Homogenization of a Contact Problem in Linear
Viscoleasticity
Abdelhamid Ainouz
Labo. AMNEDP, USTHB
february 1st, 2009
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
1 / 18
The micro-model
Let Y = [0, 1]3 be the generic cell of periodicity divided as
Y = Ym [ Yinc [ Σ where Ym is the viscoelastic material, Yinc the
rigid material and Σ is the interface barrier that separates them.
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
2 / 18
The micro-model
Let Y = [0, 1]3 be the generic cell of periodicity divided as
Y = Ym [ Yinc [ Σ where Ym is the viscoelastic material, Yinc the
rigid material and Σ is the interface barrier that separates them.
Let Ω be a bounded domain of R3 with a smooth boundary Γ.
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
2 / 18
The micro-model
Let Y = [0, 1]3 be the generic cell of periodicity divided as
Y = Ym [ Yinc [ Σ where Ym is the viscoelastic material, Yinc the
rigid material and Σ is the interface barrier that separates them.
Let Ω be a bounded domain of R3 with a smooth boundary Γ.
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
2 / 18
The micro-model
Let Y = [0, 1]3 be the generic cell of periodicity divided as
Y = Ym [ Yinc [ Σ where Ym is the viscoelastic material, Yinc the
rigid material and Σ is the interface barrier that separates them.
Let Ω be a bounded domain of R3 with a smooth boundary Γ. For all
ε > 0, set
ε
Ωm
= fx 2 Ω;
ε
Ωinc
= fx 2 Ω;
Σ
ε
=
ε
∂Ωm
x
χm ( ) = 1g,
ε
x
χinc ( ) = 1g,
ε
(1)
ε
\ ∂Ωinc
where χm (y ) (resp. χinc (y )) denotes the characteristic function of
Ym (resp. Yinc ) extended to R3 by Y -periodicity.
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
2 / 18
The micro-model
Let Y = [0, 1]3 be the generic cell of periodicity divided as
Y = Ym [ Yinc [ Σ where Ym is the viscoelastic material, Yinc the
rigid material and Σ is the interface barrier that separates them.
Let Ω be a bounded domain of R3 with a smooth boundary Γ. For all
ε > 0, set
ε
Ωm
= fx 2 Ω;
ε
Ωinc
= fx 2 Ω;
Σ
ε
=
ε
∂Ωm
x
χm ( ) = 1g,
ε
x
χinc ( ) = 1g,
ε
(1)
ε
\ ∂Ωinc
where χm (y ) (resp. χinc (y )) denotes the characteristic function of
Ym (resp. Yinc ) extended to R3 by Y -periodicity.
ε \ Γ = ∅.
We assume that Ωinc
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
2 / 18
The balance equation reads :
ε
∂2tt um
ε
∂2tt uinc
ε
divσ m
ε
divσ inc
ε
= fmε in Ωm
,
ε
ε
= finc in Ωinc
(2)
ε (resp. uε ) is the displacement of the matrix (resp.
where um
inc
ε (resp. σ ε ) is the stress tensor in Ωε (resp. Ωε ).
inclusions), σ m
m
inc
inc
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
3 / 18
The balance equation reads :
ε
∂2tt um
ε
∂2tt uinc
ε
divσ m
ε
divσ inc
ε
= fmε in Ωm
,
ε
ε
= finc in Ωinc
(2)
ε (resp. uε ) is the displacement of the matrix (resp.
where um
inc
ε (resp. σ ε ) is the stress tensor in Ωε (resp. Ωε ).
inclusions), σ m
m
inc
inc
ε :
The Kelvin-Voigt model for the viscoelastic material Ωm
ε
ε
ε
ε
ε
σm
= Am
e (um
)+Bm
e (vm
)
(3)
ε = ∂ uε the velocity vector of uε .
where vm
t m
m
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
3 / 18
The balance equation reads :
ε
∂2tt um
ε
∂2tt uinc
ε
divσ m
ε
divσ inc
ε
= fmε in Ωm
,
ε
ε
= finc in Ωinc
(2)
ε (resp. uε ) is the displacement of the matrix (resp.
where um
inc
ε (resp. σ ε ) is the stress tensor in Ωε (resp. Ωε ).
inclusions), σ m
m
inc
inc
ε :
The Kelvin-Voigt model for the viscoelastic material Ωm
ε
ε
ε
ε
ε
σm
= Am
e (um
)+Bm
e (vm
)
(3)
ε = ∂ uε the velocity vector of uε .
where vm
t m
m
ε
The Navier model for the inclusions Ωinc
ε
ε
ε
σ inc
= Ainc
e (uinc
)
(4)
where e ( ) is the linearized strain tensor.
