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Fractional Order Thermo-viscoelastic Half-space Problem with Temperature Dependent Modulus of Elasticity a

a

a

Sandeep Singh Sheoran , Kapil Kumar Kalkal & Sunita Deswal a

Department of Mathematics, G. J. University of Science and Technology, Hisar-125001, Haryana, India. Accepted author version posted online: 14 Nov 2014.

Click for updates To cite this article: Sandeep Singh Sheoran, Kapil Kumar Kalkal & Sunita Deswal (2014): Fractional Order Thermo-viscoelastic Half-space Problem with Temperature Dependent Modulus of Elasticity, Mechanics of Advanced Materials and Structures, DOI: 10.1080/15376494.2014.981621 To link to this article: http://dx.doi.org/10.1080/15376494.2014.981621

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ACCEPTED MANUSCRIPT

Downloaded by [Guru Jambheshwar University ] at 05:15 14 April 2015

Fractional Order Thermo-viscoelastic Half-space Problem with Temperature Dependent Modulus of Elasticity Sandeep Singh Sheoran, Kapil Kumar Kalkal, Sunita Deswal∗ Department of Mathematics, G. J. University of Science and Technology, Hisar-125001, Haryana, India. Abstract In this paper the propagation of waves in an isotropic homogeneous thermo-viscoelastic half-space due to thermal loading is studied under the purview of fractional order generalized thermoelasticity theory. State-space approach together with Laplace transform technique is used to obtain the general solution. Expressions for displacement, temperature and stress in the physical domain are computed numerically using a method based on Fourier expansion technique. Effects of fractional parameter, viscosity and temperature dependent modulus of elasticity on field variables are shown in figures. Some particular cases of special interest have been deduced from the present investigation.

1

INTRODUCTION

The theory of thermoelasticity concerns with the effect of thermal and mechanical disturbances in an elastic solid. The thermoelastic wave propagation is of much importance in different fields such as earthquake engineering, nuclear reactors, solid dynamics, aeronautics, astronautics etc. Generalized thermoelasticity theories have been developed with the objective of removing the paradox of infinite speed of heat propagation inherent in conventional coupled theory of thermoelasticity. ∗ corresponding

author, email: spannu [email protected]

ACCEPTED MANUSCRIPT 1

ACCEPTED MANUSCRIPT Mainly, two theories of generalized thermoelasticity are being extensively used, the first is due to Lord and Shulman [1] and second is due to Green and Lindsay [2]. These theories predict finite speed of propagation for heat waves. Lord and Shulman [1] introduced a flux rate term into Fourier’s law of heat conduction and formulated a generalized theory advocating finite speed of thermal waves in solids and suggested one relaxation time, while Green and Lindsay [2] modified both the energy equation and the equation of motion and suggested two relaxation times. Dhaliwal


and Sherief [3] extended the Lord-Shulman theory for anisotropic media. Hetnarski and Ignaczak [4] presented a survey article of various representative theories in the range of generalized thermoelasticity theory. Sherief [5] solved a problem of generalized thermoelasticity theory by using state space approach. Sherief and Hamza [6] studied a two dimensional problem under generalized thermoelasticity theory. El-Maghraby and Youssef [7] employed state space approach to solve a thermo-mechanical problem. In recent years, due to rapid development of polymer science and plastic industry, the thermoviscoelasticity theories play an important role in various engineering and technological applications. The viscoelastic nature of a medium has special significance in wave propagation. Ilioushin and Pobedria [8] formulated the theory of thermo-viscoelasticity and obtained solutions of some boundary value problems of thermo-viscoelasticity. Tanner [9] and Huilgol and Phan-Thien [10] had contributed in finding solutions of boundary value problems for linear viscoelastic materials including temperature variations in both quasi-static and dynamic problems. Othman et al. [11] studied wave propagation in thermo-viscoelastic plane with two relaxation times. Ezzat et al. [12] investigated one dimensional model of generalized thermo-viscoelasticity with two relaxation times by using state space approach. Ezzat and El-Karamany [13] proved the uniqueness and reciprocity theorems for generalized thermo-viscoelasticity with thermal relaxation. Othman [14] studied a two-dimensional thermal shock problem on generalized electro-magneto-thermo-viscoelasticity based on L-S theory. Sharma and Chand [15] studied the transient thermo-viscoelastic waves in half space due to thermal loads. Ezzat [16] and Ezzat and El-Karamany [17] solved


