Rayleigh waves in an orthotropic half space coated by a thin orthotropic layer with sliding contact

Rayleigh waves in an orthotropic half-space coated by a thinorthotropic layer with sliding contact Pham Chi Vinh⇑, Vu Thi Ngoc Anh Faculty of Mathematics, Mechanics and Informatics, Hano

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Rayleigh waves in an orthotropic half-space coated by a thin

orthotropic layer with sliding contact

Pham Chi Vinh⇑, Vu Thi Ngoc Anh

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

a r t i c l e i n f o

Article history:

Received 23 August 2013

Received in revised form 18 October 2013

Accepted 1 November 2013

Available online 11 December 2013

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

In the present paper, we are interested in the propagation of Rayleigh waves in an orthotropic elastic half-space coated with a thin orthotropic elastic layer The contact between the layer and the half space is assumed to be smooth The main aim of the paper

is to establish an approximate secular equation of the wave By using the effective bound-ary condition method, an approximate secular equations of third-order in terms of the dimensionless thickness of the layer is derived It is shown that this approximate secular equation has high accuracy From the secular equation obtained, an approximate formula

of third-order for the Rayleigh wave velocity is derived and it is a good approximation

1 Introduction

The structures of a thin ﬁlm attached to solids, modeled as half-spaces coated by a thin layer, are widely applied in modern technology The measurement of mechanical properties of thin ﬁlms deposited on half-spaces before and during loading plays an important role in health monitoring of these structures in applications, seeMakarov, Chilla, and Frohlich (1995) and Every (2002)and references therein Among various measurement methods, the surface/guided wave method

is most widely used (Every, 2002), because it is non-destructive and it is connected with reduced cost, less inspection time, and greater coverage (Hess, Lomonosov, & Mayer, 2014) For the surface/guided, wave method the Rayleigh wave is a versatile and convenient tool (Kuchler & Richter, 1998; Hess et al., 2014)

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 main purpose of the investigations of Rayleigh waves propagating in half-spaces covered with a thin layer Taking the assumption of a thin layer, explicit secular equations can be derived by replacing approximately the entire effect of the thin layer on the half-space by the so-called effective boundary conditions which relate the displacements with the stresses of the half-space at its surface

For obtaining the effective boundary conditionsAchenbach and Keshava (1967) and Tiersten (1969)replaced the thin layer by a plate modeled by different theories: Mindlin’s plate theory and the plate theory of low-frequency extension and ﬂexure, whileBovik (1996)expanded the stresses at the top surface of the layer into Taylor series in its thickness The Taylor expansion technique was then developed byBenveniste (2006), Niklasson, Datta, and Dunn (2000), Rokhlin and Huang (1992, 1993), Shuvalov and Every (2002), Steigmann (2007), Ting (2009) and Vinh and Linh (2012, 2013)

⇑ Corresponding author Tel.: +84 4 35532164; fax: +84 4 38588817.

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

Contents lists available atScienceDirect

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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Bovik (1996), Tiersten (1969) and Tuan (2008), assumed that the layer and the substrate are both isotropic and derived approximate secular equations of second-order (these equations do not coincide totally with each other).Steigmann (2007)

considered a transversely isotropic layer with residual stress overlying an isotropic half-space and he obtained an approx-imate second-order dispersion relation.Wang, Du, Lu, and Mao (2006)considered a isotropic half-space covered with a thin electrode layer and he obtained an approximate secular equation of ﬁrst-order InVinh and Linh (2012)the layer and the half-space were both assumed to be orthotropic and an approximate secular equation of third-order was obtained InVinh and Linh (2013)the layer and the half-space are both subjected to homogeneous pre-stains and an approximate secular equation of third-order was established which is valid for any pre-strain and for a general strain energy function

In all investigations mentioned above, the contact between the layer and the half-space is assumed to be perfectly bonded For the case of sliding contact, there exists only one approximate secular equation of third-order in the literature, for the case when the layer and the half-space are both isotropic, obtained byAchenbach and Keshava (1967) However, this approximate secular equation includes the shear coefﬁcient, originating from Mindlin’s plate theory (Mindlin, 1951), whose usage should be avoided as noted by, Muller and Touratier (1996), Stephen (1997) and Touratier (1991)

