DYNAMIC CONTACT PROBLEMS WITH SLIP-DEPENDENT FRICTION IN VISCOELASTICITY

DYNAMIC CONTACT PROBLEMS WITH SLIP-DEPENDENT FRICTION IN VISCOELASTICITY

I

OAN

R. IONESCU

^∗

, Q

UOC

-L

AN

NGUYEN

^∗∗

∗

Laboratoire de Mathématiques, CNRS and Université de Savoie, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France,

e-mail:

ionescu@univ-savoie.fr

∗∗

National Polytechnic Institute, Ho-Shi-Minh,

143/2D Xo Viet nghe Tinh (he m), P25, Q.BT, Ho-Shi-Minh, Vietnam, e-mail:

quoc-lan.nguyen@hcm.fpt.vn

The dynamic evolution with frictional contact of a viscoelastic body is considered. The assumptions on the functions used in modelling the contact are broad enough to include both the normal compliance and the Tresca models. The friction law uses a friction coefficient which is a non-monotone function of the slip. The existence and uniqueness of the solution are proved in the general three-dimensional case.

Keywords: slip-dependent friction, dynamic viscoelasticity, Tresca contact, normal compliance, existence and uniqueness

1. Introduction

Duvaut and Lions (1976) obtained the first existence and uniqueness results for contact problems with friction in elastodynamics. Some years later, the non-penetrability of mass was relaxed by Martins and Oden (1987) by consid- ering the normal compliance model of contact with fric- tion. In order to obtain existence and uniqueness results they considered only the viscous case (see also Kutller, 1997).

All the above results involve a fixed friction coeffi- cient µ. In the study of many frictional processes (stick- slip motions, earthquakes modelling, etc.) the friction co- efficient has to be considered variable during the slip. The simplest variation of µ is the discontinuous jump from a

‘static’ value µ

s

down to a ‘dynamic’ or ‘kinetic’ value µ

d

. Three current models of such a variation are con- sidered in mechanics and geophysics. The first one, dis- cussed latter, corresponds to a smooth dependence of the friction coefficient on the slip u

T

, i.e. µ = µ(|u

T

|). The second one considers a slip rate dependence of the fric- tion coefficient (Oden and Martins, 1985; Scholz, 1990), i.e. µ = µ(| ˙ u

_T

|). For this model the solution of the math- ematical problem in dynamic elasticity is not uniquely de- termined and presents shocks (Ionescu and Paumier, 1993;

1994). However, the problem is well posed in dynamic viscoelasticity (Ionescu, 2001; Kuttler and Shillor, 1999).

The third model, called the Dieterich and Ruina model, uses a rate- and state-dependent friction law (see, e.g., Di-

eterich, 1994; Perrin et al., 1995; Rice and Ruina, 1983;

Ruina, 1983). Though it tries to accommodate both slip and slip rate dependences, the qualitative behaviour of the solution is very close to the slip rate friction model (Favreau et al., 1999b).

The physical model of slip-dependent friction was in- troduced by Rabinowicz (1951) in the geophysical con- text of earthquakes’ modelling to explain the stick-slip phenomenon. Generally speaking, the dependence of the friction forces upon the surface displacements is usually accepted when the slip is very small on laboratory scales (see, e.g., Ohnaka et al., 1987; Scholz, 1990). Ohnaka et al. (1987) pointed out the good agreement of this model with experimental data. More recently, the slip weaken- ing model (i.e. the decrease of the friction force with slip) was intensively used in the description of earthquake ini- tiation (Campillo and Ionescu, 1997; Dascalu et al., 2000;

Favreau et al., 1999a; Ionescu and Campillo; 1999). In- deed, since the model is rate independent, it can describe a large variation of the slip rate during the initiation phase.

The first mathematical results for the slip weakening model of friction in elastostatics were obtained by Ionescu and Paumier (1996). They proved the existence of a so- lution and gave sufficient conditions for uniqueness and stability. Moreover, they analyzed the bifurcation points between different branches of solutions. More recently, the quasi-static evolution of an elastic body with slip- dependent friction was studied in Corneschi et al. (2001).

An existence result for a sufficiently small friction coeffi-

cient was proved. As far as we know, there is no existence and uniqueness result in dynamic elasticity involving slip- dependent friction.

The aim of this paper is to study the dynamic evolu- tion of a viscoelastic body which is in frictional contact with a rigid foundation. The assumptions on the functions used in modelling the contact are general enough to in- clude both the normal compliance and the Tresca models.

For a constant normal stress (displacement) the friction force may exhibit a slip weakening behaviour. The main result is the existence and uniqueness of the solution in the general three-dimensional case. The proof, based on the Galerkin method, is constructive.

2. Problem Statement

Let Ω ⊂ R

^d

(d = 2, 3) be a bounded domain, repre- senting the interior of a viscoelastic body, with a smooth boundary Γ = ∂Ω divided into three disjoint parts Γ = Γ ¯

_d

∪ ¯ Γ

_c

∪ ¯ Γ

_f

with meas (Γ

d

) > 0. The mechanical problem (MP) consists in finding the displacement field u : [0, T ] × Ω −→ R

^d

such that

σ(t) = A u(t)
+ ηC ˙u(t)
in Ω, (1) ρ¨ u(t) = div σ(t) + r(t) in Ω (2)

u(t) = 0 on Γ

d

, (3)

σ(t)n = F (t) on Γ

f

, (4)

σ

N

(t) = −m

N

(u

⁺_N

(t)) on Γ

c

(5) σ

T

(t) = −m

T

u

⁺_N

(t)
µ |u

T

(t)|
u ˙

T

(t)

| ˙u

_T

(t)|

if ˙ u

T

(t) 6= 0 on Γ

c

,

(6)

 σ

_T

(t) 

 ≤ m

_T

(u

⁺_N

(t))µ |u

_T

(t)|
if ˙ u

T

(t) = 0 on Γ

c

,

(7)

u(0) = u

0

, in Ω, (8)

˙

u(0) = u

₁

in Ω, (9)

where η > 0 is a viscosity coefficient, ρ > 0 is the den- sity, A, C are fourth-order tensors, σ is the stress tensor,

(u) = (1/2)(∇u + ∇

^T

u) is the small strain tensor, n is the unit outward normal vector on Γ, σ

_N

= σn · n is the normal stress, σ

_T

= σn − σ

_N

n is the tangential stress, u

N

= u · n is the normal displacement, u

⁺_N

is its positive part, and u

T

= u − u

N

n is the tangential displacement.

