This paper shows the existence of a solution of the quasi-static unilateral contact problem with nonlocal friction law for nonlinear elastic ma- terials

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

EXISTENCE OF SOLUTIONS FOR QUASISTATIC PROBLEMS OF UNILATERAL CONTACT WITH NONLOCAL FRICTION

FOR NONLINEAR ELASTIC MATERIALS

TOUZALINE AREZKI, MIGNOT ALAIN

Abstract. This paper shows the existence of a solution of the quasi-static unilateral contact problem with nonlocal friction law for nonlinear elastic materials. We set up a variational incremental problem which admits a solution, when the friction coefficient is small enough, and then by passing to the limit with respect to time we obtain a solution.

1. Introduction

We consider a Signorini’s quasistatic contact problem with nonlocal friction in nonlinear elasticity. In linear elasticity the quasistatic problem of unilateral contact using a normal compliance law has been solved in [1] by considering incremental problems and in [9] by an other method using a regularisation relative to time. The quasistatic contact problem with local or nonlocal friction has been solved respec- tively in [10] and in [4] by using a time-discretization method. In [2] the quasistatic contact problem with Coulomb friction was solved by the aid of an established shifting technique used to obtain increased regularity at the contact surface and by the aid of auxiliary problems involving regularized friction terms and a so-called normal compliance penalization technique. Signorini ’s problem with friction for nonlinear elastic materials or viscoelastic materials has been solved in [5] by using the fixed point’s method. In viscoelasticity, the quasistatic contact problem with a normal compliance law and friction has been solved in [11] by the same fixed point arguments. The book [8] introduces generally readers to a mathematical theory of contact problems involving deformable bodies. In carrying out the variational analysis, the authors systematically use results on elliptic and evolutionary variational inequalities, convex analysis, nonlinear equations with monotone operators, and fixed points of operators.

In this paper we propose a variational formulation using a classical regularization [6] of the normal stress characterizing the notion of nonlocal friction as in the linear case. The variational formulation is written in the form of two variational inequalities as in [4]. By using an implicit scheme as in [4, 10], we are led to solve

2000Mathematics Subject Classification. 35J85, 49J40, 47J20, 58E35.

Key words and phrases. Nonlinear elasticity; nonlocal friction; incremental;

variational inequality; monotone.

c

2005 Texas State University - San Marcos.

Submitted June 11, 2004. Published September 19, 2005.

1

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a sequence of incremental static problems and by passing to the limit, we show the existence of a solution of the quasistatic contact problem for a small enough friction coefficient.

2. Variational formulation

Let Ω⊂R^d (d= 2,3), be the domain initially occupied by the nonlinear elastic body. Ω is supposed to be open, bounded, with a sufficiently regular boundary Γ.

Γ is decomposed into three parts Γ = ¯Γ1∪Γ¯2∪Γ¯3where Γ1,Γ2,Γ3are disjoint open sets and meas(Γ1)>0. LetT >0 and let [0, T] denote the time interval of interest.

The body is clamped on Γ1 and thus the displacement field vanishes there. The body is acted upon by a volume force of densityφ1 on Ω and a surface traction of densityφ2 on Γ2. On Γ3the body is in unilateral contact with a rigid support and the conditions of contact are supposed to be as in [4].

Under these conditions the classical formulation of the mechanical problem is the following.

Problem P1. . Find a displacement field u: Ω×[0, T]→R^d such that

divσ(u) +φ1= 0 in Ω×(0, T), (2.1) σ(u) =z(ε(u)) in Ω×(0, T), (2.2)

u= 0 on Γ1×(0, T), (2.3)

σn(u) =φ2 on Γ2×(0, T), (2.4)

σ_N(u)≤0, u_N ≤0, σ_N(u).u_N = 0 on Γ₃×(0, T), (2.5)







|σT| ≤µ|RσN(u)|

|σT|< µ|RσN(u)|=⇒u˙T = 0

|σ_T|=µ|Rσ_N(u)|=⇒σ_T =−λu˙_T, λ≥0

on Γ₃×(0, T), (2.6)

u(0) =u0 in Ω (2.7)

We adopt the following notations as in [7]: Vectorn= (ni) is the outer unit normal vector to Γ;u= (ui) is the displacement field;uN =uiniis the normal displacement on Γ;uT =u−uNnis the tangential displacement on Γ. The strain tensor is

ε(u) = (ε_ij(u)) = (1

2(u_i,j+u_j,i)), i, j∈ {1, . . . , d};

the stress tensor isσ= (σij); divσ= (σij,j) is the divergence of σ, σN = (σn).n is the normal stress;σT =σn−σNnis the tangential stress. We denote bySd the space of second order symmetric tensors onR^d (d= 2,3).

