PERGAMON Nonlinear Analysis 40 (2000) 185-212 www.elsevier.nl/locate/na Homogenizing the acoustic properties of the seabed: Part I * R.P. Gilbert”, *, A. MikeliCb “Deptrtnwnt oj’ Muthmwticul Scirncus, University of Driuwwr, Newark, DE 19716, USA bAnulyse NurnCrique, CNRS UMR 5585. Bit. 101. Uniwrsitc~ Lyon I. 43, Bd. 6lu onzr nowmhrr, 69622 Villeurhunnr C&.u, Fruncr Keywords: Monophasic Biot law; Poroelastic medium; Homogenization 1. Introduction One of the pressing problems in underwater acoustics today is formulating and then solving a model for interaction of acoustic waves in a shallow ocean with the seabed. Shallow-water/seabed waveguide, direct and inverse wave propagation problems are ubiquitous in applied science and technology. One such application is for inverse imag- ing of objects submerged in the ocean or the seabed. As much of the acoustic energy passes into the seabed, this imagery is possible only if the sea environment (water, sediment, interfaces), in the absence of the object, is properly characterized beforehand. This means that a suitable model of the sediment and of propagation of sound therein must be developed and a method be proposed for solving the inverse problem of the identification of the mechanical parameters involved in this model. This model, as well as the sediment parameter and object identification scheme, must be able to take into account sound speed and density variations in the water as well as the behavior of sound in the seabed. In general, either an acoustic pulse, or a monochromatic signal with frequency o is used. Consequently, not only acoustic signals with acoustic frequencies spread about a central frequency, but time-harmonic solutions are of interest. There have been several acoustic models of the seabed [9,10,16]; however, the primary one in usage goes back *This work was partially supported by NATO Research Grant CRG 970261. Part of the work by A.M. was done during his sabbatical stay at IWR, SFB 359, Universitlt Heidelberg, Germany. * Corresponding author. 0362-546X/00/$-see front matter 0 2000 Elsevier Science Ltd. All rights reserved. PII: SO362-546X(99)00367-3
Transcript
PERGAMON Nonlinear Analysis 40 (2000) 185-212
www.elsevier.nl/locate/na
Homogenizing the acoustic properties of the seabed: Part I *
R.P. Gilbert”, *, A. MikeliCb
“Deptrtnwnt oj’ Muthmwticul Scirncus, University of Driuwwr, Newark, DE 19716, USA
bAnulyse NurnCrique, CNRS UMR 5585. Bit. 101. Uniwrsitc~ Lyon I. 43, Bd. 6lu onzr nowmhrr,
One of the pressing problems in underwater acoustics today is formulating and then solving a model for interaction of acoustic waves in a shallow ocean with the seabed. Shallow-water/seabed waveguide, direct and inverse wave propagation problems are ubiquitous in applied science and technology. One such application is for inverse imag- ing of objects submerged in the ocean or the seabed. As much of the acoustic energy passes into the seabed, this imagery is possible only if the sea environment (water, sediment, interfaces), in the absence of the object, is properly characterized beforehand. This means that a suitable model of the sediment and of propagation of sound therein must be developed and a method be proposed for solving the inverse problem of the identification of the mechanical parameters involved in this model. This model, as well as the sediment parameter and object identification scheme, must be able to take into account sound speed and density variations in the water as well as the behavior of sound in the seabed.
In general, either an acoustic pulse, or a monochromatic signal with frequency o is used. Consequently, not only acoustic signals with acoustic frequencies spread about a central frequency, but time-harmonic solutions are of interest. There have been several acoustic models of the seabed [9,10,16]; however, the primary one in usage goes back
*This work was partially supported by NATO Research Grant CRG 970261. Part of the work by A.M. was done during his sabbatical stay at IWR, SFB 359, Universitlt Heidelberg, Germany.
* Corresponding author.
0362-546X/00/$-see front matter 0 2000 Elsevier Science Ltd. All rights reserved.
to work of Biot [7,8], and Stall [25,26]. Biot’s work was rather heuristic; hence, several authors have sought to put this on a more solid footing, namely Sanchez-Palencia [23] (Chapter S), [6,24], Levy [18-201, Burridge and Keller [12]. Biot’s research relates to flow through a porous media as well as acoustic phenomena. Several authors after Biot have sought to characterize porous media. For example, Stoll [25] using a variant on the Biot model showed that energy dissipation due to the skeletal frame dominate at low frequencies whereas at higher frequencies interstitial fluids become dominant. For geophysical exploration frequencies used range from 1 Hz to over 100 kHz; hence, the concept of high and low frequency is relative to the particular sediment under consideration. Cheng et al. [ 131 showed for the time-harmonic case there exists an analogy between the poroelastic equations and the thermoelastic equations, for which the fundamental singular solutions are known. Hence, there is an interest in treating the time-harmonic case for practitioners in geophysics.
In the present paper we are interested in formulating a mathematical model to de- scribe the acoustic behavior of the seabed; however, the coupling between pore pressure diffusion and elastic matrix deformation in fluid impregnated rocks spawns a variety of time dependent phenomena [l]. This fact has also been mentioned by Auriault [5], who discusses the vibrational behavior of a porous media and offers a possible char- acterization according to reference values of certain physical quantities.
