DSpace at VNU: Rayleigh waves in an orthotropic elastic half-space overlaid by an elastic layer with spring contact

DSpace at VNU: Rayleigh waves in an orthotropic elastic half-space overlaid by an elastic layer with spring contact tài...

Rayleigh waves in an orthotropic elastic half-space overlaid

by an elastic layer with spring contact

Pham Chi Vinh.Vu Thi Ngoc Anh

Received: 12 October 2015 / Accepted: 30 May 2016

 Springer Science+Business Media Dordrecht 2016

Abstract In this paper, the propagation of Rayleigh

waves in an orthotropic elastic half-space overlaid by

an orthotropic elastic layer of arbitrary uniform

thickness is investigated The layer and the half-space

are both compressible and they are in spring contact

with each other The main aim of the paper is to derive

explicit exact secular equation of the wave This

equation has been derived by using the effective

boundary condition method From the obtained

secu-lar equation, the secusecu-lar equations for the welded and

sliding contacts can be derived immediately as special

cases For the welded contact, the obtained secular

equation recovers the secular equations previously

obtained for the isotropic and orthotropic materials

Since the obtained secular equation is totally explicit it

is a good tool for nondestructively evaluating the

adhesive bond between the layer and half-space as

well as their mechanical properties

Keywords Rayleigh waves A half-space coated by

a layer Spring contact  Exact explicit secular

equation The effective boundary condition method

1 Introduction

An elastic half-space overlaid by an elastic layer is a model (structure) finding a wide range of applications such as those in seismology, acoustics, geophysics, materials science and micro-electro-mechanical systems The measurement of mechanical properties of supported layers therefore plays an important role in understanding the behaviors of this structure in applications, see for examples Makarov et al [1] and references therein Among various measurement methods, the surface/ guided wave method is most widely used [2], because

it is non-destructive and it is connected with reduced cost, less inspection time, and greater coverage [3] Among surface/guided waves, the Rayleigh wave is a versatile and convenient tool [3,4] Since the explicit dispersion relations of Rayleigh waves are employed as theoretical bases for extracting the mechanical properties of layers and half-spaces and the adhesion between them from experimental data, they are therefore the main purpose of any investigation of Rayleigh waves propagating in elastic half-spaces covered by an elastic layer

When the layer is thin (i.e., its thickness is small in comparison with the wavelength), approximate secu-lar equations of the wave are derived by the effective boundary condition method that replaces the entire effect of the thin layer on the half-space by the so-called (approximate) effective boundary conditions The effective boundary conditions are established by either replacing approximately the layer by a plate

P C Vinh ( &)  V T N Anh

Faculty of Mathematics, Mechanics and Informatics,

Hanoi University of Science, 334, Nguyen Trai Str.,

Thanh Xuan, Hanoi, Vietnam

DOI 10.1007/s11012-016-0464-5

of the layer into Taylor series of the layer thickness

[7 13] Wang et al [8] derived the first-order

approx-imate secular equation for piezoelectric materials,

Tiersten [6], Bovik [7], Steigmann and Ogden [9]

obtained the second-order approximate secular

equa-tions, Vinh and Linh [10,11], Vinh and Anh [12,13]

obtained the third-order and fourth-order approximate

secular equaations for elastic solids

For the case in which the layer thickness is arbitrary,

the results are limited When the half-space and the layer

are both isotropic, the explicit secular equation of

Rayleigh waves was derived by Haskell [14],

Ben-Menahem and Singh [15] [Eq (3.113), p 117] For the

orthotropic case, the explicit secular equation of Rayleigh

waves was derived by Sotiropoulos [16] For the case

when the half-space and the layer are both subjected pure

pre-strains, the explicit secular equation of Rayleigh

waves was derived by Ogden and Sotiropoulos [17] for

incompressible materials and by Sotiropoulos [18] for

compressible materials In all mentioned investigations,

the contact between the layer and the half-space is

perfectly bonded and the secular equations are derived by

directly expanding a six-order determinant that is

established by the traction-free conditions at the

upper-surface of the layer and the continuity conditions for

displacements and stresses through the interface

As is well known, bonded interfaces are often

compromised due to imperfect bonding conditions and

degradation over time caused by various mechanical/

thermal loadings and environmental factors [19,20]

