Effect of Rotation and Magnetic field on Generalized ...m-hikari.com/atam/atam2011/atam1-4-2011/abdallaATAM1-4-2011.pdf · Effect of Rotation and Magnetic field on Generalized Thermo-Viscoelastic

Adv. Theor. Appl. Mech., Vol. 4, 2011, no. 1, 15 - 42

Effect of Rotation and Magnetic field on Generalized

Thermo-Viscoelastic In an infinite Circular Cylinder

A. M. Abd-Alla*, S. M. Abo-Dahab* and F. S. Bayones **

*Mathematics Department, Faculty of Science Taif University, Saudi Arabia

**Mathematics Department, Faculty of Science

Umm Al-Qura University, P. O. Box 10109, Makkah, Saudi Arabia

Abstract

The present paper deals with the propagation of waves in a homogeneous isotropic rotating generalized thermoelastic solid under influences of the magnetic field, initial stress and viscosity. The effect of rotation, magnetic field, thermal relaxation times and pressure on wave propagation in a generalized viscoelastic medium under influence of time-harmonic source are point out. Effects of rotation, thermal relaxation, magnetic field, viscosity and pressure on the propagation of waves in the transformed domain are investigated analytically by using the inverse Hankel transform. The solution in the transformed domain obtained by using Hankel transform. Numerical inversion of both transforms is carried out to obtain the temperature, stress and displacement distribution in the physical domain. Numerical results are represented graphically. Comparison are made with the results predicted by the theory of generalized magneto-thermoelasticity with one relaxation time in the absence of rotation.

Keywords: Magneto-thermoelasticity, relaxation times, thermal stress,

viscoelasticity, rotating cylinder, energy dissipation

1-Introduction The dynamical problem of magneto-thermoelasticity has received attention in the literature during the past decade. In recent years the theory of magneto-thermoelasticity which deals with the interactions among strain, temperature and electromagnetic field has drawn the attention of many researchers .The linear theory

16 A. M. Abd-Alla, S. M. Abo-Dahab and F. S. Bayones of elasticity of paramount importance in the stress analysis of steel, which is the commonest engineering structural material. To a lesser extent, linear elasticity describes the mechanical behavior of the other common solid materials, e. g. concrete, wood and coal. Thermoelasticity theories, which admit a finite speed for thermal signals, have been receiving a lot of attention for the past four decades. In contrast to the conventional coupled thermoelasticity theory based on a parabolic heat equation, Biot [1], which predicts an infinite speed for the propagation of heat, these theories involve a hyperbolic heat equation and are referred to as generalized thermoelasticity theories. In recent years, wide spread attention has been taken to the generalized thermoelastic theories due to its utilitarian aspects in determination of the finite speed of heat transportation. The problem of rotating disks or cylinders has its application in high-speed cameras, steam and gas turbines, planetary landings and in many other domains. Various authors have formulated these generalized theories on different grounds. Lord and Shulman [2] have developed a theory on the basis of a modified heat conduction law which involves heat-flux rate, Green and Lindsay [3] have developed a theory by including temperature-rate among the constitutive variables. Lebon [4] has formulated a theory by considering heat-flux as an independent variable. Also some problems in thermoelastic rotating media are due to Roychoudhuri and Debnath [5-6] and Othman [7-9]. These problems are based on more realistic elastic model since earth, moon and other planets have angular velocity. The solution to the problems of homogeneous isotropic rotating cylinders may be found in Love [10] and Sokolnikoff [11]. Abd-Alla and Abo-Dahab [12] and Sharma et al. [13] studies effect of the time-harmonic source in a generalized thermoelastic. Chandrasekharaiah [14], Green and Naghdi [15], Hossen and Mallet [16], Chandrasekharaiah and Srinath [17] and Quintanila [18] discussed the problem of thermoelasticity without energy dissipation. Nayfeh and Nasser [19] used the Cagnird- De Hoop [20] method to develop the displacement and temperature fields in a homogeneous isotropic generalized thermoelastic half-space subjected on the free surface to an instantaneously applied heat source. Some problems of generalized thermoelasticity are discussed by Nayfeh and Nasser [21], Noda and Furukawa [22] and Schoenberg and Censor [23]. Some problems in thermoelasticity without energy dissipation of materials with microstructure or voids are illustrated [24-25]. Interaction due thermoelasticity and another effects without energy dissipation is pointed out by Ciarletta et al. [26] and Quintanilla [27]. Roychoudhuri and Mukhopadhyay [28] studied the effect of rotation and relaxation times on plane waves in generalized thermo-viscoelasticity. Roychoudhuri and Banerjee [29] investigated the magneto-thermoelastic interactions in an infinite viscoelastic cylinder of temperature rate dependent material subjected to a periodic loading. Kraus [30] pointed out the electromagnetics. Song et al. [31] studied the transient disturbance in a half space under thremoelasticity with tow relaxation time due to moving internal heat source. Abd-Alla and Mahmoud [32] investigated magneto-thermoelastic problem in rotation non-homogeneous orthotropic hollow cylinder under the hyperbolic heat conduction model. Body wave propagation in rotating thermoelastic media was investigated by

