ar ticle

Journal of Advanced Physics Vol. 7, pp. 58–69, 2018 (www.aspbs.com/jap)

Copyright © 2018 by American Scientific Publishers All rights reserved. Printed in the United States of America

Plane Waves on a Gravitational Rotating Fibre-Reinforced Thermoelastic Medium with Thermal Shock Problem Kh. Lotfy1, ∗ , S. M. Abo-Dahab2 , and A. D. Hobiny3 1

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt Department of Mathematics, Faculty of Science, SVU, Qena 83523, Egypt 3 Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

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2

A general model of the equations of coupled theory (CD), Lord-Shulman (LS) theory with one relaxation time and Green-Lindsay theory GL with two relaxation times are applied to study the influence reinforcement on the total deformation of rotating thermoelastic half-space and the interaction with each other under the influence of gravity subjected to a thermal shock, it considered in two-dimensional space. The material is homogeneous isotropic elastic half space. The normal mode analysis and the eigenvalue approach techniques are used to solve the resulting non-dimensional coupled field equations for the three theories to obtain the exact expressions for the displacement components, force stresses and temperature. Some particular cases are also discussed in the context of the problem. Numerical results for the temperature, displacements and thermal stresses distribution are presented graphically and discussed. Some comparisons are made with the results obtained in the presence and absence of the rotation and the gravity field effect under three theories. The results presented in this paper should prove useful for researchers in material science, designers of new materials. Study of the phenomenon of relaxation time and gravity is also used to improve the conditions of oil extractions. KEYWORDS: Fibres, Particle-Reinforced Composites, Thermomechanical Properties, Computational Mechanics, Elastic Properties.

1. INTRODUCTION The dynamical problem of propagation of surface waves in a homogeneous and non-homogeneous elastic and thermoplastic media are of considerable importance in earthquake, engineering and seismology on account of the occurrence of non-homogeneities in the earth’s crust, as the earth is made up of different layers. Fiber-reinforced composites are widely used in engineering structures, due to their superiority over the structural materials in applications requiring high strength and stiffness in lightweight components. A continuum model is used to explain the mechanical properties of such materials. A reinforced concrete member should be designed for all conditions of stresses that may occur and in accordance with the principles of mechanics. The characteristic property of a reinforced concrete member is that its components, namely ∗

Author to whom correspondence should be addressed. Email: [email protected] Received: 14 January 2018 Accepted: 15 February 2018

58

J. Adv. Phys. 2018, Vol. 7, No. 1

concrete and steel, act together as a single unit as long as they remain in the elastic condition, i.e., the two components are bound together so that there can be no relative displacement between them. In the case of elastic solid reinforced by a series of parallel fibers it is usual to assume transverse isotropy. In the linear case, the associated constitutive relations, relating infinitesimal stress and strain components, have five material constants. In the last three decades, the analysis of stress and deformation of fiber-reinforced composite materials has been an important research area of solid mechanics. The idea of introducing a continuous self reinforcement at every point of an elastic solid was given by Belfied et al.1 The model was later applied to the rotation of a tube by Verma and Rana.2 Verma3 has also discussed the magneto elastic shear waves in self-reinforced bodies. Singh4 showed that, for wave propagation in fibre-reinforced anisotropic media, this decoupling can not be achieved by the introduction of the displacement potential. Sengupta and Nath5 discussed the problem of the surface waves in fibre-reinforced anisotropic elastic media. Hashin and Rosen6 gave the 2168-1996/2018/7/058/012

doi:10.1166/jap.2018.1397

Lotfy et al.

Plane Waves on a Gravitational Rotating Fibre-Reinforced Thermoelastic Medium with Thermal Shock Problem

Roy Choudhuri and Debnath15 16 and Othman.17–19 Lotfy20 studied the A novel solution of fractional order heat equation for photothermal waves in a semiconductor medium with a spherical cavity. The propagation of plane harmonic waves in a rotating elastic medium without thermal field has been studied. It was shown there that the rotation causes the elastic medium to be dispersive and an isotropic. These problems are based on more realistic elastic model since earth, moon and other plants have angular velocity. Some researchers in past have investigated different problem of rotating media. Chand et al.21 presented an investigation on the distribution of deformation, stresses and magnetic field in a uniformly rotating homogeneous isotropic, thermally and electrically conducting elastic half-space. Many authors (Schoenberg and Censor,22 Clarke and Burdness,23 Destrade24 ) studied the effect of rotation on elastic waves. Ting25 investigated the interfacial waves in a rotating anisotropic elastic half space by J. Adv. Phys., 7, 58–69, 2018

