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

IMA Journal of Applied Mathematics 2014 Page 1 of 14doi:10.1093/imamat/hxu023 Non-principal Rayleigh waves in deformed transversely isotropic incompressible non-linearly elastic solids P

IMA Journal of Applied Mathematics (2014) Page 1 of 14

doi:10.1093/imamat/hxu023

Non-principal Rayleigh waves in deformed transversely isotropic incompressible

non-linearly elastic solids

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

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 8 April 2014; accepted on 25 April 2014]

An explicit expression of the secular equation for non-principal Rayleigh waves in incompressible,

trans-versely 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.

1 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 byAdams et al.(2007)

The Rayleigh wave existence and uniqueness problem has been resolved with the aid of the Stroh

formalism (seeStroh 1958,1962), even for an anisotropic elastic half-space (seeBarnett 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.

Barnett & Lothe 1974,1985;Chadwick & Smith 1977;Ting 1996) Based on an identity for the

surface-impedance 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

byFu & 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 inTing(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

lim-ited 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

sec-ular equations, are very significant in various practical applications They have been given byVinh &

Ogden(2004b) andMalischewsky(2004) for isotropic solids,Ogden & Vinh(2004) andVinh & Ogden

(2004a,2005) for orthotropic solids andVinh(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 toKiselev & 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 byHayes & 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) andDestrade(2007), to name a few

Here, we obtain the explicit secular equation of non-principal Rayleigh waves propagating in

incom-pressible, 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; seeTing(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

struc-tures 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) andHu 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

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 asVinh

& 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 (seeHolzapfel 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

super-imposed on a finite deformation consisting of a pure homogeneous strain are outlined Surface waves

are studied in Section 3 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

2 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 B0and a finitely deformed

equilibrium configurationB t The deformation gradient tensor associated with the deformationB0→ Bt

is denoted by F.

In the literature, several composite materials (seeMerodio & Ogden 2005aand 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

for instance,Merodio & Ogden,2005b,c) The invariants of C most commonly used are the principal

invariants, defined by

I1= tr C, I2=1

2[(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 · (C2M). (2)

It follows that, for incompressible materials,Ω = Ω(I1, I2, I4, I5) since I3= 1 at all times

For the considered incompressible material the Cauchy stress is

σ = F ∂Ω

∂F − pI =

5



i =1,i |= 3

Ω iF∂I i

where p is a Lagrange multiplier associated with the incompressibility constraint and I is the 3× 3

identity matrix The principal values are denoted by σ , i= 1, 2, 3 The Cauchy stress tensor can be

written as

σ = 2Ω1B+ 2Ω2(I1I− B)B + 2Ω4m⊗ m + 2Ω5(m ⊗ Bm + Bm ⊗ a) − pI, (4)

where m = FM Hence, m gives the fibre direction in the deformed configuration It is clear that

princi-pal directions of stress and strain do 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 (seeMerodio & Ogden,2005a)

Ω = μ(I1− 3)

where f (I4) is the reinforcing model There are other reinforcing models (seeMerodio & Neff,2006),

in particular, some strain energy functions also have the form

Ω = μ(I1− 3)

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

sec-tions Let(X1, X2, X3) be a fixed rectangular coordinate system Consider an incompressible transversely

isotropic semi-infinite bodyB in its unstrained state B0that occupies the region X2 0 Fibres run

par-allel to each other and perpendicular to the depth direction X2, i.e M2= 0 (see Fig.1) The body is

subjected to a finite pure homogeneous strain with principal directions given by the X i-axes A finitely

deformed (pre-stressed) equilibrium stateB e is obtained A small time-dependent motion is

superim-posed upon this pre-stressed equilibrium configuration to reach a final material stateB t, called current

configuration The position vectors of a representative particle are denoted by X i , x i (X) and ˜x i (X, t) in

B0,B eandB t, respectively The deformation gradient tensor associated with the deformationsB0→ Bt

andB0→ Beare denoted by F and ¯ F, respectively, and are given in component form by

