Rayleigh waves with impedance boundary conditions in incompressible anisotropic half spaces

Rayleigh waves with impedance boundary conditions in incompressible anisotropic half spaces tài liệu, giáo án, bài giảng...

International Journal of Engineering Science 85 (2014) 175–185 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci Rayleigh waves with impedance boundary conditions in incompressible anisotropic half-spaces Pham Chi Vinh a,⇑, Trinh Thi Thanh Hue b a b Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam Faculty of Civil and Industrial Construction, National University of Civil Engineering, 55 Giai Phong Str., Hanoi, Viet Nam a r t i c l e i n f o Article history: Received 14 March 2014 Accepted 11 August 2014 Keywords: Rayleigh waves Impedance boundary conditions Incompressible Orthotropic Monoclinic Explicit secular equation a b s t r a c t In this paper, the propagation of Rayleigh waves in an incompressible elastic half-space with impedance boundary conditions is investigated The half-space is assumed to be orthotropic and monoclinic with the symmetry plane x3 ¼ The main aim of the paper is to derive explicit secular equations of the wave For the orthotropic case, the secular equation is obtained by employing the traditional approach It is an irrational equation For the monoclinic case, the method of polarization vector is used for deriving the secular equation This is an algebraic equation of eighth-order When the impedance parameters vanish, the equations obtained coincide with the corresponding secular equations of Rayleigh waves with traction-free boundary conditions Ó 2014 Elsevier Ltd All rights reserved Introduction Elastic surface waves, discovered by Rayleigh (1885) more than 120 years ago for compressible isotropic elastic solids, have been studied extensively and exploited in a wide range of applications in seismology, acoustics, geophysics, telecommunications industry and materials science, for example For Rayleigh waves their explicit secular equation are important in practical applications They can be used for solving the direct (forward) problems: evaluating the dependence of the wave velocity on material parameters, especially for solving the inverse problems: to determine material parameters from measured values of wave velocity Therefore, explicit secular equations are always the main purpose for any investigation of Rayleigh waves In the context of Rayleigh waves, it is almost always assumed that the half-spaces are free of traction As mentioned in Godoy, Durn, and Ndlec (2012), in many fields of physics such as acoustics and electromagnetism, it is common to use impedance boundary conditions, that is, when a linear combination of the unknown function and their derivatives is prescribed on the boundary See, for examples, Antipov (2002), Zakharov (2006), Yla-Oijala and Jarvenppa (2006), Mathews and Jeans (2007), Castro and Kapanadze (2008) and Qin and Colton (2012), for the acoustics case and Senior (1960), Asghar and Zahid (1986), Stupfel and Poget (2011) and Hiptmair, Lopez-Fernandez, and Paganini (2014) for the electromagnetism one, and the references therein In the other hand, when studying the propagation of Rayleigh waves in a half-space coated by a thin layer, the researchers often replace the effect of the thin layer on the half-space by the effective boundary conditions on the surface of the half-space, see, for examples, Achenbach and Keshava (1967), Tiersten (1969), Bovik (1996), Steigmann and Ogden (2007), Vinh and Khanh Linh (2012, 2013), Vinh and Anh (2014a, 2014b) and Vinh, Anh, and Thanh (2014) These ⇑ Corresponding author Tel.: +84 35532164; fax: +84 38588817 E-mail address: pcvinh@vnu.edu.vn (P.C Vinh) http://dx.doi.org/10.1016/j.ijengsci.2014.08.002 0020-7225/Ó 2014 Elsevier Ltd All rights reserved 176 P.C Vinh, T.T Thanh Hue / International Journal of Engineering Science 85 (2014) 175–185 conditions lead to the impedance boundary conditions on the surface The Rayleigh is then considered as a surface wave that propagates in a half-space without coating whose surface is not traction-free but is subjected the impedance boundary conditions As addressed in Makarov, Chilla, and Frohlich (1995) and Niklasson, Datta, and Dunn (2000), a thin layer attached to a half-space is a model finding a broad range of applications in modern technology Rayleigh waves with impedance boundary conditions are therefore needed to be investigated However, very few investigations on Rayleigh waves with impedance boundary conditions have been done Malischewsky (1987) considered the propagation of Rayleigh waves with Tiersten’s impedance boundary conditions and provided a secular equation Recently, Godoy et al (2012) investigated the existence and uniqueness of Rayleigh waves with impedance boundary conditions which are a special case of Tiersten’s impedance boundary conditions In Godoy et al (2012) and Malischewsky (1987), the half-space is assumed to be isotropic Note that the Tiersten impedance boundary conditions are not accurate ones The main purpose of this paper is to study the propagation of Rayleigh waves with Tiersten’s impedance boundary conditions (Malischewsky, 1987) in anisotropic incompressible elastic half-spaces Two cases of anisotropy are considered: orthotropic materials and monoclinic ones with the symmetry plane x3 ¼ For the orthotropic case, the secular equation is obtained by employing the traditional techniques It is an irrational equation For the monoclinic case, for obtaining the secular equation we use the method of polarization vector The secular equation obtained is an algebraic equation of eighth-order When the impedance parameters vanish, the obtained equations coincide with the corresponding secular equation of Rayleigh waves with traction-free boundary conditions Orthotropic half-spaces Consider an elastic half-space which occupies the domain x2 P We are interested in the plane strain such that: ui ¼ ui ðx1 ; x2 ; tị; i ẳ 1; 2; u3  ð1Þ where t is the time Suppose that the half-space is made of incompressible orthotropic elastic material, then the strain–stress relations are (Nair & Sotiropoulos, 1997): > < r11 ỵ p ẳ c11 u1;1 ỵ c12 u2;2 r22 þ p ¼ c12 u1;1 þ c22 u2;2 > : r12 ẳ c66 u1;2 ỵ u2;1 ị 2ị where rij and cij are respectively the stresses and the material constants, p ẳ px1 ; x2 ; tị is the hydrostatic pressure associated with the incompressibility constraint, commas indicate differentiation with respect to spatial variables xk The elastic constants c11 ; c22 ; c12 ; c66 satisfy the inequalities: cii > 0; i ẳ 1; 2; 6; c11 ỵ c22 À 2c12 > ð3Þ which are necessary and sufficient conditions for the strain energy of the material to be positive semi-denite For an incompressible material, we have: u1;1 ỵ u2;2 ẳ 4ị from which we deduce the existence of a scalar function, denoted wðx1 ; x2 ; tÞ, such that: u1 ẳ w;2 ; u2 ẳ w;1 5ị In the absence of body forces, equations of motion are: & r11;1 ỵ r12;2 ẳ qu1 r12;1 ỵ r22;2 ẳ qu€2 ð6Þ where q is the mass density, a superposed dot signifies differentiation with respect to t Introducing Eqs (2) and (5) into Eq (6) and eliminating p from the resulting equations lead to an equation for w, namely:   ;11 ỵ w ;22 c66 w;1111 ỵ c11 2c12 ỵ c22 2c66 ịw;1122 ỵ c66 w;2222 ẳ q w 7ị Consider the propagation of a Rayleigh wave, traveling with velocity cð> 0Þ and wave number kð> 0Þ in the x1 -direction and decaying in the x2 -direction, i e.: ui ! ði ¼ 1; 2ị as x2 ! ỵ1 8ị Suppose that the surface x2 ¼ is subjected to impedance boundary conditions such that (Godoy et al., 2012; Malischewsky, 1987): r12 ỵ xZ1 u1 ẳ 0; r22 ỵ xZ2 u2 ẳ at x2 ẳ 9ị P.C Vinh, T.T Thanh Hue / International Journal of Engineering Science 85 (2014) 175–185 177 where x ¼ kc is the wave circular frequency, Z1 ; Z2 ð2 RÞ are impedance parameters whose dimension is of stress/velocity (Godoy et al., 2012; Malischewsky, 1987) Using Eqs (2) and (5) and the first of (6), the impedance boundary conditions (9) is expressed in term of w as: c66 w;22 w;11 ị ỵ xZ1 w;2 ẳ 0; at x2 ¼ 0; € ;2 ¼ at x2 ¼ c66 w;222 w;112 ị ỵ c11 2c12 þ c22 Þw;112 þ xZ2 w;11 À qw ð10Þ From (5) and the decay condition (8) it is required that: wðx1 ; x2 ; tÞ ! as x2 ! þ1 ð11Þ According to Ogden and Vinh (2004) wðx1 ; x2 ; tÞ is given by: wðx1 ; x2 ; tị ẳ /yịeikx1 ctị 12ị where y ẳ kx2 Substitution of Eq (12) into Eq (7) yields: 0000 / d xị/00 ỵ xị/ ¼ ð13Þ where a prime indicates differentiation with respect to y, d ẳ c11 ỵ c22 2c12 ị=c66 ; x ¼ c2 =c22 , c22 ¼ c66 =q Note that due to (3), d > In terms of / the impedance boundary conditions (10) become: pffiffiffi /00 0ị ỵ d1 x/0 0ị ỵ /0ị ẳ p /000 0ị ỵ d ỵ xị/0 0ị d2 x/0ị ẳ 14ị p where dn ẳ Z n = qc66 Rị; n ẳ 1; 2, are dimensionless impedance parameters From Eqs (11) and (12) it follows: /x2 ị ! as x2 ! ỵ1 15ị Thus, the problem is reduced to solving Eq (13) with the boundary conditions (14) and (15) The general solution for /ðyÞ that satisfies the