homogenization of linearized elasticity systems.pdf

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Electronic Journal of Differential Equations, Vol. 1999(1999), No. 41, pp. 111.

ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu

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HOMOGENIZATION OF LINEARIZED ELASTICITY SYSTEMS

WITH TRACTION CONDITION IN PERFORATED DOMAINS

MOHAMED EL HAJJI

Abstract. In this paper, we study the asymptotic behavior of the linearizedelasticity system with nonhomogeneous traction condition in perforated do-

mains. To do that, we use the H0e

-convergence introduced by M. El Hajji in[4] which generalizes - in the case of the linearized elasticity system - the notion

ofH0-convergence introduced by M. Briane, A. Damlamian and P. Donato in[1]. We give then some examples to illustrate this result.

1. Introduction

The notion of H0e -convergence was introduced by M. El Hajji in [4] for thestudy of the asymptotic behavior of the linearized elasticity system with homoge-neous traction condition in perforated domains. It translates the notion of H0-convergence introduced by M. Briane, A. Damlamian and P. Donato in [1] for thestudy of the diffusion system problem with homogeneous Neumann condition inperforated domains which generalizes in the case of perforated domains the H-convergence introduced by F. Murat and L. Tartar in [13], and the G-convergence

for the symmetric operator introduced by S. Spagnolo in [14].This paper is devoted to giving an application of the H0e - convergence to study

the asymptotic behavior of the linearized elasticity system with nonhomogeneoustraction condition in perforated domains by using the convergence of a distributiondefined from data on the boundaries of the holes. This result is the analogue for thelinearized elasticity of Theorem 1 given by P. Donato and M. El Hajji in [3] as anapplication of the H0-convergence to the study of the nonhomogeneous Neumannproblem.

In Section 2, we recall the definition ofH0e -convergence introducing a definition ofe-admissible set similar to that given by M. Briane, A. Damlamian and P. Donatoin [1] for the H0-convergence and by F. Murat and L. Tartar in [13] for the H-convergence. In Section 3, we introduce the linearized elasticity problem and wegive the main result. We establish then the proof making use of some preliminary

results. In Section 4, we give some applications of this result - first for the caseof periodic perforated domains by holes of size r = where we use the resultsgiven by F. Lene in [11] and D. Cioranescu and P. Donato in [2]. Then we applythe results of section 3 and those of C. Georgelin in [8] and S. Kaizu in [10] when

1991 Mathematics Subject Classification. 73C02, 35B27, 37B27, 35J55, 35J67.

Key words and phrases. Homogenization, Elasticity, H0e

-convergence, Double periodicity.c1999 Southwest Texas State University and University of North Texas.

Submitted April 6, 1999. Published October 11, 1999.1


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2 MOHAMED EL HAJJI EJDE1999/41

r . Finally, we apply section 3 to the case of a perforated domain with doubleperiodicity (introduced by T. Levy in [12]) using the results given by M. El Hajjiin [9] and P. Donato and M. El Hajji in [3].

2. Recall of H0e -convergence

In this section, we recall the definition ofH0e -convergence introduced in [4]. Firstlet us introduce the following notations.

Let be a bounded open subset ofRN, the general term of a positive sequence,andcdifferent positive constants independent of. We introduce the following sets:

Ms = {symmetric linear operatorsl : RN RN

2

},

L(Ms) ={linear operatorsp : Ms Ms},

Ls(Ms) = {symmetric operatorsp L(Ms)},

Me(, ; ) =

A L(, Ls(Ms)), A(x) ||2,

A1

(x) ||2

, Ls(Ms), x a.e. .In what follows, we use the Einstein summation convention, that is, we sum over

repeated indices. We denote by e() the symmetric tensor of elasticity defined by

e(u) = (eij (u))ij where eij =1

2

uixj

+ ujxi

.

We denote by S a compact subset of . We denote the perforated domain by = \ S. We denote by the characteristic function of and we set

V =

v [H1()]N, v| = 0

, (1)

which equipped with the H1-norm forms a Hilbert space.Definition 1 (e-admissible set). The set S is said to be admissible (in ) for thelinearized elasticity if

every function in L() weak of is positive almost everywhere in , (2)and for each there is an extension operator P from V to [H10 ()]

N and thereexists a real positiveCsuch that

i) P L

V, [H10 ()]

N

,

ii) (Pv)|=v, v V,

iii) e(Pv)[(L2()]N2 Ce(v)[(L2()]N2 , v V.

