Rayleigh Surface Waves on a Kelvin-Voigt
Viscoelastic Half-Space
Stan Chiriţă, Michele Ciarletta &
Vincenzo Tibullo
Journal of Elasticity
The Physical and Mathematical Science
of Solids
ISSN 0374-3535
J Elast
DOI 10.1007/s10659-013-9447-0
1 23
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1 23
Author's personal copy
J Elast
DOI 10.1007/s10659-013-9447-0
Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic
Half-Space
Stan Chiriţă · Michele Ciarletta · Vincenzo Tibullo
Received: 21 December 2012
© Springer Science+Business Media Dordrecht 2013
Abstract In this paper we consider the propagation of Rayleigh surface waves in an exponentially graded half-space made of an isotropic Kelvin-Voigt viscoelastic material. Here
we take into account the effect of the viscoelastic dissipation energy upon the corresponding wave solutions. As a consequence we introduce the damped in time wave solutions and
then we treat the Rayleigh surface wave problem in terms of such solutions. The explicit
form of the secular equation is obtained in terms of the wave speed and the viscoelastic
inhomogeneous profile. Furthermore, we use numerical methods and computations to solve
the secular equation for some special homogeneous materials. The results sustain the idea,
existent in literature on the argument, that there is possible to have more than one surface
wave for the Rayleigh wave problem.
Keywords Seismic Rayleigh waves · Kelvin-Voigt viscoelastic half-space · Secular
equation · Damped in time wave solutions · Exponentially graded viscoelastic half-space
Mathematics Subject Classification (2000) 74D05 · 74J05 · 74J15
1 Introduction
Rayleigh surface waves [1], which model the propagation of a seismic wave at the Earth’s
surface, play an important role since they can be the most destructive type of seismic wave
produced by earthquakes. With the advent of electronics, many other applications have been
found in the industrial world. These include the manufacturing of high frequency acoustic
S. Chiriţă ()
Faculty of Mathematics, Al. I. Cuza University of Iaşi, 700506 Iaşi, Romania
e-mail: [email protected]
S. Chiriţă
Octav Mayer Mathematics Institute, Romanian Academy, 700505 Iaşi, Romania
M. Ciarletta · V. Tibullo
University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, SA, Italy
Author's personal copy
S. Chiriţă et al.
wave filters and transducers (used in everyday electronic device such as global positioning
systems, cell phones, miniature motors, detectors, sensors, etc.), the health monitoring of
elastic structures (non-destructive evaluation in the automotive or aeronautic industry), the
acoustic determination of elastic properties of solids (physics, medicine, engineering, etc.),
ultrasonic imaging techniques (medicine, oil prospection, etc.), and so on. Particularly, there
appear of great geophysical interest to see how the memory effects influence the propagation
of waves in an elastic medium.
On the other hand, the coupling between the memory and the strain effects influences
both the form of the surface wave and its velocity of propagation. Plane waves in isotropic
homogeneous linear viscoelastic materials have been studied intensively (see, for example,
Hunter [2], Lockett [3] and Hayes and Rivlin [4–7]).
The memory effects upon the propagation of Rayleigh surface waves in a semi-infinite
viscoelastic isotropic solid have been investigated in many papers (see, for example, Currie
et al. [8], Currie and O’Leary [9] and O’Leary [10]). Carcione [11] investigated the anelastic
characteristics of the Rayleigh waves with depth and as a function of the frequency from the
standpoint of balance energy. Romeo [12] considered the harmonic solutions to the general
viscoelastic problem and he used a result by Nkemzi [13] for the study of the corresponding
secular equation. These studies are all based upon a representation of the homogeneous wave
solution in terms of the slowness vector/frequency couple, that is
ur (x, t) = Ar eiω(sp xp −t) ,
(1.1)
where sp are the components of the constant complex slowness vector, ω is the assigned
real angular frequency and Ar are the components of the constant complex amplitude vector
and this is in contrast with elastic materials where the representation is based upon the
wave vector/speed couple. On this basis, it was shown by Currie et al. [8] that, among other
characteristics, there is possible more than one surface wave of the type described in (1.1)
and the wave speed may be greater than the body-wave speeds. Recently, the effects of
viscous or viscous and thermal properties on the propagation waves of an assigned frequency
or of an assigned wavelength have been studied by Ivanov and Savova [14–16] and Savova
and Ivanov [17]. In [14] it is shown that there exists a unique viscoelastic surface wave
of an assigned wavelength that satisfies some appropriate criteria for behavior at infinity.
