On some frictional contact problems with velocity condition for elastic and visco-elastic materials

We study the evolution of a class of quasistatic problems, which 
describe frictional contact between a body and a foundation. The 
constitutive law of the materials is elastic, or visco-elastic: 
with short or long memory, and the contact is modelled by a 
general subdifferential condition on the velocity. We derive weak 
formulations for the models and establish existence and 
uniqueness results. The proofs are based on evolution variational 
inequalities, in the framework of monotone operators and $fi$xed 
point methods. We show the approach of the viscoelastic solutions 
to the corresponding elastic solutions, when the viscosity tends 
to zero. Finally we also study the approach to short memory 
visco-elasticity when the long memory relaxation coefficients 
vanish.


1.
Introduction. Situations involving contact between deformable bodies take place frequently in industry and everyday life, such as impact problems, the tire with the road, or a shoe with the floor, which are just but three simple examples. The contact conditions could be very various, thus the modelling of the phenomena may be very complex, which is a source of richness of the domain of research, and the mathematical as well as engineering literature concerning this topic is rather extensive. An early attempt to study contact problems for elastic and viscoelastic materials within the framework of variational inequalities was made in the pioneering reference works [11,13,23]. Excellent references on analysis and numerical approximation of variational inequalities arising from contact problems are [20] and [17]. The mathematical, mechanical and numerical state of the art can be found in the proceedings [26].
Quasistatic processes arise when the forces applied to a system vary slowly in time so that acceleration is negligible. A number of papers investigating quasistatic frictional contact problems with viscoelastic materials have been published in [2,9,1,27]. In [9] the frictional contact was modelled by a general velocity dependent dissipation functional. In [2], frictional contact with normal damped Here and below, the indices i and j run between 1 and 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.
Let Ω ⊂ R d be a bounded domain with a Lipschitz continuous boundary Γ, which will represent the body in contact with a foundation. We need also the following notations of spaces and operator, to describe the displacement field u = (u i ), the stress field σ = (σ ij ) and the linearized strain tensor ε(u) = (ε ij (u)) , in the framework of small deformations.
The spaces H, H, H 1 and H 1 are real Hilbert spaces endowed with the inner products given by The associated norms on the spaces H, H, H 1 and H 1 are denoted by | · | H , | · | H , | · | H1 and | · | H1 , respectively.
Since the boundary Γ is Lipschitz continuous, the unit outward normal vector ν on the boundary is defined a.e. For every vector field v ∈ H 1 we use the notation v to denote the trace γ v of v on Γ and we denote by v ν and v τ the normal and the tangential components of v on the boundary given by For a regular (say C 1 ) stress field σ, the application of its trace on the boundary to ν is the Cauchy stress vector σν. We define, similarly, the normal and tangential components of the stress on the boundary by the formulas and we recall that the following Green's formula holds: Finally, for every real Hilbert space X we use the classical notation for the spaces L p (0, T, X) and W k,p (0, T, X), 1 ≤ p ≤ +∞, k = 1, 2, ...

Mechanical problems and weak formulations.
3.1. The mechanical problems. Here we describe three mathematical models for the quasistatic process of frictional contact between a deformable body and an obstacle, the so-called foundation. In the first problem QP ve we assume that the body has a viscoelastic behavior of short memory. In the second one QP ver , we assume that the body is viscoelastic of long memory. In the third one QP e , the body is supposed to be purely elastic.
For the viscoelastic body we use a Kelvin-Voigt constitutive law, i.e. of the type where G is the elasticity operator and A is the viscosity operator. Here and everywhere in the sequel the dot above represents the time derivative.
For technical reason, we model the behavior of the elastic body with the linear elastic constitutive law σ ij = g ijkl ε kl (u).
In other words, the operator G is defined as an elastic tensor. Finally, we model the frictional contact with a velocity boundary condition into the following general form Here U represents the set of contact admissible test functions, σν denotes the Cauchy stress vector on the contact boundary, and ϕ is a given convex function.
The above inequality holds almost everywhere on the contact surface. Examples and detailed explanations of inequality problems in contact mechanics which lead to boundary conditions of this form can be found in the monograph [24]. The physical setting is as follows. A deformable body occupies the domain Ω ⊂ R d and is acted upon by given forces and tractions and, as a result, its mechanical state evolves over the time interval [0, T ], T > 0. We assume that the boundary Γ of Ω is partitioned into three disjoint measurable parts Γ 1 , Γ 2 and Γ 3 , such that meas (Γ 1 ) > 0, where meas denotes the (d − 1)−dimensional Lebesgue measure in R d . The body is clamped on Γ 1 × (0, T ) and surface tractions of density f 2 act on Γ 2 × (0, T ). The solid is in frictional contact with a rigid obstacle on Γ 3 × (0, T ) and this is where our main interest lies. Moreover, a volume force of density f 0 acts on the body in Ω × (0, T ). We assume a quasistatic process and use a boundary velocity contact conditions. With these assumptions, our three initial mechanical problems may be stated as follows. and Div

Functional framework.
To obtain variational formulations of these mechanical problems QP , we suppose that We define the space of admissible displacements Let j : V → R be the contact functional defined by We suppose everywhere in the sequel that V is a closed subspace of H 1 , and j is a convex, proper and lower semicontinuous function on V.

