A QUASISTATIC UNILATERAL CONTACT PROBLEM WITH NORMAL COMPLIANCE AND NONLOCAL FRICTION

WITH NORMAL COMPLIANCE AND NONLOCAL FRICTION

AREZKI TOUZALINE

We consider a mathematical model which describes the quasistatic unilateral and frictional contact of a linear elastic body with a foundation. The contact is model- led with a normal compliance condition of such a type that the penetration is restricted with unilateral constraint and associated to the nonlocal friction law.

We present the classical formulation of the problem, the variational formulation and establish the existence of a weak solution, when the coefficient of friction is sufficiently small. The proofs are based on arguments of time-discretization, compactness and lower semicontinuity.

AMS 2010 Subject Classification: 49J40, 74 M10, 74 M15.

Key words: linear elastic, quasistatic process, frictional contact, incremental, nor- mal compliance.

1. INTRODUCTION

Contact problems involving deformable bodies are quite frequent in the industry as well as in daily life and play an important role in structural and mechanical systems. Recall that unilateral contact problems including Sig- norini’s condition were studied by many authors, we refer for instance the reader to [2, 4, 5, 6, 7, 9, 10, 12, 13, 15, 18, 19]. A first attempt to study frictional contact problems within the framework of variational inequalities was made in [8]. The mathematical, mechanical and numerical state of the art can be found in [17]. In [13] we find a detailed analysis and numerical ap- proximations of contact problems concerning elastic materials with or without friction. In this paper we consider a frictional unilateral contact between a linear elastic body and a foundation. We assume that the contact is modelled with a normal compliance condition in such a way that the penetration is allowed, restricted with unilateral constraint and associated to the nonlocal Coulomb law. This normal compliance condition is similar to the one in [12, 18] where was respectively studied a dynamic frictionless contact problem for elastic-visco-plastic materials and a quasistatic frictional contact problem for

REV. ROUMAINE MATH. PURES APPL.,56(2011),3, 235–251

nonlinear elastic materials. We assume that the forces applied to the body vary slowly in time so that the acceleration in the system is negligible. In this case we can study a quasistatic approach of the process. We recall that for linear elastic materials the quasistatic contact problem using a normal compliance law has been studied in [1] by considering incremental problems and in [14] by another method using a time-regularization. The quasistatic contact problem with local or nonlocal friction has been solved respectively in [15] and in [4] by using a time-discretization. In [2] the quasistatic contact problem with Coulomb friction was solved by 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. In the static case Signorini ’s problem with friction for nonlinear elastic materials has been solved in [6] by using a fixed point method. In the quasistatic case it was solved in [19] by using a time- discretization method. Also, in viscoelasticity, the quasistatic contact problem with normal compliance and friction has been solved in [16] for nonlinear vis- coelastic materials by the same fixed point argument. In [11] the authors resolve the quasistatic contact problems in viscoelasticity and viscoplasticity.

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 work we propose a variational formulation written in the form of two variational inequalities. By means of Euler’s implicit scheme as in [18, 19], the quasistatic contact problem leads us to solve a well-posed variational inequality at each time step. Finally, if the coefficient of friction is small enough we prove by using lower semicontinuity and compactness arguments that the limit of the discrete solution is a solution to the continuous problem.

2. PROBLEM STATEMENT AND VARIATIONAL FORMULATION

We consider a linear elastic body which occupies a regular domain Ω of R^d (d = 2,3) with boundary Γ that is partitioned into three parts such that Γ = ¯Γ₁∪Γ¯₂∪Γ¯₃ where Γ₁, Γ₂, Γ₃ are disjoint open sets and meas(Γ₁)

> 0. The body is subjected to volume forces of density ϕ1, prescribed zero displacements and tractionsϕ₂ on the part Γ₁ and Γ₂, respectively. On Γ₃the body is in unilateral and frictional contact with a foundation. We denote by u= (ui), the displacement field,

ε(u) = (ε_ij(u)) = 1

2(u_i,j+u_j,i),

the strain tensor, σ = Eε(u), the stress tensor where E denotes the fourth order tensor of elasticity coefficients. Letν be the unit outward normal vector on Γ. We denote the normal and tangential components of the displacement vector and stress vector by

u_ν =u·ν, u_τ =u−u_νν, σν = (σν)·ν, στ =σν−σνν.

With these preliminaries, the classical formulation of the problem of frictional contact we consider is the following.

Problem P1.Find a displacement field u: Ω×[0, T]→R^d such that σ =Eε(u) in Ω×(0, T),

(2.1)

divσ+ϕ₁= 0 in Ω×(0, T), (2.2)

u= 0 on Γ1×(0, T), (2.3)

σν =ϕ2 on Γ2×(0, T), (2.4)

uν ≤g, σν +p(uν)≤0, (σν+p(uν))(uν −g) = 0 on Γ3×(0, T), (2.5)







|σ_τ| ≤µ|Rσ_ν|

|σ_τ|< µ|Rσ_ν| ⇒u˙_τ = 0 on Γ₃×(0, T),

|σ_τ|=µ|Rσ_ν| ⇒ ∃λ≥0 s.t. σ_τ =−λu˙_τ (2.6)

u(0) =u₀ in Ω.