To proceed with the variational formulation, we need the function spaces:

H =L²(Ω)^d, H1= (H¹(Ω))^d,

Q={τ = (τ_ij);τ_ij =τ_ji∈L²(Ω)}=L²(Ω)^d×d_s , H(div; Ω) ={σ∈Q; divσ∈H}

Note thatH andQare Hilbert spaces equipped with the respective scalar products (u, v)H =

Z

Ω

uividx, hσ, τiQ= Z

Ω

σijτijdx.

We recall that Green’s formula holds: forσ∈H(div; Ω) hσ, ε(v)iQ+ (divσ, v)H=hσn, vi

H⁻¹2(Γ)^d×H^1/2(Γ)^d∀v∈H1.

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LetV be the closed subspace ofH1 given by

V ={v∈H1;v= 0 on Γ1} andK be the set of admissible displacements given by K={v∈V;v_N ≤0 on Γ₃}.

Since meas Γ1>0, the following Korn’s inequality holds (see [5]):

kε(v)kQ≥cΩkvkH₁ ∀v∈V (2.8) where the constant c_Ω depends only on Ω and Γ₁. We equip V with the scalar product

(u, v)_V =hε(u), ε(v)i_Q

andk · kV is the associated norm. It follows from Korn’s inequality (2.8) that the normsk · kH1 andk · kV are equivalent onV. Then (V,k · kV) is a Hilbert space.

Moreover, by the Sobolev’s trace theorem, there exists a positive constant d_Ω depending only on the domain Ω, Γ₁and Γ₃such that

kvk_L2(Γ3)^d≤d_ΩkvkV ∀v∈V (2.9) Forp∈[1,∞] , we use the standard norm ofL^p(0, T;V). We also use the Sobolev spaceW^1,∞(0, T;V) equipped with the norm

kvkW^1,∞(0,T;V)=kvkL^∞(0,T;V)+kvk˙ L^∞(0,T;V).

For every real Banach space (X,k · kX) andT >0 we use the notationC([0, T];X) for the space of continuous functions from [0, T] toX; recall thatC([0, T];X) is a real Banach space with the norm

kxk_C([0,T];X)= max

t∈[0,T]kx(t)kX. The forces and tractions are assumed to satisfy

φ1∈W^1,∞(0, T;H), φ2∈W^1,∞(0, T;L²(Γ2)^d) (2.10) Letf : [0, T]→V be given by

(f(t), v)V = Z

Ω

φ1(t).vdx+ Z

Γ2

φ2(t).vda ∀v∈V, t∈[0, T].

We note that conditions (2.10) implyf ∈W^1,∞(0, T;V). Let H^1/2(Γ3) ={w

_Γ

3:w∈H^1/2(Γ) andw= 0 on Γ1}.

Leth·,·idenote the duality pairing betweenH^−1/2(Γ₃) andH^1/2(Γ₃).

The normal stressσN (u(t))∈H^−1/2(Γ3) associated tou(t)∈V is defined by

∀w∈H^1/2(Γ₃) :

hσN(u(t)), wi=hz(ε(u(t))), ε(v)iQ−(f(t), v)V

∀v∈V :v_N =w, w_T = 0 on Γ₃

(2.11)

R:H^−1/2(Γ₃)→L²(Γ₃) is a linear compact mapping which respects the positivity (see [6]).

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Hypotheses on the nonlinear elasticity operator. As in [5] we assume z : Ω×Sd→Sd satisfies the following conditions:

(a) There existsL1>0 such that|z(., ε1)−z(., ε2)| ≤L1|ε1−ε2|for allε1, ε2

inSd, a.e in Ω.

(b) There existsL2>0 such that (z(., ε1)−z(., ε2)).(ε1−ε2)≥L2|ε1−ε2|² for allε₁, ε₂∈S_d, a.e in Ω.

(c) For anyε∈S_d, the mapping x7→z(x, ε) is measurable on Ω.

(d) (x,0_d) = 0 for allxin Ω,

Remark 2.1. z(x, τ(x))∈Q, for allτ ∈Qand thus it is possible to consider z as an operator defined fromQtoQ.

We assume that the friction coefficient satisfies

µ≥0 a.e. on Γ₃ andµ∈L^∞(Γ₃) (2.12) Also we assume that the initial datau₀∈K satisfies

hz(ε(u0)), ε(v)−ε(u0)iQ+j(u0, v−u0)≥(f(0), v−u0)V ∀v∈K. (2.13) Now assuming that the solution is sufficiently regular, we formally multiply the equilibrium equation (2.1) by v−u(t) and by using techniques similar to those˙ exposed in [7] , we show that the problem (P1) has the following variational formulation.

