Rayleigh waves in compressible orthotropic half-space overlaid by a thin un-coaxial orthotropic layer

Nội dung text: Rayleigh waves in compressible orthotropic half-space overlaid by a thin un-coaxial orthotropic layer

1. Journal of Science and Technology in Civil Engineering, HUCE (NUCE), 2021, 15 (4): 54–64 RAYLEIGH WAVES IN COMPRESSIBLE ORTHOTROPIC HALF-SPACE OVERLAID BY A THIN UN-COAXIAL ORTHOTROPIC LAYER Trinh Thi Thanh Huea,∗, Phan Thi Thu Phuonga, Pham Hong Anha aFaculty of Faculty of Building and Industrial Construction, Hanoi University of Civil Engineering, 55 Giai Phong road, Hai Ba Trung district, Hanoi, Vietnam Article history: Received 09/08/2021, Revised 30/08/2021, Accepted 06/09/2021 Abstract The problem of Rayleigh waves in compressible orthotropic elastic half-space overlaid by a thin elastic layer of which principal material axes are coincident have been researched by many scientists. However, the problem with the conditions that the half-space and the layer have only one common principal material axis that per- pendicular to the layer while the remains are not identical has not gotten enough attention. This paper presents a traditional approach to obtain an approximate secular equation by approximately replacing the thin layer by effective boundary conditions of third-order. The wave then is considered as a Rayleigh wave propagating in an orthotropic half-space, without coating, subjected to the effective boundary conditions. This explicit approxi- mate secular equation is potentially useful in non-damage assessment studies. Keywords: orthotropic; monoclinic; principal material axis; thin layer; effective boundary condition; traditional approach. © 2021 Hanoi University of Civil Engineering (HUCE) 1. Introduction Nowadays an elastic half-space overlaid by an elastic layer is a structure which has a lot of prac- tical applications such as in acoustics, materials science, seismology, geophysics and microelectro- mechanical systems (MEMS). The measurement of mechanical properties of the layer deposited on the half-space before and during using is very important and necessary see [1] and references therein. Among various measurement methods, the surface wave method is used most widely [2], and for which the Rayleigh wave is a versatile and convenient tool. By applying the Rayleigh wave tool, the explicit secular equation of Rayleigh waves are used as theoretical bases for taking out the mechanical properties of the thin layer for experimental data. Therefore, the main idea of the investigations of Rayleigh waves propagating in half-space coated by a thin layer is finding them. Taking the assumption of a thin layer, the explicit secular equation is derived by replacing approximately the full effect of the thin layer on the half-space by the so-called effective boundary conditions which relate the displacements with the stresses of the half-space at its surface. To obtain the effective boundary conditions, we have ways either by replacing approximately the layer by a plate, see [3,4], or by expanding the stresses at the upper surface of the layer into ∗Corresponding author. E-mail address: huettt@nuce.edu.vn (Hue, T. T. T.) 54