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
3 / 18
The micro-model
ε , Aε and Bε are given by
The tensors Am
m
inc
x
x
x
ε
ε
ε
Am (x ) = Am ( ), Ainc (x ) = Ainc ( ), Bm
(x ) = Bm ( ) (5)
ε
ε
ε
where Am (y ), Ainc (y ) and Bm (y ) are Y -periodic smooth tensors
satisfying the classical conditions of symmetry and coerciveness.
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
4 / 18
The micro-model
ε , Aε and Bε are given by
The tensors Am
m
inc
x
x
x
ε
ε
ε
Am (x ) = Am ( ), Ainc (x ) = Ainc ( ), Bm
(x ) = Bm ( ) (5)
ε
ε
ε
where Am (y ), Ainc (y ) and Bm (y ) are Y -periodic smooth tensors
satisfying the classical conditions of symmetry and coerciveness.
The Dirichlet boundary condition on the exterior boundary:
ε
= 0 on Γ.
um
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
(6)
february 1st, 2009
4 / 18
The micro-model
ε , Aε and Bε are given by
The tensors Am
m
inc
x
x
x
ε
ε
ε
Am (x ) = Am ( ), Ainc (x ) = Ainc ( ), Bm
(x ) = Bm ( ) (5)
ε
ε
ε
where Am (y ), Ainc (y ) and Bm (y ) are Y -periodic smooth tensors
satisfying the classical conditions of symmetry and coerciveness.
The Dirichlet boundary condition on the exterior boundary:
ε
= 0 on Γ.
um
On the interface
friction:
A. Ainouz (Labo. AMNEDP, USTHB)
Σε ,
(6)
we prescribe a linear contact condition with
CIMPA EGYPT ’09
february 1st, 2009
4 / 18
The micro-model
ε , Aε and Bε are given by
The tensors Am
m
inc
x
x
x
ε
ε
ε
Am (x ) = Am ( ), Ainc (x ) = Ainc ( ), Bm
(x ) = Bm ( ) (5)
ε
ε
ε
where Am (y ), Ainc (y ) and Bm (y ) are Y -periodic smooth tensors
satisfying the classical conditions of symmetry and coerciveness.
The Dirichlet boundary condition on the exterior boundary:
ε
= 0 on Γ.
um
On the interface
friction:
1
Σε ,
(6)
we prescribe a linear contact condition with
Normal compliance law
x
ε ε
ε
σm
ν = σ inc νε = εg ( )
ε
A. Ainouz (Labo. AMNEDP, USTHB)
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ε ε
ε(um
ν
ε
uinc
ν ε ),
february 1st, 2009
(7)
4 / 18
The micro-model
ε , Aε and Bε are given by
The tensors Am
m
inc
x
x
x
ε
ε
ε
Am (x ) = Am ( ), Ainc (x ) = Ainc ( ), Bm
(x ) = Bm ( ) (5)
ε
ε
ε
where Am (y ), Ainc (y ) and Bm (y ) are Y -periodic smooth tensors
satisfying the classical conditions of symmetry and coerciveness.
The Dirichlet boundary condition on the exterior boundary:
ε
= 0 on Γ.
um
On the interface
friction:
Σε ,
we prescribe a linear contact condition with
1
Normal compliance law
2
x
ε ε
ε
σm
ν = σ inc νε = εg ( )
ε
Linear viscous friction law
ε ε
ε
σm
τ = σ inc τ ε =
A. Ainouz (Labo. AMNEDP, USTHB)
(6)
ε ε
ε(um
ν
ε
ε(vm
CIMPA EGYPT ’09
τε
ε
uinc
ν ε ),
ε
vinc
τ ε ).
february 1st, 2009
(7)
(8)
4 / 18
Finally, the initial conditions
ε
(x, 0) = 0;
um
ε
uinc
(x, 0) = 0;
A. Ainouz (Labo. AMNEDP, USTHB)
ε
ε
vm
(x, 0) = 0 in Ωm
,
ε
ε
vinc
(x, 0) = 0 in Ωinc
.
