A uniqueness criterion for the Signorini problem with Coulomb friction

Transcription

1 A uniqueness criterion for the Signorini problem with Coulomb friction Yves REARD 1 Abstract he purpose of this paper is to study the solutions to the Signorini problem with Coulomb friction (the so-called Coulomb problem) Some optimal a priori estimates are given and a uniqueness criterion is exhibited Recently nonuniqueness examples have been presented in the continuous framework It is proven here that if a solutions satisfies a certain hypothesis on the tangential displacement and if the friction coefficient is small enough it is the unique solution to the problem In particular this result can be useful for the search of multi-solutions to the Coulomb problem because it eliminates a lot of uniqueness situations Keywords : unilateral contact Coulomb friction uniqueness of solution Introduction he so-called Signorini problem with Coulomb friction (or simply the Coulomb problem) has been introduced by Duvaut and Lions [4] It does not exactly represent the equilibrium of a solid which encounters an obstacle because when the equilibrium is reached (or any steady state solution) the friction condition is no longer an irregular law he interest of this problem is in fact to be very close to a time semidiscretization of an evolutionary problem by an implicit scheme he fact that several solutions could co-exist to an implicit scheme (independently of the size of the time step) may be an indication that the evolutionary problem has a dynamical bifurcation he first existence results for this problem were obtained by ečas Jarušek and Haslinger in [14] for a two-dimensional elastic strip assuming that the coefficient of friction is small enough and using a shifting technique previously introduced by Fichera and later applied to more general domains by Jarušek [10] Eck and Jarušek [5] give a different proof using a penalization method We emphasize that most results on existence for frictional problems involve a condition of smallness for the friction coefficient (and a compact support on Γ C ) Recently examples of nonunique solutions have been given by P Hild in [7] and [8] for a large friction coefficient As far as we know for a fixed geometry it is still an open question to know whether or not there is uniqueness of the solution for a sufficiently small friction coefficient In the finite element approximation framework the presence of bifurcation has been studied in [9] he present paper gives the first (partial) result of uniqueness of a solution to the Coulomb problem he summary is the following Section 1 introduces strong and weak formulations of the Coulomb problem Section 2 gives optimal estimates on the solutions In particular a comparison is made with the solution to the frictionless contact problem Section 3 gives an additional estimate for the resca problem ie the problem with given friction threshold And finally Section 4 gives the partial uniqueness result It is proven in proposition 5 for bidimensional problems and a friction coefficient less than one that there in no multi-solutions with one of the solutions having a tangential displacement with a constant sign he major result is given by Proposition 6 using the notion of multiplier in a pair of Sobolev spaces 1 MIP CRS UMR 5640 ISA Complexe scientifique de Rangueil oulouse France YvesRenard@insa-toulousefr 1

2 1 he Signorini problem with Coulomb friction Let Ω R d (d = 2 or 3) be a bounded Lipschitz domain representing the reference configuration of a linearly elastic body Figure 1: Elastic body Ω in frictional contact It is assumed that this body is submitted to a eumann condition on a part of its boundary Γ to a Dirichlet condition on another part Γ D and a unilateral contact with static Coulomb friction condition on the rest of the boundary Γ C between the body and a flat rigid foundation his latter part Γ C is supposed to be of nonzero interior in the boundary Ω of Ω he problem consists in finding the displacement field u(t x) satisfying div σ(u) = f in Ω (1) σ(u) =A ε(u) in Ω (2) σ(u)n = F on Γ (3) u = 0 on Γ D (4) where σ(u) is the stress tensor ε(u) is the linearized strain tensor n is the outward unit normal to Ω on Ω F and f are the given external loads anda is the elastic coefficient tensor which satisfies classical conditions of symmetry and ellipticity On Γ C it is usual to decompose the displacement and the stress vector in normal and tangential components as follows: u = un u = u u n σ (u) = (σ(u)n)n σ (u) = σ(u)n σ (u)n o give a clear sense to this decomposition we assume Γ C to have thec 1 regularity he unilateral contact condition is expressed by the following complementary condition: u g σ (u) 0 (u g)σ (u) = 0 (5) where g is the normal gap between the elastic solid and the rigid foundation in reference configuration (see Fig 2) 2

