Electronic Journal of Differential Equations, Vol. 2010(2010), No. 129, pp. 1–15.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
DYNAMIC EVOLUTION OF DAMAGE IN
ELASTIC-THERMO-VISCOPLASTIC MATERIALS
ABDELBAKI MEROUANI, FARID MESSELMI
Abstract. We consider a mathematical model that describes the dynamic
evolution of damage in elastic-thermo-viscoplastic materials with displacementtraction, and Neumann and Fourier boundary conditions. We derive a weak
formulation of the system consisting of a motion equation, an energy equation,
and an evolution damage inclusion. This system has an integro-differential
variational equation for the displacement and the stress fields, and a variational inequality for the damage field. We prove existence and uniqueness of
the solution, and the positivity of the 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
[5, 8, 21, 22, 23, 29, 30] and references therein for the case of hardening, temperature and other internal state variables and the references [12, 13, 14, 22, 25, 27] for
the case of damage field.
The aim of this paper is to study the dynamic evolution of damage in elasticthermo-viscoplastic materials. For this, we consider a rate-type constitutive equation with two internal variables of the form
σ(t) = A(ε(u̇(t))) + E(ε(u(t)))
Z t (1.1)
+
G σ(s) − A(ε(u̇(s))), ε(u(s)), θ(s), ς(s) ds,
0
in which u, σ represent, respectively, the displacement field and the stress field
where the dot above denotes the derivative with respect to the time variable, θ
represents the absolute temperature, ς is the damage field, A and E are nonlinear
operators describing the purely viscous and the elastic properties of the material,
2000 Mathematics Subject Classification. 74H20, 74H25, 74M15, 74F05, 74R20.
Key words and phrases. Damage field; temperature; elastic-thermo-viscoplastic;
variational inequality.
c
2010
Texas State University - San Marcos.
Submitted July 20, 2010. Published September 8, 2010.
1
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A. MEROUANI, F. MESSELMI
EJDE-2010/129
respectively, and G is a nonlinear constitutive function which describes the viscoplastic behavior of the material.
Examples and mechanical interpretation of elastic-viscoplastic can be found in
[9, 16]. Dynamic and quasistatic contact problems are the topic of numerous papers,
e.g. [1, 2, 4, 11, 26], and the comprehensive references [15, 28]. However, the
mathematical problem modelled the quasi-static evolution of damage in thermoviscoplastic materials has been studied in [22].
The paper is organized as follows. In Section 2 we present the mechanical problem of the dynamic evolution of damage in elastic-thermo-viscoplastic 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. Statement of the Problem
Let Ω ⊂ Rn (n = 2, 3) be a bounded domain with a Lipschitz boundary Γ,
partitioned into two disjoint measurable parts Γ1 and Γ2 such that meas(Γ1 ) > 0.
We denote by Sn the space of symmetric tensors on Rn . 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 .
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).
Here ε : H1 → H and Div : H1 → H are the deformation and divergence operators,
respectively, defined by
1
ε(u) = (εij (u)), εij (u) = (ui,j + uj,i ), Div(σ) = (σij,j ).
2
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 . 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.
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DYNAMIC EVOLUTION OF DAMAGE
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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 }.
HΓ0
We also denote by
the dual of HΓ . 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.
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
Z
(σ, ε(v))H + (Div(σ), v)H =
σν · γvdγ
∀v ∈ H1 .
∀v ∈ H1 .
Γ
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,
0
στ = σν − σν ν.
0
Moreover, we denote by V and V 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.
For the rest of this article, we will denote by c possibly different positive constants
depending only on the data of the problem.
The physical setting is the following. An elastic-thermo-viscoplastic body occupies the domain Ω. We assume that the body is clamped on Γ1 × (0, T ), (T > 0)
and therefore the displacement field vanishes there. Surface tractions of density f0
acts on Γ2 × (0, T ) and a volume forces of density f is applied in Ω × (0, T ). In
addition, we admit a possible external heat source applied in Ω × (0, T ), given by
the function q.
The mechanical problem may be formulated as follows.