extending the Stroh26 formalism. Othman and Song27 28 presented the effect of rotation in magneto thermoelastic medium. Othmam and Abbas29 discussed the effect of rotation on plane waves at the free surface of a fiber-reinforced thermoelastic half-space. Ailawalia and Budhiraja30 studied the effect of hydrostatic initial stress and rotation in Green-Naghdi (Type III) thermoelastic half-space with two-temperature. In classical theory of elasticity the gravity effect is generally neglected. The effect of gravity in the problem of propagation of waves in solids, in particular on an elastic globe, was first studied by Bromwich in Ref. [31]. Subsequently, investigation of the effect of gravity was considered by Love in Ref. [32] who showed that the velocity of Rayleigh waves is increased to a significant extent by the gravitational field when wavelengths are large. De and Sengupta in Refs. [33–35] studied the effect of gravity on surface waves, on the propagation of waves in an elastic layer and Lamb’s problem on a plane. Sen-gupta and Acharya36 studied the influence of gravity on the propagation of waves in a thermoelastic layer. Dey et al.37 derived the velocities of longitudinal and transverse waves in an initially stressed medium. Das et al.38 investigated surface waves under the influence of gravity in a non-homogeneous elastic solid medium. Abd-Alla and Ahmed in Ref. [39] investigated Rayleigh waves in an orthotropic thermoelastic medium under gravity field and initial stress. Abd-Alla and Ahmed40 discussed wave propagation in a non-homogeneous orthotropic elastic medium under the influence of gravity. Ailawalia and Narah41 depicted the effects of rotation and gravity in generalized thermoelastic medium. The problem has been solved numerically using the normal mode analysis and many works in generalized magneto-thermoelasticity with effect of rotation and other fields can be found in the literatures42–51 for a half space fiber-reinforced. Numerical results for the conductive temperature, thermodynamic temperature, displacement components and the stresses are represented graphically and the results are analyzed. In the present work we shall formulate a fiberreinforced two-dimensional problem under the effect of rotation and gravity field. The normal mode analysis is used to obtain the exact expression for the temperature, displacement components, and stress components. A comparison is carried out between the considered variables as calculated from the generalized thermoelasticity based on the rotation and influence of gravity order generalized Lord and Shulman (L-S), Green and Lindsay by including the thermal relaxation time in constitutive relations coupled theories of thermoelasticity (C-D) for different values of the rotation and gravity field. A comparison also is made between the three theories in the absence and presence of gravity field. Such problems are very important in many dynamical systems. 59

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elastic moduli for fibre-reinforced materials. The problem of reflection of plane waves at the free surface of a fibre-reinforced elastic half-space was discussed by Singh et al.7 Chattopadhyay and Choudhury8 have discussed the problem of propagation, reflection and transmission of magneto elastic shear waves in a self-reinforced medium. The reflection and transmission of plane SH wave through a self-reinforced elastic layer sandwiched between two homogeneous visco-elastic solid half-spaces has been studied by Chaudhary et al.9 Chattopadhyay and Chaudhary10 studied the propagation of magneto-elastic shear waves in an infinite self-reinforced plate. Pradhan et al.11 studies the dispersion of Loves waves in a self-reinforced layer over an elastic non-homogenous half space. The propagation of plane waves in fibre-reinforced media is discussed by Chattopadhyay et al.12 The theory of couple thermoelasticity was extended by Lord and Shulman13 and Green and Lindsay14 by including the thermal relaxation time in constitutive relations. These theories eliminate the paradox of infinite velocity of heat propagation and are termed generalized theories of thermoelasticity. This exist in the following differences between the two theories. 1. The Lord-Shulman ¸ theory (L-S) involves one relaxation time of thermoelastic process (0 . The Green and Lindsay (G-L) involves two relaxation times (0 0 . 2. The (L-S) energy equation involves first and second time derivatives of strain, whereas the corresponding equation in (G-L) theory needs only the first time derivative of strain. 3. In the linearised case according to the approach of (G-L) theory the heat cannot propagate with finite speed unless the stresses depend on the temperature velocity, whereas according to (L-S) theory the heat can propagate with finite speed even though the stresses there are independent of the temperature velocity. 4. The Lord-Shulman ¸ theory (L-S) can not be obtained from Green and Lindsay (G-L) theory.


2. FORMULATION OF THE PROBLEM, BASIC EQUATIONS AND THEIR SOLUTIONS

Lotfy et al.

gravity. A rectangular coordinate system x y z having origin on the surface y = 0 and y-axis pointing vertically into the medium is considered.

The constitutive equations for a fibre-reinforced linearly elastic anisotropic medium with respect to the reinforcement direction a are (Belfied et al.1 ) ij = ekk ij + 2 T eij + ak am ekm ij + ai aj ekk + 2 L − T ai ak ekj + aj ak eki + ak am ekm ai aj T − T0 ij − 1 + 0 (1) t

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Where ij are components of stress; eij are the components of strain; T are elastic constants; L − T are reinforcement parameters, = 3+2 t , t thermal expansion coefficient and a ≡ a1 a2 a3 , a21 +a22 +a23 = 1. We choose the fibre-direction as a ≡ 1 0 0. The strains can be expressed in terms of the displacement ui as 1 eij = ui j + uj i 2

(2)

For plane strain deformation in the xy-plane (displacement u = u v 0, w = 0. Equation (1) then yields: xx = A11 ux + A12 vy − 1 + 0 T − T0 t yy = A22 vy + A12 ux − 1 + 0 T − T0 t zz = A12 ux + vy − 1 + 0 T − T0 t xy = L uy + vx

zx = zy = 0

(3) (4) (5) (6)

Where A11 = + 2 + T + 4 L − T +

A12 = +

A22 = + 2 T (7) Since the medium is rotating uniformly with an angular velocity = n, where n is a unit vector representing the direction of the axis of rotation. All quantities considered are functions of the time variable t and the coordinates x and y. The displacement equation of motion in the rotating frame of reference has two additional terms: centripetal acceleration, × × u due to time-varying motion only and the Corioli’s acceleration 2 × u˙ where u is the dynamic displacement vector and angular velocity is = 0 0 . We consider a normal source acting at the plane surface of generalized thermoelastic halfspace under hydrostatic initial stress under the influence of 60