In the acoustic regime we tend to think frequently of monochromatic, plane waves and their refraction and scattering. If the region through which the wave passes is only slightly dispersive, that is, the viscous part of the fluid stress tensor is negligible in comparison to the pressure, monochromatic waves tend to remain monochromatic. This occurs for fluids which are slightly viscid, or inviscid. It is meaningful, therefore, for us to consider the case of time-harmonic waves of frequency o. In order to model this acoustic phenomena it is helpful to introduce dimensionless coordinates which are defined in terms of characteristic lengths and reference values of the physical parameters. Let 1 be the characteristic length of a microscopic cell, whereas L is the characteristic length at the macroscopic scale. We assume that the wavelength of the acoustic signal, /1, is related to the macroscopic length L by 2rcL = 2. The relation between I and L is given by the small parameter E, EL= 1. In terms of the characteristic lengths 1 and L we introduce the dimensionless coordinates y=X/I, and x=X/L, where X is taken to be a physical space variable. Then x = sy, and y is referred to as the
fast variable. The solid density is given by p,, the solid displacement by us, and the solid stress
tensor by 6;. In physical coordinates the equations of motion in the solid matrix are
given by
(1.1)
where 5; := Uijk/Dkl(US), &l being the components of the strain tensor. We assume that the magnitude of the coefficients in Hooke’s law for the solid matrix are of the same order of magnitude a with regard to a small parameter E, which will subsequently be defined. consequently, we have 6; = aoii, 6, = pops, Xj = lyj, uj = luj. Here po is a
R.P. Gilbrrt. A. MikrliL I Nonlineur Anul~sis 40 (2000) 1X-212 187
reference density, and a a reference stress. By introducing the dimensionless variables cr = (A/u)&(u), yi, and Uj into the system (1 .l ) yields in the time-harmonic case
(1.2)
where PI = pow2 12/a, and o is the vibrational frequency. Since poo2/a= 0(X2), where the wavelength 3, is large with respect to cell size, we see that PI = O(E~). In what follows, we shall write o = 0~07 where wg is a reference frequency. Eq. (1.2) is then resealed as
c 2 = -&Gp+;. (1.3) j ‘I
In the pore space, the momentum equation is given by
(1.4)
where 6 is the fluid velocity, j the pressure and the stress tensor is 6;=2@;j(fi)- jI;,i. Introducing dimensionless physical variables, and setting fi= c/au and j = lppo, leads to the equation
We note that due to the continuity of displacements at the interfaces between the elastic matrix and the viscous fluid, and under assumption of absence of the time-like oscillations, it is natural to look for the velocity field in the form v = lduf/dt. Here uf stands for the dimensionless fluid displacement. Then (r?V)t?= Ui12(&f/dtV)&f/& and it is small with respect to %/lat only if Usl(auf/dtV)&f/& = o( 1). We suppose it for the moment, but it should be verified a posteriori. Then the momentum equation transforms into
c uopofl a2uf
1 i
ay’ - PO% + ~2PfPoUoluf = 0. J 1
Suppose we write Q, := Uopo~/pol = O(pLw/aE); then
Q/ c $ - $ - T,pfuf = 0, i J
(1.5)
where T, := - w2p~U~l/p~ = O(E). The value of Ql is directly related to the contrast of property number C := ,u~o/a [5].
The contrast of property number is a measure of how degenerate the fluid stress tensor gets. In general, a non-degenerate fluid tensor corresponds to a monophasic situation, whereas a degenerate tensor corresponds to the diphasic case. Hence, when effective p is small we tend to have a Biot-like regime. The contrast of property number, moreover, is related to the Strouhal number Sh, by Sh = 0(sP2C). Recall, also, that
188 R.P. Gilbert, A. MikrliL I Nonlineur Anulysis 40 (2000) 185-212
Ql = O(E-’ C) = 0(&Y). Consequently, after introducing the global displacement u”, equal to uf in the fluid part and us in the elastic part, our system becomes in terms of the dimensionless physical quantities
as,’ = AD(u”) in 0: x IO, T[,
P,$ - div(dD(u”)) = Fp, in 0: x IO, T[,
(g,i: = in Rf x IO, T[,
. pfp - div(c’,“) = Fpf at2 in flf x 10 T[ 1. ) 7
ad . divx = 0 m Qi x IO, T[,
(1.6)
[d] = 0 on I?,: x IO, T[,
P . v = ~7~~‘: . v on I?[: x IO, T[,
(~8, p”} is L-periodic,
ad: u"(O)= -(O)=O in R. at
There are three cases of potential interest; however, in this paper we only consider Model C:
Model A: C = O(E~), Sh = 0( 1 ), r = -2, diphasic macroscopic behavior of the fluid and solid matrix. This case was considered previously by Biot [7,8], Sanchez-Hubert [22], Sanchez-Palencia [24,23], Levy [ 181, Buchanan-Gilbert-Lin [9,10,16].
Model B: C = O(E), Sh = 0( E- ’ ), r = - 1, monophasic elastic macroscopic behavior. Model C: C=O( I), Sh=O(c2), Y = 0, monophasic viscoelastic, macroscopic behav-
ior. The case is discussed in great detail in the present paper, along with the special case of a slightly compressible fluid component. This monophasic, viscoelastic case has been also suggested by Buckingham [l 11, who derived a similar equation primarily through heuristic considerations.
We intend to investigate Models A and B in a subsequent paper.
