DSpace at VNU: Homogenized equations of the linear elasticity in two-dimensional domains with very rough interfaces

Mechanics Research Communications 37 (2010) 285–288

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Mechanics Research Communications
journal homepage: www.elsevier.com/locate/mechrescom

Homogenized equations of the linear elasticity in two-dimensional domains
with very rough interfaces
Pham Chi Vinh *, Do Xuan Tung
Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam

a r t i c l e

i n f o

Article history:
Received 19 November 2009
Received in revised form 8 February 2010
Available online 20 February 2010
Keywords:
Homogenization
Homogenized equations
Very rough interfaces

a b s t r a c t
The main purpose of the present paper is to find homogenized equations in explicit form of the linear
elasticity theory in a two-dimensional domain with a very rough interface. In order to do that, equations
of motion and continuity conditions on the interface are first written in matrix form. Then, by an appropriate asymptotic expansion of the solution and using standard techniques of the homogenization
method, we have derived explicit homogenized equations and associate continuity conditions. Since
these equations are in explicit form, they are significant in practical applications.

Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction
Boundary-value problems in domains with rough boundaries or
interfaces appear in many fields of natural sciences and technology
such as: scattering of waves on rough boundaries (Zaki and Neureuther, 1971; Waterman, 1975; Belyaev et al., 1992; Abboud
and Ammari, 1996; Bao and Bonnetier, 2001), transmission and
reflection of waves on rough interfaces (Talbot et al., 1990; Singh
and Tomar, 2007, 2008), mechanical problems concerning the
plates with densely spaced stiffeners (Cheng and Olhoff, 1981),
the flows over rough walls (Achdou et al., 1998), the vibrations
of strongly inhomogeneous elastic bodies (Belyaev et al., 1998)
and so on. When the amplitude (height) of the roughness is much
small comparison with its period, the problems are usually analyzed by perturbation methods. When the amplitude is much large
than its period, i.e. the boundaries and interfaces are very rough,
the homogenization method is required (see for instance, Kohler
et al., 1981; Kohn and Vogelius, 1984; Brizzi, 1994; Nevard and
Keller, 1997; Chechkin et al., 1999; Amirat et al., 2004, 2007,
2008; Blanchard et al., 2007; Madureira and Valentin, 2007;
Mel’nik et al., 2009).
In Nevard and Keller (1997), the authors applied the homogenization method to the equations of the theory of linear anisotropic
elasticity, in a three-dimensional domain with a very rough interface. The authors have derived homogenized equations. However,
these equations are still in the implicit form, in particular, their
coefficients are determined by functions which are the solution
of a boundary-value problem on the periodic cell (called ‘‘cell
* Corresponding author. Tel.: +84 4 35532164; fax: +84 4 38588817.
E-mail address: (P.C. Vinh).
0093-6413/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.mechrescom.2010.02.006

problem”), that includes 27 partial differential equations. This
problem can in general only be solved numerically. However, when
the interface is in two-dimensions, the cell problem consists of
eight ordinary differential equations, rather than partial differential equations, so that, hopefully, it can be solved analytically,
and as a consequence, the homogenized equations in the explicit
form will then be obtained.
Actually, in the present paper we have derived the explicit
homogenized equations of the theory of linear elasticity in a
two-dimensional domain with a very rough interface, for the case
of isotropic material. We first write equations of motion and continuity conditions on the interface in matrix form. Then, by an
appropriate asymptotic expansion of the solution, and using the
homogenization method (see for example, Bensoussan et al.,
1978; Sanchez-Palencia, 1980; Bakhvalov and Panasenko, 1989),
the explicit homogenized equation in matrix form and associate
continuity conditions have been derived. From these we obtain explicit homogenized equations and associate continuity conditions
in components.
Since there is a large class of practical problems leading to
boundary-value problems in two-dimensional domains with very
rough interfaces, deriving their explicit homogenized equations is
significant, and is of theoretical and practical interest as well.
It is noted that, the mentioned above eight ordinary differential
equations which come from (9.10) in Nevard and Keller (1997) can
be solved analytically, and the explicit homogenized equations will
be then derived calculating their coefficients. However, these two
procedures are not as simple as those based on the matrix formulation (to be used in this paper), for the isotropic case and the
anisotropic case as well. Further, if starting from the component
formulation (corresponding to Nevard and Keller’s approach),


