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Meccanica DOI 10.1007/s11012-016-0464-5 Rayleigh waves in an orthotropic elastic half-space overlaid by an elastic layer with spring contact Pham Chi Vinh Vu Thi Ngoc Anh Received: 12 October 2015 / Accepted: 30 May 2016 Ó Springer Science+Business Media Dordrecht 2016 Abstract In this paper, the propagation of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer of arbitrary uniform thickness is investigated The layer and the half-space are both compressible and they are in spring contact with each other The main aim of the paper is to derive explicit exact secular equation of the wave This equation has been derived by using the effective boundary condition method From the obtained secular equation, the secular equations for the welded and sliding contacts can be derived immediately as special cases For the welded contact, the obtained secular equation recovers the secular equations previously obtained for the isotropic and orthotropic materials Since the obtained secular equation is totally explicit it is a good tool for nondestructively evaluating the adhesive bond between the layer and half-space as well as their mechanical properties Keywords Rayleigh waves Á A half-space coated by a layer Á Spring contact Á Exact explicit secular equation Á The effective boundary condition method P C Vinh (&) Á V T N Anh Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam e-mail: pcvinh@vnu.edu.vn Introduction An elastic half-space overlaid by an elastic layer is a model (structure) finding a wide range of applications such as those in seismology, acoustics, geophysics, materials science and micro-electro-mechanical systems The measurement of mechanical properties of supported layers therefore plays an important role in understanding the behaviors of this structure in applications, see for examples Makarov et al [1] and references therein Among various measurement methods, the surface/ guided wave method is most widely used [2], because it is non-destructive and it is connected with reduced cost, less inspection time, and greater coverage [3] Among surface/guided waves, the Rayleigh wave is a versatile and convenient tool [3, 4] Since the explicit dispersion relations of Rayleigh waves are employed as theoretical bases for extracting the mechanical properties of layers and half-spaces and the adhesion between them from experimental data, they are therefore the main purpose of any investigation of Rayleigh waves propagating in elastic half-spaces covered by an elastic layer When the layer is thin (i.e., its thickness is small in comparison with the wavelength), approximate secular equations of the wave are derived by the effective boundary condition method that replaces the entire effect of the thin layer on the half-space by the socalled (approximate) effective boundary conditions The effective boundary conditions are established by either replacing approximately the layer by a plate [5, 6], or by expanding the stresses at the upper-surface 123 Meccanica of the layer into Taylor series of the layer thickness [7–13] Wang et al [8] derived the first-order approximate secular equation for piezoelectric materials, Tiersten [6], Bovik [7], Steigmann and Ogden [9] obtained the second-order approximate secular equations, Vinh and Linh [10, 