DSpace at VNU: Homogenization of Rough Two-Dimensional Interfaces Separating Two Anisotropic Solids

Pham Chi e-mail; pcvinh@vnu.edu.vn Do Xuan Tung Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Street, Thanh Xuan, Hanoi 10000, Vietnam Homogenization of Rough Two-Dimensional Interfaces Separating Two Anisotropie Soiids In this paper we have derived homogenized equations in explicit form of the linear elasticity theory in a two-dimensional domain with an interface highly oscillating between two straight lines, by using the homogenization method First, the homogenized equation in the matrix form for generally anisotropic materials is obtained Then, it is written down in the component form for specific cases when the materials are orthotropic, monoclinic with the symmetry plane at X, =0 and X2 =0 Since these equations are in explicit form, they are significant in practical applications [DOI: 10.1115/1.4003722] Keywords: homogenization, homogenized equations, very rough interfaces, anisotropic materials 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 [1-4]; transmission and reflection of waves on rough interfaces [5-7]; mechanical problems concerning the plates with densely spaced stiffeners [8]; the flows over rough walls [9]; nearly circular hole, inclusion problems in plane elasticity and thermoelasticily [10-12]; and so on When the amplitude (height) of the roughness is much small in comparison with its period (see, for example, Ref [12]), the problems are usually analyzed by perturbation methods [13] When the amplitude is much large than its period, i.e., the boundaries and interfaces are very rough (see, for instance, Refs [14,15]), the homogenization method [16-18] is required, in which the very rough boundaries or interfaces are replaced with equivalent layers within which homogenized equations hold (see Ref [15]) The main aim of the homogenization of very rough boundaries or interfaces is to determine these homogenized equations Nevard and Keller [15] examined the homogenization of very rough three-dimensional interfaces separating two linear anisotropic solids The authors have derived the homogenized equations, but these equations are still implicit In particular, their coefficients are the solution of a boundary-value problem on the periodic cell (called "cell problem"), which includes 27 partial differential equations This problem can, in general, only be solved numerically Recently, Vinh and Tung [19] considered the homogenization of two-dimensional very rough interfaces separating two isotropic linear elastic solids, and they have derived the explicit homogenized equations, by using the standard techniques of the homogenization method along with the matrix formulation In this paper, following the procedure carried out in Ref [19], we have derived the explicit homogenized equation in the matrix form for the case of generally anisotropic materials Then, it is written down in the component form for the specific cases when the materials are orthotropic, and monoclinic with the symmetry plane at X|=0 and X2=0 Since these equations are in explicit Corresponding author Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPI.IKD MKCHANICS Manu.script received Augu.st 2010; final manuscript received January 22, 2011; accepted manuscript posted February 28 2011; published online April 14, 2011 Assoc Editor: Krishna Garikipati Journai of Appiied Mechanics form, i.e., their coefficients are explicit functions of given material and interface parameters, they are convenient in use We also note that homogenized equation (9.24) in Ref [ 15] obtained by Nevard and Keller is incorrect at least in the two-dimensional case (Remark 2) Before going to Sec 2, we recall once again the main point of the homogenization of a very rough interface oscillating between two straight lines and separating two elastic solids Due to great difficulties caused by its rapid oscillation, the interface (or the composite material layer containing the interface) is replaced with an equivalent elastic plane layer Then the actual problem is replaced with the problem of two dissimilar elastic half-spaces separated by an elastic plane layer with homogenized propenies This material layer is governed by the homogenized equations, which need to be determined Basic Equations in Matrix Form Consider a linear elastic ix)dy that occupies two-dimensionai domains fl* and fl" of the plane X\Xy, their interface L oscillates between two straight lines Xj=-A (i4>0) and X3=0, and is expressed by the equation X3=/4/i(X|/\) ( \ > ) , where h{y) {y = X | / \ ) is a periodic function of period whose maximum and minimum values are and - , respectively, as described in Fig Suppose that in the domain < X | < \ , i.e., < v < l , any straight line X3=X^=const (-/1 cii J» Cll/ \ '•' Cu •'- J / J J,3 (45) f|3 £•11/ *ll( ).i,A-/\-/-\ -^1.33 + (C55.- + ô|4-) '1,1 J.3 '-(^;iK, X3 < - A ^2 X3 < - / (46) V'3- X3 {c^S){c,^S) Note that when il* and iiir are made of the same mal material, ¿-¡j =c¡j; thus systems (44)-(46) are identical to each other other where + ^56^2 4.2 Monoclinie Materials With the Symmetrie Plane X2 the mon monoclinie materials with the symmetry plane Xo ~^- ^°^r ^^^ =0, We •have [20] (48) Substituting Eq (50) into Eq (6) yields || A,,= | Cl5 C,3 C66 0 C4, C|5 C55 C55 f,5 15 C55 C55 0 C45 0 C44 ^Cl3 [f35 (•33_ (51) A,,= and ij = 5,6 , „^ ' , C-35J A33= On the use of Eq (51 ), we can write Eqs (35)-(37) in the component form as J I + V'i,, + 2C|5^V| ,3 = p^.V|, X > 3,|, = p,V>3, X > -35+V, 33 -1- c¡¡^V¡j, -1- 2035.^^3 5+V|,n -I- (f55+ + c n (52) 2^ = P+V'2 X3 > V,,,3 + [(c, ,( 3] (53) C, | _ V , | , 5_V3,,, ClS-V'l.l + (C55- -H C13 ^b Vi, V3, a-23, 033 are continuous on X3 = - j =0 5_V3 33 + f, = p_V„ X3 < -^ C55_V3 ,, ,,3 -I- C33_V3 33 -I-/3 = p_V>3, X3 < „V* (54) X3 (55) where in which V3,,) -h (C|5(a) c¡j = {c¡j/d}/d, i,j= 1,5, d = = (C, 1(0) + C|5