International Journal of Non-Linear Mechanics 84 (2016) 23–30 Contents lists available at ScienceDirect International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm The secular equation for non-principal Rayleigh waves in deformed incompressible doubly fiber-reinforced nonlinearly elastic solids Nguyen Thi Nam a, Jose Merodio a,n, Pham Chi Vinh b a b Department of Continuum Mechanics and Structures, E.T.S Ing Caminos, Canales y Puertos, Universidad Politecnica de Madrid, 28040 Madrid, Spain Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam art ic l e i nf o a b s t r a c t Article history: Received 29 August 2014 Received in revised form 14 April 2016 Accepted 15 April 2016 Available online 16 April 2016 The explicit and implicit secular equations for the speed of a (surface) Rayleigh wave propagating in a pre-stressed, doubly fiber-reinforced incompressible nonlinearly elastic half-space are obtained Hence, the anisotropy is associated with two preferred directions, thereby modelling the effect of two families of fiber reinforcement One of the principal planes of the primary pure homogeneous strain coincides with the free surface while the surface wave is not restricted to propagate in a principal direction Results are illustrated with numerical examples In particular, an isotropic material reinforced with two families of fibers is considered Each family of fibers is characterized by defining a privileged direction Furthermore, the fibers of each family are located throughout the half space and run parallel to each other and perpendicular to the depth direction, i.e the free surface is a plane of symmetry of the anisotropy The wave speed depends strongly on the anisotropic character of the material model as well as the direction of propagation & 2016 Elsevier Ltd All rights reserved Keywords: Rayleigh waves Explicit and implicit secular equations Orthotropic half-spaces Fiber reinforcements Introduction The purpose of this paper is to extend the analysis of [1] dealing with Rayleigh waves for materials reinforced with one family of fibers to materials reinforced with two families of fibers in the framework of nonlinear elasticity This is motivated by several factors First, the use of doubly fiber-reinforced elastic composites is common in engineering applications In addition, there is a lot of interest in the acoustics of biological soft tissues (see for example, Destrade et al [2]) Soft biological tissues have been recognized as highly anisotropic due to the presence of collagen fibers [3] and are modeled as orthotropic materials with two families of fibers The Rayleigh wave existence and uniqueness problem has been resolved with the aid of the Stroh formalism [4] Fu and Mielke [5] and Mielke and Fu [6] also have shown the uniqueness of the surface wave speed based on an identity for the surface-impedance matrix The surface-wave speed can also be obtained from secular equations of implicit as well as explicit form The explicit secular equations 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 applications of the explicit secular equations n Corresponding author E-mail address: jose.merodio@upm.es (J Merodio) http://dx.doi.org/10.1016/j.ijnonlinmec.2016.04.006 0020-7462/& 2016 Elsevier Ltd All rights reserved are not limited to numerically determine the surface-wave speed 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 (see for instance [7,8]) Explicit secular equations have been given by Malischewsky [7] for isotropic solids, Ting [9,10], Ogden and Vinh [11], Vinh and Ogden [12,13], Vinh et al [1] for anisotropic solids and Vinh [14,15] for pre-stressed media, among others We establish a procedure to obtain both the explicit and implicit secular equations of non-principal Rayleigh waves propagating in incompressible, doubly fiber-reinforced, pre-stressed