Effect of rotation 17 Sharma and Grover [33]. Dai and Wang [34] studied the magneto-elastodynamic stress and perturbation of magnetic field vector in an orthotropic laminated hollow cylinder. Abd-Alla et al., [35] studied thermal stresses in a non-homogeneous orthotropic elastic multilayered cylinder. Sharma [36] investigated the wave propagation in anisotropic generalized thermoelastic media, In this paper, we investigated the effect of magnetic field, rotation, thermal relaxation time and pressure on wave propagation in a generalized viscoelastic medium under influence of rotation. Using Hankel transform, inverse Hankel transform to find the general solution and determined the displacements, temperature and stress components. Some special cases are studied. Also, the study of thermoelasticity without energy dissipation and discuses some special cases from this case. Finally, representation this cases by graphical. 2- Formulation of the problem and boundary conditions Let us consider the electromagnetic field governs by Maxwell equations, under consideration that the medium is a perfect electric conductor taking into account absence of the displacement current (SI) in the form as in Roychoudhuri and Mukhopadhyay [31]

,

,

0,

0,

e

e

J curlh

h curlEt

divh

divE

uE Ht

μ

μ

=

∂− =

∂=

=

⎛ ⎞∂= − ×⎜ ⎟

∂⎝ ⎠

ur r

rur

r

ur

rur uur

(2.1) where,

0 0( ), .h curl u H H H h= × = +r r uuur uur uuur r

(2.2)

where, h is the perturbed magnetic field over the primary magnetic field, E is the electric intensity, J is the electric current density, eμ is the magnetic permeability,

H is the constant primary magnetic field and u is the displacement vector. Let us consider a homogeneous isotropic thermoelastic half-space in an undisturbed state, initially at uniform temperature derivation T0 and upon a primary

magnetic field H acting on θ -direction. We take a Kelvin-Voigt type as viscoelastic medium. We take the origin of cylindrical coordinates system ),,( zr θ on the surface

0=z and z-axis pointing vertically into the medium and the half-space occupies the region 0≥z . A normal force or heat source acting at a point on the surface 0=z of the medium and hence, all the quantities are independent of the θ coordinate. The

18 A. M. Abd-Alla, S. M. Abo-Dahab and F. S. Bayones

elastic medium is rotating uniformly with an angular velocity ,nΩ =Ω

ur r where n is a

unit vector representing the direction of the axis of rotation. All quantities considered will be functions of the time variable t and of the coordinates r and z. The displacement equation of motion in the rotating frame has two additional term in Schoenberg and Censor [23], centripetal acceleration, uΩ×Ω×

ur ur r due to time varying

motion only where, ),0,( zr uuu = is the dynamic displacement vector and (0, ,0)Ω = Ω

ur. These terms do not appear in non-rotating media.

The equations of motion and heat conduction in the context of generalized thermoelasticity for a Kelvin-Voigt type are given the forms Roychoudhuri and Mukhopadhyay [31] with some slight changes in symbols

, ( ) , , 1,2,3,ji j i i iF u u i jσ ρ••⎡ ⎤+ = + Ω×Ω× =⎢ ⎥⎣ ⎦

ur ur r (2.3)

, ,, 1 0 1( ) ( ), 1,2,3j j j jjj vKT c T T T u u jρ τ γ δτ• •• • ••

= + + + = , (2.4)

3,2,1,,2))(( 2 =++−=•

jieTTe ijmijkkmij μτδτγλτσ (2.5)

where, ρ is density of the material, K is thermal conductivity, vc is specific heat of

the material per unit mass, 21 ττ and are thermal relaxation parameter, tα is

coefficient of linear thermal expansion, μλ and are Lame elastic constants, θ is the absolute temperature, )23( μλαγ += t , 0T is reference temperature solid, T is temperature difference )( 0T−θ , 0τ is the mechanical relaxation time due to the

viscosity, 01m t

τ τ ∂⎛ ⎞= +⎜ ⎟∂⎝ ⎠, F is the Lorenz’s body forces and ijσ is the mechanical

stress. Maxwell's electro-magnetic stress tensor ijτ is given by

( . ) , , 1, 2,3.ij e i j j i ijH h H h H h i jτ μ δ⎡ ⎤= + − =⎣ ⎦uur r

(2.6)

The basic equations are studied for LS model 2 1( . ., 0, 0, 1)i e τ τ δ= > = GL model

2 1( . ., 0, 0)i e τ τ δ≥ > = and magneto-thermo-viscoelasticity without energy dissipation. The non-vanishing displacement components are

( ,0, ), ( , , ) ( , , )r z r r z zu u u u u r z t and u u r z t= = =r

(2.7) so that

1, , , ( ).2

r r z z rrr zz rz

u u u u ue e e er r z r zθθ

∂ ∂ ∂ ∂= = = = +∂ ∂ ∂ ∂

(2.8)

The dynamic equation of motion under the effect of the rotation and magnetic field are given by Roychoudhuri and Banerjee [29]

Effect of rotation 19

].[1

],[)(1

22

2

22

2

zz

zrzzzrz

rr

rrrrzrr

utuf

rzr

utuf

rzr

Ω−∂∂

=++∂∂

+∂∂

Ω−∂∂

=+−+∂∂

+∂∂

ρτστ

ρσστσθθ

(2.9)

