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By using the effective boundary condition method, an explicit third-order approximate secular equation of the wave has been derived that is valid for any pre-strains and for a general st

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An approximate secular equation of generalized Rayleigh waves in

pre-stressed compressible elastic solids

Pham Chi Vinha,n

a

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

b Department of Engineering Mechanics, Water Resources University of Viet Nam, 175 Tay Son Str., Hanoi, Viet Nam

a r t i c l e i n f o

Article history:

Received 11 July 2012

Received in revised form

1 November 2012

Accepted 4 November 2012

Available online 10 November 2012

Keywords:

Rayleigh waves

A pre-stressed compressible elastic

half-space

A thin pre-stressed compressible elastic

layer

Approximate secular equation

a b s t r a c t

The present paper is concerned with the propagation of Rayleigh waves in a pre-stressed elastic half-space coated with a thin pre-stressed elastic layer The half-half-space and the layer are assumed to be compressible and in welded contact with each other By using the effective boundary condition method,

an explicit third-order approximate secular equation of the wave has been derived that is valid for any pre-strains and for a general strain-energy function When the pre-strains are absent, the secular equation obtained coincides with the one for the isotropic case Numerical investigation shows that the approximate secular equation obtained is a good approximation Since explicit dispersion relations are employed as theoretical bases for extracting pre-stresses from experimental data, the secular equation obtained will be useful in practical applications

1 Introduction

A pre-stressed elastic layer on a pre-stressed elastic half-space

is a model ﬁnding a broad range of applications[1], including: the

Earth’s crust in seismology, the foundation/soil interaction in

geotechnical engineering, tissue structures in biomechanics,

coated solids in material science, and micro-electro-mechanical

systems In all these situations, the presence of pre-stresses

strongly inﬂuences the mechanical characteristics of the

struc-ture, particularly the dynamic behavior, so that the evaluation of

pre-stresses appearing in the layer and the half-space is necessary

and signiﬁcant Among various non-destructive measurement

methods, the surface/guided wave method [2] is used most

extensively, and for which the guided Rayleigh wave is most

convenient For the Rayleigh-wave approach, the explicit

disper-sion relations of Rayleigh waves supported by pre-stressed layer/

substrate interactions are employed as the theoretical bases for

extracting mechanical properties and pre-stresses of the structure

from experimental data They are therefore the main factor, the

main purpose of the investigations of Rayleigh waves propagating

in half-spaces covered by a pre-stressed layer

The propagation of Rayleigh waves in a compressible

pre-stressed elastic half-space overlaid by a compressible pre-pre-stressed

elastic layer was investigated by Ogden and Sotiropoulos[3] In

that investigation, for simplicity, pre-strains corresponding to

plane isotropic deformations were considered, and an explicit dispersion relation was obtained that is valid for any strain-energy function For the case of arbitrary pre-strains, and for the strain-energy function of Murnaghan’s form, this problem was considered recently by Akbarov and Ozisik [4] and an implicit secular equation was derived

The main aim of this paper is to ﬁnd a third-order approximate secular equation for the Rayleigh waves when the layer is assumed to be thin By using the effective boundary condition method[5– ], which replaces approximately the entire effect of the layer on the half-space by a boundary condition, an approx-imate secular equation of third-order has been derived that is valid for arbitrary pre-strains, and for a general isotropic strain-energy function

When the thickness of the layer vanishes, the derived secular equation becomes the dispersion relation of Rayleigh waves traveling along the traction-free surface of a pre-stressed isotro-pic elastic half-space (see [9,10]) Note that the propagation of surface Rayleigh waves in a half-space under the effect of pre-stress was examined also by Hayes and Rivlin[11], Chadwick and Jarvis[12], Murphy and Destrade[13], Dowaikh and Ogden[14], Vinh[15], Murdoch[16], and Ogden and Steigmann[17] Refer-ences to other works can be found in these papers When pre-stresses are absent, from the obtained secular equation we immediately arrive at the approximate secular equation of sec-ond- and third-order obtained recently by Vinh and Linh[8]for the isotropic case

A numerical investigation is carried out for a special strain-energy function, and it is shown that the approximate secular

Contents lists available atSciVerse ScienceDirect

journal homepage:www.elsevier.com/locate/nlm International Journal of Non-Linear Mechanics

n

Corresponding author Tel.: þ84 4 5532164; fax: þ84 4 8588817.

