Rayleigh waves with impedance boundary conditions in incompressible anisotropic half spaces

Rayleigh waves with impedance boundary conditions in incompressible anisotropic half spaces tài liệu, giáo án, bài giảng...

Rayleigh waves with impedance boundary conditions

in incompressible anisotropic half-spaces

a

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

b

Faculty of Civil and Industrial Construction, National University of Civil Engineering, 55 Giai Phong Str., Hanoi, Viet Nam

Article history:

Received 14 March 2014

Accepted 11 August 2014

Keywords:

Rayleigh waves

Impedance boundary conditions

Incompressible

Orthotropic

Monoclinic

Explicit secular equation

a b s t r a c t

In this paper, the propagation of Rayleigh waves in an incompressible elastic half-space with impedance boundary conditions is investigated The half-space is assumed to be orthotropic and monoclinic with the symmetry plane x3¼ 0 The main aim of the paper

is to derive explicit secular equations of the wave For the orthotropic case, the secular equation is obtained by employing the traditional approach It is an irrational equation For the monoclinic case, the method of polarization vector is used for deriving the secular equation This is an algebraic equation of eighth-order When the impedance parameters vanish, the equations obtained coincide with the corresponding secular equations of Rayleigh waves with traction-free boundary conditions

Ó 2014 Elsevier Ltd All rights reserved

1 Introduction

Elastic surface waves, discovered byRayleigh (1885)more than 120 years ago for compressible isotropic elastic solids, have been studied extensively and exploited in a wide range of applications in seismology, acoustics, geophysics, telecom-munications industry and materials science, for example For Rayleigh waves their explicit secular equation are important in practical applications They can be used for solving the direct (forward) problems: evaluating the dependence of the wave velocity on material parameters, especially for solving the inverse problems: to determine material parameters from mea-sured values of wave velocity Therefore, explicit secular equations are always the main purpose for any investigation of Rayleigh waves

In the context of Rayleigh waves, it is almost always assumed that the half-spaces are free of traction As mentioned in

Godoy, Durn, and Ndlec (2012), in many fields of physics such as acoustics and electromagnetism, it is common to use imped-ance boundary conditions, that is, when a linear combination of the unknown function and their derivatives is prescribed on the boundary See, for examples, Antipov (2002), Zakharov (2006), Yla-Oijala and Jarvenppa (2006), Mathews and Jeans (2007), Castro and Kapanadze (2008) and Qin and Colton (2012), for the acoustics case andSenior (1960), Asghar and Zahid (1986), Stupfel and Poget (2011) and Hiptmair, Lopez-Fernandez, and Paganini (2014)for the electromagnetism one, and the references therein In the other hand, when studying the propagation of Rayleigh waves in a half-space coated by

a thin layer, the researchers often replace the effect of the thin layer on the half-space by the effective boundary conditions

on the surface of the half-space, see, for examples,Achenbach and Keshava (1967), Tiersten (1969), Bovik (1996), Steigmann and Ogden (2007), Vinh and Khanh Linh (2012, 2013), Vinh and Anh (2014a, 2014b) and Vinh, Anh, and Thanh (2014) These

http://dx.doi.org/10.1016/j.ijengsci.2014.08.002

⇑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

conditions lead to the impedance boundary conditions on the surface The Rayleigh is then considered as a surface wave that propagates in a half-space without coating whose surface is not traction-free but is subjected the impedance boundary con-ditions As addressed inMakarov, Chilla, and Frohlich (1995)andNiklasson, Datta, and Dunn (2000), a thin layer attached to a half-space is a model finding a broad range of applications in modern technology Rayleigh waves with impedance boundary conditions are therefore needed to be investigated However, very few investigations on Rayleigh waves with impedance boundary conditions have been done.Malischewsky (1987)considered the propagation of Rayleigh waves with Tiersten’s impedance boundary conditions and provided a secular equation Recently,Godoy et al (2012)investigated the existence and uniqueness of Rayleigh waves with impedance boundary conditions which are a special case of Tiersten’s impedance boundary conditions InGodoy et al (2012)andMalischewsky (1987), the half-space is assumed to be isotropic Note that the Tiersten impedance boundary conditions are not accurate ones

