Rayleigh waves in an orthotropic half space coated by a thin orthotropic layer with sliding contact

International Journal of Engineering Science 75 (2014) 154–164 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci Rayleigh waves in an orthotropic half-space coated by a thin orthotropic layer with sliding contact Pham Chi Vinh ⇑, Vu Thi Ngoc Anh Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334 Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam a r t i c l e i n f o Article history: Received 23 August 2013 Received in revised form 18 October 2013 Accepted November 2013 Available online 11 December 2013 Keywords: Rayleigh waves An orthotropic elastic half-space A thin orthotropic elastic layer Approximate secular equation Approximate formula for the velocity a b s t r a c t In the present paper, we are interested in the propagation of Rayleigh waves in an orthotropic elastic half-space coated with a thin orthotropic elastic layer The contact between the layer and the half space is assumed to be smooth The main aim of the paper is to establish an approximate secular equation of the wave By using the effective boundary condition method, an approximate secular equations of third-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 Rayleigh wave velocity is derived and it is a good approximation Ó 2013 Elsevier Ltd All rights reserved 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 measurement of mechanical properties of thin films deposited on half-spaces before and during loading plays an important role in health monitoring of these structures in applications, see Makarov, Chilla, and Frohlich (1995) and Every (2002) and references therein Among various measurement methods, the surface/guided wave method is most widely used (Every, 2002), because it is non-destructive and it is connected with reduced cost, less inspection time, and greater coverage (Hess, Lomonosov, & Mayer, 2014) For the surface/guided, wave method the Rayleigh wave is a versatile and convenient tool (Kuchler & Richter, 1998; Hess et al., 2014) 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 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 and Keshava (1967) and Tiersten (1969) 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 (1996) expanded the stresses at the top surface of the layer into Taylor series in its thickness The Taylor expansion technique was then developed by Benveniste (2006), Niklasson, Datta, and Dunn (2000), Rokhlin and Huang (1992, 1993), Shuvalov and Every (2002), Steigmann (2007), Ting (2009) and Vinh and Linh (2012, 2013) ⇑ Corresponding author Tel.: +84 35532164; fax: +84 38588817 E-mail address: pcvinh@vnu.edu.vn (P.C Vinh) 0020-7225/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.ijengsci.2013.11.004 P.C Vinh, V.T Ngoc Anh / International Journal of Engineering Science 75 (2014) 154–164 155 Bovik (1996), Tiersten (1969) and Tuan (2008), 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) Steigmann (2007) considered a transversely isotropic layer with residual stress overlying an isotropic half-space and he obtained an approximate second-order dispersion relation Wang, Du, Lu, and Mao (2006) considered a isotropic half-space covered with a thin electrode layer and he obtained an approximate secular equation of first-order In Vinh and Linh (2012) the layer and the half-space were both assumed to be orthotropic and an approximate secular equation of third-order was obtained In Vinh and Linh (2013) the layer and the half-space are both subjected to homogeneous pre-stains and an approximate secular equation of third-order 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 perfectly bonded For the case of sliding contact, there exists only one approximate secular equation of third-order in the literature, for the