- Email: [email protected]
Homogenization of very rough interfaces for the micropolar elasticity theory
Accepted Manuscript
Homogenization of very rough interfaces for the micropolar elasticity theory P.C. Vinh, V.T.N. Anh, D.X. Tung, N.T. Kieu PII: DOI: Reference:
S0307-904X(17)30592-9 10.1016/j.apm.2017.09.039 APM 11984
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
30 March 2017 17 August 2017 19 September 2017
Please cite this article as: P.C. Vinh, V.T.N. Anh, D.X. Tung, N.T. Kieu, Homogenization of very rough interfaces for the micropolar elasticity theory, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.09.039
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlights • The homogenization of a very rough interface of two micropolar solids is investigated. • The homogenized equations in explicit form have been obtained.
CR IP T
• They are very useful in solving various practical problems.
• The reflection and transmission of waves at a very rough interface is considered.
AC
CE
PT
ED
M
AN US
• The formulas for the reflection and transmission coefficients have been derived.
1
ACCEPTED MANUSCRIPT
CR IP T
Homogenization of very rough interfaces for the micropolar elasticity theory D. X. Tung2 , P. C. Vinh 1 ∗, V. T. N. Anh1 , N. T. Kieu2 1 Faculty of Mathematics, Mechanics and Informatics Hanoi University of Science 334, Nguyen Trai Str., Thanh Xuan, Hanoi,Vietnam 2
AN US
Faculty of Civil Engineering Hanoi Architectural University Km 10 Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam
M
Abstract
AC
CE
PT
ED
In this paper, the homogenization of a very rough two-dimensional interface separating two dissimilar isotropic micropolar elastic solids is investigated. The interface is assumed to oscillate between two parallel straight lines. The main aim is to derive homogenized equations in explicit form. These equations are obtained by the homogenization method along with the matrix formalism of the theory of micropolar elasticity. Since obtained homogenized equations are totally explicit, they are a powerful tool for solving various practical problems. As an example, the reflection and transmission of a longitudinal displacement plane wave at a very rough interface of toothcomb type is investigated. The closed-form formulas for the reflection and transmission coefficients have been derived. Based on these formulas, some numerical examples are carried out to show the dependence of the reflection and transmission coefficients on the incident angle and the geometry parameter of the interface.
Key words: Homogenized equations, Very rough interfaces, Theory of micropolar elasticity, Reflection and transmission of waves. ∗
Corresponding author: Tel:+84-4-35532164; Fax:+84-4-38588817; E-mail address: [email protected] (P. C. Vinh)
2
pcv-
ACCEPTED MANUSCRIPT
1
Introduction Boundary-value problems in domains with rough boundaries or interfaces ap-
CR IP T
pear in many fields of natural sciences and technology such as: scattering of waves at rough boundaries [1], reflection and transmission of waves at rough interfaces [2], nearly circular holes and inclusions in the plane elasticity and the thermoelasticity [3], and so on. When the amplitude (height) of the roughness is much small in com-
AN US
parison with its period, the problems are usually analyzed by perturbation methods [4]. When the amplitude is much larger than its period, i.e. the boundaries and interfaces are very rough, the homogenization method is required, see for instance: [5, 6, 7]. Nevard and Keller [8] examined the homogenization of a very rough three-
M
dimensional interface that oscillates between two parallel planes and separates two
ED
linear anisotropic solids. By applying the homogenization method [6], the authors have derived the homogenized equations, but these equations are still implicit. In
PT
some recent papers by Vinh and Tung [9, 10, 11], the explicit homogenized equations
CE
of the linear elasticity in two-dimensional domains with interfaces rapidly oscillating between two parallel straight lines and between two concentric circles have been
AC
obtained.
Modern engineering structures are often made of materials possessing an internal
structure such as reinforced soils [12, 13], bone [14, 15], cellular materials such as foams [16, 17], masonry structural elements [18, 19, 20], granular or cracked media [21], etc, and these materials are modeled by the micropolar elasticity theory [22, 23]. 3
ACCEPTED MANUSCRIPT
The consideration the boundary-value problems of the micropolar elasticity theory in domains with very rough boundaries or interfaces is therefore significant and of great theoretical and practical as well interest.
CR IP T
In this paper, the homogenization of a very rough two-dimensional interface separating two dissimilar isotropic micropolar elastic solids is investigated. It is assumed that the interface oscillates between two parallel straight lines. The main aim of
AN US
the investigation is to derive explicit homogenized equations and explicit associate continuity conditions. First, the basic equations and the continuity conditions of the linear theory of micropolar elasticity are written in matrix form. Then, by using an appropriate asymptotic expansion of the solution and following standard techniques
M
of the homogenization method, the explicit homogenized equation and the explicit associate continuity conditions in matrix form are derived. They are then written
ED
down in component form for the orthotropic case.
PT
Note that, while the continuity condition on the interface of the theory of elasticity [9, 10] and the theory of piezoelasticity [24] contains only the derivatives of the
CE
solution, the one of the micropolar elasticity theory is expressed in terms of both
AC
solution and its derivatives. This leads to requirement that the matrix equation must be written in a relevant form so that the problems on the periodicity cell can be solved easily. 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. As an example
4
ACCEPTED MANUSCRIPT
proving this, the reflection and transmission of a longitudinal displacement plane wave at a very rough interface of tooth-comb type is considered. The closed-form analytical expressions of the reflection and transmission coefficients are obtained.
some parameters is investigated numerically.
2
CR IP T
Based on them the dependence of the reflection and transmission coefficients on
Basic equations in matrix form
AN US
We consider a micropolar elastic body. The equations governing its motion are given by Eringen [22]:
tki,k + ρfi = ρ¨ ui , mki,k + εirs trs + ρli = ρj φ¨i , i = 1, 2, 3
(1)
M
where tki is the stress tensor, mki is the couple stress tensor, ui is the displacement
ED
vector, φi is the microrotation vector, ρ is the mass density, j is the micro-inertia, fi is the body force vector, li is the body couple vector, εirs is the alternating
PT
symbol (also called Levi-Civita symbol), commas denote differentiation with respect to spatial variables xk and a superposed dot signifies differentiation with respect to
CE
the time t.
AC
For linear isotropic micropolar elastic materials, the constitutive equations are: tki = λur,r δki + µ(uk,i + ui,k ) + κ(ui,k − εkir φr )
(2)
mki = αφr,r δki + βφk,i + γφi,k
where δki is the Kronecker symbol, λ, µ, κ, α, β and γ are the material moduli. Suppose that the motion of the micropolar elastic body creates a plane strain [25] for which all quantities are independent of the variable x3 and the displacement 5
ACCEPTED MANUSCRIPT
vector and the microrotation vector are of the form: u1 = u1 (x1 , x2 , t),
φ3 = φ(x1 , x2 , t), u3 ≡ φ1 ≡ φ2 ≡ 0.
u2 = u2 (x1 , x2 , t),
(3)
to:
CR IP T
Then, the equations of motion (1) and the constitutive equations (2) are simplified
t11,1 + t21,2 + ρf1 = ρ¨ u1 , t12,1 + t22,2 + ρf2 = ρ¨ u2 ,
(4)
m13,1 + m23,2 + t12 − t21 + ρl3 = ρj φ¨
AN US
and:
t11 = (λ + 2µ + κ)u1,1 + λu2,2 , t22 = λu1,1 + (λ + 2µ + κ)u2,2 , t12 = µu1,2 + (µ + κ)u2,1 − κφ, t21 = (µ + κ)u1,2 + µu2,1 + κφ , m13 = γφ,1 , m23 = γφ,2
(5)
M
Let the linear isotropic micropolar body occupy two-dimensional domains Ω+
ED
and Ω− of the plane x1 x2 whose interface is the curve L expressed by equation: x2 = h(y), y = x1 /ε, where h(y) is a periodic function of period 1 whose minimum
PT
and maximum values are 0 and H, respectively, ε is assumed to be much more small than H, i. e. the curve L is a very rough interface oscillating between two straight
CE
lines x2 = 0 and x2 = H, see Fig.1. We also assume that, in the domain 0 < x1 < ε,
AC
i.e. 0 < y < 1, any straight line x2 = x02 = const (0 < x02 < H) has exactly two intersections with the curve L. Note that in Eqs. (4) and (5), the material parameters λ, µ, κ, γ, ρ and j take
(different) constant values in Ω+ and Ω− : ( λ+ , µ+ , κ+ , γ + , ρ+ , j + for (x1 , x2 ) ∈ Ω+ λ, µ, κ, γ, ρ, j = λ− , µ− , κ− , γ − , ρ− , j − for (x1 , x2 ) ∈ Ω− 6
(6)
ACCEPTED MANUSCRIPT
where λ+ , µ+ , κ+ , γ + , ρ+ , j + , λ− , µ− , κ− , γ − , ρ− , j − are constants. Suppose Ω+ and Ω− are in welded contact with each other along their interface
CR IP T
L, then the continuity conditions, namely: [φ]L = 0, [uk ]L = 0, [t1k n1 + t2k n2 ]L = 0, [m1k n1 + m2k n2 ]L = 0, k = 1, 2
(7)
must be satisfied, where nk is the unit normal to the curve L and the notation [.]L defined as:
(8)
PT
ED
M
AN US
[f ]L = f + − f −
AC
CE
Figure 1: Two-dimensional domains Ω+ and Ω− have a very rough interface L expressed by equation x2 = h(x1 /ε)=h(y), where h(y) is a periodic function with period 1. The curve L highly oscillates between the parallel straight lines x2 = 0 and x2 = H. It is not difficult to verify that in matrix form Eqs. (4) and (5) can be written
as:
(A11 u,1 + A12 u,2 + Gu),1 + (A21 u,1 + A22 u,2 + Hu),2 + Bu,1 + Du,2 + Eu + F = ρ¨ u (9) 7
ACCEPTED MANUSCRIPT
A11
A21
λ + κ + 2µ 0 0 0 κ + µ 0 , = 0 0 γ
0 µ 0 = λ 0 0 , 0 0 0
A22
0 0 0 0 , E = 0 0 0 0 −2κ ρf2
ρl3 ]T ,
κ+µ 0 0 λ + κ + 2µ 0 , = 0 0 0 γ
0 0 0 D = 0 0 0 , −κ 0 0
(10)
0 0 0 G = 0 0 −κ , 0 0 0 u = [u1
u2
0 0 κ H = 0 0 0 , 0 0 0
φ]T , ρ = diag[ρ ρ ρj]
M
F = [ρf1
0 λ 0 = µ 0 0 , 0 0 0
AN US
0 0 0 B = 0 0 0 , 0 κ 0
A12
CR IP T
where:
The continuity conditions (7) are expressed in matrix form as: # " # A11 u,1 +A21 u,2 +Gu n1 + A12 u,1 +A22 u,2 +Hu n2 = 0 (11)
ED
[u]L = 0,
"
L
L
PT
Remark 1: Appearance of terms Gu and Hu in the brackets of Eq. (9) makes this
CE
equation compatible with the continuity condition (11). This compatibility makes the solving of the cell problems (17)-(19) more simple and the expression of solution
AC
is compact.
Expressing nk (k = 1, 2) in terms of the function h, the continuity condition (11)2
is written as: ε−1
"
" # # 0 h (y) A11 u,1 + A21 u,2 + Gu − A12 u,1 + A22 u,2 + Hu = 0 (12) L
L
8
ACCEPTED MANUSCRIPT
We introduce the vectors Σ1 = [t11
m13 ]T and Σ2 = [t21
t12
t22
m23 ]T . One
can see that these vectors are given by: (13)
CR IP T
Σ1 = A11 u,1 + A12 u,2 + Gu, Σ2 = A21 u,1 + A22 u,2 + Hu In term of Σk , Eq. (9) is written as:
Σ1,1 + Σ2,2 + Bu,1 + Du,2 + Eu + F = ρ¨ u
(14)
AN US
Note that, according to (6), the material parameters in the above equations are attached respectively to the symbol ”+ ” and ”− ” corresponding to the half-space Ω+ and the half-space Ω− .
The explicit homogenized equation in matrix form
M
3
ED
Following Bensoussan et al. [6], Sanchez-Palencia [26], Bakhvalov and Panasenko
PT
[5] we suppose that: u(x1 , x2 , ε, t)=U(x1 , y, x2 , ε, t), and we express U as follows (see, Vinh and Tung [9, 10]):
CE
U = V + ε N1 V + N11 V,1 + N12 V,2 + ε2 N2 V + N21 V,1 + N22 V,2
AC
+ N211 V,11 + N212 V,12 + N222 V,22 + O(ε3 )
(15)
where V=V(x1 , x2 , t) (being independent of y), N1 , N11 , N12 , N2 , N21 , N22 , N211 , N212 , N222 are 4×4-matrix valued functions of y and x2 (not depending on x1 and t), and they are y-periodic with period 1. Remark 2: 9
ACCEPTED MANUSCRIPT
From (15) one can see that u approaches V when ε tends to zero, V is therefore the leading term of u. When ε is very small, we find V instead of u. Thus, we need
Since y = x1 /ε, therefore we have: u,1 = U,1 + ε−1 U,y
CR IP T
to derive the equation for V. This equation is called the homogenized equation.
(16)
Substituting (15) into (9), (12) and (11)1 , and taking into account (16) yield equa-
AN US
tions that we call equations (e1 ), (e2 ) and (e3 ), respectively. In order to find the explicit homogenized equation and the explicit continuity conditions, we carry out the following steps.
