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

Meccanica
DOI 10.1007/s11012-016-0464-5

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
Ó 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 secular equation, the secular 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 Á A half-space coated by
a layer Á Spring contact Á Exact explicit secular
equation Á The effective boundary condition method

P. C. Vinh (&) Á V. T. N. Anh
Faculty of Mathematics, Mechanics and Informatics,
Hanoi University of Science, 334, Nguyen Trai Str.,
Thanh Xuan, Hanoi, Vietnam
e-mail:

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 secular 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 socalled (approximate) effective boundary conditions.
The effective boundary conditions are established by
either replacing approximately the layer by a plate
[5, 6], or by expanding the stresses at the upper-surface

123


Meccanica

of the layer into Taylor series of the layer thickness
[7–13]. Wang et al. [8] derived the first-order approximate secular equation for piezoelectric materials,
Tiersten [6], Bovik [7], Steigmann and Ogden [9]
obtained the second-order approximate secular equations, 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], BenMenahem 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 uppersurface 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 x2 ¼ 0, then the spring boundary conditions
enforced on the imperfect interface x2 ¼ 0 are [21, 23]
rk2 ðx2 ¼ 0þ Þ ¼ rk2 ðx2 ¼ 0À Þ;
k ¼ 1; 2; 3;


r22 ¼ KN u2 ðx2 ¼ 0þ Þ À u2 ðx2 ¼ 0À Þ ;


r12 ¼ KT1 u1 ðx2 ¼ 0þ Þ À u1 ðx2 ¼ 0À Þ ;


r23 ¼ KT2 u3 ðx2 ¼ 0þ Þ À u3 ðx2 ¼ 0À Þ

123

ð1Þ

where KT1 ð [ 0Þ, KT2 ð [ 0Þ and KN ð [ 0Þ are shear
and normal spring stiffnesses. The sliding contact [26]
r12 ðx2 ¼ 0þ Þ ¼ r12 ðx2 ¼ 0À Þ ¼ 0;
r23 ðx2 ¼ 0þ Þ ¼ r23 ðx2 ¼ 0À Þ ¼ 0;
u2 ðx2 ¼ 0þ Þ ¼ u2 ðx2 ¼ 0À Þ;

ð2Þ

r22 ðx2 ¼ 0þ Þ ¼ r22 ðx2 ¼ 0À Þ
is obtained directly from the spring boundary conditions (1) by letting KT1 , KT2 approach to zero and KN to
go to þ1. The perfectly bonded (welded) contact [27]
uk ðx2 ¼ 0þ Þ ¼ uk ðx2 ¼ 0À Þ;
rk2 ðx2 ¼ 0þ Þ ¼ rk2 ðx2 ¼ 0À Þ;

k ¼ 1; 2; 3

ð3Þ

is derived directly from the spring model (1) by letting
KT1 , KT2 and KN all approach to þ1.
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 secular equation, the secular 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 BenMenahem 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 layer Àh 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


Meccanica

ui ¼ ui ðx1 ; x2 ; tÞ; ui ¼ ui ðx1 ; x2 ; tÞ;
i ¼ 1; 2; u3 ¼ u3  0

ð4Þ

where ui , ui are components of the displacement vector, t is the time. Since the layer is made of orthotropic
elastic materials, the strain-stress relations are
r11 ¼ c11 u1;1 þ c12 u2;2 ; r22 ¼ c12 u1;1 þ c22 u2;2 ;
r12 ¼ c66 ð
u1;2 þ u2;1 Þ

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
ckk [ 0;

k ¼ 1; 2; 6; c11 c22 À 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
1 ; r12;1 þ r22;2 ¼ qu€2
r11;1 þ r12;2 ¼ qu€

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
c11 u1;11 þ c66 u1;22 þ ð
c12 þ c66 Þ

u2;12 ¼ qu€1 ;
ð
c12 þ c66 Þ
u1;12 þ c66 u2;11 þ c22 u2;22 ¼ qu€2

ð8Þ

where p1 ; p2 are complex in general and no requirements are imposed on their real and imaginary parts.
Using (9)–(11) into (5) we have
1 ðyÞeikðx1 ÀctÞ ; r22 ¼ kR
2 ðyÞeikðx1 ÀctÞ
r12 ¼ kR

