DSpace at VNU: Formulas for the speed and slowness of Stoneley waves in bonded isotropic elastic half-spaces with the same bulk wave velocities

Formulas for the speed and slowness of Stoneley waves in bondedisotropic elastic half-spaces with the same bulk wave velocities a Faculty of Mathematics, Mechanics and Informatics, Hanoi

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Formulas for the speed and slowness of Stoneley waves in bonded

isotropic elastic half-spaces with the same bulk wave velocities

a

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

b

Institute for Geosciences, Friedrich-Schiller University Jena, Burgweg 11, 07749 Jena, Germany

a r t i c l e i n f o

Article history:

Received 5 August 2011

Received in revised form 20 March 2012

Accepted 12 May 2012

Available online 21 June 2012

Keywords:

Stoneley waves

The wave velocity

The wave slowness

Two bonded isotropic elastic half-spaces

Holomorphic function

a b s t r a c t

This paper is concerned with the propagation of Stoneley waves in two bonded isotropic elastic half-spaces with the same bulk wave velocities Our main purpose is to ﬁnd formu-las for the wave velocity and the wave slowness By applying the complex function method, the exact formulas for the wave velocity and the wave slowness have been derived The derivation of these formulas also shows that there always exists a unique Stoneley wave for the case under consideration

1 Introduction

Interfacial waves traveling along the welded plane boundary of two different isotropic elastic half-spaces were ﬁrst inves-tigated byStoneley (1924) He derived the secular equation of the wave, and showed by means of examples that such inter-facial waves do not always exist Subsequent studies bySezawa and Kanai (1939)andScholte (1942, 1947)focused on the range of existence of Stoneley waves.Scholte (1947)found the equations expressing the boundaries of the existence domain and they are in complete agreement with the corresponding curves numerically obtained bySezawa and Kanai (1939)for the case of Poisson solids (for which the corresponding Lame constants of the half-spaces are the same) Their studies showed that the restriction on material constants that permit the existence of Stoneley waves are rather severe However,Sezawa and Kanai (1939)and Scholte did not prove the uniqueness of Stoneley waves This question was settled byBarnett, Lothe, Gavazza, and Musgrave (1985)for general anisotropic half-spaces with bonded interface The propagation of Stoneley waves

in anisotropic media was also studied byStroh (1962)andLim and Musgrave (1970) Much of the early attentions to Stone-ley waves was directed toward geophysical applications Latter studies have indicated that interfacial waves may prove to be useful probes for the non-destructive evaluations (seeLee & Corbly, 1977, Rokhlin, Hefet, & Rosen, 1980)

The propagation of Stoneley waves in two isotropic elastic half-spaces with the loosely bonded interface was studied by

Murty (1975b, 1975a) The author has derived the secular equation of the wave and obtained a lot of numerical values of the Stoneley wave velocity by directly solving that equation for the case of Poisson solids The existence and uniqueness of Stoneley waves in two half-spaces in sliding contact was investigated byBarnett, Gavazza, and Lothe (1988) The authors showed that for the isotropic elastic half-spaces, if a Stoneley exists, then it is unique, while for the anisotropic half-spaces, the possibility of a new slip-wave mode, called the second slip-wave mode, arises

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

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

Contents lists available atSciVerse ScienceDirect

International Journal of Engineering Science

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

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For the Stoneley wave, its velocity is of great interest to researchers in various ﬁelds of science The formulas for the Stoneley wave velocity are 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 Recently, the exact formulas for the velocity of Stoneley waves propagating along the loosely bonded inter-face of two isotropic elastic half-spaces have been derived byVinh and Giang (2011)

The main aim of this paper is to ﬁnd exact formulas for the velocity and the slowness of Stoneley waves propagating in two bonded isotropic elastic half-spaces which have the same bulk wave velocities By employing the complex function method that is based on the properties of Cauchy integrals and the generalized Liouville theorem, the authors derive the ex-act formulas for the velocity and for the slowness The derivation of these formulas shows that: if a Stoneley wave exists then it

is unique; and there always exits a Stoneley wave propagating along the bonded interface of two bonded isotropic elastic half-spaces which have the same bulk wave velocities The former was proved byBarnett et al (1985)by another method, and the latter was shown byStoneley (1924)expanding the corresponding secular equation into Taylor series