Problem P2. Find a displacement field u: [0, T]→V, verifying u(0) =u₀ in Ω andu(t)∈K a.e. t∈[0, T], and such that a.e. t∈[0, T]:

hz(ε(u(t))), ε(v)−ε( ˙u(t))iQ+j(u(t), v)−j(u(t),u(t))˙

≥(f(t), v−u(t))˙ _V +hσN(u(t)), v_N −u˙_N(t)i ≥0 ∀v∈V (2.14) hσN(u(t)), zN−uN(t)i ≥0, ∀z∈K (2.15) where

j(u, v) = Z

Γ3

µ|RσN(u)| |vT|da.

The aim of this paper is to show the following result.

Theorem 2.2. Let (2.10),(2.11),(2.12)and (2.13)hold. Then problem (P2) has at least a solution u ∈ W^1,∞(0, T;V) for a small enough friction coefficient µ.

Moreover, there exists a constantC >0 such that

kuk_W^1,∞_(0,T_;V₎≤Ckfk_W^1,∞_(0,T_;V₎

For the proof of this theorem, we carry a time-discretization of problem (P2).

For n ∈ N^∗, we set ∆t = ^T_n, and ti =i∆t, i = 0, . . . ,(n−1); denote by u^tⁱ the approached solution of the solution u at the time ti and ∆u^tⁱ =u^tⁱ⁺¹ −u^tⁱ. By using an implicit scheme, we obtain a sequence of incremental problems, foru⁰∈K, define as

Problem (P_n^tⁱ). Findu^tⁱ⁺¹∈K such that

hz(ε(u^tⁱ⁺¹)), ε(w)−ε(u^tⁱ⁺¹)iQ+j(u^tⁱ⁺¹, w−u^tⁱ⁺¹)−j(u^tⁱ⁺¹, u^tⁱ⁺¹−u^tⁱ)

≥(f^tⁱ⁺¹, w−u^tⁱ⁺¹)V +hσN(u^tⁱ⁺¹), wN −u^t_Nⁱ⁺¹i, ∀w∈V hσN(u^tⁱ⁺¹), w_N −u^t_Nⁱ⁺¹i ≥0, ∀w∈K whereu⁰=u₀, andf^tⁱ⁺¹ =f(t_i+1).

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3. Existence of a solution of the incremental problem Lemma 3.1. Problem (P_n^tⁱ)is equivalent to the problem(Q^t_nⁱ)stated below.

Problem (Q^t_nⁱ). Findu^tⁱ⁺¹∈K such that

hz(ε(u^tⁱ⁺¹)), ε(w)−ε(u^tⁱ⁺¹)iQ+j(u^tⁱ⁺¹, w−u^tⁱ)−j(u^tⁱ⁺¹, u^tⁱ⁺¹−u^tⁱ)

≥(f^tⁱ⁺¹, w−u^tⁱ⁺¹)V ∀w∈K (3.1)

For the proof of Lemma 3.1 in the linear case, see see [4].

Proposition 3.2. There existsµ0>0 such that if kµk_L^∞_(Γ₃₎< µ0, then problem (Q^t_nⁱ)admits a unique solution.

To show proposition 3.2 we introduce an intermediate problem. We define the convex set

C₊^∗ ={g∈L²(Γ3);g≥0 a.e. on Γ3}, and the function

ϕ(w) = Z

Γ₃

µg|wT|da.

Then, we introduce an intermediate problem by replacing in RσN(u^tⁱ⁺¹) in (3.1) by forg∈C₊^∗ as follows:

Problem (Q^t_ngⁱ ). Findug inK such that

hz(ε(ug)), ε(w)−ε(ug)iQ+ϕ(w−u^tⁱ)−ϕ(ug−u^tⁱ)

≥(f^tⁱ⁺¹, w−u_g)_V ∀w∈K . (3.2)

Now, we have the following lemma.

Lemma 3.3. For any g ∈C₊^∗ problem (Q^t_ngⁱ ) has a unique solutionu_g. Further- more, there exists constantsc_i>0,i= 1,2, such that

ku_gk_V ≤c₁kµk_L∞(Γ₃)kgk_L2(Γ₃)+c₂kf^tⁱ⁺¹k_V (3.3) Proof. Using Riesz’s representation theorem we define the nonlinear operatorA : V →V by

(Av, w)V =hz(ε(v)), ε(w)iQ.

Then hypotheses (a) and and (b) onzimply thatAis a strictly monotone, coercive and lipschitzian operator; on the other hand the functionalϕis proper, convex and lower continuous. There results from the theory of elliptic variational inequalities [3]

that the inequality (3.1) has an unique solutionu_g. Settingw= 0 in the inequality (3.2) and using both the hypothesis (b) onzand the inequality

|(ug−u^tⁱ)T| − u^t_Tⁱ

≤ |ugT| we see that there exist constantsci>0,i= 1,2, such that

kugk²_V ≤c1kµkL^∞(Γ₃)kgkL²(Γ₃)kugkV +c2kf^tⁱ⁺¹kVkugkV.