elastic half-spaces For transversely isotropic materials the explicit secular equation was given in [1] while the implicit one was given in [16] We build upon these results and use the polarization vector method to get the secular equation in explicit form The implicit secular equation is obtained from the so-called propagation condition The latter equation is used to eliminate the spurious roots that arise in the explicit secular equation The study of the propagation of Rayleigh-type surface waves in an elastic half-space subject to pre-stress goes back to the pioneering work of Hayes and Rivlin [17] and since then it has attracted the attention of many researchers There is a lot of interest in using the equations governing infinitesimal motions superimposed on a finite deformation of a nonlinear elastic half-space because it is applicable to several topics These include: the nondestructive evaluation of prestressed structures before and during loading (see, for example, Makhort [18,19], Hirao et al [20], 24 N.T Nam et al / International Journal of Non-Linear Mechanics 84 (2016) 23–30 Husson [21], Delsanto and Clark [22], Dyquennoy et al [23,24], Hu et al [25]), the acoustics of soft solids with particular attention to the analysis of biological soft tissues (see, for instance, Destrade et al [2,26,27], Vinh and Merodio [28,29] and references therein), and the (incremental) stability of the free surface of a deformed material (see, for instance, Destrade et al [30–32]), among others) Indeed, surface waves have been studied extensively in seismology, acoustics, geophysics, telecommunications industry and materials science (see Adams et al [33]) In Section 2, the basic constitutive equations associated with this study are presented This includes the material model as well as the corresponding equations for infinitesimal waves superimposed on a finite deformation consisting of a pure homogeneous strain In Section 3, the Stroh formalism is applied to the analysis of infinitesimal surface waves propagating in a statically, finitely and homogeneously deformed doubly fiber-reinforced half-space The free surface is assumed to coincide with one of the principal planes of the primary strain, but a propagating surface wave is not restricted to a principal direction (see [34] for a parallel work that enlightens this analysis) The implicit and explicit secular equations are presented In Section 4, the results are illustrated numerically in respect of a strain–energy function used to model soft tissue (see [3]) Basic equations 2.1 Kinematics Consider an elastic body whose reference configuration is denoted by )0 and a finitely deformed equilibrium configuration The deformation gradient tensor associated with the deformation is denoted by F In addition, let (X1, X2 , X3) be a fixed rectangular coordinate system in )0 The precise notation necessary for the analysis will be introduced later on Composite materials and some soft tissues are modeled as incompressible isotropic elastic solids reinforced with preferred directions (see [35,36] and references therein) Each preferred direction is associated with a family of parallel fibers Here, two families of fibers are considered We denote by M with components (M1, M2, M3) and N with components (N1, N2, N3) the unit vectors in these directions in )0 The invariants of the right Cauchy–Green deformation tensor, C = FTF , where the symbolT indicates the transpose of a matrix, most commonly used are the principal invariants (see, for instance [37]), defined by I1 = tr C, I2 = (I12 − tr(C2)), I3 = det C (1) The (anisotropic) invariants associated with M and C are usually taken as I4 = M·(CM), I5 = M·(C2M) (2) For N and C, the associated invariants are I6 = N ·(CN), I7 = N ·(C2N) (3) I3 = If M and N are perpendicular then the number