International Journal of Non-Linear Mechanics 50 (2013) 91–96 Contents lists available at SciVerse ScienceDirect International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm An approximate secular equation of generalized Rayleigh waves in pre-stressed compressible elastic solids Pham Chi Vinh a,n, Nguyen Thi Khanh Linh b a b Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334 Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam Department of Engineering Mechanics, Water Resources University of Viet Nam, 175 Tay Son Str., Hanoi, Viet Nam a r t i c l e i n f o abstract Article history: Received 11 July 2012 Received in revised form November 2012 Accepted November 2012 Available online 10 November 2012 The present paper is concerned with the propagation of Rayleigh waves in a pre-stressed elastic halfspace coated with a thin pre-stressed elastic layer The half-space and the layer are assumed to be compressible and in welded contact with each other By using the effective boundary condition method, an explicit third-order approximate secular equation of the wave has been derived that is valid for any pre-strains and for a general strain-energy function When the pre-strains are absent, the secular equation obtained coincides with the one for the isotropic case Numerical investigation shows that the approximate secular equation obtained is a good approximation Since explicit dispersion relations are employed as theoretical bases for extracting pre-stresses from experimental data, the secular equation obtained will be useful in practical applications & 2012 Elsevier Ltd All rights reserved Keywords: Rayleigh waves A pre-stressed compressible elastic halfspace A thin pre-stressed compressible elastic layer Approximate secular equation Introduction A pre-stressed elastic layer on a pre-stressed elastic half-space is a model finding a broad range of applications [1], including: the Earth’s crust in seismology, the foundation/soil interaction in geotechnical engineering, tissue structures in biomechanics, coated solids in material science, and micro-electro-mechanical systems In all these situations, the presence of pre-stresses strongly influences the mechanical characteristics of the structure, particularly the dynamic behavior, so that the evaluation of pre-stresses appearing in the layer and the half-space is necessary and significant Among various non-destructive measurement methods, the surface/guided wave method [2] is used most extensively, and for which the guided Rayleigh wave is most convenient For the Rayleigh-wave approach, the explicit dispersion relations of Rayleigh waves supported by pre-stressed layer/ substrate interactions are employed as the theoretical bases for extracting mechanical properties and pre-stresses of the structure from experimental data They are therefore the main factor, the main purpose of the investigations of Rayleigh waves