Simplifying by the normkugkV we have the inequality (3.3).

Lemma 3.4. Let Ψ : C₊^∗ → C₊^∗ be the mapping Ψ(g) = −RσN(ug) there exists µ0 >0 such that if kµk_L∞(Γ3)< µ0, then Ψadmits a fixed point g^∗ and ug^∗ is a solution of problem(Q^t_nⁱ).

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Proof. Using (2.8), (2.11), the hypothesis (b) on z and the continuity of R, we deduce that there exists a constantC >0 such that

kΨ(g₁)−Ψ(g₂)k_L2(Γ₃)≤Cku_g₁−u_g₂k_V

On the other hand by settingv =ug₁ in (Q^t_ngⁱ₂) andv =ug₂ in (Q^t_ngⁱ₁) and then adding, we obtain by using the (b) on zand (2.8), that there exists a constant C1>0 such that

kug₁−ug₂kV ≤C1kµk_L^∞_(Γ₃₎kg1−g2k_L2(Γ3)

Hence there exists a constantC2>0 such that

kΨ(g₁)−Ψ(g₂)k_L2(Γ₃)≤C₂kµk_L∞(Γ₃)kg₁−g₂k_L2(Γ₃),

and whenµ0= 1/C2, we have forkµk_L∞(Γ3)< µ0, the mapping Ψ is a contraction, thus it has a fixed pointg^∗ andu_g∗ is the solution of problem (Q^t_nⁱ).

4. Existence of a solution of the quasistatic problem

Lemma 4.1. We have the following estimates: For a positive constant µ₁ > 0, whenkµkL^∞(Γ₃)< µ₁, there existsd_i>0,i= 1,2, such that

ku^tⁱ⁺¹kV ≤d1kf^tⁱ⁺¹kV (4.1) k∆u^tⁱk_V ≤d₂k∆f^tⁱk_V (4.2) Proof. By settingw= 0 in the inequality (3.1) and using hypothesis (b) onzand the properties of j, there exists c1 >0 such that for kµk_L^∞_(Γ₃₎ < c1, we deduce that there existsd1>0 such that (4.1) is satisfied.

To show the inequality (4.2) we consider inequality of (3.1) translated at the timeti that is:

hz(ε(u^tⁱ)), ε(w)−ε(u^tⁱ)iQ+j(u^tⁱ, v−u^tⁱ⁻¹)−j(u^tⁱ, u^tⁱ−u^tⁱ⁻¹)

≥(f^tⁱ, w−u^tⁱ)V ,∀w∈K (4.3)

By settingw=u^tⁱ in (3.1) andw=u^tⁱ⁺¹ in (4.3) and add them up, we obtain the inequality

− hz(ε(u^tⁱ⁺¹))−z(ε(u^tⁱ)), ε(∆u^tⁱ)iQ−j(u^tⁱ⁺¹,∆u^tⁱ) +j(u^tⁱ, u^tⁱ⁺¹−u^tⁱ⁻¹)−j(u^tⁱ, u^tⁱ−u^tⁱ⁻¹)

≥(−∆f^tⁱ,∆u^tⁱ)V

furthermore using the inequality

|u^t_Tⁱ⁺¹−u^t_Tⁱ⁻¹| − |u^t_Tⁱ−u^t_Tⁱ⁻¹|

≤ |u^t_Tⁱ⁺¹−u^t_Tⁱ| We have

j(u^tⁱ, u^tⁱ⁺¹−u^tⁱ⁻¹)−j(u^tⁱ, u^tⁱ−u^tⁱ⁻¹)≤ j(u^tⁱ,∆u^tⁱ). Therefore,

− hz(ε(u^tⁱ⁺¹))−z(ε(u^tⁱ)), ε(∆u^tⁱ)i_Q+j(u^tⁱ,∆u^tⁱ)−j(u^tⁱ⁺¹,∆u^tⁱ)

≥(−∆f^tⁱ,∆u^tⁱ)V . Using the properties ofj we have

−j(u^tⁱ,∆u^tⁱ) +j(u^tⁱ⁺¹,∆u^tⁱ)≤j(∆u^tⁱ,∆u^tⁱ).

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As a consequence we obtain the inequality

hz(ε(u^tⁱ⁺¹))−z(ε(u^tⁱ)), ε(∆u^tⁱ)iQ−j(∆u^tⁱ,∆u^tⁱ)−(∆f^tⁱ,∆u^tⁱ)V ≤0. (4.4) Using the relation (2.11), there exists a constantc3>0 such that

kσN(∆u^tⁱ)k

H⁻¹2(Γ₃)≤c3(k∆u^tⁱkV +k∆f^tⁱkV).

