JOURNAL OF APPLIED PHYSICS 102, 074906 共2007兲
Plane waves in a rotating micropolar porous elastic solid
Jaswant Singha兲
Department of Mathematics, Government Post Graduate College, Nalagarh-174 101, Himachal Pradesh,
India
S. K. Tomarb兲
Department of Mathematics, Panjab University, Chandigarh-160 014, India
共Received 1 June 2007; accepted 12 August 2007; published online 3 October 2007兲
Propagation of plane waves in a linear, homogeneous, and isotropic micropolar porous elastic solid
rotating with a uniform angular velocity has been investigated. It has been found that there can exist
three longitudinal waves and two sets of coupled transverse waves in a linear homogeneous
micropolar porous elastic solid rotating with uniform angular velocity. Out of the three longitudinal
waves, one is a longitudinal microrotational wave, second is a longitudinal displacement wave
already encountered in the theory of micropolar elasticity, and the third one is a longitudinal void
volume fractional wave carrying a change in void volume fraction. The phase speeds of each set of
coupled transverse waves are found to be affected by the rotation of the body. However, the presence
of voids does not affect the phase speeds of coupled transverse waves. The phase speed of
longitudinal microrotational waves is found to be independent of the rotation of the body and the
presence of the voids. In general, the phase speed, energy loss, and decay coefficient of the
remaining two longitudinal waves are found to be influenced by the micropolarity, the presence of
voids, and the rotation of the body. At high and low frequencies, the rotation of the body has a
significant effect on the longitudinal displacement wave speed, while the phase speed of the wave
carrying a change in void volume fraction remains unaffected. The results of some earlier workers
have also been reduced from the present formulation. © 2007 American Institute of Physics.
关DOI: 10.1063/1.2784973兴
I. INTRODUCTION
Eringen and co-worker1,2 developed a nonlinear theory
of simple microelastic solids. Later, Eringen3,4 developed a
linear theory of micropolar elasticity, which is a subclass of
the theory developed earlier in Refs. 1 and 2 and is a generalization of the classical theory of elasticity. The basic difference between the Eringen’s theory of micropolar elasticity
and that of the classical elasticity is the introduction of an
independent microrotation vector. In classical elasticity, the
motion is described by a displacement vector only, and hence
there are three degrees of freedom, while in the micropolar
elasticity, the motion is described not only by a displacement
vector but also by a microrotation vector, and hence thereby
six degrees of freedom. The force at a point on a surface
element of a micropolar material is completely known by a
force stress tensor and by a couple tensor at that point. Physically speaking, a micropolar elastic material is a continuum,
in which the dumbbell-shaped particles are uniformly distributed in an elastic body. Parfitt and Eringen5 have shown that
there can exist four waves in a homogeneous isotropic micropolar elastic material, two of which disappear below a
critical frequency whose value depends upon the property of
the medium.
Cowin and co-worker6,7 developed the theories of nonlinear and linear elastic material with voids. The linear
theory of elastic material with voids is a special class of the
a兲
Electronic mail: [email protected]
Electronic mail: [email protected]
b兲
0021-8979/2007/102共7兲/074906/7/$23.00
nonlinear theory,6 in which the change in void volume fraction and the strain are taken as independent kinematic variables. Material having small distributed pores 共voids兲 containing nothing may be called porous material. Puri and
Cowin8 explored the possibility of plane wave propagation in
a linear elastic material with voids. They showed that there
can exist two dilational 共longitudinal兲 waves in a porous
elastic material with voids: One of them is predominantly the
dilational wave of classical elasticity and the other is predominantly a wave carrying a change in the void volume
