DSpace at VNU: Non-principal Rayleigh waves in deformed transversely isotropic incompressible non-linearly elastic solids

IMA Journal of Applied Mathematics Advance Access published May 29, 2014 IMA Journal of Applied Mathematics (2014) Page of 14 doi:10.1093/imamat/hxu023 Non-principal Rayleigh waves in deformed transversely isotropic incompressible non-linearly elastic solids Jose Merodio Department of Continuum Mechanics and Structures, E.T.S Ing Caminos, Canales y Puertos, Universidad Politecnica de Madrid, 28040 Madrid, Spain Trinh Thi Thanh Hue Faculty of Civil and Industrial Construction, National University of Civil Engineering, 55, Giai Phong Street, Hanoi, Vietnam and Nguyen Thi Nam Department of Continuum Mechanics and Structures, E.T.S Ing Caminos, Canales y Puertos, Universidad Politecnica de Madrid, 28040 Madrid, Spain [Received on 15 December 2013; revised on April 2014; accepted on 25 April 2014] An explicit expression of the secular equation for non-principal Rayleigh waves in incompressible, transversely isotropic, pre-stressed elastic half-spaces is obtained This generalizes previous works dealing with isotropic incompressible materials The free surface coincides with one of the principal planes of the primary pure homogeneous strain, but the surface wave is not restricted to propagate in a principal direction The free surface is taken to be the horizontal plane, while the fibres that give the transversely isotropic character of the material are located throughout the whole half-space and run parallel to each other and perpendicular to the depth direction Results are given, for illustration, in respect of the so-called reinforcing models It is shown that the wave velocity depends strongly on the anisotropic character of the material model Keywords: Rayleigh waves; explicit secular equation; transversely isotropic half-spaces Introduction Elastic surface waves, discovered by Rayleigh (see, for instance, Rayleigh, 1885) nearly 130 years ago for compressible isotropic elastic solids have been studied extensively and exploited in a wide range of applications including seismology, acoustics, geophysics, telecommunications industry and materials science, among others The study of surface waves travelling along the free surface of an elastic half-space affects many aspects of modern life, stretching from mobile phones through to the study of earthquakes, as addressed by Adams et al (2007) The Rayleigh wave existence and uniqueness problem has been resolved with the aid of the Stroh formalism (see Stroh 1958, 1962), even for an anisotropic elastic half-space (see Barnett et al 1973; c The authors 2014 Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications All rights reserved Downloaded from http://imamat.oxfordjournals.org/ at University of California, San Francisco on December 7, 2014 Pham Chi Vinh∗ Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam ∗ Corresponding author: pcvinh@vnu.edu.vn of 14 P C VINH ET AL Downloaded from http://imamat.oxfordjournals.org/ at University of California, San Francisco on December 7, 2014 Barnett & Lothe 1974, 1985; Chadwick & Smith 1977; Ting 1996) Based on an identity for the surfaceimpedance matrix, Fu & Mielke (2002), Mielke & Fu (2004) also have given a direct uniqueness proof (free from the Stroh formalism) for this problem Once the existence has been established, there remains to determine the surface-wave velocity from secular equations of implicit as well as explicit form A number of approaches have been suggested based on the Stroh formalism (see, for example, Barnett & Lothe, 1985; Chadwick & Wilson, 1992) that evaluate the surface-wave velocity from implicit secular equations Although all these approaches