DSpace at VNU: On a technique for deriving the explicit secular equation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer

DSpace at VNU: On a technique for deriving the explicit secular equation of Rayleigh waves in an orthotropic half-space...

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Waves in Random and Complex Media

ISSN: 1745-5030 (Print) 1745-5049 (Online) Journal homepage: http://www.tandfonline.com/loi/twrm20

On a technique for deriving the explicit secular equation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer

P C Vinh, V T N Anh & N T K Linh

To cite this article: P C Vinh, V T N Anh & N T K Linh (2016): On a technique

for deriving the explicit secular equation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer, Waves in Random and Complex Media, DOI:

10.1080/17455030.2015.1132859

To link to this article: http://dx.doi.org/10.1080/17455030.2015.1132859

Published online: 20 Jan 2016

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On a technique for deriving the explicit secular equation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer

P C Vinha, V T N Anhaand N T K Linhb

aFaculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, Hanoi, Vietnam;

bDepartment of Engineering Mechanics, Water Resources University of Vietnam, Hanoi, Vietnam

ABSTRACT

The secular equation of Rayleigh propagating in an orthotropic

half-space coated by an orthotropic layer has been obtained by

Sotiropolous [Sotiropolous, D A (1999), The e®ect of anisotropy on

guided elastic waves in a layered half-space, Mechanics of Materials

31, 215–233] and by Sotiropolous & Tougelidis [Sotiropolous, D A

and Tougelidis, G (1998), Guided elastic waves in orthotropic surface

layer, Ultrasonics 36, 371–374] However, it is not totally explicit

and some misprints have occurred in this secular equation in both

papers This secular equation was derived by expanding directly a

six-order determinant originated from the traction-free conditions at

the top surface of the layer and the continuity of displacements and

stresses through the interface between the layer and the half-space

Since the expansion of this six-order determinant was not shown in

both two papers, it has been difficult to readers to recognize these

misprints This paper presents a technique that provides a totally

explicit secular equation of the wave The technique makes clear the

way from the traction-free and continuity conditions to the secular

equation and enables us to recognize the misprints appearing in the

reported secular equation The technique can be employed to obtain

explicit secular equations of Rayleigh waves for many other cases

Moreover, the paper introduces a transfer matrix in explicit form for an

orthotropic layer that is much simpler in form than the one obtained

previously

ARTICLE HISTORY

Received 18 August 2015 Accepted 12 December 2015

1 Introduction

An elastic half-space overlaid by an elastic layer is a model (structure) finding a wide range of applications such as those in seismology, acoustics, geophysics, materials science, and micro-electro-mechanical systems The measurement of mechanical properties of supported layers therefore plays an important role in understanding the behaviors of this structure in applications, see for examples [1] and references therein Among various measurement methods, the surface/guided wave method is most widely used [2] because

it is non-destructive and it is connected with reduced cost, less inspection time, and greater coverage.[3] Among surface/guided waves, the Rayleigh wave is a versatile and convenient

CONTACT P C Vinh pcvinh@vnu.edu.vn

tool.[3,4] Since the explicit dispersion relations of Rayleigh waves are employed as theo-retical bases for extracting the mechanical properties of the layers from experimental data, they are therefore the main purpose of any investigation of Rayleigh waves propagating in elastic half-spaces covered by an elastic layer

The secular equation of Rayleigh propagating in an orthotropic half-space coated by

an orthotropic layer has been obtained by Sotiroplous and Tougelidis [5] [Equation (8)] and Sotiropolous [6] [Equation (16)] However, this secular equation is not totally explicit because it contains an implicit factor Furthermore, some misprints have been occurred in this secular equation in both papers In particular

(i) In the expression for A (η, η∗) (Equation (17) in Ref [6]): 2r

 1−c2c3−1/2 

1−c∗

2c∗

3−1/2

must be replaced by 2r



1− c2c −1/23 η 1− c∗

2c∗

3−1/2 η∗

(ii) In the expression for C (η, η∗) (Equation (19) in Ref [6]): c∗

3−1/2must be replaced by

c∗

31/2.