condition (15) is (Ogden & Vinh, 2004): /yị ẳ Aes1 y ỵ Bes2 y 16ị where A and B are constants to be determined, s1 and s2 are the roots of equation: s4 À ðd À xịs2 ỵ xị ẳ 17ị with positive real parts It follows from Eq (17): s21 ỵ s22 ¼ d À À x :¼ S; s21 :s22 ẳ x :ẳ P 18ị It is not difficult to verify that if a Rayleigh wave exists (! s1 ; s2 having positive real parts), then: 0 < r22 ỵ p ẳ c12 u1;1 ỵ c22 u2;2 ỵ c26 u1;2 ỵ u2;1 ị > > : r12 ẳ c16 u1;1 ỵ c26 u2;2 ỵ c66 u1;2 ỵ u2;1 ị 27ị in which p ẳ px1 ; x2 ; tÞ is the hydrostatic pressure associated with the incompressibility constraint In the absence of body forces, equations of motion are: & r11;1 ỵ r12;2 ẳ q u1 r12;1 ỵ r22;2 ẳ q u2 28ị The incompressibility constraint reads as: u1;1 ỵ u2;2 ẳ 29ị Solving Eqs (27)3 and (29) for u1;2 and u2;2 we have: ( u1;2 ẳ b1 u1;1 u2;1 ỵ c166 r12 30ị u2;2 ¼ Àu1;1 where b1 ¼ c26 À c16 : c66 ð31Þ Using the first of (28) and taking into account (27)1, (27)2 and (30) yield: r12;2 ¼ qu€1 À a1 u1;11 À b1 r12;1 À r22;1 ð32Þ in which a1 ẳ c11 2c12 ỵ c22 c16 c26 Þ2 c66 ð33Þ From the second of (28) it follows: r22;2 ẳ qu2 r12;1 34ị In matrix form Eqs (30), (32) and (34) are written as: f0 ¼ Mf ð35Þ where u1 u 7 f¼6 7; r12 r22 M¼ M1 M2 M3 M4 ! ð36Þ P.C Vinh, T.T Thanh Hue / International Journal of Engineering Science 85 (2014) 175–185 179 in which the matrices (operators) Mk are given by: " M1 ¼ M3 ¼ b1 @ À@ À@ # " M2 ¼ ; c66 0 # ð37Þ q@ 2t À a1 @ 21 0 q@ 2t 5; M4 ¼ MT1 the symbol ‘‘T’’ indicates transpose of a matrix, the prime signifies derivative with respect to x2 and we use the notations @ ¼ @=@x1 ; @ 21 ¼ @ =@x21 ; @ 2t ¼ @ =@t Eq (35) is the matrix formulation of the plane strain for incompressible monoclinic solids with the symmetry plane x3 ¼ (see also Vinh & Seriani, 2009, 2010) From Eq (35) we immediately arrive at Stroh’s formulation (Stroh, 1962) 3.2 Rayleigh waves Stroh’s formulation Now we consider the propagation of a Rayleigh wave, traveling with velocity c ð> 0Þ and wave number k ð> 0Þ in the x1 direction and decaying in the x2 -direction Then, the displacements and stresses of the Rayleigh wave are sought in the form: un ẳ U n yịeikx1 ctị ; rn2 ẳ iktn yịeikx1 ctị ; n ẳ 1; 2; y ẳ kx2 ð38Þ Substituting (38) into Eq (35) leads to: n0 ẳ iNn; y < ỵ1 39ị where the prime signifies differentiation with respect to y and: n¼ u ! ; t U1 u¼ U2 ! ; t¼ t1 t2 ! ; Nẳ N1 N2 N3 N4 ! 40ị in which the matrices Nk are defined by: N1 ¼ b1 À1 À1 " ! N2 ¼ ; c66 0 # ; N3 ¼ X À a1 ! ; X N4 ẳ NT1 41ị X ¼ qc2 In addition to Eq (39) are required the decay condition: nỵ1ị ẳ 42ị Suppose the half-space is subjected to the impedance boundary conditions (9) In terms of U k and t k , it is expressed as follows: t1 ¼ id1 pffiffiffiffiffiffiffiffiffiffi c66 X U ; t ¼ id2 pffiffiffiffiffiffiffiffiffiffi c66 X U at y ẳ 43ị that can be written in matrix form as: " t ¼ Au at y ¼ 0; A¼ pffiffiffiffiffiffiffiffiffiffi id1 c66 X 0 pffiffiffiffiffiffiffiffiffiffi id2 c66 X # 44ị  T ẳ A, the bar indicates complex conjugate By the transformation: Note that A r ẳ t Au 45ị Eq (39), the decay condition (42) and the boundary condition (44) become: w0 ¼ iQw; y < ỵ1 46ị r0ị ẳ 47ị and wỵ1ị ẳ 0; where wẳ u r ! ; Qẳ Q1 Q2 K Q4 ! 