(2.7)

Equation (2.1) represents the linear elastic constitutive law. Equation (2.2) represents the equilibrium equation and (2.3), (2.4) are the displacement- traction boundary conditions, respectively. Condition (2.5) represents the con- tact condition with finite penetration, introduced in [12], in which g≥0 rep- resents the maximum value of the penetration and pis a prescribed function.

Condition (2.6) represents the nonlocal friction law in which µ denotes the coefficient of friction, R is a regularization operator (see [7]) and ˙u_τ is the tangential part of velocity field. The tangential shear cannot exceed the max- imal frictional resistance µ|Rσ_ν|. Then, if the strict inequality is satisfied, the surface adheres to the foundation and is in the so-called stick state, and when equality is satisfied there is relative sliding, the so-called slip state.

In the study of the mechanical problemP₁ we adopt the following nota- tions and hypotheses:

We denote by S^d the space of second order symmetric tensors on R^d. The canonical inner products and the corresponding norms on R^d and S^dare given by

u·v=u_iv_i, |v|= (v·v)^1/2 ∀u= (u_i), v= (v_i)∈R^d, σ·τ =σ_ijτ_ij, |τ|= (τ·τ)^1/2 ∀σ = (σ_ij), τ = (τ_ij)∈S^d.

respectively. In (2.6) and below, a dot above a variable represents its derivative with respect to time. To proceed with the variational formulation, we need some function spaces

H = (L²(Ω))^d, Q={τ = (τ_ij)∈(L²(Ω))^d×d:τ_ij =τ_ji, 1≤i, j≤d}, H₁= (H¹(Ω))^d.

H, Q are real Hilbert spaces equipped with the respective inner products hu, vi_H =

Z

Ω

uividx, hσ, τi_Q= Z

Ω

σijτijdx.

Let V be the closed subspace of H1 defined by V ={v∈H₁:v = 0 on Γ₁}.

and the set of admissible displacements fields given by K={v∈V :v_ν ≤g on Γ₃}.

Since meas(Γ₁)>0, the following Korn’s inequality holds [8], (2.8) kε(v)k_Q≥cΩkvk_H1 ∀v∈V,

wherec_Ω >0 is a constant which depends only on Ω and Γ₁.We equipV with the inner product given by

(u, v)_V =hε(u), ε(v)i_Q

and let k · k_V be the associated norm. It follows from (2.8) that the norms k·k_H1 andk·k_V are equivalent and (V,k·k_V) is a real Hilbert space. Moreover, by the Sobolev trace theorem, there exists a constant d_Ω >0 depending only on the domain Ω,Γ1 and Γ3 such that

(2.9) kvk_(L2(Γ3))^d≤d_Ωkvk_V ∀v∈V.

Recall that it is well known that, if σ is a regular function, then the following Green’s formula holds

hσ, ε(v)i_Q+hdivσ, vi_H = Z

Γ

σν·vda ∀v∈H1. Let now

H¹²(Γ₃) =

w.|_Γ3 :w∈H¹²(Γ),w= 0 on Γ₁

and let the brackets h·,·i_Γ3 stand for the duality between the dual spaces H⁻¹²(Γ₃) and H¹²(Γ₃). Before we start with the variational formulation of Problem P1 let us state in which sense the duality pairing h·,·i_Γ3 is taken.

Indeed, let σ∈Q1 ={τ ∈Q: divτ ∈H}andh∈(L²(Γ2))^dsuch thatσν =h on Γ₂.Then as in [19] the normal stressσ_ν is defined by

∀w∈H¹²(Γ3) :hσ_ν, wi_Γ3 =hσ, ε(v)i_Q+hdivσ, vi_H − Z

Γ2

h·vda (2.10)

∀v∈V; vν =wand vτ = 0 on Γ3.

We suppose that R : H⁻¹²(Γ3) → L²(Γ3) is a compact linear operator (see [7]). The linearity of R implies that there exists a constantC_R>0 such that (2.11) kRτk_L2(Γ3)≤C_Rkτk

H⁻¹²(Γ3) ∀τ ∈H⁻¹²(Γ₃).

Next, in the study of the mechanical problem P1, we assume that the linear elasticity operator E= (E_ijkl) : Ω×S^d→S^d satisfies the following conditions:

(2.12)

.

^{Eijkl∈L∞(Γ3),}

.

^{Eijkl=Ejikl =Eijlk, 1≤i, j, k, l≤d,}

.

^{∃α >}0 such that∀ξ= (ξij)∈R^d×d withξij =ξji, 1≤i, j, k, l≤d, E_ijklξ_ijξ_kl≥α|ξ|².

We define the bilinear form a(·,·) on V ×V by a(u, v) =

Z

Ω

E_ijklε_ij(u)ε_kl(v)dx.