2. Hue, T. T. T., et al. / Journal of Science and Technology in Civil Engineering Taylor series of the thickness of the layer, see [5–10]. Before the year 2010, there are some studies on Rayleigh waves in an elastic half-space coated by a thin layer such as Achenbach et al. [3], Bovik [5], Steigmann et al. [9], Tiersten [4], Tuan [11], and Wang et al. [12]. However, these studies are limited to isotropic materials. After the year 2010, Vinh et al. had many research into the structures of half-space overlaid by a thin layer with less symmetry substrates such as orthotropic [13–15] and monoclinic substrates [16, 17]. In all investigations mentioned above, the principal material axes of the half-space and of the thin layer were assumed to be coincide. Therefore, the main purpose of this paper is to present an explicit approximate secular equation of Rayleigh waves in a compressible orthotropic elastic half- space coated by a thin un-coaxial orthotropic layer. Specifically, the half-space and the thin layer have only one common principal material axis that perpendicular to the layer, the remains of the half-space are inclined at an angle θ to the remains of the layer. Here, the thin layer is assumed to be perfectly bonded to the half-space. From the relationship between principal material axes of the half- space and of the thin layer, we first transfer the original problem into the problem of Rayleigh waves propagation in a compressible orthotropic elastic half-space coated by a thin coaxial monoclinic layer with the symmetrical plane x3 = 0. After that, the thin layer is approximately replaced by effective boundary conditions of third order. Then, the wave is considered as a Rayleigh wave propagating in the half-space, without the coated layer, subjected to the effective boundary conditions. By employing the traditional approach, the approximate secular equation is derived. This equation is then simplified for the case when the principal material axes of the half-space and of the layer were coincide. It is shown that this result coincides with the one previously obtained. 2. Formulation of problem Consider an orthotropic half-space coated with a thin elastic orthotropic layer. Although the thin elastic layer is also orthotropic but its principal material axes do not coincide with those of the half- space. Suppose the principal material axes of the half-space are denoted by Ox1, Ox2, Ox3 and the principal material axes of the layer are denoted by OX1, OX2, OX3. However, the axes Ox1, Ox2 of the half-space are inclined at an angle θ to the axes OX1, OX2 of the layer and Ox3 ≡ OX3. We have    cos θ − sin θ 0    xi = Ωi jX j, Ωi j =  sin θ cos θ 0  (1) Journal of Science and Technology in Civil Engineering NUCE 2021  ISSN 1859-2996  0 0 1  orthotropic: O monoclinic O O orthotropic: orthotropic: FigureFigure 1. 1. Model Model of of Problem Problem 72 at an angle to the axes of the layer and .