Rayleigh waves with impedance boundary conditions in anisotropic solids

Rayleigh waves with impedance boundary conditions in anisotropic solids tài liệu, giáo án, bài giảng , luận văn, luận án...

Accepted Manuscript Rayleigh waves with impedance boundary conditions in anisotropic solids Pham Chi Vinh, Trinh Thi Thanh Hue PII: DOI: Reference: S0165-2125(14)00064-X http://dx.doi.org/10.1016/j.wavemoti.2014.05.002 WAMOT 1843 To appear in: Wave Motion Received date: January 2014 Accepted date: May 2014 Please cite this article as: P.C Vinh, T.T Thanh Hue, Rayleigh waves with impedance boundary conditions in anisotropic solids, Wave Motion (2014), http://dx.doi.org/10.1016/j.wavemoti.2014.05.002 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain *Manuscript Click here to view linked References Rayleigh waves with impedance boundary conditions in anisotropic solids Pham Chi Vinha ∗ and Trinh Thi Thanh Hueb a Faculty of Mathematics, Mechanics and Informatics Hanoi University of Science 334, Nguyen Trai Str., Thanh Xuan, Hanoi,Vietnam b Faculty of Civil and Industrial Construction National University of Civil Engineering 55 Giai Phong Str., Hanoi, Vietnam Abstract The paper is concerned with the propagation of Rayleigh waves in an elastic half-space with impedance boundary conditions The half-space is assumed to be orthotropic and monoclinic with the symmetry plane x3 = The main aim of the paper is to derive explicit secular equations of the wave For the orthotropic case, the secular equation is obtained by employing the traditional approach It is an irrational equation From this equation, a new version of the secular equation for isotropic materials is derived For the monoclinic case, the method of polarization vector is used for deriving the secular equation and it is an algebraic equation of eighth-order When the impedance parameters vanish, this equation coincides with the secular equation of Rayleigh waves with traction-free boundary conditions Key words: Rayleigh waves, Impedance boundary conditions, Orthotropic, Monoclinic, Explicit secular equation Corresponding author: Tel:+84-4-35532164; Fax:+84-4-38588817; E-mail address: inh@vnu.edu.vn (P C Vinh) ∗ pcv- Introduction Elastic surface waves, discovered by Rayleigh [1] more than 120 years ago for compressible isotropic elastic solids, have been studied extensively and exploited in a wide range of applications in seismology, acoustics, geophysics, telecommunications industry and materials science, for example It would not be far-fetched to say that Rayleigh’s study of surface waves upon an elastic half-space has had fundamental and far-reaching effects upon modern life and many things that we take for granted today, stretching from mobile phones through to the study of earthquakes, as addressed by Adams et al [2] A huge number of investigations have been devoted to this topic As written in [3], one of the biggest scientific search engines, Google Scholar returns more than a million links for request ”Rayleigh waves” and almost millions for ”Surface waves” This data is really amazing! It shows a tremendous scale of scientific and industrial interests in this area For Rayleigh waves their explicit secular equation are important in practical applications They can be used for solving the direct (forward) problems: evaluating the dependence of the wave velocity on material parameters, especially for solving the inverse problems: to determine material parameters from measured values of wave velocity Therefore, explicit secular equations are