Effect of rotation on the surface wave propagation in magneto-thermoelastic materials with voids

Accepted Manuscript

Effect of rotation on the surface wave propagation in magneto-thermoelastic materials with voids A.M. Farhan , A.M. Abd-Alla PII: DOI: Reference:

S2468-0133(18)30134-7 https://doi.org/10.1016/j.joes.2018.10.003 JOES 90

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Journal of Ocean Engineering and Science

Received date: Revised date: Accepted date:

18 September 2018 18 October 2018 20 October 2018

Please cite this article as: A.M. Farhan , A.M. Abd-Alla , Effect of rotation on the surface wave propagation in magneto-thermoelastic materials with voids, Journal of Ocean Engineering and Science (2018), doi: https://doi.org/10.1016/j.joes.2018.10.003

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ACCEPTED MANUSCRIPT

Effect of rotation on the surface wave propagation in magnetothermoelastic materials with voids A. M. Farhan1,2 and A.M.Abd-Alla3 1 Physics Department, Faculty of Science, Jazan University, Saudi Arabia 2

Physics Department, Faculty of Science, Zagzig University, Egypt. Mathematics Department, Faculty of Science, Sohag University, Egypt Abstract. The present investigation is focused on studying the surface wave propagation in a generalized magneto-thermoelastic materials taking (Green Lindsay) model with voids. The general surface wave speed is derived to study the effects of magnetic field and rotation on surface wave in the presence of voids. In addition the present study can be comprehensively applied to the magneto-thermoelastic medium with voids. The possible non-dimensional frequency of surface wave propagation has been obtained as the solution of an algebraic equation involving a determinant whose elements contain the material parameters, the direction of applied magnetic field and rotation of surface wave propagation. The comparisons for the physical quantities are established numerically and represented graphically in different cases with respect to the used effects..

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Keywords: Green Lindsay theory, Thermoelasticity, Magnetic field, Surface

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waves, Rotation,Voids, Half-space.

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Introduction

Theory of linear elastic materials with voids is an important generalization of the classical theory of elasticity. The theory is used for investigating various types of

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geological and biological materials for which classical theory of elasticity is not adequate. The theory of linear elastic materials with voids deals the materials with a

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distribution of small pores or voids, where the volume of void is included among the kinematics variables. The theory reduces to the classical theory in the limiting case of

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the volume of void tending to zero. Such problems have attracted much attention and have undergone a certain development (in Refs. [1–4]). Surface waves have been well recognized in the study of earthquake, seismology, geophysics and Geodynamics. These waves usually have greater amplitudes as compared with body waves and travel more slowly than body waves. There are many types of surface waves but we only discussed Stoneley and Rayleigh waves. In earthquake the movement is due to the surface waves. These are also used for detecting cracks and other defects in materials. Lord Rayleigh [3] was the first to observe such kind of waves in 1885. That’s why we called it Rayleigh waves. Sengupta and Nath [5] investigated surface waves in 1

ACCEPTED MANUSCRIPT fibre-reinforced anisotropic elastic media, but their decomposition of displacement vector was not correct due to which some errors are found in their investigations [6]. The idea of continuous self-reinforcement at every point of an elastic solid was introduced by Belfield et al. [7]. The superiority of fibre-reinforced composite materials over other structural materials attracted many authors to study different types of problems in this field. Fibre-reinforced composite structures are used due to their low weight and high strength. Two important components, namely concrete

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and steel of a reinforced medium are bound together as a single unit so that there can be no relative displacement between them i.e. they act together as a single anisotropic unit. The artificial structures on the surface of the earth are excited during an earthquake, which give rise to violent vibrations in some cases. Engineers and architects are in search of such reinforced elastic materials for the structures

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that resist the oscillatory vibration. The propagation of waves depends upon the ground vibration and the physical properties of the material structure. Surface wave propagation in fiber reinforced media was discussed by various authors.