The generalized equation of heat conduction is given by Roychoudhuri and Banerjee [29]

21 0 1( ) .( )vK T c T T T u uρ τ γ δτ

• •• • ••

∇ = + + ∇ +r r

(2.10) where,

2

20 2

2

( 2 ) ( ),

( 2 ) ( ),

( 2 ) ( ),

.

r z rrr m m m

r z rm e m m

z r rzz m m m

r zrz m

u u u T Tr z r

u u uH T Tr z r

u u u T Tz r r

u uz r

θθ

σ τ λ μ τ λ τ λ γ τ

σ τ λ μ μ τ λ τ λ γ τ

σ τ λ μ τ λ τ λ γ τ

τ τ μ

•

•

•

∂ ∂= + + + − +

∂ ∂∂ ∂⎡ ⎤= + + + + − +⎣ ⎦ ∂ ∂

∂ ∂= + + + − +

∂ ∂∂ ∂⎛ ⎞= +⎜ ⎟∂ ∂⎝ ⎠

(2.11) Substituting from Eqs. (2.11) into Eqs. (2.9), we obtain

22 2 20 0 02 2

2 2 22 20 22 2

1( 2 ) ( 2 ) ( 2 )

( ) ( )

r r rm e m e m e

z r rm e m r

u u uH H Hr r r ru u uH T T u

r z z r t

τ λ μ μ τ λ μ μ τ λ μ μ

τ λ μ μ μτ γ τ ρ•

∂ ∂⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ + + + + − + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦∂ ∂⎛ ⎞∂ ∂ ∂∂⎡ ⎤+ + + + − + = − Ω⎜ ⎟⎣ ⎦ ∂ ∂ ∂ ∂ ∂⎝ ⎠

(2.12) where, zr fandf are Lorenz's force components that defined by Kraus [30] as:

2 22

2

2 22

2

1 ,r z r zr e

r zz e

u u u uf Hr r z r r z

u uf Hr z z

φ

φ

μ

μ

⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞= + + +⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦⎛ ⎞∂ ∂

= +⎜ ⎟∂ ∂ ∂⎝ ⎠

(2.13)

and the heat conduction equation is

2 2 2 2

1 0 12 2 2 2

1 r r zv

u u uK T c T Tr r r z t t t t r r z

ρ τ γ δτ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎛ ⎞+ + = + + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(2.14)

20 A. M. Abd-Alla, S. M. Abo-Dahab and F. S. Bayones 3- Boundary conditions The boundary conditions are given by

0 ( ) , 0 ,2

0 , 0

i t

z z z z r z r zp r e

rT h T o n zz

ωδσ τ τ τπ

−−′ ′+ = + =

∂+ = =

∂

(3.1a)

this boundary conditions represent that body elastic on drag stresses under linear heat boundary on the surface

0

0 , 0 ,

( ) , 02

z z z z r z r zi tp r eT h T o n z

z r

ω

σ τ τ τ

δπ

−

′ ′+ = + =

∂+ = =

∂

(3.1b)

these boundary conditions represent that body elastic no contain drag stresses under linear heat boundary on the surface. 4- Solution of the problem The displacement vector u can be written in the form

, (0, , 0)u φ ψ ψ ψ= ∇ + ∇ × = −r ur ur ur ur

(4.1) where, the two functions ψφ and are known in the theory of elasticity, by Lam'e potentials resulting irrotational and rotational part of the displacement vector u , respectively, which may reduce to

, .r zu ur z z r rφ ψ φ ψ ψ∂ ∂ ∂ ∂

= + = − −∂ ∂ ∂ ∂

(4.2)

Substituting from Eq. (4.2) into (2.12) and (2.14), and taken the solution of the considered physical variable can be decomposed in term of normal modes as the following form

tiezrTtzrT ωψφψφ −= ),](,,[),,](,,[ ***

(4.3) where, ω is the frequency, we get

2 * *1 2( ) ,a a Tφ∇ + = (4.4)

2 *32

1( ) 0,ar

ψ∇ − + = (4.5) *2

2*

12 )( φεε ∇−=+∇ T (4.6)

where,


* *2 2 2 22 1

1 2 3 12 * 2 2 * 2 * 21 1 2

* 2* *0 1 0

2 0 2 22 21

* * 2 21 1 1 1 1 2

2 22 20

21

, , , ,[1 ] [1 ]

, , 1 , 1 ,[1 ]

21 , 1 , , ,

, ,

m H m H m

mv H

e HH H

v m

ia a ac R c R c a

i T T i iK c c R

i i c c

H RKa R Rc c

δ

δ

γτ ω τω ω ετ ρ τ τ

γ ω τ γε ε τ τ ω τ τ ωρ

λ μ μτ ωδτ τ ω τρ ρ

μρ ρ τ

Ω + Ω += = = =

′ ′+ +

= = = − = −′+

+= − = − = =

′= = =2 2

2* 2 2

1,r r r z∂ ∂ ∂

∇ = + +∂ ∂ ∂

(4.7) where, ε the thermoelastic coupling constant on the surface z=0. General solution of Eqs. (4.3)-(4.5) can be found, if we introduce the Hankel transform which is defined as

drrqJrtzrftzqf )(),,(),,( 00∫∞

= (4.8)