case when the layer and the half-space are both isotropic, obtained by Achenbach and Keshava (1967) However, this approximate secular equation includes the shear coefficient, originating from Mindlin’s plate theory (Mindlin, 1951), whose usage should be avoided as noted by , Muller and Touratier (1996), Stephen (1997) and Touratier (1991) It should be note 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 Bovik (1996) and Tiersten (1969) In contrast, for the case of welded contact, the effective boundary conditions were immediately obtained The main aim of this paper is to derive an approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half-space covered with a thin orthotropic elastic layer The layer and the half-space are in sliding contact with each other By using the effective boundary condition method, an approximate effective boundary condition of third-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 third-order in terms of the dimensionless thickness of the layer is obtained We will show that the approximate secular equation obtained has high accuracy From this secular equation, an approximate formula of third-order for the Rayleigh wave velocity is established and it is a good approximation Effective boundary condition of third-order Consider an elastic half-space x2 P coated by a thin elastic layer Àh x2 The layer and the half-space are both homogeneous, compressible, orthotropic and they are in sliding contact with each other 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 We are interested in the plane strain such that: i ¼ u  i ðx1 ; x2 ; tÞ; u ui ¼ ui ðx1 ; x2 ; tÞ;   0; i ẳ 1; 2; u3 ẳ u 1ị  i are components of the displacement vector, t is the time Since the layer is made of orthotropic elastic materials, where ui ; u the strain–stress relations are: r 11 ẳ c11 u1;1 ỵ c12 u2;2 ; r 22 ẳ c12 u1;1 ỵ c22 u2;2 ; r 12 ẳ c66 u1;2 ỵ u2;1 ị; 2ị  ij are the stresses, the material constants where commas indicate differentiation with respect to spatial variables xk ; r c11 ; c22 ; c12 ; c66 satisfy the inequalities: ckk > 0; k ¼ 1; 2; 6; c11 c22 À c212 > 0; ð3Þ which are necessary and sufficient conditions for the strain energy of the material to be positive definite (see Ting, 1996) In the absent of body forces, the equations of motion for the layer is: r 11;1 ỵ r 12;2 ẳ q u1 ; r 12;1 ỵ r 22;2 ẳ q u€2 ; ð4Þ  is the mass density of the layer, a dot signifies differentiation with respect to t Following Vinh and Seriani (2009, where q 2010), from Eqs (2) and (4) we arrive at: " 0 U T # ¼ M1 M3 !"  # U ; M4 T M2 5ị where:  ẳ ẵu 1 U  T ; u  12 T ẳ ẵ r r 22 ŠT ; the symbol ‘‘ T ’’ indicate the transpose of a matrix, the prime signifies differentiation with respect to x2 and: 156 P.C Vinh, V.T Ngoc Anh / International Journal of Engineering Science 75 (2014) 154–164 " M1 ¼  M3 ¼ # À@ À cc12 @1 22 c2 Àc11 c22 12 c22 " ;  @ 2t @ 21 ỵ q M2 ¼ ;  q@ 2t c66 0 c22 # ; ð6Þ M ¼ M T1 ; here we use the notations @ ¼ @=@x1 ; @ 21 ¼ @ =@x21 ; @ 2t ¼ @ =@t2 We call Eq (5) the matrix equation for the plane strain (1)– (4) From Eq (5) we immediately arrive at Stroh’s formulation (Stroh, 1962) It follows from (5): "  ðnÞ U T nị # " ẳ Mn #  U ; T M¼ M1 M2 M3 M4 ! ; n ¼ 1; 2; 3; ; x2 ½Àh; 0Š: ð7Þ  Let h be small (i e the layer is thin), then expanding into Taylor series TðÀhÞ at x2 ¼ up to the third-order of h we have: 1   Thị ẳ T0ị T 0ịh ỵ T 00 0ịh T 000 ð0Þh : 2! 3! ð8Þ  Suppose that surface x2 ¼ Àh is free of traction, i e Thị ẳ Introducing (7) with n ẳ 1; 2; at x2 ¼ into (8) yields: ! ! 