Geometry of the problem The equations of motion in a rotating frame of reference in the context of generalized thermo elasticity without gravity are: ˙ i = ij j u¨ i + × × ui + 2 × u

i j = 1 2 3 (8)

The heat conduction equation 2 2 kTii = n1 + 0 2 CE T + T0 n1 + n0 0 2 ui i t t t t (9) Where is the density, k is the thermal conductivity, CE is specific heat at constant strain and T is temperature above reference temperature T0 . Using the summation convection. From (3)–(6), we note that the third equation of motion in (8) identically satisfied and first two equations under the influence of gravity become: 2 2 u 2 v 2 u u 2 + B − u − 2 v ˙ = A + B 11 2 1 t 2 x 2 xy y 2 v T + g − 1 + 0 t x x (10) 2 v 2 v 2 u 2 v 2 + B − v + 2 u ˙ = A + B 22 2 1 t 2 y 2 xy x 2 u T − g − 1 + 0 t y x (11) Where B1 = L , B2 = + + L. Equations (3)–(6) and (9)–(11) are the field equations of the generalized thermo-elasticity elastic solid, applicable to the (L-S) theory, the (G-L) theory, J. Adv. Phys., 7, 58–69, 2018

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as well as the classical coupled theory (CD), as follows: 1. The equations of the coupled thermo-elasticity (CD) theory, when n0 = 0

n1 = 1

0 = 0 = 0

Where

2. Lord-Shulman (L-S) theory, when n1 = n0 = 1

0 = 0

n0 = 0

0 > 0

0 ≥ 0 > 0

3. PARTICULAR CASES

For convenience, the following non-dimensional variables are used:

t = c12 t

y = c1 y 0 = c12 0

T − T0 = + 2 T

u = c1 u =

ij

c12

ij = T

v = c1 v 0 = c12 0

i j = 1 2

T xy = L uy + vx

u v ij x y t = u∗ x v∗ x ∗ x ij∗ x × expt + iay

c12 =

h11 D2 − A1 u∗ + 2 + h3Dv∗ = QD ∗

(21)

h1 D2 − A2 v∗ + −2 + h4Du∗ = iaQ ∗

(22)

∗

In terms of non-dimensional quantities defined in Eq. (12), the above governing equations reduce to (dropping the dashed for convenience) 2 u 2 u 2 v 2 u + h1 2 − 2 u − 2v˙ = h11 2 + h2 2 t x xy y v +g (13) − 1 + 0 t x x 2 v 2 v 2 u 2 v 2 + h − v + 2 u ˙ = h + h 22 2 1 t 2 y 2 xy x 2 u −g (14) − 1 + 0 t y x J. Adv. Phys., 7, 58–69, 2018

∗

A4 Du + iaA4 v = D − A3

+ 2 T

(20)

√ Where is the (complex) time constant. i = −1, a is the wave number in the y-direction and u∗ x, v∗ x, ∗ x and ij∗ x are the amplitude of the field quantities. Using Eqs. (20), (13)–(19) take the form

Where CE k

(19)

The solution of the considered physical variable can be decomposed in terms of normal modes as the following form

(12)

=

zx = zy = 0

2

∗

(23)

∗ = A11 Du∗ + iaA12 v∗ − A22 Q ∗ T xx

(24)

∗ T yy

(25)

∗

∗

= A12 Du + iaA22 v − A22 Q

∗

T zz∗ = A12 Du∗ + iav∗ − A22 Q ∗ ∗ T xy = L iau∗ + Dv∗

∗ zx = zy∗ = 0

(26) (27)

Where A1 = 2 − 2 + h1 a2 Q = 1 + 0

A2 = 2 − 2 + h22 a2

A3 = a2 + n1 + 0

d dx h4 = iah2 − g

A4 = n1 + n0 0

D=

h3 = iah2 + g

Eliminating ∗ x and v∗ x between Eqs. (21)–(23), we obtain the partial differential equation satisfied by u∗ x (28) D6 − AD4 + BD2 − C u∗ x = 0 61

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(I) Neglecting angular velocity (i.e., = 0 in Eqs. (10) and (11), we obtain transformed components of displacement, stress forces and temperature distribution in a non-rotating generalized thermoelasticity medium under hydrostatic initial stress and gravity. (II) Neglecting gravitational effect (i.e., g = 0 in Eqs. (10) and (11), the expressions for displacements, force stresses and temperature distribution reduces in a rotating generalized thermoelasticity medium under hydrostatic initial stress.

x = c1 x

2 T0 A11 A22 B1 B2 = +2 T CE +2 T (16) T xx = A11 ux + A12 vy − A22 1 + 0 t T yy = A22 vy + A12 ux − A22 1 + 0 (17) t T zz = A12 ux + vy − A22 1 + 0 (18) t

h11 h22 h1 h2 =

3. Green-Lindsay (G-L) theory, when n1 = 1

2 2 2 2 + + n + n + = n 1 0 1 0 0 x 2 y 2 t t 2 t t 2 u v × + (15) x y


Where A=

Thus, we have

1 h A + h1 A1 + h1 A4 Q + h1 h11 A3 + h3 h4 a2 h1 h11 11 2 (29)

v∗ x =

∗ xx =

(31) ∗ yy =

(32)

zz∗ =

− k32 u∗ x

=0

k6 − Ak4 + Bk2 − C = 0

(34)

The solution of Eq. (33) which is bounded as x → , is given by u∗ x =

Mn a exp−kn x

(35)

n=1

3

3

H3n Mn a exp−kn x

(44)

H4n Mn a exp−kn x

(45)

H5n Mn a exp−kn x

(46)

H6n Mn a exp−kn x

(47)

3

3

3 n=1

where H3n = −A11 kn + iaA12 H1n − A22 QH2n / T

n = 1 2 3 (48)

H4n = −A12 kn + iaA22 H1n − A22 QH2n / T

n = 1 2 3 (49)

H5n = −A12 kn + iaH1n − A22 QH2n / T

n = 1 2 3 (50)

H6n = L ia − kn H1n / T

n = 1 2 3

(51)

Mn a exp−kn x

(36)

The normal mode analysis is, in fact, to look for the solution in Fourier transformed domain. Assuming that all the field quantities are sufficiently smooth on the real line such that normal mode analysis of these functions exists.

