On finite element uniqueness studies for Coulomb s frictional contact model

Transcription

1 On finite element uniqueness studies for Coulomb s frictional contact model Patrick HILD We are interested in the finite element approximation of Coulomb frictional unilateral contact problem in linear elasticity. Using a mixed finite element method and an appropriate regularization, it becomes possible to prove existence and uniqueness when the friction coefficient is lower than Cε log(h where h and ε denote the discretization and the regularization parameters respectively. This bound converging very slowly towards 0 when h decreases (in comparison with the already known results of the non-regularized case suggests a minor dependence of the mesh-size on the uniqueness conditions, at least for practical engineering computations. Then we study the solutions of a simple finite element example in the non-regularized case. It can be shown that one, multiple or an infinity of solutions can occur and that, for a given loading, the number of solutions can eventually decrease when the friction coefficient increases. Keywords : Coulomb friction law, finite elements, mesh-size dependent uniqueness conditions, non-uniqueness example 000 Mathematics Subject Classification. 6530, 74M0. Introduction and problem set-up The Coulomb friction model is currently chosen in the numerical approximation of contact problems arising in structural mechanics. From a mathematical point of view, the study of the continuous model in elastostatics using the associated variational formulation obtained in [6] leads to existence results when the friction coefficient is sufficiently small (see [6, 3, 4, 7]. Concerning the associated finite element model, it has been proved in [8, 9] that it admits always a solution and that the solution is unique provided that the friction coefficient is lower than a positive value vanishing as the discretization parameter decreases. Also in reference [8], a convergence result of the finite element model towards the continuous model was established. Besides, in the finite dimensional context, numerous studies and examples of non-uniqueness using truss elements have been exhibited, proving that the problem is generally not well-posed (see in particular the contributions of [, 5, ]. Laboratoire de Mathématiques, Université de Savoie, CRS EP 067, Le Bourget-du-Lac, FRACE. Tel.: ; fax: ; hild@univ-savoie.fr

2 Uniqueness studies for Coulomb frictional contact Our first aim in this paper is to study the influence of a specific regularization (i.e.; the smoothing of the absolute value involved in the friction model on the uniqueness conditions for the discrete problem. We consider a mixed finite element method in section and, denoting by h and ε the discretization and the regularization parameters respectively, we show in section 3 that the problem admits a unique solution if the friction coefficient is lower than Cε log(h, and we notice that a bound of only Ch can be obtained in the case of the exact model (i.e.; when ε = 0. As a consequence, we note that if ε is chosen slowly decreasing towards zero (as h decreases, then the bound of the non-regularized case becomes more satisfactory than the one arising from the exact model. Our second aim, in section 4, is to choose a particular case of a finite dimensional problem in the non-regularized case: a simple example using finite elements. We study this problem and we show that it can admit one, multiple or an infinity of solutions. Such an example completes and illustrates the already known results using truss elements, especially [5]. Let us now consider an elastic body occupying in the initial configuration a bounded subset Ω of R. The boundary Ω of the domain Ω is supposed to be Lipschitz and consists of three nonoverlapping parts Γ D, Γ and. The unit outward normal on Ω is denoted n = (n,n and we set t = (n, n. The body is submitted to volume forces f = (f,f (L (Ω on Ω and to surface forces F = (F,F (L (Γ on Γ. The part Γ D is embedded and we suppose that the surface measure of Γ D does not vanish. Initially, the body is in contact with a rigid foundation on the straight line segment. The unilateral contact problem with Coulomb friction consists of finding the displacement field u = (u i, i and the stress tensor field σ = (σ ij, i,j, satisfying the following conditions (. (.4: div σ(u + f = 0 in Ω, σ(un = F on Γ, u = 0 on Γ D, (. where (div σ(u i = σ ij,j, i, the notation,j denotes the j th partial derivative and the summation convention of repeated indices is adopted. The stress tensor field is linked to the displacement field by the constitutive law of linear elasticity σ ij (u = λε kk (uδ ij + µε ij (u, (. where λ and µ are the positive Lamé coefficients and where ε ij (u = /(u i,j + u j,i denotes the linearized strain tensor field. On the boundary Ω, we write σ(un = σ n (un+σ t (ut and u = u n n+u t t. Let F > 0 stand for the friction coefficient on. The conditions on the contact zone are as follows: u n 0, σ n (u 0, σ n (u u n = 0, (.3 σ t (u F σ n (u, ( σ t (u F σ n (u u t = 0, σ t (u u t 0. (.4 The conditions (.3 express unilateral contact and conditions (.4 represent Coulomb friction. The closed convex cone K of admissible displacements is the subset in the

