DYNAMIC CONTACT PROBLEMS WITH VELOCITY CONDITIONS O

DYNAMIC CONTACT PROBLEMS WITH VELOCITY CONDITIONS

O

ANH

CHAU

^∗

, V

IORICA

V

ENERA

MOTREANU

^∗

∗

Laboratoire de Théorie des Systèmes, Université de Perpignan, 52 Avenue de Villeneuve, 66860 Perpignan Cedex, France,

e-mail:

{chau,viorica}@univ-perp.fr

We consider dynamic problems which describe frictional contact between a body and a foundation. The constitutive law is viscoelastic or elastic and the frictional contact is modelled by a general subdifferential condition on the velocity, including the normal damped responses. We derive weak formulations for the models and prove existence and uniqueness results.

The proofs are based on the theory of second-order evolution variational inequalities. We show that the solutions of the viscoelastic problems converge to the solution of the corresponding elastic problem as the viscosity tensor tends to zero and when the frictional potential function converges to the corresponding function in the elastic problem.

Keywords: viscoelastic, elastic, subdifferential boundary condition, dynamic process, nonlinear hyperbolic variational

inequality, maximal monotone operator, weak solution

1. Introduction

Contact problems arise in many situations, for instance, in crack and impact mechanics, or in earthquake phenom- ena. Despite the importance of their practical applications and the considerable literature devoted to these topics, many problems involving contact phenomena still re- main open.

A number of papers investigating quasistatic fric- tional contact problems with viscoelastic materials have recently been published (see, e.g., Awbi et al., 2000; Chau et al.; 2001a; 2001b; Han and Sofonea, 2000; 2001).

In (Chau et al., 2001b) frictional contact was modelled by a general velocity-dependent dissipation functional, in (Chau et al., 2001a) a bilateral contact with Tresca’s fric- tion law was analysed, while in (Han and Sofonea, 2001) frictional contact with normal compliance was studied, and in (Awbi et al., 2000; Han and Sofonea, 2000) fric- tional contact with normal damped response was consid- ered. Dynamic contact problems with normal compliance were considered in (Andrews et al., 1997a; 1997b; Kuttler and Shillor, 1999; Martins and Oden, 1987).

This paper constitutes a contribution to the study of second-order evolution contact problems. Our aim is to give versions of the results obtained in (Chau et al., 2001b) to a dynamic process. We investigate models for dynamic frictional contact between a body and an ob- stacle, in which Kelvin-Voigt viscoelastic or elastic con- stitutive laws are considered. The frictional contact is modelled by a general subdifferential boundary condition.

Further examples and detailed explanations concerning the boundary conditions of this form can be found in the monograph by Panagiotopoulos (1985) and more recently in (Chau et al., 2001b). Here, the study of viscoelastic or elastic materials in dynamic processes with subdifferential boundary conditions leads to a non-standard new mathe- matical model, implying nonlinear second-order evolution equations.

We prove the existence and uniqueness of weak solu- tions to the mechanical problems. We also show the con- tinuous dependence of these solutions on the viscosity and frictional potential function, both of which may vary be- cause of simultaneous changes in the viscosity of the body and in the roughness of the surface.

The outline of the paper is as follows. In Section 2

we introduce the notation and a preliminary material. In

Section 3 we formulate the dynamic mechanical problems

with a subdifferential frictional contact condition. Then,

after specifying the assumptions on the data, we derive

variational formulations for the problems, and we prove

an existence and uniqueness result. The proof is based

on second-order evolutionary inequalities with maximal

monotone operators. In Section 4 we prove a convergence

result which shows that the solutions to the viscoelastic

problems converge to the solution to the elastic problem

when the viscosity tends to zero and when the frictional

potential function converges to the corresponding one in

the elastic problem. Finally, in Section 5 we provide some

examples of specific subdifferential conditions to which

our results apply.

2. Notation and Preliminaries

Let S

d

be the space of second-order symmetric tensors on R

^d

(d = 2, 3), and denote the inner product and the Eu- clidean norm on R

^d

and S

_d

by ‘ · ’ and | · |, respectively.

Thus,

u · v = u

i

v

i

, |v| = (v · v)

^1/2

, ∀ u, v ∈ R

^d

, σ · τ = σ

ij

τ

ij

, |τ | = (τ · τ )

^1/2

, ∀ σ, τ ∈ S

d

. Here and below, the indices i and j run between 1 and d, the summation convention over repeated indices is adopted and the index that follows a comma indicates a partial derivative with respect to the corresponding com- ponent of the independent variable.

Let Ω ⊂ R

^d

be a bounded domain with a Lipschitz boundary Γ. We shall use the notation

H = L

²

(Ω)

^d

= {u = (u

i

) | u

i

∈ L

²

(Ω)}, H = {σ = (σ

ij

) | σ

ij

= σ

ji

∈ L

²

(Ω)}, H

₁

= {u = (u

_i

) | u

_i

∈ H

¹

(Ω)}, H

1

= {σ ∈ H | σ

ij,j

∈ L

²

(Ω)}.

The spaces H, H, H

₁

and H

₁

are real Hilbert spaces endowed with the inner products given by

(u, v)

H

= Z

Ω

u

i

v

i

dx,

(σ, τ )

_H

= Z

Ω

σ

_ij

τ

_ij

dx,

(u, v)

_H1

= (u, v)

_H

+ (ε(u), ε(v))

_H

, (σ, τ )

_H1

= (σ, τ )

_H

+ (Div σ, Div τ )

H

, respectively, where ε : H

1

→ H and Div : H

1

→ H are the deformation and the divergence operators, respec- tively, defined by

ε(u) = (ε

ij

(u)), ε

ij

(u) = 1

2 (u

i,j

+ u

j,i

), Div σ = (σ

ij,j

).