fraction. Both the waves are found to attenuate in their directions of propagation. At large frequency, the predominantly elastic wave propagates with the classical elastic dilation wave speed, but at low frequency it propagates at a
speed less than the classical wave speed. Various problems of
waves and vibrations based on the above theories of elasticity have been attempted by the researchers and they have
appeared in the open literature. Some notable research has
been performed by Parfitt and Eringen,5 Chandrasekharaiah,9
Wright,10 Iesan and Nappa,11 Golamhossen,12 Dey et al.,13
Midya,14 Tomar and Gogna,15,16 and Tomar and Singh.17,18
Propagation of plane waves in a rotating elastic solid
with voids has been investigated by Chandrasekharaiah.19 He
was motivated by the idea that most of the large bodies such
as the Earth, the Moon, and other planets have an angular
velocity. Therefore, the problems of rotating bodies are more
important than the corresponding problems of nonrotating
bodies. Inspired by this idea, we have investigated the propagation of plane waves in a linear, homogeneous, and isotro-
102, 074906-1
© 2007 American Institute of Physics
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074906-2
J. Appl. Phys. 102, 074906 共2007兲
J. Singh and S. K. Tomar
pic micropolar porous elastic solid rotating with a uniform
angular velocity. It has been found that there can exist three
longitudinal waves and two sets of coupled transverse waves,
each consisting of a transverse shear displacement wave and
a transverse microrotational wave perpendicular to each
other. Out of the three longitudinal waves, one is a longitudinal microrotational wave, the second is a longitudinal displacement wave already encountered in the theory of micropolar elasticity, while the third is a longitudinal void
volume fractional wave carrying a change in the void volume. It has been found that, in general, the rotation of the
body does influence the phase speed, energy loss, and decay
coefficient of the waves. The longitudinal microrotational
wave is found to be unaffected by the rotation of the solid. At
limitedly low and high frequencies, the angular rotation of
the body has no effect on the wave carrying a change in void
volume fraction.
The field equations in a rotating micropolar elastic solid
with voids, in the absence of body force and body couple
densities, are given by
共1兲
共␣ + ␤ + ␥兲 ⵜ 共ⵜ · ⌽兲 − ␥ ⵜ ⫻ 共ⵜ ⫻ ⌽兲 + K ⵜ ⫻ u
␣ⴱⵜ2␺ − ␰ⴱ␺ − ␻ⴱ␺˙ − ␤ⴱ ⵜ · u = ␳Kⴱ␺¨ ,
共2兲
共3兲
where ␭ and ␮ are the Lame’s parameters; K, ␣, ␤, and ␥ are
the micropolar constants; ␣ⴱ, ␤ⴱ, ␰ⴱ, ␻ⴱ, and Kⴱ are the void
parameters; u共x , t兲 and ⌽共x , t兲 are the displacement and microrotation vectors, respectively; ␺ is the change in void volume fraction from the reference volume fraction; J is the
microinertia, ␳ is the density, and ⌰ is the uniform angular
velocity. A superimposed dot represents the temporal derivative. We note that, in Eq. 共1兲, there are two additional terms
that do not appear in the corresponding equation of motion
for a nonrotating micropolar body. The term ⌰ ⫻ 共⌰ ⫻ u兲
represents the centripetal acceleration, while the term 2⌰
⫻ u̇
represents
the
Coriolis
acceleration
共see
Chandrasekharaiah19兲. It can be seen that by neglecting these
terms, one can recover the equations of motion for a nonrotating micropolar elastic body with voids given by Iesan.20
In order to discuss the propagation of plane waves in a
linear micropolar elastic solid body with voids and rotating
with uniform angular velocity, we consider a plane wave
propagating in the positive direction of a unit vector n as
follows:
兵u,⌽, ␺其 = 兵a,b,c其exp兵ı共⍀t − kn · r兲其,
− ıc23k共n ⫻ b兲 − 兵共⌰ · a兲⌰ + 2ı⍀共⌰ ⫻ a兲其 = 0,
共5兲
兵⍀2 − 2␻2o − c24k2其b − c25k2共n · b兲n − ␫␻2ok共n ⫻ a兲 = 0,
共6兲
ı␤ⴱk共n · a兲 − 共␣ⴱk2 + ␰2 + ␫␻ⴱ⍀ − ␳Kⴱ⍀2兲c = 0,
共7兲
where c21 = 共␭ + 2␮兲 / ␳, c22 = ␮ / ␳, c23 = K / ␳, c24 = ␥ / ␳J, c25 = 共␣
+ ␤兲 / ␳J, and ␻2o = K / ␳J.
Multiplying Eqs. 共5兲 and 共6兲 scalarly with vectors a and
b, respectively, we obtain
⫻共n · a兲 − ıc23k共n ⫻ b兲 · a − 兵共⌰ · a兲2 + 2ı⍀共⌰ ⫻ a兲 · a其
共8兲
= 0,
兵⍀2 − 2␻2o − c24k2其b2 − c25k2共n · b兲2 − ␫␻2ok共n ⫻ a兲 · b = 0.
