VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY
INSTITUTE OF MECHANICS
———– * ———–
NGUYEN THỊ KHANH LINH
SURFACE WAVES AND WAVES
IN THIN STRUCTURES
Major: Engineering mechanics
Code: 62 52 01 01
SUMMARY OF DOCTORAL THESIS
HANOI – 2013
Supervisor:
Assoc. Prof. Dr. Pham Chi Vinh
Referee 1:
Referee 2:
Referee 3:
The thesis is protected to the Council assessing
doctoral dissertation level Institute, meeting at the
Institute of Mechanics, 264 Doi Can - Ba Dinh - Ha Noi.
At hours and minutes, , 20
1
Chapter 1. Survey
Actuality of the thesis
Problems of elastic wave propagation, especially the ones of Rayleigh
wave propagration, are the foundation of various practical applications
in science and technology.
Elastic surface waves, discovered by Rayleigh more than 120 years
ago for compressible isotropic elastic solids, have been studied exten-
sively and exploited in a wide range of applications in seismology, acous-
tics, geophysics, telecommunications industry and materials science, for
example. It would not be far-fetched to say that Rayleigh’s study of
surface waves upon an elastic half-space has had fundamental and far-

The main purposes of the thesis
• Applying new methods to develop some problems on Rayleigh
waves that were investigated previously.
• Deriving approximate secular equations of Rayleigh waves prop-
agating in elastic half-spaces coated an elastic thin layer.
• Investigating SH wave and Lamb wave propagting in periodical
and thin structures.
Research objects
Waves propagating in elastic half-spaces, waves propagating in elas-
tic half-spaces coated with a thin elastic layer, waves propagating in
periodical, thin structures.
Research range
Exact and approximate secular equations of Rayleigh waves and
formulas for the Rayleigh wave velocity.
Research methods
The method of cubic equation, the method of least squares, the
perturbation method, the method of effective boundary condition, the
method of polarization vector and the asymptotic method.
Chapter 2. Rayleigh waves
2.1 Rayleigh waves in incompressible elastic media
subjected to gravity
2.1.1 Secular equation
Consider the problem of Rayleigh wave propagation in an incompress-
ible isotropic elastic half-space x
3
≥ 0 (Figure 2.1) subjected to gravity.
3
Figure 2.1. The problem model
The secular equation is:
(2 − x)

=
2(4 + δ)
3
−
3


16(δ + 11)(δ
2
+ 4)/27 + (δ
3
+ 12δ
2
+ 12δ + 136)/27
+
8 − 8δ − δ
2
9
3


16(δ + 11)(δ
2
+ 4)/27 + (δ
3
+ 12δ
2
+ 12δ + 136)/27
, δ ∈ [0 , 1).
(3)

−
1
3




2
.
(4)
4
Approximate formulas for the velocity of Rayleigh waves
By applying the method of least squares, we drive approximate
formulas for the velocity of Rayleigh waves:
x
1
=
B −
√
B
2
− 4AC
2A
, (5)
with A = −(5.1311 + 2δ), B = −(21.2576 + 8δ + δ
2
), C = −(15.1266 +
8δ).
x
2


26
27
+
2
3

11
3

1/3
−
8
9

26
27
+
2
3

11
3

−1/3
−
1
3



66
− ρc
2
)[B
2
12
− B
22
(B
11
− ρc
2
)]
+ρc
2

B
22
B
66

(B
11
− ρc
2
)(B
66
− ρc
2
) = 0 (8)

1
1 − ν
12
ν
21
=
ν
12
E
2
1 − ν
12
ν
21
, B
66
= G
12
,
(9)
Remark: The secular equation (8) is much more simple than the sec-
ular equations of Cerv and valid for any orthotropic elastic materials.
2.2.1.2 Formulas for the velocity
Exact formula
6
Following the same procedure carried out in [Pham Chi Vinh and
Ogden, R. W., Ach. Mech., 56 (3) (2004), 247-265], formula for the
velocity of Rayleigh waves is derived:
ρc
2

