Homogenization of very rough three dimensional interfaces for the poroelasticity theory with biot’s model

Vietnam Journal of Mechanics, VAST, Vol 41, No (2019), pp 273 – 285 DOI: https://doi.org/10.15625/0866-7136/13786 HOMOGENIZATION OF VERY ROUGH THREE-DIMENSIONAL INTERFACES FOR THE POROELASTICITY THEORY WITH BIOT’S MODEL Nguyen Thi Kieu1,∗ , Pham Chi Vinh2 , Do Xuan Tung1 Hanoi Architectural University, Vietnam VNU University of Science, Hanoi, Vietnam ∗ E-mail: kieumt@gmail.com Received: 25 April 2019 / Published online: 20 August 2019 Abstract In this paper, we carry out the homogenization of a very rough threedimensional interface separating two dissimilar generally anisotropic poroelastic solids modeled by the Biot theory The very rough interface is assumed to be a cylindrical surface that rapidly oscillates between two parallel planes, and the motion is time-harmonic Using the homogenization method with the matrix formulation of the poroelasicity theory, the explicit homogenized equations have been derived Since the obtained homogenized equations are totally explicit, they are very convenient for solving various practical problems As an example proving this, the reflection and transmission of SH waves at a very rough interface of tooth-comb type are considered The closed-form analytical expressions of the reflection and transmission coefficients have been derived Based on them, the effect of the incident angle and some material parameters on the reflection and transmission coefficients are examined numerically Keywords: homogenization; homogenized equations; very rough interfaces; fluid-saturated porous media INTRODUCTION The homogenization of very rough interfaces and boundaries is used to analyze the asymptotic behavior of various theories of the continuum mechanics in domains including a very rough interface or a very rough boundary [1] It is shown that such an interface and a boundary can be replaced by an equivalent layer within which homogenized equations hold [2] The main aim of the homogenization of very rough boundaries or very rough interfaces is to determine these homogenized equations Nevard and Keller [2] considered the homogenization of three-dimensional interfaces separating two generally anisotropic solids The homogenized equations have been derived, however, they are still implicit Gilbert and Ou [3] investigated the homogenization of a very rough three-dimensional interface that separates two dissimilar isotropic c 2019 Vietnam Academy of Science and Technology 274 Nguyen Thi Kieu, Pham Chi Vinh, Do Xuan Tung poroelastic solids and rapidly oscillates between two parallel planes The motion of the solids is assumed to be time-harmonic The homogenized equations have been obtained, but they are also still in implicit form It should be noted that, for deriving the homogenized equations, Nevard and Keller [2], Gilbert and Ou [3] start from basic equations in component form of the elasticity theory and the poroelasticity theory, respectively Using the matrix formulation (not the component formulation) of theories, Vinh and his coworkers carried out the homogenization of two-dimensional very rough interfaces and the explicit homogenized equations have been obtained for the elasticity theory [4–7], for the