are straightforward, their use requires familiarity with the Stroh formalism and a considerable amount of numerical work Another approach, based on the surface-impedance matrix, has been proposed recently by Fu & Mielke (2002) This method is very practical and efficient Nevertheless, one can also find the surface-wave velocity by directly solving explicit secular equations A large number of such secular equations have been derived by employing different methods such as the polarization vector method (see Taziev, 1989; Ting, 2004), the method of first integrals (see Mozhaev, 1995; Destrade, 2001) and the cofactor method (Ting, 2002) These methods were concisely presented in Ting (2005) The explicit secular equations obtained by the methods just mentioned often admit spurious roots that have to be carefully eliminated, as opposed to the numerical methods based on the Stroh formulation or on the surface-impedance matrix However, the application of the explicit secular equations is not limited to numerically determine the surface-wave velocity They are also convenient tools to solve the inverse problem that deals with measured values of the wave speed and their agreement with material parameters Exact formulas for the Rayleigh wave velocity, i.e analytical solutions of the explicit secular equations, are very significant in various practical applications They have been given by Vinh & Ogden (2004b) and Malischewsky (2004) for isotropic solids, Ogden & Vinh (2004) and Vinh & Ogden (2004a, 2005) for orthotropic solids and Vinh (2010, 2011) for pre-stressed media, among others Most of the investigations deal with harmonic surface waves travelling along the traction-free flat surface of half-spaces Surface waves with arbitrary profile as well as surface waves guided by topography and those travelling along forced surfaces have also attracted attention and we refer the reader for further details to Kiselev & Rogerson (2009), Kiselev & Parker (2010), Prikazchikov (2013), Parker (2013), Adams et al (2007), Fu et al (2013), Kaplunov et al (2010) and references therein We also have to note that the propagation of surface waves in an initially isotropic half-space under the effect of pre-stress has been examined in a variety of contexts by Hayes & Rivlin (1961), Chadwick & Jarvis (1979), Murdoch (1976), Dowaikh & Ogden (1990, 1991), Ogden & Steigmann (2002), Destrade & Ogden (2005), Destrade et al (2005), Murphy & Destrade (2009), Vinh (2010, 2011), Vinh & Giang (2010, 2012) and Destrade (2007), to name a few Here, we obtain the explicit secular equation of non-principal Rayleigh waves propagating in incompressible, transversely isotropic, pre-stressed elastic half-spaces To obtain such an equation, one can apply either the polarization vector method or the cofactor method since the method of first integrals is not applicable in this case; see Ting (2005) We choose the polarization vector method, which seems to be more simple The propagation of Rayleigh waves under the conditions at hand was investigated by Prikazchikov & Rogerson (2004), who derived the secular equation in implicit form The motivation behind this analysis can be summarized as follows First, the explicit secular equation is a convenient tool for (among others) the non-destructive evaluation of pre-stressed structures before and during loading; see, for example, Makhort (1978), Makhort et al (1990), Hirao et al (1981), Husson (1985), Delsanto & Clark (1987), Duquennoy et al (1999, 2006) and Hu et al (2009) Secondly, the use of fibre-reinforced elastic composites is common in engineering applications because these materials have a higher strength-to-weight ratio than classical isotropic materials used in the past