Gen. Math. Notes, Vol. 19, No. 1, November, 2013, pp.60-77
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2013
www.i-csrs.org
Available free online at http://www.geman.in
Analysis of a Dynamic Frictional Contact Problem for
Thermo-Elastic-Viscoplastic Materials with Damage
S. Boutechebak1 and A. Azeb Ahmed2
1
2
Department of Mathematics, University of Setif1, 19000, Algeria
E-mail: sou-maths19@yahoo.fr
Department of Mathematics, University of Eloued, 39000, Algeria
E-mail: aziz-azebahmed@univ-eloued.dz
(Received: 13-6-13 / Accepted: 11-9-13)
Abstract
We consider a dynamic frictional contact problem with normal domped response.The material is elastic-viscoplastic with two internal variables which
may describe the damage of the system caused by plastic deformations and a
temperature parameter. We derive a weak formulation of the system consisting
of a motion equation, an energy equation, and an evolution damage inclusion.
We prove existence and uniqueness of the weak solution, and the positivity of
the temperature. The proof is based on arguments of nonlinear evolution equations with monotone operators, a classical existence and uniqueness result on
parabolic type inequalities, fixed-point arguments and differential equations.
Keywords: Damage field, dynamic process; normal domped response, fixed
point, temperature.
1
Introduction
The constitutive laws with internal variables has been used in various publications in order to model the effect of internal variables in the behavior of real
bodies like metals, rocks polymers and so on, for which the rate of deformation
depends on the internal variables. Some of the internal state variables considered by many authors are the spatial display of dislocation, the work-hardening
of materials, the absolute temperature and the damage field, see for examples
and details [2, 7, 8, 9, 15] and references there in for the case of hardening,
61
Analysis of a Dynamic Frictional Contact Problem for...
temperature and other internal state variables and the references [5, 7, 12, 13]
for the case of damage field. Analysis of models for adhesive contact can be
found in [4, ?], and recently in the monograph [16]. We refer the reader to the
extensive bibliography on the subject in [11, 14].
The novelty in all the above papers is the introduction of an absolute
temperature. Then In this paper we extend a part of the results in [8, 14] to
more general contact conditions for elastic-thermo- viscoplastic materials. In
this paper we deal with the study of a dynamic problem of frictional contact
for general elastic-thermo-viscoplastic materials. Examples and mechanical
interpretation of elastic-viscoplastic can be found in [3].
In this paper, we consider that the material is elastic-viscoplastic with
two internal variables which may describe the damage of the system caused by
plastic deformations and a temperature parameter. The contact is with normal
domped response. Dynamic and quasistatic contact problems are the topic of
numerous papers, e.g. [1]. However, the mathematical problem modelled
the quasi-static evolution of damage in thermo-viscoplastic materials has been
studied in [7]. The paper is organized as follows. In Section 2 we present
the mechanical problem of the dynamic evolution of damage in thermo-elasticviscoplastic materials. We introduce some notations and preliminaries and
we derive the variational formulation of the problem. We prove in Section 3
the existence and uniqueness of the solution as well as the positivity of the
temperature.
2
Notation and Preliminaries
We denote by Sn the space of symmetric tensors on Rn (n = 2, 3). We define
the inner product and the Euclidean norm on Rn and Sn , respectively, by
∀u, v ∈ Rn ,
u · v = ui vi
1/2
|u| = (u · u)
n
∀u ∈ R ,
σ · τ = σij τij
∀σ, τ ∈ Sn ,
1/2
|σ| = (σ · σ)
∀σ ∈ Sn .
Let Ω ⊂ Rn be a bounded domain with a Lipschitz boundary Γ, partitioned
into three disjoint measurable parts Γ1 , Γ2 and Γ3 such that measΓ1 > 0. Here
and below, the indices i and j run from 1 to n and the summation convention
over repeated indices is used. We shall use the notation
H = L2 (Ω)n = {u = {ui } : ui ∈ L2 (Ω)},
H = {σ = {σij } : σij = σji ∈ L2 (Ω)},
H1 = {u ∈ H : ε(u) ∈ H},
H1 = {σ ∈ H : Div(σ) ∈ H},
V = H 1 (Ω).
62
S. Boutechebak et al.
Here ε : H1 → H and Div : H1 → H are the deformation and divergence
operators, respectively, defined by
ε(u) = (εij ( u)),
1
εij (u) = (ui,j + uj,i ),
2
Div(σ) = (σij,j ).
The sets H, H, H1 , H1 and V are real Hilbert spaces endowed with the canonical inner products:
Z
Z
(u, v)H =
ui vi dx, (σ, τ )H =
σij τij dx,
Ω
Ω
(u, v)H1 = (u, v)H + (ε(u), ε(v))H ,
(σ, τ )H1 = (σ, τ )H + (Div(σ), Div(τ ))H ,
(f, g)V = (f, g)L2 (Ω) + (fxi , gxi )L2 (Ω) .
The associated norms are denoted by k · kH , k · kH , k · kH1 , k · kH1 and k · kV .
Let HΓ = (H 1/2 (Γ))n and γ : H1 → HΓ be the trace map. We denote by V the
closed subspace of H1 defined by
V = {v ∈ H1 : γv = 0 on Γ1 }.
On the space V we consider the inner product given by
(u, v)V = (ε(u), ε(v))H ,
and let k · kV be the associated norm, defined by
kvkV = kε(v)kH .
(1)
It follows from Korn’s inequality that k · kH1 and k · kV are equivalent norms
on V . Therefore (V, k · kV ) is a real Hilbert space. Moreover, by the Sobolev
trace theorem there exists a positive constant C0 which depends only on Ω, Γ1
and Γ3 such that
kvkL2 (Γ3 )d ≤ C0 kvkV ∀v ∈ V
(2)
The associated norms are denoted by k · kH , k · kH , k · kH1 , k · kH1 and k · kV .
Since the boundary Γ is Lipschitz continuous, the unit outward normal vector
field ν on the boundary is defined a.e. for every vector field v ∈ H1 we denote
by vν and vτ the normal and tangential components of v on the boundary
given by vν = v · ν, vτ = v−vν ν. Moreover, since meas (Γ1 ) > 0, Korn’s