M
• The first step (deriving local problems):
In order to make the coefficients of ε−1 of equations (e1 ) and (e2 ), and the
ED
coefficient of ε of equation (e3 ) zero, the functions N1 , N11 , N12 are chosen so that:
PT
h i A11 N1,y + G = 0, 0 < y < 1, y 6= y1 , y2 ; ,y
AC
CE
[A11 N1,y + G]L = 0, [N1 ]L = 0 at y1 , y2 ; N1 (0) = N1 (1)
[A11
h i 11 A11 I + N,y = 0, 0 < y < 1, y 6= y1 , y2 ;
(17)
,y
11 11 11 I + N11 ,y ]L = 0, [N ]L = 0 at y1 , y2 ; N (0) = N (1) A11 N12 ,y + A12
,y
(18)
= 0, 0 < y < 1, y 6= y1 , y2 ;
12 12 12 [ A11 N12 ,y + A12 ]L = 0, [N ]L = 0, at y1 , y2 ; N (0) = N (1) 10
(19)
ACCEPTED MANUSCRIPT
where I is the identity 4×4-matrix, y1 , y2 (0 < y1 < y2 < 1) are two roots in the interval (0 , 1) of the equation h(y) = x2 for y, in which x2 belongs to the interval (0 H). The problems (17)-(19) are called the local problems (or the problems on the
proposition. Proposition 1: Let qk be defined as follows:
CR IP T
periodicity cell). Using these local problems it is not difficult to prove the following
AN US
q1 = hBN1,y i, q2 = hA11 N1,y + Gi, q3 = hA21 N1,y i
11 11 q4 = hB I + N11 ,y i, q5 = hA11 I + N,y i, q6 = hA21 I + N,y i
(20)
12 12 q7 = hBN12 ,y i, q8 = hA11 N,y + A12 i, q9 = hA21 N,y i
Z
M
where:
f dy
(21)
0
ED
hf i =
1
then they are given by:
PT
−1 −1 −1 −1 −1 −1 −1 q1 = hBA−1 11 ihA11 i hA11 Gi − hBA11 Gi, q2 = hA11 i hA11 Gi
CE
−1 −1 −1 −1 q3 = hA21 A−1 11 ihA11 i hA11 Gi − hA21 A11 Gi
−1 −1 −1 −1 −1 −1 −1 q4 = hBA−1 11 ihA11 i , q5 = hA11 i , q6 = hA21 A11 ihA11 i
(22)
AC
−1 −1 −1 −1 −1 −1 −1 q7 = hBA−1 11 ihA11 i hA11 A12 i − hBA11 A12 i, q8 = hA11 i hA11 A12 i −1 −1 −1 −1 q9 = hA21 A−1 11 ihA11 i hA11 A12 i − hA21 A11 A12 i
Note that in Eq. (22): hf i = (y2 − y1 )f − + (1 − y2 + y1 )f + 11
(23)
ACCEPTED MANUSCRIPT
To make the coefficient of ε0 of Eq.(e2 ) zero we take: h i h i A11 N2,y + A12 N1,2 + GN1 = A21 N1,y + H /h0 at y1 , y2 L
L
L
CR IP T
i h i 11 11 11 [A11 N1 + N21 + A N + GN = A I + N /h0 at y1 , y2 12 ,2 21 ,y ,y L
i i h h 12 12 1 12 22 = A22 + A21 N,y /h0 at y1 , y2 A11 N,y + A12 N + N,2 + GN L
L
h
A11 N11 + N211 ,y
i
L
= 0 at y1 , y2
h i 11 A11 N12 + N212 + A N = 0 at y1 , y2 12 ,y
AN US
L
h i 12 A11 N222 + A N = 0 at y1 , y2 12 ,y L
(24)
• The second step (deriving homogenized equation in explicit form):
M
Equating to zero the coefficient ε0 of Eq.(e1 ) provides:
AC
CE
PT
ED
o i A11 N2,y + A12 N1,2 + GN1 + BN1,y + E V + ,y i nh o 11 1 11 11 1 21 + A11 N,y + G + B I + N,y V,1 A11 N + N,y + A12 N,2 + GN ,y nh i o 1 12 12 12 + A11 N22 + A N + N + GN + BN + D V,2 12 ,y ,2 ,y ,y nh i o 11 + A11 N11 + N211 + A I + N V,11 11 ,y ,y ,y nh i o 11 12 + A11 N12 + N212 + A N + A N + A V,12 11 12 12 ,y ,y ,y h i 1 11 12 + A21 N,y + H V + A21 I + N,y V,1 + A21 N,y + A22 V,2 ,2 h i 12 ¨ + A11 N222 V,22 + F = ρV (25) ,y + A12 N nh
,y
By integrating equation (25) along the line x2 = const, 0 < x2 < H from y = 0 to
12
ACCEPTED MANUSCRIPT
y = 1, and taking into account Eqs. (24) we arrive at:
CR IP T
n o n o hBN1,y i + hEi V + hA11 N1,y + Gi + hB I + N11 ,y i V,1 n o n n o o 11 12 + hBN12 i + hDi V + hA I + N i V + hA N + A i V,12 ,2 11 ,11 11 12 ,y ,y ,y 1 1 + − ¨ +hΦ,2 i + Φ − Φ − + hFi = hρiV (26) h0 (y2 ) h0 (y1 ) where:
AN US
12 Φ = A21 N1,y + G V + A21 I + N11 ,y V,1 + A21 N,y + A22 V,2
(27)
It is not difficult to prove the following result. Proposition 2:
1 1 − = hΦi,2 h0 (y2 ) h0 (y1 )
M
hΦ,2 i + Φ+ − Φ−
(28)
ED
According to Proposition 2, Eq. (26) is simplified to:
CE
PT
o n n o hBN1,y i + hEi V + hA11 N1,y + Gi + hB I + N11 ,y i V,1 o n o n n o 12 11 i V + hA N + A i V,12 i + hDi V + hA I + N + hBN12 ,2 11 ,11 11 ,y 12 ,y ,y h + hA21 N1,y i + hHi V + hA21 I + N11 ,y iV,1 i 12 ¨ + hA21 N,y i + hA22 i V,2 + hFi = hρiV (29)
AC
,2
Substituting the results (22) into Eq.(29) we obtain the homogenized equation in explicit form, namely:
13
ACCEPTED MANUSCRIPT
CR IP T
h i −1 −1 −1 −1 −1 −1 −1 hA−1 i V + hA i hA A iV + hA A ihA i V ,11 12 ,12 21 ,1 11 11 11 11 11 ,2 hn o i −1 −1 −1 −1 + hA22 i + hA21 A−1 11 ihA11 i hA11 A12 i − hA21 A11 A12 i V,2 ,2 h i −1 −1 −1 −1 + hBA−1 + hA−1 11 ihA11 i 11 i hA11 Gi V,1 h i −1 −1 −1 −1 + hDi + hBA−1 ihA i hA A i − hBA A i V,2 12 12 11 11 11 11 h i −1 −1 −1 −1 −1 + hA21 A11 ihA11 i hA11 Gi − hA21 A11 Gi + hHi V ,2 i h −1 −1 −1 −1 ¨ + hEi + hBA−1 11 ihA11 i hA11 Gi − hBA11 Gi V + hFi = hρiV
(30)
(9) in which u is replaced by V. • Continuity condition
AN US
Note that for the domains x2 < 0 and x2 > H the homogenized equations are Eq.
M
By Σ02 we denote the leading term of the Σ2 defined by (13)2 . It is required that
ED
V and Σ02 are continuous on the lines x2 = 0 and x2 = H: [V]L∗ = 0,
[Σ02 ]L∗ = 0,
L∗ : x2 = 0, x2 = H
(31)
PT
Introducing the asymptotic expansion (15) into (13)2 one can see that:
CE
12 Σ02 = A21 N1,y + H V + A21 N11 ,y + I V,1 + A21 N,y + A22 V,2
(32)
AC
Integrating equation (32) along the line x2 = const, 0 < x2 < H from y = 0 to y = 1 and using the expressions of qk (k = 3, 6, 9) given in (22) we arrive at: −1 −1 −1 −1 −1 −1 −1 Σ02 = h A21 A−1 11 ihA11 i hA11 Gi − hA21 A11 Gi + hHi V + hA21 A11 ihA11 i V,1 −1 −1 −1 −1 + hA22 i + hA21 A−1 ihA i hA A i − hA A A i V,2 12 21 12 11 11 11 11 14
(33)
ACCEPTED MANUSCRIPT
Analogously, the leading term Σ01 of the vector Σ1 is given by: −1 −1 −1 −1 −1 −1 −1 Σ01 = hA−1 11 i hA11 GiV + hA11 i V,1 + hA11 i hA11 A12 iV,2
CR IP T
Remark 3:
(34)
i) In terms of Σ0k , Eq. (30) becomes:
(35)
CE
PT
ED
M
AN US
−1 −1 Σ01,1 + Σ02,2 + hBA−1 11 ihA11 i V,1 h i −1 −1 −1 −1 + hDi + hBA−1 ihA i hA A i − hBA A i 12 12 V,2 11 11 11 11 h i −1 −1 −1 −1 + hEi + hBA−1 ihA i hA Gi − hBA Gi V + hFi = hρiV¨ 11 11 11 11
Figure 2: The interface of tooth-comb type.
AC
ii) When the interface L is of tooth-comb type (see Fig.2), all h•i’s do not depend
15
ACCEPTED MANUSCRIPT
where:
1 (a1 φ+ + a2 φ− ) a1 + a2
(36)
(37)
M
hφi =
AN US
CR IP T
on x2 , the homogenized equation (30) is therefore simplified to: −1 −1 −1 −1 −1 −1 −1 −1 V,12 hA11 i V,11 + hA11 i hA11 A12 i + hA21 A11 ihA11 i −1 −1 −1 −1 + hA22 i + hA21 A−1 11 ihA11 i hA11 A12 i − hA21 A11 A12 i V,22 −1 −1 −1 −1 −1 −1 + hBA11 ihA11 i + hA11 i hA11 Gi V,1 −1 −1 −1 −1 −1 −1 −1 −1 + hDi + hBA−1 11 ihA11 i hA11 A12 i − hBA11 A12 i + hA21 A11 ihA11 i hA11 Gi −1 − hA21 A11 Gi + hHi V,2 −1 −1 −1 −1 −1 + hEi + hBA11 ihA11 i hA11 Gi − hBA11 Gi V + hFi = hρiV¨
The explicit homogenized equations in component form
PT
4
ED
Note that all coefficients of Eq (36) are constant.
CE
• The homogenized equations:
AC
Using (10) one can write the homogenized equation (30) in component form.
16
ACCEPTED MANUSCRIPT
AN US
CR IP T
They are (0 < x2 < H): −1 −1 −1 −1 −1 2 2 −1 ha i V1,11 + hbi + hb i hµb i − hµ b i V1,2 + ha−1 i−1 hλa−1 iV2,12 ,2 −1 −1 −1 −1 −1 −1 −1 −1 + hb i hµb iV2,1 + hκi + hκµb i − hb i hκb ihµb i Φ ,2 ,2 ¨ +hρf1 i = hρiV1 −1 −1 −1 −1 −1 −1 hb i hµb iV1,12 + ha i hλa iV1,1 + hb−1 i−1 V2,11 ,2 + hai + ha−1 i−1 hλa−1 i2 − hλ2 a−1 i V2,2 − hb−1 i−1 hκb−1 iΦ,1 + hρf2 i = hρiV¨2 ,2 hb−1 i−1 hκb−1 ihµb−1 i − hκµb−1 i − hκi V1,2 + hb−1 i−1 hκb−1 iV2,1 + hγ −1 i−1 Φ,11 ¨ + hγiΦ,2 + hκ2 b−1 i − hb−1 i−1 hκb−1 i2 − 2hκi Φ + hρl3 i = hρjiΦ ,2 where:
b=µ+κ
(39)
M
a = λ + 2µ + κ;
(38)
ED
For the domains x2 < 0 and x2 > H the homogenized equations in component form
CE
PT
are: ¨ (λ + 2µ + κ)V1,11 + (µ + κ)V1,22 + (λ + µ)V2,12 + κΦ,2 + ρf1 = ρV1 (λ + µ)V1,12 + (µ + κ)V2,11 + (λ + 2µ + κ)V2,22 − κΦ,1 + (ρf2 ) = ρV¨2 ¨ −κV1,2 + κV2,1 + γΦ,11 + γΦ,22 − 2κΦ + (ρl3 ) = ρj Φ
(40)
AC
• The continuity conditions:
[Vk ]L∗ = 0, [t02k ]L∗ = 0, k = 1, 2, [Φ]L∗ = 0, [m023 ]L∗ = 0
(41)
where t021 , t022 and m023 are the components of the vector Σ02 and L∗ are the straight
lines: x2 = 0 and x2 = H. By using (33) and (34) it is not difficult to see that Eqs. (38) can be also written
17
ACCEPTED MANUSCRIPT
as follows: t011,1 + t021,2 + hρf1 i = hρiV¨1 t012,1 + t022,2 + hρf2 i = hρiV¨2
(42)
where:
t011 = ha−1 i−1 V1,1 + ha−1 i−1 hλa−1 iV2,2
CR IP T
¨ m013,1 + m023,2 + t012 − t021 + hρl3 i = hρjiΦ
m023 = hγiΦ,2 ;
AN US
t012 = hb−1 i−1 hµb−1 iV1,2 + hb−1 i−1 V2,1 − hb−1 i−1 hκb−1 iΦ 0 −1 −1 −1 2 2 −1 t21 = hbi + hb i hµb i − hµ b i V1,2 + hb−1 i−1 hµb−1 iV2,1 −1 −1 −1 −1 −1 + hκi + hκµb i − hb i hκb ihµb i Φ 0 −1 −1 −1 −1 −1 −1 2 2 −1 t22 = ha i hλa iV1,1 + hai + ha i hλa i − hλ a i V2,2
(43)
m013 = hγ −1 i−1 Φ,1
M
Equations (42) and (43) are a convenient tool for studying the reflection and refrac-
ED
tion of waves at a very rough interface separating two dissimilar micropolar elastic half-spaces.
PT
Remark 4: According to (23), the coefficients of the homogenized equations (38)
CE
and (43) are totally explicit functions in terms of x2 and the material parameters, provided that hϕi is explicitly expressed in terms of x2 and constants ϕ+ (value of
AC
ϕ in Ω+ ), ϕ− (value of ϕ in Ω− ) for any ϕ. There are many interfaces for which this is satisfied such as the tooth-comb interface (Fig.2), the tooth-saw interface (Fig.3a) and the interface of sine type (Fig. 3b). For the tooth-comb interface hϕi is calculated by (37). It is given by: hϕi = (1 −
x2 + x2 − )ϕ + ϕ H H 18
(44)
ACCEPTED MANUSCRIPT
for the tooth-saw interface and: 1 2x2 + 1 2x2 − hϕi = 1 − arccos(1 − ) ϕ + arccos(1 − )ϕ π H π H
M
AN US
CR IP T
for the interface of sine type.
(45)
Reflection and transmission of a longitudinal displacement plane wave at a very rough interface of tooth-comb type
CE
PT
5
ED
Figure 3: The interfaces of tooth-saw type (a) and sin type (b) .