It should be note that for the case of smooth contact, one could not arrive at the effective boundary conditions from the relations between the displacements and the stresses at the bottom surface of the layer which were derived byBovik (1996)

and Tiersten (1969) In contrast, for the case of welded contact, the effective boundary conditions were immediately obtained

The main aim of this paper is to derive an approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half-space covered with a thin orthotropic elastic layer The layer and the half-space are in sliding contact with each other By using the effective boundary condition method, an approximate effective boundary condition of third-order which relates the normal displacement with the normal stress at the surface of the half space is derived Using this condition along with the vanishing of the shear stress at the surface of the half-space, an approximate secular equation of third-order in terms of the dimensionless thickness of the layer is obtained We will show that the approximate secular equation obtained has high accuracy From this secular equation, an approximate formula of third-order for the Rayleigh wave velocity is estab-lished and it is a good approximation

2 Effective boundary condition of third-order

Consider an elastic half-space x2P0 coated by a thin elastic layer h 6 x260 The layer and the half-space are both homogeneous, compressible, orthotropic and they are in sliding contact with each other Note that the same quantities re-lated 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 plane strain such that:

where ui; uiare components of the displacement vector, t is the time Since the layer is made of orthotropic elastic materials, the strain–stress relations are:

r11¼ c11u1;1þ c12u2;2;

r22¼ c12u1;1þ c22u2;2;

r12¼ c66ðu1;2þ u2;1Þ;

ð2Þ

where commas indicate differentiation with respect to spatial variables xk; rij are the stresses, the material constants

c11; c22; c12; c66satisfy the inequalities:

ckk>0; k ¼ 1; 2; 6; c11c22 c2

which are necessary and sufﬁcient conditions for the strain energy of the material to be positive deﬁnite (seeTing, 1996) In the absent of body forces, the equations of motion for the layer is:

r11;1þ r12;2¼ q€u1;

where qis the mass density of the layer, a dot signiﬁes differentiation with respect to t FollowingVinh and Seriani (2009, 2010), from Eqs.(2) and (4)we arrive at:

U0

T0

" #

M3 M4

T

" #

where:

U ¼ ½u1 u2T; T ¼ ½r12 r22T;

the symbol ‘‘ T ’’ indicate the transpose of a matrix, the prime signiﬁes differentiation with respect to x and:

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M1¼ 0 @1

c12

c 22@1 0

; M2¼

1

c66 0

c 22

;

M3¼

c 2

12 c 11 c 22

c 22 @2þ q@2t 0

2

4

3 5; M4¼ MT1;

ð6Þ

here we use the notations @1¼ @=@x1; @2¼ @2=@x2; @2t ¼ @2=@t2 We call Eq.(5)the matrix equation for the plane strain(1)– (4) From Eq.(5)we immediately arrive at Stroh’s formulation (Stroh, 1962) It follows from(5):

UðnÞ

TðnÞ

" #

¼ Mn

U

T

" #

M3 M4

Let h be small (i e the layer is thin), then expanding into Taylor series TðhÞ at x2¼ 0 up to the third-order of h we have:

TðhÞ ¼ Tð0Þ T0ð0Þh þ1

2!T

00ð0Þh21 3!T

Suppose that surface x2¼ h is free of traction, i e TðhÞ ¼ 0 Introducing(7)with n ¼ 1; 2; 3 at x2¼ 0 into(8)yields:

I hM4þ1

2

M61 6

3

M8

Tð0Þ ¼ hM31

2h

2

M5þ1

6h

3

M7

where I is the identity matrix of order 2, M3;M4are deﬁned by(6)and

M5¼ M3M1þ M4M3;

M6¼ M3M2þ M24;

M7¼ M3M21þ M4M3M1þ M3M2M3þ M24M3;

M8¼ M3M1M2þ M4M3M2þ M3M2M4þ M3:

ð10Þ

Taking into account(6)and(10), the relation(9)in component form is of the form:

r12þ h r1r22;1 r3u1;11 q€u1

þh

2

2 r2r12;11þ q

c66

€

r12 r3u2;111 qð1 þ r1Þ€u2;1

þh

3

6 r4r22;111þ qr5r€22;1 r6u1;1111 qr7u€1;11q2

c66

€

u1;tt

r22þ h r12;1 qu€2

þh

2

2 r1r22;11þ q

c22

€

r22 r3u1;111 qð1 þ r1Þ€u1;1

þh

3

6 r2r12;111þ qr8r€12;1 r3u2;1111 qð1 þ 2r1Þ€u2;11q2

c22

€

u2;tt

where

r1¼c12

c22

; r2¼ r1þr3

c66

; r3¼c

2

12 c11c22

c22

; r4¼ r1r2þr3

c22

r5¼1 þ r 1

c22

þr1

c66

; r6¼ ðr1þ r2Þr3; r7¼ r2þ 2r2; r8¼1 þ r 1

c66

þ1

c22

:

ð13Þ

It should be noted that, if the contact between the layer and the half-space is welded, i e the displacements and the stresses are continuous through the interface of the layer and the half-space, we immediately obtain the effective boundary conditions from Eqs.(11) and (12)by replaced u1; u2; r12and r22by u1;u2;r12andr22, respectively These effective boundary conditions are valid not only for the displacements and the stresses of Rayleigh waves but also for those of any dynamic problem However, for the sliding contact, the situation is rather different The horizontal displacement is not required to

be continuous through the interface, the effective boundary conditions are therefore not immediately obtained from Eqs

(11) and (12) As shown below, the effective boundary conditions for this case are valid for only the displacements and the stresses of Rayleigh waves

Now we consider the propagation of a Rayleigh wave, travelling (in the coated half-space) with velocity cð> 0Þ and wave number kð> 0Þ in the x1-direction and decaying in the x2-direction The displacements and the stresses of the wave are sought in the form:

u1¼ U1ðyÞeikðx 1 ctÞ; u2¼ U2ðyÞeikðx 1 ctÞ;

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for the layer, and

u1¼ U1ðyÞeikðx 1 ctÞ; u2¼ U2ðyÞeikðx 1 ctÞ;

for the half-space, where y ¼ kx2 Substituting(14) into (11) and (12)yields:

iT1ð0Þ 1 þe2

2 r2þ

qc2

c66

þ T2ð0Þ er1þe3

6r4 qc2r5

þ U1ð0Þ er3þ qc2

þe3

6 r6 qc2r7q2c4

c66

þ i U2ð0Þ e2

2r3þ qc2ð1 þ r1Þ

¼ 0; T1ð0Þ eþe3

6r2 qc2r8

þ iT2ð0Þ 1 þe2

2 r1þ

qc2

c22

þ i U1ð0Þ e2

2r3þ qc2ð1 þ r1Þ

þ U2ð0Þ e qc2þe3

6 r3 qc2ð1 þ 2r1Þ q2c4

c22

wheree¼ kh is the dimensionless thickness of the layer Let the contact between the layer and the half-space be sliding, we have (Barnett, Gavazza, & Lothe, 1988):

or, in view of the relations(14), equivalently:

Introducing the second of(18)into(16)and eliminating U1we have:

iT2ð0Þða1þ a2e2

Þ ¼ ða3eþ a4e3

where

a1¼ xr2

v ed; a2¼1

6 edðed 2e2e3Þ þ r2

vx e2e2þ 2e2e3 3e1e2 2ed

þ r4

vx2ð1 þ 3e2Þ

; a3¼ c66r2

vxðr2

vx edÞ;

a4¼c66

12 e

2

dþ r2

vx 2edðe2e3 ed 1Þ þ r2

vx 1 2e 2e3 e2e2þ 4edþ 2e1e2

2r4

vx2ð1 þ e2Þ

;

in which

x ¼c

2

c2; e1¼c11

c66

; e2¼c66

c22

; e3¼c12

c66

; ed¼ e1 e2e2;

rl¼c66

c66

; rv¼c2

c2

; c2¼

ffiffiffiffiffiffiffi

c66

q

r

; c2¼

ffiffiffiffiffiffiffi

c66

q

s :

ð21Þ

From the last two equations of(18) and (19)it follows:

From the ﬁrst of(18) and (22)we see that the surface x2¼ 0 of the half-space is subjected to the following conditions:

T1ð0Þ ¼ 0;

T2ð0Þða1þ a2e2

Þ ¼ iU2ð0Þða3eþ a4e3

The second of(23)is the desired approximate effective boundary condition (of third-order) The total effect of the layer on the half-space is replaced approximately by this condition