ACCEPTED MANUSCRIPT different problems describing the phenomenon that characterizes different theories of generalized thermo-viscoelasticity for half space of an electrically conducting viscoelastic material. Ezzat et al. [18] presented a problem on generalized thermoelasticity theory for isotropic media with temperature dependent modulus of elasticity under L-S, G-L and coupled theories. Othman [19] employed state space approach to study two dimensional problem of generalized thermoelasticity with modulus of elasticity depending on reference temperature. Othman and Lotfy [20] examined


the effect of temperature dependent properties for different theories in two dimensional problem of generalized thermoelasticity. The theory of fractional derivative and integral was established in the second half of nineteenth century. Recently, by using fractional calculus several different models have been developed to study the physical processes, mainly, in the field of viscoelasticity, heat conduction, diffusion, mechanics of solid etc. The first application of the fractional derivative was given by Abel, who applied fractional calculus in the solution of an integral equation that arises in the formulation of the tautochrone problem. The generalization of the concept of fractional calculus has been subjected to several approaches and some various alternative definitions of fractional derivatives have been explained in Oldham and Spanier [21] and Miller and Ross [22]. A quasi-static theory of thermoelasticity based on fractional heat conduction equation was proposed by Povstenko [23]. Recently, Sherief et al. [24] introduced a new model of fractional order generalized thermoelasticity in which heat conduction equation takes the following form qi + τ0

∂α qi = −ki j θ, j ∂tα

,

(1)

where qi are the components of the heat flux vector, θ is the temperature, τ0 is the thermal relaxation time parameter, ki j is thermal conductivity tensor and α is a fractional parameter such that 0 < α ≤ 1. Youssef [25] constructed another model of thermoelasticity in the context of a new consideration of heat conduction with a fractional order. Ezzat [26, 27] established a model of fractional heat conduction equation by using new Taylor’s series expansion of time-fractional


ACCEPTED MANUSCRIPT order developed by Jumarie [28]. Youssef [29] solved one dimensional problem of fractional order two-temperature generalized thermoelastic medium by using state space approach subjected to moving heat source. Sherief and El-Latief [30] established one dimensional problem on fractional order theory of thermoelasticity subjected to two different type of functions representing the thermal shock and solved the problem by using Laplace transform technique. The present paper is an attempt to study a problem on thermo-viscoelastic interactions in an


isotropic and homogeneous elastic medium with temperature dependent modulus of elasticity under fractional order theory of thermoelasticity developed by Sherief et al. [24]. We employ state space approach developed by Bahar and Hetnarski [31] on the formulation. Laplace transform technique is used to obtain the general solution. The inverse Laplace transform is carried out using a numerical inversion method developed by Honig and Hirdes [32]. Finally, effects of fractional parameter, viscosity and temperature dependent modulus of elasticity on field variables are displayed graphically.

2

BASIC EQUATIONS AND PROBLEM FORMULATION

The constitutive equations and field equations for an isotropic, homogeneous elastic solid in the absence of body forces under the fractional order theory of generalized thermo-viscoelasticity with temperature dependent modulus of elasticity can be written in the following form (Sherief et al. [24]) (i) Equation of motion ρ¨ui = σ ji, j .

