DSpace at VNU: Homogenization of very rough interfaces separating two piezoelectric solids

Acta Mech 224, 1077–1088 (2013) DOI 10.1007/s00707-012-0804-z Pham Chi Vinh · Do Xuan Tung Homogenization of very rough interfaces separating two piezoelectric solids Received: 28 August 2012 / Revised: December 2012 / Published online: 12 January 2013 © Springer-Verlag Wien 2013 Abstract The main aim of this paper is to derive homogenized equations in explicit form of the linear piezoelectricity in two-dimensional domains separated by an interface which highly oscillates between two parallel straight lines First, the basic equations of the linear theory of piezoelectricity are written down in matrix form Then, following the techniques presented recently by these authors, the explicit homogenized equation and the associate continuity condition, for generally anisotropic piezoelectric materials, are derived They are then written down in component form for some specific cases Since the obtained equations are totally explicit, they are significant in practical applications 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 [1–7], transmission and reflection of waves on rough interfaces [8–15], mechanical problems concerning plates with densely spaced stiffeners [16], flows over rough walls [17], vibrations of strongly inhomogeneous elastic bodies [18], propagations of surface waves in half-spaces with cracked surfaces [19–21], nearly circular holes and inclusions in plane elasticity and thermoelasticity [22–25] and so on When the amplitude (height) of the roughness is very small in comparison with its period, the problems are usually analyzed by perturbation methods [26] When the amplitude is much larger than its period, that is, the boundaries and interfaces are very rough, the homogenization method is required, see for instance [27–30] The aforementioned boundary-value problems originate from various physical theories such as the elasticity theory, the thermoelasticity theory, the theory of electromagnetic fields, the piezoelectricity theory, the electro-magneto-elasticity and so on For the elasticity theory, Nevard and Keller [31] examined the homogenization of a very rough threedimensional interface that oscillates between two parallel planes and separates two linear anisotropic solids By applying the homogenization method, the authors have derived the homogenized equations, but these equations are still implicit In some recent papers [32–34], the explicit homogenized equations of linear elasticity in P C Vinh (B) · D X Tung Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam E-mail: pcvinh@vnu.edu.vn Tel.: +84-4-5532164 Fax: +84-4-8588817 D X Tung Faculty of Civil Engineering, Hanoi Architectural University, Km 10 Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam 1078 P C Vinh, D X Tung Fig Two-dimensional domains + and − have a very rough interface L expressed by the equation x3 = h(x1 /ε) = h(y), where h(y) is a periodic function with period The curve L highly oscillates between the parallel straight lines x = and x3 = −A (A > 0) two-dimensional domains with interfaces rapidly oscillating