ð9Þ

ð12Þ

in which
1 ðyÞ ¼ b1 A1 shðp1 yÞ þ b1 A2 chðp1 yÞ þ b2 A3 shðp2 yÞ
R
þ b2 A4 chðp2 yÞ;
Â
2 ðyÞ ¼ i c1 A1 chðp1 yÞ þ c1 A2 shðp1 yÞ þ c2 A3 chðp2 yÞ
R
Ã
þ c2 A4 shðp2 yÞ
ð13Þ
with
bn ¼ c66 ðpn À an Þ; cn ¼ ð
c12 þ c22 pn an Þ;

Now we consider the propagation of a Rayleigh wave,
traveling along the interface between the layer and the
half-space with velocity c ð [ 0Þ and wave number
k ð [ 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
 1 ðyÞeikðx1 ÀctÞ ; u2 ¼ U
 2 ðyÞeikðx1 ÀctÞ
u1 ¼ U

pj ð
c12 þ c66 Þ
; j ¼ 1; 2;
2
2
c22 pj À c66 þ qc
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2



S þ S À 4P
S À S À 4P
; p2 ¼
;
p1 ¼

2
2
 2 À c11 Þ þ c66 ðqc
 2 À c66 Þ
ð
c12 þ c66 Þ2 þ c22 ðqc
S ¼ À
;
c22 c66
À
ÁÀ
Á
 2 c66 À qc
2
c11 À qc
P ¼
c22 c66
ð11Þ
aj ¼ À

n ¼ 1; 2
ð14Þ

Suppose that surface x2 ¼ Àh is free of traction, i. e
r12 ¼ 0; r22 ¼ 0 at x2 ¼ Àh

ð15Þ

Using (12) and (13) into (15) leads to
b1 A1 shðe1 Þ À b1 A2 chðe1 Þ þ b2 A3 shðe2 Þ À b2 A4 chðe2 Þ ¼ 0;

c1 A1 chðe1 Þ À c1 A2 shðe1 Þ þ c2 A3 chðe2 Þ À c2 A4 shðe2 Þ ¼ 0

where y ¼ kx2 and
 1 ðyÞ ¼A1 chðp1 yÞ þ A2 shðp1 yÞ þ A3 chðp2 yÞ
U
þ A4 shðp2 yÞ;
Â
 2 ðyÞ ¼i a1 A1 shðp1 yÞ þ a1 A2 chðp1 yÞ þ a2 A3 shðp2 yÞ
U
Ã
þ a2 A4 chðp2 yÞ
ð10Þ
A1 ; A2 ; A3 ; A4 are constants and for simplicity we use
the notations shð:Þ :¼ sinhð:Þ; chð:Þ :¼ coshð:Þ, the
quantities aj and pj are determined by

ð16Þ
where en ¼ pn e; n ¼ 1; 2; e ¼ kh. Putting x2 ¼ 0 in
(10) and (13), we deduce
 1 ð0Þ ¼A1 þ A3 ; U
 2 ð0Þ ¼ ið
U
a1 A2 þ a2 A4 Þ;
1 ð0Þ ¼b1 A2 þ b2 A4 ; R
2 ð0Þ ¼ ið
R
c1 A1 þ c2 A3 Þ
ð17Þ
Solving the system (17) for A1 ; A2 ; A3 ; A4 , we obtain

123


Meccanica

2.

i 
c2 
U 1 ð0Þ þ R
2 ð0Þ;
½
cŠ
½
cŠ
ib2 
a2 
U 2 ð0Þ þ
A2 ¼

 R1 ð0Þ;
½
a; bŠ
½
a; bŠ

A1 ¼

i 
c 

R2 ð0Þ;
A3 ¼ À 1 U
1 ð0Þ À
½
cŠ
½
cŠ
ib 
a1 
A4 ¼ À 1 U
2 ð0Þ À
 R1 ð0Þ
½
a; bŠ
½
a; bŠ

