CC BY
8
УДК 517.958
Homogenization of Acoustic Equations for a Partially Perforated Elastic Material with Slightly Viscous Fluid
Alexey S. Shamaev* Vladlena V. Shumilova^
Institute for Problems in Mechanics of RAS Vernadskogo, 101-1, Moscow, 119526
Russia
Received 15.04.2015, received in revised form 10.05.2015, accepted 25.06.2015 In this paper a mathematical model describing small oscillations of a heterogeneous medium is considered. The medium consists of a partially perforated elastic material and a slightly viscous compressible fluid filling the pores. For the given model the corresponding homogenized problem is constructed by using the two-scale convergence method. The boundary conditions connecting equations of the homogenized model on the boundary between the continuous elastic material and the porous elastic material with fluid are found.
Keywords: homogenization, two-scale convergence, heterogeneous medium. DOI: 10.17516/1997-1397-2015-8-3-356-370
In 1989, Nguetseng [1] introduced the notion of two-scale convergence, which provides a new approach in the homogenization theory. The method of two-scale convergence was further developed by Allaire [2] and generalized by other authors (see, e.g., [3-6]). As it turns out, this method is especially useful for studying homogenization problems whose solutions do not have a limit in the classical sense (for example, in the L2-norm). In applications, such problems describe some physical processes in heterogeneous media, for example, a diffusion process in highly heterogeneous media [2] or a joint motion of an elastic skeleton and a slightly viscous fluid [7]. Recently, the method of two-scale convergence is widely applied in the homogenization of various mathematical problems that arise in mechanics of heterogeneous media (see, e.g., [8-14]).
In this paper, we consider a mathematical problem that describes small oscillations of a heterogeneous medium consisting of a partially perforated elastic material and a slightly viscous compressible fluid filling the pores. We assume that the elastic material is inhomogeneous with e-periodic microstructure, and the structure of the perforation in the porous part of the elastic material is also e-periodic. The mathematical problem under consideration involves the linear elasticity system describing the motion of the elastic material, and the Stokes system describing the motion of the fluid. Finally, the problem is complemented by homogeneous boundary and initial conditions. Using the method of two-scale convergence and the Laplace transforms, we construct the corresponding homogenized problem and find the boundary conditions which connect equations of the homogenized problem on the boundary between the continuous elastic material and the porous elastic material with fluid. In addition, using the notion of strong two-scale convergence, we establish some corrector-type results under suitable smoothness assumptions on the solution of the homogenized problem and on the external force. When an
* sham@rambler.ru tv.v.shumilova@mail.ru © Siberian Federal University. All rights reserved
elastic part of the heterogeneous medium is completely perforated, the corresponding homogenization problem was analyzed in [7,9,12] and [15]. Namely, the first research of this problem was carried out in [15], and later the homogenized model was mathematically rigorously justified in [7] by using the method of two-scale convergence. In [9] and [12], this homogenized problem was derived in the form that is known from the classical physical papers such as [16] and [17].
1. Statement of the problem
Let Q be a bounded domain in R3 with smooth boundary dQ, and let Y = (0,1)3 be the unit cube in R3. We suppose that Q = Qo U Qi U S and Y = Yh U Ys U r, where Qo, Qi, Yh, and Ys are open connected sets in R3, S is the smooth surface that separates Q0 and Q1, and r is the smooth surface that separates Yh and Ys. In addition, we denote by Ypher (respectively, Ypser) the Y-periodic repetition of the set Yh U (dYh n dY) (respectively, Ys U (dYs n dY)) and suppose that both sets Ypher and Yser are connected in R3.
For a sufficiently small e > 0 we divide the domain Q into two subdomains Qh and Q® as follows:
QS = Qi n eYper, Qeh = Qo U Qh® U (dQh® n S), Q^ = Qi n eY^.
We suppose that the set Q^ is occupied by an elastic material, whereas the set QS is occupied by a slightly viscous compressible fluid. In the sequel, the sets Q^- and Q® are called the elastic and the fluid parts of Q, respectively.