(iii) The same misprints have been occurred in the secular Equation (8) in Ref [5] The secular equation reported in Refs [5,6] was derived by expanding directly a six-order determinant originated from the traction-free conditions at the top surface of the layer and the continuity of displacements and stresses through the interface between the layer and the half-space Since the expansion of this six-order determinant was not shown in both two papers, it has been really difficult to readers to discover these misprints

This paper introduces a technique that leads to a totally explicit secular equation of the wave Moreover, it provides a clear way from the traction-free and continuity conditions

to the explicit secular equation and enables us to find the misprints mentioned above This technique is based on the expressions of the traction amplitude vector in terms of the displacement amplitude vector of Rayleigh waves at two sides of the welded interface between the layer and the half-space [Equations (25) and (36)]

Note that, when the half-space and the layer are both isotropic, the explicit secular equation of Rayleigh waves was derived by Ben-Menahem and Singh [7], and for the pre-stressed case (the half-space and the layer are both pre-pre-stressed), the secular equations of Rayleigh waves were obtained by Ogden and Sotiropoulos [8,9] All these secular equations were derived by the same technique as that was employed to the orthotropic case, i.e directly expanding a six-order determinant established by the traction-free conditions at the surface and the continuity of displacements and stresses through the interface The expansion of this six-order determinant was also not shown Therefore, the technique presented in this paper can be used to detail clearly the derivation of the explicit secular equations mentioned above Furthermore, this technique can be employed to derive explicit dispersion relations of Rayleigh waves for other cases, for example, the cases when

the layer is monoclinic (with the symmetry plane x1 = 0, x2 = 0 or x3 = 0) and the half-space is orthtropic or pre-stressed (the explicit secular equations are still not available for these cases) This technique is also applicable for the case when the half-space and the layer are in the sliding contact.[10]

The paper also introduces a transform matrix for orthotropic layer (defined by Equation (17)) that is much compact in form than the one derived by Solyanik This matrix will be useful in computing the Rayleigh wave fields for an elastic half-space overlaid by an arbitrary number of different homogeneous layers

The paper is organized as follows In Section2, a transfer matrix in explicit form for

an orthotropic layer is derived This matrix will be employed in Section 3to obtain the expression of the traction amplitude vector in terms of the displacement amplitude vector at the layer side of the interface It is worth to note that this layer transfer matrix is much simpler

in form than the one obtained previously by Solyanik [11] In Section3, two expressions of the traction amplitude vector in terms of the displacement amplitude vector at two sides

of the interface are established Using them in the continuity condition at the interface leads to the explicit secular equation of Rayleigh waves In Section4, this secular equation

is converted to the one obtained by Sotiropolous [6] and from that the misprints are found

2 Explicit transfer matrix for an orthotropic layer

Consider a compressible orthotropic elastic layer with uniform thickness h occupying the domain a ≤ x2≤ b, b − a = h We are interested in the plane strain such that

¯u i = ¯u i (x1, x2, t ), i = 1, 2, ¯u3≡ 0 (1) where ¯u i are displacement components of the layer, t is the time In the absence of body

forces, the equations of motion are

¯σ11,1+ ¯σ12,2= ¯ρ ¨¯u1, ¯σ12,1+ ¯σ22,2= ¯ρ ¨¯u2 (2) where ¯σ ijare stress components of the layer, commas signify differentiation with respect

to x k , a dot indicates differentiation with respect to t For an orthotropic material, the

strain–stress relation is of the form

¯σ11= ¯c11¯u1,1+ ¯c12¯u2,2, ¯σ22= ¯c12¯u1,1+ ¯c22¯u2,2, ¯σ12= ¯c66(¯u1,2+ ¯u2,1) (3) where¯c ijare material constants of the layer Substituting (3) into (2) and taking into account (1) yield

¯c11¯u1,11+ ¯c66¯u1,22+ (¯c12+ ¯c66)¯u2,12= ¯ρ ¨¯u1

(¯c12+ ¯c66)¯u1,12+ ¯c66¯u2,11+ ¯c22¯u2,22= ¯ρ ¨¯u2 (4)

Now we consider the propagation of a plane wave traveling in the x1-direction with velocity

c ( > 0) and wave number k ( > 0) Then, the displacement components of the wave are

sought in the form

¯u1= ¯U1(x2)e ik(x1−ct), ¯u2= ¯U2(x2)e ik(x1−ct) (5)

Substituting (5) into (4) leads to two second-order linear differential equations for ¯U1(x2)

and ¯U2(x2), namely

k2(¯c11− ¯ρc2) ¯U1− ¯c66¯U

1− ik(¯c12+ ¯c66) ¯U2 = 0

k2(¯c66− ¯ρc2) ¯U2− ¯c22¯U

2− ik(¯c12+ ¯c66) ¯U1 = 0 (6)