11], Vinh and Anh [12, 13] obtained the third-order and fourth-order approximate secular equaations for elastic solids For the case in which the layer thickness is arbitrary, the results are limited When the half-space and the layer are both isotropic, the explicit secular equation of Rayleigh waves was derived by Haskell [14], BenMenahem and Singh [15] [Eq (3.113), p 117] For the orthotropic case, the explicit secular equation of Rayleigh waves was derived by Sotiropoulos [16] For the case when the half-space and the layer are both subjected pure pre-strains, the explicit secular equation of Rayleigh waves was derived by Ogden and Sotiropoulos [17] for incompressible materials and by Sotiropoulos [18] for compressible materials In all mentioned investigations, the contact between the layer and the half-space is perfectly bonded and the secular equations are derived by directly expanding a six-order determinant that is established by the traction-free conditions at the uppersurface of the layer and the continuity conditions for displacements and stresses through the interface As is well known, bonded interfaces are often compromised due to imperfect bonding conditions and degradation over time caused by various mechanical/ thermal loadings and environmental factors [19, 20] Therefore, the imperfectly bonded interface is actual contact between two solids There has been a number of approaches to model imperfect interfaces and probably the most commonly used approach is the so-called spring contact model [21–25] In the spring contact model, the displacements are discontinuous through the interface, the stresses are continuous and they are proportional to the jumps of displacements In particular, let the interface be the plane x2 ¼ 0, then the spring boundary conditions enforced on the imperfect interface x2 ẳ are [21, 23] rk2 x2 ẳ 0ỵ Þ ¼ rk2 ðx2 ¼ 0À Þ; k ¼ 1; 2; 3; r22 ¼ KN u2 ðx2 ¼ 0ỵ ị u2 x2 ẳ ị ; r12 ẳ KT1 u1 x2 ẳ 0ỵ ị u1 x2 ẳ ị ; r23 ẳ KT2 u3 x2 ẳ 0ỵ ị u3 x2 ẳ 0À Þ 123 ð1Þ where KT1 ð [ 0Þ, KT2 ð [ 0Þ and KN ð [ 0Þ are shear and normal spring stiffnesses The sliding contact [26] r12 ðx2 ẳ 0ỵ ị ẳ r12 x2 ẳ ị ẳ 0; r23 x2 ẳ 0ỵ ị ẳ r23 x2 ẳ ị ẳ 0; u2 x2 ẳ 0ỵ ị ẳ u2 x2 ẳ ị; 2ị r22 x2 ẳ 0ỵ Þ ¼ r22 ðx2 ¼ 0À Þ is obtained directly from the spring boundary conditions (1) by letting KT1 , KT2 approach to zero and KN to go to ỵ1 The perfectly bonded (welded) contact [27] uk x2 ẳ 0ỵ Þ ¼ uk ðx2 ¼ 0À Þ; rk2 ðx2 ¼ 0ỵ ị ẳ rk2 x2 ẳ ị; k ẳ 1; 2; ð3Þ is derived directly from the spring model (1) by letting KT1 , KT2 and KN all approach to ỵ1 This paper is concerned with the propagation of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer of arbitrary uniform thickness The layer and the half-space are both compressible and they are in spring contact with each other The main aim of the paper is to derive explicit exact secular equation of the wave This equation has been derived by using the effective boundary condition method From the obtained secular equation, the secular equations for the welded and sliding contacts can be derived immediately as special cases For the welded contact, the obtained secular equation recovers the one derived by Sotiropolous [16] for orthotropic materials, and the one obtained by BenMenahem and Singh [15] for isotropic materials Since the obtained secular equation is totally