Here r represents given body forces and F is the load on Γ

f

.

Equations (5)–(7) represent the contact with slip- dependent friction along a potential surface Γ

c

with a

rigid and fixed body. If there exists a normal gap g

gap

between the viscoelastic body and the foundation, mea- sured in the undeformed configuration, then u

⁺_N

has to be replaced by (u

N

− g

gap

)

⁺

.

In (5) the normal stress σ

N

is a function of the pen- etration u

⁺_N

. Two cases are often used in the literature.

In the first one, the Tresca model, the normal stress is given, i.e. m

_N

(s) = m

_T

(s) = S

_N

, hence the contact surface is known. The second one is the normal compli- ance model often characterized by a power-law relation- ship, i.e. m

N

(s) = |s|

^hn

, m

T

(s) = |s|

^hT

.

Equations (6) and (7) assert that if there is con- tact, the tangential (friction) stress is bounded by a func- tion of the penetration u

⁺_N

multiplied by the value of the ‘friction coefficient’ µ(|u

T

(t)|). If such a limit is not attained, sliding does not occur. Otherwise the fric- tion stress is opposite to the slip rate and its absolute value depends on the slip. As a matter of fact, if we put m

T

(s) = R(m

_N

(s))m

_N

(s), then we get ν =

|σ

_T

|/|σ

_N

| = R(|σ

_N

|)µ(|u

_T

|), which corresponds to a generalization of Coulomb’s friction law. Indeed, in this case the coefficient of friction ν is no more constant, and it accommodates the dependence on the normal stress and on the slip.

Since µ is a function of u

T

, the friction model con- sidered here is slip dependent. Indeed, for a constant nor- mal stress (displacement) the friction force may have a slip-weakening behaviour. The physical model of slip- dependent friction was introduced in the geophysical con- text of earthquake modelling. In this context it is usual to suppose that the slip rate ˙ u

_T

(on the fault) has a single direction and a single sense during the slip, i.e. there ex- ists a tangential vector T and a scalar ˙ U , with ˙ U ≥ 0 (or U ≤ 0) such that ˙u ˙

T

= ˙ U T . Even in this case, only the sequence ‘stick-slip-stick’ (i.e. ˙ U = 0; ˙ U > 0; ˙ U = 0) has to be considered. Indeed, without an explicit load- ing/unloading criterion, the slip-dependent friction model (in the form used here) is more related to a surface po- tential than to a friction law, except for local monotonic loading.

3. Assumptions, Notation and Preliminaries

In the study of the problem (1)–(9) the following assump- tions are used: A and C are symmetric and positive- definite fourth-order tensors, i.e.

A

ijkl

, C

ijkl

∈L

^∞

(Ω), A(x) : σ = A(x)σ : , C(x) : σ = C(x)σ : , (10)

∃ α > 0 such that

A(x) :  ≥ α||

²

, C(x) :  ≥ α||

²

, (11)

a.e. x ∈ Ω, ∀ i, j, k, l = 1, d and for all σ,  ∈ R

^d×dS

.

Let us suppose that the friction coefficient µ: Γ

c

× R

+

−→ R

+

is differentiable with respect to the second variable, and there exist M

1

, µ

0

> 0 such that

0 ≤ µ(x, u) ≤ µ

0

a.e. x ∈ Γ

f

, ∀ u ∈ R

+

, (12)  ∂

u

µ(x, u) 

 ≤ M

1

, ∀ u ∈ [0, +∞[ a.e. x ∈ Γ

c

, (13) and the functions x → µ(x, u) and x → ∂

u

µ(x, u) are measurable for all u ∈ R

+

. As for the functions m

n

and m

_T

, we suppose that

m

_N

(x, u) ≥ m

_N

(x, 0), ∀ u ∈ R

+

, a.e. x ∈ Γ

_c

, (14) x → m

_i

(x, u) is measurable for all u ∈ R

+

, u → m

_i

(x, u) is differentiable, and there exist C

i

, D

_i

, E

_i

≥ 0 and p

_i

≥ 1 such that

 m

_i

(x, u) 

 ≤ C

_i

+ D

_i

|u|

^pi

, (15)  m

i

(x, u

1

) − m

i

(x, u

2

) 



≤E

_i

1 + |u

₁

|

^pi−1

+ |u

₂

|

^pi−1

|u

1

− u

₂

|, (16) a.e. x ∈ Γ

_c

, and for all u, u

₁

, u

₂

∈ R

+

, with i = N or i = T . Set q

N

= P

N

+ 1, q

T

= q

T

+ 1, q = max {q

N

, q

T

} and suppose that

q < 3 if d = 3. (17) We also suppose that the density ρ ∈ L

^∞

(Ω) is posi- tive, i.e. there exists ρ

0

such that ρ(x) ≥ ρ

0

> 0. Finally, the load F and the body forces r are assumed to satisfy

F ∈ W

^1,2

0, T, [L

²

(Γ

f

)]

^N

, (18) r ∈ W

^1,2

0, T, [L

²

(Ω)]

^N

. (19) Set H := [L

²

(Ω)]

^d

endowed with the inner product

(u, v) :=

Z

Ω

ρu · v dx, ∀ u, v ∈ H,

which generates an equivalent norm denoted by | · |. De- note by | · |

q,Γc

the norm in L

^q

(Γ

_c

) and by k · k the norm in [H

¹

(Ω)]

^d

. Let V

0

be the closed subspace of [H

¹

(Ω)]

^N

given by

V

0

:= v ∈ [H

¹

(Ω)]

^d

; v = 0 on Γ

d

, and suppose that

u

0

, u

1

∈ V

0

. (20) If we denote by a, c: V

0

× V

0

→ R the following bilinear and symmetric applications:

a(u, v) :=

Z

Ω

A(u) : (v),

c(u, v) :=

Z

Ω

C(u) : (v), ∀ u, v ∈ V

0

,

then from (10) we find M > 0 such that

|a(u, v)| ≤ M kuk kvk,

|c(u, v)| ≤ M kuk kvk, ∀ u, v ∈ V.