It is not difficult to verify that the general solution of the system (6) is

¯U1(x2) = A1ch ¯b1y + A2sh ¯b1y + A3ch ¯b2y + A4sh ¯b2y

¯U2(x2) = iα1



A1sh ¯b1y + A2ch ¯b1y

+ α2



A3sh ¯b2y + A4ch ¯b2y

(7)

where y = k(x2− b), A1, A2, A3, A4are constants,¯α k and ¯b kare given by

¯α k = − (¯c12+ ¯c66)¯b k

¯c22¯b2

k − ¯c66+ ¯X , k = 1, 2, ¯X = ¯ρc

2

¯b1=



¯S + ¯S2− 4¯P

2 , ¯b2=



¯S − ¯S2− 4¯P

2

¯S = ¯ c22(¯c11− ¯X) + ¯c66(¯c66− ¯X) − (¯c12+ ¯c66)2

¯c22¯c66

¯P = (¯ c11− ¯X)(¯c66− ¯X)

¯c22¯c66

(8)

Note that ¯b1and ¯b2are complex in general and no requirements are imposed on their real and imaginary parts On use of Equations (5)–(8) into (3) we have

¯σ12= k ¯1(x2)e ik(x1−ct), ¯σ22= k ¯2(x2)e ik(x1−ct) (9)

where

¯1(x2) = ¯β1



A1sh ¯b1y + A2ch ¯b1y

+ ¯β2(A3sh ¯b2y + A4ch ¯b2y)

¯2(x2) = i¯γ1



A1ch ¯b1y + A2sh ¯b1y

+ ¯γ2



A3ch ¯b2y + A4sh ¯b2y) (10)

and

¯β n = ¯c66(¯b n − ¯α n ), ¯γ n = ¯c12+ ¯c22¯b n ¯α n, n = 1, 2 (11)

Remark 1: For the wave propagation problem c is the wave velocity (to be determined)

of Rayleigh, Stoneley or Lamb wave and k = ω/c is the wave number (ω is the given wave circular frequency), while for the reflection and/or transmission problem c = c0/sinθ0(is

given) where c0is the velocity of incident wave,θ0 (0 < θ0≤ π/2) is the incident angle and

k = k0sinθ0, k0= ω/c0,ω is also given.

Putting x2= b in Equations (7) and (10) leads to

¯U1(b) = A1+ A3, ¯U2(b) = i( ¯α1A2+ ¯α2A4)

¯1(b) = ¯β1A2+ ¯β2A4, ¯2(b) = i( ¯γ1A1+ ¯γ2A3) (12) Solving the system (12) for A1, A2, A3, A4we have

A1 = ¯γ2

[ ¯γ ] ¯U1(b) + i

[ ¯γ ] ¯2(b), A2= i ¯β2

[ ¯α; ¯β] ¯U2(b) + ¯α2

[ ¯α; ¯β] ¯1(b)

A3 = − ¯γ1

[ ¯γ ] ¯U1(b) − i

[ ¯γ ] ¯2(b), A4= − i ¯β1

[ ¯α; ¯β] ¯U2(b) − ¯α1

[ ¯α; ¯β] ¯1(b) (13)

here, for the seeking of simplicity, we use the notations

[f ; g] := f2g1− f1g2,[f ; g] (+) := f2g1+ f1g2,[f ] := f2− f1, [f ] (+) := f2+ f1 (14) The relation

[f ; g][h] − [f ; h][g] = [f ][h; g] (15)

is derived directly from (14) and is useful in calculations By substituting the expressions

of A m (m = 1, 2, 3, 4) given by (13) into (7), (10) and taking x2 = a we obtain the linear

relations of ¯U1(a), ¯U2(a), ¯1(a), and ¯2(a) in terms of ¯U1(b), ¯U2(b), ¯1(b), and ¯2(b) In

matrix form they are of the form

whereξ(.) = [ ¯U1(.) ¯U2(.) ¯1(.) ¯2(.)] T and

T=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

[ ¯γ ; chε]

[ ¯γ ]

−i[ ¯β; shε]

[ ¯α; ¯β]

−[ ¯α; shε]

[ ¯α; ¯β]

−i[chε]

[ ¯γ ]

−i[ ¯γ ; ¯αshε]

[ ¯γ ]

[ ¯αchε; ¯β]

[ ¯α; ¯β]

−i ¯α1¯α2[chε]

[ ¯α; ¯β]

−[ ¯αshε]

[ ¯γ ]