of independent invariants is six (see [38] for details) The Cauchy stress is σ=F ∂W − pI = ∂F ∑ Wi F i = 1, i ≠ σ = 2W1B + 2W2(I1I − B)B + 2W4m ⊗ m + 2W5(m ⊗ Bm + Bm ⊗ m) + 2W6n ⊗ n + 2W7(n ⊗ Bn + Bn ⊗ n) + W8(m ⊗ n + n ⊗ m)M·N − pI, W= μ (I1 − 3) + f1(I4 ) + f2 (I6) + g1(I5) + g2(I7) + G(I8) (7) in order to illustrate the results This strain energy function captures the essential features of the analysis that follows We want to establish results related to the kinematical properties of the invariants I4 and I6 as well as the invariants I5 and I7 The results allow us to distinguish the effects of the different invariants The invariant I8 is also considered so as to evaluate its influence For specific details and analysis of the reinforcing models we refer to [35,36] Here, we just mention that the energy function and the stress must vanish in the reference configuration In Section 4, we will further make this clear since a certain strain–energy function is used 2.3 Linearized incremental equations of motion Consider an incompressible, doubly fiber-reinforced, semi-infinite body ) in its unstrained state )0 that occupies the region X2 ≥ Fibers of each family run parallel to each other and perpendicular to the depth direction X2, i.e M2 = and N2 = The body is subjected to a finite pure homogeneous strain with principal directions given by the Xi-axes A finitely deformed (prestressed) equilibrium state )e is obtained A small time-dependent motion is superimposed upon this pre-stressed equilibrium configuration to reach a final material state )t , called current configuration The vector components of a representative particle are denoted by Xi, x i(X), x˜ i(X, t ) in )0 , )e and )t , respectively The deformation gradient tensor associated with the deformations )0 → )[ and )0 → )L is denoted by F¯ and F, respectively, and are given in component form by ∂∼ xi ∂xi F¯iA = , FiA = ∂XA ∂XA (8) It is clear from (8) that F¯iA = (δij + ui, j )FjA, The anisotropic nonlinear elastic strain–energy function W depends on F through the invariants of the right Cauchy–Green deformation tensor For incompressible materials, the strain energy function can be written as W = W (I1, I2, I4, I5, I6, I7, I8) since (6) where B = FFT , m ¼FM, and n ¼FN It follows that, in general, the principal directions of stress and strain not coincide In the biomechanics literature, several strain energy functions given by an isotropic elastic material augmented with the socalled reinforcing models can be found We extend the reinforcing models for one family of fibers (see [36] for complete details) to I8 = M·(CN)(M·N) 2.2 Material model (5) where p is a Lagrange multiplier associated with the incompressibility constraint, the shorthand notations Wi = ∂W /∂Ii, i = 1, 2, 4, 5, 6, 7, have been used and I is the  identity tensor The Cauchy stress tensor can be written as Finally, the invariant related to the combination of M, N, and C is (4) ∂Ii − pI, ∂F (9) where δij is the Kronecker operator, ui(X , t) denotes the small time-dependent displacement associated with the deformation )L → )[ and a comma indicates differentiation with respect to the indicated spatial coordinates in )L The necessary equations including the linearized equations of motion for anisotropic incompressible materials are summarized The incremental components of the nominal stress tensor Sji are N.T Nam et al / International Journal of Non-Linear Mechanics 84 (2016) 23–30 related to the incremental displacement gradients uk, l by (see [14,1]) ⁎ Spi = ( piqjuj, q + Pup, i − p δpi, i, j, p, q = 1, 2, 3, ( piqj = μδijBpq + 2f1′(I4 )mpmq + 2f2′ (I6)npnq] + 4f1″(I4 )mi mj mpmq + 4f2″(I6)ni nj npnq + 2g1′(I5) [δij(mpBqr mr + mqBpr mr ) + Bijmpmq + mi mj Bpq + Biqmj mp (10) + Bpjmi mq] + 2g2′(I7) where P is the value of p in )e , p⁎ = p − P is the