Vandana

JOURNAL OF THERMOELASTICITY ISSN VOL.1 NO. 1 MARCH 2013

21

Plane wave propagation in an anisotropic

thermoelastic medium with fractional order

derivative and void

Rajneesh Kumar and Vandana Gupta

AbstractThe present paper deals with the study of plane wave propagation in anisotropic thermoelastic medium having

fractional order derivative and void in the context of the theory

of three-phase-lag model and two-phase-lag model of

thermoelasticity. It is found that there exist quasi-longitudinal

wave qP1 , quasi-longitudinal thermal wave qP2 quasi-

longitudinal volume fractional wave qP3 and two quasi-

transverse waves (qS1 , qS2). The governing equations for

homogeneous transversely isotropic three-phase-lag are reduced

as a special case. It is noticed that when plane waves propagate in

one of the planes of transversely isotropic thermoelastic solid

having fractional order derivative and void, one purely quasi-

transverse wave decouples from the rest of the motion and is not

affected by the thermal and void vibrations. On the other hand,

when plane waves propagate along the axis of the solid, two

quasi-transverse wave modes qS1 , qS2 decouple from the rest of

the motion and are not affected by the thermal and void

vibrations. From the obtained results the different characteristics

of waves like phase velocity, attenuation coefficient, specific loss

and penetration depth are computed numerically and presented

graphically.

Index terms Anisotropic, Fractional calculus, Plane wave, transversely isotropic.

I. INTRODUCTION

The study of dynamic properties of elastic solids is

significant in the ultrasonic inspection of materials, vibrations

of engineering structures, in seismology, geophysical and

various other fields. Such materials are usually described by

equations of linear elastic solids; however there are materials

of a more complex microstructure (composite materials,

granular materials, soils etc.) depict specific characteristic

response to applied load.

There are a number of theories which describe mechanical

properties of porous materials and one of them is a Biot

consolidation theory of fluid-saturated porous solids [1, 2].

These theories reduce to classical elasticity when the pore

fluid is absent. Goodman and Cowin [3] established a

continuum theory for granular materials, whose matrix

material (or skeletal) is elastic and interstices are voids.

R. Kumar, Department of Mathematics, Kurukshetra University, Kurukshetra-

136119, India (E-mail [email protected])

V. Gupta Department of Mathematics, Kurukshetra University, Kurukshetra-136119, India (E-mail: [email protected])

They formulated this theory from the formal arguments

of continuum mechanics and introduced the concept of

distributed theory, which represents a continuum model for

granular materials (sand, grain, powder etc.) as well as porous

materials (rock, soil, sponge, pressed powder, cork etc.).

The basic concept underlying this theory is that the

bulk density of the material is written as the product of two

fields, the density field of the matrix material and the volume

fraction field (the ratio of the volume occupied by grains to the

bulk volume at a point of the material). This representation of

the bulk density of the material introduces an additional

kinematic variable in the theory. This idea of such

representation of the bulk density was employed by Nunziato

and Cowin [4] to develop a nonlinear theory of elastic material

with voids. They developed the constitutive equations for solid

like material which are nonconductor of heat and discussed the

restrictions imposed on these constitutive equations by

thermodynamics. They showed that the change in the volume

fraction causes an internal dissipation in the material which is

similar to that associated with viscoelastic materials. They also

considered the dynamic response and derived the general

propagation conditions on acceleration waves.

Later on Cowin and Nunziato [5] developed a theory of linear

elastic materials with voids for the mathematical study of the

mechanical behavior of porous solids. They considered several

application of the linear theory by investigating the response

of the materials to homogeneous deformations, pure bending

of beam and small amplitudes of acoustic waves. The small

acoustic waves in an infinite elastic medium with voids

showed that two distinct types of longitudinal waves and a

transverse wave can propagate without affecting the porosity

of the material and without attenuation. The two types of

longitudinal waves are attenuated and dispersed; one

longitudinal wave is associated with elastic property of the

material and the second associated with the property of the

change in porosity of the material. These longitudinal acoustic


22

waves are both attenuated and dispersed due to the change in

the material porosity.

The generalized theory of thermoelasticity is one of the

modified versions of classical uncoupled and coupled theory

of thermoelasticity and has been developed in order to remove

the paradox of physical impossible phenomena of infinite

velocity of thermal signals in the classical coupled

thermoelasticity. Hetnarski and Ignaczak [6] examined five

generalizations of the coupled theory of thermoelasticity.

The first generalization is due to Lord and Shulman [7]

who formulated the generalized thermoelasticity theory

involving one thermal relaxation time.

The second generalization is given by Green and

Lindsay [8], they developed a temperature rate-dependent

thermoelasticity that includes two thermal relaxation times.

One can refer to Hetnarski and Ignaczak [9] for a review and

presentation of generalized theories of thermoelasticity.

Chadwick and Sheet [10] and Chadwick [11] discussed

propagation of plane harmonic waves in transversely isotropic

and homogeneous anisotropic heat conduction solids

respectively. Banerjee and Pao [12] studied the thermoelastic

waves in anisotropic solids. Sharma [13] discussed the

existence of longitudinal and transverse waves in anisotropic

thermoelastic media.

The third generalization of the coupled theory of

thermoelasticity is developed by Hetnarski and Ignaczak and

is known as low temperature thermoelasticity. The fourth

generalization to the coupled theory of thermoelasticity

introduced by Green and Nagdhi and this theory is concerned

with the thermoelasticity theory without energy dissipation.