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 and the damage field
ς : Ω × (0, T ) → R such that
Z t σ(t) = A(ε(u̇(t))) + E(ε(u(t))) +
G σ(s)
0
(2.1)
− A(ε(u̇(s))), ε(u(s)), θ(s), ς(s) ds in Ω a.e. t ∈ (0, T ),
ρü = Div(σ) + f
in Ω × (0, T ),
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)
u = 0 on Γ1 × (0, T ),
σν = f0
on Γ2 × (0, T ),
in Ω × (0, T ),
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
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A. MEROUANI, F. MESSELMI
∂θ
+ βθ = 0 on Γ × (0, T ),
∂ν
∂ς
= 0 on Γ × (0, T ),
∂ν
u̇(0) = w, θ(0) = θ0 , ς(0) = ς0
EJDE-2010/129
k0
u(0) = u0 ,
(2.7)
(2.8)
in Ω.
(2.9)
This problem represents the dynamic evolution of damage in elastic-thermoviscoplastic materials. Equation (2.1) 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. (2.2) 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 (2.3) represents the
energy conservation where ψ is a nonlinear constitutive function which represents
the heat generated by the work of internal forces and q is a given volume heat
source. Inclusion (2.4) describes the evolution of damage field, governed by the
source damage function φ, where ∂K ϕ(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 (2.5) and (2.6) are the displacement-traction boundary conditions,
respectively. (2.7), (2.8) represent, respectively on Γ, a Fourier boundary condition
for the temperature and an homogeneous Neumann boundary condition for the
damage field on Γ. Finally the functions u0 , w, θ0 and ς0 in (2.9) are the initial
data.
In the study of the mechanical problem (P), we consider the following hypotheses
A : Ω × Sn → Sn satisfies the following properties:
(a) There exists an LG > 0 such that |A(x, ε1 ) − A(x, ε2 )| ≤
LA |ε1 − ε2 | for all ε1 , ε2 ∈ Sn a.e. x ∈ Ω;
(b) There exists an 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.
(2.10)
E : Ω × Sn → Sn satisfies the following properties:
(a) There exists an 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.
(2.11)
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DYNAMIC EVOLUTION OF DAMAGE
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G : Ω × Sn × Sn × R × R → Sn satisifes the following properties:
(a) There exists an 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;
(c) The mapping x → G(x, 0, 0, 0, 0) ∈ H.
(2.12)
ψ : Ω × Sn × Sn × R × R → R satisfies the following properties:
(a) There exists an 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;
(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).
(2.13)
φ : Ω × Sn × Sn × R × R → R satisfies the following properties:
(a) There exists an 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;
(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).
ρ ∈ L∞ (Ω),
f ∈ L2 (0, T ; H),
(2.14)
ρ ≥ ρ∗ > 0.
f0 ∈ L2 (0, T ; L2 (Γ2 )n ).
2
(2.15)
2
q ∈ L (0, T ; L (Ω)).
u0 ∈ V,
w0 ∈ H,
ki > 0,
θ0 ∈ V,
ς0 ∈ K.
i = 0, 1.
(2.16)
(2.17)
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
∀v ∈ V, t ∈ (0, T ).
(2.18)
The use of (2.15) permits to verify that
F ∈ L2 (0, T ; V 0 ).
(2.19)
We introduce the following continuous functionals
Z
Z
a0 : V × V → R, a0 (ζ, ξ) = k0
∇ζ · ∇ξdx + β ζξdγ,
Ω
Γ
Z
a1 : V × V → R, a1 (ζ, ξ) = k1
∇ζ · ∇ξdx.
(2.20)
(2.21)
Ω
Using the above notation and Green’s formula, we derive the following variational
formulation of mechanical problem (P).
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A. MEROUANI, F. MESSELMI
EJDE-2010/129
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)))
Z t
+
G(σ(s) − A(ε(u̇(s))), ε(u(s)), θ(s), ς(s))ds
a.e. t ∈ (0, T ),
(2.22)
0
hρü(t), viV 0 ×V + (σ(t), ε(v))H = hF(t), viV 0 ×V
∀v ∈ V, a.e. t ∈ (0, T ),
(2.23)
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)
(2.24)
∀ω ∈ V, a.e. t ∈ (0, T ),
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
(2.25)
∀ξ ∈ K, a.e. t ∈ (0, T ), ς(t) ∈ K,
u(0) = u0 ,
u̇(0) = w,
θ(0) = θ0 ,
ς(0) = ς0
in Ω.
