Acta Mech DOI 10.1007/s00707-014-1090-8 Chi Vinh Pham · Thi Ngoc Anh Vu An approximate secular equation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer Received: 28 October 2013 / Revised: 25 December 2013 © Springer-Verlag Wien 2014 Abstract In this paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic halfspace coated with a thin isotropic elastic layer The contact between the layer and the half-space is assumed to be welded 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 secular equation of fourth order in terms of the dimensionless thickness of the layer is derived It is shown that this approximate secular equation has high accuracy From the secular equation obtained, an approximate formula of third order for the velocity of Rayleigh waves is established Introduction The structures of a thin film attached to solids, modeled as half-spaces coated by a thin layer, are widely applied in modern technology The determination of mechanical properties of thin films deposited on halfspaces before and during loading plays an important role in health monitoring of these structures [1,2] Among various measurement methods, the surface/guided wave method is most widely used [2], and for this method, the Rayleigh wave is a versatile and convenient tool [3,4] 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 main purpose of the investigations of Rayleigh waves propagating in half-spaces covered with a thin layer Taking the assumption of a thin layer, explicit secular equations can be derived by replacing 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 and Keshava [5], and Tiersten [6] 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, while Bovik [7] expanded the stresses at the top surface of the layer into Taylor series in its thickness The Taylor expansion technique was then developed by Rokhlin and Huang [8,9], Niklasson [10], Benveniste [11], Steigmann and Ogden [12], Ting [13], Vinh and Linh [14,15], Vinh and Anh [16] and Vinh et al [17] Malischewsky [18] puts Tiersten’s theory into a broader context by formulating very general impedance conditions Godoy et al [19] used Malischewsky’s formalism for a similar problem with vanishing normal traction on the surface Tiersten [6] and Bovik [7] assumed that the layer and the substrate are both isotropic and derived approximate secular equations of second order In Vinh and Linh [14], the layer and the half-space were both assumed C V Pham (B) · T N A Vu Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam E-mail: pcvinh@gmail.com; pcvinh@vnu.edu.vn Tel.: +84-4-35532164 Fax: +84-4-38588817 C V Pham, T N A Vu to be orthotropic, and an approximate secular equation of third order was obtained In order to get more accurate solutions of the inverse problem (evaluating the mechanical properties of the thin films from experimental data), we need approximate secular equations of higher order The main purpose of this paper is to establish an approximate secular equation of fourth order of Rayleigh waves propagating in an isotropic half-space coated with a thin isotropic elastic layer The contact between the layer and the half-space is assumed to be welded Starting from the basic equations in matrix form and employing the Taylor expansion technique, the effective boundary conditions are derived With these boundary conditions, the fourth-order approximate secular equation is obtained by considering the propagation of Rayleigh waves in the