DSpace at VNU: Exact secular equations of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer

DSpace at VNU: Exact secular equations of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic...

Accepted Manuscript Exact secular equations of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer Pham Chi Vinh, Vu Thi Ngoc Anh, Nguyen Thi Khanh Linh PII: DOI: Reference: S0020-7683(16)00003-2 10.1016/j.ijsolstr.2015.12.032 SAS 9013 To appear in: International Journal of Solids and Structures Received date: Revised date: Accepted date: 20 April 2015 28 December 2015 30 December 2015 Please cite this article as: Pham Chi Vinh, Vu Thi Ngoc Anh, Nguyen Thi Khanh Linh, Exact secular equations of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer, International Journal of Solids and Structures (2016), doi: 10.1016/j.ijsolstr.2015.12.032 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain ACCEPTED MANUSCRIPT Highlights • The propagation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer is investigated CR IP T • The half-space and the layer may be compressible or incompressible and they are in welded contact • For the compressible/compressible case, the exact secular equation is derived by using the effective boundary condition method AC CE PT ED M AN US • For three remaining cases, the exact secular equations are obtained by the incompressible limit technique ACCEPTED MANUSCRIPT CR IP T Exact secular equations of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer AN US Pham Chi Vinha ∗, Vu Thi Ngoc Anha and Nguyen Thi Khanh Linhb a Faculty of Mathematics, Mechanics and Informatics Hanoi University of Science 334, Nguyen Trai Str., Thanh Xuan, Hanoi,Vietnam b M Department of Engineering Mechanics Water Resources University of Vietnam 175 Tay Son Str., Hanoi, Vietnam ED Abstract AC CE PT 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 may be compressible or incompressible and they are in welded contact with each other The main aim of the paper is to derive explicit exact secular equations of the wave for four possible combinations of a (compressible/incompressible) half-space coated by a (compressible/incompressible) layer For the compressible/compressible case, the explicit secular equation is derived by using the effective boundary condition method For three remaining cases, the explicit secular equations are derived from the secular equation for the compressible/compressible case by using the incompressible limit technique along with the expressions of the reduced elastic compliances in terms of the elastic stiffnesses Based on the obtained secular equations, the effect of incompressibility on the Raleigh wave propagation is considered numerically It is shown that the incompressibility strongly affects the Raleigh wave velocity and it makes the Raleigh wave velocity increasing ∗ Corresponding author: Tel:+84-4-35532164; Fax:+84-4-38588817; E-mail address: inh@vnu.edu.vn (P C Vinh) pcv- ACCEPTED MANUSCRIPT Key words: Rayleigh waves, A half-space coated by a layer, Exact secular equations, The effective boundary condition method, The incompressible limit technique Introduction CR IP T 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 AN US properties of supported layers therefore plays an important role in understanding the behaviors of this structure in applications, see