DSpace at VNU: An approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half-space coated by a thin orthotropic elastic layer

Contents lists available atSciVerse ScienceDirectWave Motion journal homepage:www.elsevier.com/locate/wavemoti An approximate secular equation of Rayleigh waves propagating in an orthotr

Contents lists available atSciVerse ScienceDirect

Wave Motion

journal homepage:www.elsevier.com/locate/wavemoti

An approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half-space coated by a thin orthotropic elastic layer

Pham Chi Vinha,∗, Nguyen Thi Khanh Linhb

aFaculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam

bDepartment of Engineering Mechanics, Water Resources University of Vietnam, 175 Tay Son Str., Hanoi, Viet Nam

a r t i c l e i n f o

Article history:

Received 12 February 2012

Accepted 18 April 2012

Available online 25 April 2012

Keywords:

Rayleigh waves

An orthotropic elastic half-space

A thin orthotropic elastic layer

Approximate secular equation

Approximate formula for the velocity

a b s t r a c t

The present paper is concerned with the propagation of Rayleigh waves in an orthotropic elastic half-space coated with a thin orthotropic elastic layer and the main purpose of the paper is to establish an approximate secular equation of the wave By using the effective boundary condition method an approximate secular equation of third-order in terms of the dimensionless thickness of the layer has been derived From this equation two different third-order approximate secular equations are obtained for the case when the half-space and the layer are both isotropic, one of which recovers the secular equation

of second-order derived by Bovik [P Bovik, A comparison between the Tiersten model and O(H) boundary conditions for elastic surface waves guided by thin layers, J Appl Mech 63 (1996) 162–167] An explicit second-order approximate formula for the Rayleigh wave velocity has been created based on the obtained approximate secular equation Since explicit dispersion relations are employed as theoretical bases for extracting the mechanical properties of the thin films from experimental data, the obtained secular equation and formula for the velocity may be useful in practical applications

© 2012 Elsevier B.V All rights reserved

1 Introduction

The structures of a thin film attached to solids, modeled as half-spaces coated with a thin layer, are widely applied

in the modern technology, measurements of mechanical properties of thin supported films play an important role in understanding the behaviors of these structures in applications; see for examples [1] and references therein Among various measurement methods, the surface/guided wave method [2] is used most extensively, and for which the guided Rayleigh wave is most convenient For the Rayleigh-wave approach, the explicit dispersion relations of Rayleigh waves supported

by thin-film/substrate interactions are employed as theoretical bases for extracting the mechanical properties of the thin films from experimental data They are therefore the key factor, the main purpose of the investigations of Rayleigh waves propagating in half-spaces covered by a thin layer Taking the assumption of thin layer, they are derived approximately by replacing the entire effect of the thin layer by the so-called effective boundary conditions either by replacing approximately the layer by a plate, see [3,4], or by expanding the displacements and stresses at the upper-surface of the layer into Taylor

series of h, the thickness of the layer, see [5–11]

Achenbach [3], Tiersten [4], Bovik [5], and Tuan [12] assumed that the layer and the substrate are both isotropic and the authors derived explicit secular equations of second-order (they do not coincide totally with each other) In [10] Steigmann considered a transversely isotropic layer with residual stress overlying an isotropic half-space and he obtained an explicit second-order dispersion relation In [13] Wang et al considered an isotropic half-space covered by a thin electrode layer

∗Corresponding author Tel.: +84 4 5532164; fax: +84 4 8588817.

E-mail addresses:pcvinh@vnu.edu.vn , pcvinh562000@yahoo.co.uk (P.C Vinh).

0165-2125/$ – see front matter © 2012 Elsevier B.V All rights reserved.

and he obtained an explicit first-order secular equation In all investigations mentioned above the substrate was assumed to

be isotropic and based on this assumption the approximate secular equations in the explicit form were derived However, for less symmetry substrates such as orthotropic and monoclinic substrates, we could not see any explicit approximate dispersion relation in the literature, to the best knowledge of the authors Therefore, the main aim of this paper is to derive an explicit approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half-space coated with a thin orthotropic elastic layer In particular, by using the effective boundary condition approach the authors derive an approximate secular equation of third-order in terms of dimensionless thickness of the layer From this equation two different third-order approximate secular equations are obtained for the case when the half-space and the layer are both isotropic, one of which recovers the secular equation of second-order derived by Bovik [5] An explicit second-order approximate formula for the Rayleigh wave velocity has been created based on the obtained approximate secular equation Since explicit dispersion relations are employed as theoretical bases for extracting the mechanical properties of the thin films from experimental data, the obtained secular equation and formula for the velocity may be useful in practical applications Note that Ting [11] considered the general case when the layer and the substrate are both generally anisotropic (see also [14]) and the author

derived dispersion relations of n-order with arbitrary n However, due to the generality, they are not totally explicit.