Fibre-reinforced elastic composites with a family of parallel fibres reinforcing a material are called NON-PRINCIPAL RAYLEIGH WAVES IN TRANSVERSELY ISOTROPIC MATERIALS of 14 Preliminaries: material model and equations of motion 2.1 Material model First, we introduce the material model For that purpose, it is sufficient to consider an elastic body whose initial geometry defines a reference configuration, which we denote by B0 and a finitely deformed equilibrium configuration Bt The deformation gradient tensor associated with the deformation B0 → Bt is denoted by F In the literature, several composite materials (see Merodio & Ogden 2005a and references therein) and also some soft tissues are modelled as incompressible transversely isotropic elastic solid with one preferred direction associated with a family of parallel fibres of collagen We denote by M the unit vector in that direction when the solid is unloaded and at rest The most general transversely isotropic non-linear elastic strain–energy function Ω depends on F through the invariants associated with M and the right Cauchy–Green strain tensor C = F F (see, for instance, Merodio & Ogden, 2005b,c) The invariants of C most commonly used are the principal invariants, defined by I1 = tr C, I2 = 12 [(tr C)2 − tr(C2 )], I3 = det C (1) The (anisotropic) invariants associated with M and C are usually taken as I4 = M · (CM), I5 = M · (C2 M) (2) It follows that, for incompressible materials, Ω = Ω(I1 , I2 , I4 , I5 ) since I3 = at all times For the considered incompressible material the Cauchy stress is σ =F ∂Ii ∂Ω − pI = − pI, Ωi F ∂F ∂F i=1,i = | (3) where p is a Lagrange multiplier associated with the incompressibility constraint and I is the × identity matrix The principal values are denoted by σi , i = 1, 2, The Cauchy stress tensor can be Downloaded from http://imamat.oxfordjournals.org/ at University of California, San Francisco on December 7, 2014 transversely isotropic elastic materials In addition, the acoustics of incompressible soft solids is being used in the analysis of biological soft tissues; see, for instance, Destrade et al (2010a) as well as Vinh & Merodio (2013a,b) Soft biological tissues were generally considered incompressible and isotropic during the early days of their analyses In more recent years, some have been recognized as highly anisotropic due to the presence of collagen fibres (see Holzapfel et al., 2000; Destrade et al., 2010b) The mechanical modelling of these materials is related to the analysis of Rayleigh waves propagating in transversely isotropic, incompressible, pre-stressed elastic half-spaces The paper is organized as follows In Section 2, the basic constitutive equations for the hyperelastic material model at hand are given Similarly, the corresponding equations for infinitesimal waves superimposed on a finite deformation consisting of a pure homogeneous strain are outlined Surface waves are studied in Section The Stroh formalism is derived and the implicit secular equation is presented It will be used to eliminate spurious roots arising in the explicit secular equation The explicit secular equation for non-principal Rayleigh waves in incompressible, transversely isotropic, pre-stressed elastic half-spaces is obtained in Section 3.2 The results are illustrated in respect of the so-called reinforcing models In Section 4, a brief discussion of the results is given 4 of 14 P C VINH ET AL written as σ = 2Ω1 B + 2Ω2 (I1 I − B)B + 2Ω4 m ⊗ m + 2Ω5 (m ⊗ Bm + Bm ⊗ a) − pI, (4) Ω= μ(I1 − 3) + f (I4 ), (5) where f (I4 ) is the reinforcing model There are other reinforcing models (see Merodio & Neff, 2006), in particular, some strain energy functions also have the form Ω= μ(I1 − 3) + g(I5 ) (6) We will