(3)

Remark 1. 1) As an example of an e-admissible set, one can consider the case ofa periodic function on a perforated domain by holes of size or r (see F. Lene[11] and C. Georgelin [8]). One can consider also a perforated domain with doubleperiodicity introduced by T. Levy in [12] (see also M. El Hajji [9]).2). Observe that ifSis admissible in the sense of definition 1, then we have a Korn

inequality in independent of, i.e.,v[L2()]N2 C()e(v)[L2()]N2 , v V.

Indeed, from the Korn inequality in and (3 iii) one has

v[L2()]N2 (Pv)[L2()]N2

c()e(Pv)[L2()]N2

C()e(v)[L2()]N2 , v V.


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the nonhomogeneous Neumann problem. We then give some examples to illustratethis result.

LetA = (aijkh) Me(, ; ) and letS be e-admissible in such that

S has boundary S of classC1. (7)

We consider the linearized elasticity system

div(Ae(u)) = 0 in ,

(A(x)e(u)) n= g on S,

u = 0 on,

(8)

where

g [H1/2(S)]N. (9)

It is well known that (8) has a unique solution. Our aim is to study the asymptoticbehavior of the solution u as approaches zero. To do that, we introduce avectorial distributiong defined in by

g , [H1()]N, [H10

()]N =g, [H1/2(S)]N,[H1/2(S)]N, [H

10 ()]

N.

(10)It is easy to check that this defines g as an element of [H

1()]N, and if

g [L2()]N, we deduce from the Riesz Theorem that g is a measure. The

following theorem shows that the convergence ofu can be deduced from the H0e -convergence of (A, S) and the convergence ofg in [H

1()]N.

Theorem 1. Let{u} be the sequence of the solutions of (8). Suppose that (7) issatisfied and that

i) (A, S)H0e A0,

ii) there exists [H1()]N such thatg strongly in [H1()]N.

(11)

Theni) Pu

u weakly in[H10()]N,

ii) Ae(u) A0e(u) weakly in

L2 ()N2

,(12)

whereu is the solution of the problem

div

A0e(u)

= in,

u= 0 on. (13)

Proof. Observe first, by using (3 ii) and (10), that

g, v[H1/2(S)]N, [H1/2(S)]N =g , Pv[H1()]N, [H1

0()]N, v V.

Hence, problem (8) is equivalent to the problem

div(Ae(u)) = P g in ,(A(x)e(u)) n= 0 on S,

u = 0 on,

since both of the two systems have the variational formulation: Find u V suchthat

Ae(v)e(v)dx= g , PvV ,V , v V. (14)


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EJDE1999/41 HOMOGENIZATION OF LINEARIZED ELASTICITY SYSTEMS 5

Let us show that there exist c independent of such that

Pu[H1

0()]N c . (15)

By takingu as a test function in the variational formulation of (14) one obtains

Ae(u)e(v)dx= g , PuV ,V .

From (3 iii) and the fact that A Me(,, ) one deduces that

e (Pu)2[L2()]N2 C

e (u) e(u)dx

C

Ae(u)e(u)dx

cg[H1()]Ne (Pv)[L2()]N2.

Hence (15) gives (11 ii). One may deduce (up to a subsequence) that

Pu

u

weakly in [H10 ()]

N

. (16)Consider now the solutionv of the problem

div(Ae(v)) = P in ,

(A(x)e(v)) n= 0 on S,

v = 0 on .

(17)

From (11 i), one deduces that

i) Pv v weakly in [H10()]

N,

ii) Ae(v) A0e(v) weakly in

L2 ()N2

,(18)

wherev is the solution to (13).On the other hand,w =u v is the solution to

div(Ae(w)) = P g in ,(A(x)e(w)) n= 0 on S,

w = 0 on.

(19)

By choosingw as a test function in the variational formulation of (19) and (3)and the fact thatA Me(,, ), one has

(Pw)2

[L2()]N2 Ce(w)2

[L2()]N2

C

Ae(w)e(w)dx

=cg , Pw[H1()]N,[H1

0()]N.

SincePw is bounded in [H10 ()]N, one deduces from (12 ii) that

g , Pw[H1()]N,[H1

0()]N 0,

which implies thatPw

0 strongly in [H10 ()]N. (20)

This, with (18) proves that in (16) one has u =u.Finally, one deduces from (20) and the fact that A Me(,, ) that

Ae(w)[L2()]N2 c e(w)[L2()]N2 c e(Pw

)[L2()]N2 0 .