In the

classical theory of elasticity, the voids is an important generalization. Nunziato and

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Cowin [8] and Cowin and Nunziato [9] discuss the theory in elastic media with voids. Puri and Cowin [10] studied the effects of voids on plane waves in linear

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elastic media and it is evident that pure shear waves remain unaffected by the presence of pores. Theory of thermoelastic material with voids is investigated by Lesan [11]. Good amount of literature on surface wave propagation in a generalized

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thermoelastic material with voids, is available in Singh and Pal [12] and references therein. Chandrasekharaiah [13] and [14] discussed the effects of voids on

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propagation of plane and surface waves. Abo-Dahab [15] investigated the propagation of P waves from stress-free surface elastic half-space with voids. The

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effect of gravity on wave propagation in an elastic solid medium was first considered by Bromwich [16]. Later on gravity effects on wave propagation were discussed by various authors [17]-[20].Surface waves in fiber-reinforced,general viscoelastic media of higher order under gravity is discussed by kakar et. al. [21] whereas Pal and Sengupta [22] studied the gravitational effects in viscoelastic media. Ren, et al. [23] investigated the coupling effects of void shape and void size on the growth of an elliptic void in a fiber-reinforced hyper-elastic thin plate. Vishwakarma et al. [24] discussed the influence of rigid boundary on the love wave propagation in elastic layer with void pores. Tvergaard [25] studied the elastic– 2

ACCEPTED MANUSCRIPT plastic void expansion in near-self-similar shapes. Fonseca, et al. [26] expressed the material voids in elastic solids with anisotropic surface energies. The extensive literature on the topic is now available and we can only mention a few recent interesting investigations in refs.[27]-[38]. The present discussion aims at investigating the propagation of surface waves in a generalized magneto-thermoplastic materials taking GL model with voids and rotation.The general surface wave speed is derived to study the effect of magnetic

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field, rotation and voids on surface waves. The results obtained in this investigation are more general in the sense that some earlier published results are obtained from our result as special cases. Moreover, influence of magnetic field (due to rotation) on the thermoelastic response is being studied in a homogeneous thermally conducting

2- Formulation of the problem

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elastic medium having a void in the half-space in the medium.

We consider a homogeneous isotropic elastic half space z  0 . We choose rectangular coordinate system (𝑥, 𝑦, 𝑧) whereas 𝑧 is pointing towards the medium. .The medium is assumed to be rotating with uniform angular velocity   (0, , 0)

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with respect to the magnetic field H  (0, H 0 , 0) is applied in the 𝑦 direction as a

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result, there would be an induced electric field E and an induced magnetic field h . Folloing Wang and Dong [38], the governing electrodynamic Maxwell equations are:

J  curl h ,

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h  curl E , t div h  0,

CE

 e

(1)

div E  0,

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 u  E   e   H ,  t  h  curl (u  H ),

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

is the magnetic

permeability, H is the constant primary magnetic field and u

is the

displacement vector. Considering an isotropic elastic solid under a magnetic

3

ACCEPTED MANUSCRIPT field H 0 acting on y-axis. The elastic medium is rotating uniformly with an angular velocity   n, where n is a unit vector representing the direction of the axis of rotation. The displacement equation of motion in the rotating frame has two additional term centripetal acceleration,   (  u ) due to time varying motion only.

Refs. [11, 12] ) given by



  

 



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The stress-strain-temperature relations, due to linearly thermoelastic medium, are (see

 ij   e kk   1   1    b    ij  2e ij , t 



..

q i   0 q i  K , i ,

(3)

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S i  ,i ,   e kk    m , g  be kk    m , .

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T0   qi ,i ,

1 1 u i . j  u j ,i  , w ij  u j .i  u i , j  .  2 2

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e ij 

(2)

(4) (5)

(6) (7) (8)

The Maxwell's electro-magnetic stress tensor  ij is given by

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 ij  e  H i h j  H j hi   H k .hk   ij  .

(9)

The basic governing dynamical equations for a homogeneous, isotropic thermoelastic

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solid after including the Coriolis and Centrifugal forces, in the absence of body forces and body couples are given [2]: ..

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 ji , j  Fi  [u i   ^ ( ^ u )],

(10)

which it tends to

 

u i , jj      u j , jj   1   1 

 t

  ,i  b  ,i  Fi  

 [u   ^ ( ^ u )]. The heat conduction equation ([12]) is given by:

4

(11)

ACCEPTED MANUSCRIPT .

..

.

.

..

C e (  0 )  T 0 u k ,k  mT 0 (  0  )  K ,ii ,

(12)

The voids equation is ..

,ii  bu k , k    m     . where

(13)

 ,b , m , are the void material parameters,  t is the coefficient of linear thermal

expansion,



constants,

e is the magnetic permeability,  is the density, q i are the components

of the heat flux vector,

 and  are Lame’s

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is the entropy per unit mass,  is the temperature,

 0 and  1 are the thermal relaxations parameters,  is

the change in the volume fraction field,



is the equilibrated inertia, C e is the

specific heat per unit mass, g is the intrinsic equibrated body forces, K is the

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thermal conductivity, T 0 is the natural temperature of thmedum, T is the absolute temperature, S i are the components of the equilibrated stress vector and F is Lorentz’s force.