∗ We have In the layer, the material constantsOX1, OX in 2 the coordinates OXOx1X32X3 OXare 3c¯i j and the material constants in the coordinates Ox1 x2 x3 are c¯i j. In the new coordinates Ox1 x2 x3, the orthotropic layer becomes the cos sin  0 55 73 (1) xi  ij X j,  ij sin cos  0 0 0 1 74 In the layer, the material constants in the coordinates are and the material OX1 X 2 X 3 cij 75 constants in the coordinates are . In the new coordinates , the Ox1 x 2 x 3 cij Ox1 x 2 x 3 76 orthotropic half-space becomes the layer made of a monoclinic material with the 77 symmetrical plane . However, in the paper, we consider the plane strain x3 0 78 problem. Hence, we only use the following material constants are given by (see Ting 79 [18]) 4 2 2 4 c11 c 11cos 2 c 12 2 c 66 cos  sin  c 22 sin  4 2 2 4 c22 c 22cos 2 c 12 2 c 66 cos  sin  c 11 sin  c c c c 2 c 4 c cos2 sin 2  80 12 12 11 22 12 66 (2) c c cos2 c sin 2  c 2 c cos 2  sin 2  cos  sin  16 11 22 12 66 c c cos2 c sin 2  c 2 c cos 2  sin 2  cos  sin  26 22 11 12 66 2 2 c66 c 66 c 11 c 22 2 c 12 4 c 66 cos sin  81 So, the initial problem consider the propagation of a Rayleigh wave in an 82 orthotropic elastic half-space coated by a thin un-coaxial orthotropic layer. Now, we 83 consider the propagation of a Rayleigh wave in an orthotropic elastic half-space coated 84 by a thin coaxial monoclinic elastic layer with the symmetrical plane . x3 0 85 3. Effective boundary conditions 86 2.1. Basic equations of the elastic layer in matrix form 87 Consider an elastic half-space overlaid by a thin elastic layer x2 0 3
3. Hue, T. T. T., et al. / Journal of Science and Technology in Civil Engineering layer made of a monoclinic material with the symmetrical plane x3 = 0. However, in the paper, we consider the plane strain problem. Hence, we only use the following material constants are given by (see Ting [18]) ∗ 4  ∗ ∗  2 2 ∗ 4 c¯11 = c¯11cos θ + 2 c¯12 + 2¯c66 cos θsin θ + c¯22sin θ ∗ 4  ∗ ∗  2 2 ∗ 4 c¯22 = c¯22cos θ + 2 c¯12 + 2¯c66 cos θsin θ + c¯11sin θ ∗  ∗ ∗ ∗ ∗  2 2 c¯12 = c¯12 + c¯11 + c¯22 − 2¯c12 − 4¯c66 cos θsin θ (2) h ∗ 2 ∗ 2  ∗ ∗   2 2 i c¯16 = − c¯11cos θ − c¯22sin θ − c¯12 + 2¯c66 cos θ − sin θ cos θ sin θ h ∗ 2 ∗ 2  ∗ ∗   2 2 i c¯26 = c¯22cos θ − c¯11sin θ − c¯12 + 2¯c66 cos θ − sin θ cos θ sin θ ∗  ∗ ∗ ∗ ∗  2 2 c¯66 = c¯66 + c¯11 + c¯22 − 2¯c12 − 4¯c66 cos θsin θ So, the initial problem consider the propagation of a Rayleigh wave in an orthotropic elastic half- space coated by a thin un-coaxial orthotropic layer. Now, we consider the propagation of a Rayleigh wave in an orthotropic elastic half-space coated by a thin coaxial monoclinic elastic layer with the symmetrical plane x3 = 0. 3. Effective boundary conditions 3.1. Basic equations of the elastic layer in matrix form Consider an elastic half-space x2 ≥ 0 overlaid by a thin elastic layer −h ≤ x2 ≤ 0. An elastic half-space is made of a orthotropic material and a thin elastic is made of a monoclinic material with the symmetrical plane x3 = 0. For such materials, in-plane motions are decoupled from anti-plane motions, therefore we can consider the plane strain such that: u j = u j (x1, x2, t) , u¯ j = u¯ j (x1, x2, t) , j = 1, 2, u3 ≡ u¯3 ≡ 0 (3) where u j and u¯ j are the displacement components of the half-space and of the thin layer, t is the time. Suppose that the layer is made of compressible monoclinic elastic material with the symmetrical plane x3 = 0. Then the strain-stress relation are