Substituting from Eq. (4.8) into Eqs. (4.3)-(4.5), the non-vanishing values of the potential functions * * *, and Tφ ψ in the transformed domain are obtained as

22 * *

1 22 ( ) ,d q a a Tdz

φ⎡ ⎤

− − =⎢ ⎥⎣ ⎦

(4.9)

22 *

32 ( ) 0,d q adz

ψ⎡ ⎤

− − =⎢ ⎥⎣ ⎦

(4.10)

2 22 * 2 *

1 22 2( ) .d dq T qdz dz

ε ε φ⎡ ⎤ ⎡ ⎤

− − = − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(4.11)

Eliminating the temperature *T from Eqs. (4.9) and (4.11), we obtain 0][ *

22

14 =Ν+Ν− φDD (4.12)

where, 2 4 2

1 1 1 2 2 2 1 1 2 2 1 12 ( ) , ( ) .q a a q a a q aε ε ε ε εΝ = − + − Ν = − + + + (4.13) Eqs. (4.9)-(4.12) tend to the solutions

1 2* ,z zAe Beξ ξφ − −= + (4.14) 3* ,zCe ξψ −= (4.15)

1 2*1 2

2

1 z zT AQ e BQ ea

ξ ξ− −⎡ ⎤= +⎣ ⎦ (4.16)

where, A, B and C are arbitrary constants we can determined by using boundary conditions and where, Re( ) 0, 1, 2,3i iξ ≥ = ,

2 2 2 21 1 1 2 2 1( ), ( )Q q a Q q aξ ξ= − − = − − (4.17)

where, 22

21 ,ξξ are the roots of equation

4 21 2 0,ξ ξ− Ν + Ν = (4.18)


2 23 3.q aξ = − (4.19)

From Eqs. (4.3), we have 1 2[ ] ,z z i tAe Be eξ ξ ωφ − − −= + (4.20) 3 ,z i tCe eξ ωψ − −= (4.21)

1 21 2

2

1 [ ] ,z z i tT AQ e BQ e ea

ξ ξ ω− − −= + (4.22)

{ }31 23[ ] ,zz z i t

ru q Ae Be C e eξξ ξ ωξ −− − −= − + − (4.23)

{ }1 2 21 2[ ] ,z z z i t

zu A e B e qCe eξ ξ ξ ωξ ξ− − − −= − + + (4.24) 1

32

2 2 21 0 1

2 2 22 0 2 3

{ [ ( )]

[ ( )] },

zi tzz e

zze

e A e M H q

B e M H q Ce M

ξω

ξξ

σ μ ξ

μ ξ

−−

−−

= − +

+ − + + (4.25) { }31 2*

4 5 6 ,zz zi trz me A e M Be M Ce Mξξ ξωτ μτ −− −−= + +

(4.26) 1 22 2 2 2 2

0 1 2{ ( ) ( )}z zi tzz eH e Ae q Be qξ ξωτ μ ξ ξ− −−′ = + + + (4.27)

where, * 2 * 2 * 2 2 2

1 1 1 2 0 12

* 2 * 2 * 2 2 22 2 2 2 0 2

2

*3 3 4 1 5 2

2 26 3 7 1 1 8 2 2

( 2 ) ( ) ,

( 2 ) ( ) ,

2 , 2 , 2 ,

( ), ( ) , ( ) .

m m e

m m e

m

M q Q H qa

M q Q H qa

M q M q M q

M q M h Q M h Q

γτ λ μ ξ λτ τ μ ξ

γτ λ μ ξ λτ τ μ ξ

μτ ξ ξ ξ

ξ ξ ξ

⎡ ⎤= + + − + +⎢ ⎥⎣ ⎦⎡ ⎤

= + + − + +⎢ ⎥⎣ ⎦

= − = =

= − = − = − (4.28) Defined the inverse Hankel transform:

dqqrJqtzqftzrf )(),,(),,( 00∫∞

=

Finally, using inverse Hankel transform into Eqs. (4.20)-(4.27), we get

1 20

0

[ ] ( ) ,z zi te Ae Be qJ qr dqξ ξωφ∞

− −−= +∫ (4.29)

30

0

( ) ,zi tCe e qJ qr dqξωψ∞

−−= ∫ (4.30)

3 1 21 2 0

2 0

1 [ ] ( ) ,z z zi tT e e AQ e BQ e qJ qr dqa

ξ ξ ξω∞

− − −−= +∫ (4.31)

{ }31 23 0

0

[ ] ( ) ,zz zi tru e q Ae Be C e qJ qr dqξξ ξω ξ

∞−− −−= − + +∫ (4.32)

{ }1 2 21 2 0

0

[ ] ( ) ,z z zi tzu e A e B e qCe qJ qr dqξ ξ ξω ξ ξ

∞− − −−= − + +∫ (4.33)


