ANALYSIS OF QUASISTATIC FRICTIONAL CONTACT PROBLEM WITH SUBDIFFERENTIAL FORM, UNILATERAL CONDITION AND LONG-TERM MEMORY

We consider a quasistatic problem which models the contact between a deformable body and an obstacle called foundation. The material is assumed to have a viscoelastic behavior that we model with a constitutive law with long-term memory, thus at each moment of time, the stress tensor depends not only on the present strain tensor, but also on its whole history. In Contact Mechanics, history-dependent operators could arise both in the constitutive law of the material and in the frictional contact conditions. The mathematical analysis of contact models leads to the study of variational and hemivariational inequalities. For this reason a large number of contact problems lead to inequalities which involve history dependent operators, called history dependent inequalities. Such inequalities could be variational or hemivariational and variational hemivariational. In this paper we derive a weak formulation of the problem and, under appropriate regularity hypotheses, we stablish an existence and uniqueness result. The proof of the result is based on arguments of variational inequalities monotone operators and Banach fixed point theorem.


Introduction
Contact mechanics still remain a rich domain of research, and the literature devoted to various aspects of the subject is growing. An early attempt at the study of contact problems for elastic viscoelastic materials within the mathematical analysis framework was introduced in the pioneering reference works [6,7,15]. Further extensions to non convex contact conditions with non-monotone and possible multi-valued constitutive laws led to the active domain of non-smooth mechanic within the framework of the so-called hemivariational inequalities, for a mathematical as well as mechanical treatment we refer to [10]. There is a growing interest in the study of history-dependent inequalities. For instance, a class of variational inequalities with history dependent operators was considered in [15], where abstract existence, uniqueness and regularity results were proved. These results were extended in [18] to a more general class of variational inequalities and were completed in [6] with error estimate and convergence results. Various results on hemivariational and variational-hemivariational inequalities with history dependent operators, formulated in Sobolev-type spaces, could befound in [7,9].
We introduce a new model of frictional contact for viscoelastic materials and to illustrate the use of history dependent variational hemivariational inequality in its variational analysis. Thus, in Section 2 we introduce the contact problem, in which the material's behavior is modeled by a nonlinear viscoelastic constitutive law with long memory, the process is quasistatic, the contact is frictional and the contact conditions are in a subdifferential form with unilateral conditions for the displacement. Then, in Section 3 we list the assumptions on the data and derive the variational formulation of the problem. It is in a form of a historydependent variational-hemivariational inequality in which the unknown is the displacement field.Next in Section 4 we state our main existence and uniqueness result, Theorem (4.2) the proof of the theorem is obtained by using arguments of elliptic variational-hemivariational inequalities and a fixed point result for history dependent operators.

The Contact Model
The physical setting we consider is the following. A deformable body occupies a domain Ω ⊂ R d (d = 1, 2, 3 in applications) with outer Lipschitz surface Γ that is divided into three disjoint measurable parts Γ i (i = 1, 2, 3) such that meas(Γ 1 ) > 0. Let [0, T ] be the time interval of interest, where T > 0. The body is clamped on Γ 1 × (0, T ) and therefore the displacement field vanishes there. A volume force of density f 0 acts in Ω ×(0, T ) and surface tractions of density f 1 act on Γ 2 ×(0, T ).
The body is in contact on Γ 3 ×(0, T ) with a rigid obstacle, the so-called foundation is in frictional contact.
We assume that the process is quasistatic with long term memory and we use (1) as constitutive law. We denote by u, σ and ε(u) the displacement field, the stress field and the linearized strain tensor, respectively, and let v be the unit outward normal vector to Γ. Here and below, we sometimes do not indicate explicitly the dependence of various functions on the spatial variable x ∈ Ω ∪ Γ. For a vector field u, we use notation u v = u ·v and u τ = u − u v v for the normal and tangential components of u on Γ. Similarly, for the stress field σ, its normal and tangential components on the boundary are defined by equalities σ v = (σv)·v and Finally, we use S d for the space of second order symmetric tensors on R d and "·" will represent the canonical inner product and the Euclidean norm on the spaces R d and S d , respectively. We also use the following notation: Here ε : H 1 → H and Div : H 1 → H are the deformation and the divergence operators, respectively, defined by: The spaces H, H, H 1 and H 1 are real Hilbert spaces endowed with the canonical inner products given by: We recall that C denotes the class of continuous functions; and C m , m ∈ N * the set of m times continuously differentiable functions.
Finally D(Ω) denotes the set of infinitely differentiable real functions with compact support in Ω; and W m,p ,m ∈ N, 1 ≤ p ≤ +∞ for the classical Sobolev spaces; and With these assumptions, the classical formulation or mathematical model which describes the equilibrium of the body in the physical setting above is the following.
Problem P. Find a displacement field u : Ω × R + → R d , a stress field σ : Ω × R + → S d and two interface First, Eq.(2.1) is the constitutive law for viscoelastic materials in which A represent the elasticity operator and B represents the relaxation tensor. Various comments and mechanical interpretation related to such kind of equations could be found in [8,16]. Equation (2.2) is the equilibrium equation that we use here since we assume that the process is quasistatic. Conditions