   σ¯ 11 = c¯11u¯1,1 + c¯12u¯2,2 + c¯16 u¯1,2 + u¯2,1
   σ¯ 22 = c¯12u¯1,1 + c¯22u¯2,2 + c¯26 u¯1,2 + u¯2,1 (4)
   σ¯ 12 = c¯16u¯1,1 + c¯26u¯2,2 + c¯66 u¯1,2 + u¯2,1 where σ¯ i j are the stress components of the layer, c¯i j are the material constants of the layer and are defined by (2). Here, commas denote differentiation with respect to spatial variables x j. Equations of motion without body forces are σ¯ 11,1 + σ¯ 12,2 = ρ¯u¯¨1 (5) σ¯ 12,1 + σ¯ 22,2 = ρ¯u¯¨2 with ρ¯ is the mass density of the layer and a dot indicates differentiation with respect to t. Solving Eqs. (4)2 and (4)3 for u¯1,2 and u¯2,2, we have u¯1,2 = n¯66σ¯ 12 + n¯26σ¯ 22 − r¯6u¯1,1 − u¯2,1 (6) u¯2,2 = n¯26σ¯ 12 + n¯22σ¯ 22 − r¯2u¯1,1 56
4. Hue, T. T. T., et al. / Journal of Science and Technology in Civil Engineering where c¯22 c¯26 c¯66 c¯12c¯26 − c¯22c¯16 n¯66 = , n¯26 = − , n¯22 = , r¯6 = − ∆ ∆ ∆ ∆ (7) c¯ c¯ − c¯ c¯ r¯ = 12 66 16 26 , ∆ = c¯ c¯ − c¯2 2 ∆ 22 66 26 Substituting (4)1 into (5)1 and taking into account (6) yields σ¯ 12,2 = ρ¯u¯¨1 − η¯u¯1,11 − r¯6σ¯ 12,1 − r¯2σ¯ 22,1 (8) in which η¯ = c¯11 − r¯6c¯16 − r¯2c¯12 (9) From the second of Eq. (5) it follows σ¯ 22,2 = ρ¯u¯¨2 − σ¯ 12,1 (10) Eqs. (6), (8) and (10) take the matrix form as follows ς¯0 = M¯ ς¯ (11) where " # " # " # " # U¯ u¯ σ¯ M¯ M¯ ζ¯ = , U¯ = 1 , Σ¯ = 12 , M¯ = 1 2 (12) Σ¯ u¯2 σ¯ 22 M¯ 3 M¯ 4 in which the matrices M¯ k are given by " # " # −r¯6∂1 −∂1 n¯66 n¯26 M¯ 1 = , M¯ 2 = −r¯2∂1 0 n¯26 n¯22 (13) " 2 − 2 # ¯ ρ∂¯ t η∂¯ 1 0 ¯ ¯ T M3 = 2 , M4 = M1 0ρ∂ ¯ t with the symbol T signifies transpose of matrix, the prime means derivative with respect to x2. Here, 2 2 2 2 2 2 we use the notations ∂1 = ∂/∂x1, ∂1 = ∂ /∂x1, ∂t = ∂ /∂t . Eq. (11) is called the matrix form of the plane strain for compressible monoclinic elastic solids with the symmetrical plane x3 = 0. From (11) and (12) it follows " # " # U¯ (n) U¯ = M¯ n , n = 1, 2, 3, , x ∈ [−h, 0] (14) Σ¯ (n) Σ¯ 2 3.2. Effective boundary conditions of third-order Since the layer is thin, h is small. Expanding Σ¯ (−h) into Taylor series about x2 = 0 up to the third-order of −h leads to h2 h3 Σ¯ (−h) = Σ¯ (0) − hΣ¯ 0 (0) + Σ¯ 00 (0) − Σ¯ 000 (0) (15) 2 6 Suppose that the surface x2 = −h is traction-free, we have Σ¯ (−h) = 0. Using Eq. (14) at x2 = 0 into Eq. (15) yields 57
5. Hue, T. T. T., et al. / Journal of Science and Technology in Civil Engineering ( ) h2   h3 h    i I − hM¯ + M¯ M¯ + M¯ 2 − M¯ M¯ + M¯ M¯ M¯ + M¯ M¯ + M¯ 2 M¯ Σ¯ (0) 4 2 3 2 4 6 3 1 4 3 2 3 2 4 4 ( ) h2   h3 h    i + −hM¯ + M¯ M¯ + M¯ M¯ − M¯ M¯ + M¯ M¯ M¯ + M¯ M¯ + M¯ 2 M¯ U¯ (0) = 0 3 2 3 1 4 3 6 3 1 4 3 1 3 2 4 3 (16) Since the thin layer is bonded perfectly to the half-space at the plane x2 = 0, we derive U (0) = U¯ (0) and Σ (0) = Σ¯ (0). Thus, from (16) we have ( ) h2   h3 h    i I − hM¯ + M¯ M¯ + M¯ 2 − M¯ M¯ + M¯ M¯ M¯ + M¯ M¯ + M¯ 2 M¯ Σ (0) 4 2 3 2 4 6 3 1 4 3 2 3 2 4 4 ( ) h2   h3 h    i + −hM¯ + M¯ M¯ + M¯ M¯ − M¯ M¯ + M¯ M¯ M¯ + M¯ M¯ + M¯ 2 M¯ U (0) = 0 3 2 3 1 4 3 6 3 1 4 3 1 3 