CIMPA EGYPT ’09
february 1st, 2009
(9)
5 / 18
Finally, the initial conditions
ε
(x, 0) = 0;
um
ε
uinc
(x, 0) = 0;
A. Ainouz (Labo. AMNEDP, USTHB)
ε
ε
vm
(x, 0) = 0 in Ωm
,
ε
ε
vinc
(x, 0) = 0 in Ωinc
.
CIMPA EGYPT ’09
february 1st, 2009
(9)
5 / 18
Finally, the initial conditions
ε
(x, 0) = 0;
um
ε
uinc
(x, 0) = 0;
ε
ε
vm
(x, 0) = 0 in Ωm
,
ε
ε
vinc
(x, 0) = 0 in Ωinc
.
(9)
Notations :
Hmε
Vmε
Vε
ε 3
ε
ε
ε
= L2 (Ωm
) , Hinc
= L2 (Ωinc
)3 , H ε = Hmε Hinc
,
1
ε 3
ε
1
ε
3
= f u 2 H (Ωm ) ; ujΓ = 0g, Vinc = H (Ωinc ) ,
ε
= Vmε Vinc
.
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
5 / 18
Finally, the initial conditions
ε
(x, 0) = 0;
um
ε
uinc
(x, 0) = 0;
ε
ε
vm
(x, 0) = 0 in Ωm
,
ε
ε
vinc
(x, 0) = 0 in Ωinc
.
(9)
Notations :
Hmε
Vmε
Vε
Uε =
ε 3
ε
ε
ε
= L2 (Ωm
) , Hinc
= L2 (Ωinc
)3 , H ε = Hmε Hinc
,
1
ε 3
ε
1
ε
3
= f u 2 H (Ωm ) ; ujΓ = 0g, Vinc = H (Ωinc ) ,
ε
= Vmε Vinc
.
8 ε
ε
< um in Ωm
:
ε in Ωε
uinc
inc
A. Ainouz (Labo. AMNEDP, USTHB)
;
ε
[Uε ] = um
CIMPA EGYPT ’09
ε
uinc
the jump over Σε .
february 1st, 2009
5 / 18
The space V ε is equipped with the norm
ε
)k2L2 (Ωε
kUε k2V ε = ke (um
m)
A. Ainouz (Labo. AMNEDP, USTHB)
3 3
ε
+ ke (uinc
)k2L2 (Ωε
CIMPA EGYPT ’09
inc )
3 3
+ ε k[Uε ]k2L2 (Σε )3 .
february 1st, 2009
6 / 18
The space V ε is equipped with the norm
ε
)k2L2 (Ωε
kUε k2V ε = ke (um
m)
3 3
ε
+ ke (uinc
)k2L2 (Ωε
inc )
3 3
+ ε k[Uε ]k2L2 (Σε )3 .
x
x
x
x
Aε (x ) = χm ( )Am ( ) + χinc ( )Ainc ( ),
ε
ε
ε
ε
x
x
ε
B ( x ) = χ m ( ) Bm ( ) .
ε
ε
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
(10)
6 / 18
The space V ε is equipped with the norm
ε
)k2L2 (Ωε
kUε k2V ε = ke (um
m)
3 3
ε
+ ke (uinc
)k2L2 (Ωε
inc )
3 3
+ ε k[Uε ]k2L2 (Σε )3 .
x
x
x
x
Aε (x ) = χm ( )Am ( ) + χinc ( )Ainc ( ),
ε
ε
ε
ε
x
x
ε
B ( x ) = χ m ( ) Bm ( ) .
ε
ε
(10)
Φ = ( ϕm , ϕinc ) 2 V ε , [Φ] = ϕm ϕinc ,
x ε
x
(x ).