for examples Makarov et al (1995) and references therein Among various measurement methods, the surface/guided M 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 ED et al., 2013) Among surface/guided waves, the Rayleigh wave is a versatile and PT convenient tool (Kuchler & Richter, 1998; Hess et al., 2013) Since the explicit dispersion relations of Rayleigh waves are employed as theoretical bases for extracting CE the mechanical properties of the layers from experimental data, they are therefore the main purpose of any investigation of Rayleigh waves propagating in elastic half- AC spaces covered by an elastic layer When the half-space and the layer are both isotropic (the isotropic case), the explicit secular equation of Rayleigh waves was derived by Haskell (1953), BenMenahem (2000) (Eq (3.113), p.117) For the orthotropic case, the explicit secular ACCEPTED MANUSCRIPT equation of Rayleigh waves was derived by Sotiropoulos (1999) by expanding directly a six-order determinant that comes from the traction-free conditions at the upper surface of the layer and the continuity conditions of the displacements and stresses CR IP T at the interface of the layer and the half-space In this investigation, the layer and the half-space are assumed to be compressible Since there are four possible combinations: the layer and the half-space are both compressible (the compress- AN US ible/compressible case) or incompressible (the incompressible/incompressible case), one is compressible and the other is incompressible (the compressible/incompressible and incompressible/compressible case), the derivation of explicit secular equations of Rayleigh waves for the remaining combinations is needed M The main purpose of this paper is to derive explicit secular equations of Rayleigh waves propagating in an orthotropic elastic half-space overlaid by an orthotropic ED elastic layer of arbitrary thickness for all possible combinations For the compress- PT ible/compressible case, the explicit secular equation is derived by employing the effective boundary condition method that proved successful in deriving approximate CE explicit secular equations of Raleigh waves in elastic half-spaces coated by a thin AC film, see Tiersten (1969), Bovik (1996), Steigmann & Ogden (2007), Vinh & Linh (2012, 2013), Vinh et al (2014a, 2014b), Vinh & Anh (2014a, 2014b) For three remaining cases, the explicit secular equations are deduced directly from the secular equation for the compressible/compressible case by using the incompressible limit approach, proposed by Destrade el al (2002), along with the expressions of the ACCEPTED MANUSCRIPT reduced elastic compliances in terms of the elastic stiffnesses in the incompressible limit, Since the obtained secular equations are exact and totally explicit, they will be CR IP T useful in practical applications Explicit secular equation of Rayleigh waves for the compressible/compressible case AN US First, the entire effect of the layer on the half-space is replaced by the exact effective boundary conditions at the interface The wave is then considered as a Rayleigh wave propagating in the half-space, without the coating layer, that is subjected to Exact effective boundary conditions ED 2.1 M the effective boundary conditions Consider an orthotropic elastic half-space x2 ≥ overlaid by an