FiA= ∂ ˜x i

∂X A

, ¯FiA= ∂x i

∂X A

It is clear from (7) that

whereδ ij is the Kronecker operator, u i (X, t) denotes the small time-dependent displacement associated

with the deformationB e → Btand a comma indicates differentiation with respect to the indicated spatial

coordinate inB e The fibre orientation inB eis 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

ten-sor S ji are related to the incremental displacement gradients u k,lby (seeOgden,1984):

Fig 1 A point in the free surface of the pre-stressed half-space: (i) the principal axes of the primary pure homogeneous strain

(x i-axes), (ii) the fibre direction in that configuration (given byα) as well as the fibres (dashed lines) along the depth direction

(x2 -axis) and (iii) the propagation direction of the wave (given byθ) Fibres are located throughout the whole half-space and run

parallel to each other and perpendicular to the depth direction.

where P = p( ¯F) is the value of p at Be (independent of time t), p∗= p − P is the time-dependent

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 alsoVinh & Merodio,2013b):

A piqj = ¯Fp α ¯Fq β ∂2Ω

∂F i α ∂F j β





F = ¯F

= 2Ω1δ ij ¯Bpq + 2Ω2(2 ¯B ip ¯Bjq − ¯Biq ¯Bjp + I1δ ij ¯Bpq − ¯Bij ¯Bpq − δij ( ¯B2) pq )

+ 2Ω4δ ijmpmq + 2Ω5[δ ij (m p ¯Bqr mr + mq ¯Bpr mr ) + ¯B ijmpmq + mimj ¯Bpq + ¯Biqmjmp + ¯Bpjmimq]+ 4Ω11¯Bip ¯Bjq + 4Ω22(I1¯Bip

− ( ¯B2) ip )(I1¯B jq − ( ¯B2) jq ) + 4Ω44m i m j m p m q + 4Ω55[m p ¯B ir m r + mi ¯Bprmr ][m q ¯Bjrmr + mj ¯Bqr mr]+ 4Ω12[ ¯Bip (I1¯Bjq − ( ¯B2) jq )

+ ¯Bjq (I1¯Bip − ( ¯B2) ip )] + 4Ω14[ ¯B ip m j m q + ¯Bjq m i m p]+ 4Ω15[ ¯B ip

× [mq ¯Bjrmr + mj ¯Bqrmr]+ ¯Bjq [m i ¯Bprmr + mp ¯Birmr]]+ 4Ω24[(I1¯Bip

− ( ¯B2) ip )m jmq + (I1¯Bjq − ( ¯B2) jq )m imp]+ 4Ω25[(I1¯Bip − ( ¯B2) ip )

× [mq ¯Bjrmr + mj ¯Bqrmr]+ (I1¯Bjq − ( ¯B2) jq )[m i ¯Bprmr + mp ¯Birmr]]

+ 4Ω [m imp [m ¯Bjrmr + mj ¯Bqrmr]+ mjmq [m ¯Bprmr + mp ¯Birmr]] (10)

where B = FFand we note that I1= ¯Bkk It is clear thatA piqj = Aqjpi Note that formula (10) holds for

any coordinate system Another form ofA piqjwas derived byPrikazchikov & Rogerson(2003,2004)

In general, the elasticity tensorA has at most 45 non-zero components.

Remark 1 If the principal directions of the pure homogeneous pre-strain are given by the Xi-axes

(or x i -axes) and the fibre direction is restricted to be one of the X i -axes (or x i-axes), it follows using

(10) that there are only 15 non-zero components of the fourth-order elasticity tensorA, namely A iijj,

A ijij (i, j = 1, 2, 3, i |= j) and A ijji (i, j = 1, 2, 3, i |= j) (see alsoVinh & Merodio,2013a)

We further specialize the elasticity tensor to some specific energy functions It follows from (5) that

Ω2=Ω5=Ω 2k=Ω 5k=Ω 1k = 0 (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 ofA piqjis (see alsoDestrade et al.,2008;Merodio