AC
As an example of application of the obtained explicit homogenized equations, in this section we consider the reflection and transmission of a longitudinal displacement plane wave at a very rough interface of tooth-comb type separating two isotropic micropolar elastic half-spaces (Fig. 2). By the meaning of homogenization, this problem is reduced to the reflection and transmission of a longitudinal
19
ACCEPTED MANUSCRIPT
displacement plane wave through a homogeneous material layer occupying the domain 0 < x2 < H (see Fig. 4). According to (42), in the absence of body forces, the
CR IP T
motion of the medium is governed by the equations: t011,1 + t021,2 = hρiV¨1 , t012,1 + t022,2 = hρiV¨2
(46)
ED
M
AN US
¨ m013,1 + m023,2 + t012 − t021 = hρjiΦ
CE
PT
Figure 4: The reflected and refracted waves created by an incident longitudinal displacement wave . For the layer, the relations between t0kn , m0kn and Vk , Φ are given by (43). For
AC
the half-spaces they are: t011 = aV1,1 + λV2,2 , t012 = µV1,2 + bV2,1 − κΦ, t021 = bV1,2 + µV2,1 + κΦ
t022
= λV1,1 + aV2,2 ,
m023
= γΦ,2 ,
m013
(47)
= γΦ,1
Note that, according to (6), the material parameters in the above constitutive equations are attached respectively to the symbol ”+ ” and ”− ” corresponding to the 20
ACCEPTED MANUSCRIPT
half-space Ω+ and the half-space Ω− . At the interfaces x2 = 0 and x2 = H the continuity condition of V1 , V2 , Φ, t021 , t022 and m023 is required to be satisfied.
Incident wave
CR IP T
5.1
Let a longitudinal displacement plane wave with the unit amplitude, the incident angle θ0 (0 < θ0 < π/2), the wave number k0 , the phase velocity v0 , propagate in the half-space Ω+ (Fig. 4). Then, its mechanical displacements V1I , V2I and the
AN US
micro-rotation ΦI are given by (see, Tomar and Gogna [27]):
V1I = sinθ0 ei(k0 x1 sinθ0 +k0 x2 cosθ0 −v0 t) , V2I = cosθ0 ei(k0 x1 sinθ0 +k0 x2 cosθ0 −v0 t) , ΦI ≡ 0 p (λ+ + 2µ+ + κ+ )/ρ+ . Note that k0 v0 = ω where ω is the circular
M
where v0 =
(48)
ED
wave frequency. The expressions (48) can be rewritten as:
PT
VmI = UmI (z)eik(x1 −ct) , m = 1, 2, ΦI = kΞI (z)eik(x1 −ct) , z = kx2
(49)
where k = k0 sinθ0 , c = v0 /sinθ0 and:
CE
U1I (z) = sinθ0 eiz cotgθ0 , U2I (z) = cosθ0 eiz cotgθ0 , ΞI (z) ≡ 0
(50)
AC
From (47) and (48) we have: t021I = ikT1I (z)eik(x1 −ct) , t022I = ikT2I (z)eik(x1 −ct) , m023I = iMI (z)eik(x1 −ct)
21
(51)
ACCEPTED MANUSCRIPT
where: T1I (z) = (2µ+ + κ+ )cosθ0 eiz cotgθ0 ,
MI (z) ≡ 0
(52)
CR IP T
T2I (z) = [λ+ sinθ0 + (λ+ + 2µ+ + κ+ )cos2 θ0 /sinθ0 ]eizcotgθ0 ,
Remark 5 (see also, Tomar and Gogna [27]): It is not difficult to prove that for an incident longitudinal displacement wave there do not exist reflected and refracted
5.2
AN US
longitudinal microrotational waves.
Stroh formalism of half-spaces and layer
As the reflected, refracted wave (propagating in the half-space Ω+ , Ω− , respectively)
M
and the reflected-refracted wave (propagating in the layer) are created and forced by
ED
the incident wave, they are therefore sought in the form (similar to (49) and (51)): V1 = U1 (z)eik(x1 −ct) , V2 = U2 (z)eik(x1 −ct) , Φ = kΞ(z)eik(x1 −ct) ,
PT
t021 = ikT1 (z)eik(x1 −ct) , t022 = ikT2 (z)eik(x1 −ct) , m023 = iM (z)eik(x1 −ct)
(53)
CE
where Um (z), Tm (z) (m = 1, 2), Ξ(z) and M (z) are unknown functions to be determined.
AC
Remark 6: Snell’s law has been taken into account of the expressions of solution (53).
From (46)-(47) and (53) one can see that the unknown functions Um (z), Tm (z) (m =
1, 2), Ξ(z) and M (z) are the solution of the differential equation: 0
ξ (z) = iNξ(z) 22
(54)
ACCEPTED MANUSCRIPT
where ξ = [U1 U2 Ξ T1 T2 M ]T , the prime indicates the differentiation with respect to z and:
N1 N2 N= ¯T N3 N 1
(55)
CR IP T
in which the bar indicates the complex conjugate. The matrices Nk are given by: 1 µ iκ 0 0 b 0 −b b 1 λ 0 N1 = − , N 2 = 0 , 0 0 a a 1 0 0 0 0 0 γk 2 λ2 2 0 0 ρc − a + a 2 µ µ (56) N3 = 0 ρc2 − b + −iκ(1 + ) b b µ κ2 0 iκ(1 + ) ρj c2 k 2 − γk 2 − 2κ + b b + − In Eq. (56) the sign ” ” (” ”) is attached to λ, µ, κ, γ, ρ and j of the half-space Ω+ (Ω− ). For the layer:
(1)
(3) (2) (1) N11 0 0 N11 0 0 N13 (3) (3) (2) N22 N23 N22 0 , N3 = 0 0 , N2 = 0 (3) (3) (2) ¯23 0 0 N N33 0 0 N33 (57)
CE
PT
0 N12 (1) N1 = N21 0 0 0
ED
M
AN US
AC
The non-zero entries of Nk (k = 1, 2, 3) for the layer are given in Appendix A. Equation (54) is called the Stroh formalism [28, 29]. It is the base tool for investigating the reflection and refraction problem. Remark 7: If we consider the propagation of a Rayleigh wave in the half-spaces Ω+ or Ω− , then Eq. (54) is its Stroh formalism in which c is the Rayleigh wave velocity 23
ACCEPTED MANUSCRIPT
that to be determined.
5.3
Characteristic equation
CR IP T
In order to find the solution of Eq. (54) we have to solve its characteristic equation: |N − pI| = 0
(58)
to calculate characteristic values p of matrix N, here I is the identity matrix of six
AN US
order.
Remark 8: In view of Remark 4 and according to Barnett and Lothe [30], there (r)
(r)
(r)
(r)
(t)
(t)
(t)
(t)
exist positive constants αk (0 < α0 < α1 < α2 ) and αk (0 < α0 < α1 < α2 ) so that:
M
For the half-space Ω+ : (r)
ED
(i) For c ≥ α2 : six roots pk of Eq. (58) are all real: pk < 0, p3+k = −pk (k = 1, 2, 3) and we order them as follows: 0 < |p1 | < |p2 | < |p3 |. (r)
(r)
≤ c < α2 : Eq. (58) has four real roots and a pair of complex
PT
(ii) For α1
conjugate roots: pk < 0, p3+k = −pk (k = 1, 2), p6 = p¯3 : 0 < |p1 | < |p2 |, Im(p3 ) < 0.
CE
(r)
(r)
(iii) For α0 ≤ c < α1 : Eq. (58) has two real roots and two pair of complex
AC
conjugate roots: p1 < 0, p4 = −p1 , p3+k = p¯k (k = 2, 3): Im(p2 ) < 0, Im(p3 ) < 0. (r)
(iv) For 0 < c < α0 : six roots of Eq. (58) are all complex: p3+k = p¯k ,
Im(pk ) < 0, (k = 1, 2, 3). For the half-space Ω− : (t)
(i) For c ≥ α2 : six roots qk of Eq. (58) are all real: qk > 0, q3+k = −qk (k = 24
ACCEPTED MANUSCRIPT
1, 2, 3) and we order them as follows: 0 < q1 < q2 < q3 . (t)
(t)
(ii) For α1 ≤ c < α2 : Eq. (58) are real has four real roots and a pair of complex conjugate roots: qk > 0, q3+k = −qk (k = 1, 2), q6 = q¯3 : 0 < q1 < q2 , Im(q3 ) > 0. (t)
(t)
CR IP T
(iii) For α0 ≤ c < α1 : Eq. (58) are real has two real roots and two pair of complex conjugate roots: q1 > 0, q4 = −q1 , q3+k = q¯k (k = 2, 3), Im(q2 ) > 0, Im(q3 ) > 0. (t)
AN US
(iv) For 0 < c < α0 : all roots of Eq. (58) are complex: q3+k = q¯k , Im(qk ) > 0, (k = 1, 2, 3).
5.4
Reflected waves
One can see that for the half-space Ω+ , the general solution of Eq. (54) is (for all
M
possibilities (i)-(iv) in Remark 7):
ED
ξ r (z) = w1r ξ 1r eip1 z + w2r ξ 2r eip2 z + w3r ξ 3r eip3 z
(59)
PT
where wkr are constants to be determined, ξ kr is the eigenvector corresponding to
CE
pk , i. e. it is a non-zero solution of equation: [N − pk I]ξ kr = 0
(60)
AC
We choose m-th component of vector ξ kr (m=1,...,6) as the cofactor of the m-th element of the first row of matrix [N − pk I]. From Remark 7 we see that: (r)
(2)
(i) For c ≥ α2 (↔ 0 < θ0 ≤ θr ): there exist three reflected waves denoted by
RWk (k = 1, 2, 3) and they are given by: RWk = wkr ξ kr eipk z eik(x1 −ct) , k = 1, 2, 3 25
(61)
ACCEPTED MANUSCRIPT
The RW1 is the reflected longitudinal displacement wave, the RW2 and RW3 are two reflected coupled transverse displacement waves (see, Parfitt and Eringen [31]; Tomar and Gogna [27]). (r)
(2)
≤ c < α2 (↔ θr
(1)
< θ0 ≤ θr ): there exist two reflected waves,
namely RW2 and RW3 , RW1 becomes a surface wave. (r)
(r)
(1)
(iii) For α0 ≤ c < α1 (↔ θr
(0)
CR IP T
(r)
(ii) For α1
< θ0 ≤ θr ): there exists one reflected waves
RW3 , the waves RW1 and RW2 convert into surface waves. (0)
AN US
(r)
(iv) For 0 ≤ c < α0 (↔ θr < θ0 ≤ π/2): no reflected wave exists, RW1 , RW2 and RW3 are all surface waves.
5.5
Refracted waves
M
For the half-space Ω− , the general solution of Eq. (54) is (for all possibilities (i)-(iv)
ED
in Remark 7):
ξ t (z) = w1t ξ 1t eiq1 z + w2t ξ 2t eiq2 z + w3t ξ 3t eiq3 z
(62)
CE
PT
where wkt are constants, ξ kt is a non-zero solution of equation: [N − qk I]ξ kt = 0
(63)
AC
and the components of ξ kt are taken as the cofactors of the first row of matrix [N − qk I]. We have conclusions similar to those corresponding to the reflected (t)
waves. For examples, for c ≥ α2 : there exist three reflected waves denoted by
TWk (k = 1, 2, 3) and they are given by: TWk = wkt ξ kt eiqk z eik(x1 −ct) , k = 1, 2, 3 26
(64)
ACCEPTED MANUSCRIPT
The TW1 is the refracted displacement longitudinal wave, the RW2 and RW3 are two refracted coupled transverse displacement waves (see, Parfitt and Eringen [31];
5.6
Transfer matrix
For the layer, the general solution of Eq. (54) is:
CR IP T
Tomar and Gogna [27]).
(65)
AN US
ξ L (z) = w1 ξ 1 eis1 z + ... + w6 ξ 6 eis6 z
where w1 ,..., w6 are constants to be determined, s1 ,..., s6 are six roots of the characteristic equation (58) corresponding to the layer, ξ k is the eigenvector corresponding
It follows from Eq. (65):
M
to sk and its components are taken as the cofactors of the first row of matrix [N−sk I].
ED
ξ L (z) = M(z)M−1 (0)ξ L (0), 0 ≤ z ≤ ε = k.H
(66)
PT
where M(z) is a square matrix of six order whose m-th column is eism z ξ m , in par-
CE
ticular:
M(z) = [eis1 z ξ 1 eis2 z ξ 2 eis3 z ξ 3 eis4 z ξ 4 eis5 z ξ 5 eis6 z ξ 6 ], 0 ≤ z ≤ ε
(67)
AC
Matrix Z = M(ε)M−1 (0) is called the transfer matrix of the layer.
5.7
The reflection and transmission coefficients
Putting z = ε in (66) we have: ξ L (ε) = Zξ L (0) 27
(68)
ACCEPTED MANUSCRIPT
From the continuity conditions at z = 0 and z = ε it follows: ξ L (ε) = ξ t (ε), ξ L (0) = ξ r (0) + ξ I (0)
It follows from (59) and (62) that:
AN US
CR IP T
where in view of (50) and (52), ξ I (0) is given by: sinθ0 cosθ0 0 ξ I (0) = + + (2µ + κ )cosθ 0 + [λ sinθ0 + (λ+ + 2µ+ + κ+ )cos2 θ0 /sinθ0 ] 0
(69)
ξ t (ε) = [ξ 1t eiε1 ξ 2t eiε2 ξ 3t eiε3 ]wt , ξ r (0) = [ξ 1r ξ 2r ξ 3r ]wr
(70)
(71)
M
where wt = [w1t w2t w3t ]T , wr = [w1r w2r w3r ]T , εk = qk ε, k = 1, 2, 3.