I. I. Geometry of the medium
The porous medium we propose to study is obtained by a periodic arrangement of the pores. The formal description goes along the following lines:
Firstly we define the geometrical structure inside the unit cell Y=]O, I[“, n=2,3. Let %Yy, (the solid part) be a closed subset of @ and 5Yr = GY \ Y/s (the fluid part). Now we make the periodic repetition of ?Vs all over R” and set “Yf = @Yy, + k, k E Z”. Obviously the obtained closed set E, = UktZn @ is a closed subset of R” and Ef = [w” \ Es in
R.P. Gilhrrt. A. Mike/it I Nonlincur Anu1ysi.s 40 (2000) 185~-212 189
an open set in R”. Following Allaire [3] we make the following assumptions on ?Yr
and Ef : (i) ?Yr is an open connected set of strictly positive measure, with a Lipschitz boundary
and -?IS has strictly positive measure in !p as well. This condition is used only for
the decompositions (2.41), (2.42). (ii) Er and the interior of E, are open sets with the boundary of class Co.‘, which are
locally located on one side of their boundary. Moreover Ef is connected. Now we see that s1= IO, L[” is covered with a regular mesh of size E, each cell being a cube ?Yi’, with 1 5 i 5 N(E) = 1011:-“[ 1 + 0( l)]. Each cube !YF is homeomorphic to 9, by linear homeomorphism II::, being composed of translation and an homothety of ratio 11.5.
We define
#j = (~~)-‘($/S) . and ‘?yY;, = (nF)-‘(~g~).
For sufficiently small E > 0 we consider the sets
r,: = {k E Z]?Y$ c n},
K,: = {k E zn]a;k n 82 # S}
and define
Obviously, (soi = X2 U S”. The domains 0: and 0; represent, respectively, the solid and fluid parts of a porous medium R. For simplicity we suppose L/c E N”.
1.2. Two-scale convergence
In order to prove the main convergence results of this paper we use the notion of two-s&e convergence which was introduced in [21] and developed further in [4].
Definition 1. The sequence {w’} C L2( Q) is said to two-scale converge to a limit
w E L2(Q x ?Y) iff for any 0 E C”(Q; CPyr(?Y)) (“per” denotes I-periodicity) one has
Lemma 1. From each bounded sequence in L2(Q) one can extract a subsequence
which two-scale converges to a limit w E L2(Q x Y).
Proof. See [21]. q
Lemma 2. (i) Let wc and EV~W” be bounded sequences in L2(Q). Then there exists u
jimction w E L2(Q; H&(Y)) and a subsequence such that both w’: und E&W” two-scale converge to w and U,,w, respectively.
190 R.P. Gilbert, A. Mikelit I Nonlineur Analysis 40 (2000) 185-212
(ii) Let w” and V,wC be bounded sequences in L2(Q). Then there exists func-
tions w E L’(Q), UE L2(Q; Hi,,(%)) an a subsequence such that both w’: and VXw” d
two-scale converge to w and V.r~(~, t) + V,v(x, y, t), respectively.
Proof. See [4,21]. 0
Remark. Let CJ E L&(W), define o’(x) = c(x/E), and let the sequence {w”} c L2(Q)
two-scale converge to a limit w E L2(Q x Vu>. Then {aCwC} two-scale converges to a limit 0w.
If we have two different estimates for gradients in the solid and in the fluid part, then the classical way to proceed is by extending the deformation from ni to fl and then passing to the limit E -+ 0. The recent results on homogenization of Neumann problems in perforated domains of general type (see [2] and also the book by Jikov et al. [ 171) suggest that it is enough to suppose the Lipschitz regularity for the pore boundaries.
In this section we use the following notations: let R :=]O,L[“, n = 2,3 denote an L
cell, 0: be the solid part of R, and C$ be the fluid part of 0. The fluid-solid interface is indicated by I?[: := 80: n 60;. We assume that in the solid part, GE, the equations of linear elasticity hold
gSJ = AD(u”), D(u’:);j := - ; (!!c+Z), (2.1)
where U’ is the displacement of the solid part and 8”’ is the stress tensor. The equations
of motion are given by (32UC
ps T - div(AD(u”)) = Fp, (2.2)
in Qi x 10, T[. In the fluid part we have the Stokes system describing the motion
(32u’:
pf at2 - div( crf9”) = Fpr
in 0; x IO, T[., whereas
(2.3)
of4 :=
div g = 0 in RF x IO, T[,
(2.4)
(2.5)
where &“/at is the fluid velocity and p’: is the Iluid pressure. The transition conditions between fluid and solid parts are given by the continuity
of displacement
[u”] = 0 on Q x 10, T[, (2.6)
R.P. Gilbrrt, A. Mikcli? I Nonlinecu Anul_vsis 40 (2000) 185-212 191
where [ . ] indicates the jump across the boundary, and the continuity of the contact force
8”: ” = gfJ \’ on I? x IO, T[. (2.7)
At the exterior boundary we
{a’:, p”} are L-periodic.
To simplify our discussion we that is
z&X, 0) = 0,
:(x,0)=0 in R.
assume periodicity, namely that the
(2.8)
assume that there is no flow or deformation at t = 0,
(2.9)
We may reformulate the system, Eqs. (2.1)-(2.9) as a variational problem, namely
find uc E C’([O, T]; Lier(0)“> n C([O, T];H~,,(fit)“) and p” E g’(]O, t[;L2(02;)) such that
s 2/D(d(t)) : II(cp)
n;
+ s
AD(d(t)) : D(q) - p” div cp = FP,: cp,
62: s f2; J’ fl
Vq E Hde,(0)” in S’(]O, T[)
where
u”(0) = 0 and g(o) = 0,
(2.10)
(2.11)
div g = 0 in fig x IO, 7’[, (2.12)
Pr: = iqPf + Xrt;Ps, (2.13)
and XA is the characteristic function of A.