explicit it is a good tool for nondestructively evaluating the adhesive bond between the layer and half-space as well as their mechanical properties Exact effective boundary condition Consider a homogeneous elastic half-space x2 ! coated by a homogeneous elastic layer Àh x2 of thickness h The half-space and the layer are both orthotropic, compressible and they are in spring contact with each other Note that 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 Meccanica ui ¼ ui ðx1 ; x2 ; tị; ui ẳ ui x1 ; x2 ; tị; i ẳ 1; 2; u3 ẳ u3 4ị where ui , ui are components of the displacement vector, t is the time Since the layer is made of orthotropic elastic materials, the strain-stress relations are r11 ¼ c11 u1;1 ỵ c12 u2;2 ; r22 ẳ c12 u1;1 ỵ c22 u2;2 ; r12 ẳ c66 u1;2 ỵ u2;1 Þ ð5Þ where commas indicate differentiation with respect to spatial variables xk , rij are the stresses, the material constants c11 , c22 , c12 , c66 satisfy the inequalities ckk [ 0; k ¼ 1; 2; 6; c11 c22 À c212 [ ð6Þ which are necessary and sufficient conditions for the strain energy to be positive definite In the absence of body forces, the equations of motion for the layer is 1 ; r12;1 ỵ r22;2 ẳ qu2 r11;1 ỵ r12;2 ẳ qu 7ị where q is the mass density of the layer, a dot signifies differentiation with respect to t Substituting (5) into (7) and taking into account (4) yield c11 u1;11 ỵ c66 u1;22 ỵ c12 ỵ c66 ị u2;12 ẳ qu1 ; c12 þ c66 Þ u1;12 þ c66 u2;11 þ c22 u2;22 ¼ qu€2 ð8Þ where p1 ; p2 are complex in general and no requirements are imposed on their real and imaginary parts Using (9)–(11) into (5) we have 1 ðyÞeikðx1 ctị ; r22 ẳ kR 2 yịeikx1 ctị r12 ẳ kR 9ị 12ị in which 1 yị ẳ b1 A1 shp1 yị ỵ b1 A2 chp1 yị ỵ b2 A3 shp2 yị R ỵ b2 A4 chp2 yị; 2 yị ẳ i c1 A1 chp1 yị ỵ c1 A2 shp1 yị ỵ c2 A3 chp2 yị R ỵ c2 A4 shp2 yị 13ị with bn ẳ c66 pn an ị; cn ẳ c12 ỵ c22 pn an Þ; Now we consider the propagation of a Rayleigh wave, traveling along the interface between the layer and the half-space with velocity c ð [ 0Þ and wave number k ð [ 0Þ in the x1 -direction and decaying in the x2 direction The displacements of the Rayleigh wave in the layer, that satisfy (8), are given by yịeikx1 ctị ; u2 ẳ U yịeikx1 ctị u1 ẳ U pj c12 ỵ c66 ị ; j ¼ 1; 2; 2 c22 pj À c66 ỵ qc s s p p 2 S ỵ S 4P S S À 4P ; p2 ¼ ; p1 ¼ 2 c11 ị ỵ c66 qc c66 ị c12 ỵ c66 ị2 ỵ c22 ðqc S ¼ À ; c22 c66 À ÁÀ Á c66 À qc 2 c11 À qc P ¼ c22 c66 ð11Þ aj ¼ À n ¼ 1; 14ị Suppose that surface x2 ẳ h is free of traction, i e r12 ¼ 0; r22 ¼ at x2 ẳ h 15ị Using (12) and (13) into (15) leads to b1 A1 shðe1 Þ À b1 A2 che1 ị ỵ b2 A3 she2 ị b2 A4 che2 ị ẳ 0; c1 A1 che1 ị c1 A2 she1 ị ỵ c2 A3 che2 ị c2 A4 she2 ị ẳ where y ẳ kx2 and yị ẳA1 chp1 yị ỵ A2 shp1 yị þ A3 chðp2 yÞ U þ A4 shðp2 yÞ;  yị ẳi a1 A1 shp1 yị ỵ a1 A2 chp1 yị ỵ a2 A3 shp2 yị U þ a2 A4 chðp2 yÞ ð10Þ A1 ; A2 ; A3 ; A4 are constants and for simplicity we use the notations sh:ị :ẳ sinh:ị; ch:ị :ẳ cosh:ị, the quantities aj and pj are determined by 16ị where en ẳ pn e; n ¼ 1; 2; e ¼ kh Putting x2 ¼ in (10) and (13), we deduce 0ị ẳA1 ỵ A3 ; U 0ị ẳ i U a1 A2 ỵ a2 A4 ị; 1 0ị ẳb1 A2 ỵ b2 A4 ; R 2 0ị ẳ i R c1 A1 ỵ c2 A3 ị 17ị Solving the system (17) for A1 ; A2 ; A3 ; A4 , we obtain 123 Meccanica i c2 U 0ị ỵ R 0ị; ẵ c ẵ c ib2 a2 U 0ị ỵ A2 ẳ R1 0ị; ẵ a; b ½ a; b A1 ¼ i c R2 0ị; A3 ẳ U 0ị ẵ c ẵ c ib a1 A4 ẳ 1 U 0ị R1 0ị ẵ a; b ½ a; b