2 Effective boundary conditions of third-order for a thin orthotropic elastic layer

Consider an elastic half-space x2 ≥0 coated by a thin elastic layer occupying the domain−h≤x2≤0 The thin layer

is assumed to be perfectly bonded to the half-space 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 We are interested in the plan strain such that:

u i =u i(x1,x2,t), u¯i= ¯u i(x1,x2,t), i=1,2, u3≡ ¯u3≡0 (1) where t is the time Suppose that the layer is made of orthotropic elastic material, then the strain–stress relations are (see [15]):

 σ ¯11= ¯c11u¯1, 1+ ¯c12u¯2, 2

¯

σ22= ¯c12u¯1, 1+ ¯c22u¯2, 2

¯

whereσ ¯ijandc¯ijare respectively the stresses and the material constants of the layer, commas indicate differentiation with

respect to spatial variables x k In the absence of body forces, equations of motion are:

 ¯ σ11 , 1+ ¯ σ12 , 2= ¯ ρ ¨¯u1

¯

whereρ ¯is the mass density of the layer, a superposed dot signifies differentiation with respect to t Taking into account(1), Eqs.(2)and(3)can be written in the matrix form as follows:

U¯′

¯

T′



=



M1 M2

M3 M4

 U¯

¯

T



(4)

whereU¯ = [ ¯u1u¯2]T,T¯ = [ ¯ σ12σ ¯22]T , the symbol ‘‘T ’’ indicates the transpose of a matrix, the prime signifies the derivative with respect to x2and:

M1=





− ¯c12

¯

c22∂1 0



 , M2=







1

¯

c66 0

¯

c22







M3=





¯

c122 − ¯c11c¯22

¯

c22 ∂2

1+ ¯ ρ∂2

t



 , M4=M1T

(5)

here we use the notations∂1= ∂/∂x1,∂2

1 = ∂2/∂x21,∂2

t = ∂2/∂t2 From(4)it follows:

U¯(n)

¯

T(n)



=M n

U¯

¯

T

 , M=



M1 M2

M3 M4

 , n=1,2,3, ,x2∈ [−h,0] (6)

Let h be small (i.e the layer be thin), then expanding T(−h)into Taylor series about x2 = 0 up to the third-order of h we

have:

¯

T(−h) = ¯T(0) −h T¯′(0) +h2

2

¯

T′′(0) −h3

6

¯

Suppose that the surface x2= −h is free from the stress, i e T¯ (−h) =0, using(6)at x2=0 into(7)yields:



I−hM4+h2

2(M3M2+M42) −h3

6[ (M3M1+M4M3)M2+ (M3M2+M42)M4]



¯

T(0) +



−hM3+h2

2(M3M1+M4M3) −h3

6[ (M3M1+M4M3)M1+ (M3M2+M42)M3]



¯

U(0)

Since the layer and the half-space are bonded perfectly to each other at the plane x2 = 0, it follows U(0) = ¯U(0)and

T(0) = ¯T(0) Thus, we have from(8):



I−hM4+h2

2(M3M2+M42) −h3

6[ (M3M1+M4M3)M2+ (M3M2+M42)M4]



T(0) +



−hM3+h2

2(M3M1+M4M3) −h3

6[ (M3M1+M4M3)M1+ (M3M2+M2)M3]



U(0)

The relation(9)between the traction vector and displacement vector of the half-space at the plane x2 = 0 is called the effective boundary condition of third-order in the matrix form It replaces (approximately) the entire effect of the thin layer

on the substrate Substituting(5)into(8)yields the effective boundary conditions in the component form, namely:

σ12+h(r1σ22 , 1−r3u1, 11− ¯ ρ ¨u1) +h2

2



r2σ12 , 11+ ρ ¯

¯

c66σ ¨12−r3u2, 111− ¯ ρ(1+r1)¨u2, 1



+h3

6



r4σ22 , 111+ ¯ ρr5σ ¨22 , 1−r6u1, 1111− ¯ ρr7u¨1, 11− ρ ¯2

¯

c66u¨1,tt



σ22+h(σ12 , 1− ¯ ρ ¨u2) +h2

2



r1σ22 , 11+ ρ ¯

¯

c22σ ¨22−r3u1, 111− ¯ ρ(1+r1)¨u1, 1



+h3

6



r2σ12 , 111+ ¯ ρr8σ ¨12 , 1−r3u2, 1111− ¯ ρ(1+2r1)¨u2, 11− ρ ¯2

¯

c22u¨2,tt



where:

r1= c¯12

¯

c22, r2=r1+ r3

¯

c66, r3= c¯2

12− ¯c11c¯22

¯

c22 , r4=r1r2+ r3

¯

c22

r5= 1+r1

¯

c22 +

r1

¯

c66, r6= (r1+r2)r3, r7=r12+2r2, r8= 1+r1

¯

c66 +

1

¯

3 An approximate secular equation of third-order of Rayleigh waves

Suppose that the elastic half-space is made of orthotropic material Then the unknown vectors U = [u1u2]T , T = [ σ12 σ22]T are satisfied Eq.(4) without the bar symbol Addition to this equation are required the effective boundary conditions(10),(11)at x2=0 and the decay condition at x2= +∞, namely:

Now we consider the propagation of a Rayleigh wave, traveling (in the coated half-space) with velocity c and wave number

k in the x1-direction and decaying in the x2-direction According to Vinh and Ogden [16] the displacement components of this Rayleigh wave which (together the stressesσ12, σ22) satisfy Eq.(4)and the decay condition(13)are given by:

u1= (B1e−kb1x2+B2e−kb2x2)e ik(x1 −ct)

u2= (α1B1e−kb1x2+ α2B2e−kb2x2)e ik(x1 −ct) (14)

where B1, B2 are constants to be determined from the effective boundary conditions(10),(11), b1,b2 are roots of the equation:

c c b4+ { (c +c )2+c (X−c ) +c (X−c )}b2+ (c −X)(c −X) =0 (15)

whose real parts are positive to ensure the decay condition, X = ρc2, and:

αk=iβk, βk= b k(c12+c66)

c22b2−c66+ ρc2 = c11− ρc2−c66b2

(c12+c66)b k , k=1,2,i=

√

From(15)we have:

b21+b22= − (c12+c66)2+c22(X−c11) +c66(X−c66)

c22c66 =S

b21·b22= (c11−X)(c66−X)

c22c66 =P

(17)

It is not difficult to verify that if a Rayleigh wave exists (→b1,b2having positive real parts), then:

and:

b1·b2=

√

P, b1+b2=



S+2

√

Substituting(14)into(2)without the bar gives:

σ12= −kc66{ (b1+ β1)B1e−kb1x2+ (b2+ β2)B2e−kb2x2}e ik(x1 −ct)

σ22=ik{ (c12−c22b1β1)B1e−kb1x2+ (c12−c22b2β2)B2e−kb2x2}e ik(x1 −ct) (20) Introducing(14)and(20)into the effective boundary conditions(10)and(11)leads to the following equations for B1, B2:

f(b1)B1+f(b2)B2=0

where:

f(b n) = −c66(b n+ βn) +kh{r3+ ¯X−r1(c12−c22b nβn)}

+k2h2

2



c66



r2+

¯

X

¯

c66

 (b n+ βn) − βn[r3+ ¯X(1+r1)]



+k3h3

6

 (c12−c22b nβn)(r4+r5X¯ ) −r6−r7X¯ −

¯

X2

¯

c66



F(b n) = (c12−c22b nβn) −kh{c66(b n+ βn) − ¯Xβn}

+k2h2

2



r3+ ¯X(1+r1) − (c12−c22b nβn)



r1+

¯

X

¯

c22



+k3h3

6



c66(b n+ βn)(r2+r8X¯ ) − βn



r3+ ¯X(1+2r1) + X¯ ¯2

c22



n=1,2, ¯X= ¯ ρc2. (22)

Due to B2+B2̸=0, the determinant of coefficients of the homogeneous system(21)must be vanish This provides:

f(b1)F(b2) −f(b2)F(b1) =0. (23) Using(22)into(23)and taking into account(16)and(19), after algebraically lengthy calculations whose details omitted we arrive at the approximate secular equation of third-order of Rayleigh waves, namely:

A0+A1ε + A2

2ε2+A3

whereε =kh called the dimensionless thickness of the layer,

A0=c66[ (c12+c22β1β2)(b2−b1) + (c12+c22b1b2)(β2− β1)]

A1=c66X¯ (β1b2− β2b1) +c22(r3+ ¯X)(β1b1− β2b2)

A2 = −



r3

¯

c66 +

¯

X

¯

c66 +

¯

X

¯

c22



A0+2X¯ (r3+ ¯X)(β2− β1) + [ ¯X(r1−1) −r3][ (c66−c12)(β2− β1) + (c66−c22β1β2)(b2−b1)] (25)

A3 = c66



3r2X¯ +3X¯2

¯

c66 +

¯

X2

¯

c22 −2r3−2

¯

X−r1X¯

 (β2b1− β1b2) +c22



r6+r7X¯ −3r2X¯ +3X¯ (r3+ ¯X)

¯

¯

X2

¯

c

 (β2b2− β1b1).

By(16)it is not difficult to prove the following equalities:

β2− β1= − (c11−X+c66b1b2)

(c12+c66)b1b2 (b2−b1)

β2b2− β1b1= −c66(b1+b2)

(c12+c66) (b2−b1), β1β2= (c11−X)

β2b1− β1b2= − (c11−X)(b1+b2)

(c12+c66)b1b2 (b2−b1).

Introducing(26)into(25)yields: A k= θ ¯A k(k=0,1,2,3), θ = [c66(b2−b1)]/[b1b2(c12+c66)], where:

¯

A0= (c122 −c11c22+c22X)b1b2+ (c11−X)X

¯

A1= [ (c11−X) ¯X+c22(r3+ ¯X)b1b2] (b1+b2)

¯

A2 = −



r3

¯

c66+

¯

X

¯

c66 +

¯

X

¯

c22



¯

A0+2[c12(r1X¯ − ¯X−r3) − (r3+ ¯X) ¯X]b1b2

+2



(r1−1) ¯X−r3+ (r3+ ¯X) X¯

c66



¯

A3 = −



3r2X¯ +3X¯2

¯

c66 +

¯

X2

¯

c22 −2r3−2

¯

X−r1X¯

 (c11−X) + c22



r6+r7X¯ −3r12X¯ +3X¯ (r3+ ¯X)

¯

c22 +

¯

X2

¯

c66



b1b2

 (b1+b2)

in which b1b2and b1+b2are given by(17)and(19) Removing the factorθ, Eq.(24)becomes:

¯

A0+ ¯A1ε +A¯2

2ε2+

¯

A3

This is the desired third-order approximate secular equation and it is fully explicit In the dimensionless form it is:

D0+D1ε +D2

2ε2+ D3

in which:

D0= (e23−e1e2+e2x)b1b2+ (e1−x)x

D1=rµ[ (e1−x)rv2x+e2(¯e2e¯23− ¯e1+rv2x)b1b2] (b1+b2)

D2 = −[¯e2¯e23− ¯e1+ (1+ ¯e2)rv2x]D0+2rµ[e3(¯e1− ¯e2e¯23) + (¯e2e¯3e3−e3−rµ¯e2¯e23+rµ¯e1)rv2x−rµrv4x2]b1b2

+2rµ[¯e1− ¯e2e¯23+ (¯e2¯e3−1+rµe¯2e¯23−rµe¯1)rv2x+rµrv4x2] (x−e1)

D3 = rµ{ (x−e1)[2(¯e1− ¯e2¯e23) + (3¯e2¯e23+2e¯2¯e3−3e¯1−2)rv2x+ (3+ ¯e2)rv4x2]

−e2[ (2¯e2¯e3+ ¯e2¯e23− ¯e1)(¯e2e¯23− ¯e1) + (¯e22e¯23+2¯e2¯e3+2e¯2¯e23−2e¯1−3¯e1¯e2)rv2x

+ (1+3e¯2)rv4x2]b1b2} (b1+b2)

b1b2=

√

P, b1+b2=



S+2

√

P

P= (1−x)(e1−x)

e2 , S= e2(e1−x) +1−x− (1+e3)2

and:

x= X

c66, e1= c11

c66, e2=c22

c66, e3= c12

c66, ¯e1= c¯11

¯

c66, ¯e2= c¯66

¯

c22, e¯3= c¯12

¯

c66

rµ= c¯66

c66, rv= c2

¯

c2, c2=



c66

ρ , c¯2=



¯

c66

¯

The squared dimensionless Rayleigh wave velocity x depend on 9 dimensionless parameters: e k,e¯k(k=1,2,3), rµ, rvand