We denote the norms on the spaces H, H, H

1

and H

1

by k · k

H

, k · k

_H

, k · k

H₁

and k · k

_H1

, respectively.

Let H

Γ

= H

^1/2

(Γ)

^d

, γ : H

1

→ H

Γ

be the trace map and ν be the outer unit normal on Γ. For every element v ∈ H

₁

we still write v for the trace γv of v on Γ, and we denote by v

ν

and v

_τ

the normal and tangential components of v on the boundary Γ given by

v

ν

= v · ν, v

τ

= v − v

ν

ν.

Let H

_Γ⁰

be the dual of H

Γ

and let h·, ·i stand for the pairing between H

_Γ⁰

and H

Γ

. For every σ ∈ H

1

, there exists an element, denoted by σν ∈ H

_Γ⁰

, such that

hσν, γvi = (σ, ε(v))

_H

+(Div σ, v)

H

, ∀v ∈ H

1

. (1) In addition, if σ is regular enough (e.g. of class C

¹

), we have

hσν, γvi = Z

Γ

σν · v da, ∀v ∈ H

1

. (2) Relations (1) and (2) imply the following Green formula:

(σ, ε(v))

_H

+ (Div σ, v)

H

= Z

Γ

σν · v da, ∀v ∈ H

1

. (3) In a similar manner, the normal and tangential compo- nents of σ are defined by

σ

ν

= (σν) · ν, σ

τ

= σν − σ

ν

ν.

Finally, for every real Hilbert space X we use the classical notation for the spaces L

^p

(0, T ; X) and W

^k,p

(0, T ; X), 1 ≤ p ≤ +∞, k = 1, 2, . . . , and we denote by C([0, T ]; X) and C

¹

([0, T ]; X) the spaces of continuous and continuously differentiable functions from [0, T ] to X, respectively. We recall now an existence and uniqueness result concerning evolution problems, taken from (Barbu, 1976, p. 268).

Theorem 1. Let V and H be two real Hilbert spaces such that V ⊂ H and the inclusion mapping of V into H is continuous and densely defined. We suppose that V is endowed with the norm k · k induced by the inner product (·, ·) and H is endowed with the norm | · |. We denote by V

⁰

the dual space of V , by h·, ·i

V⁰×V

the du- ality pairing between an element of V and an element of V

⁰

, and H is identified with its own dual H

⁰

. We assume that M is a maximal monotone set in V × V

⁰

and A is a linear, continuous and symmetric operator from V to V

⁰

satisfying the following coerciveness condition:

hAu, ui

V⁰×V

+ α|u|

²

≥ ωkuk

²

, ∀ u ∈ V, (4) where α ∈ R and ω > 0. Let g be given in W

^1,1

(0, T ; H) and u

0

, v

0

be given with

u

0

∈ V, v

0

∈ D(M ), {Au

0

+ M v

0

} ∩ H 6= ∅. (5) Then there exists a unique solution u to the following problem:



 

  d

²

u

dt

²

+ Au + M  du dt



3 g(t) a.e. on (0, T ) u(0) = u

0

, du

dt (0) = v

0

,

which satisfies

u ∈ W

^1,∞

(0, T ; V ) ∩ W

^2,∞

(0, T ; H).

We use Theorem 1 in Section 3 to prove the exis- tence and the uniqueness of the solution to the variational problem associated with our mechanical model.

3. Problem Statement. Existence and Uniqueness Result

In this section we describe the mechanical contact prob- lem, derive its variational formulation and prove an exis- tence and uniqueness result.

The physical setting is the following: We consider a body that occupies a bounded domain Ω ⊂ R

^d

with a Lipschitz continuous boundary divided into three disjoint measurable parts Γ

1

, Γ

2

and Γ

3

such that the measure of Γ

1

, denoted by |Γ

1

|, is positive. Let T > 0 and [0, T ] be the time interval of interest. Let ρ : Ω → R

+

be the mass density of the body and f

₀

: Ω × (0, T ) → R

^d

the volume force density acting in Ω × (0, T ). The body is clamped on Γ

₁

× (0, T ) and therefore the displacement field vanishes there. A surface traction of density f

₂

: Γ

2

→ R

^d

assumed to be time-independent acts on Γ

2

. On Γ

3

× (0, T ) the body may come in contact with an obstacle, the so-called foundation, and we suppose that the contact condition may be described by a subdifferential- type inequality.

We denote by u = (u

_i

) : Ω × [0, T ] → R

^d

the displacement field, by σ = (σ

_ij

) : Ω × [0, T ] → S

_d

the stress field, and ε(u) = (ε

ij

(u)) represents the lin- earized strain tensor. Moreover, dots above a function will represent the derivative with respect to the time variable, i.e. ˙ u = du/dt or ¨ u = d

²

u/dt

²

.

We now describe the mechanical model for the pro- cess of frictional contact between the body and the obsta- cle. We use a Kelvin-Voigt constitutive law of the form

σ = cAε( ˙ u) + Gε(u),

where A is the viscosity operator, G ≡ (g

ijkh

), the elas- ticity tensor, c ≥ 0 is the viscosity coefficient. When c is positive, the body exhibits a viscoelastic behavior, while for c = 0 the body is elastic. We model the frictional contact with a subdifferential boundary condition on Γ

3

of the form

u ∈ U, ϕ(v) − ϕ( ˙ u) ≥ −σν · (v − ˙ u), ∀v ∈ U, where U ⊂ H

1

represents the set of contact admissible test functions, σν denotes the Cauchy stress vector on the contact boundary and ϕ : Γ

3

× R

^d

→ R is a given

function. The initial displacement field u

0

and the initial velocity field v

0

are given.