3 Uniqueness studies for Coulomb frictional contact 3 Sobolev space (H (Ω of the displacement fields satisfying the embedding and the non-penetration conditions K = {v = (v,v V, v n 0 on, (.5 where V = {v = (v,v (H (Ω, v = 0 on Γ D. As in [6], we consider the mapping Φ : M M with { M = α H (ΓC, α 0, defined for all g M as follows: Φ(g = σ n (u(g, where u(g K is the unique solution of the variational inequality: u(g K, σ ij (u(g ε ij (v u(g dω + < Fg, v t u t (g > ΓC Ω f i (v i u i (g dω + F i (v i u i (g dγ, v K, (.6 Ω Γ where <.,. > ΓC denotes the duality pairing between the fractional Sobolev space H ( (see [] and its dual space H (. Following [6, 0], a weak solution of the unilateral contact problem with Coulomb friction is a pair (u,γ where γ is a fixed point of Φ and u is the unique solution of Problem (.6 with g = γ. The first existence result for the unilateral contact problem with Coulomb friction in the case of a sufficiently small friction coefficient F has been proved in [6]. Generalizations and/or improvements are established in [3, 4, 7]. The uniqueness seems to remain an open problem.. The discrete problem We discretize the domain Ω with a family of triangulations (T h h where the notation h > 0 stands for the discretization parameter representing the greatest diameter of a triangle in T h. The chosen space of finite elements of degree one is: V h = {v h ; v h (C(Ω, v h T (P (T T T h, v h = 0 on Γ D, where C(Ω and P (T denote the space of continuous functions on Ω and the space of polynomial functions of degree one on T respectively. We assume that the families of monodimensional traces of triangulations on are quasi-uniform in order to use inverse inequalities (see [4]. Let W h be the range of V h by the normal trace operator on : W h = {µ h ; µ h = v h ΓC.n, v h V h. Clearly, the space W h involves continuous and piecewise of degree one functions. We define M h as the closed convex cone of Lagrange multipliers expressing nonnegativity: { M h = µ h W h, µ h 0.

4 For any u and v in (H (Ω, define a(u,v = σ(u : ε(v dω, L(v = Ω Uniqueness studies for Coulomb frictional contact 4 Ω f.v dω + F.v dγ. Γ Finally, let us mention that we still keep the notation v h = v hn n + v ht t on the boundary Ω, for any v h V h. For approximating Coulomb frictional contact problem, we choose a mixed finite element method with a nonnegative parameter ε regularizing the absolute value (the case ε = 0 corresponds to the non-regularized problem. As in the continuous framework (.6, the approximated problem requires the introduction of an intermediate setting with a given slip limit g h M h. It consists of finding u h V h and λ h M h such that a(u h, v h u h + λ h (v hn u hn dγ Γ ( C + Fg h vht + ε u ht + ε dγ L(v h u h, v h V h, (. (µ h λ h u hn dγ 0, µ h M h. Henceforth, problem (. will be denoted P ε (g h. Remark. It can be checked that if (u h,λ h solves (. then u h is also solution of the variational inequality which consists of finding u h K h satisfying ( a(u h,v h u h + Fg h vht + ε u ht + ε dγ L(v h u h, for all v h K h. Here, K h stands for a finite dimensional approximation of K defined in (.5: K h = { v h V h, µ h v hn dγ 0, µ h M h. Problem P ε (g h is also equivalent to find a saddle-point (u h,λ h V h M h verifying L(u h,µ h L(u h,λ h L(v h,λ h, v h V h, µ h M h, where L(v h,µ h = a(v h,v h + µ h v hn dγ + Fg h vht + ε dγ L(v h. From results concerning saddle-point problems obtained by Haslinger, Hlaváček and ečas in [0], p.338, the existence of such a saddle-point follows. Moreover, the V ellipticity of a(.,. implies that the first argument u h is unique. Besides, if for any µ h W h, one has: µ h v hn dγ = 0, v h V h, = µ h = 0, (.