Therefore, the imperfectly bonded interface is actual

contact between two solids

There has been a number of approaches to model

imperfect interfaces and probably the most commonly

used approach is the so-called spring contact model

[21–25] In the spring contact model, the displacements

are discontinuous through the interface, the stresses are

continuous and they are proportional to the jumps of

displacements In particular, let the interface be the

plane x2¼ 0, then the spring boundary conditions

enforced on the imperfect interface x2¼ 0 are [21,23]

rk2ðx2¼ 0þÞ ¼ rk2ðx2¼ 0Þ; k¼ 1; 2; 3;

r22 ¼ KN



u2ðx2¼ 0þÞ  u2ðx2¼ 0Þ

;

r12 ¼ KT1



u1ðx2 ¼ 0þÞ  u1ðx2 ¼ 0Þ

;

r23¼ KT2



u3ðx2 ¼ 0þÞ  u3ðx2 ¼ 0Þ

ð1Þ

where KT 1ð [ 0Þ, KT2ð [ 0Þ and KNð [ 0Þ are shear and normal spring stiffnesses The sliding contact [26]

r12ðx2¼ 0þÞ ¼ r12ðx2¼ 0Þ ¼ 0;

r23ðx2¼ 0þÞ ¼ r23ðx2¼ 0Þ ¼ 0;

u2ðx2¼ 0þÞ ¼ u2ðx2 ¼ 0Þ;

r22ðx2¼ 0þÞ ¼ r22ðx2¼ 0Þ

ð2Þ

is obtained directly from the spring boundary condi-tions (1) by letting KT1, KT 2approach to zero and KNto

go toþ1 The perfectly bonded (welded) contact [27]

ukðx2 ¼ 0þÞ ¼ ukðx2¼ 0Þ;

rk2ðx2 ¼ 0þÞ ¼ rk2ðx2¼ 0Þ; k¼ 1; 2; 3 ð3Þ

is derived directly from the spring model (1) by letting

KT 1, KT2and KNall approach toþ1

This paper is concerned with the propagation of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer of arbitrary uniform thickness The layer and the half-space are both compressible and they are in spring contact with each other The main aim of the paper is to derive explicit exact secular equation of the wave This equation has been derived by using the effective boundary condition method From the obtained secu-lar equation, the secusecu-lar equations for the welded and sliding contacts can be derived immediately as special cases For the welded contact, the obtained secular equation recovers the one derived by Sotiropolous [16] for orthotropic materials, and the one obtained by Ben-Menahem and Singh [15] for isotropic materials Since the obtained secular equation is totally explicit it is a good tool for nondestructively evaluating the adhesive bond between the layer and half-space as well as their mechanical properties

2 Exact effective boundary condition

Consider a homogeneous elastic half-space x2 0 coated by a homogeneous elastic layerh  x2 0 of thickness h The half-space and the layer are both orthotropic, compressible and they are in spring contact with each other Note that same quantities related to the half-space and the layer have the same symbol but are systematically distinguished by a bar if pertaining to the layer We are interested in the plane strain such that

ui¼ uiðx1; x2; tÞ; ui¼ uiðx1; x2; tÞ;

where ui, uiare components of the displacement

vec-tor, t is the time Since the layer is made of orthotropic

elastic materials, the strain-stress relations are



r11 ¼ c111;1þ c122;2; r22 ¼ c121;1þ c222;2;



r12 ¼ c66ðu1;2þ u2;1Þ ð5Þ

where commas indicate differentiation with respect to

spatial variables xk, rij are the stresses, the material

constants c11, c22, c12, c66 satisfy the inequalities

kk[ 0; k¼ 1; 2; 6; c1122 c212[ 0 ð6Þ

which are necessary and sufficient conditions for the

strain energy to be positive definite In the absence of

body forces, the equations of motion for the layer is



r11;1þ r12;2¼ q€1; r12;1þ r22;2¼ q€2 ð7Þ

where q is the mass density of the layer, a dot signifies

differentiation with respect to t Substituting (5) into

(7) and taking into account (4) yield

111;11þ c661;22þ ðc12þ c66Þu2;12¼ q€1;

ðc12þ c66Þu1;12þ c662;11þ c222;22¼ q€2

ð8Þ

Now we consider the propagation of a Rayleigh wave,

traveling along the interface between the layer and the

half-space with velocity cð [ 0Þ and wave number

kð [ 0Þ in the x1-direction and decaying in the x2

-direction The displacements of the Rayleigh wave in

the layer, that satisfy (8), are given by

1¼ U1ðyÞeikðx1 ctÞ; 2¼ U2ðyÞeikðx1 ctÞ ð9Þ

where y¼ kx2and



U1ðyÞ ¼A1chðp1yÞ þ A2shðp1yÞ þ A3chðp2yÞ

þ A4shðp2yÞ;