,)(})]([

)]([{

032

222

02

21

2201

0

32

1

dqqrqJMCeqHMeB

qHMeAez

ez

ezti

zz

ξξ

ξω

ξμ

ξμσ−−

−∞

−

++−+

+−= ∫

(4.34) { } ,)(0654

0

* 321 dqqrqJMCeMBeMeAe zzzm

tirz

ξξξω μττ −−−∞

− ++= ∫ (4.35)

.)()}()({ 02

222

12

0

20

21 dqqrqJqBeqAeeH zztiezz ξξμτ ξξω +++=′ −−

∞−

∫ (4.36)

Now discuss some special cases from above equations to determine the constants at these cases. 5- Special cases 5.1- Case (I) normal point load acting on the surface Using Eqs. (4.25)- (4.27) into the boundary condition (3.1a), we obtain

6 8 6 7

1 1

4 8 5 7

1

, ,M M M MA p B p

M M M MC p

= = −Δ Δ

−= −

Δ (5.1) where, are values

81 MM − in Eq. (4.28) and

01 6 1 8 2 7 3 4 8 5 7, ( ) ( ).

2pp M M M M M M M M M Mπ

= Δ = − − + − (5.2)


Fig. (1a) Effect of the magnetic field Fig. (1b) Effect of the initial stress

10 50 1 0

5 60 0

10 , 0.2, 0.1, 2(10 ) __,

5(10 ) , 10 ......

p H

H H

τ τ= Ω= = = =

= −−− =

5 100 0 1

10 10

10 , 0.2, 0.1, 10 __

3(10 ) , 5(10 )......

H p

p p

τ τ= Ω= = = =

= −−− =


Fig. (1c) Effect of the rotation Fig. (1d) Effect of the viscosity

5 10

0 0 110 , 10 , 0.1, 0.2__,0.3 , 0.4......

H p τ τ= = = = Ω=

Ω= −−− Ω=

5 100 1 0

0 0

10 , 10 , 0.1, 0.1__,0.2 , 0.3......

H p τ ττ τ

= = = =

= −−− =


5.2- Case (II) thermal point source acting on the surface Using Eqs. (4.25)-(4.27) into the boundary condition (3.1b), we obtain

2 6 3 5 3 4 1 6

1 1

1 5 2 4

1

[ ] [ ], ,

[ ] .

M M M M M M M MA p B p

M M M MC p

+ −= − =

Δ Δ−

=Δ

(5.3)

where, are values 1 8 1,M M p and− Δ in Eqs. (4.28) and (5.2).

Fig. (1e) Effect of the relaxation time

5 100 0 1

1 1

10 , 10 , 0.1, 0.2, 0.1__,0.2 , 0.3......

H p τ ττ τ

= = = Ω= == −−− =



10 50 1 0

5 60 0

10 , 0.2, 0.1, 2(10 ) __,

5(10 ) , 10 ......

p H

H H

τ τ= Ω= = = =

= −−− =

5 100 0 1

10 10

10 , 0.2, 0.1, 10 __

3(10 ) , 5(10 )......

H p

p p

τ τ= Ω= = = =

= −−− =



5 100 0 110 , 10 , 0.1, 0.2__,

0.3 , 0.4......H p τ τ= = = = Ω=

Ω= −−− Ω=

5 100 1 0

0 0

10 , 10 , 0.1, 0.1__,0.2 , 0.3......

H p τ ττ τ

= = = =

= −−− =



6- Thermoelasticity without energy dissipation The basic governing equations for motion and heat conduction with the body forces in the absence of the heat source for a homogeneous magneto-thermo-viscoelastic half-space for the case of thermoelasticity without energy dissipation are given as the form in Abd-Alla and Abo-Dahab [12]

( )2 202 ( . ) [ ( )],m m eu H u T u uμτ τ λ μ μ ν ρ

••⎡ ⎤∇ + + + ∇ ∇ − ∇ = + Ω×Ω×⎣ ⎦

rr r ur ur r

(6.1) ••••

∇+=∇ uTTcTK v .02* νρ . (6.2)

We define the non-dimensional quantities

( ) ( )

* *** 1 1

01 0 0 0

2 22 20 02 2 2 2 *0

2 1 2 *1

22 ** 2 * * 02

0 0 1 2* 2 2 2 21 1 1

, , , , , (1 ),

2 ( 2 ), , , , ,

, , , , .

r zr z m

m e v m em e HH H

m

v v

cu cur Tr t t u u Tc T T T t

H c HH Rc c R Rc K

v Tc Kc cc cc

ρω ρωω ω τ τγ γ

τ λ μ μ τ λ μ μτ μ μ ωρ ρ ρ τ

τ ωτ δ β βω ρ ρ

∂′ ′ ′ ′ ′ ′= = = = = = +′∂

+ + + +′= = = = =

Ω′ ′= Ω= = = =

(6.3) Upon using quantities (6.3) in Eqs. (6.1) and (6.2), we obtain

2 2 ( . ) ,u u T u uδ••

∇ +∇ ∇ −∇ = −Ωr

r r r (6.4)

Fig. (2e) Effect of the relaxation time

5 10

0 0 1

1 1

10 , 10 , 0.1, 0.2, 0.1__,0.2 , 0.3......

