and damage 13 Since Moreover, from (4.1) we infer that a.e. on (0, T) We integrate this equality with respect to time.We use the initial conditions v 1 (0) = v 2 (0) = v 0 and (3.14) to find that Then, using the inequality 2ab ≤ a 2 γ + γb 2 we obtain On the other hand, from the Cauchy problem adhesion we can write and then

Using the definition of R(|u
Next, we apply Gronwall's inequality to deduce and from (3.13) we obtain , a.e.∈ (0, T).
Integrating the inequality with respect to time, using the initial conditions α 1 (0) = α 2 (0) = α 0 and the inequality a(α 1 − α 2 , α 1 − α 2 ) ≥ 0 we find  Thus for m sufficiently large, Λ m is a contraction on the Banach space L 2 (0, T; V × L 2 (Ω)) and so Λ has a unique fixed point.Now, we have all the ingredients needed to prove Theorem 3.1

Finally, we denote
by | • | V the norm on the dual space V .Assumptions (3.20) allow us, for a.e.t ∈ (0, T) to define f