DSpace at VNU: Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact

Wave Motion 51 (2014) 496–504 Contents lists available at ScienceDirect Wave Motion journal homepage: www.elsevier.com/locate/wavemoti Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact Pham Chi Vinh ∗ , Vu Thi Ngoc Anh, Vu Phuong Thanh Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam highlights • • • • • The propagation of Rayleigh waves in an elastic half-space coated by a thin elastic layer is considered The half-space and the layer are both isotropic and the contact between them is smooth By using the effective boundary condition method an approximate secular equation of fourth-order has been derived From it, an explicit third-order approximate formula for the Rayleigh wave velocity has been established The approximate secular equation and the formula for the velocity will be useful in practical applications article info Article history: Received May 2013 Received in revised form October 2013 Accepted 24 November 2013 Available online December 2013 Keywords: Rayleigh waves An elastic half-space coated with a thin elastic layer Approximate secular equations Approximate formulas for the velocity abstract In the present paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer The contact between the layer and the half space is assumed to be smooth The main purpose of the paper is to establish an approximate secular equation of the wave By using the effective boundary condition method, an approximate, yet highly accurate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived From the secular equation obtained, an approximate formula of third-order for the velocity of Rayleigh waves is established The approximate secular equation and the formula for the velocity obtained in this paper are potentially useful in many practical applications © 2013 Elsevier B.V All rights reserved Introduction The structures of a thin film attached to solids, modeled as half-spaces coated with a thin layer, are widely applied in modern technology Measurement of mechanical properties of thin supported films is therefore very significant [1] Among various measurement methods, the surface/guided wave method [2] is used most extensively in which the Rayleigh wave is a most convenient tool For the Rayleigh-wave approach, the explicit dispersion relations of Rayleigh waves supported by thin-film/substrate interactions are employed as theoretical bases for extracting the mechanical properties of the thin films from experimental data They are therefore the key factor of the investigations of Rayleigh waves propagating in halfspaces covered by a thin layer Taking the assumption of a thin layer, explicit secular equations can be derived by replacing approximately the entire 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 For obtaining the effective boundary conditions Achenbach [3] and Tiersten [4] replaced the thin layer by a plate modeled by different theories: Mindlin’s plate theory and the plate theory of low-frequency extension and flexure (classical plate theory), while Bovik [5] expanded the stresses at the