The fifth generalization to the coupled theory of

thermoelasticity is developed by Tzou [14] and

Chandrasekhariah [15] and is referred to dual phase-lag

thermoelasticity. Tzou[14] proposed a generalized heat

conduction law, referred as heat conduction law with dual-

phase-lags, in which microstructural effects in the heat transfer

mechanism have been considered in the macroscopic

formulation by taking into account that photon-electron

interactions on the macroscopic level causes a delay in the

increase of the lattice temperature. A corresponding

thermoelastic model with two phase lag was reported by

Chandrasekharaiah [15]. In the models [14, 15], two different

phase lags i.e., one for the heat flux vector and other for the

temperature gradient have been introduced in the Fouriers law. The phase-lag of heat flux vector is interpreted as the

relaxation time due to fast transient effects of thermal inertia

and the phase-lag of temperature gradient is interpreted as the

delay time caused due to the microstructural interactions, a

small scale effect of heat transport in space, such as photon-

electron interaction or photon scattering. One dimensional

thermoelastic wave propagation in an elastic half-space in the

context of dual phase model was studied by Roychoudhary

[16].The stability of the three-phase-lag heat conduction

equation is discussed by Quintanilla and Racke [17].

Quintanilla has studied the spatial behavior of solutions of the

three-phase-lag heat conduction equation.

During recent years, several interesting models have

been developed by using fractional calculus to study the

physical processes particularly in the area of heat conduction,

diffusion, viscoelasticity, mechanics of solids, control theory,

electricity etc. It has been realized that the use of fractional

order derivatives and integrals leads to the formulation of

certain physical problems which is more economical and

useful than the classical approach. The first application of

fractional derivatives was given by Abel [18] who applied

fractional calculus in the solution of an integral equation that

arises in the formulation of the tautochrone problem.

Caputo[19] gave the definition of fractional derivatives of

order (0,1] of absolutely continuous function. Caputo and Mainardi [20,21] and Caputo[22] found good agreement with

experimental results when using fractional derivatives for

description of viscoelastic materials and established the

connection between fractional derivatives and the theory of

linear viscoelasticity.

Oldham and Spanier[23] studied the fractional calculus

and proved the generalization of the concept of derivative and

integral to a non-integer order. A theoretical basis for the

application of fractional calculus to viscoelasticity was given

by Bagley and Torvik[24]. Applications of fractional calculus

to the theory of viscoelasticity was given by Koeller[25].

Kochubei[26] studied the problem of fractional order

diffusion. Rossikhin and Shitikova[27] presented applications

of fractional calculus to various problems of mechanics of

solids. Gorenflo and Mainardi[28] discussed the integral

differential equations of fractional orders, fractals and

fractional calculus in continuum mechanics.

Mainardi and Gorenflo[29] investigated the problem of

Mittag-Leffler-type function in fractional evolution process.

Povstenko[30] proposed a quasi-static uncoupled theory of

thermoelasticity based on the heat conduction equation with a

time-fractional derivative of order . Because the heat

conduction equation in the case 12 interpolates the parabolic equation (=1) and the wave equation (=2), this theory interpolates a classical thermoelasticity and a

thermoelasticity without energy dissipation. He also obtained

the stresses corresponding to the fundamental solutions of a

cauchy problem for the fractional heat conduction equation for

one-dimensional and two-dimensional cases.

Povstenko[31] investigated the nonlocal generalizations

of the Fourier law and heat conduction by using time and

space fractional derivatives. Youssef[32] proposed a new

model of thermoelasticity theory in the context of a new

consideration of heat conduction with fractional order and

proved the uniqueness theorem. Jiang and Xu[33] obtained a

fractional heat conduction equation with a time fractional

derivative in the general orthogonal curvilinear coordinate and

also in other orthogonal coordinate system. Povstenko[34]

investigated the fractional radial heat conduction in an infinite


23

medium with a cylindrical cavity and associated thermal

stresses.

Ezzat [35] constructed a new model of the magneto-

thermoelasticity theory in the context of a new consideration

of heat conduction with fractional derivative. Ezzat[36]

studied the problem of state space approach to thermoelectric

fluid with fractional order heat transfer. The Laplace transform

and state-space techniques were used to solve a one-

dimensional application for a conducting half space of

thermoelectric elastic material. Povstenko[37] investigated the

generalized Cattaneo-type equations with time fractional

derivatives and formulated the theory of thermal stresses.

Biswas and Sen[38] proposed a scheme for optimal control

and a pseudo state space representation for a particular type of

fractional dynamical equation.

In the present investigation, we studied the propagation

of plane waves in the context of three-phase-lag and two-

phase-lag model of thermoelasticity with fractional order

derivative and void, for anisotropic thermoelastic medium. As

a special case the basic equations for homogeneous

transversely isotropic thermoelastic three-phase-lag with

fractional order derivative and void are reduced. The

numerical results for the different characteristics of waves like

phase velocity, attenuation coefficient, specific loss and

penetration depth are computed numerically and presented

graphically.

II. FUNDAMENTAL EQUATIONS

Following Ezzat, El-Karamany and Fayik [39],

Ciarletta and scalia [40] the basic equations of homogeneous,

anisotropic diffusive generalized thermoelastic with fractional

order derivative and three-phase-lag model in the absence of

body forces and heat sources are

Equation of motion

ij ijkl kl ij ij c e B T (1)

0 E ij 0 ij 0ST C T T e bT (2) Equations of motion in the absence of body force

ij, j i u (3) The energy equation (without extrinsic heat supply) is

0 i,iST q (4)

The Fourier law (for thermoelastic three-phase-lag model) is

given as

t i ij , j ij , j

q K 1 T K 1 T

t t

(5)

Balance of equilibrated forces

ij , j ij iji A B e bT (6)

The general system of equations for anisotropic material are

obtained by using equation (1) in equation (3) and equations

(2) and (5) in equation (4), the equation of motion and heat

conduction are

Equations of motion

ijkl kl, j ij , j ij , j ic e B T u (7)

Equation of heat conduction

t

ij , ji ij , ji

2 2

q q

E ij 0 ij 0 2

K 1 T K 1 T

t t

1 C T T e bT

2t t

(8)

where

ijkm kmij ijkm ijmkc c c c are elastic parameters, iu are components of displacement vector, ij are the tensor of

thermal respectively, 0T is the reference temperature such that

0

T1

T is the density and

EC is the specific heat at

constant strain, ij ji ij ji ij ji ij K K K K e are,

respectively, the components of stress, thermal conductivity,

material constant characteristic of the theory and strain tensor,

1 2 3T x x x t is the temperature distribution from the

reference temperature T0, T and q are respectively, the

phase lag of the heat flux, the phase lag of the temperature

gradient and the phase lag of the thermal displacement, is the

fractional order derivative, is the volume fraction field, b is

the measure of diffusion effects, Aij , Bij and are void

material parameters, is the equilibrated inertia.