(2)

(ii) Heat conduction equation ! ! ∂α ∂θ ∂α ∂ui,i + βT 0 1 + τ0 α kθ,ii = ρcE 1 + τ0 α . ∂t ∂t ∂t ∂t

(3)


ACCEPTED MANUSCRIPT In the above equation, the fractional order derivative proposed by Caputo is defined as: Z t n 1 α n−α−1 d f (τ) (t − τ) dτ, n − 1 < α < n. D f (t) = Γ(n − α) 0 dτn (iii) Constitutive relations σi j = 2μei j + (λekk − βθ)δi j ,

(4)


1 ei j = (ui, j + u j,i ). 2

Here ui are the components of displacement vector ~u, τ0 is the thermal relaxation time, θ = T −T 0 , T is absolute temperature, T 0 is reference temperature assumed to obey the inequality |θ/T 0 | 1, σi j are the components of the stress tensor, ei j are the components of strain tensor, δi j is the Kronecker delta function, ekk is the cubical dilation, ρ is the density of the medium, cE is the specific heat, k is the thermal conductivity, β = (3λ + 2μ)αt , αt is the coefficient of linear thermal expansion, D

∂ ∂t .

We assume that ! ! ! ∂ ∂ ∂ λ = λ e 1 + α0 , μ = μ e 1 + α1 , β = β e 1 + β0 , ∂t ∂t ∂t

βe = (3λe + 2μe )αt , β0 = (3λe α0 + 2μe α1 )αt /βe ,

(5)

and E = E0 f (θ), λe = E0 λ0 f (θ), μe = E0 μ0 f (θ), βe = E0 βe0 f (θ),

(6)

where λe and μe are Lame’s constants, α0 and α1 are viscoelastic relaxation times, E0 is constant modulus of elasticity. The displacement components for one dimensional medium in cartesian co-ordinates have the form u x = u(x, t),

uy = uz = 0.

(7)


ACCEPTED MANUSCRIPT The strain components are given by


e = e xx =

∂u , ∂x

eyy = ezz = e xy = eyz = ezx = 0.

(8)

Thus, Eqns. (2) and (3) can be written as " ! !# " # ∂2 u ∂ ∂ ∂2 u ∂ f ∂u ρ 2 = E 0 λ0 1 + α 0 + 2μ0 1 + α1 f (θ) 2 + ∂t ∂t ∂x ∂x ∂t ∂x " #" # ∂ ∂θ ∂f f (θ) + θ , −E0 βe0 1 + β0 ∂t ∂x ∂θ ! ! ! ∂α ∂θ ∂ ∂α ∂2 u ∂2 θ + ε1 f (θ) 1 + β0 1 + τ0 α , k 2 = ρcE 1 + τ0 α ∂t ∂t ∂t ∂t ∂t∂x ∂x

(9) (10)

where ε1 = T 0 βe0 E0 . The stress tensor components in Eqn. (4) take the following forms "( ! !) ∂ ∂ ∂u σ xx = E0 λ0 1 + α0 + 2μ0 1 + α1 ∂t ∂t ∂x ! # ∂ θ f (θ), −βe0 1 + β0 ∂t

(11)

σyy = σzz = σ xy = σyz = σzx = 0.

Now, we will introduce the following non-dimensional variables ω∗ x ρω∗ c1 θ , u0 = u, t0 = ω∗ t, τ00 = ω∗ τ0 , θ0 = , c1 βe T 0 T0 0 0 0 σ σ0 = , α 1 = ω ∗ α 1 , α 0 = ω ∗ α 0 , β0 = ω ∗ β 0 , βe T 0 x0 =

(12)

where ω∗ = c E

(λe + 2μe ) , k

c21 =

(λe + 2μe ) . ρ

Using these non-dimensional variables, Eqns. (9)-(11) take the form (dropping primes for convenience)


ACCEPTED MANUSCRIPT # ∂2 u ∂ f ∂u f (θ) 2 + ∂x ∂x ∂x !" # ∂ ∂f ∂θ θ + f (θ) , − 1 + β0 ∂t ∂x ∂x

∂2 u ∂ f (θ) 2 = 1 + δ0 ∂t ∂t


α ∂2 θ ∗ ∂ = 1 + τ 0 ∂tα ∂x2

!"

