A class of implicit evolution inequalities and applications to dynamic contact problems

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to dynamic contact problems

Marius Cocou

To cite this version:

Marius Cocou. A class of implicit evolution inequalities and applications to dynamic contact problems.

Annals of the University of Bucharest. Mathematical series, București : Editura Universității din

București, 2013, 4 (LXII) (1), pp.181-192. �hal-00949617�

applications to dynamic contact problems

Marius Cocou

Abstract - This paper deals with the analysis of a class of implicit vari- ational inequalities which generalizes some dynamic contact problems cou- pling adhesion and friction between two viscoelastic bodies of Kelvin-Voigt type. Existence and uniqueness results are proved for a general system of evolution inequalities that constitutes a unified approach to study some complex dynamic surface interactions, including rebonding, debonding and friction conditions. The proofs are based on incremental formulations, sev- eral estimates, compactness arguments and a fixed point technique. These results are applied to dynamic frictional contact conditions with reversible adhesion and the coefficient of friction depending on the slip velocity.

Key words and phrases : Implicit inequalities, dynamic problems, variational solutions, adhesion, friction, viscoelasticity.

Mathematics Subject Classification (2010) : 35K85, 35R35, 49J40, 74A55, 74D05, 74H20

1 Introduction

The aim of this paper is to study a system of evolution inequalities that generalizes some interaction laws including dynamic contact, recoverable adhesion and friction between two viscoelastic bodies, when the coefficient of friction depends on the slip velocity, which represents a more realistic model than the one described in [11].

An interface model coupling unilateral contact, irreversible adhesion and local friction for elastic bodies was considered in the quasistatic case in [26]

and its mathematical analysis has been provided in [9].

Nonlocal friction laws, given by appropriate regularizations of the nor- mal component of the stress vector which occurs in the Coulomb friction conditions, and normal compliance models have been considered by several authors in the (quasi)static case, see, e.g. [17, 27] and references therein.

Dynamic frictional contact problems with normal compliance laws for a viscoelastic body have been studied in [23, 17, 18, 5] and dynamic unilateral or bilateral contact problems with friction for viscoelastic bodies have been considered in [16, 12, 19, 20, 10].

1

More recently, dynamic frictionless problems with adhesion were studied in [6, 21, 29] and a dynamic viscoelastic problem coupling unilateral contact, recoverable adhesion and nonlocal, mathematically consistent, friction was analyzed in [11].

Some interesting relaxed unilateral contact condition with pointwise fric- tion have been proposed in [25] in the static case and extended in [8] to the quasistatic case.

This paper is organized as follows. First, existence and uniqueness results are proved for a general system of evolution inequalities that constitutes a unified approach to study some complex dynamic surface interactions, including rebonding, debonding and friction conditions.

Second, the classical formulation of a dynamic contact problem with adhesion and friction is presented and a corresponding variational formu- lation is given as a system which contains an implicit evolution variational inequality coupled with a parabolic variational inequality.

Finally, based on the previous abstract results, the existence and unique- ness of variational solutions are analyzed.

2 Analysis of a system of evolution inequalities

Let (H

, | . | , (. , .)), (V

, k . k , h . , . i ), (U

, k . k

) and (Π

, | . |

, (. , .)

) be four Hilbert spaces such that V

⊂ U

⊆ H

, the imbedding from V

into U

is compact and V

dense in H

.

Let Λ

be a closed convex set in Π

such that 0 ∈ Λ

. Suppose also that Λ

is bounded, to simplify the estimates.

Define two bilinear and symmetric forms, a

, b

: V

× V

→ R and the mapping γ

: V

× Π

× Π

→ R such that

∃ m

, m

> 0 a

(u, v) ≤ m

k u k k v k , b

(u, v) ≤ m

k u k k v k , (2.1)

∃ A, B > 0 a

(v, v) ≥ A k v k

, b

(v, v) ≥ B k v k

∀ u, v ∈ V

, (2.2)

∀ u ∈ V

, γ

(u, · , · ) is a bilinear and symmetric form, (2.3)

∃ m

> 0 such that ∀ u

∈ V

, ∀ δ

∈ Λ

, ∀ η ∈ Π

,

| γ

(u

, δ

, η) − γ

(u

, δ

, η) | ≤ m

( k u

− u

k + | δ

− δ

|

) | η |

, (2.4)

γ

(u, η, η) ≥ 0 ∀ u ∈ V

, ∀ η ∈ Π

. (2.5)