(21)

From (11) and the Korn inequality, we deduce that there exists D > 0 such that

a(v, v) ≥ Dkvk

²

, c(v, v) ≥ Dkvk

²

, ∀ v ∈ V. (22) Finally, we define M : V

₀

→ V

₀⁰

, j: V

₀

× V

₀

× V

₀

→ R and f : V

₀

→ V

₀⁰

as follows:

M (w), v = Z

Γ_c

m

_N

s, [w

_N

]

⁺

v

_N

ds, v ∈ V

0

, (23)

j(u, v, w) = Z

Γ_c

m

_T

s, [u

_N

]

⁺

µ s, |u

_T

|
|w

_T

| ds, u, v, w ∈ V

0

, (24) hf (t), vi = (r(t), v) +

Z

Γ_f

F (t) · v ds, v ∈ V

0

. (25)

Using this notation, one can easily deduce that any solution of (1)–(8) satisfies the following variational problem:

(VP) Find u : [0, T ] −→ V

0

such that ¨ u(t), v − ˙ u(t) + a u(t), v − ˙u(t)

+ηc ˙ u(t), v − ˙ u(t)
+ M (u(t)), v − ˙u(t) +j u(t), u(t), v
− j u(t), u(t), ˙u(t)

≥ f (t), v − ˙u(t), (26)

u(0, x) = u

₀

(x), u(0, x) = u ˙

₁

(x). (27)

4. Existence and Uniqueness of the Solution

The main result of this section is the following:

Theorem 1. There exists a unique solution of (VP) with the following regularity:

u ∈ W

^1,∞

(0, T, V ) ∩ W

^2,2

(0, T, H). (28)

We recall here from (Ionescu, 2001) the following lemma, which will be useful in the proof of the theorem:

Lemma 1. Let Ω ⊂ R

^d

be as above and let α ∈ [2, 2(d−

1)/(d − 2)] if d ≥ 3 and α ≥ 2 if d = 2. Then, for

β = [d(α − 2) + 2]/2α if d ≥ 3 or if d = 2 and α = 2,

and for all β ∈ ](α − 1)/α, 1[ if d = 2 and α > 2, there exists a constant C = C(β) such that

kvk

_Lα(Γ)

≤ Ckvk

^1−β_L2(Ω)

kvk

^β_H1(Ω)

, ∀ v ∈ H

¹

(Ω). (29)

Proof of Theorem 1. (Uniqueness) Let u

1

and u

2

be two solutions of (26)–(27) with regularity (28) and write w =: u

1

− u

2

. If we write the variational inequality (VP) successively for u

1

and u

2

taking v = ˙ u

2

(t) in the first inequality and v = ˙ u

1

(t) in the second one, and add the resulting inequalities, we obtain

¨ w(t), ˙ w(t) + a ˙ w(t), w(t)
+ ηc ˙ w(t), ˙ w(t)
+ M (u

₁

(t)) − M (u

₂

(t)), ˙ w(t)  +

Z

Γ_c

m

_T

[u

_1N

(t)]

⁺

µ |u

_1T

(t)|

− m

_N

[u

_2N

(t)]

⁺

µ |u

_2T

(t)|


× | ˙u

_1T

(t)| − | ˙ u

_2T

(t)| ds ≤ 0.

Since the integrand of the last integral can be majorized by  m

_T

([u

_1N

(t)]

⁺

) − m

_T

([u

_2N

(t)]

⁺

) 

µ(|u

_1T

(t)|) + m

_T

([u

_2T

(t)]

⁺

) 

 µ(|u

_1T

(t)|) − µ(|u

_2T

(t)|)  , we deduce that

1 2

d

dt | ˙ w(t)|

²

+ a w(t), w(t)
 + ηc ˙ w(t), ˙ w(t)

≤E

_N

Z

Γ_c

|u

_1N

(t)|

^pn−1

+|u

_2N

(t)|

^pn−1

|w

_N

(t)| | ˙ w

_N

(t)| ds

+µ

_o

E

_T

Z

Γ_c

|u

_1N

(t)|

^pT−1

+|u

_2N

(t)|

^pT−1

×|w

_N

(t)| | ˙ w

_T

(t)| ds + Z

Γ_c

(C

_T

+ D

_T

|u

_1N

(t)|

^pT

)

×|w

_T

(t)| | ˙ w

_T

(t)| ds. (30) The first two integrals on the right-hand side of (30) can be majorized using the Hölder inequality for (q/(q − 2); q; q) as follows:

Z

Γ_c

|u

_1N

(t)|

^pn−1

|w

_N

(t)| | ˙ w

_N

(t)| ds

≤|u

₁

(t)|

^q−2

q,Γc

|w(t)|

q,Γc

| ˙ w(t)|

q,Γc

≤C

1

ku

₁

k

^q−2

L∞ (0,T ;V0)

kw(t)k k ˙ w(t)k, which implies

Z

Γ_c

|u

_1N

(t)|

^pn−1

|w

_N

(t)| | ˙ w

_N

(t)| ds

≤ C

2

η kw(t)k

²

+ η

6 k ˙ w(t)k

²

.