共␭ + ␮兲 ⵜ 共ⵜ · u兲 + 共␮ + K兲ⵜ2u + K ⵜ ⫻ ⌽ + ␤ⴱ ⵜ ␺
− 2K⌽ = ␳J⌽̈,
兵⌰2 + ⍀2 − 共c22 + c23兲k2其a − 兵共c21 − c22兲k2共n · a兲 + 共␫␤ⴱk/␳兲c其n
兵⌰2 + ⍀2 − 共c22 + c23兲k2其a2 − 兵共c21 − c22兲k2共n · a兲 + 共␫␤ⴱk/␳兲c其
II. WAVE PROPAGATION
= ␳关ü + ⌰ ⫻ 共⌰ ⫻ u兲 + 2⌰ ⫻ u̇兴,
the real parts of solution 共4兲 are physically relevant. Solution
共4兲 corresponds to the waves for which ⍀ is the frequency,
2␲ / ␩ is the wavelength, Vⴱ = ⍀ / ␩ is the phase speed, and ␦
is the attenuation coefficient. The specific loss S associated
with the waves is given by S = 4␲␦ / ␩ 共see Kolsky21兲. On
substituting Eq. 共4兲 into Eqs. 共1兲–共3兲, we obtain
共4兲
where a and b are vector constants and c is a scalar constant
representing the amplitudes, ⍀ is a positive real number, k is
a complex number, r is the position vector, and ı = 冑−1. If we
set k = ␩ − ı␦, it can be seen that for the waves to be physically acceptable, we must have ␩ ⬎ 0 and ␦ ⱖ 0, and that only
共9兲
For transverse waves, we must have n · a = 0 and n · b = 0.
This yields 兩n ⫻ a兩 = a and 兩n ⫻ b兩 = b, where a and b are the
magnitude of vectors a and b, respectively. Using these relations in 共8兲 and 共9兲, we obtain
兵⍀2 + ⌰2 sin2 ␪ − 共c22 + c23兲k2其a − ␫c23kb = 0,
共10兲
共⍀2 + 2␻2o − c24k2兲b − ␫␻2oka = 0.
共11兲
where ␪ is the angle between vector ⌰ and vector a. Eliminating a and b from Eqs. 共10兲 and 共11兲, we obtain
共12兲
AV4 + BV2 + C = 0,
where
V = ⍀/k,
L2 = 1 +
⌰2
sin2 ␪,
⍀2
A = 共1 − X兲L2 ,
B = − 兵c24L2 + c22 + c23 − X共c22 + c23/2兲其,
C = c24共c22 + c23兲
X = 2␻2o/⍀2 ,
The roots of Eq. 共12兲 are given by
2
=
V3,4
1
兵L2c24 + c22 + c23 − 共c22 + c23/2兲X
2共1 − X兲L2
± 关兵L2c24 − c22 − c23 + 共c22 + c23/2兲X其2 + 2L2c23c24X兴1/2其.
共13兲
For a nonrotating micropolar body, the angular velocity vanishes, i.e., ⌰ = 0. In this case, the expressions of velocities
given in Eq. 共13兲 reduce to those expressions of coupled
transverse waves given earlier in Parfitt and Eringen.5 From
Eq. 共13兲, it is clear that there is no term corresponding to
voids, showing that the speeds of transverse coupled waves
are not affected by the presence of voids.
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074906-3
J. Appl. Phys. 102, 074906 共2007兲
J. Singh and S. K. Tomar
For longitudinal waves, we should have n · a = a and
n · b = b. This means 兩n ⫻ a兩 = 兩n ⫻ b兩 = 0. Using these relations
in Eqs. 共5兲–共7兲, we obtain
兵⍀ + ⌰ sin ␪ −
2
2
2
共c21
+
c23兲k2其a
␫␤ⴱk
−
c = 0,
␳
兵⍀2 − 2␻2o − 共c24 + c25兲k2其b2 = 0,
ⴱ
ⴱ 2
ⴱ
ⴱ
␫␤ ka − 共␣ k + ␰ + ␫␻ ⍀ − ␳K ⍀ 兲c = 0.