From the first two of (23) we have 18ị here we use the notations ẵf ; g ẳ f2 g1 f1 g2 ; ẵf ẳ f2 f1 When KN ! ỵ1 and KT ! ỵ1 the contact between the layer and the half-space becomes perfectly bonded ð19Þ Substituting (18) into (16) yields 1 0ị ia12 R 2 0ị ỵ b11 U 0ị ib12 U 0ị ẳ 0; a11 R ð0Þ À ib22 U 0ị ẳ a21 R1 0ị ia22 R2 0ị ỵ b21 U 20ị 0ị ẳ k R1 0ị ỵ U1 0ị ; U KT 0ị ẳ k R2 0ị ỵ U2 ð0Þ U KN ð24Þ Introducing the last two of (23) and (24) into (20) leads to k k b11 R1 ð0Þ À i a12 À b12 R2 0ị a11 KT KN ỵ b11 U1 0ị ib12 U2 0ị ẳ 0; k k a21 À b21 R1 ð0Þ À i a22 b22 R2 0ị KT KN ỵ b21 U1 0ị ib22 U2 0ị ẳ 25ị where a11 a22 ½ a; bche ½bshe ½ cshe; a ; a ; a21 ¼ ¼ ¼ À 12 ; ½ c ½ a; b ½ a; b b b ẵche ẵ cche ẵbshe; c ; b11 ẳ ; b12 ẳ ; ẳ ẵ c ½ c ½ a; b b21 ¼ À cshe ½b; c1 c2 ½che ; b22 ¼ ½ c ½ a; b ð21Þ Since the layer and the half-space are in spring contact to each other at the plane x2 ¼ 0, we have from (1) (see also [21, 23, 24]) r12 ẳ KT u1 u1 ị; r22 ¼ KN ðu2 À u2 Þ; r12 ¼ r12 ; r22 ẳ r22 at x2 ẳ 22ị This is the desired exact effective boundary conditions that replace exactly the entire effect of the layer on the half-space Explicit secular equation Now we consider the propagation of a Rayleigh wave, traveling along surface x2 ¼ of the half-space with velocity c and wave number k in the x1 -direction, decaying in the x2 -direction, and satisfying the exact effective boundary conditions (25) According to Vinh and Ogden [28], the displacements of the Rayleigh wave in the half-space x2 [ are given by u1 ẳ U yịeikx1 ctị ; u2 ẳ U yịeikx1 ctị ; y ¼ kx2 or equivalently due to (9), (10), (12) and (13) 0ị ; k kR1 0ị ẳ KT ½U1 ð0Þ À U ð0Þ ; R2 ð0Þ ẳ KN ẵU2 0ị U 1 0ị ; R2 0ị ẳ R 2 0ị R1 0ị ẳ R 26ị where ð23Þ Un ð0Þ and Rn ð0Þ (n=1,2) are the displacement and traction amplitudes of the half-space at the interface x2 ẳ KT [ 0ị and KN [ 0Þ are the shear and normal spring stiffnesses, respectively From (23) it implies that When KT ¼ and KN ! ỵ1 the half-space and the layer are in sliding contact 123 U1 yị ẳ B1 eb1 y ỵ B2 eb2 y ; U2 yị ẳ ia1 B1 eb1 y ỵ a2 B2 eb2 y ị 27ị B1 and B2 are constants to be determined, and ak ¼ c12 ỵ c66 ịbk ; c22 b2k c66 ỵ X k ẳ 1; 2; X ẳ qc2 28ị b1 and b2 are two roots having positive real part (in order to make the decay condition satisfied) of the following equation Meccanica b4 Sb2 ỵ P ẳ 29ị f b1 ịFb2 ị f b2 ịFb1 ị ẳ ð38Þ S and P are calculated by (11) without bars It has been shown that if a Rayleigh wave exists, then [28] Substituting (37) into (38) and after some calculations we arrive at 0\X\minfc66 ; c11 g n ð30Þ and [29] q p p P[0; S ỵ P[0; b1 b2 ẳ P; b1 ỵ b2 ẳ S ỵ P ð31Þ Introducing (26) and (27) into the strain-stress relation (5) without bars leads to r12 ẳ kR1 yịeikx1 ctị ; r22 ẳ kR2 yịeikx1 ctị 32ị a11 a22 a12 a21 Þ À k ða11 b22 À a21 b12 Þ KN k ða12 b21 À a22 b11 Þ KT o k2 b11 b22 b12 b21 ị ẵc; b a11 b21 a21 b11 ịẵb ỵ KT KN o n k b11 b22 b12 b21 ị ẵa; b ỵ a11 b22 a21 b12 ị KT o n k b11 b22 b12 b21 ị ẵc a12 b21 a22 b11 ị ỵ KN ỵ a12 b22 a22 b12 ịẵa; c ỵ b11 b22 b12 b21 ịẵa ẳ ỵ 39ị in which R1 yị ẳ b1 B1 eb1 y ỵ b2 B2 eb2 y ; 33ị R2 yị ẳ ic1 B1 eb1 y ỵ c2 B2 eb2 y ị and bj ẳ c66 bj ỵ aj ị; cj ẳ c12 c22 bj aj ; j ẳ 1; 34ị Putting x2 ¼ in (27) and (33) gives U1 ð0Þ ¼B1 ỵ B2 ; U2 0ị ẳ ia1 B1 ỵ a2 B2 ị; R1 0ị ẳb1 B1 ỵ b2 B2 ; R2 0ị ẳ ic1 B1 ỵ c2 B2 ị 35ị With the help of (28) and (34), it is not difficult to verify that o n ẵc;b ẳ c66 c212 c22 c11 Xị b1 b2 ỵ Xc11 Xị h; ẵa;b ẳ c66 c11 Xịb1 ỵ b2 ịh; ẵa;c ẳ c66 c11 X c12 b1 b2 ịh; ẵa ẳ X c11 c66 b1 b2 ịh; ẵb ẳ ẵa;c; ẵc ẳ c22 c66 b1 b2 b1 ỵ b2 ịh 40ị where h ẳ b2 b1 ị=c12 ỵ c66 ịb1 b2 ị Introducing the expressions of aij and bij given by (21) into (39) and using the equalities (40) yield Substituting (35) into (25) leads to two linear equations for B1 and B2 , namely A1 ỵ B1 she1 she2 ỵ C1 she1 che2 ỵ D1 she2 che1 f b1 ịB1 ỵ f b2 ịB2 ẳ 0; Fb1 ịB1 ỵ Fb2 ịB2 ẳ where the coefficients A1 , B1 , C1 , D1 , E1 are given by (54) in the Appendix Equation (41) in which A1 , , E1 are determined by (54) in the Appendix is the desired exact secular equation It is totally explicit When e ! (the layer is absent), from (41) and the last of (54) it implies n opffiffiffi ð42Þ c212 À c22 c11 Xị P ỵ Xc11 Xị ẳ ð36Þ where k k f ðbn ị ẳ a11 b11 bn ỵ a12 b12 cn KT KN ỵ b11 ỵ b12 an ; k k b21 bn ỵ a22 b22 cn Fbn ị ẳ a21 KT KN ỵb21 ỵ b22 an n ẳ 1; 2ị 37ị For a non-trivial solution, the determinant of the matrix of the system (36) must vanishes, i.e., ỵ E1 che1 che2 ẳ ð41Þ This equation is the secular equation of Rayleigh waves propagating along the traction-free surface of a compressible orthotropic half-space [28] When the layer and the substrate are both isotropic we have 123 Meccanica c11 ẳ c22 ẳ k ỵ 2l; c12 ¼ k; c66 ¼ l; c66 ¼ l c12 ¼ k; c11 ¼ c22 ¼ k þ 2l; ð43Þ From (41) and (54) and taking into account (43) we obtain the secular equation for the isotropic case, namely A2 ỵ B2 she1 she2 ỵ C2 she1 che2 ỵ D2 she2 che1 ỵ E2 che1 che2 ẳ ð44Þ in which the coefficients A2 , B2 , C2 , D2 , E2 are given by (55) in the Appendix Taking the limit of Eq (44) when cN and cT both go to zero we obtain the secular equation of Rayleigh waves in an isotropic half-space coated by an isotropic layer with welded contact By multiplying two side this secular equation by k8 =ðÀb1 b2 Þ we arrive immediately at the well-known secular equation of Rayleigh waves for the isotropic case, Eq (3.113), p 117 in Ref [15] for the welded contact Remark From Eq (41) one can easily arrive at the explicit secular equations for two special cases: the welded contact and the sliding contact by: Taking the limit of two sides of Eq (41) when KN and KT both approach to ỵ1, for the welded contact Multiplying two sides of Eq (41) by KT =kc66 and then taking the limit of the resulting equation when KT ! and KN ! ỵ1, for the sliding contact cT and cN are dimensionless shear and normal spring compliances Note that cN ! 0, cT ! 0, rv [ 0, rl [ and according to (6): ek [ 0, ek [ ðk ¼ 1; 2; 3Þ, e1 e2 À e23 [ 0, e1 À e2 e23 [ Equation (41) can be rewritten as h ep ỵ p ị i h ep p ị i 2 ỵ E1 B1 ịsh2 2 C1 ỵ D1 C1 D1 ỵ shẵep1 ỵ p2 ị ỵ shẵep1 p2 ị 2 46ị ỵ A1 ỵ E1 ẳ E1 þ B1 Þsh2 Using (54) we derive the expressions of the coefficients E1 ỵ B1 , E1 B1 , C1 ỵ D1 , C1 D1 and A1 ỵ E1 Introducing these expressions into (46) yields h eðp þ p Þ i h eðp À p Þ i 2 sh2 2 Aðg; gÞ À Ag; gị p1 ỵ p2 ị2 p1 p2 ị2 shẵep1 ỵ p2 ị shẵep1 p2 ị ỵ Bg; gị Bg; gị p1 ỵ p2 p1 p2 ỵ Cg; gị ẳ sh2 47ị where p1 , p2 are defined by (11) in which P and S are expressed in terms of the dimensionless parameters as follows S ¼ ð e1 À rv2 xị ỵ e2 rv2 x e3 þ 1Þ2 ; ð48Þ e1 À rv2 xÞð1 À rv2 xị P ẳ e2 and Dimensionless secular