Effect of a magnetic field on the propagation of waves in a homogeneous isotropic thermoelastic half-space Текст научной статьи по специальности «Физика»

УДК 537.632 : 539.37

Влияние магнитного поля на распространение волн в однородном изотропном термоупругом полупространстве

A.M. Abd-Alla1, S.M. Abo-Dahab23, S.M. Ahmed4, M.M. Rashid1

1 Сохагский университет, Сохаг, 82524, Египет 2 Университет Южной долины, Кена, 83523, Египет 3 Таифский университет, Таиф, 888, Саудовская Аравия 4 Университет Эль-Ариш, Северный Синай, 31111, Египет

В работе проведен анализ распространения термоупругих волн в однородном изотропном упругом полубесконечном пространстве под воздействием магнитного поля при начальной температуре T0, на граничную поверхность которого действуют движущийся источник тепла и подвижная нагрузка с конечной скоростью. Найдено распределение температуры и напряжений в ходе нагрева и охлаждения для заданных граничных условий. Результаты численного анализа свидетельствуют о сильном влиянии магнитного поля. Проведено сравнение с результатами, полученными в рамках теории термоупругости при отсутствии магнитного поля. Результаты работы могут быть использованы в геофизических приложениях, при разработке датчиков поверхностных волн, а также для усовершенствования акустических волновых устройств и волноводов.

Ключевые слова: движущийся источник тепла, подвижная нагрузка, тепловые напряжения, термоупругость, распространение волн, магнитное поле

DOI 10.24411/1683-805X-2019-11008

Effect of a magnetic field on the propagation of waves in a homogeneous isotropic thermoelastic half-space

A.M. Abd-Alla1, S.M. Abo-Dahab23, S.M. Ahmed4, and M.M. Rashid1

1 Mathematical Department, Faculty of Science, Sohag University, 82524, Egypt 2 Mathematical Department, Faculty of Science, South Valley University, Qena, 83523, Egypt

3 Mathematical Department, Faculty of Science, Taif University, Taif, 888, Saudi Arabia 4 Mathematical Department, Faculty of Science, Al-Arish University, North Sinai, 31111, Egypt

The prime objective of the present paper is to analyze the propagation of thermoelastic waves in a homogeneous isotropic elastic semiinfinite space that is exposed to a magnetic field at initial temperature T0 and whose boundary surface is subjected to the moving heat source and load moving with finite velocity. Temperature and stress distribution occurring due to heating or cooling have been determined using certain boundary conditions. Numerical results indicate that the effect of the magnetic field is very pronounced. Comparison is made with the results predicted by the theory of thermoelasticity in the absence of a magnetic field. Apart from geophysical applications, the consequences of the present study offer a better platform to design a surface wave sensor by means of its established results. The obtained results may be also used for acquiring a better performance in surface acoustic wave devices and waveguides.

Keywords: moving heat source, moving load source, thermal stresses, thermoelasticity, wave propagation, magnetic field

1. Introduction

Thermoelastic problems play an important part in different branches of technical sciences. In engineering practice, most structures contain internal interfaces when heat flow in the structure is disturbed by some defects, such as

© Abd-Alla A.M., Abo-Dahab S.M, Ahmed S.M., Rashid M.M., 2019

holes and cracks, the local temperature gradient around the defects is increased and the temperature field is often discontinuous across the defects. Thermal disturbances of this type may produce material failure. Therefore, thermal analysis of such structures is very important. As far as loading is concerned, generally two types of loading—mechanical and

