Thermal-Shock problem in Magneto-Thermoelasticity with ...researchpub.org/journal/jot/number/vol1-no3/vol1-no3-3.pdf · Mohamed F. Abbas, Magdy A. Ezzat, A. S. Sabbahand A. A. El-Barya

JOURNAL OF THERMOELASTICITY VOL.1 NO. 3 SEPTEMBER 2013

ISSN 2328-2401 (Print) ISSN 2328-241X (Online) http://www.researchpub.org/journal/jot/jot.html

16

Abstract— In this work we study the one-dimensional problem in

magneto-thermoelasticity for half space with thermal relaxation

time in a perfectly conducting medium. The Laplace transform

technique is used to solve the problem. Inverse Laplace transforms

are obtained in an approximation manner using asymptotic

expansions valid for small values of time. Numerical results for the

temperature distribution, thermal stress, displacement, the

induced magnetic, and the electric field are represented

graphically.

Index Terms — Magneto-Thermoelasticity; Perfectly

Conducting Medium; Maxwell's equation; Laplace transform

I. INTRODUCTION

Due to its many applications in the fields of geophysics,

plasma physics and related topics, increasing attention is being

devoted to the interaction between magnetic fields and strain in

a thermoelastic solid. In the nuclear field, the extremely high

temperatures and temperature gradients as well as the magnetic

fields originating inside nuclear reactors influence their design

and operations, [1].

Sherief and Yosef [2] studied a problem in

electromagneto thermoelasticity with thermal relaxation for a

half-space whose surface is subjected to a thermal shock and is

laid on a rigid foundation. In their work, Laplace transform

techniques are used to obtain the solution by a direct approach.

Wave propagation in the elastic medium and in the free space,

bounding it, is investigated. The solution of the problem is

obtained analytically using asymptotic expansions valid for

short times. The temperature, displacement, stress, the induced

magnetic and electric field distributions are obtained

Manuscript received July 21, 2013.

M. F. Abbas, Department of Basic and Applied Science, Arab Academy for

Science, Technology and Maritime Transport, P.O.BOX 1029 Alexandria,

Egypt; (e-mail: [email protected]).

M. A. Ezzat, Department of Mathematics, Faculty of Education, Alexandria

University, Egypt.

A. S. Sabbah, Department of Mathematics, Faculty of Science, Zagazig

University, Egypt.

A. A. El-Bary, Department of Basic and Applied Science, Arab Academy

for Science, Technology and Maritime Transport, P.O.BOX 1029

Alexandria, Egypt; (e-mail: [email protected]).

analytically. The numerical values of these functions are

represented graphically.

Sherief and Ezzat [3] studied the problem of an infinitely

long annular cylinder whose inner and outer surfaces are

subjected to known surrounding temperatures and are

traction-free is considered in the presence of an axial uniform

magnetic field. The problem is in the context of generalized

magneto-thermoelasticity theory with one relaxation time. The

Laplace transform with respect to time is used. A numerical

method based on a Fourier-series expansion is used for the

inversion process. Numerical computations for the temperature,

displacement and stress distributions as well as for the induced

magnetic and electric fields are carried out and represented

graphically.

Sherief and Ezzat [4] studied the one-dimensional problem

of distribution of thermal stresses and temperature is considered

in a generalized thermoelastic electrically conducting

half-space permeated by a primary uniform magnetic field when

the bounding plane is suddenly heated to a constant temperature.

The Laplace transform technique is used to solve the problem.

Inverse transforms are obtained in an approximate manner using

asymptotic expansions valid for small values of time.

Lord and Shulmann [5] introduced the theory of

generalized thermoelasticity with one relaxation time by

postulating a new law of heat conduction to replace the classical

Fourier law. This law contains the heat flux vector as well as its

time derivative. It contains also a new constant that acts as a

relaxation time. The heat equation of this theory is of the

wave-type, ensuring finite speeds of propagation for heat and

elastic waves. The remaining governing equations for this

theory, namely, the equations of motion and the constitutive

relations remain the same as those for the coupled and the

uncoupled theories. This theory was extended by Dhaliwal and

Sherief [6] to general anisotropic media in the presence of heat

sources.

Among the authors who considered the generalized

magneto-thermoelastic equations are Nayfeh and Nasser [7]

who studied the propagation of plane waves in a solid under the

influence of an electromagnetic field. They have obtained the

governing equations in the general case and the solution for

some particular cases.

Thermal-Shock problem in

Magneto-Thermoelasticity with Thermal

Relaxation for a Perfectly Conducting

Medium

Mohamed F. Abbas, Magdy A. Ezzat, A. S. Sabbah and A. A. El-Bary

a

mailto:[email protected]



17

Choudhuri [8] extended the results in [7] to the case of

rotating media. In dealing with generalized or coupled

thermoelastic problems the solution is usually obtained by

introducing a potential function. This approach has its

limitations. It is preferable in general to formulate the problem

in terms of the physical quantities involved since the boundary

and initial conditions are directly expressed in terms of these

quantities.

A new model of the magneto-thermoelasticity theory has

been constructed in the context of a new consideration of heat

conduction with fractional derivative by Ezzat [9]. A

one-dimensional application for a conducting half-space of

thermoelectric elastic material, which is thermally shocked in

the presence of a magnetic field, has been solved using Laplace

transform and state-space techniques. According to the

numerical results and its graphs, a conclusion about the new

theory of magneto-thermoelasticity has been constructed. The

theories of coupled magneto-thermoelasticity and of

generalized magneto-thermoelasticity with one relaxation time

follow as limited cases. The result provides a motivation to

investigate conducting thermoelectric materials as a new class

of applicable materials.

Ezzat and A. A. El-Bary [10] studied a one-dimensional

model of the two-temperature generalized

magneto-thermoelasticity theory with one relaxation time in a

perfect conducting medium. The state space approach

developed in [11] is adopted for the solution of one-dimensional

problems for any set of boundary conditions. The resulting

formulation together with the Laplace transform techniques are

applied to a specific problem of a half-space subjected to

thermal shock and traction-free surface. The inversion of the

Laplace transforms is carried out using a numerical approach.

Numerical results are given and illustrated graphically for the

problem. Some comparisons have been shown in figures to

estimate the effects of the two-temperature parameter and the

applied magnetic field.

Ezzat [12] studied the state space formulation for

one-dimensional problems of generalized

magneto-thermoelasticity with two relaxation times in a

perfectly conducting medium. The Laplace transform technique

is used. The resulting formulation is applied to a thermal shock

problem, a problem of a layer medium and a problem for the

infinite space in the presence of heat sources. A numerical

method is employed for the inversion of the Laplace transforms.

Numerical results are given and illustrated graphically for the

problem considered.

I. MATHEMATICAL FORMULATION OF THE

PROBLEM

We shall consider a homogeneous, isotropic, thermoelastic

solid of infinite conductivity o occupying the region 0x ,

where the x-axis is taken perpendicular to the bounding plane of

the half-space pointing inwards. A constant magnetic field with

component )0,h,0(h and electric field ).E,0,0(E It is

assumed that the state of the medium depends only on x and t

and that the displacement vector has

components )0,0),t,x(u(u .

Maxwell's equations in vector form can be written as

tcurl

DJh , (2.1)

t

curl

BE , (2.2)

,0div B

0div E , (2.3)

o 0B (H h) ,

ED o , (2.4)

where J is the electric current density, oo and are the

magnetic and electric permeabilities, respectively and B ,

D are the magnetic and electric induction vectors, respectively.

These equations are supplemented by Ohm's law

)(

t0oo hH

uEJ , (2.5)

where o is the electric conductivity and u is the displacement

vector.

This equation can be linearized by neglecting small quantities of

the second order giving

0

tJ

00H

uE . (2.6)

For the thermoelasticty medium of prefect conductivity, the

induced electric field E will be

tH

00

uE . (2.7)

From which,

txHcurl

2

00

uE . (2.8)

Substitution from (2.8) into (2.2) yields

tx

uH

t

h 2

0

. (2.9)

From which,

x

uHh 0

. (2.10)

The Lorentz force F is given by



18

BJF . (2.11)


2

2

000t

uH

x

hJ

. (2.12)


2

2

00000xt

uH

x

hHF . (2.13)

The strain components are given by

x

uexx

, 0eeeee xyyzxzzzyy . (2.14)

Stress components are given by

ij0ijijij )TT(ee2 , (2.15)

where and are Lamé’s modulii, T is the absolute

temperature of the medium, is the material constant given by

)23( where is the coefficient of linear thermal

expansion and 0T is a reference temperature assumed to be such

that 1T/)TT( 00 .

From which, the constitutive equation is given by

)TT(x

u)2( 0xx

. (2.16)

The equation of motion is given by

iij,ji uF , (2.17)

where is the density.