From the first two of (23) we have

ð18Þ

here we use the notations
½f ; gŠ ¼ f2 g1 À f1 g2 ; ½f Š ¼ f2 À f1

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

ð19Þ

Substituting (18) into (16) yields
1 ð0Þ À ia12 R
2 ð0Þ þ b11 U
 1 ð0Þ À ib12 U
 2 ð0Þ ¼ 0;
a11 R


 1 ð0Þ À ib22 U
 2 ð0Þ ¼ 0
a21 R1 ð0Þ À ia22 R2 ð0Þ þ b21 U
ð20Þ

 1 ð0Þ ¼ À k R1 ð0Þ þ U1 ð0Þ ;
U
KT
 2 ð0Þ ¼ À k R2 ð0Þ þ U2 ð0Þ
U
KN

ð24Þ

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




k
k

b11 R1 ð0Þ À i a12 À
b12 R2 ð0Þ
a11 À
KT
KN
þ b11 U1 ð0Þ À ib12 U2 ð0Þ ¼ 0;




k
k
a21 À
b21 R1 ð0Þ À i a22 À
b22 R2 ð0Þ
KT
KN
þ b21 U1 ð0Þ À ib22 U2 ð0Þ ¼ 0
ð25Þ

where
a11
a22



½
a; bcheŠ
½bsheŠ
½

cshe; aŠ
;
a
; a21 ¼
¼
¼
À
12

 ;
½
cŠ
½
a; bŠ
½
a; bŠ

b b ½cheŠ
½
ccheŠ
½bshe;
cŠ
; b11 ¼
; b12 ¼ 1 2  ;
¼
½
cŠ
½
cŠ
½

a; bŠ

b21 ¼ À

 csheŠ
½b;
c1 c2 ½cheŠ
; b22 ¼

½
cŠ
½
a; bŠ

ð21Þ

Since the layer and the half-space are in spring contact
to each other at the plane x2 ¼ 0, we have from (1) (see
also [21, 23, 24])
r12 ¼ KT ðu1 À u1 Þ; r22 ¼ KN ðu2 À u2 Þ;
r12 ¼ r12 ; r22 ¼ r22 at x2 ¼ 0

ð22Þ

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 x2 ¼ 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
u1 ¼ U 1 ðyÞeikðx1 ÀctÞ ; u2 ¼ U 2 ðyÞeikðx1 ÀctÞ ; y ¼ kx2

or equivalently due to (9), (10), (12) and (13)
 1 ð0ފ ; k
kR1 ð0Þ ¼ KT ½U1 ð0Þ À U
 2 ð0ފ ;
R2 ð0Þ ¼ KN ½U2 ð0Þ À U
1 ð0Þ ; R2 ð0Þ ¼ R
2 ð0Þ
R1 ð0Þ ¼ R

ð26Þ
where
ð23Þ

Un ð0Þ and Rn ð0Þ (n=1,2) are the displacement and
traction amplitudes of the half-space at the interface
x2 ¼ 0. KT ð [ 0Þ and KN ð [ 0Þ are the shear and
normal spring stiffnesses, respectively. From (23) it
implies that
1.

When KT ¼ 0 and KN ! þ1 the half-space and
the layer are in sliding contact.

123

U1 ðyÞ ¼ B1 eÀb1 y þ B2 eÀb2 y ;
U2 ðyÞ ¼ iða1 B1 eÀb1 y þ a2 B2 eÀb2 y Þ

ð27Þ

B1 and B2 are constants to be determined, and
ak ¼

ðc12 þ c66 Þbk
;
c22 b2k À c66 þ X

k ¼ 1; 2; X ¼ qc2

ð28Þ

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


Meccanica

b4 À Sb2 þ P ¼ 0

ð29Þ

f ðb1 ÞFðb2 Þ À f ðb2 ÞFðb1 Þ ¼ 0

ð38Þ

S and P are calculated by (11) without bars. It has been
shown that if a Rayleigh wave exists, then [28]

Substituting (37) into (38) and after some calculations
we arrive at

0\X\minfc66 ; c11 g

n

ð30Þ

and [29]
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
pffiffiffi
P[0; S þ P[0; b1 b2 ¼ P; b1 þ b2 ¼ S þ 2 P
ð31Þ
Introducing (26) and (27) into the strain-stress relation
(5) without bars leads to
r12 ¼ kR1 ðyÞeikðx1 ÀctÞ ; r22 ¼ kR2 ðyÞeikðx1 ÀctÞ