3 An approximate secular equation of third-order

Now we can ignore the layer and consider the propagation of Rayleigh waves in the orthotropic elastic half-space whose surface x2¼ 0 is subjected to the boundary conditions (23) According to Vinh and Ogden (2004b), the displacement components of a Rayleigh wave travelling with velocity c and wave number k in the x1-direction and decaying in the

x2-direction are determined by(15)1,2in which U1ðyÞ and U2ðyÞ are given by:

U1ðyÞ ¼ B1eb 1 yþ B2eb 2 y;

where B1and B2are constant to be determined and b1;b2are roots of the equation

c22c66b4þ ½ðc12þ c66Þ2þ c22ðX c11Þ þ c66ðX c66Þb2þ ðc11 XÞðc66 XÞ ¼ 0; ð25Þ whose real parts are positive to ensure to the decay condition, X ¼qc2, and:

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ak¼ ibk; bk¼ bkðc12þ c66Þ

c22b2k c66þqc2¼c11qc2 c66b2k

ðc12þ c66Þbk

; k ¼ 1; 2; i ¼ ffiffiffiffiffiffiffi

1

p

From(25)we have:

b21þ b22¼ ðc12þ c66Þ

2

þ c22ðX c11Þ þ c66ðX c66Þ

c22c66

¼ S;

b21b22¼ðc11 XÞðc66 XÞ

c22c66 ¼ P:

ð27Þ

Note that the material constants c11;c22;c12;c66satisfy the inequalities(3)without bars It is not difﬁcult to verify that if the Rayleigh wave exists (! b1;b2having positive real parts), then:

and:

b1b2¼ ffiffiffi

P

p

; b1þ b2¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

S þ 2 ffiffiffi P p q

Using(15)1,2and(24)into the stress–strain relations(2)without bars yields expressions ofr12andr22that are given by

(15)3,4in which

T1ðyÞ ¼ ic66ðb1þ b1ÞB1eb 1 yþ ðb2þ b2ÞB2eb 2 y

;

T2ðyÞ ¼ ðc 12 c22b1b1ÞB1eb 1 yþ ðc12 c22b2b2ÞB2eb 2 y

Introducing(24), (30)into Eqs.(23)provides a homogeneous system of two linear equations for B1;B2, namely:

f ðb1ÞB1þ f ðb2ÞB2¼ 0

where

FðbnÞ ¼ ðc12 c22bnbnÞða1þ a2e2Þ bnða3eþ a4e3Þ;

For a non-trivial solution, the determinant of the matrix of the system(31)must vanish, i.e.:

Introducing(32)into(33)leads to the dispersion equation of the wave:

where

A0¼ a1½ðc12þ c22b1b2Þðb2 b1Þ þ ðc12þ c22b1b2Þðb2 b1Þ;

A1¼ a3ðb1b2 b2b1Þ;

A2¼ a2½ðc12þ c22b1b2Þðb2 b1Þ þ ðc12þ c22b1b2Þðb2 b1Þ;

A3¼ a4ðb1b2 b2b1Þ:

ð35Þ

By(26)it is not difﬁcult to prove the following equalities:

b1b2 b2b1¼ðc11 XÞðb1þ b2Þ

ðc12þ c66Þb1b2

ðb2 b1Þ;

b1b2¼c11 X

c22b1b2

; b2 b1¼ c11 X þ c66b1b2

ðc12þ c66Þb1b2

ðb2 b1Þ:

ð36Þ

Substituting(36)into(35)yields: Ak¼ hAk;ðk ¼ 0; 1; 2; 3Þ; h ¼ b 2 b 1

b1b2ðc 12 þc 66 Þ, where

A0¼ a1ðc2

12 c11c22þ c22XÞb1b2þ ðc11 XÞX

;

A1¼ a3ðc11 XÞðb1þ b2Þ;

A2¼ a2ðc2

12 c11c22þ c22XÞb1b2þ ðc11 XÞX

;

A3¼ a4ðc11 XÞðb1þ b2Þ;

ð37Þ

in which b1b2and b1þ b2are given by(29) Removing the factor h, Eq.(34)becomes:

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This is the desired third-oder approximate secular equation and it is fully explicit In the dimensionless form Eq.(38)is:

where

E0¼ xr2

v ed

e2x ed

ð Þb1b2þ eð 1 xÞx

E1¼ rlr2

vx xr2

v ed

e1 x

ð Þ bð 1þ b2Þ;