Substitution from (2.13) and (2.16) into (2.17) yields

2 2

2 2

2 2

0 0 0 0 0 02 2

u u T( 2 )

t x x

u uH H H

x t

. (2.18)

The energy equation in the absence of heat sources can be

written as

x

uTTC

ttx

Tk 0E2

2

02

2

, (2.19)

where 0 is the relaxation time, k is the thermal conductivity of

the medium , Ec the specific heat at constant strain.

For simplification we shall use the following non-dimensional

variables:

,xcx 0* ,tct 2

0* ,ucu 0

* ,2

)TT( 0

,H

hh

0

* ,cH

EE

000

*

,c 0

20

*0 ,xx*

xx

(2.20)

where k/cE , /)2(c20 .

Using the above non-dimensional variables, the governing

equations (2.7), (2.10), (2.16) and (2.18)-(2.19) will reduce to

,t

uE

(2.21)

,x

uh

(2.22)

,)x

u(2

xx

(2.23)

,t

u)V1(

xx

u)1(

2

22

22

2

2

(2.24)

,x

u)

tt(

x12

2

02

2

(2.25)

together with the boundary conditions

(i) 0xat0)t,x(xx ,

(ii) 0xat )t,x( 0 , (2.26)

where

,)2(c

T

E

20

1

,

)2(

H200

2

,

c

cV 0

,1

c00

2

.

)2(2

II. FORMULATION AND SOLUTION IN THE LAPLACE

TRANSFORM DOMAIN

Taking the Laplace transform of both sides of the Equations

(2.21)-(2.25) defined by the relation

0

st dt)t,x(fe)s,x(f ,

(3.1)



19

we get

,usE (3.2)

,x

uh

(3.3)

,x

u2xx

(3.4)

,x

us)V1(x

u)1( 22

22

2

2

(3.5)

)ss(

xx

u)ss( 2

02

22

01 . (3.6)

The boundary conditions (2.26) will be

(i) ,0xat0xx

(ii) 0xats

0

. (3.7)

Eliminating u and between Equations (3.5)-(3.6), we

obtain the following forth-order partial differential Equations

satisfied by u and ,

,0uCx

uB

x

u2

2

4

4

(3.8)

,0Cx

Bx 2

2

4

4

(3.9)

where )1(

)ss(s)V1()1)(ss(B

2

201

2222

20

,

)1(

s)V1)(ss(C

2

222

20

.

Equations (3.8)-(3.9) can be factorized as

,0u)kD)(kD( 22

221

2 (3.10)

,0)kD)(kD( 22

221

2 (3.11)

where x

D

and

21k and

22k are the roots of the

characteristic equation

.0CBkk 24 (3.12)

The solutions of (3.10)-(3.11) are

,ececuxk

2xk

121

(3.13)

.ececxk

2xk

121 (3.14)

Substitution from (3.13)-(3.14) into (3.6) and equating the

coefficients of xk1e

and

xk2e

, we get

)ss(k

ck)ss(c

20

21

112

011

, .

)ss(k

ck)ss(c

20

22

222

012

(3.15)

Substitution from (3.15) into (3.14), we get

1

2

k x1 1

2 2

1 02

1 0

k x2 2

2 2

2 0

k ce

k (s s )(s s )

k ce

k (s s )

. (3.16)

Substitution from (3.13) and (3.16) into (3.4) yields

1 2

1

2

k x k x

1 1 2 2

k x1 12 2 2

xx 1 02

1 0

k x2 2

2 2

2 0

k c e k c e

k ce

k (s s )(s s )

k ce

k (s s )

. (3.17)

Substitution from (3.7) into (3.16)-(3.17), we get the following

system of linear Equations

11 2 2

1 0

022 2 2 2

2 0 1 0

kc

k (s s )

kc .

k (s s ) s(s s )

(3.18)

2

1 0 11 12 2

1 0

2

1 0 22 22 2

2 0

(s s )kc k

k (s s )

(s s )kc k 0.

k (s s )

(3.19)

Solving the above system of linear Equations, we can get the

values of 1c and 2c .

III. INVERSION OF THE LAPLACE TRANSFORMS

We shall outline the numerical inversion method used to find the

solution in the physical domain.

Let )s(f be the Laplace transform of a function )t(f . The

inversion formula for Laplace transforms can be written as

ds (s)f ei2

1 = f(t) ts

i + d

i - d

, (4.1)



20

where d is an arbitrary real number greater than all the real parts

of the singularities of )s(f .