The main purpose of this paper is to study the propagation of Rayleigh waves with Tiersten’s impedance boundary con-ditions (Malischewsky, 1987) in anisotropic incompressible elastic half-spaces Two cases of anisotropy are considered: orthotropic materials and monoclinic ones with the symmetry plane x3¼ 0 For the orthotropic case, the secular equation

is obtained by employing the traditional techniques It is an irrational equation For the monoclinic case, for obtaining the secular equation we use the method of polarization vector The secular equation obtained is an algebraic equation of eighth-order When the impedance parameters vanish, the obtained equations coincide with the corresponding secular equation of Rayleigh waves with traction-free boundary conditions

2 Orthotropic half-spaces

Consider an elastic half-space which occupies the domain x2P0 We are interested in the plane strain such that:

where t is the time Suppose that the half-space is made of incompressible orthotropic elastic material, then the strain–stress relations are (Nair & Sotiropoulos, 1997):

r11þ p ¼ c11u1;1þ c12u2;2

r22þ p ¼ c12u1;1þ c22u2;2

r12¼ c66ðu1;2þ u2;1Þ

8

>

whererijand cijare respectively the stresses and the material constants, p ¼ pðx1;x2;tÞ is the hydrostatic pressure associated with the incompressibility constraint, commas indicate differentiation with respect to spatial variables xk The elastic con-stants c11; c22; c12; c66satisfy the inequalities:

cii>0; i ¼ 1; 2; 6; c11þ c22 2c12>0 ð3Þ

which are necessary and sufficient conditions for the strain energy of the material to be positive semi-definite For an incom-pressible material, we have:

from which we deduce the existence of a scalar function, denoted wðx1;x2;tÞ, such that:

In the absence of body forces, equations of motion are:

r11;1þr12;2¼q€u1

r12;1þr22;2¼q€u2



ð6Þ

whereqis the mass density, a superposed dot signifies differentiation with respect to t Introducing Eqs.(2) and (5)into Eq

(6)and eliminating p from the resulting equations lead to an equation forw, namely:

c66w;1111þ ðc11 2c12þ c22 2c66Þw;1122þ c66w;2222¼q w€;11þ €w;22

ð7Þ

Consider the propagation of a Rayleigh wave, traveling with velocity cð> 0Þ and wave number kð> 0Þ in the x1-direction and decaying in the x2-direction, i e.:

Suppose that the surface x2¼ 0 is subjected to impedance boundary conditions such that (Godoy et al., 2012; Malischewsky,

1987):

wherex¼ kc is the wave circular frequency, Z1; Z2ð2 RÞ are impedance parameters whose dimension is of stress/velocity (Godoy et al., 2012; Malischewsky, 1987) Using Eqs.(2) and (5)and the first of(6), the impedance boundary conditions(9)is expressed in term ofwas:

c66ðw;22 w;11Þ þxZ1w;2¼ 0; at x2¼ 0;

c66ðw;222 w;112Þ þ ðc11 2c12þ c22Þw;112þxZ2w;11qw€;2¼ 0 at x2¼ 0 ð10Þ

From(5)and the decay condition(8)it is required that:

According toOgden and Vinh (2004)wðx1;x2;tÞ is given by:

where y ¼ kx2 Substitution of Eq.(12)into Eq.(7)yields:

where a prime indicates differentiation with respect to y, d ¼ ðc11þ c22 2c12Þ=c66; x ¼ c2=c2, c2¼ c66=q Note that due to

(3), d > 0 In terms of / the impedance boundary conditions(10)become:

/00ð0Þ þ d1 ffiffiffi

x

p

/0ð0Þ þ /ð0Þ ¼ 0 /000ð0Þ þ ð1  d þ xÞ/0ð0Þ  d2 ffiffiffi

x p

where dn¼ Zn= ffiffiffiffiffiffiffiffiffiffiqc

66

p

ð2 RÞ; n ¼ 1; 2, are dimensionless impedance parameters

From Eqs.(11) and (12)it follows:

Thus, the problem is reduced to solving Eq.(13)with the boundary conditions(14) and (15) The general solution for /ðyÞ that satisfies the condition(15)is (Ogden & Vinh, 2004):