always the main purpose for any investigation of Rayleigh waves In the context of Rayleigh waves, it is almost always assumed that the half-spaces are free of traction As mentioned in [4], in many fields of physics such as acous2 tics and electromagnetism, it is common to use impedance boundary conditions, that is, when a linear combination of the unknown function and their derivatives is prescribed on the boundary In the other hand, when studying the propagation of Rayleigh waves in a half-space coated by a thin layer, the researchers often replace the effect of the thin layer on the half-space by the effective boundary conditions on the surface of the half-space (see, for examples, Achenbach and Keshava [5], Tiersten [6], Bovik [7], Steigmann and Ogden [8], Vinh and Linh [9, 10], Vinh and Anh [11], Vinh et al [12]) These conditions lead to the impedance-like boundary conditions on the surface As addressed in [13, 14], a thin layer on a half-space is a model finding a broad range of applications, including: the Earth’s crust in seismology, the foundation/soil interaction in geotechnical engineering, thermal barrier coatings, tissue structures in biomechanics, coated solids in material science, and micro-electro-mechanical systems Rayleigh waves with impedance boundary conditions are therefore significant in many fields of science and technology However, very few investigations on Rayleigh waves with impedance boundary conditions have been done In [15] Malischewsky considered the propagation of Rayleigh waves with Tiersten’s impedance boundary conditions and provided a secular equation Recently, Godoy et al [4] investigated the existence and uniqueness of Rayleigh waves with impedance boundary conditions which are a special case of Tiersten’s impedance boundary conditions In works [4] and [15] the half-space is assumed to be isotropic Nowadays, anisotropic materials are widely used in various fields of modern tech- nology The investigations of Rayleigh waves with impedance boundary conditions in anisotropic solids are therefore significant in practical applications The main purpose of this paper is to study the propagation of Rayleigh waves with Tiersten’s impedance boundary conditions [15] in anisotropic elastic half-spaces Two cases of anisotropy are considered: orthotropic materials and monoclinic ones with the symmetry plane x3 = (see [16]) For the orthotropic case the secular equation is obtained by employing the traditional approach It is an irrational equation and it provides a new version of the secular equation for isotropic materials For the monoclinic case, for obtaining the secular equation we use the method of polarization vector and the secular equation obtained is an algebraic equation of eighth-order When the impedance parameters vanish, this equation coincides with the secular equation of Rayleigh waves with traction-free boundary conditions Orthotropic half-spaces Consider an elastic half-space which occupies the domain x2 ≥ We are interested in the plane strain such that: ui = ui (x1 , x2 , t), i = 1, 2, u3 ≡ (1) where t is the time Suppose that the half-space is made of compressible orthotropic elastic material, then the strain-stress relations are [16]:   σ11 = c11 u1,1 + c12 u2,2 σ22 = c12 u1,1 + c22 u2,2   σ12 = c66 (u1,2 + u2,1) (2) where σij and cij are respectively the stresses and the material constants, commas indicate differentiation with respect to spatial variables xk The elastic constants c11 , c22 , c12 , c66 satisfy the inequalities: cii > 0, i = 1, 2, 