Plane waves in a magneto-thermoelastic solids with voids and microtemperatures due to hall current and rotation

Results in Physics 7 (2017) 4253–4263

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Plane waves in a magneto-thermoelastic solids with voids and microtemperatures due to hall current and rotation Mohamed I.A. Othman a,b,⇑, Elsyed M. Abd-Elaziz b a b

Department of Mathematics, Faculty of Science, Taif University 888, Taif, Saudi Arabia Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt

a r t i c l e

i n f o

Article history: Received 5 August 2017 Received in revised form 6 September 2017 Accepted 24 October 2017 Available online 1 November 2017 Keywords: Hall current Magnetic field Rotation Microtemperatures Voids

a b s t r a c t In this study, the effect of hall current and rotation on a magneto-thermoelastic solid with microtemperatures and voids are investigated. The medium is permeated by a strong transverse magnetic field imposed perpendicularly on the displacement plane, the induced electric field being neglected. The normal mode analysis is used to obtain the exact expressions for the considered variables. Numerical simulated results are depicted graphically to show the effect of hall current, voids parameter and rotation on resulting quantities. The results indicate that the effect of hall current, voids parameter and rotation are very pronounced. Some particular cases of special interest have been deduced from the present investigation. Ó 2017 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).

Introduction Theory of elastic materials with voids is one of the most important generalizations of the classical theory of elasticity. This theory is concerned with elastic materials consisting of a distribution of small porous (voids) in which the void volume is included among the kinematic variables. Practically, this theory is useful for investigating various types of geological and biological materials for which elastic theory is inadequate. A nonlinear theory of elastic material with voids was developed by Nunziato and Cowin [1]. Cowin and Nunziato [2] developed a theory of linear elastic materials with voids. Iesan [3] established a linear theory of thermoelastic materials with voids. He presented the basic field equations and discussed the conditions of propagation of acceleration waves in a homogeneous isotropic thermo-elastic material with voids. He showed that transverse wave propagates without affecting the temperature and the porosity of the material. Iesan [4] extended the thermoelastic theory of elastic material with voids to include initial stress and the initial heat flux effects. Dhaliwal and Wang [5] also formulated a thermoelasticity theory for elastic material with voids to include heat flux among the consecutive variables and assumed an evolution equation for the heat-flux. Chiritßa˘, and

⇑ Corresponding author at: Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt. E-mail addresses: [email protected] (M.I.A. Othman), sayed_nr@yahoo. com (E.M. Abd-Elaziz).

Scalia [6] and Pompei and Scalia [7] studied the spatial and temporal behavior of the transient solutions for the initial-boundary value problems associated with the linear theory of the thermoelastic materials with voids by using the time-weighted surface power function method. Scalia et al. [8] considered the steady time harmonic oscillations within the context of linear thermoelasticity for materials with voids and derived the spatial decay results for the amplitude of harmonic variations in a cylinder. Ciarletta and Straughan [9] investigated a model for coupled elasto-acoustic waves, thermal waves, and waves associated with the voids, in a porous medium. In the theory of micromorphic bodies formulated by Eringen and Suhubi [10,11] the material particle is endowed with three definable directors and the theory introduces nine extra degrees of freedom over the classical theory. Iesan and Quintanilla [12] developed a linear theory for elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess micro-temperatures, where an existence theorem proved and the continuous dependence of solutions of the initial data and body loads was established. Iesan [13] presented the mathematical model of the theory of micromorphic elastic solids with microtemperatures in which the microelements possess microtemperatures and can stretch and contract independently of their translations. Iesan and Quintanilla [14] presented a linear theory of thermoelastic bodies with microstructure and micro-temperatures which permits the transmission of heat as thermal waves at finite speed. The exponential stability of

https://doi.org/10.1016/j.rinp.2017.10.053 2211-3797/Ó 2017 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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solutions of equations in this theory was established by Casas and Quintanilla [15]. Othman and Hilal [16] discussed the effect of initial stress and rotation on magneto-thermoelastic material with voids and energy dissipation. Quintanilla [17] proved uniqueness theorems in the dynamical theory thermoelasticity of porous media with microtemperatures. Chiritßa˘ et al. [18] studied the linear theory of thermoelastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. Othman et al. [19] studied the effect of initial stress on a porous thermoelastic medium with micro-temperatures. The theory of magneto-thermoelasticity is concerned with the influence of magnetic field on the elastic and thermoelastic deformations of solid materials. Investigation of the interaction between the magnetic field and stress and strain in a thermoelastic solid is very important due to its many applications in the fields of geophysics and plasma physics. Especially in nuclear fields, the extremely high temperature and temperature gradients, as well as the magnetic fields originating inside nuclear reactors influence their design and operations. Nayfeh and Nasser [20] studied the propagation of plane waves in a thermoelastic solid with electric and the magnetic permeability. Othman and others [21–24] have studied much interest applications dealing of magnetic field with thermoelasticity. Abo-Dahab et al. [25] investigated the effect of rotation and magnetic field on the general model of the equations of generalized thermoelasticity. Bayones and Abd-Alla [26] studied the thermoelastic wave propagation in a rotating medium with magnetic field effects and a time-dependent heat source effect due to thermomechanical source. In all these investigations, the effects of hall currents are not considered. It was emphasized by Cowling [27] that when the strength of the magnetic field is sufficiently large, Ohm’s law needs to be modified to include hall currents. The hall effect is due merely to the sideways magnetic force on the drifting free charges. The electric field has to have a component transverse to the direction of the current density to balance this force. In many works on plasma physics, the hall effect is ignored. But if the strength of the magnetic field is high and the number of density of electrons is small, the hall effect cannot be disregarded as it has a significant effect on the flow pattern of an ionized gas. Hall effect results in a development of an additional potential difference between the opposite surfaces of a conductor for which a current is induced perpendicular to both the electric and magnetic field. The current is termed as hall currents. Zakaria [28] investigated the problem of the two-dimensional magnetic micropolar generalized thermoelastic medium in the presence of the combined effect of hall currents subjected to ramp-type. In the present research work, we have studied the effect of rotation and hall current on a thermoelastic solid with microtemperatures and voids in the presence of a uniform strong magnetic field acting on z-direction. The normal mode analysis is used to solve the resulting nondimensional coupled equations. Numerical results for the displacement component, temperature, the stresses, the microtemperature component, the change in the volume fraction field, the normal and transverse conduction currents density field and the heat flux moments are represented graphically and discussed. Comparisons are made with the results in the presence and absence of the hall current, rotation and voids parameter of the thermoelastic half-space.