Let φ

: [0, T ] × Λ

× V

→ R be a mapping such that

φ

(t, η, · , · , · ) is sequentially weakly continuous, (2.6) φ

(t, η, u, v, w

+ w

) ≤ φ

(t, η, u, v, w

) + φ

(t, η, u, v, w

), (2.7) φ

(t, η, u, v, θw) = θ φ

(t, η, u, v, w), (2.8)

φ

(0, 0, 0, 0, w) = 0, (2.9)

∀ t ∈ [0, T ], ∀ η ∈ Λ

, ∀ u, v, w, w

∈ V

, ∀ θ ≥ 0,

∃ m

> 0 such that ∀ t

∈ [0, T ], ∀ η

∈ Λ

, ∀ u

, v

, w

∈ V

,

| φ

(t

, η

, u

, v

, w

) − φ

(t

, η

, u

, v

, w

) + φ

(t

, η

, u

, v

, w

) − φ

(t

, η

, u

, v

, w

) |

≤ m

( | t

− t

| + | η

− η

|

+ k u

− u

k

+ k v

− v

k

) k w

− w

k . (2.10)

Assume that L

∈ W

(0, T ; V

), u

, u

∈ V

, β

∈ Λ

and that the following compatibility condition holds: ∃ l

∈ H

such that ∀ w ∈ V

(l

, w) + a

(u

, w) + b

(u

, w) + φ

(0, β

, u

, u

, w) = h L

(0), w i . (2.11) Consider the following problem.

Problem Q : Find u ∈ W

(0, T ; H

) ∩ W

(0, T ; V

), β ∈ W

(0, T ; Π

) such that u(0) = u

, ˙ u(0) = u

, β(0) = β

, β(τ ) ∈ Λ

for all τ ∈ ]0, T [, and a.e. t ∈ ]0, T [

(¨ u, v − u) + ˙ a

(u, v − u) + ˙ b

( ˙ u, v − u) ˙ (2.12) +φ

(t, β, u, u, v) ˙ − φ

(t, β, u, u, ˙ u) ˙ ≥ h L

, v − u ˙ i ∀ v ∈ V

,

( ˙ β, η − β)

+ γ

(u, β, η − β) ≥ 0 ∀ η ∈ Λ

. (2.13) Define the set

X

= { η ∈ C

([0, T ]; Π

) ; η(0) = β

, η(t) ∈ Λ

∀ t ∈ ]0, T ] } , where the Banach space C

([0, T ]; Π

) is endowed with the norm

k η k

= max

t∈[0,T]

[exp( − kt) | η(t) |

] for all η ∈ C

([0, T ]; Π

), k ≥ 0.

The existence and uniqueness of the solution of the problem Q will be proved

by using the following lemmas and a fixed point argument, see also [11] for

the particular case when the coefficient of friction is slip rate independent.

Lemma 2.1 For each β ∈ X

there exists a unique u

∈ W

(0, T ; H

) ∩ W

(0, T ; V

), solution of the inequality (2.12) such that u

(0) = u

,

˙

u

(0) = u

.

The proof is based on some incremental formulations, see, e.g. [10], and on a useful estimate, see [22] or [28], which, when applied to the spaces V

⊂ U

⊆ H

, implies the following result: for every > 0 there exists C

> 0 such that

k u k

≤ k u k + C

| u | ∀ u ∈ V

. (2.14) The full proof will be presented in a forthcoming paper.

Lemma 2.2 Let β

, β

∈ X

and let u

, u

be the corresponding solu- tions of (2.12) with the same initial conditions u

, u

, respectively. Then there exists a constant C

> 0, independent of β

, β

, u

, u

, such that for all t ∈ [0, T ]

| u ˙

(t) − u ˙

(t) |

+ k u

(t) − u

(t) k

≤ C

Z

0

| β

(s) − β

(s) |

ds. (2.15) Proof. Let u

, u

be the solutions of (2.12) corresponding to β

, β

∈ X

. Taking in each inequality v = ˙ u

and v = ˙ u

, respectively, for a.e.

s ∈ ]0, T [ it follows that

(¨ u

− u ¨

, u ˙

− u ˙

) + a

(u

− u

, u ˙

− u ˙

) + b

( ˙ u

− u ˙

, u ˙

− u ˙

)

≤ φ

(s, β

, u

, u ˙

, u ˙

) − φ

(s, β

, u

, u ˙

, u ˙

) +φ

(s, β

, u

, u ˙

, u ˙

) − φ

(s, β

, u

, u ˙

, u ˙

)

≤ m

( | β

− β

|

+ k u

− u

k

+ k u ˙

− u ˙

k

) k u ˙

− u ˙

k , where the second inequality follows by (2.10).