Moreover, we note that more elaborate viscoelastic models for materials with microstructure
have been recently developed by Ieşan [18] and Passarella et al. [19].
However, the damped elliptically-polarized sinusoidal wave train with real angular frequency considered in [8] leads to solutions of the Rayleigh surface wave problem that do
not possess the property of the exponential decay when time tends to infinity. Such a property is now well-established in literature on the models of continuum media with dissipation
energy. On the other hand, the wave solutions of the form (1.1) lead to an infinite energy
with respect to the spatial variables.
The present paper discusses the effect of the memory as well as of the strain upon the
propagation of Rayleigh surface waves in an isotropic viscoelastic half-space made of a
material of Kelvin-Voigt type. The presence of the dissipation energy implies that the wave
solutions have to decay to zero when the time tends to infinity. On this basis we introduce the
class of damped in time plane waves solutions in an isotropic homogeneous Kelvin-Voigt
viscoelastic space. Thus, a homogeneous wave solution is represented in terms of the wave
vector/speed couple as
ur (x, t) = Ar eiκ(ns xs −vt) ,
(1.2)
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Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic Half-Space
where Ar are the components of the constant complex amplitude vector, κ is the real wave
number and ns are the components of a real unit vector giving the propagation direction and
v is a complex parameter so that Re(v) > 0 represents the wave speed and exp[κ Im(v)t]
represents the damping in time of the wave and hence we assume that Im(v) ≤ 0. The representation (1.2) of a wave solution allows the wave speed to be complex and, moreover,
this kind of wave solution is possible if the viscosity coefficients satisfy some limiting constraints with respect to the elastic coefficients. How it can be observed, the solutions of the
form (1.2) tend to zero when the time goes to infinity and the corresponding energy is finite
in time and space.
We solve the Rayleigh surface wave problem for an exponentially graded half-space
in terms of the wave solutions (1.2). The secular equation is written in an explicit form
which outlines the effects of the elastic and viscosity coefficients. The analytic results are
accompanied by numerical simulations and computations inducing the idea of existence of
more than one surface wave solution for the Rayleigh surface wave problem. In fact, for
the Kelvin-Voigt viscoelastic materials for which the real parts of the Lamé moduli are
equal, the numerical computations furnish results which are in a strongly agreement with
those established by Currie et al. [8], relating to the existence of more than one surface
wave solution for the Rayleigh surface wave problem. However, within the context of the
particular viscoelastic material considered in Sect. 5 of [8], for the two cases described by
(5.11) and (5.12) of [8] we found three surface wave solutions for the Rayleigh surface wave
problem, in a noticeable contrast with results reported in [8] and [9], where it is stated that
they have found no case of more than two wave solutions.
2 Basic Equations
Throughout this paper, we refer the motion of a continuum to a fixed system of rectangular Cartesian axes Oxk (k = 1, 2, 3). We shall employ the usual summation and differentiation conventions: Latin subscripts are understood to range over the integers (1, 2, 3),
whereas Greek subscripts are confined to the range (1, 2), summation over repeated subscripts is implied, subscripts preceded by a comma denote partial differentiation with respect
to the corresponding Cartesian coordinate, and a superposed dot denotes time differentiation.
Throughout this section we suppose that a regular region B is filled by a homogeneous and
isotropic Kelvin-Voigt viscoelastic material. Considering that the natural state is unstressed,
then in the absence of the body force, the system of field equations for the linear theory
consists of (cf. Eringen [20], p. 325)
• the equations of motion
trl,r = ül
in B × (0, ∞),
(2.1)
where is the density mass at time t = 0, trl are the components of the stress tensor, ul
are the components of displacement vector field;
• the constitutive equations
trl = λ0 emm δrl + 2μ0 erl + λ∗ ėmm δrl + 2μ∗ ėrl ,
(2.2)
where δrl is the Kronecker delta, λ0 and μ0 are the Lamé coefficients, λ∗ and μ∗ are the
coefficients of viscosity and erl are defined by
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S. Chiriţă et al.
• the geometrical equations
1
erl = (ur,l + ul,r ).
2
(2.3)
The dissipation inequality [20]
λ∗ ėmm ėnn + 2μ∗ ėrl ėrl ≥ 0,
(2.4)
2
λ∗ + μ∗ ≥ 0.
3
(2.5)
2
λ0 + μ0 > 0.