5 Uniqueness studies for Coulomb frictional contact 5 then the second argument λ h is unique and P ε (g h admits a unique solution. ote that condition (. is fulfilled because the space W h coincides with the space obtained from V h by the normal trace operator on. It becomes then possible to define two maps: the first one denoted Ψ εh yielding the first component (i.e.; Ψ εh (g h = u h and the second one denoted Φ εh such that Φ εh :M h M h g h λ h, where (u h,λ h is the solution of P ε (g h. The introduction of the latter map allows the defining of a solution to Coulomb discrete frictional contact problem. Definition. A solution of Coulomb discrete regularized (resp. non-regularized frictional contact problem is a solution of P ε (λ h with ε > 0 (resp. ε = 0 where λ h M h is a fixed point of Φ εh. Set Ṽ h = {v h V h, v ht = 0 on. It is easy to check that the definition of.,h given by ν,h = sup v h Ṽh νv hn dγ v h, (.3 is a norm on W h (since condition (. holds. The notation. represents the (H (Ω -norm. 3. Existence and uniqueness studies We are now interested in the existence and uniqueness study for the discrete problem. In order to establish existence, it suffices to show that the mapping Φ εh admits a fixed point in M h by using Brouwer s theorem. Uniqueness is ensured if the mapping is contractive. Such a technique has been already used in the non-regularized case with discontinuous and piecewise constant Lagrange multipliers [8, 9]. Our aim is to study the regularized case (and also the non-regularized case when using continuous piecewise of degree one Lagrange multipliers. Theorem 3. Let ε > 0. The following results hold: (Existence For any positive F, there exists a solution to the Coulomb discrete regularized frictional contact problem. (Uniqueness Assume that Γ D =. If F Cε log(h then the problem admits a unique solution. The positive constant C depends neither of h nor of ε. Proof. Let (u h,λ h be the solution of P ε (g h. Taking v h = 0 in the equation of (. gives ( a(u h,u h + λ h u hn dγ Fg h ε u ht + ε dγ L(u h. (3.

6 Since g h 0, ε u ht + ε 0, and according to λ h u hn dγ = 0, Uniqueness studies for Coulomb frictional contact 6 it follows from (3., the V ellipticity of a(.,. and the continuity of L(. that: α u h a(u h,u h L(u h C u h, where α stands for the ellipticity constant of a(.,.. Here, the constant C depends on the external loads f and F. Therefore, using the trace theorem yields u ht H (ΓC C u h CC α. (3. Besides, the equality in (. implies a(u h,v h + λ h v hn dγ = L(v h, v h Ṽh. Denoting by M the continuity constant of a(.,. yields λ h v hn dγ M u h v h + C v h, v h Ṽh. As a consequence So, we come to the conclusion that ( M λ h,h M u h + C α + C. Φ εh (g h,h C, g h M h, (3.3 where C only depends on the applied loads f,f and on the continuity and ellipticity constant of a(.,.. The existence result of Theorem 3. consists now to show that the mapping Φ εh is continuous. Let (u h,λ h and (u h,λ h be the solutions of P ε (g h and P ε (g h respectively (where g h M h and g h M h. From (., we get a(u h,v h + λ h v hn dγ =L(v h, v h Ṽh, and a(u h,v h + λ h v hn dγ =L(v h, v h Ṽh,

