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

Nội dung text: Homogenization of very rough three-dimensional interfaces for the poroelasticity theory with biot’s model

1. Vietnam Journal of Mechanics, VAST, Vol.41, No. 3 (2019), pp. 273 – 285 DOI: HOMOGENIZATION OF VERY ROUGH THREE-DIMENSIONAL INTERFACES FOR THE POROELASTICITY THEORY WITH BIOT’S MODEL Nguyen Thi Kieu1,∗, Pham Chi Vinh2, Do Xuan Tung1 1Hanoi Architectural University, Vietnam 2VNU 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 three- dimensional interface separating two dissimilar generally anisotropic poroelastic solids modeled by the Biot theory. The very rough interface is assumed to be a cylindrical sur- face 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 prob- lems. 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 ef- fect 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. 1. 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 includ- ing a very rough interface or a very rough boundary [1]. It is shown that such an inter- face 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 inter- faces separating two generally anisotropic solids. The homogenized equations have been derived, however, they are still implicit. Gilbert and Ou [3] investigated the homogeniza- tion of a very rough three-dimensional interface that separates two dissimilar isotropic c 2019 Vietnam Academy of Science and Technology
2. 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 homoge- nized 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 poroelastic- ity 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 homog- enization of a such interface, called a very rough cylindrical interface, is therefore neces- sary 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 poroelas- tic 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 gov- erning 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 deter- mine the coefficients of governing equations (homogenized equations) we have to solve numerically the corresponding cell problem, and then apply the homogenization tech- niques. 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 poroe- lasticity 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 conve- nience 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 an- alytical 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.
3. Homogenization of very rough three-dimensional interfaces for the poroelasticity theory with Biot’s model 275 2. 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 time- harmonic motion of the poroelastic medium are: 2  div Σ + f = −ω ρu + ρLw , (1) i w = Kˆ [−iωρ u + gradp], (2) L ω Σ = Ce(u) − αp, (3) divw = −α : e(u) − βp, (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 (UL − u) is the displacement of the fluid relative to the solid skeleton, w = (wm), UL is 1 the displacement of the fluid part, e(u) = (e ) is the strain tensor: e = (u + u ), mn mn 2 m,n n,m commas indicate differentiation with respect to spatial variables xm, f