D

(10)
where b
1
= B
22
/B
11
, b
2
= 1 −B
2
12
/(B
11
B
22
), b
3
= B
11
/B
66
, R and D
are given by:
R = −
1
54
h(b

(1 − b
2
b
3
)
2
+ 4

,
h(b
1
, b
2
, b
3
) =
√
b
1
[2b
1
(1 − b
2
b
3
)
3
+ 9(3b
2
− b

12
E
1
, b
3
=
E
2
1
G
12
(E
1
ν
2
12
− E
2
5)
(12)
Approximate formulas
Using the method of least squares, we obtain approximate formulas:
x
1
=
B
1
−

B

b
3
[0.6(b
1
b
3
− 1) − b
1
b
2
b
2
3
(b
2
b
3
+ 2)],
C
1
=0.05b
1
b
3
(b
1
b
3
− 1) − b
2

3
[b
3
(1 + 0.5b
1
− 2b
1
b
2
b
3
) − 1.5],
B
2
=b
1
b
3
[0.5625(b
1
b
3
− 1) − b
1
b
2
b
2
3
(b

the half-space x
2
≥ 0 whose principal material axes are X, Y, Z (hình
2.9). Suppose that the Z axis coincides with the x
3
axis and (x
1
, x
2
)
Figure 2.9. The model of non-principal Rayleigh waves
is the rotated one from (X, Y ) by counter clockwise angle θ. Suppose
that the panel is subjected to the plane stress state.
Using the method of first integrals, we derive the secular equation:
F (X, θ) ≡dX
2
[(d + d
2
)X −d
3
][d
2
2
− Q
66
(dX −d
3
)]
+ (dX −d
3

) − d
1
d
2
] = 0 (15)
where X = ρc
2
and
Q
11
= B
11
c
4
θ
+ 2(B
12
+ 2B
66
)c
2
θ
s
2
θ
+ B
22
s
4
θ

− 4B
66
)c
2
θ
s
2
θ
+ B
12
(c
4
θ
+ s
4
θ
), (16)
Q
66
= (B
11
+ B
22
− 2B
12
− 2B
66
)c
2
θ

66
)c
θ
s
3
θ
,
Q
26
= −(B
11
− B
12
− 2B
66
)c
θ
s
3
θ
− (B
12
− B
22
+ 2B
66
)c
3
θ
s

3
= Q
11
d + Q
16
d
1
− Q
12
d
2
.
with c
θ
:= cosθ, s
θ
:= sinθ (0 ≤ θ ≤ π).
8
2.2.3 Conclusion
In this chapter we obtain the secular equation for principal Rayleigh
waves that is valid for all orthotropic elastic materials and much more
simple than the ones obtained recently by Cerv. Exact and approx-
imate formulas for the velocity of principal Rayleigh-edge waves are
also established and they are a powerful tool for analyzing the effect of
material parameters on the Rayleigh wave velocity. For non-principal
Rayleigh waves a secular equation in explicit form is obtained by us-
ing the method of first integrals. They are new results and have been
published "Vietnam Journal of Mechanics, 34 (2) (2012), 123 – 134"
Chapter 3. Rayleigh waves in elastic half-
spaces underlying a water layer

=

µ/ρ , δ = g/kc
2
2
, ε = kh, r =
ρ

ρ
, f (x, δ, ε) =
(δ −xthε)/(x −δthε), c is velocity, µ and ρ are Lame contants and the
mass density of the medium, g is the acceleration due to gravity, k is
the wave number, ρ

is the mass density of the water
9
Remark: To the best knowledge of the authors the exact secular equa-
tion (17) did not appear in the literature. It is shown the approximate
secular equation obtained by Bromwich is a special case of (17).
3.1.2. On existence of Rayleigh waves
Theorem 3.1.
i) A Rayleigh wave is impossible if either {δ ≥ 1, 0 < δthε ≤ 1,
r ≥ 1 + 2/δ} or {δ ≥ 1, δthε > 1, r ∈ (0, m] ∪[1 + 2/δ, +∞)}.
ii) There exists a unique Rayleigh wave, namely CRW, if {0 < δ < 1,
r ≥ 1 + 2/δ}.
iii) There exists a unique Rayleigh wave, namely GRW, if either {δ ≥
1, 0 < δthε ≤ 1, 0 < r < 1 + 2/δ} hoặc {δ ≥ 1, δthε > 1,
m < r < 1 + 2/δ}.
iv) There exist exactly two Rayleigh waves, one CRW and one GRW,
if {0 < δ < 1, 0 < r < 1 + 2/δ}.