7 Uniqueness studies for Coulomb frictional contact 7 which implies by subtraction that (λ h λ h v hn dγ = a(u h u h,v h M u h u h v h, v h Ṽh, where the continuity of the bilinear form a(.,. has been used. So, we get the following estimate λ h λ h,h M u h u h. (3.4 ext, we show that Ψ εh is continuous from M h into V h. We consider again (u h,λ h and (u h,λ h, the solutions of P ε (g h and P ε (g h respectively. We have a(u h,v h u h + λ h (v hn u hn dγ Γ C ( + Fg h vht + ε u ht + ε dγ L(v h u h, v h V h, and a(u h,v h u h + λ h (v hn u hn dγ Γ C ( + F g h vht + ε u ht + ε dγ L(v h u h, v h V h. Choosing v h = u h in the first inequality and v h = u h in the second one, we obtain thanks to (.: ( a(u h,u h u h + Fg h u ht + ε u ht + ε dγ L(u h u h, and Thus So ( a(u h,u h u h + F g h u ht + ε u ht + ε dγ L(u h u h. ( a(u h u h,u h u h F(g h g h u ht + ε u ht + ε dγ. (3.5 α u h u h F g h g h H u ( ht + ε u ht + ε H (ΓC. (3.6 The next step consists of estimating the H -norm term in (3.6. To attain our ends, we need to use two lemmas which follow hereafter.

8 Uniqueness studies for Coulomb frictional contact 8 Lemma 3. There exists a positive constant C satisfying for all f and g in H (Γ C L ( : ( fg C H (ΓC f g H (ΓC L ( + f L ( g H (ΓC. (3.7 Proof. From the definition of the H ( norm (see [], we have fg = H ( fg L ( ΓC + (f(xg(x f(yg(y dγdγ. (x y Let us begin with bounding (roughly the first term: fg L ( = f (xg (x dγ f L ( g L (. (3.8 The second term is handled as follows: (f(x(g(x g(y + g(y(f(x f(y ΓC dγdγ (x y f ΓC (x(g(x g(y + g (y(f(x f(y dγdγ (x y (x y ( f L ( g + H ( f H ( g L (. (3.9 Putting together (3.8 and (3.9 establishes (3.7. Lemma 3.3 For any real number p [, [, the following inequality holds: f L p ( C p f H (ΓC, f H (ΓC, (3.0 where C is independent of p. Proof. see [3], Lemma A.. Proof of Theorem 3.. We consider the H -norm term in (3.6. Employing estimate (3.7 gives: u ht + ε u ht + H ε = (ΓC (u u ht + u ht ht u ht u ht + ε + u ht H + ε ( u ht + u ht C u ht u ht L ( u ht + ε + u ht H + ε ( u ht + u +C u ht u ht H (ΓC ht u ht + ε + u ht + (3. L (ΓC. ε In the previous estimate, we leave the third term unchanged whereas the last one is bounded by. It remains then to bound the first two terms which is performed hereafter. We begin with the first one: u ht u ht L ( Ch p uht u ht L p ( C ph p uht u ht H (ΓC, (3.