2
共16兲
We note that Eqs. 共14兲 and 共16兲 are coupled in a and c, while
Eq. 共15兲 is uncoupled in b. From Eq. 共15兲, we can obtain
V2 =
+ c25
,
2␻2o/⍀2
1−
which gives the speed of the longitudinal microrotational
wave already discussed by Parfitt and Eringen,5 and does not
depend upon the rotation of the body.
Eliminating the quantities a and c from Eqs. 共14兲 and
共16兲, we obtain
冉
⍀ 2L 2
cⴱ2
p
k2 −
冊冉
k2 −
冊
⍀ 2 1 ␫ ␻ ⴱ⍀
Hⴱ 2
+
−
−
k = 0,
␣ⴱ
cⴱ2
l22
l21
3
共17兲
where
cⴱ2
p =
␭ + 2␮ + K
,
␳
l22 =
ⴱ
␣
,
␰ⴱ
Hⴱ =
cⴱ2
3 =
␣ⴱ
,
␳Kⴱ
l21 =
␣ⴱ
,
␤ⴱ
ⴱ
␤
,
␭ + 2␮ + K
cⴱp is the speed of the longitudinal displacement wave discussed in detail earlier by Parfitt and Eringen,5 cⴱ3 is the
speed of wave carrying a change in the void volume fraction
and investigated earlier by Puri and Cowin,8 and Hⴱ is a
coupling dimensionless number similar to that introduced
earlier by Puri and Cowin8 and reduces to it in the absence of
micropolarity. Equation 共17兲 can be written as
冉
k2 −
⍀ 2L 2
cⴱ2
p
ⴱ
冊冉
k2 −
冊
⍀ 2 1 ␫ ␻ ⴱ⍀
Nⴱ 2
+
−
−
k = 0,
␣ⴱ
cⴱ2
l22
l22
3
共18兲
共=共l22 / l21兲Hⴱ兲 ⬍ 1.
where 0 ⬍ N
In the absence of rotation and micropolarity, i.e., when
⌰ = K = 0, the dispersion relation 共18兲 reduces to
冉
⍀
c21
2
k2 −
冊冉
ⴱ
k2 −
冊
1 ␫␻ ⍀
⍀
N
− 2 k2 = 0.
ⴱ2 + 2 −
ⴱ
␣
c3
l2
l2
2
k2 −
冉
冊
which gives the phase speed of the dilatational wave in the
classical elastic rotating body. We rewrite the dispersion relation 共18兲 as
共20兲
共21兲
where
⍀ 2L 2 ⍀ 2 1 N ⴱ ␫ ␻ ⴱ⍀
+ ⴱ2 − 2 + 2 +
,
␣ⴱ
c3
l2 l2
cⴱ2
p
B,C =
冑2⍀L
cⴱp
冋再冉
⍀2 1
− 2
c*2
l2
3
再 冎册
⍀2 1
− 2
cⴱ2
l2
3
±
冊 冉 冊冎
2
+
␻ ⴱ⍀
␣ⴱ
2 1/2
.
Here, the plus “+” sign corresponds to the quantity B, while
the minus “−” sign corresponds to the quantity C. The general solution of Eq. 共20兲 is complex valued, but it may admit
real valued solutions for limitedly high and limitedly low
frequencies. For limitedly high frequency 共l2⍀ 1兲, we obtain the following two roots of Eq. 共20兲, given as
冉 冊
k1 =
ⴱ3 ⴱ ⴱ
cⴱ3
1
⍀L
p c3 N ␻
ⴱ2 + O
2
ⴱ2 2
ⴱ −␫
3 ,
2
ⴱ
⍀
2⍀ l2␣ L共c3 L − c p 兲
cp
k2 =
1
⍀ ␫␻ⴱcⴱ3
,
ⴱ −
ⴱ +O
2␣
⍀
c3
冉冊
共22a兲
共22b兲
where k1 corresponds to the longitudinal displacement wave
and k2 corresponds to the longitudinal void volume fractional
wave.