top surface of ∗ Corresponding author Tel.: +84 35532164; fax: +84 38588817 E-mail addresses: pcvinh562000@yahoo.co.uk, pcvinh@vnu.edu.vn (P.C Vinh) 0165-2125/$ – see front matter © 2013 Elsevier B.V All rights reserved http://dx.doi.org/10.1016/j.wavemoti.2013.11.008 P.C Vinh et al / Wave Motion 51 (2014) 496–504 497 the layer into Taylor series in its thickness The Taylor expansion approach was then employed by Niklasson [6], Rokhlin [7,8], Benveniste [9], Steigmann and Ogden [10], Steigmann [11], Ting [12], Vinh and Linh [13,14], Kaplunov and Prikazchikov [15] to establish the effective boundary conditions Achenbach [3], Tiersten [4], Bovik [5], Tuan [16] assumed that the layer and the substrate are both isotropic and derived approximate secular equations of second-order (these equations not coincide totally with each other) In [10] Steigmann and Ogden considered a transversely isotropic layer with residual stress overlying an isotropic half-space and the authors obtained an approximate second-order dispersion relation In [17] Wang et al considered an isotropic half-space covered by a thin electrode layer and the authors obtained an approximate secular equation of first-order In [13] the layer and the half-space were both assumed to be orthotropic and an approximate secular equation of third-order was obtained In [14] the layer and the half-space were both subjected to homogeneous pre-strains and an approximate secular equation of thirdorder was established which is valid for any pre-strain and for a general strain energy function In all investigations mentioned above, the contact between the layer and the half-space is assumed to be welded For the case of smooth contact, there exists only one approximate secular equation of third-order in the literature established by Achenbach and Keshava [3] This approximate secular equation includes the shear coefficient, originating from Mindlin’s plate theory [18], whose usage should be avoided as noted by Muller and Touratier [19], Touratier [20] This remark was also mentioned in [21] It should be noted that for the case of smooth contact, one could not arrive at the effective boundary conditions from the relations between the displacements and the stresses at the bottom surface of the layer which were derived by Tiersten [4] and Bovik [5] In contrast, for the case of welded contact, the effective boundary conditions were immediately obtained The main purpose of the paper is to establish an approximate secular equation of Rayleigh waves propagating in an isotropic elastic half-space coated with a thin isotropic elastic layer for the case of smooth contact By using the effective boundary condition method, an approximate effective boundary condition of fourth-order which relates the normal displacement with the normal stress at the surface of the half space is derived Using this condition along with the vanishing of the shear stress at the surface of the half-space, an approximate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived We will show that the approximate secular equation obtained is a very good approximation Based on