3
(2.6)
implies
μ∗ ≥ 0,
In what follows we will also assume that
μ0 > 0,
By substituting the relations (2.3) and (2.2) into (2.1), we obtain the following system of
differential equations
μ0 ul + (λ0 + μ0 )um,ml + μ∗ u̇l + (λ∗ + μ∗ )u̇m,ml = ül .
(2.7)
It is now well-known in literature that the existence of the dissipation energy leads to
solutions of (2.7) which decay exponentially at zero when the time tends to infinity. In fact,
from the above equations we can deduce
1 u̇l (t)u̇l (t) + λ0 emm (t)enn (t) + 2μ0 erl (t)erl (t) dv
2 B
t
+
λ∗ ėmm (s)ėnn (s) + 2μ∗ ėrl (s)ėrl (s) dv ds
=
1
2
0
B
u̇l (0)u̇l (0) + λ0 emm (0)enn (0) + 2μ0 erl (0)erl (0) dv,
so that we can conclude that
∞
λ∗ ėmm (s)ėnn (s) + 2μ∗ ėrl (s)ėrl (s) dv ds < ∞.
0
(2.8)
B
(2.9)
B
Under appropriate hypotheses, the relation (2.9) induces the idea that the solutions decay
to zero when time tends to infinity. For more precise information upon such a subject we
recommend the paper by Appleby et al. [21] and the book by Amendola et al. [22], Chap. 21,
and the papers cited therein.
In the present paper we will take into consideration this important feature of the viscoelastic solutions.
3 Damped in Time Plane Waves in an Isotropic Homogeneous Kelvin-Voigt
Viscoelastic Space
Throughout this section we consider damped in time plane waves of the form
ur (x, t) = Re Ar eiκ(ns xs −vt) ,
(3.1)
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Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic Half-Space
where Re{·} is the real part, κ > 0 is the real wave number, A = (A1 , A2 , A3 ) is a non-zero
complex vector, n is a real unit vector giving the propagation direction and
v = Re(v) + i Im(v)
(3.2)
Re(v) > 0
(3.3)
is a complex parameter so that
represents the wave speed and exp[κ Im(v)t] corresponds to the damping in time of the
wave and hence we assume that
Im(v) ≤ 0.
(3.4)
If it happens that the imaginary part of v vanishes, that is Im(v) = 0, then we have an
undamped harmonic in time wave. Otherwise, that is when Im(v) < 0, we have a damped in
time wave.
In what follows we are searching for displacements of the form (3.1) satisfying the basic
equations (2.7). To this end we substitute (3.1) into (2.7) to obtain
(3.5)
μ − v 2 δrl + (λ + μ)nr nl Ar = 0,
where
λ = λ0 − iκvλ∗ ,
μ = μ0 − iκvμ∗ .
(3.6)
From the condition to have non-trivial solutions of the form (3.1), it follows from (3.5) the
following characteristic equation
det μ − v 2 δrl + (λ + μ)nr nl = 0,
(3.7)
that is,
μ − v 2
2 λ + 2μ − v 2 = 0.
(3.8)
Thus, the solutions of the characteristic equation (3.8), satisfying (3.3) and (3.4), are
1
4c12 − κ 2 c1∗4 − iκc1∗2 ,
(3.9)
v1 =
2
and
v2 =
where
1
4c22 − κ 2 c2∗4 − iκc2∗2 ,
2
c1 =
c1∗
=
λ0 + 2μ0
,
c2 =
λ∗ + 2μ∗
,
c2∗ =
(3.10)
μ0
,
(3.11)
μ∗
.
(3.12)
It becomes clear from the above relations that a damped in time wave in the viscoelastic
space is possible if
2c1
2c2
,
c2∗ <
.
(3.13)
c1∗ <
κ
κ
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For v = v1 , the system (3.5) reduces to
(−δrl + nr nl )Ar = 0,
(3.14)
Al = (Am nm )nl
(3.15)
that is,
and hence we have a longitudinal wave of the form
t
t 2 ∗2
2
2 c∗4
u(L)
(x,
t)
=
Re
(A
n
)
exp
iκ
n
x
−
4c
−
κ
n r e − 2 κ c1 .
l l
s s
r
1
1
2
(3.16)
Thus, we have a damped longitudinal wave whose amplitude of oscillation decreases exponentially to zero as time limits to infinity. The decay rate is given by the exponential
t 2 ∗2
e − 2 κ c1 .