forces, heat sources and first heat source moment, taking into account the Lorentz force, can be written as:

Basic equations

Jx ¼

Following (Iesan [13]; Quintanilla [17]; Cowin and Nunziato [2] and Cowling [27], the field equations in a homogeneous, isotropic thermoelastic solid with voids, micro-temperatures, without body

Jy ¼

rij;j þ l0 eijr Jj Hr ¼ q½ui;tt þ fX  ðX  uÞgi þ ð2X  u;t Þi ;

ð1Þ

a/;jj  k0 ui;i  n1 /  x0 /;t þ mT  l1 wi;i ¼ qw/;tt ;

ð2Þ



k T ;jj  qC  T ;t  bT 0 ui;it  mT 0 /;t þ k1 wi;i ¼ 0;

ð3Þ

k6 wi;jj þ ðk4 þ k5 Þwj;ji þ l1 /;t  k2 wi  bwi;t  k3 T ;i ¼ 0:

ð4Þ

And the constitutive relations are

rij ¼ 2leij þ ðkerr þ k0 /  bTÞdij ; qij ¼ k4 wr;r dij  k5 wi;j  k6 wj;i ;

ð5Þ i; j; r ¼ 1; 2; 3:

ð6Þ

The strain-displacement relation is;

eij ¼

1 ðui;j þ uj;i Þ; 2

ð7Þ

where rij are the components of the stresses, eijr is the permutation symbol, l0 is the magnetic permeability, Jj is the conduction current density, Hr is the intensity tensor of magnetic field, q is the density, a; k0 ; n1 ; x0 ; m; w are the material constants due to presence of  voids, / is the change in volume fraction field, k is the thermal con ductivity, C is the specific heat, ui is the displacement vector, T is the absolute temperature, l1 ; b; ki ði ¼ 1; 2; . . . ; 6Þ are the constitutive coefficients, T 0 is the reference temperature chosen so that jðT  T 0 Þ=T 0 j  1, wi is the microtemperature vector, dij is the Kronecker delta, k; l are the lame’ constants, b ¼ ð3k þ 2 lÞ at ; such that at is the coefficient of thermal expansion, qi is the heat flux moment and t is the time variable. When the strength of the magnetic field is very large we include the hall current so that the generalized Ohm’s law is modified to

J i þ xe t e eiLk J L Hk ¼ r0 ðEi þ l0 eijr uj;t Hr Þ;

ð8Þ

where xe ð¼ el0 H0 =me Þ is the electron frequency, e is the charge of the electron, me is the mass of the electron, t e is the electron collision time, r0 ð¼ ne e2 t e =me Þ is the electrical conductivity, ne is the electron number density and Ei is the intensity tensor of the electric field. Formulation of the problem and solution We consider a homogeneous, isotropic, thermoelastic material with voids and microtemperatures with a half space ðx P 0Þ the rectangular Cartesian coordinates system ðx; y; zÞ having originated on the surface z ¼ 0. For two dimensional problem we assume the dynamic displacement vector as ui ¼ ðu; v ; 0Þ and the microtemperature vector wi ¼ ðw1 ; w2 ; 0Þ. The elastic material is rotating uniformly with an angular velocity X ¼ X n, since n being a unit vector representing the direction of the axis of rotation. All quantities considered will be a function of the time variable t and of the coordinates x and y. The displacement equation of motion in a rotating frame has two additional terms, according to Schoenberg and Censor [29] centripetal acceleration, X  ðX  uÞ due to time varying motion only and 2X  u;t , where X ¼ ð0; 0; XÞ. These terms, do not appear in non-rotating media. A uniform very strong magnetic field of strength H0 is assumed to be applied in the positive z-direction and the electric field Ei ¼ ð0; 0; 0Þ. Under these assumptions, Eq. (8) reduces to




 @u ; @t

ð9Þ


 @ v @u ; m  0 @t @t 1 þ m20

ð10Þ

l0 r0 H 0 @ v 1 þ m20

l0 r0 H 0

@t

þ m0

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J z ¼ 0;

ð11Þ

where m0 ð¼ xe te Þ is the hall parameter. With the help of Eqs. (5)–(11), Eqs. (1)–(4) take the form:


 @e @/ @T l r0 @ v @u lr2 u þ ðk þ lÞ 1 þ k0  b þ m  0 @x @x 1 þ m20 @x @t @t ! 2 @ u @v ; ¼q  X2 u  2X @t @t2 2 2 0 H0


 @e @/ @T l r0 @u @ v þ lr2 v þ ðk þ lÞ 1 þ k0  b  m 0 @y @y 1 þ m20 @t @t @y ! 2 @ v @u ; ¼q  X2 v þ 2X @t @t 2

ð12Þ

ð13Þ

ð14Þ


 @ @e @2/ @T k6 r2  b  k2 w1 þ ðk4 þ k5 Þ 1  l1  k3 ¼ 0; @t @x@t @x @x

ð16Þ

ð17Þ

where g0 ¼

ðt0 ; t 0e Þ ¼ g0 ðt; te Þ; q c c21 k

X0 ¼

X

g0

;

M2 ¼

v¼

@ N1 @ w1  ; @y @x

w1 ¼

a14 ¼

, a17 ¼ a2 a16  

N 1 ;

i

0

l1 g20 w , ac21

a11 ¼ mTa0 w, a12 ¼

i mc41 n k g30 w

bc21 n

k6 g0

N 2 ;

þ

w1

, a15 ¼ kkT1 0 , a16 ¼ k4 þkk56 þk6 , a18 ¼ k2 c21 k6 g20

, a20 ¼ a2 

i

bc21 n

k6 g0

þ



k2 c21 k6 g20

il1 c41 n k6 g30 w

,

d , D ¼ dy .