ð32Þ

ða11 a22 À a12 a21 Þ À

k

ða11 b22 À a21 b12 Þ
KN

k
ða12 b21 À a22 b11 Þ
KT
o
k2
ðb11 b22 À b12 b21 Þ ½c; bŠ À ða11 b21 À a21 b11 Þ½bŠ
þ
KT KN
o
n
k
ðb11 b22 À b12 b21 Þ ½a; bŠ
þ ða11 b22 À a21 b12 Þ À
KT
o
n
k
ðb11 b22 À b12 b21 Þ ½cŠ
À ða12 b21 À a22 b11 Þ þ
KN
þ ða12 b22 À a22 b12 Þ½a; cŠ þ ðb11 b22 À b12 b21 Þ½aŠ ¼ 0
þ

ð39Þ

in which
R1 ðyÞ ¼ b1 B1 eÀb1 y þ b2 B2 eÀb2 y ;

ð33Þ

R2 ðyÞ ¼ iðc1 B1 eÀb1 y þ c2 B2 eÀb2 y Þ
and
bj ¼ Àc66 ðbj þ aj Þ; cj ¼ c12 À c22 bj aj ;

j ¼ 1; 2
ð34Þ

Putting x2 ¼ 0 in (27) and (33) gives
U1 ð0Þ ¼B1 þ B2 ; U2 ð0Þ ¼ iða1 B1 þ a2 B2 Þ;
R1 ð0Þ ¼b1 B1 þ b2 B2 ; R2 ð0Þ ¼ iðc1 B1 þ c2 B2 Þ
ð35Þ

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

o
n
½c;bŠ ¼ c66 c212 À c22 ðc11 À XÞ b1 b2 þ Xðc11 À XÞ h;
½a;bŠ ¼ c66 ðc11 À XÞðb1 þ b2 Þh;
½a;cŠ ¼ c66 ðc11 À X À c12 b1 b2 Þh;
½aŠ ¼ ðX À c11 À c66 b1 b2 Þh; ½bŠ ¼ ½a;cŠ;
½cŠ ¼ c22 c66 b1 b2 ðb1 þ b2 Þh

ð40Þ

where h ¼ ðb2 À b1 Þ=ððc12 þ c66 Þb1 b2 Þ. Introducing
the expressions of aij and bij given by (21) into (39)

and using the equalities (40) yield

Substituting (35) into (25) leads to two linear equations for B1 and B2 , namely

A1 þ B1 she1 she2 þ C1 she1 che2 þ D1 she2 che1

f ðb1 ÞB1 þ f ðb2 ÞB2 ¼ 0; Fðb1 ÞB1 þ Fðb2 ÞB2 ¼ 0

where the coefficients A1 , B1 , C1 , D1 , E1 are given by
(54) in the Appendix 1.
Equation (41) in which A1 ,..., E1 are 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
opffiffiffi
ð42Þ
c212 À c22 ðc11 À XÞ P þ Xðc11 À XÞ ¼ 0

ð36Þ
where




k
k
f ðbn Þ ¼ a11 À
b11 bn þ a12 À

b12 cn
KT
KN
þ b11 þ b12 an ;




k
k
b21 bn þ a22 À
b22 cn
Fðbn Þ ¼ a21 À
KT
KN
þb21 þ b22 an

ðn ¼ 1; 2Þ

ð37Þ

For a non-trivial solution, the determinant of the
matrix of the system (36) must vanishes, i.e.,

þ E1 che1 che2 ¼ 0

ð41Þ

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

123


Meccanica

c11 ¼ c22 ¼ k þ 2l; c12 ¼ k; c66 ¼ l;
 c66 ¼ l
 c12 ¼ k;
c11 ¼ c22 ¼ k þ 2l;

ð43Þ

From (41) and (54) and taking into account (43) we
obtain the secular equation for the isotropic case,
namely
A2 þ B2 she1 she2 þ C2 she1 che2 þ D2 she2 che1
þ E2 che1 che2 ¼ 0

ð44Þ

in which the coefficients A2 , B2 , C2 , D2 , E2 are given
by (55) in the Appendix 2.
Taking the limit of Eq. (44) when cN and cT both 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 =ðÀb1 b2 Þ we arrive immediately 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.

