DSpace at VNU: An approximate secular equation of generalized Rayleigh waves in pre-stressed compressible elastic solids

International Journal of Non-Linear Mechanics 50 (2013) 91–96

Contents lists available at SciVerse ScienceDirect

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

An approximate secular equation of generalized Rayleigh waves in
pre-stressed compressible elastic solids
Pham Chi Vinh a,n, Nguyen Thi Khanh Linh b
a
b

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

abstract

Article history:
Received 11 July 2012
Received in revised form
1 November 2012
Accepted 4 November 2012
Available online 10 November 2012

The present paper is concerned with the propagation of Rayleigh waves in a pre-stressed elastic halfspace coated with a thin pre-stressed elastic layer. The 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.
& 2012 Elsevier Ltd. All rights reserved.

Keywords:
Rayleigh waves
A pre-stressed compressible elastic halfspace
A thin pre-stressed compressible elastic
layer
Approximate secular equation

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 structure, 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 dispersion 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 prestressed elastic half-space overlaid by a compressible pre-stressed
elastic layer was investigated by Ogden and Sotiropoulos [3]. In
that investigation, for simplicity, pre-strains corresponding to

n

Corresponding author. Tel.: þ84 4 5532164; fax: þ84 4 8588817.
E-mail addresses: , (P. Chi Vinh).

0020-7462/$ - see front matter & 2012 Elsevier Ltd. All rights reserved.
/>
plane isotropic deformations were considered, and an explicit
dispersion relation was obtained that is valid for any strainenergy 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 [5–8], which replaces approximately the entire effect of
the layer on the half-space by a boundary condition, an approximate secular equation of third-order has been derived that is
valid for arbitrary pre-strains, and for a general isotropic strainenergy 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 isotropic elastic half-space (see [9,10]). Note that the propagation of
surface Rayleigh waves in a half-space under the effect of prestress 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]. References to other works can be found in these papers. When prestresses are absent, from the obtained secular equation we
immediately arrive at the approximate secular equation of second- and third-order obtained recently by Vinh and Linh [8] for

the isotropic case.
A numerical investigation is carried out for a special strainenergy function, and it is shown that the approximate secular


92

P. Chi Vinh, N. Thi Khanh Linh / International Journal of Non-Linear Mechanics 50 (2013) 91–96

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 thickness 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 x2 ¼ 0. A rectangular Cartesian coordinate system
ðx1 ,x2 ,x3 Þ is employed with its axes coinciding with the principal
directions of the pure strain. The layer occupies the domain
Àh ox2 o0 and the half-space corresponds to the region x2 4 0.
The principal stretches are denoted by l1 , l2 , l3 and l 1 , l 2 , l 3 in
the half-space and in the layer, respectively. They are positive
constants. The layer is assumed to be perfectly bonded to the halfspace . Note that 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.
An incremental (infinitesimal) motion in the ðx1 ,x2 Þ-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 ,u2 Þ and ðu 1 ,u 2 Þ,
respectively.
For the layer, in the absence of body forces the equations of
motion governing infinitesimal motion are [9,18]
s 11,1 þs 21,2 ¼ r u€ 1 , s 12,1 þs 22,2 ¼ r u€ 2 ,

ð1Þ

where r 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
s ji ¼ A jilk u k,l :

ð2Þ

A ijkl are components of the fourth-order elasticity tensor defined
as follows [9,18]:

(which are different from those defined by [9] by a factor J). In
terms of these notations Eq. (2) becomes
8
s 11 ¼ a 11 u 1,1 þ a 12 u 2,2 ,
>
>
>
>
< s 22 ¼ a 12 u 1,1 þ a 22 u 2,2 ,
ð8Þ
s 12 ¼ g 1 u 2,1 þ g n u 1,2 ,
>

>
>
>
: s 21 ¼ g u 2,1 þ g u 1,2 :
n
2
From the strong-ellipticity condition, a ik and g k are required to
satisfy the inequalities [9,18]

a 11 4 0, a 22 40, g 1 4 0, g 2 4 0:

