We are interested in the finite element approximation of Coulomb’s frictional unilateral contact problem in linear elasticity.

(1)

ON FINITE ELEMENT UNIQUENESS STUDIES FOR COULOMB’S FRICTIONAL CONTACT MODEL

P

ATRICK

HILD

^∗

∗

Laboratoire de Mathématiques, Université de Savoie, CNRS UMR 5127, 73376 Le Bourget-du-Lac, France, e-mail:

hild@univ-savoie.fr

We are interested in the finite element approximation of Coulomb’s 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 less than Cε

²

| log(h)|

⁻¹

, where h and ε denote the discretization and 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 may occur and that, for a given loading, the number of solutions may eventually decrease when the friction coefficient increases.

Keywords: Coulomb’s friction law, finite elements, mesh-size dependent uniqueness conditions, non-uniqueness example

1. Introduction and Problem Set-Up

Coulomb’s 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 (Du- vaut and Lions, 1972) leads to existence results when the friction coefficient is sufficiently small (Eck and Jarušek, 1998; Jarušek, 1983; Kato, 1987; Neˇcas et al., 1980). As regards the associated finite element model, it was proved in (Haslinger, 1983; 1984) that it always admits 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 (Haslinger, 1983), 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 were ex- hibited, proving that the problem is in general not well posed (Alart, 1993; Janovský, 1981; Klarbring, 1990).

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 2 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 less than Cε

²

| log(h)|

⁻¹

, and we notice that a bound of

only Ch

¹²

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 as a parameter 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 par- ticular case of a finite dimensional problem in the non- regularized case: a simple example using finite elements.

We study this problem and show that it may admit one, multiple or an infinity of solutions. Such an example completes and illustrates the already known results using truss elements, especially (Klarbring, 1990).

Let us now consider an elastic body occupying in the initial configuration a bounded subset Ω of R

²

. The boundary ∂Ω of the domain Ω is supposed to be Lips- chitz and consists of three non-overlapping parts Γ

D

, Γ

N

and Γ

C

. The unit outward normal on ∂Ω is denoted by n = (n

1

, n

2

) and we set t = (n

2

, −n

1

). The body is submitted to volume forces f = (f

1

, f

2

) ∈ (L

²

(Ω))

²

on Ω and to surface forces F = (F

1

, F

2

) ∈ (L

²

(Γ

N

))

²

on Γ

N

. 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 Γ

C

.

The unilateral contact problem with Coulomb’s fric-

tion consists in finding the displacement field u =

(u

i

), 1 ≤ i ≤ 2 and the stress tensor field σ =

(σ

ij

), 1 ≤ i, j ≤ 2, satisfying the following condi-

(2)

tions (1)–(4):

div σ(u) + f = 0 in Ω, σ(u)n = F on Γ

N

, u = 0 on Γ

D

,

(1)

where (div σ(u))

i

= σ

ij,j

, 1 ≤ i ≤ 2, the notation

,j

denotes the j-th partial derivative and the summation con- vention 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

(u)δ

ij

+ 2µε

ij

(u), (2) where λ and µ are positive Lamé coefficients and ε

_ij

(u) = (1/2)(u

_i,j

+ u

_j,i

) denotes the linearized strain tensor field.

On the boundary ∂Ω, we write σ(u)n = σ

n

(u)n + σ

t

(u)t and u = u

n

n + u

t

t. Let F > 0 stand for the friction coefficient on Γ

C

. The conditions on the contact zone Γ

C

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) Conditions (3) express unilateral contact and conditions (4) represent Coulomb’s friction. The closed convex cone K of admissible displacements is a subset in the Sobolev space (H

¹

(Ω))

²

of the displacement fields satisfying the embedding and the non-penetration conditions

K = n

v = (v

1

, v

2

) ∈ V , v

n

≤ 0 on Γ

C

o , (5) where

V = n

v = (v

1

, v

2

) ∈ H

¹

(Ω)

2

, v = 0 on Γ

D

o . As is done in (Neˇcas et al., 1980), we consider the mapping Φ : M → M with

M = n

α ∈ H

⁻¹²

(Γ

C

), α ≥ 0 o ,

defined for all g ∈ M as Φ(g) = −σ

n

(u(g)), where u(g) ∈ K is the unique solution of the variational inequality

u(g) ∈ K, Z

Ω

σ

ij

u(g) ε

ij

v − u(g) dΩ + hF g, |v

t

| − |u

t

(g)|i

Γ_C

≥ Z

Ω

f

i

v

i

− u

i

(g) dΩ

+ Z

Γ_N

F

_i

v

_i

−u

_i

(g) dΓ, ∀v ∈ K, (6)

where h·, ·i

Γ_C

denotes the duality pairing between the fractional Sobolev space H

¹²

(Γ

_C

) (Adams, 1975) and its dual space H

⁻¹²

(Γ

C

). Following (Neˇcas et al., 1980;

Haslinger et al., 1996), a weak solution of the unilateral contact problem with Coulomb’s friction is a pair (u, γ), where γ is a fixed point of Φ and u is the unique solution of the problem (6) with g = γ.