inequality holds and thus, there exists a positive constant C0 depending only
on Ω, Γ1 such that
kε(v)kH ≥ C0 kvkH1 ∀v ∈ V.
By the Sobolev trace theorem there exists a positive constant C0 which depends
only on Ω, Γ1 and Γ3 such that
kvkL2 (Γ3 )d ≤ C0 kvkV
∀v ∈ V
(3)
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Analysis of a Dynamic Frictional Contact Problem for...
Furthermore, if σ ∈ H1 there exists an element σν ∈ HΓ0 such that the following
Green formula holds
(σ, ε(v))H + (Div(σ), v)H = (σν, γv)HΓ0 ×HΓ
In addition, if σ is sufficiently regular (say C 1 ), then
R
(σ, ε(v))H + (Div(σ), v)H = Γ σν · γvdγ
∀v ∈ H1 .
∀v ∈ H1 .
(4)
where dγ denotes the surface element. Similarly, for a regular tensor field
σ : Ω → Sn we define its normal and tangential components on the boundary
by σν = σν · ν, στ = σν − σν ν. Moreover, we denote by V 0 and V 0 the dual of
the spaces V and V , respectively. Identifying H, respectively L2 (Ω), with its
own dual, we have the inclusions
V ⊂ H ⊂ V 0,
V ⊂ L2 (Ω) ⊂ V 0 .
We use the notation h·, ·iV 0 ×V , h·, ·iV 0 ×V to represent the duality pairing between V 0 , V and V 0 , V , respectively.
Let T > 0. For every real space X, we use the notation C(0, T ; X), and
1
C (0, T ; X) for the space of continuous an continuously differentiable functions
from [0, T ] to X respectively, C(0, T ; X) is a real Banach space with the norm
|f |C(0,T ;X) = max |f (t)|X .
t∈[0,T ]
While C 1 (0, T ; X) is a real Banach space with the norm
|f |C 1 (0,T ;X) = max |f (t)|X + max |f˙(t)|X .
t∈[0,T ]
t∈[0,T ]
Finally, for k ∈ N and p ∈ [1, ∞], we use the standard notation for the Lebesgue
space Lp (0, T ; X) and for the Sobolev spaces W k,p (0, T ; X). Moreover for a
real number r, we use r+ to represent its positive part that is r+ = max(0, r),
and if X1 and X2 are real Hilbert spaces, than X1 × X2 denotes the product
Hilbert space endowed with the canonical inner product (., .)X1 ×X2 . For the
rest of this article, we will denote by C possibly different positive constants
depending only on the data of the problem, and whose value may change from
place to place. Here and in the sequel C denotes a positive constant which
may depend on Γ1 , Γ2 , Γ3 , but not depend on t or the initial data.
3
Problem Statement
We consider an elastic–termo-viscoplastic body which occupies the domain
Ω ⊂ Rn with the boundary Γ divided into three disjoint measurable parts Γ1 ;
64
S. Boutechebak et al.
Γ2 and Γ3 such that meas (Γ1 ) > 0. The time interval of interest is (0; T )
where T > 0. The body is clamped on Γ1 and so the displacement field
vanishes there. A volume force of density f acts in Ω × (0, T ) and surface
tractions of density f0 act on Γ2 ×(0, T ). On the part Γ3 . Moreover, the
process is dynamic, and thus the inertial terms are included in the equation of
motion. We use an elastic-thermo- viscoplastic constitutive law with damage
to model the material’s behaviour and an ordinary differential equation to
describe the evolution of the bonding field. The mechanical problem of the
dynamic contact with normal domped response is then written as follows.
3.1
Problem P
Find the displacement field u : Ω×(0, T ) → Rn , the stress field σ : Ω×(0, T ) →
Sn , the temperature θ : Ω × (0, T ) → R, the damage field ς : Ω × (0, T ) → R
and the adhesion field β : Ω × (0, T ) → R such that
Z
t
G σ(s)
− A(ε(u̇(s))), ε(u(s)), θ(s), ς(s) ds in Ω a.e. t ∈ (0, T ),
σ(t) = A(ε(u̇(t))) + E(ε(u(t))) +
ρü = Div(σ) + f
0
(5)
in Ω × (0, T ),
(6)
ρθ̇ − k0 ∆θ = ψ(σ, ε(u̇), θ, ς) + q in Ω × (0, T ),
ρς˙ − k1 ∆ς + ∂K ϕ(ς) 3 φ(σ, ε(u), θ, ς) in Ω × (0, T ),
u = 0 on Γ1 × (0, T ),
σν = f0 on Γ2 × (0, T ),
−σν = pν (u̇ν ) on Γ3 × (0, T ),
στ = pτ (u̇τ ) on Γ3 × (0, T ),
∂θ
+ αθ = 0 on Γ × (0, T ),
k0
∂ν
∂ς
= 0 on Γ × (0, T ),
∂ν
u(0) = u0 , u̇(0) = w0 , θ(0) = θ0 , ς(0) = ς0 in Ω.
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
This problem represents the dynamic evolution of damage and adhesion in
elastic-thermo-viscoplastic materials. Equation (5) is the elastic-thermo-viscoplastic
constitutive law where A and E are nonlinear operators describing the purely
viscous and the elastic properties of the material, respectively, and G is a nonlinear constitutive function which describes the viscoplastic behavior of the
material. (6) represents the equation of motion in which the dot above denotes the derivative with respect to the time variable and ρ is the density of
mass. Equation (7) represents the energy conservation where ψ is a nonlinear
Analysis of a Dynamic Frictional Contact Problem for...
65
constitutive function which represents the heat generated by the work of internal forces and q is a given volume heat source. Inclusion (8) describes the
evolution of damage field, governed by the source damage function φ, where
∂ϕK (ς) is the subdifferential of indicator function of the set K of admissible
damage functions given by
K = {ξ ∈ V : 0 ≤ ξ(x) ≤ 1 a.e. x ∈ Ω},
in such a way that the damage function ς varied between 0 and 1. If ς = 1
there is no damage in the material, if ς = 0 the material is completely damaged
and if 0 < ς < 1 the material is partially damaged.
Equalities (9) and (10) are the displacement-traction boundary conditions,
respectively. Condition (11) represents the normal domped response condition,
pν and pτ are given positive functions which will be described below, where σν
and στ are the normal and tangential stress, uν denotes the normal displacement. Next, equations (13) and (14) represent, respectively a Fourier boundary
condition for the temperature and an homogeneous Neumann boundary condition for the damage field on Γ. Finally the functions u0 , w0 , θ0 and ς0 in
(15) are the initial data.
In the study of the mechanical problem (P), we consider the following
hypotheses
The viscosity operator A : Ω × Sn → Sn satisfies the following properties:

(a)There exists a constant LA > 0 such that




|A(x, ε1 ) − A(x, ε2 )| ≤ LA |ε1 − ε2 | for all ε1 , ε2 ∈ Sn , a.e. x ∈ Ω.




(b)There exists a constant mA such that



(A(x, ε1 ) − A(x, ε2 )).( ε1 − ε2 ) ≥ mA |ε1 − ε2 |2
 for all ε1 , ε2 ∈ Sn a.e. x ∈ Ω.



(c) The mapping x 7→ A(x, ε) is Lebesgue measurable on Ω




for all ε ∈ Sn .



(d) The mapping x 7→ A(x, 0) ∈ H.
(16)
The elasticity operator E : Ω × Sn → Sn satisfies the following properties:

(a) There exists a constant LE > 0 such that




 |E(x, ε1 ) − E(x, ε2 )| ≤ LE |ε1 − ε2 | for all ε1 , ε2 ∈ Sn , a.e. x ∈ Ω.
(b)The mapping x 7→ E(x, ε)is Lebesgue measurable on Ω


for all ε ∈ Sn ,



(c)The mapping x 7→ E(x, 0) ∈ H.
(17)
The viscoplasticity operator G : Ω×Sn ×Sn ×R×R → Sn satisifes the following
66
S. Boutechebak et al.
properties:

(a)There exists a constant LG > 0 such that |G(x, σ1 , ε1 , θ1 , ς1 )−




G(x, σ2 , ε2 , θ2 , ς2 )| ≤ LG (|σ1 − σ2 | + |ε1 − ε2 | + |θ1 − θ2 | + |ς1 − ς2 |)




 for all σ1 , σ2 ∈ Sn , for all ε1 , ε2 ∈ Sn for all θ1 , θ2 ∈ R,
for all ς1 , ς2 ∈ R a.e. x ∈ Ω;


(b)The
mapping x → G(x, σ, ε, θ, ς) is Lebesgue measurable on Ω




for all σ, ε ∈ Sn , for all θ, ς ∈ R,



(b)The mapping x → G(x, 0, 0, 0, 0) ∈ H.
(18)
The nonlinear constitutive function ψ : Ω × Sn × Sn × R × R → R satisfies the
following properties:

(a)There exists a constant Lψ > 0 such that |ψ(x, σ1 , ε1 , θ1 , ς1 )−




ψ(x, σ2 , ε2 , θ2 , ς2 )| ≤ Lψ (|σ1 − σ2 | + |ε1 − ε2 | + |θ1 − θ2 | + |ς1 − ς2 |)




 for all σ1 , σ2 ∈ Sn , for all ε1 , ε2 ∈ Sn , for all θ1 , θ2 ∈ R,
for all ς1 , ς2 ∈ R a.e. x ∈ Ω,
(19)


(b)The mapping x → ψ(x, σ, ε, θ, ς) is Lebesgue measurable on Ω




for all σ, ε ∈ Sn , for all θ, ς ∈ R,



(c)The mapping x → ψ(x, 0, 0, 0, 0) ∈ L2 (Ω).
The damage source function φ : Ω × Sn × Sn × R × R → R satisfies the
following properties:

(a)There exists a constant Lφ > 0 such that




|φ(x,
σ1 , ε1 , θ1 , ς1 ) − φ(x, σ2 , ε2 , θ2 , ς2 )| ≤ Lφ (|σ1 − σ2 | + | ε1 − ε2 |




 +|θ1 − θ2 | + |ς1 − ς2 |) for all σ1 , σ2 ∈ Sn , for all ε1 , ε2 ∈ Sn ,
for all θ1 , θ2 ∈ R, for all ς1 , ς2 ∈ R a.e. x ∈ Ω
(20)


(b)The mapping x 7→ φ(x, σ, ε, θ, ς) is Lebesgue measurable on Ω




for all σ, ε ∈ Sn , for all θ, ς ∈ R



(c)The mapping x 7→ φ(x, 0, 0, 0, 0) ∈ L2 (Ω).
The normal compliance function pα : Γ3 × R −→ R+ , α = ν, τ satisfies:

(a) There exists a constant Lp > 0 such that




 |pα (x, r1 ) − pα (x, r2 )| ≤ Lp |r1 − r2 | ∀ r1 , r2 ∈ R, a.e.x ∈ Γ3 ,
(b) (pα (x, r1 ) − pα (x, r2 )) (r1 − r2 ) ≥ 0


(c) The mapping x −→ pα (x, r) is measurable on Γ3 , ∀r ∈ R,



(d)The mapping x −→ pα (x, r) = 0 for any r ≤ 0, a.e x ∈ Γ3 .
(21)
The mass density satisfies:
ρ ∈ L∞ (Ω), there exists ρ∗ > 0 such that
ρ ≥ ρ∗ a.e. x ∈ Ω.
(22)
67
Analysis of a Dynamic Frictional Contact Problem for...
The body forces, surface tractions and the volume heat source have the regularity
f ∈ L2 (0, T ; H), f0 ∈ L2 (0, T ; L2 (Γ2 )n ),
q ∈ L2 (0, T ; L2 (Ω)).
u0 ∈ V,
w0 ∈ H, θ0 ∈ V,
ki > 0, i = 0, 1.
(23)
ς0 ∈ K.
(24)
(25)
∀v ∈ V, t ∈ (0, T ).
(26)
We denote by F(t) ∈ V 0 the following element
hF(t), viV 0 ×V = (f (t), v)H + (f0 (t), γv)L2 (Γ2 )n
The use of (23) permits to verify that
F ∈ L2 (0, T ; V 0 ).
(27)
We introduce the following continuous functionals
Z
Z
a0 : V × V → R, a0 (ζ, ξ) = k0 ∇ζ · ∇ξdx + α ζξdγ,
Ω
Γ
Z
a1 : V × V → R, a1 (ζ, ξ) = k1 ∇ζ · ∇ξdx.
(28)
(29)
Ω
Finally, we consider the functional j : V × V → R defined by
Z
j(u, v) =
(pυ (uυ )vυ + pτ (uτ )vτ )da, ∀u, υ ∈ V
(30)
Γ3
Keeping in mind (21) and (22), we observe that integrals in (30) are well
defined. Using standard arguments based on Green’s formula (4), we can derive
the following variational formulation of the frictional problem with normal
domped response. (5)−(15) as follows.
3.2
Problem PV
Find the displacement field u : Ω×(0, T ) → Rn , the stress field σ : Ω×(0, T ) →
Sn , the temperature θ : Ω×(0, T ) → R and the damage field ς : Ω×(0, T ) → R
such that
σ(t) = A(ε(u̇(t))) + E(ε(u(t)))
Rt
(31)
+ 0 G(σ(s) − A(ε(u̇(s))), ε(u(s)), θ(s), ς(s))ds a.e. t ∈ (0, T ),
hρü(t), viV 0 ×V + (σ(t), ε(v))H + j(u̇(t), v) = hF(t), viV 0 ×V
∀v ∈ V, a.e. t ∈ (0, T ),
(32)
hρθ̇(t), ωiV 0 ×V + a0 (θ(t), ω) = hψ(σ(t), ε(u̇(t)), θ(t), ς(t)), ωiV 0 ×V
+(q(t), ω)L2 (Ω) ∀ω ∈ V, a.e. t ∈ (0, T ),
(33)
hρς(t),
˙
ξ − ς(t)iV 0 ×V + a1 (ς(t), ξ − ς(t)) ≥ hφ(σ(t), ε(u(t)), θ(t), ς(t)),
ξ − ς(t)iV 0 ×V ∀ξ ∈ K, a.e. t ∈ (0, T ), ς(t) ∈ K,
(34)
u(0) = u0 ,
u̇(0) = w0 ,
θ(0) = θ0 ,
ς(0) = ς0
(35)
68
4
S. Boutechebak et al.
Main Results
The existence of the unique solution to Problem PV is proved in the next
section. To this end, we consider the following remark which is used in different
places of the paper.
Theorem 4.1 (Existence and uniqueness) Under assumptions (16)-(25),
there exists a unique solution {u, σ, θ, ς} to problem (PV). Moreover, the solution has the regularity
u ∈ C 0 (0, T ; V) ∩ C 1 (0, T ; H),
u̇ ∈ L2 (0, T ; V),
ü ∈ L2 (0, T ; V 0 ),
σ ∈ L2 (0, T ; H),
θ ∈ L2 (0, T ; V ) ∩ C 0 (0, T ; L2 (Ω)),
(36)
(37)
(38)
(39)
(40)
θ̇ ∈ L2 (0, T ; V 0 ),
ς ∈ L2 (0, T ; V ) ∩ C 0 (0, T ; L2 (Ω)),