2 Exact formula for the velocity

Let us consider two isotropic elastic solidsXandX⁄occupying the half-space x2P0 and x260, respectively Suppose that these two elastic half-spaces are in welded contact with each other at the plane x2= 0 Then, the component of the par-ticle displacement vector and the component of the stress tensor are continuous across the interface x2= 0 Note that same quantities related toXandX⁄have the same symbol but are systematically distinguished by an asterisk if pertaining toX⁄ Suppose that the two half-spaces have the same bulk wave velocities, i.e ck¼ cðk ¼ 1; 2Þ, where c1, c

1are the longitudinal wave velocities and c2, c

2are the transverse wave velocities of the half-spaces These conditions appear to be satisﬁed at the Wiechert surface of discontinuity within the Earth, as indicated byStoneley (1924) Since these two half-spaces become the same ifq=q⁄, we therefore assume that the mass densities of the half-spaces are different from each other, i.e.q–q⁄ According toStoneley (1924)the secular equation of Stoneley waves for the case under consideration is (see Eq (3) in

Stoneley, 1924):

x2 ðqqÞ2ðqþqÞ2 ffiffiffiffiffiffiffiffiffiffiffi

1 x

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cx p

þ 4ðqqÞ2 ð1 xÞ 2 ð cxÞ þ ffiffiffiffiffiffiffiffiffiffiffi

1 x

1 cx p

x 2

wherec¼ c2=c2ð0 <c<1Þ and x ¼ c2=c2, c is the velocity of Stoneley waves Since 0 < c < c2, it follows that x 2 (0, 1) The secular Eq.(1)can be rewritten as follows:

f ðxÞ q1ðxÞ q2ðxÞ ffiffiffiffiffiffiffiffiffiffiffi

1 x

1 cx

p

where

q1ðxÞ ¼ b2ð1 þ 4cÞx2 4ð2 þcÞx þ 8

in which

Note that, since b2< a2it follows:

q2ðxÞ ¼ a2 x 2b2=a22

þ 4b22 b2=a2

=a2

and according toStoneley (1924)we have:

lim

x!0

f ðxÞ

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

f ðzÞ q1ðzÞ þ q2ðzÞ ffiffiffiffiffiffiffiffiffiffiffi

z 1

cz 1

p

where qk(z) are given by(3), and ffiffiffiffiffiffiffiffiffiffiffi

z 1

p

; ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cz 1 p are chosen as the principal branches of the corresponding square roots With the fact 0 <c< 1 it is easy to see that for z 2 (0, 1), Eq.(7)becomes Eq.(2) We will prove the following theorem: Theorem 1 If a Stoneley wave exists, then it is unique, and its squared dimensionless velocity xs¼ c2=c2is deﬁned by

xs¼ 1 b

2

a2

1 ffiffiffi

c 4

p 2p4ffiffiffic

where

I0¼1

p

Z 1= c

1

hðtÞdt; hðtÞ ¼ atan q1ðtÞ

q2ðtÞ ffiffiffiffiffiffiffiffiffiffiffi

t 1

1 ct p

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Proof Denote L = [1,1/c], S = {z 2 C, z R L}, N(z0) = {z 2 S: 0 < jz z0j <e},eis a sufﬁcient small positive number, z0is some point of the complex plane C If a function /(z) is holomorphic inX C we write /(z) 2 H(X) In order to solve Eq.(2)in the interval (0, 1) we will ﬁnd zeros of f(z) in the domain S ((0, 1))

Theorem 1is proved as follows:

Step 1: To express f(z) by product of two functions, the ﬁrst function is non-zero in S, and the second one is a polynomial of third-order (see Eq.(16))

Sine the ﬁrst factor is non-zero in S, the zeros of f(z) are identical with those of the third-order polynomial in S Due to(6), z1= 0, z2= 0 are zeros of the third-order polynomial in S

Step 2: Finding the coefficients of z3and z2of the third-order polynomial (see(26) and (27)) by expanding f(z) and the inverse of the first factor into Laurent series at infinity Knowing these coefficients we obtain immediately the expression of the third zero, given by(8), of the third-order polynomial

Step 3: Showing that: (i) if a Stoneley wave exists then it is unique (ii) if a Stoneley wave exists then its squared dimension-less velocity is the third zero of the third-order polynomial