time-dependent increment of p and the components of the fourth-order elasticity tensor ( for W = W (I1, I2, I4, I5, I6, I7, I8) are given by (see also Vinh and Merodio [29]) ( piqj = FpαFqβ = FpαFqβ [δij(npBqr nr + nqBpr nr ) + Bijnpnq + ni nj Bpq + Biqnj np + Bpjni nq] + 4g1′(I5)(mpBir mr + mi Bpr mr ) (mqBjr mr + mj Bqr mr ) + 4g2′(I7)(npBir nr + ni Bpr nr ) (nqBjr nr + nj Bqr nr ) + G′δij(mpnq + mqnp)MkNk ∂ 2W ∂Fiα ∂Fjβ ∑ Wr r∈0 + G″(mpni + npmi )(mqnj + nqmj )MkNkMtNt ∂ 2Ir ∂I ∂I + FpαFqβ ∑ Wrs r ⊗ s , ∂Fiα ∂Fjβ ∂Fiα ∂Fjβ r, s ∈ (11) where Wr = ∂W /∂Ir , Wrs = ∂ W /∂Ir ∂Is and is the index set {1, 2, 4, 5, 6, 7, 8} The components of the elasticity tensor are given in Appendix A It is clear that ( piqj = ( qjpi In general, the elasticity tensor ( has at most 45 non-zero components The fibers M and N in )0 make angles γ and δ, respectively, with OX1 and the angles are measured in opposite senses relative to that axes Since the deformation gradient F is F = diag(λ1, λ2, λ3), where λk are the principal stretches of the deformation, it follows that the components of m and n are m1 = λ1 cos γ , n1 = λ1 cos δ, m3 = λ3 sin γ , n3 = − λ3 sin δ m2 = M2 = m2 = N2 = 0, Sij, i = uă j , (15) where a dot indicates differentiation with respect to time t The incremental version of incompressibility is (see [39]) ui, i = (16) In what follows, we rewrite these equations for the case of a Rayleigh wave using the Stroh formulation In addition, using (11) we point out that there are only 25 non-zero components of the elasticity tensor ( , namely ( iiii , ( iijj , ( ijij , ( ijji , ( ii13, ( ii31, (2312, (2321, (3212, (3221 (i, j = 1, 2, 3, i ≠ j ) Surface waves (12) The vectors M and N in )e make angles ψ and ϕ, respectively, with Ox1, which, using (12), are given by tan ϕ = λ3/λ1 tan δ (14) In the absence of body forces, the incremental equations of motion are (see [39]) tan ψ = λ3/λ1 tan γ , 25 (13) Given F, one can assume that either the set of angles ψ and ϕ or the set of angles γ and δ is known (see Fig 1) We further particularize the elasticity tensor to the strain–energy function (7), i.e using (7) and (11) we write The analysis is particularized for Rayleigh waves propagating in a principal plane of the pre-strain, the plane x2 = 0, but not in general in a principal direction The incremental equation of motion can be cast as a homogeneous linear system of six first-order differential equations 3.1 The Stroh formulation We consider a Rayleigh wave traveling with speed c and with its wave vector k lying in the (x1, x3) plane The wave makes an angle θ with the x1-direction and decays in the x2-direction Then, Fig The figure on the left shows at a point O in the free surface of the pre-stressed half space: (i) the principal axes of the primary pure homogeneous strain (xi-axes) (ii) the two directions in that configuration characterizing the two families of fibers (given by ψ and ϕ) as well as the fibers of each family (dashed lines) along the depth direction (x2-axis) and (iii) the propagation direction of the wave (given by θ) Fibers of each family are located throughout the whole half space and run parallel to each other and perpendicular to the depth direction The figure on the right is a view from the top It further clarifies that the angles ψ and ϕ are measured in opposite senses relative to the x1-axis 26 N.T Nam et al / International Journal of Non-Linear Mechanics 84 (2016) 23–30 the displacements and stresses of the Rayleigh wave are written (see [40]) as un = Un(y)e ik(x1cθ + x3sθ − ct ) S2n = ikz n(y)e , ik(x1cθ + x3sθ − ct ) , (17) n = 1, 2, 3, respectively, where y = kx2, cθ = cos θ , sθ = sin θ , and k = |k| is the wave number Using (17), together with (10), (15) and (16), one can write ξ′ = iNξ, ≤ y < + ∞, (18) where the prime now signifies differentiation with respect to