For all t ∈ [0, T ], as the solutions u

, u

verify the same initial con- ditions, integrating between 0 and t yields

1

2 | u ˙

(t) − u ˙

(t) |

+ 1

2 a

(u

(t) − u

(t), u

(t) − u

(t)) +

Z

0

b

( ˙ u

− u ˙

, u ˙

− u ˙

) ds ≤ m

Z

0

| β

− β

|

k u ˙

− u ˙

k ds +m

Z

0

( k u

− u

k

k u ˙

− u ˙

k + k u ˙

− u ˙

k

k u ˙

− u ˙

k ) ds.

Using (2.14), Young’s inequality for the last three terms with appropriate constants, V

- ellipticity of a

, b

and Gronwall’s inequality, the estimate

(2.15) follows.

Now, for every element u ∈ W

(0, T ; V

), consider the inequality (2.13) with the initial condition β

, the solution of which is denoted by β

. The ex- istence and uniqueness results for this parabolic inequality follow by classical references, see, e.g. [4], [1], [2], and a direct proof is presented in [11].

Lemma 2.3 For each u ∈ W

(0, T ; V

) there exists a unique solution β

∈ X

∩ W

(0, T ; Π

) of the inequality (2.13).

Lemma 2.4 Let u

, u

∈ W

(0, T ; V

) and let β

, β

∈ X

be the corresponding solutions of (2.13) with the same initial condition β

, respec- tively. Then there exists a constant C

> 0, independent of u

, u

, β

, β

, such that for all t ∈ [0, T ]

| β

(t) − β

(t) |

≤ C

Z

0

k u

(s) − u

(s) k

ds. (2.16) Proof. Let β

, β

be the solutions of (2.13) corresponding to u

, u

. Taking in each inequality η = β

, η = β

, respectively, for all t ∈ ]0, T [, integrating over [0, t], using (2.4) and some elementary inequality yield

1

2 | β

(t) − β

(t) |

≤ Z

0

[γ

(u

, β

, β

− β

) − γ

(u

, β

, β

− β

)] ds

= Z

0

[γ

(u

, β

, β

− β

) − γ

(u

, β

, β

− β

)] ds +

Z

0

[γ

(u

, β

, β

− β

) − γ

(u

, β

, β

− β

)] ds

≤ m

Z

0

| β

− β

|

ds + m

Z

0

k u

− u

k | β

− β

|

ds

≤ m

2

Z

0

k u

(s) − u

(s) k

ds + 3 m

2

Z

0

| β

(s) − β

(s) |

ds.

By Gronwall’s inequality the estimate (2.16) is established.

Now we can prove the following existence and uniqueness result for the abstract problem Q.

Theorem 2.1 Assume that conditions (2.1)-(2.11) hold. Then there exists a unique solution of the problem Q.

Proof. For every β ∈ X

let u

∈ W

(0, T ; H

) ∩ W

(0, T ; V

) be the

solution of the inequality (2.12) corresponding to β such that u

(0) = u

,

˙

u

(0) = u

and let β

∈ X

∩ W

(0, T ; Π

) be the solution of the inequality (2.13) corresponding to u

. Define the mapping T : X

→ X

as ∀ β ∈ X

T β = β

. We shall prove that T : X

→ X

has a unique fixed point, which is equally the solution of the problem Q.

For all β

, β

∈ X

, for all t ∈ [0, T ], (2.16) and (2.15) imply that

|T β

(t) − T β

(t) |

≤ C

Z

0

k u

(s) − u

(s) k

ds

≤ C

C

Z

0

Z

0

exp( − 2kr) · exp(2kr) | β

(r) − β

(r) |

dr

ds

≤ C

C

k β

− β

k

Z

0

exp(2ks)

2k ds

≤ C

C

4k

· exp(2kt) k β

− β

k

. Then

kT β

− T β

k

= max

t∈[0,T]

[exp( − kt) |T β

(t) − T β

(t) |

]

≤

√ C

C

2k k β

− β

k

. Hence, for all β

, β

∈ X

kT β

− T β

k

≤

√ C

C

2k k β

− β

k

,

so that if k is sufficiently large it follows that T is a contraction and its

fixed point is the solution of the problem Q.

