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Mathematical modelling and analysis of bulk waves in rotating generalized thermoelastic media with voids
Applied Mathematical Modelling 35 (2011) 3396–3407
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Mathematical modelling and analysis of bulk waves in rotating generalized thermoelastic media with voids J.N. Sharma a,⇑, D. Grover a, D. Kaur b a b
Department of Mathematics, National Institute of Technology, Hamirpur 177005, India Department of Mathematics, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar 144011, India
a r t i c l e
i n f o
Article history: Received 7 May 2009 Received in revised form 6 January 2011 Accepted 11 January 2011 Available online 18 January 2011 Keywords: Thermal relaxation Volume fraction Kibel number Thermoelastic waves t-test Rotation
a b s t r a c t The present paper deals with the propagation of body waves in a homogenous isotropic, rotating, generalized thermoelastic solid with voids. The complex quartic secular equation has been solved by employing Descartes’ algorithm and perturbation method to obtain phase velocities, attenuations and specific loss factors of four attenuating and dispersive waves, which are possible to exist in such media. These wave characteristics have been computed numerically for magnesium crystal and presented graphically. Statistical analysis has been performed to compare the obtained computer simulated result in order to have estimate on the suitability of the method to compute various characteristics of the waves. This work may find applications in geophysics and gyroscopic sensors. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The theory of elastic materials with voids is concerned with the elastic materials consisting of a distribution of small pores (voids) which contains nothing of mechanical or energetic significance. Goodman and Cowin [1] established a general continuum theory for a fluid like material with voids and applied it to flow of granular materials. The nonlinear theory of elastic materials with voids was proposed by Nunziato and Cowin [2] and the linearized version was deduced by Cowin and Nunziato [3] where voids have been included as an additional kinematics’ variable. This theory reduces to the classical theory of elasticity in the limiting case when void volume vanishes. Puri and Cowin [4] presented a complete analysis of the frequency equation for plane waves in a linear elastic material with voids and observed that there are two dispersive and attenuated dilatational waves in this theory, one is predominantly the dilatational wave of classical linear elasticity and other is predominantly a wave carrying a change in void volume fraction. Chandrasekharaiah [5] established the uniqueness of solution of an initial boundary value problem formulated completely in terms of stress and volume fraction fields for homogenous and isotropic materials with voids. Chandrasekharaiah [6] studied the propagation of plane waves in a homogenous and isotropic unbounded elastic solid with voids rotating with a uniform angular velocity. Moreover, the effect of voids on surface waves propagating in thermoelastic media has got its due importance where the situation so demands. The cooling and heating of the medium also results in the expansion and contraction of the voids along with the core material which contributes towards thermal stress and vibration developments in solids. Sharma and Kaur [7] studied the plane harmonic waves in generalized thermoelastic materials with voids and Sharma et al. [8] studied an exact free vibration analysis of simply supported, homogenous isotropic, cylindrical panel with voids in the three dimensional ⇑ Corresponding author. Tel.: +91 1972 254122; fax: +91 1972 223834. E-mail addresses: [email protected] (J.N. Sharma), [email protected] (D. Grover), [email protected] (D. Kaur). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.01.014
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generalized thermoelasticity. Singh and Tomar [9] explored the possibility of plane wave propagation in an infinite thermoelastic medium with voids using the theory developed by Iesan and investigated the reflection phenomenon of a set of coupled longitudinal waves from a free plane boundary of a thermoelastic half-space with voids. Auriault [10] and Sharma and Grover [11] studied the body wave propagation in an infinite homogenous isotropic elastic and thermoelastic media respectively rotating with uniform angular velocity. During the last three decades, non-classical theories, Lord and Shulman [12], Green and Lindsay [13] have been developing to alleviate the paradox inherited in the classical theory of heat conduction which predicts an infinite speed of heat transportation and contradicts the physical facts. Chandrasekharaiah [14] referred a wave-like thermal disturbance as ‘second sound’. Ackerman and Overtone [15] and Ackerman et al. [16] have supported the experimental exhibition of the actual occurrence of ‘second sound’ at low temperatures and small intervals of time in solid helium. In the present study, wave propagation in a homogenous isotropic, thermoelastic media with voids which is rotating with ~, about a fixed axis ~ uniform angular rotation X e3 , the wave propagation of a perturbation displacement in the plane ð~ e1 ; ~ e2 Þ perpendicular to ~ e3 may be affected by the Coriolis force. The wave propagation is investigated as a function of the Kibel number C ¼ x X . The quartic complex polynomial characteristic equation yields us four complex roots, in general. These roots can be associated with four dispersive waves namely quasi-longitudinal (QL), quasi-transverse (QT), volume fractional (/-mode) and thermal wave (T-mode). The general complex characteristics equation has been solved by using Descartes’ algorithm along with irreducible case of Cardano’s and series (perturbation) expansion methods with the help of DeMoivre’s theorem in order to obtain phase speeds, attenuation coefficients and specific loss factors of the waves. The numerical solution of the secular equations is carried out for magnesium crystal like material in order to illustrate the analytical developments. The computer simulated results have also been presented graphically for velocities, attenuation and phase velocity with respect to Kibel number. 2. Wave equations in rotating media We consider a homogenous isotropic, thermoelastic solid with voids at uniform temperature T0 and initial volume frac~. with respect to an tion /0 in the undisturbed state. The medium is assumed to be rotating with uniform angular velocity X inertial frame. The Coriolis effect is caused by the rotation of an object and the inertia of mass experiencing the effect. When Newton’s laws are transformed to a rotating frame of reference, the Coriolis and Centrifugal forces appear. Both forces are proportional to the mass of the object. The Coriolis force is proportional to the rotation rate. The Coriolis force acts in a direction perpendicular to the rotation axis and to the velocity of the body in the rotating frame and is proportional to the object’s speed in rotating frame. This force is termed as inertial force (pseudo force). The Coriolis force is quite small, only the horizontal component of Coriolis force is generally important. The basic governing dynamical equations of linear generalized thermoelastic interactions after including the Coriolis and centripetal forces, in absence of body forces and heat sources, Sharma and Kaur [7] and Sharma et al. [8] are
~ ~ ~ ~ ~ X lr2~ u þ br/ brT ¼ q ~ u þ2 X u_ ; u€ þ X u þ ðk þ lÞrr ~
ð1Þ
€ ar2 / n1 / n2 /_ br ~ u þ mT ¼ qv/;
ð2Þ
€ ; K r2 T qC e T_ þ t0 T€ ¼ bT 0 r ~ u_ þ t 0~ u€ þ mT 0 /_ þ t 0 /
ð3Þ
where t0 is the thermal relaxation time, ~ uðx1 ; x2 ; x3 ; tÞ ¼ ðu1 ; u2 ; u3 Þ is the displacement vector; T(x1, x2, x3, t) is the temperature change; k, l are Lamé’s parameters; K is thermal conductivity; q and Ce are respectively, the density and specific heat at constant strain; / is change in Volume fraction, b = (3k + 2l)aT, aT is the linear thermal expansion. a, b, n1, n2, m and v are material constants due to the presence of voids. Here superposed dot represents time differentiation. We define the non-dimensional quantities:
x0i ¼
x xi c1
;
t 00 ¼ x t0 ;
t 0 ¼ x t;
X0 ¼
a5 ¼
mc41 ; K vx3
x ¼
C e ðk þ 2lÞ ; K
X
x
u0i ¼
qx c1 ui
d2 ¼
;
n ¼ n2 x ; n1
bT 0 c22 ; c21 0
k ¼
/0 ¼
;
x2 v/ c21
;
T0 ¼
2
a1 ¼ c1 k
x
;
bc1 bT 0 vx
x0 ¼
2
;
a2 ¼
T ; T0
bvbT 0 ; aqc21
eT ¼
b2 T 0 ; qC e ðk þ 2lÞ
a3 ¼
n1 c21
ax2
;
a4 ¼
d21 ¼ mT 0 v
a
c23 ; c21 ;
ð4Þ
x c ; c0 ¼ ; x c1
where
c21 ¼
k þ 2l
q
;
c22 ¼
l a ; c23 ¼ qv q
Upon using quantities (4) in Eqs. (1)–(3), we obtain
~ X ~ ~ ~ ~ d2 r2~ uÞ þ a1 r/ rT ¼ ~ u€ þ X u þ2 X u_ ; u þ 1 d2 rðr ~
ð5Þ
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r2 / a2 r ~ u a3 / þ n/_ þ a4 T ¼
€ / d21
ð6Þ
;
€ ¼ 0: u€ a5 /_ þ t0 / r2 T T_ þ t0 T€ eT r ~ u_ þ t0~
ð7Þ
Here primes have been suppressed for convenience. 3. Dispersion equation and its solution *
*
Let ð~ e1 ; ~ e2 ; ~ e3 Þ be the rotating orthonormal basis and we take X ¼ X~ u that is collinear to X is not afe3 . A perturbation ~ fected by Coriolis or convective accelerations. Therefore, we limit the analysis to displacements in the plane ð~ e1 ; ~ e2 Þ which remain constant in the direction ~ e3 . On applying the successive application of divergence and curl operators to Eq. (5) leads ~ as to a system of following two coupled differential equations for e and w
"
r2 þ X2
# @2 _ 3 þ a1 r2 / r2 T ¼ 0; e þ 2Xw @t 2
ð8Þ
! @2 d r þ X 2 w3 2Xe_ ¼ 0; @t 2
2
2
ð9Þ
~ ¼ r ~ where e ¼ r ~ u; w u ¼ w3~ e3 . Clearly the longitudinal, transverse, volume-fractional and thermal waves get coupled with each other here in contrast to wave propagation in an inertial medium. We consider waves that propagate in the direction ~ e1 of the form
ðe; w3 ; T; /Þ ¼ ðA1 ; A2 ; A3 ; A4 Þ exp fiðkx1 xtÞg;
i2 ¼ 1
ð10Þ
Upon using wave solution (10) in Eqs. (6)–(9) we obtain
h i 1 m2 ð1 þ C2 Þ A1 þ 2im2 C1 A2 A3 þ a1 A4 ¼ 0;
ð11Þ
h i 2im2 C1 A1 d2 m2 1 þ C2 A2 ¼ 0;
ð12Þ
" a2 m2 A1 a4 m2 A3 þ x2
m2
a3 n0
d1
x2
þ1þ 2
#
m2 A4 ¼ 0;
ð13Þ
eT m2 s0 A1 ð1 s0 m2 ÞA3 þ a5 m2 s0 A4 ¼ 0;
ð14Þ
n; v ¼ xk Here C ¼ x where s0 = ix + t0, n0 ¼ 1 ix X is called Kibel number. The condition for existence of non-trivial solutions forA1, A2, A3 and A4 of system of Eqs. (11)–(14) yields the dispersion equation 1
4 Y
1 m2 n2i ¼ 0:
ð15Þ
i¼1
Here n2i ; i ¼ 1; 2; 3; 4 are the roots of the quartic equation
n8 Ln6 þ Mn4 Nn2 þ P ¼ 0;
ð16Þ
where
L ¼ s0 þ
M¼
1 d21
a3 n0
x
ð1 C2 Þ2 d
2
þ C2 Þ
2
þ
s0 þ
ð1 þ d2 Þð1 þ C2 Þ
þ
d2 1
d21
s0 d2
þ
a3 n0
x
eT s0 d2
!
