DSpace at VNU: Explicit secular equations of Stoneley waves in a non-homogeneous orthotropic elastic medium under the influence of gravity

DSpace at VNU: Explicit secular equations of Stoneley waves in a non-homogeneous orthotropic elastic medium under the in...

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Explicit secular equations of Stoneley waves in a non-homogeneous

orthotropic elastic medium under the inﬂuence of gravity

Pham Chi Vinha,*, Géza Serianib

a

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

b

Istituto Nazionale di Oceanograﬁa e di Geoﬁsica Sperimentale, Borgo Grotta Gigante 42/C, 34100 Sgonico, Trieste, Italy

a r t i c l e i n f o

Keywords:

Stoneley waves

Stoneley wave velocity

Orthotropic

Secular equation

Non-homogeneous

Gravity

a b s t r a c t

The problem of Stoneley waves in a non-homogeneous orthotropic elastic medium under the influence of gravity was studied recently by Abd-Alla and Ahmed [A.M Abd-Alla, S.M Ahmed, Stoneley waves and Rayleigh waves in a non-homogeneous orthotropic elastic medium under the influence of gravity, Appl Math Comput 135 (2003) 187–200], who derived the secular equation of the wave in the implicit form In this paper, by using an appropriate representation of the solution, we obtain the secular equation of the wave in the explicit form Moreover, considering its special cases, we derive explicit secular equa-tions for a number of investigaequa-tions of Stoneley waves under the influence of gravity, for which only the implicit dispersion equations were previously obtained

1 Introduction

The propagation of Stoneley waves under the effect of gravity is a signiﬁcant problem in Seismology and Geophysics, which has attracted the attention of researchers such as De and Sengupta[1], Dey and Sengupta[2], Das et al.[3] These authors, following Biot[4], all assumed the force of gravity to create a type of initial stress of hydrostatic nature, and derived the secular equation of the wave in the implicit form De and Sengupta[1]assumed the material is isotropic elastic, while Dey and Sengupta[2]considered the case of transversely isotropic elastic materials All supposed that the material is homo-geneous However, because any realistic model of the earth must take into account continuous changes of elastic properties

in the vertical direction, the problem was extended to the (exponentially) non-homogeneous case by Das et al.[3], who as-sumed that the material is isotropic Recently, Abd-Alla and Ahmed[5]extended the problem to the orthotropic case; these authors employed two displacement potentials for expressing the solution, and derived the secular equation of the wave in the implicit form

For Rayleigh and Stoneley waves, dispersion equations in the explicit form are very signiﬁcant in practical applications They can be used for solving direct (forward) problems, i.e studying the effects of material parameters on the wave velocity; and especially the inverse problems, i.e determining material parameters from the measured values of the wave speed The main purpose of this paper is to obtain the explicit secular equation of Stoneley waves under the effect of gravity for inhomo-geneous orthotropic elastic materials This equation is identiﬁed in the explicit form by using an appropriate representation of solution From this we derive the explicit secular equations for the particular cases investigated by De and Sengupta[1], Dey and Sengupta[2], Das et al.[3], and Pal and Acharya[6], in which only implicit dispersion equations were obtained Note that a secular equation F ¼ 0 is called explicit if F is an explicit function of the wave velocity c, the wave number k, and parameters characterizing materials and external effects (see for example,[7–9]) Otherwise we call it an implicit secular equation

* Corresponding author.

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

Contents lists available atScienceDirect Applied Mathematics and Computation

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 / a m c

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Also note that, due to Abd-Alla and Ahmed’s incorrect representation of the solution (see[12]), the secular equation in the implicit form derived by them in[5]for Stoneley waves is not valid

2 Basic equations

Let us consider the two non-homogeneous orthotropic elastic bodies,XandX, occupying the half-space x3P0; x360, respectively, subject to gravity They are in welded contact with each other at the plane x3¼ 0 These two media extend to an inﬁnitely great distance from the origin andXis to be taken aboveX Same quantities related toXandXhave the same symbol but are systematically distinguished by an asterisk if pertaining toX

We are interested in planar motion in the ðx1; x3Þ-plane with displacement components u1; u2; u3such that:

Then the components of the stress tensorrij; i; j ¼ 1; 3 are related to the displacement gradients by the following equations

[5]:

r11¼ c11u1;1þ c13u3;3;

r33¼ c13u1;1þ c33u3;3;

r13¼ c55ðu1;3þ u3;1Þ;

ð2Þ

where cijare the material constants

Equations of motion are of the form[5]:

r11;1þr13;3þqgu3;1¼q€u1;

r13;1þr33;3qgu1;1¼q€u3

ð3Þ

in whichqis the mass density of the medium, and g is the acceleration due to gravity, a superposed dot denotes differen-tiation with respect to t, commas indicate differendifferen-tiation with respect to spatial variables xi In matrix (operator) form, fol-lowing the Stroh formalism (see[10,11]), the Eqs.(2) and (3)are written as follows:

u0

r0

¼ N u

r

where u ¼ ½u1;u3T;r¼ ½r13;r33T, the symbol T indicates the transpose of a matrix, the prime indicates the derivative with respect to x3and:

N ¼ N1 N2

K N3

; N1¼ 0 @1

ðc13=c33Þ@1 0

; N2¼ 1=c55 0

0 1=c33

;

K ¼ q@2t þ ½ðc2

13 c11c33Þ=c33@2 qg@1

; N3¼ NT1:

ð5Þ

Here we use the notations: @1¼ @=ð@x1Þ; @2¼ @2=ð@x2Þ; @2

t¼ @2=ð@t2Þ In addition to Eq.(4), the displacement vector u and the traction vectorrare required to satisfy the decay condition:

ForXwe have equations similar to(1)–(5)in which the quantities are asterisked, and the decay condition(6)is replaced by:

Since the half-spaces are in welded contact with each other at the plane x3¼ 0, the displacement vector and the traction vector must satisfy the continuous condition:

3 Explicit secular equation

Assume that the half-spacesXandXare made of materials with an exponential depth proﬁle:

cij¼ c0e2mx3; q¼q0e2mx3; c

ij¼ c0

ije2m x3; q¼q0e2m x3; ð9Þ

where c0; q0; m c0

ij; q0; mare constants

Now we consider the propagation of a wave, travelling with velocity c and wave number k in the x1-direction, being mostly conﬁned to the neighbourhood of the interface x3¼ 0 Then the components u1; u3 of the displacement vector andr ; r of the traction vector at the planes x ¼ const are found in the form (see[13]):

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fuk;rk3gðx1;x3;tÞ ¼ femx 3Ukðx3Þ; iemx 3Rkðx3Þgeikðx 1 ctÞ; k ¼ 1; 3: ð10Þ

Substituting(10)into(4)yields:

U0

R0

¼ iM U

R

where U ¼ U½ 1 U3T; R¼½R1 R3T, and:

M ¼ M1 M2

Q M3

; M1¼ iðm=kÞ 1

D iðm=kÞ

; M2¼ ð1=kÞ 1=c

0

55 0

0 1=c0

33

;

Q ¼ kðX dÞ ia

ia kX

; M3¼ iðm=kÞ D

1 iðm=kÞ

;

ð12Þ

here d ¼ c0

11 ðc0

13Þ2Þ=c0

33; D¼ c0

13=c0

33; a ¼q0g; X ¼q0c2, the prime indicates the derivative with respect to y ¼ kx3 Following the approach employed in[9,14,15], by eliminating U from(11), the traction vectorRðyÞ is the solution of the equation:

where the matricesa; b; care given by:

a¼ Q1¼ 1

kd

X ia=k ia=k ðX dÞ

b¼ M1Q1

þ Q1M3¼ 1

kd

0 g1

g1 0

c¼ M1Q1M3 M2¼ 1

kd

h0 img0=k þ iag2=k

img0=k iag2=k h1

ð16Þ

in which

g0¼ d ð1 DÞX; g2¼Dm

2

k2;

h0¼m

2X

k2 þ ðX dÞ

d

c0 55

þ2ma

k2 ;

h1¼m

2

k2ðX dÞ þD

2

X d

c0 33

2maD

k2 :