From Eqs. (1) and (8), and following the same procedure as
carried out in [19] we have
" 0# "
#" #
M1 M2
U
U
, x2 A ½Àh,0Š,
ð10Þ
¼
0
M3 M4
T
T
where U ¼ ½u 1 u 2 ŠT , T ¼ ½s 21 s 22 ŠT , the symbol ‘‘T’’ indicates the
transpose of a matrix, the prime signifies partial derivative with
respect to x2 and
2
3

1
"
#
0 7
6
0
Àr 2 @1
6 g2
7
, M2 ¼ 6
M1 ¼
7,
1 5
Àr 1 @1
0
4
0
"
M3 ¼

r 3 @21 þ r @2t

0

0

r 4 @21 þ r @2t

r1 ¼

@ W
@l i @l j

,

From (10) it follows that
" #
" ðnÞ #
"
M1
U
U
, M¼
¼ Mn
ðnÞ
M
3
T
T

J A ijji ¼ J A jiij ¼ J A ijij Àl i

@W
@l i

ði aj, l i a l j Þ,
ð4Þ
ði aj, l i ¼ l j Þ,

ði ajÞ

ð5Þ

for i,j A 1,2,3,W ¼ W ðl 1 , l 2 , l 3 Þ is the strain-energy function per
unit volume in unstressed state, J ¼ l 1 l 2 l 3 (noting that l k 4 0),
all other components being zero. In the stress-free configuration
(3)–(5) reduce to
A iiii ¼ l þ 2m ,

A iijj ¼ l ðia jÞ,

A ijij ¼ A ijji ¼ m ði a jÞ,

ð6Þ

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

a 11 ¼ A 1111 , a 22 ¼ A 2222 , a 12 ¼ a 21 ¼ A 1122 ,
g 1 ¼ A 1212 , g 2 ¼ A 2121 , g n ¼ A 2112 ,

ð7Þ

M 4 ¼ M T1 :

ð11Þ

M2
M4

#
, n ¼ 1,2,3, . . . ,x2 A ½Àh,0Š:
ð13Þ

Let h be small (i.e. the layer is thin), then expanding T ðÀhÞ then
expanding T ðÀhÞ into Taylor series about x2 ¼ 0 up to the thirdorder of h gives

ð3Þ

8
!
2
>
>
@W
@W
li
>
>
li
Àl j
>
2
>
<
@l i
@l j l Àl 2
i
j
J A ijij ¼

!
>
>
1
@W
>
>
J A iiii ÀJ A iijj þ l i
>
>
:2
@l i

,

a 12
g
a 11 a 22 Àa 212
g g Àg 2
, r2 ¼ n , r3 ¼ À
, r 4 ¼ À 1 2 n : ð12Þ
a 22
g2
a 22
g2

0

J A iijj ¼ l i l j

a 22

#

Here we use the notations @1 ¼ @=@x1 , @21 ¼ @2 =@x21 , @2t ¼ @2 =@t 2 and

T ðÀhÞ ¼ T ð0ÞÀhT ð0Þ þ

2

ð9Þ

2

3

h 00
h 000
T ð0ÞÀ T ð0Þ:
2
6

ð14Þ

Suppose that the surface x2 ¼ Àh is free from the stress, i.e.
T ðÀhÞ ¼ 0. Using (13) at x2 ¼ 0 in (14) yields
(
2
h
ðM 3 M2 þ M 24 Þ

IÀhM 4 þ
2
)
3
h
À ½ðM 3 M 1 þM 4 M3 ÞM 2 þ ðM3 M 2 þM 24 ÞM 4 Š T ð0Þ
6
(
2
h
ðM 3 M 1 þ M 4 M 3 Þ
þ ÀhM 3 þ
2
)
3
h
2
À ½ðM 3 M 1 þM 4 M3 ÞM 1 þ ðM3 M 2 þM 4 ÞM 3 Š U ð0Þ
6
¼ 0:

ð15Þ

Since the layer and the half-space are bonded perfectly to each
other at the plane x2 ¼ 0, it follows that Uð0Þ ¼ U ð0Þ and
Tð0Þ ¼ T ð0Þ. Thus, we have from (15)
(
2
h
ðM 3 M2 þ M 24 Þ