In order to estimate the third integral, we use C

_T

Z

Γ_c

|w

_T

(t)| | ˙ w

_T

(t)| ds ≤ C

3

η kw(t)k

²

+ η

6 k ˙ w(t)k

²

, and the Hölder inequality for ((q + 1)/(q − 1); q + 1;

q + 1) Z

Γ_c

|u

_1N

(t)|

^pT

|w

_T

(t)| | ˙ w

_T

(t)| ds

≤|u

₁

(t)|

^q−1

q+1,Γc

|w(t)|

q+1,Γc

| ˙ w(t)|

q+1,Γc

≤C

4

ku

₁

k

^q−1

L∞ (0,T ;V0)

kw(t)k k ˙ w(t)k to obtain

D

_T

Z

Γ_c

|u

_1N

(t)|

^pT

|w

_T

(t)| | ˙ w

_T

(t)| ds

≤ C

₅

η kw(t)k

²

+ η

6 k ˙ w(t)k

²

. From the above inequalities and (30), we get 1

2 d

dt | ˙ w(t)|

²

+a(w(t), w(t))] + ηc( ˙ w(t), ˙ w(t))

≤ C

6

kw(t)k

²

+ ηk ˙ w(t)k

²

. If we integrate this inequality from 0 to t, and use the coercivity of the bilinear applications a(·, ·) et c(·, ·) and the initial conditions w(0) = ˙ w(0) = 0, then

|w(t)|

²

+kw(t)k

²

≤ C

6

Z

t 0

|w(τ )|

²

+kw(τ )k

²

dτ (31) By using the Gronwall lemma in (31), the uniqueness fol- lows.

(Existence) In order to prove the existence of the solu- tion u to (VP), we shall use the Faedo-Galerkin method.

For this let us consider φ

i

∈ V as a sequence of linearly independent functions such that V = S

∞

m=1

V

m

, where V

m

= Span {φ

1

, φ

2

, . . . , φ

m

}. Since u

0

, v

0

∈ V , let u

^m₀

, v

₀^m

∈ V

m

be such that

u

^m₀

−→ u

0

, u

^m₁

−→ u

1

strongly in V. (32) If we consider the family of convex and differentiable functions Ψ

_ε

: R

^d

→ R given by

Ψ

_ε

(v) = p|v|

²

+ ε

²

− ε, v ∈ R

^d

for all positive ε, then we have

0 ≤ Ψ

ε

(v) ≤ |v|, ∀ v ∈ R

^d

, (33)

|Ψ

⁰_ε

(v)(w)| ≤ |w|, ∀ (v, w) ∈ R

^d

× R

^d

, (34)

|Ψ

ε

(v) − |v|| ≤ ε, ∀ v ∈ R

^d

. (35)

Next we define j

_ε

: V

0

× V

0

× V

0

→ R, a family of regularized frictional functionals depending on  > 0,

j

_ε

(u, v, w) = Z

Γ_c

m

_T

(s, [u

_N

]

⁺

)µ(s, |v

_T

|)Ψ

ε

(w

_T

) ds,

∀u, v, w ∈ V

0

. The functional j

_ε

is Gâteaux-differentiable with respect to the third argument and represents an approximation of j, i.e. there exists a constant C such that

|j

_ε

(u, v, w) −j(u, v, w)|

≤ Cε 1 + kuk

^q−1

, ∀ u, v, w ∈ V

0

. (36) We denote by J

_ε

: V

0

× V

0

× V

0

→ V

₀⁰

the derivative of j

_ε

with respect to the third variable given by

hJ_ε(u, v, w), zi

= Z

Γ_c

m_T(s, [u_N]⁺)µ(s, |v_T|)Ψ⁰_ε(w_T)(z_T) ds, u, v, w, z ∈ V0.

We can introduce now the following variational prob- lem with regularized friction in the finite-dimensional space V

m

:

(VP

^m_

) : Find u

^m_

: [0, T ] −→ V

m

such that

h¨ u

^m_ε

(t), vi + a(u

^m_ε

(t), v) + ηc( ˙ u

^m_ε

(t), v) +hM (u

^m

ε

(t), vi

+hJ

_ε

(u

^m_ε

(t), u

^m_ε

(t), ˙ u

^m_ε

(t)), vi

= hf (t), vi, (37)

u

^m

ε

(0) = u

^m₀

, u ˙

^m

ε

(0) = u

^m₁

. (38) Since (u; v) → J

_ε

(u, v, v) is a locally Lips- chitz continuous function on V

m

× V

m

, we deduce that (37)–(38) has a unique maximal solution u

^m_ε

∈ C

²

([0, T

_ε^m

]; V

_m

).

The continuation of the proof is divided into three parts. We begin by proving that each problem has a unique solution u

^m_

for all  > 0 and all m ∈ N. To do this, we need some a priori estimates, which will be deduced in the first two parts of the proof. Only after that shall we prove that when  → 0 and m −→ +∞, the limit of u

^m_

, in an appropriate sense, is the solution to (VP).

In order to simplify the notation, we shall omit the indices ε and m in the first two parts of the proof.

(i) A priori estimates I

Since hJ

_ε

(u, v, w), wi ≥ 0 for all u, v, w ∈ V

0

, setting v = ˙ u(t) in (37) we obtain

d dt

 1

2 | ˙u(t)|

²

+ 1

2 a(u(t), u(t))



+ η c( ˙ u(t), ˙ u(t)) + hM (u(t)), ˙ u(t)i ≤ hf (t), ˙ u(t)i. (39)

Let us introduce the following notation:

˜

m

_N

(s, u) = m

_N

(s, u) − m

_N

(s, 0), ∀ u ∈ R

+

, P (s, u) =

Z

u 0

˜

m

_N

(s, v) dv,

ˆ m(u) =

Z

Γ_c

P (s, u(s)) ds, ∀ u ∈ L

²

(Γ

_c

)

a.e. s ∈ Γ

_c

. From (14) we find that the energy associated with the normal compliance ˆ m([u

_N

]

⁺

) is positive, i.e.

