Fractional Order Generalized Thermoelastic Infinite Medium with Cylindrical Cavity Subjected to Harmonically Varying Heat

Abstract

Hamdy M. Youssef, Eman A. Al-Lehaibi Faculty of Engineering-Umm Al-Qura University, Makkah, Saudi Arabia E-mail: [email protected] Received October 23, 2010; revised November 26, 2010; accepted December 16, 2010

In this work, a Mathematical Model of an elastic material with cylindrical cavity will be constructed. The governing equations will be taken into the context of the fractional order generalized thermoelasticity theory (Youssef 2010). Laplace transform and direct approach will be used to obtain the solution when the boundary of the cavity is exposed to harmonically heat with constant angular frequency of thermal vibration. The inverse of Laplace transforms will be computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to present the effect of the fractional order parameter and the angular frequency of thermal vibration on all the studied felids.

Keywords: Thermoelasticity, Generalized Thermoelasticity, Fractional Order, Cylindrical Cavity, Harmonically Heat

1. Introduction

Recently, a considerable research effort has been expended to study anomalous diffusion, which is characterized by the time-fractional diffusion-wave equation by Kimmich [1]:

c I c,ii ,

(1)

where is the mass density, c is the concentration, is the diffusion conductivity, i is the coordinate symbol which takes the values 1, 2 and 3, the subscript "," means the derivative with respect to xi and notion I is the Riemann-Liouville fractional integral is introduced as a natural generalization of the well-known n-fold repeated integral I f t written in a convolution-type form as in [2,3]:

I

f

t

1

t t 0

1

f

d

f t

for

0

2

,

for 0

(2)

where is the gamma function.

According to Kimmich [1] Equation (1) describes different cases of diffusion where 0 1 correspond to

weak diffusion (sub diffusion), 1 correspond to normal diffusion, 1 2 correspond to strong diffusion (super diffusion) and 2 correspond to ballistic diffusion. It should be noted that the term diffusion is often used in a more generalized sense including various transport phenomena. Equation (1) is a mathematical model of a wide range of important physical phenomena, for example, the sub-diffusive transport occur in widely different systems ranging from dielectrics and semiconductors through polymers to fractals, glasses, porous, and random media. Super diffusion is comparatively rare and has been observed in porous glasses, polymer chain, biological systems, transport of organic molecules and atomic clusters on surface [4]. One might expect the anomalous Heat conduction in media where the anomalous diffusion is observed. Fujita [5,6] considered the heat wave equation for the case of 1 2 :

C T k I T,ii ,

(3)

where C is the specific heat, k is the Thermal conductivity and the subscript "," means the derivative with respect to the coordinates xi . Equation (3) can be obtained as a consequence of the

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H. M. YOUSSEF ET AL.

33

non local constitutive equation for the heat flux compo-

nents qi is in the form

qi k I 1 T,i , 1 2 .

(4)

Povstenko [4] used the Caputo heat wave equation de-

fines in the form:

qi k I 1 T,i , 0 2 ,

(5)

to get the stresses corresponding to the fundamental so-

lution of a Cauchy problem for the fractional heat con-

duction equation in one-dimensional and two-dimen-

sional cases.

Some applications of fractional calculus to various

problems of mechanics of solids are reviewed in the lite-

rature [7,8].

2. Theory of Fractional Order Generalized Thermoelasticity

The classical thermoelasticity is based on the principles

of the theory of heat conduction which is called Fourier law, which relates the heat flux components qi to the temperature gradient as follows:

qi k T,i .

(6)

In combination with the energy conservative law, this

leads to the parabolic heat conduction equation which is

considered by Povstenko [4]:

C T k T,ii ,

(7)

where dotted above T means the derivative with respect to the time t. Recently, in the non classical thermoelasticity theories, Fourier law (6) and heat conduction (7) are replaced by more general equations, have been formulated. The first well-known generalized of such a type of Lord and Shulman [9] and it takes the form:

qi oqi k T,i ,

(8)

which leads to the hyperbolic differential equation of heat conduction of Lord and Shulman [9]:

C T oT k T ,ii ,

(9)

where o is non-negative constant and is called relaxation time.