(2.26)
3. Main Results
The main results are stated by the following theorems.
Theorem 3.1 (Existence and uniqueness). Under assumptions (2.10)-(2.17), 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),
2
u̇ ∈ L (0, T ; V),
(3.2)
0
2
ü ∈ L (0, T ; V ),
(3.3)
2
σ ∈ L (0, T ; H),
2
0
(3.1)
(3.4)
2
θ ∈ L (0, T ; V ) ∩ C (0, T ; L (Ω)),
(3.5)
θ̇ ∈ L2 (0, T ; V 0 ),
(3.6)
2
0
2
ς ∈ L (0, T ; V ) ∩ C (0, T ; L (Ω)),
2
0
ς˙ ∈ L (0, T ; V ).
(3.7)
(3.8)
The proof will be done in several steps. Based on classical arguments of functional analysis concerning variational problems, and Banach fixed point theorem.
First step. Take an arbitrary element
(η, λ, µ) ∈ L2 (0, T ; V 0 × V 0 × V 0 ),
(3.9)
and consider the auxiliary problem.
Problem PV1(η,λ,µ) . Find the displacement field uη : Ω × (0, T ) → Rn , the
temperature θλ : Ω × (0, T ) → R and the damage field ςµ : Ω × (0, T ) → R which
are solutions of the variational system
hρüη (t), viV 0 ×V + (A(ε(u̇η (t))), ε(v))H + hη(t), viV 0 ×V = hF(t), viV 0 ×V
∀v ∈ V, a.e. t ∈ (0, T ),
(3.10)
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DYNAMIC EVOLUTION OF DAMAGE
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hρθ̇λ (t), ωiV 0 ×V + a0 (θλ (t), ω) = hλ(t) + q(t), ωiV 0 ×V
∀ω ∈ V, a.e. t ∈ (0, T ),
hρς˙µ (t), ξ − ςµ (t)iV 0 ×V + a1 (ςµ (t), ξ − ςµ (t))
≥ hµ, ξ − ςµ (t)iV 0 ×V
uη (0) = u0 ,
∀ξ ∈ K, a.e. t ∈ (0, T ), ςµ (t) ∈ K,
u̇η (0) = w,
θλ (0) = θ0 ,
ςµ (0) = ς0
in Ω.
(3.11)
(3.12)
(3.13)
Lemma 3.2. For all (η, λ, µ) ∈ L2 (0, T ; V 0 ×V 0 ×V 0 ), there exists a unique solution
{uη , θλ , ςµ } to the auxiliary problem PV1(η,λ,µ) satisfying (3.1)-(3.3) and (3.5)(3.8).
Proof. Let us introduce the operator A : V → V 0 ,
hAu, viV 0 ×V = (A(ε(u)), ε(v))H .
(3.14)
It follows from hypothesis (2.10) that
kAu − AvkV 0 ≤ LA ku − vkV
∀u, v ∈ V.
Which proves that A is bounded and hemi-continuous on V.
On the other hand, by (2.10) and Korn’s inequality, we find for every v ∈ V,
hAv, viV 0 ×V
≥ C02 mA kvkV .
kvkV
The passage to the limit in this inequality when kvkV → +∞ implies that A is
coercive in V.
Next, by definition of A, the use of (2.10) and Korn’s inequality permits also to
obtain
hAu − Av, u − viV 0 ×V > C02 mA ku − vkV if u 6= v.
Then A is strict monotone. Therefore, (3.10) 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 ),
(3.15)
where
Fη (t) = F(t) − η(t) ∈ V 0 .
We recall that by (2.19) we have Fη ∈ L2 (0, T ; V 0 ). Kipping 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
[7, 19] we can easily prove the existence and uniqueness of wη satisfying
wη ∈ L2 (0, T ; V) ∩ C 0 (0, T ; H),
(3.16)
ẇη ∈ L2 (0, T ; V 0 ),
(3.17)
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 ).
(3.18)
(3.19)
(3.20)
0
It follows from (3.18) and (3.19) that uη is a solution of the equation (3.15) and it
satisfies (3.1)-(3.3).