ε Note that e k > 0, ¯e k > 0(k = 1,2), e1e2−e2 > 0 and¯e1/¯e2− ¯e2 > 0 due to the fact that: c ii > 0(i = 1,2,6),

c11c22−c2 >0 andc¯11c¯22− ¯c2 >0 that ensues the strain energies to be positive definite (see [15])

4 An isotropic case

When the layer and the substrate are both isotropic:

c11=c22= λ +2µ, c12= λ, c66= µ, ¯c11= ¯c22= ¯ λ +2µ, ¯ c¯12= ¯ λ, c¯66= ¯ µ (32) With the help of(16)and(32)one can show that:

b1= 1− γx, b2=

√

where:

x= c2

c2, c2=

 µ

ρ , γ =

µ

Introducing(33)into(25)leads to:

A0= − µ2

b2[ (2−x)2−4b1b2] , A1= −X

b2[b1

¯

X+b2(r3+ ¯X)]

A2= −



r3

¯

µ +

¯

X

¯

µ +

¯

X

¯

λ +2µ ¯



A0−2X¯ (r3+ ¯X)



b1− 1

b2



−2µ[ ¯X(r1−1) −r3]



2b1− (2−x)1

b2



(35)

A3= µx



3r2X¯ + 3X¯2

¯

µ +

¯

X2

¯

λ +2µ ¯ −2r3−2X¯ −r1X¯



b1

b2+



r6+r7X¯ −3r12X¯ +3X¯ (r3+ ¯X)

¯

λ +2µ ¯ +

¯

X2

¯ µ



where:

r1=1−2γ , ¯ r2=2γ − ¯ 3, r3=4µ( ¯γ − ¯ 1), r6= −2r3, r7=4γ ¯2−5, ¯γ = ¯ µ ¯

λ +2µ ¯ . (36)

By using(35)and(36)into Eq.(24), after some manipulations we arrive at:

ˆ

A0+ ˆA1ε + ˆA2ε2+ ˆA3ε3+O(ε4) =0 (37) where:

ˆ

A0= (2−x)2−4

√

1−x1− γx

ˆ

A1=rµx[ (rv2x−4+4γ ) ¯ √1−x+rv2x



1− γx] ˆ

A2 = −1

2[ (1+ ¯ γ )rv2x−4+4γ ]ˆ ¯ A0+rµ2rv2x(4−4γ − ¯ rv2x)(1−

√

1−x1− γx) +rµ(2

√

1−x1− γx−2+x)(4−4γ − ¯ 2rv2γ ¯x) ˆ

A3 = −rµx

6 {[r

4

vx2(1+3γ ) + ¯ 4rv2x( ¯γ2−2) +8(1− ¯ γ )] √1−x

+ [rv4x2(3+ ¯ γ ) +4rv2x(2γ − ¯ 3) +8(1− ¯ γ )]1− γx} (38) here we use the notations:

rµ= µ ¯

µ , rv=

c2

¯

c2, c¯2=



¯ µ

¯

Eq.(37)is the approximate secular equation of third-order for the isotropic case From(37)we arrive immediately at the second-order approximate secular equation, namely:

ˆ

A0+ ˆA1ε + ˆA2ε2+O(ε3) =0 (40) that coincides with Eq (3.16) in Ref [12] Since A0[x(0)] =0, it implies that A0[x(ε)] =O(ε) This facts leads to:

ˆ

A2ε2 = [rµ2rv2x(4−4γ − ¯ rv2x)(1−

√

+rµ(2

√

Eq.(40)is therefore simplified to:

ˆ

Fig 1 Dependence onε =k·h of the dimensionless Rayleigh wave velocity√x=c/c2 that is calculated by the exact secular equation, Eq (6) in Ref [ 5 ] or

Eq (3.9) in Ref [ 12 ], (dashed-dot line), by the approximate secular equations of second-order (43) , (47) (dashed line), by the approximate secular equations

of third-order (37) , (46) (solid line) Hereµ =¯ 2.85×10 −10 N/m 2 ,λ =¯ 15×10 −10 N/m 2 ,c¯2=1200 m/s,c¯1=3240 m/s,µ =3.12×10 −10 N/m 2 ,

λ =1.61×10 −10 N/m 2, c2=3764 m/s, c1=5968 m/s.