U2ðyÞ ¼i



a1A1shðp1yÞ þ a1A2chðp1yÞ þ a2A3shðp2yÞ

þ a2A4chðp2yÞ

ð10Þ

A1; A2; A3; A4are constants and for simplicity we use

the notations shð:Þ :¼ sinhð:Þ; chð:Þ :¼ coshð:Þ, the

quantities ajand pjare determined by



aj¼  pjðc12þ c66Þ

22p2

j c66þ qc2; j¼ 1; 2;

p1¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

 4 P p 2

s

; p2¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

 4 P p 2

s

;

¼ ðc12þ c66Þ

2

þ c22ð qc2 c11Þ þ c66ð qc2 c66Þ



P¼



11 qc2

66 qc2

2266

ð11Þ where p1; p2 are complex in general and no require-ments are imposed on their real and imaginary parts Using (9)–(11) into (5) we have



r12¼ k R1ðyÞeikðx 1 ctÞ; r22¼ k R2ðyÞeikðx 1 ctÞ ð12Þ

in which



R1ðyÞ ¼ b1A1shðp1yÞ þ b1A2chðp1yÞ þ b2A3shðp2yÞ

þ b2A4chðp2yÞ;



R2ðyÞ ¼ i



c1A1chðp1yÞ þ c1A2shðp1yÞ þ c2A3chðp2yÞ

þ c2A4shðp2yÞ

ð13Þ with



bn¼ c66ðpn anÞ; cn¼ ðc12þ c22pnanÞ; n¼ 1; 2

ð14Þ Suppose that surface x2¼ h is free of traction, i e



Using (12) and (13) into (15) leads to



1A1shðe1Þ  b1A2chðe1Þ þ b2A3shðe2Þ  b2A4chðe2Þ ¼ 0;



c1A1chðe1Þ  c1A2shðe1Þ þ c2A3chðe2Þ  c2A4shðe2Þ ¼ 0

ð16Þ where en¼ pne; n¼ 1; 2; e ¼ kh Putting x2¼ 0 in

(10) and (13), we deduce



U1ð0Þ ¼A1þ A3; U2ð0Þ ¼ iða1A2þ a2A4Þ;



R1ð0Þ ¼ b1A2þ b2A4; R2ð0Þ ¼ iðc1A1þ c2A3Þ

ð17Þ Solving the system (17) for A1; A2; A3; A4, we obtain

½cU1ð0Þ þ

i

½cR2ð0Þ;

A2¼ i b2

½a; b



U2ð0Þ þ a2

½a; b



R1ð0Þ;

A3¼ c1

½cU1ð0Þ 

i

½cR2ð0Þ;

A4¼  i b1

½a; b



U2ð0Þ  a1

½a; b



here we use the notations

½f ; g ¼ f2g1 f1g2; ½f  ¼ f2 f1 ð19Þ

Substituting (18) into (16) yields

a11R1ð0Þ  ia12R2ð0Þ þ b11U1ð0Þ  ib12U2ð0Þ ¼ 0;

a21R1ð0Þ  ia22R2ð0Þ þ b21U1ð0Þ  ib22U2ð0Þ ¼ 0

ð20Þ where

a11¼½a; bche

½a; b ; a12¼ 

½ bshe

½c ; a21¼

½cshe; a

½a; b ;

a22¼½cche

½c ; b11¼

½ bshe; c

½c ; b12¼



b1b2½che

½a; b ;

b21¼ c1c2½che

½c ; b22 ¼

½ b; cshe

Since the layer and the half-space are in spring contact

to each other at the plane x2¼ 0, we have from (1) (see

also [21,23,24])

r12¼ KTðu1 u1Þ; r22 ¼ KNðu2 u2Þ;



r12¼ r12; r22 ¼ r22 at x2 ¼ 0 ð22Þ

or equivalently due to (9), (10), (12) and (13)

kR1ð0Þ ¼ KT½U1ð0Þ  U1ð0Þ ; k

R2ð0Þ ¼ KN½U2ð0Þ  U2ð0Þ ;

R1ð0Þ ¼ R1ð0Þ ; R2ð0Þ ¼ R2ð0Þ

ð23Þ

Unð0Þ and Rnð0Þ (n=1,2) are the displacement and

traction amplitudes of the half-space at the interface

x2¼ 0 KTð [ 0Þ and KNð [ 0Þ are the shear and

normal spring stiffnesses, respectively From (23) it

implies that

1 When KT¼ 0 and KN! þ1 the half-space and

the layer are in sliding contact

2 When KN ! þ1 and KT ! þ1 the contact between the layer and the half-space becomes perfectly bonded

From the first two of (23) we have



U1ð0Þ ¼  k

KT

R1ð0Þ þ U1ð0Þ ;