2.

Taking the limit of two sides of Eq. (41) when KN
and KT both approach to þ1, for the welded
contact.
Multiplying two sides of Eq. (41) by KT =kc66 and
then taking the limit of the resulting equation
when KT ! 0 and KN ! þ1, for the sliding
contact.

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 [ 0 ðk ¼ 1; 2; 3Þ,
e1 e2 À e23 [ 0, e1 À e2 e23 [ 0.
Equation (41) can be rewritten as
h eðp þ p Þ i
h eðp À p Þ i
1
2
1
2
þ ðE1 À B1 Þsh2

2
2
C1 þ D1
C1 À D1
þ
sh½eðp1 þ p2 ފ þ
sh½eðp1 À p2 ފ
2
2
ð46Þ
þ A1 þ E1 ¼ 0

ðE1 þ B1 Þsh2

Using (54) we derive the expressions of the coefficients E1 þ B1 , E1 À B1 , C1 þ D1 , C1 À D1 and A1 þ E1 .
Introducing these expressions into (46) yields
h eðp þ p Þ i
h eðp À p Þ i
1
2
1
2
sh2
2
2
Aðg; gÞ
À Aðg; À
gÞ
ðp1 þ p2 Þ2
ðp1 À p2 Þ2

sh½eðp1 þ p2 ފ
sh½eðp1 À p2 ފ
þ Bðg; gÞ
À Bðg; À
gÞ
p1 þ p2
p1 À p2
þ Cðg; gÞ ¼ 0
sh2

ð47Þ
where p1 , p2 are defined by (11) in which P and S are
expressed in terms of the dimensionless parameters as
follows


S ¼ ð
e1 À rv2 xÞ þ e2 1 À rv2 x À ð
e3 þ 1Þ2 ;
ð48Þ
e1 À rv2 xÞð1 À rv2 xÞ
P ¼ e2 ð
and

4 Dimensionless secular equation

Aðg; gÞ ¼ 2

fð
gÞ n

fð
gÞ
À1=2
ð1 þ ge2 Þ
2
1 À g
1 À g2
À1=2

It is useful to convert the secular Eq. (41) into
dimensionless form. For this aim we introduce
dimensionless parameters
c11
e1 ¼
;
c66
c66
e2 ¼
;
c22
c66
rl ¼
;
c66
kc66
cT ¼
KT

123

c22
c12
c11
e2 ¼
; e3 ¼
; e1 ¼
;
c66
c66
c66
c12
e3 ¼
;
c66
rffiffiffiffiffiffi
rffiffiffiffiffiffi
c2
c66
c66
; c2 ¼
rv ¼ ; c 2 ¼
;
q
c2
q
kc66
; cN ¼
KN

þ 2rlÀ1 ð1 À e3 e2

1=2

þ rlÀ2 ð1 þ ge2 Þ

1=2
gÞð1 À e3 e2 gÞ

f ðgÞ
1 À g2

fð
gÞ h
f ðgÞ io
1=2
ðcT þ cN e2 gÞðb1 þ b2 Þ þ cT cN
;
2

1Àg
1 À g2
fð
gÞ À1 n
1=2
À1=2
Bðg; gÞ ¼
ðb1 þ b2 Þ½e2 g þ e2 gŠ
r
1 À g2 l
f ðgÞ o
À1=2

þ ½cT þ cN e2 gŠ
;
1 À g2
f ðgÞ
À1=2
Cðg; gÞ ¼ 2rlÀ2 e2 g
1 À g2
þ

ð45Þ

ð49Þ


Meccanica

here

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g ¼ ð1 À xÞ=ðe1 À xÞ; g ¼ ð1 À rv2 xÞ=ð
e1 À rv2 xÞ;
f ðgÞ ¼