9 Uniqueness studies for Coulomb frictional contact 9 for any p [, [. In (3., we used an easily recoverable inverse inequality (see also [4] as well as (3.0. The second term of (3. is bounded thanks to (3.7: u ht + u ht u ht + ε + u ht H + C u ht + u ht L ( ε ( u ht + ε + u ht H + ε ( +C u ht + u ht H (ΓC u ht + ε + u ht L + ε (ΓC C ph p uht + u ht H (ΓC u ht + ε + u ht + + H ε ε u ht + u ht, H (ΓC ( (3.3 where the first L norm term is bounded as in (3. whereas the other one is roughly bounded by /ε. ext, we develop the first H norm term in (3.3 u ht + ε + u ht + = ε H ( u ht + ε + u ht + ε L ( ( + (y x u ht (x + ε + u ht (x + ε u ht (y + ε + u ht (y + dγdγ. ε It is easy to check that the L norm term is lower than meas( /4ε. Developing the previous integral, bounding then the denominator and using estimate (a + b a + b furnishes the following upper bound: 8ε 4 ( u ht (x + ε u ht (y + ε (y x + ( u ht (x + ε u ht (y + ε (y x dγdγ. We use estimate a + ε b + ε a b in the previous expression so that u ht + ε + u ht + meas( + ( u ε 4ε 8ε 4 ht + u ht. H ( H ( H ( Therefore, we deduce from (3. that a positive constant C exists verifying ( u ht + ε + u ht H + C ε ε +. (3.4 ε ( Putting estimate (3.4 in (3.3, using (3. and (3. gives u ht + ε u ht + H ε (ΓC ( C u ht u ht H (ΓC + ph p( ε + ph p( ε + ε.

10 Uniqueness studies for Coulomb frictional contact 0 Choosing p = log(h (h is assumed sufficiently small in the previous estimate, we obtain u ht + ε u ht + H ε (ΓC ( log h C u ht u ht H (ΓC + ε + log h ε + log h. (3.5 ε The inequality (3.6 together with (3.5 and the trace theorem becomes: ( log h u h u h CF g h g h H + + log h + log h, (3.6 ( ε ε ε which proves that the mapping Ψ εh is continuous. This together with (3.4 implies that Φ εh is continuous. Then, from (3.3 and according to Brouwer fixed point theorem, we conclude to the existence of at least one solution to Coulomb discrete regularized frictional contact problem. We now consider uniqueness. Under the assumption Γ D =, it has been proved in [5], Proposition 3., that there exists a positive constant β (independent of h satisfying β µ h H ( µ h,h, µ h W h. (3.7 Assembling this result with (3.6 and (3.4 yields ( log h λ h λ h H CF g ( h g h H + ( ε + log h ε + log h. ε Supposing h and ε small enough, we deduce that the mapping Φ εh is contractive if the friction coefficient F is lower than Cε log(h. That ends the proof of the theorem. The non-regularized case (i.e.; ε = 0 is handled in the following proposition. Proposition 3.4 Let ε = 0. The following results hold: (Existence For any positive F, there exists a solution to the Coulomb discrete frictional contact problem. (Uniqueness Assume that Γ D =. If F Ch then the problem admits a unique solution. The positive constant C is independent of h. Proof. Estimates (3.3 and (3.4 remain still valid when ε = 0. The starting point of the analysis is (3.5: a(u h u h,u h u h F(g h g h ( u ht u ht dγ F g h g h L ( u ht u ht L ( CF h gh g h H ( u ht u ht L ( C F h gh g h H ( u h u h,

11 Uniqueness studies for Coulomb frictional contact where an inverse inequality between L ( and H ( has been used. From the latter bound, combined with (3.4 and (3.7, we deduce Hence the result. λ h λ h H ( CFh gh g h H (. Remark 3.5. In the proof of proposition 3.4, we are not able to remove the mesh dependent uniqueness condition also when avoiding the L ( -norms and using only H ( -norms and H ( -norms. More precisely, there does not exist a positive constant C independent of h such that g h g h H ( C g h g h H ( or u ht u ht H (ΓC C u ht u ht H (ΓC.. The use of inverse inequalities in the proofs of Theorem 3. and Proposition 3.4 implies that it is not possible to generalize the calculus to the continuous problem. 4. The study of a simple finite element example We consider the triangle Ω of vertexes A = (0, 0, B = (l, 0 and C = (0,l with l > 0. We define Γ D = [B,C], Γ = [A,C], = [A,B] and {X,X denotes the canonical orthonormal basis (see Figure. We suppose that the volume forces f are absent and that the surface forces denoted F = F X + F X are such that F and F are constant on Γ. C X F Γ Γ D X Ω t A Γ C n B Figure : Setting of the problem We suppose that Ω is discretized with a single finite element of degree one. Consequently, the finite element space becomes: V h = { v h = (v h,v h (P (Ω, v h ΓD = 0. In this case M h = { g h P (, g h 0, g h (B = 0.

