Contents lists available atScienceDirectWave Motion journal homepage:www.elsevier.com/locate/wavemoti Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic
Trang 1Contents lists available atScienceDirect
Wave Motion journal homepage:www.elsevier.com/locate/wavemoti
Rayleigh waves in an isotropic elastic half-space coated by a
thin isotropic elastic layer with smooth contact
Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Viet Nam
h i g h l i g h t s
• The propagation of Rayleigh waves in an elastic half-space coated by a thin elastic layer is considered
• The half-space and the layer are both isotropic and the contact between them is smooth
• By using the effective boundary condition method an approximate secular equation of fourth-order has been derived
• From it, an explicit third-order approximate formula for the Rayleigh wave velocity has been established
• The approximate secular equation and the formula for the velocity will be useful in practical applications
a r t i c l e i n f o
Article history:
Received 3 May 2013
Received in revised form 9 October 2013
Accepted 24 November 2013
Available online 2 December 2013
Keywords:
Rayleigh waves
An elastic half-space coated with a thin
elastic layer Approximate secular
equations
Approximate formulas for the velocity
a b s t r a c t
In the present paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer The contact between the layer and the half space is assumed to be smooth The main purpose of the paper is to establish
an approximate secular equation of the wave By using the effective boundary condition method, an approximate, yet highly accurate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived From the secular equation obtained, an approximate formula of third-order for the velocity of Rayleigh waves is established The approximate secular equation and the formula for the velocity obtained in this paper are potentially useful in many practical applications
© 2013 Elsevier B.V All rights reserved
1 Introduction
The structures of a thin film attached to solids, modeled as half-spaces coated with a thin layer, are widely applied in modern technology Measurement of mechanical properties of thin supported films is therefore very significant [1] Among various measurement methods, the surface/guided wave method [2] is used most extensively in which the Rayleigh wave
is a most convenient tool For the Rayleigh-wave approach, the explicit dispersion relations of Rayleigh waves supported
by thin-film/substrate interactions are employed as theoretical bases for extracting the mechanical properties of the thin films from experimental data They are therefore the key factor of the investigations of Rayleigh waves propagating in half-spaces covered by a thin layer Taking the assumption of a thin layer, explicit secular equations can be derived by replacing
approximately the entire effect of the thin layer on the half-space by the so-called effective boundary conditions which relate the
displacements with the stresses of the half-space at its surface For obtaining the effective boundary conditions Achenbach [3] and Tiersten [4] replaced the thin layer by a plate modeled by different theories: Mindlin’s plate theory and the plate theory
of low-frequency extension and flexure (classical plate theory), while Bovik [5] expanded the stresses at the top surface of
∗Corresponding author Tel.: +84 4 35532164; fax: +84 4 38588817.
E-mail addresses:pcvinh562000@yahoo.co.uk , pcvinh@vnu.edu.vn (P.C Vinh).
0165-2125/$ – see front matter © 2013 Elsevier B.V All rights reserved.
Trang 2the layer into Taylor series in its thickness The Taylor expansion approach was then employed by Niklasson [6], Rokhlin [7,8
Benveniste [9], Steigmann and Ogden [10], Steigmann [11], Ting [12], Vinh and Linh [13,14], Kaplunov and Prikazchikov [15]
to establish the effective boundary conditions
Achenbach [3], Tiersten [4], Bovik [5], Tuan [16] assumed that the layer and the substrate are both isotropic and derived approximate secular equations of second-order (these equations do not coincide totally with each other) In [10] Steigmann and Ogden considered a transversely isotropic layer with residual stress overlying an isotropic half-space and the authors obtained an approximate second-order dispersion relation In [17] Wang et al considered an isotropic half-space covered