2 4 3 (17) Eq. (17) expresses the relation between the traction vector and the displacement vector of the half-space at the plane x2 = 0. It is called the effective boundary condition of third-order in the matrix form. It replaces approximately the all influence of the thin layer on the half-space. Note that, when the principal material axes of the layer coincide with those of the half-space i.e. θ = 0, M¯ k (k = 1, 2, 3, 4) in Eq. (17) are equivalent to Mk (k = 1, 2, 3, 4) in Eq. (9) of Ref. [13]. Substituting (13) into (17) yields the effective boundary conditions of third-order in the compo- nent form, namely:
   σ12 + h r¯6σ12,1 + r¯2σ22,1 + η¯u1,11 − ρ¯u¨1 h2 h   + ρ¯n¯ σ¨ + r¯2 + r¯ − η¯n¯ σ + ρ¯n¯ σ¨ + (r¯ r¯ − η¯n¯ ) σ 2 66 12 6 2 66 12,11 26 22 2 6 26 22,11 h3 −2¯ρr¯ u¨ + 2¯ηr¯ u − ρ¯ (1 + r¯ ) u¨ + η¯u  + ρ¯ [n¯ (2 + r¯ ) + 3¯n r¯ ] σ¨ 6 1,1 6 1,111 2 2,1 2,111 6 26 2 66 6 12,1  3  + r¯6 + 2¯r2r¯6 − 2¯ηn¯26 − 3¯ηn¯66r¯6 σ12,111 + ρ¯ [n¯22 + (n¯22 + n¯66) r¯2 + 2¯n26r¯6] σ¨ 22,1  2 2  2 h 2 i + r¯2 + r¯2r¯6 − η¯n¯22 − η¯n¯66r¯2 − 2¯ηn¯26r¯6 σ22,111 − ρ n¯66u¨1,tt − ρ¯ 3¯r6 + (2 + r¯2) r¯2 − 2¯ηn¯66 u¨1,11 h 2 i 2   o + η¯ 3¯r6 + 2¯r2 − η¯n¯66 u1,1111 −ρ n¯26u¨2,tt − ρ¯ (2 + r¯2) r¯6 − η¯n¯26 u¨2,11 + 2¯ηr¯6u2,1111 = 0 at x2 = 0 (18) h2 σ + h σ − ρ¯u¨
   + ρ¯n¯ σ¨ + r¯ σ + ρ¯n¯ σ¨ + r¯ σ − ρ¯ (1 + r¯ ) u¨ + η¯u  22 12,1 2 2 26 12 6 12,11 22 22 2 22,11 2 1,1 1,111 h3 n   + ρ¯ (n¯ + n¯ + n¯ r¯ + n¯ r¯ ) σ¨ + r¯2 + r¯ − η¯n¯ σ + ρ¯n¯ (1 + 2¯r ) σ¨ 6 22 66 66 2 26 6 12,1 6 2 66 12,111 26 2 22,1 2   + (r¯2r¯6 − η¯n¯26) σ22,111 − ρ n¯26u¨1,tt + ρ¯ η¯n¯26 − (2 + r¯2) r¯6 u¨1,11 + 2¯ηr¯6u1,1111 2 o −ρ n¯22u¨2,tt − ρ¯ (1 + 2¯r2) u¨2,11 + η¯u2,1111 = 0 at x2 = 0 (19) 58
6. Hue, T. T. T., et al. / Journal of Science and Technology in Civil Engineering 4. An approximate third-order secular equation of Rayleigh waves For the compressible orthotropic half-space, the strain-stress relations are σ11 = c11u1,1 + c12u2,2 σ22 = c12u1,1 + c22u2,2 (20)
   σ12 = c66 u1,2 + u2,1 with σi j and ci j are respectively the stress components and the material constants of half-space. For the strain energy of the material to be positive definite, the material constants c11, c22, c12, c66 satisfy 2 cii > 0, i = 1, 2, 6, c11c22 − c12 > 0 (21) When the body forces are vanished, equations of motion have the following form σ11,1 + σ12,2 = ρu¨1 (22) σ12,1 + σ22,2 = ρu¨2 where ρ is the mass density of the half-space. Introducing (20) into (22) leads to c11u1,11 + c66u1,22 + (c12 + c66) u2,12 = ρu¨1 (23) c66u2,11 + c22u2,22 + (c12 + c66) u1,12 = ρu¨2 At x2 = +∞ the decay conditions are U (+∞) = Σ (+∞) = 0 (24) T T So, the unknown vectors U = [u1 u2] , Σ = [σ12 σ22] must satisfy Eq. (23), the effective boundary conditions (18), (19) and the decay condition (24). Now we consider the propagation of Rayleigh wave, travelling with velocity c (> 0) and wave number k (> 0) in the x1-direction and decaying in the x2-direction. According to Vinh and Ogden [19] the displacement components of this Rayleigh wave which satisfy Eq. (23) and the decay condition (24) are given by   −kb1 x2 −kb2 x2 ik(x1−ct) u1 = B1e + B2e e   (25) −kb1 