!"

(13)

! # ∂θ ∂ ∂2 u + ε f (θ) 1 + β0 , ∂t ∂t ∂t∂x

(14)

! ! ∂ ∂u ∂ σ = σ xx = 1 + δ0 − 1 + β0 θ, ∂t ∂x ∂t

(15)

where δ0 =

ε1 βe0 (λ0 α0 + 2μ0 α1 ) τ0 , τ∗0 = ∗(1−α) , ε = . λ0 + 2μ0 ρcE (λ0 + 2μ0 ) ω

We consider a special case when |T − T 0 | 1 and f (θ) = (1 − α∗ )T 0 = α10 , where α∗ is empirical h i material constant having dimension K1 . Then Eqns. (13) and (14) take the form ! ! ∂2 u ∂ ∂2 u ∂ ∂θ = 1 + δ0 − 1 + β0 , ∂t ∂x2 ∂t ∂x ∂t2

α

0∂

2θ

∂x2

=

α ! " ∂θ ∗ ∂ 1 + τ0 α α0 ∂t ∂t

(16)

! # ∂ ∂2 u + ε 1 + β0 . ∂t ∂t∂x

(17)

Taking the Laplace transform of Eqns. (15)-(17) by using homogeneous initial conditions, defined and denoted as fˉ(s) =

Z

∞

e−st f (t)dt,

s > 0,

0

we obtain ∂ˉu ˉ − (1 + β0 s)θ, ∂x ∂θˉ ∂2 uˉ = M u ˉ + M , 1 2 ∂x ∂x2 ∂2 θˉ ˉ + L2 ∂ˉu , = L θ 1 ∂x ∂2 x σ ˉ = (1 + δ0 s)

(18) (19) (20)


ACCEPTED MANUSCRIPT where the Laplace transform rule for the Caputo fractional order derivative is given by [23] as: α

α

L{D f (t)} = s L{ f (t)} −

n−1 X

f k (0+ )sα−k−1 ,

k=0

n − 1 < α < n,

since all the state functions have zero initial values, we get L{Dα f (t)} = sα L{ f (t)},


and L1 = (s + τ∗0 sα+1 ),

3

L2 =

ε(1 + β0 s)(s + τ∗0 sα+1 ) α0

, M1 =

s2 1 + β0 s , M2 = . 1 + δ0 s 1 + δ0 s

STATE SPACE FORMULATION

ˉ s), uˉ (x, s), By considering the quantities θ(x,

ˉ ∂θ(x,s) ∂ˉu(x,s) ∂x , ∂x

as state variables, the Eqns. (19) and

(20) can be written in the matrix form as d ˉ ˉ s), V(x, s) = A(s)V(x, dx    0 0   0 0  where A(s) =   L  1 0   0 M

1

(21) 1

0

0

1

0

L2

M2

0

    ˉ   θ(x, s)     uˉ (x, s)  ˉ   , V(x, s) =  ˉ   ∂θ(x,s)   ∂x     ∂ˉu(x,s) ∂x

      .    

The solution of Eqn. (21) can be expressed in the form

ˉ s) = exp[A(s)x]V(0, ˉ s), V(x,     ˉ  θ(0, s)     uˉ (0, s)   . ˉ s) =  where V(0,   ∂θ(0,s) ˉ  ∂x     ∂ˉu(0,s) 

(22)

∂x


ACCEPTED MANUSCRIPT The characteristic equation of matrix A(s) is given by λ4 − (L1 + M1 + L2 M2 )λ2 + L1 M1 = 0.

(23)

The characteristic roots of Eqn. (23), namely λ21 and λ22 , satisfy the following relations λ21 + λ22 = L1 + M1 + L2 M2 ,

λ21 λ22 = L1 M1 .