To summarize, the frictional mechanical problem can be formulated as follows.

Problem P

^ccc

: Find a displacement field u : Ω×[0, T ] → R

^d

and a stress field σ : Ω × [0, T ] → S

d

such that

ρ ¨ u = Div σ + f

₀

in Ω × (0, T ), (6) σ = cAε( ˙ u) + Gε(u) in Ω × (0, T ), (7) u = 0 on Γ

1

× (0, T ), (8) σν = f

₂

on Γ

2

× (0, T ), (9) u ∈ U, ϕ(v) − ϕ( ˙ u) ≥ −σν · (v − ˙ u),

∀ v ∈ U on Γ

3

× (0, T ), (10) u(0) = u

₀

, u(0) = v ˙

₀

in Ω. (11)

To obtain the variational formulation of Problem P

^c

, we consider the set

V = {v ∈ H

₁

| v = 0 on Γ

1

} ∩ U. (12) Let us define the functional j : V → R ∪ {+∞} by

j(v) =



 

  Z

Γ3

ϕ(v) da if ϕ(v) ∈ L

¹

(Γ

3

),

+∞ otherwise.

(13)

In the sequel, we suppose that:

V is a closed linear subspace in H

1

,

is dense in H and contains D(Ω)

^d

; (14)

j is a proper, convex and lower

semicontinuous functional on V. (15) Since |Γ

1

| > 0, Korn’s inequality implies that there exists a constant C

K

> 0, depending only on Ω and Γ

1

, such that

kε(v)k

_H

≥ C

K

kvk

H₁

, ∀ v ∈ V. (16) A proof of Korn’s inequality can be found in (Neˇcas and Hlavaˇcek, 1981, p. 79). We consider the inner product on V given by

(u, v)

V

= (ε(u), ε(v))

_H

, ∀u, v ∈ V, (17) and let k · k

V

be the norm associated with the inner prod- uct (17), i.e.

kvk

V

= kε(v)k

_H

, ∀v ∈ V. (18)

From (16) it follows that k · k

H₁

and k · k

V

are equivalent norms on V . Therefore, by (14), (V, k · k

V

) is a real Hilbert space. Moreover, by combining Sobolev’s trace theorem and (16), there exists a constant C

0

, depending only on Ω, Γ

1

and Γ

3

, such that

kvk

_L2(Γ3)^d

≤ C

0

kvk

V

, ∀v ∈ V. (19) We suppose that the viscosity operator A : Ω×S

d

→ S

d

satisfies the following conditions:

(i

1

) A(x, ·) is monotone on S

d

, i.e.

(A(x, τ

1

) − A(x, τ

2

)) · (τ

1

− τ

2

) ≥ 0,

∀ τ

1

, τ

₂

∈ S

d

, a.e. x ∈ Ω;

(i

₂

) there exist r ∈ L

^∞

(Ω) and s ∈ L

²

(Ω) such that

|A(x, τ )| ≤ r(x)|τ | + s(x),

∀ τ ∈ S

d

, a.e. x ∈ Ω;

(i

₃

) A(x, ·) is continuous on S

d

, a.e. x ∈ Ω;

(i

₄

) A(·, τ ) is Lebesgue measurable on Ω for all τ ∈ S

d

. The elasticity tensor G : Ω × S

d

→ S

d

is assumed to satisfy the usual properties of ellipticity and symmetry, i.e.

(j

1

) there exists a constant m

G

> 0 such that G(x, τ ) · τ ≥ m

G

|τ |

²

, ∀ τ ∈ S

d

, a.e. x ∈ Ω;

(j

2

) G(x, τ )·σ = τ ·G(x, σ), ∀ τ , σ ∈ S

d

, a.e. x ∈ Ω;

(j

3

) g

ijkl

∈ L

^∞

(Ω) for all i, j, k, l.

We suppose that the mass density satisfies ρ ∈ L

^∞

(Ω) and there exists ρ

^∗

> 0

such that ρ(x) ≥ ρ

^∗

a.e. x ∈ Ω. (20) In the sequel, we define a new inner product on H given by

((u, v))

_H

= (ρu, v)

_H

, ∀u, v ∈ H, (21) and let ||| · |||

H

be the associated norm, i.e.

|||v|||

_H

= (ρ v, v)

^1/2_H

, ∀ v ∈ H. (22) Using assumption (20), from (22) it follows that ||| · |||

_H

and k·k

_H

are equivalent norms on H. Moreover, by (14), the inclusion mapping of (V, k · k

_V

) into (H, ||| · |||

H

) is continuous and dense. We denote by V

⁰

the dual space of V . Identifying H with its own dual, we can write V ⊂ H ⊂ V

⁰

. We use the notation h·, ·i

V⁰×V

to represent the duality pairing between V

⁰

and V . We have

hu, vi

V⁰×V

= ((u, v))

H

, ∀ u ∈ H, ∀ v ∈ V. (23)

We assume that the volume forces and tractions satisfy f

₀

∈ W

^1,1

(0, T ; H) and f

2

∈ L

²

(Γ

2

)

^d

. (24) Let us define the functional J : V → R ∪ {+∞} by

J (v) = j(v) − Z

Γ2

f

₂

· v da, ∀ v ∈ V. (25) We note that by (24) the integral in (25) is well defined.

We suppose that the initial data of Problem P

^c

sat- isfy

u

0

∈ V, v

0

∈ D(∂J ), (26) where ∂J denotes the subdifferential of J and D(∂J ) represents its domain.