ˆ

m([u

_N

(t)]

⁺

) ≥ 0, ∀ u ∈ L

²

(0, T ; V

₀

).

For all v ∈ L

²

(0, T, V

₀

), we have hM (v(t)), ˙v(t)i −

Z

Γ_c

m

_N

(s, 0) ˙v

_N

(t) dx

= Z

Γ_c

˜

m

_N

(s, [v

_N

(t)]

⁺

) ˙v

_N

(t) dx,

and after differentiation of the associated energy ˆ m(v), we get

d

dt m([v ˆ

_N

(t)]

⁺

) = Z

Γ_c

d

dt P (s, [v

_N

(t)]

⁺

) dx

= Z

Γ_c

˜

m

_N

(s, [v

_N

(t)]

⁺

)H(v

_N

(t)) ˙v

_N

(t) dx,

where H(x) is the Heaviside function. Since ˜ m

_N

(s, 0)

= 0, we obtain

˜

m

_N

([v

_N

(t)]

⁺

)H(v

_N

(t)) ˙v

_N

(t) = ˜ m

_N

([v

_N

(t)]

⁺

) ˙v

_N

(t), and the following equality follows:

hM (v(t)), ˙v(t)i

= d

dt m([v ˆ

_N

(t)]

⁺

) + Z

Γ_c

m

_N

(s, 0) ˙v

_N

(t, s) ds, v ∈ L

²

(0, T ; V

₀

).

Bearing in mind that ˆ m([v

_N

(t)]

⁺

) ≥ 0, we integrate this equation to deduce that

Z

t 0

hM (v(τ )), ˙v(τ )i dτ

≥ Z

Γ_c

m

_N

(s, 0)v

_N

(t, s) ds

− Z

Γ_c

m

_N

(s, 0)v

_N

(0, s) ds − ˆ m(v

_N

(0)),

for all v ∈ L

²

(0, T ; V

0

). If we integrate (39) over (0, t) and use the last inequality, then we obtain

| ˙u(t)|

²

+Dku(t)k

²

+ 2η Z

t

0

k ˙u(τ )k

²

dτ

≤ |u

₁

|

²

+ a(u

₀

, u

₀

) + ˆ m([u

_0N

]

⁺

) +2

Z

Γ_c

m

_N

(s, 0)u

_0N

ds − 2 Z

Γ_c

m

_N

(s, 0)u

_N

(t) ds +2hf (t), u(t)i − 2hf (0), u

₀

i

−2 Z

t

0

h ˙ f (τ ), u(τ )i dτ. (40) In order to estimate the normal displacement, we have

   Z

Γ_c

m

_N

(s, 0)u

_N

(t) ds  

 ≤ C

_T

mes(Γ

_c

)|u(t)|

2,Γc

≤ C D + D

8 ku(t)k

²

. By using the last inequality in (40), we deduce that

| ˙u(t)|

²

+ku(t)k

²

+ η Z

t

0

k ˙u(τ )k

²

dτ

≤ C + C Z

t

0

| ˙u(τ )|

²

+ ku(τ )k

²

dτ. (41) From the Gronwall lemma we obtain that the solution t → (u

^m

ε

(t); ˙ u

^m

ε

(t)) of (37)–(38) is bounded on its in- terval of existence, and hence t → (u

^m_ε

(t); ˙ u

^m_ε

(t)) is a global solution, i.e. T

_ε^m

= T . Moreover, we have

u

^m_ε

m,ε

is bounded in L

^∞

(0, T ; V

0

), (42)

 ˙u

^m_ε

m,ε

is bounded in L

^∞

(0, T ; H)∩L

²

(0, T ; V

0

). (43) (ii) A priori estimates II

If we let v = ¨ u(t) in (37) and notice that hJ

_ε

(u(t), u(t), ˙ u(t)), ¨ u(t)i

= Z

Γ_c

m

_T

([u

_N

(t)]

⁺

)µ(|u

_T

(t)|)Ψ

⁰_ε

( ˙ u

_T

(t))(¨ u

_T

(t)) ds

= Z

Γ_c

m

_T

([u

_N

(t)]

⁺

)µ(|u

_T

(t)|) d

dt {Ψ

ε

( ˙ u

_T

(t))} ds, then we get

|¨ u(t)|

²

+a(u(t), ¨ u(t)) + hM (u(t)), ¨ u(t)i + η

2 d

dt {c( ˙u(t), ˙u(t))}

+ Z

Γ_c

m

_T

([u

_N

(t)]

⁺

)µ(|u

_T

(t)|) d

dt {Ψ

ε

( ˙ u

_T

(t))} ds

= hf (t), ¨ u(t)i.

After integration from 0 to t, we obtain

Z t

0

|¨u(τ )|²dτ +η 2k ˙u(t)k²

≤ C + Z t

0

a( ˙u(τ ), ˙u(τ )) dτ − a(u(t), ˙u(t))

− Z t

0

hM (u(t)), ¨u(t)i+hf (t), ˙u(t)i−

Z t 0

h ˙f (τ ), ˙u(τ )i dτ

− Z t

0

Z

Γ_c

m_T([u_N(τ )]⁺)µ(|u_T(τ )|) d

dτ{Ψε( ˙u_T(τ )} ds dτ.

(44) The virtual power of the normal displacement can be written as

Z

t 0

hM (u(t)), ¨ u(t)i

= Z

t

0

Z

Γ_c

m

_N

(s, [u

_N

(τ )]

⁺

)¨ u

_N

(τ ) ds dτ

= hM (u(t)), ˙ u(t)i − hM (u

₀

), u

₁

i

− Z

t

0

Z

Γ_c

∂m

_N

∂u ([u

_N

(τ )]

⁺

)H(u

_N

(τ ))( ˙ u

_N

(τ ))

²

ds dτ.