ς˙ ∈ L2 (0, T ; V 0 ).
(41)
(42)
(43)
(44)
(u, σ, θ, ς) which satisfies (31)-(35) is called a weak solution to the compliance
contact Problem P. We conclude that under the stated assumptions, problem
(5)-(15) has a unique weak solution satisfying (36)-(43).
We turn now to the proof of theorem 4.1 which will be carried out in several steps and is based on arguments of nonlinear equations with monotone
operators, a classical existence and uniqueness result on parabolic inequalities
and fixed-point arguments. To this end, we assume in the following that (16)(25) hold. Below, C denotes a generic positive constant which may depend
on Ω, Γ1 , Γ2 , Γ3 , A, E, G, φ, p and T but does not depend on t nor of the rest
of input data, and whose value may change from place to place. Moreover,for
the sake of simplicity we suppress in what follows the explicit dependence of
various functions on x ∈ Ω ∪ Γ.
Let η ∈ L2 (0, T ; V 0 ) be given. In the first step we consider the following variational problem.
4.1
Problem PVη
Find the field uη : Ω × (0, T ) → Rd , such that ∀v ∈ V, p.p.t ∈ (0, T )
hρüη (t) , viV 0 ×V + (A(ε (u̇η (t)), ε(v)))H + j (u̇η (t) , v) + (η(t), v)V 0 ×V = (F (t), v)V 0 ×V
(45)
uη (0) = u0 ,
u̇η (0) = w0 in Ω.
(46)
Analysis of a Dynamic Frictional Contact Problem for...
69
Lemma 4.2 For all η ∈ L2 (0, T ; V 0 ), there exists a unique solution uη to
the auxiliary problem PVη satisfying (36)-(38).
Let us introduce the operator A : V → V 0 ,
hA u, viV 0 ×V = (A(ε(u)), ε(v))H + j (uη (t) , v) .
(47)
Therefore, (45)can be rewritten, making use the operator A, as follows
ρüη (t) + A(u̇η (t)) = Fη (t)
on V 0 a.e. t ∈ (0, T ),
(48)
where Fη (t) = F(t)−η(t) ∈ V 0 . We recall that by(27) we have Fη ∈ L2 (0, T ; V 0 ).
Keeping in mind that the operator A is strict monotone, hemi-continuous,
bounded and coercive, then by using classical arguments of functional analysis concerning parabolic equations [6] we can easily prove the existence and
uniqueness of wη satisfying
wη ∈ L2 (0, T ; V) ∩ C 0 (0, T ; H),
ẇη ∈ L2 (0, T ; V 0 ),
ρẇη (t) + A(wη (t)) = Fη (t) on V 0 a.e. t ∈ (0, T ),
wη (0) = w0 .
Consider now the function uη : (0, T ) → V defined by
Z t
uη (t) =
wη (s)ds + u0 ∀t ∈ (0, T ).
(49)
(50)
(51)
(52)
(53)
0
It follows from (51) and (52) that uη is a solution of the equation (48) and it
satisfies (36)-(38).
In the second step we use the displacement field uη obtained in Lemma 4.2
and we consider the following initial-value problem.
4.2
Problem PVλ
Find the temperature θλ : Ω × (0, T ) → R such that
hρθ̇λ (t), ωiV 0 ×V + a0 (θλ (t), ω) = hλ(t) + q(t), ωiV 0 ×V
∀ω ∈ V, a.e. t ∈ (0, T ),
(54)
θλ (0) = θ0 in Ω.
(55)
Lemma 4.3 For all λ ∈ L2 (0, T ; V 0 ), there exists a unique solution θλ to
the auxiliary problem PVλ satisfying (40) and (41).
By an application of the Poincaré-Friedrichs inequality, we can prove that
a0 is V −elliptic. Consequently, based on classical arguments of functional
analysis concerning parabolic equations, the variational equation (54) has a
unique solution θλ satisfies (40) and (41).
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S. Boutechebak et al.
4.3
Problem PVµ
Find the damage field ςµ : Ω × (0, T ) → R such that
hρς˙µ (t), ξ − ςµ (t)iV 0 ×V + a1 (ςµ (t), ξ − ςµ (t))