−[ ¯γ ; ¯βshε]

[ ¯γ ]

−i ¯β1¯β2[chε]

[ ¯α; ¯β]

[ ¯α; ¯βchε]

[ ¯α; ¯β]

i[ ¯βshε]

[ ¯γ ]

−i ¯γ1¯γ2[chε]

[ ¯γ ]

[ ¯β; ¯γ shε]

[ ¯α; ¯β]

−i[ ¯α; ¯γ shε]

[ ¯α; ¯β]

[ ¯γ chε]

[ ¯γ ]

⎤

⎥

⎥

⎥

⎥

⎥

⎥

(17)

hereε n = ε¯b n , n = 1, 2, ε = kh and [chε] = chε2− chε1, [ ¯αchε] = ¯α2chε2− ¯α1chε1,

[ ¯α; ¯βshε] = ¯α2¯β1shε1− ¯α2¯β1shε1, … Matrix T given by (17) is the transfer matrix for a compressible orthotropic layer It is not difficult to prove the equalities

t11= t33, t12= t43, t14= t23, t21= t34, t22= t44, t32= t41 (18)

where t ijare components of the transfer matrix T Analogously, using the solution (5), (7), (9), (10) with y = k(x2− a) provides

where ˆTis given by (17) in which shε is replaced by −shε In particular, it is

ˆT =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

[ ¯γ ; chε]

[ ¯γ ]

i[ ¯β; shε]

[ ¯α; ¯β]

[ ¯α; shε]

[ ¯α; ¯β]

−i[chε]

[ ¯γ ]

i[ ¯γ ; ¯αshε]

[ ¯γ ]

[ ¯αchε; ¯β]

[ ¯α; ¯β]

−i ¯α1¯α2[chε]

[ ¯α; ¯β]

[ ¯αshε]

[ ¯γ ] [ ¯γ ; ¯βshε]

[ ¯γ ]

−i ¯β1¯β2[chε]

[ ¯α; ¯β]

[ ¯α; ¯βchε]

[ ¯α; ¯β]

−i[ ¯βshε]

[ ¯γ ]

−i ¯γ1¯γ2[chε]

[ ¯γ ]

−[ ¯β; ¯γ shε]

[ ¯α; ¯β]

i[ ¯α; ¯γ shε]

[ ¯α; ¯β]

[ ¯γ chε]

[ ¯γ ]

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

(20)

One can see that the following equalities are valid

ˆt11= ˆt33, ˆt12= ˆt43, ˆt14= ˆt23, ˆt21= ˆt34, ˆt22= ˆt44, ˆt32= ˆt41 (21)

where ˆt ijare components of the transfer matrix ˆT From (16) and (19), it implies: ˆT = T−1.

Remark 2:

(i) From (19) and (20) it follows

whereη(.) = [¯v1(.) ¯v2(.) ¯σ22(.) ¯σ12(.)] Tand

A=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

[ ¯γ ; chε]

[ ¯γ ]

i[ ¯β; shε]

[ ¯α; ¯β]

−c[chε]

[ ¯γ ]

−ic[ ¯α; shε]

[ ¯α; ¯β]

i[ ¯γ ; ¯αshε]

[ ¯γ ]

[ ¯αchε; ¯β]

[ ¯α; ¯β]

−ic[ ¯αshε]

[ ¯γ ]

−c ¯α1¯α2[chε]

[ ¯α; ¯β]

¯γ1¯γ2[chε]

c[ ¯γ ]

−i[ ¯β; ¯γ shε]

c[ ¯α; ¯β]

[ ¯γ chε]

[ ¯γ ]

i[ ¯α; γ shε]

[ ¯α; ¯β]

i[ ¯γ ; ¯βshε]

c[ ¯γ ]

¯β1¯β2[chε]

c[ ¯α; ¯β]

−i[ ¯βshε]

[ ¯γ ]

[ ¯α; ¯βchε]

[ ¯α; ¯β]

⎤

⎥

⎥

⎥

⎥

⎥

⎥

(23)

¯v1 = −iω¯u1, ¯v2 = −iω¯u2are the components of the particle velocity From (21) it implies

A24= A13, A33= A22, A34= A12, A42= A31, A43= A21, A44= A11 (24)

where A ijare components of the transfer matrix A These relations were mentioned

in [12]

Comparing the matrix A with the layer transfer matrix reported Ref [11] reveals that