IÀhM 4 þ
2
)
3
h
À ½ðM 3 M 1 þM 4 M3 ÞM 2 þ ðM3 M 2 þM 24 ÞM 4 Š Tð0Þ
6


P. Chi Vinh, N. Thi Khanh Linh / International Journal of Non-Linear Mechanics 50 (2013) 91–96

(

2

þ ÀhM 3 þ

whose real parts must be positive to ensure the decay condition
(20), X ¼ rc2 , and

h
ðM3 M 1 þM 4 M 3 Þ
2

)
3
h
2
À ½ðM 3 M 1 þ M 4 M 3 ÞM 1 þðM 3 M 2 þ M4 ÞM 3 Š :Uð0Þ ¼ 0:
6

ð16Þ

The relation (16) between the traction vector and displacement
vector of the half-space at the plane x2 ¼ 0 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
s21 þ hðr 1 s22,1 Àr 3 u1,11 Àr u€ 1 Þ

2
h
r
r 6 s21,11 þ s€ 21 Àr 7 u2,111 Àr r 5 u€ 2,1
þ
2
g2
3

þ

h
6

2

r €
¼0
u
g 2 1,tt

t 1 s22,111 þ r t 2 s€ 22,1 Àt 3 u1,1111 Àr t 4 u€ 1,11 À

!

r2 €
u
¼0
a 22 2,tt

at x2 ¼ 0,

ð18Þ

t1 ¼
t4 ¼

r7

a 22
r3

g2

r3

þ r1 r6 ,

t2 ¼

þr 1 r 5 þ r 6 ,

t7 ¼ r2 r7 þ r4 r8 ;

þ r1 r2 ,
r5

t5 ¼

t8 ¼

þ

a 22

r7

g2

r4

a 22

r 7 ¼ r 2 r 3 þr 1 r 4 ,
r1

g2

,

r8 ¼

r4

a 22

þ r1 r2 ,

t3 ¼ r1 r7 þ r3 r6 ,

þ r2 r8 ,

t6 ¼

r5

g2

þ

r2

a 22

bk ¼ g2 bk þ gn ak , Zk ¼ a12 Àa22 ak bk , k ¼ 1,2:

ð23Þ

From (22) we have

g2 ðXÀg1 Þ þ a22 ðXÀa11 Þ þða12 þ gn Þ2
:¼ S,
g2 a22
ðXÀa11 ÞðXÀg1 Þ
2
2
:¼ P:
b1 Á b2 ¼
g2 a22
2

2

b1 þb2 ¼ À

ð24Þ

ð25Þ

and
pffiffiffi
S þ 2 P 4 0,

b1 b2 ¼

pffiffiffi
P,

b1 þb2 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
S þ 2 P:

ð26Þ

f ðb1 ÞB1 þf ðb2 ÞB2 ¼ 0,
Fðb1 ÞB1 þFðb2 ÞB2 ¼ 0,

þ r8 þ r2 r5 :

ð19Þ

where

Fðbn Þ ¼ Zn þkhfðr 4 þX Þan Àr 2 bn g
(
!)
2 2
k h
X
r 7 þ X r 5 ÀZn r 8 þ
þ
2
a 22
(
"
#)
2
3 3

k h
X
,
bn ðt 5 þt 6 X ÞÀan t7 þ X t8 þ
þ
6
a 22
n ¼ 1,2,

Suppose that the pre-stressed elastic half-space is compressible. Then the unknown vectors U ¼ ½u1 u2 ŠT , T ¼ ½s21 s22 ŠT satisfy
Eq. (10) without the bar symbol. In addition to this equation are
required the effective boundary conditions (17), (18) at x2 ¼ 0 and
the decay condition at x2 ¼ þ 1, namely
at x2 ¼ 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 ½u1 u2 ŠT ,
T ¼ ½s21 s22 ŠT are given by

ð28Þ

B21 þ B22 a 0,

the determinant of coefficients of the homoSince
geneous system (27) must vanish. This yields
f ðb1 ÞFðb2 ÞÀf ðb2 ÞFðb1 Þ ¼ 0:

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
A0 þ A1 e þ

A2 2 A3 3
e þ e þOðe4 Þ ¼ 0,
2
6

ð30Þ

where e ¼ kh (the dimensionless thickness of the layer) and
A1 ¼ g2 ðr 4 þX Þða1 b2 Àa2 b1 Þ þ a22 ðr 3 þX Þða1 b1 Àa2 b2 Þ,

u2 ¼ iða1 B1 eÀkb1 x2 þ a2 B2 eÀkb2 x2 Þeikðx1 ÀctÞ ,
s21 ¼ Àkfb1 B1 eÀkb1 x2 þ b2 B2 eÀkb2 x2 geikðx1 ÀctÞ ,

r4

A2 ¼ À

a 22

ð21Þ

where B1 and B2 are constants to be determined from the effective
boundary conditions (17), (18), b1 , b2 are two roots of the equation

g2 a22 b4 þ fg2 ðXÀg1 Þ þ a22 ðXÀa11 Þ þ ða12 þ gn Þ2 Þgb2
þðXÀa11 ÞðXÀg1 Þ ¼ 0,

X ¼ r c2 :

A0 ¼ ðg2 a12 þ gn a22 a1 a2 Þðb2 Àb1 Þ þ ðgn a12 þ g2 a22 b1 b2 Þða2 Àa1 Þ,

u1 ¼ ðB1 eÀkb1 x2 þ B2 eÀkb2 x2 Þeikðx1 ÀctÞ ,

s22 ¼ ikfZ1 B1 eÀkb1 x2 þ Z2 B2 eÀkb2 x2 geikðx1 ÀctÞ ,

ð27Þ

,

3. Approximate secular equation of third-order

U¼T ¼0

pffiffiffiffiffiffiffi
a11 ÀXÀg2 b2k
, k ¼ 1,2, i ¼ À1,
ða12 þ gn Þbk

f ðbn Þ ¼ Àbn þ khfr 3 þ X Àr 1 Zn g
(
)

2 2
k h
X
ðr 6 þ Þbn Àan ½r 7 þ X r 5 Š
þ
2
g2
(
)
2
3 3
k h
X
,
Zn ðt1 þ t2 X ÞÀt3 Àt4 X À
þ
6
g2

where

g2

¼

Substituting (21) into the effective boundary conditions (17) and
(18) provides two linear equations for B1 and B2, namely

t 5 s21,111 þ r t 6 s€ 21,1 Àt 7 u2,1111 Àr t 8 u€ 2,11 À

r6 ¼

a22 b2k Àg1 þ X

P 4 0,

3

r 5 ¼ r 1 þr 2 ,

ða12 þ gn Þbk

0 o X o minfa11 , g1 g
ð17Þ

s22 þ hðr 2 s21,1 Àr 4 u2,11 Àr u€ 2 Þ

2
h
r €
r 8 s22,11 þ
þ
s 22 Àr 7 u1,111 Àr r 5 u€ 1,1
2
a 22
h
6

ak ¼

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

!

at x2 ¼ 0,

þ

93

g2

þ

X

g2

þ

X

!

a 22

A0 þ 2ðr 4 þ X Þðr 3 þ X Þða2 Àa1 Þ

þ½r 1 r 4 Àr 2 r 3 þX ðr 1 Àr 2 ފ½ðgn Àa12 Þða2 Àa1 Þ

þðg2 Àa22 a1 a2 Þðb2 Àb1 ފ,
"
A3 ¼ g2

ð22Þ

þ

r3

r8 þ

X

a 22

þ 3 r6 þ

X

g2

!!

#
ðr 4 þ X ÞÀ2r 2 ðr 7 þ r5 X Þ ða2 b1 Àa1 b2 Þ


94

P. Chi Vinh, N. Thi Khanh Linh / International Journal of Non-Linear Mechanics 50 (2013) 91–96

""
þ a22

r6 þ

X

g2

þ 3 r8 þ

X

a 22

!#

#

ðr 3 þ X ÞÀ2r 1 ðr 7 þ r 5 X Þ ða2 b2 Àa1 b1 Þ:

ð31Þ
By (23) one can prove the following equalities

where
b1 b2 ¼
S¼

ða11 ÀX þ g2 b1 b2 Þ
ðb2 Àb1 Þ,
ða12 þ gn Þb1 b2
g ðb þ b Þ
a ÀX
a2 b2 Àa1 b1 ¼ À 2 1 2 ðb2 Àb1 Þ, a1 a2 ¼ 11
,
ða12 þ gn Þ
a22 b1 b2

pffiffiffi
P,

b1 þb2 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
S þ2 P,

P¼

ðe1 ÀxÞð1ÀxÞ
,
e2 e5

e2 ðe1 ÀxÞ þ e5 ð1ÀxÞÀðe3 þe4 Þ2
,
e2 e5

ð37Þ

a2 Àa1 ¼ À

and

a2 b1 Àa1 b2 ¼ À

X
a11
a22
a12
g
g
, e1 ¼
, e2 ¼
, e3 ¼
, e4 ¼ n , e5 ¼ 2 ,
g1
g1
g1
g1
g1
g1
a 11
g1
a 12
gn
g1
e1 ¼
, e2 ¼

, e3 ¼
, e4 ¼
, e5 ¼ ,
g1
a 22
g1
g1
g2

ða11 ÀXÞðb1 þ b2 Þ
ðb2 Àb1 Þ:
ða12 þ gn Þb1 b2

x¼

ð32Þ

Introducing (32) into (31) yields
Ak ¼ yA k ðk ¼ 0,1,2,3Þ,

rm ¼

y ¼ ðb2 Àb1 Þ=½ða12 þ gn Þb1 b2 Š

and
A 0 ¼ g2 ½a212 Àa22 ða11 ÀXފb1 b2 þða11 ÀXÞ½g2n Àg2 ðg1 ÀXފ,
A 1 ¼ g2 ½ðr 4 þ X Þða11 ÀXÞ þ a22 ðr 3 þ X Þb1 b2 Šðb1 þb2 Þ,
!
r4
r3

X
X
A2 ¼ À
þ
þ
þ
A 0 þ 2ðr 3 þX Þðr 4 þX ÞðXÀa11 Àg2 b1 b2 Þ

a 22

g2

g2

a 22

g1
c2
, r v ¼ , c2 ¼
g1
c2

sffiffiffiffiffiffi

rffiffiffiffiffi
g1
,

c2 ¼

r

g1
:
r

ð38Þ

The squared dimensionless Rayleigh wave speed x depends on 13
dimensionless parameters: ek, e k ðk ¼ 1,2,3,4,5Þ, r m , rv and e.
As D0 ½xðeފ ¼ OðeÞ, the second-order approximate secular equation is either
D0 þ D1 e þ

D2 2
e þ Oðe3 Þ ¼ 0,
2

ð39Þ

^2
D
e2 þOðe3 Þ ¼ 0,
2

ð40Þ

or
þ 2½r 1 r 4 Àr 2 r 3 þ ðr 1 Àr 2 ÞX Š½g2 a12 b1 b2 þ gn ðXÀa11 ފ,
("
#

X
X
A 3 ¼ Àg2 ða11 ÀXÞ r 8 þ
þ 3ðr 6 þ Þ ðr 4 þ X Þ

a 22

D0 þ D1 e þ

g2

where

É
À2r 2 ðr 7 þ r 5 X Þ ðb1 þ b2 Þ
("
#
X
X
þ3ðr 8 þ
Þ ðr 3 þX Þ
Àg2 a22 r 6 þ

g2

^ 2 ¼ À2r m e 5 fr m ðe 2 e 2 Àe 1 þr 2 xÞðe 2 e 5 À1 þ r 2 xÞÀe3 ½e 2 e 3 ðe 2 e 5 À1Þ
D
v
v
3

4
4
Àe 4 e 5 ðe 2 e 23 Àe 1 Þ þ ðe 2 e 3 Àe 4 e 5 Þr 2v xŠgb1 b2

a 22

É
À2r 1 ðr 7 þ r 5 X Þ b2 b1 ðb1 þ b2 Þ

in which b1 b2 and b1 þb2 are given by (24) and (26). After
removing the factor y, Eq. (30) becomes
A0 þ A1e þ

A2 2 A3 3
e þ e þ Oðe4 Þ ¼ 0:
2
6

D2 2 D3 3
e þ e þOðe4 Þ ¼ 0
2
6

ð41Þ

is simpler than D2.