The phase speed Vⴱ, the attenuation coefficient ␦, and
the specific loss S of the longitudinal displacement wave are
respectively given as
Vⴱ =
共19兲
Q2 sin2 ␪
⍀2
1
+
= 0,
⍀2
c21
冊
1
k = ± 关兵A + 冑B − ␫C其1/2 ± 兵A − 冑B − ␫C其1/2兴1/2 ,
2
S=
It can be seen that the dispersion relation 共19兲 does match
exactly with the dispersion relation already discussed by Puri
and Cowin.8
In the absence of micropolarity and voids, i.e., when K
= ␤ⴱ = 0, the dispersion relation 共18兲 reduces to
冉
⍀ 2L 2 ⍀ 2 1 ␫ ␻ ⴱ⍀
− 2+
= 0.
␣ⴱ
cⴱ2
l2
cⴱ2
3
p
This is a biquadratic equation in k, whose roots are given by
A=
c24
冊
⍀ 2L 2 ⍀ 2 1 N ⴱ ␫ ␻ ⴱ⍀ 2
+ ⴱ2 − 2 + 2 +
k
␣ⴱ
c3
l2 l2
cⴱ2
p
+
共14兲
共15兲
ⴱ
冉
k4 −
cⴱp
,
L
␦=
ⴱ4 ⴱ ⴱ
cⴱ3
p c3 N ␻
ⴱ2 ,
2
2
2⍀2l2␣ⴱL共cⴱ2
3 L − cp 兲
ⴱ4 ⴱ ⴱ
2␲cⴱ4
p c3 N ␻
ⴱ2
2
L2⍀3l22␣ⴱ共cⴱ2
3 L − cp 兲
.
共23兲
Similarly, for a limitedly low frequency case 共l2⍀ 1兲, we
obtain the following two roots of Eq. 共20兲, given as
k1 =
k2 =
⍀L
cⴱp冑1 − Nⴱ
l 2␻ ⴱ⍀
−
2␣ⴱ冑1 − Nⴱ
␫Nⴱl32⍀2L␻ⴱ2
+ O共⍀3兲,
2␣ⴱcⴱp共1 − Nⴱ兲3/2
−
␫冑1 − Nⴱ
+ O共⍀兲.
l2
共24兲
共25兲
In this case, the phase speed, attenuation coefficient, and
specific loss of the longitudinal displacement wave are respectively given as follows:
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074906-4
J. Appl. Phys. 102, 074906 共2007兲
J. Singh and S. K. Tomar
FIG. 1. Effect of rotation on the phase speed of coupled transverse waves
共when ⌰ = ⍀兲.
Vⴱ =
cⴱp冑1 − Nⴱ
,
L
␦=
L␻ⴱ2l32Nⴱ⍀2
2␣ⴱcⴱp共1 − Nⴱ兲
,
3/2
S=
2␲␻ⴱ2l32Nⴱ⍀
.
␣ⴱ共1 − Nⴱ兲
共26兲
From Eqs. 共22b兲 and 共25兲, it is clear that the rotation of the
body has no effect on the phase speed of the wave corresponding to the void volume fraction. From Eqs. 共23兲 and
共26兲, we see that the phase speed and attenuation coefficient
of the longitudinal displacement wave are influenced by the
rotation of the body, however, the specific loss factor depends on the rotation of the body only for the limitedly high
frequency case. When the quantities K and ⌰ are neglected,
one can see that the quantities in Eqs. 共23兲 and 共26兲 reduce to
those obtained earlier by Puri and Cowin.8
III. NUMERICAL RESULTS AND DISCUSSION
In order to discuss the problem in greater detail and to
find out the effect of the rotation of the body on the phase
speed, attenuation coefficient, and specific loss of the waves,
we have computed them by taking the following numerical
values of the relevant micropolar elastic parameters 共Chiroiu
and Munteanu22兲, while the numerical values of the relevant
parameters corresponding to voids are taken arbitrarily: ␭
= 7.55⫻ 1011 dyn/ cm2, ␮ = 6.19⫻ 1011 dyn/ cm2, K = 14.50
⫻ 108 dyn/ cm2, ␥ = 2.86⫻ 1011 dyn, J = 0.0212 cm2, ␳
= 1.16 g / cm3,
␰ⴱ = 12⫻ 1011 dyn/ cm2,
␤ⴱ = 10⫻ 1011
2
ⴱ
8
2
ⴱ
dyn/ cm , ␻ = 10⫻ 10 dyn/ cm s, ␣ = 0.001⫻ 1011 dyn,
Kⴱ = 10⫻ 1011 cm2.