Eliminating / , and T among Eqs. (21)–(25) we obtain the following tenth order differential equation satisfied by / ðyÞ; N 1 ðyÞ; N 2 ðyÞ; w1 ðyÞ and T  ðyÞ.

ð27Þ

" 5 Y

#

ðD2  a2n Þ f/ ðyÞ; N1 ðyÞ; N2 ðyÞ; w1 ðyÞT  ðyÞg ¼ 0;

ð28Þ

where a2n ðn ¼ 1; 2; 3; 4; 5Þ are the roots of the characteristic equation of the Eq. (27). The solution of the Eq. (27), bound at y ! 1, is given by 5 X ½1; H1n ; H2n ; H3n ; H4n Mn ean y :

ð29Þ

The solution of Eq. (26) bound as y ! 1, can be written as: 2 2 0 H0

l

r0 ; qg0

ð18Þ

@ N2 @ w2 þ ; @x @y

ð30Þ

where a26 is the root of the characteristic equation of the Eq. (26) and Mn ðn ¼ 1; 2; . . . ; 6Þ are some constants. We can obtain the general solutions of the displacements u, v and the micro-temperature vectors w1 , w2 , bound as y ! 1, we are substituting Eq. (20) into Eq. (19) and with the aid of Eqs. (29) and (30):

½u; v  ðx; y; tÞ ¼

5 X ½H5n ; H6n Mn ean yþi ða xn tÞ ;

ð31Þ

n¼1

ð19Þ

" # 5 X an y  a6 y ½w1 ;w2 ðx; y;tÞ ¼ ði a; an ÞH4n Mn e  ða6 ;iaÞM 6 e ei ða xn tÞ : n¼1

To get the exact solution for the physical quantities, consider the solution in the form of the normal mode as:

½N1 ; N2 ; w1 ; w2 ; /; T ðx; y; tÞ ¼ ½N1 ; N2 ; w1 ; w2 ; / ; T   ðyÞ ei ða xn tÞ ; ð20Þ where

ibc21 n , k g0

a10 ¼

w2 ðyÞ ¼ M6 ea6 y ;

2 l , c21 ¼ kþ2 q and M is the square of Hartmann num-

@ N1 @ w2 þ ; @x @y @ N2 @ w2 w2 ¼  : @y @x

a19 ¼

k3 c21 T 0 k6 g20

a13 ¼

qc21 n2 w ix0 c21 n a  ag0 ,



n¼1

ber or magnetic parameter. Assuming the potential functions w1 ðx; y; tÞ, w2 ðx; y; tÞ, N 1 ðx; y; tÞ, and N 2 ðx; y; tÞ, on the dimensionless form

u¼

iqc21 c n , k g0

ag20

½N1 ; w1 ; / ; T  ; N2 ðyÞ ¼

w g2 c1 ðx0i ; u0i Þ ¼ ðxi ; ui Þ; /0 ¼ 2 0 /; w0i ¼ w; c1 g0 i c1 g0 1 q ; ðT 0 ; p01 Þ ¼ ðT; p1 Þ; q0ij ¼ T0 bT 0 c21 ij ;

0

n1 c21

n¼1

g0

rij

ð26Þ

 2 2 k0 bT 0 M m0 M2 in where a1 ¼ a2  n2  1þm 2  X , a2 ¼ g2 wq, a3 ¼ qc2 , a4 ¼ in 1þm2 þ 0 0 1 0   2 2 m0 M 2 in 2XÞ, a5 ¼ qlc2 , a6 ¼ a5 a2  n2  1þm a7 ¼ in M þ 2X , 2  X , 1þm2

where A, B; C; F and G are defined in the Appendix I. Eq. (27) can be factored as

To facilitate the solution, the following non-dimension quantities are introduced

bT 0

½D2  a20  w2 ¼ 0;

½D10  AD8 þ BD6  C D4 þ F D2  Gf/ ðyÞ; N1 ðyÞ;N2 ðyÞ;w1 ðyÞ; T  ðyÞg ¼ 0;

ð15Þ

r0ij ¼

ð25Þ

a2 


 @ @e1 @/  k r2  q c  T  bT 0 þ k1 e1 ¼ 0;  mT 0 @t @t @t


 @ @e @2/ @T  k3 ¼ 0: k6 r2  b  k2 w2 þ ðk4 þ k5 Þ 1  l1 @t @y@t @y @y

½a16 D2  a17  N2 þ a18 /  a19 T  ¼ 0;

1

!