2
þ
þ
a3 e12
x2
þ eT s0 ;
s0 ð1 þ eT Þ þ ð1 þ C2 Þ 1 þ
a3 e12
x2 d 2
;
1 2
d
þ
a3 s0
x2
ðe12 þ e25 e45 þ e14 eT Þ þ ð1
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
s0 ð1 C2 Þ2
N¼
d2 þ
P¼
þ
a3 n0 2 2 x d1 1
!( ð1 C2 Þ2 d2
) 1 eT s0 ð1 þ C2 Þ þ s0 ð1 þ C2 Þ 1 þ 2 þ d d2
ð1 þ C2 Þa3 s0
s0 ð1 C2 Þ2 1
e12 ¼
where
e25 ð1 þ d2 Þe45 þ e12 þ e14 eT ;
x2 d 2 d2
3399
d21
a3 n0
a1 a2 ; a3
x2
e14 ¼
a3 e45
x2
a1 a4 ; a3
! ð17Þ
;
e25 ¼
a2 a5 ; a3
e45 ¼
a4 a5 : a3
ð18Þ
In general, the Eq. (16) is a quartic equation in n2 with complex coefficients and there is no general arithmetic or algebraic method of finding the exact value of the roots of imaginary quantities. Therefore this equation can be solved by Descartes’ algorithm along with irreducible case of Cardano’s method with help of De Moivre’s theorem to obtain its four roots in order to know the effect of rotation on various characteristics of waves which are significantly influenced by volume fraction and thermal fields. This gives us four pairs of values of complex phase velocity and hence four distinct types of waves are possible to propagate in such materials. Thus there are, a dilatational, an equivoluminal or a shear, a volume fraction (/-mode) and a thermal (T-mode) wave which get modified due to rotation and are possible to propagate in the thermo-elastic materials with voids rotating about an axis normal to its plane. Upon obtaining the complex roots n2i ði ¼ 1; 2; 3; 4Þ of Eq. (16), we get
mj ¼
1 nj
j ¼ 1; 2; 3; 4:
ð19Þ
The four pairs of the values of complex phase velocity in Eq. (19) give us four distinct types of attenuated and dispersive waves which are possible to propagate in thermoelastic materials with voids rotating about an axis normal to its plane. We write 1 v 1 ¼ V 1 j j þ ix Q j ;
j ¼ 1; 2; 3; 4
ð20Þ
so that kj = Rj + iQj and Rj ¼ Vxj , where Vj and Qj are real quantities. The exponent in the plane wave solution (10) becomes Qjx + iRj(x Vjt). This shows that Vj is the propagation speed and Qj is the attenuation coefficient of the waves. Upon using representation (20) in Eq. (19), we obtain
Vj ¼
1 ; Reðnj Þ
Q j ¼ xImðnj Þ;
j ¼ 1; 2; 3; 4
ð21Þ
The four distinct real values of phase velocity Vj j = 1, 2, 3, 4 corresponds to four distinct waves namely quasi-longitudinal (QL), quasi-transverse (QT), volume fractional ð/ modeÞ and thermal (T-mode) waves which are possible to propagate in such materials. These waves are attenuated in space having attenuation coefficients Qj j = 1, 2, 3, 4 and also get modified due to rotation as well as volume-fractional and thermal variations. In case X – 0, the QL-wave, QT-wave, / mode and T-mode are coupled dilatational-shear, dilatational-volume fractional and dilatational-thermal waves. The coupling is measured by the amplitude ratios
2iC1 m2i A2 ; i ¼ 1; 2; 3; 4; ¼ 2 A1 i d m2i ð1 þ C2 Þ h n oi a3 e25 m2i þ eT x2 d12 ax3 n20 m2i 1 s0 m2i A3 1 n o; ¼ A1 i a3 e45 s0 m4 þ 1 s0 m2 x2 1 a3 n0 m2 1 2 2 i i i x d1 a4 eT s0 m2i þ a2 1 s0 m2i m2i A4 n o; ¼ A1 i a3 e45 s0 m4 þ 1 s0 m2 x2 1 a3 n0 m2 1 2 2 i i i x d
i ¼ 1; 2; 3; 4;
ð22Þ
i ¼ 1; 2; 3; 4:
1
4. Perturbation solution of dispersive waves As we know, there is no direct method to solve complex quartic equations except Descartes’ algorithm along with irreducible case of Cardano’s method available in the literature, however, series approximation (perturbation) method has been widely used by authors [11,17,18] to study the wave propagation problems in classical (coupled) and non-classical (generalized) thermoelastic continua. Therefore, our aim is to solve the instant problem with Descartes’ algorithm and approximation (perturbation) method in order to bring out the comparison and check the suitability of these methods to compute various characteristics of the waves under consideration. The modified secular Eq. (16) can also be rewritten as
f ðn2 Þ eT
s0 þ
a3 e214
x2 e45
gðn2 Þ ¼ 0;
ð23Þ
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where
f ðn2 Þ ¼ ðn2 k1 Þðn2 k2 Þðn2 k3 Þðn2 k4 Þ; gðn2 Þ ¼ ðn2 k^1 Þðn2 ^k2 Þðn2 ^k3 Þ;
ð24Þ
2 2
k2 ¼
k1 ;
k3 ¼ s0 ; X
1þC 2d2
k4 ¼
^k1 ¼ M 2 ; L2
L2 ¼ s0 þ
N2 ¼
P2 ¼
1 d21
3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u 2 u 6 7 t 2 2 2 2 1C 41 þ d ð1 þ d Þ 4d 5; 1 þ C2 þ
X
a3 e214
x2 e45
a3 n0
x2
;
M 2 ¼ s0
;
s0 ð1 þ C2 Þ d2
X
^k1 ^k2 ¼ N2 ; L2
!(
1 d21
1
a3 n0
d1
x2
2
^k1 ^k2 ^k3 ¼ P2 ; L2
a3 n0
!
x2
þ
! þ
a3 e14
x2
1 þ C2 d2
!