3 Ω n g Rigid foundation Figure 2: ormal gap between the elastic solid Ω and the rigid foundation Denoting byf 0 the friction coefficient the static Coulomb friction condition reads as: if u = 0 then σ (u) F σ (u) (6) if u 0 then σ (u) =F σ (u) u u (7) he friction force satisfies the so-called maximum dissipation principle 11 Classical weak formulation σ (u)u = sup ( µ u ) (8) µ R d 1 µ F σ (u) We present here the classical weak formulation proposed by G Duvaut [3] [4] Let us introduce the following Hilbert spaces V = {v H 1 (Ω;R d )v = 0 on Γ D } = {v ΓC : v V } H 1/2 (Γ C ;R d ) = {v ΓC : v V } = {v ΓC : v V } and their topological dual spaces V and It is assumed that Γ C is sufficiently smooth such that H 1/2 (Γ C ) H 1/2 (Γ C ;R d 1 ) H 1/2 (Γ C ) and H 1/2 (Γ C ;R d 1 ) Classically H 1/2 (Γ C ) is the space of the restriction on Γ C of traces on Ω of functions of H 1 (Ω) and H 1/2 (Γ C ) is the dual space of H 1/2 (Γ 00 C ) which is the space of the restrictions on Γ C of functions of H 1/2 ( Ω) vanishing outside Γ C We refer to [1] and [11] for a detailed presentation of trace operators ow the set of admissible displacements is defined as K = {v Vv g ae on Γ C } (9) he following maps Z a(uv) = Ω A ε(u) : ε(v)dx Z Z l(v) = fvdx+ FvdΓ Ω Γ j(f λ v ) = F λ v 3

4 represent the virtual work of elastic forces the external load and the virtual work of friction forces respectively Standard hypotheses are: a( ) is a bilinear symmetric V-elliptic and continuous form on V V : α > 0 M > 0a(uu) α u 2 a(uv) M u v V V V uv V (10) l( ) linear continuous form on V ie L > 0 l(v) L v V v V (11) g F M being a nonnegative multiplier in (13) he latter condition ensure that j(f λ v ) is linear continuous on λ and convex lower semi-continuous on v when λ is a nonpositive element of (see for instance [2]) o satisfy condition (10) it is necessary that Γ D is of nonzero interior in the boundary of Ω and that the elastic coefficient tensor is uniformly elliptic (see [4]) (12) We refer to Maz ya and Shaposhnikova [13] for the theory of multipliers he set M space of multipliers from into ie the space of function f : Γ C R of finite norm denote the f M v v 0 f v v his is the norm of the linear mapping v ( f v) Of course iff is a constant on Γ C one has F M =F From the fact that Ω is supposed to be a bounded Lipschitz domain and Γ C is supposed to have thec 1 regularity it is possible to deduce that for d = 2 the space H 1/2+ε (Γ C ) is continuously included in M for any ε > 0 and for d = 3 the space H 1 (Γ C ) L (Γ C ) is included in M continuously for the norm f + f H 1 (ΓC (see [13]) In particular the space of Lipschitz continuous functions is ) L (Γ C ) continuously included in M Condition (10) implies in particular that a( ) is a scalar product on V and the associated norm v a = (a(vv)) 1/2 is equivalent to the usual norm of V : α v V v a M v V v V he continuity constant of l( ) can also be given with respect to a L a > 0 l(v) L a v a v V Constants L and L a can be chosen such that αla L ML a he classical weak formulation of Problem (1) - (7) is given by: Find u K satisfying a(uv u)+ j(f σ (u)v ) j(f σ (u)u ) l(v u) v K (14) he major difficulty about (14) is due to the coupling between the friction threshold and the contact pressure σ (u) he consequence is that this problem does not represent a variational inequality in the sense that it cannot be derived from an optimization problem 4