3   ẳ hM3 h M ỵ h M7 U0ị; I hM ỵ h M h M8 Tð0Þ 6 ð9Þ where I is the identity matrix of order 2, M ; M are defined by (6) and M5 ¼ M3 M1 þ M4 M3 ; M ¼ M M ỵ M 24 ; 10ị M ẳ M M 21 ỵ M M M1 ỵ M M2 M3 ỵ M24 M3 ; M ẳ M M M2 ỵ M M3 M2 ỵ M3 M2 M ỵ M34 : Taking into account (6) and (10), the relation (9) in component form is of the form: h q €  12;11 ỵ r2 r r r3 u2;111 q ỵ r1 ịu2;1 c66 12  3 2 h € 22;1 À r6 u € 1;11 À q u € 1;tt ¼ 0; at x2 ¼ 0;  22;111 ỵ q  r5 r  1;1111 q  r7 u r4 r ỵ c66 r 12 ỵ h r1 r 22;1 r3 u1;11 q u1 ỵ h2 q   22;11 ỵ r 22 ỵ h r 12;1 q u2 ỵ r1 r r22 r3 u1;111 q ỵ r1 ịu1;1  ! 11ị ! c22 ! 2 h € 12;1 À r u € 2;11 À q u € 2;tt ¼ at x2 ẳ 0;  12;111 ỵ q 2;1111 q  r8 r  ỵ 2r1 ịu r2 r c22 ỵ 12ị where c12 c2 c11 c22 r3 r3 ; r2 ẳ r1 ỵ ; r ẳ 12 ; r4 ẳ r1 r2 ỵ c22 c66 c22 c22 ỵ r1 r1 ỵ r1 r5 ẳ ỵ ; r ẳ r ỵ r ịr3 ; r ẳ r 21 ỵ 2r ; r ẳ ỵ : c22 c22 c66 c66 r1 ẳ 13ị It should be noted that, 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 1 ; u 2 ; r  12 and r  22 by u1 ; u2 ; r12 and r22 , respectively These effective boundary conditions from Eqs (11) and (12) by replaced u 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 sliding 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 (11) and (12) As shown below, the effective boundary conditions for this case 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 cð> 0Þ and wave number kð> 0Þ in the x1 -direction and decaying in the x2 -direction The displacements and the stresses of the wave are sought in the form:  ðyÞeikðx1 ÀctÞ ; u  yịeikx1 ctị ; 1 ẳ U 2 ¼ U u r 12 ¼ ÀikT ðyÞeikðx1 ÀctÞ ; r 22 ẳ ikT yịeikx1 ctị 14ị P.C Vinh, V.T Ngoc Anh / International Journal of Engineering Science 75 (2014) 154–164 157 for the layer, and u1 ¼ U yịeikx1 ctị ; u2 ẳ U yịeikx1 ctị ; ikx1 ctị r12 ẳ ikT yịe 15ị r22 ẳ ikT yịeikx1 ctị ; for the half-space, where y ¼ kx2 Substituting (14) into (11) and (12) yields:  !  ! ! Á À Á e3 e2 q c2 e3 À q c4  0ị e r3 ỵ q  c2 r5 ỵ U  c2 ỵ  c2 r ỵ T 0ị er ỵ iT 0ị þ r2 þ Àr À q Àr À q c66 c66 6  ! & ' ! 3À 2  à Á  e e qc  0ị e r ỵ q  c2 ỵ r ị ẳ 0; T 0ị e ỵ  c2 r ỵ iT 0ị ỵ r q r1 ỵ þ iU c22 & !' & '  à  e qc  0ị e r ỵ q  0ị eq  c2 ỵ r ị ỵ U  c2 ỵ  c2 ỵ 2r1 ị ẳ 0; r q ỵ iU c22 16ị where e ẳ kh is the dimensionless thickness of the layer Let the contact between the layer and the half-space be sliding, we have (Barnett, Gavazza, & Lothe, 1988): r12 ¼ 0; r 12 ¼ 0; u2 ¼ u2 ; r22 ¼ r 22 at x2 ¼ 0; ð17Þ or, in view of the relations (14), equivalently: T 0ị ẳ 0; T 0ị ẳ 0;  0ị; U 0ị ẳ U T 0ị ¼ T ð0Þ: ð18Þ  we have: Introducing the second of (18) into (16) and eliminating U  0ị; iT 0ịa1 ỵ a2 e2 ị ẳ a3 e ỵ a4 e3 ịU 19ị where ẫ 1ẩ ed ed 2e2 e3 ị ỵ r2v x e22 e23 ỵ 2e2 e3 3e1 e2 2ed ỵ r 4v x2 ỵ 3e2 ị ; a3 ẳ c66 r 2v xr2v x ed ị; ẫ c66 ẩ e ỵ r 2v x 2ed ðe2 e3 À ed À 1Þ þ r 2v x À 2e2 e3 À e22 e23 ỵ 4ed ỵ 2e1 e2 2r 4v x2 ỵ e2 ị ; a4 ẳ 12 d a1 ¼ xr 2v À ed ; a2 ¼ in which x¼ c2 ; c22 rl ¼ e1 ¼ c66 ; c66 c11 ; c66 rv ¼ c2 ; c2 c66 c12 ; e3 ¼ ; ed ¼ e1 À e2 e23 ; c22 c66 sffiffiffiffiffiffiffi rffiffiffiffiffiffiffi c66 c66 : c2 ¼ ; c2 ¼ q q e2 ¼ ð21Þ From the last two equations of (18) and (19) it follows: T 0ịa1 ỵ a2 e2 ị ẳ ia3 e þ a4 e3 ÞU ð0Þ: ð22Þ From the first of (18) and (22) we see that the surface x2 ¼ of the half-space is subjected to the following conditions: T 0ị ẳ 0; 23ị T 0ịa1 ỵ a2 e2 ị ẳ iU 0ịa3 e ỵ a4 e3 Þ: The second of (23) is the desired approximate effective boundary condition (of third-order) The total effect of the layer on the half-space is replaced approximately by this condition An approximate secular equation of third-order Now we can ignore the layer and consider the propagation of Rayleigh waves in the orthotropic elastic half-space whose surface x2 ¼ is subjected to the boundary conditions (23) According to Vinh and Ogden (2004b), the displacement components of a Rayleigh wave travelling with velocity c and wave number k in the x1 -direction and decaying in the x2 -direction are determined by (15)1,2 in which U ðyÞ and U yị are given by: U yị ẳ B1 eb1 y ỵ B2 eb2 y ; 24ị U yị ẳ a1 B1 eb1 y ỵ a2 B2 eb2 y where B1 and B2 are