The first existence result for the unilateral contact problem with Coulomb’s friction in the case of a sufficiently small friction coefficient F was proved in (Neˇcas et al., 1980). Generalizations and/or improvements were established in (Eck and Jarušek, 1998; Jarušek, 1983;

Kato, 1987). The uniqueness seems to remain an open problem.

2. 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

= n

v

h

; v

h

∈ C (Ω)

²

, v

h

|

T

∈ (P

1

(T ))

²

∀T ∈ T

h

, v

h

= 0 on Γ

D

o ,

where C (Ω) and P

1

(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 fam- ilies of monodimensional traces of triangulations on Γ

C

are quasi-uniform in order to use inverse inequalities (Cia- rlet, 1991). Let W

h

be the range of V

h

by the normal trace operator on Γ

C

:

W

h

= n

µ

h

; µ

h

= v

h

|

Γ_C

· n, v

h

∈ V

h

o . Clearly, the space W

h

involves functions which are continuous and piecewise of degree one. We define M

h

as the closed convex cone of Lagrange multipliers express- ing non-negativity:

M

h

= n

µ

h

∈ W

h

, µ

h

≥ 0 o . For any u and v in (H

¹

(Ω))

²

, define

a(u, v) = Z

Ω

σ(u) : ε(v) dΩ,

L(v) = Z

Ω

f · v dΩ + Z

ΓN

F · v dΓ.

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

.

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To approximate Coulomb’s frictional contact problem, we choose a mixed finite element method with a non- negative 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 in finding u

_h

∈ V

h

and λ

h

∈ M

h

such that



 



 



a(u

h

, v

h

− u

h

) + Z

Γ_C

λ

h

(v

hn

− u

hn

) dΓ +

Z

ΓC

F g

_h

q

v

_ht²

+ ε

²

− q

u

²_ht

+ ε

²

dΓ

≥ L(v

h

− u

h

), ∀v

h

∈ V

h

, Z

ΓC

(µ

_h

− λ

_h

)u

_hn

dΓ ≤ 0, ∀µ

_h

∈ M

_h

.

(7)

In what follows, the problem (7) will be denoted by P

ε

(g

h

).

Remark 1. It can be checked that if (u

h

, λ

h

) solves (7), then u

h

is also a solution of the variational inequality which consists in finding u

h

∈ K

h

satisfying

a(u

h

, v

h

− u

h

) +

Z

ΓC

F g

h

q

v

_ht²

+ε

²

− q

u

²_ht

+ε

²

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

= n

v

_h

∈ V

_h

, Z

Γ_C

µ

_h

v

_hn

dΓ ≤ 0, ∀µ

_h

∈ M

_h

o .

Problem P

ε

(g

_h

) is also equivalent to finding a saddle-point (u

h

, λ

_h

) ∈ V

_h

× M

h

satisfying

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

) = 1

2 a(v

h

, v

h

) + Z

Γ_C

µ

h

v

hn

dΓ

+ Z

Γ_C

F g

_h

q

v

_ht²

+ ε

²

dΓ − L(v

_h

).

From the results concerning saddle-point problems obtained in (Haslinger et al., 1996), the existence of such a saddle-point follows. Moreover, the V -ellipticity of a(·, ·) implies that the first argument u

h

is unique. Be- sides, if for any µ

h

∈ W

h

one has

Z

Γ_C

µ

_h

v

_hn

dΓ = 0, ∀v

_h

∈ V

_h

=⇒ µ

_h

= 0, (8)

then the second argument λ

h

is unique and P

ε

(g

h

) admits a unique solution. Note that condition (8) is fulfilled because the space W

h

coincides with the space obtained from V

h

by the normal trace operator on Γ

C

.

It becomes then possible to define two maps: the first one denoted by Ψ

εh

yielding the first component (i.e.

Ψ

_εh

(g

_h

) = u

_h

), and the other denoted by Φ

εh

such that

Φ

_εh

: M

_h

−→ M

_h

, g

h

7−→ λ

h

,

where (u

h

, λ

_h

) is the solution to P

ε

(g

_h

). The introduction of this map allows us to define a solution to Coulomb’s discrete frictional contact problem.

Definition 1. A solution to Coulomb’s discrete regularized (resp. non-regularized) frictional contact problem is a solution to P

ε

(λ

h

) with ε > 0 (resp. ε = 0), where λ

h

∈ M

h

is a fixed point of Φ

εh

.