E-mail addresses: pcvinh@vnu.edu.vn, pcvinh@gmail.com (P Chi Vinh).

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equation obtained has high accuracy Therefore, it will be a good

tool for extracting pre-stresses appearing in the structure from

experimental data

2 Effective boundary condition of third-order

We consider a homogeneous surface layer of uniform

thick-ness h overlying a homogeneous half-space, both being

pre-stressed compressible isotropic elastic materials with the

underlying deformations corresponding to pure homogeneous

strains The principal directions of strain in the two solids are

aligned, one direction being normal to the planar interface

deﬁned by x2¼0 A rectangular Cartesian coordinate system

ðx1,x2,x3Þis employed with its axes coinciding with the principal

directions of the pure strain The layer occupies the domain

hox2o0 and the half-space corresponds to the region x240

The principal stretches are denoted byl1,l2,l3andl1,l2,l3in

the half-space and in the layer, respectively They are positive

constants The layer is assumed to be perfectly bonded to the

half-space Note that quantities related to the half-half-space and the layer

have the same symbol but are systematically distinguished by a

bar if pertaining to the layer

An incremental (inﬁnitesimal) motion in the ðx1,x2Þ-plane is

now superimposed on the underlying deformations, with its

displacement components in the half-space and the layer

being independent of x3 and denoted by ðu1,u2Þ and ðu1,u2Þ,

respectively

For the layer, in the absence of body forces the equations of

motion governing inﬁnitesimal motion are[9,18]

s11,1þs21,2¼ru€1, s12,1þs22,2¼ru€2, ð1Þ

wherer is the mass density of material at the static deformed

state, a superposed dot signiﬁes differentiation with respect to t, k

indicates differentiation with respect to spatial variables xk, and

sji¼Ajilkuk,l: ð2Þ

Aijklare components of the fourth-order elasticity tensor deﬁned

as follows[9,18]:

J Aiijj¼lilj

@2W

@li@lj

J Aijij¼

li

@W

@li

lj

@W

@lj

!

l2i

l2il2j

ðiaj,lialjÞ, 1

2 J AiiiiJ Aiijjþli

@W

@li

!

ðiaj,li¼ljÞ,

8

>

<

>

:

ð4Þ

J Aijji¼J Ajiij¼J Aijijli

@W

@li

for i,j A 1,2,3,W ¼ W ðl1,l2,l3Þ is the strain-energy function per

unit volume in unstressed state, J ¼l1l2l3(noting thatlk40),

all other components being zero In the stress-free conﬁguration

(3)–(5)reduce to

Aiiii¼lþ2m, Aiijj¼lðiajÞ, Aijij¼Aijji¼mðiajÞ, ð6Þ

where l:m are the Lame moduli For simplicity, we use the

notations

a11¼A1111, a22¼A2222, a12¼a21¼A1122,

(which are different from those deﬁned by[9] by a factor J) In terms of these notations Eq.(2)becomes

s11¼a11u1,1þa12u2,2,

s22¼a12u1,1þa22u2,2,

s12¼g1u2,1þgnu1,2,

s21¼gnu2,1þg2u1,2:

8

>

ð8Þ

From the strong-ellipticity condition,aik and gkare required to satisfy the inequalities[9,18]

From Eqs (1) and (8), and following the same procedure as carried out in[19]we have

U0

T0

" #

¼ M1 M2

M3 M4

" #

U T

" # , x2A½h,0, ð10Þ

where U ¼ ½u1u2T, T ¼ ½s21s22T, the symbol ‘‘T’’ indicates the transpose of a matrix, the prime signiﬁes partial derivative with respect to x2and