Mn a exp−kn x

(37)

4. APPLICATION

Similarly v∗ x =

∗ = xy

(33)

where, kn2 n = 1 2 3 are the roots of the following characteristic equation

3

(43)

n=1

The above equation can be factorized as D

H2n Mn a exp−kn x

n=1

D6 − AD4 + B D2 − Cv∗ x ∗ x = 0

− k22 D2

3

n=1

In a similar manner, we get

− k12 D2

(42)

Substitution of Eqs. (35), (42) and (43) into Eqs. (24)–(27), we get

(30)

2

H1n Mn a exp−kn x

n=1

+A2 A4 −2h3 a2 A4 +42 2 +h3 h4 a2 A3

1 C= A A A + A1 A4 a2 Q + 42 2 A3 h1 h11 1 2 3

3 n=1

∗ x =

1 A A +h11 A2 A3 +h1 A1 A3 +Qh11 a2 A4 B= h1 h11 1 2

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n=1

∗ x =

3 n=1

Where Mn , Mn and Mn are some parameters depending on a and . Substituting from Eqs. (35)–(37) into Eqs. (21)–(23), we have Mn a = H1n Mn a

n = 1 2 3

(38)

Mn a = H2n Mn a

n = 1 2 3

(39)

x y t = f y t

Where −iah11 kn2 +iaA1 +2kn +iah4 kn2 H1n = 2ia+h3kn a2 +h1 kn3 −A2 kn

n = 123 (40)

−A4 kn + iaA4 H1n H2n = kn2 − A3 62

4.1. Thermal Shock Problem A time-dependent heat punches across the surface of semiinfinite thermo-elastic half space. In the physical problem, we should suppress the positive exponential that are unbounded at infinity. The constants M1 , M2 and M3 have to be chosen such that the boundary conditions on the surface x = 0 take the form. (1) Thermal boundary condition is that the surface of the half-space subjects to a thermal shock problem

n = 1 2 3

(41)

at x = 0

(52)

where f is a given function of y and t (the magnitude of the constant temperature applied on the boundary). (2) Mechanical boundary condition is that the surface of the half-space is traction free yy x y t = 0

at x = 0

(53)

xy x y t = 0

at x = 0

(54)

J. Adv. Phys., 7, 58–69, 2018

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Substituting the expressions of the variables considered into the above boundary conditions, we get H2n Mn a = f y t

(55)

n=1 3

H4n Mn a = 0

(56)

n=1 3

H6n Mn a = 0

(57)

n=1

61

62

63

–0.04 Ω = 0.2 and g = 9.8

–0.06 –0.08

T = 189 × 109 N/m2

CD LS GL

Ω = 0.0 and g = 0.0

–0.1 –0.12 –0.14

0

2

4

6

8

10

12

14

Distance x Fig. 2. Horizontal displacement distribution u at different values of rotation and gravity.

L = 245 × 109 N/m2

= −128 × 109 N/m2

= 032 × 109 N/m2

t = 178 × 10−5 N/m2 a = 1

= 0 + i

With a view to illustrating the analytical procedure presented earlier, we now consider a numerical example for which computational results are given. To compare these in the context of various theories of thermoelasticity and to study the effect of rotation and gravity field on wave propagation in a conducting fiber-reinforcement. We now present some numerical results for the physical constants as Singh,46 we now present some numerical results for the physical constants.

= 7800 kg/m2 k = 386

0 = 2

= 1 10

CE = 3831

∗

T0 = 293 k

= 386 × 10

f = 1

0 = 003

CE = 3831 J/kgk kg/ms2

The computations were carried out for a value of time t = 01. The numerical technique, outlined above, was used for the distribution of the real part of the thermal temperature , the displacement u and v, the stresses xx , yy and xy distribution for the problem. The field quantities, temperature, displacement components u v and stress components xx , yy and xy depend not only on space x and time t but also on the thermal relaxation time 0 and 0 . Here all the variables are taken in non dimensional

0.5

0.15 Ω = 0.2 and g = 9.8

CD LS GL

0.3

0.1 Vertical displacement v

0.4

Temperature θ

0 –0.02

0 = 002

5. NUMERICAL RESULTS

= 759 × 109 N/m2

0.02

0.2 0.1 0 –0.1

CD LS GL

Ω = 0.0 and g = 0.0

0.05 0 –0.05 –0.1

–0.2

–0.4

Ω = 0.2 and g = 9.8

–0.15

–0.3

Ω = 0.0 and g = 0.0 0

2

4

6

8

10

12

Distance x Fig. 1. The temperature distribution at different values of rotation and gravity. J. Adv. Phys., 7, 58–69, 2018

–0.2

0

2

4

6

8

10

12

14

Distance x

Fig. 3. Vertical displacement distribution v at different values of rotation and gravity.