Step 1

From(3) and (7)it is not difﬁcult to show that the function f(z) has the properties:

(f1) f(z) 2 H(S)

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

(f3) f(z) = O(z3) as jzj ? 1

(f4) f(z) is continuous on L from the left and from the right (seeMuskhelishvili, 1953) 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:

fþðtÞ ¼ q1ðtÞ þ iq2ðtÞ ffiffiffiffiffiffiffiffiffiffiffi

t 1

1 ct

p

where the bar indicates the complex conjugate

Now we deﬁne the function g(t) (t 2 L) as follows:

gðtÞ ¼f

þðtÞ

then it is obvious that:

Consider the functionC(z) deﬁned as:

CðzÞ ¼ 1

2pi

Z

L

log gðtÞ

The functionC(z) is an integral of the Cauchy type whose properties are examined in detail inMuskhelishvili (1953)(chaps 2–4) It is not difﬁcult to verify that:

(c1)C(z) 2 H(S)

(c2)C(1) = 0

(c3) Forc–1/2:C(z) =X0(z), z 2 N(1),C(z) =X1(z), z 2 N(1/c), whereX0(z) (X1(z)) bounded in N(1) (N(1/c)) and takes a deﬁned value at z = 1 (z = 1/c)

(c4) Forc= 1/2:C(z) =X2(z), z 2 N(1),C(z) = (1/2) log (z 2) +X3(z), z 2 N(2), whereX2(z) (X3(z)) bounded in N(1) (N(2)) and takes a deﬁned value at z = 1 (z = 2)

It is noted that (c3) and (c4) come respectively from the facts (seeMuskhelishvili, 1953, chap 4, Section 29): log g(1) = log g(1/c) = 0 forc–1/2 and logg(1) = 0, log g(2) = 1 forc= 1/2

We now consider the function Y(z) deﬁned by

From (f1) (f3), (c1) (c4),(12), (14)and the Plemelj formula (seeMuskhelishvili, 1953, Chapter 2, Section 17), it is not dif-ﬁcult to assert that (see alsoVinh & Giang, 2011, 2012):

(y1) Y(z) 2 H(S)

(y2) Y(z) = O(z3) as jzj ? 1

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

(y) Y+(t) = Y(t),t 2 L

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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/c 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 (seeMuskhelishvili, 1963) Thus, by the generalized Liouville theorem (Muskhelishvili, 1963) and taking account into (y2) we have:

where P(z) is a third-order polynomial From(14) and (15)we have:

Since eC (z)

–0 "z 2 S (by (c1)), from(16)it follows that:

From(6), (16)and eC(z)–0 "z 2 S, it follows that z = 0 is a double root of Eq P(z) = 0

Step 2

Also from(16)we have:

From(5), (10) and (11)it implies:

where h(t) is given by(9)2 From(13) and (19)it follows (see alsoNkemzi, 1997):

CðzÞ ¼X1

n¼0

In

in which:

In¼ 1

2p

Z1= c

1

tn

On use of(20)one can express e C (z)as follows:

eCðzÞ¼ 1 þa1

zþ

a2

z2þa3

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

eCðzÞ

0

and substituting(20), (22)into(23)yield:

a1¼ I0;a2¼I

2 0

2þ I1; a3¼

I30

By expanding ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 1=z

p

; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 1=ðczÞ p

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

f ðzÞ ¼ A3z3þ A2z2þ A1z þ A0þ O z1

where A2, A3are given by

A2¼ b21 2pffiffiffic

ð Þ2 pffiffiffic

þp1ffiffiffic

2; A3¼

ffiffiffi

c

p

Substituting(22) and (25)into(18)yields:

PðzÞ ¼ A3z3þ z2ðA2þ A3a1Þ þ z Að 1þ A3a2þ A2a1Þ þ A3a3þ A2a2þ A1a1þ A0: ð27Þ

Since z = 0 is a double root of Eq P(z) = 0, as mentioned above., from(19), (21), (24)1and(27), the third root of Eq P(z) = 0, denoted by xs, is given by

xs¼ A2

A31

2

1

c 1

where A2, A3are given by(26)and I0is calculated by(9)