4. Special cases

ð35Þ

in which Dk ðk ¼ 0,1,2,3Þ are given by

4.1. Unstressed case
When the pre-strains are absent, lk ¼ l k ¼ 1 ðk ¼ 1,2,3Þ, and
the elastic constants Aijkl are given by (6). For this case, from (6)
and (7) we have

g1 ¼ g2 ¼ gn ¼ m, a11 ¼ a22 ¼ l þ2m, a12 ¼ l,
g 1 ¼ g 2 ¼ g n ¼ m , a 11 ¼ a 22 ¼ l þ 2m , a 12 ¼ l

r m e5 ½ðe 24 e 5 À1 þ r 2v xÞðe1 ÀxÞ þe2 ðe 2 e 23 Àe 1 þ r 2v xÞb1 b2 Šðb1 þ b2 Þ,

D2 ¼ À½e 2 ðe 24 e 5 À1Þ þ e 5 ðe 2 e 23 Àe 1 Þ þ ðe 2 þ e 5 Þr 2v xŠD0

e1 ¼ e2 ¼

Àe 4 e 5 ðe 2 e 23 Àe 1 Þ þ ðe 2 e 3 Àe 4 e 5 Þr 2v xŠgb1 b2
þ 2r m ðxÀe1 Þfr m ðe 2 e 23 Àe 1 þ r 2v xÞðe 24 e 5 À1þ r 2v xÞ þ e4 ½e 2 e 3 ðe 24 e 5 À1Þ

1

1

g

,

e3 ¼

, e2 ¼ g,
g
m
c2
rm ¼ , rv ¼ ,
c2
m
e1 ¼

À2r m e 5 fr m ðe 2 e 23 Àe 1 þr 2v xÞðe 24 e 5 À1 þ r 2v xÞÀe3 ½e 2 e 3 ðe 24 e 5 À1Þ

1

g

À2,

e4 ¼ e5 ¼ 1,

1
e 3 ¼ À2,

g

c2 ¼

e 4 ¼ e 5 ¼ 1,

rffiffiffiffi
m

,

r

x¼
sffiffiffiffi

c2 ¼

c2
,
c22

m
,
r
ð43Þ

where g ¼ m=ðl þ 2mÞ and g ¼ m =ðl þ 2m Þ. Introducing (43) into
(36) yields Dk ¼ D k =gðk ¼ 0,1,2,3Þ, where D k are calculated by

Àe 4 e 5 ðe 2 e 23 Àe 1 Þ þ ðe 2 e 3 Àe 4 e 5 Þr 2v xŠg,
D3 ¼ r m e5 ðxÀe1 Þfðe 24 e 5 À1 þr 2v xÞ½e 2 ðe 24 e 5 À1Þ

D 0 ¼ ½4ðgÀ1Þ þ xŠb1 b2 þð1ÀgxÞx,

þ 4e 2 e 3 e 4 e 5 þ ðe 2 þ 3e 5 Þr 2v x þ3e 5 ðe 2 e 23 Àe 1 ފ

D 1 ¼ r m ½r 2v xð1ÀgxÞ þð4g À4 þ r 2v xÞb1 b2 Šðb1 þ b2 Þ,

À2e 4 e 5 ½e 2 e 3 ðe 24 e 5 À1Þ þ e 4 e 5 ðe 2 e 23 Àe 1 Þ þðe 2 e 3
þ e 4 e 5 Þr 2v xŠgðb1 þ b2 ÞÀr m e2 e5 fðe 2 e 23 Àe 1 þr 2v xÞ½3e 2 ðe 24 e 5 À1Þ

D 2 ¼ À½4ðg À1Þ þ ð1þ g Þr 2v xŠD0
þ 2r m ½ð2gÀ1Þð4g À4 þ 2g r 2v xÞÀgr m r 2v xð4g À4 þ r 2v xފb1 b2

þ 4e 2 e 3 e 4 e 5 þ ð3e 2 þ e 5 Þr 2v x þe 5 ðe 2 e 23 Àe 1 ފ
À2e 2 e 3 ½e 2 e 3 ðe 24 e 5 À1Þ þ e 4 e 5 ðe 2 e 23 Àe 1 Þ
þ ðe 2 e 3 þ e 4 e 5 Þr 2v xŠgb1 b2 ðb1 þ b2 Þ,