orthotropic elastic PT layer with uniform thickness h occupying the domain −h ≤ x2 ≤ The layer is assumed to be perfectly bonded to the half-space and they are both compressible CE Note that same quantities related to the half-space and the layer have the same AC symbol but are systematically distinguished by a bar if pertaining to the layer We are interested in the plane strain such that: ui = ui (x1 , x2 , t), u¯i = u¯i (x1 , x2 , t), i = 1, 2, u3 = u¯3 ≡ (1) ACCEPTED MANUSCRIPT where ui and u¯i are displacement components, t is the time For the layer, in the absence of body forces the equations of motion are: (2) CR IP T 11,1 + 12,2 = uă1 , 12,1 + 22,2 = uă2 where ij are stress components, commas signify differentiation with respect to xk , a dot indicates differentiation with respect to t For an orthotropic material the strain-stress relation is of the form (Ting, 1996): AN US σ ¯11 = c¯11 u¯1,1 + c¯12 u¯2,2 , σ ¯22 = c¯12 u¯1,1 + c¯22 u¯2,2 , σ ¯12 = c¯66 (¯ u1,2 + u¯2,1 ) (3) where c¯ij are material constants satisfying the inequalities: (4) M c¯ii > 0, i = 1, 2, 6, c¯11 c¯22 − c¯212 > which are necessary and sufficient conditions for the strain energy of the material ED to be positive definite Substituting (3) into (2) and taking into account (1) yield: PT c¯11 u¯1,11 + c¯66 u¯1,22 + ( c12 + c66 ) u2,12 = uă1 (5) CE (¯ c12 + c¯66 )¯ u1,12 + c¯66 u¯2,11 + c22 u2,22 = uă2 Now we consider the propagation of a Rayleigh wave, traveling along the interface AC between the layer and the half-space with velocity c and wave number k in the x1 -direction and decaying in the x2 -direction It is not difficult to verify that the displacement components of the Rayleigh wave in the layer, that satisfy Eqs (5), are given by: u¯1 = U¯1 (y)eik(x1 −ct) , u¯2 = U¯2 (y)eik(x1 −ct) , y = kx2 (6) ACCEPTED MANUSCRIPT where: U¯1 (y) = A1 ch¯b1 y + A2 sh¯b1 y + A3 ch¯b2 y + A4 sh¯b2 y (7) CR IP T U¯2 (y) = i α ¯ A1 sh¯b1 y + A2 ch¯b1 y + α ¯ A3 sh¯b2 y + A4 ch¯b2 y A1 , A2 , A3 , A4 are constants, α ¯ k and ¯bk are determined by: (¯ c12 + c¯66 )¯bk c¯11 − ρ¯c2 − c¯66¯b2k ¯ = ρ¯c2 = − , k = 1, 2, X ¯ c¯22¯b2k − c¯66 + X (¯ c12 + c¯66 )¯bk √ √ ¯ + S¯2 − 4P¯ ¯ − S¯2 − 4P¯ S S ¯b1 = , ¯b2 = 2 ¯ + c¯66 (¯ ¯ − (¯ c¯22 (¯ c11 − X) c66 − X) c12 + c¯66 )2 S¯ = c¯22 c¯66 ¯ c66 − X) ¯ (¯ c − X)(¯ 11 P¯ = c¯22 c¯66 AN US α ¯k = − (8) M Note that ¯b1 and ¯b2 are complex in general and no requirements are imposed on ED their real and imaginary parts On use of Eqs (6) and (7) into (3) we have: where: (9) PT ¯ (y)eik(x1 −ct) , σ ¯ (y)eik(x1 −ct) σ ¯12 = k Σ ¯22 = k Σ CE ¯ (y) = β¯1 A1 sh¯b1 y + A2 ch¯b1 y + β¯2 (A3 sh¯b2 y + A4 ch¯b2 y) Σ (10) AC ¯ (y) = i γ¯1 A1 ch¯b1 y + A2 sh¯b1 y + γ¯2 A3 ch¯b2 y + A4 sh¯b2 y) Σ and: β¯n = c¯66 (¯bn − α ¯ n ), γ¯n = (¯ c12 + c¯22¯bn α ¯ n ), n = 1, (11) Suppose that the surface x2 = −h is traction-free, then we have: σ ¯12 = σ ¯22 = at x2 = −h (12) ACCEPTED MANUSCRIPT Introducing (9) and (10) into (12) provides: β¯1 A1 shε1 − β¯1 A2 chε1 + β¯2 A3 shε2 − β¯2 A4 chε2 = (13) γ¯1 A1 chε1 − γ¯1 A2 shε1 + γ¯2 A3 chε2 − γ¯2 A4 shε2 = CR IP T where εn = ε¯bn , n = 1, 2, ε = kh, and for simplicity, we use the notations sh(.) := sinh(.), ch(.) := cosh(.) Putting x2 = in Eqs (7) and (10) leads to: U¯1 (0) = A1 + A3 , U¯2 (0) = i(¯ α1 A2 + α ¯ A4 ) (14) ¯ (0) = β¯1 A2 + β¯2 A4 , Σ ¯ (0) = i(¯ Σ γ1 