Step 3

Now we suppose that there exist two different Stoneley waves with the 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 0 < x(1), x(2)< 1 From(17)it follows P(x(1)) = P(x(2)) = P(0) = 0 But this is impossible because P(z) is a third-order polynomial and z = 0 is a double root of Eq P(z) = 0 Thus, if a Stoneley wave exists, it

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must be unique This fact was also proved by Barnett et al (1985)for the generally anisotropic case using the surface impedance method

Suppose that there exists a (unique) Stoneley wave Then by above arguments, its squared dimensionless velocity x is a zero of the third-order polynomial P(z), and x – 0 This implies that x must be xsgiven by(28), i.e by(8) The proof of Theorem 1 is ﬁnished h

Fig 1demonstrates the dependence of the Stoneley-wave velocity oncand r as a 3D-picture

3 Exact formula for the slowness

In this section we derive the formula for the squared dimensionless slowness y = 1/x of Stoneley waves that satisﬁes the equation:

FðyÞ p1ðyÞ p2ðyÞ ffiffiffiffiffiffiffiffiffiffiffiffi

y 1

p ffiffiffiffiffiffiffiffiffiffiffi

y c

p

where

p1ðyÞ ¼ b2 8y2

4ð2 þcÞy þ 1 þ 4c

y; p2ðyÞ ¼ 8b2y2

Note that p2(y) > 0 "y Eq.(29)is derived from Eq.(2)replacing x by 1/y In the complex plane C, Eq.(29)takes the form:

FðzÞ p1ðzÞ p2ðzÞ ffiffiffiffiffiffiffiffiffiffiffi

z 1

p pffiffiffiffiffiffiffiffiffiffiffiz c

where pk(z) are given by(30), and ffiffiffiffiffiffiffiffiffiffiffi

z 1

p

;pffiffiffiffiffiffiffiffiffiffiffiz c are chosen as the principal branches of the corresponding square roots For

z 2 (1 + 1) Eq.(31)becomes Eq.(29) Following the same procedure carried out in the previous section we have:

Theorem 2 If a Stoneley wave exists, then it is unique, and its squared dimensionless slowness ys¼ c2=c2is given by

ys¼2a

2þ b2cð1 cÞ 1 2ð cÞ

where

I0¼1

p

Z 1

c

hðtÞdt; hðtÞ ¼ atan p1ðtÞ

p2ðtÞpffiffiffiffiffiffiffiffiffiffiffit c ffiffiffiffiffiffiffiffiffiffiffi

1 t p

Table 1shows the numerical values of the squared dimensionless velocity of Stoneley waves which are calculated by directly solving the secular Eq (2)(for x⁄), by using the exact formulas(8) (for xs) and(32) (for ys) for some given values of r with

c= 1/3 It is seen fromTable 1that the corresponding values of x, x and 1/y totally coincide with each other

0.1 0.2 0.3 0.4 0.5 2 4 6 8 10

r

0.94 0.96 0.98

c

c 2

0.1 0.2 0.3 0.4 γ

Fig 1 Dependence of ffiffiffi

x p

¼ c=c 2 oncand r.

Table 1

Values of the squared dimensionless velocity of Stoneley waves calculated by directly solving the secular Eq (2) (x ⁄ ), by the exact formulas (8) (x s ) and (32) (y s ) for some given values of r Herec= 1/3.

x ⁄ 0.9901 0.9686 0.9501 0.9357 0.9246 0.9158 0.9087 0.9029

x s 0.9901 0.9686 0.9501 0.9357 0.9246 0.9158 0.9087 0.9029

y s 1.0100 1.0324 1.0525 1.0687 1.0815 1.0919 1.10043 1.1075 1/y s 0.9901 0.9686 0.9501 0.9357 0.9246 0.9158 0.9087 0.9029

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4 On the existence of Stoneley waves

We will prove the following theorem:

Theorem 3 There always exists a (unique) Stoneley wave propagating along the bonded interface of two isotropic elastic half-spaces with the same bulk wave velocities