It follows from (2.12) and (2.8) thatais continuous and coercive, that is, (2.13)

(

.

there existsM >0 such that|a(u, v)| ≤Mkuk_Vkvk_V ∀u, v∈V;

.

there existsm >0 such thata(v, v)≥mkvk²_V ∀v∈V.

For every real Banach space (X,k · k_X) and T > 0 we use the notation C([0, T];X) for the space of continuous functions from [0, T] toX; recall that C([0, T];X) is a real Banach space with the norm

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

t∈[0,T]kx(t)k_X.

For p ∈ [1,∞] we use the standard notation of L^p(0, T;V). We also use the Sobolev space W^1,∞(0, T;V) equipped with the norm

kvk_W1,∞(0,T:V)=kvk_L∞(0,T;V)+kvk˙ _L∞(0,T;V),

where a dot now represents the weak derivative with respect to the time vari- able.

The forces are assumed to satisfy

(2.14) ϕ1∈W^1,∞(0, T;H), ϕ2 ∈W^1,∞(0, T; (L²(Γ2))^d).

Let f : [0, T]→V given by (f(t), v)_V =

Z

Ω

ϕ₁·vdx+ Z

Γ2

ϕ₂·vda ∀v∈V, t∈[0, T].

The assumption (2.14) implies that

f ∈W^1,∞(0, T;V).

As in [12] we assume that the contact function psatisfies

(2.15)















(a)p: ]− ∞, g]→R;

(b) there existsLp>0 such that

|p(r₁)−p(r2)| ≤Lp|r₁−r2|for all r1,r2≤g;

(c) (p(r₁)−p(r₂))(r₁−r₂)≥0 for allr₁,r₂ ≤g;

(d)p(r) = 0 for allr <0.

We note that when uν < 0, i.e., when there is separation between the body and the obstacle then the condition (2.5) combined with hypotheses (2.15) shows that the reaction of the foundation vanishes (σν = 0). When 0≤u_ν < gthen−σ_ν =p(u_ν) which means that the reaction of the foundation is uniquely determined by the normal displacement. Whenuν =gthen−σ_ν ≥ p(g) and σ_ν is not uniquely determined. We note that when g = 0 then the condition (2.5) becomes the classical Signorini contact condition without a gap

u_ν ≤0, σ_ν ≤0, σ_νu_ν = 0,

and when g > 0 and p = 0, condition (2.5) becomes the classical Signorini contact condition with a gap.

uν ≤g, σν ≤0, σν(uν−g) = 0.

The last two conditions are used to model the unilateral conditions with a rigid foundation. We recall that examples of normal compliance functions can be found in [1,11,12,16,17].Next, we define the set

V0 ={v∈H1 : divσ(v)∈H}

and the functionals

jc :V ×V →R, jf :V0×V →R by

jc(v, w) = Z

Γ3

p(vν)wνda ∀v, w∈V, j_f(v, w) =

Z

Γ3

µ|Rσ_ν(v)| |w_τ|da ∀(v, w)∈V₀×V,

where the coefficient of friction µis assumed to satisfy (2.16) µ∈L^∞(Γ₃) andµ≥0 a.e.on Γ₃. We also assume that the initial data u₀ satisfies

(2.17) u0 ∈K∩V0

and the compatibility condition

(2.18) a(u₀, v−u₀) +j(u₀, v−u₀)≥(f(0), v−u₀)_V ∀v∈K, where

j=j_c+j_f.

Now, in order to establish the weak formulation of Problem P₁ , we assume that u is a smooth function satisfying (2.1)–(2.7). Indeed, let v ∈ V and multiply the equilibrium of forces (2.2) byv−u(t), integrate the result over˙ Ω and use Green’s formula to obtain

Z

Ω

σ(t)·(ε(v)−ε( ˙u(t)))dx= Z

Ω

ϕ₁(t)·(v−u(t))dx˙ + Z

Γ

σ(t)ν·(v−u(t))da.˙ Taking into account the boundary conditions (2.3) andv= 0 on Γ1, we see that

Z

Γ

σ(t)ν·(v−u(t))da˙ = Z

Γ2

ϕ₂(t)·(v−u(t))da˙ + Z

Γ3

σ(t)ν·(v−u(t))da.˙ Moreover, we have

Z

Γ3

σ(t)ν·(v−u(t))da˙ =

= Z

Γ3

σν(u(t))(vν−u˙ν(t))da+ Z

Γ3

στ(t)·(vτ−u˙τ(t))da, and

Z

Γ3

σν(u(t))(vν−u˙ν(t))da=

= Z

Γ3

(σν(u(t)) +p(uν(t)))(vν −u˙ν(t))da− Z

Γ3

p(uν(t))(vν−u˙ν(t))da.

The law of friction (2.6) leads to the following relation

σ_τ·(v_τ−u˙_τ) +µ|Rσ_ν(u)|(|v_τ| − |u˙_τ|)≥0 ∀v_τ, from which we deduce that the function u satisfies the inequality

a(u(t), v−u(t)) +˙ j(u(t), v)−j(u(t),u(t))˙ ≥ (2.19)

(f(t), v−u(t))˙ _V + Z

Γ3

(σ_ν(u(t)) +p(u_ν(t)))(v_ν−u˙_ν(t))da ∀v∈V.