thermal—are applied to the structure surface. Mechanical loading consists of pressure and shear tractions, and thermal loading is caused by frictional heating and results in thermoelastic stress. The knowledge of the thermoelastic stress field in the structure is essential for failure prevention and life prediction because the total stress consists of the thermoelastic and elastic stresses. Due to this phenomenon, frictional heating significantly influences the failure of components in contact under relative motion, e.g. ther-mocracking of breaks and face sears, and scuffing in gear. The linear dynamic theory of thermoelasticity introduced by Biot [1] holds that the governing equations for displacement and temperature fields consist of the two coupled partial differential equations. He and Cao [2] presented a valuable investigation based on the generalized thermoelastic theory with thermal relaxation in the context of Lord and Shulman's theory, which is used to investigate the mag-neto-thermoelastic problem of a thin slim strip placed in the magnetic field and subjected to the moving plane of the heat source. Deformation of a rotating generalized thermo-elastic medium subjected to ramp-type heating and loading with a hydrostatic initial stress was studied by Ailawalia and Narah [3]. Disturbances in a homogeneous isotropic elastic medium subjected to the moving source with generalized thermoelastic diffusion were considered by Deswal and Choudhary [4]. Abd-Alla and Ahmed [5] investigated the influence of both the gravity field and initial stress on the propagation of Rayleigh waves in an orthotopic thermoelastic medium. A linear temperature ramp function used to model more realistically thermal loading of the half-space surface subjected to the moving heat source was considered by Amin et al. [6]. Abouelregal [7] investigated the induced temperature, displacement, and stress field in an infinite transversely isotropic unbounded medium with a cylindrical cavity due to the moving heat source and harmonically varying heat. Abd-Alla et al. [8] studied the influence of rotation and initial stress on the propagation of Ray-leigh waves in an orthotropic elastic half-space with gravity. The microrotation effect of a load applied normal to the boundary and moving at a constant velocity along one of the coordinate axes in the generalized thermoelastic halfspace was discussed by Kumar and Deswal [9]. Abd-Alla et al. [10] solved the problem of transient coupled thermo-elasticity of an annular fin using the finite difference method. Abd-Alla et al. [11] studied the influence of the magnetic and gravity fields on the generalized thermoelastic Rayleigh waves in a granular medium with initial stress. Ailawalia and Narah [12] obtained expressions for displacement, force, stress and temperature distributions in a rotating generalized thermoelastic medium with a hydrostatic initial stress by applying Laplace and Fourier transforms subjected to ramp-type heating and loading. Kumar and Gupta [13] studied the general plane strain problem of an orthotopic micropolar thermoelastic half-space with one relaxation time. The

effect of the load velocity, nonlocal parameter, and ramp-ing-time parameter on the dynamic deflection, temperature, and bending moment of the nanobeam was investigated by Abouelregal and Zenkour [14]. Abd-Alla et al. [15] established the influence of the thermal stress and magnetic field in the thermoelastic half-space without energy dissipation. Abo-Dahabet et al. [16] investigated the effect of rotation and gravity on the reflection of P-waves from thermo-mag-netomicrostretch medium in the context of the three-phase-lag model with initial stress. Xia et al. [17] solved the generalized thermoelastic problem of an isotropic semi-infinite plate subjected to the moving heat source using the finite element method directly in the time domain. Wang and Dong [18] studied the magneto-thermodynamic stress and perturbation of the magnetic field vector in an inhomogeneous thermoelastic cylinder. Sherief and Allam [19] investigated the electromagnetic interaction of a two-dimensional generalized thermoelastic problem for an infinitely long solid conducting circular cylinder.

In contrast to all these investigations, the present paper studies the propagation of a thermoelastic wave in the halfspace made of homogeneous isotropic materials exposed to the magnetic field and in contact with the coordinate system moving with the load point by shifting the origin to the position of the load point. Here the Lame potential method is proposed to analyze the problem and to obtain numerical solutions for displacement components, stresses, and temperature distribution. The effects of the magnetic field, moving heat source, and load moving with finite velocity on the components of displacement, stress, tangential stress, and temperature are considered. 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. The results obtained are presented graphically with concluding remarks.

2. Formulation of the problem

Following Wang and Dong [18], the governing electro-dynamic Maxwell equations are

J = curlh, dh = curlE, divh = 0, divE = 0, dt

E = -l{

dus dt'

(1)

H I, h = curl(uxH),

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 magnetic field, and u is the displacement vector. A substitution of the initial magnetic field vector H(0, H0, 0) in Cartesian coordinates (x, y, z) into Eq. (1) gives u = (u,0, w), H = (0,H0,0),

E = -e *-H0 f ■»• H0 £

h = (-H0w, 0, H0U),

,, , du dw j = - H 01 —+ — J 01 dx dz

, f = we( j xH).