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rS

Figure 1. A periodic perforated domain

With the convergence (18 ii), it follows then that (12 ii) holds. Remark 3. As in the case of Theorem 1 of [9], from the linearity of the equation and

the definition of the H0e - convergence, the choice of an nonhomogeneous right-hand

side of the equation (8) is not restrictive.

4. Some applications of the main result

The case of a periodic p erforated domain. Let Y = [0, l1[.. [0, lN[ bethe representative cell, San open set ofY with smooth boundary S such thatS Y . Let r be the general term of a positive sequence which converge to zeroand satisfyingr . One denote by (rS) the set of all the translated ofrSofthe form (kl+ rS), k ZN, kl = (k1l1, . . ,kNlN). It represents the holes in RN.

One suppose that the holes (rS) do not intersect the boundary . If Sdesign the holes contained in , it follows that

S is a finite union of the holes, i.e S = kKr(kl+ S).

Set = \ S, by this construction, is a periodic perforated domain by holesof sizer (see Figure 1)

We propose to study the asymptotic behavior of the solution v of the system

div (Ae(v)) = 0 in ,

(A(x)e(v)) n= h on S,

v = 0 on ,

(21)

whereh(x) =h(

x

), h [L2(S)]N Y-periodic. (22)

We suppose that

lim0

N

rN2= 0, (23)

and thatA = (aijkh) satisfies

aijkh(x) = aijkh(x

), aijkh Me(, ; Y

). (24)

In this case of a periodic perforated domain, the homogenization of system (21) hasbeen studied by F. Lene in [11] for the case r = , and C. Georgelin in [8] for thecaser . The results obtained allow us to deduce that

(A, S)H0e A0, (25)


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whereA0 = (a0ijkh) is defined by

A0ijkh = 1

|Y| Y\S Aijkhekh(kh Pkh)eij (ij P

ij )dy, (26)

where Pij is the vector all of whose components are equal to zero except the ith

one, i.e., (Pij )k = yj ki, and for all k, h= 1, . . ,N, kh [H1(Y \ S)]N Y-periodic,

and is a solutin to

div(Ae(kh Pkh)) = 0 in Y \ S,

(Ae(kh Pkh)) n= 0 on S,

ifr = and

A0ijkh = 1

|Y|

Y

aijkhekh(kh Pkh)eij (ij P

ij)dy, (27)

wherePij is the vector all of whose components are equal to zero except the ith one

which is equal toyj , i.e., (P

ij

)k = yjki, and for anyk, h= 1, . . ,N,

kh

[H

1

(Y)]

N

Y-periodic is a solution to

div(Ae(kh Pkh)) = 0 in Y ,

ifr .On the other hand, from the results obtain by D. Cioranescu and P. Donato in

[2] for the caser = , and S. Kaizu in [10] for the case r , we can deduce thefollowing lemma.

Lemma 1 ([2],[10]). Let h be defined by (10). We suppose that (23) is satisfiedand that the reference hole S is star-shaped ifr . Then

N

rN1h inH

1() strongly, (28)

with, vH1(),H1

0() = Ih

v dx v H10 (), (29)

andIh= 1|Y|

Sh ds.

Consequently, we can apply Theorem 1 to u =Nv/rN1 andg =Nh/rN1

to obtain the following theorem.

Theorem 2. Letv be a solution of (21). Suppose that (22) and (23) are satisfiedand that S is star-shaped if r . Then there exists P an extension operatorsatisfying (3) such that

P( N

rN1v) v0 weakly in [H10()]

N,

Ae( N

rN1v) A0e(v0) weakly in L2 ()N2 ,

wherev 0 is the solution to

div

A0e(v0)

= in,

v0 = 0 on,(30)

withdefined by (29).


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,

,

,

,

,

,

,

,

,

,

,

,

,

P

P

P

P

P

Pi

3

:

FYr(Z+ kz

0)/

r(S+ kz0)/

Figure 2. The reference cell

4.1. The case of a perforated domain with double periodicity. We considerthe perforated domain defined as = \ S, where S is a set with a doubleperiodicity defined below. We adopt here the geometrical framework introduced in[5] and [6].

