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

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
1
Hanoi Architectural University, Vietnam
2
VNU University of Science, Hanoi, Vietnam
∗

E-mail:

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.

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 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

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 timeharmonic motion of the poroelastic medium are:
div Σ + f = −ω 2 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
1
the displacement of the fluid part, e(u) = (emn ) is the strain tensor: emn = (um,n + un,m ),
2
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 2 ρˆ mn un + αˆ mn p,n + f m = 0, m = 1, 2, 3
= iωαmn um,n + iωβp,
kˆ mn p,n − ω 2 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 1 f 2 f 3 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 0
c16 c12 c14 0
c16 c66 c56 0 
c66 c26 c46 0 



A11 = 
c15 c56 c55 0  , A12 = c56 c25 c45 0  ,
0
0

0 kˆ 11
0
0
0 kˆ 12


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 
,


=
c55 c45 c35 0  , A14 =  0
0
0

−α13 
iω αˆ 11 iω αˆ 12 iω αˆ 13
0
0
0
0 kˆ 13


A13




c66 c26 c46 0
c16 c66 c56 0


c12 c26 c25 0 
 , A22 = c26 c22 c24 0  ,
=
c46 c24 c44 0 
c14 c46 c45 0 
ˆ
0
0
0 kˆ 22
0
0
0 k12


A21




c56 c46 c36 0
0
0
0
−α12
 0
c25 c24 c23 0 
0
0
−α22 
,


=
c45 c44 c34 0  , A24 =  0
0
0
−α23 
iω αˆ 12 iω αˆ 22 iω αˆ 23
0
0
0
0 kˆ 23


A23



A31




c15 c56 c55 0
c56 c25 c45 0
c14 c46 c45 0 


 , A32 = c46 c24 c44 0  ,
=
c13 c36 c35 0 
c36 c23 c34 0 
ˆ
0
0
0 k13
0
0
0 kˆ 23

(9)



A33






c55 c45 c35 0
0
0
0
−α13
c45 c44 c34 0 
 0
0
0
−α23 


,
=
c35 c34 c33 0  , A34 =  0
0
0
−α33 
iω αˆ 13 iω αˆ 23 iω αˆ 33
0
0
0

0 kˆ 33


0
0
0
αˆ 11
 0
0
0
αˆ 12 
, G =
B=
 0
0
0
αˆ 13 
−iωα11 −iωα12 −iωα13 0


0
0
0
αˆ 12
 0
0
0
αˆ 22 
,


 0
0
0
αˆ 23 
−iωα12 −iωα22 −iωα23 0







ρˆ 11 ρˆ 12 ρˆ 13
0
0
0
0
αˆ 13
 0

0
0
αˆ 23 
0 
 , E = ω 2 ρˆ 12 ρˆ 22 ρˆ 23
.
D=
ρˆ 13 ρˆ 23 ρˆ 33
 0
0

0
αˆ 33 
0 
0
0
0 −iβ/ω
−iωα13 −iωα23 −iωα33 0
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 equation x3 = h(y), y = x1 / ( > 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 < (i.e. 0 < y < 1), any straight


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

277

...
.
.
... .. ..
.
..
. .. .. .. .. ..
..
x
+
..

....... ....
.. .
.
.
.
x
.
.
.
.
.
.
.
.
0
.. .. .. .. .. .. n .. ..L .... .. .. .. .. .. .. .
.. .... .... .... . .. .... .. .. .... .. ... .............
....
. . ...... ...........
.
... .-. ..... .......-A
.
. . . . . . . .. .. .. .. .. .. .. .. .. .. ..
x3

2

1

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 / , h(y) is a periodic function of period 1

line x3 = x30 = const (− A < x30 < 0) has exactly two intersections with the right section L.
Let 0 <
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
do 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)

= 0.

4. 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 +

2

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( 3 ),

(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 × 4-matrix valued functions of y and x3 (not depending on x1 , x2 ), and they are y-periodic with period 1. 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(+) = 0.
- For x3 < − A:
(−)

(−)

(−)

Ahk V,kh + A14 + B(−) V,1 + A24 + G(−) V,2

(16)

(−)

+ A34 + D(−) V,3 + E(−) V + F(−) = 0.
- 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 = 0.