is the porosity, ρ = (1 − f )ρs + f ρL is the composite mass density, ρL is the mass density of the pore −1 −1 −1 fluid, ρs is the mass density of the skeleton, Kˆ = (kˆmn) = [K + iωρwI] , ρw = f ρL, K = (kmn) is the generalized Darcy permeability tensor, symmetric and ω-dependent, f = ( fm) is the volume force acting on the solid part. From (2), we have i w = −αˆ u + kˆ p , αˆ = iωρ kˆ = αˆ . (5) m mn n ω mn ,n mn L mn nm Substitution of Eq. (5) into Eqs. (1) and (4) leads to four equations for unknowns u1, u2, u3 and p, namely 2 σmn,n + ω ρˆmnun + αˆ mn p,n + fm = 0, m = 1, 2, 3 (6) ˆ − 2
    = + kmn p,n ω ρLun ,m iωαmnum,n iωβp, (7) where ρˆmn = ρδmn − ρLαˆ mn = ρˆnm 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 (A11v,1 + A12v,2 + A13v,3 + A14v) + (A21v,1 + A22v,2 + A23v,3 + A24v) ,1 ,2 (8) + (A31v,1 + A32v,2 + A33v,3 + A34v),3 + Bv,1 + Gv,2 + Dv,3 + Ev + F = 0, T T where v = [u1 u2 u3 p] , F = [ f1 f2 f3 0] , the symbol “T” indicates the transpose of a matrix and matrices Ahk, B, G, D and E are given by     c11 c16 c15 0 c16 c12 c14 0 c16 c66 c56 0  c66 c26 c46 0  A11 =   , A12 =   , c15 c56 c55 0  c56 c25 c45 0  0 0 0 kˆ11 0 0 0 kˆ12
4. 276 Nguyen Thi Kieu, Pham Chi Vinh, Do Xuan Tung     c15 c14 c13 0 0 0 0 −α11 c56 c46 c36 0   0 0 0 −α12 A13 =   , A14 =   , c55 c45 c35 0   0 0 0 −α13 0 0 0 kˆ13 iωαˆ 11 iωαˆ 12 iωαˆ 13 0     c16 c66 c56 0 c66 c26 c46 0 c12 c26 c25 0  c26 c22 c24 0  A21 =   , A22 =   , c14 c46 c45 0  c46 c24 c44 0  0 0 0 kˆ12 0 0 0 kˆ22     c56 c46 c36 0 0 0 0 −α12 c25 c24 c23 0   0 0 0 −α22 A23 =   , A24 =   , c45 c44 c34 0   0 0 0 −α23 0 0 0 kˆ23 iωαˆ 12 iωαˆ 22 iωαˆ 23 0     c15 c56 c55 0 c56 c25 c45 0 c14 c46 c45 0  c46 c24 c44 0  A31 =   , A32 =   , (9) c13 c36 c35 0  c36 c23 c34 0  0 0 0 kˆ13 0 0 0 kˆ23     c55 c45 c35 0 0 0 0 −α13 c45 c44 c34 0   0 0 0 −α23 A33 =   , A34 =   , c35 c34 c33 0   0 0 0 −α33 0 0 0 kˆ33 iωαˆ 13 iωαˆ 23 iωαˆ 33 0     0 0 0 αˆ 11 0 0 0 αˆ 12  0 0 0 αˆ   0 0 0 αˆ  B =  12 , G =  22 ,  0 0 0 αˆ 13  0 0 0 αˆ 23 −iωα11 −iωα12 −iωα13 0 −iωα12 −iωα22 −iωα23 0     0 0 0 αˆ 13 ρˆ11 ρˆ12 ρˆ13 0  0 0 0 αˆ  ρˆ ρˆ ρˆ 0  D =  23 , E = ω2  12 22 23  .  0 0 0 αˆ 33 ρˆ13 ρˆ23 ρˆ33 0  −iωα13 −iωα23 −iωα33 0 0 0 0 −iβ/ω 3. 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 equa- tion x3 = h(y), y = x1/e (e > 0), where h(y) is a periodic function of period 1 (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 0 < x1 < e (i.e. 0 < y < 1), any straight
5. Homogenization of very rough three-dimensional interfaces for the poroelasticity theory with Biot’s model 277 x 3 x + 2 x . 0 1 . . . . n . .L . . . . . . . . - -A. Fig. 1. 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/e, h(y) is a periodic function of period 1 0 0 line x3 = x3 = const (−A h( )  ij ij 3 e cij, kij, α, β, f , ρs, ρw, ρL = (10) x1 cij−, kij−, α−, β−, f−, ρs−, ρw−, ρL−, x3 h( )  kh 3 e Akh, B, G, D, E = (11) (−) (−) (−) (−) (−) x1 A , B , G , D , E for x3 < h( ) kh e (+) (+)  (−) (−) where Akh , , E Akh , , E are expressed by (9) in which cij, , ρL are re-