On the other hand, we have Z

Γ3

(σν(u(t)) +p(uν(t))(zν−uν(t))da=

= Z

Γ3

(σ_ν(u(t)) +p(u_ν(t)))((z_ν−g)−(u_ν(t)−g))da=

= Z

Γ3

(σ_ν(u(t)) +p(u_ν(t)))(z_ν−g)da− Z

Γ3

(σ_ν(u(t)) +p(u_ν(t))(u_ν(t)−g)da.

Using the conditions (2.5) it follows that Z

Γ3

(σν(u(t)) +p(uν(t))(zν−g)da≥0 ∀z∈K,

and Z

Γ3

(σν(u(t)) +p(uν(t)))(uν(t)−g)da= 0.

Hence we deduce that (2.20)

Z

Γ3

(σ_ν(u(t)) +p(u_ν(t)))(z_ν −u_ν(t))≥0 ∀z∈K.

Finally we combine (2.7), (2.19) and (2.20) to derive the variational formula- tion of ProblemP₁.

Problem P₂. Find a displacement field u ∈ W^1,∞(0, T;V) such that u(0) =u0,u(t)∈K∩V0, for all t∈[0, T], and for almost allt∈[0, T],

a(u(t), v−u(t)) +˙ j(u(t), v)−j(u(t),u(t))˙ ≥(f(t), v−u(t))˙ V

(2.21)

+hσ_ν(u(t)), vν−u˙ν(t)i_Γ3 + (p(uν(t)), vν −u˙ν(t))_L²_(Γ3) ∀v∈V, (2.22) hσ_ν(u(t)), zν−uν(t)i_Γ3+ (p(uν(t)), zν−uν(t))_L²_(Γ3)≥0 ∀z∈K.

Our main result is contained in the following theorem.

Theorem 2.1. Let (2.12),(2.14),(2.15), (2.16) , (2.17)and (2.18)hold.

Then,there exists at least one solution of Problem P₂ if kµk_L∞(Γ3) < m/C_RM d_Ω.

Remark 2.2. It is interesting to note that as in [4] any element u such that u(t)∈K for all t∈[0, T] and satisfies the inequality (2.22) verifies

σ_ν(u(t)),u_ν(t+ ∆t)−u_ν(t)

∆t

Γ3

+

p(u_ν(t)),u_ν(t+ ∆t)−u_ν(t)

∆t

L²(Γ3)

≥0 and

σ_ν(u(t)),u_ν(t−∆t)−u_ν(t)

−∆t

Γ3

+

p(u_ν(t)),u_ν(t−∆t)−u_ν(t)

−∆t

L²(Γ3)

≤0

for all ∆t > 0. Moreover, using the assumption (2.15) (b) on p, one obtains that when ∆t→0,

(2.23) hσ_ν(u(t)),u˙ν(t)i_Γ3 + (p(uν(t)),u˙ν(t))_L²_(Γ3)= 0 a.e. t∈(0, T).

3. INCREMENTAL FORMULATION

In order to solve Problem P2, we consider an incremental formulation obtained by using an implict time discretization scheme for (2.21) and (2.22).

For n∈N^∗, we need a partition of the time interval [0, T],with 0 =t0 < t1<

· · · < t_n = T, where t_i = i∆t, i = 0,1, . . . , n, with step size ∆t = T /n. We denote by uⁱ the approximation of u at time ti and we set ∆uⁱ = uⁱ⁺¹−uⁱ, i = 0, . . . , n−1. For a continuous function w ∈ C([0, T];X) where X is a Banach space we use the notation wⁱ =w(t_i). Then we obtain the following sequence of incremental problems P_nⁱ defined for u⁰ =u0 by

Problem P_nⁱ.Find uⁱ⁺¹∈K∩V₀ such that (3.1)







a(uⁱ⁺¹, w−uⁱ⁺¹)+j(uⁱ⁺¹, w−uⁱ)−j(uⁱ⁺¹,∆uⁱ)≥(fⁱ⁺¹, w−uⁱ⁺¹)_V +hσ_ν(uⁱ⁺¹), wν−uⁱ⁺¹_ν i_Γ3+(p(uⁱ⁺¹_ν ), wν−uⁱ⁺¹_ν )_L²_(Γ3) ∀w∈V,

hσ_ν(uⁱ⁺¹), w_ν−uⁱ⁺¹_ν i_Γ3+(p(uⁱ⁺¹_ν ), w_ν−uⁱ⁺¹_ν )_L²_(Γ3)≥0 ∀w∈K.

As in [4] Problem P_nⁱ is equivalent to Problem Qⁱ_n defined as follows.

Problem Qⁱ_n.Find uⁱ⁺¹∈K∩V0 such that (3.2)

( a(uⁱ⁺¹, w−uⁱ⁺¹) +j(uⁱ⁺¹, w−uⁱ)−j(uⁱ⁺¹,∆uⁱ)

≥(fⁱ⁺¹, w−uⁱ⁺¹)V ∀w∈K.

