Mechanics Research Communications 37 (2010) 285–288 Contents lists available at ScienceDirect 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 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 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 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 35532164; fax: +84 38588817 E-mail address: pcvinh@vnu.edu.vn (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 not start from the cell problem (9.10)–(9.11) in their paper Equations of motion and continuity conditions in matrix form Consider a linear elastic body that occupies two-dimensional domains Xỵ ; X of the plane x1 x3 The interface of Xỵ and XÀ is denoted by L, and it is expressed by the equation x3 ẳ hyị; y ẳ x1 =e, where hðyÞ is a periodic function of period 1, as described in Figs 1, and The minimum value of h is ÀA ðA > 0Þ, and its maximum value is zero We also assume that, in the domain < x1 < e, i.e < 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 < e ( 1, then the curve L is called very rough interface (or highly oscillating interface) of Xỵ and X Let two parts Xỵ ; X 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 ẳ kỵ ; lỵ ; qỵ for x3 > hðx1 =eÞ kÀ ; lÀ ; qÀ for x3 < hx1 =eị 1ị where kỵ ; lỵ ; qỵ ; k ; l ; q 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 ð3Þ where commas indicate differentiation with respect to the spatial variables xi Equations of motion are of the form (Love, 1944): r11;1 ỵ r13;3 ỵ f1 ẳ qu1 ; r13;1 ỵ r33;3 ỵ f3 ẳ qu€3 in which f1 ; f 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: Ahk u;k ị;h ỵ F ẳ qu 4ị 5ị where u ẳ ẵu1 u3 T ; F ẳ ẵf1 f T , the symbol T indicates the transpose of matrices, the indices h, k take the values 1, 3, and: A11 ẳ k ỵ 2l ð2Þ The components of the stress tensor rij ; i; j ¼ 1; are related to the displacement gradients by the following equations (Love, 1944): r11 ¼ k ỵ 2lịu1;1 ỵ ku3;3 ; r33 ẳ ku1;1 ỵ k ỵ 2lịu3;3 ; r13 ẳ lu1;3 ỵ u3;1 ị Fig The curve L in the space 0yx3 ; y1 ; y2 ð0 < y1 < y2 < 1Þ are two roots in the interval (0, 1) of equation x3 hyị ẳ for y; A < x3 ¼ const < 0; y1 ¼ y1 ðx3 Þ; y2 ¼ y2 ðx3 Þ l A13 ¼ k l A31 ¼ l k A33 ¼ l k ỵ 2l 6ị Note that, since k ỵ 2l > 0; l > (see Ting, 1996), the matrix A11 is invertible Since Xỵ ; X are perfectly welded to each other along L, the continuity for the displacement vector and the traction vector must be satisfied Thus we have: ẵuL ẳ 0; ẵA11 u;1 ỵ A13 u;3 ịn1 ỵ A31 u;1 ỵ A33 u;3 ịn3 L ẳ ð7Þ where nk is xk -component of the unit normal to the curve L, by the symbol ½uL we denote the jump of u through L Expressing