by a thin electrode layer and the authors obtained an approximate secular equation of first-order In [13] the layer and the half-space were both assumed to be orthotropic and an approximate secular equation of third-order was obtained In [14] the layer and the half-space were both subjected to homogeneous pre-strains and an approximate secular equation of third-order was established which is valid for any pre-strain and for a general strain energy function
In all investigations mentioned above, the contact between the layer and the half-space is assumed to be welded For the case of smooth contact, there exists only one approximate secular equation of third-order in the literature established
by Achenbach and Keshava [3] This approximate secular equation includes the shear coefficient, originating from Mindlin’s plate theory [18], whose usage should be avoided as noted by Muller and Touratier [19], Touratier [20] This remark was also mentioned in [21]
It should be noted that for the case of smooth contact, one could not arrive at the effective boundary conditions from the relations between the displacements and the stresses at the bottom surface of the layer which were derived by Tiersten [4] and Bovik [5] In contrast, for the case of welded contact, the effective boundary conditions were immediately obtained The main purpose of the paper is to establish an approximate secular equation of Rayleigh waves propagating in
an isotropic elastic half-space coated with a thin isotropic elastic layer for the case of smooth contact By using the effective boundary condition method, an approximate effective boundary condition of fourth-order which relates the normal displacement with the normal stress at the surface of the half space is derived Using this condition along with the vanishing of the shear stress at the surface of the half-space, an approximate secular equation of fourth-order in terms
of the dimensionless thickness of the layer is derived We will show that the approximate secular equation obtained is a very good approximation Based on it, an approximate formula of third-order for the velocity of Rayleigh waves is established
2 Effective boundary condition of fourth-order
Consider an elastic half-space x3 ≥ 0 coated by a thin elastic layer−h ≤ x3 ≤ 0 Both the layer and half-space are homogeneous, isotropic and linearly elastic The layer is assumed to be thin and has a smooth contact with the half-space
In particular, the normal component of the particle displacement vector and the normal component of the stress tensor are
continuous, while the shearing stress vanishes across the interface x3=0, see Achenbach [3] and Murty [22] Note that the 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
If it is assumed that a state of plane strain exists, whereby the x2component of displacement vanishes and the x1and x3 components are functions of x1, x3and t only, i.e.
where t is the time Since the layer is made of isotropic elastic materials, the strain–stress relations take the form
¯
σ11= (¯λ +2µ)¯ ¯ u1, 1+ ¯ λ¯u3, 3,
¯
σ33= ¯ λ¯u1, 1+ (¯λ +2µ)¯ ¯ u3, 3,
¯
σ13= ¯ µ(¯u1, 3+ ¯u3, 1)
(2)
whereσ ¯ij is the stress of the layer, commas indicate differentiation with respect to spatial variables x k, ¯λandµ ¯ are Lame constants In the absent of body forces, the equations of motion for the layer is
¯
σ11 , 1+ ¯ σ13 , 3= ¯ ρ ¨¯u1,
¯
where a dot signifies differentiation with respect to t From Eqs.(2),(3)we have
U¯′
¯
T′
=
U¯
¯
T
(4) where
¯
U= ¯u1 u¯3T
, T¯ = ¯ σ13 σ ¯33
T
Trang 3the symbol ‘‘T ’’ indicate the transpose of a matrix, the prime signifies differentiation with respect to x3and
− ¯ λ
¯
λ +2µ ¯ ∂1 0
1
¯
¯
λ +2µ ¯
,
− (¯λ +2µ) ¯ 2− ¯ λ2
¯
λ +2µ ¯ ∂
2
1+ ¯ ρ∂2
t
(5)
here we use the notations∂1= ∂/∂x1, ∂2= ∂2/∂x1 , ∂2
t = ∂2/∂t2 From(4)it follows
U¯(n)
¯
T(n)
=M n
U¯
¯
T
Let h be small (i.e the layer is thin), then expanding into Taylor series T¯ (−h)at x3=0 up to the fourth-order of h we have
¯
T(−h) = ¯T(0) + ¯T′(0)(−h) + 1
2!
¯
T′′(0)h2− 1
3!
¯
T′′′(0)h3+ 1
Suppose that surface x3= −h is free of traction, i.e T¯ (−h) =0 Introducing(6)with n=1,2,3,4 at x3=0 into(7)yields
2h
2M6− 1
6h
3M8+ 1
24h
4M10
¯
T(0) =
2h
2M5+1
6h
3M7− 1
24h
4M9
¯
where I is the identity matrix of order 2, M3,M4are defined by(5)and
M5=M3M1+M4M3,
M6=M3M2+M2,
M7=M3M12+M4M3M1+M3M2M3+M42M3,
M8=M3M1M2+M4M3M2+M3M2M4+M43,
M9=M3M13+M4M3M12+M3M2M3M1+M42M3M1+M3M1M2M3+M4M3M2M3+M3M2M4M3+M43M3,
M10=M3M2M2+M4M3M1M2+M3M2M3M2+M2M3M2+M3M1M2M4+M4M3M2M4+M3M2M2+M4.