12 Uniqueness studies for Coulomb frictional contact Clearly, V h is of dimension two and M h belongs to the space W h of linear functions on vanishing at B, which is of dimension one. Moreover, since (. or equivalently (.3 is satisfied, it follows that existence is ensured for all ε 0 according to Theorem 3. and Proposition 3.4. Let v h V h and µ h M h. Then we denote by (V T,V the value of v h (A corresponding to the tangential and the normal displacements at point A respectively (in our example, we have V T = v h (A and V = v h (A. We also denote by Θ the value of µ h at point A. Then, for any v h V h and µ h M h, one obtains ( VT V T + V ε(v h = l V T + V V and Therefore σ(v h = l a(u h,v h = ( (λ + µvt + λv µ(v T + V µ(v T + V (λ + µv + λv T. ( (λ + 3µ(U T V T + U V + (λ + µ(u T V + U V T and L(v h = l(f V T + F V. Besides µ h v hn dγ = ΘV l 3 and Fµ h v ht dγ = FΘ V T l 3 Let (u h,λ h be a solution of the discrete unilateral contact problem with Coulomb friction and without regularization (i.e.; with ε = 0 in (.. As above-mentioned, the notation (U T,U stands for the value of u h (A (U T = u h (A and U = u h (A. We also denote by Λ the value of λ h at point A. To simplify the notations and the forthcoming calculations, we set Λ = Λ /3. The discrete unilateral contact problem with Coulomb friction and without regularization issued from (. and Definition. consists then of finding (U T,U, Λ R 3 such that (λ + 3µ(U T V T + U V + (λ + µ(u T V + U V T + ΛlV + FΛl V T l(f V T + F V, V T R, V R,. (λ + 3µ(U T + U + (λ + µ(u TU + FΛl U T = l(f U T + F U, Λ 0, U 0, ΛU = 0,

13 Uniqueness studies for Coulomb frictional contact 3 or equivalently (λ + 3µU + (λ + µu T + Λl = lf, (λ + µu + (λ + 3µU T + FΛl lf, (λ + µu + (λ + 3µU T FΛl lf, (λ + 3µ(UT + U + (λ + µ(u TU + FΛl U T = l(f U T + F U, Λ 0, U 0, ΛU = 0. (4. Let us now search the solutions to the equations (4.. Clearly, a solution of (4. satisfies either U = 0 or Λ = 0. (i Case : U = 0. Equations (4. become: (λ + µu T + Λl = lf, (λ + 3µU T + FΛl lf, (λ + 3µU T FΛl lf, (λ + 3µUT + FΛl U T = lf U T, Λ 0. Suppose that U T = 0. Then Λ = F, F 0, F F F. Suppose that U T > 0. Then (λ + µu T + Λl = lf, (λ + 3µU T + FΛl = lf, Λ 0. Assume that F λ+3µ λ+µ. Then U T = l(ff F (λ + 3µ F(λ + µ > 0, Λ = (λ + µf (λ + 3µF (λ + 3µ F(λ + µ Assume that F = λ+3µ λ+µ. Then If F = FF, the solutions are (λ + µu T + Λl = lf, U T > 0, Λ 0. If F FF then there are no solutions. 0.