s0 þ
a3 e214
x2 e45
þ
s0 a3 e14 e14 e14 ; 2þ x2 e45 e12
) e14 2 e14 ; 2þ ð1 þ d Þ
e45
e12
s0 a3 e214 ð1 C2 Þ2 : x2 d2 e12
ð25Þ
For most of the materials the thermo-mechanical coupling parameter (eT) is very small and therefore, we develop series expansions in terms of eT for the roots n2i ði ¼ 1; 2; 3; 4Þ of the Eq. (23) in order to explore the effect of various interacting fields on the waves. Thus, we obtain
2 n21 ðeT Þ ¼ k1 41 þ 2
e ¼ k2 41 þ
n22 ð T Þ
2 n23 ðeT Þ ¼ k3 41 þ 2
e ¼ k4 41 þ
n24 ð T Þ
L2 k1 ^k1 k1 ^k2 k1 ^k3 k1 ðk1 k2 Þðk1 k3 Þðk1 k4 Þ L2 k2 ^k1 k2 ^k2 k2 ^k3 k2 ðk2 k1 Þðk2 k3 Þðk2 k4 Þ L2 k3 ^k1 k3 ^k2 k3 ^k3 k3 ðk3 k1 Þðk3 k2 Þðk3 k4 Þ L2 k4 ^k1 k4 ^k2 k4 ^k3 k4 ðk4 k1 Þðk4 k2 Þðk4 k3 Þ
3
eT þ 5; 3
eT þ 5; 3
ð26Þ
eT þ 5; 3
eT þ 5:
Upon using the representation (18), the corresponding phase velocities and attenuation coefficients are obtained as
1 wi V i ¼ pffiffiffiffi sec ; ri 2
pffiffiffiffi wi Q i ¼ x r i sin ; 2
i ¼ 1; 2; 3; 4;
ð27Þ
where
ri ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2i þ S2i ;
wi ¼ tan1
Si ; Ri
i ¼ 1; 2; 3; 4:
ð28Þ
The quantities Ri and Si are defined in Appendix. The phase velocity and attenuation coefficients of various modes of wave propagation in case of coupled theory of thermoelasticity can be obtained from (27) and (28) by setting t0 = 0. 5. Specific loss The specific loss is the most direct method of defining internal friction for a material. The specific loss is the ratio of the amount of energy (DE) dissipated in a specimen through a stress cycle to the elastic energy (E) stored in that specimen at maximum strain. According to Kolsky [19] in case of sinusoidal plane wave of small amplitude the specific loss equals 4p times the absolute value of imaginary part of k to the real part of k. Therefore, we have
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
DE ImðkÞ ¼ 4p V i Q i ; SLi ¼ ¼ 4p x E ReðkÞ
i ¼ 1; 2; 3; 4:
3401
ð29Þ
6. Special cases 6.1. In absence of voids (thermoelastic media) In the absence of voids effect (m=0 = b) the system of Eq. (17) reduces to
Rn21 ¼ s0 ð1 þ eT Þ þ Rn21 n22 ¼ n21 n22 n23 n24 ¼
¼
1 d21
ð1 þ d2 Þð1 þ C2 Þ
ð1 C2 Þ2 d
2
d2 þ
s0 ð1 C2 Þ2 d2
a3 n0
x2
;
s0 ð1 þ C2 Þð1 þ d2 þ eT Þ d2
; ð30Þ
;
:
The root n24 is not influenced by rotation and remains independent of rest of the motion. The discussion of remaining roots n2i ði ¼ 1; 2; 3Þ has been already done by Sharma and Grover [11] in case of generalized thermoelastic rotating body. 6.2. In absence of thermal effect (elastic media with voids) In the absence of voids effect (m = 0 = b) the system of Eq. (17) reduces to
Rn21 ¼ R
n21 n22
1 d21 ¼
a3 n0
þ
x2
ð1 þ d2 Þð1 þ C2 Þ
ð1 C2 Þ2 d2
d2 þ
2 2
n21 n22 n24 ¼
ð1 þ C2 Þ
"
d2
ð1 C Þ
1
d2
d21
þ
1 d21
a3 n0
a3 e12
x2 a3 n0
x2
; ! 2
ð1 þ d Þ þ
a3 e12
x2
# ;
!
x2
ð31Þ
;
n23 ¼ s0 : Clearly, the root n23 is not influenced by rest of the motion. 6.3. Elastic media If we ignore thermal effect in Eq. (30) and voids effect in Eq. (31) above
Rn21 ¼
ð1 þ d2 Þð1 þ C2 Þ
n23 ¼ s0 ;
2
d n24 ¼
1
a3 n0
d1
x2
2
Rn21 n22 ¼
;
ð1 C2 Þ2 d2
ð32Þ
Clearly, the root n23 and n24 are not influenced by rest of the motion and we obtained the results for elastic media by Auriault [10]. 7. Numerical results and discussion In this section, we perform some numerical calculations in order to illustrate the analytical results for magnesium like material whose physical data, Eringen [20], Puri and Cowin [4], Kumar and Rani [21], Sharma and Kaur [7], is given as:
q ¼ 1:74 103 kg m3 ; k ¼ 2:17 1010 N m2 ; l ¼ 1:639 1010 N m2 ; C e ¼ 1040 J kg1 deg1 1
K ¼ 170 Wm1 deg ;
1
b ¼ 2:68 106 N m2 deg ; 1
m ¼ 2:0 106 N m2 deg b ¼ 1:13849 1010 N m2 ;
t 0 ¼ 0:05 s; X ¼ 4 rps; a ¼ 3:688 105 N;
v ¼ 1:753 1015 m2 :
3402
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
Fig. 1. Velocities in elastic medium versus Kibel number.
ph Fig. 1 shows the phase velocity profiles of cph 1 and c2 versus Kibel number for magnesium like material. In case of uncoupled thermoelasticity (elastic medium) we note that the dimensionless phase velocity of classical longitudinal wave shows resonance at the point C = 1(i.e. x = X). The profile of phase velocity (cph 1 Þ increases sharply in the range 0 < C < 1 to become asymptotic to its resonant value at C = 1 and for C > 1, it decreases to become asymptotically close to unity at large Kibel increases with increasing Kibel number to ultimately become close to number. We observe that the phase velocity cph 2 ph its the limiting value d as Kibel number tends to infinity. The profiles of cph 1 and c 2 and their variations agree with those ph of Auriault [10] except for c1 in the Kibel number range 0 6 C < 1. In Figs. 2 and 3, we observe that the phase velocity of QL wave slightly depends on the elastic property of the medium. An important fact is that QL-wave shows resonance at the point C = 1(i.e. x = X), where velocity tends to infinity. It increases in the Kibel number range 0 6 C < 1, decreases sharply for 1 < C 6 3 and then becomes steady and stable C P 3 to attain value unity in a dispersion less manner. Also, we observe that, the phase velocity of QT-wave increases with increasing Kibel number and becomes steady to attain the limiting value d at large values of Kibel number. The velocity of the volume fractional wave (/-mode) increases in 0 6 C < 1 and becomes steady for large value of Kibel number and attains the limiting value 0.62. From Fig. 3, we observe that the velocity of thermal wave (T-mode) also increases with increasing Kibel number and becomes steady to attain the limiting value 4.2 at large values of Kibel number. Fig. 4 reveals that the attenuation coefficient profile of QL-wave increases sharply and tends to infinity in the Kibel number range 0 < C < 0.5 and then decreases in the Kibel number range 0.5 < C < 1 and becomes steady for large value of Kibel number. The attenuation of the QT-wave has quite small magnitude as compare to QL-wave and volume fractional wave (/-mode). The attenuation coefficient of volume fractional wave (/-mode) has maximum value at C = 0, and decreases monotonically to become steady and stable at extreme large Kibel number. Here, the actual values of attenuations of volume-fractional 1000 times of magnitude as actually considered in graphical interpretation. Fig. 5 shows that the attenuation of waves interlace as Q1 < Q2 < Q4 < Q3 with the exceptions that Q2 < Q3 < Q1 and Q2 < Q1 in the ranges 0 < C < 0.5 and C P 1.5, respectively. The attenuation of thermal wave increases in 0 < C 6 6. The value of Q3 approaches 0.021 at C = 6. The attenuation of the volume-fractional wave increases for 0 < C < 1.5 and then it becomes steady thereafter to attain its limiting
Fig. 2. Wave velocity in elastic medium with voids versus Kibel number.
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
3403
Fig. 3. Wave velocity in thermoelastic medium with voids versus Kibel number.
Fig. 4. Attenuation in elastic medium with voids versus Kibel number.
Fig. 5. Attenuation in thermoelastic medium with voids versus Kibel number.
value 0.01. The attenuation coefficient profile of QL-wave increases in range 0 < C < 0.5, decreases for 0.5 < C < 1 to become close to zero at C = 1. Then it observes slight increase in 1 < C < 1.5 and then becomes steady thereafter. The attenuation of
3404
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
Fig. 6. Specific loss factor in elastic media with voids versus Kibel number.
Fig. 7. Specific Loss in thermoelastic medium with voids versus Kibel number.
the QT-wave observes Gaussian like behaviour in the range 0.5 < C < 1.5 and has maximum value at C = 1 and almost vanishes elsewhere. Fig. 6, shows that the specific loss factor of QL-wave observes the Gaussian like behaviour in the Kibel number range 0 6 C 6 1 and attains value zero for entire values of Kibel number. The specific loss factor of QT-wave has small increase in the range 0 6 C 6 0.5 to attain its maximum value at C = 0.5 and it becomes steady after C P 0.5. The specific loss factor of volume-fractional wave (/-mode) have maximum value at C = 0 and decreases sharply in the Kibel number range 0 < C < 0.5 and having value extremely small for large value of Kibel number. Fig. 7 reveals that the specific loss factors of QL-wave and QT-waves have quite small magnitude as compare to that of thermal wave. The specific loss factor of volume-fractional wave (/-mode) exhibits Gaussian like behaviour in the range 0 6 C 6 1.5 which decreases exponentially for C P 1.5 to becomes steady there after. The specific loss factor of thermal wave has maximum value at C = 0, and decreases monotonically to become steady and stable at extreme large Kibel number. Here, the actual values of attenuations and specific loss factors of volume-fractional and thermal waves are respectively, 10 and 100 times of their magnitudes as actually considered in graphical interpretation for distinction purpose.