ED
From (68), (69) and (71) we have:
(72)
P = Z[ξ 1r ξ 2r ξ 3r 0 0 0 ] − [0 0 0 ξ 1t eiε1 ξ 2t eiε2 ξ 3t eiε3 ]
(73)
PT
Pw = −Zξ I (0)
CE
where w = [wr wt ]T and:
AC
The solution of Eq. (72) is: w = −P−1 Zξ I (0)
(74)
It is clear that Eq. (74) is a closed-form analytical formula of vector w. By ∆kr and δkr (∆kt and δkt ) we denote the cofactor of the first and second element, respectively,
28
ACCEPTED MANUSCRIPT
∗ of the first row of matrix N − Ipk (N − Iqk ). Then the reflection coefficient wkr (k = ∗ (k = 1, 2, 3) are given by: 1, 2, 3) and refraction coefficients wkt
q q ∗ 2 2 2 |, k = 1, 2, 3 = |wkr ∆kr + δkr |, wkt = |wkt ∆2kt + δkt
(75)
CR IP T
∗ wkr
The formulas (75) in which wkr and wkt calculated by (72) are desired formulas. They are closed-form formulas. From (75) one can see that the reflection and refraction coefficients depend on 14 dimensionless parameters:
(76)
M
AN US
λ+ + κ+ + γ+ , e , e = = 2 3 µ+ µ+ j + µ+ − − λ κ γ− − − − − 2 e− , e , e , e = j k = = = 0 1 0 µ− 2 µ− 3 j − µ− − − − ρ µ j a1 rρ = + , rµ = + , rj = + , θ0 , f = , e = Hk0 ρ µ j a1 + a2 + + 2 e+ 0 = j k0 , e1 =
As an example, we use the formulas (72) and (75) to examine the dependence of
ED
the reflection coefficients and the refraction coefficients on the incident angle θ0 , the
PT
dimensionless wave number e of the incident wave, the geometry parameter f of the interface L and the dimensionless micropolar internal parameter e− 2 . They are
CE
presented graphically in Fig. 5, 6, 7, 8. The dependence of the reflection angles θk (of RWk , k = 1, 2, 3) and the refraction angles θ3+k (of TWk , k = 1, 2, 3) on
AC
the incident angle θ0 is also considered. It is illustrated in Fig. 9. Note that the reflection and refraction angles are calculated by: θk = atan
1 1 , θ3+k = atan , k = 1, 2, 3 |pk | qk
It is shown from these figures that: 29
ACCEPTED MANUSCRIPT
(i) The RW1 , RW3 and TW1 are prominent, the remains are almost zero (Figs. 5-8). ∗ ∗ (ii) The reflection coefficient w1r and the refraction coefficient w1t are almost
CR IP T
∗ is quickly constant for θ0 ∈ (0, 45o ) but vary fast for θ0 ∈ (45o , 90o ) (Fig. 5): w1r ∗ increasing while w1t is fast decreasing.
(iii) The reflection and refraction coefficients depend slightly on e and e− 2 (Fig.
AN US
6, 8).
(iv) The reflection and refraction coefficients at f = 0 and f = 1 are the same (Fig. 7). It is reasonable because these two cases are indeed the case of two halfspaces with the plane interface x2 = 0.
M
(v) The reflection and refraction angles depend almost linearly on the incident
Conclusions
PT
6
ED
angle (Fig. 9).
In this paper is investigated the homogenization of two-dimensional interfaces that
CE
separates two isotropic micropolar elastic solids and highly oscillates between two parallel straight lines. The explicit homogenized equation in matrix form for the mi-
AC
cropolar elasticity theory in two-dimension domains with a very rough interface has been derived by applying the homogenization method. Then, it is written down in component form. Since the obtained homogenized equations are fully explicit, they are a powerful tool for investigating various practical problems. As an example, the
30
ACCEPTED MANUSCRIPT
1.4 * w1t
1
w* 2t
0.8
w* 3t
0.6
w* 1r
0.4
w* 2r w* 3r
0.2 20
40
θ0
60
80
AN US
0 0
CR IP T
1.2
ED
M
∗ Figure 5: The dependence of the reflection coefficients wkr and the refraction coef+ ∗ ficients wkt on the incident angle θ0 (of degree). Here we take: e+ 0 = 0.1569, e1 = 2, + − − − − e+ 2 = 0.08, e3 = 3.0303, e0 = 0.2092, e1 = 1.4865, e2 = 0.0824, e3 = 1.6892, rρ = 0.6885, rµ = 0.6727, rj = 1.3333, f = 0.3, e = 0.5.
1.4
PT
1.2 1
w* 1t w* 2t w* 3t
0.6
w* 1r
0.4
w* 2r
0.2 0 0
w* 3r
0.2
0.4
e
AC
CE
0.8
0.6
0.8
1
∗ Figure 6: The dependence of the reflection coefficients wkr and the refraction coeffi∗ cients wkt on the dimensionless incident wave number e. Here we take: e+ 0 = 0.1569, − + + + − − e1 = 2, e2 = 0.08, e3 = 3.0303, e0 = 0.2092, e1 = 1.4865, e2 = 0.0824, o e− 3 = 1.6892, rρ = 0.6885, rµ = 0.6727, rj = 1.3333, f = 0.3, θ0 = 46 .
31
ACCEPTED MANUSCRIPT
1.4 1.2
w* 2t w* 3t w* 1r
0.6
w* 2r
0.4
w* 3r
0.2
0.2
0.4
0.6
0.8
f
1
AN US
0 0
CR IP T
w* 1t
1 0.8
ED
M
∗ Figure 7: The dependence of the reflection coefficients wkr and the refraction coeffi∗ cients wkt on the geometry parameter f of the interface. Here we take: e+ 0 = 0.1569, + + + − − − e1 = 2, e2 = 0.08, e3 = 3.0303, e0 = 0.2092, e1 = 1.4865, e2 = 0.0824, o e− 3 = 1.6892, rρ = 0.6885, rµ = 0.6727, rj = 1.3333, e = 1.0, θ0 = 46 .
1.4 1.2
PT
1
* w1t
w* 2t
0.8
w* 3t
0.6
w* 1r
0.4
w* 2r w* 3r
0.2 0.1
0.12
0.14
AC
CE
0 0.08
Figure
8:
0.16
0.18
e− 2
0.2
∗ The dependence of the reflection coefficients wkr and the refraction
+ + + + ∗ coefficients wkt on e− 2 . Here we take: e0 = 0.1569, e1 = 2, e2 = 0.08, e3 = 3.0303, − − e− 0 = 0.2092, e1 = 1.4865, e3 = 1.6892, rρ = 0.6885, rµ = 0.6727,
rj = 1.3333, f = 0.3, e = 0.5, θ0 = 46o .
32
ACCEPTED MANUSCRIPT
90 θ1
80
θ2
70
θ3 θ4
60
θ5
50
30 20 10 0 0
10
20
30
40
50
60
70
80
90
AN US
θ0
CR IP T
θ6
40
Figure 9: The dependence of the reflection and refraction angles on the incident + + + − angle θ0 . Here we take: e+ 0 = 0.1569, e1 = 2, e2 = 0.08, e3 = 3.0303, e0 = 0.2092, − − − e1 = 1.4865, e2 = 0.0824, e3 = 1.6892, rρ = 0.6885, rµ = 0.6727, rj = 1.3333, f = 0.3, e = 0.5.
M
reflection and transmission of longitudinal waves at a very rough interface of toothcomb type is considered. The closed-form analytical expressions of the reflection
ED
and transmission coefficients have been obtained. Based on them the dependence
PT
of the reflection and transmission coefficients on some parameters is investigated numerically. It is shown that for a incident longitudinal displacement wave, the
CE
reflection and refraction longitudinal displacement waves are prominent.
AC
Acknowledgments The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 107.02-2014.17. The work was partly supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM). 33
ACCEPTED MANUSCRIPT
Appendix A. The non-zero entries of matrices Nk for the layer (1)
hµb−1 ihb−1 i−1 hbi + hµb−1 i2 hb−1 i−1 − hµ2 b−1 i i hκi + hκµb−1 i − hκb−1 ihµb−1 ihb−1 i−1 =− hbi + hµb−1 i2 hb−1 i−1 − hµ2 b−1 i hλa−1 iha−1 i−1 =− hai + hλa−1 i2 ha−1 i−1 − hλ2 a−1 i
(1)
N13
(1)
N21
1
(2)
N11 =
hbi +
hµb−1 i2 hb−1 i−1
hai + 1 = 2 k hγi
hλa−1 i2 ha−1 i−1
(2)
N33
1
− hµ2 b−1 i
− hλ2 a−1 i
AN US
(2)
N22 =
CR IP T
N12 = −
2 hλa−1 iha−1 i−1 = ρc − ha i + hai + hλa−1 i2 ha−1 i−1 − hλ2 a−1 i 2 hµb−1 ihb−1 i−1 (3) 2 −1 −1 N22 = ρc − hb i + hbi + hµb−1 i2 hb−1 i−1 − hµ2 b−1 i −1 −1 −1 −1 −1 −1 −1 −1 hκi + hκµb i − hκb ihµb ihb i hµb ihb i (3) N23 = i hbi + hµb−1 i2 hb−1 i−1 − hµ2 b−1 i − hκb−1 ihb−1 i−1 (3) N11
−1 −1
(77)
ED
M
2
(3)
CE
PT
N33 = (ρjc2 − hγ −1 i−1 )k 2 + hκi + hκµb−1 i − hκb−1 ihµb−1 ihb−1 i−1 2 hκi + hκµb−1 i − hκb−1 ihµb−1 ihb−1 i−1 −1 −1 −1 − hκb ihb i + hbi + hµb−1 i2 hb−1 i−1 − hµ2 b−1 i
AC
References
[1] K.A. Zaki, A. R. Neureuther, Scattering From a Perfectly Conducting Surface With a Sinusoidal Height Profile: TE Polarization. IEEE Trans, Antennas Propag. 19 (1971) 208-214.
34
ACCEPTED MANUSCRIPT
[2] S. S. Singh, S. K. Tomar, qP-Wave at a Corrugated Interface Between Two Dissimilar Pre-Stressed Elastic Half-Spaces, J. Sound Vib. 317 (2008) 687-708.
CR IP T
[3] T. C. Ekneligoda, R. W. Zimmermam, Boundary Pertubation Solution for Nearly Circular Holes and Rigid Inclusions in an Infnite Elastic Medium, ASME J. Appl. Mech. 75 (2008) 011015.
[4] M. van Dyke, Pertubation Methods in Fluid Mechanics, Parabolic, Stanford,
AN US
CA, 1975.
[5] N. Bakhvalov, G. Panasenko, Homogenisation: averaging of processes in periodic media: mathematical problems of the mechanics of composite materials,
M
Kluwer Acad. Publ, Dordrecht Boston London, 1989.
ED
[6] A. Bensoussan, J. B. Lions, J. Papanicolaou, Asymptotic analysics for periodic
PT
structures, North Holland, Amsterdam, 1978. [7] W. Kohler, G. C. Papanicolaou and S. Varadhan, Boundary and Interface Prob-
CE
lems in Regions With Very Rough Boundaries. Multiple Scattering and Waves in Random Media, P. Chow, W. Kohler and G. Papanicolaou, eds., North-
AC
Holland, Amsterdam 1981, 165-197.
[8] J. Nevard, J. B. Keller, Homogenization of rough boundaries and interface, SAM J. Appl. Math. 57 (1997) 1660-1686.
35
ACCEPTED MANUSCRIPT
[9] P. C. Vinh, D. X. Tung, Homogenized Equations of the Linear Elasticity in Two-Dimensional Domains With Very Rough Interfaces, Mech. Res. Commun.,
CR IP T
37 (2010) 285-288. [10] P. C. Vinh, D. X. Tung, Homogenization of rough two-dimensional interfaces separating two anisotropic solids, ASME J. Appl. Mech. 78 (2011) 041014-1 (7 pages).
AN US
[11] P. C. Vinh, D. X. Tung, Homogenized equations of the linear elasticity theory in two-dimensional domains with interfaces highly oscillating between two circles, Acta Mech. 218 (2011) 333-348.
M
[12] G. Hassen, P. de Buhan, A two-phase model and related numerical tool for the design of soil structures reiforced by stiff linear inclusions, Eur. J. Mech.
ED
A/Solids 24 (2005) 987-1001.
PT
[13] P. de Buhan, G. Hassen, Maltiphase approach as a generalized homogenization procedure for modelling the macroscopic behaviour of soils reinforced by linear
CE
inclusions, Eur. J. Mech. A/Solids 27 (2008) 662-679.
AC
[14] J.F.C. Yang, R.S. Lakes, Experimental study of micropolar and couple stress elasticity in compact bone in bending, J. Biomech. 15 (1982) 91-98.
[15] J. Fatemi, P.R. Onck, G. Poort, F. van Leulen, Cosserat modeli of anisotropic cancellous bone: a micromechanical analysis, J. Phys. IV Fr. 105 (2003) 273280. 36
ACCEPTED MANUSCRIPT
[16] W.B. Anderson, R.S. Lakes, Size effects due to Cosserat elasticity and surface damage in closed-cell polymethacrylimide foam, J. Mat. Sci. 29 (1994) 6413-
CR IP T
6419. [17] S. Diebels, H. Steeb, The size effect in foams and its theoretical and numerical investigatio, Proc. R. Soc. A 458 (2002) 2869-2883.
[18] R. Masiani, P. Trovalusci, Cosserat and Cauchy materials as continuum models
AN US
of brick masonry Meccanica 31 (1996) 421-432.
[19] D. Addessi, E. Sacco, A. Paolone, Cosserat model for periodic masonry deduced by nonlinear homogenization, Eur. J. Mech. A Solids 29 (2010) 724-737.
M
[20] D. Baraldi, A. Cecchi, A. Tralli, Continuous and discrete models for masonry
ED
like material: a critical comparative study, Eur. J. Mech. A/Solids 50 (2015)
PT
39-58.
[21] N.J. Pagano, G.C. Sih, Stress singularities around a crack in a Cosserat plate.
CE
Int. J. Solids struct. 4 (1968) 531-553.
AC
[22] A.C. Eringen, Linear theory of micropolar elasticity, Math. Mech. J. 15 (1966) 909-924.
[23] A.C. Eringen, Microcontinuum Field Theories, I. Foundations and Solids, Springer, 1999.
37
ACCEPTED MANUSCRIPT
[24] P.C. Vinh, D.X. Tung, Homogenized equations of very rough interfaces separating two piezoelectric solids, Acta Mech. 224 (2013) 1077-1088.
CR IP T
[25] D. Iesan, The plane micropolar strain of orthotropic elastic solids, Archives of Mechanics 25 (1973), 547-561.
[26] E. Sanchez-Palencia, Nonhomogeneous media and vibration theory, Lecture
AN US
Notes in Phys. 127, Springer-Verlag, Heidelberg, 1980.
[27] S. K. Tomar, M. L. Gogna, Reflection and refraction of longitudinal wave at an interface between two micropolar elastic solids in welded contact, J. Acoust. Soc. Am. 97 (1995) 822-830.
M
[28] A. N. Stroh, Dislocations and cracks in anisotropic elasticity, Philos. Mag. 3
ED
(1958) 625-646.
[29] A. N. Stroh, Steady state problems in anisotropic elasticity, J. Math. Phys. 41
PT
(1962) 77-103.
CE
[30] D.M. Barnett, J. Lothe, Free surface (Rayleigh waves) in anisotropic elastic
AC
half-spaces: the surface impedance method, Proc. R. Soc. Lond. A 402 (1985) 135-152.