constant to be determined and b1 ; b2 are roots of the equation c22 c66 b ỵ ẵc12 ỵ c66 ị2 ỵ c22 X c11 ị ỵ c66 X c66 ịb ỵ c11 Xịc66 Xị ẳ 0; whose real parts are positive to ensure to the decay condition, X ẳ qc , and: 25ị 158 P.C Vinh, V.T Ngoc Anh / International Journal of Engineering Science 75 (2014) 154–164 ak ¼ ibk ; bk ¼ bk ðc12 þ c66 Þ c22 bk À c66 þ qc2 ¼ c11 À qc2 À c66 bk ; ðc12 ỵ c66 ịbk k ẳ 1; 2; i ẳ p 1: 26ị From (25) we have: c12 ỵ c66 ị2 þ c22 ðX À c11 Þ þ c66 ðX À c66 ị ẳ S; c22 c66 c11 Xịc66 Xị 2 ẳ P: b1 b2 ẳ c22 c66 2 b1 ỵ b2 ẳ 27ị Note that the material constants c11 ; c22 ; c12 ; c66 satisfy the inequalities (3) without bars It is not difficult to verify that if the Rayleigh wave exists (! b1 ; b2 having positive real parts), then: < X < minfc11 ; c66 g; 28ị and: b1 b2 ẳ p P; q p S ỵ P: b1 ỵ b2 ẳ 29ị Using (15)1,2 and (24) into the stressstrain relations (2) without bars yields expressions of r12 and r22 that are given by (15)3,4 in which  à T yị ẳ ic66 b1 ỵ b1 ịB1 eb1 y þ ðb2 þ b2 ÞB2 eÀb2 y ;  à T yị ẳ c12 c22 b1 b1 ịB1 eb1 y ỵ c12 c22 b2 b2 ịB2 eÀb2 y : ð30Þ Introducing (24), (30) into Eqs (23) provides a homogeneous system of two linear equations for B1 ; B2 , namely: & f b1 ịB1 ỵ f b2 ịB2 ẳ Fb1 ịB1 ỵ Fb2 ịB2 ẳ 31ị where Fbn ị ẳ c12 c22 bn bn ịa1 ỵ a2 e2 ị bn a3 e ỵ a4 e3 ị; f bn ị ẳ c66 bn ỵ bn ị; n ẳ 1; 2: 32ị For a non-trivial solution, the determinant of the matrix of the system (31) must vanish, i.e.: f ðb1 ÞFðb2 Þ À f b2 ịFb1 ị ẳ 0: 33ị Introducing (32) into (33) leads to the dispersion equation of the wave: A0 ỵ A1 e ỵ A2 e2 ỵ A3 e3 ẳ 0; 34ị where A0 ẳ a1 ẵc12 ỵ c22 b1 b2 ịb2 b1 ị ỵ c12 ỵ c22 b1 b2 Þðb2 À b1 ފ; A1 ¼ a3 ðb1 b2 À b2 b1 ị; A2 ẳ a2 ẵc12 ỵ c22 b1 b2 ịb2 b1 ị ỵ c12 ỵ c22 b1 b2 ịb2 b1 ị; 35ị A3 ẳ a4 b1 b2 À b2 b1 Þ: By (26) it is not difcult to prove the following equalities: c11 Xịb1 ỵ b2 ị b2 b1 ị; c12 ỵ c66 ịb1 b2 c11 X c11 X ỵ c66 b1 b2 ; b2 À b1 ¼ À ðb2 À b1 ị: b1 b2 ẳ c22 b1 b2 c12 ỵ c66 Þb1 b2 b1 b2 À b2 b1 ¼ ð36Þ b2 b1  k ; k ẳ 0; 1; 2; 3ị; h ¼ Substituting (36) into (35) yields: Ak ¼ hA , where b1 b2 c12 ỵc66 ị  ẳ a1 c2 c11 c22 ỵ c22 Xịb1 b2 ỵ c11 XịX ; A 12  ẳ a3 c11 Xịb1 ỵ b2 ị; A  ẳ a2 c2 c11 c22 ỵ c22 Xịb1 b2 ỵ c11 XịX ; A 37ị 12  ẳ a4 c11 Xịb1 ỵ b2 ị; A in which b1 b2 and b1 ỵ b2 are given by (29) Removing the factor h, Eq (34) becomes: 0 ỵ A 1e ỵ A  e2 ỵ A  e3 ẳ 0: A 38ị P.C Vinh, V.T Ngoc Anh / International Journal of Engineering Science 75 (2014) 154–164 159 This is the desired third-oder approximate secular equation and it is fully explicit In the dimensionless form Eq (38) is: E0 ỵ E1 e ỵ E2 e2 ỵ E3 e3 ¼ 0; ð39Þ where À Á E0 ¼ xr 2v ed ẵe2 x ed ịb1 b2 ỵ e1 xịx; E1 ẳ r l r 2v x xr 2v ed e1 xịb1 ỵ b2 Þ; Â Ã É 1È E2 ¼ À r4v x2 ỵ 3e2 ị ỵ r 2v x e22 e23 þ 2e2 e3 À 2ed À 3e1 e2 À ed 2e2 e3 ed ị ẵe2 x ed ịb1 b2 ỵ e1 xịx; ẫ ẩ r l ed ỵ r 2v x 2ed e2 e3 ed 1ị ỵ r 2v x 2e2 e3 e22 e23 ỵ 4ed þ 2e1 e2 À 2r 4v x2 ð1 þ e2 ị e1 xịb1 ỵ b2 ị; E3 ẳ 12 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi ð1 À xÞðe1 À xÞ ; b1 b2 ẳ P ; b1 ỵ b2 ẳ S ỵ P ; P ¼ e2 S¼ e2 ðe1 À xị ỵ x ỵ e3 ị2 ; e2 e1 ẳ c11 ; c66 40ị and e2 ¼ c22 ; c66 e3 ¼ c12 ; c66 ed ¼ e1 e2 À e23 : ð41Þ It is clear from (39) and (40) that the squared dimensionless Rayleigh wave velocity x ¼ c2 =c22 depends on dimensionless parameters, namely, ek ; ek k ẳ 1; 2; 3ị; r l ; rv and e Note that ek > 0; ek > k ẳ 1; 2ị; ed > 0; ed > according to the inequalities (3) From (39) and the first of (40) it follows that, when e ¼ 0: pffiffiffi either ðe2 x À ed Þ P ỵ e1 xịx ẳ or x ẳ ed c22 =c22 ẳ x : 42ị The rst of (42) is the secular equation of Rayleigh waves propagating in an orthotropic elastic half-space (see Chadwick, 1976; Vinh & Ogden, 2004) and the second is the dimensionless speed of the longitudinal waves of the layer (considered as a plate) These facts