14 Suppose that U T < 0. Then (λ + µu T + Λl = lf, (λ + 3µU T FΛl = lf, Λ 0, which gives U T = (ii Case : Λ = 0. so that Uniqueness studies for Coulomb frictional contact 4 l(ff + F (λ + 3µ F(λ + µ < 0, Λ = (λ + µf + (λ + 3µF (λ + 3µ F(λ + µ (λ + 3µU + (λ + µu T = lf, (λ + µu + (λ + 3µU T = lf, U 0, U T = l((λ + µf (λ + 3µF, U = l((λ + µf (λ + 3µF 4µ(λ + µ 4µ(λ + µ All the results are reported in the following proposition. There are three cases which consist of comparing the friction coefficient F with the critical value λ+3µ = λ+µ 3 4ν (ν denotes Poisson ratio with 0 < ν < /. The results are also depicted in Figures, 3 and Proposition 4.. If F < λ+3µ then the problem (4. admits a unique solution: λ+µ (Separation If F > λ+µ F λ+3µ, then U T = l((λ + µf (λ + 3µF 4µ(λ + µ (Stick If F F F and F 0 then (Right slip If F λ+µ λ+3µ F, FF + F > 0 then U T =, U = l((λ + µf (λ + 3µF, Λ = 0. (4. 4µ(λ + µ U T = 0, U = 0, Λ = F. (4.3 l(ff + F (λ + 3µ F(λ + µ, U = 0, Λ = (λ + µf + (λ + 3µF (λ + 3µ F(λ + µ. (4.4 (Left slip If F λ+µ λ+3µ F, FF F > 0 then U T = l(ff F (λ + 3µ F(λ + µ, U = 0, Λ = (λ + µf (λ + 3µF (λ + 3µ F(λ + µ. (4.5

15 Uniqueness studies for Coulomb frictional contact 5. If F = λ+3µ then, depending on the loadings, the problem (4. admits either a λ+µ unique solution or an infinity of solutions: (Separation If F > λ+µ F λ+3µ, then the solution is given by (4.. (Stick If ( F F < F F F and F 0 or F = F = 0 then the solution is given by (4.3. (Right slip If F λ+µ F λ+3µ, FF + F > 0 then the solution is given by (4.4. (From stick to left slip If F = FF and F < 0, then there exists an infinity of solutions: U T = l(f + β, U = 0, Λ = β, for all 0 β F. λ + µ 3. If F > λ+3µ then, depending on the loadings, the problem (4. admits one, two λ+µ or three solutions: (Separation If F > λ+µ F λ+3µ and FF F > 0 then the solution is given by (4.. (Stick If ( λ+3µ F λ+µ < F F F and F 0 or F = F = 0 then the solution is given by (4.3. (Right slip If F λ+µ F λ+3µ, FF + F > 0 then the solution is given by (4.4. (Separation and stick If F = FF and F < 0, then there are two solutions given by (4. and (4.3. (Stick and left slip If F = λ+3µ F λ+µ and F < 0, then there are two solutions given by (4.3 and (4.5. (Separation, stick and left slip If F F < F < λ+3µ F λ+µ and F 0 then there are three solutions given by (4., (4.3 and (4.5. F Separation U < 0 Right slip U = 0 U < 0 T F Left slip U = 0 U > 0 T Stick U =0 U =0 T Figure : Case F < λ+3µ λ+µ = 3 4ν. Problem (4. admits a unique solution.

16 Uniqueness studies for Coulomb frictional contact 6 F Separation U < 0 Right slip Infinity of solutions from stick to left slip Stick U =0 U =0 T U = 0 U < 0 T F Figure 3: Case F = λ+3µ λ+µ infinity of solutions. = 3 4ν. Problem (4. admits either a unique or an solutions: stick and separation 3 solutions stick, left slip and separation solutions: stick and left slip F Separation U < 0 Stick U =0 U =0 T Right slip U = 0 U < 0 T F Figure 4: Case F > λ+3µ λ+µ solutions. = 3 4ν. Problem (4. admits a unique, two or three The study of sufficient conditions of non-uniqueness for Coulomb frictional contact problem in the continuous framework is actually under consideration in []. REFERECES [] Adams R.A. (975: Sobolev spaces. ew-york: Academic Press.

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