make use of these reinforcing models to illustrate the results 2.2 Linearized equations of motion: incremental equations We introduce now the precise notation that is convenient for the analysis developed in the next sections Let (X1 , X2 , X3 ) be a fixed rectangular coordinate system Consider an incompressible transversely isotropic semi-infinite body B in its unstrained state B0 that occupies the region X2 Fibres run parallel to each other and perpendicular to the depth direction X2 , i.e M2 = (see Fig 1) The body is subjected to a finite pure homogeneous strain with principal directions given by the Xi -axes A finitely deformed (pre-stressed) equilibrium state Be is obtained A small time-dependent motion is superimposed upon this pre-stressed equilibrium configuration to reach a final material state Bt , called current configuration The position vectors of a representative particle are denoted by Xi , xi (X) and x˜ i (X, t) in B0 , Be and Bt , respectively The deformation gradient tensor associated with the deformations B0 → Bt ¯ respectively, and are given in component form by and B0 → Be are denoted by F and F, FiA = ∂ x˜ i , ∂XA F¯ iA = ∂xi ∂XA (7) It is clear from (7) that FiA = (δij + ui,j )F¯ jA , (8) where δij is the Kronecker operator, ui (X, t) denotes the small time-dependent displacement associated with the deformation Be → Bt and a comma indicates differentiation with respect to the indicated spatial ¯ coordinate in Be The fibre orientation in Be is in particular related to M by m = FM The necessary equations including the linearized equations of motion for transversely isotropic incompressible materials are now summarized The incremental components of the nominal stress tensor Sji are related to the incremental displacement gradients uk,l by (see Ogden, 1984): Sji = Ajilk uk,l + Puj,i − p∗ δij , (9) Downloaded from http://imamat.oxfordjournals.org/ at University of California, San Francisco on December 7, 2014 where m = FM Hence, m gives the fibre direction in the deformed configuration It is clear that principal directions of stress and strain not coincide, in general In the biomechanics literature several strain energy functions given by an isotropic elastic material augmented with the so-called reinforcing model can be found It is common to consider functions of the form (see Merodio & Ogden, 2005a) NON-PRINCIPAL RAYLEIGH WAVES IN TRANSVERSELY ISOTROPIC MATERIALS of 14 ¯ is the value of p at Be (independent of time t), p∗ = p − P is the time-dependent where P = p(F) increment of p and the components of the (pushed forward ) fourth-order elasticity tensor A for Ω = Ω(I1 , I2 , I4 , I5 ) are given by (see also Vinh & Merodio, 2013b): Apiqj = F¯ pα F¯ qβ ∂ 2Ω ∂Fiα ∂Fjβ F=F¯ = 2Ω1 δij B¯ pq + 2Ω2 (2B¯ ip B¯ jq − B¯ iq B¯ jp + I1 δij B¯ pq − B¯ ij B¯ pq − δij (B¯ )pq ) + 2Ω4 δij mp mq + 2Ω5 [δij (mp B¯ qr mr + mq B¯ pr mr ) + B¯ ij mp mq + mi mj B¯ pq + B¯ iq mj mp + B¯ pj mi mq ] + 4Ω11 B¯ ip B¯ jq + 4Ω22 (I1 B¯ ip − (B¯ )ip )(I1 B¯ jq − (B¯ )jq ) + 4Ω44 mi mj mp mq + 4Ω55 [mp B¯ ir mr + mi B¯ pr mr ][mq B¯ jr mr + mj B¯ qr mr ] + 4Ω12 [B¯ ip (I1 B¯ jq − (B¯ )jq ) + B¯ jq (I1 B¯ ip − (B¯ )ip )] + 4Ω14 [B¯ ip mj mq + B¯ jq mi mp ] + 4Ω15 [B¯ ip × [mq B¯ jr mr + mj B¯ qr mr ] + B¯ jq [mi B¯ pr mr + mp B¯ ir mr ]] + 4Ω24 [(I1 B¯ ip − (B¯ )ip )mj mq + (I1 B¯ jq − (B¯ )jq )mi mp ] + 4Ω25 [(I1 B¯ ip − (B¯ )ip ) × [mq B¯ jr mr + mj B¯ qr mr ] + (I1 B¯ jq − (B¯ )jq )[mi B¯ pr mr + mp B¯ ir mr ]] + 4Ω45 [mi mp [mq B¯ jr mr + mj B¯ qr mr ] + mj mq [mi B¯ pr mr + mp B¯ ir mr ]] (10) Downloaded from http://imamat.oxfordjournals.org/ at University of California, San Francisco on December 7, 2014 Fig A point in the free surface of the pre-stressed half-space: (i) the