6, c11 c22 − c212 > (3) which are necessary and sufficient conditions for the strain energy to be positive definite In the absence of body forces, equations of motion are: 11,1 + 12,2 = uă1 12,1 + 22,2 = uă2 (4) where is the mass density, a superposed dot signifies differentiation with respect to t Introducing (2) into (4) leads to the equations governing infinitesimal motion, expressed in terms of the displacement components, namely: c11 u1,11 + c66 u1,22 + (c12 + c66 )u2,12 = ă u1 (5) c66 u2,11 + c22 u2,22 + (c12 + c66 )u1,12 = ă u2 Now we consider the propagation of a Rayleigh wave, travelling with velocity c (> 0) and wave number k (> 0) in the x1 -direction and decaying in the x2 -direction, i e.: ui → (i = 1, 2) as x2 → +∞ (6) Suppose that the surface x2 = is subjected to impedance boundary conditions such that [4, 15]: σ12 + ωZ1 u1 = 0, σ22 + ωZ2 u2 = at x2 = (7) where ω = kc is the wave circular frequency, Z1 , Z2 (∈ R) are impedance parameters whose dimension is of stess/velocity (see [4, 15]) According to Vinh and Ogden [17], the displacement components of this Rayleigh wave which satisfy Eqs.(5) and the decay condition (6) are given by: u1 = (B1 e−kb1 x2 + B2 e−kb2 x2 )eik(x1 −ct) (8) −kb1 x2 u2 = (α1 B1 e −kb2 x2 + α2 B2 e ik(x1 −ct) )e where B1 , B2 are constants to be determined from the impedance boundary conditions (7), b1 , b2 are roots of the equation: c22 c66 b4 + {(c12 + c66 )2 + c22 (X − c11 ) + c66 (X − c66 )}b2 (9) + (c11 − X)(c66 − X) = whose real parts are positive to ensure the decay condition, X = ρc2 , and: αk = iβk , βk = √ bk (c12 + c66 ) c11 − ρc2 − c66 b2k = , k = 1, 2, i = −1 c22 bk − c66 + ρc2 (c12 + c66 )bk (10) From (9) we have: (c12 + c66 )2 + c22 (X − c11 ) + c66 (X − c66 ) + =− := S c22 c66 (c11 − X)(c66 − X) := P b21 b22 = c22 c66 b21 b22 (11) It is not difficult to verify that if a Rayleigh wave exists (→ b1 , b2 having positive real parts), then: < X < min{c66 , c11 } (12) and: b1 b2 = √ P , b1 + b2 = √ S+2 P (13) Substituting (8) into (2) yields: σ12 = −kc66 {(b1 + β1 )B1 e−kb1 x2 + (b2 + β2 )B2 e−kb2 x2 }eik(x1 −ct) σ22 = ik{(c12 − c22 b1 β1 )B1 e−kb1 x2 + (c12 − c22 b2 β2 )B2 e−kb2 x2 }eik(x1 −ct) (14) Substituting Eqs (8) and (14) into the impedance boundary conditions (7) gives: √ √ δ1 x − b1 − β1 B1 + δ1 x − b2 − β2 B2 = √ √ (15) δ2 β1 x + e3 − e2 β1 b1 B1 + δ2 β2 x + e3 − e2 β2 b2 B2 = where x = c2 /c22 , c22 = c66 /ρ, is squared dimensionless velocity of Rayleigh waves, √ δn = Zn / ρc66 (∈ R), n = 1, 2, are dimensionless impedance parameters Since B12 + B22 = 0, the determinant of the system (15) must be vanish This fact yields: (δ1 δ2 x + e3 + e2 b1 b2 )(β2 − β1 ) + (e3 + e2 β1 β2 )(b2 − b1 ) √ √ −δ1 e2 (β2 b2 − β1 b1 ) x − δ2 (β2 b1 − β1 b2 ) x = (16) e1 = c11 /c66 , e2 = c22 /c66 , e3 = c12 /c66 (17) where are dimensionless material parameters According to the inequalities (3): e1 > 0, e2 > and e1 e2 − e23 > By (10) it is not difficult to prove the following equalities: (e1 − x + b1 b2 ) (b2 − b1 ) (e3 + 1)b1 b2 (b1 + b2 ) (e1 − x) β2 b2 − β1 b1 = − (b2 − b1 ), β1 β2 = (e3 + 1) e2 b1 b2 (e1 − x)(b1 + b2 ) (b2 − b1 ) β2 b1 − β1 b2 = − (e3 + 1)b1 b2 β2 − β1 = − (18) Substituting (18) into (16) and removing the factor (b2 − b1 ) provide: √ x(e1 − x)(1 − δ1 δ2 ) + [e23 − e2 (e1 − x) − δ1 δ2 x] P √ √ √ = [δ1 e2 P + δ2 (e1 − x)] x S + P (19) in which S and P are given by: P = (e1 − x)(1 − x) e2 (e1 − x) + − x − (1 + e3 )2 , S= e2 e2 (20) Equation (19) is the (dimensionless) secular equation of Rayleigh waves propagating in an orthotropic elastic