   placed by cij+, , ρL+ cij−, , ρL− , respectively. Note that matrices Akh, B, G, D, E do not depend on x2. Suppose that Ω+, Ω− are perfectly welded to each other along L. Then, the continu- ity condition is of the form [ ui ]L = 0, i = 1, 2, 3, [ p ]L = 0, (12) [σiknk]L = 0, i = 1, 2, 3, [iωwknk]L = 0,
6. 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, A11v,1 + A12v,2 + A13v,3 + A14v n1 L (13) + + + +
    = A31v,1 A32v,2 A33v,3 A34v n3 L 0. 4. EXPLICIT HOMOGENIZED EQUATION IN MATRIX FORM Following Bensoussan et al. [15] we suppose that v(x1, x2, x3, e) = U(x1, y, x2, x3, e), and we express U as follows (see Vinh et al. [4–6,8])  1 11 12 13  2 2 21 22 23 U = V + e N V + N V,1 + N V,2 + N V,3 + e N V + N V,1 + N V,2 + N V,3 (14) 211 212 213 222 223 233
   3 + N V,11 + N V,12 + N V,13 + N V,22 + N V,23 + N V,33 + O(e ), 1 11 12 13 2 21 22 23 where V = V(x1, x2, x3) (being independent of y), N , N , N , N , N , N , N , N , 211 212 213 222 223 233 N , N , N , N , N , N are 4 × 4-matrix valued functions of y and x3 (not de- pending on x1, x2), and they are y-periodic with period 1. Since y = x1/e, we have −1 v,1 = U,1 + e 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: (+) (+) (+)
   (+) (+)
   A V,kh + A + B V,1 + A + G V,2 hk 14 24 (15) (+) (+)
   (+) (+) + A34 + D V,3 + E V + F = 0. - For x3 < −A: (−) (−) (−)
   (−) (−)
   A V,kh + A + B V,1 + A + G V,2 hk 14 24 (16) (−) (−)
   (−) (−) + A34 + D V,3 + E V + F = 0. - For −A < x3 < 0: −1 −1 h −1 −1 −1 −1 −1 −1i −1 −1 −1 hA11 i V,11 + hA11 i hA11 A12i + hA21A11 ihA11 i V,12 + hA11 i hA11 A13iV,13 h −1 −1 −1 i h −1 −1 −1 −1 −1 i + hA31A11 ihA11 i V,1 + hA21A11 ihA11 i hA11 A12i − hA21A11 A12i + hA22i V,22 ,3 h −1 −1 −1 −1 −1 i h −1 −1 −1 −1 + hA21A11 ihA11 i hA11 A13i − hA21A11 A13i + hA23i V,23 + hA31A11 ihA11 i hA11 A12i −1  i h −1 −1 −1 −1 −1  i − hA31A11 A12i + hA32i V,2 + hA33i + hA31A11 ihA11 i hA11 A13i − hA31A11 A13i V,3 ,3 ,3 (17) h −1 −1 −1 −1 −1 −1 i h −1 −1 −1 −1 −1 + hBA11 ihA11 i + hA11 i hA11 A14i V,1 + hA21A11 ihA11 i hA11 A14i − hA21A11 A14i −1 −1 −1 −1 −1 i h −1 −1 −1 −1 + hA24i + hBA11 ihA11 i hA11 A12i − hBA11 A12i + hGi V,2 + hDi + hBA11 ihA11 i hA11 A13i −1 i h −1 −1 −1 −1 −1  i − hBA11 A13i V,3 + hA31A11 ihA11 i hA11 A14i − hA31A11 A14i + hA34i V ,3 h −1 −1 −1 −1 −1 i + hEi + hBA11 ihA11 i hA11 A14i − hBA11 A14i V + hFi = 0.
7. Homogenization of very rough three-dimensional interfaces for the poroelasticity theory with Biot’s model 279 The associate continuity conditions are of the form 0 ∗ [ V ]L∗ = 0, [ Σ3 ]L∗ = 0, L : x3 = 0, x3 = −A, (18) where 0 h −1 −1 −1 −1 −1 i Σ3 = hA31A11 ihA11 i hA11 A14i − hA31A11 A14i + hA34i V −1 −1 −1 h −1 −1 −1 −1 + hA31A11 ihA11 i V,1 + hA32i + hA31A11 ihA11 i hA11 A12i (19) −1 i h −1 −1 −1 −1 −1 i − hA31A11 A12i V,2+ hA33i + hA31A11 ihA11 i hA11 A13i−hA31A11 A13i V,3, and Z 1 + − hϕi = ϕdy = (y2 − y1)ϕ + (1 − y2 + y1)ϕ . (20) 0 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. 5. 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 ≤ 0 (see Fig.2). For orthotropic poroelastic materials, we have [16] ck4 = ck5 = ck6 = 0, k = 1, 2, 3, c45 = c46 = c56 = 0, (21) α12 = α13 = α23 = 0, k12 = k13 = k23 = 0. In view of (21), from (5) we have αˆ 12 = αˆ 13 = αˆ 23 = 0, kˆ12 = kˆ13 = kˆ23 = 0, ρˆ12 = ρˆ13 = ρˆ23 = 0. (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 −1 h i hc66 i V2,11 + hc44iV2,33 + hrei − ihimi V2 = 0, for −A < x3 < 0 (25)