Then using the hypothesis (b) on zand the properties of j we deduce that there existsd3>0 such that

L2k∆u^tⁱk²_V ≤d3kkkL^∞(Γ₃)k∆u^tⁱk²_V +k∆f^tⁱkVk∆u^tⁱkV

Settingc2=_2d^L²

3, we deduce that ifkµkL^∞(Γ₃)< c2, there existsd4>0 such that k∆u^tⁱkV ≤d₄k∆f^tⁱkV

It suffices to takeµ₁= min(c₁, c₂) and the lemma is proved.

The proof of Theorem 2.2 is done as in [4], but inL^∞. For the next proposition, we define the continuous functionuⁿ: [0, T]→V by

uⁿ(t) =u^tⁱ+(t−t_i)

∆t ∆u^tⁱ on [t_i, t_i+1], i= 0, . . . , n−1.

Proposition 4.2. From the sequence (uⁿ) we can extract a subsequence still denoted(uⁿ)such that(uⁿ)converges weakly ∗in W^1,∞(0, T;V)to a functionu.

Proof. From (4.1) we deduce that the sequence (uⁿ) is bounded inC([0, T];V) and there exists a constantc₃>0 such that

0≤t≤Tmaxkuⁿ(t)k_V ≤c₃kfk_C([0,T];V₎

From (4.2) we deduce that the sequence ( ˙uⁿ) is bounded inL^∞(0, T;V) and there existsc4>0 such that

ku˙ⁿk_L^∞_(0,T;V₎= max

0≤i≤n−1k∆u^tⁱ

∆t kV ≤c4kf˙k_L^∞_(0,T;V₎

Then the sequence (uⁿ) is uniformly bounded inW^1,∞(0, T;V), and we thus can extract from it a subsequence still denoted (uⁿ) such thatuⁿ→uinW^1,∞(0, T;V) weakly∗ asn→ ∞and satisfying

kukW^1,∞(0,T;V)≤CkfkW^1,∞(0,T .V)

withC= max(c3, c4).

As in [10] let’s introduce the piecewise constant functions euⁿ : [0, T] →V and feⁿ: [0, T]→V defined by

ueⁿ(t) =u^tⁱ⁺¹, feⁿ(t) =f(ti+1) ∀t∈(ti, ti+1], i= 0, . . . , n−1.

Lemma 4.3. From the sequence (˜uⁿ) we can extract a subsequence still denoted (˜uⁿ)which satisfies the convergence results:

(i) ueⁿ→uweak∗ inL^∞(0, T;V)asn→ ∞ (ii) ueⁿ(t)→u(t)weakly inV a.e. t∈[0, T]

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Proof. From (4.1) we deduce that the sequence (ueⁿ) is uniformly bounded in L^∞(0, T;V). Thus, we can extract from it a subsequence still denoted (euⁿ) which converges weakly∗inL^∞(0, T;V). On the other hand as in [8] we thus deduce for anyt∈(0, T) the inequality

keuⁿ(t)−uⁿ(t)kV ≤T

nku˙n(t)kV, . (4.5) Since the sequence ( ˙uⁿ) is bounded inL^∞(0, T;V), we thus deduce from (4.5) that ueⁿ→uweak ∗in L^∞(0, T;V) asn→ ∞, whence (i).

For the proof of (ii), since W^1,∞(0, T;V)⊂C([0, T];V), we have uⁿ(t)→u(t) weakly inV, for allt∈[0, T], and from (4.5) we immediately we have the conclusion.

Remark 4.4. Sinceueⁿ(t)∈K a.e. t in [0, T] thenu(t)∈K a.e. t in [0, T]. On the other hand, sincef ∈W^1,∞(0, T;V), we deduce that

feⁿ→f strongly inL²(0, T;V) (4.6) Proposition 4.5. The sequence(˜uⁿ)converges strongly touinL²(0, T;V)andu is a solution of problem (P2) for a small enough friction coefficient.