≥ hµ, ξ − ςµ (t)iV 0 ×V ∀ξ ∈ K, a.e. t ∈ (0, T ), ςµ (t) ∈ K,
ςµ (0) = ς0
in Ω.
(56)
(57)
Lemma 4.4 For all µ ∈ L2 (0, T ; V 0 ), there exists a unique solution ςµ to
the auxiliary problem PVµ satisfying (42)-(43).
We know that the form a1 is not V -elliptic. To solve this problem we
introduce the functions
ς˜µ (t) = e−k1 t ςµ (t),
˜ = e−k1 t ξ(t).
ξ(t)
We remark that if ςµ , ξ ∈ K then ς˜µ , ξ˜ ∈ K. Consequently, (56) is equivalent
to the inequality
·
hρς˜µ (t), ξ˜ − ς˜µ (t)iV 0 ×V + a1 (˜
ςµ (t), ξ˜ − ς˜µ (t)) + k1 (ρ˜
ςµ , ξ˜ − ς˜µ (t))L2 (Ω)
≥ he−k1 t µ, ξ˜ − ς˜µ (t)iV 0 ×V ∀ξ˜ ∈ K, a.e. t ∈ (0, T ), ς˜µ ∈ K.
(58)
The fact that
˜ ξ)
˜ + k1 (ρξ,
˜ ξ)
˜ L2 (Ω) ≥ k1 min(ρ∗ , 1)kξk
˜2
a1 (ξ,
V
∀ξ˜ ∈ V,
(59)
and using classical arguments of functional analysis concerning parabolic inequalities, implies that (56) has a unique solution ς˜µ having the regularity (42)
and (43).
Let us consider now the auxiliary problem.
4.4
Problem PVη,λ,µ
Find the stress field ση,λ,µ : Ω × (0, T ) → Sn which is a solution of the problem
Z
ση,λ,µ (t) = E(ε(uη (t))) +
t
G ση,λ,µ (s)
(60)
0
− A(ε(u̇η (s))), ε(uη (s)), θλ (s), ςµ (s) ds
a.e. t ∈ (0, T ),
Lemma 4.5 There exists a unique solution of Problem PVη,λ,µ and it satisfies (39). Moreover, if uηi , θλi , ςµi and σηi ,λi ,µi represent the solutions of
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Analysis of a Dynamic Frictional Contact Problem for...
problems PVηi , PVλi , PVµi and PVηi ,λi ,µi , respectively, for i = 1, 2, then there
exists C > 0 such that
kση1 ,λ1 ,µ1 (t) − ση2 ,λ2 ,µ2 (t)k2H
Z t
ku̇η1 (s) − u̇η2 (s)k2V + kuη1 (s) − uη2 (s)k2V
≤C
0
+ kθλ1 (s) − θλ2 (s)k2V + kςµ1 (s) − ςµ2 (s)k2V ds.
(61)
Let Ση,λ,µ : L2 (0, T ; H) → L2 (0, T ; H) be the mapping given by
Ση,λ,µ σ(t)
Z
t
G(σ(s) − A(ε(u̇η (s))), ε(uη (s)), θλ (s), ςµ (s))ds.
= E(ε(uη (t))) +
(62)
0
Let σi ∈ L2 (0, T ; H), i = 1, 2 and t1 ∈ (0, T ). We find be using hypothesis (18)
and Hölder’s inequality
Z t1
2
2
kΣη,λ,µ σ1 (t1 ) − Ση,λ,µ σ2 (t1 )kH ≤ LG T
kσ1 (s) − σ2 (s)k2H ds.
(63)
0
By reapplication of mapping Ση,λ,µ , it follows that
2
Ση,λ,µ σ1 (t1 − Σ2η,λ,µ σ2 (t1 )2 ≤ L4G T02 t01 t2 kσ1 (s) − σ2 (s)k2 dsdt2
H
H
Reiterating this inequality n times leads to
n
2
n t1 t2 tn
Ση,λ,µ σ1 (t1 ) − Σnη,λ,µ σ2 (t1 )2 ≤ L2n
G T0 0 ...0 kσ1 (s) − σ2 (s)kH dsdtn ...dt2 .
H
Integration on the time interval (0, T ), it follows that
2n
n
L2n
G T
Ση,λ,µ σ1 − Σnη,λ,µ σ2 2 2
≤
kσ1 − σ2 k2L2 (0,T ;H)
L (0,T ;H)
n!
(2.3.28)
ItP
follows from this inequality that for n large enough, a power n of the mapping (n)
is a contraction on the space L2 (0, T ; H) and, therefore, from the
η,λ,µ
Banach fixed point theorem, there exists a unique element ση,λ,µ ∈ L2 (0, T ; H)
such that Ση,λ,µ ση,λ,µ = ση,λ,µ , which represents the unique solution of the problem PVη,λ,µ . Moreover, if uηi , θλi , ςµi and σηi ,λi ,µi represent the solutions of the
problems PVηi , PVλi , PVµi and PVηi ,λi ,µi , respectively, for i = 1, 2, then we
use (16)−(18) and Young’s inequality to obtain
kση1 ,λ1 ,µ1 (t) − ση2 ,λ2 ,µ2 (t)k2H
≤ Ckση1 ,λ1 ,µ1 (t) −
ση2 ,λ2 ,µ2 (t)k2H
+C
Z t
ku̇η1 (s) − u̇η2 (s)k2V
0
+ kuη1 (s) −
uη2 (s)k2V
+ kθλ1 (s) − θλ2 (s)k2V + kςµ1 (s) − ςµ2 (s)k2V ds.