The equations (11) to (13) are written in x-z plane as

(14)

 2u 3  2u 1 2    2  e H  z 2     2  e H 0  x z   2u 3  2u 1   2u 3 1   ( 2  )   b  [ 2   2u 3 ], x x z z z t

(15)

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CE

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2 0

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 2u 1  2u 3 2    2    H   e 0  x 2 x z  2u 1  2u 3   2u 1 1   ( 2  )   b  [ 2   2u 1 ], z x z x x t

   2  e H 02 

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C e  0

  2u 1  2u 3     T0    t z t   x t

  2  2    m  0T 0 K   , 2 t z 2   x   2  2    u1 u 3    b   .     m    2 2  z  z  t  x  x where



0

 1 0

 , t

 1  1  1

Now vector and scalar potentials 𝜓 𝑎𝑛𝑑 𝜙 are defined

5

 . t

(16)

(17)

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   , x z

u1 

u3 

   . z x

(18)

Using equation (18) in equations. (14)-(17), we get

CT22   1  b  2   ,

(19) ..

CS2  2   2   , )

(19a)

 2  2      1    2   , .

.





(20)

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.

..

2     b  2  m     . where

b 

b



  2   e H  ,

2

  ,   ,   T 0 mT 0 1  , 2  0 C e C e C S2 

,

K   , C e 0

1.Solution of the problem

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C T2 

(21)

    .  

(22)

In order to examine the possibility of a plane wave propagation in the medium under consideration, we shall assume a solution of governing equations (19-21) in the form

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 , ,   x , z , t   1 (z ), 1 (z ), 1 (z ) exp ik (x

 ct ) .

(23)

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Substituting from equation (23), into equations (19)-(21), we obtain the following set of equations

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(CT2 ( D 2  k 2 )   2   2 )1 ( z )   1  1 ( z )  b1 ( z )  0,

(C S2 (D 22  k 2 )   2  2 ) 1 (z )  0,

(24)

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(24a)

i 1 (D 2  k 2 )1 (z )   (D 2  k 2 )  i   1 (z )  i  21 (z )  0,

(25)

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b (D 2  k 2 ) 1 (z )  m  1 (z )   (D 2  k 2 )     2  1 (z )  0. (26)

where

d2 D  , dz 2 2

The elimination of

  kc .

(27)

1 , 1 and 1 from Eqs. (24)-(26), we get

L  D 2   M  D 2   N  D 2   Q  0. 3

2

6

(28)

ACCEPTED MANUSCRIPT where

L  C T2 , M  C T2 [ (  ( 2   2 )   k 2   )   (i    k 2 )]    ( 2   2  C T2 k 2 )  i  1 1    bb , N  C T2 (i    k 2 )  i  1 1   (  ( 2   2 )   k 2   )   ( 2   2  C T2 k 2 )  (  ( 2   2 )   k 2   )   (i    k 2 )  

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 i   1 (bm   1  k 2 )   2 (b 1   C T2 m )  ,

(29)

Q  ( 2   2  C T2 k 2 ) (i    k 2 )( ( 2   2 )   k 2   )  i  2 m    i  k 2  1 1  (  ( 2   2 )   k 2   )   2b 1    1bm    bbk 2 (i    k 2 ).

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Thus the solution of equation (28) which are bound as z   have the following form

  x , z , t   A1e  m z  A 2e  m z  A3e  m z  e ik ( x ct ) ,

(30)

  x , z , t   1A1e  m1z  2 A 2e  m 2z  3A3e  m3z ]e ik ( x ct ) ,

(31)

2

3

M

1

  x , z , t   1A1e  m1z  2 A 2e  m 2z  3A3e  m3z ]e ik ( x ct )

(32)

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where m1 , m 2 , m 3 are the positive roots of the equation (28),

 n   C (m  k )     1

2 T

2 n

2

2

b

CE

n 

PT

i 2 CT2 (mn2  k 2 )   2   2   i 1 b(mn2  k 2 ) n  ,  (mn2  k 2 )  i b  i 2 1  2

,

(33)

n  1, 2, 3

AC

The general solutions  of the equation (19a) are written as

  x , z , t   B1e  m z e ik ( x ct )