It is trivial to establish the following a priori estimates starting with Eqs. (2.1)- (2.9), using the C’([O, T]; Hder(0)“) short-time regularity and setting cp = du”/dt as the test function in (2.10):
azP
II-II at 5 C IlFll~~~~o,~~xn)~ =C(F), (2.14) LZ(O T.L2(I2)“) . 9
(2.15)
I C(F). L*(o.T;L*(R;.)“‘)
(2.16)
192 R.P. Gilbert, A. Mikelib I Nonlinrur Anulysis 40 (2000) 185-212
On the other hand by writing D(u”(t))=~‘D((&P/d~)(n))dy with Schwarz’s inequality we have
from which
IID(u”)Ill”(o,r;Lz(n~2)I: C(F).
Moreover, if F E Hk(O, T; L2(0>n) then
ak+lUti
II II i?tk+' 5 C IlFll W(IO,T[~R~ =@9.
L-(O,T;L*(R)")
Likewise
(2.17)
(2.18)
(2.19)
For future discussions, we assume, for simplicity of exposition, that
F E H’(0, T; L2(Q”). (2.20)
It is necessary for us to extend the definition of the pressure field which is initially defined only on 0; x 10, T[, to all of 0” x 10, T[. This poses no difficulty, as we set
$(x, t) - h R, p”(x, t) dx,
j’(x, t) :=
I--
1; x E Cl;,
IAl Q!. s p”(x, t) dx, x E 0’: S’
f
Then we may replace p” by 5” in (2.1)-(2.9) and, moreover, we get
(2.21)
II i” IIfP(o,r;L;(n))l c (2.22)
IIVF IIH’(O,T;H,P,‘(12)“)I c. (2.23)
2. I. Two-scale convergence
The a priori estimates (2.14), (2.23) and elementary properties of two-scale conver- gence imply that for Vt E [0, T] there are subsequences {u’:, p”} such that
U” + u”(x, t) in the two-scale sense (2.24)
VU” ---f V_I~o(x, t) + V,U’(X, y, t) in the two-scale sense (2.25)
0 j” -+ p (x,y, t) in the two-scale sense (2.26)
R. P. Gilbert, A. Mikelit I Nonlineur Anulysis 40 (2000) 165-212 193
where
pa(t) E C( [O, T]; L2(C2 x Y)),
u” E C2([0, T]; H;,,(R)“)
and
u’(t) E C’([O, T]; P(n,H;,,(!Y)/R)“).
The convergence statements (2.24)-(2.26) mean that Yq E C~(~1; CEJ!Y)) we have
~l$‘~%+.~) dxi~~Jjip’(x.“.‘)~(“,y)dgdr. In what follows we obtain further properties of the limit functions:
Lemma 3. The limit functions u’, u’, und p” satisjj
div,!$(x,t)+div,,.$(x,y,t)=O /
in IO, T[ x R x 3f; moreover,
p” = - fl h I
~~I p’(x, t) dx =: B(t)
on IO, T[ x R x !?I,, und
./I p” dy dx = 0 on IO, T[.
(1 j
Proof. Let cp E C,“(Q C,yJ !y)), supp cp E R x !Vr. Then
(2.27)
which implies (2.27). Now let the test function be cp(x,y) = cpl(x)q~(y), cp1 E Cc(f2) and (~2 E C,“(!Y,).
Then
194 R.P. Gilbert. A. Mike& I Nonlineur Andy.+ 40 (2000) 185-212
On the other hand
I FVI(x)(P2 (;) = l xq@P,(x)cpz (5)
=1-~(1,p”~x).(x)~;(y)
+- (E&Ji,, p’dx) ~~,p+Pz(Y)dY.
Therefore
P’(x, Y> t> = - ;% h s
“; p”(x,t> dx = B(t)
on all components of IO, T[ x R x +YS. 0
We next pass to the limit as E -+ 0 in the variational formulation (2.10)-(2.12). We assume, in what follows, that cp E Cpyr(D>n and $ E Co-(0; C,$(@))“. Then it follows that
~“~~~,)(,(x)+Ei(x,~)) dx+$(r)~dx
where p := l”Y,lpf + Igy,lp,. Moreover, we have
(2.28)
s, Xn~~mfw) : D (q + 4 (x, ;>>
-JJ 4ab"w> +wdm : (&(cp)+qd$(x, Y>)> (2.30) R I,
+- JJ PO div, $(x, Y I- cl !Y
l (l,p’dy) div,cp, (2.31)
(2.32)
R. P. Gilbert, A. Mikelit I Nonlineur Anu/_vsis 40 (2000) 185-212 195
Finally
u”(0) --+ u’(O) = 0 in the two-scale sense,
$0) ;lo + -(0) = 0 in the two-scale sense,
D(u’:(O)) -+ &(u”(0)) + DJu’(O)) in the two-scale sense.
(2.33)
Lemma 4. The triplet {u’,u’, p”} E H3(0, T;L2(d2)“) n H*(O, T;H&(G)") x H2(0, T;
L2(QH'(?Y)/R)") x H'(0,T;L2(R;L2(T3))) sati$ies the system
(2.34)
V$ E L2(0, T; H;,,(Yy ), ‘dt > 0, (2.35)
p” = B(t) (unknown) on (0, T) x R x TYy,, (2.36)
UO(0) = $0) = 0, u’(O)=0 (2.37)
div, u’(x, t) + div, u’(x, y, t) = 0 in R x IO, T[ x CC!/‘. (2.38)
We note that the two-scale limit of the Laplace transform of the system (2.10)-(2.12) was obtained in [15]. They have considered the uniqueness for the Laplace transform of (2.34)-(2.38) with U’ eliminated and not the full system (2.34)-(2.38).
2.2. Uniqueness
Lemma 5. The system (2.34)-(2.38) has a unique solution.
196 R.P. Gilbert, A. Mikelit I Nonlineur Anulysis 40 (2000) 185-212
Proof. It is sufficient to prove that for F = 0 we have only the trivial solution u” = 0, U’ = 0 and p” = 0. First we set cp = duo/& and $ = au’/& as the test functions in
hence, J,# ~‘dy=: C(t). Since u” E 0 and U’ E 0 we have so sjV p”=O which implies C(t) G 0 which is uniqueness. Cl
2.3. Derivation of the effective equations of’ u”
The system (2.34)-(2.38) is too complicated to be used directly. Since we have established uniqueness above, we seek solutions to (2.35) in a particular form involving separation of scales.
In particular, we seek for {u’ , p”} in the form
(2.41)
s I
+ n’j(y, t - ~)(D~(u’))~(x, z)dt 0
+c~'(v>(Dx(uO>>;j(x,t)
1
for y E +YY,. (2.42)
Let us now derive equations for new functions in representation formulas. The incompressibility condition (2.38) implies
+ J’ t 1 d5’j - dtv,dt(yTt - r)(Dx(u”))lj(x,r)d~
0 2P
+ $ div, t”(y, O)(Dx(u”)),(x, t) = 0.
Consequently, we choose
div ,Zij = -6.. ? ‘J
in g f>
(2.43)
(2.44)
198 R.P. Gilbert, A. Mikrli? I Nonlineur Anulysis 40 (2000) X85-212
divy cj(y, 0) = 0 in Yr, (2.45)
div, g(y,t) = 0 in YY~ x 10, T[. (2.46)
Now we substitute the decompositions (2.41) and(2.42) into (2.35) to obtain
Therefore, a natural choice is to determine {Z’j,b’j} in Y/f by the Stokes system
.I(
ei @ ej + ej C3 f?i
2 + D,(Z’j) - b”Z
> : Dy($) = 0, V$ E H;,,(Y)”
!(/f (2.47)
div, Z” = -6, in Yu~, Z’j E HA,(Y)/R {Zij, b’j} is Y-periodic
In Y/S, Z’j is determined by the following linear elastic system:
div, A {(
ei ’ ej i ej ’ ei + Dy (Zij)) } = 0 in Y,
ZVl d!V,\&
= ziil clYf\?!Y ’
z” E H;er(Y)” (2.48)
Next we formulate the auxiliary problem for the initial value of the kernel t’j:
/.I (DJ i”‘(O)) - c’jl) : Dy( $)
. Ii !Yf
ei C3 ej + ej 63 ei
2 + D,(Z’j) : Dy($) = 0 V’Ic/ E H;,,(Y)
div,<ii(0) = 0 in Yv~; c”(O) E Hd,,(Yf)
R. P. Gilbert, A. Mikelit I Nonlitwrr Anulysis 40 (2000) 185-212 199
Using (2.48) we rewrite it as a problem in ?Yf:
J’ (D,( (“(0)) - C’jZ) : Dy( I))
!//,
-.I’ ( A ei C3 ej + ej 8 ei
2 + DJZ’i) vll/ = 0 t/lj E H&(Y)
i !V, \; !Y >
OK
-kA,t”(O) + VC~ = 0 in ?Ir
div, <‘j(O) = 0 in ?Y?;/f
(o,.(tij(O)) _ $1). \, = A (
e~@c$r~@e~ + D,(zli) >
. \,
on ZYf \ a?!/, 4” is g-periodic
Then the kernel functions <‘j and zij are given by
(2.49)
.II,, (UI (G) -n”i) :D,v(i)+$~,, AD,@):
D,.($)=O Vt,bEH;,,(Y)
ej(O) is given in ?Yr by (2.49)
(2.50)
div, $ = 0 in ‘2,; {(‘I, r&j} is Y-periodic.
By choosing the mean of Zij in an appropriate way we can get s,Y u’ dy = 0. However,
since we need only D,(u’) we assume the above equality without fixing the mean 0f .F.
Now, (2.34) becomes
1 f
+z 0 .I Dx(uo)&, r)AD,@(t, t - T)) = jiF.
200 R.P. Gilbert, A. Mikeli? I Nonlineur Anulysis 40 (2000) 185-212
Let US define the symmetric tensors ~2, &J’, and %? by
(2.51)
(2.52)
+-4!-l,S (2 + :e) (y,t)dY. (2.53)
Then the function u” is determined by
- divX{.gDX(uo))
-div, ’ V?(t - r)D.r(uo(r))dr = DF in R x 10, T[, (2.54)
u’(0) = 0 in R, g&O)=0 in R. (2.55)
Problem (2.54), (2.55) is a linear evolution equation for u” only and much simpler than the two-scale system (2.34)-(2.38). It remains to prove that (a) the fourth order tensors &, 98 and % are well-defined, i.e. that the auxiliary prob-
lems (2.47)-(2.50) are well posed. (b) the problem (2.54) (2.55) admits a unique solution. We answer all those questions in Section 2.4.
2.4. Well-posedness for auxiliary problems
Lemma 6. There exists a unique Z’j E Hd,,(%f)“/R and bli E L2(‘Yf) which satis-j
i( ei @ ej + ej 63 ei
. d, 2 + D,y(Z’j) - b’jZ
> : D,($) = 0,
V$ E H;,,(Y)” (2.56)
div, zij = -6li in JYf,
i.e. the problem (2.47) is uniquely solvable.