Fε (x ) = χm ( )fmε (x ) + χinc ( )finc
ε
ε
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The weak formulation associated to the problem (2)-(9) is :
Find Uε 2 W ε , Uε (x, 0) = 0, ∂t Uε (x, 0) = 0 such that
Z
Z
x
a (Uε , Φ ) =
Fε Φ + ε
g ( )(νε [Φ])
ε
Ω
Σε
ε
for all Φ 2 V where
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
(11)
7 / 18
The weak formulation associated to the problem (2)-(9) is :
Find Uε 2 W ε , Uε (x, 0) = 0, ∂t Uε (x, 0) = 0 such that
Z
Z
x
a (Uε , Φ ) =
Fε Φ + ε
g ( )(νε [Φ])
ε
Ω
Σε
ε
for all Φ 2 V where
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
(11)
7 / 18
The weak formulation associated to the problem (2)-(9) is :
Find Uε 2 W ε , Uε (x, 0) = 0, ∂t Uε (x, 0) = 0 such that
Z
Z
x
a (Uε , Φ ) =
Fε Φ + ε
g ( )(νε [Φ])
ε
Ω
Σε
ε
for all Φ 2 V where
(11)
W ε = fV = (vm , vinc ) 2 L2 (0, T ; V ε ); ∂t V 2 L2 (0, T ; H ε )g.
equipped with the norm
ε 2
ε
kUε k2W ε = kum
kL2 (0,T ;H 1 (Ωmε )3 ) + kuinc
k2L2 (0,T ;H 1 (Ωε
3
inc ) )
2
2
ε
ε
kvm
kL2 (0,T ;L2 (Ωmε )3 ) + kvinc
kL2 (0,T ;L2 (Ωε )3 )
inc
ε 2
+ε k[U ]kL2 (0,T ;L2 (Σε )3 ) .
a (Uε , Φ ) =
Z
Ω
+
A. Ainouz (Labo. AMNEDP, USTHB)
+
[< ∂2tt Uε , Φ > +(Aε e (Uε ) + B ε e (∂t Uε ))e (Φ)]
Z
Σε
ε [ ∂ t [ Uε ] τ ε [ Φ ]
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[Uε ]νε (νε [Φ])] .
february 1st, 2009
7 / 18
Theorem
ε )3 , f ε 2 L2 ( Ωε )3 and g (y )
Assume that fmε 2 L2 (Ωm
0 Y periodic
inc
inc
smooth function on Σ such that
p
ε
kfmε kL2 (Ωmε )3 + kfinc
kL2 (Ωε )3 + ε kg (x /ε)kL2 (Σε ) C .
inc
Then the problem (11) admits a unique solution Uε 2 W ε . Furthermore
we have the a priori estimate :
ε
jjUε (t )jjW ε + ke (vm
)kL2 (0,T ;L2 (Ωmε )3
3)
C (T )
(12)
where C (T ) is a postive constant independent of ε.
Proof.
Faedo-Galerkin method.
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8 / 18
Two scale convergence
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9 / 18
De…nition
w ε 2 L2 ([0, T ]
Ω) two-scale converges to w 0 2 L2 ([0, T ]
∞ (Y )) we have
for all test function ψ 2 D(]0, T [ Ω; C#
Z TZ
0
!
ε !0
A. Ainouz (Labo. AMNEDP, USTHB)
Ω
Y ) if
x
w ε (t, x )ψ(t, x, )dxdt
ε
Ω
Z TZ
0
Ω Y
w 0 (t, x, y )ψ(t, x, y )dxdydt.
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Theorem
1
For each bounded sequence in L2 ([0, T ]
two-scale convergent subsequence.
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
Ω), one can extract a
february 1st, 2009
10 / 18
Theorem
1
2
For each bounded sequence in L2 ([0, T ]
two-scale convergent subsequence.
Ω), one can extract a
If (w ε )is a bounded sequence in L2 ([0, T ]; H 1 (Ω)) then, up to a
subsequence, there exist w 2 L2 ([0, T ]; H 1 (Ω)) and w 1 2
1 (Y ) /R) such that
L2 ([0, T ] Ω; H#
w ε two scale converges to w
rw ε two scale converges to rw + ry w 1 .
and
ε
Z TZ
0
!