[31] V. R. Parfitt, A. C. Eringen, Reflection of Plane Waves from the Flat Boundary of a Micropolar Elastic Half-Space, (1969) J. Acoust. Soc. Am. 45 (1969) 12581272. 38
Homogenization of very rough interfaces for the micropolar elasticity theory P.C. Vinh, V.T.N. Anh, D.X. Tung, N.T. Kieu PII: DOI: Reference:
S0307-904X(17)30592-9 10.1016/j.apm.2017.09.039 APM 11984
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
30 March 2017 17 August 2017 19 September 2017
Please cite this article as: P.C. Vinh, V.T.N. Anh, D.X. Tung, N.T. Kieu, Homogenization of very rough interfaces for the micropolar elasticity theory, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.09.039
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlights • The homogenization of a very rough interface of two micropolar solids is investigated. • The homogenized equations in explicit form have been obtained.
CR IP T
• They are very useful in solving various practical problems.
• The reflection and transmission of waves at a very rough interface is considered.
AC
CE
PT
ED
M
AN US
• The formulas for the reflection and transmission coefficients have been derived.
1
ACCEPTED MANUSCRIPT
CR IP T
Homogenization of very rough interfaces for the micropolar elasticity theory D. X. Tung2 , P. C. Vinh 1 ∗, V. T. N. Anh1 , N. T. Kieu2 1 Faculty of Mathematics, Mechanics and Informatics Hanoi University of Science 334, Nguyen Trai Str., Thanh Xuan, Hanoi,Vietnam 2
AN US
Faculty of Civil Engineering Hanoi Architectural University Km 10 Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam
M
Abstract
AC
CE
PT
ED
In this paper, the homogenization of a very rough two-dimensional interface separating two dissimilar isotropic micropolar elastic solids is investigated. The interface is assumed to oscillate between two parallel straight lines. The main aim is to derive homogenized equations in explicit form. These equations are obtained by the homogenization method along with the matrix formalism of the theory of micropolar elasticity. Since obtained homogenized equations are totally explicit, they are a powerful tool for solving various practical problems. As an example, the reflection and transmission of a longitudinal displacement plane wave at a very rough interface of toothcomb type is investigated. The closed-form formulas for the reflection and transmission coefficients have been derived. Based on these formulas, some numerical examples are carried out to show the dependence of the reflection and transmission coefficients on the incident angle and the geometry parameter of the interface.
Key words: Homogenized equations, Very rough interfaces, Theory of micropolar elasticity, Reflection and transmission of waves. ∗
Corresponding author: Tel:+84-4-35532164; Fax:+84-4-38588817; E-mail address: [email protected] (P. C. Vinh)
2
pcv-
ACCEPTED MANUSCRIPT
1
Introduction Boundary-value problems in domains with rough boundaries or interfaces ap-
CR IP T
pear in many fields of natural sciences and technology such as: scattering of waves at rough boundaries [1], reflection and transmission of waves at rough interfaces [2], nearly circular holes and inclusions in the plane elasticity and the thermoelasticity [3], and so on. When the amplitude (height) of the roughness is much small in com-
AN US
parison with its period, the problems are usually analyzed by perturbation methods [4]. When the amplitude is much larger than its period, i.e. the boundaries and interfaces are very rough, the homogenization method is required, see for instance: [5, 6, 7]. Nevard and Keller [8] examined the homogenization of a very rough three-
M
dimensional interface that oscillates between two parallel planes and separates two
ED
linear anisotropic solids. By applying the homogenization method [6], the authors have derived the homogenized equations, but these equations are still implicit. In
PT
some recent papers by Vinh and Tung [9, 10, 11], the explicit homogenized equations
CE
of the linear elasticity in two-dimensional domains with interfaces rapidly oscillating between two parallel straight lines and between two concentric circles have been
AC
obtained.
Modern engineering structures are often made of materials possessing an internal
structure such as reinforced soils [12, 13], bone [14, 15], cellular materials such as foams [16, 17], masonry structural elements [18, 19, 20], granular or cracked media [21], etc, and these materials are modeled by the micropolar elasticity theory [22, 23]. 3
ACCEPTED MANUSCRIPT
The consideration the boundary-value problems of the micropolar elasticity theory in domains with very rough boundaries or interfaces is therefore significant and of great theoretical and practical as well interest.
CR IP T
In this paper, the homogenization of a very rough two-dimensional interface separating two dissimilar isotropic micropolar elastic solids is investigated. It is assumed that the interface oscillates between two parallel straight lines. The main aim of
AN US
the investigation is to derive explicit homogenized equations and explicit associate continuity conditions. First, the basic equations and the continuity conditions of the linear theory of micropolar elasticity are written in matrix form. Then, by using an appropriate asymptotic expansion of the solution and following standard techniques
M
of the homogenization method, the explicit homogenized equation and the explicit associate continuity conditions in matrix form are derived. They are then written
ED
down in component form for the orthotropic case.
PT
Note that, while the continuity condition on the interface of the theory of elasticity [9, 10] and the theory of piezoelasticity [24] contains only the derivatives of the
CE
solution, the one of the micropolar elasticity theory is expressed in terms of both
AC
solution and its derivatives. This leads to requirement that the matrix equation must be written in a relevant form so that the problems on the periodicity cell can be solved easily. 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. As an example
4
ACCEPTED MANUSCRIPT
proving this, the reflection and transmission of a longitudinal displacement plane wave at a very rough interface of tooth-comb type is considered. The closed-form analytical expressions of the reflection and transmission coefficients are obtained.
some parameters is investigated numerically.
2
CR IP T
Based on them the dependence of the reflection and transmission coefficients on
Basic equations in matrix form
AN US
We consider a micropolar elastic body. The equations governing its motion are given by Eringen [22]:
tki,k + ρfi = ρ¨ ui , mki,k + εirs trs + ρli = ρj φ¨i , i = 1, 2, 3
(1)
M
where tki is the stress tensor, mki is the couple stress tensor, ui is the displacement
ED
vector, φi is the microrotation vector, ρ is the mass density, j is the micro-inertia, fi is the body force vector, li is the body couple vector, εirs is the alternating
PT
symbol (also called Levi-Civita symbol), commas denote differentiation with respect to spatial variables xk and a superposed dot signifies differentiation with respect to
CE
the time t.
AC
For linear isotropic micropolar elastic materials, the constitutive equations are: tki = λur,r δki + µ(uk,i + ui,k ) + κ(ui,k − εkir φr )
(2)
mki = αφr,r δki + βφk,i + γφi,k
where δki is the Kronecker symbol, λ, µ, κ, α, β and γ are the material moduli. Suppose that the motion of the micropolar elastic body creates a plane strain [25] for which all quantities are independent of the variable x3 and the displacement 5
ACCEPTED MANUSCRIPT
vector and the microrotation vector are of the form: u1 = u1 (x1 , x2 , t),
φ3 = φ(x1 , x2 , t), u3 ≡ φ1 ≡ φ2 ≡ 0.
u2 = u2 (x1 , x2 , t),
(3)
to:
CR IP T
Then, the equations of motion (1) and the constitutive equations (2) are simplified
t11,1 + t21,2 + ρf1 = ρ¨ u1 , t12,1 + t22,2 + ρf2 = ρ¨ u2 ,
(4)
m13,1 + m23,2 + t12 − t21 + ρl3 = ρj φ¨
AN US
and:
t11 = (λ + 2µ + κ)u1,1 + λu2,2 , t22 = λu1,1 + (λ + 2µ + κ)u2,2 , t12 = µu1,2 + (µ + κ)u2,1 − κφ, t21 = (µ + κ)u1,2 + µu2,1 + κφ , m13 = γφ,1 , m23 = γφ,2
(5)
M
Let the linear isotropic micropolar body occupy two-dimensional domains Ω+
ED
and Ω− of the plane x1 x2 whose interface is the curve L expressed by equation: x2 = h(y), y = x1 /ε, where h(y) is a periodic function of period 1 whose minimum
PT
and maximum values are 0 and H, respectively, ε is assumed to be much more small than H, i. e. the curve L is a very rough interface oscillating between two straight
CE
lines x2 = 0 and x2 = H, see Fig.1. We also assume that, in the domain 0 < x1 < ε,
AC
i.e. 0 < y < 1, any straight line x2 = x02 = const (0 < x02 < H) has exactly two intersections with the curve L. Note that in Eqs. (4) and (5), the material parameters λ, µ, κ, γ, ρ and j take
(different) constant values in Ω+ and Ω− : ( λ+ , µ+ , κ+ , γ + , ρ+ , j + for (x1 , x2 ) ∈ Ω+ λ, µ, κ, γ, ρ, j = λ− , µ− , κ− , γ − , ρ− , j − for (x1 , x2 ) ∈ Ω− 6
(6)
ACCEPTED MANUSCRIPT
where λ+ , µ+ , κ+ , γ + , ρ+ , j + , λ− , µ− , κ− , γ − , ρ− , j − are constants. Suppose Ω+ and Ω− are in welded contact with each other along their interface
CR IP T
L, then the continuity conditions, namely: [φ]L = 0, [uk ]L = 0, [t1k n1 + t2k n2 ]L = 0, [m1k n1 + m2k n2 ]L = 0, k = 1, 2
(7)
must be satisfied, where nk is the unit normal to the curve L and the notation [.]L defined as:
(8)
PT
ED
M
AN US
[f ]L = f + − f −
AC
CE
Figure 1: Two-dimensional domains Ω+ and Ω− have a very rough interface L expressed by equation x2 = h(x1 /ε)=h(y), where h(y) is a periodic function with period 1. The curve L highly oscillates between the parallel straight lines x2 = 0 and x2 = H. It is not difficult to verify that in matrix form Eqs. (4) and (5) can be written
as:
(A11 u,1 + A12 u,2 + Gu),1 + (A21 u,1 + A22 u,2 + Hu),2 + Bu,1 + Du,2 + Eu + F = ρ¨ u (9) 7
ACCEPTED MANUSCRIPT
A11
A21
λ + κ + 2µ 0 0 0 κ + µ 0 , = 0 0 γ
0 µ 0 = λ 0 0 , 0 0 0
A22
0 0 0 0 , E = 0 0 0 0 −2κ ρf2
ρl3 ]T ,
κ+µ 0 0 λ + κ + 2µ 0 , = 0 0 0 γ
0 0 0 D = 0 0 0 , −κ 0 0
(10)
0 0 0 G = 0 0 −κ , 0 0 0 u = [u1
u2
0 0 κ H = 0 0 0 , 0 0 0
φ]T , ρ = diag[ρ ρ ρj]
M
F = [ρf1
0 λ 0 = µ 0 0 , 0 0 0
AN US
0 0 0 B = 0 0 0 , 0 κ 0
A12
CR IP T
where:
The continuity conditions (7) are expressed in matrix form as: # " # A11 u,1 +A21 u,2 +Gu n1 + A12 u,1 +A22 u,2 +Hu n2 = 0 (11)
ED
[u]L = 0,
"
L
L
PT
Remark 1: Appearance of terms Gu and Hu in the brackets of Eq. (9) makes this
CE
equation compatible with the continuity condition (11). This compatibility makes the solving of the cell problems (17)-(19) more simple and the expression of solution
AC
is compact.
Expressing nk (k = 1, 2) in terms of the function h, the continuity condition (11)2
is written as: ε−1
"
" # # 0 h (y) A11 u,1 + A21 u,2 + Gu − A12 u,1 + A22 u,2 + Hu = 0 (12) L
L
8
ACCEPTED MANUSCRIPT
We introduce the vectors Σ1 = [t11
m13 ]T and Σ2 = [t21
t12
t22
m23 ]T . One
can see that these vectors are given by: (13)
CR IP T
Σ1 = A11 u,1 + A12 u,2 + Gu, Σ2 = A21 u,1 + A22 u,2 + Hu In term of Σk , Eq. (9) is written as:
Σ1,1 + Σ2,2 + Bu,1 + Du,2 + Eu + F = ρ¨ u
(14)
AN US
Note that, according to (6), the material parameters in the above equations are attached respectively to the symbol ”+ ” and ”− ” corresponding to the half-space Ω+ and the half-space Ω− .
The explicit homogenized equation in matrix form
M
3
ED
Following Bensoussan et al. [6], Sanchez-Palencia [26], Bakhvalov and Panasenko
PT
[5] we suppose that: u(x1 , x2 , ε, t)=U(x1 , y, x2 , ε, t), and we express U as follows (see, Vinh and Tung [9, 10]):
CE
U = V + ε N1 V + N11 V,1 + N12 V,2 + ε2 N2 V + N21 V,1 + N22 V,2
AC
+ N211 V,11 + N212 V,12 + N222 V,22 + O(ε3 )
(15)
where V=V(x1 , x2 , t) (being independent of y), N1 , N11 , N12 , N2 , N21 , N22 , N211 , N212 , N222 are 4×4-matrix valued functions of y and x2 (not depending on x1 and t), and they are y-periodic with period 1. Remark 2: 9
ACCEPTED MANUSCRIPT
From (15) one can see that u approaches V when ε tends to zero, V is therefore the leading term of u. When ε is very small, we find V instead of u. Thus, we need
Since y = x1 /ε, therefore we have: u,1 = U,1 + ε−1 U,y
CR IP T
to derive the equation for V. This equation is called the homogenized equation.
(16)
Substituting (15) into (9), (12) and (11)1 , and taking into account (16) yield equa-
AN US
tions that we call equations (e1 ), (e2 ) and (e3 ), respectively. In order to find the explicit homogenized equation and the explicit continuity conditions, we carry out the following steps.