(34)

4

where

m4  k 2 

 2 (   2 ) 

  x , z , t   A1e  m z  A 2e  m 1

(35) 2z

 A3e  m3z  e ik ( x ct ) ,

7

(36)

ACCEPTED MANUSCRIPT   x , z , t   1A1e  m1z  2 A 2e  m 2z  3A3e  m3z  e ik ( x ct ) ,

(37)

  x , z , t   1A1e  m1z  2 A 2e  m 2z  3A3e  m3z  e ik ( x ct ) ,

(38)

4. Boundary conditions If the thermoelastic interactions are caused by a uniform step in temperature applied to the boundary of the half-space which is held in stress-free state, then the following boundary conditions hold:

  0, z u 1  0, u 3  0,

  0, z   0,   0,

at

     z  0 

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 zx   zx  0,

(39)

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 zz   zz  0

Substitution from equations (2), (9), (30-32), (34) and (35-38) into the boundary condition equations (39), we get

a16 a26 a36 a46 a56 a66 a76 a86

a17 a27 a37 a47 a57 a67 a77 a87

M

a15 a25 a35 a45 a55 a65 a75 a85

ED

a14 a24 a34 a44 a54 a64 a74 a84

PT

a13 a23 a33 a43 a53 a63 a73 a83

AC

where

a12 a22 a32 a42 a52 a62 a72 a82

CE

a11 a21 a31 a41 a51 a61 a71 a81

8

a18 a28 a38 a48 0 a58 a68 a78 a88

(40)

ACCEPTED MANUSCRIPT

2

a12     e H

2

a13     e H

2

 m  m  m

a14  2i  km 4 , a15    '  e' H

2

a16    '  e' H

2

a17    '  e' H

2

2 1

k

2

2 2

k

2

2 3

k

2

 m  m  m

  2 m   2 m   2 m

2 1

  1 1  b1 ,

2 2

  1 2  b 2 ,

2 3

  1 3  b3 ,

a18  2i  ' km 4' ,

'2 1

k

2

'2 2

k

2

'2 3

k

2

  2 m   2 m   2 m

'2 1

  1 ' 1'  b '1' ,

'2 2

  1 ' 2'  b ' 2' ,

'

'2 3

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a11     e H

  1 ' 3'  b '3' ,

a21  2i  km1 ,

a22  2i  km 2 ,

a23  2i  km 3 ,

a24     k 2  m 42  ,

a25  2i  ' km1'

a26  2i  'km 2'

a27  2i  'km 3' ,

a28    '  k 2  m 4'2  ,

a32  m 22 , a36  0,

a33  m 33 , a37  0,

a34  a38  0,

a41  m11 , a45  0,

a42  m 2 2 , a46  0,

a43  m 33 , a47  0,

a44  a48  0,

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a31  m11 , a35  0,

a62  m 2 , a66  m 2' ,

a63  m 3 , a67  m 3' ,

a71  1 , a75  1' ,

a72  2 , a76  2' ,

ED

a61  m1 , a65  m1' ,

a81  1 , a85  1' ,

a82   2 , a86   2' ,

a83  3 , a87  3' .

a73  3 , a77  3' ,

PT

CE

a54  m 4 , a58  m 4' ,

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a51  a52  a53  a55  a56  a57  ik

a64  a68  ik ,

a74  a78  0,

a84  a88  0,

If the magnetic field is neglected, the results obtained are deduced to the relevant results

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obtained by Abo-Dahab and Abd-Alla [3]. 5. Numerical results and discussion The present study focuses on the effects of rotation, magnetic field and void pores of the medium on the propagation of plane waves in a solid. For numerical discussion, we have considered three sets of values of relevant parameters from the works of [31] as given below;

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10

  2.7 Kg / m ,   2.17  10 Nm   3.688  10

5

Nm

2

,

10 k  0.04, b  1.13849  10 ,

2

10 2 3 ,   3.278  10 Nm , Cv  1.04  10 J / kg .K , 6

  2.68  10 Nm 6 m  2  10 ,

2

,

2 1 1 1 K  1.7  10 Jm deg s , 10

  1.475  10 ,

  1.753  10

15

,

10

  1.475  10 .

For numerical computational purposes Matlab simulation technique is adopted. The variations of non-dimensional frequency of the magneto-thermoelastic material is

a uniform rotation. T 0  298K

.