5 12 eumann to Dirichlet operator In this section the eumann to Dirichlet operator on Γ C is introduced together with its basic properties his will allow to restrict the contact and friction problem to Γ C and obtain useful estimates Let λ = (λ λ ) then under hypotheses (10) and (11) the solution u to Find u V satisfying a(uv) = l(v)+ λv v V (15) is unique (see [4]) So it is possible to define the operator E : λ u ΓC his operator is affine and continuous Moreover it is invertible and its inverse is continuous It is possible to express E 1 as follows: for w let u be the solution to the Dirichlet problem Find u V satisfying u ΓC = w and (16) a(uv) = l(v) v Vv ΓC = 0 then E 1 (w) is equal to λ defined by λv = a(uv) l(v) v V Remark 1 In a weak sense one has the relation E 1 (u) = σ(u)n on Γ C ow under hypotheses (10) and (11) one has E(λ 1 ) E(λ 2 ) C2 1 α λ1 λ 2 (17) where C 1 is the continuity constant of the trace operator on Γ C and α the coercivity constant of the bilinear form a( ) One can verify it as follows Let λ 1 and λ 2 be given in and u1 u 2 the corresponding solutions to (15) then α u 1 u 2 2 V a(u1 u 2 u 1 u 2 ) = λ 1 λ 2 u 1 u 2 C 1 λ 1 λ 2 u 1 u 2 V (18) and consequently Conversely one has u 1 u 2 V C 1 α λ1 λ 2 (19) E 1 (u 1 ) E 1 (u 2 ) MC 2 2 u 1 u 2 (20) where M is the continuity constant of a( ) and C 2 > 0 is the continuity constant of the homogeneous w Dirichlet problem corresponding to (16) (ie with l(v) 0 and C 2 V v where w v ΓC = v and 5

6 a(w z) = 0 z V ) his latter estimate can be performed as follows E 1 (u 1 ) E 1 (u 2 ) E 1 (u 1 ) E 1 (u 2 )v v v ( v inf {w V :w ΓC =v} a(u 1 u 2 w) v Mγ u 1 u 2 V (21) ) w where γ inf V is the continuity constant of the homogeneous Poisson problem with respect to a Dirichlet condition on Γ C v {w V :w v =v} ΓC Using γ C 2 this gives (20) It is also possible to define the following norms on Γ C relatively to a( ) v aγc = inf w a w V w =v ΓC λv λ aγc v v aγc which are equivalent respectively to the norms in and v V λv v a α C 1 v v aγc Mγ v 1 Mγ λ λ aγc C 1 α λ With these norms the estimates are straightforward since the following lemma holds Lemma 1 Let λ 1 and λ 2 be two elements of and u 1 = E(λ 1 ) u 2 = E(λ 2 ) then under hypotheses (10) and (11) one has u 1 u 2 aγc = u 1 u 2 a = λ 1 λ 2 aγc Proof On the one hand one has u 1 u 2 2 aγ C = inf w V w Γc =u 1 u 2 w 2 a = u 1 u 2 2 a because u 1 u 2 is the minimum of 1 2 w 2 a under the constraint w ΓC = u 1 u 2 his implies u 1 u 2 2 aγ C = a(u 1 u 2 u 1 u 2 ) = λ 1 λ 2 u 1 u 2 λ 1 λ 2 aγc u 1 u 2 aγc and finally u 1 u 2 aγc λ 1 λ 2 aγc 6

7 On the other hand one has λ 1 λ 2 aγc v λ 1 λ 2 v v aγc v inf w V w ΓC =v a(u 1 u 2 w) v aγc sup v inf w V w ΓC =v u 1 u 2 a w a v aγc = u 1 u 2 a = u 1 u 2 aγc which ends the proof of the lemma 13 Direct weak inclusion formulation Let K = {v : v 0 ae on Γ C } be the (translated) set of admissible normal displacements on Γ C he normal cone in to K at v is defined as K (v ) = {µ : µ w v 0 w K } In particular K (v ) = /0 if v is given by / K he sub-gradient of j(f λ u ) with respect to the second variable 2 j(f λ u ) = {µ : j(f λ v ) j(f λ u )+ µ v u v } With this notations Problem (14) is equivalent to the following problem Find u Vλ and λ satisfying (u u ) = E(λ λ ) λ K (u g) in λ 2 j(f λ u ) in (22) More details on this equivalence can be found in [12] Remark 2 Inclusion λ K (u g) is equivalent to the complementarity relations u g λ v 0 v K λ u g = 0 which is the weak analogous to the strong complementarity relations (5) for the contact conditions Similarly the second inclusion λ 2 j(f λ u ) represents the friction condition 14 Hybrid weak inclusion formulation We will now consider the sets of admissible stresses he set of admissible normal stresses on Γ C can be defined as Λ = {λ : λ v 0 v K } 7