between two parallel straight lines and between two concentric circles have been obtained Because piezoelectric materials exhibit the electromechanical coupling phenomenon, they have been widely used in various fields of modern engineering, such as electroacoustics, transducers and control of structure vibrations (see [36]) The consideration of boundary-value problems of the piezoelectricity theory in domains with very rough boundaries or interfaces is therefore significant and of great theoretical and practical interest as well The main aim of this paper is to find explicit homogenized equations of the linear theory of piezoelectricity in two-dimensional domains including very rough interfaces We will consider the case when the interfaces highly oscillate between two parallel straight lines In order to that, first, the basic equations of linear theory of piezoelectricity are written down in matrix form Then, following the techniques presented recently by these authors [32–34], the homogenized equation and the associated continuity condition in explicit form, for generally anisotropic piezoelectric materials, are derived They are then written down in component form for several specific cases Since the obtained homogenized equations are explicit, that is, their coefficients are explicit functions of given material and interface parameters, they are useful in practical applications Basic equations Consider a linear piezoelectric body occupying two-dimensional domains + and − of the plane x1 x3 whose interface is the curve L expressed by the equation x3 = h(x1 /ε) = h(y), where h(y) is a periodic function of period whose minimum and maximum values are −A (A > 0) and zero, respectively, and ε is assumed to be much smaller than A (i.e., the curve L is a very rough interface) The interface L lies in the strip −A < x3 < (Fig 1) We also assume that, in the domain < x1 < , that is, < y < 1, any straight line x3 = x30 = const(−A < x30 < 0) has exactly two intersections with the curve L We consider the generalized plane strain (see [37]) for which the displacement components u , u , u and the electric potential φ are of the form u = u (x1 , x3 , t), u = u (x1 , x3 , t), u = u (x1 , x3 , t), φ = φ(x1 , x3 , t) (1) The strain εi j and the components of the electric field vector E i are expressed as follows [38–40]: εi j = ∂u j ∂u i , + ∂x j ∂ xi Ei = − ∂φ ∂ xi (2) The stress σi j and the components of the electric displacement vector Di are related to the strains εi j and the components E i of the electric field vector by the following relations [38,40,41]: σi j = ci jkl εkl − eli j El , Di = eikl εkl + il E l , (3) Homogenization of very rough interfaces 1079 where commas indicate differentiation with respect to xi , and ci jkl , ei jk and i j are, respectively, the elastic (measured in a constant electric field), piezoelectric (measured at a constant strain or electric field) and the dielectric (measured at a constant strain) moduli which have the following classical properties of symmetry: c jikl = ci jkl = ckli j , eki j = ek ji , ij = (4) ji and they are defined as: ci jkl , ei jk , ij = ci jkl+ , ei jk+ , i j+ for (x1 , x3 ) ∈ ci j− , ei jk− , i j− for (x1 , x3 ) ∈ +, (5) −, where ci jkl+ , ei jk+ , i j+ , ci jkl− , ei jk− , i j− are constant Using the Voigt contracted notations, the relations (3) are written in matrix form as (see [38]): ⎡ σ11 ⎤ ⎡ c11 c12 c13 c14 c15 c16 e11 e21 c22 c23 c24 c25 c26 e12 e22 c23 c33 c34 c35 c36 e13 e23 c24 c34 c44 c45 c46 e14 e24 c25 c35 c45 c55 c56 e15 e25 c26 c36 c46 c56 c66 e16 e26 ⎢ ⎥ ⎢ ⎢σ22 ⎥ ⎢c12 ⎢ ⎥ ⎢ ⎢σ ⎥ ⎢c ⎢ 33 ⎥ ⎢ 13 ⎢ ⎥ ⎢ ⎢σ23 ⎥ ⎢c14 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢σ13 ⎥ = ⎢c15 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢σ12 ⎥ ⎢c16 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ D1 ⎥ ⎢e11 ⎢ ⎥ ⎢ ⎢ D ⎥ ⎢e ⎣ ⎦ ⎣ 21 D3 e31 e12 e13 e14 e15 e16 − 11 − 12 e22 e23 e24 e25 e26 − 12 − 22 e32 e33 e34 e35 e36 − 13 − 23 e31 ⎤⎡ ε11 ⎤ ⎥⎢ ⎥ e33 ⎥ ⎢ ε22 ⎥ ⎥⎢ ⎥ ⎥ ⎢ e33 ⎥ ⎥ ⎢ ε33 ⎥ ⎥⎢ ⎥ ⎥ ⎢ e34 ⎥ ⎥ ⎢2ε23 ⎥ ⎥⎢ ⎥ ⎥ ⎢ e35 ⎥ ⎥ ⎢2ε31 ⎥ ⎥⎢ ⎥ e36 ⎥ ⎢2ε12 ⎥ ⎥⎢ ⎥ ⎥⎢ ⎥ − 13 ⎥ ⎢ φ,1 ⎥ ⎥⎢ ⎥ ⎢φ ⎥ − 23 ⎥ ,2 ⎦⎣ ⎦ − 33 φ,3 (6) The equations of motion and Gauss’s law are [38,40]: σ11,1 + σ13,3 + f σ12,1 + σ23,3 + f σ13,1 + σ33,3 + f D1,1 + D3,3 q = uă , = uă , = uă , =0 (7) / L, where ρ is the mass density (taking different constants ρ + , ρ − in + and − , respectively), for (x1 , x3 ) ∈ f , f and f are the components of body forces, q is the electric charge density, and a dot indicates differentiation with respect to the time t In addition to Eqs (7) the continuity condition on the interface L is required, namely: [u k ] L = (k = 1, 2, 3), [φ] L = 0, [ nk ] L = (k = 1, 2, 3), [Dn ] L = 0, (8) where [w] L = w+ − w− , and nk = σk1 n + σk3 n (k = 1, 2, 3), Dn = D1 n + D3 n , (9) and n k are the components of the unit normal to the curve L Explicit homogenized equation in matrix form Using (6) in (7) and taking into account (1) and (2) yields a system of equations for the displacement components and the electric potential whose matrix form is as follows: Ahk u,k ,h ă + F = ρIu, (10) 1080 P C Vinh, D X Tung where u = [u , u , u , φ]T , F = [ f , f , f , −q]T and: ⎤ ⎡ ⎡ c11 c16 c15 e11 c15 ⎥ ⎢ ⎢ ⎢c16 c66 c56 e16 ⎥ ⎢c56 ⎥ ⎢ A11 = ⎢ ⎢c c c e ⎥ , A13 = ⎢c ⎣ 15 56 55 15 ⎦ ⎣ 55 e11 e16 e15 − 11 e15 ⎤ ⎡ ⎡ c15 c56 c55 e15 c55 ⎥ ⎢ ⎢ ⎢c14 c46 c45 e14 ⎥ ⎢c45 ⎥ ⎢ A31 = ⎢ ⎢c c c e ⎥ , A33 = ⎢c ⎣ 13 36 35 13 ⎦ ⎣ 35 e31 e36 e35 − 13 e35 ⎡ ⎤ 1000 ⎢ ⎥ ⎢0 0⎥ ⎢ ⎥ I=⎢ ⎥ ⎣0 0⎦ 0000 c14 c13 e31 ⎤ ⎥ c46 c36 e36 ⎥ ⎥, c45 c35 e35 ⎥ ⎦ e14 e13 − 13 ⎤ c45 c35 e35 ⎥ c44 c34 e34 ⎥ ⎥, c34 c33 e33 ⎥ ⎦ e34 e33 − 33 (11) Here, the symbol “T ” indicates the transpose of a matrix In addition to Eq (10) the continuity condition on L is required, namely: [u] L = 0, (12) A11 u,1 + A13 u,3 n + A31 u,1 + A33 u,3 n L = 0, (13) which originate from (8) Following Bensoussan et al [28], Sanchez-Palencia [29], Bakhvalov and Panasenko [30] and Kohler et al [27], we suppose that u(x1 , x3 , t, ) = U(x1 , y, x3 , t, ε), and we express U as follows (see [32–35]): U = V + ε N1 V + N11 V,1 + N13 V,3 + ε2 N2 V + N21 V,1 + N23 V,3 + N211 V,11 + N213 V,13 + N233 V,33 + O(ε3 ), (14) where V = V(x1 , x3 , t) (being independent of y), N1 , N11 , N13 , N2 , N21 , N23 , N211 , N213 , N233 are × 4matrix