We also assume that there exists h ∈ H such that (Gε(u

₀

) + cAε(v

₀

), ε(v) − ε(v

₀

))

_H

+ J (v)

−J (v

₀

) ≥ ((h, v − v

₀

))

_H

, ∀ v ∈ V. (27) For instance, in the case when we have

(σ

^c₀

, ε(v) − ε(v

0

))

_H

+ J (v) − J (v

0

)

≥ ((f

₀

(0), v − v

0

))

H

, ∀ v ∈ V, with σ

^c₀

:= Gε(u

0

) + cAε(v

0

), the condition (27) is sat- isfied.

We turn now to derive a variational formulation for the mechanical problem P

^c

. To this end, let us fix c ≥ 0.

We suppose in the following that {u, σ} are regular func- tions satisfying (6)–(11) and such that ϕ( ˙ u) ∈ L

¹

(Γ

3

).

Let w ∈ V with ϕ(w) ∈ L

¹

(Γ

3

) and t ∈ [0, T ]. Ap- plying (3) to σ for v = w − ˙ u(t) and using (6), we get

(ρ ¨ u(t) − f

₀

(t), w − ˙ u(t))

H

+ (σ(t), ε(w)

− ε( ˙ u(t)))

H

= Z

Γ

σ(t)ν · (w − ˙ u(t)) da.

Using (8), (9), (21) and (23), we obtain

h¨ u(t), w − ˙ u(t)i

_V⁰_×V

+ (σ(t), ε(w) − ε( ˙ u(t)))

_H

= (f

₀

(t), w − ˙ u(t))

H

+ (f

₂

, w − ˙ u(t))

_L2(Γ₂)^d

+ Z

Γ₃

σ(t)ν · (w − ˙ u(t)) da. (28)

Combining (28), (10) and (13), we conclude that h¨ u(t), w − ˙ u(t)i

V⁰×V

+ (σ(t), ε(w) − ε( ˙ u(t)))

H

+j(w) − j( ˙ u(t))

≥ (f

₀

(t), w− ˙ u(t))

H

+(f

₂

, w− ˙ u(t))

_L2(Γ₂)^d

. (29)

Taking into account (13), we observe that (29) remains true for all w ∈ V . Consequently, combining (29) and (25), we deduce that

h¨ u(t), w − ˙ u(t)i

_V⁰_×V

+ (σ(t), ε(w) − ε( ˙ u(t)))

_H

+ J (w) − J ( ˙ u(t)) ≥ (f

₀

(t), w − ˙ u(t))

H

,

∀ w ∈ V, a.e. t ∈ (0, T ).

Therefore, keeping in mind (7) and (11), we are led to the following variational formulation of the mechanical problem P

^c

, for each c ≥ 0:

Problem P

_VVV^ccc

: Find a displacement field u : [0, T ] → V and a stress field σ : [0, T ] → H

1

such that

σ(t) = cAε( ˙ u(t)) + Gε(u(t)) a.e. t ∈ (0, T ), (30) h¨ u(t), w − ˙ u(t)i

V⁰×V

+ (σ(t), ε(w) − ε( ˙ u(t)))

_H

+J (w) − J ( ˙ u(t)) ≥ (f

₀

(t), w − ˙ u(t))

H

,

∀ w ∈ V, a.e. t ∈ (0, T ), (31) u(0) = u

₀

, u(0) = v ˙

₀

in Ω. (32)

We state now our existence and uniqueness result.

Theorem 2. Assume that (14), (15), (i

1

)–(i

4

), (j

1

)–(j

3

), (20), (24), (26) and (27) hold. Then for each c ≥ 0 there exists a unique solution {u, σ} to Problem P

_V^c

such that

u ∈ W

^1,∞

(0, T ; V ) ∩ W

^2,∞

(0, T ; H), (33) σ ∈ L

²

(0, T ; H), Div σ ∈ L

^∞

(0, T ; H). (34)

We conclude that, under the assumptions of Theo- rem 2, Problem P

^c

has a unique weak solution {u, σ}

having the regularity (33), (34).

Proof. Let us fix c ≥ 0. We consider the Hilbert spaces H = L

²

(Ω)

^d

and V given by (12). We introduce the operator A : V → V

⁰

defined by

hAu, vi

V⁰×V

= (Gε(u), ε(v))

_H

, ∀ u, v ∈ V. (35) Using (18), (j

2

) and (j

3

), we see that A ∈ L(V, V

⁰

), and (j

1

) implies that A satisfies the condition (4) with α = 0 and ω = m

G

.

Define now the set-valued operator M

c

: V → V

⁰

by

M

c

= B

c

+ ∂J, (36)

where B

_c

: V → V

⁰

is given by

hB

c

u, vi

V⁰×V

= c(Aε(u), ε(v))

_H

, ∀ u, v ∈ V. (37)

From (37) and (i

1

), we have hB

c

u − B

c

v, u − vi

V⁰×V

= c(Aε(u)−Aε(v), ε(u)−ε(v))

_H

≥ 0, ∀ u, v ∈ V, so the operator B

c

is monotone. Using (37) and (18), we have

kB

_c

u−B

_c

vk

_V⁰

≤ ckAε(u)−Aε(v)k

_H

, ∀ u, v ∈ V, and, keeping in mind (i

₂

), (i

3

), (i

4

) and Krasnoselski’s theorem (see Kavian, 1993, p. 60), we find that B

c

: V → V

⁰

is a continuous operator. Using again (37) and (i

2

), we find that B

c

is bounded.

From (15) and (25) we deduce that J is proper, con- vex and lower semicontinuous, which implies that ∂J is maximal monotone. Consequently, since B

_c

is mono- tone, bounded and hemicontinuous from V to V

⁰

, we conclude (Barbu, 1976, p. 39) that M

c

= B

c

+ ∂J is maximal monotone.