A1 + γ¯2 A3 ) AN US Solving the system (14) for A1 , A2 , A3 , A4 and then substituting their expressions into (13) yield: ¯ (0) − ia12 Σ ¯ (0) + b11 U¯1 (0) − ib12 U¯2 (0) = a11 Σ (15) ¯ (0) − ia22 Σ ¯ (0) + b21 U¯1 (0) − ib22 U¯2 (0) = a21 Σ in which the coefficients aij and bij are given by: M ¯ ¯ [βshε] [¯ γ shε; α ¯] [¯ γ chε] [¯ α; βchε] , a = − , a21 = , a22 = 12 ¯ ¯ [¯ γ] [¯ γ] [¯ α; β] [¯ α; β] ¯ γ¯ shε] ¯ [chε] [β; [βshε; γ¯ ] [chε] , b12 = β¯1 β¯2 , b = = , b = −¯ γ γ ¯ 22 21 ¯ ¯ [¯ γ] [¯ γ] [¯ α; β] [¯ α; β] b11 ED a11 = (16) PT here, for the seeking of simplicity, we use the notations: (17) CE [f ; g] := f2 g1 − f1 g2 , [f ] := f2 − f1 Since the layer and half-space are welded at the interface x2 = 0, it follows u¯k = AC uk (k = 1, 2) and σ ¯k2 = σk2 (k = 1, 2) at x2 = 0, or equivalently by (6) and (9): ¯ k (0) = Σk (0) (k = 1, 2) With these facts, it follows U¯k (0) = Uk (0) (k = 1, 2) and Σ from (15): a11 Σ1 (0) − ia12 Σ2 (0) + b11 U1 (0) − ib12 U2 (0) = a21 Σ1 (0) − ia22 Σ2 (0) + b21 U1 (0) − ib22 U2 (0) = (18) ACCEPTED MANUSCRIPT where Uk (0) and Σk (0) are the amplitudes of displacement and stress of the halfspace at the interface x2 = This is the desired exact effective boundary condition that replaces exactly the entire influence of the layer on the half-space Exact secular equation CR IP T 2.2 Now we consider the propagation of a Rayleigh wave, traveling along surface x2 = of the half-space with velocity c and wave number k in the x1 -direction, decaying AN US in the x2 -direction, and satisfying the exact effective boundary condition (18) According to Vinh & Ogden (2004), the displacements of the Rayleigh wave in the half-space x2 > are given by: (19) M u1 = U1 (y)eik(x1 −ct) , u2 = U2 (y)eik(x1 −ct) , y = kx2 ED where: U1 (y) = B1 e−b1 y + B2 e−b2 y , U2 (y) = i(α1 B1 e−b1 y + α2 B2 e−b2 y ) (20) PT B1 and B2 are constants to be determined, and: CE αk = (c12 + c66 )bk , k = 1, 2, X = ρc2 c22 b2k − c66 + X (21) b1 and b2 are two roots having positive real part (in order to make the decay condition AC satisfied) of the following equation: b4 − Sb2 + P = (22) S and P are calculated by (8) without the bar It follows from (22) that: b21 + b22 = S, b21 b22 = P (23) ACCEPTED MANUSCRIPT 4.1 The incompressible/incompressible case Proposition 3: Let the orthotropic elastic material of the layer approaches the CR IP T incompressible limit Then we have: lim P¯ = − rv2 x, lim S¯ = e¯δ − rv2 x − 2, β¯ γ¯ lim α ¯ k = − ¯ , lim β¯k = c66 ¯ k , lim γ¯k = c66 ¯k , k = 1, bk bk bk AN US where: (59) β¯k = rµ (1 + ¯b2k ), γ¯k = rµ (rv2 x − e¯δ + + ¯b2k ), k = 1, e¯δ = δ¯ ¯ , δ = c¯11 + c¯11 − 2¯ c12 c¯66 (60) M here ¯bk are calculated by (8) in which P¯ and S¯ are given by the first two of Eqs (59) ED Proof: From (8)4 we have: (61) PT ¯ c11 − ∆ ¯ X)(¯ ¯ c66 − X) ¯ (∆¯ ¯ = ρ¯c2 , X P¯ = ¯ c22 c¯66 ∆¯ CE Taking the limit of the equality (61) and using (56) give the first of (59) Note that AC ¯X ¯ = From (8)3 and (58) it follows: lim ∆ ¯ c22 c¯66 S¯ = − X( ¯ ∆¯ ¯ c22 + ∆¯ ¯ c66 ) − 2¯ ¯ c12 ∆¯ c66 ∆¯ (62) Taking the limit of the equality (62) and using (50), (56) give the second of (59) In view of the first of Eqs (8) we have: ¯ c12 + ∆¯ ¯ c66 )pk (∆¯ α ¯k = − ¯ ¯ c66 + ∆ ¯X ¯ ∆¯ c22 p2k − ∆¯ 20 (63) ACCEPTED MANUSCRIPT Taking the limit of the equality (63) and using (56) give the third of (59) From (11)1 and (59)3 we derive the fourth of Eqs (59) Using (8)2 and (11)2 and taking into account (58) provide: ¯ c12 c¯66 + ∆¯ ¯ c22 X ¯ + ∆¯ ¯ c22 c¯66 p2 −1 + ∆¯ k ¯ c12 + ∆¯ ¯ c66 ∆¯ CR IP T γ¯k = (64) Taking the limit of the equality (64) and using (56) give the last of (59) Proposition 4; Let the orthotropic elastic material of the half-space approaches AN US the incompressible limit Then the following equalities hold: lim P = − x, lim S = eδ − x − (65) M where eδ = δ/c66 , δ = c11 + c22 − 2c12 Proof: The proof of (65) are similar to the one of (59)1,2 ED In order to derive the explicit secular equation for the incompressible/incompressible case, first we multiply two sides of Eq (32) by ∆ (> 0), then take the limit two PT side of the obtained equation as the materials of both the layer and the half-space CE approach to the incompressible limit and take into account (59) and (65) After multiplying two sides of the resulting equation by the factor ¯b21¯b22 /(c566 s11 ) we arrive AC at the equation: A1 + B1 chε1 chε2 + C1 shε1 shε2 + D1 chε1 shε2 + E1 chε2 shε1 = 21 (66) ACCEPTED MANUSCRIPT where: ¯ γ¯2 ) − γ¯1 γ¯2 (β¯1 + β¯2 ) α2 γ¯1 + α + β¯1 β¯2 (¯ √ B1 = −A1 − [β¯ ][¯ α ; γ¯ ] (x − eδ ) P + x , √ P − − 2β¯1 β¯2 γ¯1 γ¯2 , √ C1 = (¯ α2 β¯1 γ¯2 + α ¯ β¯2 γ¯1 ) (x − eδ ) P + x √ (67) P − − (β¯1 )2 (¯ γ2 )2 − (β¯2 )2 (¯ γ1 )2 , √ D1 = β¯1 γ¯2 [¯ α ; γ¯ ] + β¯2 γ¯1 [β¯ ] P √ S + P, √ E1 = − β¯2 γ¯1 [¯ α ; γ¯ ] + β¯1 γ¯2 [β¯ ] P AN US + β¯1 γ¯2 (¯ α2 β¯1 − γ¯2 ) + β¯2 γ¯1 (¯ α1 β¯2 − γ¯1 ) CR IP T √ A1 = (¯ α2 β¯1 γ¯1 + α ¯ β¯2 γ¯2 ) (x − eδ ) P + x in which: √ S + P, (68) M α ¯ k = ¯bk , k = 1, 2, S = eδ − − x, P = − x Equation (66) with the coefficients A1 , B1 , C1 , D1 , E1 calculated by (67) is the ex- ED plicit exact secular equation of Rayleigh waves in dimensionless form for the incompressible/incompressible case CE PT When ε = (the layer is absent), from (66) and (67) we have: √ (eδ − x) P − x = (69) AC This is the dimensionless secular equation of Rayleigh waves propagating in an incompressible orthotropic half-space (see Ogden & Vinh, 2004) As an example, we use Eq (66) to compute the dimensionless wave velocity x (of mode ”0”) with given dimensionless parameters as follows: eδ = 3, e¯δ = 2.8, rµ = 1, rv = 2.8 22 AN US CR IP T ACCEPTED MANUSCRIPT Figure 2: Rayleigh wave velocity curves draw by solving the exact secular equation (66) (solid line) and the third-order approximate secular equation (3.14) in Vinh et al (2014a) (dashed line) Here we take eδ = 3, e¯δ = 2.8, rµ = 1, rv = 2.8 M and then comparing the exact velocity curve with the corresponding third-order ED approximate velocity curve (dashed line) draw by solving the approximate secular equation (3.14) in Vinh et al (2014a) in the interval ε ∈ [0; 1.5] It is seen from Fig PT that in this interval the exact and approximate curves are almost coincide with each other This asserts that the secular equation Eq (66) as well as the relations The compressible/incompressible case AC 4.2 CE (59) and (65) are correct Analogously, to derive the explicit secular equation for the compressible/incompressible case, we multiply two sides of Eq (32) by ∆ (> 0), then take the limit two side of the resulting equation as the materials of the half-space approach to the incompressible limit and take into account (65) After multiplying