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A. MEROUANI, F. MESSELMI
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Furthermore, by an application of the Poincaré-Friedrichs inequality, we can find
a constant β 0 > 0 such that
Z
Z
Z
β
|∇ζ|2 dx +
|ζ|2 dγ ≥ β 0
|ζ|2 dx ∀ζ ∈ V.
k
0
Ω
Γ
Ω
Thus, we obtain
a0 (ζ, ζ) ≥ c1 kζk2V ∀ζ ∈ V,
(3.21)
0
where c1 = k0 min(1, β )/2, which implies that a0 is V −elliptic. Consequently,
based on classical arguments of functional analysis concerning parabolic equations,
the variational equation (3.11) has a unique solution θλ satisfies (3.5)-(3.6).
On the other hand, we know that the form a1 is not V -elliptic. To solve this
problem we introduce the functions
˜ = e−k1 t ξ(t).
ς˜µ (t) = e−k1 t ςµ (t), ξ(t)
We remark that if ςµ , ξ ∈ K then ς˜µ , ξ˜ ∈ K. Consequently, (3.12) 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.
(3.22)
The fact that
˜ ξ)
˜ + k1 (ρξ,
˜ ξ)
˜ L2 (Ω) ≥ k1 min(ρ∗ , 1)kξk
˜2
a1 (ξ,
V
∀ξ˜ ∈ V,
(3.23)
and using classical arguments of functional analysis concerning parabolic inequalities [7, 10], implies that (3.22) has a unique solution ς˜µ having the regularity
(3.7)-(3.8). This completes the proof .
Let us consider now the auxiliary problem.
Problem PV2(η,λ,µ) . Find the stress field ση,λ,µ : Ω × (0, T ) → Sn which is a
solution of the problem
Z t ση,λ,µ (t) = E(ε(uη (t))) +
G ση,λ,µ (s)
0
(3.24)
− A(ε(u̇η (s))), ε(uη (s)), θλ (s), ςµ (s) ds a.e. t ∈ (0, T ),
Lemma 3.3. There exists a unique solution of Problem PV2(η,λ,µ) and it satisfies
(3.4). Moreover, if {uη , θλi , ςµi } and ση,λ,µ represent the solutions of problems
PV1(ηi ,λi ,µi ) and PV2(ηi ,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
≤c
ku̇η1 (s) − u̇η2 (s)k2V + kuη1 (s) − uη2 (s)k2V
(3.25)
0
+ kθλ1 (s) − θλ2 (s)k2V + kςµ1 (s) − ςµ2 (s)k2V ds.
Proof. Let Ση,λ,µ : L2 (0, T ; H) → L2 (0, T ; H) be the mapping given by
Ση,λ,µ σ(t)
Z
= E(ε(uη (t))) +
t
G(σ(s) − A(ε(u̇η (s))), ε(uη (s)), θλ (s), ςµ (s))ds.
0
(3.26)
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Let σi ∈ L2 (0, T ; H), i = 1, 2 and t1 ∈ (0, T ). We find be using hypothesis (2.12)
and Hölder’s inequality
Z t1
2
2
kσ1 (s) − σ2 (s)k2H ds.
(3.27)
kΣη,λ,µ σ1 (t1 ) − Ση,λ,µ σ2 (t1 )kH ≤ LG T
0
Integration on the time interval (0, t2 ) ⊂ (0, T ), it follows that
Z t2 Z t1
Z t2
kσ1 (s) − σ2 (s)k2H dsdt1 .
kΣη,λ,µ σ1 (t1 ) − Ση,λ,µ σ2 (t1 )k2H dt1 ≤ L2G T
0
0
0
Using again (3.27), it follows that
kΣη,λ,µ σ1 (t2 ) −
Ση,λ,µ σ2 (t2 )k2H
≤
L4G T 2
Z
t2
t1
Z
kσ1 (s) − σ2 (s)k2H dsdt1 .
0
0
For t1 , t2 , . . . , tn ∈ (0, T ), we generalize the procedure above by recurrence on n.
We obtain the inequality
kΣη,λ,µ σ1 (tn ) − Ση,λ,µ σ2 (tn )k2H
Z tn
Z t2 Z t1
n
≤ L2n
T
.
.
.
kσ1 (s) − σ2 (s)k2H ds dt1 . . . dtn−1 .
G
0
0
0
Which implies
kΣη,λ,µ σ1 (tn ) −
Ση,λ,µ σ2 (tn )k2H
L2n T n+1
≤ G
n!