say that in the limit e ! two modes are possible, one of which approaches the classical Rayleigh wave in the orthotropic half-space and the other approaches the longitudinal wave of the layer with the velocity given by (42)2 (see also, Achenbach & Keshava, 1967 for the isotropic case), provided xà < Fig presents the dependence on e ẳ k:h ẵ0; of the dimensionless Rayleigh wave velocity x ¼ c2 =c22 that is calculated by the exact secular equation (solid line) and by the approximate secular equation (39) (dashed line) Here we take e1 ¼ 2:5; e2 ¼ 3; e3 ¼ 1:5; e1 ¼ 1:8;  e2 ¼ 1;  e3 ¼ 0:6; rl ¼ 0:5; rv ¼ Note that the exact secular equation is of the determinant form, that is similar in form to Eq (30) in Achenbach and Keshava (1967) and is not reproduced here It is seen from Fig that the exact velocity curve and the third-order approximate one almost totally coincide with each other for the values of e ½0; 1Š 0.8 x 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ε Fig Plots of the dimensionless Rayleigh wave velocity xeị in the interval ẵ0; that is calculated by the exact secular equation (solid line) and by the approximate secular Eq (39) (dashed line) Here we take e1 ¼ 2:5; e2 ¼ 3; e3 ¼ 1:5;  e1 ¼ 1:8;  e2 ¼ 1; e3 ¼ 0:6; r l ¼ 0:5; r v ¼ 160 P.C Vinh, V.T Ngoc Anh / International Journal of Engineering Science 75 (2014) 154–164 Isotropic case When the layer and the half-space are both isotropic: c11 ¼ c22 ¼ k ỵ 2l; c12 ẳ k; c66 ẳ l; c11 ẳ c22 ẳ k ỵ 2l ; c12 ẳ k; c66 ẳ l : 43ị From (26), (27) and (43) one can see that: b1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi À cx; b2 ¼ pffiffiffiffiffiffiffiffiffiffiffi À x; b1 ¼ b1 ; b2 ẳ ; b2 44ị where c2 ẳ r l c2 l : ; x¼ 2; c¼ q k þ 2l c2 ð45Þ Introducing (20), (43) and (44) into (35) and after some manipulations, the equation (34) is equivalent to: ^0 ỵ A ^1e ỵ A ^ e2 ỵ A ^ e3 ẳ 0; A 46ị where i Ãh ^ ¼ Âr x À 4ð1 À c Þ ðx À 2Þ2 À 4b1 b2 ; A v à ^ ¼ r l r x2 b1 Âr x À 4ð1 À c Þ ; A v v i Ãh ^ ¼ À 81 c ị ỵ 4r 2v xc 2 2ị ỵ r 4v x2 ỵ 3c ị ðx À 2Þ2 À 4b1 b2 ; A n o ^ ị2 ỵ r 2v x 82 þ 3c Àc 2 Þ þ 2r 2v xð4 À 2c c 2 ị r 4v x2 ỵ c ị ; A3 ẳ rl xb1 81 c 47ị ẳl  =l; r v ẳ c2 =c2 ; c  =ð  Þ Eq (46) is the approximate secular equation of third-order for the isotropic case It k ỵ 2l here r l ẳ l shows that for this case the squared dimensionless Rayleigh wave velocity x ¼ c2 =c22 depends on four dimensionless param; r l and rv eters c; c  With the help of (40)–(44) it is not difficult verify that Ek ¼ Eck ; k ẳ 0; 1; 2; 3ị, where: 0 ẳ 4c  1ị ỵ r 2v x fẵ4c 1ị ỵ xb1 b2 ỵ cxịxg; E 1 ẳ r l r x 4c  1ị ỵ r 2v x cxịb1 ỵ b2 ị; E v 2 ẳ 81 c ị ỵ 4r 2v xc 2 2ị ỵ r 4v x2 ỵ 3c ị fẵ4c 1ị ỵ xb1 b2 ỵ cxịxg; E n o 3 ẳ rl 81 c ị2 ỵ r2v x 82 ỵ 3c c 2 ị ỵ 2r2v x4 2c c 2 ị r 4v x2 ỵ c ị cxịb1 ỵ b2 ị; E 48ị 0 þ E 1 e þ E  e2 þ E 3 e3 ẳ 0; E 49ị p p and b1 ¼ À cx; b2 ¼ À x Therefore, we have an alternative approximate secular equation of third-order for the isotropic case, namely: k ; ðk ¼ 0; 1; 2; 3Þ are given by (48) in which E 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 xðeÞ that is of the form: xeị ẳ x0ị ỵ x0 0ị e þ x00 ð0Þ x000 ð0Þ e þ e þ Oðe4 Þ; ð50Þ where xð0Þ is the squared dimensionless velocity of Rayleigh waves propagating in an orthotropic elastic half-space that is given by (see Vinh & Ogden, 2004b): x0ị ẳ p p s1 s2 s3 = s1 =3ịs2 s3 ỵ 2ị ỵ q! q p p Rỵ Dỵ R D ; 51ị e23 where s1 ¼ ee21 ; s2 ¼ À e1 e2 ; s3 ¼ e1 and R, D are given by: hðs1 ; s2 ; s3 Þ; 54 pffiffiffiffiffi Dẳ ẵ2 s1 s2 ịhs1 ; s2 ; s3 ị ỵ 27s1 s2 ị2 ỵ s1 s2 s3 ị2 ỵ 4; 108 Rẳ 52ị P.C Vinh, V.T Ngoc Anh / International Journal of Engineering Science 75 (2014) 154–164 161 in which hðs1 ; s2 ; s3 ị ẳ p s1 ẵ2s1 s2 s3 ị3 ỵ 93s2 s2 s3 2ị; 53ị and the roots in (51) taking their principal values From (39) it follows that:   E1  2E2 E20x 2E0x E1 E1x ỵ E0xx E21  00 ; x 0ị ẳ ;   E0x xẳx0ị E30x xẳx0ị h   i  x000 0ị ẳ 6E3 ỵ 6E2x x0 0ị ỵ 3E1xx x0 0ị ỵ 3E1x x00 0ị ỵ 3E0xx x0 0ịx00 0ị ỵ E0xxx x0 0ị =E0x  x0 0ị ẳ 54ị xẳx0ị where E1 ; E2 and E3 are given by (40) and: " # ( ) pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi ðe2 x À ed Þ À x e1 À x 4e2 x2 ẵ2ed ỵ 3e2 ỵ e1 ịx ỵ 2e1 e2 ỵ ed ỵ e1 ị  p p E0x ẳ r v ỵ e1 2x ; ỵ e1 xịx ỵ r v x À ed Þ pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi e2 e2 À x e1 À x E0xx E0xxx ( ) 4e2 x2 ẵ2ed ỵ 3e2 ỵ e1 ịx þ 2e1 e2 þ ed ð1 þ e1 Þ þ e1 À 2x ¼ 2r v pffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi e2 À x e1 À x > > < 2 8e2 x 12e2 ỵ e1 ịx ỵ 3e2 ỵ 6e1 ỵ e1 ịx þ ed ð1 À e1 Þ À 4e1 e1 ð1 þ e1 Þ= qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; þ ðr2v x À ed ị ỵ p > > : ; e2 ð1 À xÞ3 ðe1 À xÞ3 > > < = 2 8e x À 12e ỵ e ịx ỵ 3e ỵ 6e þ e Þx þ e ð1 À e Þ À 4e e ỵ e ị 2 d 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 3r 2v ỵ p > > 3 : ; e2 xị e1 xị ỵ 3r2v x ed ị1 e1 ị2 fẵe2 e1 þ 1Þ À 2ed Šx À 2e1 e2 þ ed ỵ e1 ịg qq ; p e2 À xÞ5 ðe1 À xÞ5  à 2x À e1 ỵ e2 ịb1 b2 E1x ẳ rl r2v ed e1 ỵ 2r 2v e1 ỵ ed ịx 3r2v x2 b1 ỵ b2 ị ỵ r l r 2v xðr2v x À ed Þðe1 À xÞ ; 2e2 b1 b2 b1 ỵ b2 ị & ' 2x e1 ỵ e2 ịb1 b2 E1xx ẳ 2r l r 2v r2v e1 ỵ ed 3r 2v xịb1 ỵ b2 ị þ Àed e1 þ 2ðr2v e1 þ ed Þx À 3r 2v x2 2e2 b1 b2 b1 ỵ b2 ị ( 2 r l r v xðr v x ed ịe1 xị ẵ2x e1 ỵ e2 ịb1 b2 ỵ ẵ4e2 b1 b2 ỵ ỵ e2 ị1 ỵ e1 2xị 2 4e22 b1 b2 b1 ỵ b2 ị b1 ỵ b2 ị ' ẵ2x e1 ỵ e2 ịb1 b2 ỵ e1 2xị ỵ ; b1 b2 r x ỵ 3e2 ị ỵ r2v xe22 e23 ỵ 2e2 e3 2ed 3e1 e2 ị þ ed ðed À 2e2 e3 Þ v ( ) 4e2 x2 ẵ2ed ỵ 3e2 ỵ e1 ịx ỵ 2e1 e2 ỵ ed ỵ e1 ị þ e1 À 2x  pffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi e2 À x e1 À x E2x ¼ À à  r 2v 2r 2v x1 ỵ 3e2 ị ỵ e22 e23 ỵ 2e2 e3 2ed 3e1 e2 ẵe2 x ed ịb1 b2 ỵ e1 xịx 55ị p p p here b1 b2 ẳ P; b1 ỵ b2 ẳ S ỵ P; P; S are given by (40) When the layer and the half-space are both isotropic, the Rayleigh wave velocity is (approximately) expressed by (50) in which:  1  2 E 2 2E 0x E 1 E 1x ỵ E 0xx E 2  2E E 0x x0 0ị ẳ   ; x00 0ị ẳ ;  3  E0x xẳx0ị E 0x xẳx0ị h   i  ỵ 6E 2x x0 0ị ỵ 3E 1xx x02 0ị ỵ 3E 1x x00 0ị ỵ 3E 0xx x0 0ịx00 0ị ỵ E 0xxx x03 0ị =E 0x  x000 0ị ẳ 6E xẳx0ị 1 ; E 2 ; E 3 are given by (48), E 0x ; E 0xx ; E 0xxx ; E 1x ; E 1xx ; E 2x are given by: E ð56Þ 162 P.C Vinh, V.T Ngoc Anh / International Journal of Engineering Science 75 (2014) 154–164 n o pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0x ¼ r2 ẵ4c 1ị ỵ x x cx ỵ cxịx E v " # 4cx2 ỵ 8c2 11c 3ịx ỵ 23 2c2 ị  1ị ỵ xr2v 2cx ỵ pp ỵ 4c ; x cx " # 4cx2 ỵ 8c2 11c 3ịx ỵ 23 2c2 ị  p ffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi E0xx ¼ 2r v À 2cx þ À x À cx 3 2  à 8c x À 12c1 ỵ cịx ỵ 3c ỵ 6c ỵ 1ịx ỵ 4c4 ỵ 3c c ị7  1ị ỵ xr 2v qq ỵ 4c 42c ỵ 5; ð1 À xÞ3 ð1 À cxÞ3 0xxx E 8c2 x3 12c1 ỵ cịx2 ỵ 3c2 ỵ 6c þ 1Þx þ 4cðÀ4 þ 3c À c2 Þ7 qq ẳ 3r 2v 42c ỵ xị3 cxị3 ỵ  1ị ỵ xr 2v ẵ8c2 7c ỵ 1ịx 22c2 1ị 3c 1ị2 ẵ4c qq ; xị5 cxị5 ẩ ẫ c b ỵ b1 1x ẳ rl r 4c  1ị ỵ 2ẵr 2v À 4cðc  À 1ފx À 3cr2v x2 ðb1 ỵ b2 ị r l r 2v xẵ4c  1ị ỵ xr2v cxị E ; v 2b1 b2 ' cb2 ỵ b1 ị  1ị 3cr 2v x b1 ỵ b2 ị 4c  1ị ỵ 2r 2v 4cc  ỵ 4cịx 3cr 2v x2 r2v À 4cðc 2b1 b2 " # 2  1ị ỵ r v x1 cxị 4cb1 b2 þ ð1 þ cÞð1 þ c À 2cxÞ À ðcb2 ỵ b1 ị r l r v xẵ4c c ỵ 2cxịcb2 ỵ b1 ị ; ỵ 2 b1 b2 b1 ỵ b2 4b1 b2 1xx ẳ 2r l r E v &   à 2x ẳ r2 r x1 ỵ 3c ị ỵ 2c 2 2ị fẵ4c 1ị ỵ xb1 b2 ỵ cxịxg E v v " # 4cx2 ỵ 8c2 11c 3ịx ỵ 23 2c2 ị 2 ị ỵ 4rv xc  2ị 8c  1ị 2cx ỵ pp ; rv x ỵ 3c x cx 57ị p p here b1 ẳ cx; b2 ẳ x; x0ị is the squared dimensionless velocity of Rayleigh waves propagating in an isotropic elastic half-space that is given by (see Vinh & Ogden, 2004a): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀ1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffi pffiffiffiffi xð0Þ ẳ 41 cị c ỵ R ỵ D ỵ R D ; 58ị 0.9 x 0.85 0.8 0.75 0.7 0.65 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ε