Using the above inequalities and

|a(u(τ ), ˙u(τ ))| ≤ Cku(τ )k k ˙u(τ )k

≤ 2 C

²

η ku(τ )k

²

+ η

8 k ˙u(τ )k

²

,

|hM (u(t)), ˙u(t)i| ≤ Z

Γ_c

(C

_N

+ D

_N

|u(t)|

^pn

)| ˙ u(t)| ds

≤ C(1 + ku(t)k

^q−1

)

η + η

8 k ˙u(t)k

²

, we deduce from (44) that

ηk ˙ u(t)k

²

+ Z

t

0

|¨ u(τ )|

²

dτ ≤ C + C Z

t

0

k ˙u(τ )k

²

dτ

+   

Z

t 0

Z

Γ_c

|u

_N

(τ )|

^pn−1

u ˙

²_N

(τ ) ds dτ   

+   

Z

t 0

Z

Γ_c

m

_T

([u

_N

(τ )]

⁺

)µ(|u

_T

(τ )|) d dτ

× {Ψ

ε

( ˙ u

_T

(τ ))} ds dτ  

 . (45)

The second integral on the right-hand side of (45) can be estimated as follows:

Z

Γ_c

|u

_N

(τ )|

^pn−1

u ˙

²_N

(τ ) ds

≤ |u(τ )|

^q−2

q,Γc

| ˙u(τ )|

²

q,Γc

≤ ku(τ )k

^q−2

k ˙u(τ )k

²

≤ Ck ˙u(τ )k

²

.

In order to estimate the last integral of (45), we re- place the non-differential term |u

_T

(τ )| with Ψ

r

(u

_T

(τ )) (r > 0) to get

Z

t 0

Z

Γ_c

m

_T

([u

_N

(τ )]

⁺

)µ(|u

_T

(τ )|) d dτ

× {Ψ

ε

( ˙ u

_T

(τ ))} ds dτ

= Z

t

0

Z

Γ_c

m

_T

([u

_N

(τ )]

⁺

)µ(Ψ

_r

(u

_T

(τ ))) d dτ

× {Ψ

_ε

( ˙ u

_T

(τ ))} ds dτ +

Z

t 0

Z

Γ_c

m

_T

([u

_N

(τ )]

⁺

)µ(|u

_T

(τ )|)

− µ(Ψ

r

(u

_T

(τ )))  d

dτ {Ψ

ε

( ˙ u

_T

(τ ))} ds dτ, and, after integration by parts, we have

Z

t 0

Z

Γ_c

m

_T

([u

_N

(τ )]

⁺

)µ(Ψ

_r

(u

_T

(τ ))) d

dτ {Ψ

_ε

( ˙ u

_T

(τ ))} ds dτ

= Z

Γ_c

m

_T

([u

_N

(t)]

⁺

)µ(Ψ

_r

(u

_T

(t)))Ψ

_ε

( ˙ u

_T

(t)) ds

− Z

Γ_c

m

_T

([u

_0N

]

⁺

)µ(Ψ

_r

(u

_0T

))Ψ

_ε

(u

_1T

) ds

− Z

t

0

Z

Γ_c

 ∂m

_T

∂u ([u

_N

(τ )]

⁺

)H(u

_N

(τ )) ˙ u

_N

(τ )

× µ(Ψ

r

(u

_T

(τ )))Ψ

ε

( ˙ u

_T

(τ )) + m

_T

([u

_N

(τ )]

⁺

) ∂µ

∂u

× (Ψ

r

(u

_T

(τ )))Ψ

⁰_r

(u

_T

(τ ))( ˙ u

_T

(τ ))Ψ

ε

( ˙ u

_T

)] ds dτ.

We now use the estimates: (12) for µ, (33) for Ψ

_ε

, (34) for Ψ

⁰_r

, and (15) for m

_T

, and obtain

  

Z

t 0

Z

Γ_c

m

_T

([u

_N

(τ )]

⁺

)µ(Ψ

_r

(u

_T

(τ ))) d dτ

× {Ψ

_ε

( ˙ u

_T

(τ ))} ds dτ   

≤ C + +C Z

Γ_c

(C

_T

+ D

_T

|u(t)|

^pT

)| ˙ u

_T

(t)| ds

+ C Z

t

0

Z

Γ_c

|u(τ )|

^pT−1

| ˙u

_T

(τ )|

²

ds dτ

+ C Z

t

0

Z

Γ_c

(C

_T

+ D

_T

|u(τ )|

^pT

)| ˙ u

_T

(τ )|

²

ds dτ.

Using (35) to estimate the difference between Ψ

r

(u

_T

(τ )) and | ˙u

_T

(τ )|, we have

  

Z

t 0

Z

Γ_c

m

_T

([u

_N

(τ )]

⁺

)[µ(|u

_T

(τ )|) − µ(Ψ

_r

(u

_T

(τ )))] d dτ {Ψ

ε

( ˙ u

_T

(τ ))} ds dτ

 

 ≤ C(ε, m

_T

, µ)r,

where C(ε, m

_T

, µ) is a constant independent of r. We pass to the limit r → 0 to obtain

  

Z

t 0

Z

Γ_c

m

_T

([u

_N

(τ )]

⁺

)µ(|u

_T

(τ )|) d

dτ {Ψ

ε

( ˙ u

_T

(τ ))} ds dτ   

≤ C + C Z

Γ_c

(C

_T

+ D

_T

|u(t)|

^pT

)| ˙ u

_T

(t)| ds

+C Z

t

0

Z

Γ_c

|u(τ )|

^pT−1

| ˙u

_T

(τ )|

²

ds dτ

+C Z

t

0

Z

Γ_c

(C

_T

+ D

_T

|u(τ )|

^pT

)| ˙ u

_T

(τ )|

²

ds dτ. (46)