propagating in half-spaces covered by a pre-stressed layer The propagation of Rayleigh waves in a compressible prestressed elastic half-space overlaid by a compressible pre-stressed elastic layer was investigated by Ogden and Sotiropoulos [3] In that investigation, for simplicity, pre-strains corresponding to n Corresponding author Tel.: ỵ84 5532164; fax: ỵ84 8588817 E-mail addresses: pcvinh@vnu.edu.vn, pcvinh@gmail.com (P Chi Vinh) 0020-7462/$ - see front matter & 2012 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.ijnonlinmec.2012.11.004 plane isotropic deformations were considered, and an explicit dispersion relation was obtained that is valid for any strainenergy function For the case of arbitrary pre-strains, and for the strain-energy function of Murnaghan’s form, this problem was considered recently by Akbarov and Ozisik [4] and an implicit secular equation was derived The main aim of this paper is to find a third-order approximate secular equation for the Rayleigh waves when the layer is assumed to be thin By using the effective boundary condition method [5–8], which replaces approximately the entire effect of the layer on the half-space by a boundary condition, an approximate secular equation of third-order has been derived that is valid for arbitrary pre-strains, and for a general isotropic strainenergy function When the thickness of the layer vanishes, the derived secular equation becomes the dispersion relation of Rayleigh waves traveling along the traction-free surface of a pre-stressed isotropic elastic half-space (see [9,10]) Note that the propagation of surface Rayleigh waves in a half-space under the effect of prestress was examined also by Hayes and Rivlin [11], Chadwick and Jarvis [12], Murphy and Destrade [13], Dowaikh and Ogden [14], Vinh [15], Murdoch [16], and Ogden and Steigmann [17] References to other works can be found in these papers When prestresses are absent, from the obtained secular equation we immediately arrive at the approximate secular equation of second- and third-order obtained recently by Vinh and Linh [8] for the isotropic case A numerical investigation is carried out for a special strainenergy function, and it is shown that the approximate secular 92 P Chi Vinh, N Thi Khanh Linh / International Journal of Non-Linear Mechanics 50 (2013) 91–96 equation obtained has high accuracy Therefore, it will be a good tool for extracting pre-stresses appearing in the structure from experimental data Effective boundary condition of third-order We consider a homogeneous surface layer of uniform thickness h overlying a homogeneous half-space, both being pre-stressed compressible isotropic elastic materials with the underlying deformations corresponding to pure homogeneous strains The principal directions of strain in the two solids are aligned, one direction being normal to the planar interface defined by x2 ¼ A rectangular Cartesian