(9)
Taking into account(5)and(9), the relation(8)in component form is of the form
¯
σ13+h (1−2γ ) ¯σ ¯ 33 , 1+4ρ¯ ¯c22(1− ¯ γ )¯u1, 11− ¯ ρ ¨¯u1
+h2
2
(2γ − ¯ 3) ¯σ13 , 11+ 1
¯
c22σ ¨¯13+4ρ¯ ¯c22(1− ¯ γ )¯u3, 111−2ρ( ¯ 1− ¯ γ )¨¯u3, 1
+h3
6
(4γ − ¯ 3) ¯σ33 , 111+
2
¯
c12(1− ¯ γ ) + ¯1
c22(1−2γ ) ¯
¨¯
σ33 , 1−8ρ¯ ¯c22(1− ¯ γ )¯u1, 1111
+ ¯ ρ(5−4γ ¯2)¨¯u1, 11− ρ ¯
¯
c22u¨¯1,tt
+ h4 24
(5−4γ ) ¯σ ¯ 13 , 1111+2
1
¯
c12(1− ¯ γ )
¯
c22(2γ ¯2− ¯ γ −2)
¨¯
σ13 , 11+ 1
¯
c24σ ¨¯13 ,tt+4ρ( ¯ 2− ¯ γ − ¯γ2)¨¯u3, 111−8ρ¯ ¯c22(1− ¯ γ )¯u3, 11111
−2ρ( ¯ 1− ¯ γ )
1
¯
c12 +
1
¯
c22
¨¯
u3,1tt
¯
σ33+h( ¯σ13 , 1− ¯ ρ ¨¯u3) +h2
2
(1−2γ ) ¯σ ¯ 33 , 11+ 1
¯
c2σ ¨¯33+4ρ¯ ¯c22(1− ¯ γ )¯u1, 111
−2ρ( ¯ 1− ¯ γ )¨¯u1, 1
+ h3 6
(2γ − ¯ 3) ¯σ13 , 111+
2
¯
c2(1− ¯ γ ) + 1
¯
c2
¨¯
σ13 , 1
Trang 4+4ρ¯ ¯c22(1− ¯ γ )¯u3, 1111− ¯ ρ(3−4γ )¨¯ ¯ u3, 11− ρ ¯
¯
c2u¨¯3,tt
+ h4
24
(4γ − ¯ 3) ¯σ33 , 1111 +
1
¯
c2(4−6γ ) + ¯ ¯1
c2(2−6γ + ¯ 4γ ¯2)
¨¯
σ33 , 11+ 1
¯
c4σ ¨¯33 ,tt+4ρ( ¯ 2− ¯ γ − ¯γ2)¨¯u1, 111
−8ρ¯ ¯c22(1− ¯ γ )¯u1, 11111−2ρ( ¯ 1− ¯ γ )
1
¯
c12 +
1
¯
c22
¨¯
u1,1tt
where
¯
c1=
¯
λ +2µ ¯
¯
¯ µ
¯
¯
c2
¯
Remark 1 If the contact between the layer and the half-space is welded, i.e the displacements and the stresses are
continuous through the interface of the layer and the half-space, we immediately obtain the effective boundary conditions from Eqs.(10)and (11)by replaced u¯1, ¯u3, ¯σ13 and σ ¯33 by u1,u3, σ13 and σ33, respectively These effective boundary conditions are valid not only for the displacements and the stresses of Rayleigh waves but also for those of any dynamic problem However, for the case of smooth contact the situation is rather different The horizontal displacement is not required to be continuous through the interface, the effective boundary conditions are therefore not immediately obtained from Eqs.(10)and(11) As shown below, the effective boundary conditions obtained for the case of smooth contact are valid for only the displacements and the stresses of Rayleigh waves