principal axes of the primary pure homogeneous strain (xi -axes), (ii) the fibre direction in that configuration (given by α) as well as the fibres (dashed lines) along the depth direction (x2 -axis) and (iii) the propagation direction of the wave (given by θ ) Fibres are located throughout the whole half-space and run parallel to each other and perpendicular to the depth direction 6 of 14 P C VINH ET AL where B = FF and we note that I1 = B¯ kk It is clear that Apiqj = Aqjpi Note that formula (10) holds for any coordinate system Another form of Apiqj was derived by Prikazchikov & Rogerson (2003, 2004) In general, the elasticity tensor A has at most 45 non-zero components We further specialize the elasticity tensor to some specific energy functions It follows from (5) that Ω2 =Ω5 =Ω2k =Ω5k = Ω1k = (k = 1, 2, 4, 5), Ω1 = μ/2, Ω4 = f (I4 ) and Ω44 = f (I4 ) for this reinforcing model Using these results and (10), the expression of Apiqj is (see also Destrade et al., 2008; Merodio & Ogden, 2002) Apiqj = δij [μB¯ pq + 2f (I4 )mp mq ] + 4f (I4 )mi mj mp mq (11) On the other hand, using (6) and (10) one can write Apiqj = μδij B¯ pq + 2g (I5 )[δij (mp B¯ qr mr + mq B¯ pr mr ) + B¯ ij mp mq + mi mj B¯ pq + B¯ iq mj mp + B¯ pj mi mq ] + 4g (I5 )(mp B¯ ir mr + mi B¯ pr mr )(mq B¯ jr mr + mj B¯ qr mr ) (12) for the other reinforcing model In the absence of body forces, incremental equations of motion are (see Ogden, 1984) Sij,i = uă j , i, j = 1, 2, 3, (13) where a dot indicates differentiation with respect to time t The incremental version of the incompressibility is (see Ogden, 1984) uk,k = (14) In summary, under the conditions at hand, the principal directions of the pre-strain are the xk axes, | j, and m2 = The half-space is maintained in this static state of deformation by the i.e B¯ ij = if i = application of stresses that are obtained using (4) Furthermore, it is easy to check that the plane x2 = is a principal plane of stress while the planes x1 = and x3 = support shear stresses It follows from (10) that, under these circumstances, there are only 25 non-zero components of the elasticity tensor | j) and Aijji (i, j = 1, 2, 3, i = | j) , Aii13 (i = 1, 2, 3), A, namely: Aiijj , (i, j = 1, 2, 3), Aijij (i, j = 1, 2, 3, i = Aii31 (i = 1, 2, 3), A2312 , A2321 , A3212 and A3221 Surface waves The analysis is specialized to Rayleigh waves propagating in a principal plane of the pre-strain, the plane x2 = 0, but not in a principal direction The incremental equation of motion can be cast as a homogeneous linear system of six first-order differential equations Downloaded from http://imamat.oxfordjournals.org/ at University of California, San Francisco on December 7, 2014 Remark If the principal directions of the pure homogeneous pre-strain are given by the Xi -axes (or xi -axes) and the fibre direction is restricted to be one of the Xi -axes (or xi -axes), it follows using (10) that there are only 15 non-zero components of the fourth-order elasticity tensor A, namely Aiijj , | j) and Aijji (i, j = 1, 2, 3, i = | j) (see also Vinh & Merodio, 2013a) Aijij (i, j = 1, 2, 3, i = NON-PRINCIPAL RAYLEIGH WAVES IN TRANSVERSELY ISOTROPIC MATERIALS 3.1 of 14 The Stroh formalism We consider a Rayleigh wave travelling with velocity c and with its wave vector k laying in the (x1 , x3 ) plane The wave makes an angle θ with the x1 -direction and decays in the x2 -direction Then, the displacements and stresses of the Rayleigh wave are written (see Destrade et al., 2005) as un = Un (y) eik(x1 cθ +x3 sθ −ct) , S2n = iktn (y) eik(x1 cθ +x3 sθ −ct) , n = 1, 2, 3, y = kx2 , (15) ξ = iNξ , y < +∞ (16) where the prime signifies differentiation with respect to y and u ξ= , t ⎡ ⎤ U1 u = ⎣U2 ⎦ , U3 ⎡ ⎤ t1 t = ⎣t2 ⎦ , t3 