Proof. From inequality (3.1) we deduce the inequality

hz(ε(u^tⁱ⁺¹)), ε(v)−ε(u^tⁱ⁺¹)iQ+j(u^tⁱ⁺¹, v−u^tⁱ⁺¹)≥(f^tⁱ⁺¹, v−u^tⁱ⁺¹)_V ∀v∈K whence

hz(ε(ueⁿ(t))), ε(v)−ε(euⁿ(t))iQ+j(ueⁿ(t), v−euⁿ(t))

≥(feⁿ(t), v−u˜ⁿ(t))V ∀v∈K, a. e. t∈[0, T], (4.7) we also have the inequality

hz(ε(ue^n+m(t))), ε(v)−ε(eu^n+m(t))i_Q+j(eu^n+m(t), v−ue^n+m(t))

≥(fe^n+m(t), v−eu^n+m(t))V∀v∈K, a.e. t∈[0, T]. (4.8) Setting v = euⁿ(t) in (4.7) and v =eu^n+m(t) in (4.8) and adding them, we obtain the inequality

hz(ε(eu^n+m(t)))−z(ε(ueⁿ(t))), ε(ueⁿ(t))−ε(eu^n+m(t))iQ

+ Z

Γ3

µ(|RσN(˜u^n+m(t))|+|RσN(˜uⁿ(t))|)|u˜^n+m_T (t)−u˜ⁿ_T(t)|da

≥ −(fe^n+m(t)−feⁿ(t),ue^n+m(t)−euⁿ(t))V

Then using (2.10) and that the mappingR is compact, we deduce that kRσN(˜uⁿ(t))k_L2(Γ3)≤CkσN(˜uⁿ(t))k

H⁻¹2(Γ3)

≤C₁( sup

t∈(0,T)

k˜uⁿ(t)k_V + sup

t∈(0,T)

kf˜ⁿ(t)k_V) Since

k˜uⁿ(t)kV ≤ kuⁿ(t)kV +T n sup

t∈(0,T)

ku˙ⁿ(t)kV, we have

sup

t∈(0,T)

k˜uⁿ(t)kV ≤ max

t∈[0,T]kuⁿ(t)kV +T sup

t∈(0,T)

ku˙ⁿ(t)kV

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So we deduce that there exists a constantC⁰>0 such that sup

t∈(0,T)

ku˜ⁿ(t)kV ≤C⁰kfkW^1,∞(0,T;V)

Whence we deduce also that there exits a constantC2>0 such that kue^n+m(t)−ueⁿ(t)k²_V

≤C2(kµk_L^∞_(Γ₃₎ku˜^n+m_T (t)−˜uⁿ_T(t)k_L2(Γ₃)^d+kfe^n+m(t)−feⁿ(t)k²_V).

Having in mind that

k˜u^n+m_T (t)−u˜ⁿ_T(t)k_L2(Γ₃)^d≤ k˜u^n+m_T (t)−u^n+m_T (t)k_L2(Γ₃)^d

+ku^n+m_T (t)−uⁿ_T(t)k_L2(Γ₃)^d+kuⁿ_T(t)−u˜ⁿ_T(t)k_L2(Γ₃)^d, since (uⁿ) is bounded inW^1,∞(0, T;V), the sequence (uⁿ|Γ₃) is relatively compact inC([0, T];L²(Γ3)^d) and there exists a subsequence still denoted (uⁿ) such that for allη >0 there existsn1∈N, so that for alln≥n1and allt∈[0, T],

ku^n+m_T (t)−uⁿ_T(t)k_L2(Γ₃)^d ≤η . On the other hand we have

kuⁿ_T(t)−u˜ⁿ_T(t)k_L2(Γ₃)^d≤ckuⁿ(t)−u˜ⁿ(t)kV ≤cT

nku˙n(t)kV,

where ( ˙uⁿ) is bounded in L^∞(0, T;V). Combining these results we obtain that there exists a positive constantC3 such that

Z T 0

kue^n+m_T (t)−euⁿ_T(t)k²_L2(Γ3)^ddt≤C₃( 1 n² +η²).

On the other hand from (4.6), we have: For all η > 0 there exists n2 in N such that for alln≥n2 and allm∈N,

Z T 0

kfe^n+m(t)−feⁿ(t)k²_Vdt≤η.

Then we obtain that there exists a constant C₄ >0 such that for all η >0 there existsn₃ inN such that for alln≥n₃= max(n₁, n₂) and allm∈N,

Z T 0

kue^n+m(t)−ueⁿ(t)k²_Vdt≤C₄(2η+1 n)

On the other hand for allη >0 there existsn₄inNsuch that for alln≥n₄, ¹_n ≤η.

We thus deduce that for all η >0 there existsn₅ = max(n₄, n₃) such that for all n≥n5,

Z T 0

kue^n+m(t)−ueⁿ(t)k²_Vdt≤3C₄η.