x2 −kb2 x2 ik(x1−ct) u2 = α1B1e + α2B2e e where B1, B2 are constants to be determined from the effective boundary conditions (18) and (19), b1, b2 are roots of the bellow equation 4 h 2 i 2 c22c66b + (c12 + c66) + c22 (X − c11) + c66 (X − c66) b + (c11 − X)(c66 − X) = 0 (26) whose real parts are positive to ensure the decay condition, X = ρc2 and √ αn = iβn, n = 1, 2, i = −1 b (c + c ) c − X − c b2 (27) β = n 12 66 = 11 66 n n 2 c22bn − c66 + X (c12 + c66) bn From Eq. (26) we have (c + c )2 + c (X − c ) + c (X − c ) b2 + b2 = − 12 66 22 11 66 66 := S 1 2 c c 22 66 (28) 2 2 (c11 − X)(c66 − X) b1b2 = := P c22c66 59
7. Hue, T. T. T., et al. / Journal of Science and Technology in Civil Engineering For a Rayleigh wave exists, b1, b2 have positive real parts. We deduce 0 < X < min {c11, c66} (29) and q √ √ b1 + b2 = S + 2 P, b1b2 = P (30) Substituting (25) into (20)2,3 yields n o −kb1 x2 −kb2 x2 ik(x1−ct) σ12 = −kc66 (b1 + β1) B1e + (b2 + β2) B2e e n o (31) −kb1 x2 −kb2 x2 ik(x1−ct) σ22 = ikc66 (c12 − c22b1β1) B1e + (c12 − c22b2β2) B2e e Introducing Eqs. (25) and (31) into the effective boundary conditions (18) and (19) gives to two equations for B1, B2 f (b1) B1 + f (b2) B2 = 0 (32) F (b1) B1 + F (b2) B2 = 0 where ε2 ε3 f (bn) = f0 (bn) + f1 (bn) ε + f2 (bn) + f3 (bn) 2 6 (33) ε2 ε3 F (b ) = F (b ) + F (b ) ε + F (b ) + F (b ) n 0 n 1 n 2 n 2 3 n 6 with f0 (bn) = −c66 (bn + βn) (34) f1 (bn) = X¯ − η¯ − r¯2 (c12 − c22bnβn) − ir¯6c66 (bn + βn) (35) h   i   ¯ 2 ¯ ¯ f2 (bn) = n¯66 X − η¯ + r¯2 + r¯6 c66 (bn + βn) − X − η¯ + r¯2X βn h   i   (36) − i n¯26 X¯ − η¯ + r¯2r¯6 (c12 − c22bnβn) + 2ir¯6 X¯ − η¯   h  i ¯ 2 ¯ 2 ¯ f3 (bn) = − X − η¯ 3¯r6 + 2¯r2 + n¯66 X − η¯ − r¯2X h   i ¯ 2 ¯ + X − η¯ (2¯r6n¯26 + r¯2n¯66 + n¯22) + r¯2 r¯6 + r¯2 + r¯2n¯22X (c12 − c22bnβn) h    i (37) − i X¯ − η¯ n¯26X¯ + 2¯r6 + r¯2r¯6X¯ βn h   i 3 ¯ ¯ + i r¯6 + 2¯r2r¯6 + r¯2n¯26X + X − η¯ (3¯r6n¯66 + 2¯n26) c66 (bn + βn) F0 (bn) = c12 − c22bnβn (38) h i F1 (bn) = − c66 (bn + βn) − X¯βn (39)   F2 (bn) = − r¯2 + n¯22X¯ (c12 − c22bnβn) + X¯ − η¯ + r¯2X¯   (40) − i r¯6 + n¯26X¯ c66 (bn + βn) h   i 2 ¯ ¯ F3 (bn) = r¯6 + r¯2 + n¯66 X − η¯ + (r¯6n¯26 + r¯2n¯66 + n¯22) X c66 (bn + βn) h   i h    i − X¯ − η¯ + 2¯r2 + n¯22X¯ X¯ βn + i X¯ − η¯ n¯26X¯ + 2¯r6 + r¯2r¯6X¯ (41) h    i − i n¯26 X¯ − η¯ + r¯2 r¯6 + n¯26X¯ (c12 − c22bnβn) and ε = kh is the dimensionless thickness of the layer, X¯ = ρ¯c2. 60
8. Hue, T. T. T., et al. / Journal of Science and Technology in Civil Engineering 2 2 Due to B1 + B2 , 0, the determinant of system (32) must be vanish. This fact yields f (b1) F (b2) − f (b2) F (b1) = 0 (42) Using (33)–(41) into (42) and taking into account (27), the approximate third-order secular equa- tion in ε of Rayleigh waves is given as follows A A   A + A ε + 2 ε2 + 3 ε3 + O ε4 = 0 (43) 0 1 2 6 where A0 = f0 (b1) F0 (b2) − f0 (b2) F0 (b1)   (44) = c66 (c12 + c22β1β2)(b2 − b1) + (c12 + c22b1b2)(β2 − β1) A1 = f0 (b1) F1 (b2) − f0 (b2) F1 (b1) + f1 (b1) F0 (b2) − f1 (b2) F0 (b1)   (45) = Xc¯ 66 (b2β1 − b1β2) − X¯ − η¯ c22 (b2β2 − b1β1) + ir¯6A0 A2 = f0 (b1) F2 (b2) − f0 (b2) F2 (b1) + f2 (b1) F0 (b2) − f2 (b2) F0 (b1) + 2 ( f1 (b1) F1 (b2) − f1 (b2) F1 (b1)) h  i   ¯ 2 ¯ ¯ ¯ = − n¯22X + r¯6 + n¯66 X − η¯ A0 + 2X X − η¯ (β2 − β1) (46) h  i   + r¯2X¯ − X¯ − η¯ (c66 − c12)(β2 − β1) + (c66 − c22β1β2)(b2 − b1)   + 2ir¯6Xc¯ 66 (b2β1 − b1β2) − 2ir¯6 X¯ − η¯ c22 (b2β2 − b1β1) A3 = f0 (b1) F3 (b2) − f0 (b2) F3 (b1) + f3 (b1) F0 (b2) − f3 (b2) F0 (b1) + 3 ( f1 (b1) F2 (b2) − f1 (b2) F2 (b1)) + 3 ( f2 (b1) F1 (b2) − f2 (b2) F1 (b1)) h      i ¯ ¯ ¯ 2 ¯ = − 3¯n66X − 2 X − η¯ + n¯22X + 2¯r2 + 3¯r6 X (b2β1 − b1β2) n  h   i o ¯ ¯ ¯ 2 2 ¯ + X − η¯ 3¯n22X + n¯66 X − η¯ + 2¯r2 + 3¯r6 − 2¯r2X c22 (b2β2 − b1β1) (47) h   i   ¯ ¯ ¯ 3 ¯ ¯ − i 3¯n66r¯6 X − η¯ + 3¯n22r¯6X − n¯26r¯2X + r¯6 A0 + 6ir¯6X X − η¯ (β2 − β1) h    i   − 2i X¯ − η¯ n¯26X¯ − r¯6 + r¯2r¯6X¯ c12 (β2 − β1) + c22β1β2 (b2 − b1) h    i + 4i 2¯r2r¯6X¯ − X¯ − η¯ n¯26X¯ + 2¯r6 c66 (b2 − b1 + β2 − β1) By Eq. (27) we deduce c11 − X + c66b1b2 c11 − X β2 − β1 = − (b2 − b1) , β1β2 = (c12 + c66) b1b2 c22b1b2 c66 (b1 + b2) b2β2 − b1β1 = − (b2 − b1) (48) c12 + c66 (c11 − X)(b1 + b2) β2b1 − β1b2 = − (b2 − b1) (c12 + c66) b1b2 Introducing (48) into Eqs. (44)–(47) yields c66 (b2 − b1) Ak = γA¯k (k = 0, 1, 2, 3) , γ = (49) (c12 + c66) b1b2 where h  i ¯ 2 A0 = c12 − c11c22 + c22X b1b2 + X (c11 − X) (50) h   i A¯1 = X¯ (c11 − X) + X¯ − η¯ c22b1b2 (b1 + b2) + ir¯6A¯0 (51) 61
9. Hue, T. T. T., et al. / Journal of Science and Technology in Civil Engineering h  i ¯ ¯ 2 ¯ ¯ A2 = − n¯22X + r¯6 + n¯66 X − η¯ A0 h    i + 2 c12 r¯2X¯ − X¯ + η¯ − X¯ X¯ − η¯ b1b2 " #     X¯ (52) − 2 r¯2X¯ − X¯ − η¯ + X¯ − η¯ (c11 − X) c66 h   i + 2ir¯6 X¯ (c11 − X) + X¯ − η¯ c22b1b2 (b1 + b2) nh    i ¯ ¯ ¯ ¯ 2 2 ¯ A3 = − X − η¯ 3¯n22X + n¯66X − n¯66η¯ + 2¯r2 + 3¯r6 − 2¯r2X c22b1b2 h      i o ¯ ¯ ¯ 2 ¯ + 3¯n66X − 2 X − η¯ + n¯22X + 2¯r2 + 3¯r6 X (c11 − X) (b1 + b2) h   i − i 3¯n r¯ X¯ − η¯ + 3¯n r¯ X¯ − n¯ r¯ X¯ + r¯3 A¯ 66 6 22 6 26 2 6 0 (53) h     i − i 2 X¯ − η¯ n¯26X¯ + 5¯r6 − 10¯r2r¯6X¯ [c12b1b2 − (c11 − X)] !   c11 − X − 6ir¯6X¯ X¯ − η¯ + b1b2 c66 in which b1b2 and b1 + b2 are defined by (28) and (30). Removing the factor γ, Eq. (43) reduces A¯ A¯   A¯ + A¯ ε + 2 ε2 + 3 ε3 + O ε4 = 0 (54) 0 1 2 6 This is the third-order approximate dispersion equation and it is entirely explicit. When the principal material axes of the layer coincide the principal material axes of the half- ∗ ∗ ∗ ∗ space, i.e. θ = 0. Then, c¯11 = c¯11, c¯22 = c¯22, c¯12 = c¯12, c¯66 = c¯66, c¯16 = c¯26 = 0. We have: 1 1 r¯6 = n¯26 = 0, n¯22 = ∗ , n¯66 = ∗ c¯22 c¯66  2 (55) ∗ c¯∗ c¯∗ − c¯∗ c¯12 11 22 12 r¯2 = ∗ := r1, η¯ = ∗ := −r3 c¯22 c¯22 r r 1 + r r r = r + 3 , r = r r + 3 , r = 1 + 1 2 1 c¯∗ 4 1 2 c¯∗ 5 c¯∗ c¯∗ 66 22 22 66 (56) 2 1 + r1 1 r6 = (r1 + r2) r3, r7 = r1 + 2r2, r8 = ∗ + ∗ c¯66 c¯22     Note that, r j j = 1, 8 in Eqs. (55) and (56) are equivalent to r j j = 1, 8 in Eq. (12) of Ref. [13]. Substituting (55) and (56) into (33) yields h i f (bn) = − c66 (bn + βn) + ε r3 + X¯ − r1 (c12 − c22bnβn) " ! # ε2 X¯   + c (b + β ) r + − β X¯ + r + Xr¯ 66 n n 2 ∗ n 3 1 (57) 2 c¯66 3 "   ¯ 2 # ε ¯ ¯ X + (c12 − c22bnβn) r4 + r5X − r6 − r7X − ∗ 6 c¯66 62