In all the above equations, a comma (,) followed by a

suffix denotes differentiation with respect to spatial coordinate

and a superposed dot (.) denotes the derivative with respect to

time.

For Two-Phase-Lag Model ijK 0

We define the dimensionless quantities: 2 2

i 0 i i 0 i 0 q 0 q

2 2 2 2

t 0 t 0 0

0

12 66 33 44E

1 2

11 11

66 44

3 4

11 11

x C x u C u t C t C

T C C C T

T

c c c cC

K c c

c c

c c

(9)

Here 0C is the longitudinal wave velocity in the isotropic

version of the medium.

III. SOLUTION OF THE PROBLEM


24

Upon introducing the quantities defined in equation

(9), in equations (6)-(8), after suppressing the primes and

assuming the solution of the resulting equations as

1 2 3 1 3

1 2 3 m m

u u u T x x t

U U U T exp x n t

(10)

where is the circular frequency and is the complex wave

number. U1, U2, U3 and T* are undetermined amplitude vectors

that are independent of time t and coordinates xi , nm is the unit

wave normal vector, we obtain 2 2 2 2 2

ijkm m j 0 ik 0 K

2 2

j ij 0 0 ij j

c n n C C U

n T C T B n 0

(11)

2 2 2 2

0 2 j ij K ij j i

2 3 2t

0 ij j i

2 2 2 2

2 E 0 2

C M n U K n n 1

C K n n 1

M C C T M b 0,

(12)

2 2

ij j i 0 0

2 2 2 2 4 2

ij j i 0 0

B n U bT T C

A n n C C 0,

(13)

where

2

q 2 q

2

M 1

2

Equations (11)-(13) is the linear system of five homogeneous

equations in five unknowns 1 2 3U U U T and .The

Christoffels tensor may be expressed as follows

ij ijkm m j i ij j ij i j

ij i j ij i j i ij j

c n n n K K n n

K K n n A A n n B B n

(14)

Using (14) in (11)-(13) yield 2 2 2

ij 0 K 0 i 1 C U T T S 0 (15)

2i i 2 3 4B U S S S T 0 (16)

2 2 2 32

2 K 2 K 2 K

1 1

2

9 7 8

R U R U R U

S S T S 0

(17)

where

2 2 2 2 2 2

1 i 0 2 0 0

3 4 0 5 2 1 6 3 1

t

2 2

7 1 0

2 2 2 2

8 2 1 0 9 2 E 1

S B / C S / C C

S A / S bT S / S /

K 1

S / C

K 1

S R b / C S R C /

The non-trivial solution of the system of equations (15)-(17) is

ensured by a determinantal equation

2 2 2

1 2 3 4 5 6

2 2 2

7 8 9 10 11 12

2 2 2

13 14 15 16 17 18

2

19 20 21 22 23 24

2

25 26 27 28 29 30

K K K K K K

K K K K K K

0K K K K K K

K K K K K K

K K K K K K

(18)

The equation (18) yields to following polynomial

characteristic equation in as

10 8 6 4 2A B C D E F 0 , (19)

where the coefficients A, B, C, D, E, F are given in appendix

A. Solving equation (19), we obtain ten roots of , that is,

1 2 3 4 and 5 . Corresponding to these roots,

there exist five waves in descending order of their velocities,

namely a P wave, a thermal wave, a volume fractional wave

and two transverse waves,

Now we derive the expressions of phase velocity and

attenuation coefficient of these types of waves as

Phase Velocity

The phase velocity is given by

i

i

V i 1 2 3 4 5

Re (20)

where iV i 1 2 3 4 5 are, respectively, the velocities of

qP1, qP2, qP3, qS1 and qS2 waves.

Attenuation Coefficient

The attenuation coefficient is defined as

i iQ Im g i 1 2 3 4 5 (21)

where iQ i 1 2 3 4 5 are, respectively, the attenuation

coefficients of qP1, qP2, qP3, qS1 and qS2 waves.


25

Specific Loss

The specific loss is the ratio of energy W dissipated in

taking a specimen through a stress cycle, to the elastic energy

(W) stored in the specimen when the strain is a maximum. The

specific loss is the most direct method of defining internal

friction for a material. For a sinusoidal plane wave of small

amplitude, Kolksy [29] shows that the specific loss W / W

equals 4 times the absolute value of the imaginary part of

to that of real part of i.e.

i

i

i i

Im WR 4 i 1 2 3 4 5

W Re

(22)

Penetration Depth

The Penetration depth is defined by

i

i

1S i 1 2 3 4 5

Im (23)

Transversely Isotropic Media

Applying the transformation

1 1 2 2 1 2

3 3

x x cos x sin x x sin x cos

x x

(24)

where is the angle of rotation in the x1-x2 plane, in the equations (3), (5) and (6), the basic equations for

homogeneous transversely isotropic three-phase-lag model are

11 1,11 12 2,21 13 3,31 66 1,22 2,12

44 1,33 3,13 1 ,1 1 ,1 1

c u c u c u c u u

c u u B T u

(25)

11 2,22 44 2,33 13 3,31 66 1,21 2,11

13 44 3,32 1 ,2 1 ,2 2

c u c u c u c u u

c c u B T u

(26)

13 44 1,13 2,23 44 3,11 3,22

33 3,33 3 ,3 3 ,3 3

c c u u c u u

c u B T u

(27)

,11 ,22 1 ,33 3

1,1 2,2 1 3,3 3

bT A A

u u B u B

(28)

t t

1 ,11 ,22 3 ,33

1 ,11 ,22 3 ,33

2 2

q q

2

E 0 1 1,1 2,2 3 3,3 0

K 1 T T K 1 T

t t

K 1 T T K 1 T

t t

1 .