piezoelectricity theory [8], for the micropolar elasticity [9] and for the poroelasticity with Auriault’s model for time-harmonic motions [10] A cylindrical surface with a very rough right section is a three-dimensional very rough interface (see Fig 1), and it appears frequently in practical problems The homogenization of a such interface, called a very rough cylindrical interface, is therefore necessary and significant in practical applications Recall that, a right section of a cylindrical surface is the intersection of it with a plane perpendicular to its generatrices In this paper, we carry out the homogenization of a very rough cylindrical interface that separates two dissimilar generally anisotropic poroelastic solids with time-harmonic motion, and it oscillates between two parallel planes When the motion of the poroelastic solids is the same along the direction perpendicular to the plane of right section of the very rough cylindrical interface, the problem is reduced to the homogenization of a two-dimensional very rough interface which is the right section (directrix) of the very rough cylindrical interface Therefore, this paper can be considered as an extension of the investigation by Vinh et al [10] There exist two models describing the motion of poroelastic solids: Biot’s model [11, 12] and Auriault’s model [13, 14] In Biot’s model, the coefficients of equations governing the motion of poroelastic solids are known Meanwhile, as Auriault’s model takes into account the detailed micro-structures of pores including fluid, in order to determine the coefficients of governing equations (homogenized equations) we have to solve numerically the corresponding cell problem, and then apply the homogenization techniques Therefore, Biot’s model is more convenient in use In this paper, the motion of poroelastic solids is assumed to be governed by the Biot theory [11, 12] To carry out the homogenization of the very rough cylindrical interface, first, the basic equations and the continuity conditions of the linear theory of anisotropic poroelasticity are written in matrix form Then, by using an appropriate asymptotic expansion of the solution and following standard techniques of the homogenization method, the explicit homogenized equation and the explicit associate continuity conditions in matrix form are derived Since the obtained homogenized equations are totally explicit, i.e their coefficients are explicit functions of given material and interface parameters, they are of great convenience in solving practical problems To prove this, the reflection and transmission of SH waves at a very rough interface of tooth-comb type are considered The closed-form analytical expressions of the reflection and transmission coefficients are obtained Based on them the dependence of the reflection and transmission coefficients on some parameters is investigated numerically Homogenization of very rough three-dimensional interfaces for the poroelasticity theory with Biot’s model 275 BASIC EQUATIONS IN MATRIX FORM Consider an anisotropic poroelastic medium in which the pore fluid is Newtonian and incompressible According to Biot [11], the basic equations governing the timeharmonic motion of the poroelastic medium are: div Σ + f = −ω ρu + ρ L w , ˆ [−iωρ L u + i