E2¼ 1

6 r

4

vx2ð1 þ 3e2Þ þ r2

vx e 2e2þ 2e2e3 2ed 3e1e2

edð2e2e3 edÞ

e2x ed

ð Þb1b2þ eð 1 xÞx

E3¼ 1

12rl e

2

dþ r2

vx 2edðe2e3 ed 1Þ þ r2

vx 1 2e2e3 e2e2þ 4edþ 2e1e2

2r4

vx2ð1 þ e2Þ

e1 x

ð Þ bð 1þ b2Þ;

b1b2¼ ffiffiffi

P

p

; b1þ b2¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

S þ 2 ffiffiffi P p q

; P ¼ð1 xÞðe1 xÞ

e2

;

S ¼e2ðe1 xÞ þ 1 x ð1 þ e3Þ

2

e2

;

ð40Þ

and

e1¼c11

c66

; e2¼c22

c66

; e3¼c12

c66

It is clear from(39) and (40)that the squared dimensionless Rayleigh wave velocity x ¼ c2=c2depends on 9 dimensionless parameters, namely, ek; ek ðk ¼ 1; 2; 3Þ; rl;rv and e Note that ek>0; ek>0 ðk ¼ 1; 2Þ; ed>0; ed>0 according to the inequalities(3) From(39)and the ﬁrst of(40)it follows that, whene¼ 0:

either ðe2x edÞ ffiffiffi

P

p

The ﬁrst of(42)is the secular equation of Rayleigh waves propagating in an orthotropic elastic half-space (seeChadwick,

1976;Vinh & Ogden, 2004) and the second is the dimensionless speed of the longitudinal waves of the layer (considered as a plate) These facts say that in the limite! 0 two modes are possible, one of which approaches the classical Rayleigh wave in the orthotropic half-space and the other approaches the longitudinal wave of the layer with the velocity given by(42)2(see also,Achenbach & Keshava, 1967for the isotropic case), provided x<1

Fig 1presents the dependence one¼ k:h 2 ½0; 1 of the dimensionless Rayleigh wave velocity x ¼ c2=c2that is calculated

by the exact secular equation (solid line) and by the approximate secular equation (39) (dashed line) Here we take

e1¼ 2:5; e2¼ 3; e3¼ 1:5; e1¼ 1:8; e2¼ 1; e3¼ 0:6; rl¼ 0:5; rv¼ 3 Note that the exact secular equation is of the determinant form, that is similar in form to Eq.(30) inAchenbach and Keshava (1967)and is not reproduced here It is seen fromFig 1

that the exact velocity curve and the third-order approximate one almost totally coincide with each other for the values of

e2 ½0; 1

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

0.2 0.3 0.4 0.5 0.6 0.7

0.8 x

ε

Fig 1 Plots of the dimensionless Rayleigh wave velocity xðeÞ in the interval ½0; 1 that is calculated by the exact secular equation (solid line) and by the approximate secular Eq (39) (dashed line) Here we take e ¼ 2:5; e ¼ 3; e ¼ 1:5; e ¼ 1:8; e ¼ 1; e ¼ 0:6; r ¼ 0:5; r ¼ 3.

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4 Isotropic case

When the layer and the half-space are both isotropic:

c11¼ c22¼ k þ 2l; c12¼ k; c66¼l; c11¼ c22¼ k þ 2l; c12¼ k; c66¼ l: ð43Þ From(26), (27) and (43)one can see that:

b1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cx

p

; b2¼ ffiffiffiffiffiffiffiffiffiffiffi

1 x

p

; b1¼ b1; b2¼1

b2

where

c2¼

ffiffiffiffi

l

q

r

; x ¼c

2

Introducing(20), (43) and (44)into(35)and after some manipulations, the equation(34)is equivalent to:

^

where

^

A0¼ r2

vx 4ð1 cÞ

ðx 2Þ2 4b1b2

; A^1¼ rlr2

vx2b1 r2

vx 4ð1 cÞ

;

^

A2¼ 1

6 8ð1 cÞ þ 4r2

vxðc2 2Þ þ r4

vx2ð1 þ 3cÞ

ðx 2Þ2 4b1b2

;