ε !0
A. Ainouz (Labo. AMNEDP, USTHB)
Σε
x
w ε (t, x )ψ(t, x, )dsε dt
ε
Z TZ
0
Ω Σ
w (t, x )ψ(t, x, y )dxdsy dt
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Thanks to (12) and the preceeding theorem, we have:
∞ (Y ))3
for all ϕm 2 D(]0, T [ Ω; C#
m
lim
Z TZ
0
=
Z TZ
0
A. Ainouz (Labo. AMNEDP, USTHB)
ε
Ωm
Ω Ym
x
ε
um
(t, x ) ϕm (t, x, )dxdt
ε
um (t, x, y ) ϕm (t, x, y )dxdydt,
CIMPA EGYPT ’09
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(13)
11 / 18
Thanks to (12) and the preceeding theorem, we have:
∞ (Y ))3
for all ϕm 2 D(]0, T [ Ω; C#
m
lim
Z TZ
0
Z TZ
=
0
ε
Ωm
Ω Ym
x
ε
um
(t, x ) ϕm (t, x, )dxdt
ε
um (t, x, y ) ϕm (t, x, y )dxdydt,
(13)
∞ (Y ))3
for all ϕinc 2 D(]0, T [ Ω; C#
inc
lim
Z TZ
0
=
Z TZ
0
A. Ainouz (Labo. AMNEDP, USTHB)
ε
Ωinc
Ω Y inc
x
ε
uinc
(t, x ) ϕinc (t, x, )dxdt
ε
uinc (t, x, y ) ϕinc (t, x, y )dxdydt.
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Thanks to (12) and the preceeding theorem, we have:
∞ (Y ))3
for all Φm 2 D(]0, T [ Ω; C#
m
lim
Z TZ
0
=
Z TZ
0
A. Ainouz (Labo. AMNEDP, USTHB)
ε
Ωm
Ω Ym
3
x
ε
e ( um
) Φm (t, x, )dxdt
ε
(e (um ) + ey u1m )Φm (t, x, y )dxdydt,
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Thanks to (12) and the preceeding theorem, we have:
∞ (Y ))3
for all Φm 2 D(]0, T [ Ω; C#
m
lim
Z TZ
0
=
Z TZ
0
ε
Ωm
Ω Ym
3
x
ε
e ( um
) Φm (t, x, )dxdt
ε
(e (um ) + ey u1m )Φm (t, x, y )dxdydt,
∞ (Y ))3
for all Φinc 2 D(]0, T [ Ω; C#
inc
lim
Z TZ
0
=
Z TZ
0
ε
Ωinc
Ω Y inc
A. Ainouz (Labo. AMNEDP, USTHB)
(15)
3
x
ε
e (uinc
) Φinc (t, x, )dxdt
ε
(e (uinc ) + ey u1inc )Φinc (t, x, y )dxdydt. (16)
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12 / 18
We also have
lim ε
=
Z TZ
0
A. Ainouz (Labo. AMNEDP, USTHB)
Z TZ
0
Ω Σ
x
ε
um
(t, x )ψ(t, x, )dsε dt
ε
Σε
um (t, x, y )ψ(t, x, y )dxdsy dt,
CIMPA EGYPT ’09
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(17)
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We also have
lim ε
=
=
Z TZ
0
Ω Σ
lim ε
Z TZ
Z TZ
0
Ω Σ
x
ε
um
(t, x )ψ(t, x, )dsε dt
ε
Σε
um (t, x, y )ψ(t, x, y )dxdsy dt,
0
0
A. Ainouz (Labo. AMNEDP, USTHB)
Z TZ
Σε
(17)
x
ε
uinc
(t, x )ψ(t, x, )dsε dt
ε
uinc (t, x, y )ψ(t, x, y )dxdsy dt.