gradp], w=K ω Σ = Ce(u) − αp, divw = −α : e(u) − βp, (1) (2) (3) (4) where Σ = (σmn ) represents the total stress tensor, C = (cmn ) is the elasticity tensor of the skeleton, α = (αij ) is the Biot effective stress coefficient (tensor), β is the inverse of the Biot modulus reflecting compressibility of the fluid and of the skeleton, p is the fluid pressure (positive for compression), u = (um ) is the displacement of the solid part, w = f (U L − u) is the displacement of the fluid relative to the solid skeleton, w = (wm ), U L is the displacement of the fluid part, e(u) = (emn ) is the strain tensor: emn = (um,n + un,m ), commas indicate differentiation with respect to spatial variables xm , f is the porosity, ρ = (1 − f )ρs + f ρ L is the composite mass density, ρ L is the mass density of the pore ˆ = (kˆ mn ) = [K−1 + iωρw I]−1 , ρw = f −1 ρ L , fluid, ρs is the mass density of the skeleton, K K = (k mn ) is the generalized Darcy permeability tensor, symmetric and ω-dependent, f = ( f m ) is the volume force acting on the solid part From (2), we have i ˆ k mn p,n , αˆ mn = iωρ L kˆ mn = αˆ nm (5) ω Substitution of Eq (5) into Eqs (1) and (4) leads to four equations for unknowns u1 , u2 , u3 and p, namely wm = −αˆ mn un + σmn,n + ω ρˆ mn un + αˆ mn p,n + f m = 0, m = 1, 2, = iωαmn um,n + iωβp, kˆ mn p,n − ω ρ L un ,m (6) (7) where ρˆ mn = ρδmn − ρ L αˆ mn = ρˆ nm and σij are expressed in terms of u1 , u2 , u3 and p by (3) Four equations {(6), (7)} can be written in matrix form as follows (A11 v,1 + A12 v,2 + A13 v,3 + A14 v),1 + (A21 v,1 + A22 v,2 + A23 v,3 + A24 v),2 + (A31 v,1 + A32 v,2 + A33 v,3 + A34 v),3 + Bv,1 + Gv,2 + Dv,3 + Ev + F = 0, (8) where v = [u1 u2 u3 p] T , F = [ f f f 0] T , the symbol “T” indicates the transpose of a matrix and matrices Ahk , B, G, D and E are given by     c11 c16 c15 c16 c12 c14 c16 c66 c56  c66 c26 c46     A11 =  c15 c56 c55  , A12 = c56 c25 c45  , 0 kˆ 11 0 kˆ 12 276 Nguyen Thi Kieu, Pham Chi Vinh, Do Xuan Tung    c15 c14 c13 0 0 −α11 c56 c46 c36   0 −α12  ,   = c55 c45 c35  , A14 =  0 −α13  iω αˆ 11 iω αˆ 12 iω αˆ 13 0 0 kˆ 13  A13    c66 c26 c46 c16 c66 c56   c12 c26 c25   , A22 = c26 c22 c24  , = c46 c24 c44  c14 c46 c45  ˆ 0 kˆ 22 0 k12  A21    c56 c46 c36 0 0 −α12  c25 c24 c23  0 −α22  ,   = c45 c44 c34  , A24 =  0 −α23  iω αˆ 12 iω αˆ 22 iω αˆ 23 0 0 kˆ 23  A23  A31    c15 c56 c55 c56 c25 c45 c14 c46 c45     , A32 = c46 c24 c44  , = c13 c36 c35  c36 c23 c34  ˆ 0 k13 0 kˆ 23 (9)  A33     c55 c45 c35 0 0 −α13 c45 c44 c34   0 −α23    , = c35 c34 c33  , A34 =  0 −α33  iω αˆ 13 iω αˆ 23 iω αˆ 33 0 0 kˆ 33  0 αˆ 11  0 αˆ 12  , G = B=  0 αˆ 13  −iωα11 −iωα12 −iωα13  0 αˆ 12  0 αˆ 22  ,   0 αˆ 23  −iωα12 −iωα22 −iωα23      ρˆ 11 ρˆ 12 ρˆ 13 0 0 αˆ 13   0 αˆ 23    , E = ω ρˆ 12 ρˆ 22 ρˆ 23  D= ρˆ 13 ρˆ 23 ρˆ 33  0 αˆ 33   0 −iβ/ω −iωα13 −iωα23 −iωα33 CONTINUITY CONDITIONS IN MATRIX FORM Consider a linear poroelastic body that occupies three-dimensional domains Ω+ , Ω− , their interface is a very rough cylindrical surface, whose generatrices are parallel to 0x2 and its right section (directrix) L, belong to the plane x2 = 0, is expressed by equation x3 = h(y), y = x1 / ( > 0), where h(y) is a periodic function of period (see Fig 1) Suppose that the interface oscillates between two planes x3 = − A (A > 0) and x3 = 0, and in the plane x2 = 0: in the domain < x1 < (i.e < y < 1), any straight Homogenization of very rough three-dimensional interfaces for the poroelasticity theory with Biot’s model 277 x + x n L .