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On formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two elastic half-spaces Faculty of Mathematics, Mechanics and Informatics, Hanoi Universi

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On formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two elastic half-spaces

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

Article history:

Received 30 December 2010

Received in revised form 1 May 2011

Accepted 5 May 2011

Available online 13 May 2011

Formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces are derived using the complex function method The derivation also shows that if a Stoneley wave exists, then it is unique By using the obtained formulas, we can easily reproduce the numerical results previously obtained by Murty [G S Murty, Phys Earth Planet Interiors 11 (1975), 65–79.] by directly solving the secular equation

Keywords:

Stoneley waves

Stoneley wave velocity

Loosely bonded interface

Holomorphic function

1 Introduction

Rayleigh surface waves[1]and Stoneley interfacial waves[2]in isotropic elastic solids discovered many years ago, in 1885 by Rayleigh and 1924 by Stoneley, respectively, have been studied extensively and exploited in a wide range of applications in seismology, acoustics, geophysics, telecommunications industry and materials science, for example

The velocities of Rayleigh waves and Stoneley waves are of great interest to researchers in variousﬁelds of science The formulas for them provide powerful tools for solving the direct (forward) problems: studying effects of material parameters on the wave velocity, and especially for the inverse problems: determining material parameters from the measured values of the wave speed The formulas for the velocity of Rayleigh waves and Stoneley waves are thus of theoretical as well as practical interest For Rayleigh waves, some formulas for the velocity have been obtained recently, see for instance[3–13], while no formulas have been derived for Stoneley waves so far, to the best of the authors' knowledge

The main aim of this paper is to establish formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces by using the complex function method It is shown from the derivation of these formulas that, if a Stoneley wave exists, then it is unique, as proved by Barnett et al.[14]by another technique

Using the obtained formulas, it is easy to recover the numerical results obtained previously by Murty[15]by directly solving the secular equation

2 Secular equation

In this section we present brieﬂy the derivation of the secular equation of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces For details, the reader is referred to the papers[15,16]by Murty

Let us consider two isotropic elastic solidsΩ and Ω⁎ occupying the half-spaces x2≥0 and x2≤0, respectively Suppose that these two elastic half-spaces are not in welded contact with each other at the plane x2= 0 (see[15,16]) In particular, the normal

Wave Motion 48 (2011) 647–657

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

E-mail address: pcvinh@vnu.edu.vn (P.C Vinh).

Contents lists available atScienceDirect

Wave Motion

j o u r n a l h o m e p a g e : w w w e l s ev i e r c o m / l o c a t e / wave m o t i

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component of the particle displacement vector and the normal component of the stress tensor are continuous, while the shearing stress vanishes across the interface x2= 0 The plane x2= 0 is then called the loosely bonded interface (see[15]) of the half-spaces

Ω and Ω⁎ Note that same quantities related to Ω and Ω⁎ have the same symbol but are systematically distinguished by an asterisk

if pertaining toΩ⁎

We are interested in planar motion in the (x1x2)-plane with the displacement components u1, u2, u3such that:

ui= uiðx1; x2; tÞ; i = 1; 2; u3≡ 0; ð1Þ where t is the time Then the equations of motion are of the form[17]:

c2Lu1;11+ c2Tu1;22+ c2L−c2

T

u2;12= ̈u1;

c2L−c2

T

u1;12+ c2Lu2;22+ c2Tu2;11= ̈u2; ð2Þ

in which a superposed dot denotes the differentiation with respect to t, commas indicate the differentiation with respect to spatial variables xi, cL= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

λ + 2μ

ð Þ = ρ

p

and cT= ffiffiffiffiffiffiffiffiffiffi

μ = ρ

p are speed of the longitudinal wave and the transverse wave, respectively,

ρ is the mass density, λ and μ are usual the Lame constants

The stress components on the planes x2= const are related to the displacement gradients by:

σ12=μ u 1 ;2+ u2;1

; σ22=λ u 1 ;1+ u2;2

In addition to Eq.(2), the decay condition is required, i.e.:

ui= 0 i = 1ð ; 2Þ; σij= 0 ið; j = 1; 2Þ at x2= +∞: ð4Þ ForΩ⁎ we have equations similar to Eqs.(1)–(3)in which the quantities are asterisked, and the decay condition(4)are replaced by:

u⁎i = 0 i = 1ð ; 2Þ; σ⁎ij = 0 ið; j = 1; 2Þ at x2=−∞: ð5Þ Since the half-spaces are not in welded contact in the meaning as mentioned above, we have:

u2= u⁎2; σ22=σ⁎22; σ12=σ⁎12= 0 at x2= 0: ð6Þ Now we consider the propagation of a wave, travelling with velocity c and wave number k (N0) in the x1-direction, being mostly conﬁned to the neighbourhood of the interface x2= 0 Then the displacement components u1, u2(of Ω) are found in the form:

u1= Ae−kbx2eik xð 1 −ct Þ

; u2= Be−kbx2eik xð 1 −ct Þ

where A, B, b are constants, and:

in order to ensure the decay condition(4) Substituting Eq.(7)into Eq.(2)leads to a system of two homogeneous linear equations for A, B, and vanishing its determinant yields the equation determining b, namely:

c2Lc2Tb4− c2

Lc2T−c2

+ c2Tc2L−c2

b2+ c 2L−c2

c2T−c2

Now we show that if Eq.(8)holds, then we have (see also[18]):

Indeed, as the coefﬁcients of the quadratic Eq.(9)for X = b2are all real, and its determinantΔ=[cL2(cT−c2)−cT(cL2−c2)]2≥0,

Eq.(9)has always two real roots denoted by X1, X2, and they both must positive, otherwise Eq.(8)doesn't hold This leads to

X1X2N0, i.e.::

c2L−c2

c2T−c2

As X1+ X2= [cL2(cT−c2) + cT(cL2

−c2)]/cL2cT, if both (cL2

−c2) and (cT−c2) are all negative, then X1+ X2b0 But this contradicts the fact that X, X are both positive Therefore both (c2−c2) and (c −c2) are all positive, i.e we have Eq.(10)

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With the condition(10), Eq.(9)has two roots satisfying Eq.(8), namely:

b1=

ffiffiffiffiffiffiffiffiffiffiffiffiffi

1−c2

c2

L

s

; b2=

1−c2

c2 T

s

thus we have:

u1= A1e−kb1 x2

+ A2e−kb2 x2

eik xð 1 −ct Þ

;

u2= −b1

i A1e−kb1 x2

+ i

b2A2e−kb2 x2

eik xð 1 −ct Þ

where A1, A2are constants to be determined

Similarly, the displacement components u1⁎, u2⁎ are expressed by:

u⁎1= A⁎ 1ekb⁎1 x 2+ A⁎2ekb⁎2 x 2

eik xð 1 −ct Þ

;

u⁎2= b⁎1

i A⁎1ekb⁎1 x2

− i b⁎2A⁎2e

kb ⁎

2 x2

!

eik xð 1 −ct Þ

where Rebk⁎N0(k=1, 2), A1⁎, A2⁎ are constants to be determined:

b⁎1=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1−c2

c⁎L2

v

u

; b⁎2=

1− c2 c⁎T2

v u

and if Reb⁎N0, then:

From Eqs.(10) and (16)we conclude that:

Proposition 1 If a Stoneley wave exists (this implies RebN0, Reb⁎N0), then its velocity is subject to:

0b c b min cT; c⁎T

n o

Making use of Eqs.(3), (13), and(14)in the boundary condition(6)yields a system of four homogeneous linear equations for