À1=2
e23 e2 g3

2

þ e1 g þ ½e2 ðe1 À 1Þ À

À1=2
e23 Šge2

À1

ð50Þ
function fð
gÞ is given by (50) in which e1 ; e2 and e3
are replaced by e1 ; 1=
e and e3 , respectively. In the
p2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
Eq. (49): ðb1 þ b2 Þ ¼ S þ 2 P in which
e2 ðe1 À xÞ þ 1 À x À ðe3 þ 1Þ2
;
e2
ðe1 À xÞð1 À xÞ
P¼
e2
S¼

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 cN and 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
fð
gÞ
Aðg; gÞ ¼ 2
1 À g2
&
fð
gÞ
À1=2
À1=2
 ð1 þ ge2 Þ
þ 2rlÀ1 ð1 À e3 e2 gÞ
1 À g2
'
f ðgÞ
1=2
1=2
Âð1 À e3 e2 gÞ þ rlÀ2 ð1 þ ge2 Þ
;
1 À g2
fð
gÞ À1
1=2
À1=2
r ðb1 þ b2 Þ½e2 g þ e2 gŠ;
Bðg; gÞ ¼
1 À g2 l
f ðgÞ
À1=2
ð52Þ

Cðg; gÞ ¼ 2rlÀ2 e2 g
1 À g2
By comparing this equation with the secular equation
derived by Sotiropoulos, Eq. (16) in Ref. [16], one can
immediately discover misprints in this secular equation (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 factor ðs1 þ s2 Þ (in the expressions
of Bðg; gÞ and Bðg; À
gÞ).
For the sliding contact the coefficients of Eq. (47)
are simplified to
h fð
gÞ i 2
Aðg; gÞ ¼ 2
ðb1 þ b2 Þ;
1 À g2
ð53Þ
fð
gÞ À1 f ðgÞ
r
; Cðg; gÞ ¼ 0
Bðg; gÞ ¼
1 À g2 l 1 À g2

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 e ¼ kh (understood as the dimensionless
thickness of the layer or the dimensionless wave
number) and on the dimensionless spring compliances
cN and cT . The material dimensionless parameters are
taken as: For the Figs. 1, 3, 5: e1 ¼ 3:0; e2 ¼ 3:5;
e3 ¼ 1:5; e1 ¼ 1:8; e2 ¼ 1:2; e3 ¼ 0:5; rv ¼ 2:8;
rl ¼ 0:5.
For the Figs. 2, 4, 6: e1 ¼ 3:2; e2 ¼ 2:8; e3 ¼ 1:5;
e1 ¼ 1:8; e2 ¼ 1:1; e3 ¼ 0:6; rv ¼ 2; rl ¼ 1.
The wave velocity curves of first modes in the
interval e 2 ½0 2:5Š 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). Figures 3, 4 show the
wave velocity curves of first four modes (0, 1, 2, 3) in
the interval e 2 ½0 2:5Š of the welded contact (solid
line) and sliding contact (dashed line). Figures 5, 6
present the effect of the dimensionless spring compliances cN (characterizing the normal imperfection) and
cT (characterizing the shear imperfection) on the wave
velocity. Recall that for the perfectly bonded contact
cN ¼ cT ¼ 0.
It is shown from these figures that:
1.

x

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

1

0.8

0.6

0.4

0.2

0

0

0.5

1

1.5

2

ε

2.5

Fig. 1 Velocity curves of first six modes in the interval e 2
½0 2:5Š for the spring contact Here we take e1 ¼ 3:0; e2 ¼
3:5; e3 ¼ 1:5; e1 ¼ 1:8; e2 ¼ 1:2; e3 ¼ 0:5; rv ¼ 2:8; rl ¼ 0:5;
cT ¼ 15; cN ¼ 8

123


Meccanica
1

x

x

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0
0

0.5

1

1.5

2

ε

0

2.5

Fig. 2 Velocity curves of first five modes in the interval e 2
½0 2:5Š for the spring contact. Here we take e1 ¼ 3:2;
e2 ¼ 2:8; e3 ¼ 1:5; e1 ¼ 1:8; e2 ¼ 1:1; e3 ¼ 0:6; rv ¼ 2; rl ¼ 1, cT ¼
12; cN ¼ 10