2t t

. C T T u u u bT

(29)

where

iij i ij ij i ij ij ij ij i ij K K K K B B i is not summed

1 11 12 1 13 3 3 13 1 33 3 c c c 2c c 1 and 3 are the coefficients of thermal linear expansion. In

the above equations (25)-(29) and using the solution defined

by (10) we obtain the following characteristic equation

10 8 6 4 2A B C D E F 0

(30)

where A B C D E and F are given in appendix B.

Case1 Let us consider plane harmonic waves propagating in a

principal plane perpendicular to the principal direction (0, 1,

0) i.e. wave normal n=(sin, 0, cos) inclined at angle to x3 axis. The characteristic equation (30) reduces to

2 2 2 23 4 sin cos 0 (31) 8 6 4 2

1 2 3 4 5F F F F F 0 (32)

1 2 3 4F F F F and 5F are given in Appendix C.

Equation (31) corresponds to purely transverse wave mode,

which is not affected by void and thermal variations.

Case11 For =900, i.e. when the wave normal n= (1,0,0) is perpendicular to the x3 axis , the characteristic equation (30)

reduces to 2 2

3 0 2 2

4 0 6 4 2

1 2 3 4S S S S 0

where 1 2 3 4S S S S are given in Appendix D.

Particular Cases

(1) If we take

11 22 33 12 13 44 66 1 2 3

1 3 1 3 1 2 3

1 2 3

c c c c c c c

K K K K K K A A A

B B B

in the equations (25)-(29), we obtain the result for the

case of cubic crystal materials.

(2) If we take

11 33 12 13 44 66

1 3 1 3 1 3

1 2 3 1 2 3

c c 2 c c c c

K K K K K K

A A A B B B

the equations (25)-(29) yield corresponding

expressions for isotropic materials.


26

(3) In the absence of void effect, the characteristic equation (32) reduces to the characteristic equation

corresponding to the transversely isotropic

generalized thermoelastic medium. 6 4 2

1 2 3 4J J J J 0

where J1,J2,J3 and J4 are given in appendix D.

IV. NUMERICAL RESULTS AND DISCUSSION

The material chosen we take the following values of relevant

parameter as 10 2 10 2

11 12

10 2 10 2

13 33

10 2 4

44 0

3 3 6 2

1

6 2 3

3

2

1 3

c 5 974 10 N / m c 2 624 10 N / m ,

c 2 17 10 N / m c 6 17 10 N / m

c 3 278 10 N / m T 0 298 10 K

1 74 10 Kg / m 2 68 10 N / m deg

2 68 10 N / m deg C 4 27 10 J / Kg deg

K 0 17 10 W / m deg K

2

T q 1 11

3 33

0 17 10 W / m deg

0 4 s 0 5 s 0 6 s K c C / 4

K c C / 4

Void and initial stress parameters are 15 2 5

1

5 10

3 1

10 5 2 1

3

0 0505655 10 m A 0 9798 10 N

A 0 92174 10 N B 0 052849 10 N

B 0 041 10 N b 3 23 10 N / m K

We can solve equation (32) with the help of the software

Matlab 7.0.4 and after solving the equation (32) and using the

formulas given by (20)-(23), we can compute the values of

phase velocity, attenuation coefficient, specific loss and

penetration depth for intermediate values of frequency () and different values of fractional order derivative i.e. =0.1, 1.0, 1.8 in theories of two phase and three phase lag model of

thermoelasticity. The dense vertical line, sparse vertical line

and dense horizontal line corresponds respectively to

=0.1(three phase lag), =1.0(three phase lag) and =1.8(three phase lag) whereas sparse horizontal line, dense squares and

sparse squares corresponds to =0.1(two phase lag), =1.0(two phase lag) and =1.8(two phase lag) respectively.

0

1

2

3

4

5

6

7

8

9

0.0

0.5

1.0

1.5

2.0

2.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Phase V

elo

city

Freq

uenc

y

Fig. 1 Variation of phase velocity (V1) w.r.t. frequency ()

0.0

0.1

0.2

0.3

0.4

0.5

0.0

0.5

1.01.5

2.02.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Ph

ase

Ve

locity

Frequ

ency


0.0

0.2

0.4

0.6

0.0

0.5

1.0

1.5

2.0

2.5

3.0

III Phase Lag

III Phase Lag

IIIPhase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Pha

se

Velo

city

Freq

uenc

y


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0

0.5

1.01.5

2.02.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Phase V

elo

city

Freque

ncy



27

0

1

2

3

4

5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Atte

nu

atio

n C

oe

ffic

ien

t

Freq

uenc

y

Fig. 5 Variation of Attenuation coefficient (Q1) w.r.t. frequency ()