(24)


By Taylor’s series expansion of exp[A(s)x], we obtain exp[A(s)x] =

∞ X [A(s)x]n n=0

n!

.

(25)

Using Cayley-Hamilton theorem, we can express A4 and higher powers of matrix A in terms of I, A, A2 and A3 , where I is a identity matrix of second order. Thus, the infinite series in Eqn. (25) can be reduced to the following form Γ(x, s) = exp[A(s)x] = a0 I + a1 A(s) + a2 A2 (s) + a3 A3 (s),

(26)

where a0 , a1 , a2 , a3 are parameters depending upon x and s. By Cayley-Hamilton theorem, the characteristic roots ±λ1 and ±λ2 of matrix A must satisfy Eqn. (26), thus we get a0 + a1 λ1 + a2 λ21 + a3 λ31 = exp[λ1 x],

(27)

a0 − a1 λ1 + a2 λ21 − a3 λ31 = exp[−λ1 x],

(28)

a0 + a1 λ2 + a2 λ22 + a3 λ32 = exp[λ2 x],

(29)

a0 − a1 λ2 + a2 λ22 − a3 λ32 = exp[−λ2 x].

(30)

By solving above system of linear equations, we obtain a0 =

λ21 cosh(λ2 x) − λ22 cosh(λ1 x)

λ21 − λ22 cosh(λ1 x) − cosh(λ2 x) a2 = , λ21 − λ22

,

λ31 sinh(λ2 x) − λ32 sinh(λ1 x)

, λ1 λ2 (λ21 − λ22 ) λ2 sinh(λ1 x) − λ1 sinh(λ2 x) . a3 = λ1 λ2 (λ21 − λ22 )

a1 =


ACCEPTED MANUSCRIPT Putting the values of a0 , a1 , a2 , a3 and I, A, A2 and A3 in Eqn. (26) and using Eqn. (24), we have exp[A(s)x] = Γ(x, s) = [Γi j ],

i, j = 1, 2, 3, 4

(31)

where the entries Γi j (x, s) are given as


Γ11 =

(λ21 − L1 ) cosh(λ2 x) − (λ22 − L1 ) cosh(λ1 x) λ21 − λ22

,

   λ2 sinh(λ1 x) − λ1 sinh(λ2 x)   , Γ12 = L2 M1  λ1 λ2 (λ21 − λ22 ) Γ13 =

λ2 (λ21 − M1 ) sinh(λ1 x) − λ1 (λ22 − M1 ) sinh(λ2 x)

Γ22 =

(λ21 − M1 ) cosh(λ2 x) − (λ22 − M1 ) cosh(λ1 x)

λ1 λ2 (λ21 − λ22 )    cosh(λ1 x) − cosh(λ2 x)   , Γ14 = L2  λ21 − λ22    λ2 sinh(λ1 x) − λ1 sinh(λ2 x)   , Γ21 = L1 M2  λ1 λ2 (λ21 − λ22 ) λ21 − λ22

,

   cosh(λ1 x) − cosh(λ2 x)   , Γ23 = M2  λ21 − λ22 Γ24 =

,

λ2 (λ21 − L1 ) sinh(λ1 x) − λ1 (λ22 − L1 ) sinh(λ2 x) λ1 λ2 (λ21 − λ22 )

,

   λ2 (λ21 − M1 ) sinh(λ1 x) − λ1 (λ22 − M1 ) sinh(λ2 x)   , Γ31 = L1  λ1 λ2 (λ21 − λ22 )    cosh(λ1 x) − cosh(λ2 x)   , Γ32 = L2 M1  λ21 − λ22