Variational analysis
To derive a variational formulation of the problem we use the spaces for the displacement field we use the we consider the space of fourth-order tensor fields which is a real Banach space with norm Finally, we use N for the set of positive integers and R + for the set of nonnegative real numbers. For a normed space X, we use the notation C(R + ; X) for the space of continuous functions defined on R + with values in X.
By the Sobolev trace theorem, we have γ being the norm of the trace operator γ : We now list the assumptions on the data and we assume that 1. the elasticity operator A : Ω × S d → S d satisfies the following properties (a) There exists L A > 0 such that for all ε 1 ,ε 2 ∈ S d ,a.e.x ∈ Ω, There exists m > 0 such that for all ε 1 ,ε 2 ∈ S d ,a.e.x ∈ Ω, 2. The relaxation tensor B is such that 3. the potential function j v : Γ 3 × R → R,assumed to satisfy the following conditions Next, we assume that the penetration bound g : Γ 3 → R, the memory function F m :Γ 3 × R → R + and the friction bound F b : Γ 3 × R → R are assumed to satisfy the following conditions.
We also assume that the densities of body forces and surface tractions have the regularity and, finally, we assume the smallness condition We now introduce the set of the admissible displacement fields U ⊂ V and the function f : Assume now that (u, σ) represents a couple of regular functions which satisfy (2.1) − (2.6) and let t ∈ R + , v ∈ U . We perform an integration by parts, split the surface integral on three integrals on Γ 1 , Γ 2 and Γ 3 , and use the equalities (2.2) − (2.4)to deduce that Next, we use the contact boundary condition (2.5), the definition (3.12) and the definition of the Clarke subdifferential to obtain that Note that here and below we use notation j 0 v (r 1 ; r 2 ) for the generalized directional derivative of j v at r 1 in the direction r 2 , see [1,2] for details.
On the other hand, the friction law (2.6) yields We now combine equality (3.13) with inequalities (3.14), (3.15) to deduce that Finally, we substitute the consitutive law (2.1) in (3.15) and use notation (3.12) to obtain the following variational formulation of Problem P, in terms of displacement.
Problem PV Find a displacement field u : R + → U such that The unique solvability of Problem PV is given by the following existence and uniqueness result, that we state here and prove in the next section.
We end this section with some remarks on the weak solvability of the contact problem P.
First, a couple of functions (u, σ) defined on the positive real line R + with values on the product space V × Q is called a weak solution to Problem P if u is a solution of the variational problem PV and σ satisfies the constitutive law (2.1).
We conclude that, under the assumption of Theorem 8.1, Problem P has a unique weak solution. Moreover, the solution has the regularity u ∈ C(R + ; U ) and σ ∈ C(R + ; Q).
Next, recall that Theorem 8.1 provides the weak solvability of the contact problem P under the smallness assumption (24) involving the friction bound F b , and the normal compliance potential j v . Finally, note that the unknowns η v and ξ v of Problem P cannot be recovered since they cannot be computed when the solution u of Problem P is known.Actually, these unknowns represent interface forces and, as usual in solving contact problems with unilateral constraints, we do not have information neither on the uniqueness of these functions and on their regularity.