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0

0.5

1.01.5

2.02.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Atte

nu

atio

n C

oe

ffic

ien

t

Freque

ncy


0.0

0.4

0.8

1.2

1.6

2.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Atte

nu

atio

n C

oe

ffic

ien

t

Freq

uenc

y


0.0

0.3

0.6

0.9

1.2

1.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Atte

nu

atio

n C

oe

ffic

ien

t

Fre

quency


0123456789

10111213

14

15

0.0

0.5

1.0

1.5

2.0

2.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Pen

tra

tio

n D

ep

th

Fre

quency

Fig. 9 Variation of Specific Loss (R1) w.r.t. frequency ()

0.0

0.1

0.2

0.3

0.4

0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Pen

tra

tio

n D

ep

th

Fre

quency



28

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.0

0.5

1.0

1.5

2.0

2.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Pen

tra

tio

n D

ep

th

Fre

quency


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.0

0.5

1.0

1.5

2.0

2.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Pen

tra

tio

n D

ep

th

Freq

uenc

y


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.00.5

1.01.5

2.02.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Sp

ecific

lo

ss

Freque

ncy

Fig. 13 Variation of Pentration depth (S1) w.r.t. frequency ()

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.00.5

1.0 1.52.0 2.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Sp

ecific

lo

ss

Frequency


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.00.51.01.52.02.53.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Sp

ecific

lo

ss

Fre

quency


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.00.5

1.01.5

2.02.5

3.0

III Phase Lag

III Phase Lag

III Phase Lag

II Phase Lag

II Phase Lag

II Phase Lag

Sp

ecific

lo

ss

Freq

uenc

y


Phase Velocity

Figs (1)-(4) show the variation of phase velocities of

different waves with respect to for different values of fractional order derivative . It is clear from fig.1 that the value of V1 increase sharply in both two and three

phase lag models (=0.1, 1.0) whereas for =1.8, V1 increases slowly. Fig.2 exhibits that the value of V2

decrease for small values of (Three phase lag model)

and increase slowly for higher values of whereas for two phase lag model and all fractional orders, the value

of phase velocity increase slowly. Fig.3 depicts that for


29

three phase lag model (=0.1, 1.0), phase velocity V3 increase sharply for 1 and then increase smoothly whereas for two phase lag model (=0.1, 1.0), phase velocity V3 increase smoothly. For the fractional order

derivative =1.8, V3 first increase and then decrease smoothly. Fig.4 shows that the value of phase velocity

V4 increase smoothly for all fractional orders and phase

lag models.

Attenuation Coefficient

Figs (5)-(8) show the variation of attenuation coefficient

of different waves with respect to for different values of fractional order derivative . Fig.5 shows that the attenuation coefficient Q1 increase continuously for all

fractional orders and both two and three phase lag

models. Fig.6 shows that the attenuation coefficient Q2

decrease but initially with a sharp decrease. For =1.8, Q2 increase but with fluctuation for three phase lag

model. Figs 7-8 shows that Q3 and Q4 increase for all

fractional orders and for both phase lag models. But

values of Q3 are higher in comparison to Q4 and three

phase model in comparison to two phase lag model.

Specific Loss

Figs (9)- (12) show the variation of specific loss with

respect to for different values of fractional order derivative . Fig.9 depicts that specific loss R1 increase with increase in frequency but magnitude values in case

of =1.8 are smaller in comparison to other fractional orders. Fig.10 shows that R2 decrease with increase in

frequency but values of R2 are higher in case of three

phase lag model. Due to small variation in values of R2 ,

it appears to be constant. Fig.11 shows that the behavior

and variation of R3 is opposite to that of R1. Fig.12

shows that specific loss R4 increase for three phase

lag(=0.1,1.0,1.8) and two phase lag model(=1.0, 1.8) whereas R4 decrease for two phase lag(=0.1).

Penetration Depth

Figs (13)-(16) shows the variation of penetration depth

with respect to for different values of fractional order derivative . Fig. 13 shows that penetration depth S1 decrease smoothly w.r.t. frequency for all fractional

orders but magnitude values in case of three phase lag

model are higher. Fig.14 shows that S2 increase initially

and then decrease slowly for three phase lag model

(=0.1, 1.8). For =1.0, S2 decrease initially and then increase smoothly. Fig.15 depicts that S3 initially

decrease sharply and then smoothly as increases. From fig.14 it is clear that S3 initially decrease sharply and

then smoothly as increases. Fig.16 shows that S4 has same behavior and variation as S1 but with different

magnitude values. Magnitude values in case of S4 are

higher than S1.

CONCLUSION

The propagation of plane wave in anisotropic medium in

the context of the theories of three phase lag model and

two phase lag model of thermoelasticity have been

studied.

It is found that their exist quasi-longitudinal wave (qP1),

quasi-longitudinal thermal wave (qP2), quasi-

longitudinal volume fractional

wave (qP3) and two

quasi-transverse waves (qS1, qS2). The governing

equations for homogeneous transversely isotropic three-

phase-lag are reduced as a special case and obtained that

four coupled quasi waves and one quasi transverse wave

which is decoupled from the rest of the motion. The

values of V1, Q1, R1 are higher in comparison with the

corresponding quantities. The values of phase velocity,

attenuation, specific loss and penetration depth are

higher for three phase lag model in comparison with two

phase lag model.

Appendix A

1 22 29 8 15 14 10

4 14 3 15 7 22 29 3 10 4 8 13 22 29

A K K K K K K K

K K K K K K K K K K K K K K


30

1 9 15 29 22 8 16 22 29

15 22 30 15 23 29 17 21 29 18 22 27

1 14 10 22 30 10 23 29 11 21 29 12 22 27

1 29 20 10 17 11 15 1 22 29

9 15 12 15 10 18 2 22 29 8 15 14 10

3 7 1

B K K K K K K K K K

K K K K K K K K K K K K


K K K K K K K K K K

K K K K K K K K K

[ (

K K K K

K K K

(

))

.