Γ33 =

(λ21 − M1 ) cosh(λ1 x) − (λ22 − M1 ) cosh(λ2 x) λ21 − λ22

,

   λ1 sinh(λ1 x) − λ2 sinh(λ2 x)   , Γ34 = L2  λ21 − λ22


ACCEPTED MANUSCRIPT    cosh(λ1 x) − cosh(λ2 x)   , Γ41 = L1 M2  λ21 − λ22    λ2 (λ21 − L1 ) sinh(λ1 x) − λ1 (λ22 − L1 ) sinh(λ2 x)   , Γ42 = M1  λ1 λ2 (λ21 − λ22 )    λ1 sinh(λ1 x) − λ2 sinh(λ2 x)   , Γ43 = M2  λ21 − λ22 Downloaded by [Guru Jambheshwar University ] at 05:15 14 April 2015

Γ44 =

4

(λ21 − L1 ) cosh(λ1 x) − (λ22 − L1 ) cosh(λ2 x) λ21 − λ22

.

APPLICATION

We consider a homogeneous isotropic thermo-viscoelastic half space x ≥ 0. The boundary of half space x = 0 is taken to be traction free and subjected to thermal shock. Mathematically, the boundary conditions can be expressed as σ(0, t) = σ0 = 0,

(32)

θ(0, t) = θ0 = θ∗ H(t),

(33)

where H(t) is the Heaviside unit step function and θ∗ is constant temperature. Laplace transform of Eqns. (32) and (33) yields σ(0, ˉ s) = σˉ0 = 0,

(34)

∗ ˉ s) = θˉ0 = θ . θ(0, s

(35)

Using Eqns. (34) and (35) in Eqn. (18), we obtain θ∗ ∂ˉu(0, s) =η , ∂x s

(36)

where η=

1 + β0 s . 1 + δ0 s


ACCEPTED MANUSCRIPT Since the solution of Eqn. (22) is unbounded at infinity (Γi j are not bounded as x → ∞), thus the initial condition should be so adjusted that the infinite terms are eliminated from Γi j which is tantamount to suppressing the positive exponentials in Γi j . Therefore each cosh(λi x) and sinh(λi x) must be replaced by 12 e−λi x and − 12 e−λi x respectively.


Now, substituting x = 0 in Eqn. (22) and using Eqns. (35) and (36) and performing necessary matrix operations, we obtain a system of linear equations in uˉ (0, s) and    (λ1 λ2 + L1 )η − L1 M2  θ∗ uˉ (0, s) = −   , s λ1 λ2 (λ21 − λ22 ) ! ∗ ˉ s) ∂θ(0, L1 (λ1 λ2 + M1 ) L2 η θ = − + . ∂x λ1 λ2 (λ1 + λ2 ) (λ1 + λ2 ) s ˉ s), uˉ (0, s), Now inserting the values of θ(0,

ˉ ∂θ(0,s) ∂ˉu(0,s) ∂x , ∂x

ˉ ∂θ(0,s) ∂x , whose solution gives

(37) (38)

from Eqns. (35)-(38) into Eqn. (22)

and using Eqn. (24) we obtain   ∗  (λ2 − L1 − L2 η)e−λ1 x − (λ2 − L1 − L2 η)e−λ2 x  θ   2 1  ˉ s) = −  θ(x,  , 2 2 s λ1 − λ 2   θ∗  ψ1 e−λ1 x − ψ2 e−λ2 x  uˉ (x, s) = −   , s λ21 − λ22

(39) (40)

where ψ1 =

η(λ21 − L1 ) + L1 M2 λ1

, ψ2 =

η(λ22 − L1 ) + L1 M2 λ2

.

Using Eqns. (39) and (40) in Eqn. (18), we obtain  ! θ∗ η(M1 − L1 + L2 M2 − L2 η) + L1 M2  e−λ1 x − e−λ2 x  σ(x, ˉ s) = −  ,  s η∗ λ21 − λ22

(41)

where η∗ =

η . 1 + β0 s

(42)



