Nicole Turbé

Annales de la Faculté des Sciences de Toulouse

ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSENICOLE TURBÉBloch waves for a solid-fluid mixtureAnnales de la faculté des sciences de Toulouse 5e série, tome 6, no 2 (1984), p. 87-101<http://www.numdam.org/item?id=AFST_1984_5_6_2_87_0>© Université Paul Sabatier, 1984, tous droits réservés.L’accès aux archives de la revue « Annales de la faculté des sciences de Toulouse »(http://picard.ups-tlse.fr/~annales/) implique l’accord avec les conditions générales d’utilisa-tion (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression sys-tématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi-chier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/BLOCH WAVES FOR A SOLID-FLUID MIXTURENicole Turbé (1)Annales Faculté des Sciences ToulouseVol VI,1984, P. 87 a 101(1 ) Laboratoire de Mécanique Théorique, Université Pierre et Marie Curie, Tour 66, 4 place Jussieu75230 Paris Cédex 05 - France.Resume : On étud ie les vibrations d’un mélange infini, a structure period ique, constitué d’un sol ideélastique et d’un fluide visqueux, barotrope. La solution des equations du mouvement admet undéveloppement en fonction d’ondes de Bloch. Cette représentation peut etre utilis~e pour trouverdes approximations. Un exemple en est fourni par Ie lien avec la théorie de l’homogénéisation.~ Summary : Vibrations of an unbounded, periodic, mixture of an elastic solid and a viscous com-pressible flu id are stud ied. The solution of the equations of motion admits an expansion in termsof Bloch waves. This representation can be used to find approximations. The connection with thehomogenization theory is such an example.I. - INTRODUCTIONWave propagation in periodic structures is of interest by a lot of applications. Geophy-sical or engineering problems, for instance, led to several works on laminated media or fiber-reinforced composites [1]. Using Floquet’s theory for ordinary linear differential equations withperiodic coefficients, effects of dispersion were analysed, allowing comparison with approximatetheories [s], [7], [~] ,88More generally, the study of periodic structures is related to an analysis of the pro-perties of differential operators with period ic coefficients. The spectral resolution of these opera-tors comes from the existence of generalized eigenfunctions, the Bloch waves. Then, it is easy toshow that the solution of linear equations, that describe effects due to the periodicity of thestructure, can be given in terms of Bloch waves. So, in particular, the solution of the equationsof motion of a periodic, unbounded medium with elastic or viscoelastic behaviour, has an expan-sion depending on eigenvectors of operators associated with the elasticity problem. These opera-tors act upon functions which are defined on the basic cell of the material. The components ofthese eigenvectors are determined by solving differential or integrodifferential systems [9] , [10] .In this work, we study linearized vibrations of an unbounded, periodic mixture ofan elastic solid and a compressible, barotropic, viscous fluid. The mechanical problem is statedin section 2 and then, we show, in section 3, that the solution admits a Bloch expansion, in termsof eigenfunctions of differential operators which are defined from the elasticity and viscositycoefficients. This expansion can lead us to find directly approximations of the solution. If theviscosity terms are small, the average method can be used. As another application, in section 4,we assume that the initial conditions are slowly varying functions of x, depending on a smallparameter e and we study the asymptotic behaviour of the solution. It is given by the functionobtained from the homogenization method ( [5] , [4] ) applied to the initial problem.2. - STATEMENT OF THE PROBLEM .2.1. - Local equationsA mixture of an elastic solid and a compressible, barotropic, viscous fluid fills thewhole space IR3, the solid in the part nS and the fluid in nf. We consider a reference state, therest, in which the pressure is a constant everywhere, and the density of mass p (p in in n6 ) is constant in each of regions S~f and We study the displacement field u, u = u(x,t),with respect to the rest