λ xzxz in the expression for a11in [11] must be replaced byλ xxzz (ii) One can see that the expressions of elements of the transfer matrix A are much

simpler in form than the corresponding expressions obtained by Solyanik [11]

3 Explicit secular equation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer

Consider a compressible orthotropic elastic half-space x2 ≥ 0 overlaid by a compressible

orthotropic elastic layer with arbitrary thickness h occupying the domain −h ≤ x2≤ 0 It is assumed that the layer and the half-space are in welded contact with each other and the

top surface of the layer x2= −h is free from traction Note that same quantities related to

the half-space and the layer have the same symbol but are systematically distinguished by

a bar if pertaining to the layer

Consider the propagation of a Rayleigh wave traveling with velocity c and wave number

k in the x1-direction, decaying in the x2-direction From the traction-free condition: ¯σ12 =

¯σ22 = 0 at x2 = −h, using (16), (17) with a = −h, b = 0 and taking into account the continuity of displacements and stresses through the interface x2= 0 we have

(0) = ¯ MU(0), ¯ M = −T−14 T3, T3=



t31t32

t41t42



, T4=



t33t34

t43t44



(25)

where(.) = [1(.) 2(.)] T, U(.) = [U1(.) U2(.)] T According to Vinh and Ogden [13], the

displacements of the Rayleigh wave in the half-space x2> 0 are given by

u1= U1(y)e ik(x1−ct) , u

2= U2(y)e ik(x1−ct) , y = kx2 (26) where

U1(y) = B1e −b1y + B2e −b2y

, U2(y) = i(α1B1e −b1y + α2B2e −b2y ) (27)

B1and B2are constants to be determined, and

α k= (c12+ c66)b k

c22b2

k − c66+ X , k = 1, 2, X = ρc

b1and b2are two roots with positive real part of the following equation

S and P are calculated by (8) without the bar symbol It has been shown that if a Rayleigh wave exists, then [13]

0 < X < min {c66, c11} (30) and [14]

P > 0, S + P > 0, b1b2=√P, b1+ b2=



S + 2√P (31) Using expressions (26) and (27) into the strain–stress relation (3) provides

σ12= k1(y)e ik(x1−ct), σ22= k2(y)e ik(x1−ct) (32)

where

1(y) = β1B1e −b1y + β2B2e −b2y, 2(y) = i(γ1B1e −b1y + γ2B2e −b2y ) (33) where

β k = −c66(b k + α k ), γ k = c12− c22b k α k , k = 1, 2 (34)

Taking x2= 0 in (27) and (33) gives

U1(0) = B1+ B2, U2(0) = i(α1B1+ α2B2)

1(0) = β1B1+ β2B2, 2(0) = i(γ1B1+ γ2B2) (35)

Eliminating B1, B2from Equation (35) yields the relation

[α]MU(0), M =

⎡

⎣[α; β] −i[β]

i[α; γ ] [γ ]

⎤

From (25) and (36) it follows



M− [α] ¯ M

U(0) = 0 ⇔ T4M+ [α]T3



U(0) = 0 (37)

Due to U(0) = 0 the determinant of the matrix of system (37) must be zero

Expanding (38) and using (15) make Equation (38) to be equivalent to

(t33t44− t34t43)[γ ; β] + i(t33t41− t43t31)[β] + (t33t42− t43t32)[α; β]

−(t34t41− t44t31)[γ ] + i(t34t42− t44t32)[α; γ ] + (t31t42− t32t41)[α] = 0 (39) With the help of (28) and (34), it is not difficult to verify that

[γ ; β] = c66



c2

12− c22(c11− X)b1b2+ X(c11− X)θ

[α; β] = c66(c11− X)(b1+ b2)θ, [α; γ ] = c66(c11− X − c12b1b2)θ

[α] = (X − c11− c66b1b2)θ, [β] = [α; γ ], [γ ] = c22c66b1b2(b1+ b2)θ (40)

where b1b2 = √P, b1+ b2 = S + 2√P and θ = (b2 − b1)/[(c12+ c66)b1b2] After multiplying two sides of Equation (39) by[ ¯γ ][ ¯α; ¯β]/θ and taking into account (40), this equation becomes

A0+ B0chε1chε2+ C0shε1shε2+ D0chε1shε2+ E0shε1chε2= 0 (41)

where A0, B0, C0, D0, and E0are given by

A0= 2 ¯β1¯β2¯γ1¯γ2(X − c11− c66

√

P)

−c66[ ¯α; ¯β ¯γ ] (+)

c2

12− c22(c11− X)√P + X(c11− X)