8 his is the opposite of K the polar cone to K he set of admissible tangential stresses on Γ C can be defined as Λ (F λ ) = {λ : λ w + F λ w 0 w } With this Problem (14) is equivalent to the following problem Find u Vλ and λ satisfying (u u ) = E(λ λ ) (u g) Λ (λ ) in u Λ (F λ )(λ ) in (23) where the two inclusions can be replaced by inequalities as follows Find u Vλ and λ satisfying (u u ) = E(λ λ ) λ Λ µ λ u g 0 µ Λ (24) λ Λ (F λ ) µ λ u 0 µ Λ (F λ ) Remark 3 he inclusion u Λ (F λ )(λ ) implies the following complementarity relation λ u = F λ u and the following weak maximum dissipation principle which is the weak formulation of (8) λ u µ Λ (F λ ) µ u 2 Optimal a priori estimates on the solutions to the Coulomb problem For the sake of simplicity a vanishing contact gap (g 0) will be considered in the following Remark 4 In the case of a nonvanishing gap it is possible to find u g V such that u g ΓC = gn and then w = u u g is solution to the problem Find w Vλ and λ satisfying a(wv) = l(v) a(u g v)+ λ w + λ w w Λ (λ ) in w Λ (F λ )(λ ) in ie a contact problem without gap but with a modified source term 8 (25)

9 Following remarks 13 and 14 a solution (uλ) to Problem (22) (ie a solution u to Problem (14)) satisfies the complementarity relations λ u = 0 λ u = F λ u his implies λu 0 which expresses the dissipativity of contact and friction conditions he first consequence of this is that solutions to Problem (14) can be bounded independently of the friction coefficient Proposition 1 Assuming hypotheses (10) (11) (13) are satisfied and g 0 let (uλ) be a solution to Problem (22) which means u solution to Problem (14) then u a L a λ aγc L a Proof One has u V L M α λ Lγ α u 2 a = a(uu) = l(u)+ λu L a u a which states the first estimates he estimate on λ aγc can be performed using the intermediary solution u to the following problem with a homogeneous eumann condition on Γ C a(u v) = l(v) v V (26) Since u a L a for the same reason as for u and using Lemma 1 one has λ 0 2 aγ = a(u u u u ) = λu u C λu L a λ aγc he two last estimates can be stated thanks to equivalence of norms introduced in section 11 It is possible to compare u a to the corresponding norm of the solution u c to the Signorini problem without friction defined as follows Find u c K satisfying a(u c v u c ) l(v u c ) v K (27) It is well known that under hypotheses (10) and (11) this problem has a unique solution (see [11]) Proposition 2 Assuming hypotheses (10) (11) (13) are satisfied and g 0 let u be a solution to Problem (14) u c be the unique solution to Problem (27) and u the solution to Problem (26) then u a u c a u a 9

10 Proof One has a(u u ) = l(u ) a(u c u c ) = l(u c ) a(uu) = l(u)+ λ u Since u c is the solution to the Signorini problem without friction it minimizes over K the energy functional 1 2 a(vv) l(v) he solution u minimizes this energy functional over V hus since u K one has 1 2 a(u u ) l(u ) 1 2 a(uc u c ) l(u c ) 1 a(uu) l(u) 2 and the following relations allow to conclude a(u c u c ) a(u u ) = l(u c u ) a(uu) a(u c u c ) = l(u u c )+ λ u because then 1 2 a(uc u c ) 1 2 a(u u ) a(uu) 1 2 a(uc u c )+ F λ u 1 2 a(uc u c ) It is also possible to estimate how far from u c is a solution u to Problem (14) Let us introduce the following norms on Γ C For v let us define v aγc = inf w a = inf z aγc w V z w =v z =v One has ow for λ let us define v aγc = inf w a = inf v aγc w V z w =v z =v v aγc v aγc v aγc v aγc λ aγc v v 0 λ v v aγc v λ v v aγc λ aγc v v 0 λ v v aγc hen the following equivalence of norms are immediate: λ v v v aγc α C 1 v v aγc γ M v 1 γ M λ λ aγc C 1 α λ And the following result can be easily deduced: α C 1 v v aγc γ M v 1 γ M λ λ aγc C 1 α λ 10