Let us consider the half-space z > 0 initially at a temperature To and in the stress-free state. Temperature, displacement, and stress fields will vary due to external loading. On this assumption, the displacement occurs along the x and z axes and is a function of the spatial coordinates x, z and time t.

The dynamic equations of motion in the magnetic field are given by

3 2u

dt2

3 w

_ ,, , . d 2u 3 2u

= fx + (X + 2^)-r +

ax dz ,, . d2w dT

+ (X + W)^ ,

axaz ox

.. .d2 w 32 w = fz + (X + 2W)~t +

dz dx

+ (X + w)

du dT -Y-

(3)

(4)

dxdz dz

The generalized equation of heat conduction is given by

KV2T = pce

dt

V &

3 2T

dt2

+ (3X + 2w)atT0

(

^2 &

V-u.

(5)

dt + T° dt2

v J

The stress-displacement relations with incremental isotropy are given by

a xx = (X + 2w) dU "Y(T - To), dx dz

a zz = (X + 2n) dw + -Y(T - To), dz dx

(6)

Txz = W

dw du — + —

dx dz

Here y = (3X + 2^)a t, T is the temperature, ce is the specific heat, p is the density, a t is the coefficient of thermal expansion, t0 is the relaxation time, K is the thermal conductivity, fx = weHo(d2u/dx2 +d2w/dxdz) and fz = We x X H<2(d2^dxdz + d2Wdz2) are the components of the Lorentz force.

We will consider a moving heat source applied to the half-space surface with the following initial conditions:

T(x, z, t) = f (x - vt),

Txz (X, Z, t) = ° Ozz (x, Z, t) = 0.

Along with the moving heat source, we consider the moving load with the following initial conditions:

°zz (x z 0 = g(x - vt),

r\T

Txz (x, z, t) = 0, — + hT = 0. dx

(8)

Here h is the surface heat transfer coefficient, f and g are the arbitrary functions, and v is the velocity of both source and load. Now, we will use the following dimensionless variables to transform the above equations into dimension-

less forms:

2 3

, ij / Cj , Cj p

t = — t, x j = — xj, u = —--1

kj j k1 k1 (3X + 2w)atT0

T' = LJ0, a';=.

aj

(9)

T0 ' 'J (3X + 2w)atT0

where cj2 = (X + 2^)/ p and kj = K/ (pce).

In terms of the dimensionless variables in (9), Eqs. (3)-(6) take the following forms:

% dr d 2r &

- + T-

V2T =

3t dt2

2

f 3 32 &

+ e Vu'

dt dt2

(10)

a2 ' a2 /

d u ^ „2n d w

dx2

(X + 2W+ We H2)— + (X + W + We H02 )—— +

2 dx dz

d2u' .. , ,3T' .. , .32u'

+ (X + = (X + 2^)—(U)

dz 2 dx dt2

2

dz'

32 w

(X + 2W+We H0 + (X + W + We H02 +

dT'

2

d u ' dx'dw'

2

d w

+ W—-- (X + 2u)—- = (X + 2u)—(12)

The stress components in the dimensionless forms are a' du 2, dw ,

axx = ^T + (1"2c ^"T ,

dx dz

/ 3w' ^ du ,

azz = ^T + (1 "2c ^"T ,

dz dx

= 2 , 3w' du'

Txz =c 1 37 + 37''

(13)

where e = (3X + 2w)2 at2T0/[pce (X + 2w)] and T = To x x cj2/kj.

In the subsequent discussions we omit primes for convenience.

3. Solution of the problem

Using Helmholtz's theorem [20], the displacement components can be written as

rhir

(14)

dé dw dé dw u = —L-—L-, w !-,

dx dz dz dx

where ^(x, z, t) and ^(x, z, t) are the scalar functions, and y = (0, 0) is the vector potential function.

The substitution of Eqs. (14) into Eqs. (10)-(12) gives

% - 8 d2 U id d2 &

d2 & C2V? - m2 d

T' = e —+t „

dt' dt'2

j \ J

( - m2 d2 &

(15)

dx

2

» = T,

V,2---

Vj c2 dx"2

w = 0,

where c2 = (X + w), C2 = (X + 2w+ weHo2)/(X + 2w).