@2 @  x0  n1 /  k0 e1  l1 e1 þ mT ¼ 0; 2 @t @t

ð24Þ

a9 ¼ k0aw, a8 ¼ a2 þ

2 2 0 H0

ar2  qw

½D2  a12  T  þ a13 ½D2  a2  N1 þ a14 / þ a15 ½D2  a2  N 2 ¼ 0;

½N1 ; N 2 ; w1 ; w2 ; / ; T   ðyÞ

are the amplitude of the physical pffiffiffiffiffiffiffi quantities, n is the angular frequency, i ¼ 1 and a is the wave number in the x-direction. Apply the Eqs. (18), (19) and (20) on Eqs. (12)–(17) respectively; we obtain (after suppressing primes)

½D2  a1  N1 þ a2 /  a3 T  þ a4 w1 ¼ 0;

ð21Þ

½a5 D2  a6  w1  a7 N1 ¼ 0;

ð22Þ

½D2  a8  /  a9 ½D2  a2  N1  a10 ½D2  a2  N2 þ a11 T  ¼ 0;

ð23Þ

ð32Þ To obtain the solutions of current density J x and J y , we substitute Eq. (31) into Eqs. (9) and (10), we get

½Jx ; J y ðx; y; tÞ ¼

5 X ½H7n ; H8n M n ean yþi ða xn tÞ :

ð33Þ

n¼1

Moreover, the general solutions of stresses and the heat flux moment, bound as y ! 1, by using Eq. (18) in Eqs. (5), (6) and with the help of Eqs. (29)–(32):

½rxx ; ryy ; rzz ðx; y; tÞ ¼

5 X ½H9n ; H10n ; H11n Mn ean yþi ða xn tÞ ;

ð34Þ

n¼1

½rxy ; ryx ; rxz ; ryz ðx;y;tÞ ¼

5 X ½H12n ;H12n ;0;0Mn ean yþi ða x n tÞ ; n¼1

ð35Þ

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½qxx ; qyy ; qzz ðx; y; tÞ ¼ " # 5 X ðH13n ; H14n ; H15n ÞM n ean y þ ðL1 ; L1 ; 0ÞM6 ea6 y ei ða xn tÞ ; n¼1

ð36Þ ½qxy ; qyx ; qxz ; qyz ðx; y; tÞ ¼ " # 5 X ðH16n ; H16n ; 0; 0ÞM n ean y  ðL2 ; L3 ; 0; 0ÞM 6 ea6 y ei ða xn tÞ ; n¼1

ð37Þ where Hsn ðs ¼ 1; 2; . . . ; 16Þ, L1 ; L2 and L3 are defined in the Appendix II.

½D2  a1  N1  a3 T  þ a4 w1 ¼ 0;

ð44Þ

½a5 D2  a6  w1  a7 N1 ¼ 0;

ð45Þ

½D2  a12  T  þ a13 ½D2  a2  N1 þ a15 ½D2  a2  N2 ¼ 0;

ð46Þ

½a16 D2  a17  N2  a19 T  ¼ 0;

ð47Þ

½D2  a20  w2 ¼ 0:

ð48Þ

Eliminating N 1 , N 2 , w1 and T  among Eqs. (44)–(48), we obtain the following eighthorder differential equation satisfied by N 1 ðyÞ, N 2 ðyÞ, w1 ðyÞ and T  ðyÞ

½D8  A1 D6 þ A2 D4  A3 D2 þ A4 fN1 ðyÞ; N 2 ðyÞ; w1 ðyÞ; T  ðyÞg ¼ 0; ð49Þ

Applications Consider the following non-dimensional boundary conditions at ðy ¼ 0Þ to determine the coefficients M n ðn ¼ 1; 2; . . . ; 6Þ. (1) The mechanical boundary conditions are (i) The normal stress condition (stress free), so that

rxx ¼ 0:

ð38Þ

ðN1 ; w1 ; T  ; N2 ÞðyÞ ¼

(ii) The tangential stress condition (stress free), then

rxy ¼ 0:

ð39Þ

(2) The normal and the tangential heat flux moments are free, so that

qxx ¼ qxy ¼ 0:

ð40Þ

(3) The condition of the change in the volume fraction field (/ is constant in y-direction), then

@/ ¼ 0: @y

ð41Þ

(4) The thermal condition (the half-space subjected to thermal shock applied to the boundary). This leads to

T ¼ p1 ei ða xn tÞ ;

where d1 ¼ a12 a16 þ a17  a15 a19 , d2 ¼ a12 a17  a a15 a19 , d3 ¼ a1 a5 þ d4 ¼ a1 a6 þa4 a7 , d5 ¼ a3 a13 a5 a16 , d6 ¼ a3 a13 ða5 a17 þa6 a16 þ a6 , a2 a5 a16 Þ, d7 ¼ a3 a13 ða6 a17 þa2 a5 a17 þa2 a6 a16 Þ, d8 ¼ a3 a13 a6 a17 a2 , A1 ¼ d3 a16 þd1 a5 d5 , A2 ¼ d4 a16 þd1 d3 þd2 a5 d6 , A3 ¼ d4 d1 þd2 d3 d7 , A4 ¼ d4 d2 d8 . The solutions of Eq. (49) are 2

ð42Þ

1 0 H91 M1 BM C B H B 2 C B 121 C B B B M3 C B H131 C B B BM C ¼ B H B 4 C B 161 C B B @ M5 A @ a1 H21 0

M6

H31

H92

H93

H94

H95

H122

H123

H124

H125

H132

H133

H134

H135

H162

H163

H164

H165

a2 H22 a3 H23 a4 H24 a5 H25 H32

H33

H34

H35

0

11 0

0 C C C L1 C C L2 C C C 0 A 0

B B B B B B B B @

0

where a2 n ðn ¼ 1; 2; 3; 4Þ are the roots of the characteristic equation of Eq. (49) and H1n ¼ a

a7 2 5 an a6

0 0 0 0

p1 ð43Þ

, H2n ¼

 a2 n a1 þa4 H1n

a3

a19 H2n 2 16 an a17

, H3n ¼ a

.