If we use the Hölder inequality in the second part of (46), then the following estimates are obtained:

Z

Γ_c

|u(t)|

^pT

| ˙u

_T

(t)| ds ≤ |u(t)|

^q−1

q,Γc

| ˙u

_T

(t)|

q,Γc

≤ ku(t)k

^q−1

k ˙u(t)k, Z

Γ_c

|u(τ )|

^pn−1

| ˙u(τ )|

²

ds ≤ |u(τ )|

^q−2

q,Γc

| ˙u(τ )|

²

q,Γc

≤ ku(τ )k

^q−2

k ˙u(τ )k

²

, Z

Γ_c

|u(τ )|

^pT

| ˙u

_T

(τ )|

²

ds ≤ |u(τ )|

^q−1

q,Γc

| ˙u

_T

(τ )|

²

q,Γc

≤ ku(τ )k

^q−1

k ˙u(τ )k

²

. Since the functions u and u are bounded in ˙ L

^∞

(0, T ; V ) and L

²

(0, T ; V )), respectively, from the above estimates and (45) we deduce that

k ˙u(t)k

²

+ Z

t

0

|¨ u(τ )|

²

dτ ≤ C + C Z

t

0

k ˙u(τ )k

²

dτ. (47) Using the Gronwall lemma, we conclude that

 ˙u

^m_ε

m,ε

is bounded in L

^∞

(0, T ; V

0

), (48)

 ¨ u

^m_ε

m,ε

is bounded in L

²

(0, T ; H). (49) (iii) Passage to the limit in m and ε

From (42), (43), (48), (49) we deduce that there exists a subsequence of u

^m_ε

m,ε

(again denoted by {u

^m_ε

}

_m,ε

), such that

u

^m

ε

* u weak* in L

∗ ^∞

(0, T ; V

₀

), (50)

˙

u

^m_ε

* ˙

^∗

u weak* in L

^∞

(0, T ; V

0

), (51)

¨

u

^m_ε

* ¨

^∗

u weak* in L

^∞

(0, T ; H), (52)

as ε → 0 and m → +∞. If we write Ω

_T

= Ω × ]0, T [, then

u

^m_ε

m,ε

,  ˙u

^m_ε

m,ε

are bounded in H

¹

(Ω

T

).

Since the embedding of H

¹

(Ω

T

) in L

²

(Ω

T

) = L

²

(0, T ; H) is compact, we find that there exists a sub- sequence of {u

^m_ε

} (again denoted by {u

^m_ε

}), such that

u

^m_ε

→ u strongly in L

²

(0, T ; H), (53)

˙ u

^m

ε

→ ˙u strongly in L

²

(0, T ; H). (54) Moreover, since the trace map from H

¹

(Ω

T

) to L

²

(∂Ω

T

) is a compact operator and ∂Ω

T

=

∂Ω×]0, T [S Ω × {0} S Ω × {T }, we deduce that u

^m

ε

(T ) → u(T ) strongly in H, (55)

˙

u

^m_ε

(T ) → ˙ u(T ) strongly in H, (56) and from (50)–(52) we have

u

^m_ε

(T ) * u(T ) weakly in V.

We only have to verify that u is the solution of (26) and (27). Let w ∈ L

²

(0, T ; V

0

) be fixed and let w

^m

∈ L

²

(0, T ; V

m

) be a sequence such that

w

^m

→ w strongly in L

²

(0, T ; V

0

).

If we let v = w

^m

(t) − ˙ u

^m_ε

(t) in (37) and use the inequality

j

_ε

(u, v, w) − j

_ε

(u, v, z) ≥ hJ

_ε

(u, v, z), w − zi,

∀ u, v, w, z ∈ V

₀

, after integration of (37) from 0 to T , we have Z

T

0

{(¨ u

^m

ε

(t), w

^m

(t) − ˙ u

^m

ε

(t))+a(u

^m

ε

(t), w

^m

(t) − ˙ u

^m

ε

(t)) + ηc( ˙ u

^m_ε

(t), w

^m

(t) − ˙ u

^m_ε

(t))

+ hM (u

^m

ε

(t)), w

^m

(t) − ˙ u

^m

ε

(t)i} dt +

Z

T 0

j

_ε

(u

^m_ε

(t), u

^m_ε

(t), w

^m

(t))

− j

_ε

(u

^m

ε

(t), u

^m

ε

(t), ˙ u

^m

ε

(t))dt

≥ Z

T

0

hf (t), w

^m

(t) − ˙ u

^m_ε

(t)i dt.

If we use the estimate (36), after some algebra we obtain

Cε(1 + ku

^m

ε

k

L2 (0,T ;V )

) + Z

T

0

(¨ u

^m

ε

(t), w

^m

) dt + 1

2 |u

₁

|

²

− 1

2 | ˙u

^m_ε

(T )|

²

+ Z

T

0

a(u

^m_ε

(t), w

^m

) dt

+a(u

₀

, u

₀

) + η Z

T

0

c( ˙ u

^m_ε

(t), w

^m

) dt

+ Z

T

0

 j(u

^m_ε

(t), u

^m

ε

(t), w

^m

)

−j(u

^m_ε

(t), u

^m_ε

(t), ˙ u

^m_ε

(t)) dt

+ Z

T

0

hM (u

^m

ε

(t)), w

^m

i dt + ˆ m([u

^m_εN

(T )]

⁺

) − ˆ m([u

_0N

]

⁺

)

− Z

Γ_c

m

_N

(s, 0) u

^m_εN

(T ) − u

_0N

 ds

≥ a(u

^m

ε

(T ), u

^m

ε

(T )) + η Z

T

0

c( ˙ u

^m

ε

(t), ˙ u

^m

ε

(t)) dt +

Z

T 0

hf (t), w

^m

(t) − ˙ u

^m_ε

(t)i dt. (57)