According to equation (9), Kaliski [10] and Lord and

Shulman [9] constructed the theory of generalized ther-

moelasticity.

In the context of the generalized thermoelasticity, the

governing equations for isotropic medium are defined as

follows:

The equation of motion

ij, j Fi ui .

(10)

The constitution relation

ij 2 eij ekk ij ,

(11)

where , are component, Fi

Lamй's constant, the body force

coumi ipsotnheentd,isplacTemeTnot

is the increment of the dynamical temperature where To is the reference temperature, 3 2 T where

T is called the thermal expansion coefficient, where ij

is the Kronecker delta symbol, ij is the stress tensor

such that ij ji and e ij is the strain tensor satisfy the

relations

eij

1 2

ui ,j

u j ,i

.

(12)

The heat flux equation

qi,i C To e .

(13)

The entropy increment equation per unit volume takes

the form

ToS C To e ,

(14)

where S is the entropy increment of the material.

The heat flux-entropy equation

qi,i To S .

(15)

The heat equation without any heat sources

qi

o

qi t

k I 1 ,i

0 2 .

(16)

By using Equations (14,15,16), we have the heat equation in the form [11]:

k

I

1 ,ii

t

o

2 t2

C

To

e,0

2

(17)

0 1

where

1

1 2

for weak conductivity

for

normal

conductivity

for strong conductivity

3. The Problem Formulation Let us consider a perfectly conducting elastic infinite body with cylindrical cavity occupies the region R r of an isotropic homogeneous medium whose state can be expressed in terms of the space variable r and the time variable t such that all of the state functions vanish at infinity. We will use the cylindrical system of coordinates (r,,z) with the z-axis lying along the axis of the cylinder. Due to symmetry, the problem is one-dimensional with all the functions considered depending on the radial distance r and the time t. The medium described above is considered to be quiescent and the surface of the cavity is subjected to

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34

H. M. YOUSSEF ET AL.

harmonically varying heat and traction free described mathematically as follow:

R,t 1 cos t ,

(18)

and

rr R,t 0 ,

(19)

where 1 is constant and is the angular frequency of thermal vibration ( 0 for a thermal shock). It is assumed that there are no body forces and no heat sources in the medium. Thus, the field equations (10), (11), (12) and (17) in cylindrical case can be set as [12]:

2

e r

r

2u t2

,

(20)

I 12

t

o

2 t2

C k

To k

e , 0

2

,

(21)

rr

2

u r

e

,

(22)

2

u r

e

(23)

zz e ,

(24)

zr r z z 0 ,

(25)

e

1 r

ru r

,

(26)

where

2

2 r2

1 r

r

For convenience, we shall use the following non-dimensional variables [12]:

r,

u

co

r,

u

,

t,

o

co2

t,

o

,

To

,

,

where

co2

2

and

C k

.

Equations (20-26) assume the form (where the primes are suppressed for simplicity)

e r

a

r

2u t2

,

(27)

I 12

t

o

2 t2

e,0

2,

(28)

rr

2

u r

2 2

u r

a

2

,

(29)

2 2

u r

2

u r

a

2

,

(30)

zz 2 2 e a 2 ,

(31)

where

a

To 2

,

C

,

1

2

2 and

3 2 T .

4. Formulation in the Laplace Transform Domain

Taking the Laplace transform for the both sides of the Equations (27-31), this is defined as follows:

L f t f s f t est d t ,

(32)

0

where the rule for the Laplace transform of the Riemann-Liouville fractional integral for zero initial function reads from Povstenko [4]:

L I

f

t

1 s

L f

t

f

s s

,

0.

(33)

Then, we have

de dr

2

d dr

1u ,

(34)

2 3 4 e ,

(35)

rr

2

du dr

2 2

u r

2

2

(36)

2 2

du dr

2

u r

22

,

(37)

zz

2

2

e

2 2

,

(38)

R, s 5 s ,

(39)

rr R, s 0 ,

(40)

where 2

d d

2 r2

1 r

d dr

, 1

s2 ,2

a

,

3 (s 0s 1) ,

4 3 ,

5

s2

s 2

,

and an over bar symbol denotes its Laplace transform

and s denotes the Laplace transform parameter. Eliminating u from the Equations (26,34,35), we get

2 1 e 22 ,

(41)

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H. M. YOUSSEF ET AL.