8. Statistical analysis In order to explore which method (Descartes’ or perturbation) gives better results in respect of different characteristics of the waves, we have performed the statistical analysis here. We employed student’s t-test, Gupta and Kapoor [22] at three
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
3405
levels of significance (1%, 5% and 10%) to the data obtained from our FORTRAN coded program in case of phase velocities (Vi), attenuations (Qi) and specific loss factors (SLi) with the help of both Descartes’ and perturbation methods. Null hypothesis
HV0 : lV1 ¼ lV2 ;
HQ0 : lQ1 ¼ lQ2 ;
SL SL HSL 0 : l1 ¼ l2
where
lVi ; lQi ; lSL i ði ¼ 1; 2Þ;
denote mean values of the data corresponding to phase velocity, attenuation coefficient and specific loss factor respectively. Here the subscript values i = 1 and i = 2 refer to perturbation and Descartes’ method computations of the respective quantity. The considered degree of freedom = n1 + n2 2 = 11 + 11 2 = 20 and tabulated values of statistic (t) at 1%, 5% and 10%, levels of significance are ttab. = 2.528, ttab. = 1.725 and ttab. = 1.325, respectively. The corresponding calculated values of t statistic in respect of phase velocities (Vi), attenuations (Qi) and specific loss (SLi), i = 1, 2, 3, 4 are obtained as (6.1324, 4.3911, 14.090, 2.5528), (3400.69, 7.4875, 3.2479, 10.8674) and (14.0484, 34.2048, 7.0429, 1.5346) respectively. From the above analysis it is observed that at three considered levels of significance namely, 1%, 5% and 10%, the null hypotheses, Q Q V V HV0 ; HQ0 and HSL 0 are rejected except for SL4 at 1% and 5% level of significance. Moreover, the mean values l1 ; l2 ; l1 ; l2 SL SL and l1 ; l2 in respect of phase velocity, attenuation coefficient and specific loss factor obey the inequalities SL lV1 i > lV2 i ; i ¼ 1; 3; lV1 i < lV2 i ; i ¼ 2; 4; lQ2 > lQ1 and lSL 2 > l1 , respectively. Thus, it is concluded that perturbation method
is a better choice than Descartes’ method for computation of the phase velocities V1 and V3. However, the latter is well suited as compare to the former one for computing attenuation coefficients and specific loss factors of waves in addition to phase velocities V2 and V4. In case of SL4 null hypothesis is accepted and hence from the coefficient of variations (CV), we conclude that Descartes’ method is better choice than perturbation method. 9. Conclusions It is found that free wave propagation in a homogenous isotropic rotating thermoelastic media with voids gives rise to four waves namely, quasi-longitudinal (QL) wave, quasi-transverse (QT) wave, volume fractional (/-mode) and thermal wave (T-mode). At isentropic (low-frequency) conditions thermal wave does not exist while at isothermal (high-frequency) conditions it becomes diffusive in character. These waves are significantly affected due to rotation, voids and thermal variations. The modified QL and QT-waves approaches their classical counterparts in elastokinetics in the absence of the volumefraction and thermal fields. The dimensionless phase velocity cph 1 ðorV 1 Þ has finite value for all value of Kibel number except at unity, where it suffers resonance. The statistical analysis revealed that the Descartes’ method is a better choice than series method for the computations of phase velocities V2 and V4, attenuation coefficients and specific loss factor of the waves. However, the latter is well suited for the computations of phase velocities V1 and V3 in case of complex secular equations governing the wave motion. Appendix A These quantities are used in Eq. (28) and given by
R1 ¼ k1 þ
" # a3 e45 ðk1 k3 Þðk1 k4 Þ t 0 ReðD1 Þ þ x1 ImðD1 Þ ; x2 ðk1 k2 Þ ðReðD1 ÞÞ2 þ ðImðD1 ÞÞ2
" # a3 e45 ðk2 k3 Þðk2 k4 Þ t 0 ReðD2 Þ þ x1 ImðD2 Þ ; R2 ¼ k2 þ x2 ðk2 k1 Þ ðReðD2 ÞÞ2 þ ðImðD2 ÞÞ2 " # a3 e45 ReðN3 ÞReðD3 Þ þ ImðN3 ÞImðD3 Þ R3 ¼ t 0 þ ; x2 ðReðD3 ÞÞ2 þ ðImðD3 ÞÞ2 R4 ¼
1
a3
d1
x
2
þ 2
" # a3 e45 ReðN4 ÞReðD4 Þ þ ImðN4 ÞImðD4 Þ ; x2 ðReðD4 ÞÞ2 þ ðImðD4 ÞÞ2
" # a3 e45 ðk1 k3 Þðk1 k4 Þ x1 ReðD1 Þ t0 ImðD1 Þ S1 ¼ ; x2 ðk1 k2 Þ ðReðD1 ÞÞ2 þ ðImðD1 ÞÞ2 " # a3 e45 ðk2 k3 Þðk2 k4 Þ x1 ReðD2 Þ t0 ImðD2 Þ ; S2 ¼ x2 ðk2 k1 Þ ðReðD2 ÞÞ2 þ ðImðD2 ÞÞ2
3406
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
" # a3 e45 ImðN3 ÞReðD3 Þ ReðN3 ÞImðD3 Þ ; x2 ðReðD3 ÞÞ2 þ ðImðD3 ÞÞ2
S3 ¼ x1 þ
" ( )# 1 ImðN 4 ÞReðD4 Þ ReðN 4 ÞImðD4 Þ ; S4 ¼ a3 x n þ e45 x ðReðD4 ÞÞ2 þ ðImðD4 ÞÞ2 1
ReðD1 Þ ¼ ðk1 t 0 Þ k1
ReðD3 Þ ¼
ReðD4 Þ ¼
1
t0
d21
1
a3 þ d21 x2
! a3
þ
x2
1
a3 t0 d21 x2
!
a3 n
x
2
ReðD2 Þ ¼ ðk2 t0 Þ k2
;
1
a3
d1
x
þ 2
!
2
a3 n
x2
;
ðt 0 k1 Þðt 0 k2 Þ x2 x2 ð1 a3 nÞð2t 0 k1 k2 Þ;
!(
1
a3
d1
x
2
!
k1 2
1
a3
d1
x
2
!
) x2 a23 n2
k2 2
( þ x2 a3 nð1 a3 nÞ 2
1
a3 d21 x2
!
) k1 k2 ;
ReðN3 Þ ¼ t 0 ðt 0 k3 Þðt0 k4 Þ x2 x2 ð2t0 k3 k4 Þ; ( ReðN4 Þ ¼ t 0
1
a3
d1
x
2
1
ImðD1 Þ ¼ x
k1
ImðD2 Þ ¼ x1 k2
!
k3 2 1 d21
þ
1
a3 k4 d21 x2
!
) x2 a23 n2
( x2 a3 n 2
1
a3
d1
x2
2
! þ na3 ðk1 t 0 Þ ; 2
x
! a3 þ þ na ðk t Þ ; 3 2 0 d21 x2 1
1 a3 ð1 a3 nÞ ðt 0 k1 Þðt 0 k2 Þ x2 þ ð2t 0 k1 k2 Þ t 0 2 þ 2 d1 x
1
) k3 k4 ;
a3
" ImðD3 Þ ¼ x
!
!# ;
" ImðD4 Þ ¼ x
1
ImðN 3 Þ ¼ x1
( ! ) ! ( ! ! )# 1 a3 1 a3 1 a3 1 a3 2 2 2 a3 n 2 2 2 k1 k2 t 0 1 a3 n k1 k2 x a3 n ; d1 x d21 x2 d21 x2 d21 x2
ðt 0 k3 Þðt 0 k4 Þ x2 þ t 0 ð2t0 k3 k4 Þ ;
"( ImðN 4 Þ ¼ x1
1
a3 k3 d21 x2
!
1
a3 k4 d21 x2
!
) x2 a23 n2
( þ t 0 a3 n 2
1
a3 d21 x2
!
)# k3 k4
:
ðA:1Þ
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
M.A. Goodman, S.C. Cowin, A continuum theory for granular materials, Arch. Rational Mech. Anal. 44 (1972) 249–266. J.W. Nunziato, S.C. Cowin, A nonlinear theory of elastic materials with voids, Arch. Rational Mech. Anal. 72 (1979) 175–201. S.C. Cowin, J.W. Nunziato, Linear elastic materials with voids, J. Elast. 13 (1983) 125–147. P. Puri, S.C. Cowin, Plane waves in linear elastic materials with voids, J. Elast. 15 (1985) 167–183. D.S. Chandrasekharaiah, A uniqueness theorem in the theory of elastic material with voids, J. Elast. 18 (1987) 173–179. D.S. Chandrasekharaiah, Plane waves in a rotating elastic solid with voids, Int. J. Eng. Sci. 25 (1987) 591–596. J.N. Sharma, D. Kaur, Generalized thermoelastic plane harmonic waves in materials with voids, Int. J. App. Mech. Eng. 13 (4) (2008). P.K. Sharma, D. Kaur, J.N. Sharma, Three-dimensional vibration analysis of a thermoelastic cylindrical panel with voids, Int. J. Solids Struct. 45 (2008) 5049–5058. J. Singh, S.K. Tomar, Plane waves in thermo-elastic material with voids, Mech. Mater. 39 (2007) 932–940. J.L. Auriault, Body wave propagation in rotating elastic media, Mech. Res. Commun. 31 (2004) 21–27. J.N. Sharma, D. Grover, Body wave propagation in rotating thermoelastic media, Mech. Res. Commun. 36 (2009) 715–721. H.W. Lord, Y. Shulman, The generalized dynamical theory of thermoelasticity, J. Mech. Phys. Sol. 15 (1967) 299–309. A.E. Green, K.A. Lindsay, Thermoelasticity, J. Elast. 2 (1972) 1–7. D.S. Chandrasekharaiah, Thermoelasticity with second sound-a review, Appl. Mech. Rev. 39 (1986) 355–376. C.C. Ackerman, W.C. Overtone, Second sound in helium-3, Phys. Rev. Lett. 22 (1969) 764–766. C.C. Ackerman, B. Bentman, H.K. Fairbank, R.A. Guyer, Second sound in helium, Phys. Rev. Lett. 16 (1966) 789–791. A. Nayfeh, S.N. Nasser, Thermoelastic waves in solids with thermal relaxations, Acta Mech. 12 (1971) 53–69.