The solution of Eq. (48) are 

w2 ðyÞ ¼ R5 ea5 y ;

ð51Þ

2 5

where a is the root of the characteristic equation of Eq. (48). The expressions for the displacement components, the microtemperature vector, the temperature field, current density the stresses and the heat flux moment in the generalized thermoelasticity with microtemperatures for rotating medium under the magnetic field and hall current effects, can be written as:

u ðx; y; tÞ ¼

4 X  H4n Rn ean yþi ða xn tÞ ;

ð52Þ

n¼1

v ðx; y; tÞ ¼

4 X  H5n Rn ean yþi ða xn tÞ ;

ð53Þ

n¼1

w1 ðx; y; tÞ ¼

4 X   i a H3n Rn ean yþi ða xn tÞ  a5 R5 ea5 yþi ða xn tÞ ;

ð54Þ

n¼1

1 C C C C C: C C C A

ð50Þ

n¼1

Since p1 being the applied constant temperature at the boundary. Substituting the expressions of the considered quantities in the above boundary conditions, to obtain the equations satisfied by the constants M n ðn ¼ 1; 2; . . . ; 6Þ. Then one can obtain a system of six equations. After applying the inverse of matrix method, we get the values of the constants M n ðn ¼ 1; 2; . . . ; 6Þ.

4 X  ð1; H1n ; H2n ; H3n ÞRn e an y ;

w2 ðx; y; tÞ ¼

4 X





 an H3n Rn ean yþi ða xn tÞ  i a R5 ea5 yþi ða xn tÞ ;

ð55Þ

n¼1

T ðx; y; tÞ ¼

4 X  H2n Rn ean yþi ða xn tÞ ;

ð56Þ

n¼1

½J x ; J y ðx; y; tÞ ¼

4 X  ½H6n ; H7n Rn ean yþi ða xn tÞ ;

ð57Þ

n¼1

Particular and special cases Neglecting the void effect Neglecting the material constants due to the presence of voids. Putting ða ¼ k0 ¼ n1 ¼ x0 ¼ m ¼ l1 ¼ w ¼ 0Þ in Eqs. (21)–(26) we get

ðrxx ; ryy ; rzz Þðx; y; tÞ ¼

4 X  ðH8n ; H9n ; H10n ÞRn ean yþi ða xn tÞ ;

ð58Þ

n¼1

ðrxy ; ryx ; rxz ; ryz Þðx; y; tÞ ¼

4 X n¼1



ð1; 1; 0; 0ÞH11n Rn ean yþi ða xn tÞ ;

ð59Þ

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M.I.A. Othman, E.M. Abd-Elaziz / Results in Physics 7 (2017) 4253–4263

ðqxx ; qyy ; qzz Þðx; y; tÞ ¼ " # 4 X   ðH12n ; H13n ; H15n ÞRn ean y þ ðL1 ; L1 ; 0ÞR5 ea5 y ei ða xn tÞ ;

Neglecting the magnetic field effect

n¼1

ð60Þ

ðqxy ; qyx ; qxz ; qyz Þðx; y; tÞ ¼ " # 4 X   an y    a y 5 ð1; 1; 0; 0ÞH16n Rn e  ðL2 ; L3 ; 0; 0ÞR5 e ei ða xn tÞ ;

Numerical results and discussion

ð61Þ

n¼1

 H5n ¼  n  iaH1n , H8n ¼ iab1 H4n  l0 H0 r0 c1 in H0 r0 c1 in   b2  ¼ 1þm2 ðm0 H4n þ H5n Þ, H7n ¼ l01þm 2 0 0 ðm0 H5n  H4n Þ, L1 ¼ ia ðb5  b6 Þ 5 , H9n ¼ iab2 H4n  b1 n H5n  H2n , 2 L2 ¼ ðb7 2 H11n ¼ H10n ¼ iab2 H4n  b2 n H5n  H2n , 5 þ b8 a Þ,       2 2 2 iab4 H5n  b4 n H4n , H12n ¼ ðb5 a  b6 n ÞH3n , L3 ¼ ðb7 a  b8 2 5 Þ,     2 2 H13n ¼ ðb6 a2  b5 2 n ÞH3n , H15n ¼ ðb6 a  b6 n ÞH3n , H16n ¼ ðb7 þ b8 Þ   n H3n . In order to determine the coefficients Rn ðn ¼ 1; 2; 3; 4; 5Þ we use

where

  n H5n

a

H4n ¼ ia  an H1n , H2n ,

a

H6n

a

a

a

a

a

a

a

a

a

a

the boundary conditions in Eqs. (56), (58), (59), (60) and (61) and by using the inverse of matrix method as following:

1 0  H81 R1 B C B  B R2 C B H111 B C B  BR C ¼ BH B 3 C B 121 B C B  @ R4 A @ H161 0

R5

H21

H82

H112 H122 H162 H22

H83

H113 H123 H163 H23

H84

H114 H124

H164  L2 H24

0

11 0

C 0C C  C L1 C C A 0

1

0 B C B0C B C B 0 C: B C B C @0A p1

Taking Hartmann number M 0 ¼ 0, in the governing equations, the corresponding expressions of the physical variables can be obtained from Eqs. (21)–(26) without magnetic field effect.

In this section we perform some numerical calculations in order to illustrate the analytical results. Following Dhaliwal and Singh [30] magnesium material was chosen for this purpose.

k ¼ 9:4  1010 N=m2 ;

l ¼ 4  1010 N=m2 ; b 

¼ 7:779  108 N; k ¼ 1:7  102 N=s  K; T 0 ¼ 298 K; q

at ¼ 7:4033  107 N=m2 ; C 

¼ 1:74  103 kg=m3 ;

¼ 1:04  103 J=kg  K; k1 ¼ 3:5 N=s; k2 ¼ 4:5 N=s; k3 ¼ 5:5 N=s K; k4 ¼ 6:5 N=s m2 ; ¼ 9:6 N=s m ; 2

k5 ¼ 7:6 N=s m2 ;

k6

9

l1 ¼ 0:0085 N; b ¼ 0:15  10 N; p1 s0 ¼ 0:02 rad=s; s1

¼ 10 K; a ¼ 1:6 m; n ¼ s0 þ is1 ; ¼ 0:3 rad=s; 0 6 y 6 5:

Since, we have n ¼ s0 þ is1 , ent ¼ es0 t ½cosðs1 tÞ þ i sinðs1 tÞ and for small values of time we can take n ¼ s0 (real).