H p τ ττ τ

= = = Ω= == −−− =


••••

∇=−∇ uTT .*2

2*1 ββ . (6.5)

Substituting from Eq. (4.3) in Eqs. (6.4) and (6.5) assuming that the disturbance and temperature to be harmonic, one may get

*2

*1

2 )( Tbb =+∇ φ , (6.6)

0)1( *32

2 =+−∇ ψbr

, (6.7)

0)( *24

*3

2 =∇++∇ φεε T (6.8) where,

2 2 2 2 21 1

1 2 32 * 2 2 *

* 22 2 22 *2

3 4 0* * 2 21 1

( ) ( ), , ,(1 )

1, , , 1 .

p m H m

m

c cb b bc R

ir r r z

ω ωτ δ τ

β ωωε ε τ τ ωβ β

Ω + Ω += = =

′ ′+

∂ ∂ ∂ ′= = ∇ = + + = −′ ′ ′ ′∂ ∂ ∂

(6.9)

By using Hankel transform, we have 2

2 * *1 22 ( )d q b b T

dzφ

⎡ ⎤− − =⎢ ⎥

⎣ ⎦ , (6.10) 2

2 *32 ( ) 0d q b

dzψ

⎡ ⎤− − =⎢ ⎥

⎣ ⎦ , (6.11) 22 2

2 * *3 42 2( )d dq T q

dz dzε ε φ

⎡ ⎤ ⎡ ⎤− − = − −⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ . (6.12) Eliminating the temperature *T from Eqs. (6.10) and (6.12), we obtain

0][ *4

23

4 =Ν+Ν− φDD , (6.13) where,

2 4 23 1 3 4 2 4 1 3 4 2 3 12 ( ) , ( )q b b q b b q bε ε ε ε εΝ = − + − Ν = − + + + . (6.14)

Eqs. (6.10)-(6.13) tend to the solutions zz eLeL 21

21* ξξφ −− += , (6.15)

zeL 33

* ξψ −= , (6.16) 1 2* * *

1 1 2 22

1 z zT L Q e L Q eb

ξ ξ− −⎡ ⎤= +⎣ ⎦ (6.17)

where, A, B and C are arbitrary constants we can determined by using boundary conditions and where, 3,2,1,0)Re( =≥ iiξ ,

* 2 2 * 2 21 1 1 2 2 1( ), ( )Q q b Q q bξ ξ= − − = − − (6.18)

where, 22

21 ,ξξ are the roots of equation

042

34 =Ν+Ν− ξξ , (6.19)


322

3 bq −=ξ . (6.20) From Eqs. (4.3), we have

tizz eeLeL ωξξφ −−− += ][ 2121 , (6.21) tizeeL ωξψ −−= 3

3 , (6.22) tizz eeQLeQL

bT ωξξ −−− += ][1

21 *22

*11

2 , (6.23) { } tizzz

r eeLeLeLqu ωξξξ ξ −−−− −+−= 3213321 ][ (6.24)

{ } tizzzz eeqLeLeLu ωξξξ ξξ −−−− ++−= 221

32211 ][ , (6.25) 1

32

* 2 2 21 1 0 1

* 2 2 22 2 0 2 3 3

{ [ ( )]

[ ( )] },

zi tzz e

zze

e L e M H q

L e M H q L e M

ξω

ξξ

σ μ ξ

μ ξ

−−

−−

= − +

+ − + + (6.26) { }31 2*

1 4 2 5 3 6 ,zz zi trz me L e M L e M L e Mξξ ξωτ μτ −− −−= + +

(6.27) 1 22 2 2 2 2

0 1 1 2 2{ ( ) ( )}z zi tzz eH e L e q L e qξ ξωτ μ ξ ξ− −−′ = + + + (6.28)

where, * * 2 * 2 * 2 2 21 1 1 0 1

2

* * 2 * 2 * 2 2 22 2 2 0 2

2

* * * *3 3 4 1 5 2

* 2 2 * * * *6 3 7 1 1 8 2 2

( 2 ) ( ) ,

( 2 ) ( ) ,

2 , 2 , 2 ,

( ), ( ) , ( ) .

m m e

m m e

m

M q Q H qb

M q Q H qa

M q M q M q

M q M h Q M h Q

γτ λ μ ξ λτ μ ξ

γτ λ μ ξ λτ μ ξ

μτ ξ ξ ξ

ξ ξ ξ

⎡ ⎤= + + − + +⎢ ⎥⎣ ⎦⎡ ⎤

= + + − + +⎢ ⎥⎣ ⎦

= − = =

= − = − = −

(6.29)

Finally, using inverse Hankel transform into Eqs. (4.22)-(4.29), we get

dqqrqJeLeLe zzti )(][ 0210

21 ξξωφ −−∞

− += ∫ , (6.30)

dqqrqJeeL zti )(00

33ξωψ −

∞− ∫= , (6.31)

dqqrqJeQLeQLeb

T zzti )(][10

*22

*11

02

21 ξξω −−∞

− += ∫ , (6.32)

{ } dqqrqJeLeLeLqeu zzztir )(][ 03321

0

321 ξξξω ξ −−−∞

− ++−= ∫ , (6.33)