63

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Invoking the boundary conditions (55)–(57) at the surface x = 0 of the plate, we obtain a system of three equations. After applying the inverse of matrix method, we have the values of the three constants Mj , j = 1 2 3. Hence, we obtain the expressions of displacements, temperature distribution and another physical quantities of the plate the muscles. ⎞ ⎛ ⎛ ⎞−1 ⎛ ⎞ 0 M1 H41 H42 H43 ⎟ ⎜ ⎜ ∗⎟ ⎜ ⎟ ⎜f ⎟ ⎜ M2 ⎟ ⎜ (58) ⎟ ⎟ = ⎝ H21 H22 H23 ⎟ ⎜ ⎠ ⎜ ⎠ ⎝ ⎠ ⎝ M3 0 H H H

0.04

Horizontal displacement u

3

0.06


0.25

0.2

0.2

CD LS GL

Ω = 0.0 and g = 0.0

0.15

0

0.05

–0.1

σxy

σxx

Ω = 0.0 and g = 0.0

0.1

0.1

0

–0.2

–0.05 –0.1

–0.3

–0.15

Ω = 0.2 and g = 9.8

Ω = 0.2 and g = 9.8

0

2

4

6

8

10

–0.5

12

0

1

2

3

Distance x

4

5

6

7

8

Distance x

Fig. 4. The distribution of stress component xx at different values of rotation and gravity.

Fig. 6. The distribution of stress component xy at different values of rotation and gravity.

forms. When there two thermal relaxation time for two different values of = 00 and = 02) and g g = 00 and g = 98 m/sec2 ) i.e., in the absence and the presence of the rotation and gravity. The results are shown in Figures 1–6. The graph shows the three curves predicted by different theories of thermoelasticity. In these figures, the solid lines represent the solution in the Coupled theory, the dotted lines represent the solution in the generalized Lord and Shulman theory and dashed lines represent the solution derived using the Green and Lindsay theory. We notice that the results for the temperature, the displacement and stresses distribution when the relaxation time is including in the heat equation are distinctly different from those when the relaxation time is not mentioned in heat equation, because the thermal waves in the Fourier’s theory of heat equation travel with an infinite speed of propagation as opposed to finite speed in the non-Fourier case. This demonstrates clearly the difference

between the coupled and the theory of generalized thermoelasticity. For the value of y, namely y = 1, were substituted in performing the computation. It should be noted (Fig. 1). It is clear from the graph that has decreases to arrive the minimum value at the beginning in two cases = 00 and = 02) and g g = 00 and g = 98. The value of temperature quantity converges to zero with increasing the distance x. The effect of rotation and gravitation on temperature decreases the value of amplitude of its nature for the medium with rotation under the influence of gravity. Figure 2 the horizontal displacement u, begins from the negative values and then increases (when = 00 and g = 00 to arrive the maximum amplitudes, also move in the wave propagation. When the rotation and gravitation are presented always starts from the positive value and terminates at the zero value, in the first smooth decreases again to reach its minimum, beyond it u falls 0.8

0.15

0.7

Ω = 0.0 and g = 0.0

0.1

CD LS GL

0.6

0.05

Temperature θ

0.5

0

σ yy

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CD LS GL

–0.4

–0.2 –0.25

Lotfy et al.

–0.05

CD LS GL

Ω = 0.2 and g = 9.8

–0.15

0.2

0

2

4

6

8

10

0 Ω = 0.2 –0.1

12

Distance x Fig. 5. The distribution of stress component yy at different values of rotation and gravity.

64

0.3

0.1

–0.1

–0.2

Ω = 0.0

0.4

–0.2

0

2

4

6

8

10

12

Distance x Fig. 7. The temperature distribution with g = 98 at different values of rotation. J. Adv. Phys., 7, 58–69, 2018

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0.14

0.1

0.1 0.08

Ω = 0.0

0.05

CD LS GL

Ω = 0.0

0

0.06

–0.05

σxx


0.12

0.04

–0.1

Ω = 0.2

0.02 0

–0.15

CD LS GL

Ω = 0.2

–0.02 –0.2 –0.04 –0.06

0

2

4

6

8

10

12

–0.25

14

0

2

4

Distance x

6

8

10

12

Distance x Fig. 10. The distribution of stress component xx with g = 98 at different values of rotation.

again to try to retain zero at infinity but the effect of rotation and gravity illustrated in the figure the value of u for = 02 and g = 98 are smaller when compared to those for = 00 and g = 00. Behavior of displacement u in (CD), (LS) and (GL) theories for two different values of rotation is similar in two cases. Figure 3, the vertical displacement v, we see that the displacement component v always starts from the negative value (when = 02 and g = 98 and terminates at the zero value but v always starts from the positives value (when = 00 and g = 00, Also v in this case reach minimum value, beyond reaching zero at the infinity (state of particles equilibrium). The displacements u and v show different behaviours, because of the elasticity of the solid tends to resist vertical displacements in the problem under investigation. The stress component, xx reach coincidence with negative value (Fig. 4) and satisfy the boundary condition at x = 0, start sharp decreases and reach

the minimum value (when = 02 and g = 98 and converges to zero with increasing the distance x, behavior of three theories are similar (i.e., different values of relaxation time). In the case, when the rotation and influence of gravity are absence, increases in the start and start decreases in the context of the three theories until reaching the zero value with increases the distance. These trends obey elastic and thermoelastic properties of the solid under investigation. The rotation and gravity caused composed the wave propagation and lessees the amplitudes of the stress value. Also Figure 5 take the same behavior of Figure 4. Figure 6, shows that the stress component xy satisfies the boundary condition at x = 0 and had a different behaviour of xx when the rotation and gravity are absence. It decreases in the start and start increases (maximum) in the context of the three theories and propagation until reaching the zero value at infinity, but when the rotation and gravity are presented (when = 02 and g = 98 it decreases in the