Now we consider the propagation of a Rayleigh wave, travelling (in the coated half-space) with velocity c(>0)and wave
number k(>0)in the x1-direction and decaying in the x3-direction The displacements and the stresses of the wave are sought in the form
¯
u1= ¯U1(y)e ik(x1 −ct), u¯3= ¯U3(y)e ik(x1 −ct),
¯
for the layer, and
u1=U1(y)e ik(x1 −ct), u3=U3(y)e ik(x1 −ct),
for the half-space, where y=kx3 Substituting(13)into(10)and(11)yields
i T¯1(0)
−1+ ε2
1
2(2γ − ¯ 3) +1
2
c2
¯
c22
+ ε4
1
24(4γ − ¯ 5) − 1
12
c2
¯
c12(1− ¯ γ ) − 1
12
c2
¯
c22(2γ ¯2
− ¯ γ −2) − 1
24
c4
¯
c24
+ ¯T3(0)
ε(1−2γ ) + ε ¯ 3
1
6(3−4γ ) − ¯ 1
3
c2
¯
c12(1− ¯ γ )
−1
6
c2
¯
c22(1−2γ ) ¯
+ ¯U1(0)
ε
4ρ¯ ¯c22( ¯γ −1) + ¯ρc2
+ ε3
4
3ρ¯ ¯c22( ¯γ −1)
+1
6ρ ¯c2(5−4γ ¯2) −1
6ρ ¯c4
¯
c22
+i U¯3(0)
ε2
2ρ¯ ¯c22( ¯γ −1) + ¯ρc2(1− ¯ γ )
+ ε4
1
6ρ ¯c2(2− ¯ γ − ¯γ2) −1
3ρ¯ ¯c22(1− ¯ γ ) − 1
12ρ ¯c4(1− ¯ γ )
1
¯
¯
c2
¯
T1(0)
ε + ε3
1
6(3−2γ ) − ¯ 1
3
c2
¯
c2(1− ¯ γ ) −1
6
c2
¯
c2
+i T¯3(0)
−1+ ε2
1
2(1−2γ ) ¯
+1
2
c2
¯
c2
+ ε4 24
(3−4γ ) − ¯ c2
¯
c2(4−6γ ) − ¯ c2
¯
c2(2−6γ + ¯ 4γ ¯2) −c4
¯
c4
Trang 5+i U¯1(0)
ε2ρ( ¯ 1− ¯ γ )c2−2c¯22 + ε
4
12ρ ¯
2c2(2− ¯ γ − ¯γ2) −4c¯22(1− ¯ γ ) −c4
1
¯
c2
¯
c2
(1− ¯ γ )
+ ¯U3(0)
ε ¯ρc2+ ε3
2
3ρ¯ ¯c22(1− ¯ γ ) − 1
6ρ ¯c2(3−4γ ) − ¯ 1
6ρ ¯c¯4
c2
=0
whereε =kh is the dimensionless thickness of the layer.
Let the contact between the layer and the half-space is smooth, i.e
or equivalently
according to(13)and(14) Introducing(17)2into(15)yields
¯
T3(0)(a1+a2ε2) + ¯U1(0)(a3+a4ε2) +i U¯3(0)(a5ε +a6ε3) =0,
i T¯3(0)(−1+a7ε2+a8ε4) +i U¯1(0)(a5ε2+a6ε4) + ¯U3(0)(a9ε +a10ε3) =0 (18)
in which
a1=1−2γ , ¯
a2= 1
6(3−4γ ) − ¯ 1
6c 2
2
¯
c2(1− ¯ γ ) + 1
¯
c2(1−2γ ) ¯
,
a3= −4ρ¯ ¯c22(1− ¯ γ ) + ¯ρc2,
a4= −4
3ρ¯ ¯c2(1− ¯ γ ) +1
6ρ ¯c2(5−4γ ¯2) −1
6ρ ¯c¯4
c2,
a5= −2ρ¯ ¯c22(1− ¯ γ ) + ¯ρc2(1− ¯ γ ),
a6= 1
6ρ ¯c2(2− ¯ γ − ¯γ2) −1
3ρ¯ ¯c22(1− ¯ γ ) − 1
12ρ( ¯ 1− ¯ γ )c4
1
¯
c2 +
1
¯
c2
,
a7= 1
2(1−2γ ) + ¯ 1
2
c2
¯
c2,
24(3−4γ ) − ¯ 1
24c 2
1
¯
c2(4−6γ ) + ¯ 1
¯
c2(2−6γ + ¯ 4γ ¯2)
24
c4
¯
c4,
a9= ¯ ρc2,
a10= 2
3ρ¯ ¯c2(1− ¯ γ ) −1
6ρ ¯c2(3−4γ ) − ¯ 1
6ρ ¯c¯4
c2.