{ } dqqrqJeqLeLeLeu zzztiz )(][ 032211

0

221 ξξξω ξξ −−−∞

− ++−= ∫ , (6.34)


,)(})]([

)]([{

0332

222

0*22

21

220

*11

0

32

1

dqqrqJMeLqHMeL

qHMeLe

ze

z

ezti

zz

ξξ

ξω

ξμ

ξμσ

−−

−∞

−

++−+

+−= ∫ (6.35)

{ } ,)(06352410

* 321 dqqrqJMeLMeLMeLe zzzm

tirz

ξξξω μττ −−−∞

− ++= ∫ (6.36)

1 22 2 2 2 20 1 1 2 2 0

0

{ ( ) ( )} ( )z zi tzz eH e L e q L e q qJ qr dqξ ξωτ μ ξ ξ

∞− −−′ = + + +∫

(6.37) Now discuss some special cases from above equations to determine the constants at these cases. 6.1- Case (I) normal point load acting on the surface Using Eqs. (6.26)- (6.28) into the boundary condition (3.1a), we obtain

* * *6 8 6 7 4 8 5 7

1 2 32 2 2

[ ], , ,M M M M M M M ML p L p L p −= = − = −

Δ Δ Δ (6.38) where, the values 63 MM − take the form as in Eq. (4.28) and

*8

*7

*2

*1 ,, MandMMM take the form as in Eq. (6.29) and

* * * * * *02 6 1 8 2 7 3 4 8 5 7, ( ) ( )

2pp M M M M M M M M M Mπ

= Δ = − − + −. (6.39)



10 5

05 6

0 0

10 , 0.2, , 2(10 )__,

5(10 ) , 10 ......

p H

H H

= Ω= =

= −−− =

5 100

10 10

10 , 0.2, 10 __

3(10 ) , 5(10 )......

H p

p p

= Ω= =

= −−− =



5 10

0 10 , 10 , 0.2__,0.3 , 0.4......

H p= = Ω=

Ω= −−− Ω=

5 100 0

0 0

10 , 10 , 0.1__,0.2 , 0.3......

H p ττ τ

= = =

= −−− =

6.2- Case (II) thermal point source acting on the surface Using Eqs. (6.26)-(6.28) into the boundary condition (3.1b), we obtain

* *2 6 3 5 3 4 1 6

1 22 2

* *1 5 2 4

32

[ ] [ ], ,

[ ]

M M M M M M M ML p L p

M M M ML p

+ −= =

Δ Δ

−=

Δ (6.40) where, are values 63 MM − in Eq.(4.28) and *

8*7

*2

*1 ,, MandMMM in Eq. (6.29) and

2Δ Eq. (6.31).



10 5

0 05 6

0 0

10 , 0.2, 0.1, 2(10 ) __,

5(10 ) , 10 ......

p H

H H

τ= Ω= = =

= −−− =

5 100 0

10 10

10 , 0.2, 0.1, 10 __

3(10 ) , 5(10 )......

H p

p p

τ= Ω= = =

= −−− =



5 10

0 010 , 10 , 0.1, 0.2__,0.3 , 0.4......

H p τ= = = Ω=

Ω= −−− Ω=

5 100 0

0 0

10 , 10 , 0.1__,0.2 , 0.3......

H p ττ τ= = =

= −−− =

38 A. M. Abd-Alla, S. M. Abo-Dahab and F. S. Bayones 7- Numerical result and discussion Numerical calculations are carried out for (i) the space and time dependent of the temperature, (ii) the displacement and the stress components along the r-direction at different values of magnetic field, initial stress, rotation and relaxation time for LS model, GL model and GN model of thermoelasticity theory. We now present some numerical results for displacement, temperature, stress components using the inverse of Hankel transform. The numerical results are obtained taking into account evaluation an Copper material, the physical data for such material are given [31].

10 1 2 10 1 2

3 1 2 6 1

2 1 3 2 2 1 2

0 0

7 .55 10 , 3.86 10 ,

8.96 10 , 17.87 10 ,

3.98 10 , 3.845 10 ,300 .

t

v

kgm s kgm s

kgm s k

K kgk m s c m k sT k

λ μ

ρ α

θ

− − − −

− − − −

− − − −

= × = ×

= × = ×

= × = ×

= =

We take into account the time .01.0,2,1.0,5.0 12 ==== hzt ττ For, magneto-thermo-viscoelasticity with energy dissipation Case (I): normal point load acting on the surface Figs. (1a-e), display the radial variations for the temperature T, displacements radial ur and axial uz, normal stress zzσ and shear stress rzσ respectively with the influences of the magnetic field H0, initial stress P, rotation Ω , viscosity 0τ and relaxation times