equation Ag; gị ẳ f gị n f gị 1=2 ỵ ge2 ị g g2 À1=2 It is useful to convert the secular Eq (41) into dimensionless form For this aim we introduce dimensionless parameters c11 e1 ¼ ; c66 c66 e2 ¼ ; c22 c66 rl ¼ ; c66 kc66 cT ¼ KT 123 c22 c12 c11 e2 ¼ ; e3 ¼ ; e1 ¼ ; c66 c66 c66 c12 e3 ¼ ; c66 rffiffiffiffiffiffi rffiffiffiffiffiffi c2 c66 c66 ; c2 ¼ rv ¼ ; c ¼ ; q c2 q kc66 ; cN ẳ KN ỵ 2rl1 e3 e2 1=2 ỵ rl2 ỵ ge2 ị 1=2 gị1 e3 e2 gÞ f ðgÞ À g2 fð gÞ h f gị io 1=2 cT ỵ cN e2 gịb1 þ b2 Þ þ cT cN ; 1Àg À g2 fð gÞ À1 n 1=2 À1=2 Bðg; gị ẳ b1 ỵ b2 ịẵe2 g ỵ e2 g r g2 l f gị o 1=2 ỵ ẵcT ỵ cN e2 g ; g2 f gị 1=2 Cg; gị ẳ 2rl2 e2 g g2 ỵ 45ị 49ị Meccanica here q p g ẳ xị=e1 xị; g ẳ rv2 xị= e1 rv2 xị; f gị ẳ 1=2 e23 e2 g3 ỵ e1 g ỵ ẵe2 e1 1Þ À À1=2 e23 ge2 À1 ð50Þ function fð gÞ is given by (50) in which e1 ; e2 and e3 are replaced by e1 ; 1= e and e3 , respectively In the p2 p Eq (49): b1 ỵ b2 ị ẳ S ỵ P in which e2 e1 xị ỵ x e3 ỵ 1ị2 ; e2 e1 xị1 xị Pẳ e2 Sẳ 51ị It is clear that the left side of (47) is an explicit function of x and ten dimensionless parameters defined by (45) Taking the limit of two sides of (47) when cN and cT both go to zero we obtain the secular equation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer with welded contact This is the Eq (47) in which fð gÞ Aðg; gị ẳ g2 & f gị 1=2 1=2 ỵ ge2 ị ỵ 2rl1 e3 e2 gÞ À g2 ' f ðgÞ 1=2 1=2 e3 e2 gị ỵ rl2 ỵ ge2 Þ ; À g2 fð gÞ À1 1=2 1=2 r b1 ỵ b2 ịẵe2 g ỵ e2 g; Bg; gị ẳ g2 l f gị 1=2 52ị Cg; gị ẳ 2rl2 e2 g g2 By comparing this equation with the secular equation derived by Sotiropoulos, Eq (16) in Ref [16], one can immediately discover misprints in this secular equation (and also in the secular Eq (8) in Ref [30]) which have been mentioned in Ref [31] Note that the eqution (47) with the coefficients given by (52) is totally explicit, while Eq (16) in Ref [16] and Eq (8) in [30] are both not totally explicit because they both contain an implicit factor ðs1 þ s2 Þ (in the expressions of Bðg; gÞ and Bðg; À gÞ) For the sliding contact the coefficients of Eq (47) are simplified to h fð gÞ i Ag; gị ẳ b1 ỵ b2 ị; g2 ð53Þ fð gÞ À1 f ðgÞ r ; Cðg; gị ẳ Bg; gị ẳ g2 l À g2 Numerical examples As an application of the obtained secular equations we use them to numerically examine the dependence of the Rayleigh wave velocity on the dimensionless parameter e ¼ kh (understood as the dimensionless thickness of the layer or the dimensionless wave number) and on the dimensionless spring compliances cN and cT The material dimensionless parameters are taken as: For the Figs 1, 3, 5: e1 ¼ 3:0; e2 ¼ 3:5; e3 ¼ 1:5; e1 ¼ 1:8; e2 ¼ 1:2; e3 ¼ 0:5; rv ¼ 2:8; rl ¼ 0:5 For the Figs 2, 4, 6: e1 ¼ 3:2; e2 ¼ 2:8; e3 ¼ 1:5; e1 ¼ 1:8; e2 ¼ 1:1; e3 ¼ 0:6; rv ¼ 2; rl ¼ The wave velocity curves of first modes in the interval e ½0 2:5 corresponding to the spring contact are presented in Fig (modes 0, 1, 2, 3, 4, 5) and Fig (modes 0, 1, 2, , 4) Figures 3, show the wave velocity curves of first four modes (0, 1, 2, 3) in the interval e ½0 2:5 of the welded contact (solid line) and sliding contact (dashed line) Figures 5, present the effect of the dimensionless spring compliances cN (characterizing the normal imperfection) and cT (characterizing the shear imperfection) on the wave velocity Recall that for the perfectly bonded contact cN ¼ cT ¼ It is shown from these figures that: x