Fig Plots of the dimensionless Rayleigh wave velocity xðeÞ in the interval ½0; 1Š that is calculated by the exact secular equation (3.9) in Tuan (2008) (solid  ¼ 2=3 line) and by the approximate formula (50) for the isotropic case (dashed line) Here we take r l ¼ 3; r v ¼ 0:6; c ¼ 1=3 and c P.C Vinh, V.T Ngoc Anh / International Journal of Engineering Science 75 (2014) 154164 163 where R ẳ 227 90c ỵ 99c2 32c3 ị=27; D ẳ 41 cị2 11 62c ỵ 107c2 64c3 ị=27; 59ị and the roots in the formula (58) taking their principal values Note that xð0Þ can be calculated by another formula derived by Malischewsky (2004), or by the approximate expressions with high accuracy obtained recently by Vinh and Malischewsky (2007, 2008) Fig presents the dependence on e ½0; 1Š of the dimensionless Rayleigh wave velocity x ¼ c2 =c22 that is calculated by the exact secular equation (3.9) in Tuan (2008) (solid line) and by the approximate formula (50) for the isotropic case (dashed  ¼ 2=3 It is shown from Fig that the exact velocity curve and the thirdline) Here we take r l ¼ 3; r v ¼ 0:6; c ¼ 1=3 and c order approximate one are very close to each other for the values of e ½0; 0:8Š Conclusions In this paper, the propagation of Rayleigh waves in an orthotropic elastic half-space covered with a thin orthotropic elastic layer is investigated The contact between the layer and half space is assumed to be sliding An approximate secular equation of third-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 An approximate formula of third-order for the velocity of Rayleigh waves is established using the obtained approximate secular equation and it is a good approximation The approximate secular equation and the approximate formula for the velocity are fully explicit, they will be useful in practical applications Acknowledgement The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) References Achenbach, J D., & Keshava, S P (1967) Free waves in a plate supported by a semi-infinite continuum Journal of Applied Mechanics, 34, 397–404 Barnett, D M., Gavazza, S D., & Lothe, J (1988) Slip waves along the interface between two anisotropic elastic half-spaces in sliding contact Proceedings of the Royal Society of London A, 415, 389–419 Benveniste, Y (2006) A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media Journal of the Mechanics and Physics of Solids, 54, 708–734 Bovik, P (1996) A comparison between the Tiersten model and O(H) boundary conditions for elastic surface waves guided by thin layers Journal of Applied Mechanics, 63, 162–167 Chadwick, P (1976) The existence of pure surface modes in elastic materials with orthorhombic symmetry Journal of Sound and Vibration, 47, 39–52 Every, A G (2002) Measurement of the near-surface elastic properties of solids and thin supported films Measurement Science and Technology, 13, R21–R39 Hess, P., Lomonosov, A M., & Mayer, A P (2014) Laser-based linear and nonlinear guided elastic waves at surfaces (2D) and wedges (1D) Ultrasonics, 54, 39–55 Kuchler, K., & Richter, E (1998) Ultrasonic surface waves for studying the properties of thin films Thin Solid Films, 315, 29–34 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, 5028–5034 Malischewsky, A P (2004) A note on Rayleigh-wave velocities as a function of the material parameters Geofisica Internasional, 43, 507–509 Mindlin, R D (1951) Influence of rotatory inertia and shear on flexural motion isoropic elastic plates Journal of Applied Mechanics, 18, 31–38 Muller, P., & Touratier, M (1996) On the so-called variational consistency of plate models, I Indefinite plates: Evaluation of dispersive behaviour Journal of Sound and Vibration, 188, 515–527 Niklasson, A J., Datta, S K., & Dunn, M L (2000) On approximating guided waves in thin anisotropic coatings by means of effective boundary