Moreover, the initial data u

0

, v

0

satisfy (5) due to (26) and (27). Thus, all the requirements of Theorem 1, with A defined by (35), M = M

c

given in (36) and g = f

₀

, are satisfied. By defining σ by (30), it follows that there exists a unique solution {u, σ} to Problem P

_V^c

satisfying (33).

It remains to show (34) for σ. From (30), (i

2

), (j

2

), (j

3

) and (33) it follows that σ ∈ L

²

(0, T ; H). Let t ∈ [0, T ] and ψ ∈ D(Ω)

^d

. Since J ( ˙ u(t) ± ψ) = J ( ˙ u(t)) <

+∞, choosing w = ˙ u(t) ± ψ ∈ V (see (14)) in (31), using (3), (21) and (23), we obtain

ρ ¨ u = Div σ + f

₀

in H.

Now, taking into account (33) and (24), we arrive at Div σ ∈ L

^∞

(0, T ; H), and thus (34) is satisfied. The uniqueness of the solution follows from Theorem 1. The proof of Theorem 2 is now complete.

4. Convergence as Viscosity Vanishes

In this section we investigate the behaviour of the solu- tion to the viscoelastic problem P

_V^c

when the viscosity operator converges to zero and when the frictional poten- tial function tends to the potential of the corresponding elastic problem. We suppose in the sequel that (14), (i

2

)–

(i

4

), (j

1

)–(j

3

), (20), (24) hold and the following addi- tional property is satisfied:

(i

5

) A(x, ·) is strongly monotone on S

d

, i.e. there exists m

_A

> 0 such that

(A(x, τ

1

) − A(x, τ

2

)) · (τ

1

− τ

2

)

≥ m

_A

|τ

1

− τ

2

|

²

, ∀ τ

1

, τ

2

∈ S

d

, a.e. x ∈ Ω.

We focus our attention on the convergence to fric- tional elasticity and the continuity with respect to the fric- tion potential function. Thus, we consider a sequence of problems P

_V^cn

obtained from Problem P

_V^c

in which we set c = c

n

, where (c

n

) is a sequence of viscosity coef- ficients such that c

n

→ 0 as n → ∞; ϕ = ϕ

n

, where ϕ

n

: Γ

3

× R

^d

→ R are given functions; u

n0

and v

n0

stand for the initial displacements and velocities, respec- tively. We have the following variational problem, for each n:

Problem P

_VVV^cccnnn

: Find a displacement field u

n

: [0, T ] → V and a stress field σ

n

: [0, T ] → H

1

such that

σ

n

(t) = c

n

Aε( ˙ u

n

(t))+Gε(u

n

(t)) a.e. t ∈ (0, T ), (38) h¨ u

n

(t), w− ˙ u

n

(t)i

V⁰×V

+(σ

n

(t), ε(w)−ε( ˙ u

n

(t)))

H

+J

n

(w) − J

n

( ˙ u

n

(t)) ≥ (f

₀

(t), w− ˙ u

n

(t))

H

,

∀ w ∈ V, a.e. t ∈ (0, T ), (39) u

_n

(0) = u

_n0

, u ˙

_n

(0) = v

_n0

in Ω. (40)

Here J

n

is defined by (25) for j = j

n

, where j

n

is given by (13) for ϕ = ϕ

n

.

Next we consider the elastic problem P

_V⁰

obtained from P

_V^c

for c = 0 and the data ϕ : Γ

3

× R

^d

→ R, u

0

, v

0

given.

Problem P

_VVV⁰⁰⁰

: Find a displacement field u : [0, T ] → V and a stress field σ : [0, T ] → H

1

such that

σ(t) = Gε(u(t)) a.e. t ∈ (0, T ), (41) h¨ u(t), w − ˙ u(t)i

V⁰×V

+ (σ(t), ε(w) − ε( ˙ u(t)))

_H

+J (w) − J ( ˙ u(t)) ≥ (f

₀

(t), w − ˙ u(t))

H

,

∀ w ∈ V, a.e. t ∈ (0, T ), (42) u(0) = u

0

, u(0) = v ˙

0

in Ω. (43)

Here, the functionals J and j are defined by (25) and (13), respectively.

We assume that j satisfies (15);

j

n

is a proper, convex and lower

semicontinuous function on V for all n; (44) there exist m ∈ [1, 2) and α > 0 such that, for all n,

|ϕ

_n

(x, y) − ϕ(x, y)|

≤ αc

n

|y|

^m

, ∀ y ∈ R

^d

, a.e. x ∈ Γ

3

; (45)

there exists h ∈ H such that

(Gε(u

0

), ε(v) − ε(v

0

))

_H

+ J (v) − J (v

0

)

≥ ((h, v − v

0

))

H

, ∀ v ∈ V ; (46) for each n there exists h

n

∈ H such that

(Gε(u

_n0

) + c

_n

Aε(v

n0

), ε(v) − ε(v

_n0

))

_H

+J

n

(v) − J

n

(v

n0

)

≥ ((h

n

, v − v

n0

))

H

, ∀ v ∈ V ; (47) and v

0

∈ D(∂J ), v

n0

∈ D(∂J

n

). Finally,

u

n0

→ u

0

in V, v

n0

→ v

0

in H as n → ∞. (48) Let us remark that if we have, for all n, v

n0

= v

0

and

(σ

n0

, ε(v) − ε(v

0

))

_H

+ J

n

(v) − J

n

(v

0

)

≥ ((f

₀

(0), v − v

₀

))

_H

, ∀v ∈ V, (σ

₀

, ε(v) − ε(v

₀

))

_H

+ J (v) − J (v

₀

)

≥ ((f

₀

(0), v − v

0

))

H

, ∀v ∈ V, with σ

n0

:= Gε(u

n0

) + c

n

Aε(v

n0

), σ

0

:= Gε(u

0

), then the assumptions (46)–(48) are satisfied.