U2ð0Þ ¼  k

KN

R2ð0Þ þ U2ð0Þ

ð24Þ

Introducing the last two of (23) and (24) into (20) leads to



a11 k

KTb11



R1ð0Þ  i

a12 k

KNb12



R2ð0Þ

þ b11U1ð0Þ  ib12U2ð0Þ ¼ 0;



a21 k

KT

b21



R1ð0Þ  i

a22 k

KN

b22



R2ð0Þ

þ b21U1ð0Þ  ib22U2ð0Þ ¼ 0 ð25Þ This is the desired exact effective boundary conditions that replace exactly the entire effect of the layer on the half-space

3 Explicit secular equation

Now we consider the propagation of a Rayleigh wave, traveling along surface x2¼ 0 of the half-space with velocity c and wave number k in the x1-direction, decaying in the x2-direction, and satisfying the exact effective boundary conditions (25) According to Vinh and Ogden [28], the displacements of the Rayleigh wave in the half-space x2[ 0 are given by

u1¼ U1ðyÞeikðx1 ctÞ; u2 ¼ U2ðyÞeikðx1 ctÞ; y¼ kx2

ð26Þ where

U1ðyÞ ¼ B1eb1 yþ B2eb2 y;

U2ðyÞ ¼ iða1B1eb1 yþ a2B2eb2 yÞ ð27Þ

B1and B2are constants to be determined, and

ak¼ ðc12þ c66Þbk

c22b2

k c66þ X; k¼ 1; 2; X ¼ qc

b1 and b2 are two roots having positive real part (in order to make the decay condition satisfied) of the following equation

b4 Sb2þ P ¼ 0 ð29Þ

S and P are calculated by (11) without bars It has been

shown that if a Rayleigh wave exists, then [28]

and [29]

P[0; Sþ P[0; b1b2¼ ffiffiffi

P

p

; b1þ b2¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Sþ 2 ffiffiffi P p q

ð31Þ Introducing (26) and (27) into the strain-stress relation

(5) without bars leads to

r12¼ kR1ðyÞeikðx1 ctÞ; r22¼ kR2ðyÞeikðx1 ctÞ ð32Þ

in which

R1ðyÞ ¼ b1B1eb1 yþ b2B2eb2 y;

R2ðyÞ ¼ iðc1B1eb1 yþ c2B2eb2 yÞ ð33Þ

and

bj¼ c66ðbjþ ajÞ; cj¼ c12 c22bjaj; j¼ 1; 2

ð34Þ Putting x2 ¼ 0 in (27) and (33) gives

U1ð0Þ ¼B1þ B2; U2ð0Þ ¼ iða1B1þ a2B2Þ;

R1ð0Þ ¼b1B1þ b2B2; R2ð0Þ ¼ iðc1B1þ c2B2Þ

ð35Þ Substituting (35) into (25) leads to two linear

equa-tions for B1 and B2, namely

fðb1ÞB1þ f ðb2ÞB2¼ 0; Fðb1ÞB1þ Fðb2ÞB2¼ 0

ð36Þ where

fðbnÞ ¼ a11 k

KT

b11

bnþ a12 k

KN

b12

cn

þ b11þ b12an;

FðbnÞ ¼ a21 k

KT

b21

bnþ a22 k

KN

b22

cn

þb21þ b22an ðn ¼ 1; 2Þ ð37Þ

For a non-trivial solution, the determinant of the

matrix of the system (36) must vanishes, i.e.,

fðb1ÞFðb2Þ  f ðb2ÞFðb1Þ ¼ 0 ð38Þ Substituting (37) into (38) and after some calculations

we arrive at

n

ða11a22 a12a21Þ  k

KN

ða11b22 a21b12Þ

þ k

KT

ða12b21 a22b11Þ

þ k

2

KTKN

ðb11b22 b12b21Þo

½c; b  ða11b21 a21b11Þ½b

þn

ða11b22 a21b12Þ  k

KT

ðb11b22 b12b21Þo

½a; b

n

ða12b21 a22b11Þ þ k

KN

ðb11b22 b12b21Þo

½c

þ ða12b22 a22b12Þ½a; c þ ðb11b22 b12b21Þ½a ¼ 0

ð39Þ With the help of (28) and (34), it is not difficult to verify that

½c;b ¼ c66

n

c212 c22ðc11 XÞ

b1b2þ Xðc11 XÞo

h;

½a;b ¼ c66ðc11 XÞðb1þ b2Þh;

½a;c ¼ c66ðc11 X  c12b1b2Þh;