M1¼ 0 r2@1

r1@1 0

" #

, M2¼

1

g2 0

0 1

a22

2 6 6

3 7

7,

M3¼ r3@2þr@2

0 r4@2þr@2t

, M4¼MT: ð11Þ

Here we use the notations @1¼@=@x1, @2¼@2=@x2, @2

t¼@2=@t2and

r1¼a12

a22

, r2¼gn

g2, r3¼

a11a22a2

12

a22

, r4¼ g1g2g2

n

g2 : ð12Þ

From(10)it follows that

UðnÞ

TðnÞ

" #

¼Mn U T

" # , M ¼ M1 M2

M3 M4

" #

, n ¼ 1,2,3, ,x2A½h,0:

ð13Þ Let h be small (i.e the layer is thin), then expanding T ðhÞ then expanding T ðhÞ into Taylor series about x2¼0 up to the third-order of h gives

T ðhÞ ¼ T ð0ÞhT0ð0Þ þh

2

2T

00

ð0Þh

3

6T

000

Suppose that the surface x2¼ h is free from the stress, i.e

T ðhÞ ¼ 0 Using(13)at x2¼0 in(14)yields

IhM4þh2

2 ðM3M2þM

2

Þ (

h3

6 ½ðM3M1þM4M3ÞM2þ ðM3M2þM

2

ÞM4

)

T ð0Þ

þ hM3þh2

2ðM3M1þM4M3Þ (

h3

2

ÞM3

)

U ð0Þ

Since the layer and the half-space are bonded perfectly to each other at the plane x2¼0, it follows that Uð0Þ ¼ U ð0Þ and Tð0Þ ¼ T ð0Þ Thus, we have from(15)

IhM4þh2

2 ðM3M2þM

2

Þ (

h3

2ÞM4 ) Tð0Þ

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þ hM3þh2

2ðM3M1þM4M3Þ

(

h3

6½ðM3M1þM4M3ÞM1þ ðM3M2þM

2

ÞM3

) :Uð0Þ ¼ 0: ð16Þ

The relation(16)between the traction vector and displacement

vector of the half-space at the plane x2¼0 is called the effective

boundary condition of third-order in matrix form It replaces

approximately the entire effect of the thin layer on the substrate

With the help of(11)we can write(16)in component form as

s21þhðr1s22,1r3u1,11ru€1Þ

þh2

2 r6s21,11þ

r

g2

€s21r7u2,111rr5u€2,1

þh3

6 t1s22,111þrt2€s22,1t3u1,1111rt4u€1,11r2

g2u€1,tt

!

¼0

s22þhðr2s21,1r4u2,11ru€2Þ

þh2

2 r8s22,11þ

r

a22

€s22r7u1,111rr5u€1,1

þh3

6 t5s21,111þrt6€s21,1t7u2,1111rt8u€2,11r2

a22

€

u2,tt

!