M
• The first step (deriving local problems):
In order to make the coefficients of ε−1 of equations (e1 ) and (e2 ), and the
ED
coefficient of ε of equation (e3 ) zero, the functions N1 , N11 , N12 are chosen so that:
PT
h i A11 N1,y + G = 0, 0 < y < 1, y 6= y1 , y2 ; ,y
AC
CE
[A11 N1,y + G]L = 0, [N1 ]L = 0 at y1 , y2 ; N1 (0) = N1 (1)
[A11
h i 11 A11 I + N,y = 0, 0 < y < 1, y 6= y1 , y2 ;
(17)
,y
11 11 11 I + N11 ,y ]L = 0, [N ]L = 0 at y1 , y2 ; N (0) = N (1) A11 N12 ,y + A12
,y
(18)
= 0, 0 < y < 1, y 6= y1 , y2 ;
12 12 12 [ A11 N12 ,y + A12 ]L = 0, [N ]L = 0, at y1 , y2 ; N (0) = N (1) 10
(19)
ACCEPTED MANUSCRIPT
where I is the identity 4×4-matrix, y1 , y2 (0 < y1 < y2 < 1) are two roots in the interval (0 , 1) of the equation h(y) = x2 for y, in which x2 belongs to the interval (0 H). The problems (17)-(19) are called the local problems (or the problems on the
proposition. Proposition 1: Let qk be defined as follows:
CR IP T
periodicity cell). Using these local problems it is not difficult to prove the following
AN US
q1 = hBN1,y i, q2 = hA11 N1,y + Gi, q3 = hA21 N1,y i
11 11 q4 = hB I + N11 ,y i, q5 = hA11 I + N,y i, q6 = hA21 I + N,y i
(20)
12 12 q7 = hBN12 ,y i, q8 = hA11 N,y + A12 i, q9 = hA21 N,y i
Z
M
where:
f dy
(21)
0
ED
hf i =
1
then they are given by:
PT
−1 −1 −1 −1 −1 −1 −1 q1 = hBA−1 11 ihA11 i hA11 Gi − hBA11 Gi, q2 = hA11 i hA11 Gi
CE
−1 −1 −1 −1 q3 = hA21 A−1 11 ihA11 i hA11 Gi − hA21 A11 Gi
−1 −1 −1 −1 −1 −1 −1 q4 = hBA−1 11 ihA11 i , q5 = hA11 i , q6 = hA21 A11 ihA11 i
(22)
AC
−1 −1 −1 −1 −1 −1 −1 q7 = hBA−1 11 ihA11 i hA11 A12 i − hBA11 A12 i, q8 = hA11 i hA11 A12 i −1 −1 −1 −1 q9 = hA21 A−1 11 ihA11 i hA11 A12 i − hA21 A11 A12 i
Note that in Eq. (22): hf i = (y2 − y1 )f − + (1 − y2 + y1 )f + 11
(23)
ACCEPTED MANUSCRIPT
To make the coefficient of ε0 of Eq.(e2 ) zero we take: h i h i A11 N2,y + A12 N1,2 + GN1 = A21 N1,y + H /h0 at y1 , y2 L
L
L
CR IP T
i h i 11 11 11 [A11 N1 + N21 + A N + GN = A I + N /h0 at y1 , y2 12 ,2 21 ,y ,y L
i i h h 12 12 1 12 22 = A22 + A21 N,y /h0 at y1 , y2 A11 N,y + A12 N + N,2 + GN L
L
h
A11 N11 + N211 ,y
i
L
= 0 at y1 , y2
h i 11 A11 N12 + N212 + A N = 0 at y1 , y2 12 ,y
AN US
L
h i 12 A11 N222 + A N = 0 at y1 , y2 12 ,y L
(24)
• The second step (deriving homogenized equation in explicit form):
M
Equating to zero the coefficient ε0 of Eq.(e1 ) provides:
AC
CE
PT
ED
o i A11 N2,y + A12 N1,2 + GN1 + BN1,y + E V + ,y i nh o 11 1 11 11 1 21 + A11 N,y + G + B I + N,y V,1 A11 N + N,y + A12 N,2 + GN ,y nh i o 1 12 12 12 + A11 N22 + A N + N + GN + BN + D V,2 12 ,y ,2 ,y ,y nh i o 11 + A11 N11 + N211 + A I + N V,11 11 ,y ,y ,y nh i o 11 12 + A11 N12 + N212 + A N + A N + A V,12 11 12 12 ,y ,y ,y h i 1 11 12 + A21 N,y + H V + A21 I + N,y V,1 + A21 N,y + A22 V,2 ,2 h i 12 ¨ + A11 N222 V,22 + F = ρV (25) ,y + A12 N nh
,y
By integrating equation (25) along the line x2 = const, 0 < x2 < H from y = 0 to
12
ACCEPTED MANUSCRIPT
y = 1, and taking into account Eqs. (24) we arrive at:
CR IP T
n o n o hBN1,y i + hEi V + hA11 N1,y + Gi + hB I + N11 ,y i V,1 n o n n o o 11 12 + hBN12 i + hDi V + hA I + N i V + hA N + A i V,12 ,2 11 ,11 11 12 ,y ,y ,y 1 1 + − ¨ +hΦ,2 i + Φ − Φ − + hFi = hρiV (26) h0 (y2 ) h0 (y1 ) where:
AN US
12 Φ = A21 N1,y + G V + A21 I + N11 ,y V,1 + A21 N,y + A22 V,2
(27)
It is not difficult to prove the following result. Proposition 2:
1 1 − = hΦi,2 h0 (y2 ) h0 (y1 )
M
hΦ,2 i + Φ+ − Φ−
(28)
ED
According to Proposition 2, Eq. (26) is simplified to:
CE
PT
o n n o hBN1,y i + hEi V + hA11 N1,y + Gi + hB I + N11 ,y i V,1 o n o n n o 12 11 i V + hA N + A i V,12 i + hDi V + hA I + N + hBN12 ,2 11 ,11 11 ,y 12 ,y ,y h + hA21 N1,y i + hHi V + hA21 I + N11 ,y iV,1 i 12 ¨ + hA21 N,y i + hA22 i V,2 + hFi = hρiV (29)
AC
,2
Substituting the results (22) into Eq.(29) we obtain the homogenized equation in explicit form, namely:
13
ACCEPTED MANUSCRIPT
CR IP T
h i −1 −1 −1 −1 −1 −1 −1 hA−1 i V + hA i hA A iV + hA A ihA i V ,11 12 ,12 21 ,1 11 11 11 11 11 ,2 hn o i −1 −1 −1 −1 + hA22 i + hA21 A−1 11 ihA11 i hA11 A12 i − hA21 A11 A12 i V,2 ,2 h i −1 −1 −1 −1 + hBA−1 + hA−1 11 ihA11 i 11 i hA11 Gi V,1 h i −1 −1 −1 −1 + hDi + hBA−1 ihA i hA A i − hBA A i V,2 12 12 11 11 11 11 h i −1 −1 −1 −1 −1 + hA21 A11 ihA11 i hA11 Gi − hA21 A11 Gi + hHi V ,2 i h −1 −1 −1 −1 ¨ + hEi + hBA−1 11 ihA11 i hA11 Gi − hBA11 Gi V + hFi = hρiV
(30)
(9) in which u is replaced by V. • Continuity condition
AN US
Note that for the domains x2 < 0 and x2 > H the homogenized equations are Eq.
M
By Σ02 we denote the leading term of the Σ2 defined by (13)2 . It is required that
ED
V and Σ02 are continuous on the lines x2 = 0 and x2 = H: [V]L∗ = 0,
[Σ02 ]L∗ = 0,
L∗ : x2 = 0, x2 = H
(31)
PT
Introducing the asymptotic expansion (15) into (13)2 one can see that:
CE
12 Σ02 = A21 N1,y + H V + A21 N11 ,y + I V,1 + A21 N,y + A22 V,2
(32)
AC
Integrating equation (32) along the line x2 = const, 0 < x2 < H from y = 0 to y = 1 and using the expressions of qk (k = 3, 6, 9) given in (22) we arrive at: −1 −1 −1 −1 −1 −1 −1 Σ02 = h A21 A−1 11 ihA11 i hA11 Gi − hA21 A11 Gi + hHi V + hA21 A11 ihA11 i V,1 −1 −1 −1 −1 + hA22 i + hA21 A−1 ihA i hA A i − hA A A i V,2 12 21 12 11 11 11 11 14
(33)
ACCEPTED MANUSCRIPT
Analogously, the leading term Σ01 of the vector Σ1 is given by: −1 −1 −1 −1 −1 −1 −1 Σ01 = hA−1 11 i hA11 GiV + hA11 i V,1 + hA11 i hA11 A12 iV,2
CR IP T
Remark 3:
(34)
i) In terms of Σ0k , Eq. (30) becomes:
(35)
CE
PT
ED
M
AN US
−1 −1 Σ01,1 + Σ02,2 + hBA−1 11 ihA11 i V,1 h i −1 −1 −1 −1 + hDi + hBA−1 ihA i hA A i − hBA A i 12 12 V,2 11 11 11 11 h i −1 −1 −1 −1 + hEi + hBA−1 ihA i hA Gi − hBA Gi V + hFi = hρiV¨ 11 11 11 11
Figure 2: The interface of tooth-comb type.
AC
ii) When the interface L is of tooth-comb type (see Fig.2), all h•i’s do not depend
15
ACCEPTED MANUSCRIPT
where:
1 (a1 φ+ + a2 φ− ) a1 + a2
(36)
(37)
M
hφi =
AN US
CR IP T
on x2 , the homogenized equation (30) is therefore simplified to: −1 −1 −1 −1 −1 −1 −1 −1 V,12 hA11 i V,11 + hA11 i hA11 A12 i + hA21 A11 ihA11 i −1 −1 −1 −1 + hA22 i + hA21 A−1 11 ihA11 i hA11 A12 i − hA21 A11 A12 i V,22 −1 −1 −1 −1 −1 −1 + hBA11 ihA11 i + hA11 i hA11 Gi V,1 −1 −1 −1 −1 −1 −1 −1 −1 + hDi + hBA−1 11 ihA11 i hA11 A12 i − hBA11 A12 i + hA21 A11 ihA11 i hA11 Gi −1 − hA21 A11 Gi + hHi V,2 −1 −1 −1 −1 −1 + hEi + hBA11 ihA11 i hA11 Gi − hBA11 Gi V + hFi = hρiV¨
The explicit homogenized equations in component form
PT
4
ED
Note that all coefficients of Eq (36) are constant.
CE
• The homogenized equations:
AC
Using (10) one can write the homogenized equation (30) in component form.
16
ACCEPTED MANUSCRIPT
AN US
CR IP T
They are (0 < x2 < H): −1 −1 −1 −1 −1 2 2 −1 ha i V1,11 + hbi + hb i hµb i − hµ b i V1,2 + ha−1 i−1 hλa−1 iV2,12 ,2 −1 −1 −1 −1 −1 −1 −1 −1 + hb i hµb iV2,1 + hκi + hκµb i − hb i hκb ihµb i Φ ,2 ,2 ¨ +hρf1 i = hρiV1 −1 −1 −1 −1 −1 −1 hb i hµb iV1,12 + ha i hλa iV1,1 + hb−1 i−1 V2,11 ,2 + hai + ha−1 i−1 hλa−1 i2 − hλ2 a−1 i V2,2 − hb−1 i−1 hκb−1 iΦ,1 + hρf2 i = hρiV¨2 ,2 hb−1 i−1 hκb−1 ihµb−1 i − hκµb−1 i − hκi V1,2 + hb−1 i−1 hκb−1 iV2,1 + hγ −1 i−1 Φ,11 ¨ + hγiΦ,2 + hκ2 b−1 i − hb−1 i−1 hκb−1 i2 − 2hκi Φ + hρl3 i = hρjiΦ ,2 where:
b=µ+κ
(39)
M
a = λ + 2µ + κ;
(38)
ED
For the domains x2 < 0 and x2 > H the homogenized equations in component form
CE
PT
are: ¨ (λ + 2µ + κ)V1,11 + (µ + κ)V1,22 + (λ + µ)V2,12 + κΦ,2 + ρf1 = ρV1 (λ + µ)V1,12 + (µ + κ)V2,11 + (λ + 2µ + κ)V2,22 − κΦ,1 + (ρf2 ) = ρV¨2 ¨ −κV1,2 + κV2,1 + γΦ,11 + γΦ,22 − 2κΦ + (ρl3 ) = ρj Φ
(40)
AC
• The continuity conditions:
[Vk ]L∗ = 0, [t02k ]L∗ = 0, k = 1, 2, [Φ]L∗ = 0, [m023 ]L∗ = 0
(41)
where t021 , t022 and m023 are the components of the vector Σ02 and L∗ are the straight
lines: x2 = 0 and x2 = H. By using (33) and (34) it is not difficult to see that Eqs. (38) can be also written
17
ACCEPTED MANUSCRIPT
as follows: t011,1 + t021,2 + hρf1 i = hρiV¨1 t012,1 + t022,2 + hρf2 i = hρiV¨2
(42)
where:
t011 = ha−1 i−1 V1,1 + ha−1 i−1 hλa−1 iV2,2
CR IP T
¨ m013,1 + m023,2 + t012 − t021 + hρl3 i = hρjiΦ
m023 = hγiΦ,2 ;
AN US
t012 = hb−1 i−1 hµb−1 iV1,2 + hb−1 i−1 V2,1 − hb−1 i−1 hκb−1 iΦ 0 −1 −1 −1 2 2 −1 t21 = hbi + hb i hµb i − hµ b i V1,2 + hb−1 i−1 hµb−1 iV2,1 −1 −1 −1 −1 −1 + hκi + hκµb i − hb i hκb ihµb i Φ 0 −1 −1 −1 −1 −1 −1 2 2 −1 t22 = ha i hλa iV1,1 + hai + ha i hλa i − hλ a i V2,2
(43)
m013 = hγ −1 i−1 Φ,1
M
Equations (42) and (43) are a convenient tool for studying the reflection and refrac-
ED
tion of waves at a very rough interface separating two dissimilar micropolar elastic half-spaces.
PT
Remark 4: According to (23), the coefficients of the homogenized equations (38)
CE
and (43) are totally explicit functions in terms of x2 and the material parameters, provided that hϕi is explicitly expressed in terms of x2 and constants ϕ+ (value of
AC
ϕ in Ω+ ), ϕ− (value of ϕ in Ω− ) for any ϕ. There are many interfaces for which this is satisfied such as the tooth-comb interface (Fig.2), the tooth-saw interface (Fig.3a) and the interface of sine type (Fig. 3b). For the tooth-comb interface hϕi is calculated by (37). It is given by: hϕi = (1 −
x2 + x2 − )ϕ + ϕ H H 18
(44)
ACCEPTED MANUSCRIPT
for the tooth-saw interface and: 1 2x2 + 1 2x2 − hϕi = 1 − arccos(1 − ) ϕ + arccos(1 − )ϕ π H π H
M
AN US
CR IP T
for the interface of sine type.
(45)
Reflection and transmission of a longitudinal displacement plane wave at a very rough interface of tooth-comb type
CE
PT
5
ED
Figure 3: The interfaces of tooth-saw type (a) and sin type (b) .