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H. Singh, J.N. Sharma, Generalized thermoelastic waves in transversely isotropic media, J. Acoust. Soc. Am. 77 (1985) 1046–1053. H. Kolsky, Stress Waves in Solids, Dover Press, New York, 1963. A.C. Eringen, Plane waves in nonlocal micropolar elasticity, Int. J. Eng. Sci. 22 (1984) 1113–1121. R. Kumar, L. Rani, Interaction due to mechanical and thermal sources in a thermoelastic half-space with voids, J. Vib. Control 11 (2005) 499–517. S.C. Gupta, V.K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand & Sons, New Delhi (India), 1970.
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Mathematical modelling and analysis of bulk waves in rotating generalized thermoelastic media with voids J.N. Sharma a,⇑, D. Grover a, D. Kaur b a b
Department of Mathematics, National Institute of Technology, Hamirpur 177005, India Department of Mathematics, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar 144011, India
a r t i c l e
i n f o
Article history: Received 7 May 2009 Received in revised form 6 January 2011 Accepted 11 January 2011 Available online 18 January 2011 Keywords: Thermal relaxation Volume fraction Kibel number Thermoelastic waves t-test Rotation
a b s t r a c t The present paper deals with the propagation of body waves in a homogenous isotropic, rotating, generalized thermoelastic solid with voids. The complex quartic secular equation has been solved by employing Descartes’ algorithm and perturbation method to obtain phase velocities, attenuations and specific loss factors of four attenuating and dispersive waves, which are possible to exist in such media. These wave characteristics have been computed numerically for magnesium crystal and presented graphically. Statistical analysis has been performed to compare the obtained computer simulated result in order to have estimate on the suitability of the method to compute various characteristics of the waves. This work may find applications in geophysics and gyroscopic sensors. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The theory of elastic materials with voids is concerned with the elastic materials consisting of a distribution of small pores (voids) which contains nothing of mechanical or energetic significance. Goodman and Cowin [1] established a general continuum theory for a fluid like material with voids and applied it to flow of granular materials. The nonlinear theory of elastic materials with voids was proposed by Nunziato and Cowin [2] and the linearized version was deduced by Cowin and Nunziato [3] where voids have been included as an additional kinematics’ variable. This theory reduces to the classical theory of elasticity in the limiting case when void volume vanishes. Puri and Cowin [4] presented a complete analysis of the frequency equation for plane waves in a linear elastic material with voids and observed that there are two dispersive and attenuated dilatational waves in this theory, one is predominantly the dilatational wave of classical linear elasticity and other is predominantly a wave carrying a change in void volume fraction. Chandrasekharaiah [5] established the uniqueness of solution of an initial boundary value problem formulated completely in terms of stress and volume fraction fields for homogenous and isotropic materials with voids. Chandrasekharaiah [6] studied the propagation of plane waves in a homogenous and isotropic unbounded elastic solid with voids rotating with a uniform angular velocity. Moreover, the effect of voids on surface waves propagating in thermoelastic media has got its due importance where the situation so demands. The cooling and heating of the medium also results in the expansion and contraction of the voids along with the core material which contributes towards thermal stress and vibration developments in solids. Sharma and Kaur [7] studied the plane harmonic waves in generalized thermoelastic materials with voids and Sharma et al. [8] studied an exact free vibration analysis of simply supported, homogenous isotropic, cylindrical panel with voids in the three dimensional ⇑ Corresponding author. Tel.: +91 1972 254122; fax: +91 1972 223834. E-mail addresses: [email protected] (J.N. Sharma), [email protected] (D. Grover), [email protected] (D. Kaur). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.01.014
3397
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
generalized thermoelasticity. Singh and Tomar [9] explored the possibility of plane wave propagation in an infinite thermoelastic medium with voids using the theory developed by Iesan and investigated the reflection phenomenon of a set of coupled longitudinal waves from a free plane boundary of a thermoelastic half-space with voids. Auriault [10] and Sharma and Grover [11] studied the body wave propagation in an infinite homogenous isotropic elastic and thermoelastic media respectively rotating with uniform angular velocity. During the last three decades, non-classical theories, Lord and Shulman [12], Green and Lindsay [13] have been developing to alleviate the paradox inherited in the classical theory of heat conduction which predicts an infinite speed of heat transportation and contradicts the physical facts. Chandrasekharaiah [14] referred a wave-like thermal disturbance as ‘second sound’. Ackerman and Overtone [15] and Ackerman et al. [16] have supported the experimental exhibition of the actual occurrence of ‘second sound’ at low temperatures and small intervals of time in solid helium. In the present study, wave propagation in a homogenous isotropic, thermoelastic media with voids which is rotating with ~, about a fixed axis ~ uniform angular rotation X e3 , the wave propagation of a perturbation displacement in the plane ð~ e1 ; ~ e2 Þ perpendicular to ~ e3 may be affected by the Coriolis force. The wave propagation is investigated as a function of the Kibel number C ¼ x X . The quartic complex polynomial characteristic equation yields us four complex roots, in general. These roots can be associated with four dispersive waves namely quasi-longitudinal (QL), quasi-transverse (QT), volume fractional (/-mode) and thermal wave (T-mode). The general complex characteristics equation has been solved by using Descartes’ algorithm along with irreducible case of Cardano’s and series (perturbation) expansion methods with the help of DeMoivre’s theorem in order to obtain phase speeds, attenuation coefficients and specific loss factors of the waves. The numerical solution of the secular equations is carried out for magnesium crystal like material in order to illustrate the analytical developments. The computer simulated results have also been presented graphically for velocities, attenuation and phase velocity with respect to Kibel number. 2. Wave equations in rotating media We consider a homogenous isotropic, thermoelastic solid with voids at uniform temperature T0 and initial volume frac~. with respect to an tion /0 in the undisturbed state. The medium is assumed to be rotating with uniform angular velocity X inertial frame. The Coriolis effect is caused by the rotation of an object and the inertia of mass experiencing the effect. When Newton’s laws are transformed to a rotating frame of reference, the Coriolis and Centrifugal forces appear. Both forces are proportional to the mass of the object. The Coriolis force is proportional to the rotation rate. The Coriolis force acts in a direction perpendicular to the rotation axis and to the velocity of the body in the rotating frame and is proportional to the object’s speed in rotating frame. This force is termed as inertial force (pseudo force). The Coriolis force is quite small, only the horizontal component of Coriolis force is generally important. The basic governing dynamical equations of linear generalized thermoelastic interactions after including the Coriolis and centripetal forces, in absence of body forces and heat sources, Sharma and Kaur [7] and Sharma et al. [8] are