coordinate system ðx1 ,x2 ,x3 Þ is employed with its axes coinciding with the principal directions of the pure strain The layer occupies the domain Àh ox2 o0 and the half-space corresponds to the region x2 The principal stretches are denoted by l1 , l2 , l3 and l , l , l in the half-space and in the layer, respectively They are positive constants The layer is assumed to be perfectly bonded to the halfspace Note that 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 An incremental (infinitesimal) motion in the ðx1 ,x2 Þ-plane is now superimposed on the underlying deformations, with its displacement components in the half-space and the layer being independent of x3 and denoted by ðu1 ,u2 Þ and ðu ,u Þ, respectively For the layer, in the absence of body forces the equations of motion governing innitesimal motion are [9,18] s 11,1 ỵs 21,2 ẳ r u , s 12,1 ỵs 22,2 ẳ r u , ð1Þ where r is the mass density of material at the static deformed state, a superposed dot signifies differentiation with respect to t, k indicates differentiation with respect to spatial variables xk, and s ji ¼ A jilk u k,l : ð2Þ A ijkl are components of the fourth-order elasticity tensor defined as follows [9,18]: (which are different from those defined by [9] by a factor J) In terms of these notations Eq (2) becomes s 11 ẳ a 11 u 1,1 ỵ a 12 u 2,2 , > > > > < s 22 ¼ a 12 u 1,1 ỵ a 22 u 2,2 , 8ị s 12 ẳ g u 2,1 ỵ g n u 1,2 , > > > > : s 21 ẳ g u 2,1 ỵ g u 1,2 : n From the strong-ellipticity condition, a ik and g k are required to satisfy the inequalities [9,18] a 11 0, a 22 40, g 0, g 0: From Eqs (1) and (8), and following the same procedure as carried out in [19] we have " 0# " #" # M1 M2 U U , x2 A ẵh,0, 10ị ẳ M3 M4 T T where U ẳ ẵu u T , T ẳ ½s 21 s 22 ŠT , the symbol ‘‘T’’ indicates the transpose of a matrix, the prime signifies partial derivative with respect to x2 and " # Àr @1 g2 , M2 ¼ M1 ¼ 7, Àr @1 " M3 ¼ r @21 ỵ r @2t 0 r @21 ỵ r @2t r1 ¼ @ W @l i @l j , From (10) it follows that " # " ðnÞ # " M1 U U , Mẳ ẳ Mn nị M T T J A ijji ¼ J A jiij ¼ J A ijij Àl i @W @l i ði aj, l i a l j Þ, ð4Þ ði aj, l i ẳ l j ị, i ajị 5ị for i,j A 1,2,3,W ¼ W ðl , l , l Þ is the strain-energy function per unit volume in unstressed state, J ¼ l l l (noting that l k 0), all other components being zero In the stress-free configuration (3)–(5) reduce to A iiii ẳ l ỵ 2m , A iijj ¼ l ðia jÞ, A ijij ¼ A ijji ¼ m ði a jÞ, ð6Þ where l :m are the Lame moduli For simplicity, we use the notations a 11 ¼ A 1111 , a 22 ¼ A 2222 , a 12 ¼ a 21 ¼ A 1122 , g ¼ A 1212 , g ¼ A 2121 , g n ẳ A 2112 , 7ị M ¼ M T1 : ð11Þ M2 M4 # , n ẳ 1,2,3, ,x2 A ẵh,0: 13ị Let h be small (i.e the layer is thin), then expanding T ðÀhÞ then expanding T ðÀhÞ into Taylor series about x2 ¼ up to the thirdorder of h gives ð3Þ ! > > @W @W li > > li Àl j > > < @l i @l j l Àl i j J A ijij ¼ ! > > @W > > J A iiii J A iijj ỵ l i > > :2 @l i , a 12 g a 11 a 22 Àa 212 g g