AC
As an example of application of the obtained explicit homogenized equations, in this section we consider the reflection and transmission of a longitudinal displacement plane wave at a very rough interface of tooth-comb type separating two isotropic micropolar elastic half-spaces (Fig. 2). By the meaning of homogenization, this problem is reduced to the reflection and transmission of a longitudinal
19
ACCEPTED MANUSCRIPT
displacement plane wave through a homogeneous material layer occupying the domain 0 < x2 < H (see Fig. 4). According to (42), in the absence of body forces, the
CR IP T
motion of the medium is governed by the equations: t011,1 + t021,2 = hρiV¨1 , t012,1 + t022,2 = hρiV¨2
(46)
ED
M
AN US
¨ m013,1 + m023,2 + t012 − t021 = hρjiΦ
CE
PT
Figure 4: The reflected and refracted waves created by an incident longitudinal displacement wave . For the layer, the relations between t0kn , m0kn and Vk , Φ are given by (43). For
AC
the half-spaces they are: t011 = aV1,1 + λV2,2 , t012 = µV1,2 + bV2,1 − κΦ, t021 = bV1,2 + µV2,1 + κΦ
t022
= λV1,1 + aV2,2 ,
m023
= γΦ,2 ,
m013
(47)
= γΦ,1
Note that, according to (6), the material parameters in the above constitutive equations are attached respectively to the symbol ”+ ” and ”− ” corresponding to the 20
ACCEPTED MANUSCRIPT
half-space Ω+ and the half-space Ω− . At the interfaces x2 = 0 and x2 = H the continuity condition of V1 , V2 , Φ, t021 , t022 and m023 is required to be satisfied.
Incident wave
CR IP T
5.1
Let a longitudinal displacement plane wave with the unit amplitude, the incident angle θ0 (0 < θ0 < π/2), the wave number k0 , the phase velocity v0 , propagate in the half-space Ω+ (Fig. 4). Then, its mechanical displacements V1I , V2I and the
AN US
micro-rotation ΦI are given by (see, Tomar and Gogna [27]):
V1I = sinθ0 ei(k0 x1 sinθ0 +k0 x2 cosθ0 −v0 t) , V2I = cosθ0 ei(k0 x1 sinθ0 +k0 x2 cosθ0 −v0 t) , ΦI ≡ 0 p (λ+ + 2µ+ + κ+ )/ρ+ . Note that k0 v0 = ω where ω is the circular
M
where v0 =
(48)
ED
wave frequency. The expressions (48) can be rewritten as:
PT
VmI = UmI (z)eik(x1 −ct) , m = 1, 2, ΦI = kΞI (z)eik(x1 −ct) , z = kx2
(49)
where k = k0 sinθ0 , c = v0 /sinθ0 and:
CE
U1I (z) = sinθ0 eiz cotgθ0 , U2I (z) = cosθ0 eiz cotgθ0 , ΞI (z) ≡ 0
(50)
AC
From (47) and (48) we have: t021I = ikT1I (z)eik(x1 −ct) , t022I = ikT2I (z)eik(x1 −ct) , m023I = iMI (z)eik(x1 −ct)
21
(51)
ACCEPTED MANUSCRIPT
where: T1I (z) = (2µ+ + κ+ )cosθ0 eiz cotgθ0 ,
MI (z) ≡ 0
(52)
CR IP T
T2I (z) = [λ+ sinθ0 + (λ+ + 2µ+ + κ+ )cos2 θ0 /sinθ0 ]eizcotgθ0 ,
Remark 5 (see also, Tomar and Gogna [27]): It is not difficult to prove that for an incident longitudinal displacement wave there do not exist reflected and refracted
5.2
AN US
longitudinal microrotational waves.
Stroh formalism of half-spaces and layer
As the reflected, refracted wave (propagating in the half-space Ω+ , Ω− , respectively)
M
and the reflected-refracted wave (propagating in the layer) are created and forced by
ED
the incident wave, they are therefore sought in the form (similar to (49) and (51)): V1 = U1 (z)eik(x1 −ct) , V2 = U2 (z)eik(x1 −ct) , Φ = kΞ(z)eik(x1 −ct) ,
PT
t021 = ikT1 (z)eik(x1 −ct) , t022 = ikT2 (z)eik(x1 −ct) , m023 = iM (z)eik(x1 −ct)
(53)
CE
where Um (z), Tm (z) (m = 1, 2), Ξ(z) and M (z) are unknown functions to be determined.
AC
Remark 6: Snell’s law has been taken into account of the expressions of solution (53).
From (46)-(47) and (53) one can see that the unknown functions Um (z), Tm (z) (m =
1, 2), Ξ(z) and M (z) are the solution of the differential equation: 0
ξ (z) = iNξ(z) 22
(54)
ACCEPTED MANUSCRIPT
where ξ = [U1 U2 Ξ T1 T2 M ]T , the prime indicates the differentiation with respect to z and:
N1 N2 N= ¯T N3 N 1
(55)
CR IP T
in which the bar indicates the complex conjugate. The matrices Nk are given by: 1 µ iκ 0 0 b 0 −b b 1 λ 0 N1 = − , N 2 = 0 , 0 0 a a 1 0 0 0 0 0 γk 2 λ2 2 0 0 ρc − a + a 2 µ µ (56) N3 = 0 ρc2 − b + −iκ(1 + ) b b µ κ2 0 iκ(1 + ) ρj c2 k 2 − γk 2 − 2κ + b b + − In Eq. (56) the sign ” ” (” ”) is attached to λ, µ, κ, γ, ρ and j of the half-space Ω+ (Ω− ). For the layer:
(1)
(3) (2) (1) N11 0 0 N11 0 0 N13 (3) (3) (2) N22 N23 N22 0 , N3 = 0 0 , N2 = 0 (3) (3) (2) ¯23 0 0 N N33 0 0 N33 (57)
CE
PT
0 N12 (1) N1 = N21 0 0 0
ED
M
AN US
AC
The non-zero entries of Nk (k = 1, 2, 3) for the layer are given in Appendix A. Equation (54) is called the Stroh formalism [28, 29]. It is the base tool for investigating the reflection and refraction problem. Remark 7: If we consider the propagation of a Rayleigh wave in the half-spaces Ω+ or Ω− , then Eq. (54) is its Stroh formalism in which c is the Rayleigh wave velocity 23
ACCEPTED MANUSCRIPT
that to be determined.
5.3
Characteristic equation
CR IP T
In order to find the solution of Eq. (54) we have to solve its characteristic equation: |N − pI| = 0
(58)
to calculate characteristic values p of matrix N, here I is the identity matrix of six
AN US
order.
Remark 8: In view of Remark 4 and according to Barnett and Lothe [30], there (r)
(r)
(r)
(r)
(t)
(t)
(t)
(t)
exist positive constants αk (0 < α0 < α1 < α2 ) and αk (0 < α0 < α1 < α2 ) so that:
M
For the half-space Ω+ : (r)
ED
(i) For c ≥ α2 : six roots pk of Eq. (58) are all real: pk < 0, p3+k = −pk (k = 1, 2, 3) and we order them as follows: 0 < |p1 | < |p2 | < |p3 |. (r)
(r)
≤ c < α2 : Eq. (58) has four real roots and a pair of complex
PT
(ii) For α1
conjugate roots: pk < 0, p3+k = −pk (k = 1, 2), p6 = p¯3 : 0 < |p1 | < |p2 |, Im(p3 ) < 0.
CE
(r)
(r)
(iii) For α0 ≤ c < α1 : Eq. (58) has two real roots and two pair of complex
AC
conjugate roots: p1 < 0, p4 = −p1 , p3+k = p¯k (k = 2, 3): Im(p2 ) < 0, Im(p3 ) < 0. (r)
(iv) For 0 < c < α0 : six roots of Eq. (58) are all complex: p3+k = p¯k ,
Im(pk ) < 0, (k = 1, 2, 3). For the half-space Ω− : (t)
(i) For c ≥ α2 : six roots qk of Eq. (58) are all real: qk > 0, q3+k = −qk (k = 24
ACCEPTED MANUSCRIPT
1, 2, 3) and we order them as follows: 0 < q1 < q2 < q3 . (t)
(t)
(ii) For α1 ≤ c < α2 : Eq. (58) are real has four real roots and a pair of complex conjugate roots: qk > 0, q3+k = −qk (k = 1, 2), q6 = q¯3 : 0 < q1 < q2 , Im(q3 ) > 0. (t)
(t)
CR IP T
(iii) For α0 ≤ c < α1 : Eq. (58) are real has two real roots and two pair of complex conjugate roots: q1 > 0, q4 = −q1 , q3+k = q¯k (k = 2, 3), Im(q2 ) > 0, Im(q3 ) > 0. (t)
AN US
(iv) For 0 < c < α0 : all roots of Eq. (58) are complex: q3+k = q¯k , Im(qk ) > 0, (k = 1, 2, 3).
5.4
Reflected waves
One can see that for the half-space Ω+ , the general solution of Eq. (54) is (for all
M
possibilities (i)-(iv) in Remark 7):
ED
ξ r (z) = w1r ξ 1r eip1 z + w2r ξ 2r eip2 z + w3r ξ 3r eip3 z
(59)
PT
where wkr are constants to be determined, ξ kr is the eigenvector corresponding to
CE
pk , i. e. it is a non-zero solution of equation: [N − pk I]ξ kr = 0
(60)
AC
We choose m-th component of vector ξ kr (m=1,...,6) as the cofactor of the m-th element of the first row of matrix [N − pk I]. From Remark 7 we see that: (r)
(2)
(i) For c ≥ α2 (↔ 0 < θ0 ≤ θr ): there exist three reflected waves denoted by
RWk (k = 1, 2, 3) and they are given by: RWk = wkr ξ kr eipk z eik(x1 −ct) , k = 1, 2, 3 25
(61)
ACCEPTED MANUSCRIPT
The RW1 is the reflected longitudinal displacement wave, the RW2 and RW3 are two reflected coupled transverse displacement waves (see, Parfitt and Eringen [31]; Tomar and Gogna [27]). (r)
(2)
≤ c < α2 (↔ θr
(1)
< θ0 ≤ θr ): there exist two reflected waves,
namely RW2 and RW3 , RW1 becomes a surface wave. (r)
(r)
(1)
(iii) For α0 ≤ c < α1 (↔ θr
(0)
CR IP T
(r)
(ii) For α1
< θ0 ≤ θr ): there exists one reflected waves
RW3 , the waves RW1 and RW2 convert into surface waves. (0)
AN US
(r)
(iv) For 0 ≤ c < α0 (↔ θr < θ0 ≤ π/2): no reflected wave exists, RW1 , RW2 and RW3 are all surface waves.
5.5
Refracted waves
M
For the half-space Ω− , the general solution of Eq. (54) is (for all possibilities (i)-(iv)
ED
in Remark 7):
ξ t (z) = w1t ξ 1t eiq1 z + w2t ξ 2t eiq2 z + w3t ξ 3t eiq3 z
(62)
CE
PT
where wkt are constants, ξ kt is a non-zero solution of equation: [N − qk I]ξ kt = 0
(63)
AC
and the components of ξ kt are taken as the cofactors of the first row of matrix [N − qk I]. We have conclusions similar to those corresponding to the reflected (t)
waves. For examples, for c ≥ α2 : there exist three reflected waves denoted by
TWk (k = 1, 2, 3) and they are given by: TWk = wkt ξ kt eiqk z eik(x1 −ct) , k = 1, 2, 3 26
(64)
ACCEPTED MANUSCRIPT
The TW1 is the refracted displacement longitudinal wave, the RW2 and RW3 are two refracted coupled transverse displacement waves (see, Parfitt and Eringen [31];
5.6
Transfer matrix
For the layer, the general solution of Eq. (54) is:
CR IP T
Tomar and Gogna [27]).
(65)
AN US
ξ L (z) = w1 ξ 1 eis1 z + ... + w6 ξ 6 eis6 z
where w1 ,..., w6 are constants to be determined, s1 ,..., s6 are six roots of the characteristic equation (58) corresponding to the layer, ξ k is the eigenvector corresponding
It follows from Eq. (65):
M
to sk and its components are taken as the cofactors of the first row of matrix [N−sk I].
ED
ξ L (z) = M(z)M−1 (0)ξ L (0), 0 ≤ z ≤ ε = k.H
(66)
PT
where M(z) is a square matrix of six order whose m-th column is eism z ξ m , in par-
CE
ticular:
M(z) = [eis1 z ξ 1 eis2 z ξ 2 eis3 z ξ 3 eis4 z ξ 4 eis5 z ξ 5 eis6 z ξ 6 ], 0 ≤ z ≤ ε
(67)
AC
Matrix Z = M(ε)M−1 (0) is called the transfer matrix of the layer.
5.7
The reflection and transmission coefficients
Putting z = ε in (66) we have: ξ L (ε) = Zξ L (0) 27
(68)
ACCEPTED MANUSCRIPT
From the continuity conditions at z = 0 and z = ε it follows: ξ L (ε) = ξ t (ε), ξ L (0) = ξ r (0) + ξ I (0)
It follows from (59) and (62) that:
AN US
CR IP T
where in view of (50) and (52), ξ I (0) is given by: sinθ0 cosθ0 0 ξ I (0) = + + (2µ + κ )cosθ 0 + [λ sinθ0 + (λ+ + 2µ+ + κ+ )cos2 θ0 /sinθ0 ] 0
(69)
ξ t (ε) = [ξ 1t eiε1 ξ 2t eiε2 ξ 3t eiε3 ]wt , ξ r (0) = [ξ 1r ξ 2r ξ 3r ]wr
(70)
(71)
M
where wt = [w1t w2t w3t ]T , wr = [w1r w2r w3r ]T , εk = qk ε, k = 1, 2, 3.