0.04

0.02

0

–0.02

–0.02

–0.04

–0.04

–0.06

–0.06 –0.08

–0.08 –0.1

–0.1

CD LS GL

Ω = 0.2

–0.12 –0.14

Ω = 0.0

0

σ yy

Vertical displacement v

0.02 Ω = 0.0

0

2

4

6

8

10

12

CD LS GL

–0.14 14

Distance x Fig. 9. Vertical displacement distribution v with g = 98 at different values of rotation. J. Adv. Phys., 7, 58–69, 2018

Ω = 0.2

–0.12

–0.16

0

2

4

6

8

10

12

Distance x Fig. 11. The distribution of stress component yy with g = 98 at different values of rotation.

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Fig. 8. Horizontal displacement distribution u with g = 98 at different values of rotation.


0.1

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0.06 with gravity

0.05

0.04


Ω = 0.0

0

σxy

–0.05 –0.1 –0.15 CD LS GL

–0.2 Ω = 0.2

–0.25 –0.3

0

1

2

3

4

5

6

7

0.02 0 –0.02 –0.04 –0.06 –0.08 –0.1

8

0

2

4

6

8

10

12

14

Distance x

Distance x Fig. 12. The distribution of stress component xy with g = 98 at different values of rotation.

Fig. 14. Horizontal displacement distribution u with = 02 at different values of gravity.

start and start increases in the context of the three theories until reaching zero value. These trends obey elastic and thermoelastic properties of the solid under investigation. Figures 7–12 show the comparison between the temperature , displacement components u, v, the force stresses components xx , yy and xz , the case of different two values of rotation and constant of gravity (g = 98 under three theories. It should be noted (Fig. 7) that in this problem. It is clear from the graph that increases to maximum value at the beginning, where it experiences smooth decreases (with minimum negative gradient). Graph lines for both values of rotation show different slopes. In other words, the temperature lines for = 00 has the highest gradient when compared with that of = 02 in all range. In addition, all lines begin to coincide when the horizontal distance x increases to reach the reference temperature of the solid. These results obey physical reality for the behaviour of fiber as a polycrystalline solid.

Figure 8, the horizontal displacement u, despite the peaks (for different values of rotation) occurs at equal value of gravity, the magnitude of the maximum displacement peak strongly depends on the rotation. It is also clear that the rate of change of u decreases with increasing the rotation. On the other hand, Figure 9 shows atonable increase of the vertical displacement v, near the beginning reach minimum value and then reaching zero value at the infinity (state of particles equilibrium). Figure 10, the horizontal stresses xx Graph lines for both values of rotation show different slopes. In other words, the xx component line for = 00 g = 98 has the highest gradient when compared with that of = 02 g = 98. In addition, all lines begin to coincide when the horizontal distance x is increases to reach zero after their relaxations at infinity. Variation of has a serious effect on both magnitudes of mechanical stresses. These trends obey elastic and thermoelastic properties of the solid under investiga-

0.6

0.02 without gravity CD LS GL

0.4 0.3

0

Vertical displacement v

0.5

Temperature θ

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CD LS GL

without gravity

without gravity

0.2 0.1 0 –0.1 –0.2

2

4

6

8

10

12

Distance x Fig. 13. The temperature distribution with = 02 at different values of gravity.

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–0.04 –0.06 –0.08 –0.1

CD LS GL

with gravity

–0.12

with gravity 0

–0.02

–0.14

0

2

4

6

8

10

12

14

Distance x Fig. 15. Vertical displacement distribution v with = 02 at different values of gravity. J. Adv. Phys., 7, 58–69, 2018

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0.2

0.05

0.15

CD LS GL

without gravity

0.1

0 –0.05

without gravity

–0.1

0

σxy

σxx

0.05

–0.05

–0.15

–0.1 –0.2

CD LS GL

–0.25

–0.2 –0.25

with gravity

with gravity

–0.15

0

2

4

6

8

10

–0.3

12

0

1

2

3

Distance x

4

5

6

7

8

Distance x

Fig. 16. The distribution of stress component xx with = 02 at different values of gravity.

the same behaviour. In other words, the temperature lines for g = 00 has the highest gradient when compared with that of g = 98 and converges to zero with an increase in distance x. In addition, all lines begin to coincide when the horizontal distance x is beyond increases to reach the reference temperature of the solid. These results appear physics harmonically for the behaviour of fiber as a polycrystalline solid. Behavior of displacement u in both three theories (CD), (L-S) and (G-L) for the two different values of gravity is opposite, as shown in Figure 14. Values of displacement for g = 98 are large compared to those for g = 00 in the ranges 0 ≤ x ≤ 1 but small in the range 1 ≤ x ≤ 14, while values are the same for the three theories at x ≥14, occurs at equal value of rotation = 02. Behaviour of displacement v for both the three theories in the two cases of gravity is similar, shows that in all three theories the values of displacement v for g = 98 are small compared to those for g = 00 as shown in Figure 15. Figure 16, the horizontal stresses xx Graph lines for both values of gravitation show different slopes at the same value of rotation

0.15 without gravity

0.1

LS 0.05 1

σyy

0

0.5

θ

–0.05 –0.1

CD LS GL

with gravity –0.15 –0.2

0

2

4

6

8

10

4 3

–0.5 –1 0 12

Distance x Fig. 17. The distribution of stress component yy with = 02 at different values of gravity. J. Adv. Phys., 7, 58–69, 2018

0

2 2

4

y

1 6

8

x

10

12 0

Fig. 19. 3D temperature distribution with constant values of rotation and gravity under LS theory.