We have the following result.

Proposition 3.1. There exists a unique solution to Problem Qⁱ_n if kµk_L^∞_(Γ3)< m/CRM dΩ.

We shall prove this proposition in several steps. In the first step, we introduce the following intermediate problem. Indeed we define the following non-empty closed subset

C+={η∈L²(Γ3); η ≥0a.e. on Γ3}

and for a fixedη∈C+let us consider the following contact problem with given friction.

Problem Qⁱ_nη. Find uⁱ⁺¹_η ∈K such that (3.3)

( (Auⁱ⁺¹_η , w−uⁱ⁺¹_η )_V +jη(w−uⁱ)−jη(uⁱ⁺¹_η −uⁱ)

≥(fⁱ⁺¹, w−uⁱ⁺¹_η )V ∀w∈K,

where the operator A:V →V and the functionaljη :V →Rare respectively defined by

(Au, v)V =a(u, v) +jc(u, v), jη(v) = Z

Γ3

µη|v_τ|da.

We now prove the following lemma.

Lemma 3.2. Problem Qⁱ_nη has a unique solution.

Proof. By the assumptions (2.13), (2.15)(b) and (2.15)(c), the operator A is strongly monotone and Lipschitz continuous; on the other hand the func- tional jη is a semi-norm continuous. Since K is a non-empty closed convex subset of V, it follows from the theory of elliptic variational inequalities [3]

that the inequality (3.3) has a unique solution.

In the second step we define the following mapping T :C₊→C₊ as η→T(η) =u_η. Then the following lemma holds.

Lemma 3.3. T has a unique fixed point η^∗ and uη^∗ is a unique solution of Problem Qⁱ_n.

Proof. We set v = u_η1 in inequality of Problem Qⁱ_nη

2 and v = u_η2 in inequality of ProblemQⁱ_nη1.Using (2.15)(c), we find after adding the resulting inequalities that

a(u_η1−u_η2, u_η1−u_η2)≤

≤jη1(uη2 −uⁱ)−jη1(uη1 −uⁱ) +jη2(uη1 −uⁱ)−jη2(uη2 −uⁱ).

Hence using (2.15)(c), (2.9), (2.10) and (2.13)(b), we get kT(η₁)−T(η₂)k_L2(Γ3)≤ M d_ΩC_R

m kµk_L∞(Γ3)kη₁−η₂k_L2(Γ3).

Then it follows that for kµk_L^∞_(Γ3) < m/M dΩCR, T is contractive; thus it admits a unique fixed point η^∗ andu_η^∗ is a unique solution of Problem Qⁱ_n.

Remark 3.4. As uη^∗ ∈V0 thenuⁱ⁺¹ ∈V0. 4. EXISTENCE RESULT

The main result of this section is to show the existence of a solution obtained as a limit of the interpolate function of the discrete solution. For thus it is necessary at first to establish the following lemma.

Lemma 4.1. There exists a constant c > 0 such that for kµk_L^∞_(Γ3) <

m/M dΩCR, we have

(4.1) kuⁱ⁺¹k_V ≤ckfⁱ⁺¹k_V, k∆uⁱk_V ≤ck∆fⁱk_V.

Proof. Take w = 0 in inequality (3.2); then, using (2.12)(b) and (2.9), one obtains that there exists a constant c > 0 such that for kµk_L∞(Γ3) <

m/M d_ΩC_R,the first inequality (4.1) hods.

To prove the second inequality, set v =uⁱ in inequality (3.2) and then v =uⁱ⁺¹ in the translated inequality satisfied byuⁱ. We find after adding the resulting inequalities that

a(∆uⁱ,∆uⁱ)−j(uⁱ, uⁱ⁺¹−uⁱ⁻¹) +j(uⁱ, uⁱ−uⁱ⁻¹) +j(uⁱ⁺¹,∆uⁱ)≤

≤(∆fⁱ,∆uⁱ)_V. On the other hand, we have

j(uⁱ, uⁱ⁺¹−uⁱ⁻¹)−j(uⁱ, uⁱ−uⁱ⁻¹)−j(uⁱ⁺¹, uⁱ⁺¹−uⁱ) =

= Z

Γ3

(p(uⁱ_ν)−p(uⁱ⁺¹_ν ))∆uⁱ_νda+

+ Z

Γ3

µ|Rσ_ν(uⁱ)|(|uⁱ⁺¹_τ −uⁱ⁻¹_τ |−|uⁱ⁻¹_τ −uⁱ_τ|)da−

Z

Γ3

µ|Rσ_ν(uⁱ⁺¹)||∆uⁱ_τ|da.

Using (2.15)(c), we have Z

Γ3

(p(uⁱ_ν)−p(uⁱ⁺¹_ν ))(∆uⁱ_ν)da≤0, and moreover as

||uⁱ⁺¹_τ −uⁱ⁻¹_τ | − |uⁱ⁻¹_τ −uⁱ_τ|| ≤ |∆uⁱ_τ|, we obtain

a(∆uⁱ,∆uⁱ)≤ Z

Γ3

µ|Rσ_ν(∆uⁱ)| |∆uⁱ_τ|da+ (∆fⁱ,∆uⁱ)V. Then using (2.11), (2.9) and (2.13)(b), this last inequality implies

mk∆uⁱk²_V ≤a(∆uⁱ,∆uⁱ)≤dΩM CRkµk_L^∞_(Γ3)k∆uⁱk²_V +k∆fⁱk_Vk∆uⁱk_V. Therefore, it follows that there exists a constantc >0 such that forkµk_L^∞_(Γ3)<

m/dΩM CR,

k∆uⁱk_V ≤ck∆fⁱk_V.

Next, we define the function uⁿ(t),continuous in [0, T]→V by uⁿ(t) =uⁱ+t−ti

∆t ∆uⁱ on [ti, ti+1], i= 0, . . . , n−1.

Then as in [19] we have the following lemmas.

Lemma 4.2.There exists a function u, such that, by selecting, a subse- quence still denoted (uⁿ), we have

uⁿ→u weak ∗ in W^1,∞(0, T;V).

On the other hand, we introduce the following piecewise constant func- tions

euⁿ: [0, T]→V, feⁿ: [0, T]→V, defined by

euⁿ(t) =uⁱ⁺¹, feⁿ(t) =f(t_i+1) ∀t∈(t_i, t_i+1] ,i= 0, . . . , n−1.

Lemma 4.3.There exists a subsequence of (euⁿ) still denoted (euⁿ) such that the following results hold

(4.2)

(i) euⁿ→uweak ∗in L^∞(0, T;V),

(ii)ueⁿ(t)→u(t) weakly in V a.e. t∈[0, T], (iii) u(t)∈K∩V0, for all t∈[0, T].

We now have the following proposition.

Proposition 4.4. The weak limit u is a solution to Problem P₂. Proof. In the first step we show the inequality (2.22). Indeed, from in- equality (3.2) we deduce the inequality

a(uⁱ⁺¹, w−uⁱ⁺¹) +j(uⁱ⁺¹, w−uⁱ⁺¹)≥(fⁱ⁺¹, w−uⁱ⁺¹)_V ∀ w∈K, from which we deduce that for almost all t∈(0, T),

a(˜uⁿ(t), w−u˜ⁿ(t)) +j(˜uⁿ(t), w−u˜ⁿ(t))≥( ˜fⁿ(t), w−u˜ⁿ(t))_V ∀w∈K.

By standard arguments passing to the limit as n→+∞, we have a(u(t), w−u(t)) +j(u(t), w−u(t))≥f(t), w−u(t)V ∀w∈K which implies by Green’s formula that for all t∈[0, T],

hσ_ν(u(t)), w_ν−u_ν(t)i_Γ3+ (p(u_ν(t)), w_ν−u_ν(t))_L2(Γ3)≥0 ∀w∈K and then (2.22) follows.

Now, in order to prove the inequality (2.21), in the first inequality (3.1) take, for v∈V, w=uⁱ+v∆tand divide by ∆t, one obtains

a

uⁱ⁺¹, v−∆uⁱ

∆t

+j(uⁱ⁺¹, v)−j

uⁱ⁺¹,∆uⁱ

∆t

≥

fⁱ⁺¹, v−∆uⁱ

∆t

V

+ +

σ_ν(uⁱ⁺¹), v_ν−∆uⁱ_ν

∆t

Γ3

+

p(uⁱ⁺¹_ν ), v_ν−∆uⁱ_ν

∆t

L²(Γ3)

∀v∈V.

Since from the second inequality (3.1) we have

σν(uⁱ⁺¹), vν −∆uⁱ_ν

∆t

Γ3

+ p uⁱ⁺¹_ν

, vν −uⁱ⁺¹_ν

L²(Γ3) ≥

≥ hσ_ν(uⁱ⁺¹), vνi_Γ3 + (p(uⁱ⁺¹_ν ), vν)_L²_(Γ3), then it follows that

a

uⁱ⁺¹, v− ∆uⁱ

∆t

+j(uⁱ⁺¹, v)−j

uⁱ⁺¹,∆uⁱ

∆t

≥

≥

fⁱ⁺¹, v− ∆uⁱ

∆t

V

+hσ_ν(uⁱ⁺¹), vνi_Γ3+ (p(uⁱ⁺¹_ν ), vν)_L²_(Γ3) ∀v∈V.

This inequality implies that for any v∈L²(0, T;V),











a(ueⁿ(t), v(t)−u˙ⁿ(t)) +j(euⁿ(t), v(t))−j(ueⁿ(t),u˙ⁿ(t))

≥(feⁿ(t), v(t)−u˙ⁿ(t))_V +hσ_ν(˜uⁿ(t)), v_ν(t)i_Γ3 +(p(˜uⁿ_ν(t)), vν(t))_L²_(Γ3) fora.a. t∈(0, T).