functions of y and x3 (not depending on x1 , t), and the they are y-periodic with the period The matrix functions N1 , , N233 are determined so that Eq (10) and the boundary conditions (12) and (13) are satisfied Our main purpose is to study asymptotic behavior of the boundary-value problem (10), (12) and (13) when ε tends to zero In particular, we want to find the explicit homogenized equation of this problem, that is, the equation for the leading term V = [V1 , V2 , V3 , ]T in the asymptotic expansion (14), and the associated continuity conditions Following the same procedure as was carried out in [32–34], the explicit homogenized equation of the problem (10), (12) and (13) in matrix form is as follows: A+ hk V,k A1 11 + ,h ă x3 > + F+ = ρ + IV, V,11 + A−1 11 A33 + −1 A31 A−1 11 A−1 11 A13 V,13 + −1 A−1 11 −1 A31 A−1 11 A11 A−1 11 A13 − −1 A31 A−1 11 A13 V,1 ,3 V,3 ă A < x3 < + F = IV, ă x3 < A A hk V,k + F− = ρ − IV, (15) ,3 ,h and the continuity conditions on the straight lines x3 = and x3 = −A have the form: −1 A31 A−1 11 A11 −1 V,1 + −1 A33 + A31 A−1 11 A11 [V] L ∗ = 0, L ∗ is the lines: x3 = 0, x3 = −A −1 −1 A−1 11 A13 − A31 A11 A13 V,3 L∗ = 0, (16) Homogenization of very rough interfaces 1081 Here, the matrices Ahk are given by (11), and g = gdy = (y2 − y1 )g+ + (1 − y2 + y1 )g− , (17) where g+ and g− are the values of g in + and − , respectively, y1 , y2 (0 < y1 < y2 < 1) are two roots in the interval (0 , 1) of the equation h(y) = x30 for y, in which x30 belongs to the interval (−A 0) The functions y1 (x3 ), y2 (x3 ) are two inverse branches of the function x3 = h(y) Note that due to the positive definiteness + and of the strain energy, detA11 = 0, the matrix A−1 11 therefore exists It is clear that when the materials of − are the same, Eqs (15) , (15) and (15) coincide with each other and with Eq (10) Explicit homogenized equations in component form for some specific cases In this section, we write down the homogenized equation (15) and the continuity conditions (16) in component form, for two specific cases when the solids are made of orthotropic crystals of class 222 and hexagonal crystals of class mm (see [38]) Following the same procedure, one can obtain the explicit homogenized equation and the associated continuity conditions, in component form, for other cases 4.1 Orthotropic-(222) crystals Consider orthotropic-(222) crystals (see [38]), for which the matrices Ahk are of the form: A11 ⎡ c11 0 ⎢ ⎢ c66 =⎢ ⎢0 c 55 ⎣ ⎡ A31 0 ⎢ ⎢0 =⎢ ⎢c ⎣ 13 0 0 c55 0 0 e36 0 ⎤ ⎡ 0 c13 ⎤ ⎥ ⎥ ⎢ ⎥ ⎢ 0 e36 ⎥ ⎥ , A13 = ⎢ ⎥ ⎢c 0 ⎥ , ⎥ ⎦ ⎦ ⎣ 55 − 11 e14 0 ⎤ ⎤ ⎡ c55 0 ⎥ ⎥ ⎢ e14 ⎥ ⎢ c44 0 ⎥ ⎥ ⎥ , A33 = ⎢ ⎥ ⎢0 c 0⎥ 33 ⎦ ⎦ ⎣ 0 0 − 33 (18) From (6) one can see that σ13 = c55 (u 1,3 + u 3,1 ), σ23 = e14 φ,1 + c44 u 2,3 , σ33 = c13 u 1,1 + c33 u 3,3 , D3 = e36 u 2,1 − 33 φ,3 (19) Introducing (18) into (15), (16) and after some manipulations, we obtain the explicit homogenized equations in component form and the associated continuity conditions They are: For x3 > 0: ⎧ ⎪ c11+ V1,11 + (c13+ + c55+ )V3,13 + c55+ V1,33 + f 1+ = + Vă1 c V + 66+ 2,11 + (e36+ + e14+ ) ,13 + c44+ V2,33 + f 2+ = Vă2 c55+ V3,11 + (c55+ + c13+ )V1,13 + c33+ V3,33 + f 3+ = + Vă3 11+ ,11 + (e14+ + e36+ )V2,13 − 33+ ,33 − q+ = (20) 1082 P C Vinh, D X Tung For −A < x3 < 0: −1 c11 −1 −1 −1 V1,11 + c13 c11 c11 −1 −1 V3,13 + c55 V3,1 ,3 = Vă1 + f1 c66 −1 −1 −1 V2,11 + e36 c66 c66 −1 ,13 + e14 −1 11 −1 + c55 −1 −1 11 ,1 ,3 −1 V1,3 + ( c44 + e14 1 )V2,3 ,3 + f = Vă2 11 −1 −1 −1 −1 −1 −1 −1 −1 c55 V3,11 + c55 V1,13 + c13 c11 c11 V1,1 ,3 + ( c13 c11 −1 − c11 c13 + c33 )V3,3 ,3 + f = Vă3 −1 −1 −1 −1 −1 −1 −1 −1 − 11 V2,13 + e36 c66 c66 V2,1 ,3 ,11 + e14 11 11 − e14 + For x3 < −A: ,3 −1 11 −1 11 −1 e36 c66 −1 c66 −1 −1 − e36 c66 − 33 ,3 ,3 −1 c11 −1 − q =0 (21) ⎧ ⎪ c11− V1,11 + (c13− + c55− )V3,13 + c55− V1,33 + f = Vă1 c V − 66− 2,11 + (e36− + e14− ) ,13 + c44 V2,33 + f = Vă2 c55− V3,11 + (c55− + c13− )V1,13 + c33− V3,33 + f = Vă3 11− ,11 + (e14− + e36− )V2,13 − 33− ,33 − q− = (22) and: 0 , σ23 , σ33 , D30 are continuous on the lines x3 = 0, x3 = −A, V1 , V2 , V3 , , σ13 (23) where σi0j , D30 are the coefficients of ε0 (i.e., they are leading terms) in their asymptotic expansions, and they are given by: −1 σ13 = c55 = σ23 −1 −1 −1 11 (V3,1 + V1,3 ), −1 11 e14 −1 −1 = c11 c13 c11 σ33 −1 D30 = c66 −1 −1 ,1 + V1,1 + −1 c66 e36 V2,1 + c44 + −1 11 e14 −1 c33 + c11 −1 c66 −1 −1 −1 c66 e36 − −1 −1 11 −1 c11 c13 2 −1 11 e14 −1 − c11 c13 −1 − c66 e36 − 33 V2,3 , V3,3 , (24) ,3 It is readily to see that when the materials of + and − are the same, Eqs (20), (21) and (22) coincide with and D become, respectively, σ and D given by (19) Also note that for this case V and each other, and σk3 k3 3 V3 are decoupled from V2 and 4.2 Hexagonal crystals of class mm In this subsection, we consider hexagonal crystals of class 6mm (see [38]), for which the matrices Ahk take the form: ⎤ ⎤ ⎡ ⎡ c11 0 0 c13 e31 ⎥ ⎥ ⎢ ⎢ ⎢ c66 0 ⎥ ⎢0 0 0⎥ ⎥ ⎥ ⎢ ⎢ A11 = ⎢ ⎥ , A13 = ⎢c 0 ⎥ , 0 c e 44 15 ⎦ ⎦ ⎣ ⎣ 44 e15 0 0 e15 − 11 ⎤ ⎤ ⎡ ⎡ 0 c44 e15 c44 0 ⎥ ⎥ ⎢ ⎢ ⎢0 0 0⎥ ⎢ c44 0 ⎥ ⎥ ⎥ ⎢ ⎢ A31 = ⎢ (25) ⎥ , A33 = ⎢ 0 c e ⎥ 33 33 ⎦ ⎣c13 0 ⎦ ⎣ 0 e33 − 33 e31 0 Homogenization of very rough interfaces 1083 For this case, σk3 (k = 1, 2, 3) and D3 are expressed in terms of u k (k = 1, 2, 3) and φ by (using (6)): σ13 = c44 (u 1,3 + u 3,1 ) + e15 φ,1 , σ23 = c44 u 2,3 , σ33 = c13 u 1,1 + c33 u 3,3 + e33 φ,3 , D3 = e31 u 1,1 + e33 u 3,3 − 33 φ,3 (26) Substituting (25) into (15), (16) and after some calculations, we arrive at the explicit homogenized equations in component form and the associated continuity conditions, namely: For x3 > 0: ⎧ ⎪ c11+ V1,11 + c44+ V1,33 + (c13+ + c44+ )V3,13 + (e31+ + e15+ ) ,13 + f 1+ = + Vă1 c V + 66+ 2,11 + c44+ V2,33 + f 2+ = Vă2 (27) ⎪ (c44+ + c13+ )V1,13 + c44+ V3,11 + c33+ V3,33 + e15+ ,11 + e33+ ,33 + f 3+ = + Vă3 (e 15+ + e31+ )V1,13 + e15+ V3,11 + e33+ V3,33 − 11+ ,11 − 33+ ,33 − q+ = For −A < x3 < 0: −1 −1 −1 V1,11 + c11 c13 c11 c11 ∗ + c44 V1,3 ,3 + f −1 −1 c66 V2,11 + c44 ∗ ∗ c44 V3,11 + e15 ,11 −1 −1 −1 V3,13 + c11 e31 c11 −1 ,13 ∗ ∗ + c44 V3,1 + e15 = Vă1 V2,3 ,3 + f = Vă2 1 + c44 V1,13 + c11 c13 c11 −1 − c11 c13 V3,3 + −1 V1,1 −1 −1 −1 e33 + c11 e31 c11 c13 c11 −1 −1 c33 + c11 c13 + ,3 −1 c11 −1 −1 −1 −1 c11 e31 c11 c13 c11 −1 −1 − c11 c13 e31 ,3 ∗ 11 ,11 ∗ + e15 V1,13 + −1 −1 c11 e31 c11 −1 −1 − c11 c13 e31 + e33 V3,3 + c11 e31 −1 c11 −1 V1,1 −1 ,3 + −1 − c11 e31 − 33 − q =0 ,3 ,3 + f = Vă3 e15 V3,11 − ,1 ,3 ,3 (28) For x3 < −A: ⎧ ⎪ c11− V1,11 + c44− V1,33 + (c13− + c44− )V3,13 + (e31− + e15− ) ,13 f = Vă1 c V 66− 2,11 + c44− V2,33 + f 2− = ρ Vă2 (c44 + c13 )V1,13 + c44 V3,11 + c33− V3,33 + e15− ,11 + e33− ,33 + f = Vă3 (e 15 + e31− )V1,13 + e15− V3,11 + e33− V3,33 − 11− ,11 − 33− ,33 − q− = (29) and: 0 , σ23 , σ33 , D30 are continuous on the lines x3 = 0, x3 = −A, V1 , V2 , V3 , , σ13 (30) where ∗ ∗ V1,3 + V3,1 + e15 = c44 σ13 ,1 , σ23 = c44 V2,3 , −1 −1 σ33 = c13 c11 c11 −1 V1,1 + −1 − c11 c13 V3,3 + −1 −1 D30 = e31 c11 c11 −1 −1 c33 + c11 c13 −1 c11 −1 −1 −1 −1 e33 + c11 e31 c11 c13 c11 V1,1 + −1 −1 −1 c11 e31 c11 c13 c11 −1 −1 − c11 c13 e31 + e33 V3,3 + c11 e31 −1 c11 −1 −1 − c11 c13 e31 ,3 , −1 −1 −1 − c11 e31 − 33 ,3 (31) 1084 P C Vinh, D X Tung Fig The tooth-comb interface L, L : x1 = a (−A < x3 < 0), L : x1 = a + b (−A < x3 < 0), L : x3 = (0 ≤ x1 ≤ a), L : x3 = −A (a ≤ x1 ≤ a + b) Here we use the notations: c44 ¯ ∗ e15 ¯ ∗ 11 ¯ /d, e15 = /d, 11 = /d, d d d e15 c44 11 + , d¯ = d = c44 11 + e15 d d d ∗ c44 = (32) ∗ = c , e∗ = e „ ∗ = One can see that when + and − are made of the same material, c44 44 11 , Eqs (27), 15 11 15 0 (28) and (29) are identical to each other, and σk3 and D3 become, respectively, σk3 and D3 given by (26) We also see that V2 is decoupled from V1 , V3 and for this case Finally, as an example, we consider the case when the interface L is of “tooth-comb” type as illustrated in Fig In this case, Eqs (28) are simplified to + f = ρ Vă1 , V2,11 + V2,33 + f = Vă2 , V1,13 + V3,11 + γ3 V3,33 + γ4 ,11 + γ5 ,33 + f = Vă3 , V1,13 + V3,11 + θ3 V3,33 − θ4 ,11 − θ5 ,33 − q = 0, α1 V1,11 + α2 V1,33 + α3 V3,13 + α4 ,13 (33) where a , ≤ n ≤ 1, a+b c13 −1 −1 e31 −1 −1 −1 −1 ∗ ∗ ∗ c11 c = c11 , α2 = c44 , α3 = + c44 , α4 = + e15 , c11 c11 11 c2 c13 −1 −1 −1 −1 ∗ = c66 , β2 = c44 , γ1 = α3 , γ2 = c44 , γ3 = c33 + c11 − 13 , c11 c11 e31 c13 −1 −1 e31 c13 e31 −1 −1 ∗ ∗ c , θ1 = c = e15 , γ5 = e33 + − + e15 , c11 c11 11 c11 c11 11 e2 e31 −1 −1 ∗ ∗ = e15 , θ3 = γ5 , θ4 = − 11 , θ5 = − 33 + c11 − 31 c11 c11 ϕ = nϕ− + (1 − n)ϕ+ , n = α1 β1 γ4 θ2 (34) (35) One can see, from Eqs (34) and (35), that the effective coefficients αk , βk , γk and θk of Eqs (33) are functions of the parameter n They are the corresponding coefficients of Eqs (27) and (29) when n = and n = 1, respectively Figures 3, 4, and show, respectively, the dependence of the effective coefficients αk (k = 1, 2, 3), γk (k = 1, 2, 3), θk (k = 1, 2, 3) and θk (k = 4, 5) on the parameter n, for which the material constants are given in Table Homogenization of very rough interfaces 1085 180 160 α1 α k (109 N/m2) 140 120 100 α3 80 α2 60 40 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 n Fig The dependence of the coefficients α1 , α2 , α3 on the parameter n The material constants are given in Table 180 160 γ γk (109 N/m2) 140 120 100 γ 80 γ 60 40 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 n Fig The dependence of the coefficients γ1 , γ2 , γ3 on the parameter n