Àg , r2 ¼ n , r3 ¼ À , r ¼ À n : ð12Þ a 22 g2 a 22 g2 J A iijj ¼ l i l j a 22 # Here we use the notations @1 ¼ @=@x1 , @21 ¼ @2 =@x21 , @2t ¼ @2 =@t and T hị ẳ T 0ịhT 0ị ỵ ð9Þ h 00 h 000 T ð0ÞÀ T 0ị: 14ị Suppose that the surface x2 ẳ h is free from the stress, i.e T hị ẳ Using (13) at x2 ¼ in (14) yields ( h M M2 ỵ M 24 ị IhM ỵ ) h ẵM M ỵM M3 ịM ỵ M3 M ỵM 24 ịM T 0ị ( h M M ỵ M M ị ỵ hM ỵ ) h ẵM M ỵM M3 ịM ỵ M3 M ỵM ịM U 0ị ẳ 0: 15ị Since the layer and the half-space are bonded perfectly to each other at the plane x2 ẳ 0, it follows that U0ị ẳ U 0ị and T0ị ẳ T 0ị Thus, we have from (15) ( h M M2 ỵ M 24 ị IhM ỵ ) h ẵM M ỵM M3 ịM ỵ M3 M ỵM 24 ịM T0ị P Chi Vinh, N Thi Khanh Linh / International Journal of Non-Linear Mechanics 50 (2013) 9196 ( ỵ hM þ whose real parts must be positive to ensure the decay condition (20), X ¼ rc2 , and h ðM3 M ỵM M ị ) h ẵM M ỵ M M ịM ỵM M ỵ M4 ÞM Š :Uð0Þ ¼ 0: ð16Þ The relation (16) between the traction vector and displacement vector of the half-space at the plane x2 ¼ is called the effective boundary condition of third-order in matrix form It replaces approximately the entire effect of the thin layer on the substrate With the help of (11) we can write (16) in component form as s21 ỵ hr s22,1 r u1,11 Àr u€ Þ  2 h r r s21,11 ỵ s 21 r u2,111 r r u 2,1 ỵ g2 ỵ h r € ¼0 u g 1,tt t s22,111 ỵ r t s 22,1 t u1,1111 Àr t u€ 1,11 À ! r2 € u ẳ0 a 22 2,tt at x2 ẳ 0, 18ị t1 ¼ t4 ¼ r7 a 22 r3 g2 r3 þ r1 r6 , t2 ¼ þr r þ r , t7 ¼ r2 r7 þ r4 r8 ; ỵ r1 r2 , r5 t5 ẳ t8 ẳ ỵ a 22 r7 g2 r4 a 22 r ẳ r r ỵr r , r1 g2 , r8 ẳ r4 a 22 ỵ r1 r2 , t3 ẳ r1 r7 ỵ r3 r6 , ỵ r2 r8 , t6 ẳ r5 g2 ỵ r2 a 22 bk ẳ g2 bk ỵ gn ak , Zk ¼ a12 Àa22 ak bk , k ¼ 1,2: 23ị From (22) we have g2 Xg1 ị ỵ a22 Xa11 ị ỵa12 ỵ gn ị2 :ẳ S, g2 a22 Xa11 ịXg1 ị 2 :ẳ P: b1 b2 ẳ g2 a22 2 b1 ỵb2 ẳ 24ị 25ị and p S ỵ P 0, b1 b2 ẳ p P, b1 ỵb2 ẳ q p S ỵ P: 26ị f b1 ịB1 ỵf b2 ịB2 ẳ 0, Fb1 ịB1 ỵFb2 ịB2 ẳ 0, ỵ r8 ỵ r2 r5 : 19ị where Fbn ị ẳ Zn ỵkhfr ỵX ịan r bn g ( !) 2 k h X r ỵ X r Zn r ỵ ỵ a 22 ( " #) 3 k h X , bn t ỵt X ịan t7 ỵ X t8 ỵ ỵ a 22 n ẳ 1,2, Suppose that the pre-stressed elastic half-space is compressible Then the unknown vectors U ẳ ẵu1 u2 T , T ẳ ẵs21 s22 ŠT satisfy Eq (10) without the bar symbol In addition to this equation are required the effective boundary conditions (17), (18) at x2 ¼ and the decay condition at x2 ẳ ỵ 1, namely at x2 ẳ ỵ 1: ð20Þ Now we consider the propagation of a Rayleigh wave, travelling (in the coated half-space) with velocity c and wave number k in the x1-direction and decaying in the x2-direction In according to Dowaikh and Ogden [9], Vinh [10] the vectors U ẳ ẵu1 u2 T , T ẳ ẵs21 s22 T are given by 28ị B21 ỵ B22 a 0, the determinant of coefficients of the homoSince geneous system (27) must vanish This yields f ðb1 ÞFðb2 ÞÀf b2 ịFb1 ị ẳ 0: 29ị Introducing (28) into (29) and taking into account (24) and (26), after algebraically lengthy calculations whose details are