+
x
2
3

ε
3
− r

δ
5
x
3
−
5δ
3
3x
+
2δ x
3

ε
4
= 0, δthε < x < 1
(18)
3.2. Approximate formulas for the velocity
Both δ and ε being small
By using the perturbation method, we obtain the second-order ap-
proximation of the squared dimensionless velocity of Rayleigh waves:


= 2(x
0
− 2) + 2(1 − x
0
)
−1/2
− δ,
10
a
2
=−

2+(1−x
0
)
−
3
2

r
2
(δ
2
− x
2
0
)
2
a
3

0
is calculated by (7).
Global approximation
Suppose that the water layer is thin: ε << 1, by using the method
of least squares, we derive the global approximation:
x =
−B +
√
B
2
− 4AC
2A
(21)
A = 5.4364(1+ε
∗
)
2
+8δ−2ε
∗
δ
2
+8ε
∗
+δ
2
−2ε
∗
2
δ
2

2
δ
4
, với ε
∗
= rε.
3.3. Conclusion
In this chapter, the exact secular equation of the wave is derived
and based on it the existence of Rayleigh waves is examined. When
the layer being thin, a fourth-order approximate secular equation is
established and using it some approximate formulas for the velocity
are established. The results of this chapter have been published in the
journal "Meccanica, Vol. 48, pp.2051-2060, 2013"
Chapter 4. Rayleigh waves in a half-space
coated by a thin layer
4.1 Rayleigh waves in a compressible orthotropic
elastic half-space coated by a thin orthotropic
elastic layer
4.1.1 Effective boundary condition of third-order
Consider an elastic half-space x
2
≥ 0 coated by a thin elastic layer −h ≤
x
2
≤ 0. Both the layer and the half-space are homogeneous, orthotropic
and elastic. The thin layer is assumed to be perfectly bonded to the
half-space.
11
Figure 4.1. The problem moldel
By using the effective boundary condition method, the entire effect of

− r
3
u
2,111
− ¯ρ(1 + r
1
)¨u
2,1

+
h
3
6

r
4
σ
22,111
+ ¯ρr
5
¨σ
22,1
− r
6
u
1,1111
− ¯ρr
7
¨u
1,11

¯c
22
¨σ
22
−r
3
u
1,111
−¯ρ(1+r
1
)¨u
1,1

+
h
3
6

r
2
σ
12,111
+ ¯ρr
8
¨σ
12,1
−r
3
u
2,1111

1
=
¯c
12
¯c
22
, r
2
= r
1
+
r
3
¯c
66
, r
3
=
¯c
2
12
− ¯c
11
¯c
22
¯c
22
, r
4
= r

7
= r
2
1
+ 2r
2
, r
8
=
1 + r
1
¯c
66
+
1
¯c
22
.
4.1.2 An approximate secular equation of third-order
Using expressions of the stresses and the displacements of the half-space
into the effective boundary conditions (22), (23) leads to the dispersion
equation of the wave, namely
D
0
+ D
1
ε +
D
2
2

− x)x,
D
1
=r
µ

(e
1
− x)r
2
v
x + e
2
(¯e
2
¯e
2
3
− ¯e
1
+ r
2
v
x)b
1
b
2

(b
1

− ¯e
2
¯e
2
3
) + (¯e
2
¯e
3
e
3
− e
3
− r
µ
¯e
2
¯e
2
3
+ r
µ
¯e
1
)r
2
v
x − r
µ
r

− r
µ
¯e
1
)r
2
v
x + r
µ
r
4
v
x
2

(x − e
1
),
D
3
= r
µ

(x−e
1
)