& Ogden,2002)

A piqj = δij[μ ¯B pq + 2f(I4)m p m q]+ 4f(I4)m i m j m p m q (11)

On the other hand, using (6) and (10) one can write

A piqj = μδij ¯Bpq + 2g(I5)[δ ij (m p ¯Bqrmr + mq ¯Bpr mr ) + ¯B ijmpmq + mimj ¯Bpq + ¯Biqmjmp + ¯Bpjmimq]

+ 4g(I5)(m p ¯Birmr + mi ¯Bprmr )(m q ¯Bjrmr + mj ¯Bqrmr ) (12)

for the other reinforcing model

In the absence of body forces, incremental equations of motion are (seeOgden,1984)

where a dot indicates differentiation with respect to time t The incremental version of the

incompress-ibility is (seeOgden,1984)

In summary, under the conditions at hand, the principal directions of the pre-strain are the x k axes,

i.e ¯B ij = 0 if i |= j, and m2= 0 The half-space is maintained in this static state of deformation by the

application of stresses that are obtained using (4) Furthermore, it is easy to check that the plane x2= 0

is a principal plane of stress while the planes x1= 0 and x3= 0 support shear stresses It follows from

(10) that, under these circumstances, there are only 25 non-zero components of the elasticity tensor

A, namely: A iijj,(i, j = 1, 2, 3), A ijij (i, j = 1, 2, 3, i |= j) and A ijji (i, j = 1, 2, 3, i |= j) , A ii13 (i = 1, 2, 3),

A ii31 (i = 1, 2, 3), A2312,A2321,A3212andA3221

3 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

3.1 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 angleθ with the x1-direction and decays in the x2-direction Then, the

dis-placements and stresses of the Rayleigh wave are written (seeDestrade et al.,2005) as

u n = Un (y) e ik (x1c θ +x3s θ −ct), S

2n = iktn (y) e ik (x1c θ +x3s θ −ct), n = 1, 2, 3, y = kx2, (15)

respectively, where c θ = cos θ, s θ = sin θ, and k = |k| is the wave number Using (15), together with

(9), (13) and (14), one can write

where the prime signifies differentiation with respect to y and

ξ =



u t



⎡

⎣U U12

U3

⎤

⎦ , t =

⎡

⎣t t12

t3

⎤

⎦ , N = N1 N2

1

within which the matrices Nkand K are defined by

⎡

⎣−c0θ f01 −s0θ

⎤

⎦ , N2=

⎡

⎣ d01 00 −d013

−d13 0 d3

⎤

⎦ , K =

⎡

⎣h01 h03 h02

h2 0 h4

⎤

where

f1= a11c θ + a13s θ, f2= a31c θ + a33s θ,

h1= ρc2− b111c2θ − b113c θ s θ − b133s2θ, h2= −b311c2θ − b313c θ s θ − b333s2θ,

h3= ρc2− e11c2θ − e13c θ s θ − e33s2θ, h4= ρc2− d311c2θ − d313c θ s θ − d333s2θ.

(19)

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

The boundary condition of zero incremental traction using the expression given for S 2n in (15) means

that

In passing, we note that the matrices N1, N2 and K in (18) particularized for isotropic materials

coin-cide, respectively, with the matrices N1, N2and N3+ XI, where X = ρc2, given by (2.9) and (2.10) in

Destrade et al.(2005)

3.2 Implicit and explicit secular equations

The implicit secular equation is given by (seeDestrade et al.,2005;Prikazchikov & Rogerson,2004for

complete details)

in which

ωI= −(s1+ s2+ s3), ωII= s1s2+ s2s3+ s3s1, ωIII= −s1s2s3, (23)

where s1, s2and s3are the eigenvalues of the Stroh matrix N with positive imaginary parts, and m and

n are defined, respectively, as

m=

f2 1

h3

− d1 h1+

f2 2

h3

− d3 h4+ 2

f1f2

h3

+ d13 h2,

n = (h1h4− h2

2)



d1d3−d3f12+ d1f22

h3

−2d13f1f2

h3

− d2 13



(24)