two sides of the resulting 23 ACCEPTED MANUSCRIPT equation by the factor 1/(c466 s11 ) we have: A1 + B1 chε1 chε2 + C1 shε1 shε2 + D1 chε1 shε2 + E1 chε2 shε1 = γ1∗ + γ¯2∗ ) ¯ β¯2∗ ) + β¯1∗ β¯2∗ (¯ α2 β¯1∗ + α A1 = γ¯1∗ γ¯2∗ (¯ √ P −1 CR IP T where: (70) √ ¯ β¯2∗ γ¯2∗ ) (x − eδ ) P + x − 2β¯1∗ β¯2∗ γ¯1∗ γ¯2∗ , − (¯ α2 β¯1∗ γ¯1∗ + α √ B1 = − A1 + [¯ γ ∗ ][¯ α; β¯∗ ] (x − eδ ) P + x , √ P −1 AN US 2 C1 = α ¯ β¯2∗ (¯ γ1∗ )2 + α ¯ β¯1∗ (¯ γ2∗ )2 + (β¯1∗ ) γ¯2∗ + (β¯2∗ ) γ¯1∗ (71) √ − (β¯1∗ γ¯2∗ )2 + (β¯2∗ γ¯1∗ )2 − (¯ α1 β¯1∗ γ¯2∗ + α ¯ β¯2∗ γ¯1∗ ) (x − eδ ) P + x , √ α; β¯∗ ] P − β¯1∗ γ¯2∗ [¯ D1 = β¯2∗ γ¯1∗ [¯ γ ∗] √ S+2 P M √ E1 = β¯2∗ γ¯1∗ [¯ γ ∗ ] − β¯1∗ γ¯2∗ [¯ α; β¯∗ ] P √ S + P, where S and P are given by (68) Equation (70) with the coefficients A1 , B1 , C1 , D1 , E1 ED calculated by (71) is the explicit exact secular equation of Rayleigh waves in dimensionless form for the compressible/incompressible case CE PT When ε = (the layer is absent), from (70) and (71) we have: √ (eδ − x) P − x = (72) AC This is the dimensionless secular equation of Rayleigh waves propagating in an incompressible orthotropic half-space (see Ogden & Vinh, 2004) 4.3 The incompressible/compressible case For obtaining the explicit secular equation for the incompressible/compressible case, we take the limit two side of Eq (32) as the materials of the layer approach to the 24 ACCEPTED MANUSCRIPT incompressible limit and take into account (59) After multiplying two sides of the resulting equation by the factor ¯b21¯b22 /c566 the secular equation is of the form: where: A1 = 2β¯1 β¯2 γ¯1 γ¯2 x − e1 − √ (73) CR IP T A1 + B1 chε1 chε2 + C1 shε1 shε2 + D1 chε1 shε2 + E1 chε2 shε1 = P √ ¯ β¯2 γ¯2 ) (e23 − e1 e2 + e2 x) P + x(e1 − x) + (¯ α2 β¯1 γ¯1 + α AN US √ ¯ γ¯2 ) − γ¯1 γ¯2 (β¯1 + β¯2 ) e1 − x − e3 P , α2 γ¯1 + α − β¯1 β¯2 (¯ √ B1 = −A1 − [β¯ ][¯ α ; γ¯ ] (e23 − e1 e2 + e2 x) P + x(e1 − x) , C1 = (β¯1 )2 (¯ γ2 )2 + (β¯2 )2 (¯ γ1 )2 x − e1 − √ P (74) M √ + (¯ α2 β¯1 γ¯2 + α ¯ β¯2 γ¯1 ) (e23 − e1 e2 + e2 x) P + x(e1 − x) √ α1 β¯2 − γ¯1 ) e1 − x − e3 P , α2 β¯1 − γ¯2 ) + β¯2 γ¯1 (¯ − β¯1 γ¯2 (¯ ED √ D1 = β¯1 γ¯2 [¯ α ; γ¯ ](e1 − x) + e2 β¯2 γ¯1 [β¯ ] P √ S + P, PT √ E1 = − β¯2 γ¯1 [¯ α ; γ¯ ](e1 − x) + e2 β¯1 γ¯2 [β¯ ] P √ S + P, where S and P are given by (38) Equation (73) with the coefficients A1 , B1 , C1 , D1 , E1 CE calculated by (74) is the explicit exact secular equation of Rayleigh waves in dimen- AC sionless form for the incompressible/compressible case When ε = (the layer is absent), from (73) and (74) it follows: e23 − e2 (e1 − x) √ P + x(e1 − x) = (75) that is the dimensionless secular equation of Rayleigh waves in a compressible orthotropic half-space (see Vinh & Ogden, 2004) 25 ACCEPTED MANUSCRIPT Remark 3: The explicit secular equations of Rayleigh waves for the incompressible cases can be derived by employing the effective boundary condition method and they are the same as Eqs (66), (70) and (73) This proves the validity of the CR IP T obtained secular equations Numerical examples In this Section, as an example of application of the obtained approximate secular AN US equations, we consider numerically the effect of the incompressibility on the Rayleigh wave velocity For this aim we consider four examples In the first example, a compressible half-space is coated either by a compressible layer or by an