it, an approximate formula of third-order for the velocity of Rayleigh waves is established Effective boundary condition of fourth-order Consider an elastic half-space x3 ≥ coated by a thin elastic layer −h ≤ x3 ≤ Both the layer and half-space are homogeneous, isotropic and linearly elastic The layer is assumed to be thin and has a smooth contact with the half-space In particular, the normal component of the particle displacement vector and the normal component of the stress tensor are continuous, while the shearing stress vanishes across the interface x3 = 0, see Achenbach [3] and Murty [22] Note that the same quantities related to the half-space and the layer have the same symbol but are systematically distinguished by a bar if pertaining to the layer If it is assumed that a state of plane strain exists, whereby the x2 component of displacement vanishes and the x1 and x3 components are functions of x1 , x3 and t only, i.e ui = ui (x1 , x3 , t ), u¯ i = u¯ i (x1 , x3 , t ), i = 1, 3, u2 = u¯ ≡ (1) where t is the time Since the layer is made of isotropic elastic materials, the strain–stress relations take the form ¯ u3 , , σ¯ 11 = (λ¯ + 2µ)¯ ¯ u1,1 + λ¯ ¯ u1,1 + (λ¯ + 2µ)¯ σ¯ 33 = λ¯ ¯ u3,3 , (2) σ¯ 13 = µ(¯ ¯ u1,3 + u¯ 3,1 ) ¯ and µ where σ¯ ij is the stress of the layer, commas indicate differentiation with respect to spatial variables xk , λ ¯ are Lame constants In the absent of body forces, the equations of motion for the layer is 11,1 + 13,3 = uă , 13,1 + 33,3 = uă (3) where a dot signifies differentiation with respect to t From Eqs (2), (3) we have  ′  M1 U¯ = M3 T¯ ′ M2 M4   U¯ T¯ (4) where U¯ = u¯  u¯ T , T¯ = σ¯ 13  σ¯ 33 T 498 P.C Vinh et al / Wave Motion 51 (2014) 496–504 the symbol ‘‘T ’’ indicate the transpose of a matrix, the prime signifies differentiation with respect to x3 and M1 =  −∂1  −λ¯ ∂1 ¯λ + 2µ ¯   µ ¯ , M2 =   (λ¯ + 2µ) ¯ − λ¯ 2 ∂1 + ρ∂ ¯ t2 − M3 =  λ¯ + 2µ ¯  ,    λ¯ + 2µ ¯ (5)   ρ∂ ¯ t2 , M4 = M1 T here we use the notations ∂1 = ∂/∂ x1 , ∂12 = ∂ /∂ x1 , ∂t2 = ∂ /∂ t From (4) it follows U¯ (n) T¯ (n)     U¯ =M ¯ , T  M1 M = M3 n  M2 , M4 n = 1, 2, 3, , x3 ∈ [−h, 0] (6) Let h be small (i.e the layer is thin), then expanding into Taylor series T¯ (−h) at x3 = up to the fourth-order of h we have T¯ (−h) = T¯ (0) + T¯ ′ (0)(−h) + ′′ 1 T¯ (0)h2 − T¯ ′′′ (0)h3 + h4 T¯ ′′′′ (0) 3! 4! (7) 2! Suppose that surface x3 = −h is free of traction, i.e T¯ (−h) = Introducing (6) with n = 1, 2, 3, at x3 = into (7) yields  I − hM4 + 2 h M6 − h M8 + 24  h M10 T¯ (0) =  hM3 − 2 h M5 + h M7 − 24  h M9 U¯ (0) (8) where I is the identity matrix of order 2, M3 , M4 are defined by (5) and M5 = M3 M1 + M4 M3 , M6 = M3 M2 + M42 , M7 = M3 M12 + M4 M3 M1 + M3 M2 M3 + M42 M3 , M8 = M3 M1 M2 + M4 M3 M2 + M3 M2 M4 + M43 , (9) M9 = M3 M13 + M4 M3 M12 + M3 M2 M3 M1 + M42 M3 M1 + M3 M1 M2 M3 + M4 M3 M2 M3 + M3 M2 M4 M3 + M43 M3 , M10 = M3 M12 M2 + M4 M3 M1 M2 + M3 M2 M3 M2 + M42 M3 M2 + M3 M1 M2 M4 + M4 M3 M2 M4 + M3 M2 M42 + M44 Taking into account (5) and (9), the relation (8) in component form is of the form   σ¯ 13 + h (1 − 2γ¯ )σ¯ 33,1 + c22 (1 )u1,11 uă h2 ă ă + (2 − 3)σ¯ 13,11 + σ¯ 13 + 4ρ¯ c¯2 (1 − γ¯ )¯u3,111 − 2ρ( ¯ − γ¯ )u¯ 3,1 c¯2    h3 (4γ¯ − 3)σ¯ 33,111 + (1 − γ¯ ) + (1 ) ă 33,1 c¯22 (1 − γ¯ )¯u1,1111 + c¯1 c¯2  ă h4 ă (5 4γ¯ )σ¯ 13,1111 + 2 (1 − γ¯ ) + ρ( ¯ − 4γ¯ )u¯ 1,11 − u¯ 1,tt + 24 c¯2 c¯1  1 + (2 2) ă 13,11 + ă 13,tt + 4( )uă 3,111 c22 (1 )u3,11111 c2 c2 1 ă 2ρ( ¯ − γ¯ ) + u¯ 3,1tt = at x3 = 0, c¯1 c¯2  h2 ă 33 + h( 13,1 u¯ ) + (1 − 2γ¯ )σ¯ 33,11 + ă 33 + c22 (1 )u1,111 c¯1     h3 − 2( )uă 1,1 + (2 3)σ¯ 13,111 + (1 − γ¯ ) + ă 13,1 c2 c1 (10) P.C Vinh et al / Wave Motion 51 (2014) 496–504  499  ρ¯ h4 + 4ρ¯ c¯22 (1 − γ¯ )¯u3,1111 − ρ( )uă 3,11 uă 3,tt + (4γ¯ − 3)σ¯ 33,1111 24 c¯1   1 + (4 − 6γ¯ ) + (2 + ) ă 33,11 + ă 33,tt + 4( )uă 1,111 c1 c2 c1 1 − 8ρ¯ c¯2 (1 − γ¯ )¯u1,11111 − 2ρ( ¯ − γ¯ ) + uă 1,1tt = at x3 = c1 c2 (11) where  c¯1 =  λ¯ + 2µ ¯ , ρ¯ c¯2 = µ ¯ , ρ¯ γ¯ = c¯22 c¯12 (12) Remark If the contact between the layer and the half-space is welded, i.e the displacements and the stresses are continuous through the interface of the layer and the half-space, we immediately obtain the effective boundary conditions from Eqs (10) and (11) by replaced u¯ , u¯ , σ¯ 13 and σ¯ 33 by u1 , u3 , σ13 and σ33 , respectively These effective boundary conditions are valid not only for the displacements and the stresses of Rayleigh waves but also for those of any dynamic problem However, for the case of smooth contact the situation is rather different The horizontal displacement is not required to be continuous through the interface, the effective boundary conditions are therefore not immediately obtained from Eqs (10) and (11) As shown below, the effective boundary conditions obtained for the case of smooth contact are valid for only the displacements and the stresses of Rayleigh waves Now we consider the propagation of a Rayleigh wave, travelling (in the coated half-space) with velocity c (>0) and wave number k (>0) in the x1 -direction and decaying in the x3 -direction The displacements and the stresses of the wave are sought in the form u¯ = U¯ (y)eik(x1 −ct ) , ik(x1 −ct ) σ¯ 13 = −ikT¯1 (y)e u¯ = U¯ (y)eik(x1 −ct ) , (13) σ¯ 33 = −ikT¯3 (y)eik(x1 −ct ) , for the layer, and u1 = U1 (y)eik(x1 −ct ) , u3 = U3 (y)eik(x1 −ct ) , σ13 = −ikT1 (y)eik(x1 −ct ) , (14) σ33 = −ikT3 (y)eik(x1 −ct ) for the half-space, where y = kx3 Substituting (13) into (10) and (11) yields     c2 1 c2 c2 ¯ (2γ¯ − 3) + + ε ( γ ¯ − ) − ( − γ ¯ ) − (2γ¯ iT1 (0) −1 + ε 2 c¯22 24 12 c¯12 12 c¯22    c4 1 c2 − γ¯ − 2) − + T¯3 (0) ε(1 − 2γ¯ ) + ε (3 − 4γ¯ ) − (1 − γ¯ ) 24 c¯2 c¯12      c2 2 − (1 − 2γ¯ ) + U¯ (0) ε 4ρ¯ c¯2 (γ¯ − 1) + ρ¯ c + ε ρ¯ c¯2 (γ¯ − 1) c¯2     c4 2 2 ¯ + iU3 (0) ε 2ρ¯ c¯2 (γ¯ − 1) + ρ¯ c (1 − γ¯ ) + ρ¯ c (5 − 4γ¯ ) − ρ¯ 6 c¯2    + ε4   T¯1 (0) ε + ε + c2 c¯12 1 12 ρ¯ c (2 − γ¯ − γ¯ ) − ρ¯ c¯22 (1 − γ¯ ) −  + ε4 24 (3 − 2γ¯ ) − c2 c¯22  (3 − 4γ¯ ) − c2 c¯12 (1 − γ¯ ) − c2 c¯12 (4 − 6γ¯ ) − c2 c¯22 ρ¯ c (1 − γ¯ )  c¯12 +   + iT¯3 (0) −1 + ε (2 − 6γ¯ + 4γ¯ ) − = 0, c¯22 c4 c¯14  (1 − 2γ¯ ) (15) 500 P.C Vinh et al / Wave Motion 51 (2014) 496–504     ε4  2 2 + iU¯ (0) ε ρ( ¯ − γ¯ ) c − 2c¯2 + ρ¯ 2c (2 − γ¯ − γ¯ ) − 4c¯2 (1 − γ¯ ) − c 12 c¯12      1 c4 2 ¯ + (1 − γ¯ ) + U3 (0) ε ρ¯ c + ε =0 ρ¯ c¯2 (1 − γ¯ ) − ρ¯ c (3 − 4γ¯ ) − ρ¯ 6 c¯1 c¯2 where ε = kh is the dimensionless thickness of the layer Let the contact between the layer and the half-space is smooth, i.e σ13 = 0, σ¯ 13 = 0, u3 = u¯ , σ33 = σ¯ 33 at x3 = (16) or equivalently T1 (0) = 0, T¯1 (0) = 0, U3 (0) = U¯ (0), T3 (0) = T¯3 (0) (17) according to (13) and (14) Introducing (17)2 into (15) yields T¯3 (0)(a1 + a2 ε ) + U¯ (0)(a3 + a4 ε ) + iU¯ (0)(a5 ε + a6 ε ) = 0, iT¯3 (0)(−1 + a7 ε + a8 ε ) + iU¯ (0)(a5 ε + a6 ε ) + U¯ (0)(a9 ε + a10 ε ) = (18) in which a1 = − 2γ¯ , a2 =  (3 − 4γ¯ ) − c 2 c¯12 (1 − γ¯ ) +  c¯22 (1 − 2γ¯ ) , a3 = −4ρ¯ c¯22 (1 − γ¯ ) + ρ¯ c , c4 a4 = − ρ¯ c¯22 (1 − γ¯ ) + ρ¯ c (5 − 4γ¯ ) − ρ¯ , 6 c¯2 a5 = −2ρ¯ c¯22 (1 − γ¯ ) + ρ¯ c (1 − γ¯ ), a6 = a7 = a8 = ρ¯ c (2 − γ¯ − γ¯ ) − ρ¯ ¯ (1 − γ¯ ) − 2 (1 − 2γ¯ ) + 24 c2 c¯12 (3 − 4γ¯ ) − 24  12 ρ( ¯ − γ¯ )c c¯12 + c¯22  , (19) ,  c22 c 1 c¯1 c¯2 (4 − 6γ¯ ) +  (2 − 6γ¯ + 4γ¯ ) − 2 c4 24 c¯14 , a9 = ρ¯ c , a10 = 1 c4 6 c¯12 ρ¯ c¯22 (1 − γ¯ ) − ρ¯ c (3 − 4γ¯ ) − ρ¯ Eliminating U¯ from (18) we have iT¯3 (0)(−a3 + a11 ε + a12 ε ) = −U¯ (0)(a3 a9 ε + a13 ε ) (20) where a11 = −a4 + a3 a7 − a1 a5 , a12 = a4 a7 + a3 a8 − a2 a5 − a1 a6 , a13 = a25 (21) + a4 a9 + a3 a10 From the last two equations of (17) and Eq (20) it follows T3 (0)(−a3 + a11 ε + a12 ε ) = iU3 (0)(a3 a9 ε + a13 ε ) (22) From the first of (17) and (22) we see that the surface x3 = of the half-space is subjected to the following conditions T1 (0) = 0, T3 (0)(−a3 + a11 ε + a12 ε ) = iU3 (0)(a3 a9 ε + a13 ε ) (23) The second of (23) is the approximate effective boundary condition (of fourth-order) The total effect of the layer on the half-space is replaced approximately by this condition P.C Vinh et al / Wave Motion 51 (2014) 496–504 501 An approximate secular equation of fourth-order Now we can ignore the layer and consider the propagation of Rayleigh waves in the isotropic elastic half-space x3 ≥ whose surface x3 = is subjected to the boundary conditions (23) According to Achenbach [23], the displacement components of a Rayleigh wave travelling with velocity c and wave number k in the x1 -direction and decaying in the x3 -direction are determined by (14)1,2 in which U1 (y) and U3 (y) are given by U1 (y) = A1 e−b1 y + A2 e−b2 y , (24) U3 (y) = α1 A1 e−b1 y + α2 A2 e−b2 y where A1 and A2 are constant to be determined and b1 = γ =  √ − γ x,  c22 , c1 = c1 − x, b2 = λ + 2µ , ρ b1 α1 = −  c2 = i µ , ρ , α2 = x= c2 c22 , i b2 , < x < (25) Substituting (14)1,2 and (24) into the stress–strain relations (2) without the bar yields that the stresses σ13 and σ33 are given by (14)3,4 in which  T1 (y) = ic22 ρ −2b1 A1 e −b y  T3 (y) = −c22 ρ c2 c22 + b2  c2 c22   −b y − A2 e  , (26)  − A1 e−b1 y − 2A2 e−b2 y Introducing (24), (26) into (23) provides a homogeneous system of two linear equations for A1 , A2 namely  f1 A1 + f2 A2 = F1 A1 + F2 A2 = (27) where f1 = −2b1 , f2 = (x − 2), F1 = F2 = −a3 + a11 ε + a12 ε b1 (x − 2) − (a3 a9 ε + a13 ε ), c2 ρ c2 ρ 2b2 c22 ρ (a3 − a11 ε − a12 ε ) − c24 ρ (28) (a3 a9 ε + a13 ε ) For a non-trivial solution, the determinant of the