2(¯e
1
−¯e

2
¯e
3
+¯e
2
¯e
2
3
−¯e
1
)(¯e
2
¯e
2
3
−¯e
1
)
+ (¯e
2
2
¯e
2
3
+ 2¯e
2
¯e
3
+ 2¯e
2

),
với b
1
b
2
=
√
P , b
1
+ b
2
=

S + 2
√
P , S =
e
2
(e
1
−x) +1 −x − (1 +e
3
)
2
e
2
,
P =
(1 − x)(e
1

¯e
1
=
¯c
11
¯c
66
, ¯e
2
=
¯c
66
¯c
22
, ¯e
3
=
¯c
12
¯c
66
, r
µ
=
¯c
66
c
66
, r
v

µ
, r
v
, e
k
, ¯e
k
, k =
1, 2, 3 and ε.
4.1.3 An approximate formula of second-order for
the velocity
Let ε be small, we have
x(ε) = x
0
+ x

(0) ε +
x

(0)
2
ε
2
+ O(ε
3
) (25)
13
where x
0
= x(0) is the squared dimensionless velocity of Rayleigh waves

√
D

(26)
s
1
= e
2
/e
1
, s
2
= 1 − e
2
3
/(e
1
e
2
), s
3
= e
1
x

(0) = −
D
1
D
0x





x=x
0
and R, D are given by (11) in which b
1
, b
2
, b
3
are substitutied by
s
1
, s
2
, s
3
.
4.2 Rayleigh waves in an incompressible orthotropic
elastic half-space coated by a thin orthotropic
elastic layer
4.2.1 Effective boundary condition of third-order
Consider an elastic half-space x
2
≥ 0 coated by a thin elastic layer
−h ≤ x
2
≤ 0. The layer and the half-space are both homogeneous,

¯
δu
2,111
− 2¯ρ¨u
2,1
)
+
h
3
6
(r
1
σ
22,111
+
¯ρ
¯c
66
¨σ
22,1
− r
2
u
1,1111
− ¯ρr
3
¨u
1,11
−
¯ρ

3
6
(r
1
σ
12,111
+
2¯ρ
¯c
66
¨σ
12,1
+
¯
δu
2,1111
− 3¯ρ¨u
2,11
) = 0 at x
2
= 0, (28)
where:
¯
δ = ¯c
11
+¯c
22
−2¯c
12
, r

D
2
2
ε
2
+
D
3
6
ε
3
+ O(ε
4
) = 0, (29)
where:
D
0
=(x − e
δ
)(
√
P −1) −S − P − 1,
D
1
=r
µ

r
2
v

2
µ
r
2
v
x

xr
2
v
− ¯e
δ

+ r
µ
¯e
δ
(x − e
δ
+ S + 2
√
P ),
D
3
= − r
µ

S + 2
√
P

,
¯e
δ
=¯e
1
+ ¯e
2
− 2¯e
3
, e
δ
= e
1
+ e
2
− 2e
3
,
P =1 − x, S = e
δ
− 2 − x
and:
x =
ρc
2
c
66
, e
1
=

¯c
66
,
¯e
3
=
¯c
12
¯c
66
, r
µ
=
¯c
66
c
66
, r
v
=
c
2
¯c
2
, c
2
=

c
66

isotropic elastic materials with the underlying deformations correspond-
ing to pure homogeneous strains (see Figure 4.5). The principal direc-
tions of strain in the two solids are aligned, ones direction being normal
to the planar interface defined by x
2
= 0. A rectangular Cartesian coor-
dinate system (x
1
, x
2
, x
3
) is employed with its axes coinciding with the
principal directions of the pure strain. The layer occupies the domain
−h < x
2
< 0 and the half-space corresponds to the region x
2
> 0.
Figure 4.5. The problem model
By using the effective boundary condition method, the entire effect of
the layer on the half-space is replaced by effective boundary conditions,
namely:
s
21
+ h(r
1
s
22,1
− r