ð42Þ

therefore

D0 ¼ e5 ½e23 Àe2 ðe1 Àxފb1 b2 þðe1 ÀxÞ½e24 Àe5 ð1Àxފ,
D1 ¼

Àe 4 e 5 ðe 2 e 23 Àe 1 Þ þ ðe 2 e 3 Àe 4 e 5 Þr 2v xŠg

ð34Þ

This is the desired third-order approximate secular equation and it
is fully explicit.
When the thickness of the layer vanishes, i.e. e ¼ 0, the secular
equation (34) becomes A 0 ¼ 0 that is equivalent to Eq. (5.11) in [9]
and Eq. (25) in [10].
In dimensionless form, Eq. (34) becomes
D0 þD1 e þ

þ 2r m ðxÀe1 Þfr m ðe 2 e 23 Àe 1 þ r 2v xÞðe 24 e 5 À1þ r 2v xÞ þe4 ½e 2 e 3 ðe 24 e 5 À1Þ

ð33Þ

þ 2r m ½4ðg À1Þ þ2ðg þ 2r m À2r m g Þr 2v xÀr m r 4v x2 Šð1ÀgxÞ,
D 3 ¼ r m f½8ð1Àg Þ þ4ð2g À3Þr 2v x þ ð3þ g Þr 4v x2 ŠðgxÀ1Þ
ð36Þ

À½8ð1Àg Þ þ4ðg 2 À2Þr 2v x þ ð1 þ3g Þr 4v x2 Šb1 b2 gðb1 þ b2 Þ,

ð44Þ


P. Chi Vinh, N. Thi Khanh Linh / International Journal of Non-Linear Mechanics 50 (2013) 91–96

pffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffi
where b1 ¼ 1Àgx, b2 ¼ 1Àx. Therefore, for the unstressed case
(i.e. for the isotropic case), the approximate secular equation of
third-order is
D 0 þD 1 e þ

D2 2 D3 3
e þ e þ Oðe4 Þ ¼ 0
2
6

D 0 þD 1 e þ

4
4

8
2r m pffiffiffi
2D0
D2 ¼ À
½ðl þ 1Þðl À3Þ þ2r 2v xŠÀ
P fr m ðl À9þ 3r 2v xÞ
3
3
4

D2 2
e þOðe3 Þ ¼ 0
2

ð46Þ

or

4

4

8

4

4

4

8

4

þ2ð2Àl Þ½ðl À3Þðl À3Þ þ ð2l À3Þr 2v xŠg,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi 8
4
4
2
D3 ¼ r m ðxÀ3Þ S þ2 P½ðl À4l þ 3 þr 2v xÞð2l À12 þ5r 2v xÞ
3
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
4
4
4
2 pffiffiffi
À2ð2Àl Þð3Àl Þðl À3þ r 2v xފÀ r m P S þ 2 P
3
8

4

4

4

4

½ðl À9 þ3r 2v xÞð3r 2v xÀ2l Þ þ 2l ðl À3Þðl À3 þr 2v xފ,

n

D 0 þD 1 e þ

4

½ðl À1Þðl À3Þ þ r 2v xŠÀ2l ½ðl À3Þðl À3Þ þ ð2l À3Þr 2v xŠg
2r m
8
4
4
ðxÀ3Þfr m ðl À9 þ3r 2v xÞ½ðl À1Þðl À3Þ þr 2v xŠ
þ
3

ð45Þ

that coincides with Eq. (46) in [8]. In the second-order, the
approximate secular equation is either

95

D2 2
e þOðe3 Þ ¼ 0,
2

ð47Þ

where D 0 , D 1 and D 2 are given by (44), and

ð55Þ

where
ð3ÀxÞð1ÀxÞ
ð6À4xÞ
, S¼
,
3
3
4
4
l þ rð1Àl Þ
R
rm ¼
, r 2v ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi:
r
4
l þ rð1Àl4 Þ