For v = v2 , the system (3.5) reduces to
(λ + μ)(Ar nr )nl = 0,
(3.17)
Am nm = 0,
(3.18)
that is,
from which we deduce a transverse wave of the form
t
t 2 ∗2
(T )
2
∗4
2
ur (x, t) = Re Ar exp iκ ns xs −
4c2 − κ c2
e − 2 κ c2 .
2
(3.19)
The transverse wave is a damped in time wave whose amplitude of oscillation decreases to
t 2 ∗2
zero like e− 2 κ c2 as time limits to infinity.
It is easy to see that when the viscosity effects are absent, then the restrictions (3.13) are
identically satisfied and the wave speeds v1 and v2 coincide with the elastic wave speeds
c1 and c2 , respectively. In such a case it is obvious that the waves are undamped in time.
Moreover, we can observe that the viscosity effects diminish the elastic wave speeds. More
precisely, we have
κ 2 c1∗4
≤ c1 ,
Re(v1 ) = c12 −
4
(3.20)
2 c∗4
κ
2
≤ c2 .
Re(v2 ) = c22 −
4
4 Rayleigh Viscoelastic Waves on an Exponentially Graded Half-Space
Throughout this section we will assume B to be the half-space x2 > 0. We also assume that
the half-space is free of body forces and its surface x2 = 0 is free of traction, that is
t2r (x1 , 0, x3 , t) = 0
for all x1 , x3 ∈ R, t ∈ R+ .
(4.1)
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Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic Half-Space
Further we consider the propagation of an inhomogeneous plane wave with wave number
κ in the x1 -direction and with attenuation in the x2 -direction in the half-space x2 > 0 made
of an isotropic Kelvin-Voigt material with an exponential depth profile in the form
λ0 (x) = e−2τ x2 ,
μ0 (x) = me−2τ x2 ,
λ∗ (x) = Le−2τ x2 ,
μ∗ (x) = Me−2τ x2 ,
(x) = Re−2τ x2 ,
(4.2)
where τ , , m, L, M and R are real constants so that
m > 0,
2
+ m > 0,
3
M ≥ 0,
2
L + M ≥ 0,
3
(4.3)
R > 0.
The Rayleigh surface wave problem for the exponentially graded half-space x2 > 0, made
of the isotropic Kelvin-Voigt viscoelastic material with the depth profile (4.2), consists of
the boundary value problem defined by the basic equations (2.1)–(2.3), with the boundary
value conditions (4.1) and the following asymptotic conditions
lim ur (x,t) = 0,
x2 →∞
lim tmn (x,t) = 0,
x2 →∞
for all x1 , x3 ∈ R, t ≥ 0.
(4.4)
To solve the Rayleigh surface wave problem for the exponentially graded viscoelastic
half-space, we first use an idea by Destrade [23] and, thus, we search for solutions of the
basic equations (2.1)–(2.3) in the form
τ
u1 (x, t) = Re V1 eiκ[x1 −vt+i(r− κ )x2 ] ,
τ
u2 (x, t) = Re V2 eiκ[x1 −vt+i(r− κ )x2 ] ,
(4.5)
u3 (x, t) = 0,
where V1 and V2 are constant complex parameters with |V1 | + |V2 | = 0, κ is the wave
number, v is a complex parameter, subject to the conditions (3.3) and (3.4), and r is a
complex parameter so that
Re(r) >
|τ |
,
κ
(4.6)
last condition being imposed by the asymptotic behavior expressed in (4.4). By a substitution
of the relation (4.5) into (2.2) and by using (4.2), we obtain
τ
τ
t11 = Re iκ c11 V1 + ic12 r −
V2 eiκ[x1 −vt+i(r+ κ )x2 ] ,
κ
τ
τ
t22 = Re iκ c12 V1 + ic11 r −
V2 eiκ[x1 −vt+i(r+ κ )x2 ] ,
κ
(4.7)
(4.8)
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τ
τ
t12 = Re iκ ic66 r −
V1 + c66 V2 eiκ[x1 −vt+i(r+ κ )x2 ] ,
κ
τ
τ )x ]
iκ[x1 −vt+i(r+ κ
2
t33 = Re iκ c12 V1 + ic12 r −
,
V2 e
κ
(4.9)
(4.10)
t13 = t23 = 0,
where
c11 = + 2m − iκv(L + 2M),
c12 = − iκvL,
c66 = m − iκvM.