Set

V ˜

_h

= n

v

_h

∈ V

h

, v

_ht

= 0 on Γ

C

o . It is easy to check that the definition of k · k

₋1

2,h

given by

kνk

₋1

2,h

= sup v

h∈

V

˜h

Z

ΓC

νv

hn

dΓ kv

h

k

1

(9)

is a norm on W

_h

(since the condition (8) holds). The notation k · k

₁

represents the (H

¹

(Ω))

²

-norm.

3. Existence and Uniqueness Studies

We are now interested in the existence and uniqueness study for the discrete problem. In order to establish the existence, it suffices to show that the mapping Φ

εh

admits a fixed point in M

h

by using Brouwer’s theorem. The uniqueness is ensured if the mapping is contractive. Such a technique was already used in the non-regularized case with discontinuous and piecewise constant Lagrange multipliers (Haslinger, 1983; 1984). Our aim is to study the regularized case (and also the non-regularized one) when using Lagrange multipliers which are piecewise continuous of degree one.

Theorem 1. Let ε > 0. The following results hold:

(Existence) For any positive F , there exists a solution to Coulomb’s discrete regularized frictional contact problem.

(Uniqueness) Assume that Γ

D

∩ Γ

C

= ∅. If F ≤

Cε

²

| log(h)|

⁻¹

, then the problem admits a unique solu-

tion. The positive constant C depends on neither h

nor ε.

(4)

Proof. Let (u

h

, λ

h

) be the solution to P

ε

(g

h

). Taking v

h

= 0 in (7) gives

a(u

h

, u

h

) + Z

Γ_C

λ

h

u

hn

dΓ

− Z

Γ_C

F g

_h

ε −

q

u

²_ht

+ ε

²

dΓ ≤ L(u

_h

). (10)

Since g

h

≥ 0, ε − pu

²_ht

+ ε

²

≤ 0, and according to Z

Γ_C

λ

h

u

hn

dΓ = 0,

it follows from (10), the V -ellipticity of a(·, ·) and the continuity of L(·) that

αku

h

k

²₁

≤ a(u

h

, u

h

) ≤ L(u

h

) ≤ Cku

h

k

1

, 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

ku

ht

k

H¹2(ΓC)

≤ C

⁰

ku

h

k

1

≤ CC

⁰

α . (11) Besides, the equality in (7) implies

a(u

_h

, v

_h

) + Z

Γ_C

λ

_h

v

_hn

dΓ = L(v

_h

), ∀v

_h

∈ ˜ V

_h

. Denoting by M

⁰

the continuity constant of a(·, ·) yields Z

ΓC

λ

h

v

hn

dΓ ≤ M

⁰

ku

h

k

1

kv

h

k

1

+Ckv

h

k

1

, ∀v

h

∈ ˜ V

h

. As a result,

kλ

h

k

₋1

2,h

≤ M

⁰

ku

h

k

1

+ C ≤ M

⁰

α + 1

C. So, we conclude that kΦ

_εh

(g

_h

)k

₋1

2,h

≤ C

⁰

, ∀g

_h

∈ M

_h

, (12) where C

⁰

only depends on the applied loads f , F , and on the continuity and ellipticity constants of a(·, ·).

The existence result of Theorem 1 consists now in showing that the mapping Φ

εh

is continuous.

Let (u

_h

, λ

_h

) and (u

h

, λ

_h

) be the solutions to P

ε

(g

h

) and P

ε

(g

_h

), respectively (where g

h

∈ M

h

and g

_h

∈ M

h

). From (7), we get

a(u

_h

, v

_h

) + Z

ΓC

λ

_h

v

_hn

dΓ = L(v

_h

), ∀v

_h

∈ ˜ V

_h

,

and

a(u

_h

, v

_h

) + Z

Γ_C

λ

_h

v

_hn

dΓ = L(v

_h

), ∀v

_h

∈ ˜ V

_h

,

which implies by subtraction that Z

Γ_C

(λ

h

− λ

h

)v

hn

dΓ = a(u

h

− u

h

, v

h

)

≤ M

⁰

ku

h

− u

h

k

1

kv

h

k

1

, ∀v

h

∈ ˜ V

h

, where the continuity of the bilinear form a(·, ·) was used.

So we get the following estimate:

kλ

h

− λ

h

k

₋1

2,h

≤ M

⁰

ku

h

− u

h

k

1

. (13) Next, 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 to P

ε

(g

_h

) and P

ε

(g

_h

), respectively. We have

a(u

h

, v

h

− u

h

) + Z

Γ_C

λ

h

(v

hn

− u

hn

) dΓ

+ Z

Γ_C

F g

h

q v

_ht²

+ ε

²

− q

u

²_ht

+ ε

²

dΓ

≥ L(v

h

− u

h

), ∀v

h

∈ V

h

, and

a(u

_h

, v

_h

− u

_h

) + Z

Γ_C

λ

_h

(v

_hn

− u

_hn

) dΓ

+ Z

ΓC

F g

_h

q

v

_ht²

+ ε

²

− q

u

²_ht

+ ε

²

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, from (7) we obtain

a(u

_h

, u

_h

−u

_h

)+

Z

Γ_C

F g

_h

q

u

²_ht

+ε

²

− q

u

²_ht

+ε

²

dΓ

≥ L(u

h

− u

h

) and

a(u

_h

, u

_h

−u

_h

)+

Z

ΓC

F g

_h

q

u

²_ht

+ε

²

− q

u

²_ht

+ε

²

dΓ

≥ L(u

h

− u

h

).