ED
From (68), (69) and (71) we have:
(72)
P = Z[ξ 1r ξ 2r ξ 3r 0 0 0 ] − [0 0 0 ξ 1t eiε1 ξ 2t eiε2 ξ 3t eiε3 ]
(73)
PT
Pw = −Zξ I (0)
CE
where w = [wr wt ]T and:
AC
The solution of Eq. (72) is: w = −P−1 Zξ I (0)
(74)
It is clear that Eq. (74) is a closed-form analytical formula of vector w. By ∆kr and δkr (∆kt and δkt ) we denote the cofactor of the first and second element, respectively,
28
ACCEPTED MANUSCRIPT
∗ of the first row of matrix N − Ipk (N − Iqk ). Then the reflection coefficient wkr (k = ∗ (k = 1, 2, 3) are given by: 1, 2, 3) and refraction coefficients wkt
q q ∗ 2 2 2 |, k = 1, 2, 3 = |wkr ∆kr + δkr |, wkt = |wkt ∆2kt + δkt
(75)
CR IP T
∗ wkr
The formulas (75) in which wkr and wkt calculated by (72) are desired formulas. They are closed-form formulas. From (75) one can see that the reflection and refraction coefficients depend on 14 dimensionless parameters:
(76)
M
AN US
λ+ + κ+ + γ+ , e , e = = 2 3 µ+ µ+ j + µ+ − − λ κ γ− − − − − 2 e− , e , e , e = j k = = = 0 1 0 µ− 2 µ− 3 j − µ− − − − ρ µ j a1 rρ = + , rµ = + , rj = + , θ0 , f = , e = Hk0 ρ µ j a1 + a2 + + 2 e+ 0 = j k0 , e1 =
As an example, we use the formulas (72) and (75) to examine the dependence of
ED
the reflection coefficients and the refraction coefficients on the incident angle θ0 , the
PT
dimensionless wave number e of the incident wave, the geometry parameter f of the interface L and the dimensionless micropolar internal parameter e− 2 . They are
CE
presented graphically in Fig. 5, 6, 7, 8. The dependence of the reflection angles θk (of RWk , k = 1, 2, 3) and the refraction angles θ3+k (of TWk , k = 1, 2, 3) on
AC
the incident angle θ0 is also considered. It is illustrated in Fig. 9. Note that the reflection and refraction angles are calculated by: θk = atan
1 1 , θ3+k = atan , k = 1, 2, 3 |pk | qk
It is shown from these figures that: 29
ACCEPTED MANUSCRIPT
(i) The RW1 , RW3 and TW1 are prominent, the remains are almost zero (Figs. 5-8). ∗ ∗ (ii) The reflection coefficient w1r and the refraction coefficient w1t are almost
CR IP T
∗ is quickly constant for θ0 ∈ (0, 45o ) but vary fast for θ0 ∈ (45o , 90o ) (Fig. 5): w1r ∗ increasing while w1t is fast decreasing.
(iii) The reflection and refraction coefficients depend slightly on e and e− 2 (Fig.
AN US
6, 8).
(iv) The reflection and refraction coefficients at f = 0 and f = 1 are the same (Fig. 7). It is reasonable because these two cases are indeed the case of two halfspaces with the plane interface x2 = 0.
M
(v) The reflection and refraction angles depend almost linearly on the incident
Conclusions
PT
6
ED
angle (Fig. 9).
In this paper is investigated the homogenization of two-dimensional interfaces that
CE
separates two isotropic micropolar elastic solids and highly oscillates between two parallel straight lines. The explicit homogenized equation in matrix form for the mi-
AC
cropolar elasticity theory in two-dimension domains with a very rough interface has been derived by applying the homogenization method. Then, it is written down in component form. Since the obtained homogenized equations are fully explicit, they are a powerful tool for investigating various practical problems. As an example, the
30
ACCEPTED MANUSCRIPT
1.4 * w1t
1
w* 2t
0.8
w* 3t
0.6
w* 1r
0.4
w* 2r w* 3r
0.2 20
40
θ0
60
80
AN US
0 0
CR IP T
1.2
ED
M
∗ Figure 5: The dependence of the reflection coefficients wkr and the refraction coef+ ∗ ficients wkt on the incident angle θ0 (of degree). Here we take: e+ 0 = 0.1569, e1 = 2, + − − − − e+ 2 = 0.08, e3 = 3.0303, e0 = 0.2092, e1 = 1.4865, e2 = 0.0824, e3 = 1.6892, rρ = 0.6885, rµ = 0.6727, rj = 1.3333, f = 0.3, e = 0.5.
1.4
PT
1.2 1
w* 1t w* 2t w* 3t
0.6
w* 1r
0.4
w* 2r
0.2 0 0
w* 3r
0.2
0.4
e
AC
CE
0.8
0.6
0.8
1
∗ Figure 6: The dependence of the reflection coefficients wkr and the refraction coeffi∗ cients wkt on the dimensionless incident wave number e. Here we take: e+ 0 = 0.1569, − + + + − − e1 = 2, e2 = 0.08, e3 = 3.0303, e0 = 0.2092, e1 = 1.4865, e2 = 0.0824, o e− 3 = 1.6892, rρ = 0.6885, rµ = 0.6727, rj = 1.3333, f = 0.3, θ0 = 46 .
31
ACCEPTED MANUSCRIPT
1.4 1.2
w* 2t w* 3t w* 1r
0.6
w* 2r
0.4
w* 3r
0.2
0.2
0.4
0.6
0.8
f
1
AN US
0 0
CR IP T
w* 1t
1 0.8
ED
M
∗ Figure 7: The dependence of the reflection coefficients wkr and the refraction coeffi∗ cients wkt on the geometry parameter f of the interface. Here we take: e+ 0 = 0.1569, + + + − − − e1 = 2, e2 = 0.08, e3 = 3.0303, e0 = 0.2092, e1 = 1.4865, e2 = 0.0824, o e− 3 = 1.6892, rρ = 0.6885, rµ = 0.6727, rj = 1.3333, e = 1.0, θ0 = 46 .
1.4 1.2
PT
1
* w1t
w* 2t
0.8
w* 3t
0.6
w* 1r
0.4
w* 2r w* 3r
0.2 0.1
0.12
0.14
AC
CE
0 0.08
Figure
8:
0.16
0.18
e− 2
0.2
∗ The dependence of the reflection coefficients wkr and the refraction
+ + + + ∗ coefficients wkt on e− 2 . Here we take: e0 = 0.1569, e1 = 2, e2 = 0.08, e3 = 3.0303, − − e− 0 = 0.2092, e1 = 1.4865, e3 = 1.6892, rρ = 0.6885, rµ = 0.6727,
rj = 1.3333, f = 0.3, e = 0.5, θ0 = 46o .
32
ACCEPTED MANUSCRIPT
90 θ1
80
θ2
70
θ3 θ4
60
θ5
50
30 20 10 0 0
10
20
30
40
50
60
70
80
90
AN US
θ0
CR IP T
θ6
40
Figure 9: The dependence of the reflection and refraction angles on the incident + + + − angle θ0 . Here we take: e+ 0 = 0.1569, e1 = 2, e2 = 0.08, e3 = 3.0303, e0 = 0.2092, − − − e1 = 1.4865, e2 = 0.0824, e3 = 1.6892, rρ = 0.6885, rµ = 0.6727, rj = 1.3333, f = 0.3, e = 0.5.
M
reflection and transmission of longitudinal waves at a very rough interface of toothcomb type is considered. The closed-form analytical expressions of the reflection
ED
and transmission coefficients have been obtained. Based on them the dependence
PT
of the reflection and transmission coefficients on some parameters is investigated numerically. It is shown that for a incident longitudinal displacement wave, the
CE
reflection and refraction longitudinal displacement waves are prominent.
AC
Acknowledgments The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 107.02-2014.17. The work was partly supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM). 33
ACCEPTED MANUSCRIPT
Appendix A. The non-zero entries of matrices Nk for the layer (1)
hµb−1 ihb−1 i−1 hbi + hµb−1 i2 hb−1 i−1 − hµ2 b−1 i i hκi + hκµb−1 i − hκb−1 ihµb−1 ihb−1 i−1 =− hbi + hµb−1 i2 hb−1 i−1 − hµ2 b−1 i hλa−1 iha−1 i−1 =− hai + hλa−1 i2 ha−1 i−1 − hλ2 a−1 i
(1)
N13
(1)
N21
1
(2)
N11 =
hbi +
hµb−1 i2 hb−1 i−1
hai + 1 = 2 k hγi
hλa−1 i2 ha−1 i−1
(2)
N33
1
− hµ2 b−1 i
− hλ2 a−1 i
AN US
(2)
N22 =
CR IP T
N12 = −
2 hλa−1 iha−1 i−1 = ρc − ha i + hai + hλa−1 i2 ha−1 i−1 − hλ2 a−1 i 2 hµb−1 ihb−1 i−1 (3) 2 −1 −1 N22 = ρc − hb i + hbi + hµb−1 i2 hb−1 i−1 − hµ2 b−1 i −1 −1 −1 −1 −1 −1 −1 −1 hκi + hκµb i − hκb ihµb ihb i hµb ihb i (3) N23 = i hbi + hµb−1 i2 hb−1 i−1 − hµ2 b−1 i − hκb−1 ihb−1 i−1 (3) N11
−1 −1
(77)
ED
M
2
(3)
CE
PT
N33 = (ρjc2 − hγ −1 i−1 )k 2 + hκi + hκµb−1 i − hκb−1 ihµb−1 ihb−1 i−1 2 hκi + hκµb−1 i − hκb−1 ihµb−1 ihb−1 i−1 −1 −1 −1 − hκb ihb i + hbi + hµb−1 i2 hb−1 i−1 − hµ2 b−1 i
AC
References
[1] K.A. Zaki, A. R. Neureuther, Scattering From a Perfectly Conducting Surface With a Sinusoidal Height Profile: TE Polarization. IEEE Trans, Antennas Propag. 19 (1971) 208-214.
34
ACCEPTED MANUSCRIPT
[2] S. S. Singh, S. K. Tomar, qP-Wave at a Corrugated Interface Between Two Dissimilar Pre-Stressed Elastic Half-Spaces, J. Sound Vib. 317 (2008) 687-708.
CR IP T
[3] T. C. Ekneligoda, R. W. Zimmermam, Boundary Pertubation Solution for Nearly Circular Holes and Rigid Inclusions in an Infnite Elastic Medium, ASME J. Appl. Mech. 75 (2008) 011015.
[4] M. van Dyke, Pertubation Methods in Fluid Mechanics, Parabolic, Stanford,
AN US
CA, 1975.
[5] N. Bakhvalov, G. Panasenko, Homogenisation: averaging of processes in periodic media: mathematical problems of the mechanics of composite materials,
M
Kluwer Acad. Publ, Dordrecht Boston London, 1989.
ED
[6] A. Bensoussan, J. B. Lions, J. Papanicolaou, Asymptotic analysics for periodic
PT
structures, North Holland, Amsterdam, 1978. [7] W. Kohler, G. C. Papanicolaou and S. Varadhan, Boundary and Interface Prob-
CE
lems in Regions With Very Rough Boundaries. Multiple Scattering and Waves in Random Media, P. Chow, W. Kohler and G. Papanicolaou, eds., North-
AC
Holland, Amsterdam 1981, 165-197.
[8] J. Nevard, J. B. Keller, Homogenization of rough boundaries and interface, SAM J. Appl. Math. 57 (1997) 1660-1686.
35
ACCEPTED MANUSCRIPT
[9] P. C. Vinh, D. X. Tung, Homogenized Equations of the Linear Elasticity in Two-Dimensional Domains With Very Rough Interfaces, Mech. Res. Commun.,
CR IP T
37 (2010) 285-288. [10] P. C. Vinh, D. X. Tung, Homogenization of rough two-dimensional interfaces separating two anisotropic solids, ASME J. Appl. Mech. 78 (2011) 041014-1 (7 pages).
AN US
[11] P. C. Vinh, D. X. Tung, Homogenized equations of the linear elasticity theory in two-dimensional domains with interfaces highly oscillating between two circles, Acta Mech. 218 (2011) 333-348.
M
[12] G. Hassen, P. de Buhan, A two-phase model and related numerical tool for the design of soil structures reiforced by stiff linear inclusions, Eur. J. Mech.
ED
A/Solids 24 (2005) 987-1001.
PT
[13] P. de Buhan, G. Hassen, Maltiphase approach as a generalized homogenization procedure for modelling the macroscopic behaviour of soils reinforced by linear
CE
inclusions, Eur. J. Mech. A/Solids 27 (2008) 662-679.
AC
[14] J.F.C. Yang, R.S. Lakes, Experimental study of micropolar and couple stress elasticity in compact bone in bending, J. Biomech. 15 (1982) 91-98.
[15] J. Fatemi, P.R. Onck, G. Poort, F. van Leulen, Cosserat modeli of anisotropic cancellous bone: a micromechanical analysis, J. Phys. IV Fr. 105 (2003) 273280. 36
ACCEPTED MANUSCRIPT
[16] W.B. Anderson, R.S. Lakes, Size effects due to Cosserat elasticity and surface damage in closed-cell polymethacrylimide foam, J. Mat. Sci. 29 (1994) 6413-
CR IP T
6419. [17] S. Diebels, H. Steeb, The size effect in foams and its theoretical and numerical investigatio, Proc. R. Soc. A 458 (2002) 2869-2883.
[18] R. Masiani, P. Trovalusci, Cosserat and Cauchy materials as continuum models
AN US
of brick masonry Meccanica 31 (1996) 421-432.
[19] D. Addessi, E. Sacco, A. Paolone, Cosserat model for periodic masonry deduced by nonlinear homogenization, Eur. J. Mech. A Solids 29 (2010) 724-737.
M
[20] D. Baraldi, A. Cecchi, A. Tralli, Continuous and discrete models for masonry
ED
like material: a critical comparative study, Eur. J. Mech. A/Solids 50 (2015)
PT
39-58.
[21] N.J. Pagano, G.C. Sih, Stress singularities around a crack in a Cosserat plate.
CE
Int. J. Solids struct. 4 (1968) 531-553.
AC
[22] A.C. Eringen, Linear theory of micropolar elasticity, Math. Mech. J. 15 (1966) 909-924.
[23] A.C. Eringen, Microcontinuum Field Theories, I. Foundations and Solids, Springer, 1999.
37
ACCEPTED MANUSCRIPT
[24] P.C. Vinh, D.X. Tung, Homogenized equations of very rough interfaces separating two piezoelectric solids, Acta Mech. 224 (2013) 1077-1088.
CR IP T
[25] D. Iesan, The plane micropolar strain of orthotropic elastic solids, Archives of Mechanics 25 (1973), 547-561.
[26] E. Sanchez-Palencia, Nonhomogeneous media and vibration theory, Lecture
AN US
Notes in Phys. 127, Springer-Verlag, Heidelberg, 1980.
[27] S. K. Tomar, M. L. Gogna, Reflection and refraction of longitudinal wave at an interface between two micropolar elastic solids in welded contact, J. Acoust. Soc. Am. 97 (1995) 822-830.
M
[28] A. N. Stroh, Dislocations and cracks in anisotropic elasticity, Philos. Mag. 3
ED
(1958) 625-646.
[29] A. N. Stroh, Steady state problems in anisotropic elasticity, J. Math. Phys. 41
PT
(1962) 77-103.
CE
[30] D.M. Barnett, J. Lothe, Free surface (Rayleigh waves) in anisotropic elastic
AC
half-spaces: the surface impedance method, Proc. R. Soc. Lond. A 402 (1985) 135-152.
[31] V. R. Parfitt, A. C. Eringen, Reflection of Plane Waves from the Flat Boundary of a Micropolar Elastic Half-Space, (1969) J. Acoust. Soc. Am. 45 (1969) 12581272. 38