−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
1

ϕ =

0

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.
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,
α12 = α13 = α23 = 0, k12 = k13 = k23 = 0.

(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 = 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
c66

−1

V2,11 + c44 V2,33 +

re − i im V2 = 0, for − A < x3 < 0

(25)


280

Nguyen Thi Kieu, Pham Chi Vinh, Do Xuan Tung

where
re+ = ω 2 ρ+ −

ω 2 ρ2L+ ρw+ k222+
ω 3 ρ2L+ k22+
,
im
=
,
+
1 + ω 2 ρ2w+ k222+
1 + ω 2 ρ2w+ k222+

re− = ω 2 ρ− −

ω 3 ρ2L− k22−
ω 2 ρ2L− ρw− k222−
,
im
=
,
−
1 + ω 2 ρ2w− k222−

1 + ω 2 ρ2w− k222−

re = ω 2 ρ −

ω 2 ρ2L ρw k222
1 + ω 2 ρ2w k222

, im =

(26)

ω 3 ρ2L k22
.
1 + ω 2 ρ2w k222

In addition to Eqs. (23)–(25), are required the continuity conditions on lines L∗ : x3 =
− A, x3 = 0, namely
V2

L∗

= 0,

0
σ23

L∗

= 0,

(27)

0
where σ23
= c44 V2,3 .

Fig. 2. 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 = −

2 − A2 )] +
−[re− − c66− ( P1I
1I

2 − A2 )]2 + [im − 2c
2
[re− − c66− ( P1I

−
66− P1I A1I ]
1I

2c44−
2
[re− − c66− ( P1I

−

A21I )] +

2
[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 3 =

−1
re − c66

−1 ( P 2
1I

−1
− A21I ) − i [ im − 2 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 3 = Pˆ3 − i Aˆ 3 where (real numbers) Pˆ3 , Aˆ 3 are given by
−1
[ re − c66

Pˆ3 =

−1 ( P 2
1I

− A21I )] +

−1
[ re − c66

−1 ( P 2
1I

−1
− A21I )]2 + [ im − 2 c66

−1 P A ] 2
1I 1I

2 c44

Aˆ 3 =

−1
im − 2 c66

−1 P A
1I 1I

2 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 3 + 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 3 + 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 = 1 +
, a2 = (2 − a1 ).
c44 ( Aˆ 3 + 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 2 ρ + A2
ρ L+
, ε2 =
, ε3 =
, ε4 =
, ε 5 = ωρ+ k22+ , ε 6 =
,

a+b
c44+
c44+
c44+
ρ+

c66−
ω 2 ρ − A2
ρ L−
ε7 =
, ε8 =
, ε 9 = ωρ− k22− , ε 10 =
, θ, f 1 , f 2 .
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. 3 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 1 , ε 2 makes the reflection coefficient increasing and makes the
transmission coefficient decreasing (see Figs. 3(b), 3(c)).
(iii) In contrast, the increasing of ε 4 makes the reflection coefficient decreasing and
makes the transmission coefficient increasing (see Fig. 3(d)).
1.6

1

|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
1

0.4

0.9
0.8

0.2

0.7
0
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
1
θ = 600; ε = 1.2; ε = 1.3; ε = 1.5; ε = 1.6; ε = 0.6;
0
2
3
4
5
6
ε = 1.8; ε = 3.5; ε = 3.1; ε = 1.2; f = 0.8; f = 0.7
7

8

(a)

9

10

1

0.8

2

(b)
(c)

(d)
|R|
|T|

|R|
|T|

1

1
0.8
|R|, |T|

|R|, |T|

0.8
0.6

0.6

0.4

0.4

0.2

0.2

0

0.9

0

0.5

1
2
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

3

0.5

(c)

1
1.5
2
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

3

(d)

Fig. 3. The dependence of the moduli | R| and | T | of the reflection
and transmission coefficients on θ0 (a), ε 1 (b), ε 2 (c), ε 4 (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 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.
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