Z
T
kσ1 (s) − σ2 (s)k2H ds.
0
Thus, we can infer, by integrating over the interval time (0, T ), that
n+2
L2n
G T
kσ1 − σ2 k2L2 (0,T ;H) .
n!
It follows from this inequality that for n large enough, a power n of the mapping Ση,λ,µ is a contraction on the space L2 (0, T ; H) and, therefore, from the 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
PV2(η,λ,µ) . Moreover, if {uη , θλi , ςµi } and ση,λ,µ represent the solutions of problem
PV1(ηi ,λi ,µi ) and PV2(ηi ,λi ,µi ) , respectively, for i = 1, 2, then we use (2.10), (2.11),
(2.12) and Young’s inequality to obtain
kΣη,λ,µ σ1 − Ση,λ,µ σ2 k2L2 (0,T ;H) ≤
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.
Which permits us to obtain, using Gronwall’s lemma, the inequality (3.25).
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) ,
(3.28)
10
A. MEROUANI, F. MESSELMI
EJDE-2010/129
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)
(3.29)
0
− A(ε(u̇η (s))), ε(uη (s)), θλ (s), ςµ (s) ds, ε(v) .
H
Lemma 3.4. The mapping Λ has a fixed point
(η ∗ , λ∗ , µ∗ ) ∈ L2 (0, T ; V 0 × V 0 × V 0 ).
Proof. Let t ∈ (0, T ) and
(η1 , λ1 , µ1 ), (η2 , λ2 , µ2 ) ∈ L2 (0, T ; V 0 × V 0 × V 0 ).
Let us start by using hypotheses (2.10), (2.11) and (2.12) to obtain
kΛ0 (η1 (t), λ1 (t), µ1 (t)) − Λ0 (η2 (t), λ2 (t), µ2 (t))kV 0
Z t
≤ LE kuη1 (t) − uη2 (t)kV + LG
kση1 ,λ1 ,µ1 (s) − ση2 ,λ2 ,µ2 (s)kH
(3.30)
0
+ LA ku̇η1 (s) − u̇η2 (s)kV + kuη1 (s) − uη2 (s)kV
+ kθλ1 (s) − θλ2 (s)kL2 (Ω) + kςµ1 (s) − ςµ2 (s)kL2 (Ω) ds a.e. t ∈ (0, T ).
On the other hand, we know that for a.e. t ∈ (0, T ),
Z t
kuη1 (t) − uη2 (t)kV ≤
ku̇η1 (s) − u̇η2 (s)kV ds.
(3.31)
0
Applying Young’s and Hölder’s inequalities, (3.30) becomes, via (3.31),
kΛ0 (η1 (t), λ1 (t), µ1 (t)) − Λ0 (η2 (t), λ2 (t), µ2 (t))k2V 0
Z t
≤c
kση1 ,λ1 ,µ1 (s) − ση2 ,λ2 ,µ2 (s)k2H + ku̇η1 (s) − u̇η2 (s)k2V
(3.32)
0
+ kuη1 (s) − uη2 (s)k2V + kθλ1 (s) − θλ2 (s)k2V + kςµ1 (s) − ςµ2 (s)k2V ds
a.e. t ∈ (0, T ). Furthermore, we find by taking the substitution η = η1 , η = η2 in
(3.10) and choosing v = u̇η1 − u̇η2 as test function
hρ(üη1 (t) − üη2 (t)) + Au̇η1 (t) − Au̇η2 (t), u̇η1 (t) − u̇η2 (t)iV 0 ×V
= hη2 (t) − η1 (t), u̇η1 (t) − u̇η2 (t)iV 0 ×V
a.e. t ∈ (0, T ).
By virtue of (2.10), this equation becomes
(ρ∗ )2 d
ku̇η1 (t) − u̇η2 (t)k2H + mA ku̇η1 (t) − u̇η2 (t)k2V
2 dt
≤ kη2 (t) − η1 (t)kV 0 ku̇η1 (t) − u̇η2 (t)kV .
Integrating this inequality over the interval time variable (0, t), Young inequality
leads to
Z t
∗ 2
2
(ρ ) ku̇η1 (t) − u̇η2 (t)kH + mA
ku̇η1 (s) − u̇η2 (s)k2V ds
0
EJDE-2010/129
DYNAMIC EVOLUTION OF DAMAGE
≤
2
mA
11
t
Z
kη1 (s) − η2 (s)k2V 0 ds.