incompressible M layer Two these layers have the same elastic constants In the second example, the compressible half-space is replaced by an incompressible In the third (fourth) ED example, two different (compressible and incompressible) half-spaces with the same PT elastic constants are covered with the same compressible (incompressible) layer In particular, in the first example, we take e1 = 2.8, e2 = 2.5, e3 = 0.6 for the CE half-space and e¯1 = 3.5, e¯2 = 0.5, e¯3 = (¯ eδ = 3.5) for the layers and rµ = 1, rv = 2.2 AC In the second example, we choose eδ = 3.2 for the half-space and e¯1 = 2.2, e¯2 = 1, e¯3 = 0.6 for the layers and rµ = 1, rv = 2.8 In the third example, the dimensionless parameters are taken as e¯1 = 2.5, e¯2 = 0.6, e¯3 = for the layer and e1 = 3.0, e2 = 1.8, e3 = 0.5 for the half-spaces and 26 ACCEPTED MANUSCRIPT rµ = 0.6, rv = 2.6 In the last one, they are e¯δ = 2.4545 for the layer and e1 = 3.3, e2 = 1, e3 = 0.8 for the half-spaces and rµ = 0.8, rv = 2.5 CR IP T The numerical results of the first, second, third and fourth examples are presented in Figures 3, 4, and 6, respectively To establish these wave velocity curves the secular equations (36), (66), (70) and (73) are employed AN US It is shown from the Figures 3-6: (i) The incompressibility affects strongly the Rayleigh wave velocity (ii) The incompressibility makes the Rayleigh wave velocity increasing Conclusions M In this paper, the explicit exact secular equations of Rayleigh waves propagating ED in an orthotropic half-space coated by an orthotropic layer with arbitrary thickness PT are obtained The contact between the layer and the half-space is perfectly bonded When the layer and the half-space are both compressible, the secular equation is CE derived by the effective boundary condition method For three remaining cases: both the layer and the half-space are incompressible, one is compressible and the AC other is incompressible, the secular equations are derived directly from the secular equation for the compressible/compressible case by using the incompressible limit technique along with the expressions of the reduced elastic compliances in terms of the elastic stiffnesses Since the obtained secular equations are exact and totally 27 ACCEPTED MANUSCRIPT explicit, they will be useful in practical applications Acknowledgement CR IP T The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant N0 107.02-2014.04 References Achenbach, J D and Keshava, S P (1967) Free waves in a plate supported by AN US a semi-infinite continuum Journal of Applied Mechanics 34, 397-404 Ben-Menahem, A and Singh, S J (2000) Seismic waves and Sources SpringerVerlag New York Inc., Second edition M Bovik, P (1996) A comparison between 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wedges (1D) Ultrasonics 54, 39-55 Makarov, S., Chilla, E and Frohlich, H J (1995) Determination of elastic AN US constants of thin films from phase velocity dispersion of different surface acoustic wave modes J Appl Phys 78, 5028-5034 Ogden, R W and Vinh, P C (2004) On Rayleigh waves in incompressible orthotropic elastic solids J Acoust Soc Am 115, 530-533 M Sotiropoulos, D A (1999) The effect of anisotropy on guided elastic waves in a layered half-space Mechanics of Materials, 31, 215-223 ED Steigmann, D J., Ogden R W (2007) Surface waves supported by