(4.11)
Further, we substitute the relations (4.5) and (4.7)–(4.10) into the basic equations (2.1) to
find
τ2
τ
τ
−c66 r 2 − 2 + c11 E1 V1 + i c66 r +
+ c12 r −
V2 = 0,
κ
κ
κ
(4.12)
τ2
τ
τ
i c66 r −
+ c12 r +
V1 + −c11 r 2 − 2 + c66 E2 V2 = 0,
κ
κ
κ
where
E1 = 1 −
Rv 2
,
c11
E2 = 1 −
Rv 2
.
c66
Thus, the propagation condition is
2τ 2
4c12 τ 2
τ4
r 4 − E1 + E2 + 2 r 2 + E1 E2 + E1 + E2 +
+ 4 = 0,
2
κ
c11 κ
κ
and we observe that its solutions are dependent on the parameter τ ; we obtain
1
2τ 2
16c12 τ 2
2
E1 + E2 + 2 ± (E1 − E2 )2 −
r1,2 =
.
2
κ
c11 κ 2
(4.13)
(4.14)
(4.15)
We denote by r1 and r2 the roots of (4.14) for which we have
Re(r1 ) >
|τ |
,
κ
Re(r2 ) >
|τ |
.
κ
For r = r1 , the corresponding solution V (1) = (V1(1) , V2(1) ) of (4.12) is given by
1 c11
τ2
−r12 + 2 + E2 ,
V1(1) =
κ c66
κ
c
i
τ
τ
12
V2(1) = −
r1 +
r1 − +
,
κ
κ c66
κ
while the solution V (2) = (V1(2) , V2(2) ) of (4.12), corresponding to r = r2 , is
c12
i
τ
τ
(2)
V1 = −
r2 −
r2 + +
,
κ
κ c66
κ
1
τ2
c11
E1 .
V2(2) =
−r22 + 2 +
κ
κ
c66
(4.16)
(4.17)
(4.18)
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Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic Half-Space
Further, from (4.8), (4.9), (4.17) and (4.18), we have
τ
(1)
t12
= Re T12(1) eiκ[x1 −vt+i(r1 + κ )x2 ] ,
τ
(1)
= Re T22(1) eiκ[x1 −vt+i(r1 + κ )x2 ] ,
t22
(4.19)
τ
(2)
= Re T12(2) eiκ[x1 −vt+i(r2 + κ )x2 ] ,
t12
τ
(2)
= Re T22(2) eiκ[x1 −vt+i(r2 + κ )x2 ] ,
t22
(4.20)
τ
τ
− 2c66 ,
T12(1) = 2(c12 + c66 )r1 − c11 Γ r1 −
κ
κ
τ τ
T22(1) = i (c12 + c66 )(1 + E2 ) − c11 Γ − 2c11 r1 −
,
κ κ
(4.21)
τ τ
= i (c12 + c66 )(1 + E2 ) + c12 Γ − 2c12 r2 −
,
κ κ
τ
τ
T22(2) = −2(c12 + c66 )r2 + c11 Γ r2 −
+ 2(2c12 + c66 )
κ
κ
(4.22)
and
where
T12(2)
and
τ2
τ2
+ E1 = r22 − 2 − E2
2
κ
κ
1
16c12 τ 2
2
=
.
E1 − E2 − (E1 − E2 ) −
2
c11 κ 2
Γ = −r12 +
(4.23)
Let us now consider the Rayleigh surface wave propagation problem. We seek a solution
of the problem as a linear combination of the above two solutions, that is we set
τ
τ
u1 (x, t) = Re δ1 V1(1) eiκ[x1 −vt+i(r1 − κ )x2 ] + δ2 V1(2) eiκ[x1 −vt+i(r2 − κ )x2 ] ,
τ
τ
u2 (x, t) = Re δ1 V2(1) eiκ[x1 −vt+i(r1 − κ )x2 ] + δ2 V2(2) eiκ[x1 −vt+i(r2 − κ )x2 ] ,
(4.24)
u3 (x, t) = 0,
where δ1 and δ2 are complex parameters so that |δ1 | + |δ2 | = 0. Further, we have
τ
τ
t12 = Re δ1 T12(1) eiκ[x1 −vt+i(r1 + κ )x2 ] + δ2 T12(2) eiκ[x1 −vt+i(r2 + κ )x2 ] ,
τ
τ
t22 = Re δ1 T22(1) eiκ[x1 −vt+i(r1 + κ )x2 ] + δ2 T22(2) eiκ[x1 −vt+i(r2 + κ )x2 ] ,
(4.25)
t23 = 0,
so that the boundary conditions (4.1) imply the following algebraic system, for determining
the unknown parameters δ1 and δ2 ,
δ1 T12(1) + δ2 T12(2) = 0,
δ1 T22(1) + δ2 T22(2) = 0.