/ðyÞ ¼ Aes1 y

þ Bes2 y

ð16Þ

where A and B are constants to be determined, s1and s2are the roots of equation:

with positive real parts It follows from Eq.(17):

s2þ s2¼ d  2  x :¼ S; s2:s2¼ 1  x :¼ P ð18Þ

It is not difficult to verify that if a Rayleigh wave exists (! s1;s2having positive real parts), then:

and

s1:s2¼ ffiffiffi

P

p

; s1þ s2¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

S þ 2 ffiffiffi P p q

ð20Þ

Substituting Eq.(16)into the boundary conditions(14), we obtain:

s2 d1s1 ffiffiffi

x p

þ 1

A þ s2 d1s2 ffiffiffi

x p

þ 1

B ¼ 0

s3 ðd  1  xÞs1þ d2 ffiffiffi

x p

A þ s3 ðd  1  xÞs2þ d2 ffiffiffi

x p

For a nontrivial solution, the determinant of coefficients of the system(21)must vanish After removal of the factor ðs2 s1Þ, this yields:

s2þ s2þ s2s2þ ðd  xÞs1s2 ðd1s1s2þ d2Þðs1þ s2Þ ffiffiffi

x p

þ d1d2x  ðd  1  xÞ ¼ 0 ð22Þ

Using Eqs.(18) and (20), Eq.(22)becomes:

ðd  xÞ ffiffiffiffiffiffiffiffiffiffiffi

1  x

p

þ ðd1d2 1Þx ¼ d1

ffiffiffiffiffiffiffiffiffiffiffi

1  x

p

þ d2

x

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

d 2  x þ 2 ffiffiffiffiffiffiffiffiffiffiffi

1  x p q

ð23Þ

Eq.(23)is the (dimensionless) secular equation of Rayleigh waves propagating in an incompressible orthotropic elastic half-space whose surface is subjected to the impedance boundary conditions(9)

Taking d1¼ d2¼ 0 in Eq.(23)we obtain the (dimensionless) secular equation of Rayleigh waves propagating along a trac-tion-free surface of incompressible orthotropic elastic half-space, namely (Ogden & Vinh, 2004):

ðd  xÞ ffiffiffiffiffiffiffiffiffiffiffi

1  x

p

When the elastic half-space is transversely isotropic (with the isotropic axis being x3-axis): c11¼ c22, c11 c12¼ 2c66, then

d¼ 4 Taking into account this fact, Eq.(23)simplifies to:

ð4  xÞ ffiffiffiffiffiffiffiffiffiffiffi

1  x

p

þ ðd1d2 1Þx ¼ d1

ffiffiffiffiffiffiffiffiffiffiffi

1  x

p

þ d2

x p

1 þ ffiffiffiffiffiffiffiffiffiffiffi

1  x p

ð25Þ

For isotropic materials, the secular equation is also of the form(25)in which x ¼qc2=l; lis the shear modulus

3 Monoclinic materials with the symmetry plane x3¼ 0

3.1 Basic equations in matrix form

Consider a linearly elastic half-space x2P0, made of a monoclinic material with the symmetry plane x3¼ 0 For such materials, in-plane motions are decoupled from anti-plane motions, therefore we can consider the plane strain such that:

u1¼ u1ðx1;x2;tÞ; u2¼ u2ðx1;x2;tÞ; u3 0 ð26Þ

where t is the time For the monoclinic material with the symmetry plane x3¼ 0, the strain–stress relations are (Nair & Sotiropoulos, 1999):

r11þ p ¼ c11u1;1þ c12u2;2þ c16ðu1;2þ u2;1Þ

r22þ p ¼ c12u1;1þ c22u2;2þ c26ðu1;2þ u2;1Þ

r12¼ c16u1;1þ c26u2;2þ c66ðu1;2þ u2;1Þ

8

>

in which p ¼ pðx1;x2;tÞ is the hydrostatic pressure associated with the incompressibility constraint In the absence of body forces, equations of motion are:

r11;1þr12;2¼q€u1

r12;1þr22;2¼q€u2



ð28Þ

The incompressibility constraint reads as:

Solving Eqs.(27)3and(29)for u1;2and u2;2we have:

u1;2¼ b1u1;1 u2;1þ 1

c 66r12

u2;2¼ u1;1

(

ð30Þ

where

b1¼c26 c16

c66

Using the first of(28)and taking into account(27)1,(27)2and(30)yield:

in which

a1¼ c11 2c12þ c22ðc16 c26Þ

2

From the second of(28)it follows:

In matrix form Eqs.(30), (32) and (34)are written as:

where

f¼

u1

u2

r12

r22

2

6

6

3

7

7; M ¼ M1 M2

M3 M4

ð36Þ

in which the matrices (operators) Mkare given by:

M1¼ b1@1 @1

@1 0

" #

; M2¼

1

c 66 0

0 0

" #

M3¼ q@2t a1@2 0

0 q@2t

2

4

3 5; M4¼ MT

1

ð37Þ

the symbol ‘‘T’’ indicates transpose of a matrix, the prime signifies derivative with respect to x2and we use the notations

@1¼ @=@x1; @2¼ @2=@x2; @2t ¼ @2=@t2 Eq.(35)is the matrix formulation of the plane strain for incompressible monoclinic solids with the symmetry plane x3¼ 0 (see alsoVinh & Seriani, 2009, 2010) From Eq.(35)we immediately arrive at Stroh’s formulation (Stroh, 1962)

3.2 Rayleigh waves Stroh’s formulation

Now we consider the propagation of a Rayleigh wave, traveling with velocity c ð> 0Þ and wave number k ð> 0Þ in the x1 -direction and decaying in the x2-direction Then, the displacements and stresses of the Rayleigh wave are sought in the form:

un¼ UnðyÞeikðx 1 ctÞ; rn2¼ iktnðyÞeikðx 1 ctÞ; n ¼ 1; 2; y ¼ kx2 ð38Þ

Substituting(38)into Eq.(35)leads to:

where the prime signifies differentiation with respect to y and:

n¼ u

t ; u ¼

U1

U2

; t ¼ t1

t2

; N ¼ N1 N2

N3 N4

ð40Þ

in which the matrices Nkare defined by:

N1¼ b1 1

1 0

; N2¼

1

c 66 0

0 0

" #

; N3¼ X  a1 0

X ¼qc2 In addition to Eq.(39)are required the decay condition:

Suppose the half-space is subjected to the impedance boundary conditions(9) In terms of Ukand tk, it is expressed as follows:

t1¼ id1

ffiffiffiffiffiffiffiffiffiffi

c66X

p

U1; t2¼ id2

ffiffiffiffiffiffiffiffiffiffi

c66X

p

that can be written in matrix form as:

t ¼ Au at y ¼ 0; A ¼ id1

ffiffiffiffiffiffiffiffiffiffi

c66X p

0

0 id2 ffiffiffiffiffiffiffiffiffiffi

c66X p

ð44Þ

Note that AT¼ A, the bar indicates complex conjugate By the transformation:

Eq.(39), the decay condition(42)and the boundary condition(44)become:

and

where

w ¼ u

Q1 Q2

K Q4

ð48Þ

in which the matrices Qkand K are expressed in terms of matrices Nkand A by:

Q1¼ N1þ N2A; Q2¼ N2

K ¼ N3þ N4A  AN1 AN2A; Q4¼ N4 AN2

ð49Þ

From(49)and taking into account the facts:

N2¼ NT

2; N3¼ NT

3; N4¼ NT

one can show that:



QT

2¼ Q2; KT

¼ K; Q4¼ QT

Eqs.(39) and (46)are referred to Stroh’s formulation (Stroh, 1962)

3.3 Fundamental equations

Proposition 1 If 2m-vector YðyÞ is a solution of the problem:

where the prime signifies differentiation with respect to y and:

P ¼ P1 P2

P3 P4

ð53Þ

m  m-matrices Pkare constant matrices (being independent of y) and they satisfy the equalities:

P2¼ PT

2; P3¼ PT

3; P4¼ PT

then:



where

I ¼ 0 I

I 0

I is m  m identity matrix We call Eq.(55)the fundamental equations

Proof:

Lemma 1 Suppose the matrix P expressed by(53)is invertible:

P1

¼ P

ð1Þ

1 Pð1Þ2

Pð1Þ

3 Pð1Þ

4

" #

ð56Þ

and the equalities(54)hold for the matrices Pk Then, these equalities also hold for the matrices Pð1Þk

Proof From PP1

¼ I it follows:

P1Pð1Þ1 þ P2Pð1Þ3 ¼ I; P1Pð1Þ2 þ P2Pð1Þ4 ¼ 0

P3Pð1Þ

1 þ P4Pð1Þ

3 ¼ 0; P3Pð1Þ

2 þ P4Pð1Þ

4 ¼ I

8

<

Taking transpose and complex conjugate two sides of the equalities(57)and using(54)yield:

Pð1Þ

3

T P2þ Pð1Þ1 T P4¼ I; Pð1Þ

4

T P2þ Pð1Þ2 T P4¼ 0

Pð1Þ

3 T P1þ Pð1Þ1 T P3¼ 0; Pð1Þ4 T P1þ Pð1Þ2 T P3¼ I

8

<

equivalently

Pð1Þ4 T Pð1Þ2 T

Pð1Þ3 T Pð1Þ1 T

2

4

3

5 P1 P2

P3 P4

That means:

P1

¼ P

ð1Þ

4 T Pð1Þ2 T

Pð1ÞT Pð1ÞT

2

4

3

From(56) and (60)and the uniqueness of P1it follows:

Pð1Þ

2 ¼ Pð1Þ2 T; Pð1Þ

3 ¼ Pð1Þ3 T; Pð1Þ

The proof is completed h

Lemma 2 Suppose the matrix P expressed by(53)is invertible and the equalities(54)hold for the matrices Pk For all n 2 Z the matrix Pnis expressed as:

Pn¼ P

ðnÞ

1 PðnÞ

2

PðnÞ3 PðnÞ4

" #

Then, the equalities(54)also hold for the matrices PðnÞk

Proof

+ Clearly, the equalities(54)hold for matrices Pð0Þ

k and Pð1Þ

k + Assume the equalities(54)hold for the matrices PðnÞ

k ; n > 1 Then, it is not difficult to show that these equalities are sat-isfied for the matrices Pðnþ1Þk That means the equalities(54)hold for PðnÞk for all n 2 Z; n P 0

+ By theLemma 1, the equalities(54)hold for PðnÞ

k for all n 2 Z; n 6 0 The proof of theLemma 2is finished h

Lemma 3 Suppose the matrix P expressed by(53)is invertible and the matrices Pksatisfy the equalities(54) Then we have:

Proof By theLemma 2, PðnÞk satisfy the equalities(54)for all n 2 Z With this fact one can see that:

^IPn

¼ P

ðnÞ

3 PðnÞ4

PðnÞ

1 PðnÞ

2

" #

! ^IPnT¼ P

ðnÞ

3 T PðnÞ1 T

PðnÞ4 T PðnÞ2 T

2 4

3

5 ¼ ^IPn



Proof of the proposition 1 Pre-multiplying two sides of the Eq.(52)1by YT^IPnwe have:



YT^I Pn

Y0

Taking transpose and complex conjugate two sides of Eq.(64)and using(63)yield:

ðY0ÞT^I Pn

From(64) and (65)it follows:

d

dy



YT^I Pn

Y

¼ 0 ! YT^I Pn

Y ¼ C 8 y 2 ½0 þ 1

where C is a constant Due to the second of(52)the constant C must be zero Therefore we have:



Taking y ¼ 0 in Eq.(66)we arrive at the fundamental Eq.(55) The proposition is proved

Remark 1

(i) Eq.(55)recover the fundamental equation (15) inCollet and Destrade (2004)when P is a real matrix

(ii) There are at most (2m  1) independent fundamental equations according to the Cayley–Hamilton theorem

3.4 Explicit secular equations

Now in Eq.(55)we take P ¼ Q that is given by(48) and (49), and Y ¼ w According to the second of(47):rð0Þ ¼ 0, Eq

(55)are therefore simplified to:



As Q is a 4  4-matrix, according toRemark 1, (ii), it is sufficiently to take three different values of n for deriving the secular equation of the wave It seems that the choice n ¼ 1; 1; 2 is the best one (Ting, 2004) Suppose U1ð0Þ – 0, then the vector uð0Þ can be written as: uð0Þ=U1ð0Þ½1 aT, wherea¼ U2ð0Þ=U1ð0Þ is a complex number,a¼ a þ ib; a; b are real Introducing the expression of uð0Þ into(67)and taking into account the fact that KðnÞis hermitian (due to(51)andLemma 2) we have:

½1 a K

ðnÞ

11 KðnÞ12

KðnÞ12 KðnÞ22

" #

1

that provides three equations:

Kð1Þ11 þ Kð1Þ12 aþ Kð1Þ12 aþ Kð1Þ22 a a¼ 0

Kð1Þ

11 þ Kð1Þ12aþ Kð1Þ12aþ Kð1Þ22a a¼ 0

Kð2Þ

11 þ Kð2Þ12aþ Kð2Þ12aþ Kð2Þ22a a¼ 0

8

>

<

>

:

ð69Þ

the elements KðnÞij of the matrices KðnÞðn ¼ 1; 1; 2Þ are given by:

Kð1Þ11 ¼ a1þ ðd2þ 1ÞX; Kð1Þ22 ¼ X; Kð1Þ12 ¼ i ffiffiffiffiffiffiffiffiffiffi

c66X

p

Kð2Þ11 ¼ 2b1½a1þ ðd21þ 1ÞX

; Kð2Þ22 ¼ 0; Kð2Þ12 ¼ a1 ðd1d2þ 2ÞX þ ib1

ffiffiffiffiffiffiffiffiffiffi

c66X

p

and Kð1Þ

ij ¼ ^Kð1Þ

ij =q where qð2 RÞ is the determinant of the matrix Q and:

^

Kð1Þ11 ¼ X; K^ð1Þ22 ¼c66ða1 d

2XÞ  ða1þ c66þ b21c66ÞX þ X2

c66

;

^

Kð1Þ12 ¼ b1X þ i ðd1 d2Þ ffiffiffiffiffiffiffiffiffiffi

c66X

p

 d1

ffiffiffiffiffiffiffiffiffiffiffiffi X=c66

p X

As the matrix KðnÞis hermitian, KðnÞ

11, KðnÞ

22 ðn ¼ 1; 2Þ; ^Kð1Þ

11 , ^Kð1Þ

22 are real and KðnÞ

12 ðn ¼ 1; 2Þ; ^Kð1Þ

12 are complex numbers whose real and imaginary parts are denoted, respectively, by Kðn;rÞ

12 and Kðn;iÞ

12 ðn ¼ 1; 2Þ; ^Kð1;rÞ

12 and ^Kð1;iÞ

12 Substitutinga¼ a þ ib,

KðnÞ

12 = Kðn;rÞ

12 þ iKðn;iÞ12 ðn ¼ 1; 2Þ, ^Kð1Þ

12 = ^Kð1;rÞ

12 þ i^Kð1;iÞ

12 into Eq.(69)we arrive at a system of three linear equations, namely:

^

Kð1;rÞ12 K^ð1;iÞ12 K^ð1Þ22

Kð1;rÞ12 Kð1;iÞ12 Kð1Þ22

Kð2;rÞ12 Kð2;iÞ12 Kð2Þ22

2

6

4

3 7 5

2a

2b

a2þ b2

2 6

3 7

5 ¼

^Kð1Þ 11

Kð1Þ11

Kð2Þ11

2 6 4

3 7

whose solution is:

where D is the determinant of the 3  3 matrix in Eq.(73), Dkare the determinants of matrices obtained by replacing this matrix’s kth column with the vector on the right-hand side of Eq.(73) It follows from Eq.(74)that:

which is the desired explicit secular equation of the waves The expansions of the determinants D; Dkare lengthy and are not displayed here, but they are easily computed by using the expressions in Eqs.(70)–(72) Also from(70)–(72)one can see that the secular Eq.(75)is an algebraic equation of eighth-order in X

Remark 2 To obtain the secular equation of the wave we can apply the fundamental Eq.(55)with the impedance boundary condition(43)in which P ¼ N However, the derivation is more complicated, especially when the size of the square matrix N

is higher than 4

When the impedance parameters vanish, i.e d1¼ d2¼ 0, from(70)–(72)it follows

Kð1;rÞ

12 ¼ Kð1;iÞ12 ¼ Kð2;iÞ12 ¼ Kð2Þ22 ¼ ^Kð1;iÞ

In view of(76) and (73)is simplified to:

^

Kð1;rÞ12 K^ð1Þ11 K^ð1Þ22

0 Kð1Þ

11 Kð1Þ 22

Kð2;rÞ

12 Kð2Þ

11 0

2

6

4

3 7 5

2a 1

a2þ b2

2 6

3 7

5 ¼

0 0 0

2 6

3

whose determinant of this system’s matrix must be zero, i.e.:

Kð1;rÞ12 K^ð1Þ11 K^ð1Þ22

0 Kð1Þ11 Kð1Þ22

Kð2;rÞ

12 Kð2Þ

11 0

2

6

4

3 7

or equivalently:

^

Kð1;rÞ

12 Kð1Þ

22Kð2Þ

11 þ Kð2;rÞ12 K^ð1Þ

22 Kð1Þ

11  Kð1Þ22K^ð1Þ

11

Again from(70)–(72)we have:

Kð1Þ11 ¼ X  a1; Kð1Þ22 ¼ X;

Kð2Þ11 ¼ 2b1ða1þ XÞ; Kð2;rÞ12 ¼ a1 2X;

^

Kð1Þ11 ¼ X; K^ð1;rÞ12 ¼ b1X;

^

Kð1Þ22 ¼c66a1 ða1þ c66þ b

2

1c66ÞX þ X2

c66

ð80Þ

Substituting(80)into(79)leads to:

2b21ða1 XÞX2þ ða1 2XÞ X2þ ða1þ XÞa1c66 ða1þ c66þ b

2

1c66ÞX þ X2

c66

This is the secular equation of Rayleigh waves propagating in an incompressible monoclinic half-space with the symmetry plane x3¼ 0 whose surface is free of traction When the material is orthotropic, c16¼ c26¼ 0, from(31)and(33)it follows:

On view of(82)and the fact that 0 < X < a1; 0 < X < c66(seeOgden & Vinh, 2004), Eq.(81)is equivalent to:

ða1 XÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1  X=c66

p

that coincides with the secular Eq (23) inOgden and Vinh (2004)of Rayleigh waves propagating in an incompressible ortho-tropic half-space with traction-free condition

Again let the half-space be orthotropic, i e equalities(82)hold Using(70), (71), (72), (75) and (82), it is not difficult to verify that for the orthotropic case: D1¼ 0 and the corresponding secular equation in dimensionless form is of the from:



D2

where D; D2and D3are given by:



D ¼ ðd2 d1Þ d 2þ 2 þ d

x  d

þ ð2d1 d2Þx2

x p



D2¼ d2þ d2x d 2x þ ðd  xÞðx  1Þ

 d2xðd  xÞ  dð2 þ dÞx þ 2dx2 x3



D3¼ ðd1 d2Þ d21x  d

þ d 31x  2d2þ d1ð2 þ dÞ

x  d1x2

x

in which d ¼ ðc11þ c22 2c12Þ=c66; d1 and d2are the dimensionless impedance parameters, x ¼ c2=c2 (c2¼ c66=q) is the squared dimensionless velocity of Rayleigh waves From(85)one can see that Eq.(84)is an algebraic equation of sixth-order

in x

Remark 3

(i) By squaring (two times) two sides of Eq.(23), we arrive at Eq.(84) Eq.(23)is therefore considered as the original ver-sion of Eq.(84)

(ii) Eq.(23)is much more simple than Eq.(84) This fact says that it is reasonable to consider separately the case of ortho-tropic materials

(iii) Eq.(23)will be useful in investigating the uniqueness of Rayleigh waves in incompressible orthotropic half-spaces with impedance boundary conditions by the complex function method (see,Vinh & Ha Giang, 2012; Vinh, 2013)

4 Conclusions

In this paper, the propagation of Rayleigh waves in anisotropic incompressible half-spaces subjected impedance bound-ary conditions is investigated Two cases of anisotropy are considered: orthotropic materials and monoclinic materials with the symmetry plane x3¼ 0 For orthotropic case, the secular equation is derived by using the traditional technique and it is

an irrational equation For the monoclinic half-spaces, first the impedance boundary condition is replaced by a traction-free-like boundary condition Then the secular equation is derived by using the method of polarization vector This equation is an

algebraic equation of eighth-order It is worth to note that, in principle, the secular equations for the incompressible mate-rials can be obtained from those for the corresponding compressible matemate-rials followingDestrade, Martin, and Ting (2002)

andDestrade and Ogden (2010) In this paper, the authors follow the traditional direct approach that was used by many other, see, for examples,Dowaikh and Ogden (1990), Nair and Sotiropoulos (1999), Chadwick (1997), Destrade (2001), Ogden and Vinh (2004), Fu (2005), Edmondson and Fu (2009), Shams, Destrade, and Ogden (2011) and Destrade, Ogden, Sgura, and Vergori (2014)

Acknowledgments

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

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4 Conclusions

In this paper, the propagation of Rayleigh waves in anisotropic incompressible half- spaces subjected impedance bound-ary conditions is investigated...

Vinh, P C., & Anh, V T N (2014b) Rayleigh waves in an orthotropic elastic half- space coated by a thin orthotropic elastic layer with smooth contact International Journal of Engineering