1τ . From Fig. (1a), it is clear that the temperature decreases with an increasing of the magnetic field, the displacements and axial stress increase with an increasing of magnetic field and the shear stress decreases with an increasing of magnetic field. From Fig. (1b), it is seen that the temperature, displacements and stresses increase with an increasing of the initial stress P. Fig. (1c), shows that increased values of the temperature and shear stress with an increasing the rotation, while values of the displacement components and normal stress are take the same values (a slight change) at different values of rotation. Fig. (1d), displays that with an increasing of viscosity parameter there is an increasing in the temperature but there is no change in the displacement components, radial and shear stresses. From Fig. (1e), it is obvious that the temperature increases with an increasing of the relaxation time while effect of the relaxation time is a slight change of remainder components. Case (II): thermal point source acting on the surface Figs. (2a-e), display the radial variations of the temperature T, displacements radial ur and axial uz, normal stress zzσ and shear stress rzτ respectively with the influences of the magnetic field H0, initial stress P, rotation Ω , viscosity 0τ and relaxation times

1τ . From Figs. (2a), it is seen that the temperature, the displacements and axial stress increase with an increasing of magnetic field and the shear stress decreases with an increasing of magnetic field. From Fig. (2b), it is seen that the temperature, the

Effect of rotation 39 displacements and stresses increase with an increasing of the initial stress. From Figs. (2c,2d), it is concluded that with the effects of the rotation and viscosity have too slight change on the temperature, the displacement and stresses (i.e, this components only increasing with decreases the values of r). From Fig. (2e), it is obvious that the temperature increase with an increasing of the relaxation time while all remainder components are take near the same values with varies values of the relaxation times. For, magneto-thermo-viscoelasticity without energy dissipation Case (I): normal point load acting on the surface Figs. (3a-d), display the radial variations for the temperature T, displacements radial ur and axial uz, normal stress zzσ and shear stress rzτ respectively with the influences of the magnetic field H0, initial stress P, rotation Ω and viscosity 0τ . From Fig. (3a), it is seen that the temperature and shear stress are decreased with an increasing of the magnetic field but the displacement components and normal stress are increased with the small values of the magnetic field and then decreased with the large values of the magnetic field. Fig. (3b) shows that there is a good increasing variation on the temperature, displacement components but the normal and shear stresses are increased with an increasing of the initial stress. From Fig. (3c), it is appear that there is an increasing in the temperature, displacement components and shear stress with an increasing of the rotation but there is a slight increase in the normal stress with an increasing of the rotation. Fig. (3d), shows a slight decreases in the temperature and the normal stress but the displacement components and shear stress decrease with an increasing of the viscosity. Case (II): thermal point source acting on the surface Figs. (4a-d), exhibit the effect of the magnetic field, initial stress, rotation and viscosity parameters respectively on the temperature, displacements, normal stress and shear stress, it is clear that T, ur, uz, zzσ and rzτ are increase with an increasing of the magnetic field, initial stress and the rotation respectively. Also, From Fig. (4d), it is clear that the temperature, displacements, stresses decreases with an increasing of the viscosity. Finally, it is concluded that the numerical results are agreement with the results and discussions obtained by Abd-Alla and Abo-Dahab [12]. 8- Conclusion The aim of this paper is to estimate the effects of the rotation, magnetic field, thermal relaxation times and pressure on wave propagation in a generalized viscoelastic medium under influence of time-harmonic source for the two cases: (i) with energy dissipation and (ii) without energy dissipation (i.e., there is no relaxation times). From the calculations and numerical results obtained, we concluded that: (1) With the variation of the radial variation r, the temperature T, displacements components ur and uz, stresses zzσ and rzτ are decreased and tend to zero with an increasing of the radial that indicates to the independence of all dependent variables upon the radial of the cylinder. Physically, it is concluded that with an increasing of

40 A. M. Abd-Alla, S. M. Abo-Dahab and F. S. Bayones the cylinder radius; the temperature, displacements, stresses tend to zero. (2) For the case if the problem in energy dissipation: (i) if normal point load acting on the surface: H0 affects decreasing on T and rzτ but affects increasing on ur, uz and zzσ , the initial stress P affects increasing on all dependent variables, angular rotation Ω and relaxation times parameter 1τ affects increasing on T, rzτ but affects decreasing on all remain components, also, it is concluded that the viscosity parameter 0τ affects only on the temperature T. (ii) if thermal point load acting on the surface: H0 and 1τ affect decreasing on rzτ but affect increasing on all remain components, the pressure P increases in all variables, also, it is concluded that Ω and 0τ don’t affect on all dependent variables. (3) For the case if the problem in without energy dissipation: (i) if normal point load acting on the surface: H0 acts decreasing on T and rzτ but frequented between increasing and decreasing on all remain components, the initial stress P affects increasing on all dependent variables, angular rotation Ω affects increasing on all variables, also, it is concluded that the viscosity parameter 0τ affects slight on the temperature T and zzσ but affects decreasing on ur, uz and rzτ . (ii) if thermal point load acting on the surface: H0, Pand Ω affect increasing on all variables, but the viscosity parameter 0τ affect decreasing on all dependent variable. Physically, it is concluded that all parameters are influenced on the temperature T, displacement components ur and uz, stresses zzσ and rzτ , especially, the magnetic field and initial stress if the problem in energy dissipation but influenced by all independent parameters if the problem without energy dissipation, that have a good effect on Biology, Volcanos, Geology, Seismic waves, Earthquakes, High-speed cameras, steam and Gas turbines, Planetary landings and in many other domains, etc. References

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Received: August, 2011

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