Now we are going to state the mathematical problem describing the joint motion of elastic and fluid parts of Q. Let us assume that u®(x, t) is the displacement vector, ekh(u®) are the components of the strain tensor, ekh(u®) = (du|/dxh + du|/dxk) /2 , and f (x,t) G H2(0, T; L2(Q)3) is a given force. The equations of motion in the elastic part Q^- are given by
= + fi(x,t) in Qh X (0,T), (1)
where Po(x) is the density of the elastic material,
Po(x) = po(e-1x), po(y) G L^er(Y), 1 < po(y) < Pi (pi = const > 1),
("per" denotes Y-periodicity), a® are the components of the stress tensor, a® = afjkh(x)ekh(ue), and ajkh(x) = ajkh(e-ix) are the elasticity coefficients such that aijkh(y) G L^r(Y) and
aijkh(y) = ajikh(y) = akhij(y) = aijhk(y), 1 < i,j, k, h < 3,
aijkh(v)iijikh ^ co^ijiij, co > 0, G R, iij = j
Here and throughout this paper, we use the convention that repeated indices imply summation from 1 to 3.
The equations of motion in the fluid part QS are given by
d2u® da®
PW = ~oj + fi(x,t) in Q® X (0,T), (2)
where ps is the fluid density, ps = const > 0, and
f 8ue\
j = Sijp® + e2(n^ijSkh + 2^5ik5jh)ekh [ -^f ) , P®(M) = —Ydivu®(x,t).
Here pe(x,t) is the fluid pressure, Sij is the Kronecker symbol, y = c2ps, c is the speed of sound in the fluid, c = const > 0, and e2n and e2p are the viscosity coefficients of the fluid satisfying the following conditions: p > 0 and n/p > -(2/3)a with 0 < a < 1 [15].
Besides, at the interface Se = dn dQSS we have the continuity of the displacement and of the normal stress:
Kk = 0, [aj nj ]se = 0, (3)
where [•k denotes the jump across the boundary Se, and nj, j = 1,2,3, are the components of the unit normal to Se.
Finally, the problem is supplemented by homogeneous initial and boundary conditions
due
ue (x, 0) = — (x, 0)=0,x G Q] ue(x, t) = 0, x G dQ,t G (0,T). (4)
The variational formulation of problem (1)-(4) is the following: find a function ue(t) with values in Hq(Q)3 such that
r d2ue idne \ C
Pevidx + e2be[ — v + ce(ue,v) = fadx Vv G H1(Q)3, (5)
/n dt2 i ' \dt
diie
ue(0) = d- (0)=0, (6)
dt
where p£(x) = p0(x) for x G p£(x) = ps for x G QS, and
b£(u,v) = (n div u div v + 2peij (u)eij (v))dx,
c£(u,v) = y div u div vdx + aijkhekh(u)eij (v)dx.
Jq%
In the same way as in [15], where the whole elastic part of Q was supposed to be porous, one can show that for any e > 0 there exists a unique solution of problem (5), (6).
Let us extend the vector function f (x,t) by zero for t < 0 and t > T. Next, we convert the evolutionary problem (1)-(4) into the stationary one by using the Laplace transform g(t) ^ g£ in time. Then the variational formulation (5), (6) becomes: for a fixed A with ReA > Ao > 0, find a function u£(x) G H^Q)3 such that
A2 f p£(u£x)ividx + Ae2b£(u£,v) + c£(u£x,v)= i (fx)ivid,x Vv G Hl0(Q)3. (7)
JQ JQ
2. Two-scale convergence
We begin this section with two basic definitions related to the theory of two-scale convergence (see [2,5]).
Let u£(x) be a bounded sequence in L2(Q).
Definition 1. A sequence u£(x) weakly two-scale converges to a function u(x,y) G L2(QxY,dxx dy) = L2(Q x Y), u£(x) u(x,y), if
lim u£(x)ip(x)^(e-1x)dx = / u(x,y)ip(x)^(y)dxdy (8)
£^0J Q JQJ Y
n JnJ Y
'^(Q) and '(y) G C£r
for any functions tp(x) G C^(Q) and '(y) G C™ (Y).
n
It should be noted that the class of test functions used in the above definition can be enlarged. For example, one can takes in (8) test functions y(x) G C(Q) and ^(y) G Lper (Y) or y(x) G L2(Q) and V(y) G Cper(Y) (see [2,5]).