N= N1 N2 K N1 , (17) within which the matrices Nk and K are defined by ⎡ N1 = ⎣−cθ f1 f2 ⎤ −sθ ⎦ , ⎡ d1 N2 = ⎣ −d13 ⎤ −d13 0 ⎦, d3 ⎡ h1 K=⎣0 h2 h3 ⎤ h2 0⎦, h4 (18) where f1 = a11 cθ + a13 sθ , f2 = a31 cθ + a33 sθ , h1 = ρc2 − b111 c2θ − b113 cθ sθ − b133 s2θ , h3 = ρc2 − e11 c2θ − e13 cθ sθ − e33 s2θ , h2 = −b311 c2θ − b313 cθ sθ − b333 s2θ , (19) h4 = ρc2 − d311 c2θ − d313 cθ sθ − d333 s2θ and the remaining coefficients are given in Appendix A Equation (16) is the Stroh formalism (Stroh, 1958, 1962) The decay condition is expressed in the following form ξ (+∞) = (20) The boundary condition of zero incremental traction using the expression given for S2n in (15) means that t(0) = (21) In passing, we note that the matrices N1 , N2 and K in (18) particularized for isotropic materials coincide, respectively, with the matrices N1 , N2 and N3 + X I, where X = ρc2 , given by (2.9) and (2.10) in Destrade et al (2005) Downloaded from http://imamat.oxfordjournals.org/ at University of California, San Francisco on December 7, 2014 respectively, where cθ = cos θ , sθ = sin θ , and k = |k| is the wave number Using (15), together with (9), (13) and (14), one can write of 14 3.2 P C VINH ET AL Implicit and explicit secular equations The implicit secular equation is given by (see Destrade et al., 2005; Prikazchikov & Rogerson, 2004 for complete details) nωI − (m − ωII )ωIII = (22) in which ωII = s1 s2 + s2 s3 + s3 s1 , ωIII = −s1 s2 s3 , (23) where s1 , s2 and s3 are the eigenvalues of the Stroh matrix N with positive imaginary parts, and m and n are defined, respectively, as f12 − d1 h1 + h3 m= n = (h1 h4 − h22 ) f22 − d3 h4 + h3 f1 f + d13 h2 , h3 d3 f12 + d1 f22 2d13 f1 f2 d1 d3 − − − d13 h3 h3 (24) Equation (22) is called the implicit secular equation (see also Destrade et al., 2005) because the expressions for the√ωI , ωII and ωIII in terms of X are not known A (subsonic) Rayleigh wave exists with velocity c = X /ρ if and only if (22) is satisfied Now we apply the method of polarization vector, see Ting (2005), to obtain the explicit secular equation of the wave Using (16), (20), (21), and that N2 and K are symmetric, one can write u¯ (0)K(n) u(0) = where K(n) is defined as Nn = ∀n ∈ Z N(n) N(n) K(n) N(n) (25) (26) From (25) the explicit secular equation is obtained |K2 , K3 , K1 |2 + 4|K2 , K3 , K4 | |K2 , K1 , K4 | = 0, where ⎡ (−1) ⎤ K11 ⎢ (1) ⎥ K1 = ⎣ K11 ⎦ , (3) K11 ⎡ (−1) ⎤ K22 ⎢ (1) ⎥ K2 = ⎣ K22 ⎦ , (3) K22 ⎡ (−1) ⎤ K33 ⎢ (1) ⎥ K3 = ⎣ K33 ⎦ , (3) K33 (27) ⎡ (−1) ⎤ K13 ⎢ (1) ⎥ K4 = ⎣ K13 ⎦ (28) (3) K13 in which Kij(n) are entries of the matrix K(n) Equation (27) is the explicit secular equation under the conditions at hand (1) (1) (1) (1) (3) (3) (3) (3) , K22 , K33 , K13 are polynomials of degree 1, 1, 1, in X ; K11 , K22 , K33 , K13 The elements K11 (−1) (−1) (−1) (−1) are polynomials of degree 2, 1, 2, in X and K11 , K22 , K33 , K13 are polynomials of degree 2, 3, 2, in X , respectively Therefore (27) is an algebraic equation of order 12 in X The numerical resolution of (27) yields a priori 12 roots for X From these, it is easy to discard the complex roots, the negative real roots and the roots corresponding to supersonic surface waves More in particular, there is only one (subsonic) Rayleigh wave (see Fu, 2005), and that one satisfies the implicit secular equation (22) Downloaded from http://imamat.oxfordjournals.org/ at University of California, San Francisco on December 7, 2014 ωI = −(s1 + s2 + s3 ), NON-PRINCIPAL RAYLEIGH WAVES IN TRANSVERSELY ISOTROPIC MATERIALS of 14 In order to illustrate the results further, we consider some particular strain–energy functions Numerical resolution of the polynomial (27) yields the wave velocity The expressions are quite lengthy and