We use the coordinate system moving with the load point by shifting the origin to the load point

2. d2 d2

V2 ^

1 dx"2 dz"2

where m = v/cl is the dimensionless loading speed, and the coordinates x" and z" move in the positive direction of the x axis with the speed m. A substitution of Eq. (16) into Eqs. (15) gives

dx"

„г " 2 d2 v, + m—- -Tm --

1 ff ff2

dx"

T ' =

%

d 2 d2 &

= e -m—- + Tm

dx dx"2

V У

2A ...2 d2Ф

' 1 V>

С2У12ф- m

dx'

2

= T

2 - mL JL &

л

V2 -

с dx'

у = 0.

(17)

(18) (19)

Replacing Vj by V to and x" by x results in

%

2 , d 2 d2 &

V + m--Tm 2

dx dx 2

v

T =

%

= e

d 2 d2 &

-m—+Tm 2 dx dx2

v y

d2 &

2 „„2 d

C2V2 - m

dx2

Ф = T,

2 Д2 &

V2 -

m d

с2 dx2

у = 0.

(20) (21) (22)

We see that Eqs. (20) and (21) are coupled with T and while Eq. (22) is independent on y. In order to solve the coupled Eqs. (20) and (21), we eliminate T or ^ to obtain the following equation

- d 2 a2 If . 2 a2 &

V2 + m— - Tm2 —2 dx dx

C2V2 - m2—2

2 dx2

v J

%

+ e

d .2 i2 & dx2

m— -Tm 2 dx dx2

v y

i = 0.

To solve Eqs. (22) and (23), we take

v ikx-az ту ikx-a'z

I = Ae , у = Be ,

(23)

(24)

where A and B are the constants, and k, a, and a' are the unknown quantities to be determined.

A substitution of ^ and y into (22) and (23) gives

a'2 -

(

2

m

- С

v

k2 = 0, a4 + aa2 + b = 0.

(25)

For a' to be real, 1 - m2/c2 > 0 and c2 > m2, and a3 =

= k-J 1 - m2/c2 is the root of Eq. (251).

Equation (252) is biquadratic in a and hence it has four roots. We shall consider the roots with positive real parts. Let them be a, and a2

т1)2 = 4-a ±4a2-4b,

' -v 2

(26)

where

a = -2k2 + Tm2k2 + imk +

m2k2 eimk eTm2k2

C9

C9

C9

b = k4 - imk3 - Tm 2k4 +

.3,3 „4,4 2,4 . ,3 „„ 2,4 im k Tm k m k eimk eTm k

C2

C2

C2

C2

C2

Thus, from relation (24) we have

У= E Ak

ikx -a3z

k=-~

• = E [Bk

ikx-a, z

1Z +Cte

ikx-a 2z -|

k=-~

2 12 \ ikx-a^i

(27)

(28)

+ Ck(C2(a2 -k2) + ], (29)

where Ak, Bk and Ck are the arbitrary constants, which will be determined by using certain boundary conditions. The stress components in terms of ^ and y are

T = E [Bk (C2(of - k2) + m2 k )ei

k=-

Txz = с

%d2у d2у + 2 d2Ф&

dx dz

dxdz

a _э2ф_2c2 T

axx = dx2 dxdz +в1 dz2 T,

a =d!i+2c2 T z dz2 dxdz 1 dx2 ' where P = X/ (X + 2|a).

By substituting Eqs. (27)-(29) into Eqs. (30)-(32), we obtain

Txz =-c2 E [Ak (a32 + k2y'fa-a3z + k

+ Bk (2ikal)eikx-ajz + Ck (2ika2) eikx-a2 z ], (33)

axx = E [Ak(2ikc2a3)eikx-a3z + k

+ Bk(a2(ft -C2) + k2(C -m2 - 1))eifa-aJz + + Ck (a2 (Pj - C2) + k2 (C - m2 - 1))eikx-a2z], (34)

azz = E [Ak(-2ikc2a3)eikx-a3z + k

+ Bk (a2 (1 - C2) + k2(C2 - m2 -ft ))eiix-aJz + + Ck(a2(1 -C2) + k2(C2 -m2 -ft))e^"^]. (35)

By substituting Eqs. (27) and (28) into Eq. (15), we obtain

u = EE [Ak (a3)eifa_a3z + Bk (ik)eikx-aJz +

(30)

(31)

(32)

k

+ Ck (ik)eikx-a2z ],

w = E [ Ak (ik)eikx-a3z - Bk ( a1)eifa-a1z ■ k

- Ck(a2)eifa-a2z].