^

A3¼1

6rlxb1 8ð1 cÞ2þ r2

vx 8ð2 þ 3c c2Þ þ 2r2

vxð4 2c c2Þ r4

vx2ð1 þ cÞ

;

ð47Þ

here rl¼ l=l;rv¼ c2=c2; c¼ l=ðkþ 2lÞ Eq(46)is the approximate secular equation of third-order for the isotropic case It shows that for this case the squared dimensionless Rayleigh wave velocity x ¼ c2=c2depends on four dimensionless param-etersc; c;rland rv

With the help of(40)–(44)it is not difﬁcult verify that Ek¼Ek

c;ðk ¼ 0; 1; 2; 3Þ, where:

E0¼ 4ðc 1Þ þ r2

vx

½4ðc 1Þ þ xb1b2þ ð1 cxÞx

E1¼ rlr2

vx 4ðc 1Þ þ r2

vx

ð1 cxÞðb1þ b2Þ;

E2¼ 1

6 8ð1 cÞ þ 4r2vxðc2

2Þ þ r4vx2

ð1 þ 3cÞ

½4ðc 1Þ þ xb1b2þ ð1 cxÞx

E3¼1

6rl 8ð1 cÞ2þ r2

vx 8ð2 þ 3c c2Þ þ 2r2

vxð4 2c c2Þ r4

vx2ð1 þ cÞ

ð1 cxÞðb1þ b2Þ;

ð48Þ

and b1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cx

p

;b2¼ ffiffiffiffiffiffiffiffiffiffiffi

1 x p Therefore, we have an alternative approximate secular equation of third-order for the iso-tropic case, namely:

in which Ek;ðk ¼ 0; 1; 2; 3Þ are given by(48)

5 An approximate formula of third-order for the velocity

In this section we establish an approximate formula of third-order for the squared dimensionless Rayleigh wave velocity

xðeÞ that is of the form:

xðeÞ ¼ xð0Þ þ x0ð0Þeþx

00ð0Þ

2 e2þx

000ð0Þ

where xð0Þ is the squared dimensionless velocity of Rayleigh waves propagating in an orthotropic elastic half-space that is given by (seeVinh & Ogden, 2004b):

xð0Þ ¼ ffiffiffiffiffis

1

p

s2s3= ð ffiffiffiffiffis

1

p

=3Þðs2s3þ 2Þ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R þ ffiffiffiffi D p

3

q

þ

R ffiffiffiffi D p

3

q

where s1¼e2

e 1;s2¼ 1 ee2

1 e 2;s3¼ e1and R, D are given by:

R ¼ 1

54hðs1;s2;s3Þ;

108½2

ffiffiffiffiffi

s1

p ð1 s2Þhðs1;s2;s3Þ þ 27s1ð1 s2Þ2þ s1ð1 s2s3Þ2þ 4;

ð52Þ

Trang 8

in which

hðs1;s2;s3Þ ¼ ffiffiffiffiffi

s1

p

and the roots in(51)taking their principal values

From(39)it follows that:

x0ð0Þ ¼ E1

E0x

x¼xð0Þ

; x00ð0Þ ¼ 2E2E

2 0x 2E0xE1E1xþ E0xxE21

E30x

x¼xð0Þ

;

x000ð0Þ ¼ 6E3þ 6E2xx0ð0Þ þ 3E1xxx0 2

ð0Þ þ 3E1xx00ð0Þ þ 3E0xxx0ð0Þx00ð0Þ þ E0xxxx0 3

ð0Þ

=E0x

x¼xð0Þ

ð54Þ

where E1;E2and E3are given by(40)and:

E0x¼ r2v ðe2x edÞ

ffiffiffiffiffiffiffiffiffiffiffi

1 x

p ffiffiffiffiffiffiffiffiffiffiffiffiffie

1 x p ffiffiffiffiffi

e2

þ ðr2vx edÞ 4e2x

2 ½2edþ 3e2ð1 þ e1Þx þ 2e1e2þ edð1 þ e1Þ

2 ffiffiffiffiffie

2

p ffiffiffiffiffiffiffiffiffiffiffi

1 x

;