A1, A2, A1⁎, A2⁎, and making its determinant zero we have:

c4T

4

1−c

2

c2

T

s ffiffiffiffiffiffiffiffiffiffiffiffiffi

1−c

2

c2 L

s

− 2−c

2

c2 T

!2

1−c

2

c2 L

ρc⁎

4 T

4

1− c

2

c⁎2

L

v

u ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1−c

2

c⁎2

T

v u

− 2−c

2

c⁎2

T

0

@

1 A

2

2 4

3 5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1− c

2

c⁎2

L

v

Eq.(18)is desired secular equation that coincides with the one derived by Murty[15,16](see also, for example,[17,19]) From the above argument it is clear that Eqs.(17) and (18)are the necessary condition for the existence of a Stoneley wave It is not difﬁcult to show that they are also the sufﬁcient condition Thus we have:

Proposition 2 For a Stoneley wave to exist, it is necessary and sufﬁcient that Eqs.(17) and (18)are both satisﬁed

3 Formulas for the velocity of Stoneley waves

Without loss of generality we can suppose that cT≤cT⁎ For the sake of simplicity we introduce the following notations:

x = c

2

c2

T

; B = c

2 T

c⁎2; D = ρ

ρ⁎; F =

c2T c⁎2; E = c

2 T

c2 L

649 P.C Vinh, P.T.H Giang / Wave Motion 48 (2011) 647–657

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In terms of these notations Eq.(18)is of the form:

2−x

ð Þ2

1−Fx

ð Þ1 = 2−4 1−xð Þ1 = 2ð1−ExÞ1 = 2ð1−FxÞ1 = 2+

1

DB2 ð2−BxÞ2

1−Ex

ð Þ1 = 2−4 1−Bxð Þ1 = 2ð1−FxÞ1 = 2ð1−ExÞ1 = 2

Due to Eq.(17)we have:

Now, in the complex plane C, we consider the equation:

2−z

ð Þ2 ffiffiffi

F

p ffiffiffiffiffiffiffiffiffiffiffi

z−1F

r

+ 4 ffiffiffiffiffiffi EF

p ffiffiffiffiffiffiffiffiffiffi

z−1

z−1E

r ffiffiffiffiffiffiffiffiffiffiffi

z−1F

r +

1

DB2 ð2−BzÞ2 ffiffiffi

E

z−1 E

r + 4 ffiffiffiffiffiffiffiffi BEF

z−1 B

z−1 F

z−1 E

r

where ffiffiffiffiffiffiffiffiffiffi

z−1

p

, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z−1 = B

p

z−1 = E

p

z−1 = F

p

are chosen as the principal branches of the corresponding square roots For the real values of z∈(01), Eq.(22)is equivalent to Eq.(20) Note that Eq.(22)always has a real root, namely z = 0

Multiplying two sides of Eq.(22)by ffiffiffiffiffiffiffiffiffiffi

z−1

p (the key technique) gives:

f zð Þ =: 2−zð Þ2 ffiffiffi

F

z−1F

r ffiffiffiffiffiffiffiffiffiffi

z−1

p + 4 ffiffiffiffiffiffi EF

p

z−1

ð Þ

ffiffiffiffiffiffiffiffiffiffiffi

z−1E

z−1F

r

ð23Þ

+ 1

DB2ð2−zBÞ2 ffiffiffi

E

z−1E

z−1

p + 4

ffiffiffiffiffiffiffiffi BEF p

DB2

z−1B

z−1F

z−1E

z−1

p

= 0: From the assumption cT≤cT⁎(→b≤1) and the fact cT⁎bcL⁎(→FbB), it follows that there are three basic possibilities:

Case 1 1NBNENFN0

Case 2 1NBNFNEN0

Case 3 1NENBNFN0

By replacing at least one inequality in a basic case by possible equalities we have a special case First, we consider the basic cases

3.1 Case 1: 1NBNENFN0

We will prove the following theorem:

Theorem 1 Let 1NBNENF If a Stoneley wave exists, then it is unique, and its squared dimensionless velocity xs= c2/cTis deﬁned by:

xs=−A2

where A2, A3given by Eq.(55)and:

I ̂0= 1

π ∫

1 = B

1

θ1ð Þdt + ∫t

1 = E

1= B

θ2ð Þdt + ∫t

1 = F

1= E

θ3ð Þdtt

0

@

1

in whichθk(t) are given by Eqs.(45)–(48)