nk in terms of h, the continuity condition (7) can be written as: ẵuL ẳ 0; e1 ẵh yịA11 u;1 ỵ A13 u;3 ịL ẵA31 u;1 ỵ A33 u;3 L ẳ ð8Þ 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 ẳ V ỵ eN1 V þ N11 V;1 þ N13 V;3 Þ þ e2 ðN2 V ỵ N21 V;1 ỵ N23 V;3 ỵ N211 V;11 þ N213 V;13 þ N233 V;33 Þ þ Oðe3 Þ Fig Two-dimensional domains Xỵ and X have a very rough interface L expressed by equation x3 ẳ hx1 =eị ¼ hðyÞ, where hðyÞ is a periodic function with period The curve L oscillates between the straight lines x3 ¼ 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-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-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 ẳ x1 =e, we have: u;1 ẳ U;1 ỵ e1 U;y ð10Þ 287 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: ẵA11 N1;y ;y ẳ 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 L ¼ 0; ½N1 L ¼ at y1 ; y2 ; N1 0ị ẳ N1 1ị ẵA11 E ỵ N11 ;y ;y ẳ 0; < y < 1; y – y1 ; y2 ; 11 11 11 ẵA11 E ỵ N11 ;y L ẳ 0; ẵN L ẳ at y1 ; y2 ; N 0ị ẳ N 1ị A11 N13 ¼ 0; < y < 1; y – y1 ; y2 ; ;y ỵ A13 ;y h i ¼ 0; ½N13 L ¼ 0; at y1 ; y2 ; A11 N13 ;y ỵ A13 11ị N13 0ị ẳ N13 1ị 13ị 12ị Z udy ẳ y2 y1 ịuỵ ỵ y2 ỵ y1 ịu D E q1 ẳ A11 E ỵ N11 ; ;y D E ; q3 ẳ A31 E ỵ N11 ;y 23ị D E q2 ẳ 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 hyị ẳ 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 ẳ hyị It is easy to see from (11) that N1;y ¼ Equating to zero the coefficient e0 of Eq.ðe1 Þ provides: i h i ỵ A13 N11 A11 N2;y ỵ A13 N1;3 V þ A11 N1 þ N21 V;1 ;y ;3 ;y ;y h i 13 ỵ A11 N23 V;3 ;y þ A13 N þ N;3 ;y h i V;11 ỵ A11 N11 ỵ N211 ỵ A11 E þ N11 ;y ;y ;y h i þ A13 N11 þ A13 þ A11 N13 V;13 þ A11 N13 ỵ N213 ;y ;y hui ẳ 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 Ề1 D E D E À1 Ầ1 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 E1 D E V ;11 ỵ A1 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 1 A A V;3 ỵ hA33 i ỵ A31 A1 A A A A 13 31 13 11 11 11 11 14ị ;3 ẳ0 ỵ hFi hqiV 26ị ;y ỵẵA11 N233 ;y 13 Therefore we have the following theorem ỵ A13 N ;y V;33 ỵ A33 V;3 ị;3 13 ỵẵA31 E þ N11 ;y V;1 þ A31 N;y V ;3 ;3 þ F À qV ¼ Making the coefficient h i Theorem Let uðx1 ; x3 ; e; tÞ satisfy (5) and (8) with Ahk are defined by (6), the curve L: x3 ẳ hyị; y ẳ x1 =e, is a very rough interface which oscillates between two lines x3 ẳ and x3 ẳ A A > 0ị and hðyÞ is a differentiable y-periodic function with period 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 ị zero gives: A11 N2;y ỵ A13 N1;3 h V L nh i h i o 21 V;1 ỵ A11 N ỵ N;y ỵ A13 N11 h A31 E ỵ N11 ;3 ;y nh L i oL 23 13 13 ỵ A11 N;y ỵ A13 N ỵ N;3 L h ẵA33 ỵ A31 N;y V;3 L 0 L h V;11 ỵ ẵA11 