E0xx¼ 2r2

v

4e2x2 2e½ dþ 3e2ð1 þ e1Þx þ 2e1e2þ edð1 þ e1Þ

2 ffiffiffiffiffie

2

1 x

þ ðr2vx edÞ 2 þ8e2x

3 12e2ð1 þ e1Þx2þ 3e2ð1 þ 6e1þ e2Þx þ edð1 e1Þ2 4e1e1ð1 þ e1Þ

4 ffiffiffiffiffie

2

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 xÞ3

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðe1 xÞ3 q

8

>

9

>

>;

E0xxx¼ 3r2

v 2 þ8e2x

3 12e2ð1 þ e1Þx2þ 3e2ð1 þ 6e1þ e2Þx þ edð1 e1Þ2 4e1e1ð1 þ e1Þ

4 ffiffiffiffiffie

2

ðe1 xÞ3 q

8

>

9

>

þ3ðr

2

vx edÞð1 e1Þ2f½e2ðe1þ 1Þ 2edx 2e1e2þ edð1 þ e1Þg

8 ffiffiffiffiffie

2

ðe1 xÞ5

E1x¼ rlr2

vede1þ 2ðr2ve1þ edÞx 3r2vx2

ðb1þ b2Þ þ rlr2

vxðr2

vx edÞðe1 xÞ2x 1 e1 ð1 þ e2Þb1b2

2e2b1b2ðb1þ b2Þ ;

E1xx¼ 2rlr2

v ðr2ve1þ ed 3r2vxÞðb1þ b2Þ þ e de1þ 2ðr2ve1þ edÞx 3r2vx2 2x 1 e1 ð1 þ e2Þb1b2

2e2b1b2ðb1þ b2Þ

þrlr

2

vxðr2

vx edÞðe1 xÞ 4e2b21b22ðb1þ b2Þ ½4e2b1b2þ ð1 þ e2Þð1 þ e1 2xÞ

½2x 1 e1 ð1 þ e2Þb1b22

ðb1þ b2Þ2 (

þ½2x 1 e1 ð1 þ e2Þb1b2ð1 þ e1 2xÞ

b1b2

;

E2x¼ 1

6 r

4

vx2ð1 þ 3e2Þ þ r2vxðe2e2

þ 2e2e3 2ed 3e1e2Þ þ edðed 2e2e3Þ

4e2x

2 ½2edþ 3e2ð1 þ e1Þx þ 2e1e2þ edð1 þ e1Þ

2 ffiffiffiffiffie

2

1 x

1

6r

2

v 2r2

vxð1 þ 3e2Þ þ e2e2

þ 2e2e3 2ed 3e1e2

ðe2x edÞb1b2þ ðe1 xÞx

here b1b2¼ ffiffiffi

P

p

;b1þ b2¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

S þ 2 ffiffiffi P p p

;P; S are given by(40) When the layer and the half-space are both isotropic, the Rayleigh wave velocity is (approximately) expressed by(50)in which:

x0ð0Þ ¼

E1

E0x

x¼xð0Þ

; x00ð0Þ ¼ 2E2

E2 0x 2E0xE1E1xþ E0xxE2

E3 0x

x¼xð0Þ

;

x000ð0Þ ¼ 6E3þ 6E2xx0ð0Þ þ 3E1xxx0 2

ð0Þ þ 3E1xx00ð0Þ þ 3E0xxx0ð0Þx00ð0Þ þ E0xxxx0 3

ð0Þ

=E0x

x¼xð0Þ

ð56Þ

E1; E2; E3are given by(48), E0x; E0xx; E0xxx; E1x; E1xx; E2xare given by:

Trang 9

E0x¼ r2

v ½4ðc 1Þ þ x ffiffiffiffiffiffiffiffiffiffiffi

1 x

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cx

p

þ ð1 cxÞx

þ 4ðc 1Þ þ xr2

v

1 2cx þ4cx2þ ð8c2 11c 3Þx þ 2ð3 2c2Þ

2 ffiffiffiffiffiffiffiffiffiffiffi

1 x

1 cx p

;

E0xx¼ 2r2

v 1 2cx þ4cx2þ ð8c2 11c 3Þx þ 2ð3 2c2Þ

1 x

1 cx p

þ 4ðc 1Þ þ xr2

v

2cþ8c2x3 12cð1 þcÞx2þ 3ðc2þ 6cþ 1Þx þ 4cð4 þ 3cc2Þ

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 xÞ3

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1 cxÞ3 q

2 6

3 7 5;