(19)
EliminatingU¯1from(18)we have
where
a11= −a4+a3a7−a1a5,
a12=a4a7+a3a8−a2a5−a1a6,
a13=a25+a4a9+a3a10.
(21)
From the last two equations of(17)and Eq.(20)it follows
From the first of(17)and(22)we see that the surface x3=0 of the half-space is subjected to the following conditions
T1(0) =0,
The second of(23)is the approximate effective boundary condition (of fourth-order) The total effect of the layer on the half-space is replaced approximately by this condition
Trang 63 An approximate secular equation of fourth-order
Now we can ignore the layer and consider the propagation of Rayleigh waves in the isotropic elastic half-space x3 ≥0
whose surface x3 = 0 is subjected to the boundary conditions(23) According to Achenbach [23], the displacement
components of a Rayleigh wave travelling with velocity c and wave number k in the x1-direction and decaying in the
x3-direction are determined by(14)1, 2in which U1(y)and U3(y)are given by
U1(y) =A1e−b1y+A2e−b2y,
where A1and A2are constant to be determined and
√
1−x, α1= −b1
b2,
γ =c2
c2, c1= λ +2µ
c2
Substituting(14)1, 2and(24)into the stress–strain relations(2)without the bar yields that the stressesσ13andσ33are given
by(14)3, 4in which
T1(y) =ic2ρ
−2b1A1e−b1y+ 1
b2
c2
c2−2
A2e−b2y
,
T3(y) = −c22ρ
c2
c2 −2
A1e−b1y−2A2e−b2y
(26)
Introducing(24),(26)into(23)provides a homogeneous system of two linear equations for A1,A2namely
f1A1+f2A2=0
where
f1= −2b1,
f2= (x−2),
F1= −a3+a11ε2+a12ε4
b1
c24ρ2(a3a9ε +a13ε3),
F2= 2b2
c2ρ (a3−a11ε2−a12ε4) − 1
c4ρ2(a3a9ε +a13ε3).
(28)
For a non-trivial solution, the determinant of the matrix of the system(27)must vanish
f1 f2
F1 F2
=0.
Expanding this determinant and using(28)lead to the dispersion equation of the wave, namely
where
A0=
rν2x−4(1− ¯ γ )(x−2)2−4b1b2
,
A1=rµrν2x2b1
rν2x−4(1− ¯ γ ),
A2= −1
6
8(1− ¯ γ ) +4rν2x( ¯γ2−2) +rν4x2(1+3γ ) ¯
(x−2)2−4b1b2,
A3= 1
6rµxb1
8(1− ¯ γ )2+rν2x8(−2+3γ − ¯γ ¯ 2) +2rν2x(4−2γ − ¯γ ¯ 2) −rν4x2(1+ ¯ γ ),
24
(x−2)2−4b1b2
4(1− ¯ γ ) +rν2x(4γ ¯2−7) +2rν4x2(1+4γ − ¯ 2γ ¯2) −rν6x3γ ( ¯ 2+ ¯ γ )
(30)
where rµ= ¯ µ/µ,rv=c/¯c Eq.(29)is the desired approximate secular equation
Trang 7Fig 1 Dependence onε =k·h∈ [0 1]of the dimensionless Rayleigh wave velocity√x=c/c2 that is calculated by the exact secular equation and by the approximate secular equations of fourth-order (29) Two corresponding curves almost totally coincide with each other Here we take rµ=0.5,r =
5, γ =1/4 andγ =¯ 2/3.
Table 1
Some values of√x, corresponding toFig 1(rµ =0.5,r =5, γ =1/4 and
¯
γ = 2/3), that are calculated by the exact secular equation (√xext ), by the approximate secular equation (29) (√xapp ).