Definition 2. A sequence u®(x) G L2(Q) strongly two-scale converges to a function u(x,y) G L2(Q x Y), u®(x) -— u(x,y), if
lim / u®(x)v®(x)dx = / / u(x, y)v(x, y)dxdy whenever v®(x) - v(x,y). ®i0./q ./q./ Y
Let us briefly recall the main properties of two-scale convergence, the proofs of which can be found in [2,4] and [5].
(i) If u®(x) is a bounded sequence in L2(Q), then there exists a function u(x, y) G L2(Q x Y) such that, up to a subsequence, u®(x) — u(x,y).
(ii) If u®(x) u(x,y), then
u®(x) — J u(x,y)dy in L2(Q), limiinf ||ue(x)||L2(fi) > ||u(x,y)||L2(QxY).
(iii) If a(y) G L^r(Y) and u®(x) -— u(x,y), then a(e-ix)u®(x) -— a(y)u(x, y).
22
(iv) If u®(x) — u(x, y) and lim®io ||u®(x)||L2(q) = ||u(x, y)||L2(nxY), then u®(x) — u(x,y).
(v) If u®(x) -— u(x, y) and u(x, y) G C(Q, L^er(Y)), then
lim ||u®(x) — u(x,e-ix)||L2(Q) = 0.
(vi) Let u®(x) be a bounded sequence in Hi(Q). Then there exist functions u(x) G Hi(Q) and ui(x,y) G L2(Q,Hper(Y)/R) such that, up to a subsequence, u®(x) -— u(x) and Vu®(x) — Vu(x) + Vyui(x, y). Moreover, u(x) G Hi(Q) if u®(x) G Hi(Q).
(vii) Let u®(x) be a sequence in Hi(Qi) such that
I|u®||l2(q1) < C, ||Vu®||i2(Qhe)3 < C, e||Vu®||i2(QS)3 < C,
where C is a positive constant which does not depend on e. Then there exist functions u(x) G Hi(Qi), ui(x,y) G L2(Qi,Hper(Yh)/R), and w(x,y) G L2(Qi,H^(Y)) with w(x, y) = 0 for y G Yh, such that, up to a subsequence,
u®(x) - u(x) + w(x,y), x(Qh®)Vu®(x) - x(Yh)(Vu(x) + Vyui(x,y)).
ex(QS)Vu®(x) - x(YS)Vyw(x,y), where x(D) denotes the characteristic function of the set D. Moreover, u(x) G Hq(Q) if
u
(x) G Hq1 (Q).
Now, using the above properties of two-scale convergence, we are going to study the asymptotic behavior of the solution «A of problem (7) when e goes to 0. Firstly, choosing in (7) v = «A, we obtain
c£(m|, «A) < C, e 26£(m|, «A) < C.
Hereinafter, C denotes various positive constants independent of e.
By using the same arguments as in [7], we deduce that the solution «A of problem (7) satisfies the a priory estimates
II«aIIlw < C, e||V(u^)i||L2(n.)a < C, ||V(ui)i||i2(fih)3 < C, || divuA||L*(ni) < C. (9)
For notational convenience we denote by VuA the 3 x 3 matrix with coefficients d(wA)i/dxj. In order to proceed we need the following crucial lemma.
Lemma 1. Let «A be a solution of problem (7). Then, up to a subsequence,
2 2 «A(x) — for x G ^0, «A(x) — «a(x)+ wa(x, y) for x G Oi, (10)
V«A(x) — VuA(x) + Vyu°(x, y) for x G O0, (11)
X(Oh£)VueA(x) - X(Yh)(VuA(x) + VyuA(x,y)) for x G ^, (12)
ex(^)V«A(x) - x(Ys)VywA(x, y) for x G ^, (13)
where
«a G FoW, «A G L2(Oo,HPer(Y)3/R3), «A G L2(Oi,Hp1er(Yh)3/R3), wa G L2(fii,Hp1er(Y)3), wA = 0 for y G Y*, divy wa = 0 for y G Ys.
Proof. Using the above estimates and properties (vi) and (vii) of two-scale convergence, we have, up to a subsequence,
«A(x) — v0(x) for x G O0, uA(x) — vA(x) + wA(x,y) for x G 01, (14)
where v0 G H 1(^o)3, vA G H 1(^i)3, and wa G L2^,^(Y)3) with wa = 0 for y G Y*. Besides, in virtue of property (vii), we also derive relation (13). Furthermore, relation (13) and the last estimate in (9) yield divy wA = 0 for y G Ys.