that is the same as one obtained by Bovik (Eq (38) in Ref [5]), here:

A∗2=rµ2rv2x(4−4γ − ¯ rv2x)(1−

√

1−x1− γx) +rµ(2

√

1−x1− γx−2+x)(4−4γ − ¯ 2rv2γ ¯x). (44) With the help of(30)–(33)it is not difficult to verify that D k= ¯D k/γ (k=0,1,2,3)where:

¯

D0= [4(γ −1) +x]b1b2+ (1− γx)x

¯

D1=rµ[rv2x(1− γx) + (4γ − ¯ 4+rv2x)b1b2] (b1+b2)

¯

D2 = −[4( ¯γ −1) + (1+ ¯ γ )rv2x]D0+2rµ[ (2γ −1)(4γ − ¯ 4+2γ ¯rv2x) − γrµrv2x(4γ − ¯ 4+rv2x)]b1b2

+2rµ[4( ¯γ −1) +2( ¯γ +2rµ−2rµγ ) ¯ rv2x−rµrv4x2] (1− γx) (45)

¯

D3 = rµ{[8(1− ¯ γ ) +4(2γ − ¯ 3)rv2x+ (3+ ¯ γ )rv4x2] (γx−1)

− [8(1− ¯ γ ) +4( ¯γ2−2)rv2x+ (1+3γ ) ¯ rv4x2]b1b2} (b1+b2)

where b1= √1− γx,b2=

√

1−x Therefore, we have an alternative approximate secular equation of third-order for the

isotropic case, namely:

¯

D0+ ¯D1ε +D¯2

2ε2+

¯

D3

in whichD¯k(k = 0,1,2,3,4)are given by(45) SinceD¯0[x(ε)] = O(ε), the second-order approximate secular equation take the form:

¯

D0+ ¯D1ε +D

∗ 2

where:

D∗2 = 2rµ[ (2γ −1)(4γ − ¯ 4+2γ ¯rv2x) − γrµrv2x(4γ − ¯ 4+rv2x)]b1b2

+2rµ[4( ¯γ −1) +2(3γ − ¯ 2)rv2x+rv4x2] (1− γx). (48)

Fig 1presents the dependence onε, the dimensionless thickness of the layer, of the dimensionless Rayleigh wave velocity

√

x(ε)that is calculated by the exact secular equation, Eq.(6)in Ref [5] or Eq (3.9) in Ref [12], by the second-order approximate secular equations(43),(47), and by the third-order approximate secular equations(37),(46) The materials are an (isotropic) gold layer on an (isotropic) fused silica substrate The material parameters are:µ = ¯ 2.85×10− 10N/m2,

¯

λ =15×10−10N/m2,c¯2=1200 m/s,c¯1=3240 m/s,µ =3.12×10−10N/m2,λ =1.61×10−10N/m2, c2=3764 m/s,

c1 =5968 m/s (see [5]) It is shown from theFig 1that the third-order approximate secular equation is really better than the one of second-order

Finally, we note that when the half-space is isotropic, two roots of Eq.(15)having positive real part are:

b1=



1−c2

c2, b2=



1−c2

c2, c1=  λ +2µ

ρ , 0<c<c2. (49)

Then one can obtain the explicit approximate secular equation of the wave by introducing(49)into the effective boundary conditions(10)and(11) When the half-space is orthotropic (or of less symmetry), the situation is quite different: the analytical expression of these two roots cannot be determined, deriving the explicit approximate secular equation is therefore more difficult That is why the previously obtained secular equations are limited only to the isotropic case

5 An approximate formula of second-order of the velocity

In this section we provide an explicit approximate formula of second-order for the squared dimensionless Rayleigh wave

velocity x(ε)that is of the form:

x(ε) =x0+x′(0) ε +x

′′(0)

where x0 :=x(0)is the squared dimensionless velocity of Rayleigh waves propagating in an orthotropic elastic half-space that is given by (see [16]):

x0= √s1s2s3



( √s1/3)(s2s3+2) + 3

R+

√

D+ 3



R−

√

D



(51)

where s1=e2/e1, s2=1−e23/(e1e2), s3=e1, R and D are given by:

R= − 1

54h(s1,s2,s3)

D= − 1

108[2

√

s1(1−s2)h(s1,s2,s3) +27s1(1−s2)2+s1(1−s2s3)2+4] (52)

in which

h(s1,s2,s3) = √s1[2s1(1−s2s3)3+9(3s2−s2s3−2)] (53) and the roots in(51)taking their principal values