Assume that Y and Zare two fixed reference cells,

Y =]0, yo1[...]0, yoN[, Z=]0, z

o1[...]0, z

oN[. (31)

We sety0 = (yo1, . . ,y

oN), z

0 = (zo1,...,zoN). (32)

LetF Y andS Zbe two closed subsets with smooth boundaries and nonemptyinteriors.

Suppose that r and are the general term of two positive sequences such thatr < and

lim0

r

= 0. (33)

We assume that for each >0 there exists a fine K ZN, such thatkK

r

(Z+ kz 0) = Y, (34)

and that

(F) ( kK

r

(S+ kz 0)) = .

This means that for any the sets Y and Y \F are exactly covered by a finitenumber of translated cells of rZand

rS respectively. Denote

SY = (Y \ F) (

kK

r

(S+ kz0))

and Y = Y \ SY. From (34) it follows that there exist a finite set K

Z

N suchthat

SY =

kK

r

(S+ kz0).

HenceSY is a subset ofY\ Fof closed sets (inclusions) periodically distributedwith periodicityr/ and of the same size as the period (see Figure 2).

We also assume that for each >0, there exists a finite set H ZN such thathZN

(SY + hy0) =

hH

(SY + hy0)

and we set

S =

hH

(SY + hy0).


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,

,

,

,

,

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,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

A

A

AU

,

,

Y r(S+ hyo) r(Z+ hy

0)

F

Figure 3. The perforated domain

Hence, >0, and are exactly covered by a finite number of translated cellsofY and SY respectively. Consequently, the structure of presents a doubleperiodicity ( and r). The zones in which the inclusions are concentrated are -

periodic and of size . The inclusions in each zone are r-periodic and of size r(see Figure 3).

Our aim is to apply Theorem 1 to this case of double periodicity with a matrixA = (aijkh) defined in (21) and satisfying

aijkh(x) =aijkh(x

, x

r)

i) aijkh Y Z periodic

ii) aijkh L(Z, C0(Y)) or aijkh L

Y, C0(Z)

iii) aijkh = aijhk =ajikh

iv) >0 s.t. aijkh(y, z)ekheij eij eij, a.e. (y, z) Y Z

(35)

for any symmetric tensoreij , andh is defined by

h= (F Q)h, h H

1/2(S) (36)

whereh is Z-periodic, h H1/2(S), and

h, 1H1/2(S),H1/2(S= 0. (37)

The operatorQ L

H1/2 (S) , H1/2 (SY)

is defined by

Qz, vH1/2(SY), H1/2(SY) =

kK

(r

)N1z, v1 H1/2(S+kz0), H1/2(S+kz0),

(38)and the operatorF L(H1/2(SY), H

1/2(S)) is defined by

Fu, H1/2(S), H1/2(S) = hH()N1u, 1 H1/2(SY+hy0), H1/2(SY+hy0),

(39)where and are the homotheties

: x

rx, : x

x

. (40)

From the result obtained in [9], we deduce that

(A, S)H0e A0, (41)


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whereA0 = (a0ijkh) is defined as follows: set

dijkh (y, z) =

F(y) + Y\F(y)Z\S(z)

aijkh(y, z), (42)

and for l , m= 1,..,N, let Rlm = Rlmk k=1,..,Nbe the vector defined byRlmk =zmkl.

We denote bylm =lm(y, .) the unique function in [H1(Z\S)]N Z-periodic whichis a solution to

zj[dijkhe

zkh(R

lm lm)] = 0 inZ\ S

dijkhezkh(R

lm lm) nj = 0 on S.

(43)

We set

qijkh = 1

|Z|

Z\S

dijrsezrs(R

kh kh)dz.

Let Plm =

Plmk k=1,..,N

be the vector defined by Plmk = ymkl, and let lm in

[H1(F)]N Y-periodic, which is a solution to

yj[qijkhe

ykh(P

lm lm)] = 0 inF,

qijkheykh(P

lm lm) nj = 0 on F\ Y .

(44)

We define the homogenizated coefficients by

a0ijkh = 1

|Y|

Y

qijrseyrs(P

kh kh)dy, (45)

wherey = (yi)i=1,..,N andz = (zi)i=1,..,N.Observe that the coefficients (a0ijkh) are obtained by applying the homogenization

process twice (see the classical methods of homogenization introduced by F. Muratand L. Tartar in [13] and S. Spagnolo in [14]). Indeed, first starting with the tensor

(dijkh) and homogenizing with respect to Z, we obtain the tensor (qijkh). Thenstarting with (qijkh) and homogenizing with respect to Y, we obtain the tensor(a0ijkh).