y and ⎡ ⎤ ⎢ u⎥ ξ = ⎢ ⎥, z ⎢ ⎥ ⎣ ⎦ ⎡z ⎤ ⎢ 1⎥ z = ⎢ z2 ⎥, ⎢ z3 ⎥ ⎣ ⎦ ⎡U ⎤ ⎢ 1⎥ u = ⎢ U2 ⎥, ⎢U ⎥ ⎣ 3⎦ ⎤ ⎡ ⎢ N1 N2 ⎥ N=⎢ , T⎥ ⎢ K N1 ⎥ ⎦ ⎣ (19) ⎡ ⎤ ⎢ d1 −d13⎥ N2 = ⎢ 0 ⎥, ⎢ ⎥ ⎣ −d13 d3 ⎦ (26) Eq (24) is called the implicit secular equation (see also [40,1]) because the expressions for the ωI, ωII, ωIII in terms of X are unknown A Rayleigh wave exists with speed c = X/ρ if and only if (24) is satisfied In this section we derive the explicit secular equation of the wave using the method of polarization vector (see [41,10,42] for instance) Using (18), (22), (23) and that N2 and K are symmetric, one can write u¯ T (0)K (n)u(0) = where K (n ) ∀ n ∈ Z, (27) is defined as ⎡ N (n) N (n) ⎤ ⎥ N n = ⎢ (n) ⎢⎣ K N 4(n) ⎥⎦ ⎡h h ⎤ 2⎥ ⎢ K = ⎢ h3 ⎥, ⎥ ⎢ ⎣ h2 h4 ⎦ (20) (28) From (27) the explicit secular equation is obtained as |K2, K3, K1|2 + 4|K2, K3, K 4||K2, K1, K 4| = 0, where d = (2121(2323 − (22123, d3 = ⎛ f2 ⎞ ⎛ f2 ⎞ ⎛ff ⎞ u = ⎜⎜ − d1⎟⎟h1 + ⎜⎜ − d3⎟⎟h4 + 2⎜⎜ + d13⎟⎟h2 , v ⎝ h3 ⎠ ⎝ h3 ⎠ ⎝ h3 ⎠ 2 ⎡ ⎤ d3f1 + d1f2 2d13f1f2 2⎥ = (h1h4 − h22)⎢ d1d3 − − − d13 h3 h3 ⎢⎣ ⎥⎦ 3.3 Explicit secular equation in which the matrices N1, N2, k are defined by ⎡ f ⎤ ⎢ ⎥ N1 = ⎢ −cθ −sθ ⎥, ⎢ ⎥ ⎣ f2 ⎦ positive imaginary parts and u, v are defined as ( d1 = 2323 , d ( d13 = 2123 , d where ⎡ K (−1) ⎤ ⎢ 11 ⎥ (1) ⎥, K1 = ⎢ K11 ⎥ ⎢ ⎢⎣ K (3) ⎥⎦ 11 (2121 , f1 = a11cθ + a13sθ , d f2 = a31cθ + a33sθ , h1 = ρc − b111cθ2 − b113cθsθ − b133sθ2, h2 = − b311cθ2 − b313cθsθ − b333sθ2, h3 = ρc − e11cθ2 − e13cθsθ − e33sθ2, h4 = ρc − d311cθ2 − d313cθsθ − d333sθ2, (21) and the rest of coefficients are given in Appendix B Eq (18) is the so-called Stroh formulation (see [4]) The decay condition is expressed as ξ( + ∞) = (22) The boundary condition of zero incremental traction using the expression given for S2n in (17) means that z(0) = (23) In passing, we note that our formulation particularized for transversely isotropic materials and isotropic materials coincides with the ones given in [1,40], respectively In particular, for instance, the matrices N1, N2 and k in (20) particularized for isotropic materials coincide, respectively, with the matrices N1, N2 and N3 + X I , where X = ρc 2, given by (2.9) and (2.10) in [40] 3.2 Implicit secular equation The implicit secular equation is given by (see [40,16] for complete details) vωI − (u − ωII )ωIII = 0, (24) in which ωI = − (s1 + s2 + s3), ωII = s1s2 + s2s3 + s3s1, ωIII = − s1s2s3, (29) (25) where s1s2, s3 are the eigenvalues of the Stroh matrix N with ⎡ K (−1) ⎤ ⎢ 22 ⎥ (1) ⎥, K2 = ⎢ K22 ⎥ ⎢ ⎢⎣ K (3) ⎥⎦ 22 ⎡ K (−1) ⎤ ⎢ 33 ⎥ (1) ⎥, K3 = ⎢ K33 ⎥ ⎢ ⎢⎣ K (3) ⎥⎦ 33 ⎡ K (−1) ⎤ ⎢ 13 ⎥ (1) ⎥, K = ⎢ K13 ⎥ ⎢ ⎢⎣ K (3) ⎥⎦ 13 (30) in which Kij(n) are entries of the matrix K (n) and are given in Appendix C Equation (29) is the explicit secular equation This is a cumbersome polynomial of degree 12 in X = ρc (see [1]) In order to illustrate the results further we consider some particular strain–energy functions Numerical results A modified version of the well known Gasser–Ogden–Holzapfel (GOH) model (see [3]) is adopted In particular, we consider that W= k μ (I1 − 3) + 2k + ∑ {exp[k2(Ii − 1)2] − 1} i = 4,5,6,7 k3 (I8 − I8(0))2 , (31) where μ, k1, k2 and k3 are positive constants and = (M·N) is the value of I8 in the reference configuration The GOH model is given by (31) with no dependence on I5, I7 and I8 Furthermore, it is assumed that the fibers contribute to the strain–energy function when these are elongated Here, we use (31) as a prototype to show the robustness of the methodology herein regardless of this last statement Nevertheless, and in passing, we mention that lately there has been some discussion regarding the tension-compression switch in these models and we refer to [43] for further details It is easy to check that the strain energy is zero in the undeformed configuration as well as