To prove that v°|S = v^|S, we extend the perforation from to by setting = One Y^. Then, up to a subsequence,
x(Oe)uA(x) - |Yh|uA(x) in L2(0)3 (see [18]), where uA G H0(O)3. On the other hand, from (14) we obtain
x(a n fi0)«A(x) - |Yh|v0(x) in L2(^)3, x(fiie)uA(x) - |Yh|vi(x) in L2^)3.
It is easy to see that uA(x) = v°°(x) if x G O0 and uA(x) = v^(x) if x G O1, therefore v°°|S = v1 |S.
Finally, relations (11) and (12) immediately follow from property (vi) of two-scale convergence.
□
3. Limiting behavior of the pressure
In this section we study the limit behavior of the fluid pressure p\(x). Namely, we state and prove the following lemma.
Lemma 2. Let u\ be a solution of problem (7), and let p\ = —y div u\ in Q6e. Then, up to a subsequence,
x(Qt)pi(x) A x(Ys)px(x), px(x) e L2(Ql). (15)
Moreover,
i \Y s\ (
divy u\(x,y)dy = \Ys\ divux(x) +--px(x) + div wx(x,y)dy. (16)
Jyh Y Jys
Proof. From (9) it follows that the sequence x№l)pX is bounded in L2(Q1). Therefore, up to a subsequence, we can assume x(^S)pX(x) ^ px(x,y) for x e
Now we take a test vector function of the form v(x) = e^(x)b(e-1x), where <f(x) e ),
b(y) e Hler (Y)3, supp b(y) c Ys. Passing to the two-scale limit in the integral identity (7) as e ^ 0, we have
/ / px(x,y)f(x)divy b(y)dxdy = 0 JnYs
or, since ¥>(x) is arbitrary,
/ px(x,y)divy b(y)dy = 0,
Jy s
which implies px(x,y) = px(x) for y e Ys. Relation (15) follows now from property (iii) of two-scale convergence.
To prove equality (16), we use Lemma 1 and get
divuex ^ divux(x) + div wx(x,y)dy in L2(Q1), (17)
JY s
x(^h)div u\ ^ \Yh \ div ux(x)+ divy ux(x,y)dy in L2(Qi).
Jy h
On the other hand, we already have proved that
Thus,
x(ttse)divux ^ —^px(x) in L2(Qi).
Y
\Y s\ f
div u\ Yh\ divux(x) — ---px(x)+ divy ux(x,y)dy in L2(Q1).
Y Jy h
Comparing the last relation with (17) yields the desired equality (16). □
4. The cell Stokes problem
Now we choose in (7) a test vector function of the form v(x) = £(x)b(e-1x), where £(x) e ) and b(y) e H1er(Y)3 with supp b(y) c Ys and divy b(y) = 0. Passing in (7) to the limit as e ^ 0, using Lemma 1, and taking into account that £(x) is arbitrary, we obtain
/ (X2pswx(x,y) — XpAyywx(x,y) — gx(x)) b(y)dy = 0, (18)
Jy s
where we denote gA (x) = /A (x) — A2psu A (x) — Vp a (x). Since the orthogonal of divergence-free functions is exactly the gradients, from (18) it follows that there exists a function $A e L2(Qi,Hper.(Ys)) such that
A2psw a (x,y) — A^Ayy wa (x,y) — gA (x) = Vy $ A(x,y), x e Oi, y e Ys. (19)
Now, as in [12], we look for a vector function wA(x,y) in the form
WA(x,y) = MI(x)N(y), x e Oi, y e Ys, (20)
where M£ e L2(O1) and N e Hper (Y)3, r = 1, 2,3, are to be specified. To do this, we substitute (20) into (19) and get
MI (x) (AV N (y) — AMAyy N (y)) — gA (x) = Vy $ a (x, y). (21)
Now we set
Mr (x) = (gA)r(x), r = 1, 2, 3; $ A(x,y) = —(gA)r(x)WAr(y). Then from (21) it follows that
(gA)r(x) (VyWAr(y) + AVN(y) — AMAyyN (y)) = (gA)r(x)er,
where er is the unit vector of the yr-axis. Finally, we define the pair (N£(y),Wj[(y)} as the solution of the following Stokes problem:
Vy WI + A2ps N — A^AyyN1 = er, divy N£ =0 in Ys, N =0 on r. (22)
Let us summarize the results of this section in the following lemma. Lemma 3. Let wA (x, y) be as in Lemma 1. Then
WA(x, y) = ((/a)r(x) — A2ps(u a)r(x) — (x)) N(y), (23)
where N£ (y), r =1, 2, 3, are the solutions of the cell Stokes problems (22).