Assumptions.
We suppose that the viscosity nonlinear operator to be Lipschitz continuous and strongly monotone: The elastic operator is linear and coercive: in Ω. (2) Then we could define the inner product on the space of admissible displacement as follows:

KHALID ADDI, OANH CHAU AND DANIEL GOELEVEN
The relaxation operator satisfies: The volume and surface densities of the body verify: This previous assumption allows us to define the following vector function: and verifying the regularity: Finally the initial displacement satisfies and Using then Green's formula, we obtain the following weak formulations for our different mechanical problems.
4. Existence and uniqueness results in elasticity and visco-elasticity. Now we are going to prove an existence and uniqueness result for each variational formulation.
Proof. From (1) we can define A : V → V by: Then A is Lipschitz continuous and strongly monotone. For any fixed η ∈ W 1,2 (0, T ; H) and t ∈ [0, T ], we deduce by general result on variational inequality of second kind the existence of v η (t) (see e.g. [18,21]).
To prove the regularity of v η , we verify that for any t 1 , t 2 ∈ [0, T ] : Using then the strong monotonicity of A, we deduce that there exists some constant c > 0 Then v η is a.e. differentiable, and the regularity η ∈ W 1,2 (0, T ; H) follows from those of f and η. Proof. Let η 1 , η 2 ∈ W, t ∈ [0, T ]. By the definition of | · | V and the linearity of G, we have: To continue, we recall that W is a Banach space endowed with the norm

KHALID ADDI, OANH CHAU AND DANIEL GOELEVEN
Then from the last two inequalities we verify that for any η 1 , η 2 ∈ W, n ∈ N, we have This shows that for some n ∈ N, Λ n is a contracting operator. Then by Banach fixed point Theorem, Λ n has a unique fixed point which is also the unique fixed point of Λ (see e.g. Tome 1 page 17 in [28]).
We have now all the ingredients to prove the Theorem 4.1. Let η * ∈ W be the fixed point of Λ. Define Using η = η * in Lemma 4.2, we see that {u, σ} is solution of the variational inequality in QV P ve . Putting then into the variational inequality, we obtain We conclude that {u, σ} is the unique solution to Problem QV P ve with the corresponding regularity. The uniqueness comes from the uniqueness in Lemma 4.2 and in Lemma 4.3. u ∈ W 2,2 (0, T ; V ), σ ∈ W 1,2 (0, T ; H 1 ).
Proof. We proceed closely as in the proof of Theorem 4.1. Let consider the set W = {η ∈ W 1,2 (0, T ; H) | η(0) = Gε(u 0 )} and the operator Λ : After some algebraic manipulations, it is not difficult to see that there exists some constant c > 0, such that for all η 1 , η 2 ∈ W, and for all t ∈ [0, T ], we have:  Proof. The result in Theorem 4.5 is a direct consequence of the following general result. Theorem.(Brézis) Let (V, (·, ·) V ) be a real Hilbert space and let j : V → (−∞, +∞] be a convex proper lower semi-continuous functional. Let f ∈ W 1,2 (0, T ; V ) and u 0 ∈ V be such that Then, there exists a unique element u ∈ W 1,2 (0, T ; V ) which satisfies for all v ∈ V, a.e. t ∈ (0, T ), and Theorem (Brézis) has been proved in [6], p.117, using arguments of evolution equations in the framework of maximal monotone operators. A version of this theorem has been considered in [18], where the proof was based on a time-discretization method.
Note here that V is our admissible displacement space, endowed with the inner product defined by the linear tensor G, which verifies (2). The vector function f from densities of applied forces has been defined after the hypothesis (4). Finally the conditions (5) and (6) on the initial displacement allow us to use Theorem (Brézis), and to conclude the existence of displacement solution field u in the problem QV P e , with u ∈ W 1,2 (0, T ; V ). We use now an argument similar to that used at the end of the proof of Theorem 1 in QP ve , to obtain σ ∈ W 1,2 (0, T ; H 1 ). This concludes the existence part of Theorem 4.5. The uniqueness part results from the uniqueness guaranteed by Theorem (Brézis).
Elastic problem. Let u ∈ W 1,2 (0, T ; V ), σ ∈ W 1,2 (0, T ; H 1 ) be the unique solution of the elastic problem QV P e . Theorem 5.1. There exists some constant c > 0 such that : We deduce then immediately the following consequence.
Corollary 1. Assume : Proof of Theorem 5.1. Put w =u(t) in QV P ve θ and w =u θ (t) in QV P e . We have for all t ∈ [0, T ] : We then add the two inequalities to obtain: Then we integrate the last inequality by using the strong monotonicity of the viscosity operator: Div σ θ (t) = Div σ(t) = −f 0 (t); we see that This gives the results stated in Theorem 5.1.
6. Approach to short memory visco-elasticity when the relaxation tends to zero. Let a family of bounded initial displacements u λ 0 ∈ V, ∀λ > 0 Family of long memory visco-elastic problems. QV P ver λ . Let any λ > 0. Define the displacement field u λ (0) = u λ 0 . Short memory visco-lastic problem. Let u ∈ W 2,2 (0, T ; V ) be the unique displacement solution field of the problem QV P ve . Theorem 6.1. There exists some constant c > 0 such that : We obtain then: Proof of Theorem 6.1. Put w =u(t) in QV P ver λ and w =u λ (t) in QV P ve . Add the inequalities and integrate by using again the strong monotonicity of the viscosity operator. We obtain for all t ∈ [0, T ] : From which we deduce that

KHALID ADDI, OANH CHAU AND DANIEL GOELEVEN
We deduce that The conclusion follows then by using the following version of Gronwall's inequality (see e.g. Lemma 2 page 10 in [4]), that we recall. Then we have ∀t ∈ [0, T ], ξ(t) ≤ c 1 e c2 T .