reference.Under the hypothesis of small perturbations, the stress perturbation tensor is given by :withIn expression (1 ), the perturbation of pressure, that depends on the density (baro-tropic fluid), is related with the velocity of sound co , in the reference state, and with the lineari-zed strain tensor êmn(U) by the continuity equation (see, for instance,[5] ).Let Y be the cube ( ]0,1 [ )3 C IR3. We assume that, in the rest reference, the mixturehas a 27rY-periodic structure : geometric positions of fluid and solid parts are 27rY-periodic,Y = Yf U Ys, and the fluid and solid properties are 27rY-periodic. In the framework of homoge-nization theory, i.e. the limit case with small period, E. Sanchez-Palencia [5] and J. Sanchez-Hubert [4] studied the vibrations of such a mixture, in a bounded region of IR3.So, the elasticity coefficients ahlmn in (1 ) are functions defined on 203C0Ys (notation of[2]), and they satisfy the usual symmetry (3) and positivity (4) conditions :(3) alhmn = amnhQwhere a is a constant and e denotes the complex conjugate of e. The viscosity coefficients X and J1, in (1 ) satisfy : .We assume that there are no given body forces, so the equations of motion are given by :with uhQ (u) defined by (1 ).Moreover,. at the interface between the solid and the fluid, we must have the continui-ty of displacement and stress : .satisfied at points obtained by periodicity from points of 27rr, r boundary between Ys and Y~.The brackets mean «jump of» and n denotes a unit vector, normal to 27rr.To these equations and boundary conditions, we must finally add initial data :2.2. - Variational formulationProblem (6), (7), (8) can be expressed as an initial value problem in (L 2{1R3))3.From (6) and by means of (7), we show that the solution u satisfies :Let us introduce the 27rY-periodic coefficients :Then, we define two sesquilinear forms on (H (IR3)) . :0 q 3htroducing a weighted inner product, with weight p , on the Hilbert space (L"(!R")) , the pro-btem has the following form :. Find u, function of t with values in (H ~ ()R~)) , such that:In (14), f and g are data such that :Let Band C be the associated operators to the forms band c. In an equivalent way,equation (13) can be written :The problem seems to look like the one which is related to the vibrations of an un-bounded, periodic medium with instantaneous memory [10]. But, the forms b and c are not coer-cive. Nevertheless, for any positive real number @, the form a’ + Q2 (with associated operatorA ~ + a , 2 ~ a defined by ( 17 ), is coercive on (H 1 (lR 3)) .So, a change of unknown function is done in (13) or (15) :And the new function v is solution of : :Under the assumptions (3), (4), (5), (15), in the classical framework of semi-grouptheory, for instance (see J. Sanchez-Hubert [4]), the solutions of problems (16), (15) and conse-quently (18), (19) exist and are unique.3. - BLOCH EXPANSION OF THE SOLUTION3.1. - Bloch wavesBy means of operatof A~, we use the technics of Bloch expansion, described byA. Bensoussan, J.L. Lions and G. Papanicolaou [2] in the scalar case, and by N. Turbe [8] forthe equations of elasticity.The periodicity of the coefficients in operator A~ leads to the definition of a set ofsesquilinear forms (or a set of operators) which act upon functions defined on the basic cell27rY. Let (HP(2~rY))3 (p for periodic) be the space of the vectors of(H~ (2?rY)) which take equalvalues on two opposite points of two opposite sides of the cell 27rY. With the help of coefficients(9) (resp. (10)), we define the set of forms b(k) (resp. c(k)) : :The space (L 9 (2?rY)) 3 is equipped with the weighted inner product, defined by po,and we denote by B(k) (resp. C(k)) the operators which are associated with the forms b(k) (resp.c(k)) : :And we introduce :The operators kEY, act upon the vectors of ( H1 p( 2~rY )) 3 for which the stressbelong to (H1 (2~rY))3 and that satisfy the transmission conditions (7).As it is done in the study of the elastic, periodic medium (see N. Turbé [8]), we provethat, for any k E Y, A~(k) + (32 is a positive, selfadjoint operator with compact resolvent. As aconsequence :for each k E Y, there exist eigenvalues 0  ~ (k)  oj*(k) ... - withcorresponding eigenfunctions c~~(x;k), ~p1 (x ;k)... of the operator A~(k) whichform an orthonormal basis in (L (27rY)) .Any function of (LZ(IR3)) , with complex values, can be then expanded by meansof the eigenvectors of the operators A~(k) (N. Turbe [8]) : :with3.2. - Representation of the solutionThe set of basis { k) } is used in equation (13). Let v and w be two vectors of(H (IR3 ))3 which components, in the Bloch expansion, are denoted and w (k). As wehave it for an elastic, periodic medium : :In the same way, we can represent c(v,w) with the help ofv(k) and w (k). First, letus suppose that v and w belong to ( ~ (IR3))3. We construct the auxiliary functions v(x;k) andw(x;k) (used in the proof of the Bloch expansion theorem [9]) : :7 is an element of (L2(2~rY))3 we expand in the basis of the eigenfunctions ~pm(. ; k) ~ :’ Let us consider then the scalar c(k) ~(. ; k), w(. ; k)). From definitions (20) and (26),it comes :We integrate this expression over k, k E Y. The functions v and w have compactsupports and the coefficients chQmn are periodic, so we have :But if decomposition (27) is used, it results :Let us define :Finally, it comes :Expression (29) holds true for v and w in (  (IR3))3. But (  (IR3))3 is denselyembedded into (H 1 (lR3))3, so (29) holds true for any v and w in (H 1 (IR3))3. Let us note thatexpression (25) can be obtained in the same way, using the properties of the functions Also,we have :The solution u of (13) has a Bloch expansion (23), the components of which dependon the time parameter t. Expressions (25), (29) and (30) are introduced into (13) and it resultsthat the functions m(k,t), for fixed k, are determined from the differntial system (31)-(32) :This coupled system, with an infinite number of unknown functions, is not easy tosolve. But it can be used to find approximations of the solution. Let us point out two examples : :from the average method if the viscosity terms are small [9], from an expansion of the solutionif the data are slowly varying functions.The parameter (3 appears in the expression of u, not only from the coefficientsby (31 ), but also from the choice of the basis But this fact is only a technics in thewriting of the solution u. An example of this fact is given in the next section.4. - CONNECTION WITH THE HOMOGENIZATION THEORY4.1. - Setting of the problemLet e be a small positive parameter. We suppose that the initial data f and g are slowlyvarying functions of x by means of e :f and gf have asymptotic power series expansions: :The period of the med ium seems to be small compared to the data scale. So theassumptions of the approximate homogenization theory are satisfied. We prove that an expansionof the solution u, in powers of e, has for first term the function given by the homogenizationtheory, which is written down in J. Sanchez-Hubert[4].In the expression, of the form (23), of the solution u, we ach ieve the change : k = eK,and we study the function E3 u defined by (35), when e tends to zero.This function u (t) is a solution to (36)-(37) :4.2. Study of the operators and C(eK)Let us investigate the behaviour of the eigenvalues and eigenfunctions of the operatorThe study which is done for the elastic, periodic material (N. Turbe[9]) is used here.For m ~ 0, the eigenvalue ~(0) being simple or multiple, it is possible to find avector such that the eigenvalue and eigenvector are written :And it results that :For m = 0, wo(0) = 0 is an eigenvalue associated with any arbitrary constantvector. This eigenvalue produces three holomorphic branches of eigenvalues. Then andadmit the expansions (([8]) :where the constant vector ~p°(x;0), which depends on K, is one of the three eigenvectors(r = 1,2,3), orthonormed vectors in (L2(2~rY))3, associated with the eigenvalues~ = ~JK), obtained from :In (39), the operators are respectively defined by (40) and (41 ).The term is function of where the vector classically introduced in the homogenization of elasticity problems ([5])are solutions of :(e th vector of the IR3 natural basis).So, equation (39) produces a linear, homogeneous system, with variables (x;0),the solution of which furnishes ~~~~(K) and (K) (r= 1,2,3).From these properties, it follows that :(without an add on r, r = 1,2,3).In the same way, it is possible to obtain an