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

In order to do that, equations of motion and continuity conditions on the interface are first written in matrix form.. Then, by an appro-priate asymptotic expansion of the solution and us

Homogenized equations of the linear elasticity in two-dimensional domains

with very rough interfaces

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 appro-priate 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

Neu-reuther, 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

ana-lyzed 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)

InNevard and Keller (1997), the authors applied the

homogeni-zation method to the equations of the theory of linear anisotropic

elasticity, in a three-dimensional domain with a very rough

inter-face 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

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 differen-tial 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 con-tinuity 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 ex-plicit 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) inNevard 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 formu-lation (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), 0093-6413/$ - see front matter Ó 2010 Elsevier Ltd All rights reserved.

* Corresponding author Tel.: +84 4 35532164; fax: +84 4 38588817.

E-mail address: pcvinh@vnu.edu.vn (P.C Vinh).

Contents lists available atScienceDirect

Mechanics Research Communications

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / m e c h r e s c o m

these procedures will become complicated for the interfaces

oscil-lating 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

rea-sons, and in order to introduce the matrix approach to the

prob-lems more complicated than the one considered byNevard 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

domainsXþ;Xof the plane x1x3 The interface ofXþandXis

de-noted by L, and it is expressed by the equation x3¼ hðyÞ; y ¼ x1=e,

where hðyÞ is a periodic function of period 1, as described inFigs 1,

and 2 The minimum value of h is A ðA > 0Þ, and its maximum

va-lue is zero We also assume that, in the domain 0 < x1<e, i.e

0 < y < 1, any straight line x3¼ x0¼ const ðA < x0<0Þ has

ex-actly two intersections with the curve L (seeFig 2) Suppose that

0 <e 1, then the curve L is called very rough interface (or highly

oscillating interface) ofXþandX Let two partsXþ;Xof 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;land the mass densityqdefined as follows:

k;l;q¼ kþ;lþ;qþfor x3>hðx1=eÞ

k;l;qfor x3<hðx1=eÞ



ð1Þ

where kþ;lþ;qþ;k;l;q are constant We consider the plane

strain for which the displacement components u1;u2;u3are of the

form:

ui¼ uiðx1;x3;tÞ; i ¼ 1; 3; u2 0 ð2Þ

The components of the stress tensorrij;i; j ¼ 1; 3 are related to

the displacement gradients by the following equations (Love,

1944):

r11¼ ðk þ 2lÞu1;1þ ku3;3;r33¼ ku1;1þ ðk þ 2lÞu3;3;

where commas indicate differentiation with respect to the spatial

variables xi Equations of motion are of the form (Love, 1944):

r11;1þr13;3þ f1¼q€u1; r13;1þr33;3þ f3¼q€u3 ð4Þ

in which f1; f3are the components of the body force, and a super-posed dot signifies differentiation with respect to the time t Substi-tuting(3)into(4)yields a system of equations for the displacement components whose matrix form is:

ðAhku;kÞ;hþ F ¼qu€ ð5Þ

where u ¼ ½u1u3T; F ¼ ½f1f3T, the symbol T indicates the trans-pose of matrices, the indices h, k take the values 1, 3, and:

A11¼ kþ 2l 0

0 l

A13¼ 0 k

 

A31¼ 0 l

k 0

 

A33¼ l 0

0 k þ 2l

ð6Þ

Note that, since k þ 2l>0; l>0 (seeTing, 1996), the matrix

A11is invertible SinceXþ; Xare perfectly welded to each other along L, the continuity for the displacement vector and the traction vector must be satisfied Thus we have:

½uL¼ 0; ½ðA11u;1þ A13u;3Þn1þ ðA31u;1þ A33u;3Þn3L¼ 0 ð7Þ

where nkis xk-component of the unit normal to the curve L, by the symbol ½uLwe denote the jump ofuthrough L Expressing nkin terms of h, the continuity condition(7)can be written as:

½uL¼ 0;e1½h0ðyÞðA11u;1þ A13u;3ÞL ½A31u;1þ A33u;3L¼ 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 þeðN1V þ N11V;1þ N13V;3Þ þe2ðN2V þ N21V;1þ N23V;3