half-space whose surface is subjected to the impedance boundary conditions (7) Taking δ1 = δ2 = in Eq (19) we obtain the (dimensionless) secular equation of Rayleigh waves propagating along a traction-free surface of an orthotropic elastic half-space, namely: (e23 − e1 e2 + e2 x) (1 − x)(e1 − x)/e2 + (e1 − x)x = (21) that is equivalent to the secular equation: √ (c66 − X)[c212 − c22 (c11 − X)] + X c22 c66 (c11 − X)(c66 − X) = (22) which was derived by Chadwick [18] (see also [17, 19]) When the elastic half-space is isotropic, i e c11 =c22 =λ + 2µ, c66 = µ, c12 = λ, λ and µ are Lame constants, one can see that two roots of Eq (9) having positive real parts are: b1 = − γx, b2 = √ − x, γ = µ/(λ + 2µ) (23) and from (10) and these facts it follows: β1 = b1 , β2 = 1/b2 (24) Introducing (23) and (24) into Eq (16) and taking into account e1 = e2 =1/γ, e3 = 1/γ − we arrive at the equation: √ √ √ √ √ (x − 2)2 − − x − γx + x x(δ1 − x + δ2 − γx) √ √ +δ1 δ2 x( − x − γx − 1) = (25) that coincides with Eq (16) in [4] and (2.7) in [15] in other notations In other hand, √ √ √ with the fact S + P = − x + − γx for the isotropic elastic half-spaces, Eq (19) becomes: √ √ x(1 − γx)(1 − δ1 δ2 ) + [x + 4(γ − 1) − γδ1 δ2 x] − x − γx √ √ √ √ −(δ1 + δ2 )(1 − γx) x − x − [δ1 (1 − x) + δ2 (1 − γx)] x − γx = (26) Equation (26) is new version of the (dimensionless) secular equation of Rayleigh waves propagating in an isotropic elastic half-space subjected to the impedance boundary conditions (7) When δ2 = Eq (25) is simplified to: √ √ √ (x − 2)2 − − x − γx + δ1 x x − x = that is identical to Eq (15) in [4] in other notations (27) 3.2 Rayleigh waves Stroh’s formulation Now we consider the propagation of a Rayleigh wave, travelling with velocity c (> 0) and wave number k (> 0) in the x1 -direction and decaying in the x2 -direction Then, the displacements and stresses of the Rayleigh wave are sought in the form: un = Un (y)eik(x1 −ct) , σn2 = iktn (y)eik(x1−ct) , n = 1, 2, y = kx2 (39) Substituting (39) into Eq (36) leads to: ξ ′ = iNξ, ≤ y < +∞ (40) where the prime signifies differentiation with respect to y and: u t ξ= , u= U1 U2 , t= t1 t2 , N= N1 N2 N3 N4 (41) in which the matrices Nk are defined by: N1 = −r6 −1 −r2 , N2 = n66 n26 n26 n22 , N3 = X−η 0 X , N4 = N1T (42) X = ρc2 In addition to Eq (40) are required the decay condition: ξ(+∞) = (43) and the impedance boundary conditions (7) that, in terms of Uk and tk , is expressed as follows: t1 = iδ1 c66 XU1 , t2 = iδ2 12 c66 XU2 at y = (44) that can be written in matrix form as: √ iδ1 c66 X √0 iδ2 c66 X t = Au at y = 0, A = (45) ¯ T = −A, the bar indicates complex conjugate By the transformation: Note that A σ = t − Au (46) equation (40), the decay condition (43) and the boundary condition (45) become: w′ = iQw, ≤ y < +∞ (47) w(+∞) = 0, σ(0) = (48) and: where: w= u σ , Q= Q1 Q2 Q3 Q4 (49) in which the matrices Qk are expressed in terms of matrices Nk and A by: Q1 = N1 + N2 A, Q2 = N2 , (50) Q3 = N3 + NT1 A − AN1 − AN2A, Q4 = NT1 − AN2 From (49) and taking into account the facts: ¯ T , N3 = N ¯ T , N4 = N ¯ T, A ¯ T = −A N2 = N (51) ¯ T = Q2 , Q ¯ T = Q3 , Q4 = Q ¯T Q (52) one can show that: Equations (40) and (47) are referred to Stroh’s formulation [20] 13 3.3 Fundamental equations Proposition: If 2m-vector Y(y) is a solution of the problem: Y ′ = iPY, ≤ y < +∞, Y(+∞) = (53) where the prime signifies differentiation with respect to y and: P= P1 P2 P3 P4 (54) m × m-matrices Pk are constant matrices (being independent of y) and they satisfy the equalities: ¯ T , P3 = P ¯ T , P4 = P ¯T, P2 = P (55) ¯ T (0)ˆIPn Y(0) = ∀ n ∈ Z Y (56) then: where: ¯I = I I I is m × m identity matrix We call Eqs (56) the fundamental equations Proof: Lemma 1: Suppose the matrix P expressed by (54) is invertible: (−1) P −1 = (−1) P1 P2 (−1) (−1) P3 P4 (57) and the equalities (55) hold for the matrices Pk Then, these equalities also hold (−1) for the matrices Pk 14 Proof: From PP−1 = I it follows: (−1) (−1) (−1) (−1) P1 P1 + P2 P3 = I, P1 P2 + P2 P4 = (−1) (−1) (−1) (−1) P3 P1 + P4 P3 = 0, P3 P2 + P4 P4 = I (58) Taking transpose and complex conjugate two sides of the equalities (58) and using (55) yield:  P(−1) T P + P(−1) T P = I, P(−1) T P + P(−1) T P = 4 T T T (−1) (−1) (−1)  (−1) T P1 + P1 P3 = 0, P4 P1 + P2 P3 = I P3 (59) equivalently:   That means: (−1) P4 (−1) T T P3 (−1) P2 (−1) T T P1  P−1 =   P1 P2 P3 P4  (−1) P4 (−1) P3 T (−1) P2 T (−1) P1 (−1) (−1) = P2 T (−1) , P3 (−1) = P3 T (60)  T T From (57), (61) and the uniqueness of P−1 it follows: P2 =I  (−1) , P4 (61) (−1) T = P1 (62) The proof is completed Lemma 2: Suppose the matrix P expressed by (54) is invertible and the equalities (55) hold for the matrices Pk For all n ∈ Z the matrix Pn is expressed as: (n) Pn = (n) P1 P2 (n) (n) P3 P4 , n = 0, P0 = I (63) (n) Then, the equalities (55) also hold for the matrices Pk (0) (1) Proof: + Clearly, the equalities (55) hold for matrices Pk and Pk 15 (n) + Assume the equalities (55) hold for the matrices Pk , n > Then, it is not (n+1) difficult to show that these equalities are satisfied for the matrices Pk That (n) means the equalities (55) hold for Pk for all n ∈ Z, n ≥ (n) for all n ∈ Z, n ≤ The + By the lemma 1, the equalities (55) hold for Pk proof of the lemma is finished Lemma 3: Suppose the matrix P expressed by (54) is invertible and the matrices Pk satisfy the equalities (55) Then we have: T ˆIPn = ˆIPn , ∀ n ∈ Z (64) (n) Proof: By the lemma 2, Pk satisfy the equalities (55) for all n ∈ Z With this fact one can see that: ˆIPn = (n) P3 (n) P1 (n) P4 (n) P2 Proof of the proposition: T  → ˆIPn =  (n) P3 (n) P4 T T (n) P1 (n) P2 T T   = ˆIPn ¯ T ˆIPn we have: Pre-multiplying two sides of the equation (53)1 by Y ¯ T ˆI Pn Y ′ = i Y ¯ T ˆI Pn+1 Y Y (65) Taking transpose and complex conjugate two sides of Eq (65) and using (64) yield: ¯ ′)T ˆI Pn Y = −i Y ¯ T ˆI Pn+1 Y (Y From (65) and (66) it follows: d ¯ Tˆ n ¯ T ˆI Pn Y = C ∀ y ∈ [0 + ∞] Y IP Y = → Y dy 16 (66) where C is a constant Due to the second of (53) the constant C must be zero Therefore we have: ¯ T (y)ˆIPn Y(y) = ∀ n ∈ Z, ∀ y ∈ [0 + ∞) Y (67) Taking y = in Eq (67) we arrive at the fundamental equations (56) The proposition is proved Remark 1: i) Equations (56) recover the fundamental equations (15) in Ref [21] when P is a real matrix ii) There are at most (2m − 1) independent fundamental equations according to the CayleyHamilton theorem 3.4 Explicit secular equations Now in Eqs (56) we take P = Q that is given by (49), (50), and Y = w According to the second of (48): σ(0) = 0, Eqs (56) are therefore simplified to: ¯ T (0)Q(n) u u(0) = ∀ n ∈ Z (n) In what follows, the elements of the matrix Q3 (68) (n) are denoted by Qij (i, j = 1, 2), without the index 3, for simplicity As Q is a 4×4-matrix, according to Remark 1, ii), it is sufficiently to take three different values of n for deriving the secular equation of the wave It seems that the choice n = −1, 1, is the best choice Suppose U1 (0) = 0, then the vector u(0) can be written as: u(0)=U1 (0)[1 α]T , where α = U2 (0)/U1 (0) 17 is a complex number, α = a + ib, a, b