ð17Þ

The displacement vector U is determined in terms ofRby:

Now we seek the solution of the Eq.(13)in the form:

whereR0is a non-zero constant vector, p is a complex number which must satisfy the condition:

in order to ensure the decay condition(6) Substituting(19)into(13)leads to:

AsR0is a non-zero vector, the determinant of the system(21)must vanish This provides an equation for determining p, namely:

where

S ¼ 2D d

c0

55

þ 1

c0 33

þ 1

c0 55

X 2m 2

k2 ;

P ¼ðc

0

11 XÞðc0

55 XÞ

c0

33c0 55

m 2

k2

1

c0 33

þ 1

c0 55

X 2D d

c0 55

þm

4

k4

a2þ 2amðc0

55 c0

13Þ

k2c0c0 :

ð23Þ

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It follows from(22)that:

where p2; p2are two roots of the quadratic Eq.(22)for p2 It is not difﬁcult to demonstrate that the vectorR0¼ ½A BT, the solution of(21), is given by:

A ¼ia

kp

2þ g1p img0

k iag2=k;

B ¼ Xp2þ h0:

ð25Þ

Let p1;p2be the two roots of(22)satisfying the condition(20) Then the general solution of the Eq.(13)is:

RðyÞ ¼c1

A1

B1

eip 1 yþc2

A2

B2

where Ak;Bkðk ¼ 1; 2Þ are given by(25), in which p is replaced by pk, and pk; c1, andc2ðc2þc2–0Þ are constants to be determined

We have the following result:

Proposition Suppose p1; p2are the two roots of(22)satisfying the condition(20) Then we have:

P > 0; 2 ffiffiffi

P

p

S > 0; p1p2¼ ffiffiffi

P

p

; p1þ p2¼ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffi P

p

S

q

where S; P are deﬁned by(23)

Indeed, from(20)it follows that ImðpiÞ > 0 If the discriminant D of the quadratic Eq.(22)for p2is non-negative, then its two roots must be negative in order that ImðpiÞ > 0 In this case, P ¼ p2p2>0 and the pair p1;p2 are of the form:

p1¼ ir1;p2¼ ir2 where r1;r2 are positive If D < 0, the Eq (22) for p2 has two conjugate complex roots, again

P ¼ p2p2>0, and in order to ensure ImðpiÞ > 0, it must be p1¼ t þ ir and p2¼ t þ ir, where r is positive In both cases,

P ¼ p2p2>0; p1p2is a negative real number and p1þ p2is a purely imaginary number with a positive imaginary part, thus

ðp1þ p2Þ2is a negative number Therefore, with the help of(24), it follows that the relations(27)are true

It is noted that the result(27)3,(27)4were obtained in[13], but without showing that P > 0; 2 ffiffiffi

P p

S > 0

From(18) and (26)we have:

UðyÞ ¼c1 E1

F1

eip1yþc2 E2

F2

where

Ek¼ e2p2

k ie1pkþ e0;

Fk¼ f3p3

kþ if2p2

where ej¼ bj=kd; j ¼ 0; 1; 2; fj¼ wj=kd; j ¼ 0; 1; 2; 3, and:

b0¼ h0ðDX am=k2Þ ð1=k2Þða þ XmÞðmg0þ ag2Þ;

b1¼ ð1=kÞ½g1ða þ XmÞ þ Xðmg0þ ag2Þ þ ah0;

b2¼ ð1=k2Þaða þ XmÞ þ g1X þ XðDX am=k2Þ;

w0¼ ð1=kÞh0½aD mðX dÞ ð1=kÞðmg0þ ag2ÞðX d þ ma=k2Þ;

w1¼ g1ðX d þ ma=k2Þ þ ð1=k2Þaðmg0þ ag2Þ þ ðX dÞh0;

w2¼ ð1=kÞaðX d þ ma=k2Þ þ ð1=kÞag1þ ðX=kÞ½aD mðX dÞ;

w3¼ d:

ð30Þ

Similarly, vectors uandrare sought in the form:

fu;r

k3gðx1;x3;tÞ ¼ fem x 3Uðx3Þ; iem x 3Rðx3Þgeikðx 1 ctÞ; k ¼ 1; 3 ð31Þ

in which

U¼c

1

E

1

F

1

eip

1 yþc 2

E 2

F 2

eip

and

R¼c

1

A1

B

eip

1 yþc 2

A2

B

eip

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1;c

2are constants to be determined, A;B;E;Fðk ¼ 1; 2Þ are deﬁned by:

A

k¼ia

kðp

Þ2þ g

1pim

g 0

k ia

g

2=k;

B

¼ XðpÞ2þ h0;

E

¼ e

2ðpÞ2 ie1pþ e

0;

Fk¼ f

3ðpÞ3þ if2ðpÞ2þ f

1pþ if0; k ¼ 1; 2

ð34Þ

in which a¼q0g; g

j;h0are deﬁned by formulas similar to those for gj;h0, and e

j ¼ bj=kd;f

j ¼ w

j=kdand bj;w

j are ex-pressed by those similar to (30); furthermore, d

ðD;d;X

Þ are given by the expressions similar to those for d ðD;d;XÞ, and p

1;p

2are two roots of the equation:

satisfying:

Ip

j <jm

j

here S

; Pare determined by those similar to(23)

Analogously as above, one can show that:

P>0; 2 ffiffiffiffiffi

P

p

S>0; p

1p

2¼ ffiffiffiffiffi

P

p

; p

1þ p

2¼ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffiffiffi

P

p

S

q

From the continuity conditions(8)we have:

Ukð0Þ ¼ Ukð0Þ; Rkð0Þ ¼Rkð0Þ; k ¼ 1; 3: ð38Þ

Eq.(38)yield a homogeneous linear system forc1;c2;c

1;c

2 The secular equation, determining the Stoneley wave velocity c,

is obtained by vanishing the determinant of the system:

E1 E2 E1 E2

F1 F2 F1 F2

A1 A2 A1 A2

B1 B2 B1 B2

After some algebraic manipulations, taking into account(25), (29) and (34)and removing the factor ðp1 p2Þðp

1 p

2Þ , the dispersion Eq.(39)is equivalent to:

q11 q12 q

11 q 12

q21 q22 q

21 q 22

q31 q32 q

31 q 32

q41 q42 q

41 q 42

where

q11¼ b2

2 ffiffiffi

P

p

S

q

b1;

q12¼ b2S þ b1

2 ffiffiffi P

p

S

q

þ 2b0;

q21¼ w3ðS ffiffiffi

P

p

Þ w2

2 ffiffiffi P

p

S

q

þ w1;

q22¼ w3ðS þ ffiffiffi

P

p Þ

2 ffiffiffi P

p

S

q

w2S w1

2 ffiffiffi P

p

S

q

2w0;

q31¼ d ða=kÞ

2 ffiffiffi P

p

S

q

þ g1

;

q32¼ d ð2=kÞðmg0þ ag2Þ ða=kÞS g1

2 ffiffiffi P

p

S q

;

q41¼ d X

2 ffiffiffi P

p

S q

; q42¼ d XS þ 2h½ 0;

ð41Þ

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11¼ b2

2 ffiffiffiffiffi

P

p

S

q

b1;

q

12¼ b2S

b1

2 ffiffiffiffiffi

P

p

S

q

þ 2b0;

q

21¼ w

3ðS ffiffiffiffiffi

P

p

Þ þ w 2

2 ffiffiffiffiffi

P

p

S

q

þ w

1;

q

22¼ w

3ðSþ ffiffiffiffiffi

P

p Þ

2 ffiffiffiffiffi

P

p

S

q

w

2Sþ w 1

2 ffiffiffiffiffi

P

p

S

q

2w 0

q

31¼ d ða=kÞ

2 ffiffiffiffiffi

P

p

S

q

þ g 1

q

32¼ d ð2=kÞðmg

0þ ag

2Þ ða=kÞSþ g

1

2 ffiffiffiffiffi

P

p

S q

;

q

41¼ d X

2 ffiffiffiffiffi

P

p

S q

; q

42¼ XS

þ 2h0

:

ð42Þ

It is clear that qij; q

ijare explicit functions of c; k; , thus, the secular Eq.(40)is fully explicit

4 Special cases

4.1 Explicit secular equation for inhomogeneous isotropic media subject to gravity

The propagation of Stoneley waves in inhomogeneous isotropic solids under the effect of gravity was investigated by Das

et al.[3], but the authors did not derived the dispersion equation in the explicit form due to the characteristic equations for p and pbeing fully quartic When the half-spaces are isotropic we have:

c0

11¼ c0

33¼ k0þ 2l0; c0

55¼l0; c13¼ k0;

c0

11¼ c033¼ k0þ 2l0; c0

55¼l0; c0

On view of(43), Eq.(40)becomes:

q11 q12 q

11 q 12

q21 q22 q

21 q 22

q31 q32 hq

31 hq 32

q41 q42 hq

41 hq 42

where h ¼l0=l0 The elements qijare given by:

q11¼ b2

2 ffiffiffi

P

p

S

q

b1;

q12¼ b2S þ b1

2 ffiffiffi P

p

S

q

þ 2b0;

q21¼ w3ðS ffiffiffi

P

p

Þ w2

2 ffiffiffi P

p

S

q

þ w1;

q22¼ w3ðS þ ffiffiffi

P

p Þ

2 ffiffiffi P

p

S

q

w2S w1

2 ffiffiffi P

p

S

q

2w0;

q31¼ d

2 ffiffiffi P

p

S

q

þ g1

;

q32¼ d 2ð mg0þg2Þ S g1

2 ffiffiffi P

p

S q

;

q41¼ d x

2 ffiffiffi P

p

S q

; q42¼ d½xS þ 2h0;

ð45Þ

where x ¼ X=l0ðx¼ X=l0Þ and:

b0¼ h0ðDx mÞ ðþ x mÞð mg0þg2Þ;

b1¼ ½g1ðþ x mÞ þ xð mg0þg2Þ þh0;

b2¼ðþ x mÞ þ g1x þ xðDx mÞ;

w0¼ h0½ D mðx dÞ ð mg0þg2Þðx d þ mÞ;

w1¼ g1ðx d þ mÞ þð mg0þg2Þ þ ðx dÞh0;

w2¼ðx d þ mÞ þg1þ x½ D mðx dÞ; w3¼ d;

g0¼ 2½2ð1 c0Þ c0x;

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g1¼ 2ð1 c0

Þð2 xÞ; g2¼ 1 2c0

m2;

h0¼ m2x þ ½x 4ð1 c0Þð1 xÞ þ2þ 2m

;

d ¼ x½x 4ð1 c0Þ 2

; S ¼ ð1 þc0Þx 2 2 m2;

P ¼ ð1 xÞð1 c0xÞ m2½2ð4c0 3Þ þ ð1 þc0Þx þ m4c022 2mð3 c0 1Þ;

c0¼ l0

k0þ 2l0; ¼q0g

kl0; m ¼ m

where

The elements q

ijare expressed by formulas similar to(45)in which ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffi P

p

S

p

is replaced by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffiffiffi

P

p

S

p

The quantities

bi;w

i;g

i;h0;d;S;P;c0;; m;D;dare given by formulas similar to(46) and (47) Eq.(44), along with(45)–(47), estab-lishes the explicit secular equation of Stoneley waves for the case of inhomogeneous isotropic elastic half-spaces subject

to gravity Note that qijand q

ijare dimensionless quantities

4.2 Explicit secular equation for homogeneous transversely isotropic half-spaces subject to gravity

When two half-spaces are homogeneous, i.e m ¼ m¼ 0, one can see that the explicit secular equation of the wave is of the form(40), in which the elements qijare simpliﬁed to:

q11¼

2 ffiffiffi

P

p

S

q

þ a=ðkc055Þ;

q12¼ S a

kc055

2 ffiffiffi P

p

S

q

þ 2Dð1 X=c0

55Þ;

q21¼ S ffiffiffi

P

p

D ðX dÞ=c055;

q22¼ Dþ ðX dÞ=c0

55 S ffiffiffi

P p

h i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffi P

p

S

q

þ 2ðaDÞ=ðkc055Þ;

q31¼ ða=kÞ

2 ffiffiffi P

p

S

q

þ d ð1 þDÞX;

q32¼ ða=kÞð2D SÞ ½d ð1 þDÞX

2 ffiffiffi P

p

S

q

;

q41¼ X

2 ffiffiffi

P

p

S

q

; q42¼ XS þ 2ðX dÞð1 X=c0

55Þ þ 2a2=ðk2c0

55Þ;