½a ¼ ðX  c11 c66b1b2Þh; ½b ¼ ½a;c;

½c ¼ c22c66b1b2ðb1þ b2Þh ð40Þ where h¼ ðb2 b1Þ=ððc12þ c66Þb1b2Þ Introducing the expressions of aij and bijgiven by (21) into (39) and using the equalities (40) yield

A1þ B1she1she2þ C1she1che2þ D1she2che1

where the coefficients A1, B1, C1, D1, E1are given by (54) in the Appendix1

Equation (41) in which A1, , E1are determined by (54) in the Appendix 1 is the desired exact secular equation It is totally explicit

When e ! 0 (the layer is absent), from (41) and the last of (54) it implies

n

c212 c22ðc11 XÞo ffiffiffi

P

p

þ Xðc11 XÞ ¼ 0 ð42Þ

This equation is the secular equation of Rayleigh waves propagating along the traction-free surface of a compressible orthotropic half-space [28]

When the layer and the substrate are both isotropic

we have

c11 ¼ c22¼ k þ 2l; c12¼ k; c66¼ l;

11 ¼ c22¼ kþ 2 l; c12¼ k; c66¼ l ð43Þ

From (41) and (54) and taking into account (43) we

obtain the secular equation for the isotropic case,

namely

A2þ B2she1she2þ C2she1che2þ D2she2che1

in which the coefficients A2, B2, C2, D2, E2are given

by (55) in the Appendix2

Taking the limit of Eq (44) when cNand cTboth go

to zero we obtain the secular equation of Rayleigh

waves in an isotropic half-space coated by an isotropic

layer with welded contact By multiplying two side

this secular equation by k8=ðb12Þ we arrive

imme-diately at the well-known secular equation of Rayleigh

waves for the isotropic case, Eq (3.113), p 117 in Ref

[15] for the welded contact

Remark 1 From Eq (41) one can easily arrive at the

explicit secular equations for two special cases: the

welded contact and the sliding contact by:

1 Taking the limit of two sides of Eq (41) when KN

and KT both approach to þ1, for the welded

contact

2 Multiplying two sides of Eq (41) by KT=kc66and

then taking the limit of the resulting equation

when KT! 0 and KN ! þ1, for the sliding

contact

4 Dimensionless secular equation

It is useful to convert the secular Eq (41) into

dimensionless form For this aim we introduce

dimensionless parameters

e1¼c11

c66; e2¼c22

c66; e3¼c12

c66; e1¼11

66;

2¼66

22

; e3¼12

66

;

rl¼66

c66

; rv¼c2

2

; c2¼

ffiffiffiffiffiffi

c66 q

r

; c2¼

ffiffiffiffiffiffi

66

 q

r

;

cT ¼kc66

KT ; cN¼kc66

KN

ð45Þ

cT and cN are dimensionless shear and normal spring compliances Note that cN 0, cT 0, rv[ 0, rl[ 0 and according to (6): ek[ 0, ek[ 0ðk ¼ 1; 2; 3Þ,

e1e2 e2

3[ 0, e1 e22

3[ 0

Equation (41) can be rewritten as

ðE1þ B1Þsh2heðp1þ p2Þ

2

i

þ ðE1 B1Þsh2heðp1 p2Þ

2 i

þC1þ D1

2 sh½eðp1þ p2Þ þC1 D1

2 sh½eðp1 p2Þ

Using (54) we derive the expressions of the coeffi-cients E1þ B1, E1 B1, C1þ D1, C1 D1and A1þ E1 Introducing these expressions into (46) yields

Aðg; gÞ

sh2h eðp1þ p2Þ

2 i

ðp1þ p2Þ2  Aðg; gÞ

sh2h eðp1 p2Þ

2 i

ðp1 p2Þ2

þ Bðg; gÞsh½eðp1þ p2Þ

p1þ p2

 Bðg; gÞsh½eðp1 p2Þ

p1 p2

þ Cðg; gÞ ¼ 0

ð47Þ where p1, p2are defined by (11) in which P and S are expressed in terms of the dimensionless parameters as follows

¼ ðe1 r2

vxÞ þ e2



1 r2

vx ðe3þ 1Þ2

;