¼0

where

r5¼r1þr2, r6¼r3

g2

þr1r2, r7¼r2r3þr1r4, r8¼ r4

a22

þr1r2,

t1¼ r7

a22

þr1r6, t2¼ r5

a22

þr1

g2, t3¼r1r7þr3r6,

t4¼r3

g2þr1r5þr6, t5¼

r7

g2þr2r8, t6¼

r5

g2þ

r2

a22

,

t7¼r2r7þr4r8; t8¼ r4

a22

þr8þr2r5: ð19Þ

3 Approximate secular equation of third-order

Suppose that the pre-stressed elastic half-space is

compressi-ble Then the unknown vectors U ¼ ½u1u2T, T ¼ ½s21s22T satisfy

Eq.(10)without the bar symbol In addition to this equation are

required the effective boundary conditions(17),(18)at x2¼0 and

the decay condition at x2¼ þ 1, namely

U ¼ T ¼ 0 at x2¼ þ 1: ð20Þ

Now we consider the propagation of a Rayleigh wave, travelling

(in the coated half-space) with velocity c and wave number k in

the x1-direction and decaying in the x2-direction In according to

Dowaikh and Ogden [9], Vinh [10] the vectors U ¼ ½u1u2T,

T ¼ ½s21s22T are given by

u1¼ ðB1ekb1x2þB2ekb2x2Þeikðx1ctÞ,

u2¼iða1B1ekb 1 x 2þa2B2ekb 2 x 2Þeikðx 1 ctÞ,

s21¼ kfb1B1ekb 1 x 2þb2B2ekb 2 x 2geikðx 1 ctÞ,

s22¼ikfZ1B1ekb 1 x 2þZ2B2ekb 2 x 2geikðx 1 ctÞ, ð21Þ

where B1and B2are constants to be determined from the effective

boundary conditions(17),(18), b1, b2are two roots of the equation

g2a22b4þ fg2ðXg1Þ þa22ðXa11Þ þ ða12þgnÞ2Þgb2

þ ðXa ÞðXg Þ ¼0, ð22Þ

whose real parts must be positive to ensure the decay condition

(20), X ¼rc2, and

ak¼ ða12þgnÞbk

a22b2g1þX¼

a11Xg2b2k

ða12þgnÞbk

, k ¼ 1,2, i ¼ ffiffiffiffiffiffiffi

1

p ,

bk¼g2bkþgnak, Zk¼a12a22akbk, k ¼ 1,2: ð23Þ From(22)we have

b21þb22¼ g2ðXg1Þ þa22ðXa11Þ þ ða12þgnÞ2

g2a22

:¼ S,

b2b2¼ðXa11ÞðXg1Þ

g2a22

One can show that if a Rayleigh wave exists (-b1,b2 having positive real parts), then (see also[10,8,19])

0oX ominfa11,g1g ð25Þ and

P 4 0, S þ 2 ffiffiffi

P p

40, b1b2¼ ffiffiffi

P p , b1þb2¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

S þ 2 ffiffiffi P p q

: ð26Þ Substituting(21)into the effective boundary conditions(17) and (18)provides two linear equations for B1and B2, namely

f ðb1ÞB1þf ðb2ÞB2¼0, Fðb1ÞB1þFðb2ÞB2¼0, ð27Þ where

f ðbnÞ ¼ bnþkhfr3þX r1Zng

þk2h2

2 ðr6þ

X

g2Þbnan½r7þX r5

þk3h3

6 Znðt1þt2X Þt3t4X X

2

g2

,

FðbnÞ ¼Znþkhfðr4þX Þanr2bng

þk2h2

2 r7þX r5Zn r8þ X

a22

!

þk3h3

6 bnðt5þt6X Þan t7þX t8þX2

a22

" #

,

n ¼ 1,2, X ¼rc2: ð28Þ Since B2þB2a0, the determinant of coefﬁcients of the homo-geneous system(27)must vanish This yields

f ðb1ÞFðb2Þf ðb2ÞFðb1Þ ¼0: ð29Þ Introducing(28)into(29)and taking into account(24) and (26), after algebraically lengthy calculations whose details are omitted,

we arrive at an approximate secular equation of third-order for the Rayleigh waves, namely

A0þA1eþA2

2 e2

þA3

6e3þOðe4

wheree¼kh (the dimensionless thickness of the layer) and

A0¼ ðg2a12þgna22a1a2Þðb2b1Þ þ ðgna12þg2a22b1b2Þða2a1Þ,

A1¼g2ðr4þX Þða1b2a2b1Þ þa22ðr3þX Þða1b1a2b2Þ,

A2¼ r4

a22

þr3

g2þ

X

g2þ

X

a22

!

A0þ2ðr4þX Þðr3þX Þða2a1Þ

þ ½r1r4r2r3þX ðr1r2Þ½ðgna12Þða2a1Þ

þ ðg2a22a1a2Þðb2b1Þ,

A3¼g2 r8þ X

a22

þ3 r6þX

g2

!!

ðr4þX Þ2r2ðr7þr5X Þ

ða2b1a1b2Þ

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þa22 r6þX

g2þ3 r8þ X

a22

!

ðr3þX Þ2r1ðr7þr5X Þ

ða2b2a1b1Þ:

ð31Þ

By(23)one can prove the following equalities

a2a1¼ ða11X þg2b1b2Þ

ða12þgnÞb1b2

ðb2b1Þ,

a2b2a1b1¼ g2ðb1þb2Þ

ða12þgnÞ ðb2b1Þ, a1a2¼ a11X

a22b1b2

,

a2b1a1b2¼ ða11XÞðb1þb2Þ

ða12þgnÞb1b2

ðb2b1Þ: ð32Þ

Introducing(32)into(31)yields

Ak¼yAkðk ¼ 0,1,2,3Þ, y¼ ðb2b1Þ=½ða12þgnÞb1b2

and

A0¼g2½a2

12a22ða11XÞb1b2þ ða11XÞ½g2

ng2ðg1XÞ,

A1¼g2½ðr4þX Þða11XÞ þa22ðr3þX Þb1b2ðb1þb2Þ,

A2¼ r4

a22

þr3

g2þ

X

g2þ

X

a22

!