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S. Boutechebak et al.
Which permits us to obtain, using Gronwall’s lemma, the inequality (61).
Second step. Let us consider the mapping
Λ : L2 (0, T ; V 0 × V 0 × V 0 ) → L2 (0, T ; V 0 × V 0 × V 0 ),
defined by
Λ(η(t), λ(t), µ(t))
= Λ0 (η(t), λ(t), µ(t)), ψ ση,λ,µ (t), ε(u̇η (t)), θλ (t), ςµ (t) ,
φ ση,λ,µ (t), ε(uη (t)), θλ (t), ςµ (t) ,
(64)
where the mapping Λ0 is given by
hΛ0 (η(t), λ(t), µ(t)), viV 0 ×V
Z t = E(ε(uη (t))) +
G ση,λ,µ (s)
0
θλ (s), ςµ (s) ds, ε(v)
∀v ∈ V.
− A(ε(u̇η (s))), ε(uη (s)),
(65)
H
Lemma 4.6 The mapping Λ has a fixed point
(η ∗ , λ∗ , µ∗ ) ∈ L2 (0, T ; V 0 × V 0 × V 0 ).
Let t ∈ (0, T ) and
(η1 , λ1 , µ1 ), (η2 , λ2 , µ2 ) ∈ L2 (0, T ; V 0 × V 0 × V 0 ).
We use the notation uηi = ui , u̇ηi = u̇i , üηi = üi , θλi = θi , ςµi = ςi and
σηi ,λi ,µi = σi , for i = 1, 2. Let us start by using (3)and hypotheses (16),
(17),(18), (20) and (21), to obtain
kΛ0 (η1 (t), λ1 (t), µ1 (t)) − Λ0 (η2 (t), λ2 (t), µ2 (t))kV 0
Z t
≤ LE ku1 (t) − u2 (t)kV + LG
kσ1 (s) − σ2 (s)kH
0
(66)
+ LA ku̇1 (s) − u̇2 (s)kV + ku1 (s) − u2 (s)kV
+ kθ1 (s) − θ2 (s)kL2 (Ω) + kς1 (s) − ς2 (s)kL2 (Ω) ds
+ Cku1 (t) − u2 (t)kV
a.e. t ∈ (0, T ).
Rt
On the other hand, since ui (t) = u0 + 0 u̇i (s)ds, we know that for a.e. t ∈
(0, T ),
Z
t
ku1 (t) − u2 (t)kV ≤
ku̇1 (s) − u̇2 (s)kV ds.
0
(67)
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Analysis of a Dynamic Frictional Contact Problem for...
Applying Young’s and Holder’s inequalities, (66) becomes, via (65) and (67)
kΛ0 (η1 (t), λ1 (t), µ1 (t)) − Λ0 (η2 (t), λ2 (t), µ2 (t))k2V 0
Z t
≤C
kσ1 (s) − σ2 (s)k2H + ku̇1 (s) − u̇2 (s)k2V + ku1 (s) − u2 (s)k2V
0
+ kθ1 (s) − θ2 (s)k2V + kς1 (s) − ς2 (s)k2V ds a.e.t ∈ (0, T ).
(68)
Furthermore, we find by taking the substitution η = η1 , η = η2 in (45) and
choosing v = u̇1 − u̇2 as test function
hρ(ü1 (t) − ü2 (t)) + Au̇1 (t) − Au̇2 (t), u̇1 (t) − u̇2 (t)iV 0 ×V
= hη2 (t) − η1 (t), u̇1 (t) − u̇2 (t)iV 0 ×V a.e. t ∈ (0, T ).
By virtue of (16) and (22), integrating this inequality over the interval time
variable (0, t) and Young inequality leads to
Z t
Z t
2
ku̇1 (s) − u̇2 (s)kV ds ≤ C
kη1 (s) − η2 (s)k2V 0 ds a.e. t ∈ (0, T ). (69)
0
0
which also implies, using a variant of (67), that
Z t
2
ku1 (s) − u2 (s)kV ≤ C
kη1 (s) − η 2 (s)k2V 0 ds
a.e.t ∈ (0, T ),
(70)
0
Moreover, if we take the substitution λ = λ1 , λ = λ2 in (54) and subtracting
the two obtained equations, we deduce by choosing ω = θλ1 −θλ2 as test function
Z t
(ρ∗ )2
2
kθ1 (t) − θ2 (t)kL2 (Ω) + C1
kθ1 (s) − θ2 (s)k2V ds
2
0
Z t
≤
kλ1 (s) − λ2 (s)kV 0 kθ1 (s) − θ2 (s)kV ds a.e. t ∈ (0, T ).
0
Employing Hölder’s and Young’s inequalities, we deduce that
Z t
2
kθλ1 (t) − θλ2 (t)kL2 (Ω) +
kθλ1 (s) − θλ2 (s)k2V ds
0
Z t
≤C
kλ1 (s) − λ2 (s)k2V 0 ds a.e. t ∈ (0, T ).
(71)
0
Substituting now {µ = µ1 , ξ = ς˜µ1 }, {µ = µ2 , ξ = ς˜µ2 } in (58) and subtracting
the two inequalities, we obtain
Z t
2
k˜
ς1 (t) − ς˜2 (t)kL2 (Ω) +
k˜
ς1 (t) − ς˜2 (t)k2V ds
0
Z t
≤C
ke−k1 t (µ1 (s) − µ2 (s))k2V 0 ds a.e. t ∈ (0, T ),
0
74
S. Boutechebak et al.
from which also follows that
Z
t
kς1 (s) − ς2 (s)k2V ds
kς1 (t) −
+
0
Z t
kµ1 (s) − µ2 (s)k2V 0 ds a.e. t ∈ (0, T ),
≤C
ς2 (t)k2L2 (Ω)
(72)
0
We can infer, using (61), (68), (69), (71), (72) and (70), that