. ]

[ (

8 22 27 17 21 29 16 22 29

15 22 30 23 29 7 14 4 22 30 23 29

5 21 29 6 22 27 7 20 29 4 17 5 15

K K K K K K K K


K

) (


7 29 22 6 15 4 18 13 3 10 22 30 23 29

3 27 6 22 3 11 21 29 3 12 22 27 9 4 22 29

8 4 23 29 22 30 4 11 20 29 5 10 20 29

4 12 22 26 6 10 22 26 19 29 5 10 14 4 11 14

5 8 15 4 8 17


K K K K K K K K K K K K K K K K



) [ (

) K K

K K K K K K

( (

3 11 15 3 10 17

25 22 6 10 14 4 12 14 6 8 15

4 8 18 3 12 15 3 10 18

K K K K K K

K K K K K K K K K K K

K K K K K K K K K

))

(

2 9 15 22 29 8 16 22 29 15 22 30

15 23 29 17 21 29 18 22 27

14 10 22 30 10 23 29 11 21 29 12 22 27

20 29 10 17 11 15 9 15 22 29

26 22 12 15 10 18 1 9 16 22 29 15 22 30

1

C ( K K K K K K K K K K K K

K K K K K K K K K

K K K K K K K K K K K K K

K K K K K K K K K K

K K K K K K (K K K

( (

)

( ) K)) ( ( K K K K

K

5 23 29 17 21 29 18 22 27

8 16 22 30 23 29 15 23 30 24 28

21 18 28 17 30 27 17 24 18 23

14 10 23 30 24 28 21 11 30 12 28

27 11 24 12 23 20 10 17 30 18 28

11 16 29 15 11 30 12

K K K K K K K K


K K K K K K K K K K



K K K K K

(

(

K

(

K

)

K

28 27 11 18 12 17

26 10 17 24 18 23 15 11 24 12 23

12 16 22 21 11 18 12 17

3 7 15 23 30 24 28 16 22 30 23 29

21 17 30 18 28 27 17 24

18 23 7 14 4 23 30 24 28 7 20 4 17 30

K K K K K


K K K K K K K K


K K K K K K K K


)

(

)]

[ (

(

K) ( K)

18 28

5 16 29 15 5 30 6 28 27 5 18 6 17

7 26 4 17 24 23 18 15 5 24 6 23

6 16 22 21 5 18 6 17

13 3 10 23 30 3 21 11 30 12 28

27 3 11 24 12 23 3 8 23 30 24 28

3 9 23 29 2

K K



K K K K K K K K


K K K K K K K

( ))

(

(

K K K K

)]

[

K K K K

(

K

K

2 30 3 21 5 30 6 28

3 27 5 24 6 23 4 20 11 30 12 28

10 20 5 30 6 28 20 27 5 12 6 11

4 26 11 24 12 23 10 26 5 24 6 23

21 26 5 12 6 11 19 3 10 17 30 18 28

3 15 11 30 12 28 3 1

K K K K K K K





K

( )

)] [ (

K K K K K K K

1 16 29 3 27 11 28 12 17

4 8 17 30 18 28 4 9 17 29 8 15 5 30 6 28

5 29 9 15 8 16 8 27 5 18 6 17

14 11 30 12 28 10 14 5 30 6 28

14 27 5 12 6 11 4 26 11 18 12 17

10 26 5 18 6 17

K K K K K K K K





K K K K

( )

( )

K K

15 26 5 12 6 11

26 3 10 17 24 18 23 3 15 11 24 12 23

3 12 16 22 3 21 11 18 12 17

4 8 17 24 18 23 4 9 18 22 8 15 5 24 6 23

6 8 16 22 8 21 5 18 6 17

K K K K K K


K K K K K K K K K K


K K K K K K K K K

)]

[ (

( )

K )]


31

2 9 16 22 15 23 17 21 29

22 15 30 18 27 8 16 22 15 23 17 21 30

16 23 29 28 18 21 15 24 27 17 24 18 23

14 10 23 30 24 28 21 11 30 12 28

27 11 24 12 23 20 10 17 30 18 28 1

D K K K K K K K K K




K K K K K K K K

[ ( (

(

(

( K K

)

K K

1 16 29

15 11 30 12 28 27 11 18 12 17

1 8 16 11 30 12 28 9 16 22 30 23 29

15 23 30 24 28 17 21 30 24 27

18 21 28 23 27 20 18 12 28 11 30

16 26 12 23 11 24 3 7 16 23 30 24

K K

K K K K K K K K K K


K K K K K K K K K K


K K K K

)

( ( ) (

)

)K K K K K K K K K) [

28

7 20 16 6 28 5 30 7 26 4 17 24 18 23

15 5 24 6 23 6 16 22 21 5 18 6 17

16 5 24 6 23 19 3 16 12 28 11 30

4 9 17 30 18 28 9 15 5 30 6 28

5 9 16 29 9 27 5 18 6 17 8 16 5 3

{

( )

)






[

K

]

K( )

0 6 28

16 26 5 12 6 11 25 3 16 12 23 11 24

9 15 8 16 5 24 6 23 6 9 16 22

9 21 5 18 6 17

K K



K K K K K K

] (

)

2 8 16 11 30 12 28 9 16 22 30

16 23 29 15 23 30 15 24 28 17 21 30

18 21 28 17 24 27 18 23 27

9 16 19 5 30 6 28 9 16 5 24 6 23

E K K K K K K K K K K K


K K K K K K K K K )


( (

)

.