half-space without the layer It is shown that this approximate secular equation has high accuracy, and it is much more better than Bovik’s second-order approximate dispersion relation From the secular equation obtained, an approximate formula of third order for the velocity of Rayleigh waves is established Note that, Achenbach and Keshava [5] derived an approximate secular of fourth order for the isotropic case However, this approximate secular equation includes the shear coefficient, originating from Mindlin’s plate theory [20], whose usage should be avoided as noted by Touratier [21], Muller and Touratier [22] and Stephen [23] Effective boundary condition of fourth order Consider an elastic half-space x3 ≥ coated by a thin elastic layer −h ≤ x3 ≤ The layer and the half-space are both homogeneous, compressible and isotropic The layer is assumed to be perfectly bonded to the halfspace Note that the respective 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 We consider a plane motion in the (x1 , x3 )-plane with displacement components (u , u , u ) such that u i = u i (x1 , x3 , t), u¯ i = u¯ i (x1 , x3 , t), i = 1, 3, u = u¯ ≡ 0, (1) where t is the time Since the layer is made of isotropic elastic materials, the strain–stress relations take the form σ¯ 11 = (λ¯ + 2μ) ¯ u¯ 1,1 + λ¯ u¯ 3,3 , ¯ u¯ 3,3 , σ¯ 33 = λ¯ u¯ 1,1 + (λ¯ + 2μ) ¯ u¯ 1,3 + u¯ 3,1 ), σ¯ 13 = μ( (2) where σ¯ i j is the stress of the layer, commas indicate differentiation with respect to spatial variables xk and λ¯ and μ¯ are Lamé constants In the absent of body forces, the equations of motion for the layer are: σ¯ 11,1 + σ¯ 13,3 = ρ¯ uă , 13,1 + 33,3 = uă , (3) where a dot signifies differentiation with respect to t From Eqs (2), (3), we have U¯ T¯ = M1 M2 U¯ M3 M4 T¯ , (4) where U¯ = u¯ u¯ T , T¯ = σ¯ 13 σ¯ 33 T , the symbol “T ” indicate the transpose of a matrix, the prime signifies differentiation with respect to x3 , and ⎡ ⎡1 ⎤ ⎤ −∂1 ⎢ ⎢ μ¯ ⎥ ⎥ M1 = ⎣ −λ¯ ⎦ , M2 = ⎣ ⎦, ∂1 0 ¯ λ¯ + 2μ¯ ⎡ ⎤λ + 2μ¯ (5) (λ¯ + 2μ) ¯ − λ¯ 2 ¯ t ⎥ ∂1 + ρ∂ ⎢− M3 = ⎣ λ¯ + 2μ¯ ⎦ , M4 = M1 T ρ∂ ¯ t2 An approximate secular equation of Rayleigh waves Here, we use the notations ∂1 = ∂/∂ x1 , ∂12 = ∂ /∂ x1 , ∂t2 = ∂ /∂t Equation (4) is called the matrix form of basic equations (see also Vinh and Seriani [24,25]) From (4), it follows that U¯ (n) T¯ (n) = Mn U¯ T¯ ⎡ , M =⎣ M1 M2 ⎤ ⎦ , n = 1, 2, 3, , x3 ∈ [−h, 0] (6) M3 M4 Let h be small (i e the layer is thin), then expanding into a Taylor series T¯ (−h) at x3 = up to the fourth order of h, we have: 1 T¯ (−h) = T¯ (0) − T¯ (0)h + T¯ (0)h − T¯ (0)h + h T¯ (0) 2! 3! 4! (7) Suppose that the surface x3 = −h is free of traction, i.e., T¯ (−h) = Introducing (6) with n = 1, 2, 3, at x3 = into (7) yields 1 I − h M4 + h M6 − h M8 + h M10 T¯ (0) 24 1 = h M3 − h M5 + h M7 − h M9 U¯ (0), 24 (8) where I is the identity matrix of order Since the layer and the half-space are bonded perfectly to each other at the plane x3 = 0, if follows that U¯ (0) = U (0), T¯ (0) = T (0) From (8), we have 1 I − h M4 + h M6 − h M8 + h M10 T (0) 24 = h M3 − h M5 + h M7 − h M9 U (0) 24 (9) 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 , M9 = M3 M13 + M4 M3 M12 + M3 M2 M3 M1 + M42 M3 M1 (10) + 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 The relation (9) between the traction vector and displacement vector of the half-space at the plane x3 = is called effective boundary condition of fourth order in the matrix form It