+
h
3
6

t
1
s
22,111
+ ¯ρt
2
¨s
22,1
− t
3
u
1,1111
− ¯ρt
4
¨u
1,11
−
¯ρ
2
¯γ
2
¨u
1,tt

= 0 at x

− r
7
u
1,111
− ¯ρr
5
¨u
1,1

(31)
+
h
3
6

t
5
s
21,111
+ ¯ρt
6
¨s
21,1
− t
7
u
2,1111
− ¯ρt
8
¨u

2
, r
6
=
r
3
¯γ
2
+ r
1
r
2
, r
7
= r
2
r
3
+ r
1
r
4
,
r
8
=
r
4
¯α
22

= r
1
r
7
+ r
3
r
6
, t
4
=
r
3
¯γ
2
+ r
1
r
5
+ r
6
, t
5
=
r
7
¯γ
2
+ r
2

¯α
22
+ r
8
+ r
2
r
5
.
(32)
16
4.3.2 An approximate secular equation of third-order
By introducing expressions of the incremental stresses and the incre-
mental displacements of the half-space into the effective boundary con-
ditions (30), (31), we arrive at the approximate secular equation of
third-order of Rayleigh waves, namely
D
0
+ D
1
ε +
D
2
2
ε
2
+
D
3
6

(1 − x)],
D
1
= r
µ
e
5

(¯e
2
4
¯e
5
− 1 + r
2
v
x)(e
1
− x) + e
2
(¯e
2
¯e
2
3
− ¯e
1
+ r
2
v

) + (¯e
2
+ ¯e
5
)r
2
v
x

D
0
− 2r
µ
¯e
5

r
µ
(¯e
2
¯e
2
3
− ¯e
1
+ r
2
v
x)(¯e
2

) + (¯e
2
¯e
3
− ¯e
4
¯e
5
)r
2
v
x


b
1
b
2
+ 2r
µ
(x − e
1
)

r
µ
(¯e
2
¯e
2

5
(¯e
2
¯e
2
3
− ¯e
1
) + (¯e
2
¯e
3
− ¯e
4
¯e
5
)r
2
v
x


,
D
3
= r
µ
e
5
(x − e

+ 3¯e
5
)r
2
v
x + 3¯e
5
(¯e
2
¯e
2
3
− ¯e
1
)

− 2¯e
4
¯e
5

¯e
2
¯e
3
(¯e
2
4
¯e
5

)
− r
µ
e
2
e
5

(¯e
2
¯e
2
3
− ¯e
1
+ r
2
v
x)

3¯e
2
(¯e
2
4
¯e
5
− 1) + 4¯e
2
¯e

¯e
3
(¯e
2
4
¯e
5
− 1)
+ ¯e
4
¯e
5
(¯e
2
¯e
2
3
− ¯e
1
) + (¯e
2
¯e
3
+ ¯e
4
¯e
5
)r
2
v

1
− x)(1 − x)
e
2
e
5
,
S =
e
2
(e
1
− x) + e
5
(1 − x) − (e
3
+ e
4
)
2
e
2
e
5
(35)
17
and:
x =
ρc
2

, e
5
=
γ
2
γ
1
,
¯e
1
=
¯α
11
¯γ
1
, ¯e
2
=
¯γ
1
¯α
22
, ¯e
3
=
¯α
12
¯γ
1
, ¯e

=

γ
1
ρ
, ¯c
2
=

¯γ
1
¯ρ
.
where α
ij
, γ
k
, ρ are respectively the material constants and the mass
density of the half-space.
It is clear that the squared dimensionless Rayleigh wave velocity x =
c
2
/c
2
2
depend on 13 dimensionless parameters: e
k
, ¯e
k
(k = 1, 2, 3, 4, 5),

5.1 SH waves in infinite, periodically layered elastic
media containing thin layers
5.1.1. Setting problem
Consider a SH propagating in infinite periodically layered of which
each periodicity cell consists of N different isotropic elastic layers (N ≥
2) with the thickness h
1
, h
2
, , h
N
. The thickness of the periodicity cell
is h = h
1
+ +h
N
. Assume that the angle between the wave propagation
direction and the x
1
-axis is θ (Figure 5.1)
2
x
N
h
k
1
x
1
h
h