−c66



¯γ1¯γ2[ ¯α; ¯β] (+) + ¯β1¯β2[ ¯γ ] (+)

(c11− X − c12

√

P)

B0= −A0+ c66[ ¯γ ][ ¯α; ¯β]

c2

12− c22(c11− X)√P + X(c11− X)

C0= [ ¯β2; ¯γ2](+) (X − c11− c66

√

P)

−c66[ ¯α ¯β; ¯γ ] (+)

c2

12− c22(c11− X)√P + X(c11− X)

−c66



[ ¯α ¯β; ¯γ2](+) + [ ¯β2; ¯γ ] (+)(c11− X − c12

√

P)

D0= c66



¯β1¯γ2[ ¯γ ](X − c11) + c22¯β2¯γ1[ ¯α; ¯β]√P

S + 2√P

E0= c66



¯β2¯γ1[ ¯γ ](c11− X) − c22¯β1¯γ2[ ¯α; ¯β]√P

S + 2√P (42) Equation (41) is the desired secular equation From (8), (11), (31) and (42) it is clear that Equation (41) is totally explicit

Whenε = 0, Equation (41) becomes A0+ B0= 0, or equivalently

(c66− X)c2

12− c22(c11− X)+ X√c22c66

(c11− X)(c66− X) = 0 (43) according to the second of (42) This equation is the secular equation of Rayleigh waves propagating along the traction-free surface of a compressible orthotropic half-space.[13]

Isotropic case

When the layer and the substrate are both isotropic

c11= c22= λ + 2μ, c12= λ, c66= μ, ¯c11= ¯c22= ¯λ + 2 ¯μ, ¯c12= ¯λ, ¯c66= ¯μ (44) With the help of (44) and Equations (8), (11), (28), and (34), one can see that

b1= 1− γ x, b2=√1− x, α1= b1, α2= 1/b2

¯b1= 1− ¯γ ¯x, ¯b2=√1− ¯x, ¯α1= −¯b1, ¯α2= −1/¯b2

β1= −2ρ c2

2b1, β2= −ρ c2

2(2 − x)/b2, γ1= −ρ c2

2(2 − x), γ2= −2 ρ c2

2

¯β1= 2 ¯ρ ¯c2

2¯b1, ¯β2= ¯ρ ¯c2

2(2 − ¯x)/¯b2, ¯γ1= − ¯ρ ¯c2

2(2 − ¯x), ¯γ2= −2 ¯ρ ¯c2

where

x = c2/c2

2, c2=√μ/ρ, γ = μ/(λ + 2μ)

¯x = c2/¯c2

2, , ¯c2=√¯μ/ ¯ρ, ¯γ = ¯μ/(¯λ + 2 ¯μ)

(46)

Introducing (45) into (42), we obtain the explicit secular equation for the isotropic case, namely

A0+ B0chε1chε2+ C0shε1shε2+ D0chε1shε2+ E0shε1chε2= 0 (47)

in which A0, B0, C0, D0, and E0are given by

A0= 4¯b1¯b2(2 − ¯x)2(2 − ¯x)(b1b2− 1) +4b1b2− (2 − x)2

r μ−2

− (4 − ¯x)(2b1b2+ x − 2)r−1μ 

B0= −A0− ¯b1¯b2¯x2

4b1b2− (2 − x)2

r μ−2

C0= 4¯b2

1¯b2 2



4b1b2(1 − r μ−1)2−2− (2 − x)r μ−12

+ (2 − ¯x)2

(2 − ¯x)2(b1b2− 1)

− 2(2 − ¯x)(2b1b2+ x − 2)r−1

μ +4b1b2− (2 − x)2

r−2

μ



D0= ¯b1¯x xb2(2 − ¯x)2− 4b1¯b2

2



r μ−1, E0= ¯b2¯x xb1(2 − ¯x)2− 4b2¯b2

1



r μ−1 (48)

where r μ = μ/ ¯μ, r v = c2/¯c2and¯x = r2

v x.

By multiplying two sides of Equation (47) by k8/( − ¯b1¯b2) we arrive immediately at the

well-known secular equation of Rayleigh waves for the isotropic case, Equation (3.113), p.117 in Ref [7]

4 Misprints in the secular equation derived by Sotiropoulos

Now we convert Equation (41) into an equation whose form is the same as the one of Equation (16) in Ref [6] or of Equation (8) in [5] It is clear that Equation (41) can be rewritten