5. Homogenized tensors
Lemma 4. Let «A be a solution of problem (7). Then
aijfchefch(uA) ^ bjjfchefeh(uA) in l2(Oo), (24)
)„E „..(uA ) -------(u a ) , ß p a in L2(
where
x(^ie)aijfchefch(uA) ^qijfchefch(u a) + AJPA in L2(^I), (25)
bijfch = (aijfch + aijimeym(Vfch)) dy, qjfch = (я^аа - ajimeym(Qfch)) dy, Jy Jy h
'Y ./Yh
ву = -/ divy Qij (y)dy. Jy h
(26)
Here Vkh(y) G Hper(Y)3/R3 and Qkh(y) G Hper(Yh)3/R3 are the solutions of the following cell problems:
d
d— K-fch + aijimefm(Vfch)) =0 in Y, (27)
dyj
d
dj iaijkh — aijlmeim(Qkh)) = 0 in Yh, (aijkh — aijimeym(Qkh)) Vj =0 on V, where Vj, j = 1, 2, 3, are the components of the unit normal to the boundary r. Proof. Using the properties of two-scale convergence, we have
a^jkhekh(uEx) ^ J aijkh(y)(ekh(ux) + ekh(u°x))dy in l2(qo), (29)
x(tthe)aFijkhekh(uEx) ^ / aijkh(y)(ekh(ux) + evkh(ux))dy in L2(ih). (30)
JY h
To prove (24), we take in (7) a test vector function v(x) = e^(x)b(e-1x), where <f(x) e C^(Q0), b(y) e Cper (Y)3. Passing to the limit as e ^ 0 and using Lemma 1, we obtain
JY aijkh (ekh(ux) + elh(ux)) e!j (b)dy = (31)
We look for a solution of (31) in the form
u0x(x,y)= Vkh(y)(x), x e n0, y e Y,, (32)
dxh
where Vkh(y) e H^er(Y)3/R3. Substituting (32) into (31) yields
J (aijkh + aijtmeym(Vkh)) ej(b)dy = 0. (33)
An integration by parts shows that (33) is a variational formulation associated to (27). Furthermore, we have
J aijkh (ekh(ux) + evkh(u°x)) dy = bjkhekh(ux).
Comparing the last equality with (29), we derive (24).
It remains to prove (25). For this purpose we choose in (7) a test vector function v(x) = eip(x)b(e-1x), where p(x) e Cq°(Q1), b(y) e C™r(Y)3. Passing to the limit as e ^ 0 and using Lemmas 1 and 2, we get
/ aijkh (ekh(ux) + evkh(ux)) efj(b)dy — px divy bdy = 0. (34)
JYh JYs
We look for a solution of (34) in the form
uxx(x,y) = —Qkh(y)(x) — px(x)Q(y), x e n1, y e Yh, (35)
dxh
where Qkh(y), Q(y) e Hper(Yh)3/R3. Substituting (35) into (34), we obtain two integral identities:
I (aijkh — a,ijlmeylm(Qkh)) ej (b)dy = 0, (36)
jy h
/ aijkhelh(Q)eyii(b)dy — divy bdy = 0. (37)
JYh J Y h
An integration by parts shows that (36) is a variational formulation associated to (28), while (37) is a variational formulation of the following cell problem:
d
d— KjfchekJQ)) =0 in Yh; aijfchekh(Q)vj = v on r. (38)
dyj
For further needs, we extend Qkh(y) and Q(y) from Yh to the entire periodicity cell Y in such a way that the extended vector functions Qkh(y) and Q(y) belong to Hper(Y)3 and
||Qkh 1 |Hper(Y )3 < C ||Qkh||Hper(Yh)3 , ||(Q||Hper (Y)3 < C ||Q||Hper (Yh)3 •
Setting b = Q in (36) and b = Qkh in (37) yields
/ aijfchekh(Q)dy = / divy Qijdy = —flj. JYh JYh
Thus, in view of (26) we have
(efch(u a) + ekh(u A)) dy = qijfchefch(u a) + flijPA.