Taking iyds , Equation (4.1) will be

dy y) i + (df e 2

e = f(t) y t i

-

td

. (4.2)

Expanding the function )t(fe)t(h td in a Fourier series in

the interval L2,0 , we obtain the approximate formula [13]

DE)t(f)t(f , (4.3)

,c + c 2

1 = (t)f k

1=k0

for L2 t 0 , (4.4)

) L / k i + d (f e Re L

e = c L t / k i

td

k . (4.5)

and DE the discretization error, can be made arbitrarily small by

choosing d large enough [13].

As the infinite series in Equation (4.4) can only be summed up to

a finite number N of terms, the approximate value of )t(f

becomes

,c + c 2

1 = ) t ( f k

N

1=k0N for L2 t 0 . (4.6)

Using (4.6) to evaluate )t(f , we introduce a truncation error LE

that must be added to the discretization error to produce the total

approximation error.

Two methods are used to reduce the total error. First, the

`Korrecktur-method is used to reduce the discretization error.

Next, the -algorithm is used to reduce the truncation error and

therefore to accelerate convergence, [14].

The Korrecktur-method uses the following formula to evaluate

the function )t(f

DLd2 E)tL2(fe)t(f)t(f

, (4.7)

where the new discretization error DD EE , [13].

Thus, the approximate value of )t(f becomes

)tL2(fe)t(f)t(f NdL2

NNK

, (4.8)

where N is an integer such that NN .

We shall describe the -algorithm that is used to accelerate the

convergence of the series in (4.6).

Let N be an odd natural number and let

c = s k

m

1=km , (4.9)

be the sequence of partial sums of (4.6). We define the

-sequence by

. . . , 3 , 2 , 1 = m , s = , 0 = mm,1m,0 . (4.10)

and

n+1,m n 1 ,m+1

n,m+1 n,m

1= + , n , m = 1 , 2 , 3 , ...

(4.11)

It can be shown that [13] the sequence

.....,,,.......,, 1,N1,31,1 (4.12)

converges to 2

cE)t(f 0

D faster than the sequence of partial

sums

.....,3,2,1m,sm (4.13)

The actual procedure used to invert the Laplace Transforms

consists of using equation (4.8) together with the -algorithm.

The values of d and L are chosen according the criteria outlined

in [13].

IV. NUMERICAL RESULTS

The copper material was chosen for purposes of numerical

evaluations. The constants of the problem were taken as

42 293T0 02.00

5)10(39.1V 8954 381cE

5t )10(78.1 386k 10)10(76.7

10)10(86.3 )36/()10( 90 7

0 )10(7.5

70 )10(4 1H0 8838

The computations were carried out for three values of time,

namely for 1.0t , 2.0t and 3.0t . The displacement,

stress, temperature, induced electric field and induced magnetic

field distributions are shown in Figures 1–5, respectively. In all

figures, it is clear that all the functions considered have a

non-zero values only in a bounded region of space at a given

instant. Outside this region the values vanish and this means that

the region has not felt thermal disturbance yet. At different

instants, the non-zero region moves forward correspondingly

with the passage of time, [15]. This indicates that heat

propagates as a wave with finite velocity in medium. It is

completely different from the case for the classical theories of

thermoelasticity where an infinite speed of propagation is

inherent and hence all the considered functions have a non-zero

(although may be very small) value for any point in the medium.

Due to the coupling between the governing equations, the

arrival of any wave front at certain position affects all the

considered functions. By numerical experimentation on the

values of the functions just before and just after the arrival of the

wave fronts and by analogy to the wave propagation in

generalized thermoelasticity Sherief and Hamza [16], it was

found that the first and second waves are mainly

thermo-mechanical in nature while the third wave affects

diffusion mainly.



21

From Fig. 1, it can be observed that the medium along the axis x

adjoining the surface undergoes expansion deformation because

of thermal shock while the others compressive deformation. The

deformation is a dynamic process. With the passage of time, the

expansion region moves insides gradually and becomes larger

and larger. Thus the displacement becomes larger and larger. At

a given instant, the non-zero region of the displacement is finite,

which is due to the wave effect of heat. It indicates that heat

transfers into the deep of the medium with a finite velocity with

the time passing. The more the considered instant, the more the

thermal disturbed region and the displacement correspondingly.