A0þ2ðr3þX Þðr4þX ÞðXa11g2b1b2Þ

þ2½r1r4r2r3þ ðr1r2ÞX ½g2a12b1b2þgnðXa11Þ,

A3¼ g2ða11XÞ r8þ X

a22

þ3ðr6þX

g2Þ

ðr4þX Þ (

2r2ðr7þr5X Þ

ðb1þb2Þ

g2a22 r6þX

g2þ3ðr8þ X

a22

Þ

ðr3þX Þ (

2r1ðr7þr5X Þ

b2b1ðb1þb2Þ ð33Þ

in which b1b2 and b1þb2 are given by (24) and (26) After

removing the factory, Eq.(30)becomes

A0þA1eþA2

2 e2

þA3

6e3þOðe4

This is the desired third-order approximate secular equation and it

is fully explicit

When the thickness of the layer vanishes, i.e.e¼0, the secular

equation(34)becomes A0¼0 that is equivalent to Eq (5.11) in[9]

and Eq (25) in[10]

In dimensionless form, Eq.(34)becomes

D0þD1eþD2

2 e2

þD3

6e3

þOðe4

in which Dkðk ¼ 0,1,2,3Þ are given by

D0¼e5½e2e2ðe1xÞb1b2þ ðe1xÞ½e2e5ð1xÞ,

D1¼rme5½ðe2e51 þ r2xÞðe1xÞ þe2ðe2e2e1þr2xÞb1b2ðb1þb2Þ,

D2¼ ½e2ðe2e51Þ þ e5ðe2e2e1Þ þ ðe2þe5Þr2xD0

2rme5frmðe2e2e1þr2xÞðe2e51 þ r2xÞe3½e2e3ðe2e51Þ

e4e5ðe2e2e1Þ þ ðe2e3e4e5Þr2xgb1b2

þ2rmðxe1Þfrmðe2e2e1þr2xÞðe2e51þ r2xÞ þ e4½e2e3ðe2e51Þ

e4e5ðe2e2e1Þ þ ðe2e3e4e5Þr2xg,

D3¼rme5ðxe1Þfðe2e51 þr2xÞ½e2ðe2e51Þ

þ4e2e3e4e5þ ðe2þ3e5Þr2x þ3e5ðe2e2e1Þ

2e4e5½e2e3ðe2e51Þ þ e4e5ðe2e2e1Þ þ ðe2e3

þe4e5Þr2xgðb1þb2Þrme2e5fðe2e2e1þr2xÞ½3e2ðe2e51Þ

þ4e2e3e4e5þ ð3e2þe5Þr2x þe5ðe2e2e1Þ

2e2e3½e2e3ðe2e51Þ þ e4e5ðe2e2e1Þ

þ ðe e þe e Þr2xgbb ðb þbÞ, ð36Þ

where

b1b2¼ ffiffiffi P

p , b1þb2¼

S þ2 ffiffiffi P p q

, P ¼ðe1xÞð1xÞ

e2e5

,

S ¼e2ðe1xÞ þ e5ð1xÞðe3þe4Þ

2

e2e5

and

x ¼X

g1, e1¼a11

g1 , e2¼a22

g1 , e3¼a12

g1 , e4¼gn

g1, e5¼g2

g1,

e1¼a11

g1 , e2¼

g1

a22

, e3¼a12

g1 , e4¼

gn

g1, e5¼

g1

g2,

rm¼g1

g1, rv¼

c2

, c2¼

ffiffiffiffiffi

g1 r

r , c2¼

ffiffiffiffiffiffi

g1 r

s

The squared dimensionless Rayleigh wave speed x depends on 13 dimensionless parameters: ek, ekðk ¼ 1,2,3,4,5Þ, rm, rvande