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tion. Figure 11, shows that the stress component yy , take the same behaviour of Figure 10. Figure 12, shows that the stress component xy satisfy the boundary condition, it sharp decreases in the start and start increases (minimum) in the context of the = 02 but when = 00 take the different behaviour. The lines for = 00 has the highest gradient when compared with that of = 02. These trends obey elastic and thermoelastic properties of the solid. Figures 13–18 show the comparison between the temperature , displacement components u, v, the force stresses components xx , yy and xz , the case of different two values of gravity and constant of rotation ( = 02) under three theories. It should be noted (Fig. 13) that in this problem. It is clear from the graph that increases to maximum value at the beginning, where it experiences smooth decreases (with minimum negative gradient). Graph lines for both values of gravity show different slopes but take

Fig. 18. The distribution of stress component xy with = 02 at different values of gravity.


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LS 0.15

LS 0.2

0.1 0.05

0.1

σyy

u

0 0

–0.05 –0.1

–0.1

–0.15

4 3

–0.2 0

2 5

1

10 15

–0.2 0

4

y

5 10

0

LS

0.15 0.1 0.05

v

0 –0.05 –0.1 –0.15

4 3 2 4

6

8

10

1 12

14

y

0

x

Fig. 21. 3D vertical displacement distribution u with constant values of rotation and gravity under LS theory.

Fig. 23. 3D stress distribution yy with constant values of rotation and gravity under LS theory.

relaxations at infinity. Variation of g has a serious effect on both magnitudes of mechanical stresses. Behavior of the stress component yy , in all three theories and two cases (g = 00 and g = 98 is similar to Figure 16, as depicted in Figure 17. Figure 18, shows that the stress component xy satisfy the boundary condition and shows that in all three theories, it sharp decreases in the start and start increases (minimum) in the context of the g = 00 and g = 98 take the same behaviors when = 02. Finally, Figures 19–24, plot in 3D the variations of the temperature , the displacement components u, v, the stresses xx , yy and xy distribution with the axes (x y). It is concluded that all the values decreases with an increasing of smaller values of y-axis and with the larger values increase. It is also clear that the temperature, displacement and shear stress component have been start from zero, decrease with the smaller values of x-axis and return to increase to tend zero as x tends to infinity but the normal stress component start also from zero, increases with

LS

LS

0.3

0.3

0.2

0.2

0.1

0.1 0

σxy

σxx

ARTICLE

( = 02. In other words, the xx component line for g = 98 has the lowers gradient when compared with that of g = 00. In addition, all lines begin to coincide when the horizontal distance x is increases to reach zero after their

2

y

x

Fig. 20. 3D horizontal displacement distribution u with constant values of rotation and gravity under LS theory.

–0.2 0

1 15 0

x

3

2

–0.1

–0.3

–0.3

4 2

4

2 6

8

10

12

0

y

x Fig. 22. 3D normal stress distribution xx with constant values of rotation and gravity under LS theory.

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–0.1 –0.2

–0.2

–0.4 0

0

–0.4 0

2

4

6

8

0

1

3

2

4

y

x Fig. 24. 3D stress distribution xy with constant values of rotation and gravity under LS theory. J. Adv. Phys., 7, 58–69, 2018

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the smaller values of x-axis and return twice to decreasing and increasing periodically to tends to zero as x tends to infinity.

6. CONCLUSIONS

References and Notes

1. A. J. Belfield, T. G. Rogers, and A. J. M. Spencer, J. Mech. Phys. Solids 31, 25 (1983). 2. P. D. S. Verma and O. H. Rana, Mech. Materials 2, 353 (1983). 3. P. D. S. Verma, Int. J. Eng. Sci. 24, 1067 (1986). 4. S. J. Singh, Sãdhanã 27, 1 (2002). 5. P. R. Sengupta and S. Nath, Sãdhanã 26, 363 (2001). 6. Z. Hashin and W. B. Rosen, J. Appl. Mech. 31, 223 (1964). 7. B. Singh and S. J. Singh, Sãdhanã 29, 249 (2004). 8. A. Chattopadhyay and S. Choudhury, Int. J. Eng. Sci. 28, 485 (1990). 9. S. Chaudhary, V. P. Kaushik, and S. K. Tomar, Acta Geophysica Polonica 52, 219 (2004). 10. A. Chattopadhyay and S. Choudhury, Int. J. Num. Anal. Methods in Geomech. 19, 289 (1995). 11. A. Pradhan, S. K. Samal, and N. C. Mahanti, Tamkang J. Sci. Eng. 6, 173 (2003).