R. l? Gilbert. A. Mikrlit I Nonlinrur Anulysis 40 (2000) 185-212 201
Proof. There exists vectors A such that
A - j&j. A E Hd,,(LVf )“/R and div,, A = - 1 in ?Yf.
Let
and let
a(u, v) := J pv(~P,v(~) - d, J b’jdiv,, v.
iv,
Let
P := {z E Hd,,(?Yf)“/R : div.,2 = O}.
Then for v E v we have a(v, u) = J!Y, (D,“(u)[‘. Jf a(v, u) = 0, then u is a periodic rigid
motion. Since Ef and ?Yr are connected, v is constant and as we suppose .[!,,, v = 0, then u = 0. Therefore it is easy to prove that there is an CI > 0 such that
Now by the Lax-Milgram lemma we have the desired result. 0
Lemma 7. There is a unique Z’J E Hd,,(Ys)” such that
div, {(
A k(e; @ <i + ei ~3 e;) + D,(Zii) = 0 E ?Vs
(2.57)
(2.58)
i.e. the problem (2.48) admits a unique solution.
Proof. Using the extension theorem for periodic Lipschitz sets from Acerbi et al. [2],
there is an extension ZIJ of Z’j to ?YS such that
1, @“I2 5 k, s,, IZ’j12,
J jVZii12 < k2 !Y J IVZ’jl2.
!Vf
(2.59)
Now set u = Z” - Z”, then a solution to the Dirichlet problem for the system of linear elasticity in ?Yy, is assumed by the Kom inequality and the Lax-Milgram lemma. 0
Remark. We note that Z’j E Hd,r(Y)‘.
202 R.P. Gilbert, A. MikeliC I Nonlinear Analysis 40 (2000) 185-212
Lemma 8. The fourth-order tensors d’, dejined by
for i,j,k,/E{l,..., n}, is positive dejinite.
Proof. Let A be any symmetric matrix. Then
Let
z” := c AijZij and b’. := c &b’j. i,j ij i<j i<j
Then we have
s (A + DY(Zi.) - b’.Z) : L&(G) = 0, \Jll/ E f$_,(%)”
I,
and
div, Z’. = -Tr A.
Consequently,
&‘AA = s
(A + DY(Zi.) - b’Z) : A !(/f
ZZ / N/f
IA + D,(Z”)12 - / (A + DJZ’) - b’Z) : &(z”) !Y<
- J’ b”1 : (A + D,(Z”)) Af
= J’ ‘!Vf
IA + D,(Z’.)12 - 1 b’.(div, Z’. + Tr A) !Yf
= s IA + D,(Z”)l* !Yf
= Aij(A” + Dy(Zij))12
i-Cj
(2.60)
The last expression has at most quadratic growth. We now prove that the last expression is not degenerate. If A + DJZ’) = 0 then aZ,“/ayi = -Aii which implies Z/ = -Aiiyi +
R. P. Gilbert. A. MikrliL I Nonlinrur Analysis 40 (2000) 185-212 203
&Yl? . . . , yi_l, yi+l,. . . , Yn) and Vb’. = 0 implies 6” is a constant, and hence, b” = 0.
In R2 we have
where we set 8Z~/~?y2=cp1(y2):=ctyz+q, and dZi/C?yl =q2(y1):=c~Yl fc4. Hence, CI + c2 = -2A12, and
Z,“=-Ally, +c1y2+c3,
Z; = -A22Y2 - (2A,2 - Cl )y, + c4. (2.62)
Z/ and Z; are periodic and Ef and fyf are connected. This implies that CI = 0, A12 = 0, A,, =O, A2? =O.
The case for R3 may be similarly treated. This implies that for n = 2,3 there exists
an LY > 0 such that
.d’hA > alA12. 0 (2.63)
Lemma 9 (Variational form of the problem (2.49)). The problem to ,find tji(0)~
H’(?gf)/R, where div,,c’j (0) = 0 in g~f, and cij ELM such that
-.I’ ( A ei @ ej + e, @ e;
2 + D,,(Z’j) . VI) = 0,
i!V, \i!V >
‘d$ E Hd,_,(?Yf )” admits a unique solution.
Proof. We note that
I ( A ei @ e, + <j 8 ei
2 + D,(Z’j) . VI)
i”/, \??I 1
=-I (
A e, C3 <j + ej C3 ei
2 + Dy(Zij) : D,( $)
I, > (2.64)
which is well-defined by Lemma 7. Now the Lax-Milgram lemma applied to the Stokes system (2.49) gives the result. 0
Lemma 10 (Variational form of the problem (2.50)). The problem to jind 5” E L”(0, 7’; H;,,(g)“) n H’(0, T; H;,,(Yf)“), and xii E L2((0, T) x Yf) such that
= 0 in ?Yf, such that (‘j(O) is given by (2.49), and J p = 0 !!I has a unique solution.
204 R.P. Gilbrrf, A. MikeliC I Nonlineur Anulysis 40 (2000) 185-212
Proof. Let
W := t+b E Hd,,(Y)“Jdiv, $ = 0 in ~J’v,,
Then (2.50) reads
AD,@) : D_,$ = 0, M,b E w, <‘j(O) E W.