P¼ e2ðe1 r2

vxÞð1  r2

vxÞ

ð48Þ

and

Aðg;  gÞ ¼ 2 fðgÞ

1   g 2

n ð1 þ ge1=22 Þ fðgÞ

1   g 2

þ 2r1l ð1  e 3 e1=22 gÞð1   e 3 e 1=22 Þ

þ r l2ð1 þ  g e1=22 Þ fðgÞ

1  g 2

þ fðgÞ

1   g 2

h

ðc T þ c N e1=22 gÞðb 1 þ b 2 Þ þ c T cN fðgÞ

1  g 2

io

; Bðg;  gÞ ¼ fðgÞ

1   g 2 r1l n

ðb 1 þ b 2 Þ½e1=22 g þ  e1=22 

þ ½c T þ c N  e1=22  fðgÞ

1  g 2

o

; Cðg;  gÞ ¼ 2r 2

l e 1=22  fðgÞ

1  g 2

ð49Þ

g¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1  xÞ=ðe1 xÞ

p

; g¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1  r2

vxÞ=ðe1 r2

vxÞ

q

;

fðgÞ ¼ e2

3e1=22 g3þ e1g2þ ½e2ðe1 1Þ  e2

3ge1=22  1 ð50Þ function fðgÞ is given by (50) in which e1; e2 and e3

are replaced by e1; 1=e2 and e3, respectively In the

Eq (49):ðb1þ b2Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Sþ 2 ffiffiffi P p p

in which

S¼e2ðe1 xÞ þ 1  x  ðe3þ 1Þ

2

e2

;

P¼ðe1 xÞð1  xÞ

e2

ð51Þ

It is clear that the left side of (47) is an explicit function

of x and ten dimensionless parameters defined by (45)

Taking the limit of two sides of (47) when cNand cT

both go to zero we obtain the secular equation of

Rayleigh waves in an orthotropic half-space coated by

an orthotropic layer with welded contact This is the

Eq (47) in which

Aðg; gÞ ¼ 2 fðgÞ

1 g2

 ð1 þ ge1=22 Þ fðgÞ

1 g2þ 2rl1ð1  e3e1=22 gÞ

ð1  e31=22 gÞ þ r 2l ð1 þ ge1=22 Þ fðgÞ

1 g2

;

Bðg; gÞ ¼ fðgÞ

1 g2r1l ðb1þ b2Þ½e1=22 gþ e1=22 g;

Cðg; gÞ ¼ 2r2l 1=22 g fðgÞ

By comparing this equation with the secular equation

derived by Sotiropoulos, Eq (16) in Ref [16], one can

immediately discover misprints in this secular

equa-tion (and also in the secular Eq (8) in Ref [30]) which

have been mentioned in Ref [31] Note that the

eqution (47) with the coefficients given by (52) is

totally explicit, while Eq (16) in Ref [16] and Eq (8)

in [30] are both not totally explicit because they both

contain an implicit factorðs1þ s2Þ (in the expressions

of Bðg; gÞ and Bðg; gÞ)

For the sliding contact the coefficients of Eq (47)

are simplified to

Aðg; gÞ ¼ 2h fðgÞ

1 g2

i2

ðb1þ b2Þ;

Bðg; gÞ ¼ fðgÞ2rl1 fðgÞ

; Cðg; gÞ ¼ 0

ð53Þ

5 Numerical examples

As an application of the obtained secular equations we use them to numerically examine the dependence of the Rayleigh wave velocity on the dimensionless parameter e¼ kh (understood as the dimensionless thickness of the layer or the dimensionless wave number) and on the dimensionless spring compliances

cNand cT The material dimensionless parameters are taken as: For the Figs.1,3,5: e1 ¼ 3:0; e2¼ 3:5;

e3¼ 1:5; e1¼ 1:8; e2¼ 1:2; e3¼ 0:5; rv¼ 2:8;

rl¼ 0:5

For the Figs 2,4,6: e1 ¼ 3:2; e2¼ 2:8; e3¼ 1:5;

1¼ 1:8; e2¼ 1:1; e3 ¼ 0:6; rv¼ 2; rl¼ 1 The wave velocity curves of first modes in the interval e2 ½0 2:5 corresponding to the spring contact are presented in Fig.1 (modes 0, 1, 2, 3, 4, 5) and Fig.2 (modes 0, 1, 2, 3 , 4) Figures3,4 show the wave velocity curves of first four modes (0, 1, 2, 3) in the interval e2 ½0 2:5 of the welded contact (solid line) and sliding contact (dashed line) Figures 5,6

present the effect of the dimensionless spring compli-ances cN(characterizing the normal imperfection) and

cT(characterizing the shear imperfection) on the wave velocity Recall that for the perfectly bonded contact

cN ¼ cT¼ 0

It is shown from these figures that:

1 The Rayleigh wave velocity decreases when the dimensionless thickness of the layer (or the

0 0.5 1 1.5 2 2.5 0

0.2 0.4 0.6 0.8 1

ε x

Fig 1 Velocity curves of first six modes in the interval e 2

½0 2:5 for the spring contact Here we take e 1 ¼ 3:0; e 2 ¼ 3:5; e 3 ¼ 1:5;  e 1 ¼ 1:8;  e 2 ¼ 1:2;  e 3 ¼ 0:5; r v ¼ 2:8; r l ¼ 0:5;

dimensionless wave number) increases (see

Figs.1,2,3,4,5and6)

2 The picture of velocity curves for the spring

contact is quite different from the one for the

sliding and welded contacts (see Figs.1,2,3and

4) In particular:

• The velocity of the modes 0 and 1 for the

spring contact decreases much more quickly

than the one corresponding to the sliding and

welded contacts

• The modes 1, 2 for the spring contact appear

welded contacts, i e the modes 1, 2 for the spring contact initiate at the values of e which are much smaller than those for the sliding and welded contacts

• For the sliding and welded contacts the velocity of modes decreases regularly, while for the spring contact there often exist inter-vals of e in which the wave velocity is almost constant, except the modes 0 and 1

3 For the same mode, the velocity curve for the sliding contact always lies above the one for the

0

0.2

0.4

0.6

0.8

1

ε x

Fig 2 Velocity curves of first five modes in the interval e2

½0 2:5 for the spring contact Here we take e 1 ¼ 3:2;

e2¼ 2:8; e 3 ¼ 1:5;  e1¼ 1:8;  e2¼ 1:1;  e3¼ 0:6; r v ¼ 2; r l ¼ 1, c T ¼

12; c N ¼ 10

0

0.2

0.4

0.6

0.8

1

ε x

Fig 3 Velocity curves of first four modes in the interval e 2

½0 2:5 for the welded contact (solid line) and the sliding contact

(dashed line) Here we take e 1 ¼ 3:0; e 2 ¼ 3:5; e 3 ¼ 1:5;

 1 ¼ 1:8;  e 2 ¼ 1:2;  e 3 ¼ 0:5; r v ¼ 2:8; r l ¼ 0:5

0 0.2 0.4 0.6 0.8 1

ε x

Fig 4 Velocity curves of first four modes in the interval e 2

½0 2:5 for the welded contact (solid line) and the sliding contact (dashed line) Here we take e 1 ¼ 3:2; e 2 ¼ 2:8; e 3 ¼ 1:5;

1¼ 1:8;  e 2 ¼ 1:1;  e 3 ¼ 0:6; r v ¼ 2; r l ¼ 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

x

c

N , c T

Mode 0: ε=0.3

: cT= 0

Mode 1: ε=0.3 Mode 2: ε=1.1

: cN= 0

Fig 5 Dependence of the wave velocity of modes 0, 1, 2 on the normal imperfection c N (solid lines) and on the shear imperfection c T (dashed lines) Here we take e 1 ¼ 3:0;

e 2 ¼ 3:5; e 3 ¼ 1:5;  e 1 ¼ 1:8;  e 2 ¼ 1:2;  e 3 ¼ 0:5; r v ¼ 2:8; r l ¼ 0:5

4 For the mode 0, the normal spring compliance (the

normal imperfection) cN affects on the wave

velocity more strongly than the shear spring

compliance (the shear imperfection) cT, while

for others modes we have the inverse (see

Figs.5,6)

Note that in order to draw the velocity curves, the

dimensionless secular equations established in

Sect.4are employed In particular, the Eq (47) is

employed for drawing the velocity curves in the

Figs.1,2,5and6; the Eqs (52) and (53) are used

for establishing the velocity curves in the Figs.3

and 4

6 Conclusions

In this paper, the explicit exact secular equation of

Rayleigh waves propagating in an orthotropic

half-space coated by an orthotropic layer with spring

contact has been obtained This equation is derived by

using the effective boundary condition method From

the obtained secular equation, the secular equations for

the welded and sliding contacts are derived as special

cases For the welded contact, the obtained secular

equation recovers the secular equations previously

obtained for the isotropic and orthotropic materials

The obtained secular equations are a good tool for

nondestructively evaluating the adhesive bond

between the layer and half-space as well as their mechanical properties

Acknowledgments The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 107.02-2014.04.