matrix of the system (27) must vanish   f1  F1  f2  = F2  Expanding this determinant and using (28) lead to the dispersion equation of the wave, namely A0 + A1 ε + A2 ε + A3 ε + A4 ε = (29) where A0 = rν2 x − 4(1 − γ¯ ) (x − 2)2 − 4b1 b2 ,    A1 = rµ rν2 x2 b1 rν2 x − 4(1 − γ¯ ) ,  A2 = − A3 = 6   8(1 − γ¯ ) + 4rν x(γ¯ − 2) + rν x (1 + 3γ¯ )     (x − 2)2 − 4b1 b2 , rµ xb1 8(1 − γ¯ )2 + rν2 x 8(−2 + 3γ¯ − γ¯ ) + 2rν2 x(4 − 2γ¯ − γ¯ ) − rν4 x2 (1 + γ¯ ) A4 = −   24 (30)  ,   (x − 2)2 − 4b1 b2 4(1 − γ¯ ) + rν2 x(4γ¯ − 7) + 2rν4 x2 (1 + 4γ¯ − 2γ¯ ) − rν6 x3 γ¯ (2 + γ¯ ) where rµ = µ/µ, ¯ rv = c2 /¯c2 Eq (29) is the desired approximate secular equation 502 P.C Vinh et al / Wave Motion 51 (2014) 496–504 √ Fig Dependence on ε = k · h ∈ [0 1] of the dimensionless Rayleigh wave velocity x = c /c2 that is calculated by the exact secular equation and by the approximate secular equations of fourth-order (29) Two corresponding curves almost totally coincide with each other Here we take rµ = 0.5, rv = 5, γ = 1/4 and γ¯ = 2/3 Table √ Some values of x, corresponding to Fig (rµ = 0.5, rv = 5, γ = 1/4 and √ γ¯ = 2/3), that are calculated by the exact secular equation ( xext ), by the √ approximate secular equation (29) ( xapp ) ε √ x √ ext xapp 0.2 0.4 0.6 0.8 1.0 0.9325 0.9325 0.6146 0.6152 0.4535 0.4555 0.3647 0.3674 0.3102 0.3125 0.2767 0.2781 √ √ From (29) and the first of (30) it follows that, when ε = either (x − 2)2 − − x − γ x = or x = 4(1 − γ¯ )¯c22 /c22 That means, in the limit ε → two modes are possible, one of which approaches the classical Rayleigh√ wave in the isotropic half-space and the other approaches the longitudinal wave of the layer with the velocity c = 2c¯2 − γ¯ , as noted by Achenbach and Keshava [3] √ Fig presents the dependence on ε = k · h ∈ [0 1] of the dimensionless Rayleigh wave velocity x = c /c2 that is calculated by the exact secular equation and by the approximate secular equations of fourth-order (29) Here we take √ rµ = 0.5, rv = 5, γ = 1/4 and γ¯ = 2/3 Some values of x are listed in Table Note that the exact secular equation is similar in form to Eq (30) in Ref [3], and is not reproduced here It is seen from Fig that the exact velocity curve and the approximate velocity curve of fourth-order almost totally coincide with each other for the values of ε ∈ [0 1] This shows that the approximate secular equation (29) is a very good approximation An approximate formula of third-order for the velocity In this section we establish an approximate formula of third-order for the squared dimensionless Rayleigh wave velocity x(ε) that is of the form x(ε) = x(0) + x′ (0) ε + x′′ (0) ε2 + x′′′ (0) ε3 + O(ε4 ) (31) where x(0) is the squared dimensionless velocity of Rayleigh waves propagating in an isotropic elastic half-space that is given by (see [24])  x(0) = 4(1 − γ ) − γ+ √  R+ D+ √  R−  −1 D (32) in which R = 2(27 − 90γ + 99γ − 32γ )/27, D = 4(1 − γ )2 (11 − 62γ + 107γ − 64γ )/27 (33) and the roots in the formula (32) taking their principal values Note that x(0) can be calculated by another formula derived by Malischewsky [25] P.C Vinh et al / Wave Motion 51 (2014) 