Proof Denote L = L1∪L2∪L3with L1= [1, 1/B], L2= [1/B, 1/E], L3= [1/E, 1/F], S = {z∈C, z∉L}, N(z0) = {z∈S:0b|z−z0|bε}, ε

is a sufﬁcient small positive number, z0is some point of the complex plane C If a functionϕ(z) is holomorphic in Ω⊂C we write ϕ(z)∈H(Ω) From Eq.(23)it is not difﬁcult to show that the function f(z) has the properties:

( f1) f(z)∈H(S)

( f2) f(z) is bounded in N(1/F) and N(1)

( f) f(z) = O(z3) as |z|→∞

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( f4) f(z) is continuous on L from the left and from the right (see[20]) with the boundary values f+(t) (the right boundary value

of f(z)), f−(t) (the left boundary value of f(z)) deﬁned as follows:

fFð Þ =t

f1Fð Þ; t ∈ Lt 1

8

>

where:

f1þð Þ = i 2−tt ð Þ2 ffiffiffi

F

p ffiffiffiffiffiffiffiffiffiffiffi 1

F−t

t−1

p +

ffiffiffi E p

DB2ð2−BtÞ2

ffiffiffiffiffiffiffiffiffiffiffi 1

E−t

t−1

p

−4

DB2

B−t

r ffiffiffiffiffiffiffiffiffiffiffi 1

F−t

E−t

t−1 p

−4pffiffiffiffiffiffiEF

t−1

ð Þ

E−t

F−t

r

;

ð27Þ

F

F−t

t−1

p +

ffiffiffi E p

DB2ð2−BtÞ2

E−t

t−1 p

−4

DB2

t−1B

F−t

E−t

t−1

p

−4pffiffiffiffiffiffiEF

t−1

ð Þ

E−t

F−t

r

;

ð28Þ

F

F−t

t−1

p + 4

DB2

t−1B

F−t

t−1E

t−1

p + 4 ffiffiffiffiffiffi EF

p

t−1

ð Þ

t−1E

F−t

r

+

ffiffiffi E p

DB2ð2−BtÞ2

t−1E

t−1

p

;

ð29Þ

fk−ð Þ = ft þ

the bar indicates the complex conjugate

Note that fk+(t) (fk −(t)) is the right (left) boundary value of f(z) on Lkand i = ffiffiffiffiffiffiffiffi

−1

p Now we introduce the function g(t) (t∈L)

as follows:

g tð Þ =

f1þð Þt

f1−ð Þ ;t t∈ L1

f2þð Þt

f3þð Þt

8

>

<

>

:

ð31Þ

From Eqs.(26) and (31)it is clear that

Consider the functionΓ(z) deﬁned by:

Γ zð Þ = 1

2πi∫L

logg tð Þ

It is not difﬁcult to verify that:

(γ1) Γ(z)∈H(S)

(γ2) Γ(∞)=0

(γ3) Γ(z)=−(1/2)log(z−1)+Ω0(z), z∈N(1), Γ(z)=Ω1(z), z∈N(1/F) where Ω0(z) (Ω1(z)) bounded in N(1) (N(1/F)) and takes a deﬁned value at z=1(z=1/F)

It is noted that (γ3) comes from the fact (see[20]):

Introduce the functionΦ(z) given by:

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It is implied from (γ1)−(γ3) that:

(ϕ1) Φ(z)∈H(S)

(ϕ2) Φ(z)≠0∀z∈S

(ϕ3) Φ(z)=O(1) as |z|→∞

(ϕ4) Φ zð Þ = z−1ð Þ− 1

expΩ0ð Þ for z ∈ N 1z ð Þ; Φ zð Þ = expΩ1ð Þ; z ∈ N 1 = Fz ð Þ:

From the Plemelj formula[20], the functionΦ(z) is seen directly to satisfy the boundary condition:

We now consider the function Y(z) deﬁned by:

From ( f1)–(f3), Eq.(32), (ϕ1)−(ϕ4) and Eq.(37), it follows that:

(y1) Y(z)∈H(S)

(y2) Y(z) = O(z3) as | z|→∞

(y3) Y(z) is bounded in N(1) and N(1/F)

(y4) Y+(t) = Y−(t), t∈L

Properties (y1) and (y4) of the function Y(z) show that Y(z) is holomorphic in entire complex plane C, with the possible exception of points: z = 1 and z = 1/F By (y3) these points are removable singularity points and it may be assumed that the function Y(z) is holomorphic in the entire complex plane C (see[21]) Thus, by the generalised Liouville theorem[21]and taking account into (y2) we have:

where P(z) is a third-order polynomial

From Eqs.(37) and (38)we have:

SinceΦ(z)≠0∀z∈S (by (ϕ2)), andΦ(z)→∞ as z→1, Φ(1/F)≠0 (by (ϕ4)), from Eq.(39)we deduce:

f zð Þ = 0 X P zð Þ = 0 in S ∪ 1f g ∪ 1 = Ff g: ð40Þ

In view of Eq.(40), instead ofﬁnding zeros of f(z) we now look for three zeros of the third-order polynomial P(z) In order to do that,ﬁrst we have to determine P(z)

From Eqs.(35)and(39)we have:

From Eqs.(27)–(31)it follows:

log g tð Þ = iϕ tð ÞÞ; ϕ tð Þ =: Arg g tð Þ; ð42Þ where:

ϕ tð Þ = ϕ1ð Þ;t t∈ L1

ϕ2ð Þ;t t∈ L2

ϕ3ð Þ;t t∈ L3

:

8

<

It is not difﬁcult to verify that:

ϕ1ð Þ = 2π + 2θt 1ð Þ; ϕt 2ð Þ = 2π + 2θt 2ð Þ; ϕt 3ð Þ = 2θt 3ð Þ;t ð44Þ where:

and:

φ1ð Þ = − 2−tt ð Þ2 ffiffiffi

F

F−t

r +

ffiffiffi E p

DB2ð2−BtÞ2

E−t

r

−4

DB2

B−t

F−t

E−t

r

= 4 ffiffiffiffiffiffi EF

t−1

E−t

F−t

r

; ð46Þ

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φ2ð Þ = −t ð2−tÞ

2 ffiffiffi F

1

q +

ffiffiffi E p

DB2ð2−BtÞ2

E−t r

4

DB2

t−1B

F−t

E−t

r + 4 ffiffiffiffiffiffi EF

t−1

E−t

F−t

φ3ð Þ =t ð2−tÞ2 ffiffiffi

F

F−t

r + 4

DB2

t−1 B

F−t

t−1 E

t−1

t−1 E

F−t

r

= ffiffiffi E p

DB2ð2−BtÞ2

t−1 E

r

: ð48Þ From Eqs.(33) and (42)it follows (see also[4]):

−Γ zð Þ = ∑∞

n = 0

In

in which:

In= 1

2π ∫

1 = F

1

On use of Eq.(49)we can express e− Γ(z)as follows:

e−Γ zð Þ= 1 + a1

z +

a2

z2 +a3

z3 + O z−4

where a1, a2, a3are constants to be determined Employing the identity:

e−Γ zð Þ

and substituting Eqs.(49)and(51)into Eq.(52)yields:

a1= I0; a2=I

2

2 + I1; a3=I

3

By expanding ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1−1 = z

p

, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1−1 = Bzð Þ

p

1−1 = Ezð Þ

p

1−1 = Fzð Þ

p

into Laurent series at inﬁnity, it is not difﬁcult to verify that:

f zð Þ = A3z3+ A2z2+ A1z + A0+ O z−1

where:

A3= ffiffiffi

F

p

+

ffiffiffi E p D

A2=−9

ffiffiffi F p

2 − 1

2 ffiffiffi F

p + 4 ffiffiffiffiffiffi

EF

p

−

ffiffiffi E p D

1

2+

1 2E +

4 B

+ 4

DB2 :

ð55Þ

Substituting Eqs.(51) and (54)into Eq.(41)yields:

P zð Þ = A3z3+ z2ðA2+ A3a1Þ + z Að 1+ A3a2+ A2a1Þ + A3a3+ A2a2+ A1a1+ A0: ð56Þ

It is clear from Eq.(23)that f(0) = 0 and f(1) = 0 hence by Eq.(40):

On the use of Eqs.(56)and(57)and taking into account theﬁrst of Eq.(53), it is easy to see that the third root of the equation P(z) = 0, denoted by xs, is:

xs=−A2

It follows from Eqs.(44) and (50)that:

I0= 1

E−1

thus x is given by Eq.(24)

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Now we suppose that there exist two different Stoneley waves with corresponding velocities x(1), x(2)(x(1)≠x(2)) Then x(1), x(2)are two different roots of Eq f(z) = 0, and 0bx(1), x(2)b1, according toProposition 1 From Eq.(40)it follows P(x(1)) = P(x(2)) = 0 This fact and Eq (57) imply that the third-order polynomial P(z) has four different roots But this is impossible Thus, if a Stoneley wave exists, it must be unique

Suppose that there exists a (unique) Stoneley wave Then by above arguments, its squared dimensionless velocity x is a root of the third-order polynomial P(z), and x≠0, x≠1 By Eq.(57), x must be xsgiven by Eq.(24) The proof ofTheorem 1is completed 3.2 Case 2: 1NBNFNEN0

Following the same procedure carried out for theCase 1, and noting that the relations (44) are still valid for this case, one can see that:

Theorem 2 Let 1NBNFNE If a Stoneley wave exists, then it is unique, and its squared dimensionless velocity xs= c2/cTis deﬁned by:

xs=−A2

where A2, A3deﬁned by Eq.(55)and:

I ̂0= 1

π ∫

1 = B

1

θ1ð Þdt + ∫t

1 = F

1 = B

θ2ð Þdt + ∫t

1 = E

1 = F

θ3ð Þdtt

0

@

1

θk(t) are given by Eq.(45)in which:

F

F−t

r +

ffiffiffi E p

DB2ð2−BtÞ2

E−t

r

−4

DB2

B−t

F−t

E−t

r

= −4pffiffiffiffiffiffiEF ffiffiffiffiffiffiffiffiffiffi

t−1

E−t

F−t

r

; ð62Þ

φ2ð Þ =t

2−t

ð Þ2 ffiffiffi F

1

q +

ffiffiffi E p

DB2ð2−BtÞ2

E−t r

−4

DB2

t−1B

F−t

E−t

r

−4pffiffiffiffiffiffiEF ffiffiffiffiffiffiffiffiffiffi

t−1

E−t

F−t

φ3ð Þ =t

ffiffiffi E p

DB2ð2−BtÞ2

E−t

t−1

E−t

t−1F

r + 4

DB2

t−1B

t−1F

E−t

r

= ð2−tÞ2 ffiffiffi

F

t−1F

r

: ð64Þ

3.3 Case 3: 1NENBNFN0

Analogously, we have:

Theorem 3 Suppose that 1NENBNF If a Stoneley wave exists, then it is unique, and its squared dimensionless velocity xs= c2/cTis

deﬁned by:

xs=−A2

A3−12 1B+ 1

E

where A2, A3given by Eq.(55)and:

I ̂0= 1

π ∫

1 = E

1

θ1ð Þdt− ∫t

1 = B

1 = E

θ2ð Þdt + ∫t

1 = F

1 = B

θ3ð Þdtt

0

@

1

θk(t) are determined by Eq.(45)in which:

F

F−t

r +

ffiffiffi E p

DB2ð2−BtÞ2

E−t

r

−4

DB2

B−t

F−t

E−t

r

= −4pffiffiffiffiffiffiEF ffiffiffiffiffiffiffiffiffiffi

t−1

E−t

F−t

r

; ð67Þ

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φ2ð Þ =t

ffiffiffi

E

p

DB2ð2−BtÞ2

t−1E

r

−4

DB2

B−t

F−t

t−1E r

2−t

ð Þ2 ffiffiffi

F

1

q + 4 ffiffiffiffiffiffi EF

t−1

t−1 E

q ffiffiffiffiffiffiffiffiffiffi

1

F

F−t

t−1

t−1E

F−t

r + 4

DB2

t−1B

F−t

t−1E

r

= ffiffiffi E p

DB2ð2−BtÞ2

t−1E

r

: ð69Þ Note that for this case, instead of Eq.(44)we have the following:

ϕ1ð Þ = 2π + 2θt 1ð Þ; ϕt 2ð Þ = π−2θt 2ð Þ; ϕt 3ð Þ = 2θt 3ð Þ:t ð70Þ

4 Special cases

As BNF and 0bE≤3/4b1, there are ﬁve special cases as follows: Case 1.1: 1=BNENFN0, Case 1.2: 1NB=ENFN0, Case 1.3:

1NBNE=FN0, Case 1.4: 1=BNE=FN0, Case 2.1: 1=BNFNEN0 The ﬁrst four cases originate in the basicCase 1, the last comes from theCase 2 The results of all these cases are deduced directly from the corresponding basic cases

In particular, for the Case 1.1, the squared dimensionless velocity xs= c2/cTof the Stoneley wave is deﬁned by Eq.(24)in which:

A2=−

ffiffiffi

F

p

2 9 +

1 F

+ ffiffiffiffiffiffi EF

p

1 + 1 D

−

ffiffiffi E p 2D 9 +

1 E

;

A3= ffiffiffi

F

p

+

ffiffiffi E p

D ; I ̂0=1

π ∫

1= E 1

θ2ð Þdt + ∫t

1= F

1 = E

θ3ð Þdtt

0

@

1 A;

ð71Þ

θ2(t),θ3(t) are deﬁned by Eqs.(45), (47) and (48)

For the Case 1.2, xsis deﬁned by Eq.(24)where:

A2=−

ffiffiffi

F

p

2 9 +

1 F

+ 4 ffiffiffiffiffiffi EF

p

−

9 E

+ 4

ffiffiffi F p

DE;

A3= ffiffiffi

F

p

+

ffiffiffi E p

D ; I ̂0= 1

π ∫

1 = E

1

θ1ð Þdt + ∫t

1 = F

1 = E

θ3ð Þdtt

0

@

1 A;

ð72Þ

θ1(t),θ3(t) are deﬁned by Eqs.(45), (46) and (48)

For the Case 1.3, after some manipulations and taking into account Eq.(28), we have:

xs=−A1

A2 +

1

2 1−1B

where:

A1= 4 B

2

D ffiffiffi

E p

−1

+ ffiffiffiffiffiffi BE p

−B

B2D ; A2= 1 + 1

I ̂0= 1

π ∫

1= B

1

atanφ1ð Þdt + ∫t

1= E

1 = B

atanφ20ð Þdtt

0

@

1

in whichφ1(t) is given by Eq.(46), and:

φ20=

4 ffiffiffi

E

t−1

1

E−t

r + 4

ffiffiffiffiffiffi BE p

DB2

t−1B

E−1 r

2−t

ð Þ2+ 1

Putting B = 1 in Eqs.(73)–(76)provides the result for the Case 1.4 In particular, for this case xsis given by:

xs= 4 1 −pffiffiffiE

−π1 ∫

1 = E

Trang 10

φ0ð Þ =t 4

ffiffiffiffiffiffiffiffiffiffiffiffi

1−Et

t−1 p

2−t

Since xsgiven by Eq.(77)is the velocity of a Rayleigh wave propagating in an isotropic elastic half-space with the bulk wave velocities cLand cT(see[4,13]), it follows 0bxsb1 This means that, according toProposition 2, a Stoneley wave is always possible for the case when two isotropic elastic half-spaces having the same bulk wave velocities This fact has also been proved by Barnett

et al.[14]

Analogously, the result for the Case 2.1 is obtained directly from the one of the Case 2 In particular, xs= c2/cTis deﬁned by