Yh
Finally, comparing the last equality with (30), we obtain (25). □
It should be noted that the homogenized coefficients bijkh and qjjkh are real and they possess the classical properties of symmetry and ellipticity (see [7] and [19]).
To conclude this section, we substitute representation (35) into equality (16). Then
+ ^Pa + divw° + afchefch(u a) = 0, x G Oi, (39)
where n = |Ys| is the porosity, aj = n^j — flj,
fl = / div, Q(y)dy, „0(*)=/«,(x,y)dy.
Yh Ys
Note that the vector function w° satisfies the boundary condition w° • Z = 0 on dO1, where Z is the unit normal to dO1. Indeed, using (10), we can easily deduce that
lim f div uAdx = f div u Adx + f div w0dx e^0./Q JQ JQi
and the desired boundary condition follows immediately by integration by parts.
6. Homogenized problem
Now we choose in (7) a test vector function v G Hq(O)3, which does not depend on e. Then passing in (7) to the limit as e ^ 0 and using Lemmas 1, 2, and 4, we obtain
A2po / (u a)jvjdx + A2 / (pi («a)i + ps(w°)j) vjdx — n / pa div vdx+ JQ0 ./Q1 ./Q1
+ / bjjfchefch(u a)eij(v)dx + / (qijfchefch(uA)+ flijPA) ej(v)dx = (f ^vjdx, ./Qo -'Q1 -'Q
where
po = Po{y)dy, pi = nps + p^n Ph = Po(v)dy. Jy Jy h
Integrating by parts in (40) and using the results of Sections 4 and 5, we conclude that the differential form of the homogenized problem corresponding to problem (7) is
X2po(ux)i = dj + (fx)i in Qo, (41)
8ax-
X2Pi(ux)i + X2ps(w0x)i = dj + (fx)i in Qi, (42)
(n + P)px + divw° + akhekh(ux) = 0 in Qi, (43)
Y
wx = fr - X2ps (ux)r - dxr) J, NX dy, (44)
w0° ■ ( = 0 on dQi, ux =0 on dQ, [ux]s = 0, [ajnj]s = 0, (45)
where nj are the components of the unit normal to S, aj = bijkhekh(ux) in Q0 and aj = qijkhekh(ux) - aijPx in Qi.
Using the general theory of elliptic problems (see, e.g., [20]), one can prove that there exists a unique solution of problem (41)-(45).
Now our aim is to derive the non-stationary homogenized problem in the original variables x and t. For this purpose, we apply the inverse Laplace transform to (23) and obtain
w(x,y,t)= fr(x,t) - ps^ur (x,t) - dp(x,t)^ * Nr(y,t), (46)
where the symbol * denotes the convolution in t,
gi(t) * g2(t)= gi(t - s)g2(s)ds.
0
Now we set
dNr r
Lr(y,t) = — (y,t), Dr(t) = J Lr(y,t)dy, r = 1, 2, 3.
Since Lr = XNr, it is easy to see that Lr (y,t) is a solution of the Stokes problem 31 r
ps— - pAyyLr + VyWr = 0, divy Lr =0 in Ys x (0, T), Lr (y, 0) = (ps)-! er, y e Ys; Lr (y,t)=0,y e r, t e (0,T). Further, we can rewrite (46) as
w(x,y,t)= fr (x,t) - ps ^ur (x,t) - dr (x,t)^ * 0 Lr (y,r)dr, (47)
and so
/■ t
w0(x,t)= fr (x,t) - ps ^ur (x,t) - dp (x,t)) * f Dr (t )dT.