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Using these values, it was found that   0.01,

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shown graphically in Figs. 1-7 for a thermo-elastic medium with voids in presence of

 0  0.05 and Ω=1.3, for the three modes

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Fig. 1: Variations of the non-dimensional frequency with respect to magnetic field at

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Fig. 2: : Variations of the non-dimensional frequency with respect to magnetic field at

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 0  0.05 and Ω=1.3, for the three modes

Fig. 3: Variations of the non-dimensional frequency with respect to magnetic field at  0  0.05 and Ω=1.3, for the three modes

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Fig. 4: Variations of the non-dimensional frequency with respect to magnetic field at  0  0.05 for different values of rotation Ω

Fig. 5: Variations of the non-dimensional frequency with respect to magnetic field at Ω=1.3 for different values the thermal relaxations  0

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Fig. 6: Variations of the non-dimensional frequency with respect to rotation  at

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 0  0.05 for different values the magnetic field

Fig. 7: Variations of the non-dimensional frequency with respect to rotation  at  0  0.05 for different values the relaxation times

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ACCEPTED MANUSCRIPT Fig. 1 illustrates the effect of magnetic field on the non-dimensional frequency, which it increases with increasing of magnetic field in the whole range of the H 0 -axis for three modes.The non-dimensional frequency smaller except those of the lower modes vibrations with increase and become larger with with an increase of the magnetic field of the two mediums. In Figs. 2, the effect of magnetic field on the non-dimensional frequency in the whole

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range of the H 0 -axis for three modes as shown in Figures (1). It's obvious that the nondimensional frequency increases with increasing of magnetic field.For a specified magnetic field , the non-dimensional frequency of the medium with the parabolically varying magnetic field are the largest. This order corresponds to that of the medium, as shown in Figure (3). is drawn to exhibit the reason for this ordering is that the increase of the

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magnetic field is more effective than that of the mass on the natural frequencies of the medium.

In Fig. 4, a graphical representation is given for variations the non-dimensional frequency with respect to the magnetic field for the medium, which it has increased with the rotation

 and magnetic field. We can also observe from from Fig. (4) that the non-dimensional

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frequency stays basically H 0 invariable when the rotation  is a large. An obvious trend in that non-dimensional frequency for higher mode number 3 is always larger than the

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corresponding one for lower 3, is also obtained. From the figure 5, it has been observed that the variation of the shows the variations of the

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non-dimensional frequency with respect to the magnetic field H 0 of the rotating medium, which it has decreased with increasing of the relaxation time and magnetic field. The figure

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shows the deflection of the vibration modes without nodal circles, where the maximum deflection rising at the free edge is taken as unity.

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From the figure 6, it is observed that the variations of the non-dimensional frequency with respect to the rotation  for the rotating medium, which it has increased

with the

increasing of the magnetic field and rotation. Since for larger rotation the non-dimensional frequency stay nearly invariable, as indicated above, only two typical values of rotation are considered. Fig, 7 delineates the effect of relaxation time associated with rotation elastic half space with the non-dimensional frequency for the medium, which it has oscillatory behavior in the whole range of the  -axis, which it decreases with increasing of relaxation time  0 .

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6. Conclusion

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This article directly used Lame’s potential method to study the surface wave propagation in magneto-thermoelastic problems of homogeneous, isotropic material with half-space based on two dimensional thermoelasticity. Due to rotation, the Coriolis and Centrifugal forces act as the cause of damping on thermoelastic voids. It is noticed that due to rotation the effect of thermo-elastic voids into the medium is much lesser at the periphery of the half-

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space. Here we are concentrating on the impact of magnetothermo-elastic voids into the homogeneous solids in very short time duration such that the influence of magnetic field as well as voids be taken into account. With the view of theoretical analysis and numerical computation,

we

can

conclude

the

following

phenomena:

1. The present article provides a detailed analysis of voids responses on surface wave

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propagation of magnetothermoelastic disturbance in an unbounded elastic medium in the presence of a magnetic field applied to the traction free boundary of the half-space.

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2. The theoretical analysis and computational results confirm that the magnetic field and rotation can increase the disturbance in surface wave propagation of the thermoelastic field with

respect

to

the

classical

coupled

thermoelasticity

model.

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3. One can select the appropriate amount of rotation to enhance the effect of magnetic field on surface wave propagation of thermoelasticity. It is encouraging that voids will also make in

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sense

comparison

other

thermoelastic.