In all the figures, the solid curve corresponds to the case
of a nonrotating body, while the dotted curve corresponds to
the case of a rotating body. Figure 1 depicts the effect of
rotation of the body on the phase speeds of the coupled transverse waves. We have plotted the graphs of the phase speed
V4 by magnifying its original value with a factor of 10. It is
clear from curves I and II that the phase speed V3 of the
FIG. 2. Effect of angular velocity of the body on the phase speed of coupled
transverse waves 共when ⍀ = 10 rad/ s and ␪ = 60°兲.
coupled transverse wave is not influenced by the rotation of
the body, while curves III and IV show that the phase speed
V4 of the coupled transverse wave is significantly affected by
the rotation of the body. The effect is maximum at a 90°
angle of rotation of the body.
Figure 2 depicts the variation of the phase speed of
coupled transverse waves with respect to the angular velocity
of the rotating body when the angle of rotation is 60° and the
wave frequency is 10 rad/ s. In this case, the graphs of the
phase speed V4 are plotted after magnifying the phase
speed’s original value with a factor of 10. Curves I and III
show the graphs of the phase speeds V3 and V4 of the
coupled transverse waves in a nonrotating body, while curves
II and IV show the graphs of the phase speeds in a rotating
body. From curves I and II, it is noted that the coupled transverse wave with phase speed V3 is not influenced with the
variation of the angular velocity of the body, while curves III
and IV show that the phase speed V4 of the coupled transverse wave decreases with an increase of the angular velocity
of the body: it approaches zero when the angular velocity of
the body takes large values.
Figure 3 depicts the variation of the phase speed with
respect to the frequency of the coupled transverse wave
propagating with speed V4 in rotating and nonrotating bodies
when the angle of rotation is 60° and the angular velocity is
10 m / s. In this case, the graphs of the phase speed V4 is
plotted after magnifying its original value with a factor of 10.
It is noted that the phase speed of this coupled transverse
wave in the rotating body is zero when the frequency of the
wave is very small, and then it increases with the increase of
the frequency of the wave and attains the value of the phase
speed of the nonrotating body. At limitedly high frequency,
the phase speed V4 is the same in the rotating and nonrotating bodies.
Figures 4–6 depict the effect of the angle of rotation of
the body, the angular velocity of the body, and the frequency
Downloaded 01 Mar 2010 to 220.227.11.194. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp
074906-5
J. Singh and S. K. Tomar
FIG. 3. Effect of frequency on the phase speed of a coupled transverse
wave.
J. Appl. Phys. 102, 074906 共2007兲
FIG. 5. Effect of angular velocity on the phase speed and attenuation coefficient of the longitudinal displacement wave at limitedly high frequency.
of the wave, respectively, on the phase speed and the attenuation coefficient of the longitudinal displacement wave for a
limitedly high frequency case. In these figures, curves I and
II show the effect on the phase speed, while curves III and
IV show the affect on the attenuation coefficient in the rotating and nonrotating bodies. In all these figures, the attenuation coefficient of the wave is plotted after magnifying its
original value with a factor of 10. In Fig. 4, we notice that
the phase speed of the wave is not affected by the rotation of
the body at 0° and 180° angles of the rotation 共that is, when
the axis of the rotation of the body coincides with the direction of propagation of the longitudinal displacement wave兲,
while the phase speed of the wave is affected at other values
of the angle of rotation: the effect is maximum at a 90° angle
of rotation. Similarly, the attenuation coefficient of the wave
is not affected at 0° and 180° angles of rotation, but at other
angles of rotation, the attenuation coefficient is significantly
affected. It is highly influenced at a 90° angle of rotation. In
Fig. 5, it is noted from curves I and II that the phase speed of
the longitudinal displacement wave decreases with an increase of the angular velocity of the body, and approaches to
zero as the angular velocity of the body approaches infinity,
whereas curves III and IV depict that with an increase of the
angular velocity of the body, the attenuation coefficient also
increases, and it approaches zero as the angular velocity of
the body approaches infinity. In Fig. 6, curves I, II, III, and
IV show that at small frequencies, both the phase speed and
FIG. 4. Effect of rotation on phase speed and attenuation coefficient of the
longitudinal displacement wave at limitedly high frequencies.