(2.65)
Let WN := span(c), , 42,. . . ,4~} where the r/j, form a basis for W. Then the Galerkin approximation for (2.65) is
Find 5’ = c,y=, G!r(t)4j(v) such that
(2.66)
Since { $,}+i ,,,, is a basis for WN, the matrix
{./ “/, Dy(&) : D.v(cb/)
k,l=I....,N
is a Gramm matrix and (2.66) has a unique C” solution on some interval (0, T(N)). After using in (2.66) the test functions tN and d(N/&, respectively we obtain
d
P dt +, J’ lDy(SN )I2 + .i’ &(rN) : Dy(tN> = 0,
!Y$ (2.67)
(2.68)
for t E (0, T(N)). Eqs. (2.67), (2.68) imply the global existence for tN. Also, using (2.67_), (2.68)
we pass to the limit N 4 o in (2.66) and obtain that every cluster point 5 satisfies (2.65). Since the problem (2.65) has at most one solution in W, we have [ = 5.
Now, existence of rcq E L2((0, T) x %Vy,) satisfying the first of Eqs. (2.50) is obvi- ous. Uniqueness of rev is a consequence of the surjectivity of the operator div and assumptions on the geometry. 0
It remains to prove that (2.54), (2.55) has a unique solution
u” E H3(0, T;L2(s2)“) n H2(0, T; H;&)“).
Study of the problem (2.54), (2.55) depends on the definiteness of the tensors d and 99 and of the convolution kernel tensor W(t). We have proved in Lemma 8 that d is positive definite. Establishing the same property for B’ and g(t) is not clear. However, we have the following result, which is sufficient for the study of (2.54), (2.55):
R.P. Gilbert. A.
Lemma 11. Let G(y) be the
order tensor
cc? = yxz + .% + @(y),
is positive d<jinitc.
Mikrlit I Nonlineur Anulysis 40 (2000) 185-212 205
Laplace transform of the tensor V(t). Then the jburth-
?;>o (2.69)
Proof. We note that for a symmetric matrix A
!YhA = 2p, I
(A + &(I’“)) : A - .I
p”TrA + A(A + &(I+)) : A, (2.70) . d, !V,
where {I?, p”} are defined by the problem (Laplace’s transform of (2.54)): Find ri E Hder( Y/)/R and p E L2( gf) such that
[ ~N/,.(A + D@)) : W,4 - li, pi div,. ti f
i .I + A(A + D@)) : L&l/b = 0 ‘V,
VI) E H,‘,,(Y) (2.71)
I div,. r” = -Tr A in Z/r.
Obviously, the problem (2.71) has a unique solution for all y > 0. Then using (2.71), (2.70) becomes
YAA = 2py J
IA +D,,(rL)/2 + A(A + oy(ri)) :(A +D).(r")) > cw(il)lAi* Y, I IV,
which proves the lemma. 0
Existence for the problem (2.54), (2.55) can be proved using the methods of Laplace’s transform or the variational methods (see e.g. the exhaustive textbook Dautray and Lions [ 141). However, we have existence by construction and only uniqueness is needed.
Since we are interested only in evolution on [0, T], we suppose F = 0 for t > T and t < 0. Then the constants in the uniform a priori estimates (2.14)-(2.16) have at most a polynomial growth in T and the domain of existence of the Hilbert-space-valued distri-
butional Laplace transform 9’(u”) is IO, +cc[. For more details on Laplace’s transform of Banach-space-valued distributions we refer to Dautray and Lions [14].
We have the following result.
Lemma 12. Let v E .@+(R, H&(fl>n), e@‘v E L2(0, fx; H&(02>n) and e&l dv/dt E
L*(O, +x;H&(~2)“), 6 > 0 be a solution for (2.54), (2.55). Then such solution is unique.
Proof. Let us consider a solution u corresponding to F = 0. Then, due to the assump- tions, we can apply Laplace’s transform to (2.54). We obtain
3. The slightly compressible monophasic elastic behavior
As we assume the fluid is slightly compressible, we are able to remove the pres-
sure from Eq. (2.4) by assuming that the variation of pressure from the rest state is
small and that Ap x -c2pf divu’:. This can be considered as a “regularized” version
of the incompressible case from Section 2 and we just give a short outline of the
homogenization procedure. Now the variational formulation (2.10) becomes: Find U” E C’([O, T]; L2(fl)n) fl
C([O, T]; Hde,(Sl>n) such that
d’ - s dt2 (1,
+&f .I’ div u” div cp = I
FP,:R VV E H;,,(W fl2/ . (1
where we again take the initial conditions to be
u”(O) = 0 and $0) = 0,
(3.1)
(3.2)
(3.3) Pr: = Xrl:Pf + Xn:Ps.
R.P. Gilbert, A. Mikeli? I Nonlineur Anulysi.s 40 (2000) 185-212 207
The estimates (2.14)-(2.16) apply once more. Passing to the two-scale limit is iden- tical, only we replace Eq. (2.3 1) in Section 2 by
c’(,rL divuCdiv(cp+c$ (x,:))
vcp E c;JG)“, vlj E C,“(O; cg(YYy). (3.4)
Following the analysis of Section 2 we have that the doublet {u”, u’} E H3(0, T; L2(fl)n) n H2(0, ?“;H&(02>n) x H2(0, T; L2(s1; ff’(?g)/R)“) satisfies the system
d2 - J’ dt2 r!
DuO(Ocp + C2Pfl% s
div, u’(t)div, qn + c2pf R
+ C2Pf .IJ’
div, u’ div, II/ dy dx = 0, R i)/f
‘d$ E Lz(O, T; Hi,,“).
uO(0) = Z(O) zz 0, u’(0) = 0.