From(29)it follows that:

x′(0) = −D1

D 0x





x=x0

, x′′(0) = −D 0xx D21−2D 1x D1D 0x+D2D20x

D30x g





x=x0

(54) whereϕx:= ∂ϕ/∂x,ϕxx:= ∂2ϕ/∂x2, D1, D2are given by(30)and:

D 0x= (e1−2x) +4e2x2+ [2δ −3(1+e1)e2]x+2e1e2− δ(1+e1)

2√e2√1−x√e1−x , δ =e23−e1e2 (55)

D 0xx= −2+8e2x3−12e2(1+e1)x2+3e2(1+6e1+e2)x− δ(1−e1)2−4e1e2(1+e1)

D 1x = rµ[ (e1−x)rv2x+e2(¯e2¯e23− ¯e1+rv2x)b1b2] [2x−1−e1− (1+e2)b1b2]

2e2b1b2(b1+b2) +rµ



rv2(e1−2x) +4rv2x2+ [2δ − ¯ 3rv2(1+e1)]x+2rv2e1− (1+e1)¯δ

2b1b2

 (b1+b2) (57) hereδ = ¯ ¯ e2e¯23− ¯e1, b1b2=

√

P,(b1+b2) = 

S+2

√

P, P, S are given by(30) It is clear from(30)and(51)–(57)that the

squared dimensionless Rayleigh wave velocity x given by(50)is an explicit function in terms of 9 dimensionless parameters,

namely: e k,¯e k(k=1,2,3), rµ, rvandε

When the layer and the half-space are both isotropic the formulas(55)–(57)are simplified to D 0x= ¯D 0x/γ, D 0xx= ¯D 0xx/γ,

D 1x= ¯D 1x/γwhere:

¯

D 0x=1−2γx+4γx2+ (8γ2−11γ −3)x+2(3−2γ2)

¯

D 0xx= −2γ +8γ2x3−12γ (1+ γ )x2+3(γ2+6γ +1)x+4γ (−4+3γ − γ2)

¯

D 1x = −rµ[ (1− γx)rv2x+ (4γ − ¯ 4+rv2x)b1b2] γb2+b1

2b1b2 +rµ



rv2(1−2γx) + 4γrv2x2+ [8γ ( ¯γ −1) −3rv2(1+ γ )]x+2rv2−4(1+ γ )( ¯γ −1)

2b b

 (b1+b2) (60)

here b1= √1− γx, b2=

√

1−x Therefore, for the isotropic case, the second-order approximation of x(ε)is expressed by

(50)in which:

x′(0) = −D¯1

¯

D 0x





x=x0

, x′′(0) = −D¯0xx D¯2−2D¯1x D¯1D¯0x+D∗

2D¯2

0x

¯

D30x g





x=x0

¯

D1, D∗2are determined by(45),(48),D¯0x,D¯0xx,D¯1xare calculated by(58)–(60)and x0is the squared dimensionless velocity of Rayleigh waves propagating in an isotropic elastic half-space that is given by (see [16]):

x0=4(1− γ )



2−4

3γ + 3

R+

√

D+ 3



R−

√

D

− 1

(62) where:

R=2(27−90γ +99γ2−32γ3)/27, D=4(1− γ )2(11−62γ +107γ2−64γ3)/27 (63) the roots in(62)taking their principal values Note that x0can be given by another exact formula obtained by Malischewsky [17], or by the approximate expressions with high accuracy obtained recently by Vinh & Malischewsky [18,19] It should

be noted that one can obtain the expressions(58)–(60)by directly taking the differentiation with respect to x of D¯0andD¯1

given by(45)

6 Conclusions

In this paper the propagation of Rayleigh waves in an orthotropic elastic half-space coated by a thin orthotropic elastic layer is investigated First, an effective boundary conditions of third-order are derived that replaces the entire effect of the thin layer on the half-space Then, by using it the authors derive an approximate secular equation of third-order of Rayleigh waves From this equation two different third-order approximate secular equations are obtained for the case when the half-space and the layer are both isotropic It is shown that one of which recovers the secular equation of second-order derived

by Bovik [5] Based on the obtained approximate secular equation, an explicit second-order approximate formula for the Rayleigh wave velocity has been created The obtained secular equation and formula for the velocity may be employed as theoretical bases for extracting the mechanical properties of the thin films from experimental data

Acknowledgment

The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED)

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