- .-A x3 Fig Three-dimensional domains Ω+ and Ω− are separated by a very rough cylindrical surface whose generatrices are parallel to 0x2 and its right section (directrix) L (belong to the plane x2 = 0) is expressed by equation x3 = h(y), y = x1 / , h(y) is a periodic function of period line x3 = x30 = const (− A < x30 < 0) has exactly two intersections with the right section L Let < 1, then the interface is called very rough interface of Ω+ and Ω− Suppose that the domains Ω+ , Ω− are occupied by different homogeneous poroelastic materials In particular, the material parameters are defined as  x1  cij+ , k ij+ , α+ , β + , f + , ρs+ , ρw+ , ρ L+ , x3 > h( ) cij , k ij , α, β, f , ρs , ρw , ρ L = (10)  cij− , k ij− , α− , β − , f − , ρs− , ρw− , ρ L− , x3 < h( x1 ) where cij+ , , ρ L+ , cij− , , ρ L− are constant Correspondingly, the matrices Akh , B, G, D, E are given by  x1 (+)  Akh , B(+) , G(+) , D(+) , E(+) for x3 > h( ) Akh , B, G, D, E = (11)  A(−) , B(−) , G(−) , D(−) , E(−) for x3 < h( x1 ) kh (+) where Akh , , E(+) (−) Akh , , E(−) are expressed by (9) in which cij , , ρ L are re- placed by cij+ , , ρ L+ cij− , , ρ L− , respectively Note that matrices Akh , B, G, D, E not depend on x2 Suppose that Ω+ , Ω− are perfectly welded to each other along L Then, the continuity condition is of the form [ ui ] L = 0, i = 1, 2, 3, [ p ] L = 0, [σik nk ] L = 0, i = 1, 2, 3, [iωwk nk ] L = 0, (12) 278 Nguyen Thi Kieu, Pham Chi Vinh, Do Xuan Tung where nk is the xk -component of the unit normal to the curve (right section) L, and we introduce the notation [ ] L , defined such as: [ f ] L = f+ − f− on L In view of (3) and (5), in matrix form the continuity condition (12) takes the form v = 0, L A11 v,1 + A12 v,2 + A13 v,3 + A14 v n1 + A31 v,1 + A32 v,2 + A33 v,3 + A34 v n3 L (13) = EXPLICIT HOMOGENIZED EQUATION IN MATRIX FORM Following Bensoussan et al [15] we suppose that v( x1 , x2 , x3 , ) = U( x1 , y, x2 , x3 , ), and we express U as follows (see Vinh et al [4–6, 8]) U = V+ N1 V + N11 V,1 + N12 V,2 + N13 V,3 + N2 V + N21 V,1 + N22 V,2 + N23 V,3 + N211 V,11 + N212 V,12 + N213 V,13 + N222 V,22 + N223 V,23 + N233 V,33 + O( ), (14) where V = V( x1 , x2 , x3 ) (being independent of y), N1 , N11 , N12 , N13 , N2 , N21 , N22 , N23 , N211 , N212 , N213 , N222 , N223 , N233 are × 4-matrix valued functions of y and x3 (not depending on x1 , x2 ), and they are y-periodic with period Since y = x1 / , we have v,1 = U,1 + −1 U,y Following the same procedure as the one carried out by Vinh et al [9], one can derive the explicit homogenized equation (equation for V) in matrix form of Eq (8), namely - For x3 > 0: (+) (+) (+) Ahk V,kh + A14 + B(+) V,1 + A24 + G(+) V,2 (15) (+) + A34 + D(+) V,3 + E(+) V + F(+) = - For x3 < − A: (−) (−) (−) Ahk V,kh + A14 + B(−) V,1 + A24 + G(−) V,2 (16) (−) + A34 + D(−) V,3 + E(−) V + F(−) = - For − A < x3 < 0: −1 A11 −1 V,11 + −1 A11 + −1 −1 A31 A11 A11 −1 + −1 −1 A21 A11 A11 −1 −1 V,1 + −1 −1 A11 ,3 + −1 −1 A21 A11 A11 + V,2 ,3 −1 −1 A11 + −1 A11 A14 −1 −1 − BA11 A13 V,3 + −1 −1 A31 A11 A11 + E + −1 V,12 + A11 −1 −1 A11 V,1 + −1 A21 A11 −1 −1 A11 A14 − −1 A11 A13 V,13 −1 −1 A31 A11 A11 −1 A11 A14 V + F = −1 A11 A12 − V V,3 ,3 −1 A21 A11 A14 −1 −1 D + BA11 A11 −1 −1 A11 A14 − A31 A11 A14 + A34 −1 BA11 A14 −1 −1 −1 A11 A13 − A31 A11 A13 −1 −1 A11 −1 −1 A11 A12 − BA11 A12 + G V,2 + −1 −1 −1 −1 A11 A12 − A21 A11 A12 + A22 V,22 −1 −1 A33 + A31 A11 A11 −1 −1 + A24 + BA11 A11 −1 BA11 −1 −1 −1 −1 A11 A13 − A21 A11 A13 + A23 V,23 + −1 − A31 A11 A12 + A32 −1 BA11 −1 −1 −1 A11 A12 + A21 A11 A11 ,3 −1 −1 A11 A13 (17) Homogenization of very rough three-dimensional interfaces for the poroelasticity theory with Biot’s model 279 The associate continuity conditions are of the form [ V ] L∗ = 0, [ Σ03 ] L∗ = 0, L∗ : x3 = 0, x3 = − A, (18) where Σ03 = −1 −1 A31 A11 A11 −1 −1 + A31 A11 A11 −1 −1 −1 −1 A11 A14 − A31 A11 A14 + A34 V −1 −1 A32 + A31 A11 A11 V,1 + −1 −1 −1 − A31 A11 A12 V,2 + A33 + A31 A11 A11 −1 −1 −1 A11 A12 (19) −1 −1 A11 A13 − A31 A11 A13 V,3 , and ϕ = ϕdy = (y2 − y1 ) ϕ+ + (1 − y2 + y1 ) ϕ− (20) It is readily to verify that, when the motion of the poroelastic solids is the same along the generatrix direction 0x2 , i.e V does not depend on x2 , the homogenized equation (17) is simplified to Eq (27) in Vinh et al [10] It should be noted that the matrices Aik , B, D and E in Eq (17) (corresponding to Biot’s model) are not equal to the matrices Aik , B, D and E, respectively, in Eq (27) in Vinh et al [10] (corresponding to Auriault’s model), in general REFLECTION AND REFRACTION OF SH WAVE WITH A VERY ROUGH INTERFACE OF TOOTH-COMB TYPE In this section we consider the reflection and transmission of SH waves (u1 ≡ u3 ≡ p ≡ 0, u2 = u2 ( x1 , x3 )) at a very rough interface of tooth-comb type separating two orthotropic poroelastic half-spaces By the meaning of homogenization, this problem is reduced to the reflection and transmission of SH waves (V1 ≡ V3 ≡ P ≡ 0, V2 = V2 ( x1 , x3 )) through a homogeneous material layer occupying the domain − A ≤ x3 ≤ (see Fig 2) For orthotropic poroelastic materials, we have [16] ck4 = ck5 = ck6 = 0, k = 1, 2, 3, c45 = c46 = c56 = 0, α12 = α13 = α23 = 0, k12 = k13 = k23 = (21) In view of (21), from (5) we have αˆ 12 = αˆ 13 = αˆ 23 = 0, kˆ 12 = kˆ 13 = kˆ 23 = 0, ρˆ 12 = ρˆ 13 = ρˆ 23 = (22) From Eqs (15)–(17) and taking into account (21), (22) (without the body forces), the motion of SH waves is governed by the equations c66+ V2,11 + c44+ V2,33 + (re+ − i im+ ) V2 = 0, for x3 > 0, (23) c66− V2,11 + c44− V2,33 + (re− − i im− ) V2 = 0, for x3 < − A, (24) −1 c66 −1 V2,11 + c44 V2,33 + re − i im V2 = 0, for − A < x3 < (25) 280 Nguyen Thi Kieu, Pham Chi Vinh, Do Xuan Tung where re+ = ω ρ+ − ω ρ2L+ ρw+ k222+ ω ρ2L+ k22+ , im = , + + ω ρ2w+ k222+ + ω ρ2w+ k222+ re− = ω ρ− − ω ρ2L− k22− ω ρ2L− ρw− k222− , im = , − + ω ρ2w− k222− + ω ρ2w− k222− re = ω ρ − ω ρ2L ρw k222 + ω ρ2w k222 , im = (26) ω ρ2L k22 + ω ρ2w k222 In addition to Eqs (23)–(25), are required the continuity conditions on lines L∗ : x3 = − A, x3 = 0, namely V2 L∗ = 0, σ23 L∗ = 0, (27) where σ23 = c44 V2,3 Fig The reflection and refraction of SH wave with the homogenized layer Assume that a homogeneous incident SH I wave with the unit amplitude, the incident angle θ, propagates in the half-space Ω+ (Fig 2) When striking at the layer it generates a