The Rayleigh wave velocity decreases when the dimensionless thickness of the layer (or the 0.8 0.6 0.4 0.2 0 0.5 1.5 ε 2.5 Fig Velocity curves of first six modes in the interval e ½0 2:5 for the spring contact Here we take e1 ¼ 3:0; e2 ¼ 3:5; e3 ¼ 1:5; e1 ¼ 1:8; e2 ¼ 1:2; e3 ¼ 0:5; rv ¼ 2:8; rl ¼ 0:5; cT ¼ 15; cN ¼ 123 Meccanica x x 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0.5 1.5 ε 2.5 Fig Velocity curves of first five modes in the interval e ½0 2:5 for the spring contact Here we take e1 ¼ 3:2; e2 ¼ 2:8; e3 ¼ 1:5; e1 ¼ 1:8; e2 ¼ 1:1; e3 ¼ 0:6; rv ¼ 2; rl ¼ 1, cT ¼ 12; cN ¼ 10 0.5 1.5 ε 2.5 Fig Velocity curves of first four modes in the interval e ½0 2:5 for the welded contact (solid line) and the sliding contact (dashed line) Here we take e1 ¼ 3:2; e2 ¼ 2:8; e3 ¼ 1:5; e1 ¼ 1:8; e2 ¼ 1:1; e3 ¼ 0:6; rv ¼ 2; rl ¼ 1 x x 0.9 0.8 0.8 Mode 2: ε=1.1 0.7 0.6 0.6 Mode 1: ε=0.3 0.5 0.4 0.4 0.3 0.2 Mode 0: ε=0.3 : cN= 0.2 0.1 0 0.5 1.5 ε :c =0 T cN, c T 2.5 Fig Velocity curves of first four modes in the interval e ½0 2:5 for the welded contact (solid line) and the sliding contact (dashed line) Here we take e1 ¼ 3:0; e2 ¼ 3:5; e3 ¼ 1:5; e1 ¼ 1:8; e2 ¼ 1:2; e3 ¼ 0:5; rv ¼ 2:8; rl ¼ 0:5 Fig Dependence of the wave velocity of modes 0, 1, on the normal imperfection cN (solid lines) and on the shear imperfection cT (dashed lines) Here we take e1 ¼ 3:0; e2 ¼ 3:5; e3 ¼ 1:5; e1 ¼ 1:8; e2 ¼ 1:2; e3 ¼ 0:5; rv ¼ 2:8; rl ¼ 0:5 dimensionless wave number) increases (see Figs 1, 2, 3, 4, and 6) The picture of velocity curves for the spring contact is quite different from the one for the sliding and welded contacts (see Figs 1, 2, and 4) In particular: welded contacts, i e the modes 1, for the spring contact initiate at the values of e which are much smaller than those for the sliding and welded contacts For the sliding and welded contacts the velocity of modes decreases regularly, while for the spring contact there often exist intervals of e in which the wave velocity is almost constant, except the modes and • • The velocity of the modes and for the spring contact decreases much more quickly than the one corresponding to the sliding and welded contacts The modes 1, for the spring contact appear much more early than those of the sliding and 123 • For the same mode, the velocity curve for the sliding contact always lies above the one for the welded contact (see Figs 3, 4) Meccanica between the layer and half-space as well as their mechanical properties 0.9 0.8 Mode 2: ε = 1.7 Acknowledgments The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 107.02-2014.04 0.7 x 0.6 Mode 1: ε = 0.4 :cN= 0.5 :c =0 T 0.4 0.3 Appendix The coefficients of the secular Eq (41) Mode 0: ε = 0.4 0.2 0.1 The coefficients are: c ,c N T Fig Dependence of the wave velocity of modes 0, 1, on the normal imperfection cN (solid lines) and on the shear imperfection cT (dashed lines) Here we take e1 ¼ 3:2; e2 ¼ 2:8; e3 ¼ 1:5; e1 ¼ 1:8; e2 ¼ 1:1; e3 ¼ 0:6; rv ¼ 2; rl ¼ For the mode 0, the normal spring compliance (the normal imperfection) cN affects on the wave velocity more strongly than the shear spring compliance (the shear imperfection) cT , while for others modes we have the inverse (see Figs 5, 6) Note that in order to draw the velocity curves, the dimensionless secular equations established in Sect are employed In particular, the