8. 280 Nguyen Thi Kieu, Pham Chi Vinh, Do Xuan Tung where 2 2 2 3 2 h ω ρ ρw+k i ω ρ k22+ re = ω2 ρ − L+ 22+ , im = L+ , + + 2 2 2 + 2 2 2 1 + ω ρw+k22+ 1 + ω ρw+k22+ 2 2 2 3 2 h ω ρ ρw−k i ω ρ k22− re = ω2 ρ − L− 22− , im = L− , (26) − − 2 2 2 − 2 2 2 1 + ω ρw−k22− 1 + ω ρw−k22− 2 2 2 3 2 h ω hρ ρwk i i ω hρ k i hrei = ω2 hρi − L 22 , himi = L 22 . 2 2 2 2 2 2 1 + ω hρwk22i 1 + ω hρwk22i ∗ In addition to Eqs. (23)–(25), are required the continuity conditions on lines L : x3 = −A, x3 = 0, namely   =  0  = V2 L∗ 0, σ23 L∗ 0, (27) 0 where σ23 = hc44iV2,3. Fig. 2. The reflection and refraction of SH wave with the homogenized layer Assume that a homogeneous incident SHI wave with the unit amplitude, the in- cident 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 inci- dent SHI wave, the reflected SHR wave, the (transmitted) refracted SHT wave are of the
9. Homogenization of very rough three-dimensional interfaces for the poroelasticity theory with Biot’s model 281 form −(A1I x1+A3I x3) −i(P1I x1+P3I x3) V2I = e e , (28) −(A1R x1+A3R x3) −i(P1R x1+P3R x3) V2R = R e e , (29) −(A1T x1+A3T x3) −i(P1T x1+P3T x3) V2T = T e e , (30) where R is the reflection coefficient, T is the refraction coefficient, PI (P1I, P3I ), PR(P1R, P3R), PT(P1T, P3T) represent the propagation vectors and AI (A1I, A3I ), AR(A1R, A3R), AT(A1T, A3T) represent the attenuation vectors of the homogeneous incident SHI wave, reflected SHR wave, refracted SHT wave, respectively and (see Vinh et al. [10]) P1I = PI sin θ, P3I = −PI cos θ, PI = |PI |, (31) A1I = AI sin θ, A3I = −AI cos θ, AI = |AI |. Substituting (28) into Eq. (23) yields v q v q u 2 2 u 2 2 u −re+ + re+ + im+ u re+ + re+ + im+ A = t , P = t . (32) I 2 2 I 2 2 2(c66+ sin θ + c44+ cos θ) 2(c66+ sin θ + c44+ cos θ) 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 θR = θ (Fig.2). Introducing Eq. (30) into Eq. (24) and using equalities (33) lead to v u q u −[re − c (P2 − A2 )] + [re − c (P2 − A2 )]2 + [im − 2c P A ]2 t − 66− 1I 1I − 66− 1I 1I − 66− 1I 1I A3T = − , 2c44− v (35) u q u [re − c (P2 − A2 )] + [re − c (P2 − A2 )]2 + [im − 2c P A ]2 t − 66− 1I 1I − 66− 1I 1I − 66− 1I 1I P3T = − . 2c44− In view of Snell’s law, one can see that the general solution of Eq. (25) is given by −iKˆ3x3 iKˆ3x3 −i(P1I −iA1I )x1 V2 = (B1e + B2e )e , (36) where B1 and B2 are constants to be determined and s −1 −1 2 2 −1 −1 hrei − hc66 i (P1I − A1I ) − i[himi − 2hc66 i P1I A1I ] Kˆ3 = . (37) hc44i