conditions Journal of the Acoustical Society of America, 108, 924–933 Rokhlin, S I., & Huang, W (1992) Ultrasonic wave interaction with a thin anisotropic layer between two anisotropic solids: Exact and asymptoticboundary-condition methods Journal of the Acoustical Society of America, 92, 1729–1742 Rokhlin, S I., & Huang, W (1993) Ultrasonic wave interaction with a thin anisotropic layer between two anisotropic solids: II Second-order asymptotic boundary conditions Journal of the Acoustical Society of America, 94, 3405–3420 Shuvalov, A L., & Every, E G (2002) Some properties of surface acoustic waves in anisotropic-coated solids, studied by the impedance method Wave Motion, 36, 257–273 Steigmann, D J (2007) Surface waves supported by thin-film/substrate interactions IMA Journal of Applied Mathematics, 72, 730–747 Stephen, N G (1997) Mindlin plate theory: Best shear coefficient and higher spectra validity Journal of Sound and Vibration, 202, 539–553 Stroh, A N (1962) Steady state problems in anisotropic elasticity Journal of Mathematical Physics, 41, 77–103 Tiersten, H F (1969) Elastic surface waves guided by thin films Journal of Applied Physics, 46, 770–789 Ting, T C T (1996) Anisotropic elasticity: Theory and applications NewYork: Oxford University Press Ting, T C T (2009) Steady waves in an anisotropic elastic layer attached to a half-space or between two half-spaces-a generalization of Love waves and Stoneley waves Mathematics and Mechanics of Solids, 14, 52–71 Touratier, M (1991) An efficient standard plate theory International Journal of Engineering Science, 29, 901–916 Tuan, T T (2008) The ellipticity (H/V -ratio) of Rayleigh surface waves PhD thesis, Friedrich-Schiller University Jena Vinh, P C., & Linh, N T K (2012) An approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half-space coated by a thin orthotropic elastic layer Wave Motion, 49, 681–689 Vinh, P C., & Linh, N T K (2013) An approximate secular equation of generalized Rayleigh waves in pre-stressed compressible elastic solids International Journal of Non-Linear Mechanics, 50, 91–96 Vinh, P C., & Malischewsky, P (2007) An improved approximation of Bergmann’s form for the Rayleigh wave velocity Ultrasonic, 47, 49–54 Vinh, P C., & Malischewsky, P (2008) Improved approximations of the Rayleigh wave velocity Journal of Thermoplastic Composite Materials, 21, 337–352 Vinh, P C., & Ogden, R W (2004a) On formulas for the Rayleigh wave speed Wave Motion, 39, 191–197 164 P.C Vinh, V.T Ngoc Anh / International Journal of Engineering Science 75 (2014) 154–164 Vinh, P C., & Ogden, R W (2004b) Formulas for the Rayleigh wave peed in orthotropic elastic solids Archives of Mechanics, 56, 247–265 Vinh, P C., & Seriani, G (2009) Explicit secular equations of Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity Wave Motion, 46, 427–434 Vinh, P C., & Seriani, G (2010) Explicit secular equations of Stoneley waves in a non-homogeneous orthotropic elastic medium under the influence of gravity Applied Mathematics and Computations, 215, 3515–3525 Wang, J., Du, J., Lu, W., & Mao, H (2006) Exact and approximate analysis of surface acoustic waves in an infinite elastic plate with a thin metal layer Ultrasonics, 44, e941–e945  ... C T (2009) Steady waves in an anisotropic elastic layer attached to a half- space or between two half- spaces -a generalization of Love waves and Stoneley waves Mathematics and Mechanics of Solids,... Consider an elastic half- space x2 P coated by a thin elastic layer Àh x2 The layer and the half- space are both homogeneous, compressible, orthotropic and they are in sliding contact with each other... University Jena Vinh, P C., & Linh, N T K (2012) An approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half- space coated by a thin orthotropic elastic layer Wave Motion,