√
√
From(29)and the first of(30)it follows that, whenε =0 either(x−2)2−4√1−x√1− γx=0 or x=4(1− ¯ γ )¯c2/c2 That means, in the limitε →0 two modes are possible, one of which approaches the classical Rayleigh wave in the isotropic
half-space and the other approaches the longitudinal wave of the layer with the velocity c = 2c¯2√1− ¯ γ, as noted by Achenbach and Keshava [3
Fig 1presents the dependence onε = k·h ∈ [0 1]of the dimensionless Rayleigh wave velocity√x = c/c2that
is calculated by the exact secular equation and by the approximate secular equations of fourth-order(29) Here we take
rµ =0.5, rv =5, γ =1/4 andγ = ¯ 2/3 Some values of√x are listed inTable 1 Note that the exact secular equation is similar in form to Eq (30) in Ref [3], and is not reproduced here It is seen fromFig 1that the exact velocity curve and the approximate velocity curve of fourth-order almost totally coincide with each other for the values ofε ∈ [0 1] This shows that the approximate secular equation(29)is a very good approximation
4 An approximate formula of third-order for the velocity
In this section we establish an approximate formula of third-order for the squared dimensionless Rayleigh wave velocity
x(ε)that is of the form
x(ε) =x(0) +x′(0) ε +x′′(0)
2 ε2+x′′′(0)
where x(0)is the squared dimensionless velocity of Rayleigh waves propagating in an isotropic elastic half-space that is given by (see [24])
x(0) =4(1− γ )
2−4
3γ + 3
√
√
D
− 1
(32)
in which
R=2(27−90γ +99γ2−32γ3)/27,
and the roots in the formula(32)taking their principal values Note that x(0)can be calculated by another formula derived
by Malischewsky [25]
Trang 8Fig 2 Plots of the dimensionless Rayleigh wave velocity√x(ε)in the interval[0 1]that is calculated by the exact secular equation and by the formula (31) Here we take rµ=4,r =0.5, γ =1/4 andγ =¯ 2/3.
Table 2
Some values of√x, corresponding toFig 2(rµ= 4,r = 0.5, γ =1/4 and
¯
γ = 2/3), that are calculated by the exact secular equation (√xext ), by the approximate formula (31) (√xapp ).
√
√
From(29)it follows that
x′(0) = −A1
A 0x
x=x( 0 ), x′′(0) = −2A2A20x−2A 0x A1A 1x+A 0xx A21
A30x
x=x( 0 ),
x′′′(0) = −
6A3+6A 2x x′(0) +3A 1xx x′2(0) +3A 1x x′′(0) +3A 0xx x′(0)x′′(0) +A 0xxx x′3(0)/A 0x
x=x( 0 )
(34)
where A1,A2and A3are given by(30)and
A 0x=rν2(x−2)2−4
√
1−x1− γx
+2
4( ¯γ −1) +rν2x
√
1−x√1− γx
,
A 0xx=4rν2
√
1−x√1− γx
+2
4( ¯γ −1) +rν2x
1+ (γ −1)2
2 (1−x)3 (1− γx)3
,
A 0xxx=6rν2
1+ (γ −1)2
2 (1−x)3 (1− γx)3
−3(1− γ )2(1+ γ −2γx)4(1− ¯ γ ) −rν2x
2 (1−x)5 (1− γx)5 ,
A 1x=
rµrν2x
4(1− ¯ γ )(5γx−4) +rν2x(6−7γx)
A 1xx= rµrν2
4 (1− γx)3
4(1− ¯ γ )(−8+24γx−15γ2x2) +rν2x(24−60γx+35γ2x2),
3r
2
ν
2( ¯γ2−2) +rν2x(1+3γ )( ¯ x−2)2−4
√
1−x1− γx
3
4rν2x( ¯γ2−2) +8(1− ¯ γ ) +rν4x2(1+3γ ) ¯
√
1−x√1− γx
Fig 2shows the plots of the dimensionless Rayleigh wave velocity√x(ε)in the interval[0 1]that is calculated by the exact secular equation and by the formula(31) Here we take rµ=4,rv=0.5, γ =1/4 andγ = ¯ 2/3 Some values of√x
are listed inTable 2 It is shown that the approximate velocity curve is close to the exact velocity curve in the interval[0 1]
Trang 95 Conclusions
In this paper the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer
is considered The contact between the layer and half space is assumed to be smooth An approximate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived using the effective boundary condition method
We have shown that the approximate secular equation obtained has high accuracy An approximate formula of third-order for the velocity of Rayleigh waves is established using the obtained approximate secular equation The approximate secular equation and the formula for the velocity are potentially useful in many practical applications
Acknowledgment
The work was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED)
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