0.0 0.4 0.8 1.2 1.6 2.0

-0.3

-0.2

-0.1

0.0

0.1

t = 0.1

t = 0.2

t = 0.3

Figure 1: Displacement distribution

0.0 0.4 0.8 1.2 1.6 2.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

t = 0.1

t = 0.2

t = 0.3

Figure 2: Stress Distribution

0.0 0.4 0.8 1.2 1.6 2.00.0

0.4

0.8

1.2

t = 0.1

t = 0.2

t = 0.3

Figure3: Temperature distribution

0.0 0.4 0.8 1.2 1.6 2.0

-0.8

-0.4

0.0

0.4

0.8

t = 0.1

t = 0.2

t = 0.3

Figure 4: Induced electric field distribution



22

0.0 0.4 0.8 1.2 1.6 2.0

-1.2

-0.8

-0.4

0.0

0.4

t = 0.1

t = 0.2

t = 0.3

Figure 5: Induced magnetic field distribution

REFERENCES

[1] J. L. Nowinski, Theory of Thermoelasticity with Applications,

Sijthoff & Noordhoff International, Alphen Aan Den Rijn, 1978.

[2] H.Sherief, H.Yosef, short time solution for a problem in

magnetothermoelasticity with thermal relaxation, Journal of

Thermal Stresses, vol.27, pp. 537–559, 2004.

[3] H.Sherief , M.Ezzat, A problem in generalized

magneto-thermoelasticity for an infinitely long annular cylinder,

Journal of Engineering Mathematics,vol.34,pp.387–402, 1998.

[4] H.Sherief, M.Ezzat, M.A., A thermal-shock problem in

magnetothermoelasticity with thermalrelaxation, Int.J. Solids

Structures, vol.33, pp. 4449-4459, 1996.

[5] H. Lord, Y. Shulman, A generalized dynamical theory of

thermo-elasticity, J. Mech. Phys. Solid, vol.15, pp.299–309, 1967.

[6] R. Dhaliwal and H. Sherief, Generalized thermoelasticity for

anisotropic media. Q. Appl. Math., vol. 33, pp.1–8, 1980.

[7] A. Nayfeh and S. Nemat-nasser, electromagneto-thermoelastisity

plane waves in solid with Thermal Relaxation J. Appl. Me&. Serie

Es., vol.39, pp.108-113, 1972.

[8] S. Choudhuri, electro-magneto-thermo-elastic Plane waves in

rotating media with thermal Relaxation. S. K. Int. J. Engng Sci.,

vol.22, pp.519-530, 1984.

[9] M. Ezzat, Magneto-thermoelasticity with thermoelectric properties

and fractional derivative heat transfer, Physica B, vol. 406, pp.

30-35, 2011.

[10] M.A.Ezzat, A.A.El-Bary, State space approach of two-temperature

magneto-thermoelasticity with thermal relaxation in a medium of

perfect conductivity, Int. J. Engng Sci.,vol. 47, pp. 618–630,2009.

[11] M.Ezzat, State space approach to unsteady two-dimensional free

convection flow through a pours medium, Can. J. Phys., vol.72, pp.

311-317, 1994.

[12] M. Ezzat, state space approach to generalized

Magneto-thermoelasticity with two relaxation Times in a medium

of perfect conductivity, Int. J. Engng Sci.,vol. 35, pp. 741-752,

1997.

[13] G. Honig and U. Hirdes, A Method For The Numerical Inversion of

the Laplace Transform, J. Comp. Appl. Math., vol.10,pp.

113-132,1984.

[14] H. Sherief, Problem in electromagnetic thermoelastisity for an

infinitely long solid conducting Circular cylinder with thermal

relaxation, Int. J. Engng Sci., vol. 32,pp.1137-1149,1994.

[15] H.Tianhu, S.Yapeng, T. Xiaogeng, A two-dimensional generalized

thermal shock problem for a half-space in

electromagneto-thermoelasticity, Int. J. Engng Sci.,vol. 42 ,pp.

809-823,2004.

[16] H. Sherief, F. Hamza, Generalized thermoelastic problem of a thick

plate under axisymmetric temperature distribution, J. Therm.

Stresses, vol.17,pp. 435–453,1994.

Documents

Thermal-Shock problem in Magneto-Thermoelasticity with ...researchpub.org/journal/jot/number/vol1-no3/vol1-no3-3.pdf · Mohamed F. Abbas, Magdy A. Ezzat, A. S. Sabbahand A. A. El-Barya