2 9 16 23 30 24 28F K K K K K K K

2 2

1 11 2 0 3 12 4 13 5 1 1

1 0 1 7 21 8 22 10 23 11 1 2

12 0 2 13 31 14 32 15 33 17 1 3

18 0 3 19 1 20 2 21 3 22 3

23 2 24 4

K K C K K K S B

K T K K K K S B

K T K K K K S B

K T K B K B K B K S

K S K S K

2 2

25 2 26 2 5

2

27 2 6 28 8 29 7 30 9

2 9 16 1 12 66 11 2 13 44 11

2 2

3 66 11 4 44 11 5 33 11 6 0 11

7 0 11 8 1 11 9 3 11 10 1 11

11 3 1

R K R S

K R S K S K S K S

K K K c c / c c c / c

c / c c / c c / c / C c

bT / c A / c A / c B / c

B / c

2 21 1 1 0 11 2 3 0 11 1 1 0 112 2 2

2 3 0 11 11 0

T / c T / c a B / C c

a B / C c c C

Appendix B

1 6 10 2 7 11 14 18

3 8 12 15 19

4 9 13 16 20 5 17 21

6

20 2 1 1 19 24 7 13 8 15

2 2

2 19 24 7 13 8 15 1 13 19 24

2

7 19 24 13 20 24 19 2

5 7 26 S g g g g g

A S S S B S S S S S

C S S S S S

D S S S S S

g g

S g

E

g g g g g g

S S S

F g g g

g g g

g g g g g g g g

5 14 18 24 15 19 23

12 8 20 24 19 25 9 18 24 10 19 23

17 24 8 14 9 13 19 22 10 13 8 15

2 2

3 1 19 24 13 20 24 19 25

14 18 24 15 19 23 7 13 20 25 26 7

2

20 24 19 25

g g g g g g

g g g g g g g g g g g g


S g g g g g g g g

g g g g g g g g g g g

g g g g g

18 14 25 15 26

23 14 7 15 20 12 8 20 25 7 26

18 9 25 10 26 23 9 7 10 20

17 8 14 25 15 26 13 9 24 10 26

2

9 24 23 9 15 10 24 22 8 14 7 15 20

2

13 9 7 10 20 10 19

g g g g

g g g g g g g g g

g g g g g g g g g



g g g g g g

18 9 15 10 14

2 2 2

13 19 24 7 19 24 13 20 24 19 25

14 18 24 15 19 23 12 8 20 24 19 25 9 18 24

10 19 23 17 24 8 14 9 13

19 22 10 13 8 15

g g g g g


g g g g g g g g g g g g g g g

g g g g g g g g g

g g g g g g

2 2 2

4 19 24 13 20 24 19 25

14 18 24 15 19 23 7 13 20 25 26 7

2

20 24 19 25 18 14 25 15 26

23 14 7 15 20 12 8 20 25 7 26

18 9 25 10 26 23 9 7 10 20

17 8 14 2

S g g g g g g g


g g g g g g g g g

g g g g g g g g g

g g g g g g g g g

g g g g

5 15 26 13 9 24 10 26

2

9 24 23 9 15 10 24 22 8 14 7 15 20

2

13 9 7 10 20 10 19

2

18 9 15 10 14 1 13 20 25 7 26

2

20 24 25 19 18 14 24 15 26

23 14 7 15 20 7

g g g g g g g


g g g g g g

g g g g g g g g g g

g g g g g g g g g

g g g g g g

20 25 7 26

17 9 24 10 26 22 9 7 10 20

g g

g g g g g g g g g


32

4 2 2

5 1 20 25 7 26 13 20 25 7 26

2

20 24 25 19 18 14 25 15 26

23 14 7 15 20 7 20 25 7 26

17 9 24 10 26 22 9 7 10 20

6 6 19 24 2 13 3 12

7 6 2 13 19 25

S g g g g g g g g

g g g g g g g g g

g g g g g g g g

g g g g g g g g g

S g g g g g g g

S g g g g g

2

20 24 19 24

14 18 24 15 19 23 12 3 19 25 20 24

4 18 24 5 19 23 17 24 3 14 4 13

19 22 5 13 3 15

2

8 6 2 13 20 25 7 26 19 25 20 24

18 14 25 15 26 23 14 7 15 20

12 3

g g g g



g g g g g g

S g g g g g d g g g g g

g g g g g g g d g g

g g

20 25 7 26 18 4 25 5 26

23 4 7 5 20 17 3 14 25 15 26

2

4 24 13 4 25 5 26

23 4 15 5 14 22 3 14 7 15 20

2

13 4 7 5 20 5 19 18 4 15 5 14

g g d g g g g g g

g g d g g g g g g g g

g g g g g g g

g g g g g g g g d g g

g g d g g g g g g g g g

2

9 6 17 4 25 5 26 2 20 25 7 26

22 4 7 5 20

10 11 19 24 2 8 3 7

11 11 2 8 19 25 20 24 18 9 24 10 19 23

7 3 19 25 20 24 4 18 24 5 19 28

2

3 19 24 17 24 3 9 4 8

19 22 5 8

S g w g g g g g g g g g

g g g g

S g g g g g g g

S g g g g g g g g g g g g g


w g g g g g g g g g

g g g g g

3 10g

12 11 2 8 20 25 7 26 18 9 25 10 26

23 9 7 10 20 7 3 19 25 20 24

18 4 25 5 26 23 4 7 5 20

2

3 19 25 20 24 4 18 24 5 19 23

17 3 9 25 10 26 8 4 25 5 26

23 4 10 5 9

S g g g g g g g g g g g

g g g g g g g g g g

g g g g g g g g g

g g g g

( ( ( )

( )) (

( ) (

g g g g g g g

g g g g g g g

))

( ( )

)

g g g g

g g g g g

22 3 9 7 10 20

8 4 7 5 20 18 4 10 5 9

2

13 11 3 20 25 7 26 18 4 25 5 26

23 4 7 5 20

14 16 24 2 8 14 9 13 7 3 14 4 13

12 3 9 4 8

15 16 2 8 14 25 15 26 13 9 25

g g g g g

g g g g g g g g g ,

S g g g g g g g g g g

g g g g ,

S g g g g g g g g g g g g

g g g g g ,

S g g g g

( ( )