48ị in which the matrices Q k and K are expressed in terms of matrices Nk and A by: Q ¼ N1 þ N2 A; Q ¼ N2 K ¼ N3 þ N4 A À AN1 À AN2 A; Q ¼ N4 À AN2 ð49Þ 180 P.C Vinh, T.T Thanh Hue / International Journal of Engineering Science 85 (2014) 175–185 From (49) and taking into account the facts: T; N2 ¼ N T; N3 ¼ N T; N4 ¼ N  T ¼ ÀA A ð50Þ one can show that: T ¼ Q ; Q 2 T Q4 ¼ Q  T ¼ K; K ð51Þ Eqs (39) and (46) are referred to Stroh’s formulation (Stroh, 1962) 3.3 Fundamental equations Proposition If 2m-vector YðyÞ is a solution of the problem: Y0 ¼ iPY; y < ỵ1; Yỵ1ị ẳ 52ị where the prime signifies differentiation with respect to y and: P¼ P1 P2 P3 P4 ! ð53Þ m  m-matrices Pk are constant matrices (being independent of y) and they satisfy the equalities: T ; P2 ¼ P T ; P3 ¼ P T P4 ¼ P ð54Þ then:  T 0ị^IPn Y0ị ẳ n Z Y 55ị where I ẳ I I ! I is m  m identity matrix We call Eq (55) the fundamental equations Proof: Lemma Suppose the matrix P expressed by (53) is invertible: " P À1 ¼ ðÀ1Þ P2 ðÀ1Þ P4 P1 P3 ðÀ1Þ # ð56Þ ðÀ1Þ ðÀ1Þ and the equalities (54) hold for the matrices Pk Then, these equalities also hold for the matrices Pk Proof From PP1 ẳ I it follows: 1ị < P1 P1ị ỵ P2 P3 ẳ I; : 1ị P3 P1 1ị ỵ P4 P3 1ị ỵ P2 P4 1ị ỵ P4 P4 P1 P2 ẳ 0; P3 P2 1ị ẳ0 1ị ẳI 57ị Taking transpose and complex conjugate two sides of the equalities (57) and using (54) yield: 1ị T < P1ị P2 ỵ P1 T P4 ẳ I; : 1ị T P3 1ị T P1 ỵ P 1ị T P2 ỵ P2 1ị T P1 ỵ P2 P4 P3 ẳ 0; P4 1ị T P4 ẳ 1ị T P3 ẳ I 58ị equivalently ðÀ1Þ T P2 ðÀ1Þ T P1 P4 P3 ðÀ1Þ T ðÀ1Þ T P1 P2 P3 P4 ! ẳI 59ị That means: P ẳ4 ðÀ1Þ T ðÀ1Þ T P4 P2 ðÀ1Þ P3 T ðÀ1Þ P1 T ð60Þ 181 P.C Vinh, T.T Thanh Hue / International Journal of Engineering Science 85 (2014) 175–185 From (56) and (60) and the uniqueness of PÀ1 it follows: 1ị P2 1ị T ẳ P2 1ị 1ị T P3 ; ẳ P3 1ị P4 ; 1ị T ẳ P1 ð61Þ The proof is completed h Lemma Suppose the matrix P expressed by (53) is invertible and the equalities (54) hold for the matrices Pk For all n Z the matrix Pn is expressed as: " Pn ẳ nị P2 nị P4 P1 P3 nị # nị n 0; P0 ẳ I ; 62ị nị Then, the equalities (54) also hold for the matrices Pk Proof ð0Þ ð1Þ + Clearly, the equalities (54) hold for matrices Pk and Pk ðnÞ + Assume the equalities (54) hold for the matrices Pk ; n > Then, it is not difficult to show that these equalities are satnỵ1ị nị ised for the matrices Pk That means the equalities (54) hold for Pk for all n Z; n P ðnÞ + By the Lemma 1, the equalities (54) hold for Pk for all n Z; n The proof of the Lemma is finished h Lemma Suppose the matrix P expressed by (53) is invertible and the matrices Pk satisfy the equalities (54) Then we have: ^IPn T ¼ ^IPn ; 8n2Z ð63Þ ðnÞ Proof By the Lemma 2, Pk satisfy the equalities (54) for all n Z With this fact one can see that: " ^IPn ¼ ðnÞ P4 ðnÞ P2 P3 P1 ðnÞ ðnÞ # nị T P ! ^IPn T ẳ nị P4 T ðnÞ T P1 ðnÞ T P2 ¼ ^IPn à  T^IPn we have: Proof of the proposition Pre-multiplying two sides of the Eq (52)1 by Y  T^I Pn Y0 ¼ i Y  T ^I Pnỵ1 Y Y 64ị Taking transpose and complex conjugate two sides of Eq (64) and using (63) yield:  ịT^I Pn Y ẳ i Y  T ^I Pnỵ1 Y Y 65ị From (64) and (65) it follows: d h  T^ n i  T^I Pn Y ¼ C Y IP Y ¼0 ! Y dy y ẵ0 ỵ where C is a constant Due to the second of (52) the constant C must be zero Therefore we have:  T yị^IPn Yyị ẳ n Z; y ẵ0 ỵ 1ị Y 66ị Taking y ẳ in Eq (66) we arrive at the fundamental Eq (55) The proposition is proved Remark (i) Eq (55) recover the fundamental equation (15) in Collet and Destrade (2004) when P is a real matrix (ii) There are at most (2m À 1) independent fundamental equations according to the Cayley–Hamilton theorem 3.4 Explicit secular equations Now in Eq (55) we take P ¼ Q that is given by (48) and (49), and Y ¼ w According to the second of (47): (55) are therefore simplied to:  T 0ịKnị u0ị ẳ n Z u r0ị ẳ 0, Eq 67ị 182 P.C Vinh, T.T Thanh Hue / International Journal of Engineering Science 85 (2014) 175–185 As Q is a  4-matrix, according to Remark 1, (ii), it is sufficiently to take three different values of n for deriving the secular equation of the wave It seems that the choice n ¼ À1; 1; is the best one (Ting, 2004) Suppose U ð0Þ – 0, then the vector uð0Þ can be written as: u0ị=U 0ịẵ1 aT , where a ẳ U 0ị=U 0ị is a complex number, a ẳ a ỵ ib; a; b are real Introducing the expression of uð0Þ into (67) and taking into account the fact that KðnÞ is hermitian (due to (51) and Lemma 2) we have: " Š ½1 a K 11 ðnÞ K 12 ðnÞ K 22 K 12 ðnÞ # ! a nị ẳ 0; n ẳ 1; 1; ð68Þ that provides three equations: ðÀ1Þ ðÀ1Þ 1ị 1ị >  ỵ K 22 > K ỵ K 12 a ỵ K 12 a aa ẳ > < 11 1ị 1ị 1ị  ỵ K 1ị  K 11 ỵ K 12 a ỵ K 12 a 22 aa ẳ > > > : 2ị 2ị 2ị 2ị  ỵ K 22 aa ẳ0 K 11 ỵ K 12 a ỵ K 12 a 69ị ðnÞ the elements K ij of the matrices KðnÞ ðn ¼ À1; 1; 2Þ are given by: ð1Þ K 11 ẳ a1 ỵ d21 ỵ 1ịX; 2ị K 11 ẳ 2b1 ẵa1 ỵ d21 ỵ 1ịX ; and 1ị K ij p 1ị K 12 ẳ i c66 X d1 d2 ị; 1ị K 22 ẳ X; 2ị K 22 ẳ 0; 70ị 2ị K 12 ẳ a1 d1 d2 ỵ 2ịX ỵ ib1 p c66 X ðd1 À d2 Þ ð71Þ ^ ðÀ1Þ =q where qð2 Rị is the determinant of the matrix Q and: ẳK ij 2 ^ 1ị ẳ c66 a1 d2 Xị a1 ỵ c66 ỵ b1 c66 ịX þ X ; K 22 c66 h pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi i ẳ b1 X ỵ i d1 d2 ị c66 X d1 X=c66 X ^ 1ị ẳ X; K 11 ^ ðÀ1Þ K 12 ð72Þ ðnÞ ðnÞ ^ ðÀ1Þ , K ^ ðÀ1Þ are real and K ðnÞ ðn ¼ 1; 2Þ; K ^ ðÀ1Þ are complex numbers whose As the matrix KðnÞ is hermitian, K 11 , K 22 n ẳ 1; 2ị; K 11 22 12 12 n;rị n;iị 1;rị ^ ^ 1;iị Substituting a ẳ a ỵ ib, real and imaginary parts are denoted, respectively, by K and K n ẳ 1; 2ị; K and K 12 12 12 12 ðn;iÞ ðnÞ ðn;rÞ ^ ðÀ1Þ = K ^ 1;rị ỵ iK ^ 1;iị into Eq (69) we arrive at a system of three linear equations, namely: K 12 = K 12 ỵ iK 12 n ¼ 1; 2Þ, K 12 12 12 ^ ðÀ1;rÞ K 12 ð1;rÞ K 12 ð2;rÞ K 12 ^ ðÀ1;iÞ K 12 ð1;iÞ K 12 ð2;iÞ K 12 3 ^ ðÀ1Þ 2a ^ ðÀ1Þ K ÀK 22 11 76 7 ð1Þ 1ị K 22 