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Choosing test functions
We take in (11) the following test functions
h
x
x i
ϕε
θ (t ) ϕ(x, ) + εϕ1 (x, )
ε
ε
ϕ(x, y ) = χm (y ) ϕm (x ) + χinc (y ) ϕinc (x )
ϕ1 (x, y ) = χm (y ) ϕ1m (x, y ) + χinc (y ) ϕ1inc (x, y )
where θ (t ) 2 D(]0, T [); ϕm , ϕinc 2 D(Ω)3 and
∞ (Y ))3 . Passing to the two-scale limit we get
ϕ1m , ϕ1inc 2 D(Ω; C#
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14 / 18
Corollary
The two scale homogenized system
jYm j∂2tt um
divx (
T (vm
where Tij = jΣjδij
A. Ainouz (Labo. AMNEDP, USTHB)
Z
Ym
1
[Am [e (um ) + ey (u1m )] + Bm [e (vm ) + ey (vm
)]])
vinc )+H (um
uinc ) = fm + ĝ in Ω,
um = 0 on Γ
R
R
Hij ; Hij = Σ (ν ν)ij ; ĝ = Σ g ν
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15 / 18
Corollary
The two scale homogenized system
jYm j∂2tt um
divx (
T (vm
Z
Ym
vinc )+H (um
jYinc j∂2tt uinc
Z
Y2
uinc ) = fm + ĝ in Ω,
um = 0 on Γ
R
R
Hij ; Hij = Σ (ν ν)ij ; ĝ = Σ g ν
where Tij = jΣjδij
T (vm
1
[Am [e (um ) + ey (u1m )] + Bm [e (vm ) + ey (vm
)]])
divx (
Z
Y inc
vinc ) + H (um
[Ainc (e (uinc ) + ey (u1inc ))])
uinc ) = finc + ĝ in Ω,
[Ainc (e (uinc ) + ey (u1inc ))] ν = 0 on Γ
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
15 / 18
1
divy (Am [e (um ) + ey (u1m )] + Bm [e (v1 ) + ey (vm
)]) = 0 in Ω
divy (Ainc [e (uinc ) + ey (u1inc )]) = 0 in Ω
1
(Am [e (um ) + ey (u1m )] + Bm [e (vm ) + ey (vm
)]) ν
1
Ainc [e (uinc ) + ey (uinc )] ν = 0 in Ω
y ! u1m , u1inc are Y -periodic.
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
Ym
Yinc
= 0 in Ω
Σ
Σ
february 1st, 2009
16 / 18
1
divy (Am [e (um ) + ey (u1m )] + Bm [e (v1 ) + ey (vm
)]) = 0 in Ω
divy (Ainc [e (uinc ) + ey (u1inc )]) = 0 in Ω
1
(Am [e (um ) + ey (u1m )] + Bm [e (vm ) + ey (vm
)]) ν
1
Ainc [e (uinc ) + ey (uinc )] ν = 0 in Ω
y ! u1m , u1inc are Y -periodic.
Ym
Yinc
= 0 in Ω
Σ
Σ
Initial Condition
um (x, 0) = vm (x, 0) = 0,
uinc (x, 0) = vinc (x, 0) = 0.
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
16 / 18
Theorem
The homogenized problem:
∂2tt um
div (Ahom
m e ( um ) +
T (vm
∂2tt u2
hom
Bm
e ( ∂ t um ) +
vinc )+H (um
Z t
0
Cmhom (t
uinc ) = fm
divx (Ahom
inc e (uinc )) + T (vm
um = 0,
s )e (um )ds )
ĝ in Ω,
vinc ) + H (um
hom
Ainc e (uinc ) ν = 0 on Γ,
uinc ) = finc
ĝ ,
um (0) = uinc (0) = 0 in Ω,
vm (0) = vinc (0) = 0 in Ω.
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
17 / 18
M. Buckingham, Wave propagation, stress relaxation and
grain-to-grain shearing in saturated, unconsolidated marine sediments,
Acous. Soc. of Am. journal, 108 (6) (2000) pp 2796-2815.
D. Cioranescu and F. Murat, un terme étrange venu d’alleurs.
Nonlinear Partial Di¤erential Equations and Their Applications (H.
Brézis and J. L. Lions, eds), Collège de France Seminar, vol. II et III,
Research Notes in Mathematics, vol. 60 and 70, Pitman, 1982, pp.
93-138 and 154-178.
P. Donato, L. Faella and S. Monsurro, Homogenization of the wave
equation in composites with imperfect interface : A memory e¤ect, J.
de Math. Pures et Appl. 87 (2007) pp 119-143
G. Francfort, P. Suquet, Homogenization and mechanical dissipation in
thermoviscoelasticity. Arch. Rational Mech. Anal. 96, (1986)265–293
R. Gilbert, A. Panchenko and X. Xie, Homogenization of a viscoelastic
matrix in linear frictional contact. Math. Meth. App. Sci., 28, (2005)
pp 309-328.
A. Ainouz (Labo. AMNEDP, USTHB)
CIMPA EGYPT ’09
february 1st, 2009
18 / 18