omitted, we arrive at an approximate secular equation of third-order for the Rayleigh waves, namely A0 þ A1 e þ A2 A3 e þ e ỵOe4 ị ẳ 0, 30ị where e ¼ kh (the dimensionless thickness of the layer) and A1 ẳ g2 r ỵX ịa1 b2 a2 b1 ị þ a22 ðr þX Þða1 b1 Àa2 b2 Þ, u2 ẳ ia1 B1 ekb1 x2 ỵ a2 B2 ekb2 x2 ịeikx1 ctị , s21 ẳ kfb1 B1 ekb1 x2 þ b2 B2 eÀkb2 x2 geikðx1 ÀctÞ , r4 A2 ¼ À a 22 ð21Þ where B1 and B2 are constants to be determined from the effective boundary conditions (17), (18), b1 , b2 are two roots of the equation g2 a22 b4 ỵ fg2 Xg1 ị ỵ a22 Xa11 ị ỵ a12 ỵ gn ị2 ịgb2 ỵXa11 ịXg1 ị ¼ 0, X ¼ r c2 : A0 ¼ ðg2 a12 ỵ gn a22 a1 a2 ịb2 b1 ị ỵ gn a12 ỵ g2 a22 b1 b2 ịa2 a1 ị, u1 ẳ B1 ekb1 x2 ỵ B2 ekb2 x2 ịeikx1 ctị , s22 ẳ ikfZ1 B1 ekb1 x2 ỵ Z2 B2 eÀkb2 x2 geikðx1 ÀctÞ , ð27Þ , Approximate secular equation of third-order U¼T ¼0 pffiffiffiffiffiffiffi a11 ÀXÀg2 b2k , k ẳ 1,2, i ẳ 1, a12 ỵ gn ịbk f bn ị ẳ bn ỵ khfr ỵ X Àr Zn g ( ) 2 k h X r ỵ ịbn an ẵr ỵ X r ỵ g2 ( ) 3 k h X , Zn t1 ỵ t2 X ịt3 t4 X ỵ g2 where g2 ¼ Substituting (21) into the effective boundary conditions (17) and (18) provides two linear equations for B1 and B2, namely t s21,111 ỵ r t s 21,1 t u2,1111 Àr t u€ 2,11 À r6 ¼ a22 b2k g1 ỵ X P 0, r ẳ r ỵr , a12 ỵ gn Þbk o X o minfa11 , g1 g ð17Þ s22 ỵ hr s21,1 r u2,11 r u Þ  2 h r € r s22,11 þ þ s 22 Àr u1,111 Àr r u€ 1,1 a 22 h ak ¼ One can show that if a Rayleigh wave exists (-b1 ,b2 having positive real parts), then (see also [10,8,19]) ! at x2 ẳ 0, ỵ 93 g2 ỵ X g2 ỵ X ! a 22 A0 ỵ 2r ỵ X ịr ỵ X ịa2 a1 ị ỵẵr r r r ỵX r r ịẵgn a12 ịa2 a1 ị ỵg2 a22 a1 a2 ịb2 b1 ị, " A3 ẳ g2 22ị ỵ r3 r8 þ X a 22 þ r6 þ X g2 !! # r ỵ X ị2r r þ r5 X Þ ða2 b1 Àa1 b2 Þ 94 P Chi Vinh, N Thi Khanh Linh / International Journal of Non-Linear Mechanics 50 (2013) 9196 "" ỵ a22 r6 þ X g2 þ r8 þ X a 22 !# # r ỵ X ị2r r þ r X Þ ða2 b2 Àa1 b1 Þ: ð31Þ By (23) one can prove the following equalities where b1 b2 ẳ Sẳ a11 X ỵ g2 b1 b2 ị b2 b1 ị, a12 ỵ gn ịb1 b2 g b ỵ b ị a X a2 b2 a1 b1 ¼ À 2 ðb2 Àb1 Þ, a1 a2 ẳ 11 , a12 ỵ gn ị a22 b1 b2 p P, b1 ỵb2 ẳ q p S ỵ2 P, Pẳ e1 xị1xị , e2 e5 e2 e1 xị ỵ e5 1xịe3 ỵe4 ị2 , e2 e5 37ị a2 a1 ¼ À and a2 b1 Àa1 b2 ¼ À X a11 a22 a12 g g , e1 ¼ , e2 ¼ , e3 ¼ , e4 ¼ n , e5 ¼ , g1 g1 g1 g1 g1 g1 a 11 g1 a 12 gn g1 e1 ¼ , e2 ¼ , e3 ¼ , e4 ¼ , e5 ¼ , g1 a 22 g1 g1 g2 a11 Xịb1 ỵ b2 ị b2 b1 ị: a12 ỵ gn ịb1 b2 xẳ 32ị Introducing (32) into (31) yields Ak ẳ yA k k ẳ 0,1,2,3ị, rm ẳ y ẳ b2 b1 ị=ẵa12 ỵ gn ịb1 b2 and A ẳ g2 ẵa212 a22 a11 Xịb1 b2 ỵa11 Xịẵg2n g2 g1 Xị, A ẳ g2 ẵr ỵ X ịa11 Xị ỵ a22 r ỵ X ịb1 b2 b1 ỵb2 ị, ! r4 r3 X X A2 ẳ þ þ þ A þ 2ðr þX Þðr ỵX ịXa11 g2 b1 b2 ị a 22 g2 g2 a 22 g1 c2 , r v ¼ , c2 ¼ g1 c2 sffiffiffiffiffiffi rffiffiffiffiffi g1 , c2 ¼ r g1 : r ð38Þ The squared dimensionless Rayleigh wave speed x depends on 13 dimensionless parameters: ek, e k k ẳ 1,2,3,4,5ị, r m , rv and e As D0 ẵxeị ẳ Oeị, the second-order approximate secular equation is either D0 ỵ D1 e ỵ D2 e ỵ Oe3 ị