It is easy to check that
d2w0 ( s d2ur dp N dDr t s d2ur dp
dt2 = / — p■-W — at J * "aT + D(0)lfr — p■-W — dtr)
Finally, we apply the inverse Laplace transform to system (41)-(43). As a result, we deduce that the homogenized problem corresponding to the original problem (1)-(4) takes the form
po idW = dx- j Sh) + 'n °o x <0.T>. <48>
n S+✓ /—✓ ds-—£ 1. dD1+PDr (0) (/r—ps ^—^ a =
dt2 ' ^ V ^ dt2 dxj ' dt ' ^ ^ dt2 dxr/
dfduk \ (49) = dx" ^¿jkhdx--a—Pj + /¿(x,t) in Oi x (0, T),
n / d2u dP \ /"4
( y + fl)P + divj /r — f/^ — dx^J * _/o Dr (t)dr + aije-(u) = 0 in Oi x (0,T), (50)
( (/r — P-^ — Jp") * / Djr(t)d^ Zj = 0 on dOi, [u]s = 0, [ct-n— ]s = 0,
(51)
du
u(x, 0) = —(x, 0) = 0, x G O; u(x,t)=0, x G dO, t G (0,T), (52)
where ct— = bjjfchefch(u) in Oo and ct— = qjjfchefch(u) — ajp in Oi.
Remark that system (48) describes the propagation of acoustic waves in the homogeneous elastic material contained in O0, while system (49), (50) corresponds to the Biot model [17] and describes the propagation of acoustic waves in the heterogeneous medium contained in Oi.
Finally, we analyze the boundary conditions, which connect equations of the homogenized problem on the boundary S between the continuous elastic material and the porous elastic material with fluid. From Section 5 it follows that these conditions depend on the following constants: bijkh, qjjkh, n, and a", where bijkh are the homogenized elasticity coefficients for the continuous elastic material, qjjkh are the homogenized elasticity coefficients for the porous elastic material without fluid, n is the porosity of the elastic material in O1, and a" are the coefficients, which characterize the compressibility of the porous elastic material.
7. Strong two-scale convergence
Our next goal is to prove the strong two-scale convergence in (10) under the additional smoothness assumptions on the solution of the homogenized problem (41)-(45) and on the external force /. Namely, in this section we suppose that /A(x) G C 1(O), uA(x) G C3(O) and p a(x) G C2(Oi).
Theorem 1. Let be a solution of problem (7), and let pA = —7 div uA in O-. Then
lim / |uA(x) — uA(x)|2dx = 0, lim / |pA(x) — pA(x)|2dx = 0, (53)
lim / |uA(x) — u A(x) — wA(x,e-1x)|2dx = 0, (54)
im |e(uX(x) - ux(x) - euX(x,e !x)) |2 dx = 0, (55)
£^0Jn 0
lim ^{uX(x) - ux(x) - euX(x,e-ix))|2 dx = 0, (56)
e^0JnhE
lim e2 ^(uuX(x) - wx(x,e-!x))|2 dx = 0. (57)
Here the triple {ux(x),px(x),w°(x)} with w°(x) = / wx(x,y)dy is the solution of the ho-
Jy s
mogenized problem (41)-(45), and uX(x,y), uX(x,y), wx(x,y) are given by (32), (35), (23), respectively.
Proof. In the integral identity (7), we take a test vector function v = u\ and pass to the limit as e ^ 0. Then
lim AM pe| ueX| 2dx + 2Aje2 | eij(u\)| 2dx + y (divueX)2dx\ + \ Jq JQi JQs J
+ lim aijkhekh(ux)eij(u\)dx = ux ■ fxdx + / (ux + w°) ■ fXdx. e^0Jnh JQo JQi
lQh
Let us introduce the vector function
r^iE(x) = euX(x,e-!x) + bXX(x) for x G Q0; r^xE(x) = 0 for x G Q0,
where bXX is a boundary layer function in a neighborhood of dQ0, such that t^1 G H^Qo)3 and Hb^Hn 1(Q0)3 ^ 0 as e ^ 0 (see [18]).
Denote by zX(x, y) the right-hand side of (35), where Qkh(y) and Q(y) are replaced by Qkh(y) and Q(y), respectively. It is easy to check that
/ divy zX dy = - divy uX dy. Jy s Jy h
Then, in view of (16), we obtain that zi satisfies the equality
f n
(div ux + divx wx + divy z^ dy =--px. (59)
Jy s Y
Let us now introduce the vector function
t2x(x) = wx(x,e-!x)+ ezX(x,e-!x) + b2x(x) + b3x(x) for x G Q^ t2x(x) =0 for x G Qi,
where b2x and b3x are boundary layer functions in a neighborhood of dQi (see [18]) such that 12X G Hl(Qi)3 and
Ub^Um^)3 ^ 0, Hb^H^Qy ^ 0, eU^UL^^ ^ 0, supp b3x c Qi.