On the other hand, using the results obtain by P. Donato and M. El Hajji in [4],we obtain

Lemma 2. Let h be defined by (10), suppose that h, 1H1/2(S),H1/2(S) = 0,and that (33) is satisfied. Then

rh strongly inH

1(),

where is given by

, H1(),H10

() = Ih

dx H10 (), (46)

with

=

|YZ|

|Y F| |S|

1, Ih= h, 1H1/2(S),H1/2(S), (47)

and is defined by

= |F|

|Y|+

|Y \ F|

|Y|

|Z\ S|

|Z| .

Hence, we can apply Theorem 1 tou =ru and obtain


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Theorem 3. Let v be the solution of (21). Then there exists P an extensionoperator satisfying (3) such that

P(rv

) u

0

weakly in [H

1

0()]

N

,Ae(rv) A

0e(u0) weakly in

L2 ()N2

,

wherev 0 is the solution of the problem

div

A0e(v0)

= in,

v0 = 0 on,(48)

whereA0 = (a0ijkh) is given by (43)-(45), and defined by (46), (47).

Acknowledgments. It is a pleasure for the author to acknowledge his indebtednessto Patrizia Donato for her help and friendly suggestions throughout this work.

References

[1] Briane M., A. Damlamian & Donato P., H-convergence in perforated domains, NonlinearPartial Differential Equations & Their Applications, Vol. 13, College de France seminar,Editors: H. Brezis & J.-L. Lions, Longman, New York, a paraitre

[2] Cioranescu D. & Donato P., Homogen eisation du probleme de Neumann non homogene dansdes ouverts perfores, Asymptotic Analysis, Vol.1 (1988), 115138

[3] Donato P. & El Hajji M., An application of theH0-convergence to some nonhomogeneousNeumann problems, Gakuto International Series, Math. Sc. and Appl., Gakkotosho, Tokyo,Japan, Vol. 9 (1997), 137156.

[4] Donato P. & El Hajji M., H0-convergence for the linearized elasticity system. Journal ofElasticity, to appear.

[5] Donato P. & Saint Jean Paulin J., Homogenization of Poisson equation in a porous mediumwith double periodicity, Japan J. Industr. Appl. Math., Vol. 10 (1993), 333349.

[6] Donato P. & Saint Jean Paulin J., Stokes Flow in a Porous Medium with Double Periodicity,

Progress in Partial Differential Equations: the Metz Surveys 3, Pitman Research Notes inMath. Longman, New York, Vol. 314 (1994), 116129.

[7] Francfort G.A. & Murat F., Homogenization and Optimal bounds in Linear Elasticity, Arch.Rational Mech. Anal., Vol. 94 (1986), 307334.

[8] Georgelin C., Contribution a letude de quelques problemes en elasti cite tridimensionnel le,

These de Doctorat, Universite de Paris IV, 1989[9] El Hajji M., Homogenization of the linearized elasticity system in a perforated domain with

double periodicity, Revista de Matematicas Aplicadas, to appear.[10] Kaizu S., The Poisson equation with nonautonomous semilinear doundary conditions in

domains with many tiny holes, Japan. Soc. for Industr. & Appl. Math., Vol. 22 (1991), 1222

1245.[11] Lene F., Comportement macroscopique de materiaux elastiques comportant des inclusions

rigides ou des trous repartis periodique ment, Compt. Rend. Acad. Sci. Paris, I, Vol. 292(1981), 7578.

[12] Levy T., Filtration in a porous fissured rock: influence of the fissures connexity, Eur. J.

Mech, B/Fluids, Vol. 4(9) (1990), 309327.[13] Murat F. & Tartar T., H-convergence in Topics in the mathematical modeling of compos-

ite materials, Editor: R.V. Kohn, Progress in Nonlinear Differential Equations and their

Applications, 1997, 2143.[14] Spagnolo S., Sulla convergenza di soluzioni di equazioni paraboliche ed el littiche, Ann. Sc.

Norm. Sup. Pisa, Vol. 22 (1968), 571597.

Mohamed El HajjiUniversite de Rouen, UFR des Sciences

UPRES-A 60 85 (Labo de Math.)76821 Mont Saint Aignan, France

E-mail address: Mohamed.Elhajjiuniv-rouen.fr

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