54 2b ẳ K 11 2ị 2ị a2 ỵ b K 11 K 22 73ị whose solution is: 2a ẳ D1 =D; 2b ẳ D2 =D; a2 ỵ b ¼ D3 =D ð74Þ where D is the determinant of the  matrix in Eq (73), Dk are the determinants of matrices obtained by replacing this matrix’s kth column with the vector on the right-hand side of Eq (73) It follows from Eq (74) that: D21 ỵ D22 4DD3 ẳ 75ị which is the desired explicit secular equation of the waves The expansions of the determinants D; Dk are lengthy and are not displayed here, but they are easily computed by using the expressions in Eqs (70)–(72) Also from (70)–(72) one can see that the secular Eq (75) is an algebraic equation of eighth-order in X Remark To obtain the secular equation of the wave we can apply the fundamental Eq (55) with the impedance boundary condition (43) in which P ¼ N However, the derivation is more complicated, especially when the size of the square matrix N is higher than When the impedance parameters vanish, i.e d1 ¼ d2 ¼ 0, from (70)–(72) it follows 1;rị 1;iị 2;iị 2ị ^ 1;iị ẳ K 12 ¼ K 12 ¼ K 12 ¼ K 22 ẳ K 12 76ị In view of (76) and (73) is simplified to: 6 ^ ðÀ1;rÞ K 12 ^ ðÀ1Þ K 11 K 11 ð2;rÞ K 12 ð2Þ K 11 ð1Þ ^ ðÀ1Þ 2a 3 K 22 76 7 1ị K 22 54 ẳ a2 ỵ b 0 whose determinant of this system’s matrix must be zero, i.e.: ð77Þ P.C Vinh, T.T Thanh Hue / International Journal of Engineering Science 85 (2014) 175–185 6 ^ ðÀ1;rÞ K 12 ^ ðÀ1Þ K 11 K 11 ð2;rÞ K 12 ð2Þ K 11 ^ ðÀ1Þ K 22 183 1ị K 22 5ẳ0 78ị h i ^ 1;rị K 1ị K 2ị ỵ K 2;rị K ^ ðÀ1Þ K ð1Þ À K ð1Þ K ^ ðÀ1Þ ¼ K 12 22 11 12 22 11 22 11 ð79Þ ð1Þ or equivalently: Again from (70)–(72) we have: 1ị 1ị K 22 ẳ 2ị K 11 ẳ 2b1 a1 ỵ Xị; ^ 1;rị ẳ ^ 1ị ẳ X; K K 11 12 K 11 ¼ X À a1 ; X; 2;rị K 12 ẳ a1 2X; 80ị b1 X; ^ 1ị ẳ c66 a1 a1 þ c66 þ b1 c66 ÞX þ X K 22 c66 Substituting (80) into (79) leads to: " 2b1 a1 XịX ỵ a1 2Xị X ỵ a1 ỵ Xị # a1 c66 a1 ỵ c66 ỵ b1 c66 ịX ỵ X ¼0 c66 ð81Þ This is the secular equation of Rayleigh waves propagating in an incompressible monoclinic half-space with the symmetry plane x3 ¼ whose surface is free of traction When the material is orthotropic, c16 ¼ c26 ¼ 0, from (31) and (33) it follows: b1 ¼ 0; a1 ẳ c11 2c12 ỵ c22 82ị On view of (82) and the fact that < X < a1 ; < X < c66 (see Ogden & Vinh, 2004), Eq (81) is equivalent to: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða1 À XÞ X=c66 X ẳ 83ị that coincides with the secular Eq (23) in Ogden and Vinh (2004) of Rayleigh waves propagating in an incompressible orthotropic half-space with traction-free condition Again let the half-space be orthotropic, i e equalities (82) hold Using (70), (71), (72), (75) and (82), it is not difficult to verify that for the orthotropic case: D1 ¼ and the corresponding secular equation in dimensionless form is of the from:  À 4D D 3 ẳ D 84ị  D  and D  are given by: where D; È ÂÀ Á à Épffiffiffi  ¼ ðd2 À d1 ị d2 ỵ ỵ d x d ỵ ð2d1 À d2 Þx2 x D  à  ẳ d2 ỵ d2 x d2 x ỵ d À xÞðx À 1Þ À d2 xðd À xÞ À d2 ỵ dịx ỵ 2dx2 x3 D 2 È À Á  à Épffiffiffi  ẳ d1 d2 ị d x d þ Àd x À 2d2 þ d1 ð2 þ dÞ x À d1 x2 x D 1 ð85Þ in which d ẳ c11 ỵ c22 2c12 ị=c66 ; d1 and d2 are the dimensionless impedance parameters, x ¼ c2 =c22 (c22 ¼ c66 =q) is the squared dimensionless velocity of Rayleigh waves From (85) one can see that Eq (84) is an algebraic equation of sixth-order in x Remark (i) By squaring (two times) two sides of Eq (23), we arrive at Eq (84) Eq (23) is therefore considered as the original version of Eq (84) (ii) Eq (23) is much more simple than Eq (84) This fact says that it is reasonable to consider separately the case of orthotropic materials (iii) Eq (23) will be useful in investigating the uniqueness of Rayleigh waves in incompressible orthotropic half-spaces with impedance boundary conditions by the complex function method (see, Vinh & Ha Giang, 2012; Vinh, 2013) Conclusions In this paper, the propagation of Rayleigh waves in anisotropic incompressible half-spaces subjected impedance boundary conditions is investigated Two cases of anisotropy are considered: orthotropic materials and monoclinic materials with the symmetry plane x3 ¼ For orthotropic case, the secular equation is derived by using the traditional technique and it is an irrational equation For the monoclinic half-spaces, first the impedance boundary condition is replaced by a traction-freelike boundary condition Then the secular equation is derived by using the method of polarization vector This equation is an 184 P.C Vinh, T.T Thanh Hue / International Journal of Engineering Science 85 (2014) 175–185 algebraic equation of eighth-order It is worth to note that, in principle, the secular equations for the incompressible materials can be obtained from those for the corresponding compressible materials following Destrade, Martin, and Ting (2002) and Destrade and Ogden (2010) In this paper, the authors follow the traditional direct approach that was used by many other, see, for examples, Dowaikh and Ogden (1990), Nair and Sotiropoulos (1999), Chadwick (1997), Destrade (2001), Ogden and Vinh (2004), Fu (2005), Edmondson and Fu (2009), Shams, Destrade, and Ogden (2011) and Destrade, Ogden, Sgura, and Vergori (2014) Acknowledgments 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 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a stratified layer of liquid and an analysis of the impedance boundary conditions Journal of Applied Mathematics and Mechanics, 70, 573–581  ... useful in investigating the uniqueness of Rayleigh waves in incompressible orthotropic half- spaces with impedance boundary conditions by the complex function method (see, Vinh & Ha Giang, 2012; Vinh,... equation of Rayleigh waves propagating in an incompressible orthotropic elastic halfspace whose surface is subjected to the impedance boundary conditions (9) Taking d1 ¼ d2 ¼ in Eq (23) we obtain the... few investigations on Rayleigh waves with impedance boundary conditions have been done Malischewsky (1987) considered the propagation of Rayleigh waves with Tiersten’s impedance boundary conditions