ẳ 0, 39ị ^2 D e2 ỵOe3 ị ẳ 0, 40ị or ỵ 2ẵr r r r ỵ r r ịX ẵg2 a12 b1 b2 ỵ gn Xa11 ị, (" # X X A ẳ g2 a11 Xị r ỵ ỵ 3r ỵ ị r ỵ X ị a 22 D0 ỵ D1 e ỵ g2 where ẫ 2r r ỵ r X ị b1 ỵ b2 ị (" # X X ỵ3r ỵ ị r ỵX ị g2 a22 r ỵ g2 ^ ẳ 2r m e fr m e e e ỵr xịe e ỵ r xịe3 ẵe e ðe e À1Þ D v v 4 Àe e ðe e 23 e ị ỵ e e Àe e Þr 2v xŠgb1 b2 a 22 ẫ 2r r ỵ r X ị b2 b1 b1 ỵ b2 ị in which b1 b2 and b1 ỵb2 are given by (24) and (26) After removing the factor y, Eq (30) becomes A0 ỵ A1e þ A2 A3 e þ e þ Oðe4 Þ ¼ 0: D2 D3 e ỵ e ỵOe4 ị ẳ 41ị is simpler than D2 Special cases ð35Þ in which Dk k ẳ 0,1,2,3ị are given by 4.1 Unstressed case When the pre-strains are absent, lk ¼ l k ¼ k ẳ 1,2,3ị, and the elastic constants Aijkl are given by (6) For this case, from (6) and (7) we have g1 ¼ g2 ¼ gn ¼ m, a11 ¼ a22 ẳ l ỵ2m, a12 ẳ l, g ẳ g ¼ g n ¼ m , a 11 ẳ a 22 ẳ l ỵ 2m , a 12 ¼ l r m e5 ½ðe 24 e À1 þ r 2v xÞðe1 ÀxÞ þe2 ðe e 23 e ỵ r 2v xịb1 b2 b1 ỵ b2 ị, D2 ẳ ẵe e 24 e 1ị þ e ðe e 23 Àe Þ þ ðe þ e Þr 2v xŠD0 e1 ¼ e2 ¼ Àe e ðe e 23 e ị ỵ e e e e ịr 2v xgb1 b2 ỵ 2r m xe1 ịfr m e e 23 e ỵ r 2v xịe 24 e 1ỵ r 2v xị þ e4 ½e e ðe 24 e 1ị 1 g , e3 ẳ , e2 ẳ g, g m c2 rm ¼ , rv ¼ , c2 m e1 ¼ À2r m e fr m e e 23 e ỵr 2v xịe 24 e ỵ r 2v xịe3 ẵe e ðe 24 e À1Þ g À2, e4 ¼ e5 ¼ 1, e ¼ À2, g c2 ¼ e ¼ e ¼ 1, rffiffiffiffi m , r x¼ sffiffiffiffi c2 ¼ c2 , c22 m , r 43ị where g ẳ m=l ỵ 2mị and g ẳ m =l ỵ 2m ị Introducing (43) into (36) yields Dk ẳ D k =gk ẳ 0,1,2,3ị, where D k are calculated by Àe e e e 23 e ị ỵ e e Àe e Þr 2v xŠg, D3 ¼ r m e5 ðxÀe1 Þfðe 24 e À1 ỵr 2v xịẵe e 24 e 1ị D ẳ ẵ4g1ị ỵ xb1 b2 ỵ1gxịx, ỵ 4e e e e ỵ e ỵ 3e ịr 2v x ỵ3e e e 23 e ị D ẳ r m ẵr 2v x1gxị ỵ4g ỵ r 2v xịb1 b2 b1 ỵ b2 ị, 2e e ẵe e e 24 e 1ị ỵ e e e e 23 e ị ỵe e ỵ e e ịr 2v xgb1 þ b2 ÞÀr m e2 e5 fðe e 23 e ỵr 2v xịẵ3e e 24 e 1ị D ẳ ẵ4g 1ị ỵ 1ỵ g ịr 2v xD0 ỵ 2r m ẵ2g1ị4g ỵ 2g r 2v xịgr m r 2v x4g ỵ r 2v xịb1 b2 ỵ 4e e e e ỵ 3e ỵ e ịr 2v x þe ðe e 23 Àe ފ À2e e ½e e ðe 24 e 1ị ỵ e e e e 23 e ị ỵ e e ỵ e e ịr 2v xgb1 b2 b1 ỵ b2 ị, 42ị therefore D0 ẳ e5 ẵe23 e2 e1 xịb1 b2 ỵe1 xịẵe24 e5 1xị, D1 ẳ e e e e 23 e ị ỵ ðe e Àe e Þr 2v xŠg ð34Þ This is the desired third-order approximate secular equation and it is fully explicit When the thickness of the layer vanishes, i.e e ¼ 0, the secular equation (34) becomes A ¼ that is equivalent to Eq (5.11) in [9] and Eq (25) in [10] In dimensionless form, Eq (34) becomes D0 ỵD1 e ỵ ỵ 2r m ðxÀe1 Þfr m ðe e 23 Àe þ r 2v xÞðe 24 e À1þ r 2v xị ỵe4 ẵe e e 24 e 1ị 33ị ỵ 2r m ẵ4g 1ị ỵ2g ỵ 2r m À2r m g Þr 2v xÀr m r 4v x2 1gxị, D ẳ r m fẵ81g ị ỵ42g 3ịr 2v x ỵ 3ỵ g ịr 4v x2 gx1ị 36ị ẵ81g ị ỵ4g 2ịr 2v x ỵ þ3g Þr 4v x2 Šb1 b2 gðb1 þ b2 Þ, ð44Þ P Chi Vinh, N Thi Khanh Linh / International Journal of Non-Linear Mechanics 50 (2013) 91–96 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi where b1 ¼ 1Àgx, b2 ¼ 1Àx Therefore, for the unstressed case (i.e for the isotropic case), the approximate secular equation of third-order