Z T
kΛ0 (η1 (s), λ1 (s), µ1 (s)) − Λ0 (η2 (s), λ2 (s), µ2 (s))k2V 0 ds
0
Z T
≤C
kη1 (s) − η2 (s)k2V 0 + kλ1 (s) − λ2 (s)k2V 0 + kµ1 (s) − µ2 (s)k2V 0 ds .
0
(73)
, implies
Z t kψ σ1 (s), ε(u̇1 (s)), θ1 (s), ς1 (s) − ψ σ2 (s), ε(u̇2 (s)), θ2 (s), ς2 (s) k2V 0 ds
0
Z t
2
≤ 3Lψ
kσ1 (s) − σ2 (s)k2H + ku̇1 (s) − u̇2 (s)k2V
0
+ kθ1 (s) − θ2 (s)k2V + kς1 (t) − ς2 (t)k2V ds a.e. t ∈ (0, T ).
Furthermore, the hypothesisis (19), (61), (69), (71) and (72), permits us to
deduce that
Z T kψ σ1 (s), ε(u̇1 (s)), θ1 (s), ς1 (s) − ψ σ2 (s), ε(u̇2 (s)), θ2 (s), ς2 (s) k2V 0 ds
0
Z T
kη1 (s) − η2 (s)k2V 0 + kλ1 (s) − λ2 (s)k2V 0 + kµ1 (s) − µ2 (s)k2V 0 ds
≤C
0
(74)
Similarly, using (61), (70), (71) and (72), we obtain the following estimate for
φ,
Z T kφ σ1 (s), ε(u1 (s)), θ1 (s), ς1 (s) − φ σ2 (s), ε(u2 (s)), θ2 (s), ς2 (s) k2V 0 ds
0
Z T
2
2
2
≤C
kη1 (s) − η2 (s)kV 0 + kλ1 (s) − λ2 (s)kV 0 + kµ1 (s) − µ2 (s)kV 0 ds.
0
(75)
From (73), (74) and (75), we conclude that there exists a positive constant
C > 0 verifying
kΛ(η1 , λ1 , µ1 ) − Λ(η2 , λ2 , µ2 )kL2 (0,T ;V 0 ×V 0 ×V 0 )
≤ Ck(η1 − η2 , λ1 − λ2 , µ1 − µ2 )kL2 (0,T ;V 0 ×V 0 ×V 0 ) ,
(76)
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Analysis of a Dynamic Frictional Contact Problem for...
and so, by reapplication of mapping Λ, yields
kΛ2 (η1 , λ1 , µ1 ) − Λ2 (η2 , λ2 , µ2 )kL2 (0,T ;V 0 ×V 0 ×V 0 )
C2
≤
k(η1 − η2 , λ1 − λ2 , µ1 − µ2 )kL2 (0,T ;V 0 ×V 0 ×V 0 ) .
2!
We generalize this procedure by recurrence on n. Then we obtain the formula
kΛn (η1 , λ1 , µ1 ) − Λn (η2 , λ2 , µ2 )kL2 (0,T ;V 0 ×V 0 ×V 0 )
Cn
k(η1 − η2 , λ1 − λ2 , µ1 − µ2 )kL2 (0,T ;V 0 ×V 0 ×V 0 ) .
≤
n!
(77)
We know that the sequence (C n /n!)n converges to 0. So, for n sufficiently large
Cn
< 1. It means that a large power n of the operator Λ is a contraction on
n!
2
L (0, T ; V 0 × V 0 × V 0 ). Hence, Banach fixed point theorem shows that Λ admits
a unique fixed point (η ∗ , λ∗ , µ∗ ) ∈ L2 (0, T ; V 0 × V 0 × V 0 ).
We can now prove the existence of a solution to problem (PV). To this aim,
it is sufficient to remark that for a.e. t ∈ (0, T ),
(η ∗ (t), v)V́,V = (E(ε(uη∗ (t))), ε(v))H
Z t G ση∗ ,λ∗ ,µ∗ (s) − A(ε(u̇η∗ (s))), ε(uη∗ (s)), θλ∗ (s), ςµ∗ (s) ds, ε(v)
+
0
H
∀v ∈ V
ψ(ε(uη∗ (t)), θλ∗ (t), ςµ∗ (t)) = λ∗ (t),
φ(ε(uη∗ (t)), θλ∗ (t), ςµ∗ (t)) = µ∗ (t),
which completes the proof.
Theorem 4.7 (Positivity of the temperature) Let the hypotheses of
Theorem 4.1 hold and suppose in addition that
ψ(σ, ε(u), θ, ς) ≥ 0 a.e. in Ω × (0, T ),
q ≥ 0 a.e. in Ω × (0, T ),
θ0 ≥ 0 a.e. in Ω × (0, T ).
(78)
(79)
(80)
Then, the solution {u, σ, θ, ς} to problem (PV) satisfies the following property
θ(x, t) ≥ 0 for a.e. (x, t) ∈ Ω × (0, T ).
(81)
We use a maximum principle argument [2]. Thus, we test the equation
(33) by the function −θ− , where f − denoting the so-called negative part of a
76
S. Boutechebak et al.
function f ; i.e., f − = max{0, −f }, and integrate over (0, T ). We can infer,
using the hypothesis (78), (79) and (80), that
1 ∗ 2 − 2
(ρ ) kθ kL∞ (0,T ;L2 (Ω)) + C1 kθ− k2L2 (0,T ;V )
2
Z Z
t
ψ(ε(u(x, s)), θ(x, s), ς(x, s))θ− (x, s) dx ds
≤−
Z 0t Z Ω
−
0
q(x, s)θ− (x, s) dx ds ≤ 0 a.e. t ∈ (0, T ).
Ω
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