3 A dynamic contact problem with adhesion and friction

Consider two viscoelastic bodies, characterized by a Kelvin-Voigt constitu- tive law, which occupy the reference domains Ω

of R

, d = 2 or 3, with Lipschitz continuous boundaries Γ

= ∂Ω

, α = 1, 2. Assume the small deformation hypothesis. Let Γ

, Γ

and Γ

be three open disjoint suffi- ciently smooth parts of Γ

such that Γ

= Γ

∪ Γ

∪ Γ

and, to simplify the estimates, meas(Γ

) > 0, α = 1, 2.

Let y

(x

, t) be the position at time t ∈ [0, T ], where T > 0, of

the material point represented by x

in the reference configuration and

u

(x

, t) := y

(x

, t) − x

be the displacement vector of x

at time t, with

the Cartesian coordinates u

= (u

, ..., u

) = (u

, u

).

Let ε

, with the Cartesian coordinates (ε

(u

)), and σ

, with the Cartesian coordinates (σ

), be the infinitesimal strain tensor and the stress tensor, respectively, corresponding to Ω

, α = 1, 2. The usual summation convention will be used for i, j, k, l = 1, . . . , d.

Assume that the displacement U

= 0 is prescribed on Γ

× ]0, T [ , α = 1, 2, and, to simplify, that the densities of both bodies are equal to 1. Let f = (f

, f

) and F = (F

, F

) denote the given body forces in Ω

∪ Ω

and tractions on Γ

∪ Γ

, respectively. The initial displacements and velocities of the bodies are denoted by u

= (u

, u

), u

= (u

, u

).

Suppose that the solids can be in contact between the potential contact surfaces Γ

and Γ

that can be parametrized by two C

functions, ϕ

, ϕ

, defined on an open subset Ξ of R

such that ϕ

(ξ) − ϕ

(ξ) ≥ 0 ∀ ξ ∈ Ξ and each Γ

is the graph of ϕ

on Ξ that is Γ

= { (ξ, ϕ

(ξ)) ∈ R

; ξ ∈ Ξ } , α = 1, 2. Let m

: Ξ → R

, with m

(ξ) := ( ∇ ϕ

(ξ), − 1), m

(ξ) :=

( −∇ ϕ

(ξ), 1), ∀ ξ ∈ Ξ, be the outward normal to Γ

, α = 1, 2. Since the displacements, their derivatives and the gap are assumed small, by using a method similar to the one presented in [3] (see also [10]) the following unilateral contact condition at time t on the set Ξ is obtained:

0 ≤ ϕ

(ξ) − ϕ

(ξ) + u

(ξ, ϕ

(ξ), t) − u

(ξ, ϕ

(ξ), t)

−∇ ϕ

(ξ) · u

(ξ, ϕ

(ξ), t) + ∇ ϕ

(ξ) · u

(ξ, ϕ

(ξ), t) ∀ ξ ∈ Ξ.

Using the definition of m

, m

, this relation can be written under the following form: for all ξ ∈ Ξ

m

(ξ) · u

(ξ, ϕ

(ξ), t) + m

(ξ) · u

(ξ, ϕ

(ξ), t) ≤ ϕ

(ξ) − ϕ

(ξ). (3.1) Let n

:= m

/ | m

| denote the unit outward normal vector to Γ

, α = 1, 2, and define the initial normalized gap between the two contact surfaces by

g

(ξ) := ϕ

(ξ) − ϕ

(ξ)

p 1 + |∇ ϕ

(ξ) |

∀ ξ ∈ Ξ.

Let the normal and tangential components of a displacement field v

, α = 1, 2, of the relative displacement corresponding to v := (v

, v

), including the initial gap g

in the normal direction, and of the stress vector σ

n

on Γ

be given by

v

:= v

(ξ, t) = v

(ξ, ϕ

(ξ), t),

v

:= v

(ξ, t) = v

(ξ, ϕ

(ξ), t) · n

(ξ), v

:= v

(ξ, t) = v

− v

n

, [v

] := [v

](ξ, t) = v

+ v

− g

, [v

] := [v

](ξ, t) = v

− v

, σ

:= σ

(ξ, t) = (σ

n

) · n

, σ

:= σ

(ξ, t) = σ

n

− σ

n

,

(3.2)

for all ξ ∈ Ξ and for all t ∈ [0, T ]. Let g := − [u

] = g

− u

− u

be the gap corresponding to the solution u := (u

, u

). Assuming that

∇ ϕ

(ξ) ' ∇ ϕ

(ξ), it follows that the unilateral contact condition (3.1) at time t can be written as

[u

] (ξ, t) = − g(ξ, t) ≤ 0 ∀ ξ ∈ Ξ. (3.3) Let β denote an internal state variable, see, e.g. [13]–[15], representing the intensity of adhesion or the adhesion field (β = 1 means that the adhesion is total, β = 0 means that there is no adhesion and 0 < β < 1 is the case of partial adhesion).