replaces (approximately) the entire effect of the thin layer on the substrate Introducing (5) and (10) into (9), we obtain the effective boundary conditions of fourth order in component form, namely C V Pham, T N A Vu σ13 + h (1 − 2γ¯ )σ33,1 + c22 (1 )u 1,11 uă + h2 )uă 3,1 (2 3)13,11 + ă 13 + c22 (1 − γ¯ )u 3,111 − 2ρ(1 c¯2 + h3 (4γ¯ − 3)σ33,111 + + ρ(5 ¯ − 4γ¯ )uă 1,11 + (1 ) + (1 ) ă 33,1 8ρ¯ c¯22 (1 − γ¯ )u 1,1111 c¯1 c¯2 uă 1,tt c22 + h4 1 (5 4γ¯ )σ13,1111 + 2 (1 − γ¯ ) + (2 2) ă 13,11 24 c1 c2 1 ă 13,tt + 4(2 )uă 3,111 c22 (1 − γ¯ )u 3,11111 − 2ρ(1 ¯ − γ¯ ) + c2 c1 c2 uă 3,1tt =0 at x3 = 0, σ33 + h(σ13,1 − ρ¯ uă ) (11) + h2 )uă 1,1 (1 )33,11 + ă 33 + 4ρ¯ c¯22 (1 − γ¯ )u 1,111 − 2ρ(1 c¯1 + h3 + 1 h4 (4γ¯ − 3)σ33,1111 + (4 − 6γ¯ ) + (2 + ) ă 33,11 + ă 33,tt 24 c1 c2 c1 (2 3)σ13,111 + ρ¯ (1 − γ¯ ) + ă 13,1 + c22 (1 )u 3,1111 (3 )uă 3,11 uă 3,tt c2 c1 c1 + 4(2 )uă 1,111 c22 (1 − γ¯ )u 1,11111 − 2ρ(1 ¯ − γ¯ ) 1 + c12 c2 uă 1,1tt =0 at x3 = 0, (12) where c¯1 = λ¯ + 2μ¯ , c¯2 = ρ¯ c¯2 μ¯ , γ¯ = 22 ρ¯ c¯1 (13) An approximate secular equation of fourth order Now, we ignore the layer and consider the propagation of a Rayleigh wave in the half-space subjected the boundary conditions (11) and (12) The Rayleigh wave travels with velocity c (>0) and wave number k (>0) in the x1 -direction and decays in the x3 -direction, i.e., its displacements and stresses vanish at x3 = +∞ According to Achenbach [26], the displacements and stresses of Rayleigh wave satisfying (4) (without bars) and the decay condition are given by u = B1 e−kb1 x3 + B2 e−kb2 x3 eik(x1 −ct) , u = α1 B1 e−kb1 x3 + α2 B2 e−kb2 x3 eik(x1 −ct) , (14) σ13 = −kc22 ρ (b1 + β1 )B1 e−kb1 x3 + (b2 + β2 )B2 e−kb2 x3 eik(x1 −ct) , 1 1 − − b1 β1 B1 e−kb1 x3 + − − b2 β2 B2 e−kb2 x3 eik(x1 −ct) , γ γ γ γ where B1 and B2 are constants to be determined, and √ b1 = − γ x, b2 = − x, αk = iβk , β1 = b1 , β2 = , k = 1, 2, b2 σ33 = kic22 ρ γ = c22 c12 , c1 = λ + 2μ , c2 = ρ μ c2 , x = , < x < ρ c2 (15) (16) An approximate secular equation of Rayleigh waves Introducing (14), (15) into the effective boundary conditions (11), (12) leads to the following equation for B1 , B2 : f (b1 )B1 + f (b2 )B2 = 0, F(b1 )B1 + F(b2 )B2 = 0, (17) where: f (bn ) = −(bn + βn ) + ε 4rμ (γ¯ − 1) + rμrv2 x + (2γ¯ − 1) ε2 ε3 + + 1 − − b n βn γ γ (2γ¯ − 3) + xrv2 (bn + βn ) − βn 4rμ (γ¯ − 1) + 2(1 − γ¯ )rμrv2 x 1 − − b n βn γ γ (4γ¯ − 3) + (1 − 2γ¯ )xrv2 − (4γ¯ − 5)rμrv2 x − 8rμ (1 − γ¯ ) − rμrv4 x + ε4 24 (4γ¯ − 5) + 2(2 − γ¯ )xrv2 − x rv4 (bn + βn ) − βn 8rμ (γ¯ − 1) + 4(2 − γ¯ − γ¯ )xrv2 rμ − 2(1 − γ¯ )rμrv4 x , −2− γ ε2 + (2γ¯ ε3 (2γ¯ + F(bn ) = + x rv4 rμ γ¯ bn βn + ε − (bn + βn ) + rμrv2 xβn γ 1 − 1) − x γ¯ rv2 − − bn βn − 4rμ (1 − γ¯ ) + 2(1 − γ¯ )xrμrv2 γ γ (18) − 3) + (2 − γ¯ )xrv2 (bn + βn ) − βn 4(γ¯ − 1)rμ + (3 − 4γ¯ )xrv2 rμ + ε4 24 (4γ¯ − 3) + 2(1 − γ¯ − γ¯ )xrv2 + x rv4 γ¯ 1 − − b n βn γ γ + 4(2 − γ¯ − γ¯ )xrv2 rμ − 8(1 − γ¯ )rμ − 2(1 − γ¯ )x rv4 rμ , μ¯ c2 , rv = μ c¯2 Due to B12 + B22 = 0, the determinant of coefficients of the homogeneous system (17) must vanish This provides: f (b1 )F(b2 ) − f (b2 )F(b1 ) = (19) n = 1, 2, ε = kh, rμ = From (18) and (19), after algebraically lengthy calculations whose detail omitted, we arrive at the approximate secular equation of fourth order of Rayleigh waves, namely A0 + A1 ε + A2 ε2 + A3 ε3 + A4 ε4 + O(ε5 ) = 0, (20) where ε = kh is called the dimensionless thickness of the layer, and A0 = (2 − x)2 − 4b1 b2 , A1 = xrμ [rv2 x − 