As D0½xðeÞ ¼OðeÞ, the second-order approximate secular equa-tion is either

D0þD1eþD2

2e2þOðe3Þ ¼0, ð39Þ or

D0þD1eþ

^

D2

2e2þOðe3Þ ¼0, ð40Þ where

^

D2¼ 2rme5frmðe2e2e1þr2xÞðe2e51 þ r2xÞe3½e2e3ðe2e51Þ

e4e5ðe2e2e1Þ þ ðe2e3e4e5Þr2xgb1b2

þ2rmðxe1Þfrmðe2e2e1þr2xÞðe2e51þ r2xÞ þe4½e2e3ðe2e51Þ

e4e5ðe2e2e1Þ þ ðe2e3e4e5Þr2xg ð41Þ

is simpler than D2

4 Special cases

4.1 Unstressed case When the pre-strains are absent, lk¼lk¼1 ðk ¼ 1,2,3Þ, and the elastic constants Aijklare given by(6) For this case, from(6)

and(7)we have

g1¼g2¼gn¼m, a11¼a22¼lþ2m, a12¼l,

g1¼g2¼gn¼m, a11¼a22¼lþ2m, a12¼l ð42Þ therefore

e1¼e2¼1

g, e3¼

1

g2, e4¼e5¼1, c2¼

ffiffiffiffi

m r

r , x ¼c2

c2,

e1¼1

g, e2¼g, e3¼1

g2, e4¼e5¼1, c2¼

ffiffiffiffi

m r

s ,

rm¼m

m, rv¼

c2

where g¼m=ðlþ2mÞ and g¼m=ðlþ2mÞ Introducing (43) into

(36)yields Dk¼Dk=gðk ¼ 0,1,2,3Þ, where Dkare calculated by

D0¼ ½4ðg1Þ þ xb1b2þ ð1gxÞx,

D1¼rm½r2xð1gxÞ þð4g4 þ r2xÞb1b2ðb1þb2Þ,

D2¼ ½4ðg1Þ þ ð1þgÞr2xD0

þ2rm½ð2g1Þð4g4 þ 2gr2xÞgrmr2xð4g4 þ r2xÞb1b2

þ2rm½4ðg1Þ þ2ðgþ2rm2rmgÞr2xrmr4x2ð1gxÞ,

D3¼rmf½8ð1gÞ þ4ð2g3Þr2x þ ð3þgÞr4x2

ðgx1Þ

½8ð1gÞ þ4ðg2

2Þr2x þ ð1 þ3gÞr4x2b b gðb þb Þ, ð44Þ

Trang 5

where b1¼ ffiffiffiffiffiffiffiffiffiffiffiffi

1gx

p

, b2¼ ffiffiffiffiffiffiffiffiffi 1x p Therefore, for the unstressed case (i.e for the isotropic case), the approximate secular equation of

third-order is

D0þD1eþD2

2 e2þD3

6 e3þOðe4Þ ¼0 ð45Þ that coincides with Eq (46) in [8] In the second-order, the

approximate secular equation is either

D0þD1eþD2

2 e2þOðe3Þ ¼0 ð46Þ

or

D0þD1eþDn

2

2 e2þOðe3Þ ¼0, ð47Þ

where D0, D1and D2are given by(44), and

Dn

2¼2rm½ð2g1Þð4g4 þ 2gr2xÞgrmr2xð4g4 þ r2xÞb1b2

þ2rm4ðg1Þ þ 2ð3g2Þr2x þ r4x2

ð1gxÞ: ð48Þ

Eq.(47)identiﬁes with Eq.(47)in[8]

4.2 In-plane isotropically pre-strained solids

In this subsection we consider the case of isotropic pre-strains

(equibiaxial deformations), where (see also[3])

l1¼l2¼l, l1¼l2¼l: ð49Þ

Using(3)–(5) and (7)and taking into account(49), from(38)we

have

e1¼e2, e5¼1, e1¼1=e2, e5¼1: ð50Þ

For this case, the approximate secular equation of third-order is of

the form as in(35), where Dkare calculated by(36)and(37)in

which the fact(50)is taken into account, and

e1¼

2 @

2W

@l2

l @

2W

@l2

@2W

@l1@l2

!