P¼

Dn2 ¼ 2r m ½ð2gÀ1Þð4g À4 þ 2g r 2v xÞÀgr m r 2v xð4g À4 þ r 2v xފb1 b2
Â
Ã
þ2r m 4ðg À1Þ þ 2ð3g À2Þr 2v x þ r 4v x2 ð1ÀgxÞ:

ð48Þ

ð56Þ

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])

l1 ¼ l2 ¼ l, l 1 ¼ l 2 ¼ l :

e5 ¼ 1,

e 1 ¼ 1=e 2 ,

e 5 ¼ 1:

ð50Þ

For this case, the approximate secular equation of third-order is of
the form as in (35), where Dk are calculated by (36) and (37) in
which the fact (50) is taken into account, and


@2 W
@2 W
@W @W
2l 2
2 l
þ
À

@l1
@l1 @l2 @l2 @l1
!
!
, e3 ¼
,
e1 ¼
@2 W
@2 W
@W
@2 W
@2 W
@W
þ
þ
l
À
l
À
2
2
@l1 @l2
@l1
@l1 @l2
@l1
@l
@l
1

e4 ¼ 1À

l

1

@W
2
@l2

@2 W
2

@l1

!
@2 W
@W
À
þ
@l1 @l2
@l1

ð51Þ

for the half-space and there are similar expressions for the layer.
Continuity of the normal stress (see [3]) implies that

ll 3

D0 þD1 e þ

ð49Þ

Using (3)–(5) and (7) and taking into account (49), from (38) we
have
e1 ¼ e2 ,

Here, in addition to l we introduce two new dimensionless
2
2
parameters r ¼ m=m and R ¼ ðmr l Þ=ðmrl Þ. In this case, the
4
squared dimensionless Rayleigh wave velocity x ¼ ðl rc2 Þ=m is
determined (approximately) by the equation

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, and e.
Fig. 1 shows the dependence on the dimensionless thickness of
the layer e A ½0, 1Š of the squared dimensionless Rayleigh wave
velocity x (with given values of l,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. 1 that (i) the approximate curves of second- and third-order
of x are very close to the exact curve for the values of e A ½0, 1Š; (ii)
for e A ½0, 0:5Š 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 prestressed compressible isotropic elastic half-space coated by a thin

ð52Þ

0.65
0.6

Now to show the accuracy of the approximate secular equations obtained we take the two materials to have strain-energy
functions of Blatz–Ko form (see [3]), namely
À2
À2
À2
ðl1 þ l2 þ l3 þ 2JÀ5Þ

x

e1 ¼ e 1 ¼ 3,

4

e3 ¼ l ,

4

e3 ¼ l ,

4

e4 ¼ 2Àl ,

4

0.45
0.4
0.35
0.3

e 4 ¼ 2Àl ,

4

l
:
l4 þ rð1Àl4 Þ

λ=0.9, r=0.3, R=2

ð53Þ

for the half-space and similarly for the layer. We set l3 ¼ l 3 ¼ 1 in
addition to the assumption (49). Then, from (3)–(5) and (51)–(53)
it follows that

l4 ¼

0.55
0.5

m
2

ð57Þ

0.7

@W
@W
¼ ll3
:
@l2
@l 2

W¼

D2 2 D3 3
e þ e ¼ 0,
2
6

ð54Þ

Introducing (50) and (54) into (36) and (37) we have
pffiffiffi
8
4
4
D0 ¼ ðl À9 þ 3xÞ P þ ð3ÀxÞ½ðl À1Þðl À3Þ þxŠ,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi 8
pffiffiffi

4
4
D1 ¼ r m S þ2 Pfð3ÀxÞ½ðl À1Þðl À3Þ þ r 2v xŠ þ P ½l À9 þ3r 2v xŠg,

0.25
0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ε

Fig. 1. Plots of xðeÞ calculated by the approximate secular equation of secondorder (dash-dot line), by the approximate secular equation of third-order (dashed
line), and by the exact secular equation (8) in Ref. [3]. Here l ¼ 0:9, r ¼ 0:3, R ¼ 2.


96

P. Chi Vinh, N. Thi Khanh Linh / International Journal of Non-Linear Mechanics 50 (2013) 91–96

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 accuracy. 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 International 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 halfplane, 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 prestressed, 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 prestressed 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 prestrained 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.