3.1 Classical formulation

Let A

, B

denote two fourth-order tensors, the elasticity tensor and the viscosity tensor corresponding to Ω

, with the components ( A

) and ( B

), respectively. Assume that these components satisfy the follow- ing classical symmetry and ellipticity conditions: C

= C

= C

∈ L

(Ω

), ∀ i, j, k, l = 1, . . . , d, ∃ α

> 0 such that C

τ

τ

≥ α

τ

τ

,

∀ τ = (τ

) verifying τ

= τ

, ∀ i, j = 1, . . . , d, where C

= A

, C = A

or C

= B

, C = B

∀ i, j, k, l = 1, . . . , d, α = 1, 2.

We choose the following state variables: the infinitesimal strain ten- sor (ε

, ε

) = (ε(u

), ε(u

)) in Ω

∪ Ω

, the relative normal displacement [u

] = u

+ u

− g

, the relative tangential displacement [u

] = u

− u

, and the intensity of adhesion β in Ξ.

Let µ = µ(ξ, [ ˙ u

]) ≥ 0 be the slip rate dependent coefficient of friction and assume that µ : Ξ × R

→ R

is a bounded function such that for a.e. ξ ∈ Ξ, µ(ξ, · ) is Lipschitz continuous with the Lipschitz constant independent of ξ and for any v ∈ R

, µ( · , v) is measurable.

Define a truncation operator ϑ = ϑ

by ϑ : R → R , ϑ(s) = − l

if s ≤ − l

, ϑ(s) = s if | s | < l

and ϑ(s) = l

if s ≥ l

, where l

> 0 is a given characteristic length (see, e.g. [26, 29]).

Let κ : R × [0, 1] → R be a bounded Lipschitz continuous function such that κ(0, 0) = 0. Note that various normal compliance conditions, friction and adhesion with damage laws can be obtained by choosing particular functions as κ, see [17, 18, 21, 27, 29].

We consider the following classical formulation of the dynamic contact problem coupling adhesion and friction.

Problem P

: Find (u

, u

), β such that u(0) = u

= (u

, u

), ˙ u(0) = u

= (u

, u

) in Ω

× Ω

, β(0) = β

in Ξ and, for all t ∈ ]0, T [,

¨

u

− div σ

(u

, u ˙

) = f

in Ω

, (3.4) σ

(u

, u ˙

) = A

ε(u

) + B

ε( ˙ u

) in Ω

, (3.5) u

= 0 on Γ

, σ

n

= F

on Γ

, α = 1, 2, (3.6)

σ

n

+ σ

n

= 0 in Ξ, (3.7)

σ

= κ([u

], β) in Ξ, (3.8)

| σ

| ≤ µ([ ˙ u

]) | σ

| in Ξ and (3.9)

| σ

| < µ([ ˙ u

]) | σ

| ⇒ [ ˙ u

] = 0 ,

| σ

| = µ([ ˙ u

]) | σ

| ⇒ ∃ θ ≥ 0, [ ˙ u

] = − θσ

,

β ∈ [0, 1] in Ξ and (3.10)

b β ˙ ≥ w if β = 0,

b β ˙ = w − C

ϑ([u

]

) β if β ∈ ]0, 1[, b β ˙ ≤ w − C

ϑ([u

]

) if β = 1,

where β

∈ [0, 1] in Ξ, C

> 0, b > 0, w > 0, σ

= σ

(u

, u ˙

), α = 1, 2, σ

:= σ

, σ

:= σ

, σ := σ

.

3.2 Variational formulation

We adopt the following notations:

H

:= [H

(Ω

)]

× [H

(Ω

)]

∀ s ∈ R ,

h v, w i

= h v

, w

i

+ h v

, w

i

∀ v = (v

, v

) ∈ H

, ∀ w = (w

, w

) ∈ H

.