þ@W

@l1

, e3¼

2 l @

2W

@l1@l2

þ@W

@l2

@W

@l1

l @

2W

@l2

@2W

@l1@l2

!

þ@W

@l1

,

e4¼1

2@W

@l2

l @

2W

@l2

@2W

@l1@l2

!

þ@W

@l1

ð51Þ

for the half-space and there are similar expressions for the layer

Continuity of the normal stress (see[3]) implies that

ll3

@W

@l2

¼ll3

@W

@l2

Now to show the accuracy of the approximate secular

equa-tions obtained we take the two materials to have strain-energy

functions of Blatz–Ko form (see[3]), namely

W ¼m

2ðl

2

1 þl22 þl23 þ2J5Þ ð53Þ

for the half-space and similarly for the layer We setl3¼l3¼1 in

addition to the assumption(49) Then, from(3)–(5)and(51)–(53)

it follows that

e1¼e1¼3, e3¼l4, e3¼l4, e4¼2l4, e4¼2l4,

l4¼ l4

l4þrð1l4Þ

Introducing(50) and (54)into(36) and (37)we have

D0¼ ðl89 þ 3xÞ ffiffiffi

P

p

þ ð3xÞ½ðl41Þðl43Þ þx,

D1¼rm

S þ2 ffiffiffi

P

p

q

fð3xÞ½ðl41Þðl43Þ þ r2x þ ffiffiffi

P p

½l89 þ3r2xg,

D2¼ 2D0

3 ½ðl

4

þ1Þðl43Þ þ2r2x2rm

3

ffiffiffi P

p

frmðl89þ 3r2xÞ

½ðl41Þðl43Þ þ r2x2l4½ðl43Þðl83Þ þ ð2l43Þr2xg

þ2rm

3 ðx3Þfrmðl

8

9 þ3r2xÞ½ðl41Þðl43Þ þr2x

þ2ð2l4Þ½ðl43Þðl83Þ þ ð2l43Þr2xg,

D3¼2

3rmðx3Þ

S þ2 ffiffiffi P p q

½ðl84 4þ3 þr2xÞð2l412 þ5r2xÞ

2ð2l4Þð3l4Þðl43þ r2xÞ2

3rm

ffiffiffi P

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

S þ 2 ffiffiffi P p q

½ðl89 þ3r2xÞð3r2x2l4Þ þ2 4ðl43Þðl43 þr2xÞ, ð55Þ where

P ¼ð3xÞð1xÞ

3 , S ¼

ð64xÞ

3 ,

rm¼l4þrð1l4Þ

r , r

2¼ R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

l4þrð1l4Þ

Here, in addition to l we introduce two new dimensionless parameters r ¼m=m and R ¼ ðmr l2Þ=ðmr l2Þ In this case, the squared dimensionless Rayleigh wave velocity x ¼ ðl4rc2Þ=m is determined (approximately) by the equation

D0þD1eþD2

2 e2

þD3

6e3

in which Dk ðk ¼ 0,1,2,3Þ are given by(55) It is clear from(55)

and (57) that x is a function in terms of four dimensionless parameters, namely:l, r, R, ande

Fig 1shows the dependence on the dimensionless thickness of the layer eA½0, 1 of the squared dimensionless Rayleigh wave velocity x (with given values ofl,r and R) that is calculated by the approximate secular equations of second- and third-order, and by the exact secular equation, Eq (8) in Ref.[3] It is shown from

Fig 1that (i) the approximate curves of second- and third-order

of x are very close to the exact curve for the values ofeA½0, 1; (ii) foreA½0, 0:5 these curves almost coincide with each other These facts show that the approximate secular equations obtained are good approximations

5 Conclusions

In this paper, the propagation of Rayleigh waves in a pre-stressed compressible isotropic elastic half-space coated by a thin

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

ε

λ=0.9, r=0.3, R=2

Fig 1 Plots of xðeÞ calculated by the approximate secular equation of second-order (dash-dot line), by the approximate secular equation of third-second-order (dashed