496–504 503 √ Fig Plots of the dimensionless Rayleigh wave velocity x(ε) in the interval [0 1] that is calculated by the exact secular equation and by the formula (31) Here we take rµ = 4, rv = 0.5, γ = 1/4 and γ¯ = 2/3 Table √ Some values of x, corresponding to Fig (rµ = 4, rv = 0.5, γ = 1/4 and √ γ¯ = 2/3), that are calculated by the exact secular equation ( xext ), by the √ approximate formula (31) ( xapp ) ε √ x √ ext xapp 0.2 0.4 0.6 0.8 1.0 0.9325 0.9325 0.9063 0.9062 0.8815 0.8806 0.8635 0.8606 0.8557 0.8515 0.8586 0.8584 From (29) it follows that   2A2 A20x − 2A0x A1 A1x + A0xx A21  ′′   , x ( ) = − ,  A0x x=x(0) A30x x=x(0)     ′′′ ′ ′2 ′′ ′ ′′ ′3 x (0) = − 6A3 + 6A2x x (0) + 3A1xx x (0) + 3A1x x (0) + 3A0xx x (0)x (0) + A0xxx x (0) /A0x  A1  x′ (0) = − (34) x=x(0) where A1 , A2 and A3 are given by (30) and √      A0x = rν2 (x − 2)2 − − x − γ x + 4(γ¯ − 1) + rν2 x + γ − 2γ x  A0xx = 4rν x−2+ √  A0xxx = 6rν  √ − x − γx (γ − 1)2 1+   (1 − x)3 (1 − γ x)3  A1x = A1xx = + 4(γ¯ − 1) + rν x  √ rµ rν2 (1 − γ x)3   + γ − 2γ x x−2+ √ √ − x − γx  ,  (1 − x)5   ,   −  (γ − 1)2 1+   (1 − x)3 (1 − γ x)3 3(1 − γ )2 (1 + γ − 2γ x) 4(1 − γ¯ ) − rν2 x rµ rν2 x 4(1 − γ¯ )(5γ x − 4) + rν2 x(6 − 7γ x) − γx  (1 − γ x)5 ,  ,   4(1 − γ¯ )(−8 + 24γ x − 15γ x2 ) + rν2 x(24 − 60γ x + 35γ x2 ) ,    √  A2x = − rν2 2(γ¯ − 2) + rν2 x(1 + 3γ¯ ) (x − 2)2 − − x − γ x   1 + γ − 2γ x − 4rν x(γ¯ − 2) + 8(1 − γ¯ ) + rν x (1 + 3γ¯ ) x − + √ √ − x − γx √ Fig shows the plots of the dimensionless Rayleigh wave velocity x(ε) in the interval [0 1] that is calculated by the √ exact secular equation and by the formula (31) Here we take rµ = 4, rv = 0.5, γ = 1/4 and γ¯ = 2/3 Some values of x are listed in Table It is shown that the approximate velocity curve is close to the exact velocity curve in the interval [0 1] 504 P.C Vinh et al / Wave Motion 51 (2014) 496–504 Conclusions In this paper the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer is considered The contact between the layer and half space is assumed to be smooth An approximate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived using the effective boundary condition method We have shown that the approximate secular equation obtained has high accuracy An approximate formula of third-order for the velocity of Rayleigh waves is established using the obtained approximate secular equation The approximate secular equation and the formula for the velocity are potentially useful in many practical applications Acknowledgment The work was supported by the Vietnam National Foundation for Science 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function of the material parameters, Geofis Int 45 (2004) 507–509  ... main purpose of the paper is to establish an approximate secular equation of Rayleigh waves propagating in an isotropic elastic half-space coated with a thin isotropic elastic layer for the case... Conclusions In this paper the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer is considered The contact between the layer and half space is assumed... 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 [14] Pham Chi Vinh, Thi Khanh Linh