35

2 3 4 e .

(42)

Eliminating e from Equations (41,42), we obtain

4 1 24 3 2 13 0 . (43)

In a similar manner, we can show that e satisfies the equation

4 1 24 3 2 13 e 0 . (44)

The bounded solutions of Equations (41,42) at infinity can be written in the form

2 Ai pi2 1 K0 pir ,

(45)

i 1

2

e Bi K0 pir ,

(46)

i 1

where Ko(.) is the modified Bessel function of the second kind of order zero. A1, A2, B1 and B2 are all parameters depending on the parameter s of the Laplace transform, p12 and p22 are the roots of the characteristic equation

p4 1 24 3 p2 13 0 ,

(47)

and satisfy the relations

p2 p2

1

2

1

24

3

,

p12 p22 13 .

Using Equation (41), we obtain

Bi 2 pi2 Ai , i=1,2.

(48)

Thus, we have

2

e 2 Ai pi2 K0 pir .

(49)

i 1

Substituting from Equation (49) into the Laplace transform of Equation (26), we obtain

2

u 2 Ai pi K1 pir ,

(50)

i 1

where K1(.) is the modified Bessel function of the second kind of order one. In deriving Equation (50), we have used the following well-known relation of the Bessel function:

z K0 z d z z K1 z ,

Finally, substituting from Equations (45,49,50) into Equations (36-38), we obtain the stress components in the form:

rr

2 2 Ai i 1

2

1

K

0

pi r

2 r

pi

K1

pi r

,

(51)

2 2 Ai i 1

2 1 2 pi2

K0

pi r

2 r

pi

K1

pi r

,

(52)

2 zz 2 21 2 pi2 Ai K0 pir .

(53)

i 1

Using the boundary conditions (39,40), we get

2 Ai pi2 1 K0 pi R 5 ,

(54)

i 1

2 Ai i 1

21K0

pi

R

2 r

pi

K1 pi R 0 .

(55)

Then, we have

A1 A2

l11 l21

l12 l22

1

5 0

,

(56)

where

l11 p12 1 K0 p1R , l12 p22 1 K0 p2R ,

l21

2

21 K 0

p1r

2 R

2

p1

K1

p1 R

,

l22

2

21

K0

p2R

2 R

2

p2

K1

p2 R

.

Then, we have

A1

5 G

2

21 K 0

p2 R

2 R

2

p2

K1

p2 R

,

and

A2

5 G

2

21

K0

p1r

2 R

2

p1

K1

p1 R

,

where

G

2

21K0

2 R

2

p2

K1

p2R p2R

p12 1

K0 p1R

2 21K0 p1r

2 R

2

p1

K1

p1R

p22 1

K0 p2 R .

Those complete the solution in the Laplace transform space.

5. Numerical Inversion of the Laplace Transform

In order to invert the Laplace transform, we adopt a nu- merical inversion method based on a Fourier series ex- pansion [13]. By this method the inverse f (t) of the Laplace

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36

H. M. YOUSSEF ET AL.

transform f s is approximated by

f

t

ect t1

1

2

f

c

R1

N

f

k 1

c

i

k t1

i k t

exp

t1

(57)

, 0 t1 2t,

where N is a sufficiently large integer representing the number of terms in the truncated Fourier series, chosen such that

ex p

ct

R1

f

c

i

N t1

ex p

i N t

t1

1,

(58)

where 1 is a prescribed small positive number that corresponds to the degree of accuracy required. The para- meter c is a positive free parameter that must be greater than the real part of all the singularities of f s . The optimal choice of c was obtained according to the criteria described in [13].

Figure 3. The stress distribution.

Figure 4. The displacement distribution.

Figure 1. The temperature distribution.

Figure 2. The radial stress distribution. Copyright © 2011 SciRes.