From Theorem 2 it follows that, for each n, Prob- lem P

_V^cn

has a unique solution {u

n

, σ

n

} with regu- larity u

n

∈ W

^1,∞

(0, T ; V ) ∩ W

^2,∞

(0, T ; H), σ

n

∈ L

²

(0, T ; H), Div σ

n

∈ L

^∞

(0, T ; H), and Problem P

_V⁰

has a unique solution {u, σ} with regularity u ∈ W

^1,∞

(0, T ; V ) ∩ W

^2,∞

(0, T ; H), σ ∈ L

²

(0, T ; H), Div σ ∈ L

^∞

(0, T ; H).

We are now in a position to formulate our conver- gence result.

Theorem 3. Let (c

n

) be a sequence in (0, +∞) such that c

n

→ 0 as n → ∞. Suppose that (14), (15), (i

2

)–(i

5

), (j

1

)–(j

3

), (20), (24), (44)–(47) and denote by {u

n

, σ

n

}, {u, σ} the unique solutions to Problems P

_V^cn

and P

_V⁰

, respectively. Then there exists a constant C > 0, depending on u and on the data, but independent of n, such that for all n we have

ku

n

− uk

C([0,T ];V )

+ ku

_n

− uk

C¹([0,T ];H)

+ kσ

n

− σk

_L2(0,T ;H)

≤ C(ku

n0

− u

0

k

V

+ kv

n0

− v

0

k

H

+ √ c

n

).

Consequently, if (48) holds, then as n → ∞, u

n

→ u in C([0, T ]; V ) ∩ C

¹

([0, T ]; H),

σ

n

→ σ in L

²

(0, T ; H).

Proof. Let t ∈ [0, T ]. Taking w = ˙ u(t) in (39) and using (38), we have

h¨ u

n

(t), ˙ u(t) − ˙ u

n

(t)i

V⁰×V

+ c

n

(Aε( ˙ u

n

(t)), ε( ˙ u(t)) − ε( ˙ u

n

(t)))

_H

+ (Gε(u

_n

(t)), ε( ˙ u(t)) − ε( ˙ u

_n

(t)))

_H

+ j

_n

( ˙ u(t)) − j

_n

( ˙ u

_n

(t))

≥ (f

₀

(t), ˙ u(t) − ˙ u

n

(t))

H

+ (f

₂

, ˙ u(t) − ˙ u

n

(t))

_L2(Γ2)^d

, and taking w = ˙ u

_n

(t) in (42), using (41), we obtain

h¨ u(t), ˙ u

n

(t) − ˙ u(t)i

V⁰×V

+ (Gε(u(t)), ε( ˙ u

_n

(t)) − ε( ˙ u(t)))

_H

+ j( ˙ u

n

(t)) − j( ˙ u(t))

≥ (f

₀

(t), ˙ u

n

(t) − ˙ u(t))

H

+ (f

₂

, ˙ u

_n

(t) − ˙ u(t))

_L2(Γ₂)^d

.

Adding the last two inequalities, we deduce that for each t ∈ [0, T ],

h¨ u

_n

(t) − ¨ u(t), ˙ u(t) − ˙ u

_n

(t)i

_V⁰_×V

+ c

n

(Aε( ˙ u

n

(t)) − Aε( ˙ u(t)), ε( ˙ u(t)) − ε( ˙ u

n

(t)))

_H

+ c

n

(Aε( ˙ u(t)), ε( ˙ u(t)) − ε( ˙ u

n

(t)))

H

+ (Gε(u

_n

(t)) − Gε(u(t)), ε( ˙ u(t)) − ε( ˙ u

_n

(t)))

_H

+ j

n

( ˙ u(t)) − j

n

( ˙ u

n

(t)) + j( ˙ u

n

(t)) − j( ˙ u(t)) ≥ 0.

Integrating this inequality on [0, t] and using (23), (j

2

), (i

₅

), (18), (40), (43), we conclude that

1

2 ||| ˙ u

_n

(t) − ˙ u(t)|||

²_H

+ c

_n

m

_A

Z

t

0

k ˙ u

_n

(s) − ˙ u(s)k

²_V

ds

+ 1

2 (Gε(u

n

(t)) − Gε(u(t)), ε(u

n

(t)) − ε(u(t)))

_H

≤ 1

2 |||v

n0

− v

0

|||

²_H

+ c

n

Z

t 0

kAε( ˙ u(s))k

H

k ˙ u(s) − ˙ u

n

(s)k

V

ds

+ 1

2 (G(ε(u

_n0

) − ε(u

₀

)), ε(u

_n0

) − ε(u

₀

))

_H

+

Z

t 0

|j

n

( ˙ u(s))−j

n

( ˙ u

n

(s))+j( ˙ u

n

(s))−j( ˙ u(s))| ds.

Using (j

1

), (j

3

) and (18), and since by (i

2

), (i

3

) and (i

4

) we have Aε( ˙ u) ∈ L

²

(0, T ; H), we deduce that

1

2 ||| ˙ u

n

(t) − ˙ u(t)|||

²_H

+c

_n

m

_A

Z

t 0

k ˙ u

_n

(s) − ˙ u(s)k

²_V

ds

+ m

_G

2 ku

n

(t) − u(t)k

²_V

≤ 1

2 |||v

n0

− v

0

|||

²_H

+ c

_n

2m

A

Z

t 0

kAε( ˙ u(s))k

²_H

ds

+ c

_n

m

_A

2

Z

t 0

k ˙ u

_n

(s) − ˙ u(s)k

²_V

ds

+ C

2 ku

_n0

− u

₀

k

²_V

+ Z

t

0

(|j

_n

( ˙ u(s)) − j( ˙ u(s))|

+|j

_n

( ˙ u

_n

(s)) − j( ˙ u

_n

(s))|) ds, (49) where C is a positive constant independent of n and may change from line to line. From the fact that ϕ