286

P.C. Vinh, D.X. Tung / Mechanics Research Communications 37 (2010) 285–288

these procedures will become complicated for the interfaces oscillating between two parallel curves such as two concentric circles
or ellipses, much complicated for the systems including more than
two unknowns. In the mean time, the matrix approach will keep
almost the same simple for these cases. With the mentioned reasons, and in order to introduce the matrix approach to the problems more complicated than the one considered by Nevard and
Keller (1997) we do not start from the cell problem (9.10)–(9.11)
in their paper.
2. Equations of motion and continuity conditions in matrix
form
Consider a linear elastic body that occupies two-dimensional
domains Xþ ; XÀ of the plane x1 x3 . The interface of Xþ and XÀ is denoted by L, and it is expressed by the equation x3 ¼ hðyÞ; y ¼ x1 =e,
where hðyÞ is a periodic function of period 1, as described in Figs. 1,
and 2. The minimum value of h is ÀA ðA > 0Þ, and its maximum value is zero. We also assume that, in the domain 0 < x1 < e, i.e.
0 < y < 1, any straight line x3 ¼ x03 ¼ const ðÀA < x03 < 0Þ has exactly two intersections with the curve L (see Fig. 2). Suppose that
0 < e ( 1, then the curve L is called very rough interface (or highly
oscillating interface) of Xþ and XÀ . Let two parts Xþ ; XÀ of the body
be perfectly welded to each other. Suppose that the body is made
of isotropic material, and it is characterized by Lame’s constants:
k; l and the mass density q defined as follows:

k; l; q ¼



kþ ; lþ ; qþ for x3 > hðx1 =eÞ
kÀ ; lÀ ; qÀ for x3 < hðx1 =eÞ

ð1Þ

where kþ ; lþ ; qþ ; kÀ ; lÀ ; qÀ are constant. We consider the plane
strain for which the displacement components u1 ; u2 ; u3 are of the
form:

ui ¼ ui ðx1 ; x3 ; tÞ;

i ¼ 1; 3; u2  0

ð3Þ

where commas indicate differentiation with respect to the spatial
variables xi . Equations of motion are of the form (Love, 1944):

r11;1 þ r13;3 þ f1 ¼ qu€1 ;

r13;1 þ r33;3 þ f3 ¼ qu€3

in which f1 ; f 3 are the components of the body force, and a superposed dot signifies differentiation with respect to the time t. Substituting (3) into (4) yields a system of equations for the displacement
components whose matrix form is:

€
ðAhk u;k Þ;h þ F ¼ qu

ð4Þ

ð5Þ

where u ¼ ½u1 u3 ŠT ; F ¼ ½f1 f 3 ŠT , the symbol T indicates the transpose of matrices, the indices h, k take the values 1, 3, and:

A11 ¼



k þ 2l 0

ð2Þ

The components of the stress tensor rij ; i; j ¼ 1; 3 are related to
the displacement gradients by the following equations (Love,
1944):

r11 ¼ ðk þ 2lÞu1;1 þ ku3;3 ; r33 ¼ ku1;1 þ ðk þ 2lÞu3;3 ;
r13 ¼ lðu1;3 þ u3;1 Þ

Fig. 2. The curve L in the space 0yx3 ; y1 ; y2 ð0 < y1 < y2 < 1Þ are two roots in the
interval (0, 1) of equation x3 À hðyÞ ¼ 0 for y; ÀA < x3 ¼ const < 0; y1 ¼
y1 ðx3 Þ; y2 ¼ y2 ðx3 Þ.

l

0



A13 ¼



0 k

l 0



A31 ¼



0

l

k 0



A33 ¼



l

0



0 k þ 2l
ð6Þ

Note that, since k þ 2l > 0; l > 0 (see Ting, 1996), the matrix
A11 is invertible. Since Xþ ; XÀ are perfectly welded to each other
along L, the continuity for the displacement vector and the traction
vector must be satisfied. Thus we have:

½uŠL ¼ 0;

½ðA11 u;1 þ A13 u;3 Þn1 þ ðA31 u;1 þ A33 u;3 Þn3 ŠL ¼ 0

ð7Þ

where nk is xk -component of the unit normal to the curve L, by the
symbol ½uŠL we denote the jump of u through L. Expressing nk in
terms of h, the continuity condition (7) can be written as:
0