• Formula for determining Ω
3
Ω
3
=

1
µ

−1
ρ
−1

sin
4
θ
3
−

1
µ

f
2
21
−f
2
12
(ρΩ
1

−1
, p
21
 = ρ − µcos
2
θ
f
1
11
=
1

0
p
12
(x
1
)dx
1
x
1

0
p
21
(x
2
)dx
2
, f

p
12
(x
1
)dx
1
x
1

0
p
21
(x
2
)dx
2
x
2

0
p
12
(x
3
)dx
3
,
f
2
22

.
• The recurrent formula for calculating Ω
2n+1
(n ≥ 2)
Ω
2n+1
=ρ
−1

1
µ

−1
g
2n
12
(1)

ρΩ
1
− µcos
2
θ

− ρ
−1
g
2n
21
(1)

1
, Ω
3
, , Ω
2n−1
whose expressions are
written in detail in the thesis.
5.2. Lamb wave in an infinite, periodically layered,
pre-strained, incompressible elastic medium
5.2.1. Setting problem
Let us consider a periodically layered, infinite elastic medium. Each
periodicity cell consists of N different layers (N ≥ 2). Suppose that the
material layers are incompressible isotropic and subject to homogeneous
initial deformations. At the initial state, we introduce the coordinate
plane Ox
1
x
3
coincides with the bottom plane of the first material layer
of the periodicity cell (see Fig. 5.2). By h
i
and h we denote, respectively,
the thickness of the i-th layer (i = 1, , N) and the thickness of the
periodicity cell at the initial state.
20
2
x
N
h
1


B
−1
2121

−1
ρ
(40)
where: B
ijkl
are material parameters, B
∗
ijkl
= B
ijkl
+ p, p is the
hydrostatic pressure.
• Formula for Ω
3
Ω
3
= −

ρs
0
13

−1

s

s
2
ij
, t
2
ij
, M are determined in terms of the material pa-
rameters and Ω
1
, whose expressions are given in detail in the
thesis.
• The recurrent formula for calculating Ω
2n+1
(n ≥ 2):
Ω
2n+1
=−

ρs
0
13

−1


t
2n
12
(1)s
0

t
]


,
(42)
∀n ≥ 1.
21
where t
2n
ij
(1), A
u
, B
v
, C
s
, D
t
(0 ≤ u, v, s, t ≤ 2n) are calculated in terms
of the material parameters and Ω
1
, Ω
3
, Ω
2n−1
, these expressions are
given in the thesis.
5.3. Conclusion
In this chapter, are obtain the formulae for Ω

when the water layer being thin. Further, of second-order of ap-
proximate formulas for the Rayleigh wave velocity are also es-
22
tablished for this case using the perturbation method and the
method of least squares.
(iv) An approximate formula of second-order for the velocity has
been derived for the case when the layer being thin and the gravity
effect being small and it recovers Bromwich’s formula.
• By using the effective boundary condition method, have been de-
rived the approximate secular equation of third-order of Rayleigh
waves propagating in compressible and incompressible, orthotropic
half-spaces coated a thin compressible and incompressible or-
thotropic layer.
• By using the effective boundary condition method, has been de-
rived the approximate secular equation of third-order of Rayleigh
waves propagating in pre-strained half-spaces coated a thin pre-
strained layer.
• Have been obtained the formulae for Ω
1
, Ω
3
and the recurrent for-
mulae for computing Ω
2n+1
, n ≥ 2 in the asymptotic expansions
of the wave velocity of:
(i) SH waves propagating in infinite, periodically layered, isotropic
elastic media containing thin layers.
(ii) Lamb waves propagating in infinite, periodically layered, pre-
strained media containing thin layers.

compressible elastic solids, International Journal of Non-Linear
Mechanics, Vol. 50, pp. 91–96, 2013.
6) Pham Chi Vinh, Nguyen Thi Khanh Linh, An explicit secular
equation of Rayleigh waves propagating along an obliquely cut
surface in a directional fiber-reinforced composite, Vietnam Jour-
nal of Mechanics, Vol. 34 (2) , pp. 123 – 134, (2012).