~ ~ ~ ~ ~ X lr2~ u þ br/ brT ¼ q ~ u þ2 X u_ ; u€ þ X u þ ðk þ lÞrr ~
ð1Þ
€ ar2 / n1 / n2 /_ br ~ u þ mT ¼ qv/;
ð2Þ
€ ; K r2 T qC e T_ þ t0 T€ ¼ bT 0 r ~ u_ þ t 0~ u€ þ mT 0 /_ þ t 0 /
ð3Þ
where t0 is the thermal relaxation time, ~ uðx1 ; x2 ; x3 ; tÞ ¼ ðu1 ; u2 ; u3 Þ is the displacement vector; T(x1, x2, x3, t) is the temperature change; k, l are Lamé’s parameters; K is thermal conductivity; q and Ce are respectively, the density and specific heat at constant strain; / is change in Volume fraction, b = (3k + 2l)aT, aT is the linear thermal expansion. a, b, n1, n2, m and v are material constants due to the presence of voids. Here superposed dot represents time differentiation. We define the non-dimensional quantities:
x0i ¼
x xi c1
;
t 00 ¼ x t0 ;
t 0 ¼ x t;
X0 ¼
a5 ¼
mc41 ; K vx3
x ¼
C e ðk þ 2lÞ ; K
X
x
u0i ¼
qx c1 ui
d2 ¼
;
n ¼ n2 x ; n1
bT 0 c22 ; c21 0
k ¼
/0 ¼
;
x2 v/ c21
;
T0 ¼
2
a1 ¼ c1 k
x
;
bc1 bT 0 vx
x0 ¼
2
;
a2 ¼
T ; T0
bvbT 0 ; aqc21
eT ¼
b2 T 0 ; qC e ðk þ 2lÞ
a3 ¼
n1 c21
ax2
;
a4 ¼
d21 ¼ mT 0 v
a
c23 ; c21 ;
ð4Þ
x c ; c0 ¼ ; x c1
where
c21 ¼
k þ 2l
q
;
c22 ¼
l a ; c23 ¼ qv q
Upon using quantities (4) in Eqs. (1)–(3), we obtain
~ X ~ ~ ~ ~ d2 r2~ uÞ þ a1 r/ rT ¼ ~ u€ þ X u þ2 X u_ ; u þ 1 d2 rðr ~
ð5Þ
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J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
r2 / a2 r ~ u a3 / þ n/_ þ a4 T ¼
€ / d21
ð6Þ
;
€ ¼ 0: u€ a5 /_ þ t0 / r2 T T_ þ t0 T€ eT r ~ u_ þ t0~
ð7Þ
Here primes have been suppressed for convenience. 3. Dispersion equation and its solution *
*
Let ð~ e1 ; ~ e2 ; ~ e3 Þ be the rotating orthonormal basis and we take X ¼ X~ u that is collinear to X is not afe3 . A perturbation ~ fected by Coriolis or convective accelerations. Therefore, we limit the analysis to displacements in the plane ð~ e1 ; ~ e2 Þ which remain constant in the direction ~ e3 . On applying the successive application of divergence and curl operators to Eq. (5) leads ~ as to a system of following two coupled differential equations for e and w
"
r2 þ X2
# @2 _ 3 þ a1 r2 / r2 T ¼ 0; e þ 2Xw @t 2
ð8Þ
! @2 d r þ X 2 w3 2Xe_ ¼ 0; @t 2
2
2
ð9Þ
~ ¼ r ~ where e ¼ r ~ u; w u ¼ w3~ e3 . Clearly the longitudinal, transverse, volume-fractional and thermal waves get coupled with each other here in contrast to wave propagation in an inertial medium. We consider waves that propagate in the direction ~ e1 of the form
ðe; w3 ; T; /Þ ¼ ðA1 ; A2 ; A3 ; A4 Þ exp fiðkx1 xtÞg;
i2 ¼ 1
ð10Þ
Upon using wave solution (10) in Eqs. (6)–(9) we obtain
h i 1 m2 ð1 þ C2 Þ A1 þ 2im2 C1 A2 A3 þ a1 A4 ¼ 0;
ð11Þ
h i 2im2 C1 A1 d2 m2 1 þ C2 A2 ¼ 0;
ð12Þ
" a2 m2 A1 a4 m2 A3 þ x2
m2
a3 n0
d1
x2
þ1þ 2
#
m2 A4 ¼ 0;
ð13Þ
eT m2 s0 A1 ð1 s0 m2 ÞA3 þ a5 m2 s0 A4 ¼ 0;
ð14Þ
n; v ¼ xk Here C ¼ x where s0 = ix + t0, n0 ¼ 1 ix X is called Kibel number. The condition for existence of non-trivial solutions forA1, A2, A3 and A4 of system of Eqs. (11)–(14) yields the dispersion equation 1
4 Y
1 m2 n2i ¼ 0:
ð15Þ
i¼1
Here n2i ; i ¼ 1; 2; 3; 4 are the roots of the quartic equation
n8 Ln6 þ Mn4 Nn2 þ P ¼ 0;
ð16Þ
where
L ¼ s0 þ
M¼
1 d21
a3 n0
x
ð1 C2 Þ2 d
2
þ C2 Þ
2
þ
s0 þ
ð1 þ d2 Þð1 þ C2 Þ
þ
d2 1
d21
s0 d2
þ
a3 n0
x
eT s0 d2
!
2
þ
þ
a3 e12
x2
þ eT s0 ;
s0 ð1 þ eT Þ þ ð1 þ C2 Þ 1 þ
a3 e12
x2 d 2
;
1 2
d
þ
a3 s0
x2
ðe12 þ e25 e45 þ e14 eT Þ þ ð1
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
s0 ð1 C2 Þ2
N¼
d2 þ
P¼
þ
a3 n0 2 2 x d1 1
!( ð1 C2 Þ2 d2
) 1 eT s0 ð1 þ C2 Þ þ s0 ð1 þ C2 Þ 1 þ 2 þ d d2
ð1 þ C2 Þa3 s0
s0 ð1 C2 Þ2 1
e12 ¼
where
e25 ð1 þ d2 Þe45 þ e12 þ e14 eT ;
x2 d 2 d2
3399
d21
a3 n0
a1 a2 ; a3
x2
e14 ¼
a3 e45
x2
a1 a4 ; a3
! ð17Þ
;
e25 ¼
a2 a5 ; a3
e45 ¼
a4 a5 : a3
ð18Þ
In general, the Eq. (16) is a quartic equation in n2 with complex coefficients and there is no general arithmetic or algebraic method of finding the exact value of the roots of imaginary quantities. Therefore this equation can be solved by Descartes’ algorithm along with irreducible case of Cardano’s method with help of De Moivre’s theorem to obtain its four roots in order to know the effect of rotation on various characteristics of waves which are significantly influenced by volume fraction and thermal fields. This gives us four pairs of values of complex phase velocity and hence four distinct types of waves are possible to propagate in such materials. Thus there are, a dilatational, an equivoluminal or a shear, a volume fraction (/-mode) and a thermal (T-mode) wave which get modified due to rotation and are possible to propagate in the thermo-elastic materials with voids rotating about an axis normal to its plane. Upon obtaining the complex roots n2i ði ¼ 1; 2; 3; 4Þ of Eq. (16), we get
mj ¼
1 nj
j ¼ 1; 2; 3; 4:
ð19Þ
The four pairs of the values of complex phase velocity in Eq. (19) give us four distinct types of attenuated and dispersive waves which are possible to propagate in thermoelastic materials with voids rotating about an axis normal to its plane. We write 1 v 1 ¼ V 1 j j þ ix Q j ;
j ¼ 1; 2; 3; 4
ð20Þ
so that kj = Rj + iQj and Rj ¼ Vxj , where Vj and Qj are real quantities. The exponent in the plane wave solution (10) becomes Qjx + iRj(x Vjt). This shows that Vj is the propagation speed and Qj is the attenuation coefficient of the waves. Upon using representation (20) in Eq. (19), we obtain
Vj ¼
1 ; Reðnj Þ
Q j ¼ xImðnj Þ;
j ¼ 1; 2; 3; 4
ð21Þ
The four distinct real values of phase velocity Vj j = 1, 2, 3, 4 corresponds to four distinct waves namely quasi-longitudinal (QL), quasi-transverse (QT), volume fractional ð/ modeÞ and thermal (T-mode) waves which are possible to propagate in such materials. These waves are attenuated in space having attenuation coefficients Qj j = 1, 2, 3, 4 and also get modified due to rotation as well as volume-fractional and thermal variations. In case X – 0, the QL-wave, QT-wave, / mode and T-mode are coupled dilatational-shear, dilatational-volume fractional and dilatational-thermal waves. The coupling is measured by the amplitude ratios
2iC1 m2i A2 ; i ¼ 1; 2; 3; 4; ¼ 2 A1 i d m2i ð1 þ C2 Þ h n oi a3 e25 m2i þ eT x2 d12 ax3 n20 m2i 1 s0 m2i A3 1 n o; ¼ A1 i a3 e45 s0 m4 þ 1 s0 m2 x2 1 a3 n0 m2 1 2 2 i i i x d1 a4 eT s0 m2i þ a2 1 s0 m2i m2i A4 n o; ¼ A1 i a3 e45 s0 m4 þ 1 s0 m2 x2 1 a3 n0 m2 1 2 2 i i i x d
i ¼ 1; 2; 3; 4;
ð22Þ
i ¼ 1; 2; 3; 4:
1
4. Perturbation solution of dispersive waves As we know, there is no direct method to solve complex quartic equations except Descartes’ algorithm along with irreducible case of Cardano’s method available in the literature, however, series approximation (perturbation) method has been widely used by authors [11,17,18] to study the wave propagation problems in classical (coupled) and non-classical (generalized) thermoelastic continua. Therefore, our aim is to solve the instant problem with Descartes’ algorithm and approximation (perturbation) method in order to bring out the comparison and check the suitability of these methods to compute various characteristics of the waves under consideration. The modified secular Eq. (16) can also be rewritten as
f ðn2 Þ eT
s0 þ
a3 e214
x2 e45
gðn2 Þ ¼ 0;
ð23Þ
3400
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
where
f ðn2 Þ ¼ ðn2 k1 Þðn2 k2 Þðn2 k3 Þðn2 k4 Þ; gðn2 Þ ¼ ðn2 k^1 Þðn2 ^k2 Þðn2 ^k3 Þ;
ð24Þ
2 2
k2 ¼
k1 ;
k3 ¼ s0 ; X
1þC 2d2
k4 ¼
^k1 ¼ M 2 ; L2
L2 ¼ s0 þ
N2 ¼
P2 ¼
1 d21
3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u 2 u 6 7 t 2 2 2 2 1C 41 þ d ð1 þ d Þ 4d 5; 1 þ C2 þ
X
a3 e214
x2 e45
a3 n0
x2
;
M 2 ¼ s0
;
s0 ð1 þ C2 Þ d2
X
^k1 ^k2 ¼ N2 ; L2
!(
1 d21
1
a3 n0
d1
x2
2
^k1 ^k2 ^k3 ¼ P2 ; L2
a3 n0
!