Figure 5. The strain distribution. 6. Numerical Results and Discussion With a view to illustrating the analytical procedure presented earlier, we now consider a numerical example for which computational results are given. The results depict the variation of temperature, stress, displacement and strain fields in the context of Youssef model [11]. For this purpose, copper is taken as the thermoelastic ENG

H. M. YOUSSEF ET AL.

37

material for which we take the following values of the different physical constants [12]:

K 386 kg m k 1 s3 , T 1.78 105 k 1 ,

8954 kg m3 , To 293 k , C 383.1 m2 k 1 s2 ,

3.86 1010 kg m1 s2

,

7.76 1010 kg m1 s2 .

From the above values we get the nondimensional values for our problem as: a 0.0104441, 1.618, 2 4 .

The results of the temperature, the stresses, the dis- placement and the strain are shown in Figures 1-5 respectively with wide range of non dimensional distance r from r = R = 1.0 up to r = 2.0, non dimensional time t = 0.08 and non dimensional relaxation time o 0.001 with different values of the parameter 0.8, 1.0,1.2 which describe the three types of conductivity (weak conductivity, normal conductivity, strong conductivity), respectively and with different values of the parameter 0, 10, 15 . We can see the significant effect of the parameter and the angular frequency of thermal vibration on all the studied fields.

7. Conclusion

We considered a perfectly conducting elastic isotropic homogeneous infinite body with cylindrical cavity in the context of the fractional order generalized thermoelasticity theory (Youssef model). The effect of the fractional parameter and the angular frequency of thermal vibration on all the studied fields are very significant. New classification of the materials must be constructed according to the fractional parameter which describes the ability of the material to conduct the heat. 8. References

[1] R. Kimmich, "Strange Kinetics, Porous Media, and NMR," Chemical Physics, Vol. 284, 2002, pp. 243-285.

doi:10.1016/S0301-0104(02)00552-9 [2] I. Podlubny, "Fractional differential equations," Academic Press, New York, 1999. [3] F. Mainardi and R. Gorenflo, "On Mittag-Lettler-Type Function in Fractional Evolution Processes," Journal of Computational and Applied MathEMATICS, Vol. 118, No. 1-2, 2000, pp. 283-299. doi:10.1016/S0377-0427(00)002 94-6 [4] Y. Z. Povstenko, "Fractional Heat Conductive and Associated Thermal Stress," Journal of Thermal Stress, Vol. 28, 2005, pp. 83-102. doi:10.1080/014957390523741 [5] Y. Fujita, "Integrodifferential Equation Which Interpolates the Heat Equation and Wave Equation (I)," Osaka Journal of Mathematics, Vol. 27, 1990, pp. 309-321. [6] Y. Fujita, "Integrodifferential Equation Which Interpolates the Heat Equation and Wave Equation (II)," Osaka Journal of Mathematics, Vol. 27, 1990, pp. 797-804. [7] Y. N. Rabotnov, "Creep of Structure Elements," Naka, Moscow, 1966. [8] Y. A. Rossikhin and M. V. Shitikova, "Applications of Fractional Calculus to Dynamic Problems of Linear and Non Linear Hereditary Mechanics of Solids," Applied Mechanics Reviews, Vol. 50, No.1, 1997, pp. 15-67. doi:10.1115/1.3101682 [9] H. W. Lord and Y. Shulman, "A Generalized Dynamical Theory of Thermoelasticity," Journal of the Mechanics and Physics of Solids, Vol. 15, No.5, 1967, pp. 299-309. doi:10.1016/0022-5096(67)90024-5 [10] S. Kaliski, "Wave Equations of Thermoelasticity," Bulletin of the Polish Academy of Sciences Technology, Vol. 13, 1965, pp. 253-260. [11] H. M. Youssef, "Fractional Order Generalized Thermoelasticity," Journal of heat transfer, Vol. 132 No. 6, 2010. doi:10.1115/1.4000705 [12] H. M. Youssef, "Generalized Thermoelastic Infinite Medium with Cylindrical Cavity Subjected to Moving Heat Source," Mechanics Research Communications, Vol. 36, No. 4, 2009, pp. 487-496. doi:10.1016/j.mechrescom. 2008.12.004 [13] G. Hanig and U. Hirdes, "A Method for the Numerical Inversion of Laplace Transform," Journal of Computational and Applied Mathematics, Vol. 10, No. 1, 1984, pp. 113-132. doi:10.1016/0377-0427(84)90075-X

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