E0xxx¼ 3r2

v 2cþ8c2x3 12cð1 þcÞx2þ 3ðc2þ 6cþ 1Þx þ 4cð4 þ 3cc2Þ

4

ð1 cxÞ3 q

2

6

3 7

þ3ðc 1Þ2½4ðc 1Þ þ xr2

v½ð8c2 7cþ 1Þx 2ð2c2 1Þ

8

ð1 cxÞ5

E1x¼ rlr2

v 4ðc 1Þ þ 2½r2

v 4cðc 1Þx 3cr2

vx2

ðb1þ b2Þ rlr2

vx½4ðc 1Þ þ xr2

vð1 cxÞcb2þ b1

2b1b2

;

E1xx¼ 2rlr2

v r2

v 4cðc 1Þ 3cr2

vx

ðb1þ b2Þ 4ðc 1Þ þ 2ðr2

v 4ccþ 4cÞx 3cr2

vx2

2b1b2

þrlr

2

vx½4ðc 1Þ þ r2

vxð1 cxÞ 4b21b22

4cb1b2þ ð1 þcÞð1 þc 2cxÞ ðcb2þ b1Þ2

b1þ b2

ðcþ 1 2cxÞðcb2þ b1Þ

b1b2

;

E2x¼ 1

3r

2

v r2

vxð1 þ 3cÞ þ 2ðc2 2Þ

4ðc 1Þ þ x

1

6 r

4

vx2ð1 þ 3cÞ þ 4r2

vxðc2 2Þ 8ðc 1Þ

1 2cx þ4cx2þ ð8c2 11c 3Þx þ 2ð3 2c2Þ

1 x

1 cx p

here b1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cx

p

;b2¼ ffiffiffiffiffiffiffiffiffiffiffi

1 x

p

;xð0Þ is the squared dimensionless velocity of Rayleigh waves propagating in an isotropic elastic half-space that is given by (seeVinh & Ogden, 2004a):

xð0Þ ¼ 4ð1 cÞ 2 4

3cþ

R þ ffiffiffiffi D p

3

q

þ

R ffiffiffiffi D p

3

q

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 0.85

0.9 x

ε

Fig 2 Plots of the dimensionless Rayleigh wave velocity xðeÞ in the interval ½0; 1 that is calculated by the exact secular equation (3.9) in Tuan (2008) (solid line) and by the approximate formula (50) for the isotropic case (dashed line) Here we take r ¼ 3; r ¼ 0:6;c¼ 1=3 and c¼ 2=3.

Trang 10

R ¼ 2ð27 90cþ 99c2 32c3Þ=27;

and the roots in the formula(58)taking their principal values Note that xð0Þ can be calculated by another formula derived by

Malischewsky (2004), or by the approximate expressions with high accuracy obtained recently byVinh and Malischewsky (2007, 2008)

Fig 2presents the dependence one2 ½0; 1 of the dimensionless Rayleigh wave velocity x ¼ c2=c2that is calculated by the exact secular equation (3.9) inTuan (2008)(solid line) and by the approximate formula(50)for the isotropic case (dashed line) Here we take rl¼ 3; rv¼ 0:6;c¼ 1=3 and c¼ 2=3 It is shown fromFig 2that the exact velocity curve and the third-order approximate one are very close to each other for the values ofe2 ½0; 0:8

6 Conclusions

In this paper, the propagation of Rayleigh waves in an orthotropic elastic half-space covered with a thin orthotropic elastic layer is investigated The contact between the layer and half space is assumed to be sliding An approximate secular equation of third-order in terms of the dimensionless thickness of the layer is derived using the effective boundary condition method It is shown that the approximate secular equation obtained has high accuracy An approximate formula of third-or-der for the velocity of Rayleigh waves is established using the obtained approximate secular equation and it is a good approximation The approximate secular equation and the approximate formula for the velocity are fully explicit, they will

be useful in practical applications

Acknowledgement

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

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Ting, T C T (2009) Steady waves in an anisotropic elastic layer attached to a half- space or between two half- spaces -a generalization of Love waves and Stoneley waves Mathematics... approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half- space coated by a thin orthotropic elastic layer Wave Motion, 49, 681–689

Vinh, P...

3 An approximate secular equation of third-order

Now we can ignore the layer and consider the propagation of Rayleigh waves in the orthotropic elastic half- space whose surface x2¼

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