0
Consequently,
Z t
Z t
2
ku̇η1 (s) − u̇η2 (s)kV ds ≤ c
kη1 (s) − η2 (s)k2V 0 ds a.e. t ∈ (0, T ).
0
(3.33)
0
which also implies, using a variant of (3.31), that
Z t
2
kuη1 (s) − uη2 (s)kV ≤ c
kη1 (s) − η2 (s)k2V 0 ds a.e.t ∈ (0, T ),
(3.34)
0
Moreover, if we take the substitution λ = λ1 , λ = λ2 in (3.11) and subtracting the
two obtained equations, we deduce by choosing ω = θλ1 − θλ2 as test function
Z t
(ρ∗ )2
kθλ1 (t) − θλ2 (t)k2L2 (Ω) + c1
kθλ1 (s) − θλ2 (s)k2V ds
2
0
Z t
≤
kλ1 (s) − λ2 (s)kV 0 kθλ1 (s) − θλ2 (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 ).
(3.35)
0
Substituting now {µ = µ1 , ξ = ς˜µ1 }, {µ = µ2 , ξ = ς˜µ2 } in (3.22) and subtracting
the two inequalities, we obtain
Z t
k˜
ςµ1 (t) − ς˜µ2 (t)k2L2 (Ω) +
k˜
ςµ1 (t) − ς˜µ2 (t)k2V ds
0
Z t
≤c
ke−k1 t (µ1 (s) − µ2 (s))k2V 0 ds a.e. t ∈ (0, T ),
0
from which also follows that
ςµ2 (t)k2L2 (Ω)
Z
t
kςµ1 (t) −
+
kςµ1 (s) − ςµ2 (s)k2V ds
0
Z t
≤c
kµ1 (s) − µ2 (s)k2V 0 ds a.e. t ∈ (0, T ),
0
We can infer, using (3.25), (3.32), (3.33), (3.35) and (3.36), that
Z t
kΛ0 (η1 (s), λ1 (s), µ1 (s)) − Λ0 (η2 (s), λ2 (s), µ2 (s))k2V 0 ds
0
Z tZ s
≤c
ku̇η1 (r) − u̇η2 (r)k2V + kθλ1 (r) − θλ2 (r)k2V
0
0
+ kuη1 (r) − uη2 (r)k2V + kςµ1 (r) − ςµ2 (r)k2V dr ds a.e. t ∈ (0, T )
Z TZ T
≤c
ku̇η1 (r) − u̇η2 (r)k2V + kθλ1 (r) − θλ2 (r)k2V
0
0
+ kuη1 (r) − uη2 (r)k2V + kςµ1 (r) − ςµ2 (r)k2V dr ds
(3.36)
12
A. MEROUANI, F. MESSELMI
Z
≤c
T
EJDE-2010/129
ku̇η1 (s) − u̇η2 (s)k2V
0
+ kuη1 (s) − uη2 (s)k2V + kθλ1 (s) − θλ2 (s)k2V + kςµ1 (s) − ςµ2 (s)k2V ds
Z T
kη1 (s) − η2 (s)k2V 0 + kλ1 (s) − λ2 (s)k2V 0 + kµ1 (s) − µ2 (s)k2V 0
≤c
0
+ kuη1 (s) − uη2 (s)k2V ds
Thus, by (3.34), we find
Z T
kΛ0 (η1 (s), λ1 (s), µ1 (s)) − Λ0 (η2 (s), λ2 (s), µ2 (s))k2V 0 ds
0
T
Z
≤c
(3.37)
kη1 (s) −
η2 (s)k2V 0
+ kλ1 (s) −
λ2 (s)k2V 0
+ kµ1 (s) −
µ2 (s)k2V 0
ds .
0
Furthermore, hypothesis (2.13) implies
Z t kψ ση1 ,λ1 ,µ1 (s), ε(u̇η1 (s)), θλ1 (s), ςµ1 (s)
0
− ψ ση2 ,λ2 ,µ2 (s), ε(u̇η2 (s)), θλ2 (s), ςµ2 (s) k2V 0 ds
Z t
≤ 3L2ψ
kση1 ,λ1 ,µ1 (s) − ση2 ,λ2 ,µ2 (s)k2H + ku̇η1 (s) − u̇η2 (s)k2V
0
+ kθλ1 (s) − θλ2 (s)k2V + kςµ1 (t) − ςµ2 (t)k2V ds a.e. t ∈ (0, T ).