So we conclude that

eu_n→ustrongly inL²(0, T;V) (4.9) Now to prove that uis a solution of problem, in the inequality of problem (P_n^tⁱ), forv∈V setw=u^tⁱ+v∆t and divide by ∆t; we obtain the inequality

hz(ε(u^tⁱ⁺¹)), ε(v)−ε(∆u^t^t

∆t )iQ+j(u^tⁱ⁺¹, v)−j(u^tⁱ⁺¹,∆u^tⁱ

∆t )

≥(f(t_i+1), v−∆u^tⁱ

∆t )_V +hσ_N(u^tⁱ⁺¹), v_N −∆u^t_Nⁱ

∆t i ∀v∈V

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Whence for anyv∈L²(0, T;V), we have hz(ε(euⁿ(t))), ε(v(t))−ε d

dtuⁿ(t)

iQ+j(ueⁿ(t), v(t))−j

euⁿ(t), d dtuⁿ(t)

≥(feⁿ(t), v(t)− d

dtuⁿ(t))_V +hσ_N(euⁿ(t)), v_N(t)− d dtuⁿ_N(t)i

Integrating both sides of the previous inequality on (0, T), we obtain the inequality Z T

0

hz(ε euⁿ(t)

), ε(v(t))−ε(d

dtuⁿ(t))iQdt +

Z T 0

j(euⁿ(t), v(t))dt− Z T

0

j(euⁿ(t), d

dtuⁿ(t))dt

≥ Z T

0

(feⁿ(t), v− d

dtuⁿ(t))Vdt+ Z T

0

hσN(euⁿ(t)), vN − d

dtuⁿ_N(t)idt

(4.10)

Lemma 4.6. For any v∈L²(0, T;V)we have the following properties:

n→∞lim Z T

0

hz(ε(euⁿ(t))), ε(v(t))−ε(d

dtuⁿ(t))iQdt

= Z T

0

hz(ε(u(t))), ε(v(t))−ε( ˙u(t))i_Qdt

(4.11)

lim inf

n→∞

Z T 0

j(euⁿ(t), d

dtuⁿ(t))dt≥ Z T

0

j(u(t),u(t))dt˙ (4.12)

n→∞lim Z T

0

j(euⁿ(t), v(t))dt= Z T

0

j(u(t), v(t))dt (4.13)

n→∞lim Z T

0

(feⁿ(t), v(t)− d

dtuⁿ(t))_Vdt= Z T

0

(f(t), v(t)−u(t))˙ _Vdt (4.14) Proof. For proving (4.11), we write

Z T 0

hz(ε(ueⁿ(t))), ε(v(t))−ε(d

dtuⁿ(t))iQdt

= Z T

0

hz(ε(ueⁿ(t)))−z(ε(u(t))), ε(v(t))−ε(d

dtuⁿ(t))iQdt +

Z T 0

hz(ε(u(t))), ε(v(t))−ε(d

dtuⁿ(t))iQdt Using (4.9) and the hypothesis (a) onz, we have

Z T 0

hz(ε(˜uⁿ(t)))−z(ε(u(t)), ε(v(t))−ε(d

dtuⁿ(t))iQdt

≤ckueⁿ−uk_L2(0,T;V)(kvk_L2(0,T;V)+ku˙ⁿk_L2(0,T;V))→0 We deduce that

n→∞lim Z T

0

hz(ε(euⁿ(t)))−z(ε(u(t))), ε(v(t))−ε(d

dtuⁿ(t))iQdt= 0.

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On the other hand, we have Z T

0

hz(ε(u(t))), ε(v(t))−ε(d

dtuⁿ(t))iQdt= Z T

0

(Au(t), v(t)− d

dtuⁿ(t))Vdt which approaches

Z T 0

(Au(t), v(t)− d

dtu(t))Vdt= Z T

0

hz(ε(u(t))), ε(v(t))−ε( ˙u(t))iQdt.

To prove (4.12) we write j(euⁿ(t), d

dtuⁿ(t)) =j(ueⁿ(t)−u(t), d

dtuⁿ(t)) +j(u(t), d dtuⁿ(t)) then we have

Z T 0

j(euⁿ(t)−u(t), d

dtuⁿ(t))dt

≤ckµk_L^∞_(Γ₃₎kRσN(ueⁿ−u)k_L2(0,T;L²(Γ3))ku˙ⁿ_Tk_L2(0,T;L²(Γ3)^d). Since the mappingR is compact, we have

n→∞lim kRσN(euⁿ−u)k_L2(0,T:L²(Γ3))= 0 (4.15) and

lim inf

n→∞

Z T 0

j(u(t), d

dtuⁿ(t))dt≥ Z T

0

j(u(t),u(t))dt ,˙

see [4]. To prove (4.13) it suffices to use (4.15). From (4.6) we deduce for any v∈L²(0, T;V):

n→∞lim Z T

0

(feⁿ(t), v(t)− d

dtuⁿ(t))Vdt= Z T

0

(f(t), v(t)−u(t))˙ Vdt.

whence (4.14) is proved.