the stress tensor We specialize (31) to some special models and compare the results with the ones obtained for the neo-Hookean material, whose energy I8(0) N.T Nam et al / International Journal of Non-Linear Mechanics 84 (2016) 23–30 function is μ W0 = (I1 − 3) (32) Hence, we treat in turn the following cases: (i) the strain energy function (31) just with the invariants I1, I4, I6 W (1)(I1, I4, I6) = k μ (I1 − 3) + 2k ∑ [e k2(Ii− 1) − 1]; (33) i = 4,6 (ii) similarly, the strain energy function (31) just with the invariants I1, I5, I7 W (2)(I1, I5, I7) = k μ (I1 − 3) + 2k ∑ i = 5,7 [e k2(Ii− 1) − 1] (34) In Fig 2, values of x = ρc 2/μ vs θ ∈ [0, π /2] obtained using (29) are plotted for different strain–energy functions under two 27 conditions, namely λ1 = 1.3, λ2 = and λ3 = 1/λ1 (right-hand plot figure) and λ1 = 1.2, λ2 = and λ3 = 1/λ1 (left-hand plot figure) In both cases, the dotted-dashed curve is associated with (31) for γ = π /6, δ = π /3, k1 = k3 = 0.5μ and k2 ¼0.5 The values of these parameters are used accordingly in the models (32)–(34) The curves associated to the neo-Hookean model have their maximum value at θ ¼0, which is expected for an isotropic model That is not the case for the non-isotropic models since the principal directions of stress and strain not coincide Other parameters could be used as well as other angles for the fibers Furthermore, the influence on the wave speed of the isotropic base model introduced by the invariants I5 and I7 (the model (34)) is stronger than the one given by the invariants I4 and I6 (the model (33)) This result was shown in [1] for the analysis of transversely isotropic materials with one family of fibers In Fig 3, the same analysis is developed for γ = δ = π /4 (perpendicular) Under these circumstances (the two families of fibers are symmetric with respect to the OX1 axis), it follows that I4 = I6 and I5 = I7 and, furthermore, the principal directions of strain and Fig In the two plots, the curves show the dependence of x = ρc 2/μ on θ ∈ [0, π /2] obtained using (29) for (31), the dotted-dashed curve, (32), the thin solid curve, as well as (33) and (34), the dashed and thick solid curves, respectively For the different calculations, we have taken, accordingly for each model, γ = π /6 , δ = π /3, k1 = k3 = 0.5μ, k2 = 0.5 The principal stretches are λ1 = λ = 1.2, λ2 = 1, λ3 = 1/λ1 (left-hand plot); λ1 = 1.3, λ2 = and λ3 = 1/λ1 (right-hand plot) Fig The curves show in the two plots the dependence of x = ρc 2/μ on θ ∈ [0, π /2] as given by (29) for (31), the dotted-dashed curve, as well as (33) and (34), the dashed and thick solid curves, respectively The parameters of the different models have been taken as γ = π /4 , δ = π /4 , k1 = k3 = 0.5μ and k2 ¼0.5 The principal stretches are λ1 = 1.2, λ2 = 1, λ3 = 1/λ1 (left-hand plot) and b) λ1 = 1.3, λ2 = 1, λ3 = 1/λ1 (right-hand plot) Results for the neo-Hookean model (32), the thin solid curve, are also shown for comparison 28 N.T Nam et al / International Journal of Non-Linear Mechanics 84 (2016) 23–30 Fig Corresponding plots to the ones given in Fig for γ = π /6 and δ = π /4 Fig Corresponding plots to the ones given in Fig for γ = π /6 and δ = π /6 stress coincide Hence, each curve in Fig has its maximum value at θ ¼0 Corresponding plots to the ones given in Fig are shown in Figs and for different angles (not perpendicular) of γ and δ In particular γ = π /6 and δ = π /4 in Fig and γ = π /6 and δ = π /6 in Fig The influence of the term in (31) that includes the invariant I8 on the surface wave speed of the isotropic model is not as significant as the influence of the other non-isotropic invariants Indeed, results may be different for other strain–energy functions and other deformations We consider now that the elastic half-space is initially under uniaxial tension along the