ð62Þ

The voids parameters are: w ¼ 1:753  1015 m2 , a ¼ 3:688 5

10

N, n1 ¼ 1:475  1010 N=m2 , k0 ¼ 1:13849  1010 N=m2 , m ¼

2  106 N=m2 deg, x0 ¼ 0:0787  103 N=m2 s. The

electric

constants:

5

Neglecting the hall current effect By taking m0 ¼ 0, in Eqs. (21)–(26) we obtain the corresponding expressions of the physical variables in generalized thermoelasticity with microtemperatures and voids under the effects of rotation, magnetic field and without hall current effect.

r0 ¼ 9:36  105 Col2 =Cal:cm:s,

H0 ¼ 10 Col=cm:s. The computations are carried out for x ¼ 0 and t ¼ 0:5. The results of the numerical evaluation of the displacements u, v , the microtemperature vector w1 , the tem-perature T, the stresses rxx and rxy , the change in the volume fraction field /, the normal conduction current density field J x , the transverse conduction current density field J y and the heat flux moments qxx and qxy are illustrated in Figs. 1–11.

Fig. 1. Horizontal displacement distribution u with distance y.

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Fig. 2. Vertical displacement distribution

v

with distance y.

Fig. 3. Distribution of microtemperature vector w1 with distance y.

Fig. 4. Distribution of temperature T with distance y.

Comparisons of all these dimensionless physical quantities are made for following different cases: (i) Magneto-thermoelasticity with microtemperatures and voids under hall current effect and rotation (MTM(VHR)).

(ii) Magneto-thermoelasticity with microtemperatures and voids under rotation effect (MTM(VR)). (iii) Magneto-thermoelasticity with microtemperatures and voids under hall current effect (MTM(VH)). (iv) Magneto-thermoelasticity with microtemperatures under hall current effect and rotation (MTM(HR)).

M.I.A. Othman, E.M. Abd-Elaziz / Results in Physics 7 (2017) 4253–4263

4259

Fig. 5. Distribution of normal conduction current density field Jx with distance y.

Fig. 6. Distribution of transverse conduction current density field Jy with distance y.

Fig. 7. Distribution of normal stresses

In the Figs. 1–11, the sold line represent results in (MTM(VHR)) when m0 ¼ 0:5, X ¼ 0:2; the small dash line represent results in (MTM(VR)) when m0 ¼ 0, X ¼ 0:2, and the large dash line represent results in (MTM(VH)) when m0 ¼ 0:5, X ¼ 0, while the small dash line with dots represent results in (MTM(HR)) when ða ¼ k0 ¼ n1 ¼ x0 ¼ m ¼ l1 ¼ w ¼ 0Þ.

rxx with distance y.

Fig. 1 depicts the space variation of the horizontal displacement u versus y. The magnitude of horizontal displacement is found to be large for MTM(VHR) model. It attains its maximum values 0:243, 0:215, 0:075 and 0:063 for MTM(VHR), MTM(VR), MTM (VH) and MTM(HR) models respectively at y  0:5. It is increasing in the range 0 6 y 6 0:5 and decreasing in the range 0:5 6 y 6 2:5

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Fig. 8. Distribution of shearing stresses

rxy with distance y.

Fig. 9. Distribution of heat flux moment qxx with distance y.

Fig. 10. Distribution of heat flux moment qxy with distance y.

and then tends to zero for y P 2:5. It can be seen that hall current, rotation and void parameters show the increasing effect on the magnitude of horizontal displacement. The variation of the normal displacement v with distance y is shown in Fig. 2. The behavior of v for all the models is almost similar. It is clear that the displacement components v decreases continuously with an increase in y. The

effect of hall current, rotation and the presence of voids is to increase the magnitude of the normal displacement. In Fig. 3 we have compared the variations of the microtemperature vector w1 for the different models considered. The values of microtemperature vector w1 initially show a sharp increase, attains a maximum value at y ¼ 0:7 and then decreases with increase in y till becomes

M.I.A. Othman, E.M. Abd-Elaziz / Results in Physics 7 (2017) 4253–4263

4261

Fig. 11. Change in volume fraction field distribution / with distance y.

Fig. 12. Distribution of the stress

rxy versus the distances at t ¼ 0:2 X ¼ 0:2, m0 ¼ 0:5, H0 ¼ 105 .

Fig. 13. Distribution of heat flux moment qxx versus the distances at t ¼ 0:2, X ¼ 0:2, m0 ¼ 0:5, H0 ¼ 105 .

constant for y P 4:5. Fig. 4 represents the changes of temperature distribution T versus y: The temperature distribution exhibits the same behavior in all the four models. The values of T increases in the case of MTM (VHR) compared to the other cases. Clearly the presence of void parameters, hall current and rotation acts to increase the magnitude of temperature. Fig. 5 reveals the distribution of normal conduction current density field J x with distance y. It is evident that, the value of J x is decreased in the MTM (VHR) as compared to the MTM(VR), MTM(VH) and MTM(HR). Fig. 6 shows the distribution of transverse conduction current density field J y with distance y. It is clear from the figure that, void parameters show increasing effect on the magnitude of J y . The presence of hall

current parameter causes a decrease in the magnitude of J y . Fig. 7 investigates the distribution of normal stresses rxx with distance y. In this figure the values of rxx for MTM(HR) are small compared to those in the other cases. The values of rxx for MTM(VH) are large compared to those in the other cases. Fig. 8 explains the distribution of tangential stress rxy versus the distance y. The values of rxy are decreasing for absence void parameters, while it increases for the presence hall parameter. Fig. 9 depicts that, the values of heat flux moment qxx for presence voids are small compared to those for absence voids. From this figure, the large value of qxx being for MTM(HR) and smallest value is for MTM (VHR). Fig. 10 clarifies that, the values of the heat flux moment qxy for presence