The material constants are given in Table 20 θ1, θ2, θ3 (C/m 2) 18 16 θ θ 14 12 10 θ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 n Fig The dependence of the coefficients θ1 , θ2 , θ3 on the parameter n The material constants are given in Table Remark Piezoelectrics have found many applications in the area of signal processing, transduction and frequency control, where the reflection and refraction of wave energy at the interface of two different piezoelectric media play an important role (see [42]) In the past two decades, the reflection/refraction problem of waves on plane interfaces between two dissimilar piezoelectric materials received considerable attention (see [43]) A large number of investigations have been made on this topic, see for example the works [42–49] and references therein However, no study has been carried out for very rough interfaces separating two dissimilar 1086 P C Vinh, D X Tung -8 -8.5 θ4 , θ5 (10-9 C 2/(Nm2)) -9 θ5 -9.5 -10 -10.5 -11 θ4 -11.5 -12 -12.5 -13 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 n Fig The dependence of the coefficients θ4 , θ5 on the parameter n The material constants are given in Table Table Material constants for the piezoelectric solids: BaTiO3 for ci j 109 ei j ij N m2 C m2 10−9 C2 N m2 −, see [43], and fully poled Pz27 for +, see [45] c11− c33− c44− c66− c13− c11+ c33+ c44+ c66+ c13+ 166 162 43 44.5 78 147 113 23 21 94 e31− −4.4 e33− 18.6 e15− 11.6 e31+ −3.1 e33+ 16 e15+ 11.6 11− 33− 11.2 12.6 11+ 10.005 33+ 8.0926 piezoelectric solids due to the mathematical difficulties resulting from the very rough interfaces With the explicit homogenized equations in component form and the associated continuity conditions, the reflection and transmission of waves on a very rough interface are simplified to the wave reflection and transmission on a layer with plane boundaries The mathematical difficulty is therefore reduced considerably The explicit homogenized equations in component form will also be employed conveniently in other practical problems Conclusions In this paper, we consider the homogenization of the generalized plane strain problem of the linear theory of piezoelectricity in two-dimensional domains with a very rough interface that rapidly oscillates between two parallel straight lines Following the same procedure as was carried out in [32–34], the explicit homogenized equation and associate continuity conditions are derived They are written down in component form for two specific cases when the solids are made of orthotropic-(222) crystals and hexagonal crystals of class 6mm Since the obtained homogenized equations are totally explicit, they are very useful in use Acknowledgments The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no.107.02-2012.12 References Zaki, K.A., Neureuther, A.R.: Scattering from a perfectly con- ducting surface with a sinusoidal height profile: TE polarization IEEE Trans Antennas Propag 19, 208–214 (1971) Waterman, P.C.: Scattering by periodic surfaces J Acoust Soc Am 57, 791–802 (1975) Homogenization of very rough interfaces 1087 Belyaev, 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