N13 þ N213 þ A13 N11 L h V;13 þ½A11 N11 þ N211 ;y ;y € x3 > Ahkþ V;kh þ Fþ ¼ qþ V; D D Ề1 D Ề1 D E ED EÀ1 À1 À1 À1 À1 V A A1 V ỵ A A A ỵ A A V;1 ;11 ;13 31 11 11 11 11 13 11 13 ỵẵA11 N233 ;y ỵ A13 N L h V;33 ẳ 15ị In order to make (15) satised we take: ẵA11 N2;y ỵ A13 N1;3 L ẳ at y1 ; y2 h i 11 ỵ A13 N11 =h at y1 ; y2 ẵA11 N1 ỵ N21 ;y ;3 L ẳ A31 E þ N;y h i h iL 13 A11 N23 ẳ A33 ỵ A31 N13 =h at y1 ; y2 ;y ỵ A13 N ỵ N;3 ;y L L h i A11 N11 ỵ N211 ẳ at y1 ; y2 ;y h L i A11 N13 ỵ N213 ỵ A13 N11 ẳ at y1 ; y2 ;y L h i 233 13 A11 N;y ỵ A13 N ẳ at y1 ; y2 L ð16Þ ð17Þ ð18Þ ð19Þ ð21Þ E D E A11 E ỵ N11 V;11 ỵ A11 N13 ;y ;y þ A13 V ;13 hD E D E i V;1 ỵ A31 N13 ỵ A31 E ỵ N11 ;y ;y V;3 ẳ0 ỵ ẵhA33 iV;3 ;3 ỵ F À hqiV € ¼ 0; ÀA < x3 < þ hFi À hqiV € x3 A AhkÀ V;kh þ FÀ ¼ qÀ V; D ED Ề1 À1 À1 A31 A11 A11 V;1 ỵ D ED E1 D E D E À1 AÀ1 V;3 AÀ1 hA33 i þ A31 AÀ1 11 11 11 A13 À A31 A11 A13 29ị 30ị L ẳ and ẵVL ẳ 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 Là ¼ 0; Là 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 ẳ L 22ị 28ị h ;3 ;3 ;3 ð20Þ By integrating Eq (14) along the line x3 ¼ const; ÀA < x3 < from y ¼ to y ¼ (see Fig 2), and taking into account (16)(21) we have: D ỵ D ED EÀ1 D E D E À1 À1 À1 A A V;3 hA33 i ỵ A31 A1 A A A A 13 31 13 11 11 11 11 ð27Þ 32ị Integrating (32) along the line L from y ẳ to y ¼ 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: 1; kỵ ỵ 2lỵ ịV 1;11 ỵ lỵ V 1;33 ỵ kỵ ỵ lỵ ịV 3;13 ỵ f1ỵ ẳ qỵ V > > > < x3 > 33ị > ỵ lỵ ịV 1;13 ỵ lỵ V 3;11 ỵ kỵ ỵ 2lỵ ịV 3;33 ỵ f3ỵ ẳ qỵ V ; k > ỵ > : x3 > D E D E 0D EÀ1 À1 1 1 V ỵ V ỵ V 1;11 1;3 3;1 l l B kỵ2l ;3 ;3 B D B E D E À1 B k B þ kþ2 V 3;13 þ hf1 i ¼ hqiV€ ; A < x3 < 0; kỵ2l l B B D B D Ề1 ED Ề1 D Ề1 ð34Þ B k V 1;13 ỵ kỵ2 V 1;1 þ l1 V 3;11 B l kþ2 l l B ;3 B D B E2 D E B À1 lkỵlị k 3; ỵ i hkỵ2l ỵ kỵ2l V 3;3 ỵ hf3 i ẳ hqiV B kỵ2 l @ ;3 ÀA < x3 < € 1; k ỵ 2l ịV 1;11 ỵ l V 1;33 ỵ k ỵ l ịV 3;13 ỵ f1 ẳ q V > > > < x3 < A > ỵ l ịV 1;13 ỵ l V 3;11 ỵ k ỵ 2l ịV 3;33 ỵ f3 ẳ q V ; 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A Treatise on the Mathematical Theory of Elasticity, fourth ed Dover Publications, New York Madureira, A.L., Valentin, F., 2007 Asymptotics of the Poisson problem in domains with curved rough. .. Equations of motion and continuity conditions in matrix form Consider a linear elastic body that occupies two-dimensional domains Xỵ ; X of the plane x1 x3 The interface of Xỵ and X is denoted by... called very rough interface (or highly oscillating interface) of Xỵ and X Let two parts Xỵ ; X of the body be perfectly welded to each other Suppose that the body is made of isotropic material,