þ N211V;11þ N213V;13þ N233V;33Þ þ Oðe3

where V ¼ Vðx1;x3;tÞ (being independent of y), N1, N11, N13, N2, N21,

N23, N211, N213, N233are 2  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  2-matrix In what follows, byu;ywe denote the derivative ofuwith respect to the variable y The matrix valued functions N1, , N233are determined so that the Eq.(5)and the continuity conditions(8)are satisfied Since y ¼ x1=e, we have:

Fig 1 Two-dimensional domains Xþ and X have a very rough interface L

expressed by equation x 3 ¼ hðx 1 =eÞ ¼ hðyÞ, where hðyÞ is a periodic function with

Fig 2 The curve L in the space 0yx 3 ; y 1 ; y 2 ð0 < y 1 < y 2 < 1Þ are two roots in the interval (0, 1) of equation x 3  hðyÞ ¼ 0 for y; A < x 3 ¼ const < 0; y 1 ¼

y 1 ðx 3 Þ; y 2 ¼ y 2 ðx 3 Þ.

Substituting(9)into(5) and (8), and taking into account(10)

yield equations which we call Eqs ðe1Þ and ðe2Þ, respectively In

or-der to make the coefficients ofe1 of Eqs ðe1Þ and ðe2Þ zero, the

functions N1, N11, N13are chosen as follows:

½A11N1;y;y¼ 0; 0 < y < 1; y – y1;y2;

½A11N1

;yL¼ 0; ½N1L¼ 0 at y1;y2; N1

ð0Þ ¼ N1ð1Þ ð11Þ

½A11E þ N11;y

;y¼ 0; 0 < y < 1; y – y1; y2;

½A11 E þ N11

;y

L¼ 0; ½N11L¼ 0 at y1; y2; N11

ð0Þ ¼ N11ð1Þ ð12Þ

A11N13;y þ A13

;y¼ 0; 0 < y < 1; y – y1; y2;

A11N13;y þ A13

L¼ 0; ½N13L¼ 0; at y1; y2;

N13

where y1;y2ð0 < 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

ðA 0Þ The functions y1ðx3Þ; y2ðx3Þ are two inverse branches of

the function x3¼ hðyÞ It is easy to see from(11)that N1

;y¼ 0

Equating to zero the coefficiente0of Eq.ðe1Þ provides:

A11N2

;yþ A13N1

;3

;yV þ A11 N1

þ N21;y

þ A13N11

;3

;yV;1

þ A11N23;y þ A13 N1þ N13;3

;yV;3

þ A11 N11þ N211;y

;yþ A11 E þ N11;y

V;11

þ A11N13þ N213;y 

þ A13N11

;yþ A13þ A11N13;y

V;13 þ½A11N233;y þ A13N13;yV;33þ ðA33V;3Þ;3

þ½A31 E þ N11

;y

V;1þ A31N13

;yV;3;3þ F qV ¼ 0€

ð14Þ

Making the coefficiente0of Eq ðe2Þ zero gives:

A11N2

;yþ A13N1

;3

Lh0V

þ A11 N1

þ N21;y

þ A13N11

;3

Lh0 A31 E þ N11

;y

L

V;1

þ A11N23

;y þ A13 N1

þ N13;3

Lh0 ½A33þ A31N13

;y

L

V;3 þ½A11 N11

þ N211;y

Lh0V;11þ ½A11 N13

þ N213;y

þ A13N11

Lh0V;13 þ½A11N233

;y þ A13N13

Lh0V;33¼ 0

ð15Þ

In order to make(15)satisfied we take:

½A11N2

;yþ A13N1

;3L¼ 0 at y1; y2 ð16Þ

½A11N1þ N21;y

þ A13N11;3L¼ Ah 31E þ N11;yi

L=h0 at y1; y2 ð17Þ

A11N23;y þ A13N1þ N13;3

L¼ Ah 33þ A31N13;yi

L=h0 at y1; y2 ð18Þ

A11N11þ N211;y 

L¼ 0 at y1; y2 ð19Þ

A11N13þ N213;y 

þ A13N11

L¼ 0 at y1; y2 ð20Þ

A11N233;y þ A13N13

L¼ 0 at y1; y2 ð21Þ

By integrating Eq (14) along the line x3¼ const; A < x3<0

from y ¼ 0 to y ¼ 1 (seeFig 2), and taking into account(16)–(21)

we have:

A11E þ N11;y

V;11þ A11N13;y þ A13

V;13

þ A31 E þ N11

;y

V;1þ A31N13

;y

V;3

;3

þ ½hA33iV;3;3þ F  hqi €V ¼ 0 ð22Þ

here:

hui ¼

Z1 0

udy ¼ ðy2 y1Þuþþ ð1  y2þ y1Þu ð23Þ

Note that if mijare elements of matrix M, then hMi ¼ ðhmijiÞ It is clear that in order to make Eq.(22)explicit we have to calculate the quantities:

q1¼ A11 E þ N11

;y

; q2¼ A11N13

;y þ A13

;

q3¼ AD 31E þ N11;yE

; q4¼ AD 31N13;yE ð24Þ

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

q1¼ A111

D E1

;q2¼ A111

D E1

A1

11A13

;q3¼ A31A1

11

A1 11

D E1

q4¼ A31A1

11

A1 11

D E1

A1

11A13

 A31A1

11A13

ð25Þ

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

A111

D E1

V;11þ A111

D E1

A111A13

V;13

þ DA31A111E

A111

D E1

V;1

;3

þ hA33i þ AD 31A111E

A111

D E1

A111A13

 A31A111A13

V;3

;3

Therefore we have the following theorem

Theorem 1 Let uðx1;x3;e;tÞ satisfy(5) and (8)with Ahkare defined

by(6), the curve L: x3¼ hðyÞ; y ¼ x1=e, is a very rough interface which oscillates between two lines x3¼ 0 and x3¼ A 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:

A1 11

D E1

V;11þ A1 11

D E1

A1

11A13

V;13þ A31A1

11

A1 11

D E1

V;1

;3

þ hA33i þ A31A1

11

A1 11

D E1

A1

11A13

 A31A1

11A13

V;3

;3

ð28Þ

þ hFi  hqi €V ¼ 0; A < x3<0

A31A1 11

A1 11

D E1

V;1þ



hA33i þ A31A1

11

A1 11

D E1

A1

11A13

 A31A1

11A13

V;3



¼ 0 and ½VL¼ 0; Lis lines x3¼ 0; x3¼ A Note that the continuity condition(30)1is originated from:

½A31u;1þ A33u;3L¼ 0;

Lis either the line x3¼ 0; or the line x3¼ A ð31Þ

Substituting(9)into(31)and taking into account(10)yield an equation denoted by Eq ðe3Þ By equating to zero the coefficient of

e0of Eq ðe3Þ we have:

A31E þ N11;y

V;1þ A 33þ A31N13;y

V;3

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

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

ðkþþ 2lþÞV1;11þlþV1;33þ ðkþþlþÞV3;13þ f1 þ¼qþV€1;

x3>0

ðkþþlþÞV1;13þlþV3;11þ ðkþþ 2lþÞV3;33þ f3 þ¼qþV€3;

x3>0

8

>

<

>

:

ð33Þ

1

kþ2l

D E1

V1;11þ 1

l

D E1

V1;3

;3

þ 1 l

D E1

V3;1

;3

þ 1

kþ2l

D E1

k þ2l

D E

V3;13þ hf1i ¼ hqi €V1; A < x3<0;

1

l

D E1

V1;13þ k

þ2l

D E

1 kþ2l

D E1

V1;1

;3

þ 1 l

D E1

V3;11

kþ2li1h k

þ2l

þ 4 lðkþlÞ kþ2l

V3;3

;3

þ hf3i ¼ hqi€V3;

A < x3<0

0

B

B

B

B

B

B

B

B

ð34Þ

ðkþ 2lÞV1;11þlV1;33þ ðkþlÞV3;13þ f1 ¼qV€1;

x3<A

ðkþlÞV1;13þlV3;11þ ðkþ 2lÞV3;33þ f3 ¼qV€3;

x3<A

8

>

<

>

:

ð35Þ

V1;V3;r0

13;r0

33 continuous on x3¼ A; x3¼ 0 ð36Þ

where

r0

13¼ h1=li1ðV1;3þ V3;1Þ;

r0

33¼ h1=ðk þ 2lÞi1hk=ðk þ 2lÞiV1;1

þ h1=ðk þ 2lÞi1hk=ðk þ 2lÞi2þ 4 lðk þlÞ

kþ 2l

V3;3 ð37Þ

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