X1-axis x1 = λX1, x2 = λ−1/2X2 , x3 = λ−1/2X3, λ > 0, λ = const than the one given by I4 and I6 in agreement with the results of Vinh et al [1] Furthermore, under the circumstances at hand, the influence of I4 and I6 on the speed of the isotropic base model is not strong in the domain of λ-values shown in the figure For λ1 = 1.2, λ2 = 1, λ3 = 1/λ1 and waves propagating along the x1-axis, Fig shows values of x = ρc 2/μ vs γ = δ ∈ [0, π /2] (the angle that each fiber family makes with the X1-axis) obtained using (29) for (31) (dotted-dashed curve), (33) (dashed curve), (34) (thick solid curve) The parameters used for the calculations are k1 = 0.5μ and k2 ¼0.5 The curve associated with the neo-Hookean model (32), thin solid one, is horizontal since it is an isotropic model and has the value x = ρc 2/μ = 1.6227 (35) In addition, the surface waves propagate in the x1-direction and the two families of fibers are symmetrically disposed with respect to the X1-axis, in particular, γ = δ = π /4 In Fig 6, values of x = ρc 2/μ vs λ obtained using (29) are shown for the neo-Hookean model (32), the solid curve, as well as for (33) and (34), the dotted and dashed curves, respectively The parameters for the different models are k1 = 0.5μ and k2 ¼ 0.5 A simple comparison among the curves establishes that the anisotropy influences the Rayleigh speed of the isotropic base model The influence of the invariants I5 and I7 on the wave speed of the isotropic base model is stronger Conclusions The explicit and implicit secular equations for the speed of a (surface) Rayleigh wave propagating in a pre-stressed, doubly fiberreinforced incompressible nonlinearly elastic half-space have been obtained 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 This generalizes previous results dealing with transversely isotropic nonlinearly elastic solids (see [1]) To illustrate the analysis, several strain– N.T Nam et al / International Journal of Non-Linear Mechanics 84 (2016) 23–30 29 the isotropic base model Furthermore, the influence on the wave speed of the isotropic base model introduced by the invariants I5 and I7 is stronger than the one given by the invariants I4 and I6 The models at hand are prototypes and have to be used with caution specially under fiber compression (see [43]) Acknowledgments PCV acknowledges support from the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the Grant no 107.02-2014.04 JM acknowledges support from the Ministerio de Ciencia in Spain under the project reference DPI2014-58885-R Appendix A Elasticity tensor ( piqj ( piqj = 2W1δijBpq + 2W2(2 BipBjq − BiqBjp + I1δijBpq − BijBpq − δij(B 2)pq ) + 2W4δijmpmq + 2W5[δij(mpmk Bkq + mq mk Bkp) + Bijmpmq + Bpqmi mj + Bqimpmj + Bpjmq mi ] + 2W6δijn pnq Fig Under uniaxial tension along the X1-axis with γ = δ = π /4 and waves propagating along the x1-axis, the Figure shows the dependence of x = ρc 2/μ on λ as given by (29) for (33), the dotted curve, and (34), the dashed curve Results for the neo-Hookean model (32), the solid curve, are also shown for comparison The parameters for the different models are k1 = 0.5μ and k2 ¼0.5 + 2W7[δij(n pnkBkq + nqnkBkp) + Bijn pnq + Bpqn in j + 2(Bqin pn j + Bpjnqn i )] + W8δij(mpnq + mq n p)MkNk + 4W11BpiBqj + 4W22(I1Bip − (B 2)ip)(I1Bjq − (B 2)jq ) + 4W44mpmq mi mj + 4W55(Birmpmr + Bprmi mr )(Bjrmq mr + Bqrmj mr ) + 4W66n pnqn in j + 4W77(Birn pn r + Bprn in r )(Bjrnqn r + Bqrn jn r ) + W88(mpn i + n pmi ) (mq n j + nqmj )MkNkMtNt + 4W12[Bip(I1Bjq − (B 2)jq ) + Bjq(I1Bip − (B 2)ip)] + 4W14(Bpimq mj + Bqjmpmi ) + 4W15[Bpi(Bjrmq mr + Bqrmj mr ) + Bqj(Birmpmr + Bprmi mr )] + 4W16(Bpinqn j + Bqjn pn i ) + 4W17[Bpi(Bjrnqn r + Bqrn jn r ) + Bqj(Birn pn r + Bprn in r )] + 2W18[Bpi(mq n j + nqmj )MkNk + Bqj(mpn i + n pmi )MkNk] + 4W24[(I1Bip − (B 2)ip)mj mq + (I1Bjq − (B 2)jq )mi mp] + 4W25[(I1Bip − (B 2)ip)[mq Bjrmr + mj Bqrmr ] + (I1Bjq − (B 2)jq )[mi Bprmr + mpBirmr ]] + 4W26[(I1Bip − (B 2)ip)n jnq + (I1Bjq − (B 2)jq )n in p] + 4W27[(I1Bip − (B 2)ip)(nqBjrn r + n jBqrn r ) + (I1Bjq − (B 2)jq )(n iBprn r + n pBirn r )] + 2W28[BipI1(mq n j + nqmj ) − BpγBγi(mq n j + nqmj ) + BqjI1(mpmi + n pmi ) − BqγBγj(mpn i + n pmi )]MkNk Fig Under uniaxial tension along the X1-axis with λ1 = 1.2, λ2 = 1, λ3 = 1/λ1 and waves propagating along the x1-axis, the figures shows values of x = ρc 2/μ vs γ = δ ∈ [0, π /2], (the angle that each family of fibers makes with the X1−direction ) as given by (29) for (31), (33) and (34), dotted-dashed, dashed and thick solid curves, respectively The values of the parameters used in the calculations are k1 = k3 = 0.5μ and k2 ¼0.5 Results for the neo-Hookean model (32), the thin solid curve, are also shown for comparison energy functions have been considered In particular, the materials under consideration are neo-Hookean models augmented with two functions, each one of them accounting for the existence of a unidirectional reinforcement The functions endow the material with its anisotropic character and each one is referred to as a reinforcing model We consider two cases for the nature of the anisotropy: on the one hand, reinforcing models that have a particular influence on the shear response of the material (I5, I7); on the other hand, reinforcing models that depend only on the stretch in the fiber direction (I4, I6) The anisotropy influences the surface wave speed of + 4W45[mpmi (Bjrmq mr + Bqrmj mr ) + mq mj (Birmpmr + Bprmi mr )] + 4W46(mpmi nqn j + n pn imq mj ) + 4W47[mpmi (Bjrnqn r + Bqrn jn r ) + mq mj (Birn pn r + Bprn in r )] + 4W48(mpmi (mq n j + nqmj )MkNk + mq mj (mpn i + n pmi )MkNk ) + 4W56[nqn i(Birmpmr + Bprmi mr ) + n pn j(Birmq mr + Bqrmj mr )] + 4W57(Birmpmr + Bprmi mr )(Bjrnqn r + Bqrn jn r ) + 2W58[(Birmpmr + Bprmi mr )(mq n j + nqmj ) + (Bjrmq mr + Bqrmj mr ) (mpn i + n pmi )]MkNk + 4W67[n pn i(Bjrnqn r + Bqrn jn r ) + nqn j(Birn pn r + Bprn in r )] + 2W68[n pn i(mq n j + nqmj ) + nqn j(mpn i + n pmi )] + 2W78[(Birn pn r + Bprn in r )(mq n j + nqmj ) + (Bjrnqn r + Bqrn jn r ) (mpn i + n pmi )], with Bij = FikFjk and I1 = Bkk (36) 30 N.T Nam et al / International Journal of Non-Linear Mechanics 84 (2016) 23–30 Appendix B The expressions of coefficients associated with the Stroh formalism ⁎ a11 = ((2123(1223 − (2323(1221 )/d , a13 = ((2123(⁎2332 − (2323(2132)/d, ⁎ a31 = ((2123(1221 − (2121(2312)/d, ⁎ a33 = ((2123(2132 − (2121(2332 )/d , ⁎ ⁎ b111 = ((1111 + (2222 − 2(1122), b113 = 2((1131 − (2231), ⁎ b133 = ((⁎2222 + (1331 + (1133 − (1122 − (3322), b133 = (3131, b311 = ((1113 − (2213), b333 = ((3133 − (3122), d311 = (1313, d313 = 2((1333 − (1322), e11 = ((1212 + ⁎ a11(1221 ⁎ d333 = ((⁎3333 + (2222 − 2(2233), + a31(1223), ⁎ e33 = ((3232 + a13(3221 + a33(2332 ), ⁎ e13 = (2(1232 + a11(3221 + a13(1221 + a31(⁎3223 + a33(1223) ⁎ Here the notation Apiqj = Apiqj + Pδijδpq has been introduced Appendix C The components of matrix K (1) K11 = h1, (1) K13 = h2 , (1) (1) K22 = h3, K33 = h4 , (3) K11 = d1h12 + d3h22 − 2d13h1h2 − 2(f1h1 + f2 h2)cθ (3) + h3cθ2, K13 = d1h1h2 − d13(h22 + h1h4 ) + d3h2h4 − (f1h1 + f2 h2)sθ − (f1h2 + f2 h4 )cθ + h3sθ cθ , (3) K22 = f12 h1 + 2f1f2 h2 − 2f1h3cθ + f22 h4 − 2f2 h3sθ , (3) K33 = d1h22 + d3h42 − 2d13h2h4 − 2(f1h2 + f2 h4 )sθ + h3sθ2, (−1) K11 = (d3h3 − f22 )(h4 cθ2 − 2h2cθsθ + h1sθ2), (−1) K13 = (d13h3 + f1f2 )(h4 cθ2 − 2h2cθsθ + h1sθ2), (−1) K22 = [2d13f1f2 + d1f22 + d13 h3 + d3(f12 − d1h3)](h1h4 − h22), (−1) K33 = (d1h3 − f12 )(h4 cθ2 − 2h2cθsθ + 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Conclusions The explicit and implicit secular equations for the speed of a (surface) Rayleigh wave propagating in a pre-stressed, doubly fiberreinforced incompressible nonlinearly elastic half-space