FIG. 6. Effect of frequency on the phase speed and attenuation coefficient of
the longitudinal displacement wave at limitedly high frequency.
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074906-6
J. Singh and S. K. Tomar
J. Appl. Phys. 102, 074906 共2007兲
FIG. 7. Effect of rotation on the phase speed and attenuation coefficient of
the longitudinal displacement wave at limitedly high frequency.
FIG. 9. Effect of frequency on the phase speed and attenuation coefficient of
the longitudinal displacement wave at limitedly high frequency.
attenuation coefficient are significantly influenced, while at
high frequency of the wave, the effect is negligible.
Figures 7–9 depict the effect of the rotation of the body,
the angular velocity of the body, and the frequency of the
wave, respectively, on the phase speed and attenuation coefficient of the longitudinal displacement wave for a limitedly
low frequency case. In these figure, curves I and II represent
the phase speed and curves III and IV represent the attenuation coefficient. The attenuation coefficient of the wave is
plotted after magnifying its original value with the factors
105, 105, and 104, respectively. The conclusion of these figures is similar to that as discussed earlier for the limitedly
high frequency case.
Figures 10–12 depict the effect of the rotation of the
body, angular velocity of the body, and frequency of the
wave, respectively, on the specific loss of the longitudinal
displacement wave for a limitedly high frequency. Figure 10
depicts that the specific loss of the wave increases from a
certain negative value with an increase of the angle of rotation and achieves its maximum value at 90°. Figure 11 depicts that, with an increase of the angular velocity of the
body, the specific loss of the wave increases from certain
negative values, and it approaches zero as the angular velocity of the body approaches infinity. Figure 12 depicts that the
specific loss of the wave is affected by the frequency in the
range 0–10. After this range of frequency, the specific loss of
FIG. 8. Effect of angular velocity on the phase speed and attenuation coefficient of the longitudinal displacement wave at limitedly high frequency.
FIG. 10. Effect of rotation on the specific loss of a longitudinal displacement wave at limitedly high frequency.
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074906-7
J. Appl. Phys. 102, 074906 共2007兲
J. Singh and S. K. Tomar
In this article, we have studied the wave propagation in
an infinite micropolar porous elastic solid rotating with an
angular velocity. It is found that there can exist five plane
waves: One is a longitudinal displacement wave, the second
is a longitudinal microrotational wave, the third is a longitudinal void volume fractional wave, and the remaining two
waves are the sets of coupled transverse waves consisting of
a transverse displacement wave and a transverse microrotational wave propagating with different speeds. The last two
waves propagate in the medium only if the value of the frequency is greater than a particular value, below which they
degenerate into a distance decaying vibrations. It can be concluded that:
共i兲 The longitudinal microrotational wave is not influenced by the rotation of the body and presence of voids.
共ii兲 The sets of the coupled transverse waves are affected
by the rotation of the body but remain independent by the
presence of the voids, as in the case of nonrotating bodies.
共iii兲 For limitedly low and high frequencies, the waves
corresponding to the void volume fraction remain independent by the effect of rotation of the body, whereas the longitudinal displacement wave is influenced by the rotation of
the body.
共iv兲 For the limitedly high frequency case, the phase
speed, attenuation coefficient, and specific loss are influenced by the rotation of the body only when the angle of
rotation is different from 0° and 180°. These quantities are
highly influenced when the angle of rotation is 90°.
共v兲 For the limitedly low frequency case, the phase speed
and the attenuation coefficient are influenced by the rotation
of the body, whereas the specific loss remains independent
by the influence of rotation of the body.
FIG. 12. Effect of frequency on the specific loss of a longitudinal displacement wave at limitedly high frequency.
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FIG. 11. Effect of the angular velocity on the specific loss of a longitudinal
displacement wave at limitedly high frequency.
the wave is not influenced by the frequency of the wave. In
Figs. 4–9, the symbol ¯␦ denotes the attenuation coefficient
for nonrotating body.
IV. CONCLUSIONS
1
2
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