Thus

a(u

h

− u

h

, u

h

− u

h

)

≤ Z

Γ_C

F (g

_h

−g

_h

) q

u

²_ht

+ε

²

− q

u

²_ht

+ε

²

dΓ. (14) Consequently,

αku

_h

− u

h

k

²₁

≤ F kg

h

− g

_h

k

H^{− 1}2(ΓC)

×

q

u

²_ht

+ ε

²

− q

u

²_ht

+ ε

²

_H¹₂_(Γ

C)

. (15)

(5)

The next step consists in estimating the H

¹²

-norm term in (15). To attain our ends, we need to use two lemmas:

Lemma 1. There exists a positive constant C satisfying kf gk

H¹2(Γ_C)

≤ C

kf k

H¹2(Γ_C)

kgk

L^∞(Γ_C)

+ kf k

_L∞(Γ_C)

kgk

H¹2(Γ_C)

. (16)

for all f and g in H

¹²

(Γ

C

) ∩ L

^∞

(Γ

C

).

Proof. From the definition of the H

¹²

(Γ

C

)-norm (Adams, 1975), we have

kf gk

²

H¹2(Γ_C)

= kf gk

²_L2(ΓC)

+ Z

Γ_C

Z

Γ_C

f (x)g(x) − f (y)g(y)

2

(x − y)

²

dΓ dΓ.

Let us begin with bounding (roughly) the first term:

kf gk

²_L2(Γ_C)

= Z

ΓC

f

²

(x)g

²

(x) dΓ

≤ kf k

²_L2(ΓC)

kgk

²_L∞(ΓC)

. (17) The second term is handled as follows:

Z

ΓC

Z

ΓC

(f (x)(g(x)−g(y))+g(y)(f (x)−f (y)))

²

(x − y)

²

dΓ dΓ

≤ 2 Z

Γ_C

Z

Γ_C

f

²

(x)(g(x) − g(y))

²

(x − y)

²

+ g

²

(y) f (x) − f (y)

2

(x − y)

²

dΓ dΓ (18)

≤ 2

kf k

²_L∞(Γ_C)

kgk

²

H¹2(Γ_C)

+kf k

²

H¹2(Γ_C)

kgk

²_L∞(Γ_C)

.

Putting together (17) and (18) establishes (16).

Lemma 2. For any real number p ∈ [1, ∞[, the following inequality holds:

kf k

_Lp(Γ_C)

≤ C √ pkf k

H¹2(Γ_C)

, ∀f ∈ H

¹²

(Γ

_C

), (19) where C is independent of p.

Proof. see (Ben Belgacem, 2000).

Proof of Theorem 1 (continued). We consider the H

¹²

- norm term in (15). Employing the estimate (16) gives

q

u

²_ht

+ ε

²

− q

u

²_ht

+ ε

²

_H¹₂_(Γ

C)

=

(u

ht

− u

ht

) u

ht

+ u

ht

p u

²_ht

+ ε

²

+ pu

²_ht

+ ε

²

_H¹₂_(Γ

C)

≤ Cku

ht

− u

ht

k

L^∞(Γ_C)

×

u

_ht

+ u

_ht

p u

²_ht

+ ε

²

+ pu

²_ht

+ ε

²

_H¹₂_(Γ

C)

+ Cku

_ht

− u

_ht

k

H¹2(Γ_C)

×

u

ht

+ u

ht

p u

²_ht

+ ε

²

+ pu

²_ht

+ ε

²

L^∞(ΓC)

. (20)

In the previous estimate, we leave the third term un- changed whereas the last one is bounded by 1. It remains then to bound the first two terms, which is performed here- after. We begin with the first one:

ku

ht

− u

ht

k

L^∞(Γ_C)

≤ Ch

⁻¹^p

ku

ht

− u

ht

k

L^p(Γ_C)

≤ C √

ph

⁻¹^p

ku

ht

−u

ht

k

H¹2(Γ_C)

, (21) for any p ∈ [1, ∞[. In (21), we used an easily recoverable inverse inequality (Ciarlet, 1991), as well as (19). The second term of (20) is bounded due to (16):

u

ht

+ u

ht

p u

²_ht

+ ε

²

+ pu

²_ht

+ ε

²

H¹²(ΓC)