(4.26)
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Consequently, the secular equation
T12(2) = 0,
T (2) (1)
T
12
(1)
T
22
(4.27)
22
can be written in the following form
τ
τ
τ
τ
− 2ξ
2r2 − (1 + ξ )Γ r2 −
− 2(2 − ξ )
2r1 − (1 + ξ )Γ r1 −
κ
κ
κ
κ
τ τ
− 1 + E2 + (1 − ξ ) Γ − 2 r2 −
κ κ
τ τ
× 1 + E2 − (1 + ξ ) Γ + 2 r1 −
= 0.
(4.28)
κ κ
Here we have used the notation
ξ=
c66
,
c12 + c66
(4.29)
and, moreover, we have set
d12 =
+ 2m
,
R
L + 2M
,
R
κd2∗2
C2 =
.
d2
m
,
R
κd1∗2
C1 =
,
d1
d1∗2 =
d22 =
d2∗2 =
M
,
R
(4.30)
Further, if we set
v = id2 ω,
(4.31)
then we have
E1 = 1 +
ω2
1 + C1 dd21 ω
d22
,
d12
E2 = 1 +
ω2
,
1 + C2 ω
(4.32)
and
ξ=
1 + C2 ω
d12
d22
− 1 + ( dd12 C1 − C2 )ω
.
(4.33)
Thus, by a substitution of the relations (4.31) to (4.33) into the relations (4.15) and (4.28),
we obtain the secular equation in an explicit form in terms of the unknown ω. We must now
solve such a secular equation for ω which satisfies the following conditions:
Re(ω) ≤ 0,
Im(ω) < 0,
(4.34)
and
Re(r1 ) >
|τ |
,
κ
Re(r2 ) >
|τ |
.
κ
(4.35)
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Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic Half-Space
Remark 1 It is worth noting that, when |τκ| = 0, the secular equation (4.28) reduces to that
of the homogeneous viscoelastic material as expressed by
2
d22
ω2
ω2
ω2
= 0.
(4.36)
− 4 1 +
1
+
2+
1 + C2 ω
1 + C2 ω
1 + C1 dd2 ω d12
1
Remark 2 When the viscosity effects are absent we have C1 = 0 and C2 = 0 and the secular
equation (4.36) reduces to the well-known elastic secular equation
c22 2
2
2
(4.37)
2+ω −4
1 + ω 2 1 + ω2 = 0.
c1
5 Some Numerical Examples
In this section we will examine in detail some particular homogeneous viscoelastic materials. The analytic results are coupled and supplemented by detailed numerical computations
that allow us to compare with the earlier work in [8].
First of all we consider the viscoelastic model studied by Currie et al. [8]. Thus, we
will assume a particular viscoelastic material for which λ0 = μ0 and λ∗ and μ∗ are small
compared with μ0 , that is, we set
v
v
,
μ = μ0 1 − iεb
,
(5.1)
λ = μ0 1 − iεa
c2
c2
where ε is a small parameter and a and b are prescribed real parameters which satisfy the
inequalities
2
(5.2)
a + b ≥ 0.
3
It was shown in [8] that, for certain ranges of the parameter a/b, more than one surface wave
of type (1.1) is possible.
With choices (5.1) and (5.2), from (4.23), (4.30) and (4.32), we obtain
b ≥ 0,
r12 = 1 +
w2
,
3 + ε(a + 2b)w
r22 = 1 +
w2
,
1 + εbw
and the secular equation (4.36) becomes
2
w2
w2
w2
−4
2+
1+
1+
= 0,
1 + εbw
3 + ε(a + 2b)w
1 + εbw
(5.3)
(5.4)
and so we have to select the solutions of the secular equation (5.4) which satisfy the conditions
Re(w) ≤ 0,
Im(w) < 0,
(5.5)
Re(r1 ) > 0,
Re(r2 ) > 0.