x2
þ
! þ
a3 e14
x2
1 þ C2 d2
!
s0 þ
a3 e214
x2 e45
þ
s0 a3 e14 e14 e14 ; 2þ x2 e45 e12
) e14 2 e14 ; 2þ ð1 þ d Þ
e45
e12
s0 a3 e214 ð1 C2 Þ2 : x2 d2 e12
ð25Þ
For most of the materials the thermo-mechanical coupling parameter (eT) is very small and therefore, we develop series expansions in terms of eT for the roots n2i ði ¼ 1; 2; 3; 4Þ of the Eq. (23) in order to explore the effect of various interacting fields on the waves. Thus, we obtain
2 n21 ðeT Þ ¼ k1 41 þ 2
e ¼ k2 41 þ
n22 ð T Þ
2 n23 ðeT Þ ¼ k3 41 þ 2
e ¼ k4 41 þ
n24 ð T Þ
L2 k1 ^k1 k1 ^k2 k1 ^k3 k1 ðk1 k2 Þðk1 k3 Þðk1 k4 Þ L2 k2 ^k1 k2 ^k2 k2 ^k3 k2 ðk2 k1 Þðk2 k3 Þðk2 k4 Þ L2 k3 ^k1 k3 ^k2 k3 ^k3 k3 ðk3 k1 Þðk3 k2 Þðk3 k4 Þ L2 k4 ^k1 k4 ^k2 k4 ^k3 k4 ðk4 k1 Þðk4 k2 Þðk4 k3 Þ
3
eT þ 5; 3
eT þ 5; 3
ð26Þ
eT þ 5; 3
eT þ 5:
Upon using the representation (18), the corresponding phase velocities and attenuation coefficients are obtained as
1 wi V i ¼ pffiffiffiffi sec ; ri 2
pffiffiffiffi wi Q i ¼ x r i sin ; 2
i ¼ 1; 2; 3; 4;
ð27Þ
where
ri ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2i þ S2i ;
wi ¼ tan1
Si ; Ri
i ¼ 1; 2; 3; 4:
ð28Þ
The quantities Ri and Si are defined in Appendix. The phase velocity and attenuation coefficients of various modes of wave propagation in case of coupled theory of thermoelasticity can be obtained from (27) and (28) by setting t0 = 0. 5. Specific loss The specific loss is the most direct method of defining internal friction for a material. The specific loss is the ratio of the amount of energy (DE) dissipated in a specimen through a stress cycle to the elastic energy (E) stored in that specimen at maximum strain. According to Kolsky [19] in case of sinusoidal plane wave of small amplitude the specific loss equals 4p times the absolute value of imaginary part of k to the real part of k. Therefore, we have
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
DE ImðkÞ ¼ 4p V i Q i ; SLi ¼ ¼ 4p x E ReðkÞ
i ¼ 1; 2; 3; 4:
3401
ð29Þ
6. Special cases 6.1. In absence of voids (thermoelastic media) In the absence of voids effect (m=0 = b) the system of Eq. (17) reduces to
Rn21 ¼ s0 ð1 þ eT Þ þ Rn21 n22 ¼ n21 n22 n23 n24 ¼
¼
1 d21
ð1 þ d2 Þð1 þ C2 Þ
ð1 C2 Þ2 d
2
d2 þ
s0 ð1 C2 Þ2 d2
a3 n0
x2
;
s0 ð1 þ C2 Þð1 þ d2 þ eT Þ d2
; ð30Þ
;
:
The root n24 is not influenced by rotation and remains independent of rest of the motion. The discussion of remaining roots n2i ði ¼ 1; 2; 3Þ has been already done by Sharma and Grover [11] in case of generalized thermoelastic rotating body. 6.2. In absence of thermal effect (elastic media with voids) In the absence of voids effect (m = 0 = b) the system of Eq. (17) reduces to
Rn21 ¼ R
n21 n22
1 d21 ¼
a3 n0
þ
x2
ð1 þ d2 Þð1 þ C2 Þ
ð1 C2 Þ2 d2
d2 þ
2 2
n21 n22 n24 ¼
ð1 þ C2 Þ
"
d2
ð1 C Þ
1
d2
d21
þ
1 d21
a3 n0
a3 e12
x2 a3 n0
x2
; ! 2
ð1 þ d Þ þ
a3 e12
x2
# ;
!
x2
ð31Þ
;
n23 ¼ s0 : Clearly, the root n23 is not influenced by rest of the motion. 6.3. Elastic media If we ignore thermal effect in Eq. (30) and voids effect in Eq. (31) above
Rn21 ¼
ð1 þ d2 Þð1 þ C2 Þ
n23 ¼ s0 ;
2
d n24 ¼
1
a3 n0
d1
x2
2
Rn21 n22 ¼
;
ð1 C2 Þ2 d2
ð32Þ
Clearly, the root n23 and n24 are not influenced by rest of the motion and we obtained the results for elastic media by Auriault [10]. 7. Numerical results and discussion In this section, we perform some numerical calculations in order to illustrate the analytical results for magnesium like material whose physical data, Eringen [20], Puri and Cowin [4], Kumar and Rani [21], Sharma and Kaur [7], is given as:
q ¼ 1:74 103 kg m3 ; k ¼ 2:17 1010 N m2 ; l ¼ 1:639 1010 N m2 ; C e ¼ 1040 J kg1 deg1 1
K ¼ 170 Wm1 deg ;
1
b ¼ 2:68 106 N m2 deg ; 1
m ¼ 2:0 106 N m2 deg b ¼ 1:13849 1010 N m2 ;
t 0 ¼ 0:05 s; X ¼ 4 rps; a ¼ 3:688 105 N;
v ¼ 1:753 1015 m2 :
3402
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
Fig. 1. Velocities in elastic medium versus Kibel number.
ph Fig. 1 shows the phase velocity profiles of cph 1 and c2 versus Kibel number for magnesium like material. In case of uncoupled thermoelasticity (elastic medium) we note that the dimensionless phase velocity of classical longitudinal wave shows resonance at the point C = 1(i.e. x = X). The profile of phase velocity (cph 1 Þ increases sharply in the range 0 < C < 1 to become asymptotic to its resonant value at C = 1 and for C > 1, it decreases to become asymptotically close to unity at large Kibel increases with increasing Kibel number to ultimately become close to number. We observe that the phase velocity cph 2 ph its the limiting value d as Kibel number tends to infinity. The profiles of cph 1 and c 2 and their variations agree with those ph of Auriault [10] except for c1 in the Kibel number range 0 6 C < 1. In Figs. 2 and 3, we observe that the phase velocity of QL wave slightly depends on the elastic property of the medium. An important fact is that QL-wave shows resonance at the point C = 1(i.e. x = X), where velocity tends to infinity. It increases in the Kibel number range 0 6 C < 1, decreases sharply for 1 < C 6 3 and then becomes steady and stable C P 3 to attain value unity in a dispersion less manner. Also, we observe that, the phase velocity of QT-wave increases with increasing Kibel number and becomes steady to attain the limiting value d at large values of Kibel number. The velocity of the volume fractional wave (/-mode) increases in 0 6 C < 1 and becomes steady for large value of Kibel number and attains the limiting value 0.62. From Fig. 3, we observe that the velocity of thermal wave (T-mode) also increases with increasing Kibel number and becomes steady to attain the limiting value 4.2 at large values of Kibel number. Fig. 4 reveals that the attenuation coefficient profile of QL-wave increases sharply and tends to infinity in the Kibel number range 0 < C < 0.5 and then decreases in the Kibel number range 0.5 < C < 1 and becomes steady for large value of Kibel number. The attenuation of the QT-wave has quite small magnitude as compare to QL-wave and volume fractional wave (/-mode). The attenuation coefficient of volume fractional wave (/-mode) has maximum value at C = 0, and decreases monotonically to become steady and stable at extreme large Kibel number. Here, the actual values of attenuations of volume-fractional 1000 times of magnitude as actually considered in graphical interpretation. Fig. 5 shows that the attenuation of waves interlace as Q1 < Q2 < Q4 < Q3 with the exceptions that Q2 < Q3 < Q1 and Q2 < Q1 in the ranges 0 < C < 0.5 and C P 1.5, respectively. The attenuation of thermal wave increases in 0 < C 6 6. The value of Q3 approaches 0.021 at C = 6. The attenuation of the volume-fractional wave increases for 0 < C < 1.5 and then it becomes steady thereafter to attain its limiting
Fig. 2. Wave velocity in elastic medium with voids versus Kibel number.
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
3403
Fig. 3. Wave velocity in thermoelastic medium with voids versus Kibel number.
Fig. 4. Attenuation in elastic medium with voids versus Kibel number.
Fig. 5. Attenuation in thermoelastic medium with voids versus Kibel number.
value 0.01. The attenuation coefficient profile of QL-wave increases in range 0 < C < 0.5, decreases for 0.5 < C < 1 to become close to zero at C = 1. Then it observes slight increase in 1 < C < 1.5 and then becomes steady thereafter. The attenuation of
3404
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
Fig. 6. Specific loss factor in elastic media with voids versus Kibel number.