not enlightening, so we omit them and just provide some numerical results We consider the following strain–energy functions (see, for instance, Merodio & Neff, 2006; Merodio & Ogden, 2005a) μγ1 (I4 − 1)2 , μγ2 (I5 − 1)2 , (29) (30) (31) where μ, γ1 and γ2 are material constants The last model is the well known neo-Hookean one The other two strain energy functions introduce reinforcing models First, for simplicity, we consider that the fibres are parallel to the X1 -direction and that the elastic half-space is initially under uniaxial tension along the X1 -axis (see Vinh & Merodio, 2013a,b), i e x1 = λX1 , x2 = λ−1/2 X2 , x3 = λ−1/2 X3 , λ > 0, λ = const (32) The wave makes an angle θ with the x1 -direction Figure shows the dependence of the squared dimensionless Rayleigh wave velocity x = ρc2 /μ obtained using (27) on θ The anisotropy strongly influences the Rayleigh wave velocity of the isotropic base model Furthermore, the influence on the wave velocity of the isotropic base model introduced by the invariant I5 is stronger than the one given by the invariant I4 As shown in Fig 2, if the surface wave propagates in a direction perpendicular to the fibre direction, then the wave velocity is associated with the one corresponding to the neo-Hookean material without reinforcement as given by Flavin (1963) Let us consider now that the fibre direction is not a principal direction The pre-strain is pure homogeneous and given by x1 = λ1 X1 , x2 = λ2 X2 , x3 = λ3 X3 , (33) where λk are the principal stretches of the deformation and obey that λ1 λ2 λ3 = We further consider that the plane x2 = is free of tractions Under these conditions, the components of the Cauchy stress are obtained, after some simple manipulations, using (4) for a given strain energy function Figure shows the dependence on θ ∈ [0 π/2] (the angle between the wave propagation direction and the x1 -axis) of x = ρc2 /μ obtained using (27) when the fibre direction makes an angle α = π/6 with the x1 -axis for (29) (dash–dot line), (30) (dashed line) and (31) (solid line) The parameters used for the computations are γ1 = γ2 = 34 , while λ1 = 32 , λ2 = and λ3 = 23 The curve associated with the NeoHookean model has a maximum for θ = On the other hand, the other two curves have a maximum at an angle θ ∈ (0 π/2) We also note that when (27) is specialized to isotropic materials, it coincides with Equation (4.5) in Destrade et al (2005), which is the secular equation for non-principal Rayleigh waves in deformed isotropic incompressible materials Exact formulas for the velocity of Rayleigh waves can be derived for special cases in which the wave propagation direction coincides with one of the principal directions of the pre-strain We not give details but show some numerical results The elastic half-space is initially under uniaxial tension along the X1 -axis (see (32)) and we consider a wave propagating in the x1 -direction, which also gives the direction of the fibre reinforcement Figure shows the dependence of the squared dimensionless Rayleigh wave velocity x = ρc2 /μ on λ ∈ [1 1.5] Downloaded from http://imamat.oxfordjournals.org/ at University of California, San Francisco on December 7, 2014 μ (I1 − 3) + μ Ω = (I1 − 3) + μ Ω = (I1 − 3), Ω1 = 10 of 14 P C VINH ET AL Fig Values of x = ρc2 /μ obtained using (27) vs θ ∈ [0 π/2] when the fibre direction makes an angle α = π/6 with the x1 -axis for (29) (dash–dot line), (30) (dashed line) and (31) (solid line), with γ1 = γ2 = 34 and (λ1 , λ2 , λ3 ) = 32 , 1, 23 for (29) (dashed line), (30) (dash–dot line) and (31) (solid line) Here, we have considered that γ1 = γ2 = 52 The curves show that the anisotropic character of the material strongly influences the Rayleigh wave velocity