Integrating both sides of the previous inequality on (0, T) we obtain the in- equality

Z _T

0

a(ueⁿ(t), v(t)−u˙ⁿ(t))dt+ Z _T

0

j(ueⁿ(t), v(t))dt−

(4.3)

− Z T

0

j(euⁿ(t),u˙ⁿ(t))dt≥ Z T

0

(feⁿ(t), v(t)−u˙ⁿ(t))Vdt+

+ Z T

0

(hσ_ν(˜uⁿ(t)), vν(t)i_Γ3 + (p(˜uⁿ_ν(t)), vν(t))_L²_(Γ3))dt.

Now, in order to pass to the limit in the previous inequality, we must at first prove the following lemmas.

Lemma 4.5. We have

lim inf

n→∞

Z T 0

a(euⁿ(t), uⁿ(t))dt≥ Z T

0

a(u(t),u(t))dt,˙ (4.4)

lim inf

n→∞

Z T 0

j(euⁿ(t),u˙ⁿ(t))dt≥ Z T

0

j(u(t),u(t))dt.^. (4.5)

Proof. For theproof of (4.4) it suffices to see [4]. To prove (4.5) we have Z T

0

j(euⁿ(t),u˙ⁿ(t))dt= Z T

0

jc(ueⁿ(t),u˙ⁿ(t))dt+ Z T

0

jf(ueⁿ(t),u˙ⁿ(t))dt.

We now see that

Z T 0

jc(euⁿ(t),u˙ⁿ(t))dt=

= Z T

0

(jc(euⁿ(t),u˙ⁿ(t))−jc(u(t),u˙ⁿ(t)))dt+ Z T

0

jc(u(t),u˙ⁿ(t))dt.

Next we have

lim inf

n→+∞

Z T 0

j_c(euⁿ(t),u˙ⁿ(t))dt≥ (4.6)

≥lim inf

n→+∞

Z T 0

(j_c(euⁿ(t),u˙ⁿ(t))−j_c(u(t),u˙ⁿ(t)))dt+ lim inf

n→+∞

Z T 0

j_c(u(t),u˙ⁿ(t))dt.

Using standard arguments, the first term of the right hand side of the previous inequality is estimated as

Z T 0

(jc(euⁿ(t),u˙ⁿ(t))−jc(u(t),u˙ⁿ(t)))dt

≤ckueⁿ_ν −uνk_L2(0,T;L²(Γ3))

which implies as ueⁿ_ν → u_ν strongly inL²(0, T;L²(Γ₃)),

(4.7) lim

n→+∞

Z T 0

(jc(euⁿ(t),u˙ⁿ(t))−jc(u(t),u˙ⁿ(t)))dt= 0.

The functionalv→RT

0 j_c(u(t), v(t))dtis convex and continuous onL²(0, T;V) so it is sequentially weakly lower semicontinuous, which implies that

(4.8) lim inf

n→+∞

Z T 0

j_c(u(t),u˙ⁿ(t))dt≥ Z T

0

j_c(u(t),u(t))dt.˙ Using now (4.7), (4.8) and keeping in mind (see [4]) that

lim inf

n→+∞

Z T 0

j_f(euⁿ(t),u˙ⁿ(t))dt≥ Z T

0

j_f(u(t),u(t))dt,˙ we deduce from (4.6) that

lim inf

n→+∞

Z T 0

j(ueⁿ(t),u˙ⁿ(t))dt≥ Z T

0

jc(u(t),u(t))dt˙ + Z T

0

jf(u(t),u(t))dt˙ =

= Z T

0

j(u(t),u(t))dt.^.

Lemma 4.6. For all v∈L²(0, T;V) the following properties hold:

(4.9) lim

n→∞

Z T 0

a(euⁿ(t), v(t))dt= Z T

0

a(u(t), v(t))dt,

(4.10) lim

n→∞

Z T 0

j(euⁿ(t), v(t))dt= Z T

0

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

(4.11) lim

n→∞

Z T 0

feⁿ(t), v(t)−u˙ⁿ(t)

V dt= Z T

0

(f(t), v(t)−u(t))^. Vdt,

n→∞lim Z T

0

(hσ_ν(˜uⁿ(t)), v_ν(t)i_Γ3+ (p(˜uⁿ_ν(t)), v_ν(t))_L2(Γ3))dt= (4.12)

= Z T

0

(hσ_ν(u(t)), v_ν(t)i_Γ3 + (p(u_ν(t)), v_ν(t))_L²_(Γ3))dt.

Proof. To prove (4.9) it suffices to see [4]. To show (4.10) we write for t∈(0, T),

j(ueⁿ(t), v(t)) = (j(euⁿ(t), v(t))−j(u(t), v(t))) +j(u(t), v(t)).

Since

j(ueⁿ(t), v(t))−j(u(t), v(t)) =j_c(ueⁿ(t), v(t))−j_c(u(t), v(t))+

+j_f(ueⁿ(t), v(t))−j_f(u(t), v(t)), using (2.15)(b) and (2.9), we find

Z T 0

(j_c(euⁿ(t), v(t))−j_c(u(t), v(t)))dt

≤

≤ckueⁿ_ν −uνk_L2(0,T;L²(Γ3))kvk_L2(0,T;V), which implies

n→∞lim Z T

0

(jc(ueⁿ(t), v(t))−jc(u(t), v(t)))dt= 0.