Eq (47) is employed for drawing the velocity curves in the Figs 1, 2, and 6; the Eqs (52) and (53) are used for establishing the velocity curves in the Figs and Conclusions In this paper, the explicit exact secular equation of Rayleigh waves propagating in an orthotropic halfspace coated by an orthotropic layer with spring contact has been obtained This equation is derived by using the effective boundary condition method From the obtained secular equation, the secular equations for the welded and sliding contacts are derived as special cases For the welded contact, the obtained secular equation recovers the secular equations previously obtained for the isotropic and orthotropic materials The obtained secular equations are a good tool for nondestructively evaluating the adhesive bond pffiffiffiÁ À A1 ¼ 2b1 b2 c1 c2 X À c11 À c66 P À c66 a2 b1 c1 ỵ a1 b2 c2 ị pffiffiffi o n  c212 À c22 ðc11 À XÞ P ỵ Xc11 Xị n o c66 c1 c2 a2 b1 ỵ a1 b2 ị ỵ b1 b2 c1 ỵ c2 ị p c11 À X À c12 P n kc kc66 pffiffiffi 66 À 2b1 b2 c1 c2 ðc11 À XÞ þ c22 P KT KN qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi k2 c66  Sỵ2 P KT KN o p c12 c22 c11 ỵ c22 Xị P ỵ Xc11 Xị p B1 ẳ b21 c22 ỵ b22 c21 X À c11 À c66 P À c66 ð a1 b1 c2 ỵ a2 b2 c1 ị n o p c212 c22 c11 Xị P ỵ Xc11 Xị c66 a2 b2 c21 ỵ a1 b1 c22 ỵ b21 c2 ỵ b22 c1 p 2  ðc11 À X À c12 PÞ À b2 c21 ỵ b1 c22 ị q & p kc66 kc66 p c11 Xị ỵ c22 P Sỵ2 P KT KN ' pffiffiffi k2 c66 ðc12 c22 c11 ỵ c22 Xị P ỵ Xc11 XÞ À KT KN n pffiffiffio P cðc11 À Xị c22 b1 c2 ẵ a; b C1 ẳ c66 b2 c1 ½ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n pffiffiffi kc66 o kc66 b c ẵ a; b S ỵ P ỵ b1 c2 ẵ c K K n pffiffiffi T oN  c12 À c22 c11 Xị P ỵ Xc11 Xị p P D1 ẳ c66 b1 c2 ẵ cX c11 ị ỵ c22 b2 c1 ẵ a; b q n pffiffiffi kc66 o kc66 À b c ½ a; b S ỵ P b2 c1 ½ c K K n pffiffiffi T oN c12 c22 c11 Xị P ỵ Xðc11 À XÞ n c2 À c22 ðc11 Xị E1 ẳ A1 ỵ c66 ẵ cẵ a; b 12 o p P ỵ Xc11 XÞ ð54Þ 123 Meccanica Appendix The coefficients of the secular Eq (44) The coefficients are: n A2 ¼ 4p1 p2 ð2 À rv2 xÞ 2ð2 À rv2 xÞðb1 b2 1ị ỵ 4b1 b2 xÞ2 rlÀ2 À ð4 À rv2 xÞ Â ð2b1 b2 þ x À 2ÞrlÀ1 À 2cT cN ð2 À rv2 xÞ Â Ã Â 4b1 b2 À ð2 À xÞ2 À 2cT ð2 À rv2 xÞb1 x À 2cN o  ð2 À rv2 xÞb2 x ; n  Ã2 o B2 ¼ 4p21 p22 4b1 b2 ð1 À rlÀ1 ị2 xịrl1 n ỵ À rv2 xÞ2 ð2 À rv2 xÞ2 ðb1 b2 À 1ị 22 rv2 xị2b1 b2 ỵ x 2ịrl1 o ỵ 4b1 b2 xị2 rl2 16p21 p22 ỵ rv2 xÞ4 n o  à  cT cN 4b1 b2 xị2 ỵ cT b1 x ỵ cN b2 x ;  à C2 ¼ p2 rv2 x2 b1 ð2 À rv2 xÞ2 À 4b2 p21 rlÀ1 ỵ p2 rv2 x cN rv2 xÞ2 À 4cT p21 à   4b1 b2 À xị2 rl1 ; D2 ẳ p1 rv2 x2 b2 ð2 À rv2 xÞ2 À 4b1 p22 rlÀ1 à  À p1 rv2 x 4cN p22 À cT ð2 À rv2 xÞ2 à   4b1 b2 xị2 rl1 ; E2 ẳ À A2 À p1 p2 rv4 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an orthotropic layer Waves Random Complex Media 26:176–188 123 ... curves in the Figs and Conclusions In this paper, the explicit exact secular equation of Rayleigh waves propagating in an orthotropic halfspace coated by an orthotropic layer with spring contact. .. propagating in an orthotropic elastic half-space coated by a thin orthotropic elastic layer Wave Motion 49:681–689 11 Vinh PC, Linh NTK (2012) An approximate secular equation of generalized Rayleigh. .. Rayleigh waves in pre-stressed compressible elastic solids Int J Non-Linear Mech 50:91–96 12 Vinh PC, Anh VTN (2014) Rayleigh waves in an orthotropic half-space coated by a thin orthotropic layer with