n

( ˙ u

_n

(s)) ∈ L

¹

(Γ

₃

), ϕ( ˙ u(s)) ∈ L

¹

(Γ

₃

), by using (45), it follows that ϕ

_n

( ˙ u(s)) ∈ L

¹

(Γ

₃

), ϕ( ˙ u

_n

(s)) ∈ L

¹

(Γ

₃

). Conse- quently, using (45), Young’s inequality and (19), we can write, for all s ∈ [0, T ],

|j

n

( ˙ u(s)) − j( ˙ u(s))| + |j

n

( ˙ u

n

(s)) − j( ˙ u

n

(s))|

≤ αc

n

Z

Γ3

| ˙ u(s)|

^m

da + αc

n

Z

Γ3

| ˙ u

n

(s)|

^m

da

≤ α(1 + 2

^m−1

)c

n

Z

Γ₃

| ˙ u(s)|

^m

da

+ α2

^m−1

c

_n

Z

Γ₃

| ˙ u

_n

(s) − ˙ u(s)|

^m

da

≤ α(1 + 2

^m−1

) c

n

Z

Γ3

| ˙ u(s)|

^m

da

+ c

n

Z

Γ₃

"

2 − m 2

 α2

^m−1

τ



_2−m²

+ m

2 τ

^m2

| ˙ u

n

(s) − ˙ u(s)|

²

i da

≤ C(c

n

k ˙ u(s)k

^m_L2(Γ₃)^d

+ c

n

) + c

n

m

2 τ

^m2

k ˙ u

n

(s) − ˙ u(s)k

²_L2(Γ₃)^d

≤ C(c

n

C

₀^m

k ˙ u(s)k

^m_V

+ c

n

) + c

n

m

2 τ

^m2

C

₀²

k ˙ u

n

(s) − ˙ u(s)k

²_V

,

where τ > 0 is a constant that will be chosen below.

Integrating this inequality on [0, t] and using Hölder’s in- equality, we obtain

Z

t 0

(|j

_n

( ˙ u(s)) − j( ˙ u(s))| + |j

_n

( ˙ u

_n

(s)) − j( ˙ u

_n

(s))|) ds

≤ Cc

n

+ c

_n

m 2 τ

^m2

C

₀²

Z

t 0

k ˙ u

_n

(s) − ˙ u(s)k

²_V

ds

≤ Cc

n

+ c

_n

m

_A

3

Z

t 0

k ˙ u

_n

(s) − ˙ u(s)k

²_V

ds, (50)

for τ chosen such that (m/2)τ

^2/m

C

₀²

≤ m

_A

/3. Com- bining (49) with (50) and using the equivalence of the norms ||| · |||

H

and k · k

H

on H, we infer that

k ˙ u

n

(t) − ˙ u(t)k

²_H

+ c

n

Z

t 0

k ˙ u

n

(s) − ˙ u(s)k

²_V

ds

+ ku

n

(t) − u(t)k

²_V

≤ C(kv

n0

− v

0

k

²_H

+ ku

_n0

− u

0

k

²_V

+ c

_n

).

From the continuity of the embedding V ⊂ H, it follows that

ku

n

− uk

²C([0,T ];V )

+ ku

n

− uk

²_C1([0,T ];H)

+c

_n

k ˙ u

_n

− ˙ uk

²_L2(0,T ;V )

≤ C(kv

_n0

− v

₀

k

²_H

+ ku

_n0

− u

₀

k

²_V

+ c

_n

). (51)

It remains to prove that for all n we have kσ

_n

− σk

_L2(0,T ;H)

≤ C(ku

n0

− u

0

k

V

+ kv

n0

− v

0

k

H

+ √

c

n

). (52)

From (38) and (41) we deduce that kσ

n

(t) − σ(t)k

²_H

≤ 2c

²_n

kAε( ˙ u

n

(t))k

²_H

+ 2kGε(u

n

(t)) − Gε(u(t))k

²_H

, a.e. t ∈ (0, T ). Condition (i

2

) implies that there exist two positive constants δ

1

and δ

2

such that kAτ k

²_H

≤ δ

1

kτ k

²_H

+ δ

2

, ∀τ ∈ H. It follows that

kσ

_n

− σk

²_L2(0,T ;H)

≤ 2c

²_n

δ

1

k ˙ u

n

k

²_L2(0,T ;V )

+ 2T δ

2

c

²_n

+ 2 Z

T

0

kGε(u

n

(t)) − Gε(u(t))k

²_H

dt.

Using again (j

3

) and (18), we deduce that kσ

n

− σk

²_L2(0,T ;H)

≤ 4c

²_n

δ

1

k ˙ u

n

− ˙ uk

²_L2(0,T ;V )

+ 4c

²_n

δ

1

k ˙ uk

²_L2(0,T ;V )

+ 2T δ

2

c

²_n

+ 2C Z

T

0

ku

n

(t) − u(t)k

²_V

dt.

Keeping in mind (51), we arrive at

kσ

_n

− σk

²_L2(0,T ;H)

≤ C(c

_n

+ 1)(ku

_n0

− u

₀

k

²_V

+ kv

n0

− v

0

k

²_H

+ c

n

) + Cc

²_n

, so, since c

n

→ 0 as n → ∞, we obtain (52). This completes the proof.

We conclude by Theorem 3 that the weak solution to the frictional elastic problem P

⁰

can be approached by the weak solution to the frictional viscoelastic problem P

^cn

when the coefficient of viscosity is small enough and the corresponding friction potential functions satisfy (45).