The Results presented in this article may be useful for researchers who are working on

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material science, mathematical physics, and thermodynamics with low temperatures as well as the development of the hyperbolic thermoelastic theory.

References [1] A.M. Abd-Alla, S. M. Abo-Dahab and F. S. Bayones, Propagation of Rayleigh waves in magneto-thermo-elastic half-space of a homogeneous orthotropic material under the effect of rotation, initial stress and gravity field, Journal of Vibration and Control, 19, 1395–1420 (2013).

15

ACCEPTED MANUSCRIPT [2] A. M. Abd-Alla and S. R. Mahmoud, Magneto-thermoelastic problem in rotating nonhomogeneous orthotropic hollow cylinder under the hyperbolic heat conduction model, Meccanica, 45, 451-462, (2010). [3] L. Rayleigh, On wave propagation along the plane surface of an elastic solid. Proc. London. Math. Soc.17: pp. 4-11 (1885). [4] R. Stoneley, The elastic waves at the surface of separation of two solids. Proc. R. Soc. London. A106: pp. 416-420 (1924).

media. Sadhana 26: pp. 363-370 (2001).

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[5] P. R. Sengupta and S. Nath. Surface waves in fibre-reinforced anisotropic elastic [6] S. Sing, Comments on “Surface waves in fibre-reinforced anisotropic elastic media” by Sengupta and Nath, Sãdhanã 27(3): pp 405-407(2002).

[7] A. J. Belfield, T. G. Rogers, A. J. M. Spencer, Stress in elastic plates reinforced by

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fibre lying in concentric circles. J. Mech. Phys. Solids. 31: pp. 25-54 (1983).

[8] J. W. Nunziato and S. C. Cowin, A Nonlinear Theory of Elastic Materials with Voids, Archive for Rational Mechanics and Analysis, Vol. 72, No. 2, pp. 175-201, 1979. [9] S. C. Cowin and J. W. Nunziato, Linear Elastic Materials with Voids, Journal of

M

Elasticity, Vol. 13, No. 2, 125-147, 1983.

[10] P. Puri and S. C. Cowin, Plane Waves in Linear Elastic Materials with Voids, Journal

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of Elasticity, Vol. 15, No. 2, pp. 167-183, 1985.

[11] D. Iesan, A Theory of Thermoelastic Materials with Voids, Acta Mechanica, Vol. 60, No. 1-2, pp. 67- 89, 1986.

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[12] B. Singh and Raj Pal, Surface wave propagation in a generalized thermoelastic material with voids, Applied Mathematics, vol. 2, 521-526, 2011.

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[13] D. S. Chandrasekharaiah, Effects of surface stresses and voids on rayleigh waves in an elastic solid, Int. J. Engng. Sci. Vol. 25 No. 2, pp.205-211, 1987.

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[14] D. S. Chandrasekharaiah, Plane waves in a rotating elastic solid with voids, Int. J. Engng. Sci. Vol. 25 No. 5, pp.591-596, 1987. [15] S.M. Abo-Dahab, Propagation of P waves from stress-free surface elastic half-space

with voids under thermal relaxation and magnetic field. Applied Mathematical Modelling, Vol. 34, pp 1798-1806, 2010. [16] J. Bromwich, On the influence of gravity on elastic waves, and, in particular, on the vibrations of an elastic globe, Proc. London Math. Soc., 30 : 98-120 (1898).

16

ACCEPTED MANUSCRIPT [17] A. M. Abd-Alla, T. A. Nofal, S. M. Abo-Dahab and A. Al-Mullise. Surface waves propagation in fibre-reinforced anisotropic elastic media subjected to gravity field. Int J. Phys. Sci. 8 (14), pp. 574-584 (2013). [18] A. M. Abd-Alla and S. M. Ahmed, Stoneley and Rayleigh waves in a nonhomogeneous orthotropic elastic medium under the influence of gravity, Applied Mathematics and Computation135:pp. 187-200 (2003). [19] S.K. De and P.R. Sengupta, Influence of gravity on wave propagation in an elastic

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layer, J.Acanst. Soc. Am. 55, 5-13 (1974).

[20] P.R. Sengupta and D. Acharya, the influence of gravity on the propagation of waves in a thermoelastic layer, Rev. Romm. Sci. Techmol. Mech. Appl., Tome 24, 395-406 (1979).