is D ỵD e ỵ D2 D3 e ỵ e ỵ Oe4 ị ẳ D ỵD e ỵ 4 2r m p 2D0 D2 ẳ ẵl þ 1Þðl À3Þ þ2r 2v xŠÀ P fr m ðl 9ỵ 3r 2v xị 3 D2 e ỵOe3 ị ẳ 46ị or 4 4 ỵ22l ịẵl 3ịl 3ị ỵ ð2l À3Þr 2v xŠg, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 4 D3 ẳ r m x3ị S ỵ2 Pẵl 4l ỵ ỵr 2v xị2l 12 ỵ5r 2v xị q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 4 pffiffiffi À2ð2Àl Þð3Àl Þðl 3ỵ r 2v xị r m P S ỵ P 4 4 ẵl ỵ3r 2v xị3r 2v x2l ị ỵ 2l l 3ịl þr 2v xފ, n D þD e þ ẵl 1ịl 3ị ỵ r 2v x2l ẵl 3ịl 3ị ỵ 2l 3ịr 2v xg 2r m 4 x3ịfr m l ỵ3r 2v xịẵl 1ịl 3ị þr 2v xŠ þ ð45Þ that coincides with Eq (46) in [8] In the second-order, the approximate secular equation is either 95 D2 e ỵOe3 ị ẳ 0, ð47Þ where D , D and D are given by (44), and ð55Þ where ð3ÀxÞð1ÀxÞ ð6À4xÞ , Sẳ , 3 4 l ỵ r1l Þ R rm ¼ , r 2v ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: r l ỵ r1l4 ị Pẳ Dn2 ẳ 2r m ẵ2g1ị4g ỵ 2g r 2v xịgr m r 2v x4g ỵ r 2v xịb1 b2 ỵ2r m 4g 1ị ỵ 23g 2ịr 2v x ỵ r 4v x2 ð1ÀgxÞ: ð48Þ ð56Þ Eq (47) identifies with Eq (47) in [8] 4.2 In-plane isotropically pre-strained solids In this subsection we consider the case of isotropic pre-strains (equibiaxial deformations), where (see also [3]) l1 ¼ l2 ¼ l, l ¼ l ¼ l : e5 ¼ 1, e ¼ 1=e , e ¼ 1: ð50Þ For this case, the approximate secular equation of third-order is of the form as in (35), where Dk are calculated by (36) and (37) in which the fact (50) is taken into account, and   @2 W @2 W @W @W 2l 2 l ỵ @l1 @l1 @l2 @l2 @l1 ! ! , e3 ¼ , e1 ¼ @2 W @2 W @W @2 W @2 W @W ỵ ỵ l l 2 @l1 @l2 @l1 @l1 @l2 @l1 @l @l e4 ¼ 1À l @W @l2 @2 W @l1 ! @2 W @W ỵ @l1 @l2 @l1 ð51Þ for the half-space and there are similar expressions for the layer Continuity of the normal stress (see [3]) implies that ll D0 ỵD1 e ỵ 49ị Using (3)(5) and (7) and taking into account (49), from (38) we have e1 ¼ e2 , Here, in addition to l we introduce two new dimensionless 2 parameters r ¼ m=m and R ẳ mr l ị=mrl ị In this case, the squared dimensionless Rayleigh wave velocity x ¼ ðl rc2 Þ=m is determined (approximately) by the equation in which Dk k ẳ 0,1,2,3ị are given by (55) It is clear from (55) and (57) that x is a function in terms of four dimensionless parameters, namely: l, r, R, and e Fig shows the dependence on the dimensionless thickness of the layer e A ½0, 1Š of the squared dimensionless Rayleigh wave velocity x (with given values of l,r and R) that is calculated by the approximate secular equations of second- and third-order, and by the exact secular equation, Eq (8) in Ref [3] It is shown from Fig that (i) the approximate curves of second- and third-order of x are very close to the exact curve for the values of e A ½0, 1Š; (ii) for e A ½0, 0:5Š these curves almost coincide with each other These facts show that the approximate secular equations obtained are good approximations Conclusions In this paper, the propagation of Rayleigh waves in a prestressed compressible isotropic elastic half-space coated