Appendix 1 The coefficients of the secular Eq (41)

The coefficients are:

A1 ¼ 2 b12c1c2

X c11 c66

ffiffiffi P

p 

 c66ða21c1þ a12c2Þ

n

c212 c22ðc11 XÞ ffiffiffi

P

p

þ Xðc11 XÞ

 c66

n



c1c2ða21þ a12Þ þ b12ðc1þ c2Þo



c11 X  c12

ffiffiffi P

p 

 2 b12c1c2n kc66

KT

ðc11 XÞ þkc66

KN

c22

ffiffiffi P p



ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Sþ 2 ffiffiffi P p q

 k

2c66

KTKN

 ðc2

12 c22c11þ c22XÞ ffiffiffi

P

p

þ Xðc11 XÞ

B1 ¼2

1c2þ b22c2

X c11 c66

ffiffiffi P

p 

 c66ða1

1c2þ a2

2c1Þ

c212 c22ðc11 XÞ

P

p

þ Xðc11 XÞ

 c6622c2þ a11c2þ b2c2þ b2c1

 ðc11 X  c12

ffiffiffi P

p

Þ  ð b22c21þ b21c22Þ

 kc66

KT

ðc11 XÞ þkc66

KN

c22

ffiffiffi P p

Sþ 2 ffiffiffi P p q

k

2c66

KTKN

ðc2

12 c22c11þ c22XÞ ffiffiffi

P

p

þ Xðc11 XÞ

C1 ¼ c66

n



2c1½cðc11 XÞ  c22

1c2½a; b ffiffiffi

P

p o



ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Sþ 2 ffiffiffi P p q

þn



1c2½a; bkc66

KT

 b2c1½ckc66

KN

o

n

c212 c22ðc11 XÞ ffiffiffi

P

p

þ Xðc11 XÞo

D1 ¼ c66





1c2½cðX  c11Þ þ c22

2c1½a; b ffiffiffi

P

p 



ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Sþ 2 ffiffiffi P p q

n



2c1½a; bkc66

KT

 b1c2½ckc66

KN

o

n

c212 c22ðc11 XÞ ffiffiffi

P

p

þ Xðc11 XÞo

E1 ¼  A1þ c66½c½a; bn

c212 c22ðc11 XÞ

 ffiffiffi P

p

þ Xðc11 XÞo

ð54Þ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

cN, cT

Mode 2: ε = 1.7

Mode 1: ε = 0.4

Mode 0: ε = 0.4

:cN= 0

: cT= 0

Fig 6 Dependence of the wave velocity of modes 0, 1, 2 on the

normal imperfection c N (solid lines) and on the shear

imperfection c T (dashed lines) Here we take e 1 ¼ 3:2; e 2 ¼

2:8; e 3 ¼ 1:5;  e 1 ¼ 1:8;  e 2 ¼ 1:1;  e 3 ¼ 0:6; r v ¼ 2; r l ¼ 1

Appendix 2 The coefficients of the secular Eq (44)

The coefficients are:

A2 ¼ 4p1p2ð2  r2

vxÞn 2ð2  r2

vxÞðb1b2 1Þ

þ

4b1b2 ð2  xÞ2

r2l  ð4  r2vxÞ

 ð2b1b2þ x  2Þr1l  2cTcNð2  r2

vxÞ



4b1b2 ð2  xÞ2

 2cTð2  r2

vxÞb1x 2cN

 ð2  r2vxÞb2xo

;

B2 ¼ 4p2

1p22n

4b1b2ð1  r1

l Þ2

2 ð2  xÞr1

l

2o

þ ð2  r2

vxÞ2n

ð2  r2

vxÞ2ðb1b2 1Þ

 2ð2  r2

vxÞð2b1b2þ x  2Þrl1

þ

4b1b2 ð2  xÞ2

r2l o



16p21p22þ ð2  r2

vxÞ4

n

cTcN

4b1b2 ð2  xÞ2

þ cTb1xþ cNb2xo

;

C2 ¼ p2r2vx2

b1ð2  r2

vxÞ2 4b2p21

r1l

þ p2r2vx

cNð2  r2

vxÞ2 4cTp21



4b1b2 ð2  xÞ2

r1l ;

D2 ¼ p1r2vx2

b2ð2  r2

vxÞ2 4b1p22

r1l

 p1r2vx

4cNp22 cTð2  r2

vxÞ2



4b1b2 ð2  xÞ2

r1l ;

E2 ¼  A2 p1p2r4vx2

4b1b2 ð2  xÞ2

rl2 ð55Þ where

b1 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cx

p

; b2 ¼ ffiffiffiffiffiffiffiffiffiffiffi

1 x

p

; p1 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 cr2x

q

;

p2 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 r2x

q

kþ 2l; c ¼

 l



kþ 2 l; rl ¼ l

l; rv ¼ c2

2

c2¼

ffiffiffi

l

q

r

; c2¼

ffiffiffi l

 q

r

; x¼c 2

c2 ð0\x\1Þ

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