11 Lemma 2 here exists C 3 > 0 such that for all λ λ aγc C 3 λ aγc λ aγc C 3 λ aγc his allow also to define an equivalent norm on M given forf M by F a v v 0 F v aγc v aγc which satisfies α C 1 γ M F a F M With these definitions the following result holds: C 1γ M F a α Lemma 3 here exists C 4 > 0 such that Proof One has Moreover it is known (see [1]) that the norm F v aγc C 4 F a v aγc v F v aγc F a v aγc v 2 1/2Γ C = v 2 L 2 (Γ C ) + Z is equivalent to the norm Γ C ZΓC v (x) v (y) 2 x y d dxdy and it is easy to verify that v 1/2ΓC v 1/2ΓC for any v hus the result can be deduced from the previously presented equivalences of norms Of course the tangential stress on Γ C corresponding to u c is vanishing he tangential stress corresponding to u can be estimated as follows As λ Λ (F λ ) one has λ aγc v v 0 λ v v aγc sup v v 0 F λ v v aγc C 4 F a λ aγc ow with the result of Proposition 1 this means and the following result holds λ aγc L a C 3 C 4 F a (28) Proposition 3 Assuming hypotheses (10) (11) (13) are satisfied and g 0 let u be a solution to Problem (14) and u c be the solution to Problem (27) then u c u a L a C 3 C 4 F a u c u V LC 3C 4 α F a 11

12 Proof With λ and λ c the corresponding stresses on Γ C because λ c K (u c ) and λ K (u g) and the fact that K is a monotone set-valued map one has λ c λ uc u 0 ow u c u a can be estimated as follows u c u 2 a = a(uc uu c u) = λ c λu c u λ aγc u c u a which gives the result taking into account (28) he latter result implies that if Problem (14) has several solutions then they are in a ball of radius L a C 3 C 4 F a centered around u c In particular if u 1 and u 2 are two solutions to Problem (14) one has u 1 u 2 a 2L a C 3 C 4 F a his is illustrated by Fig 3 u a u c a 0 F a Figure 3: Admissibility zone for u a Remark 5 For a friction coefficientf constant on Γ C the graph on Fig 3 can be more precise for F = F a small since from the proof of Proposition 2 and the continuity result given by the latter proposition one can deduce u 2 a uc 2 a +F λc uc at least if λ c uc < 0 Of course if λ c uc = 0 the solution u c to the Signorini problem without friction is also solution to the Coulomb problem for any friction coefficient 3 Elementary estimates on the resca problem What is usually called the resca problem is the friction problem with a given friction threshold Let θ be given Find u K satisfying (29) a(uv u)+ j(θv ) j(θu ) l(v u) v K 12

13 It is well known that under standard hypothesis (10) (11) (13) this problem has a unique solution (see [11]) which minimizes the functional 1 a(uu)+ j(θu) l(u) 2 In fact it is not difficult to verify that all the estimates given in the latter section for the solutions to the Coulomb problem are still valid for the solution to the resca problem Moreover the solution to the resca problem continuously depends on the friction threshold θ his result is stated in the following lemma: Lemma 4 Assuming hypotheses (10) (11) (13) are satisfied if u 1 u 2 are the solutions to Problem (29) for a friction threshold θ 1 Λ and θ 2 Λ respectively then there exists a constant C 5 > 0 independent of θ 1 and θ 2 such that the following estimate holds u 1 u 2 2 a C 5 θ 1 θ 2 aγc Proof One has which implies a(u 1 u 2 u 1 ) l(u 2 u 1 )+ j(θ 1 u 2 ) j(θ 1 u 1 ) 0 a(u 2 u 1 u 2 ) l(u 1 u 2 )+ j(θ 2 u 1 ) j(θ 2 u 2 ) 0 u 1 u 2 2 a θ 1 θ 2 u 1 u2 which gives the estimate using Proposition 1 (in fact C 5 2C 4 L a ) Remark 6 It seems not to be possible to establish a Lipschitz continuity with respect to the friction threshold θ Such a result would automatically imply the uniqueness of the solution to the Coulomb problem for a sufficiently small friction coefficient 4 A Uniqueness criterion P Hild in [7 8] exhibits some multi-solutions for the Coulomb problem on triangular domains hese solutions have been obtained for a large friction coefficient (F > 1) and for a tangential displacement having a constant sign For the moment it seems that no multi-solution has been exhibited for an arbitrary small friction coefficient in the continuous case although such a result exists for finite element approximation in [6] but for a variable geometry As far as we know no uniqueness result has been proved even for a sufficiently small friction coefficient he result presented here is a partial uniqueness result which determines some cases where it is possible to say that a particular solution of the Coulomb problem is in fact the unique solution A contrario this result can be used to search multi-solutions for an arbitrary small friction coefficient by the fact that it eliminates a lot of situations he partial uniqueness results we present in this section are deduced from the estimate given by the following lemma Lemma 5 Assuming hypotheses (10) (11) (13) are satisfied and g 0 if u 1 and u 2 are two solutions to Problem (14) and λ 1 and λ 2 are the corresponding contact stresses on Γ C then one has the following estimate u 1 u 2 2 a = λ 1 λ 2 2 aγ ζ λ 2 C u1 u2 ζ 2 j(f λ 1 u2) 13