are real Introducing the expression of u(0) (n) into (68) and taking into account the fact that Q3 is hermitian (see (52)) we have: (n) [1 α] ¯ (n) Q11 Q12 (n) Q12 (n) Q22 α = 0, n = −1, 1, (69) that provides three equations:  (−1) (−1) (−1) (−1)  ¯ + Q22 αα ¯=0  Q11 + Q12 α + Q12 α (1) (n) (1) (1) (n) The elements Qij of the matrices Q3 (n = −1, 1, 2) are given by: = (1 + c66 δ12 n66 )X − η, Q22 = (1 + c66 δ22 n22 )X, (1) = c66 δ1 δ2 n26 X + i c66 X(δ1 − δ2 r2 ), Q12 (1) (71) (2) = −2[c66 δ12 (n26 + n66 r6 )X + r6 (X − η)], Q22 = −2c66 δ22 n26 r2 X, (2) = η − c66 δ1 δ2 (n22 + n66 r2 + n26 r6 )X − (1 + r2 )X − i Q11 Q12 (−1) (70) (1) Q11 Qij (1) ¯ + Q22 αα ¯=0 Q11 + Q12 α + Q12 α    (2) (2) (2) (2) ¯ + Q22 αα ¯=0 Q11 + Q12 α + Q12 α (2) c66 Xδ2 ηn26 , (72) ˆ (−1) /q where q ∈ R is the determinant of the matrix Q(1) and =Q ij ˆ (−1) = X[(X − η)n22 − r ] − c66 δ (n22 + n2 X − n22 n66 X)X, Q 11 26 ˆ (−1) Q = η[1 + c66 δ22 (n226 − n22 n66 )X − n66 X] + X(n66 X − − r62 ), 22 ˆ (−1) = c66 δ1 δ2 (n22 r6 − n26 r2 )X + X[(η − X)n26 + r2 r6 ] Q 12 + i (n) As the matrix Q3 c66 X{δ1 (r2 − n66 r2 X + n26 r6 X) + δ2 [(X − η)n22 − r22 ]} (73) (n) (n) ˆ (−1) , Q ˆ (−1) are real and is hermitian, Q11 , Q22 (n = 1, 2), Q 11 22 (n) (n) ˆ (−1) , Q ˆ (−1) are complex numbers whose real and imaginary Q12 , Q12 (n = 1, 2), Q 12 12 18 (n,r) parts are denoted, respectively, by Q12 (n) (n,r) Substituting α = a + ib, Q12 = Q12 into Eqs (70) we arrive  ˆ (−1,r) Q 12  (1,r)  Q12 (2,r) Q12 whose solution is: (n,i) and Q12 (n,i) + iQ12 ˆ (−1,r) ˆ (−1,i) (n = 1, 2), Q and Q 12 12 ˆ (−1) = Q ˆ (−1,r) + iQ ˆ (−1,i) (n = 1, 2), Q 12 12 12 at a system of three linear equations, namely:    ˆ (−1)  (−1) ˆ (−1,i) ˆ −Q11 Q Q 2a 12 22   (1,i) (1)    =  −Q(1) −2b Q12 Q22  11  (2) (2,i) (2) a2 + b2 −Q11 Q12 Q22 2a = D1 /D, −2b = D2 /D, a2 + b2 = D3 /D (74) (75) where D is the determinant of the × matrix in Eq (74), Dk are the determinants of matrices obtained by replacing this matrix’s kth column with the vector on the right-hand side of Eq (74) From Eq (75) it follows: D12 + D22 − 4DD3 = (76) that is the desired explicit secular equation of the wave The expressions of the determinants D, Dk are lengthy and are not displayed here, but they are easily computed by using the expressions in Eqs (71)-(73) Also from (71)-(73) one can see that the secular equation (76) is an algebraic equation of eighth-order in X Remark 2: To obtain the secular equation of the wave we can apply the fundamental equations (56) with the impedance boundary condition (44) in which P = N However, the derivation is more complicated, especially when the size of the square matrix N is higher than When the impedance parameters vanish, i e δ1 = δ2 = 0, from (71)-(73) we 19 have: (1,i) (2,i) ˆ (−1,i) = Q(1,r) = Q(2) = Q12 = Q12 = Q 12 12 22 (77) With (77) Eq (74) is simplified to:   (−1,r) ˆ ˆ (−1) Q ˆ (−1)  2a    Q Q 12 11 22  (1) (1)  =  Q11 Q22    2 (2,r) (2) a +b Q12 Q11 (78) whose determinant of this system’s matrix must be zero, i e.: ˆ (−1,r) Q ˆ (−1) Q ˆ (−1) Q 12 11 22 (1) (1) Q11 Q22 = (2,r) (2) Q12 Q11 (79) or equivalently: (−1,r) Q12 (1) (2) (2,r) Q22 Q11 + Q12 (−1) (1) (1) (−1) Q22 Q11 − Q22 Q11 =0 (80) Again from (71)-(73) we have: (1) Q11 = X − η, (1) Q22 = X, Q11 = −2r6 (−η + X), (2) Q12 = η − (1 + r2 )X, (−1) Q11 r22 ], (−1,r) Q12 (−1) Q22 = X[(X − η)n22 − (2,r) (81) = X(ηn26 + r2 r6 − n26 X), = η(1 − n66 X) + X(n66 X − − r62 ) Substituting (81) into (80) leads to: [η − (1 + r2 )X]{(η − X)[(η − X)(n66 X − 1) + r62 X] + X [(η − X)n22 + r22 ]} (82) + 