Define the Hilbert spaces (H , | . | ) with the associated inner product denoted by (. , .), (V , k . k ) with the associated inner product (of H

) denoted by h . , . i and the set Λ as follows:

H := H

=

L

(Ω

)

×

L

(Ω

)

, V = V

× V

, where V

= { v

∈

H

(Ω

)

; v

= 0 a.e. on Γ

} , α = 1, 2, Λ = { η ∈ L

(Ξ) ; η ∈ [0, 1] a.e. in Ξ } .

Assume that F

∈ W

(0, T ; [L

(Γ

)]

), f

∈ W

(0, T ; [L

(Ω

)]

), α = 1, 2, u

, u

∈ V and β

∈ Λ.

Let a, b be two bilinear, continuous and symmetric mappings defined on H

× H

→ R as

a(v, w) = a

(v

, w

) + a

(v

, w

), b(v, w) = b

(v

, w

) + b

(v

, w

)

∀ v = (v

, v

), w = (w

, w

) ∈ H

, where, for α = 1, 2, a

(v

, w

) =

Z

Ω^α

A

ε(v

) · ε(w

) dx, b

(v

, w

) = Z

Ω^α

B

ε(v

) · ε(w

) dx.

Consider L as an element of W

(0, T ; H

) such that ∀ t ∈ [0, T ] h L, v i = X

α=1,2

Z

Ω^α

f

· v

dx + X

α=1,2

Z

Γ^α2

F

· v

ds ∀ v = (v

, v

) ∈ H

.

We define the following mappings:

J : L

(Ξ) × (H

)

→ R , J(β, u, v, w) =

Z

Ξ

µ([v

]) | κ([u

], β) | | [w

] | dξ

∀ β ∈ L

(Ξ), ∀ (u, v, w) ∈ (H

)

, γ : H

× [L

(Ξ)]

→ R , γ(u, δ, η) =

Z

Ξ

C

b ϑ([u

]

) δ η dξ,

$ : L

(Ξ) → R , $(η) = Z

Ξ

w

b η dξ ∀ u ∈ H

, ∀ δ, η ∈ L

(Ξ).

Assume the following compatibility condition: ∃ l ∈ H such that (l, v) + a(u

, v) + b(u

, v) − (κ([u

], β

), v

)

+J(β

, u

, u

, v) = h L(0), v i ∀ v ∈ V . (3.11) A variational formulation of the problem P

is the following.

Problem P

: Find u ∈ W

(0, T ; H) ∩ W

(0, T ; V ), β ∈ W

(0, T ; L

(Ξ)) such that u(0) = u

, ˙ u(0) = u

in Ω

∪ Ω

, β(0) = β

in Ξ, β (τ ) ∈ Λ for all τ ∈ ]0, T [ and a.e. t ∈ ]0, T [

(¨ u, v − u) + ˙ a(u, v − u) + ˙ b( ˙ u, v − u) ˙ − (κ([u

], β), v

− u ˙

)

(3.12) +J (β, u, u, ˙ v) − J(β, u, u, ˙ u) ˙ ≥ h L, v − u ˙ i ∀ v ∈ V ,

( ˙ β, η − β)

+ γ (u, β, η − β) ≥ $(η − β) ∀ η ∈ Λ. (3.13) The formal equivalence between the variational system (3.12),(3.13) and the classical problem (3.4)–(3.10) can be easily proved by using Green’s formula.

3.3 Existence and uniqueness of variational solutions The following existence and uniqueness result holds.

Theorem 3.1 Under the above assumptions, there exists a unique solution of the Problem P

.

Proof. We apply Theorem 2.1 to H

= H , V

= V , U

= H

, where 1/2 > ι > 0, Π

= L

(Ξ), Λ

= Λ, u

= u

, u

= u

, a

= a, b

= b, L

= L, γ

= γ − $ and

φ

(t, η, u, v, w) = − (κ([u

], η), w

)

+ J(η, u, v, w)

∀ t ∈ [0, T ], ∀ η ∈ L

(Ξ), ∀ u, v, w ∈ V .

One can easily verify the properties (2.1)-(2.11). Thus, by Theorem 2.1 there exists a unique solution of the problem P

.

Note that the same method can be used to study the dynamic contact problem with irreversible adhesion (see, e.g. [29]), for which the evolution of the intensity of adhesion is governed by a differential equation.

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Marius Cocou

Aix-Marseille University, CNRS, LMA UPR 7051, Centrale Marseille Address:

Laboratoire de M´ecanique et d’Acoustique

31 chemin Joseph Aiguier, 13402 Marseille Cedex 20, France E-mail: cocou@lma.cnrs-mrs.fr