Trang 6

a pre-stressed compressible isotropic elastic layer has been

investigated An effective boundary conditions of third-order are

established that replaces approximately the entire effect of the

layer on the half-space Then, by using it an explicit third-order

approximate secular equation of the wave has been derived that

is valid for any pre-strains and for a general strain-energy

function This approximate secular equation recovers the one for

the isotropic case where the pre-strains are absent It is shown

that the approximate secular equation obtained has high

accu-racy Therefore, it will be helpful in practical applications

Acknowledgement

The work was supported by the Viet Nam National Foundation

for Science and Technology Development (NAFOSTED) under

Grant no 107.02-2012.12 and partly by the Abdus Salam

Inter-national Center for Theoretical Physics (ICTP)

References

[1] D Bigoni, M Gei, A.B Movchan, Dynamics of a pre-stressed stiff layer on an

elastic half space: ﬁltering and band gap characteristics of periodic structural

models derived from long-wave asymptotics, Journal of the Mechanics and

Physics of Solids 56 (2008) 2494–2520.

[2] A.G Every, Measurement of the near-surface elastic properties of solids and

thin supported ﬁlms, Measurement Science and Technology 13 (2002)

R21–R39.

[3] R.W Ogden, D.A Sotiropoulos, The effect of pre-stresses on guided ultrasonic

waves between a surface layer and a half-space, Ultrasonics 34 (1996)

491–494.

[4] S.D Akbarov, M Ozisik, The inﬂuence of the third order elastic constants to

the generalized Rayleigh wave dispersion in a pre-stressed stratiﬁed

half-plane, International Journal of Engineering Science 41 (2003) 2047–2061.

[5] H.F Tiersten, Elastic surface waves guided by thin ﬁlms, Journal of Applied Physics 40 (1969) 770–789.

[6] P Bovik, A comparison between the Tiersten model and O boundary conditions for elastic surface waves guided by thin layers, ASME Journal of Applied Mechanics 63 (1996) 162–167.

[7] P.G Malischewsky, Seismological implications of impedance-like boundary conditions, in: Proceedings of the Days on Diffraction, St Petersburg, Russia,

2011, pp 137–140.

[8] Pham Chi Vinh, Nguyen Thi Khanh Linh, An approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half-space coated by a thin orthotropic elastic layer, Wave Motion 49 (2012) 681–689.

[9] M.A Dowaikh, R.W Ogden, On surface waves and deformations in a compressible elastic half-space, Stability and Applied Analysis of Continuous Media 1 (1) (1991) 27–45.

[10] Pham Chi Vinh, On formulas for the Rayleigh wave velocity in pre-stressed compressible solids, Wave Motion 48 (2011) 614–625.

[11] M Hayes, R.S Rivlin, Surface waves in deformed elastic materials, Archive for Rational Mechanics and Analysis 8 (1961) 358–380.

[12] P Chadwick, D.A Jarvis, Surface waves in pre-stressed elastic bodies, Proceedings of the Royal Society of London Series A 366 (1979) 517–536 [13] J.C Murphy, M Destrade, Surface waves and surface stability for a pre-stressed, unconstrained, non-linearly elastic half-space, International Journal

of Non-Linear Mechanics 44 (2009) 545–551.

[14] M.A Dowaikh, R.W Ogden, On surface waves and deformations in a pre-stressed incompressible elastic solid, IMA Journal of Applied Mathematics 44 (1990) 261–384.

[15] Pham Chi Vinh, On formulas for the velocity of Rayleigh waves in pre-strained incompressible elastic solids, Transactions of ASME Journal of Applied Mechanics 77 (2) (2010) 021006 (9 pages).

[16] A.I Murdoch, The propagation of surface waves in bodies with material boundaries, Journal of the Mechanics and Physics of Solids 24 (1976) 137–146.

[17] R.W Ogden, D.J Steigmann, Plane strain dynamics of elastic solids with intrinsic boundary elasticity and application to surface wave propagation, Journal of the Mechanics and Physics of Solids 50 (2002) 1869–1896 [18] R.W Ogden, Non-linear Elastic Deformations, Ellis Horwood, Chichester, 1984.

[19] Pham Chi Vinh, Explicit secular equations of Rayleigh waves in elastic media under the inﬂuence of gravity and initial stress, Applied Mathematics and Computation 215 (2009) 395–404.

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