ð48Þ

where

S ¼ 2D d

c0

55

þ 1

c0 33

þ 1

c0 55

X;

P ¼ðc

0

11 XÞðc0

55 XÞ

c0

33c0 55

a 2

k2c0

33c0 55 :

ð49Þ

Elements q

ij; i ¼ 1; 2; 3; 4; j ¼ 1; 2 are defined by formulas similar to(48)in which ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffi P

p

S

p

is replaced by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffiffiffi

P

p

S

p

Note that, in this case, it can be shown that the Rayleigh wave velocity is limited by:

0 < c2<minðc0

55=q0; c0

11=q0; c0=q0; c0=q0Þ: ð50Þ

Indeed, in view of(49)1we have:

S ¼ c0

33ðX c0

11Þ þ c0

55ðX c0

55Þ þ ðc0

13þ c0

55Þ2

=ðc0

33c0

It follows from(27)1and(49)2that ðc0

11 XÞ and ðc0

55 XÞ must have the same sign This yields:

0 < X < minðc0

11;c0

55Þ or X > maxðc0

11;c0

Using(51)we see that the discriminant D ¼ S2

4P of Eq.(22)is given by:

D ¼ ðc0

13þ c055Þ4þ 2ðc013þ c055Þ2c0

33ðX c011Þ þ c055ðX c055Þ

þ c033ðX c011Þ c055ðX c055Þ2

=ðc033c0

55Þ2þ 4a

2

k2c0

33c0 55 : ð53Þ

Now, if(52)2exists, then it follows from(53)that D P 0, so Eq.(22)for this case has two real roots p2; p2with the same sign, according to(27)1 On the other hand, it is clear from(51) and (52)2that S ¼ p2þ p2>0 Thus, both p2and p2are positive This contradicts the fact that p; p must have a positive imaginary part

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Thus, it must be:

0 < X ¼q0c2<minðc0

55;c0

Similarly, we have:

0 < X

Then, from(54) and (55)we deduce(50)

Eq.(40)in which qij; q

ijgiven by(48)is the explicit secular equation of Stoneley waves for orthotropic homogeneous elas-tic half-spaces subject to gravity It provides the explicit secular equation of Stoneley waves for the transversely isotropic case which was investigated by Dey and Sengupta[2]

When the media are isotropic we have the relations(43) From(40), (43), (48) and (49)one can see that the explicit sec-ular equation for this case is:

q11 q12 q

11 q 12

q21 q22 q

21 q 22

q31 q32 q

31 q 32

q41 q42 q

41 q 42

where qijare given by:

q11¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ð1 þ ffiffiffi

P

p

Þ ð1 þc0Þx

q

þ;

q12¼ ð1 c0

Þð4 3xÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1 þ ffiffiffi

P

p

Þ ð1 þc0Þx

q

;

q21¼ 1 2c0þc0x ffiffiffi

P

p

;

q22¼ ð2c0 1 c0x ffiffiffi

P

p Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ ffiffiffi

P

p

Þ ð1 þc0Þx

q

þ 2ð1 2c0Þ;

q31¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1 þ ffiffiffi

P

p

Þ ð1 þc0Þx

q

þ 2ð1 c0Þð2 xÞ;

q32¼ ½4ð1 c0Þ ð1 þc0Þx 2ð1 c0Þð2 xÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ ffiffiffi

P

p

Þ ð1 þc0Þx

q

;

q41¼ x

2ð1 þ ffiffiffi

P

p

Þ ð1 þc0Þx

q

;

q42¼ x½ð1 þc0Þx 2 þ 2½x 4ð1 c0Þð1 xÞ þ 22

ð57Þ

in which:

Elements q

ijare determined by similar formulas to(57), in which

2ð1 þ ffiffiffi

P

p

Þ ð1 þc0Þx

q

is replaced by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ ffiffiffiffiffi

P

p

Þ ð1 þc0Þx

q

:

Eqs.(56)–(58)establish the explicit secular equation for the investigation[1]

4.3 Explicit secular equation for inhomogeneous transversely isotropic half-spaces

When gravity is absent, i.e a ¼ a¼ 0, the dispersion equation of Stoneley waves is Eq.(40), in which qijare reduced to (q

ij determined by the similar formulas):

q11¼

2 ffiffiffi

P

p

S

q

þ 2 m;

q12¼ S 2 m

2 ffiffiffi P

p

S

q

þ 2Dð1 X=c055Þ þ 2 m2;

q21¼ S ffiffiffi

P

p

þ m

2 ffiffiffi P

p

S

q

Dþ m2 ðX dÞ=c0

55;

q22¼ D m2

þ ðX dÞ=c055 S ffiffiffi

P p

h i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ffiffiffi P

p

S

q

þ mS þ ð2 mÞDþ m2

ðX dÞ=c055

;

q31¼ d ð1 þDÞX;

q32¼ 2 m½d ð1 DÞX ½d ð1 þDÞX

2 ffiffiffi P

p

S

q

;

q41¼ X

2 ffiffiffi

P

p

S

q

; q42¼ XS þ 2ðX dÞð1 X=c055Þ þ 2 m2X;

ð59Þ

where S is calculated by(23) and:

Trang 9

P ¼ðc

0

11 XÞðc0

55 XÞ

c0

33c0

55

m2 1

c0 33

þ 1

c0 55

X 2D d

c0 55

Eqs.(40), (59) and (60)provide the explicit secular equation for the investigation[6]

4.4 Explicit secular equation for homogeneous isotropic half-spaces

Now we consider the case when two half-spaces are homogeneous isotropic elastic and they are not subject to gravity For this case we have:

c11¼ c33¼ k þ 2l; c55¼l; c13¼ k;

c

11¼ c

33¼ kþ 2l; c

55¼l; c

13¼ k; m ¼ m¼ a ¼ a¼ 0; ð61Þ

where k; l; k; lare Lame constants of the half-spaces It is easy to verify that the roots pj; p

j ðj ¼ 1; 2Þ of the character-istic equations(22) and (35)are:

p1¼ i

ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 c

2

c2

s

; p2¼ i

ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 c 2

c2

s

; p

1¼ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 c 2

c2 1

s

; p

2¼ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 c 2

c2 2

s

ð62Þ

in which c2¼ ðk þ 2lÞ=q; c2¼l=q; c2

1 ¼ ðkþ 2lÞ=q; c2

2 ¼l=q It is clear that the secular equation(39)is equivalent to:

F1=ðiE1Þ F2=ðiE2Þ F1=ðiE1Þ F2=ðiE2Þ

A1=ðiklE1Þ A2=ðiklE2Þ A1=ðiklE

1Þ A2=ðiklE

2Þ

B1=ðklE1Þ B2=ðklE2Þ B1=ðklE

1Þ B2=ðklE

2Þ

By employing(25), (29), (34), (61), (62)one can show that:

F1=E1¼ p1; A1=ðkE1Þ ¼ 2lp1; B1=ðkE1Þ ¼ lð2 c2=c2Þ;

F2=E2¼ 1=p2; A2=ðkE2Þ ¼ lð2 c2=c2Þ=p2; B2=ðkE2Þ ¼ 2l;

F

1=E

1¼ p

1; A

1=ðkE1Þ ¼ 2lp

1; B

1=ðkE1Þ ¼ lð2 c2=c2

2Þ;

F

2=E

2¼ 1=p

2; A

2=ðkE2Þ ¼ lð2 c2=c2

2Þ=p

2; B

2=ðkE2Þ ¼ 2l:

ð64Þ

On use of(64)into(63)and setting pj¼ bj=ðikÞ; p

j ¼ bj=ðikÞ; j ¼ 1; 2 ðbj¼ k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 c2=c2 j

q

; bj ¼ k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 c2=c2 j

q

Þ we have:

2b1=k ð2 c2=c2Þðk=b2Þ 2ðl=lÞðb1=kÞ ðl=lÞð2 c2=c2

2Þðk=b2Þ

2 c2=c2 2 ðl=lÞð2 c2=c2

2Þ 2ðl=lÞ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.65

0.7 0.75 0.8

2 /c

0 55

Fig 1 Dependence of the dimensionless Stoneley wave velocity x ¼q0 c 2 =c 0

55 on m Here D ¼ 0:5; d 1 ¼ 2; d 2 ¼ 0:7; D

¼ 0:4; d

1 ¼ 2:5; d

2 ¼ 0:6;

Trang 10

This is the secular equation of Stoneley waves for the case of two homogeneous isotropic elastic half-spaces without the ef-fect of gravity, that coincides with the one published in[16]

5 Numerical results and discussion

It is readily to see from(40)that the dimensionless Stoneley wave velocity x ¼q0c2=c0

55depends on 11 dimensionless parameters, namely,D¼ c0

13=c0

33; d1¼ c0

11=c0

55;d2¼ c0

55=c0

33; m ¼ m=k; / ¼ g=ðkc2Þ; D¼ c0=c0; d1¼ c0=c0; d2¼ c0=c0;

m¼ m=k; d3¼q0=q0; d4¼ c0=c0

55, here c2¼ c0

55=q0 Given 11 these dimensionless parameters, it is not difﬁcult to numerically calculate the dimensionless Stoneley wave velocity x using the dispersion Eq.(40) It is well known that, for the homogeneous isotropic half-spaces not being subject to the gravity, the Stoneley wave exists if shear velocities

c2¼l=qand c

2¼l=qdiffer only slightly (see, for example,[16,17]), we therefore take d3¼ 0:25; d4¼ 0:3

Fig 1shows the dependence of x on m.Fig 2shows the variation of x with min three cases:Xis isotropic with

D¼ 0:5; d1¼ 4; d2¼ 0:25 ðk¼ 2lÞ; X is orthotropic with D¼ 0:5; d1¼ 6; d2¼ 0:25; X is orthotropic with

D¼ 0:5; d1¼ 2; d2¼ 0:25.Fig 3presents the dependence of x on m for different values of the parameter /, whileFig 4

shows the variation with / of x for distinct values of the parameter m

It is clear fromFigs 1, 3 and 4that the dimensionless Stoneley wave velocity x depends strongly on the inhomogeneity and the gravity Especially, theFig 2shows that the orthotropy also strongly affects on the dimensionless Stoneley wave

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.55

0.6 0.65 0.7 0.75 0.8 0.85 0.9

2 /c

0 55

m* Fig 2 Dependence of the dimensionless Stoneley wave velocity x ¼q0 c 2 =c 0

55 on m for three cases: X

is isotropic (dashed line) with

D

¼ 0:5; d1 ¼ 4; d2 ¼ 0:25 ðk

¼ 2l Þ;X

is orthotropic with D

¼ 0:5; d1 ¼ 6; d2 ¼ 0:25 (solid line);X

is orthotropic with D

¼ 0:5; d1 ¼ 2; d2 ¼ 0:25 (dash-dot line) For all casesXis orthotropic with D ¼ 0:5; d 1 ¼ 2; d 2 ¼ 0:7, and m ¼ 0:1; / ¼ 0; d 3 ¼ 0:25; d 4 ¼ 0:3.

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.65

0.7 0.75 0.8

2/c

0 55

m Fig 3 Dependence of the dimensionless Stoneley wave velocity x ¼q0 c 2 =c 0

55 on m for different values of the parameter / : / ¼ 0 (dash-dot line), / ¼ 0:2 (solid line), / ¼ 0:5 (dashed line) For three cases: D ¼ 0:5; d ¼ 2; d ¼ 0:7; D

¼ 0:5; d

¼ 4; d

¼ 0:25; m ¼ 0:1; d ¼ 0:25; d ¼ 0:3.

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