Fig. 7. Specific Loss in thermoelastic medium with voids versus Kibel number.
the QT-wave observes Gaussian like behaviour in the range 0.5 < C < 1.5 and has maximum value at C = 1 and almost vanishes elsewhere. Fig. 6, shows that the specific loss factor of QL-wave observes the Gaussian like behaviour in the Kibel number range 0 6 C 6 1 and attains value zero for entire values of Kibel number. The specific loss factor of QT-wave has small increase in the range 0 6 C 6 0.5 to attain its maximum value at C = 0.5 and it becomes steady after C P 0.5. The specific loss factor of volume-fractional wave (/-mode) have maximum value at C = 0 and decreases sharply in the Kibel number range 0 < C < 0.5 and having value extremely small for large value of Kibel number. Fig. 7 reveals that the specific loss factors of QL-wave and QT-waves have quite small magnitude as compare to that of thermal wave. The specific loss factor of volume-fractional wave (/-mode) exhibits Gaussian like behaviour in the range 0 6 C 6 1.5 which decreases exponentially for C P 1.5 to becomes steady there after. The specific loss factor of thermal wave has maximum value at C = 0, and decreases monotonically to become steady and stable at extreme large Kibel number. Here, the actual values of attenuations and specific loss factors of volume-fractional and thermal waves are respectively, 10 and 100 times of their magnitudes as actually considered in graphical interpretation for distinction purpose.
8. Statistical analysis In order to explore which method (Descartes’ or perturbation) gives better results in respect of different characteristics of the waves, we have performed the statistical analysis here. We employed student’s t-test, Gupta and Kapoor [22] at three
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
3405
levels of significance (1%, 5% and 10%) to the data obtained from our FORTRAN coded program in case of phase velocities (Vi), attenuations (Qi) and specific loss factors (SLi) with the help of both Descartes’ and perturbation methods. Null hypothesis
HV0 : lV1 ¼ lV2 ;
HQ0 : lQ1 ¼ lQ2 ;
SL SL HSL 0 : l1 ¼ l2
where
lVi ; lQi ; lSL i ði ¼ 1; 2Þ;
denote mean values of the data corresponding to phase velocity, attenuation coefficient and specific loss factor respectively. Here the subscript values i = 1 and i = 2 refer to perturbation and Descartes’ method computations of the respective quantity. The considered degree of freedom = n1 + n2 2 = 11 + 11 2 = 20 and tabulated values of statistic (t) at 1%, 5% and 10%, levels of significance are ttab. = 2.528, ttab. = 1.725 and ttab. = 1.325, respectively. The corresponding calculated values of t statistic in respect of phase velocities (Vi), attenuations (Qi) and specific loss (SLi), i = 1, 2, 3, 4 are obtained as (6.1324, 4.3911, 14.090, 2.5528), (3400.69, 7.4875, 3.2479, 10.8674) and (14.0484, 34.2048, 7.0429, 1.5346) respectively. From the above analysis it is observed that at three considered levels of significance namely, 1%, 5% and 10%, the null hypotheses, Q Q V V HV0 ; HQ0 and HSL 0 are rejected except for SL4 at 1% and 5% level of significance. Moreover, the mean values l1 ; l2 ; l1 ; l2 SL SL and l1 ; l2 in respect of phase velocity, attenuation coefficient and specific loss factor obey the inequalities SL lV1 i > lV2 i ; i ¼ 1; 3; lV1 i < lV2 i ; i ¼ 2; 4; lQ2 > lQ1 and lSL 2 > l1 , respectively. Thus, it is concluded that perturbation method
is a better choice than Descartes’ method for computation of the phase velocities V1 and V3. However, the latter is well suited as compare to the former one for computing attenuation coefficients and specific loss factors of waves in addition to phase velocities V2 and V4. In case of SL4 null hypothesis is accepted and hence from the coefficient of variations (CV), we conclude that Descartes’ method is better choice than perturbation method. 9. Conclusions It is found that free wave propagation in a homogenous isotropic rotating thermoelastic media with voids gives rise to four waves namely, quasi-longitudinal (QL) wave, quasi-transverse (QT) wave, volume fractional (/-mode) and thermal wave (T-mode). At isentropic (low-frequency) conditions thermal wave does not exist while at isothermal (high-frequency) conditions it becomes diffusive in character. These waves are significantly affected due to rotation, voids and thermal variations. The modified QL and QT-waves approaches their classical counterparts in elastokinetics in the absence of the volumefraction and thermal fields. The dimensionless phase velocity cph 1 ðorV 1 Þ has finite value for all value of Kibel number except at unity, where it suffers resonance. The statistical analysis revealed that the Descartes’ method is a better choice than series method for the computations of phase velocities V2 and V4, attenuation coefficients and specific loss factor of the waves. However, the latter is well suited for the computations of phase velocities V1 and V3 in case of complex secular equations governing the wave motion. Appendix A These quantities are used in Eq. (28) and given by
R1 ¼ k1 þ
" # a3 e45 ðk1 k3 Þðk1 k4 Þ t 0 ReðD1 Þ þ x1 ImðD1 Þ ; x2 ðk1 k2 Þ ðReðD1 ÞÞ2 þ ðImðD1 ÞÞ2
" # a3 e45 ðk2 k3 Þðk2 k4 Þ t 0 ReðD2 Þ þ x1 ImðD2 Þ ; R2 ¼ k2 þ x2 ðk2 k1 Þ ðReðD2 ÞÞ2 þ ðImðD2 ÞÞ2 " # a3 e45 ReðN3 ÞReðD3 Þ þ ImðN3 ÞImðD3 Þ R3 ¼ t 0 þ ; x2 ðReðD3 ÞÞ2 þ ðImðD3 ÞÞ2 R4 ¼
1
a3
d1
x
2
þ 2
" # a3 e45 ReðN4 ÞReðD4 Þ þ ImðN4 ÞImðD4 Þ ; x2 ðReðD4 ÞÞ2 þ ðImðD4 ÞÞ2
" # a3 e45 ðk1 k3 Þðk1 k4 Þ x1 ReðD1 Þ t0 ImðD1 Þ S1 ¼ ; x2 ðk1 k2 Þ ðReðD1 ÞÞ2 þ ðImðD1 ÞÞ2 " # a3 e45 ðk2 k3 Þðk2 k4 Þ x1 ReðD2 Þ t0 ImðD2 Þ ; S2 ¼ x2 ðk2 k1 Þ ðReðD2 ÞÞ2 þ ðImðD2 ÞÞ2
3406
J.N. Sharma et al. / Applied Mathematical Modelling 35 (2011) 3396–3407
" # a3 e45 ImðN3 ÞReðD3 Þ ReðN3 ÞImðD3 Þ ; x2 ðReðD3 ÞÞ2 þ ðImðD3 ÞÞ2
S3 ¼ x1 þ
" ( )# 1 ImðN 4 ÞReðD4 Þ ReðN 4 ÞImðD4 Þ ; S4 ¼ a3 x n þ e45 x ðReðD4 ÞÞ2 þ ðImðD4 ÞÞ2 1
ReðD1 Þ ¼ ðk1 t 0 Þ k1
ReðD3 Þ ¼
ReðD4 Þ ¼
1
t0
d21
1
a3 þ d21 x2
! a3
þ
x2
1
a3 t0 d21 x2
!
a3 n
x
2
ReðD2 Þ ¼ ðk2 t0 Þ k2
;
1
a3
d1
x
þ 2
!
2
a3 n
x2
;
ðt 0 k1 Þðt 0 k2 Þ x2 x2 ð1 a3 nÞð2t 0 k1 k2 Þ;
!(
1
a3
d1
x
2
!
k1 2
1
a3
d1
x
2
!
) x2 a23 n2
k2 2
( þ x2 a3 nð1 a3 nÞ 2
1
a3 d21 x2
!
) k1 k2 ;
ReðN3 Þ ¼ t 0 ðt 0 k3 Þðt0 k4 Þ x2 x2 ð2t0 k3 k4 Þ; ( ReðN4 Þ ¼ t 0
1
a3
d1
x
2
1
ImðD1 Þ ¼ x
k1
ImðD2 Þ ¼ x1 k2
!
k3 2 1 d21
þ
1
a3 k4 d21 x2
!
) x2 a23 n2
( x2 a3 n 2
1
a3
d1
x2
2
! þ na3 ðk1 t 0 Þ ; 2
x
! a3 þ þ na ðk t Þ ; 3 2 0 d21 x2 1
1 a3 ð1 a3 nÞ ðt 0 k1 Þðt 0 k2 Þ x2 þ ð2t 0 k1 k2 Þ t 0 2 þ 2 d1 x
1
) k3 k4 ;
a3
" ImðD3 Þ ¼ x
!
!# ;
" ImðD4 Þ ¼ x
1
ImðN 3 Þ ¼ x1
( ! ) ! ( ! ! )# 1 a3 1 a3 1 a3 1 a3 2 2 2 a3 n 2 2 2 k1 k2 t 0 1 a3 n k1 k2 x a3 n ; d1 x d21 x2 d21 x2 d21 x2
ðt 0 k3 Þðt 0 k4 Þ x2 þ t 0 ð2t0 k3 k4 Þ ;
"( ImðN 4 Þ ¼ x1
1
a3 k3 d21 x2
!
1
a3 k4 d21 x2
!
) x2 a23 n2
( þ t 0 a3 n 2
1
a3 d21 x2
!
)# k3 k4
:
ðA:1Þ
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