≤ Cku

ht

+ u

ht

k

_L^∞_(Γ_C₎

×

1 p u

²_ht

+ ε

²

+ pu

²_ht

+ ε

²

_H¹₂_(Γ

C)

+ Cku

_ht

+ u

_ht

k

H¹2(Γ_C)

×

1 p u

²_ht

+ ε

²

+ pu

²_ht

+ ε

²

_L∞(ΓC)

≤ C √

ph

⁻¹^p

ku

ht

+ u

ht

k

H¹2(ΓC)

×

1 p u

²_ht

+ ε

²

+ pu

²_ht

+ ε

²

_H¹₂_(Γ

C)

+ 1

2ε ku

ht

+ u

ht

k

H¹2(Γ_C)

, (22)

(6)

where the first L

^∞

-norm term is bounded as in (21), whereas the other is roughly bounded by 1/2ε. Next, we develop the first H

¹²

-norm term in (22):

1 p u

²_ht

+ ε

²

+ pu

²_ht

+ ε

²

2

H¹2(ΓC)

=

1 p u

²_ht

+ ε

²

+ pu

²_ht

+ ε

²

2

L²(Γ_C)

+ Z

ΓC

Z

ΓC

1 (y − x)

²

× 1

p u

²_ht

(x) + ε

²

+ pu

²_ht

(x) + ε

²

− 1

p u

²_ht

(y) + ε

²

+ pu

²_ht

(y) + ε

²

!

²

dΓ dΓ.

It is easy to check that the L

²

-norm term is less than meas(Γ

_C

)/4ε

²

. Developing the previous integral, bounding then the denominator and using the estimate (a + b)

²

≤ 2a

²

+ 2b

²

furnishes the following upper bound:

1 8ε

⁴

Z

Γ_C

Z

Γ_C

p u

²_ht

(x) + ε

²

− p

u

²_ht

(y) + ε

²

(y − x)

²

+

pu

²_ht

(x) + ε

²

− pu

²_ht

(y) + ε

²

(y − x)

²

dΓ dΓ.

We use the estimate | √

a

²

+ ε

²

− √

b

²

+ ε

²

| ≤ |a − b| in the previous expression so that

1 p u

²_ht

+ ε

²

+ pu

²_ht

+ ε

²

2

H¹2(Γ_C)

≤ meas(Γ

C

) 4ε

²

+ 1

8ε

⁴

ku

ht

k

²

H¹2(Γ_C)

+ ku

ht

k

²

H¹2(Γ_C)

.

Therefore we deduce from (11) that there exists a positive constant C satisfying

1 p u

²_ht

+ ε

²

+ pu

²_ht

+ ε

²

_H¹₂_(Γ

C)

≤ C 1 ε + 1

ε

²

. (23)

Applying (23) to (22) and using (11) and (20), we get

q

u

²_ht

+ ε

²

− q

u

²_ht

+ ε

²

_H¹₂_(Γ

C)

≤ Cku

_ht

− u

_ht

k

H¹2(Γ_C)

× 1 + √

ph

⁻^p¹

1 ε + √

ph

⁻¹^p

1 ε + 1

ε

²

! .

Choosing p = − log(h) (h is assumed to be sufficiently small) in the previous estimate, we obtain

q

u

²_ht

+ ε

²

− q

u

²_ht

+ ε

²

_H¹₂_(Γ

C)

≤ Cku

ht

− u

ht

k

H¹2(ΓC)

× 1 +

√ − log h

ε + − log h

ε + − log h ε

²

! . (24)

Inequality (15) together with (24) and the trace theorem becomes

ku

h

− u

h

k

1

≤ CF kg

h

− g

_h

k

H^{− 1}2(ΓC)

× 1 +

√ − log h

ε + − log h

ε + − log h ε

²

! , (25)

which proves that the mapping Ψ

_εh

is continuous. This, together with (13), implies that Φ

_εh

is continuous. Then, from (12) and the Brouwer fixed point theorem, we conclude the existence of at least one solution to Coulomb’s discrete regularized frictional contact problem.

We now consider the uniqueness. Under the assump- tion that Γ

D

∩ Γ

C

= ∅, it was proved in (Coorevits et al., 2002) that there exists a positive constant β (independent of h) satisfying

βkµ

h

k

H^{− 1}²(ΓC)

≤ kµ

h

k

₋1

2,h

, ∀µ

h

∈ W

h

. (26) Assembling this result with (25) and (13) yields

kλ

_h

− λ

_h

k

H^{− 1}2(Γ_C)

≤ CF kg

_h

− g

_h

k

H^{− 1}2(Γ_C)

× 1 +

√ − log h

ε + − log h

ε + − log h ε

²

! .

Supposing that h and ε are small enough, we deduce that

the mapping Φ

εh

is contractive if the friction coefficient

(7)

F is less than Cε

²

| log(h)|

⁻¹

. This completes the proof of the theorem.