Eq.(60)in which:

A2=−

ffiffiffi

F

p

2 9 +

1 F

+ ffiffiffiffiffiffi EF

p

1 + 1 D

−

1 E

;

A3= ffiffiffi

F

p

+

ffiffiffi E p

D ; I ̂0= 1

π ∫

1 = F

1

θ2ð Þdt + ∫t

1 = E

1= F

θ3ð Þdtt

0

@

1 A;

ð79Þ

whereθ2(t),θ3(t) are deﬁned by Eqs.(45), (63) and (64)

Remark 1

i) FromTheorems 1, 2, 3, it is shown that there is at most one Stoneley wave can travel along the interface of two isotropic elastic half-spaces in sliding contact This fact was also proved by Barnett et al.[14]by another method

ii) FromProposition 2, the necessary and sufﬁcient condition for the existence of a Stoneley wave is 0bxsb1, where xsis given

by Eq.(24) or (60) or (65) That means, for given material parameters, by simply calculating the quantity xswe can know whether a Stoneley wave exists or not, and its velocity in the case of existence This will make the obtained formulas useful

in practical applications

iii) When BN1, i.e cTNcT⁎, the formulas(24), (60) and (65)are still applicable, but in which B, D, E, F, xsmust be replaced by B⁎ = 1/B, D⁎ = 1/D, E⁎ = F/B, F⁎ = E/B, xs⁎=c2/cT⁎2, respectively

iv) By using the obtained formulas we easily obtain the numerical results obtained by Murty[15]by solving directly the secular

Eq.(20) For examples:

- Taking B = 1/2 ; D = 3.4 (→R=BD=1.7);E=1/3;F=1/6, and using the formula(24)give c2/cT= 0.9738 that coincides with the one corresponding to R = 1.7, D = 3.4 in Table 1 of paper[15]

- Taking B = 0.3 ; D = 3.0 (→R=BD=0.9);E=1/3;F=0.1, and using the formula(65)give c2/cT= 0.9996 that coincides with the one corresponding to R = 0.9, D = 3.0 in Table 1 of paper[15]

- For B = 1/2 ; D = 0.2 (→R=BD=0.1);E=1/3;F=1/6, the formula(24)gives xs= 1 The Stoneley wave therefore does not exist for this case This fact was also shown by Murty[15]

- Taking B = 0.7 ; D = 3 ; F = 0.45 ; E = 0.25, and using the formula(60)provide xs= 0.8735 that coincides with the one obtained directly from the secular Eq.(20)

- Consider the case of Poisson solids (i.e whenλ=λ⁎, μ=μ⁎, see[15]) with D = 1 ; R = 1.5 (E = 1/3, F = 1/2) As B = R/

D = 1.5N1 we have to employ the above obtained formulas in which B, D, E, F, xsare replaced by B⁎ = 1/B = 1/1.5, D⁎ = 1/

D = 1, E⁎ = F/B = 1/3, F⁎ = E/B = 1/4.5, x⁎=cs 2/cT⁎2, respectively Applying the formula(24)with asterisked parameters yields c/cT⁎=0.9893 that coincides with the result shown in[15]

5 Conclusions

In this paper, by using the complex function method we have found formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces From the derivation of these formulas, it is shown that if a Stoneley wave exists, then it is unique Using the obtained formulas it is easy to know, for given material parameters, whether a Stoneley wave exists or not, and its velocity when it exists These formulas will therefore be useful in practical applications The method presented in this paper may be applicable to the case of perfectly bonded interfaces, as well as to Stoneley waves propagating along a material interface between two half spaces (see, for instance,[22,23])

Acknowledgments

The work was supported by the Vietnam National Foundation For Science and Technology Development (NAFOSTED) under Grant No 107.02-2010.07

References

[1] L Rayleigh, On waves propagating along the plane surface of an elastic solid, Proc R Soc Lond A17 (1885) 4–11.

[2] R Stoneley, Elastic waves at the surface of separation of two solids, Proc Roy Soc Lond A106 (1924) 416–428.

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