10. 282 Nguyen Thi Kieu, Pham Chi Vinh, Do Xuan Tung It is easy to verify that Kˆ3 = Pˆ3 − iAˆ 3 where (real numbers) Pˆ3, Aˆ 3 are given by v u q u [hrei − hc−1i−1(P2 − A2 )] + [hrei − hc−1i−1(P2 − A2 )]2 + [himi − 2hc−1i−1P A ]2 t 66 1I 1I 66 1I 1I 66 1I 1I Pˆ3 = , 2hc44i −1 −1 himi − 2hc66 i P1I A1I Aˆ3 = . 2hc44iPˆ3 (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, c44+(A3I + iP3I )(1 − R) B1 − B2 = − , hc44i(Aˆ 3 + iPˆ3) (39) −(Aˆ3+iPˆ3)A (Aˆ3+iPˆ3)A (A3T +iP3T )A B1e + B2e = Te , c (A + iP ) −(Aˆ3+iPˆ3)A (Aˆ3+iPˆ3)A 44− 3T 3T (A3T +iP3T )A B1e − B2e = − Te . hc44i(Aˆ 3 + iPˆ3) Solving the system (39) for R and T we obtain closed-form analytical expressions for the reflection and transmission coefficients, namely pr − sn ms − pq R = , T = , (40) mr − qn mr − qn where −(Aˆ3+iPˆ3)A (Aˆ3+iPˆ3)A (A3T +iP3T )A m = a1e + a2e , n = −2e , −(Aˆ3+iPˆ3)A (Aˆ3+iPˆ3)A −(Aˆ3+iPˆ3)A (Aˆ3+iPˆ3)A p = −{a2e + a1e }, q = a1e − a2e , c (A + iP ) 44− 3T 3T (A3T +iP3T )A −(Aˆ3+iPˆ3)A (Aˆ3+iPˆ3)A (41) r = 2 e , s = −{a2e − a1e }, hc44i(Aˆ 3 + iPˆ3) c44+(A3I + iP3I ) a1 = 1 + , a2 = (2 − a1). hc44i(Aˆ 3 + iPˆ3) From (40) and (41) one can see that R and T depend on 13 dimensionless parameters, namely 2 2 a c44− c66+ ω ρ+ A ρL+ ε1 = , ε2 = , ε3 = , ε4 = , ε5 = ωρ+k22+, ε6 = , a + b c + c + c + ρ+ 44 44 44 (42) 2 2 c66− ω ρ− A ρL− ε7 = , ε8 = , ε9 = ωρ−k22−, ε10 = , θ, f1, f2. c44− c44− ρ− 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.3 that:
11. 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| |T0| in which |R|, |T|,(|R0|, |T0|) are the reflection, refrac- tion coefficients with the rough interface, (without the rough interface) (see Fig. 3(a)). (ii) The increasing of ε1, ε2 makes the reflection coefficient increasing and makes the transmission coefficient decreasing (see Figs. 3(b), 3(c)). (iii) In contrast, the increasing of ε4 makes the reflection coefficient decreasing and makes the transmission coefficient increasing (see Fig. 3(d)). 1 1.6 |R| |R| |T| 1.5 |T| |R | 1.4 0.8 0 |T | 1.3 | 0 0 0.6 1.2 |, |T 0 1.1 |R|, |T| 0.4 1 |R|, |T|, |R 0.9 0.2 0.8 0.7 0 0 10 20 30 40 θ 50 60 70 80 90 0.1 0.2 0.3 0.4 0.5 ε 0.6 0.7 0.8 0.9 0 0 1 ε = 0.3; ε = 1.6; ε = 1.3; ε = 1.5; ε = 1.6; ε = 0.6; θ = 60 ; ε = 1.2; ε = 1.3; ε = 1.5; ε = 1.6; ε = 0.6; 1 2 3 4 5 6 0 2 3 4 5 6 ε =1.8; ε =3.5; ε =1.1; ε =1.2; f = 0.1; f = 0.2 ε = 1.8; ε = 3.5; ε = 3.1; ε = 1.2; f = 0.8; f = 0.7 7 8 9 10 1 2 7 8 9 10 1 2 (a) (b) (c) (d) |R| |R| |T| 1 |T| 1 0.8 0.8 0.6 0.6 |R|, |T| |R|, |T| 0.4 0.4 0.2 0.2 0 0 0.5 1 ε 1.5 2 2.5 3 0.5 1 ε 1.5 2 2.5 3 2 4 ε =0.7; ε =1.3; ε =0.2; ε =1.6; ε =0.6; ε =1.8; ε =0.7; ε =1.3; ε =1.5; ε =2.6; ε =0.6; ε =1.8; 1 3 4 5 6 7 1 2 3 5 6 7 ε =3.5; ε =1.1; ε =0.8; f = 0.1; f = 0.2; θ=60o ε =2.1; ε =0.1; ε =1.2; f = 0.3; f = 0.2; θ=30o 8 9 10 1 2 8 9 10 1 2 (c) (d) Fig. 3. The dependence of the moduli |R| and |T| of the reflection and transmission coefficients on θ0 (a), ε1 (b), ε2 (c), ε4 (d) 6. 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 inves- tigating various practical problems. As an example, the reflection and transmission of
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