The non-regularized case (i.e. ε = 0) is handled in the proposition that follows.

Proposition 1. Let ε = 0. The following results hold:

(Existence) For any positive F , there exists a solution to Coulomb’s discrete frictional contact problem.

(Uniqueness) Assume that Γ

D

∩ Γ

C

= ∅. If F ≤ Ch

¹²

, then the problem admits a unique solution. The positive constant C is independent of h.

Proof. Estimates (12) and (13) remain still valid when ε = 0. The starting point of the analysis is (14):

a(u

h

− u

h

, u

h

− u

h

)

≤ Z

Γ_C

F (g

_h

− g

_h

)(|u

_ht

| − |u

_ht

|) dΓ

≤ F kg

_h

− g

h

k

_L2(ΓC)

k |u

ht

| − |u

ht

| k

_L2(ΓC)

≤ CF h

⁻¹²

kg

_h

− g

h

k

H^{− 1}2(Γ_C)

ku

ht

− u

ht

k

L²(Γ_C)

≤ C

⁰

F h

⁻¹²

kg

_h

− g

h

k

H^{− 1}2(ΓC)

ku

h

− u

h

k

1

, where an inverse inequality between L

²

(Γ

_C

) and H

⁻¹²

(Γ

C

) was used. From the last bound, combined with (13) and (26), we deduce that

kλ

h

− λ

h

k

H^{− 1}2(ΓC)

≤ CF h

⁻¹²

kg

h

− g

_h

k

H^{− 1}2(ΓC)

. This proves the proposition.

Remark 2. 1. In the proof of Proposition 1, we are not able to remove the mesh dependent uniqueness condition, also when avoiding the L

²

(Γ

_C

)-norms and using only H

¹²

(Γ

C

)-norms and H

⁻¹²

(Γ

C

)-norms. More precisely, there does not exist a positive constant C independent of h such that

k |g

_h

− g

h

| k

H^{− 1}2(ΓC)

≤ Ckg

_h

− g

h

k

H^{− 1}2(ΓC)

or

k |u

ht

| − |u

ht

| k

H¹2(Γ_C)

≤ Cku

ht

− u

ht

k

H¹2(Γ_C)

. 2. The use of inverse inequalities in the proofs of Theo- rem 1 and Proposition 1 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 vertices A = (0, 0), B = (`, 0) and C = (0, `) with ` > 0. We define Γ

D

= [B, C], Γ

_N

= [A, C], Γ

C

= [A, B], and {X

1

, X

₂

} denotes the canonical orthonormal basis (see Fig. 1). We suppose that the volume forces f are absent and that the surface forces denoted by F = F

1

X

1

+ F

2

X

2

are such that F

1

and F

2

are constant on Γ

N

.

N

C D

A B

C

n t

F

X

1 2

Γ

Γ Γ Ω

Fig. 1. Problem setting.

We suppose that Ω is discretized with a single finite element of degree one. Consequently, the finite element space becomes

V

h

= n

v

h

= (v

h1

, v

h2

) ∈ P

1

(Ω)

²

, v

h

|

Γ_D

= 0 o . In this case, we have

M

_h

= n

g

_h

∈ P

₁

(Γ

_C

), g

_h

≥ 0, g

_h

(B) = 0 o . Clearly, V

_h

is of dimension two and M

_h

belongs to the space W

h

of linear functions on Γ

C

vanishing at B, which is of dimension one. Moreover, since (8), or equivalently (9), is satisfied, it follows that the existence is ensured for all ε ≥ 0 according to Theorem 1 and Proposi- tion 1.

Let v

h

∈ V

h

and µ

h

∈ M

h

. Then we denote by (V

T

, V

N

) 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

_h1

(A) and V

_N

= −v

_h2

(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

, we obtain

ε(v

h

) = 1 2`

2V

_T

V

_T

+ V

_N

V

T

+ V

N

2V

N

!

and

σ(v

_h

) = 1

`





(λ+2µ)V

T

+λV

N

µ(V

T

+V

N

) µ(V

T

+V

N

) (λ+2µ)V

N

+λV

T



 .

(8)

Therefore

a(u

h

, v

h

) = 1 2

(λ + 3µ)(U

T

V

T

+ U

N

V

N

) + (λ + µ)(U

_T

V

_N

+ U

_N

V

_T

) and

L(v

h

) = − 1

2 `(F

1

V

T

+ F

2

V

N

).

Besides,

Z

Γ_C

µ

_h

v

_hn

dΓ = ΘV

_N

` 3 and

Z

Γ_C

F µ

h

|v

ht

| dΓ = F Θ|V

T

|`

3 .