10. Hue, T. T. T., et al. / Journal of Science and Technology in Civil Engineering h i F (bn) = c12 − c22bnβn − ε c66 (bn + βn) − βnX¯ " !# ε2 X¯ + X¯ + r + Xr¯ − (c − c b β ) r + 3 1 12 22 n n 1 ∗ (58) 2 c¯22 3 "   ¯ 2 !# ε ¯ ¯ ¯ X + c66 (bn + βn) r2 + r8X − βn X + r3 + 2r1X + ∗ 6 c¯22 This result is equivalent the result which was derived by Vinh et al. [13] (see Eq. (22)). So, the approximate third-order secular equation of Rayleigh waves of this case coincide with the approximate secular equation of third-order which was obtained by Vinh et al. [13]. 5. Conclusions In this paper, the authors researched on the Rayleigh waves in the elastic half-space coated by the thin elastic layer. Here, both the half-space and the thin layer are assumed to be orthotropic and compressible. However, the principal material axes of the half-space do not coincide with those of the half-space. By transforming the coordinate system, we bring the original problem to the problem of Rayleigh waves propagation in the orthotropic elastic half-space coated by a thin coaxial monoclinic layer with the symmetrical plane x3 = 0. After that, the effective boundary conditions of third order are given that replaces the full influence of the thin layer on the half-space. For using the effective boundary conditions, an approximate dispersion equation of third order of Rayleigh waves is derived in this paper. This is a completely new result. When the principal material axes of the half-space and of the layer were coincide, the secular equation in this paper reduces to those given by Vinh et al. [13]. Because the obtained secular equation is entirely explicit, it will be useful in practical applications. Acknowledgements This study was carried out within the project supported by National University of Civil Engineer- ing, grant number: 49-2021/KHXD. References [1] Makarov, S., Chilla, E., Frohlich,¨ H.-J. (1995). Determination of elastic constants of thin films from phase velocity dispersion of different surface acoustic wave modes. Journal of Applied Physics, 78(8): 5028–5034. [2] Every, A. G. (2002). Measurement of the near-surface elastic properties of solids and thin supported films. Measurement Science and Technology, 13(5):R21. [3] Achenbach, J. D., Keshava, S. P. (1967). Free waves in a plate supported by a semi-infinite continuum. The American Society of Mechanical Engineers, 34:397–404. [4] Tiersten, H. F. (1969). Elastic surface waves guided by thin films. Journal of Applied Physics, 40(2): 770–789. [5] Bovik, P. (1996). A comparison between the Tiersten model and O(H) boundary conditions for elastic surface waves guided by thin layers. The American Society of Mechanical Engineers, 63:162–167. [6] Niklasson, A. J., Datta, S. K., Dunn, M. L. (2000). On approximating guided waves in plates with thin anisotropic coatings by means of effective boundary conditions. The Journal of the Acoustical Society of America, 108(3):924–933. [7] Rokhlin, S. I., Huang, W. (1992). Ultrasonic wave interaction with a thin anisotropic layer between two anisotropic solids: Exact and asymptotic-boundary-condition methods. The Journal of the Acoustical Society of America, 92(3):1729–1742. 63
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