(5.6)
and
Author's personal copy
S. Chiriţă et al.
Table 1 The values of v, r1 and r2 for the particular viscoelastic material considered in [8]
εa
εb
0.02
0.04
0.06
0.04
v/c2
r1
r2
0.919257 − 0.0165285i
0.847474 + 0.000882399i
0.393329 − 0.000882473i
1.77425 − 0.0437257i
0.0231564 − 0.223912i
0.0232984 − 1.46535i
1.99904 − 0.099745i
0.037905 + 0.580987i
0.0226496 + 1.73464i
0.393296 + 0.000881931i
0.919243 − 0.0172835i
0.847507 − 0.000882109i
1.77284 − 0.0824545i
0.0234998 + 0.225511i
0.0235534 + 1.46697i
2.00066 − 0.0602629i
0.0382061 − 0.577997i
0.0228953 − 1.73286i
Numerical results will be considered for the two cases as given in [8], that is,
εa = 0.02,
εb = 0.04,
(5.7)
εa = 0.06,
εb = 0.04.
(5.8)
and
The computation is made with the software package Wolfram Mathematica version 7.0.1.0.
In agreement with the analytic results given by the relations (5.4) to (5.6), the admissible
solutions for v and r, for the cases expressed in (5.7) and (5.8), are given in the Table 1.
Concluding, for the cases considered in [8], we have found three surface wave solutions
of the type (1.2) for the Rayleigh surface wave problem. This is in a noticeable contrast with
the results reported in [8] for the wave solutions of type (1.1), where it is found that there
exist two wave solutions.
In the general case of the Kelvin-Voigt viscoelastic homogeneous material, from the
relations (3.13), we deduce that
0 ≤ C1 < 2,
0 ≤ C2 < 2.
(5.9)
When λ0 = μ0 and the viscosity coefficients range into the set defined by (5.9), the secular
equation is
2
w2
w2
w2
−4
1+
1+
=0
(5.10)
2+
√
1 + C2 w
1 + C2 w
3 + 3C1 w
with
r12 = 1 +
w2
,
√
3 + 3C1 w
r22 = 1 +
w2
.
1 + C2 w
(5.11)
Equation (5.10), with the conditions (5.5) and (5.6), can have either a solution or it can
have more than an admissible solution for w. In fact, as it can be seen from Fig. 1, in the
class of viscoelastic materials with the viscosity coefficients so that (C1 , C2 ) ∈ A1 there is a
unique surface wave solution, for (C1 , C2 ) ∈ A2 there are two surface wave solutions, while
for (C1 , C2 ) ∈ A3 there are three surface wave solutions. Table 2 presents some examples
of viscoelastic materials from each class A1 to A3 , characterized by λ0 = μ0 , C1 and C2 ,
together with the values of the parameters v, r1 and r2 of the corresponding sets of Rayleigh
surface wave solutions. From Table 1 and Table 2 it can be observed that, in the classes A2
and A3 , there is a Rayleigh surface wave with the characteristics (wave speed and decay
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Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic Half-Space
Fig. 1 The domains of
viscoelastic materials (with
λ0 = μ0 and characterized by C1
and C2 ) with the number of
Rayleigh wave solutions
Table 2 The values of v, r1 and r2 for the set of the Rayleigh wave solutions for some specified viscoelastic
materials characterized by λ0 = μ0 , C1 and C2
C1
C2
v/c2
r1
r2
0.5
1.0
0.860958 − 0.423783i
0.862209 + 0.116357i
0.346807 − 0.0847438i
1.0
0.8
1.75
1.0
0.492535 − 1.66342i
1.16655 + 1.35667i
0.392526 + 2.17912i
0.864529 − 0.326975i
0.840434 + 0.0303701i
0.39616 − 0.0314553i
0.0090599 − 1.69235i
0.38025 − 1.97115i
0.0141401 − 0.513189i
0.743741 − 1.60562i
0.0234002 + 0.265052i
0.0215112 + 1.5048i
0.816203 − 0.423236i
0.848123 − 0.00121024i
0.3926790 + 0.00120344i
rate) close to those of the corresponding classical elastic wave. The remaining other waves
are characterized by a lower as well as a higher wave speed with respect to this one, but they
are endowed with a higher decay rate in time.
6 Particle Paths
The components of the displacement field corresponding to the Rayleigh problem for a
homogeneous viscoelastic half-space are
1 Re D1 e−κσ1 x2 +iκ(x1 −vt) − iσ2 D2 e−κσ2 x2 +iκ(x1 −vt) ,
κ
1 u2 (x, t) = Re iσ1 D1 e−κσ1 x2 +iκ(x1 −vt) + D2 e−κσ2 x2 +iκ(x1 −vt) ,
κ
u1 (x, t) =
(6.1)
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S. Chiriţă et al.
where D1 and D2 are parameters which satisfy
2iσ2
1 + σ22
D1
=
=−
,
2
D2
2iσ1
1 + σ2
(6.2)
and
σ12 = 1 +
w2 ( cc21 )2
1 + C1 cc21 w
σ22 = 1 +
,
w2
.