Now let us verify the convergence of the terms of the left-hand side of (57). First, we prove that for m → +∞

and ε → 0 we have Z

T

0

j(u

^m

ε

(t), u

^m

ε

(t), ˙ u

^m

ε

(t)) dt

→ Z

T

0

j(u(t), u(t), ˙ u(t)) dt. (58) Indeed, after some algebra, we get

|j(u

^m_ε

(t), u

^m_ε

(t), ˙ u

^m_ε

(t)) − j(u(t), u(t), ˙ u(t))|

≤ C h

|u

^m_ε

(t)|

^q−2

q,Γc

+ |u(t)|

^q−2

q,Γc

 |u

^m_ε

(t) − u(t)|

q,Γc

× | ˙u

^m

ε

(t)|

q,Γc

+ 

1 + |u(t)|

^q−1

q+1,Γc



× |u

^m_ε

(t) − u(t)|

q+1,Γc

| ˙u

^m_ε

(t)|

q+1,Γc

+ 

1 + |u(t)|

^q−1

q,Γc

 | ˙u

^m

ε

(t) − ˙ u(t)|

q,Γc

i .

If we use now Lemma 1 for β = 3(q − 1) + 2

2(q + 1) < 1 if d = 3

and for

β ∈ ] q

q + 1 , 1[ if d = 2, we obtain

|u

^m_ε

(t) − u(t)|

q+1,Γc

≤ C|u

^m_ε

(t) − u(t)|

^1−β

× (ku

^m_ε

k

L∞ (0,T ;V0)

+ kuk

L∞ (0,T ;V0)

)

^β

,

| ˙u

^m_ε

(t) − ˙ u(t)|

q+1,Γc

≤ C| ˙u

^m_ε

(t) − ˙ u(t)|

^1−β

× (k ˙u

^m_ε

k

L∞ (0,T ;V0)

+ k ˙ uk

L∞ (0,T ;V0)

)

^β

. From the last three inequalities we deduce that

Z

T 0

 j(u

^m

ε

(t), u

^m

ε

(t), ˙ u

^m

ε

(t)) − j(u(t), u(t), ˙ u(t))   dt

≤ C



ku

^m_ε

− uk

^1−β

L2 (0,T ;H)

+ k ˙ u

^m_ε

− ˙uk

^1−β

L2 (0,T ;H)

 , and, by using (53) as m → +∞ and ε → 0, we obtain (58). In a similar way, we conclude that

Z

T 0

hM (u

^m_ε

(t)), w

^m

(t)i dt → Z

T

0

hM (u(t)), w(t)i dt, and therefore

ˆ m([u

^m

εN

(T )]

⁺

) → ˆ m([u

_N

(T )]

⁺

)

as m → +∞ and ε → 0.

Indeed, we have the estimate 

 m([u ˆ

^m

εN

(T )]

⁺

) − ˆ m([u

_N

(T )]

⁺

)  

≤ Z

Γ_c

 P (s, [u

^m

εN

(T )]

⁺

) − P (s, [u

_N

(T )]

⁺

)   ds

≤ Z

Γ_c

|u

^m_ε

(T )|

^q−1

+ |u(T )|

^q−1

|u

^m_ε

(T ) − u(T )| ds.

Using the Hölder inequality and Lemma 1, we deduce that Z

Γ_c

|u

^m_ε

(T )|

^q−1

|u

^m_ε

(T ) − u(T )| ds

≤ ku

^m_ε

k

^q−1

L∞ (0,T ;V )

|u

^m_ε

(T ) − u(T )|

q,Γc

≤ Cku

^m_ε

k

^q−1

L∞ (0,T ;V )

|u

^m_ε

(T ) − u(T )|

^1−β

× (ku

^m_ε

k

L∞ (0,T ;V )

+ kuk

L∞

(0,T ;V )

)

^β

,

and from (55) we get the strong convergence of the asso- ciated energy.

If we pass to the limit in (57) as m → +∞ and ε → 0, and we bear in mind the strong convergence proved above, we obtain

Z

T 0

(¨ u(t), w(t)) dt + 1

2 |u

₁

|

²

− 1

2 | ˙u(T )|

²

+ a(u

₀

, u

₀

) +

Z

T 0

a(u(t), w(t)) dt + η Z

T

0

c( ˙ u(t), w(t)) dt

+ Z

T

0

[j(u(t), u(t), w(t)) − j(u(t), u(t), ˙ u(t))] dt

+ Z

T

0

hM (u(t)), w(t)i dt + ˆ m([u

_N

(T )]

⁺

)

− ˆ m([u

_0N

]

⁺

) dt − Z

Γ_c

m

_N

(s, 0) [u

_N

(T ) − u

_0N

] ds

≥ lim inf

m→+∞,ε→0



a(u

^m_ε

(T ), u

^m_ε

(T ))

+ η Z

T

0

c( ˙ u

^m_ε

(t), ˙ u

^m_ε

(t)) dt

 +

Z

T 0

hf (t), w(t) − ˙u(t)i dt

≥ a(u(T ), u(T )) + η Z

T

0

c( ˙ u(t), ˙ u(t)) dt

+ Z

T

0

hf (t), w(t) − ˙u(t)i dt.

Finally, for all w ∈ L

²

(0, T ; V

0

) we have Z

T

0

[ (¨ u(t), w(t) − ˙ u(t)) + a(u(t), w(t) − ˙ u(t)) + ηc( ˙ u(t), v − ˙ u(t)) + hM (u(t)), v − ˙ u(t)i + j(u(t), u(t), w(t)) − j(u(t), u(t), ˙ u(t))] dt

≥ Z

T

0

hf (t), w(t) − ˙u(t)i dt, and the pointwise inequality (26) follows.

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