Proof It is clear that in order a Stoneley wave to exist it is sufﬁcient that ys> 1, where ysis given by(32) Since Eq F(z) = 0 has no solution in the interval (c, 1] due to the discontinuity of F(z) in (0, 1), it follows ysR(c, 1] Therefore if ys>cthen ys> 1 This yields if ys>c, then a Stoneley wave can be propagate along the bonded interface of two isotropic elastic half-spaces with the same bulk wave velocities Now we prove that ys>c Indeed, from(33)it implies that I0P(1 c)/2 Taking into account this fact, from(32)we have:

yscP

a2ð1 cÞ þ b2chð1 cÞ2þ 2c2i

due toc63/4 < 1 and a2> b2 The proof ofTheorem 3is ﬁnished Note that the existence of Stoneley waves for the case under consideration was also asserted byStoneley (1924)by another technique h

5 Conclusions

In this paper, the exact formulas for the velocity and the slowness of Sroneley waves propagating along the bonded inter-face of two isotropic elastic half-spaces having the same bulk wave velocities are derived using the complex function

meth-od By using the obtained exact formulas, it is shown that there always exists a unique Stoneley wave propagating in two bonded isotropic elastic half-spaces with the same bulk wave velocities Since the obtained formulas are explicit they are useful in practical applications

Acknowledgments

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

by the DAAD

References

Barnett, D M., Gavazza, S D., & Lothe, J (1988) Slip waves along the interface between two anisotropic elastic half-spaces in sliding contact Proceedings of the Royal Society of London Series A – Mathematical and Physical Sciences, 415, 389–419.

Barnett, D M., Lothe, J., Gavazza, S D., & Musgrave, M J P (1985) Consideration of the existence of interfacial (Stoneley) waves in bonded anisotropic elastic half-spaces Proceedings of the Royal Society of London Series A – Mathematical and Physical Sciences, 412, 153–166.

Lee, D A., & Corbly, D M (1977) Use of interface waves for nondestructive inspection IEEE Transactions on Sonics and Ultrasonics, 24, 206–212 Lim, T C., & Musgrave, M J P (1970) Stoneley waves in anisotropic media Nature, 225, 372.

Murty, G S (1975a) A theoretical model for the attenuation and dispersion of stoneley waves at the loosely bonded interface of elastic half spaces Physics of the Earth and Planetary Interiors, 11, 65–79.

Murty, G S (1975b) Wave propagation at an unbounded interface between two elastic half-spaces Journal of the Acoustical Society of America, 58, 1094–1095.

Muskhelishvili, N I (1953) Singular integral equations Noordhoff-Groningen.

Muskhelishvili, N I (1963) Some Basuc problems of mathematical theory of elasticity Netherlands: Noordhoff.

Nkemzi, D (1997) A new formula for the velocity of Rayleigh waves Wave Motion, 26, 199–205.

Rokhlin, S., Hefet, M., & Rosen, M (1980) An elastic interface wave guided by a thin ﬁlm between two solids Journal of Applied Physics, 51, 3579–3582 Scholte, J G (1942) On the Stoneley wave equation Proceedings/ Koninklijke Nederlandsche Akademie van Weten-schappen, 45, 159–164.

Scholte, J G (1947) The range of existence of Rayleigh and Stoneley waves Monthly Notices of the Royal Astronomical Society Geophysical Supplement, 5, 120–126.

Sezawa, K., & Kanai, K (1939) The range of possible existence of Stoneley waves, and some related problems Bulletin of the Earthquake Research Institute Tokyo University, 17, 1–8.

Stoneley, R (1924) Elastic waves at the surface of separation of two solids Proceedings of the Royal Society of London Series A – Mathematical and Physical Sciences, 106, 416–428.

Stroh, A N (1962) Steady state problems in anisotropic elasticity Journal of Mathematical Physics, 41, 77–103.

Vinh, Pham Chi, & Giang, Pham Thi Ha (2011) On formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two elastic half-spaces Wave Motion, 48, 646–656.

Vinh, Pham Chi, & Giang, Pham Thi Ha (2012) Uniqueness of Stoneley waves in pre-stressed incompressible elastic media International Journal of Non-Linear Mechanics, 47, 128–134.

Acknowledgments... another technique h

5 Conclusions

In this paper, the exact formulas for the velocity and the slowness of Sroneley waves propagating along the bonded inter-face of two isotropic elastic. .. class="page_container" data-page="6">

4 On the existence of Stoneley waves< /p>

We will prove the following theorem:

Theorem There always exists a (unique) Stoneley wave propagating along the bonded

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