½uŠL ¼ 0; eÀ1 ½h ðyÞðA11 u;1 þ A13 u;3 ފL À ½A31 u;1 þ A33 u;3 ŠL ¼ 0

ð8Þ

3. Explicit homogenized equations
Following Bensoussan et al. (1978), Sanchez-Palencia (1980),
Bakhvalov and Panasenko (1989), Kohler et al. (1981) we suppose
that: uðx1 ; x3 ; t; eÞ ¼ Uðx1 ; y; x3 ; t; eÞ, and we express U as follows:

U ¼ V þ eðN1 V þ N11 V;1 þ N13 V;3 Þ þ e2 ðN2 V þ N21 V;1 þ N23 V;3
þ N211 V;11 þ N213 V;13 þ N233 V;33 Þ þ Oðe3 Þ

Fig. 1. Two-dimensional domains Xþ and XÀ have a very rough interface L
expressed by equation x3 ¼ hðx1 =eÞ ¼ hðyÞ, where hðyÞ is a periodic function with

period 1. The curve L oscillates between the straight lines x3 ¼ 0 and x3 ¼ ÀA.

ð9Þ

where V ¼ Vðx1 ; x3 ; tÞ (being independent of y), N1, N11, N13, N2, N21,
N23, N211, N213, N233 are 2 Â 2-matrix valued functions of y and x3
(not depending on x1 , t), and they are y-periodic with period 1, E
is the identity 2 Â 2-matrix. In what follows, by u;y we denote the
derivative of u with respect to the variable y. The matrix valued
functions N1, . . ., N233 are determined so that the Eq. (5) and the
continuity conditions (8) are satisfied. Since y ¼ x1 =e, we have:

u;1 ¼ U;1 þ eÀ1 U;y

ð10Þ


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P.C. Vinh, D.X. Tung / Mechanics Research Communications 37 (2010) 285–288

Substituting (9) into (5) and (8), and taking into account (10)
yield equations which we call Eqs. ðe1 Þ and ðe2 Þ, respectively. In order to make the coefficients of eÀ1 of Eqs. ðe1 Þ and ðe2 Þ zero, the
functions N1, N11, N13 are chosen as follows:

here:

½A11 N1;y Š;y ¼ 0; 0 < y < 1; y – y1 ; y2 ;

Note that if mij are elements of matrix M, then hMi ¼ ðhmij iÞ. It is

clear that in order to make Eq. (22) explicit we have to calculate
the quantities:

½A11 N1;y ŠL ¼ 0; ½N1 ŠL ¼ 0 at y1 ; y2 ; N1 ð0Þ ¼ N1 ð1Þ


½A11 E þ N11
;y Š;y ¼ 0; 0 < y < 1; y – y1 ; y2 ;


11
11
11
½A11 E þ N11
;y ŠL ¼ 0; ½N ŠL ¼ 0 at y1 ; y2 ; N ð0Þ ¼ N ð1Þ


A11 N13
¼ 0; 0 < y < 1; y – y1 ; y2 ;
;y þ A13
;y
h
i
¼ 0; ½N13 ŠL ¼ 0; at y1 ; y2 ;
A11 N13
;y þ A13

ð11Þ

N13 ð0Þ ¼ N13 ð1Þ

ð13Þ

ð12Þ

Z
0

1

udy ¼ ðy2 À y1 Þuþ þ ð1 À y2 þ y1 ÞuÀ

D 
E
q1 ¼ A11 E þ N11
;
;y
D 
E
;
q3 ¼ A31 E þ N11
;y

ð23Þ

D
E
q2 ¼ A11 N13
;y þ A13 ;
D

E
q4 ¼ A31 N13
;y

ð24Þ

From (11)–(13), it is not difficult to verify that:

L

where y1 ; y2 ð0 < y1 < y2 < 1Þ are two roots in the interval (0, 1) of
the equation hðyÞ ¼ x3 for y, in which x3 belongs to the interval
ðÀA 0Þ. The functions y1 ðx3 Þ; y2 ðx3 Þ are two inverse branches of
the function x3 ¼ hðyÞ. It is easy to see from (11) that N1;y ¼ 0.
Equating to zero the coefficient e0 of Eq.ðe1 Þ provides:

i
h 

i
þ A13 N11
A11 N2;y þ A13 N1;3 V þ A11 N1 þ N21
V;1
;y
;3
;y
;y
h

i

1
13
þ A11 N23
V;3
;y þ A13 N þ N;3
;y
h 
i


V;11
þ A11 N11 þ N211
þ A11 E þ N11
;y
;y
;y
h 


i
þ A13 N11 þ A13 þ A11 N13
V;13
þ A11 N13 þ N213
;y
;y

hui ¼

h

D
EÀ1
D
EÀ1 D
E
D
ED
EÀ1
À1
q1 ¼ AÀ1
; q2 ¼ AÀ1
AÀ1
AÀ1
11
11
11 A13 ; q3 ¼ A31 A11
11
D
ED
EÀ1 D
E D
E
À1
AÀ1
q4 ¼ A31 AÀ1
AÀ1
11
11
11 A13 À A31 A11 A13
ð25Þ

On use of (24) and (25) into (22) we have:

D

D
EÀ1 D
E
V ;11 þ AÀ1
AÀ1
11
11 A13 V;13
D
ED
EÀ1 
À1
þ A31 AÀ1
V ;1
A
11
11

AÀ1
11

EÀ1

;3


D

ED
EÀ1 D
E D
E 
À1
À1
À1
A
À
A
V;3
þ hA33 i þ A31 AÀ1
A
A
A
A
13
31
13
11
11
11
11

ð14Þ

;3

€ ¼0
þ hFi À hqiV

ð26Þ

;y

þ½A11 N233
;y

13

Therefore we have the following theorem.

þ A13 N Š;y V;33 þ ðA33 V;3 Þ;3


13
€
þ½A31 E þ N11
;y V;1 þ A31 N;y V ;3 Š;3 þ F À qV ¼ 0
Making the coefficient

h

i

Theorem 1. Let uðx1 ; x3 ; e; tÞ satisfy (5) and (8) with Ahk are defined
by (6), the curve L: x3 ¼ hðyÞ; y ¼ x1 =e, is a very rough interface which
oscillates between two lines x3 ¼ 0 and x3 ¼ ÀA ðA > 0Þ and hðyÞ is a
differentiable y-periodic function with period 1. In addition, suppose
u ¼ Uðx1 ; y; x3 ; e; tÞ and Uðx1 ; y; x3 ; e; tÞ has asymptotic form (9). Then,

Vðx1 ; x3 ; tÞ is a solution of the problem:

e0 of Eq. ðe2 Þ zero gives:

0

A11 N2;y þ A13 N1;3 h V
L
nh 

i
h 
i o
0
1
21
V;1
þ A11 N þ N;y þ A13 N11
h À A31 E þ N11
;3
;y
nh

 L
i oL
0
23
1
13
13

þ A11 N;y þ A13 N þ N;3 ŠL h À ½A33 þ A31 N;y
V;3



 L
0
0
ŠL h V;11 þ ½A11 N13 þ N213
þ A13 N11 ŠL h V;13
þ½A11 N11 þ N211
;y
;y

€ x3 > 0
Ahkþ V;kh þ Fþ ¼ qþ V;
D
D
EÀ1
D
EÀ1 D
E
ED
EÀ1 
À1
À1
À1
À1
V
A

AÀ1
V
þ
A
A
A
þ
A
A
V;1
;11
;13
31 11
11
11
11 13
11


0

13
þ½A11 N233
;y þ A13 N ŠL h V;33 ¼ 0

ð15Þ
In order to make (15) satisfied we take:

½A11 N2;y þ A13 N1;3 ŠL ¼ 0 at y1 ; y2



h 
i
0
11
þ A13 N11
=h at y1 ; y2
½A11 N1 þ N21
;y
;3 ŠL ¼ A31 E þ N;y
h

i
h
iL
0
1
13
A11 N23
¼ A33 þ A31 N13
=h at y1 ; y2
;y þ A13 N þ N;3
;y
L
L
h 
i
A11 N11 þ N211
¼ 0 at y1 ; y2
;y

h 
 L
i
A11 N13 þ N213
þ A13 N11 ¼ 0 at y1 ; y2
;y
L
h
i
233
13
A11 N;y þ A13 N
¼ 0 at y1 ; y2
L

ð16Þ
ð17Þ
ð18Þ
ð19Þ

ð21Þ


E
D
E
A11 E þ N11
V;11 þ A11 N13
;y
;y þ A13 V ;13

hD 
E
D
E i
V;1 þ A31 N13
þ A31 E þ N11
;y
;y V;3
€ ¼0
þ ½hA33 iV;3 Š;3 þ F À hqiV