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1. The curves in the context of the (CD), (LS) and (GL) theories decrease exponentially with increasing x, this indicate that the thermoelastic waves are unattenuated and nondispersive, where purely thermoelastic waves undergo both attenuation and dispersion. 2. The curves of the physical quantities with (CD) theory in most of figures are lower in comparison with those under (LS) theory and (GL) theory, due to the relaxation times. The thermal relaxation time have significant effects on all the field quantities. 3. Analytical solutions based upon normal mode analysis for themoelastic problem in solids have been developed and utilized. 4. It can be concluded that a change of volume is attended by a change of the temperature while the effect of the deformation upon the temperature distribution is the subject of the theory of thermoelasticity. 5. The value of all the physical quantities converges to zero with an increase in distance x and all functions are continuous. 6. The fibre-reinforced has an important role on the distribution of the field quantities. 7. The method which used in the present article is applicable to a wide range of problems in thermodynamics and thermoelasticity. 8. The presence of rotation and influence of gravity plays a significant role in all the physical quantities. The amplitude of all the physical quantities decreases while the rotation and gravity increase. Therefore, the presence of rotation and influence of gravity in the current model is of significance. 9. Deformation of a body depends on the nature of forced applied as well as the type of boundary conditions.

12. A. Chattopadhyay, R. L. K. Venkateswarlu, and S. Saha, Sãdhanã 27, 613 (2002). 13. H. W. Lord and Y. Shulman, ¸ J. Mech. Phys. Solid 15, 299 (1967). 14. A. E. Green and K. A. Lindsay, J. Elasticity 2, 1 (1972). 15. S. K. Roy Choudhuri and L. Debnath, Int. J. Eng. Sci. 21, 155 (1983a). 16. S. K. Roy Choudhuri and L. Debnath, J. Appl. Mech. 50, 283 (1983b). 17. M. I. A. Othman, Int. J. Solids Struct. 41, 2939 (2005b). 18. M. I. A. Othman, Acta Mechanica. 174, 129 (2005). 19. M. I. A. Othman and B. Singh, Int. J. Solids and Structures 44, 2748 (2007). 20. Kh. Lotfy, Chaos, Solitons and Fractals 99, 233 (2017). 21. D. Chand, J. N. Sharma, and S. P. Sud, International Journal of Engineering Science 28, 547 (1990). 22. M. Schoenberg and D. Censor, Quarterly of Applied Mathematics 31, 115 (1973). 23. N. S. Clarke and J. S. Burdness, Journal of Applied Mechanics 61, 724 (1994). 24. M. Destrade, Proceeding of the Royal Society 460, 653 (2004). 25. T. C. T. Ting, Wave Motion 40, 329 (2004). 26. A. N. Stroh, Journal of Mathematics and Physics 41, 77 (1962). 27. M. I. A. Othman and Y. Song, International Journal of Engineering Sciences 46, 459 (2008). 28. M. I. A. Othman and Y. Song, Multidiscipline Modeling in Mathematics and Structure 5, 43 (2009). 29. M. I. A. Othman and I. A. Abbas, Meccanica 46, 413 (2011). 30. P. Ailawalia and S. Budhiraja, International Journal of Applied Mathematics and Mechanics 7, 93 (2011). 31. T. J. J. A. Bromwich, Proc. London. Math. Soc. 30, 98 (1898). 32. A. E. H. Love, Some Problems of Geodynamics, Dover, New York (1911). 33. S. N. De and P. R. Sengupta, Gerlands. Beitr. Geophys. (Leipzig) 82, 421 (1973). 34. S. N. De and P. R. Sengupta, J. Acoust. Soc. Am. 55, 919 (1974). 35. S. N. De and P. R. Sengupta, Gerlands. Beitr. Geophys. (Leipzig) 85, 311 (1976). 36. P. R. Sengupta and D. Acharya, Rev. Roum. Sci. Technol. Mech. Appl. 24, 395 (1979). 37. S. C. Das, D. P. Acharya, and P. R. Sengupta, Rev. Roum. Des. Sci. Tech. 37, 539 (1992). 38. S. Dey, N. Roy, and A. Dutta, Indian. J. Pure. Appl. Math. 15, 795 (1984). 39. A. M. Abd-Alla and S. M. Ahmed, J. Earth Moon Planets 75, 185 (1996). 40. A. M. Abd-Alla and S. M. Ahmed, Appl. Math. Comput. 135, 187 (2003). 41. P. Ailawalia and N. S. Narah, Appl. Math. Mech. 30, 1505 (2009). 42. Kh. Lotfy, Chin. Phys. B 21, 014209 (2012). 43. Kh. Lotfy, Chin. Phys. B 21, 064214 (2012). 44. Kh. Lotfy and M. E. Gabr, Optics and Laser Technology 97, 198 (2017). 45. M. Othman and Kh. Lotfy, Mechanics of Materials 60, 120 (2013). 46. A. M. Abd-Alla, S. M. Abo-Dahab, and Aftab Khan, Computers, Materials and Continua 53, 49 (2017). 47. S. M. Abo-Dahab, Mechanics of Advanced Materials and Structures 25, 319 (2018). 48. A. M. Abd-Alla, T. A. Nofal, S. M. Abo-Dahab, and A. Al-Mullise, International Journal of Physical Sciences 8, 574 (2013). 49. Ibrahim A. Abbas and S. M. Abo-Dahab, J. Comput. Theor. Nanosci. 11, 607 (2014). 50. S. Abo-Dahab and Kh. Lotfy, Waves in Random and Complex Media 27, 67 (2017). 51. Kh. Lotfy and S. M. Abo-Dahab, J. Comput. Theor. Nanosci. 12, 1709 (2015).