(36)

(37)

4. Case I: moving heat source

Let us consider the boundary conditions for the moving heat source given by Eq. (7) to determine the arbitrary con-

stants Ak, Bk, and Ck. Then the surface of the medium at z = 0 assumes the conditions T(x, 0, t) = f (x - vt),

t xz ( x, 0, t ) = 0, a zz ( x, 0, t ) = 0.

In the first equation in (38), we assume

T(x,0, t) = f (x - vt) = £ akejk(x-vt\ k

where

ak = 1/n ] f (x - vt) ék(x-vt)dx, f (x - vt) = e"(x-vt)2.

-n

From Eqs. (29), (33), (35), and (38), we obtain

(38)

(39)

£ [Bk (C2(af - k2) + m2 k2) +

k

+ Ck (C2(a2 - k ) + m k )] = £ ak,

k

(40)

£ [Ak(a2 + k2) + Bk(2jkaj) + Ck(2ika2) = 0, (41)

k

X [4 (-2ikc2 03) + 5k (o?(J - C2) +

k

+ k2(C2 -m2-Pi)) + Ck(o2(l-C2) + + k2(C2 - m2 -Pl))] = 0. (42)

We find the solution of Eqs. (40)-(42) by Cramer's rule (Appendix I).

5. Case II: moving load source

Let us consider the boundary conditions for the moving load source given by Eq. (8) to determine the arbitrary constants Ak, Bfk, and Ck for the medium surface with z = 0: azz (x, 0, t) = G(x - vt), txz (x, 0, t) = 0,

^ + hT = 0,

dx

where h is the coefficient of heat transfer. The load function in (43) will be given by

azz (x,0, t) = G(x - vt) = £ bke

k

ik ( x-vt )

(43)

(44)

Fig. 1. Case I. Variation of T, u, w, axx, azz, and txz at different relaxation times t0 with respect to horizontal distance z

U.UUU1-1-1--U.UU1-1-1-

0.0 0.5 l.O z 0.0 0.5 l.O z

Fig. 2. Case I. Variation of T, u, w, axx, azz, and txz at different wave numbers k with respect to horizontal distance z

bk =1 J g (x - vt )eik (x-vt W, g (x - vt ) = e"(x-vt > .

From Eqs. (29), (33), (35), and (43), we obtain

£ [Ak(-2ikc2a3) + Bk(o?(l-C2) +

k=-»

+ k2(C2 - m2 - Pi)) + C'(a2(1 - C2) +

+ k2 (C2 - m2-Pi))] = £ bke~ikvt, (45)

k

£ [A'(g32 + k2) + B'(2ikg1 ) + C'k (2ika2)] = 0, (46)

k

£ [ Bk (ik + h )(C2( a3 - k2) + m2 k2) +

k=-~

+ C' (ik + h)(C2 (a3 - k2 ) + m2k2 )] = 0. (47)

We find the solution of Eqs. (45)-(47) by Cramer's rule (Appendix II).

6. Numerical results and discussion

To illustrate the analytical procedure presented earlier, we now consider a numerical example for which computational results are given.

The material chosen for this purpose is stainless steel, the physical data for which are the following: X = 9.3 x x 1010 N m-1, = 8.4 x 1010 N m-1, at = 13.2 x 10-6 deg-1, p = 7.97 x 103 kg m-3, T0 = 298 K, K = 50 kg m s"3 K-1, k, = = 0.112 cm2 s-1, ce = 0.56 x 103 K"1 m2 s2, x = 0.01, c, = 20, m = 4.25.

Considering the above physical data, dimensionless field variables are evaluated, and the results are presented in the form of graphs at different positions of z at x = 1.0. The range of motion of the heat source and load is taken to be 0 < z < 0.2. The graphical results for the displacement components, normal stress, tangential stress, and temperature are shown in Figs. 1-6.