Let (u

h

, λ

h

) be a solution to the discrete unilateral contact problem with Coulomb’s friction and without regularization (i.e. with ε = 0 in (7)). As was mentioned above, the notation (U

T

, U

_N

) stands for the value of u

h

(A) (U

_T

= −u

_h1

(A) and U

N

= −u

_h2

(A)). We also denote by Λ

⁰

the value of λ

_h

at point A. To simplify the notation and the forthcoming calculations, we set Λ = 2Λ

⁰

/3.

The discrete unilateral contact problem with Coulomb’s friction and without regularization issued from (7) and Definition 1 consists then in finding (U

T

, U

N

, Λ) ∈ R

³

such that



 



 



(λ+3µ)(U

T

V

T

+U

N

V

N

)+(λ + µ)(U

T

V

N

+U

N

V

T

) + Λ`V

N

+ F Λ`|V

T

|

≥ −`(F

₁

V

_T

+ F

₂

V

_N

), ∀V

_T

∈ R, ∀V

_N

∈ R, (λ + 3µ)(U

_T²

+ U

_N²

) + 2(λ + µ)(U

T

U

N

) + F Λ`|U

T

|

= −`(F

₁

U

_T

+ F

₂

U

_N

),

Λ ≥ 0, U

N

≤ 0, ΛU

N

= 0, or equivalently,



 



 



(λ + 3µ)U

_N

+ (λ + µ)U

_T

+ Λ` = −`F

₂

, (λ + µ)U

N

+ (λ + 3µ)U

T

+ F Λ` ≥ −`F

1

, (λ + µ)U

N

+ (λ + 3µ)U

T

− F Λ` ≤ −`F

1

,

(λ + 3µ)(U

_T²

+ U

_N²

) + 2(λ + µ)(U

T

U

N

) + F Λ`|U

T

|

= −`(F

1

U

T

+ F

2

U

N

), Λ ≥ 0, U

_N

≤ 0, ΛU

_N

= 0.

(27) Let us now look for solutions to (27). Clearly, a solution to (27) satisfies either U

N

= 0 or Λ = 0.

(i) Case 1: U

N

= 0. Equations (27) become



 

 

 

 

(λ + µ)U

_T

+ Λ` = −`F

₂

, (λ + 3µ)U

T

+ F Λ` ≥ −`F

1

, (λ + 3µ)U

T

− F Λ` ≤ −`F

1

, (λ + 3µ)U

_T²

+ F Λ`|U

_T

| = −`F

₁

U

_T

, Λ ≥ 0.

• Suppose that U

T

= 0. Then

Λ = −F

2

, F

2

≤ 0, |F

1

| ≤ F |F

2

|.

• Suppose that U

T

> 0. Then



 



 



(λ + µ)U

T

+ Λ` = −`F

2

, (λ + 3µ)U

_T

+ F Λ` = −`F

₁

, Λ ≥ 0.

– Assume that F 6= (λ + 3µ)/(λ + µ). Then

U

T

= `(F F

2

− F

1

)

(λ + 3µ) − F (λ + µ) > 0,

Λ = (λ + µ)F

₁

− (λ + 3µ)F

2

(λ + 3µ) − F (λ + µ) ≥ 0.

– Assume that F = (λ + 3µ)/(λ + µ). Then

∗ If F

1

= F F

2

, the solutions are

(λ + µ)U

T

+ Λ` = −`F

2

, U

T

> 0, Λ ≥ 0.

∗ If F

1

6= F F

2

, then there are no solutions.

• Suppose that U

T

< 0. Then



 



 



(λ + µ)U

_T

+ Λ` = −`F

₂

, (λ + 3µ)U

_T

− F Λ` = −`F

1

, Λ ≥ 0,

which gives

U

T

= `(F F

2

+ F

1

)

−(λ + 3µ) − F (λ + µ) < 0,

Λ = −(λ + µ)F

1

+ (λ + 3µ)F

2

−(λ + 3µ) − F (λ + µ) ≥ 0.

(9)

(ii) Case 2: Λ = 0.



 

 

 

 

(λ + 3µ)U

N

+ (λ + µ)U

T

= −`F

2

, (λ + µ)U

N

+ (λ + 3µ)U

T

= −`F

1

, U

N

≤ 0,

so that

U

T

= ` (λ + µ)F

2

− (λ + 3µ)F

1

4µ(λ + 2µ) ,

U

N

= ` (λ + µ)F

1

− (λ + 3µ)F

2

4µ(λ + 2µ) ≤ 0.

All the results are reported in the proposition that follows. There are three cases which consist in compar- ing the friction coefficient F with the critical value (λ + 3µ)/(λ + µ) = 3 − 4ν (ν denotes Poisson’s ratio with 0 < ν < 1/2). The results are also depicted in Figs. 2–4.