0

0.5

1

1.5

2

ε

2.5

Fig. 4 Velocity curves of first four modes in the interval e 2
½0 2:5Š for the welded contact (solid line) and the sliding contact
(dashed line). Here we take e1 ¼ 3:2; e2 ¼ 2:8; e3 ¼ 1:5;
e1 ¼ 1:8; e2 ¼ 1:1; e3 ¼ 0:6; rv ¼ 2; rl ¼ 1

1

1

x

x 0.9

0.8

0.8

Mode 2: ε=1.1

0.7

0.6

0.6

Mode 1: ε=0.3

0.5

0.4

0.4
0.3

0.2

Mode 0: ε=0.3

: cN= 0

0.2
0.1

0

0

0.5

1

1.5

2

ε

:c =0
T

0

1

2

3

4

5

6

cN, c T

2.5

Fig. 3 Velocity curves of first four modes in the interval e 2
½0 2:5Š for the welded contact (solid line) and the sliding contact
(dashed line). Here we take e1 ¼ 3:0; e2 ¼ 3:5; e3 ¼ 1:5;
e1 ¼ 1:8; e2 ¼ 1:2; e3 ¼ 0:5; rv ¼ 2:8; rl ¼ 0:5

Fig. 5 Dependence of the wave velocity of modes 0, 1, 2 on the
normal imperfection cN (solid lines) and on the shear
imperfection cT (dashed lines). Here we take e1 ¼ 3:0;

e2 ¼ 3:5; e3 ¼ 1:5; e1 ¼ 1:8; e2 ¼ 1:2; e3 ¼ 0:5; rv ¼ 2:8; rl ¼ 0:5

dimensionless wave number) increases (see
Figs. 1, 2, 3, 4, 5 and 6).
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, 3 and
4). In particular:

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 intervals of e in which the wave velocity is almost
constant, except the modes 0 and 1.

2.

•

•

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
much more early than those of the sliding and

123

•

3.

For the same mode, the velocity curve for the
sliding contact always lies above the one for the
welded contact (see Figs. 3, 4).


Meccanica
1

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

0.9
0.8

Mode 2: ε = 1.7

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

0.7

x

0.6

Mode 1: ε = 0.4
:cN= 0

0.5

:c =0
T

0.4
0.3

Appendix 1 The coefficients of the secular Eq. (41)

Mode 0: ε = 0.4

0.2
0.1

The coefficients are:
0

1

2

3

4

5
c ,c
N

6
T

Fig. 6 Dependence of the wave velocity of modes 0, 1, 2 on the
normal imperfection cN (solid lines) and on the shear
imperfection cT (dashed lines). Here we take e1 ¼ 3:2; e2 ¼
2:8; e3 ¼ 1:5; e1 ¼ 1:8; e2 ¼ 1:1; e3 ¼ 0:6; rv ¼ 2; rl ¼ 1

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. 4 are employed. In particular, the Eq. (47) is
employed for drawing the velocity curves in the
Figs. 1, 2, 5 and 6; 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 halfspace 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

pffiffiffiÁ
À
A1 ¼ 2b1 b2 c1 c2 X À c11 À c66 P À c66 ð
a2 b1 c1 þ a1 b2 c2 Þ
pffiffiffi
o
n
 c212 À c22 ðc11 À XÞ P þ Xðc11 À XÞ
n
o
À c66 c1 c2 ð
a2 b1 þ a1 b2 Þ þ b1 b2 ð
c1 þ c2 Þ
pffiffiffiÁ
À
 c11 À X À c12 P



n kc
kc66 pffiffiffi
66
À 2b1 b2 c1 c2
ðc11 À XÞ þ
c22 P
KT
KN
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi k2 c66
 Sþ2 PÀ
KT KN

o
pffiffiffi
2
 ðc12 À c22 c11 þ c22 XÞ P þ Xðc11 À XÞ
pffiffiffiÁ
ÁÀ
À
B1 ¼ b21 c22 þ b22 c21 X À c11 À c66 P
À c66 ð
a1 b1 c2 þ a2 b2 c1 Þ
nÀ
o
Ápffiffiffi
c212 À c22 ðc11 À XÞ P þ Xðc11 À XÞ
À
Á