€ ¼ 0; ÀA < x3 < 0
þ hFi À hqiV
€ x3 6 A
AhkÀ V;kh þ FÀ ¼ qÀ V;
D
ED
EÀ1
À1
À1
A31 A11 A11
V;1 þ

D
ED
EÀ1 D
E D
E 
À1
AÀ1

V;3
AÀ1
hA33 i þ A31 AÀ1
11
11
11 A13 À A31 A11 A13

ð29Þ

ð30Þ
LÃ

¼ 0 and ½VŠLà ¼ 0; L is lines x3 ¼ 0; x3 ¼ ÀA

Note that the continuity condition (30)1 is originated from:

½A31 u;1 þ A33 u;3 ŠLà ¼ 0;
LÃ is either the line x3 ¼ 0; or the line x3 ¼ ÀA

ð31Þ

Substituting (9) into (31) and taking into account (10) yield an
equation denoted by Eq. ðe3 Þ. By equating to zero the coefficient of
e0 of Eq. ðe3 Þ we have:




 i
13

A31 E þ N11
;y V ;1 þ A33 þ A31 N;y V;3 Ã ¼ 0
L

ð22Þ

ð28Þ

Ã

h

;3

;3

;3

ð20Þ

By integrating Eq. (14) along the line x3 ¼ const; ÀA < x3 < 0
from y ¼ 0 to y ¼ 1 (see Fig. 2), and taking into account (16)–(21)
we have:

D

þ

D
ED

EÀ1 D
E D
E 
À1
À1
À1
A
À
A
V;3
hA33 i þ A31 AÀ1
A
A
A
A
13
31
13
11
11
11
11

ð27Þ

ð32Þ

Integrating (32) along the line LÃ from y ¼ 0 to y ¼ 1 and using
the results obtained above we derive equation (30)1.

288

P.C. Vinh, D.X. Tung / Mechanics Research Communications 37 (2010) 285–288

On use of (6), we can write (27)–(30) in the component form as:

8
€ 1;
ðkþ þ 2lþ ÞV 1;11 þ lþ V 1;33 þ ðkþ þ lþ ÞV 3;13 þ f1þ ¼ qþ V
>
>
>
<
x3 > 0
ð33Þ
>
þ
lþ ÞV 1;13 þ lþ V 3;11 þ ðkþ þ 2lþ ÞV 3;33 þ f3þ ¼ qþ V€ 3 ;
ðk
>
þ
>
:
x3 > 0
D E

D E

0D

EÀ1
À1
À1
1
1
1
V
þ
V
þ
V
1;11
1;3
3;1
l
l
B kþ2l
;3
;3
B
D

B
E
D
E
À1
B
1
k

B
þ kþ2
V 3;13 þ hf1 i ¼ hqiV€ 1 ; ÀA < x3 < 0;
kþ2l
l
B
B
D

B D EÀ1
ED
EÀ1
D EÀ1
ð34Þ
B 1
k
1
V 1;13 þ kþ2
V 1;1 þ l1
V 3;11
B l
kþ2
l
l
B
;3
B
D

B

E2
D
E
B
À1
lðkþlÞ
1
k
€ 3;
þ
i hkþ2l þ 4 kþ2l
V 3;3 þ hf3 i ¼ hqiV
B
kþ2
l
@
;3
ÀA < x3 < 0
8
€ 1;
ðkÀ þ 2lÀ ÞV 1;11 þ lÀ V 1;33 þ ðkÀ þ lÀ ÞV 3;13 þ f1À ¼ qÀ V
>
>
>
<
x3 < ÀA
>
þ
lÀ ÞV 1;13 þ lÀ V 3;11 þ ðkÀ þ 2lÀ ÞV 3;33 þ f3À ¼ qÀ V€ 3 ;
ðk

>
À
>
:
x3 < ÀA
V 1 ; V 3 ; r013 ; r033 continuous on x3 ¼ ÀA;

x3 ¼ 0

ð35Þ

ð36Þ

where

r013 ¼ h1=liÀ1 ðV 1;3 þ V 3;1 Þ;
r033 ¼ h1=ðk þ 2lÞiÀ1 hk=ðk þ 2lÞiV 1;1




lðk þ lÞ
V 3;3
þ h1=ðk þ 2lÞiÀ1 hk=ðk þ 2lÞi2 þ 4
k þ 2l

ð37Þ

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