of the initial Neo-Hookean material Furthermore, the invariant I5 has a greater influence than the invariant I4 Downloaded from http://imamat.oxfordjournals.org/ at University of California, San Francisco on December 7, 2014 Fig The curves show the dependence of x = ρc2 /μ obtained using (27) on θ ∈ [0 π/2], which is the angle that the propagating wave makes with the x1 -direction Fibres are parallel to the X1 -direction, the elastic half-space is initially under uniaxial tension along the X1 -axis and the strain–energy function is given by (29) (dashed line), (30) (dash–dot line) and (31) (solid line) for γ1 = γ2 = 34 and λ = 1.3 NON-PRINCIPAL RAYLEIGH WAVES IN TRANSVERSELY ISOTROPIC MATERIALS 11 of 14 Conclusions In this paper, the propagation of non-principal Rayleigh waves in deformed transversely isotropic incompressible solids has been investigated The explicit secular equation for Rayleigh waves is obtained employing the method of polarization vector This generalizes previous analyses dealing with isotropic incompressible pre-stressed materials It has been shown that the wave velocity depends strongly on the anisotropic character of the material model Funding The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) JM acknowledges support from the Ministerio de Ciencia in Spain under the project reference DPI2011-26167 References Adams, S D M., Craster, R V & Williams, D P (2007) Rayleigh waves guided by topography Proc R Soc A, 463, 531–550 Barnett, D M & Lothe, J (1974) Consideration of the existence of surface wave (Rayleigh wave) solutions in anisotropic elastic crystals J Phys F Metal Phys., 4, 671–686 Barnett, D M & Lothe, J (1985) Free surface (Rayleigh) waves in 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A2121 A2323 − A22123 , b111 = (A∗1111 + A∗2222 d1 = A2323 /d, − 2A1122 ), b311 = (A1113 − A2213 ), d3 = A2121 /d, b113 = 2(A1131 − A2231 ), b313 = (A∗2222 + A∗1331 (A.1) d13 = A2123 /d, b133 = A3131 , + A1133 − A1122 − A3322 ), (A.2) b333 = (A3133 − A3122 ), d311 = A1313 , d313 = 2(A1333 − A1322 ), e11 = (A1212 + a11 A∗1221 + a31 A1223 ), + a31 A∗3223 + a33 A1223 ), d333 = (A∗3333 + A∗2222 − 2A2233 ), e13 = (2A1232 + a11 A3221 + a13 A∗1221 e33 = (A3232 + a13 A3221 + a33 A∗2332 ) Here the notation A∗piqj = Apiqj + P has been used (A.3) (A.4) Downloaded from http://imamat.oxfordjournals.org/ at University of California, San Francisco on December 7, 2014 Ting, T C T (2004) The polarization vector and secular equation for surface waves in an anisotropic elastic half-space Int J Solids Struct., 41, 2065–2083 Ting, T C T (2005) Explicit secular equation for surface waves in an anisotropic elastic half-space from Rayleigh to today Surface Waves in Anisotropic and Laminated Bodies and Defects 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Vinh, P C & Ogden, R W (2004a) Formulas for the Rayleigh wave peed in orthotropic elastic solids Arch Mech., 56, 247–265 Vinh, P C & Ogden, R W (2004b) On formulas for the Rayleigh wave speed Wave Motion, 39, 191–197 Vinh, P C & Ogden, R W (2005) On the Rayleigh wave speed in orthotropic elastic solids Meccanica, 40, 147–161  ... propagation of non-principal Rayleigh waves in deformed transversely isotropic incompressible solids has been investigated The explicit secular equation for Rayleigh waves is obtained employing the... differentiation with respect to the indicated spatial ¯ coordinate in Be The fibre orientation in Be is in particular related to M by m = FM The necessary equations including the linearized equations... secular equation for non-principal Rayleigh waves in incompressible, transversely isotropic, pre-stressed elastic half-spaces is obtained in Section 3.2 The results are illustrated in respect