( ) ))

( ( )

( ))

(

)

g g g g( ( g g

10 26

2

9 24 23 9 15 10 14 7 3 14 25 15 26

2

4 24 13 4 25 5 26 23 4 15 5 14

2

24 3 14 4 13 12 3 9 25 10 26

8 4 25 5 26 23 4 10 5 6

22 3 9 15 10 14 8 4 15 5 14

13 4 10 5

g g

g g g g g g g g g g g g g



g g g g g g g g

) (

g g


g

(

(

g g g g

)

)

9 ),

16 16 2 9 25 10 26 7 4 25 5 26

3 14 25 15 26 13 4 25 5 26

2

4 24 23 4 15 5 14 22 4 10 5 9

4

17 16 4 25 5 26

18 21 19 2 10 13 8 15 7 5 13 3 15

12 5 8 3 10

S g g g g g g g g g g g

g g g g g g g g g g

g g g g g g g g g g g g ,

S g g g g g ,

S g g g g g g g g g g g g

g g g g

(

(

),

)

g

19 21 2 8 14 7 15 20 13 9 7 10 20

2

18 9 15 10 14 9 19 7 3 14 7 15 20

2

13 4 7 5 20 5 19 18 4 15 5 14

2

19 5 13 3 15 12 3 9 7 10 20

8 4 7 5 20 18 4 10 5 9

17

S g g g g g g g g g g



(

g g

( ( ) ( )

) ( ( )

( ) )

( ( )

(

g g g g g g g g

g g g g g g g g g) )

g

3 9 15 10 14 8 4 15 5 14

13 4 10 5 9

2

20 21 2 9 7 10 20 7 4 7 5 20

2

3 14 7 15 20 13 4 7 5 20 5 19

18 4 15 5 14 17 4 10 5 9

4

4

21 21 7 5 20

g g g g g g g g g g

g g g g g ,

S g g g g g g g g g

g g g g g g

(

))

( ( ) ( )

g g g g

g g g g g g g g

( ( ) ( )

( )) ),g g

gS g ( ), g g


33

2 2 2

1 1 2 3 3 4 2 1 2 1

3 1 3 2 4 1 1 5 1 1

2 2 2

6 1 2 1 7 2 1 3 3 4

8 2 3 2 9 2 1 10 2 1

2 2 2

11 1 3 2 12 2 3 2 13 1 4 2 4 2 5

14 3 2 15 3 2

16 1 10 17

g n n d n d g n n d

g n n d g in g g in a

g n n d g n n d n d

g n n d g in g g in a

g n n d

g

g n n d g n d n d n d

g in g g in a

=n g =

2 2 2

2 10 18 3 11 19 8 1 2 9 3

2 2 2 2

20 6 21 1 5 22 2 5 23 3 3

2 2 2 2 2 2

24 1 2 3 1 1 2 2 3 3

2 2 2 4 2

25 4 26 1 0 1 t

2

1 3 1 2 1 0 1 t

3 3

n g =n g = n n + n

g = - g =in q g =in q g =in q

g n n n q n n q n q

g q g R b / C K

q K / K q K / C K

q K / C

20 1 t 4 1 1 t

5 1 1 1 t

6 1 3 1 t

K q R C / K

q R / K

q R / K

Appendix C

1 2 5 1 6 11 14

2 2

2 6 11 14 1 7 11 13 8 10 14 11 14

5 4 10 14 3 11 13 9 14 4 6 2 8

11 12 2 7 3 6

3 1 6 7 26 20 25 10 7 26 8 25

13 7 20 8 7 5 2 7 26 20 25 10 3

( (( )

)

F r r r r r r ,

F w r r r r r r r r r r w r r

r r r r r r r r r r r r r

r r r r r r ,

F r r d g g g r r g r g

r r g

)

( ( ( )

( )r d r) (r d g r r( )g g

26 4 25

13 3 20 4 7 9 2 7 26 8 25 6 3 26 4 25

2

4 14 13 3 8 4 7 12 2 7 20 8 7

2

3 11 6 3 20 4 7 10 3 8 4 7

2 2

7 11 13 8 10 14 11 14

2

4 1 20 25 7 26

6 7 2

g r g

r r g r d r r r g r g r r g r g

w r r r r r r r r r r g r d

w r r r r g r d r

( )) (

) ( ( )

( ) )

( )

r r r r

w r r r r r r w r r ,

F

)

( (w r g g d g

r d

)

( ( g

6 20 25 10 7 26 8 25

13 7 20 8 7 9 3 26 4 25 12 3 20 4 7

g g r r g r g

r r g r d r r g

)

r g r r g( )) ( ))r d ,

4

5 7 26 20 25

2 2

1 4 2 2 3 1

2 2

4 1 5 2 6 4 5

7 2

2 2

12

8 2 9 10

2 2

10 11

3

11 9

5

8

13

F w d g g g ,

r sin cos , r sincos, r i sin,

r ia sin, r sincos, r sin cos ,

r g cos, r a cos, r sin,

r cos, r sin c

( )

r q sin r q co

os ,

2 2 2 214 1 2 3

s

r sin q cos q sin q cos

Appendix D

2

1 8 2 2 8 2

2

1 10 2 1 8 5

2

3 7 26 20 25 10 1 26 1 25 5 1 20 1 7

2

4 7 26 20 25

1 1 6 14 2 5 14

2 2

2 1 6 25 14 6 14 5 2 25 3 13 12 2 7 3 6

2

3 3 12

S q S q

a q q

S g g g g a g q g a

S g g g

J r r r r r r

J r r g r r r r r g r r r r r r r

J r r

2 46 25 14 1 25 4 25r g r r g J g

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