14 Proof One has u 1 u 2 2 a = λ1 λ 2 2 aγ = λ 1 C λ2 u1 u2 + λ 1 λ2 u1 u2 Because K is a monotone set-valued map one has λ 1 λ2 u1 u2 0 hus u 1 u 2 2 a (λ1 ζ)+(ζ λ2 )u1 u2 ζ 2 j(f λ 1 u2 ) But 2 j(f λ u ) is also a monotone set-valued map with respect to its second variable which implies the result (and also the fact that u 1 u 2 2 a inf ζ λ 2 ζ 2 j(f λ 1 u2 ) aγ C ) An immediate consequence of this lemma is the following result for a vanishing tangential displacement Proposition 4 Assuming hypotheses (10) (11) (13) are satisfied and g 0 if u is a solution to Problem (14) such that u = 0 ae on Γ C and if C 3 C 4 F a < 1 then u is the unique solution to Problem (14) Proof Let us assume that u is another solution to Problem (14) hen from Lemma 5 one has u u 2 a ζ λ u u ζ 2 j(f λ u ) but because u = 0 and due to the complementarity relations λ u = F λ u and ζu = F λ u it implies using lemma 3 u u 2 a F (λ λ ) u C 3 C 4 F a λ λ aγc u u 2 aγ C C 3 C 4 F a u u 2 a which allows to conclude In the case d = 2 it is possible to give a result to a solution having a tangential displacement with a constant sign on Γ C We will say that a tangential displacement u is strictly positive if µ u > 0 for all µ such that µ 0 (ie µ v 0 for all v v 0 ae on Γ C ) and µ 0 Proposition 5 Assuming hypotheses (10) (11) (13) are satisfied g 0 and d = 2 if u is a solution to Problem (14) such that u > 0 and C 3 F a < 1 then u is the unique solution to Problem (14) (WhenF is constant over Γ C the condition reduces to C 3 F < 1) Proof Let us assume that u is another solution to Problem (14) with λ and λ the corresponding contact stresses on Γ C hen from Lemma 5 one has u u 2 a ζ λ u u ζ 2 j(f λ u ) Because u > 0 one has λ =F λ and 2 j(f λ u ) containsf λ hus taking ζ =F λ one obtains u u 2 a F (λ λ )u u λ λ aγc F (u u) a F a u u 2 a 14