This permits us to deduce, via (3.25), (3.33), (3.35) and (3.36), that
Z T
kψ ση1 ,λ1 ,µ1 (s), ε(u̇η1 (s)), θλ1 (s), ςµ1 (s)
0
− ψ ση2 ,λ2 ,µ2 (s), ε(u̇η2 (s)), θλ2 (s), ςµ2 (s) k2V 0 ds
(3.38)
Z T
≤c
kη1 (s) − η2 (s)k2V 0 + kλ1 (s) − λ2 (s)k2V 0 + kµ1 (s) − µ2 (s)k2V 0 ds
0
Similarly, using (3.25), (3.34), (3.35) and (3.36), we obtain the following estimate
for φ,
Z T kφ ση1 ,λ1 ,µ1 (s), ε(uη1 (s)), θλ1 (s), ςµ1 (s)
0
− φ ση2 ,λ2 ,µ2 (s), ε(uη2 (s)), θλ2 (s), ςµ2 (s) k2V 0 ds
(3.39)
Z T
≤c
kη1 (s) − η2 (s)k2V 0 + kλ1 (s) − λ2 (s)k2V 0 + kµ1 (s) − µ2 (s)k2V 0 ds.
0
From (3.37), (3.38) and (3.39), 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 )
(3.40)
≤ Ck(η1 − η2 , λ1 − λ2 , µ1 − µ2 )kL2 (0,T ;V 0 ×V 0 ×V 0 ) ,
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 )
EJDE-2010/129
DYNAMIC EVOLUTION OF DAMAGE
≤
13
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!
(3.41)
We know that the sequence (C n /n!)n converges to 0. So, for n sufficiently large
Cn
n! < 1. It means that a large power n of the operator Λ is a contraction on
L2 (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 ),
Z t ∗
E(ε(uη (t))) +
G ση∗ ,λ∗ ,µ∗ (s) − A(ε(u̇η∗ (s))), ε(uη∗ (s)), θλ∗ (s), ςµ∗ (s) ds
0
= η ∗ (t),
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 3.5 (Positivity of the temperature). Let the hypotheses of Theorem 3.1
hold and suppose in addition that
ψ(σ, ε(u), θ, ς) ≥ 0
a.e. in Ω × (0, T ),
(3.42)
q≥0
a.e. in Ω × (0, T ),
(3.43)
θ0 ≥ 0
a.e. in Ω × (0, T ).
(3.44)
Then, the solution {u, σ, θ, ς} to problem (PV) is such that
θ(x, t) ≥ 0
for a.e. (x, t) ∈ Ω × (0, T ).
(3.45)
Proof. We use a maximum principle argument [5]. Thus, we test the equation
(2.24) by the function −θ− , where f − denoting the so-called negative part of a
function f ; i.e., f − = max{0, −f }, and integrate over (0, T ). We can infer, using
the hypothesis (3.42), (3.43) and (3.44), that
1 ∗ 2 − 2
(ρ ) kθ kL∞ (0,T ;L2 (Ω)) + c1 kθ− k2L2 (0,T ;V )
2
Z tZ
≤−
ψ(ε(u(x, s)), θ(x, s), ς(x, s))θ− (x, s) dx ds
0
Ω
Z tZ
−
q(x, s)θ− (x, s) dx ds ≤ 0 a.e. t ∈ (0, T ).
0
Ω
Consequently
kθ− kL2 (0,T ;V )∩L∞ (0,T ;L2 (Ω)) ≤ 0,
which eventually gives (3.45).
14
A. MEROUANI, F. MESSELMI
EJDE-2010/129
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Abdelbaki Merouani
Departement de Mathematiques, Univerisite de Bordj Bou Arreridj, Bordj Bou Arreridj 34000, Algeria
E-mail address: badri merouani@yahoo.fr
Farid Messelmi
Departement de Mathematiques, Univerisite Zian Achour de Djelfa, Djelfa 17000, Algeria
E-mail address: foudimath@yahoo.fr