[21] R. Kakar, S. Kakar and K. Kaur, Rayleigh, Love and Stoneley waves in fiber-

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reinforced,general viscoelastic media of higher order under gravity. International Journal of Physical and Mathematical Sciences, Vol.4 No.1 pp.53-61 (2013).

[22] K. C. Pal and P. R. Sengupta, Surface waves in viscoelastic media of general type in the presence of thermal field and gravity. Proc. Indian Natl. Sci. Acad. A53: pp. 353-372

M

(1987).

[23] J. Ren, H. Li, C. Cheng and X. Yuan, Coupling effects of void shape and void size on

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the growth of an elliptic void in a fiber-reinforced hyper-elastic thin plate, Acta Mechanica Solida Sinica, Vol. 25, pp.312-320 (2012). [24] S. K. Vishwakarma, S. Gupta and D. K. Majhi, Influence of rigid boundary on the

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love wave propagation in elastic layer with void pores, Acta Mechanica Solida Sinica, Vol. 26, pp. 551-558 (2013).

Elastic–plastic void expansion in near-self-similar shapes,

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[25] V. Tvergaard, Computational

Materials

Science,

Vol.

50,

pp.

3105-3109

(

2011).

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[26] I. Fonseca, N. Fusco, G. Leoni and V. Millot, Material voids in elastic solids with anisotropic surface energies, Journal de Mathématiques Pures et Appliquées, Vol. 96, pp. 591-639 (r2011). [27] S. A. Yahia,A. A. Hassen, H. M. Sid Ahmed and A. Tounsi, Wave propagation in

functionally graded plates with porosities using various higher-order shear deformation plate theories, Struct. Eng. Mech., 53(6), 1143-1165 (2015). [28] M. Benadouda,

A. A. Hassen, A. Tounsi andS.R.Mahmoud ,An efficient shear

deformation theory for wave propagation in functionally graded material beams with porosities”, Earthquakes and Structures, 13(3), 255-265 2017). 17

ACCEPTED MANUSCRIPT [29] A. A. Bousahla, S. Benyoucef, A. Tounsi and S.R.Mamoud, On thermal stability of plates with functionally graded coefficient of thermal expansion, Struct. Eng. Mech., 60(2), 313-335 (2016). [30] B. Bouderba, Houari Mohammed Sid Ahmed, Abdelouahed Tounsi and S.R.Mahmoud, Thermal stability of functionally graded sandwich plates using a simple shear deformation theory”, Struct. Eng. Mech., 58(3), 397-422 2016). [31] F. El-Haina, A. Bakora, A. A. Bousahla and S.R.Mahmoud, A simple analytical

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approach for thermal buckling of thick functionally graded sandwich plates, Struct. Eng. Mech., 63(5), 585-595 2017).

[32] A. Menasria, A. Bouhadra, A. Tounsi and S.R.Mahmoud, A new and simple HSDT for thermal stability analysis of FG sandwich plates, Steel and Composite Structures, 25(2), 157-175 2017).

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[33] A. Attia, A. A.Bousahla, A.Tounsi and A.S. Alwabli, A refined four variable plate theory for thermoelastic analysis of FGM plates resting on variable elastic foundations, Structural Engineering and Mechanics, Vol. 65, No. 4 (2018) 453-464 (2018). [34] B. Bouderba, H. M. Sid Ahmed and A. Tounsi, Thermomechanical bending response Structures, 14(1), 85 – 104 (2013.

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of FGM thick plates resting on Winkler–Pasternak elastic foundations, Steel and Composite

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[35] Y. Beldjelili, A. Tounsi and S.R.Mahmoud,Hygro-thermo-mechanical bending of SFGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory, Smart Structures and Systems, 18(4), 755-786 (2016).

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[36] C. Abdelbaki, A. Tounsi, H. Habib and S.R.Mahmoud, Thermal buckling analysis of cross-ply laminated plates using a simplified HSDT, Smart Structures Systems, 19(3), 289-

CE

297 (2017).

[37] A. Hamidi, H. M. Sid Ahmed, S.R.Mahmoud and A. Tounsi, -Hamidi et al. (2015),

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“A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates”, Steel and Composite Structures, 18(1), 235 – 253 (2015). [38] X.Wang and K.Dong, Magneto-thermo-dynamic stress and perturbation of magnetic field vector in a non-homogeneous thermoelastic cylinder, Eur.J. Mech. A Solids, 25, 98109 (2006).

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