4(1 − γ¯ )]b2 + rv2 xb1 , A2 = − A0 xrv2 (1 + γ¯ ) − 4(1 − γ¯ ) + 2rμ2 rv2 x 4(1 − γ¯ ) − rv2 x (1 − b1 b2 ) + 2rμ (2b1 b2 − + x) 4(1 − γ¯ ) − 2rv2 x γ¯ , A3 = − xrμ [(γ¯ + 3)rv4 x + 4(2γ¯ − 3)rv2 x + 8(1 − γ¯ )]b1 + [(3γ¯ + 1)rv4 x + 4(γ¯ − 2)rv2 x + 8(1 − γ¯ )]b2 , (21) C V Pham, T N A Vu 0.95 0.9 0.85 x 1/2 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.5 ε 1.5 √ Fig Plots of the dimensionless Rayleigh wave velocity x = c/c2 in the interval ε ∈ [0 1.5] that are calculated by the exact secular equation (6) in Ref [7] (solid line), by Bovik’s second-order approximate secular equation (Eq 38, Ref [7]) (dash-dot line) and by the approximate secular equation of fourth-order (24) (dashed line) Here we take rμ = 1.9266, rv = 1.6070, γ = 0.3556 and γ¯ = 0.2258 A4 = A0 8(1 − γ¯ ) − 8xrν2 − 12xrν2 γ¯ + 12xrν2 γ¯ + x rν4 + 6x rν4 γ¯ 24 +x rν4 γ¯ + (1 − b1 b2 ) − 8rμ2 (γ¯ − 1)2 + 8rμ2 rν2 x(2 − 3γ¯ + γ¯ ) + 2rμ2 rν4 x (γ¯ + 2γ¯ − 4) + rμ2 rν6 x (1 + γ¯ ) + 16rμ γ¯ (γ¯ − 1) + b1 b2 rμrν2 rν2 x (γ¯ + 4γ¯ − 1) + 2x(1 − 8γ¯ + 5γ¯ ) + rμ x rν4 (γ¯ + 4γ¯ − 1)(x − 2x) + 2rν2 (1 − 8γ¯ + 5γ¯ )(x − 2) − 8γ¯ (γ¯ − 1) (22) Since A0 [x(ε)] = O(ε), from (22), it follows that A4 ε4 = A∗4 ε4 + O(ε5 ), where A∗4 = (1 − b1 b2 ) − 8rμ2 (γ¯ − 1)2 + 8rμ2 rv2 x(2 − 3γ¯ + γ¯ ) +2rμ2 rv4 x (γ¯ + 2γ¯ − 4) + rμ2 rv6 x (1 + γ¯ ) + 16rμ γ¯ (γ¯ − 1) + b1 b2 rμrv2 rv2 x (γ¯ + 4γ¯ − 1) + 2x(1 − 8γ¯ + 5γ¯ ) + rμ x rv4 (γ¯ + 4γ¯ − 1)(x − 2x) + 2rv2 (1 − 8γ¯ + 5γ¯ )(x − 2) − 8γ¯ (γ¯ − 1) Therefore, Eq (20) can be written as A0 + A1 ε + A2 ε2 + A3 ε3 + A∗4 ε4 + O(ε5 ) = (23) (24) Equations (20) and (24) are the desired approximate secular equations Up to the third order of ε, Eqs (20) and (24) coincide with Eq (38) in Ref [14] Taking into account the fact that A0 [x(ε)] = O(ε), one can see that up to the second order, Eqs (20), (24) and Bovik’s second-order approximation (Eq 38 in Ref [7]) are identical to each other and they coincide with Eq (43) in Ref [14] √ Figure presents the dependence on ε ∈ [0 1.5] of the dimensionless Rayleigh wave velocity x = c/c2 that is calculated by the exact secular equation (6) in Ref [7] by Bovik’s second-order approximate secular equation (Eq 38, Ref [7]) and by the approximate secular equation of fourth order (24) Here, we take rμ = 1.9266, rv = 1.6070, γ = 0.3556 and γ¯ = 0.2258 It is seen from Fig that the exact velocity curve and the approximate velocity curve of fourth order almost totally coincide with each other That means the fourth-order approximate secular equation (24) has high accuracy Figure also shows that it is much better than Bovik’s second-order approximate secular equation An approximate secular equation of Rayleigh waves 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) x (0) ε + ε + O(ε4 ), (25) where x(0) is the squared dimensionless velocity of Rayleigh waves propagating in an isotropic elastic halfspace that is given by [27] x(0) = 4(1 − γ ) − γ + 3 R+ √ D+ R− √ −1 D , (26) in which R = 2(27 − 90γ + 99γ − 32γ )/27, (27) D = 4(1 − γ )2 (11 − 62γ + 107γ − 64γ )/27, the roots in the formula (26) taking their principal values From (20), it follows that x (0) = − A1 A0x , x (0) = − A2 A20x − A0x A1 A1x + A0x x A21 x=x(0) A30x , x=x(0) (28) x (0) = − 6A3 + 6A2x x (0) + 3A1x x x (0) + 3A1x x (0) + 3A0x x x (0)x (0) + A0x x x x (0) /A0x x=x(0) , where A1 , A2 and A3 are given by (21) and + γ − 2γ x , A0x x = + A0x = x − + √ √ 1− x 1−γx + γ − 2γ x , A0x x x = 3(γ − 1)2 (1 − x)5 (1 − γ x)5 (γ − 1)2 (1 − x)3 (1 − γ x)3 , rv2 x(4 − 5x) + 12x(1 − γ¯ ) − 8(1 − γ¯ ) rv2 x(4 − 5γ x) , + √ √ 1−γx 1−x r (8 − 24x + 15x ) + 4(1 − γ¯ )(4 − 3x) rv2 (8 − 24γ x + 15γ¯ x ) , = rμ v + (1 − γ x)3 (1 − x)3 1 = − rv2 (1 + γ¯ )A0 − (1 + γ¯ )rv2 x − 4(1 − γ¯ ) A0x 2 √ + 2rμrv − − x − γ x rμ (2 − 2γ¯ − rv2 x) + 2γ¯ − γ¯ x A1x = rμ A1x x A2x (29) + γ − 2γ x + rμ2 rv2 x 4(1 − γ¯ ) − rv2 x √ √ 1− x 1−γx + γ − 2γ x 2(1 − γ¯ ) − rv2 x γ¯ + 2rμ − √ √ 1− x 1−γx Note that, x(0) can be calculated by another exact formula derived by Malischewsky [28] or by the approximate expressions with high accuracy obtained recently by Vinh and Malischewsky [29–31] √ Figure presents the dependence on ε ∈ [0 0.8] of the dimensionless Rayleigh wave velocity x = c/c2 that is calculated by the exact secular equation (6) in Ref [7] and by the approximate formula (25) Here, we take rμ = 0.1, rv = 3.85, γ = 0.39 and γ¯ = 0.13 It is seen that the approximate velocity curve is very close to the exact one C V Pham, T N A Vu 0.95 0.9 0.85 x½ 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 √ Fig Plots of the dimensionless Rayleigh wave velocity x(ε) in the interval [0 0.8] that are calculated by the exact secular equation (6) in Ref [7] (solid line) and by the formula (25) (dashed line) Here we take rμ = 0.1, rv = 3.85, γ = 0.39 and γ¯ = 0.13 Conclusions In this paper, the propagation of Rayleigh waves in an isotropic elastic half-space covered with a thin isotropic elastic layer is investigated The contact between the layer and the half-space is welded An approximate secular equation of fourth order in terms of the dimensionless thickness of the layer is derived using the effective boundary condition method It is shown that the approximate secular equation obtained has high accuracy From the approximate secular equation obtained, an approximate formula of third order for the velocity of Rayleigh waves is established Since the secular equation and the formula for the Rayleigh wave velocity obtained are totally explicit, they are good tools for evaluating the mechanical properties of thin films deposited to half-spaces Acknowledgments The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) References Makarov, S., Chilla, E., Frohlich, H.J.: Determination of elastic constants of thin films from phase velocity dispersion of different surface acoustic wave modes J Appl Phys 78, 5028–5034 (1995) Every, A.G.: Measurement of the near-surface elastic properties of solids and thin supported films Meas Sci Technol 13, R21–R39 (2002) Kuchler, K., Richter, E.: Ultrasonic surface waves for studying the properties of thin films Thin Solid Films 315, 29–34 (1998) Hess, P., Lomonosov, A.M., Mayer, A.P.: Laser-based linear and nonlinear guided elastic waves at surfaces (2D) and wedges (1D) 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Stoneley waves Math Mech Solids 14, 52–71 (2009) An approximate secular equation of Rayleigh waves 14 Vinh, P.C., Linh, N.T.K.: An approximate secular equation of Rayleigh waves propagating in an orthotropic... approximate secular equation (24) has high accuracy Figure also shows that it is much better than Bovik’s second-order approximate secular equation An approximate secular equation of Rayleigh waves. .. order of Rayleigh waves propagating in an isotropic half-space coated with a thin isotropic elastic layer The contact between the layer and the half-space is assumed to be welded Starting from