by a thin ð52Þ 0.65 0.6 Now to show the accuracy of the approximate secular equations obtained we take the two materials to have strain-energy functions of Blatz–Ko form (see [3]), namely 2 l1 ỵ l2 ỵ l3 ỵ 2J5ị x e1 ẳ e ẳ 3, e3 ¼ l , e3 ¼ l , e4 ¼ 2Àl , 0.45 0.4 0.35 0.3 e ¼ 2Àl , l : l4 þ rð1Àl4 Þ λ=0.9, r=0.3, R=2 ð53Þ for the half-space and similarly for the layer We set l3 ¼ l ¼ in addition to the assumption (49) Then, from (3)–(5) and (51)–(53) it follows that l4 ¼ 0.55 0.5 m 57ị 0.7 @W @W ẳ ll3 : @l2 @l Wẳ D2 D3 e ỵ e ẳ 0, 54ị Introducing (50) and (54) into (36) and (37) we have pffiffiffi 4 D0 ẳ l ỵ 3xị P ỵ 3xịẵl 1ịl 3ị ỵx, q p p 4 D1 ẳ r m S ỵ2 Pf3xịẵl 1ịl 3ị ỵ r 2v x ỵ P ẵl ỵ3r 2v xg, 0.25 0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ε Fig Plots of xðeÞ calculated by the approximate secular equation of secondorder (dash-dot line), by the approximate secular equation of third-order (dashed line), and by the exact secular equation (8) in Ref [3] Here l ¼ 0:9, r ¼ 0:3, R ¼ 96 P Chi Vinh, N Thi Khanh Linh / International Journal of Non-Linear Mechanics 50 (2013) 91–96 a pre-stressed compressible isotropic elastic layer has been investigated An effective boundary conditions of third-order are established that replaces approximately the entire effect of the layer on the half-space Then, by using it an explicit third-order approximate secular equation of the wave has been derived that is valid for any pre-strains and for a general strain-energy function This approximate secular equation recovers the one for the isotropic case where the pre-strains are absent It is shown that the approximate secular equation obtained has high accuracy Therefore, it will be helpful in practical applications Acknowledgement The work was supported by the Viet Nam National Foundation for Science and Technology Development (NAFOSTED) under Grant no 107.02-2012.12 and partly by the Abdus Salam International Center for Theoretical Physics (ICTP) References [1] D Bigoni, M Gei, A.B Movchan, Dynamics of a pre-stressed stiff layer on an elastic half space: filtering and band gap characteristics of periodic structural models derived from long-wave asymptotics, Journal of the Mechanics and Physics of Solids 56 (2008) 2494–2520 [2] A.G Every, Measurement of the near-surface elastic 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Plots of xðeÞ calculated by the approximate secular equation of secondorder (dash-dot line), by the approximate secular equation of third-order (dashed line), and by the exact secular equation. .. 137–140 [8] Pham Chi Vinh, Nguyen Thi Khanh Linh, An approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half-space coated by a thin orthotropic elastic layer, Wave... explicit third-order approximate secular equation of the wave has been derived that is valid for any pre-strains and for a general strain-energy function This approximate secular equation recovers