Now we denote tX (x) = ux (x) + t^(x) + tX£(x). By construction, the vector function tX belongs to Hi(Q)3. Setting v = tX in (7) and then passing to the two-scale limit as e ^ 0, we obtain ^
Yjn = ux ■ f xdx + / (ux + w°) ■ f xdx, (60)
n=i J Qo J QI
where
Ii = A2po / |u A |2dx, /2 = A2ph / |u A |2dx + A2// / |u a + wa |2 ./Qo •/ Qi JQi JYs
/3 = / aijfch(y)(efch(uA) + ekh(uA))(eij (u A)+ eyj (u A))dxdУ,
Qo Y
-ijk^V-kM«a; T ey (u0))(e .(u ) + ey (u0)
/ Qo ^ Y
/4 = / aijfch(y)(efch(u A) + ekh(u A))(e»j (u a ) + eyj (uA ))dxdy,
./Qi JYh
/5 = A^ i i |Vy wA |2dxdy, /5 = — i pA dx (according to (59)). ./q^Y 7 ./Qi
By the property of lower semicontinuity (ii), the left-hand side of (58) is greater than or equal
to the left-hand side of (60). Since the right-hand sides of (58) and (60) coincide, we have
A2 lim i pe|uA|2dx = /1, A2 lim i pe|uA|2dx = /2, ^ «/ Qo ^ J Qi
lim / afjfchefch(uA)eij(uA)dx = /з, lim / «fjfchefch(uA)eij(uA)dx = /4,
Ei0hlo Ei0 hlhle
2A^ lim / e2|ejj(uA)|2dx = /5, 7 lim / (divuA)2dx =
Now, using properties (iii)-(v) of two-scale convergence leads to (53) and (54). Moreover, we have
e(uA (x)) — e(u a (x)) + ey (uA (x,y)), x G O°, (61)
x(OhJe(uA(x)) - x(Yh) (e(uA (x)) + ey (uA (x,y))) , x G Oi, (62)
ex(O-)e(uA(x)) - x(Ys)ey(wa(x,y)), x G Oi. (63)
Under the above smoothness assumptions, (55)-(57) follow immediately from (61)-(63). This completes the proof of Theorem 1. □
Finally, applying the inverse Laplace transform to (53)-(57), we deduce the following result. Theorem 2. Let ue(x,t) be a solution of problem (1)-(4). Then
lim / |ue(x,t) — u(x,t)|2dx = 0, lim / |pe(x,t) — p(x,t)|2dx = 0, £i0 JQh ei0./Q|
lim / |ue(x,t) — u(x,t) — w(x, e-1x, t)|2dx = 0, lim / |e (ue(x,t) — u(x, t) — eu°(x, e-1 x, t))|2 dx = 0, lim / |e(ue(x, t) — u(x, t) — eu1(x, e-1x, t))|2 = 0, lim / e2 |e (ue(x,t) — w(x,e-1 x, t))^ dx = 0.
Here the pair {u(x, t),p(x, t)} is the solution of the homogenized problem (48)-(52), w(x, y,t) is given by (47), and
u°(x,y,t) = V kh(y) (x,t), u1(x,y,t) = —Qkh(y) ^ (x,t) — p(x, t)Q(y), xh xh
where the vector functions Vkh(y), Qkh(y), and Q(y) are the solutions of the cell problems (27), (28), and (38), respectively.
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Усреднение уравнений акустики для частично перфорированного упругого материала со слабовязкой жидкостью
Алексей С. Шамаев Владлена В. Шумилова
Рассмотрена математическая модель, описывающая малые колебания гетерогенной среды, состоящей из частично перфорированного упругого материала и слабовязкой сжимаемой жидкости, заполняющей поры. Для данной модели с помощью метода двухмасштабной сходимости построена соответствующая усредненная модель и найдены граничные условия, связывающие уравнения усредненной модели на границе между сплошным упругим материалом и пористым упругим материалом с жидкостью.
Ключевые слова: усреднение, двухмасштабная сходимость, гетерогенная среда.