Passaging to the limit in inequality (4.8), we obtain the inequality Z T

0

hz(ε(u(t))), ε(v(t))−ε( ˙u(t))iQdt +

Z T 0

j(u(t), v(t))dt− Z T

0

(j(u(t),u(t)))dt˙

≥ Z T

0

(f(t), v(t)−u(t))˙ Vdt+ Z T

0

hσN(u(t)), vN(t)−u˙N(t)idt

(4.16)

In this inequality we set v(s) =

(z fors∈(t, t+λ)

˙

u(s) elsewhere to obtain the inequality

1 λ

Z t+λ t

(hz(ε(u(s))), ε(z)−ε( ˙u(s))i_Q+j(u(s), z)−j(u(s),u(s)))ds˙

≥ 1 λ

Z t+λ t

(f(s), z−u(s))˙ Vds+1 λ

Z t+λ t

hσN(u(s)), zN −u˙N(s)ids .

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Passing to the limit, one obtains thatusatisfies the inequality (2.14). To complete the proof, integrate on (0, T) both sides of (3.1); that is,

Z T 0

hz(ε(ueⁿ(t))), ε(v(t))−ε(ueⁿ(t))iQdt+ Z T

0

j(ueⁿ(t), v(t)−ueⁿ(t)))dt

≥ Z T

0

(feⁿ(t), v(t)−ueⁿ(t))_Vdt ∀v∈L²(0, T;V)

such that v(t)∈K, a.e. t ∈[0, T]. Passaging to the limit in the above inequality, and using (4.6), (4.9), we obtain the inequality

Z T 0

(hz(ε(u(t))), ε(v(t))−ε(u(t))iQ+j(u(t), v(t)−u(t))))dt

≥ Z T

0

(f(t), v(t)−u(t))_V ∀v∈L²(0, T;V);v(t)∈K, a.e. t∈[0, T] Following the same reasoning as previously done, we deduce that u satisfies the inequality

hz(ε(u(t))), ε(w)−ε(u(t))iQ+j(u(t), w−u(t))

≥(f(t), w−u(t))_V ∀w∈K,a.e. t∈[0, T].

Using Green’s formula in the above inequality, as in [4], we obtain thatusatisfies the inequality (2.15) and consequentlyuis a solution of problem (P2).

Conclusion. In this article we have shown the existence of a solution to the quasistatic unilateral contact problem with nonlocal friction for nonlinear elastic materials for a small enough friction coefficient . As well known the problem of unique- ness of the solution still remains open.

References

[1] L.-E. Andersson; A quasistatic frictional problem with normal compliance, Nonlinear Anal.

Th . Appl.16, 4, 347-369, 1991.

[2] L.-E. Andersson;Existence Results for Quasistatic Contact Problems with Coulomb friction, Appl.Math.Optim.42: 169-202, 2000.

[3] H. Brezis; Equations et inéquations non linéaires dans les espaces vectoriels en dualité, Annales Inst. Fourier, 18, 115-175, 1968.

[4] M. Cocou, E. Pratt, M. Raous; Formulation and approximation of quasistatic frictional contact, Int. J. Engng Sc.,34, 7, 783-798, 1996.

[5] S. Drabla, M. Sofonea;Analysis of a Signorini problem with friction, IMA Journal of Applied Mathematics, 63, 2, 113-130, 1999.

[6] G. Duvaut;Equilibre d’un solide ´elastique avec contact unilat´eral et frottement de Coulomb, Cr Acad. Sci. Paris, Ser A, 290, 263-265, 1980.

[7] G. Duvaut, J.-L. Lions;les in´equations en m´ecanique et en physique, Dunod, Paris, 1972.

[8] W. Han, M. Sofonea; Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity.

Studies in advanced Mathematics 30, Americal Mathematical Society and International Press, 2002.

[9] A. Klarbring, A. Miklic, M. Shillor;A global existence result for the quasistatic problem with normal compliance. Internat. Ser. Numer. Math. 101. Birkh¨auser Verlag. basel, 85-111, 1991.

[10] R. Rocca;Existence of a solution for a quasistatic problem of unilateral contact with local friction, CR. Acad. Sci Paris. Ser1, t. 328, 1253-1258, 1999.

[11] M. Rochi, M. Schillor, M. Sofonea;Quasistatic viscoplastic contact with normal compliance and friction, J. Elasticity, 51, 105-126, 1998.

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Touzaline Arezki

Faculté de Mathématiques, USTHB BP 32 EL Alia, Bab-Ezzouar, 16111, Algérie E-mail address:[email protected]

Mignot Alain

IRMAR Math´ematiques, Campus de Beaulieu, Universit´e de Rennes1, 35042, Rennes Cedex, France

E-mail address:[email protected]