Figure 1 shows variations of the absolute values of temperature |T|, displacement components |u|, |w|, normal stresses |axx|, |azz |, and shear stress |txz| with respect to the z axis at different relaxation times t0 due to the moving heat source. It is obvious that the absolute values of displacement component |w|, normal stress |azz |, temperature and shear stress |txz | increase with increasing relaxation time, while the absolute value of normal stress and shear stress have an oscillatory behavior for generalized thermoelastic material in the whole range of axis z due to the moving heat source. We observe larger oscillation in the left side and smaller oscillation in the right side. There is no effect of the relaxation time on the absolute value of the displacement component |u|, and it is noticed that the temperature |T|, shear stress |t xz and normal stress |a zz| satisfy the boundary conditions. As a result, each location in the thermally disturbed region receives less amount of energy as the relaxation time increases. This in turn leads

to a reduction in the local temperature distribution at the half-space surface.

Figure 2 displays variations of the absolute values of temperature |T|, displacement components |u|, |w|, normal stresses |axx|, |azz |, and shear stress |txz| with respect to the z axis at different wave numbers k due to the moving heat source. It is obvious that the absolute values of displacement components and temperature decrease with increasing wave numbers, while the absolute value of normal stress |a xx| increases with increasing wave number. The displacement components, normal stress |azz| and shear stress |txz| have an oscillatory behavior for generalized thermoelastic material in the whole range of axis z due to the moving heat source. It is noticed that the temperature |T|, shear stress |txz| , and normal stress |a zz| satisfy the boundary conditions. The heat source is moving in the z direction, and the curves also tend to move in the same direction. From the variation of stress components txz,

\u\

0.6985-

0.6980-

0.6975-

0.6970

____^ - ^

/ ^ \ s \ %

t0 = 0.01- \ \ X \

0.02---

0.03--

0.00

0.05

0.10

0.15

0.00

0.05

0.10

0.15

Fig. 4. Case II. Variation of T, u, w, axx, azz, and Txz at different relaxation times t0 with respect to horizontal distance z

axx, azz due to the moving heat source, we see that stress components txz, axx, azz vary less in the left side and more in the right side.

Figure 3 shows variations of the absolute values of temperature |T|, displacement components |u|, |w|, normal stresses |axx|, |azz |, and shear stress |txz| with respect to the z axis at different magnetic fields H0 due to the moving load source. It is obvious that the absolute values of temperature and normal stresses axx, azz decrease with increasing magnetic field, while the absolute values of shear stress and displacement component |u| increases with increasing magnetic field. The absolute values of displacement components |u| and |w| have an oscillatory behavior due to the moving heat source. We see larger oscillation in the left side and smaller oscillation in the right side. Here we find that the magnetic field causes the displacement components to oscillate, and it increases with increasing

axis z in the whole range of axis z. It is also seen that the stress distribution has a local extremum at the position of the magnetic field increase.

Figure 4 demonstrates variations of the absolute values of temperature |T|, displacement components |u|, \w\, normal stresses |axx|, |azz and shear stress |txz| with respect to axis z at different relaxation times t0 due to the moving load source. It is obvious that the absolute values of temperature, displacement components, shear stress, and normal stresses increase with increasing relaxation time. It is noticed that the temperature |T|, shear stress |txz| and normal stress |a zz | satisfy the boundary conditions. The relaxation time parameter also has a significant effect on the temperature distributions.

Figure 5 illustrates variations of temperature |T|, displacement components |u|, |w|, normal stresses |axx|, |azz |, and shear stress |txz | with respect to the z axis at different

Fig. 5. Case II. Variation of T, u, w, axx, azz, and txz at different wave numbers k with respect to horizontal distance z

wave numbers k due to the moving load source. It is obvious that the absolute values of temperature, displacement component |w|, shear stress and normal stresses increase with increasing wave number, while the displacement component decreases with increasing wave number. In case of the moving load source, the wave number effect is more significant for displacement components and stress components in comparison with the temperature field. It is noticed that the temperature |T|, shear stress |t xz| , and normal stress |a zz | satisfy the boundary conditions.