(3.5)
div, $ dy )
(3.6)
(3.7)
As before to show uniqueness it is sufficient to prove that for F = 0 we have only the trivial solution u” = 0, u’ = 0. Taking qn = du’/dt, and II/ = du’ldt as test functions in (3.5), (3.6), we obtain after adding the two equations together that
208 R.P. Gilbert. A. MikeliC I Nonlinrar Analysis 40 (2000) 185-212
Now as in the Section 2 we argue that duo(t)/& = 0, which because of the initial conditions implies u” = 0. Again we have that DJu’(t)) = 0, which implies by the previous arguments that u’(t) = 0. This is uniqueness.
As before we construct the solution U’ in a special form, namely
J t
u’(x, y, t) = c ~~(y)(&(~“))ij(~, t) + B”(JJ, t - Z) (D~(u”))tj(X, t) dr
i,j 0
(3.9)
Then
(3.10)
Now we substitute into (3.6) to get
+ (DxU”);jDy(Bi(0)> 1 : Dy($)
+m/ .I’ ADy(B’j(t - ~>)(Dx(u”))ij: Dy(lc/) i,i f 0
+ C {l I,,(D,z)ijA ( ei ’ ej 1 ei @ ei + D,(RY)) : Dy(t+!/) i,i
+ C2Pf JJ (divy$){S;j(D,(Uo)), + (divYA;‘)(Dx(Uo));j) n ?Yf
J/J t
C2Pf div,(B’j(t - Z))(Dx(u”));j divy$ dz = 0, V$ E Hd,,(%‘4/)“. R ‘!)/f 0
Therefore a natural choice is
e; @ ej + ej 6% ei
2 + D,(A’j) : D,($) = 0, V’rcI E f&(Y)“.
(3.11)
R.P. Gilbert, A. Mikelit I Nonlinwr Anulysis 40 (2000) 185-212 209
In TS, Ali is described by
div ‘; A CC
e, 8 ej + ej 8 f?i
2 + D,(A’j)
1) = 0 in Yy,,
(3.12)
AiiJ+,,\,:,,/ = Aii~~,v,\~y, AIJ E Hd,,(YJ.
We now compute the initial values for the kernel B’j,
I 2P s &w(O)): D,.($)
!1/,
+ c2pr .I’
,~ (div,A’j + &,) div,.$ = 0, ‘Y’$ E Hi,,(Y) f
@(O) E H;,,(Yf).
(3.13)
As in the previous section we rewrite (3.13) as a problem in %‘r:
i
-pAJU(0) = c2prOF divJA” in Y/f
(2@,,(B’i(0)) + c2pr(div,A” + S,)Z) . v = 0 on ZYr \ &??I.
We now construct the kernel function for B’i:
Gl,,D, (3D.M +~,,AD,(B”):D?.(i)
fC2 Pf s
divl.Biidiv,.$ = 0, !Yr
‘J$ E Hd,,(Y),
where B’j(0) is given in ‘?Yf by (3.14). Now, (3.5) becomes
Uniqueness for (3.20), (3.21) is proved as for the analogous equation in Section 2. It is reduced to proving that the effective tensor composed of the Laplace transforms y&.yIco + B.s[CO + +co is positive definite. Establishing the required positive definiteness is along the same lines as in the proof of Lemma 11, with - J!Y, p”Tr A replaced by
c2pr sfY, (Tr h $ div, I’“) Tr A. In fact, the slightly incompressible case can be considered as a penalization of the
incompressible case. Even if the equation seems to be a bit simpler, in fact it is not the case since c2 is big and the terms like divyd’j + bkl are likely to be small. Therefore, its correct study requires knowledge of the incompressible case.
4. Concluding remarks
The homogenized behavior of the incompressible viscous fluid in the elastic porous medium is described by Eqs. (2.54) and (2.55). This equation enters the equations of the linear viscoelusticity, i.e. our mixture consisting of an incompressible viscous fluid and an elastic solid has an overall viscoelastic behavior.
This confirms the observation made by Burridge and Keller [ 121, who found by means of an asymptotic expansion the effective transformed equations. They observed dependence of the tensors in effective constitutive laws on y, i.e. viscoelastic effects.
R.P. Gilbert, A. Mikrlit I Nonlinrur Anu1.W 40 (2000) 185-212 211
However, they did not get the problem (2.54), (2.55). Also their argument was only formal. We should add that Biot [25] also introduced viscoelastic effects into his system of equations, but in a rather heuristic manner; i.e. he assumes certain coefficients after one has transformed to the frequency domain are actually dependent on the frequency CO. In the time domain these coefficients become nonlinear differential operators of d/at. He then linearizes these operator using the method of Kryloff and Bogliuboff. Hence, in a sense the present paper may also be thought as a justification of Biot’s heuristic approach.
We note that the effective stress in (2.54) has three constitutive parts. The first is .&‘&(c%~/&). d is a viscosity tensor and represents a contribution coming from the effective elementary stresses in the fluid.
The second part is S?&(u”). 99 is an instantaneous elasticity tensor and it contains the combined effects of the effective elementary stresses from the elastic structure with the effective response of the fluid to the stress from the elastic part caused by the flow.
Finally, we have the term sd %‘(t - r)Dx(uO(r))dr corresponding to the long memory effects. It is a spatial average of the time evolution of the stresses due to the interaction fluid/structure, which is initially caused by the flow effects.
We note that the &xtive physical dejbrmation is lu’, due the scaling in the introduction. Also, it is important to note that since the gradient of u’. is uniformly bounded in L*, the inertial term coming from the fluid part is much smaller than the first time derivative of the fluid velocity. This justifies neglecting the inertia and dealing with the Stokes system.
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