Equation (22) is called the implicit secular equation (see alsoDestrade et al.,2005) because the

expres-sions for theωI,ωII andωIII 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, seeTing(2005), to obtain the explicit secular

equation of the wave Using (16), (20), (21), and that N2and K are symmetric, one can write

¯u(0)K (n)u(0) = 0 ∀n ∈ Z (25)

where K(n)is defined as

2

4

From (25) the explicit secular equation is obtained

|K2, K3, K1|2+ 4|K2, K3, K4| |K2, K1, K4| = 0, (27) where

⎡

⎢K

(−1)

11

K (1)

11

K (3)

11

⎤

⎥

⎦ , K2=

⎡

⎢K

(−1)

22

K (1)

22

K (3)

22

⎤

⎥

⎦ , K3=

⎡

⎢K

(−1)

33

K (1)

33

K (3)

33

⎤

⎥

⎦ , K4=

⎡

⎢K

(−1)

13

K (1)

13

K (3)

13

⎤

in which K (n)

ij are entries of the matrix K(n) Equation (27) is the explicit secular equation under the

conditions at hand

The elements K (1)

11, K (1)

22, K (1)

33, K (1)

13 are polynomials of degree 1, 1, 1, 0 in X ; K (3)

11, K (3)

22, K (3)

33, K (3)

13

are polynomials of degree 2, 1, 2, 2 in X and K (−1)

11 , K (−1)

22 , K (−1)

33 , K (−1)

13 are polynomials of degree 2, 3,

2, 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 (seeFu,2005), and that one satisfies the implicit

secular equation (22)

In order to illustrate the results further, we consider some particular strain–energy functions

Numer-ical 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)

Ω1=μ

2(I1− 3) + μγ1

Ω2=μ

2(I1− 3) + μγ2

Ω3=μ

whereμ, γ1 andγ2 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 (seeVinh & Merodio,2013a,b), i e

x1= λX1, x2= λ −1/2 X

2, x3= λ −1/2 X

3, λ > 0, λ = const. (32) The wave makes an angleθ with the x1-direction Figure2shows 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 I5is 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 byFlavin(1963)

Let us consider now that the fibre direction is not a principal direction The pre-strain is pure

homo-geneous and given by

x1= λ1X1, x2= λ2X2, x3= λ3X3, (33) whereλ kare the principal stretches of the deformation and obey thatλ1λ2λ3= 1 We further consider

that the plane x2= 0 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

Figure3shows the dependence onθ ∈ [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 angleα = π/6 with

the x1-axis for (29) (dash–dot line), (30) (dashed line) and (31) (solid line) The parameters used for the

computations areγ1= γ2=3

4, whileλ1=3

2,λ2= 1 and λ3=2

3 The curve associated with the Neo-Hookean model has a maximum forθ = 0 On the other hand, the other two curves have a maximum at

an angleθ ∈ (0 π/2).

We also note that when (27) is specialized to isotropic materials, it coincides with Equation (4.5)

inDestrade 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 do 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 Figure4

shows the dependence of the squared dimensionless Rayleigh wave velocity x = ρc2/μ on λ ∈ [1 1.5]

Fig 2 The curves show the dependence of x = ρc2/μ obtained using (27) onθ ∈ [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 = 3 andλ = 1.3.

Fig 3 Values of x = ρc2/μ obtained using (27) vsθ ∈ [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), withγ1= γ2 = 3 and(λ1 ,λ2 ,λ3) =3 , 1, 2 

.

for (29) (dashed line), (30) (dash–dot line) and (31) (solid line) Here, we have considered thatγ1=

γ2=5

2 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 I5has a greater influence

than the invariant I

in< i>Destrade et al.(2005), which is the secular equation for non-principal Rayleigh waves in deformed

isotropic incompressible. .. 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 I5has... 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