Similarly, we have

n→∞lim Z T

0

(j_f(ueⁿ(t), v(t))−j_f(u(t), v(t)))dt= 0, as

Z T 0

(j_f(euⁿ(t), v(t))−j_f(u(t), v(t)))dt

≤

≤ckRσ_ν(ueⁿ_ν −u_ν)k_L2(0,T;L²(Γ3))kvk_L2(0,T;V), where (see [4])

n→+∞lim kRσ_ν(ueⁿ_ν −uν)k_L2(0,T;L²(Γ3))= 0.

Then (4.10) follows. To show (4.11), we use (4.2)(i) and that feⁿ → f strongly inL²(0, T;V). To prove (4.12), we use (2.10),(4.2)(ii) and that ˜uⁿ_ν → uν strongly inL²(0, T;L²(Γ3)).

Next, by going to the limit in inequality (4.5) using Lemmas 4.5 and 4.6, we obtain the following inequality

Z T 0

(a(u(t), v(t)−u(t)) +˙ j(u(t), v(t))−j(u(t),u(t)))dt^. ≥ (4.13)

≥ Z T

0

(f(t), v(t)−u(t))^. _Vdt+ Z T

0

hσ_ν(u(t)), v_ν(t)i_Γ3dt+

+ Z T

0

(p(u_ν(t)), v_ν(t))_L²_(Γ3)dt ∀v∈L²(0, T;V).

Finally, we plug (2.23) in (4.13), and by a standard argument from the in- equality obtained one obtains (2.21).

REFERENCES

[1] L.-E. Andersson, A quasistatic frictional problem with normal compliance. Nonlinear Anal. Th. Appl.16(1991),4, 347–369.

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

[3] H. Brezis,Equations et in´equations non lin´eaires dans les espaces vectoriels en dualit´e.

Ann. Inst. Fourier18(1968), 115–175.

[4] M.Cocu, E. Pratt and M. Raous, Formulation and approximation of quasistatic fric- tional contact. Int. J. Engng Sc.34(1996),7, 783–798.

[5] M. Cocou and R. Rocca, Existence results for unilateral quasistatic contact problems with friction and adhesion. Math. Model. Num. Anal.34(2000), 981–1001.

[6] S. Drabla and M Sofonea,Analysis of a Signorini problem with friction. IMA Journal of Applied Mathematics63(1999),2, 113–130.

[7] G. Duvaut, Equilibre d’un solide ´elastique avec contact unilat´eral et frottement de Coulomb. C.R. Acad. Sci. Paris, Ser. A290(1980), 263–265.

[8] G. Duvaut and J.L. Lions,Les in´equations en m´ecanique et en physique. Dunod, Paris, 1972.

[9] C. Eck, J. Jaruˇsek and M. Krbeˇc, Unilateral Contact Problems: Variational Methods and Existence Theorems.Pure and Appl. Math.270, CRC Press, New York, 2005.

[10] C. Eck, J. Jaruˇsek and M. Sofonea,A dynamic elastic-visco-plastic unilateral contact problems with normal damped response and Coulomb friction. European J. Appl. Math.

21(2010),3, 229–251.

[11] W. Han and M. Sofonea,Quasistatic Contact Problems in Viscoelasticity and Viscoplas- ticity. Studies in Advanced Mathematics30, Amer. Math. Soc. and International Press, 2002.

[12] J. Jarusˇek and M. Sofonea, On the solvability of dynamic elastic-visco-plastic contact problems. ZAMM Z. Angew. Math. Mech.88(2008), 3–22.

[13] N. Kikuchi and T.J. Oden, Contact problems in elasticity: a study of variational in- equalities and finite element methods. SIAM, Philadelphia, 1988.

[14] A. Klarbring, A. Mikelic and 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.

[15] R. Rocca, Existence of a solution for a quasistatic problem of unilateral contact with local friction. C.R. Acad. Sci. Paris, Ser. 1328(1999), 1253–1258.

[16] M. Rochdi, M. Schillor and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction. J. Elasticity51(1998), 105–126.

[17] M. Sofonea, W. Han and M. Shillor,Analysis and Approximation of Contact Problems with Adhesion or Damage. Pure and Appl. Math.276, Chapman & Hall / CRC Press, Boca Raton, Florida, 2006.

[18] A. Touzaline,A quasistatic frictional contact problem with normal compliance and finite penetration for elastic materials. Glasnik Mathemati´cki45(65)(2010), 109–124.

[19] A. Touzaline and D.J. Teniou,A quasistatic unilateral contact problem with friction for nonlinear elastic materials. Math. Model. Anal.12(2007), 497–514.

Received 14 February 2011 Facult´e de Math´ematiques, USTHB Laboratoire de Syst`emes Dynamiques

BP 32 EL ALIA Bab-Ezzouar, 16111, Algeria