Figure 6 presents variations of the absolute values of temperature |T|, displacement components |u|, |w|, normal stresses |axx|, |azz |, and shear stress |txz| with respect to the z axis at different magnetic fields Ho due to the moving load source. It is obvious that the absolute values of temperature, displacement components |u|, |w|, shear stress, and normal stress |a xx| increase with increasing magnetic field, while the normal stress |a zz | decreases with increas-

ing magnetic field. It is noticed that the temperature |Г|, shear stress |txz| , and normal stress |g zz| satisfy the boundary conditions. The load is moving in the z direction, and the curves also tend to move in the same direction. From the variation of stress components тxz , gxx , and azz due to the moving load, we see that stress components тxz , axx , and azz vary less in the left side and more in the right side.

7. Conclusion

Propagation of waves in an isotropic thermoelastic halfspace subjected to the moving heat source, moving load source, and magnetic field is investigated based on the generalized thermoelastic theory. From the above discussions, we can arrive at the following conclusions:

Variation in the magnetic field, wave number, and relaxation time has a significant effect on the temperature, displacement components, and stress components.

Fig. 6. Case II. Variation of T, u, w, axx, azz, and t at different magnetic fields H0 with respect to horizontal distance z

Analysis of displacement components, stress compo- AAk ABk ACk

nents, and temperature in a body due to the moving load Ak = _> Bk = > Ck = > (AI)

source and moving heat source reveals that the relaxation where

= i

( ajia32 a23 + a12 a2la33 + al3 a31a22)'

time play a significant role in all the analyzed quantities. A = a11a22a33 + a12a31a23 + a13a21a32 -

a11 = 0, a12 = C2(a? -k2) + m2k2,

Variations of temperature, displacement components, and stress components depend on the nature of this moving

heat source and the moving load source. AAk = ak(a22a33 - a23a32),

The magnetic field, relaxation time, and wave number ABk = ak (a23a31 - a21a33),

of the half-space give the same effect in case I and case II, ACk = ak (a21a32 - a22a31), as mentioned above in the results.

The present theoretical results may provide interesting 2 2 2 2

information for experimental scientists, researchers, and a13 = C2(a2 - k ) + m k ,

seismologists working on this subject. a21 = a2 + k2, a22 = 2ika1, a23 = 2ika2,

2 k 2

Appendix I 31 2 3' 2 2

The constants Ak, Bk and Ck determined from a32 = a(1 C2) + k (C2 m P1)'

Eqs. (40)-(42) are obtained using Cramer's rule: a33 = a2(1 - C2) + k2 (C2 - m2 -P1).

Appendix II

The constants A'k, B'k and C'k determined from Eqs. (45)-(47) are obtained using Cramer's rule:

a' _M B _ABk C _ACk Ak _ , Bk , Ck _ ! :

(AII)

where

dn _ —likc2^,

du _o2(1 -C2) + k2(C2— ш2 — ßi), d13 _ст^(1 — C2) + k2(C2 — ш2 —ß1),

¿21 + k2, ¿22 _ 2ik, ¿23 _ 2ik^2,

¿31 _ 0, ¿32 _ (ik + A)(C2(ct^ — k2) + ш2k2), ¿33 _ (ik + h)(C2(^2 — k2) + ш2k2),

A — ¿11 ¿22 ¿33 + ¿12 ¿31 ¿23 + ¿13 ¿21 ¿32

— (¿11¿32¿23 + ¿12¿21¿33 + ¿13¿31¿22)' AAk _ ^¿33 — ¿32¿23)bke~lkU> ABk _ (¿31¿23 — ¿21¿33)bke~ik> ACk _ (¿21¿32 — ¿3^22)^^•

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Received 29.05.2018, revised 19.09.2018, accepted 26.09.2018

CeedeHua 06 aemopax

Abd-Elmooty Mohamed Abd-Alla, Prof., Sohag University, Egypt, mohmrr@yahoo.com Sayed Mohamed Abo-Dahab, Prof., South Valley University, Egypt, sdahb@yahoo.com Said Mohamed Ahmed, Dr., Al-Arish University, Egypt, said_m_ahmad@yahoo.com Mohamed Mahrous Rashid, Dr., Sohag University, Egypt, mohammed_rashid2013@yahoo.com