thin- PT film/substrate interactions IMA Journal of Applied Mathematics 72, 730-747 Tiersten, H F (1969) Elastic Surface Waves Guided by Thin Films Journal CE of Applied Physics 46, 770-789 AC Ting, T C T (1996) Anisotropic Elasticity: Theory and applications Oxford University Press, NewYork Vinh, P C., Ogden, R W (2004) Formulas for the Rayleigh wave speed in orthotropic elastic solids Ach Mech 56, 247-265 Vinh, P C., Linh, N T K (2012) An approximate secular equation of Rayleigh 29 ACCEPTED MANUSCRIPT 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 general- CR IP T ized Rayleigh waves in pre-stressed compressible elastic solids International Journal of Non-Linear Mechanics 50, 91-96 Vinh, P C., Linh, N T K., Anh, V T N (2014a) Rayleigh waves in an AN US incompressible orthotropic elastic half-space coated by a thin elastic layer Archives of Mechanics 66, 173-184 Vinh, P C., Anh, V T N., Thanh, V P (2014b) Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact Wave M Motion 51, 496-504 Vinh, P C., Anh, V T N (2014a) An approximate secular equation of Rayleigh ED waves in an isotropic elastic half-space coated with a thin isotropic elastic layer Acta PT Mechanica 225, 2539-2547 Vinh, P C., Anh, V T N (2014b) Rayleigh waves in an orthotropic half-space CE coated by a thin orthotropic layer with sliding contact Int J Eng Sci 75, AC 154-164 30 CE PT ED M AN US CR IP T ACCEPTED MANUSCRIPT AC Figure 3: A compressible half-space coated by a compressible layer (solid line drawn by solving Eq (36)), by an incompressible layer (dashed line drawn by solving (73)) Here we take e1 = 2.8, e2 = 2.5, e3 = 0.6 for the half-space, e¯1 = 3.5, e¯2 = 0.5, e¯3 = (¯ eδ = 3.5) for the layers and rµ = 1, rv = 2.2 31 CE PT ED M AN US CR IP T ACCEPTED MANUSCRIPT AC Figure 4: An incompressible half-space coated by a compressible layer (solid line drawn by solving (70)), by an incompressible layer (dashed line drawn by solving (66)) Here we take eδ = 3.2 for the half-space and e¯1 = 2.2, e¯2 = 1, e¯3 = 0.6 for the layers and rµ = 1.0, rv = 2.8 32 CE PT ED M AN US CR IP T ACCEPTED MANUSCRIPT AC Figure 5: A compressible layer coats a compressible half-space (solid line drawn by solving Eq (36)), coats an incompressible half-space (dashed line drawn by solving (70)) Here we take e¯1 = 2.5, e¯2 = 0.6, e¯3 = 1.0 for the layer, e1 = 3.0, e2 = 1.8, e3 = 0.5 for the half-spaces and rµ = 0.6, rv = 2.6 33 CE PT ED M AN US CR IP T ACCEPTED MANUSCRIPT AC Figure 6: An incompressible layer coats a compressible half-space (solid line drawn by solving (73)), an incompressible half-space (dashed line drawn by solving (66) Here we take e¯δ = 2.4545 for the layer, e1 = 3.3, e2 = 1, e3 = 0.8 for the half-spaces and rµ = 0.8, rv = 2.5 34 ... remaining cases, the exact secular equations are obtained by the incompressible limit technique ACCEPTED MANUSCRIPT CR IP T Exact secular equations of Rayleigh waves in an orthotropic elastic half-space. .. M The main purpose of this paper is to derive explicit secular equations of Rayleigh waves propagating in an orthotropic elastic half-space overlaid by an orthotropic ED elastic layer of arbitrary... the Rayleigh wave velocity increasing Conclusions M In this paper, the explicit exact secular equations of Rayleigh waves propagating ED in an orthotropic half-space coated by an orthotropic layer