15 which implies u = u when F a < 1 Of course the same reasoning is valid for u < 0 Let us now define the space of multipliers M( ) of the functions ξ : Γ C R d such that ξn = 0 ae on Γ C and such that the two following equivalent norms are finite: ξ M( ) v v 0 ξv v and ξ a v v 0 ξv aγc v aγc Because Γ C is assumed to have thec 1 regularity M( ) is isomorphic to (M ) d 1 It is possible to give a more general result assuming that λ =F λ ξ with ξ M( ) It is easy to see that this implies ξ 1 ae on the support of λ and more precisely ξ Dir (u ) ae on the support of λ where Dir () is the sub-derivative of the convex map R d x x his means that it is reasonable to assume that ξ Dir (u ) ae on Γ C Proposition 6 Assuming hypotheses (10) (11) (13) are satisfied and g 0 if u is a solution to Problem (14) such that λ =F λ ξ with ξ M( ) ξ Dir (u ) ae on Γ C and C 3 F a ξ a < 1 then u is the unique solution to Problem (14) Proof Let us assume that u is another solution to Problem (14) with λ and λ the corresponding contact stresses on Γ C hen from Lemma 5 one has u u 2 a ζ λ u u ζ 2 j(f λ u ) hen a possible choice is ζ =F λ ξ which gives together with the fact that F ξ a F a ξ a u u 2 a F ξ(λ λ )u u C 3 F a ξ a λ λ aγc u u a C 3 F a ξ a u u 2 a which implies u = u when C 3 F a ξ a < 1 Remark 7 Using equivalence of norms a more restrictive condition than C 3 F a ξ a < 1 is the condition α F M ξ < M( ) C 1 C 3 γ M As illustrated on Fig 4 for d = 2 the multiplier ξ as to vary from 1 to +1 each time the sign of the tangential displacement changes from negative to positive he set M( ) does not contain any multiplier having a discontinuity of the first kind his implies that in order to satisfy the assumptions of Proposition 6 the tangential displacement of the solution u cannot pass from a negative value to a positive value being zero only on a single point of Γ C 15

16 +1 ξ 0-1 Γ C u 0 Γ C Figure 4: Example of tangential displacement u and a possible corresponding multiplier ξ for d = 2 Perspectives As far as we know the result given by propositions 4 5 and 6 are the first results dealing with the uniqueness of the solution to the Coulomb problem without considering a regularization of the contact or the friction law In the future it may be interesting to investigate the following open problems: Is it possible to prove that for a sufficiently regular domain and a sufficiently regular loading a solution of the Coulomb problem is necessarily such that λ =F λ ξ with ξ M( )? his could be a way to prove a uniqueness result for a sufficiently small friction coefficient and regular loadings he more the tangential displacement u is oscillating around 0 (ie the more u changes it sign for d = 2) the more the multiplier ξ varies and thus the more ξ M( is great Does it mean that a ) multi-solutions for an arbitrary small friction coefficient and a fixed geometry has to be searched with very oscillating tangential displacement (necessarily for all the solutions)? Finally it should be interesting to study the convergence of finite element methods in the uniqueness framework given by Proposition 6 References [1] RA ADAMS Sobolev spaces Academic Press 1975 [2] L-E ADERSSO Existence Results for Quasistatic Contact Problems with Coulomb Friction Applied Mathematics and Optimisation 42 (2): pp [3] G DUVAU Problèmes unilatéraux en mécanique des milieux continus in: Actes du congrès international des mathématiciens (ice 1970) ome 3 Gauthier-Villars Paris pages pp [4] G DUVAU JL LIOS Les inéquations en mécanique et en physique Dunod Paris 1972 [5] C ECK J JARUŠEK Existence results for the static contact problem with Coulomb friction Mathematical Models and Methods in Applied Sciences LV 8 : pp

17 [6] R HASSAI P HILD I IOESCU -D SAKKI A mixed finite element method and solution multiplicity for Coulomb frictional contact Comput Methods Appl Mech Engrg pages (2003) [7] P HILD An example of nonuniqueness for the continuous static unilateral contact model with Coulomb friction C R Acad Sci Paris pages (2003) [8] P HILD on-unique slipping in the Coulomb friction model in two-dimensional linear elasticity Q Jl Mech Appl Math pages (2004) [9] P HILD Y REARD Local uniqueness and continuation of solutions for the discrete coulomb friction problem in elastostatics to appear in Quarterly of Applied Mathematics 2005 [10] J JARUŠEK Contact problems with bounded friction Coercive case Czechoslovak Math J 33 : pp [11] KIKUCHI J ODE Contact problems in elasticity SIAM 1988 [12] P LABORDE Y REARD Fixed points strategies for elastostatic frictional contact problems submitted 2003 [13] VG MAZ YA O SHAPOSHIKOVA heory of multipliers in spaces of differentiable functions Pitman 1985 [14] J EČAS J JARUŠEK J HASLIGER On the solution of variational inequality to the Signorini problem with small friction Bolletino UMI 5 17-B : pp