À c66 a2 b2 c21 þ a1 b1 c22 þ b21 c2 þ b22 c1
pffiffiffi
2
2
 ðc11 À X À c12 PÞ À ðb2 c21 þ b1 c22 Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
&
pffiffiffi
kc66
kc66 pffiffiffi
Â
ðc11 À XÞ þ
c22 P
Sþ2 P
KT
KN
'
pffiffiffi
k2 c66  2
ðc12 À c22 c11 þ c22 XÞ P þ Xðc11 À XÞ
À
KT KN
n
pffiffiffio
 P
cŠðc11 À XÞ À c22 b1 c2 ½
a; bŠ
C1 ¼ c66 b2 c1 ½
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n
pffiffiffi

kc66 o
 kc66 À b c ½
a; bŠ
 S þ 2 P þ b1 c2 ½
2 1 cŠ
K
K
n
pffiffiffi T
oN
2
 c12 À c22 ðc11 À XÞ P þ Xðc11 À XÞ

pffiffiffi
 P
D1 ¼ c66 b1 c2 ½
cŠðX À c11 Þ þ c22 b2 c1 ½
a; bŠ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n
pffiffiffi
kc66 o
 kc66 À b c ½
a; bŠ
 S þ 2 P À b2 c1 ½
1 2 cŠ
K
K
n
pffiffiffi T
oN

2
 c12 À c22 ðc11 À XÞ P þ Xðc11 À XÞ
n

 c2 À c22 ðc11 À XÞ
E1 ¼ À A1 þ c66 ½
cŠ½
a; bŠ
12
o
pffiffiffi
 P þ Xðc11 À XÞ

ð54Þ

123


Meccanica

Appendix 2 The coefficients of the secular Eq. (44)
The coefficients are:
n
A2 ¼ 4p1 p2 ð2 À rv2 xÞ 2ð2 À rv2 xÞðb1 b2 À 1Þ
Ã
Â
þ 4b1 b2 À ð2 À xÞ2 rlÀ2 À ð4 À rv2 xÞ
 ð2b1 b2 þ x À 2ÞrlÀ1 À 2cT cN ð2 À rv2 xÞ
Â
Ã

 4b1 b2 À ð2 À xÞ2 À 2cT ð2 À rv2 xÞb1 x À 2cN
o
 ð2 À rv2 xÞb2 x ;
n
Â
Ã2 o
B2 ¼ 4p21 p22 4b1 b2 ð1 À rlÀ1 Þ2 À 2 À ð2 À xÞrlÀ1
n
þ ð2 À rv2 xÞ2 ð2 À rv2 xÞ2 ðb1 b2 À 1Þ
À 2ð2 À rv2 xÞð2b1 b2 þ x À 2ÞrlÀ1
Â
à o
þ 4b1 b2 À ð2 À xÞ2 rlÀ2
Â
Ã
À 16p21 p22 þ ð2 À rv2 xÞ4
n
o
Â
Ã
 cT cN 4b1 b2 À ð2 À xÞ2 þ cT b1 x þ cN b2 x ;
Â
Ã
C2 ¼ p2 rv2 x2 b1 ð2 À rv2 xÞ2 À 4b2 p21 rlÀ1
Â
Ã
þ p2 rv2 x cN ð2 À rv2 xÞ2 À 4cT p21
Ã
Â
 4b1 b2 À ð2 À xÞ2 rlÀ1 ;

Â
Ã
D2 ¼ p1 rv2 x2 b2 ð2 À rv2 xÞ2 À 4b1 p22 rlÀ1
Ã
Â
À p1 rv2 x 4cN p22 À cT ð2 À rv2 xÞ2
Ã
Â
 4b1 b2 À ð2 À xÞ2 rlÀ1 ;
Â
Ã
E2 ¼ À A2 À p1 p2 rv4 x2 4b1 b2 À ð2 À xÞ2 rlÀ2
ð55Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi
1 À cx; b2 ¼ 1 À x; p1 ¼ 1 À crv2 x;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p2 ¼ 1 À rv2 x
b1 ¼

l
c2
l
l
; c ¼ 
; rl ¼ ; rv ¼
k þ 2l
l

c2
k þ 2l
rffiffiffi
rffiffiffi
2
l
c
l
; c2 ¼
c2 ¼
; x ¼ 2 ð0\x\1Þ
q
q
c2
c ¼

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