2r6 X (η − X)[(η − X)n26 + r2 r6 ] = Equation (82) is the secular equation of Rayleigh waves propagating in a monoclinic half-space with the symmetry plane x3 = whose surface is free of traction This 20 equation obtained by Destrade [22] using the method of first integrals (Eq (28)) and by Ting [23] using the cofactor-matrix approach (Eq (4.12b)) Note that, from the first three equations of (77) it implies D ≡ D1 ≡ D3 ≡ Therefore Eq (76) becomes D2 = 0, i e it simplifies to Eq (79) that is equivalent to Eq (82) Thus, Eq (82) is a special case of Eq (76) When the material is orthotropic, c16 = c26 = (see [16]) From (32) we have r6 = n26 = 0, r2 = c12 1 c11 c22 − c212 , n66 = , n22 = , η= c22 c66 c22 c22 (83) Using (71)-(73), (76) and (83), it is not difficult to verify that for the case of orthotropic materials, the secular equation is D1 = In dimensionless form it is of the from: ¯ − 4D ¯D ¯3 = D (84) ¯ D ¯ and D ¯ are given by: where D, √ ¯ = x x δ22 + e2 [e3 δ1 (1 − x) + δ2 (x − e1 )]+ D √ x (δ1 e2 − e3 δ2 ) {e23 (1 − x) + (e1 − x)[δ22 x + e2 (x − 1)]}, ¯ = −x2 e2 + δ [δ (x − 1) + (x − e1 )]+ D − e1 e2 + e23 + δ12 e2 x + xe2 {e23 (x − 1) − (e1 − (85) x)[δ22 x + e2 (x − 1)]}, √ ¯ = x x (e3 δ2 − δ1 e2 ) [δ − δ x + (e1 − x)]+ D 1 √ x e1 e2 − e23 − δ12 e2 x − xe2 [e3 δ1 (1 − x) + δ2 (x − e1 )] in which the dimensionless material parameters ek (k = 1, 2, 3) are defined by (17), δ1 and δ2 are the dimensionless impedance parameters, x = c2 /c22 , c22 = c66 /ρ From (85) one can see that Eq (84) is an algebraic equation of sixth-order in x 21 Remark 3: i) By squaring (two times) two sides of Eq (19) and then dividing the resulting equation by (e1 − x)2 /e22 we arrive at Eq (84) Equation Eq (19) is therefore considered as the original version of Eq (84) ii) Equation (19) is much more simple than Eq (84) This fact says that it is reasonable to consider separately the case of orthotropic materials iii) Equation (19) is will be useful in investigating the uniqueness of Rayleigh waves in orthotropic half-spaces with impedance boundary conditions by the complex function method [24, 25] Conclusions In this paper, the propagation of Rayleigh waves in anisotropic half-spaces subjected impedance boundary conditions is investigated Two cases of anisotropy are considered: orthotropic materials and monoclinic materials with the symmetry plane x3 = For orthotropic case, the secular equation is derived by using the traditional technique and it is an irrational equation From this equation we obtain a new secular equation of the wave for the isotropic case For the monoclinic half-spaces, first the impedance boundary condition is replaced by a traction-free-like boundary condition Then the secular equation is derived by using the method of polarization vector This equation is an algebraic equation of eighth-order 22 Acknowledgments The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) References [1] L Rayleigh, On waves propagating along the plane surface of an elastic solid, Proc R Soc Lond A17 (1885) 4-11 [2] S D M Adams, R V Craster and D P Williams, Rayleigh waves guided by topography, Proc R Soc A 463 (2007) 531-550 [3] V Voloshin, Moving load on elastic structures: passage through the wave speed barriers, PhD thesis, Brunel University 2010 [4] E Godoy, M Durn, J-C Ndlec, On the existence of surface waves in an elastic half-space with impedance boundary conditions, Wave Motion 49 (2012), 585594 [5] J D Achenbach and S P Keshava, Free waves in a plate supported by a semi-infinite continuum, J Appl Mech 34 (1967), 397-404 [6] H.F Tiersten, Elastic surface waves guided by thin films, J Appl Phys 40 (1969) 770-789 23 [7] P Bvik, A comparison between the Tiersten model and O(h) boundary conditions for elastic surface waves guided by thin layers, Trans ASME 63 (1996) 162-167 [8] D J Steigmann, R W Ogden, Surface waves supported by thin-film/substrate interactions, IMA J Appl Math 72 (2007), 730-747 [9] Pham Chi Vinh, Nguyen Thi Khanh Linh, An approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half-space coated by a thin orthotropic elastic layer, Wave Motion 49 (2012), 681-689 [10] Pham Chi Vinh, Nguyen Thi Khanh Linh, An approximate secular equation of generalized Rayleigh waves in pre-stressed com-pressible elastic solids, International Journal of Non-Linear Mechanics 50 (2013), 91-96 [11] Pham Chi Vinh, Vu Thi Ngoc Anh, Rayleigh waves in an orthotropic elastic half-space coated by a thin orthotropic elastic layer with smooth contact Int J Eng Sci 75 (2014), 154-164 [12] Pham Chi Vinh, Vu Thi Ngoc Anh, Vu Phuong Thanh, Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact, Wave Motion (2014), http://dx.doi.org/10.1016/j.wavemoti.2013.11.008 24 In press, [13] A J Niklasson, S K Datta, and M L Dunn, On approximating guided waves in plates with thin anisotropic coatings by means of effective boundary conditions, J Acoust Soc Am 108 (2000), 924-933 [14] S Makarov, E Chilla and H J Frohlich, Determination of elastic constants of thin films from phase velocity dispersion of dierent surface acoustic wave modes, J Appl Phys 78 (1995), 5028-5034 [15] P.G Malischewsky, Surface Waves and Discontinuities, Elsevier, Amsterdam, 1987 [16] T C T Ting, Anisotropic Elasticity: Theory and Applications Oxford University Press, New York, 1996 [17] P C Vinh, R W Ogden, Formulas for the Rayleigh wave speed in orthotropic elastic solids, Ach Mech 56 (3) (2004) 247-265 [18] P Chadwick, The existence of pure surface modes in elastic materials with orthorhombic symmetry, J Sound Vib 47 (1976), 39-52 [19] P C Vinh, R W Ogden, On the Rayleigh wave speed in orthotropic elastic solids, Meccanica 40 (2005), 147-161 [20] A N Stroh, Steady state problems in anisotropic elasticity, Journal of Mathematical Physics 41 (1962), 77103 25 [21] B Collet and M Destrade, Explicit secular equations for piezoacoustic surface waves: Shear-horizontal modes, J Acoust Soc Am 116 (2004), 3432-3442 [22] M Destrade, The explicit secular equation for surface waves in monoclinic elastic crystal, J Acoust Soc Am 109 (2001), 1398-1402 [23] T C T Ting, Explicit secular equations for surface waves in monoclinic materials with the symmetry plane at x1 = 0, x2 = or x3 = 0, Proc R Soc Lond A 458 (2002), 1017-1031 [24] Pham Chi Vinh and Pham Thi Giang, Uniqueness of Stoneley waves in pre-stressed incompressible elastic media, Int J Non-Liner Mech 47 (2012) 128-134 [25] Pham Chi Vinh, Scholte-wave velocity formulae, Wave Motion 50 (2013), 180190 26 ... few investigations on Rayleigh waves with impedance boundary conditions have been done In [15] Malischewsky considered the propagation of Rayleigh waves with Tiersten’s impedance boundary conditions. .. here to view linked References Rayleigh waves with impedance boundary conditions in anisotropic solids Pham Chi Vinha ∗ and Trinh Thi Thanh Hueb a Faculty of Mathematics, Mechanics and Informatics... isotropic Nowadays, anisotropic materials are widely used in various fields of modern tech- nology The investigations of Rayleigh waves with impedance boundary conditions in anisotropic solids are therefore