estimate ofC(eK) ’Pm(x;eK). The operatorC(eK) is expanded in powers of e : :where C and C,pq are defined by similar expressions to (40) and (41), with the coefficientschlmn instead of bhlmn + Then it comes :4.3. - Approximate solutionIn the initial conditions (37), we adopt the data scale and we change y into y / e.Then, we use the following property : if h(x,y) is smooth on IR3 X 27rY and has compact supportin xThe basis ~ J is orthonormal and the vectors ~p°(x;0) are constant, so wededuce :where P o is the mean value of the densityA Aand and g"(K) denote the Fourier transforms of the vectors f°(x) and g°(x).Let us now find the first term of the expansion of e3 u (t). We replace expression(35) of u (t) into (36) and we use (38), (44) and (45). Then, by taking the scalar product withwe have the equations satisfied by the coefficients uom(t)Accord ing to the initial cond itions (46) we single out two cases: For m =1,2,... it comes :which has the solution 0. Therefore, as for as the elastic, periodic material ([8]l, the first termof the solution comes from the contribution m = 0.nFor m = 0, the components uo(r)(t) are solutions of (48)-(49) :where the coefficients 03B3sr(K) are given by :The system (48) is a linear differential system, with the following characteristicequation :that has six roots pv = (v = 1,6). Therefore, the solutions of (48) are of the formexp(pt), where the coefficients satisfy :The solution of the linear, homogeneous system (52) depends on six constants, ob-tained from the initial conditions (49). So, the first term of the expansion of u is :the coefficients pv satisfy (51) and A~) (52), (54).By using (54), we easily show that :4.4. - Homogenized constitutive lawWe prove that the function u° in (53) satisfies :where the homogenized stress tensor Qh has the expression obtained by J. Sanchez-Hubert 4 .The term is deduced from (39) with the scalar product of (39) by e~, IR3natural basis vector. From definition (50), it comes 03B3rs 03C8(s)j. With an evident notation, we obtain :Let us note again that, in (55), ~(r) is an eigenvector of the operator B(0) + ~C(O),13 any positive real number. The term - E2 K p K , in (55), is associated with a x x - derivative.As for the time dependance, it is given by the expression of the Laplace transform, for any point ~i.With the definition of the classical homogenized coefficients (see E. Sanchez-Palencia~S ]), we obtain :where £(u) denotes the Laplace transform of the function u.Therefore, the homogenized constitutive law is given by a convolution product,associated with a non-instantaneous memory.101REFERENCES[1] J.D. ACHENBACH. «Waves and vibrations in directionally reinforced composites».Composite materials, vol. 2, Mechanics of composite materials L.J. Broutman, R.M.Kroch (ed.). Academic Press, New-York (1975), pp. 309-351.[2] A. BENSOUSSAN, J.L. LIONS, G. PAPANICOLAOU. «Asymptotic analysis forperiodic structures». North-Holland, Amsterdam (1978).[3] T.J. DELPH, G. HERRMANN, R.K. KAUL. «Harmonic wave propagation in a perio-dically layered, infinite elastic body, antiplane strain». J. Appl. Mech., 45 (1978),pp. 343-349.[4] J. SANCHEZ-HUBERT. «Asymptotic study of the macroscopic behaviour of a solid-fluid mixture». Math. Meth. Appl. Sci., 2 (1980) pp. 1-11.[5] E. SANCHEZ-PALENCIA. «Non-homogeneous media and vibration theory». Springer,Berlin (1980).[6] C.T. SUN, J.D. ACHENBACH, G. HERRMANN. «Continuum theory foralaminatedmedium». J. Appl. Mech. 35 (1968), pp. 467-473.[7] C. SVE. «Time harmonic waves traveling obliquely in a periodically laminated me-dium». J. Appl. Mech. 38 (1971), pp. 477-482.[8] N. TURBE. «Applications of Bloch expansion to periodic elastic and viscoelasticmedia». Math. Meth. Appl. Sci. 4 (1982) pp. 433-449.[9] N. TURBE. Thèse de Doctorat, Université Pierre et Marie Curie (1982).[10] N. TURBE. Communication colloque du Groupe Français de Rhéologie (1982).[11] C.H. WILCOX. «Theory of Bloch waves». J. Anal. Math. 33 (1978) pp. 146-167.(Manuscrit reçu le 6 juin 1983)

Bloch waves for a solid-fluid mixture

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Bloch waves for a solid-fluid mixture

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Annales de la Faculté des Sciences de Toulouse