An Existence and Uniqueness Result
We present in this section an abstract result on history-dependent variational-hemivariational inequalities that we shall use to prove the unique solvability of Problem PV. For more details on the material presented in this section, we send the reader to [1,2].
Theorem 4.1.Let X be a reflexive Banach space and Y be a normed space. We denote by X the dual of X and by ·, · X ×X the duality pairing of X and X . Let K be a subset of X and A : X → X , Ψ : C(R + ; X) → C(R + ; Y ) be given operators ,consider also a function φ : Y × K × K → R, a locally Lipschitz function j : X → R and a function f : R + → X . With these data we consider the problem of finding a function u : R + → U such that, for each t ∈ R + , the following inequality holds: In the study of (4.1), we assume the following hypotheses.
K is a nonempty, closed and convex subset of X.
A : X → X is an operator such that (a)A is pseudomonotone and there exist α A > 0, β A , γ A ∈ R and u 0 ∈ K such that: A is strongly monotone,i.e.,there exists m A > 0 such that (a) φ(y, u, ·) is convex and l.s.c. on K, for all y ∈ Y, u ∈ K For any n ∈ N , there exists s n > 0 such that (4.5) f ∈ C(R + ; X * ) (4.7) Note that an operator Ψ which satisfies condition (4.5) is called a history dependent operator. Inequality (4.1) is governed both by the function φ which is assumed to be convex with respect its second argument and by the function j which is locally Lipschitz and could be nonconvex. Therefore, this inequality is a variational-hemivariational inequality. In addition, the function φ in (4.1) depends on the operator Ψ , assumed to be history-dependent.For this problem we have the following existence and uniqueness result.
Theorem 4.2. Let X be a reflexive Banach space, Y a normed space, and assume that (4.2)-(4.7) hold.
The proof of is obtained by using arguments of elliptic variational-hemivariational inequalities and a fixed point result for history dependent operators.
Proof (Theorem 4.1) We start by defining the operators A : V → V ,F : C(R + ; V ) → C(R + ; Q×L 2 (Γ 3 )) and the functions φ : Then, it is easy to see that Problem PV is equivalent to the problem of finding a function u : R + → U such that for each t ∈ R + , the following inequality holds: To solve this problem, we use Theorem 4.1 with X = V , Y = L 2 (Γ 3 ) and K = U and, to this end, we check in what follows that assumptions (4.2)-(4.7) hold. We use arguments similar to those used in our previous works [8,9] and, for this reason, we skip the details and we resume the proof as follows. First, we note that assumption (3.7) and definition (3.12) imply (4.2). Next, a simple calculation based on the definition (4.5) of the operator A and the properties (3.4) of the elasticity operator show that (4.2) holds with m A = α A = m F . Moreover, using assumption (3.9) and the trace inequality (3.2), it is easy to see that the function φ defined by (4.5) satisfies condition (4.3) with α φ = L F b γ . On the other hand, assumption (3.6) on the function j v and definition (4.8) show that condition (4.4) holds with α j = α jv . And, a simple calculation based on assumptions (3.5), (3.9) imply that the operator (4.9) is a history-dependent operator, i.e., it satisfies condition (4.5). Now, keeping in mind that m A = α A = m F ,α φ =L F b γ and α j = α jv , we easily deduce that the smallness assumption (3.11) shows that conditions (4.6) hold, too. Finally, we note that regularity (3.9) on the densites of the body forces and tractions combined with definition (3.12) show that condition (4.7) is satisfied. We are now in a position to use Theorem 4.2 to deduce the existence of a unique function u ∈ C(R + ; U ) such that (4.12) holds, for each t ∈ R + . And, using notation (4.8)-(4.12), we deduce that u is the unique solution to Problem PV which concludes the proof.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.