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solutions without any assumed restrictions on the actual physical quantities that appear in the governing equations of the problem considered. The results concluded from the above analysis can be summarized as follows:

voids and rotation are large compared to those for absence voids. In all cases, it starts with zero value initially. Fig. 11 exhibits the values of change in the volume fraction field / against the distance y. the values of / decreases continuously with an increase in y, the values in the case of MTM(VHR) are higher than in case of MTM (VH). Figs. 12, 13 are giving 3D surface curves for the physical quantities, i.e., the stress component rxy and the heat flux moment qxx for the case of MTM(VHR). These figures are very important to study the dependence of these physical quantities on the vertical component of distance.

1. The hall current and rotation play a significant role in the distribution of all the physical quantities. The parameters of all the physical quantities vary (increase or decrease). 2. All the physical quantities satisfy the boundary conditions. 3. The results are graphically described for the medium of magnesium. 4. The microtemperature is a very useful theory in the field of geophysics, earth-quake engineering and seismologist working in the field of mining tremors and drilling into the earth’s crust. 5. The value of all physical quantities converges to zero with an increase in the distance y and all functions are continuous.

Conclusions The normal mode analysis technique used in this study provides a quite successful approach in dealing with a wide range of problems in thermodynamics. This approach provides us with exact

Appendix I s1 ¼ a1 a5 þ a6 , s2 ¼ a1 a6 þ a4 a7 , s3 ¼ a12 a16 þ a17  a15 a19 , s4 ¼ a12 a17  a15 a19 a2 , s5 ¼ a11 ða14 a16  a15 a18 Þ, s6 ¼ a11 ða17 a14  a15 a18 a2 Þ, s7 ¼ a10 a18 , s8 ¼ a10 ða12 a18  a14 a19 þ a18 a2 Þ, s9 ¼ a2 a10 ða18 a12  a14 a19 Þ, s10 ¼ a2 a5 , s11 ¼ a2 ða6 þ a5 a2 Þ, s12 ¼ a2 a6 a2 ; s13 ¼ a9 a12  a11 a13 , s14 ¼ a17 ða9 a15  a10 a13 Þ, s15 ¼ a5 a3 , s16 ¼ a3 ða6 þ a5 a2 Þ, s17 ¼ a3 a6 a2 , s18 ¼ a8  a9 a14 , s19 ¼ a18 ða10 a13  a9 a15 Þ, s20 ¼ a8 a16 þ s3  s7 , s21 ¼ a8 s3 þ s4  s5  s8 , s22 ¼ a8 s4  s6  s9 , s23 ¼ a16 a9 , s24 ¼ a9 a14 þ a16 a13  s14 , s25 ¼ a13 a14 þ s14 a2 , s26 ¼ a17  s19 þ a16 a18 , r ¼ a51a16 , s27 ¼ a17 s18  s19 a2 , A ¼ rða5 s20 þ s1 a16  s10 s23  s15 a16 Þ, G ¼ rðs2 s22  s25 s12 þ s17 s27 Þ, B ¼ rða5 s21 þ s1 s20 þ a16 s2  s10 s24  s11 s23  s15 s26  s16 a16 Þ, F ¼ rðs1 s22 þ s2 s21  s11 s25  s12 s24 þ s16 a27  s17 a26 Þ, C ¼ rða5 s22 þ s1 s21 þ s2 s20  s10 s25  s11 s24  s12 s23 þ s15 a27  s16 a26  a16 s17 Þ.

Appendix II r 1 ¼ a3 a15 þ a2 a10 , r 2 ¼ a5 a8 a15  a2 a10 a12  a3 a10 a14  a2 a11 a15 , r3 ¼ a1 a10 þ a10 a12  a11 a15 þ a3 a9 a15  a3 a10 a13 , r4 ¼ a3 a15 a9 þ a1 a10 a12  a1 a11 a15  a3 a10 a13 a2 , r5 ¼ a4 a10 , r 6 ¼ a4 a10 a12  a4 a11 a15 , L1 ¼ iaa6 ðb5  b6 Þ, L2 ¼ b7 a26 þ a2 b8 , L3 ¼ b8 a26 þ a2 b7 , a7 2 5 an a6

H1n ¼ a H7n ¼

,

4

2

2

3 an þr 4 þr 5 an r 6 H2n ¼ a10 an rr , a2 þr

l0 H0 r0 c1 in ðm0 H5n 1þm20

1 n

2

þ H6n Þ, H8n ¼

2

H3n ¼ an a1 þa2aH3 2n þa4 H1n ,

l0 H0 r0 c1 in ðm0 H6n 1þm20

18 H 2n H4n ¼ a19aH3naa , 2 a 16 n

17

H5n ¼ ia  an H1n ,

H6n ¼ an  iaH1n ,

 H5n Þ, H9n ¼ iab1 H5n  an b2 H6n þ b3 H2n  H3n , H10n ¼ iab2 H5n  an b1 H6n þ b3 H2n  H3n ,