In addition to the mathematical interest in the convergence properties proved in Theorem 3, this is of importance from the mechanical point of view, because the frictional elas- ticity appears as a limit case of frictional viscoelasticity.

Remark 1. A similar argument to the one used in the proof of Theorem 3 shows that Theorem 3 remains true when we replace the condition (45) with the following property: there exist an integer p ≥ 1 and numbers m

1

, . . . , m

p

∈ [1, 2), α

1

, . . . , α

p

> 0 such that for ev- ery n one has

|ϕ

n

(x, y) − ϕ(x, y)| ≤ c

n p

X

i=1

α

i

|y|

^mi

,

∀ y ∈ R

^d

a.e. x ∈ Γ

3

. (53)

5. Examples of Subdifferential Contact Condition

In this section we present some examples of contact fric- tion laws which lead to an inequality of the form (10) and for which (14) and (15) hold. Using Theorem 2, we con- clude that the boundary value problem P

_V^c

corresponding to each of the examples below has a unique weak solution.

Moreover, the convergence result given in Theorem 3 is applicable in all the concrete examples below.

Example 1. We consider the bilateral contact with

Tresca’s friction law. This contact condition can be found

in (Duvaut and Lions, 1976; Panagiotopoulos, 1985) and,

more recently, in (Amassad et al., (1999); Chau et al., 2001b). We use the following boundary condition:



 

 

u

_ν

= 0, |σ

τ

| ≤ g,

|σ

τ

| < g =⇒ ˙ u

τ

= 0,

|σ

τ

| = g =⇒ ˙ u

_τ

= −λσ

_τ

, for some λ ≥ 0, on Γ

3

× (0, T ).

Here g represents the friction bound, i.e. the magnitude of the limiting friction traction at which slip begins. We assume that g ∈ L

^∞

(Γ

₃

), g ≥ 0 a.e. on Γ

3

.

We set U = {v ∈ H

1

| v

ν

= 0 on Γ

3

} and deduce from (12) that

V = {v ∈ H

₁

| v = 0 on Γ

1

, v

_ν

= 0 on Γ

3

}.

We see that (10) holds with the choice

ϕ(x, y) = g(x)|y

_{τ (x)}

|, ∀ x ∈ Γ

3

, y ∈ R

^d

, where y

_{τ (x)}

:= y − y

ν(x)

ν(x), y

ν(x)

:= y · ν(x), with ν(x) as the unit normal at x ∈ Γ

3

. In the convergence result, denote by g the friction bound for the elastic prob- lem P

_V⁰

and by g

n

the one for the viscoelastic problem P

_V^cn

. If the functions g

_n

, g ∈ L

^∞

(Γ

₃

) satisfy, for all n,

kg

n

− gk

_L^∞_(Γ3)

≤ αc

n

,

for some α > 0, the assumption (45) is satisfied with

m = 1. 

Example 2. We model the bilateral contact with a vis- cous friction condition defined by a tangential damped re- sponse. We use the following boundary condition:

u

ν

= 0, σ

τ

= −µ| ˙ u

τ

|

^q−1

u ˙

τ

on Γ

3

× (0, T ), where 0 < q < 1 and µ ∈ L

^∞

(Γ

3

), µ ≥ 0 a.e. on Γ

3

.

We let U = {v ∈ H

1

| v

ν

= 0 on Γ

3

}, and then V = {v ∈ H

1

| v = 0 on Γ

1

, v

ν

= 0 on Γ

3

}.

Thus (10) is satisfied with ϕ(x, y) = 1

q + 1 µ(x) |y

_{τ (x)}

|

^q+1

, ∀ x ∈ Γ

₃

, y ∈ R

^d

. In the convergence result denote by µ the coefficient of friction for the elastic problem P

_V⁰

and by µ

n

the one for the viscoelastic problem P

_V^cn

. If the positive functions µ

n

, µ ∈ L

^∞

(Γ

3

) satisfy, for all n,

kµ

_n

− µk

_L∞(Γ₃)

≤ α(q + 1)c

_n

,

for some α > 0, the assumption (45) is satisfied with m = q + 1. 

Example 3. We consider a model of damped response contact with Tresca’s friction law (see, e.g., Jarušek and Eck, 1999; Rochdi and Shillor, 2001c). We use the fol- lowing boundary condition:



 

 

−σ

ν

= k| ˙ u

ν

|

^q−1

u ˙

ν

, |σ

τ

| ≤ g,

|σ

τ

| < g =⇒ ˙ u

τ

= 0,

|σ

_τ

| = g =⇒ ˙ u

_τ

= −λσ

_τ

, for some λ ≥ 0, on Γ

₃

× (0, T ). Here 0 < q < 1 and g, k ∈ L

^∞

(Γ

₃

), g, k ≥ 0.

We have U = H

¹

(Ω)

^d

and

V = {v ∈ H

¹

(Ω)

^d

| v = 0 on Γ

1

}.

Thus (10) is satisfied with ϕ(x, y) = 1

q + 1 k(x) |y

_ν(x)

|

^q+1

+ g(x)|y

_τ

(x)|,

∀x ∈ Γ

₃

, y ∈ R

^d

.

In the convergence result denote by g, k the parame- ters for the elastic problem P

_V⁰

and by g

n

, k

_n

those for the viscoelastic problem P

_V^cn

. If the positive functions g

_n

, k

_n

, g, k ∈ L

^∞

(Γ

₃

) satisfy, for all n,

kk

n

− kk

_L^∞_(Γ3)

≤ α

1

(q + 1)c

n

, kg

n

− gk

L^∞(Γ₃)

≤ α

2

c

_n

,

for some α

1

, α

₂

> 0, then (53) is satisfied with p = 2, m

₁

= q + 1, m

2

= 1. 

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