1 + C2 w
(6.3)
In terms of σ1 and σ2 the secular equation (4.36) is
2
4σ1 σ2 = 1 + σ22 .
(6.4)
Otherwise, we can write (6.1) in the form
1
A11 cos κ x1 − Re(v)t − A12 sin κ x1 − Re(v)t eκ Im(v)t ,
κ
1
u2 =
A21 cos κ x1 − Re(v)t − A22 sin κ x1 − Re(v)t eκ Im(v)t ,
κ
u1 =
(6.5)
where
A11 = Re D1 e−κσ1 x2 − iσ2 D2 e−κσ2 x2 ,
A21 = Re iσ1 D1 e−κσ1 x2 + D2 e−κσ2 x2 ,
A12 = Im D1 e−κσ1 x2 − iσ2 D2 e−κσ2 x2 ,
(6.6)
A22 = Im iσ1 D1 e−κσ1 x2 + D2 e−κσ2 x2 .
Consequently, every point in the viscoelastic half-space will move in a plane elliptical
orbit given by
(A21 u1 − A11 u2 )2 + (A22 u1 − A12 u2 )2 =
1
(A11 A22 − A12 A21 )2 e2κ Im(v)t .
κ2
(6.7)
Moreover, the ellipse is described in a retrograde (direct) sense if
A11 A22 − A12 A21 > 0 (< 0).
(6.8)
At the top surface x2 = 0, we have
iσ2
2
D 2 1 − σ2 ,
A11 = Re(D1 − iσ2 D2 ) = Re
1 + σ22
iσ2
2
D 2 1 − σ2 ,
A12 = Im(D1 − iσ2 D2 ) = Im
1 + σ22
1 2
A21 = Re(iσ1 D1 + D2 ) = Re D2 1 − σ2 ,
2
1 2
A22 = Im(iσ1 D1 + D2 ) = Im D2 1 − σ2 ,
2
(6.9)
so that, if we set
D2 =
2
,
1 − σ22
(6.10)
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Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic Half-Space
we get
2iσ2
A11 = Re
,
1 + σ22
2iσ2
A12 = Im
,
1 + σ22
A21 = 1,
A22 = 0.
(6.11)
Consequently, we obtain the following sufficient conditions: the viscoelastic wave is retrograde if
2iσ2
> 0;
(6.12)
Im
1 + σ22
the viscoelastic wave is direct if
2iσ2
Im
1 + σ22
< 0.
(6.13)
η2 > 0,
(6.14)
With the notation
σ2 = η2 + iθ2 ,
we can say that the viscoelastic wave is retrograde if
η22 − 3θ22 + 1 > 0,
(6.15)
η22 − 3θ22 + 1 < 0.
(6.16)
while it is direct if
Now it is easy to see from the Table 2 that the Rayleigh surface waves corresponding to
the classical elastic waves are retrograde, while the others are direct at the surface.
7 Conclusions
We studied the propagation of surface waves over an exponentially graded half-space of
isotropic Kelvin-Voigt viscoelastic material by means of wave solutions with spatial and
temporal finite energy. The secular equation (4.28) is written in terms of the wave speed
and the viscoelastic inhomogeneous profile. It is further specialized for the case of a homogeneous viscoelastic half-space and some numerical examples are considered that allow a
comparison with the earlier work on the subject. Our calculation confirms the existence of
more than one surface wave as demonstrated in [8], but for the same viscoelastic material we
have found three possible wave solutions with finite energy, instead of two wave solutions
with spatial infinite energy found in [8]. This is in contrast with the results of the analysis
developed in [9] where it is concluded that the authors have found no case of more than two
possible surface viscoelastic waves.
We have shown that when there is just one wave solution this is found to be retrograde
at the free surface. While when there is more than one viscoelastic surface wave, one is
retrograde and the others are direct at the free surface. Thus, the existence of the direct
surface waves appears as a consequence of the viscous dissipation energy.
Acknowledgement We express our gratitude to the referees for their helpful suggestions and comments.
The work by the author SC was supported by the Romanian Ministry of Education and Research and Innovation through the CNCS grant PN-II-ID-PCE-2012-4-0068.
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S. Chiriţă et al.
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