Proposition 2. 1. If F < (λ + 3µ|/(λ + µ), then the problem (27) admits a unique solution:

(Separation) If F

2

> ((λ + µ)/(λ + 3µ))F

1

, then

U

T

= ` (λ + µ)F

2

− (λ + 3µ)F

1

4µ(λ + 2µ) , U

_N

= ` (λ + µ)F

₁

− (λ + 3µ)F

₂

4µ(λ + 2µ) , Λ = 0.

(28)

(Stick) If |F

1

| ≤ F |F

2

| and F

2

≤ 0, then

U

T

= 0, U

N

= 0, Λ = −F

2

. (29) (Right slip) If F

2

≤ ((λ + µ)/(λ + 3µ))F

1

, F F

2

+F

1

>

0, then

U

T

= `(F F

2

+ F

1

)

−(λ + 3µ) − F (λ + µ) ,

U

N

= 0, Λ = −(λ + µ)F

1

+ (λ + 3µ)F

2

−(λ + 3µ) − F (λ + µ) .

(30)

(Left slip) If F

2

≤ ((λ + µ)/(λ + 3µ))F

1

, F F

2

− F

1

>

0, then

U

_T

= `(F F

2

− F

1

) (λ + 3µ) − F (λ + µ) ,

U

_N

= 0, Λ = (λ + µ)F

1

− (λ + 3µ)F

2

(λ + 3µ) − F (λ + µ) .

(31)

F F

Right slip

Stick Left slip

Separation

U = 0 U < 0

U =0 U =0 U = 0

U > 0

U < 0

_N

N

N N

T

T T

1 2

Fig. 2. Case F <

^λ+3µ_λ+µ

= 3 − 4ν. Problem (27) admits a unique solution.

F F

Separation U < 0

_N

Right slip U = 0

_N

U < 0

_T

Infinity of solutions from stick to left slip

Stick U =0

_T

U =0

_N

2

1

Fig. 3. Case F =

^λ+3µ_λ+µ

= 3 − 4ν. Problem (27) admits either a unique or an infinity of solutions.

F

Stick U =0

_T

U =0

_N

Right slip U = 0 U < 0

_T^N F

2 solutions:

stick and left slip 3 solutions

stick, left slip and separation

Separation U < 0

_N

2 solutions:

stick and separation

1 2

Fig. 4. Case F >

^λ+3µ_λ+µ

= 3 − 4ν. Problem (27)

admits a unique, two or three solutions.

(10)

2. If F = (λ + 3µ)/(λ + µ), then, depending on the loadings, the problem (27) admits either a unique solution or an infinity of solutions:

(Separation) If F

2

> ((λ + µ)/(λ + 3µ))F

1

, then the solution is given by (28).

(Stick) If (−F |F

2

| < F

1

≤ F |F

2

| and F

2

≤ 0) or F

1

= F

2

= 0, then the solution is given by (29).

(Right slip) If F

2

≤ ((λ + µ)/(λ + 3µ))F

1

, F F

2

+F

1

>

0, then the solution is given by (30).

(From stick to left slip) If F

1

= F F

₂

and F

2

< 0, then there exists an infinity of solutions:

U

T

= −`(F

2

+ β) λ + µ , U

_N

= 0, Λ = β,

for all 0 ≤ β ≤ −F

₂

.

3. If F > (λ + 3µ)/(λ + µ), then, depending on the loadings, the problem (27) admits one, two or three solutions:

(Separation) If F

2

> ((λ + µ)/(λ + 3µ))F

1

and F F

2

− F

1

> 0, then the solution is given by (28).

(Stick) If (−((λ + 3µ)/(λ + µ))|F

2

| < F

1

≤ F |F

2

| and F

2

≤ 0) or F

1

= F

2

= 0, then the solution is given by (29).

(Right slip) If F

2

≤ ((λ + µ)/(λ + 3µ))F

1

, F F

2

+F

1

>

0, then the solution is given by (30).

(Separation and stick) If F

1

= F F

2

and F

2

< 0, then there are two solutions given by (28) and (29).

(Stick and left slip) If F

1

= ((λ + 3µ)/(λ + µ))F

2

and F

₂

< 0, then there are two solutions given by (29) and (31).

(Separation, stick and left slip) If −F |F

2

| < F

1

<

−((λ + 3µ)/(λ + µ))|F

₂

| and F

₂

≤ 0 then there are three solutions given by (28), (29) and (31).

The study of sufficient conditions of non-uniqueness for Coulomb’s frictional contact problem in the continuous framework is actually under consideration in (Hassani et al., 2001).

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ⁿ

dans lui même: application à la mécanique du contact. — Math. Model. Numer. Anal., Vol. 27, No. 2, pp. 203–222.

Ben Belgacem F. (2000): Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element method. — SIAM J. Numer. Anal., Vol. 37, No. 4, pp. 1198–1216.

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Haslinger J. (1983): Approximation of the Signorini problem with friction, obeying the Coulomb law. — Math. Meth.

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