reflected SHR wave propagating in the half-space Ω+ and a refracted SHT wave traveling in the half-space Ω− Following Borcherdt [17], the homogeneous incident SH I wave, the reflected SHR wave, the (transmitted) refracted SHT wave are of the Homogenization of very rough three-dimensional interfaces for the poroelasticity theory with Biot’s model 281 form V2I = e−( A1I x1 + A3I x3 ) e−i( P1I x1 + P3I x3 ) , (28) V2R = R e−( A1R x1 + A3R x3 ) e−i( P1R x1 + P3R x3 ) , (29) V2T = T e−( A1T x1 + A3T x3 ) e−i( P1T x1 + P3T x3 ) , (30) where R is the reflection coefficient, T is the refraction coefficient, P I ( P1I , P3I ), PR ( P1R , P3R ), PT ( P1T , P3T ) represent the propagation vectors and A I ( A1I , A3I ), AR ( A1R , A3R ), AT ( A1T , A3T ) represent the attenuation vectors of the homogeneous incident SH I wave, reflected SHR wave, refracted SHT wave, respectively and (see Vinh et al [10]) P1I = PI sin θ, P3I = − PI cos θ, PI = |P I |, (31) A1I = A I sin θ, A3I = − A I cos θ, A I = |A I | Substituting (28) into Eq (23) yields AI = re2+ + im2+ −re+ + 2(c66+ sin2 θ + c44+ cos2 θ ) , re+ + PI = re2+ + im2+ 2(c66+ sin2 θ + c44+ cos2 θ ) (32) Snell’s law gives immediately P1I = P1R = P1T , A1I = A1R = A1T (33) Substituting Eq (29) into Eq (23) and using equalities (33) yield P3R = − P3I , A3R = − A3I (34) Equalities (31), (33) and (34) say that the refracted SHR wave is a homogeneous wave with the reflection angle θ R = θ (Fig 2) Introducing Eq (30) into Eq (24) and using equalities (33) lead to A3T = − P3T = − − A2 )] + −[re− − c66− ( P1I 1I − A2 )]2 + [im − 2c [re− − c66− ( P1I − 66− P1I A1I ] 1I 2c44− [re− − c66− ( P1I − A21I )] + [re− − c66− ( P1I , (35) − A21I )]2 + [im− − 2c66− P1I A1I 2c44− ]2 In view of Snell’s law, one can see that the general solution of Eq (25) is given by ˆ ˆ V2 = ( B1 e−iK3 x3 + B2 eiK3 x3 )e−i( P1I −iA1I )x1 , (36) where B1 and B2 are constants to be determined and Kˆ = −1 re − c66 −1 ( P 1I −1 − A21I ) − i [ im − c66 c44 −1 P A ] 1I 1I (37) 282 Nguyen Thi Kieu, Pham Chi Vinh, Do Xuan Tung It is easy to verify that Kˆ = Pˆ3 − i Aˆ where (real numbers) Pˆ3 , Aˆ are given by −1 [ re − c66 Pˆ3 = −1 ( P 1I − A21I )] + −1 [ re − c66 −1 ( P 1I −1 − A21I )]2 + [ im − c66 −1 P A ] 1I 1I c44 Aˆ = −1 im − c66 −1 P A 1I 1I c44 Pˆ3 , (38) Using (28)–(30), (36) and the continuity conditions (27) yields a system of four equations for B1 , B2 , R and T, namely B1 + B2 = R + 1, B1 − B2 = − c44+ ( A3I + iP3I )(1 − R) , c44 ( Aˆ + i Pˆ3 ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ (39) B1 e−( A3 +i P3 ) A + B2 e( A3 +i P3 ) A = Te( A3T +iP3T ) A , B1 e−( A3 +i P3 ) A − B2 e( A3 +i P3 ) A = − c44− ( A3T + iP3T ) ( A3T +iP3T ) A Te c44 ( Aˆ + i Pˆ3 ) Solving the system (39) for R and T we obtain closed-form analytical expressions for the reflection and transmission coefficients, namely R= pr − sn ms − pq , T= , mr − qn mr − qn (40) where ˆ ˆ ˆ ˆ m = a1 e−( A3 +i P3 ) A + a2 e( A3 +i P3 ) A , n = −2e( A3T +iP3T ) A , ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ p = −{ a2 e−( A3 +i P3 ) A + a1 e( A3 +i P3 ) A }, q = a1 e−( A3 +i P3 ) A − a2 e( A3 +i P3 ) A , c44− ( A3T + iP3T ) ( A3T +iP3T ) A ˆ ˆ ˆ ˆ e , s = −{ a2 e−( A3 +i P3 ) A − a1 e( A3 +i P3 ) A }, ˆ ˆ c44 ( A3 + i P3 ) c44+ ( A3I + iP3I ) a1 = + , a2 = (2 − a1 ) c44 ( Aˆ + i Pˆ3 ) r=2 (41) From (40) and (41) one can see that R and T depend on 13 dimensionless parameters, namely ε1 = a c44− c66+ ω ρ + A2 ρ L+ , ε2 = , ε3 = , ε4 = , ε = ωρ+ k22+ , ε = , a+b c44+ c44+ c44+ ρ+ c66− ω ρ − A2 ρ L− ε7 = , ε8 = , ε = ωρ− k22− , ε 10 = , θ, f , f c44− c44− ρ− (42) Using formulas (40), (41) we consider the dependence of the moduli | R| and | T | of the reflection and refraction coefficients on some dimensionless parameters It can be seen from Fig that: Homogenization of very rough three-dimensional interfaces for the poroelasticity theory with Biot’s model 283 (i) When the incident angle θ0 increases, moduli | R|, | R0 | increase and moduli | T |, | T0 | decrease, | R| < | R0 |, | T | > | T0 | in which | R|, | T |, (| R0 |, | T0 |) are the reflection, refraction coefficients with the rough interface, (without the rough interface) (see Fig 3(a)) (ii) The increasing of ε , ε makes the reflection coefficient increasing and makes the transmission coefficient decreasing (see Figs 3(b), 3(c)) (iii) In contrast, the increasing of ε makes the reflection coefficient decreasing and makes the transmission coefficient increasing (see Fig 3(d)) 1.6 |R| |T| |R0| |R| |T| 1.5 1.4 |T0| 1.3 0.6 |R|, |T| |R|, |T|, |R0|, |T0| 0.8 1.2 1.1 0.4 0.9 0.8 0.2 0.7 0 10 20 30 40 60 70 80 θ050 ε1 = 0.3; ε2 = 1.6; ε3 = 1.3; ε4 = 1.5; ε5 = 1.6; ε6 = 0.6; ε7=1.8; ε8=3.5; ε9=1.1; ε10=1.2; f1 = 0.1; f2 = 0.2 90 0.1 0.2 0.3 0.4 0.5 ε 0.6 0.7 θ = 600; ε = 1.2; ε = 1.3; ε = 1.5; ε = 1.6; ε = 0.6; ε = 1.8; ε = 3.5; ε = 3.1; ε = 1.2; f = 0.8; f = 0.7 (a) 10 0.8 (b) (c) (d) |R| |T| |R| |T| 1 0.8 |R|, |T| |R|, |T| 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.9 0.5 2.5 ε2 1.5 ε1=0.7; ε3=1.3; ε4=0.2; ε5=1.6; ε6=0.6; ε7=1.8; o ε8=3.5; ε9=1.1; ε10=0.8; f1 = 0.1; f2 = 0.2; θ=60 0.5 (c) 1.5 2.5 ε4 ε1=0.7; ε2=1.3; ε3=1.5; ε5=2.6; ε6=0.6; ε7=1.8; o ε8=2.1; ε9=0.1; ε10=1.2; f1 = 0.3; f2 = 0.2; θ=30 (d) Fig The dependence of the moduli | R| and | T | of the reflection and transmission coefficients on θ0 (a), ε (b), ε (c), ε (d) CONCLUSIONS In this paper the homogenization of a very rough cylindrical interface that separates two dissimilar generally anisotropic poroelastic solids with time-harmonic motion, and oscillates rapidly between two parallel planes is investigated The explicit homogenized equation in matrix form has been derived by applying the homogenization method Since the obtained homogenized equations are fully explicit, they are a powerful tool for investigating various practical problems As an example, the reflection and transmission of 284 Nguyen Thi Kieu, Pham Chi Vinh, Do Xuan Tung SH waves at a very rough interface of tooth-comb type are considered The closed-form analytical expressions of the reflection and transmission coefficients have been obtained Employing them, the effect of the incident angle and the material parameters on the reflection and transmission coefficients is investigated numerically ACKNOWLEDGMENTS The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 107.02-2017.07 REFERENCES [1] W Kohler, G Papanicolaou, and S Varadhan Boundary and interface problems in regions with very rough boundaries In Multiple Scattering and Waves in Random 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