H11n ¼ iab2 H5n  an b2 H6n þ b3 H2n  H3n , H12n ¼ iab4 H6n  an b4 H5n , H13n ¼ ða2 b5  b6 a2n ÞH4n , H14n ¼ ða2 b6  b5 a2n ÞH4n , H15n ¼ ða2 b6  b6 a2n ÞH4n , k c2

l 0 1 H16n ¼ ia ðb8 þ b7 Þ an H4n , b1 ¼ kþ2 , b2 ¼ bTk 0 , b3 ¼ g2 wbT , b4 ¼ bTl0 , b5 ¼ bT 0 0

0

Appendix B. Supplementary data Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.rinp.2017.10.053. References [1] Nunziato JW, Cowin SC. A non-linear theory of elastic materials with voids. Arch Ration Mech Anal 1979;72:175–201. [2] Cowin SC, Nunziato JW. Linear theory of elastic materials with voids. J Elast 1983;13:125–47. [3] Iesan D. A theory of thermoelastic materials with voids. Acta Mech 1986;60:67–89. [4] Iesan D. A theory of initially stressed thermoelastic material with voids. An Stiint Univ Ai I Cuza Lasi Sect I a Mat 1987;33:167–84. [5] Dhaliwal RS, Singh A. Dynamic coupled thermoelasticity. New Delhi: Hindustan Publ Corp; 1980. [6] Chiritßa˘ S, Scalia A. On the spatial and temporal behavior in linear thermoelasticity of materials with voids. J Therm Stresses 2001;24:433–55. [7] Pompei A, Scalia A. On the asymptotic spatial behavior in the linear thermoelasticity of materials with void. J Therm Stresses 2002;25:183–93. [8] Scalia A, Pompei A, Chirita S. On the behavior of steady time harmonic oscillations thermoelastic materials with voids. J Therm Stresses 2004;27:209–26.

g20 qc ðk4 þk5 þk6 Þ bk T 0 c21

, b6 ¼

g20 qc k4 bk T 0 c21

, b7 ¼

g20 qc k5 bk T 0 c21

, b8 ¼

g20 qc k6 bk T 0 c21

, n ¼ 1; 2; 3; 4; 5.

[9] Ciarletta M, Straughan B. Thermo-poroacoustic acceleration waves in elastic materials with voids. J Math Anal Appl 2007;333:142–50. [10] Eringen AC, Suhubi ES. Nonlinear theory of simple microelastic solids. Int J Eng Sci 1964;2:389–403. [11] Eringen AC. Microcontinuum field theories, I. Foundations and solids. New York, Berlin, Heidelberg: Springer-Verlag; 1999. [12] Iesan D, Quintanilla R. On a theory of thermoelasticity with microtemperatures. J Therm Stresses 2000;23:199–215. [13] Iesan D. On a theory of micromorphic elastic solids with microtemperatures. J Therm Stresses 2001;24:737–52. [14] Iesan D, Quintanilla R. On thermoelastic bodies with inner structure and micro-temperatures. J Math Anal Appl 2009;354:12–23. [15] Casas PS, Quintanilla R. Exponential stability in thermoelasticity with microtemperatures. Int J Eng Sci 2005;43:33–47. [16] Othman MIA, Hilal MIM. Effect of initial stress and rotation on magnetothermo- elastic material with voids and energy dissipation. Multidiscipline Model Mater Struct 2017;13(2):331–46. [17] Quintanilla R. Uniqueness in thermoelasticity of porous media with microtemperatures. Arch Mech Arch Mech Stos 2009;61:371–82. [18] Chiritßa˘ S, Ciarletta M, D’Apice C. On the theory of thermoelasticity with microtemperatures. J Math Anal Appl 2013;397:349–61. [19] Othman MIA, Tantawi RS, Abd-Elaziz EM. Effect of initial stress on a thermoelastic medium with voids and microtemperatures. J Porous Media 2016;19:155–72. [20] Nayfeh A, Nasser SN. Electromagneto-thermoelastic plane waves in solids with thermal relaxation. J Appl Mech 1972;39:108–13.

M.I.A. Othman, E.M. Abd-Elaziz / Results in Physics 7 (2017) 4253–4263 [21] Othman MIA, Song YQ. Effect of rotation on plane waves of the generalized electromagneto-thermo-viscoelasticity with two relaxation times. Appl Math Modell 2008;32:811–25. [22] Othman MIA, Tantawi RS, Abd-Elaziz EM. Effect of initial stress on a porous thermoelastic medium with microtemperatures. J Porous Media 2016;19 (2):155–72. [23] Othman MIA, Song YQ. The effect of thermal relaxation and magnetic field on generalized micropolar thermoelastic medium. J Appl Mech Tech Phys 2016;57(1):108–16. [24] Othman MIA, Hasona WM, Abd-Elaziz EM. The effect of rotation on fiberreinforced under generalized magneto-thermoelasticity subject to thermal loading due to laser pulse: comparison of different theories. Can J Phys 2014;92(3):1–14.

4263

[25] Abo-Dahab SM, Abd-Alla AM, Alqarni AJ. A two-dimensional problem with rotation and magnetic field in the context of four thermoelastic theories. Results Phys 2017;7:2742–51. [26] Bayones FS, Abd-Alla AM. Eigenvalue approach to two dimensional coupled magneto-thermoelasticity in a rotating isotropic medium. Results Phys 2017;7:2941–9. [27] Cowling TG. Magneto hydrodynamics. New York: Wiley Inter Science; 1957. [28] Zakaria M. Effect of hall current on generalized magneto-thermoelasticity micropolar solid subjected to ramp-type heating. Int Appl Mech 2014;50:92–104. [29] Schoenberg M, Censor D. Elastic waves in rotating media. Quart Appl Math 1973;31:115–25. [30] Dhaliwal RS, Wang J. A heat-flux dependent theory of thermoelasticity with voids. Acta Mech 1995;110:33–9.