LIMITING CASES 1. To obtain the expressions of displacement, temperature and stress in the context of generalized thermo-viscoelasticity theory with temperature dependent modulus of elasticity under applied boundary conditions, we shall neglect the effect of fractional parameter α from the assumed theory. This can be achieved by substituting α = 1 in the basic equations and per-


forming necessary modifications in the Eqns. (39)-(41). 2. Expressions for the field variables under fractional order generalized thermoelasticity theory with temperature dependent modulus of elasticity can be established by neglecting the viscosity effect. For this purpose, we shall set α0 = 0 and α1 = 0, which implies λ = λe , μ = μe , β0 = 0, δ0 = 0, and β = βe . By considering these changes, the problem reduces to fractional order generalized thermoelasticity theory with temperature dependent modulus of elasticity and the expressions for displacement, temperature and stress can be obtained from the Eqns. (39)-(41). 3. If we neglect the effect of temperature dependent modulus of elasticity then we are left with the relevant problem under fractional order generalized thermo-viscoelasticity theory. Hence, we substitute α∗ = 0 in the expression of f (θ) as a result of which we obtain α0 = 1 and L2 = ε(1 + β0 s)(s + τ∗0 sα+1 ). By considering these changes, we can get the mathematical expressions for the field variables displacement, temperature and stress in the context of fractional order generalized thermo-viscoelasticity theory from Eqns. (39)-(41).



NUMERICAL INVERSION OF LAPLACE TRANSFORM

We shall now outline the numerical inversion method used to find the solution in the physical domain. The inversion formula of the Laplace transform is defined as Z c+ι∞ 1 f (t) = e st fˉ(s)ds, 2πι c−ι∞

(43)


where fˉ(s) is the Laplace transform of function f (t). In order to invert Laplace transforms in the equations given in Section 4, we apply a numerical inversion method based on Fourier series expansion explained by Honig and Hirdes [32]. In this method, the inverse transform f (t) of the Laplace transform fˉ(s) is approximated by the relation as  ! N ιkπt X  ect  1 ˉ ιkπ  f (c) + Re  , 0 ≤ t < 2t1 , f (t) = e t1 fˉ c + t1 2 t 1 k=1

(44)

where N is a sufficiently large integer representing the number of terms in the truncated Fourier series chosen such that " !# ιNπt ιNπ ct t1 ˉ e Re e f c + ≤ ε1 , t1

(45)

where ε1 is a prescribed small positive value that corresponds to the degree of accuracy to be achieved and c is a positive constant and must be greater than the real parts of all the singularities of fˉ(s). The optimal choice of c was obtained according to the criteria described by Honig and Hirdes [32].

7

NUMERICAL RESULTS AND DISCUSSION

To illustrate and compare the theoretical results obtained in the Section 4, we now present some numerical results which depict the variations of displacement, temperature and stress fields. The material chosen for the purpose of numerical evaluation is copper, for which we take the following


ACCEPTED MANUSCRIPT values of the physical constants E0 = 36.9 × 1010 kgm−1 s−2 , k = 386 Wm−1 K −1 , T 0 = 293 K, v = 0.33, ρ = 8954 kgm−3 , αt = 1.78 × 10−5 K −1 , cE = 383.1 Jkg−1 K −1 , α0 = 0.06 s, α1 = 0.09 s, α∗ = 0.0051 K −1 . The constants λ0 and μ0 are given as


λ0 =

v , (1 + v)(1 − 2v)

μ0 =

1 , 2(1 + v)

(46)

where v is the Poisson ratio. The computations are carried out for t = 0.1 and θ∗ = 10. The numerical technique, outlined in previous section was used to invert the Laplace transform in Eqns. (39)-(41), providing the displacement, temperature and stress distributions in the physical domain. The results are represented graphically for different positions of x. Figures 1-3 exhibit the space variation of the displacement u, temperature θ and stress σ for different values of fractional parameter α. In figure 1, we have plotted the displacement u against distance x at two different mentioned values of fractional parameter α. It can be seen that u starts with values 0.000628 and 0.000575 at the boundary of half space for α = 0.5 and α = 1.0 respectively and then decreases to zero value but u for α = 1.0 again attains small magnitude in the range 0.9 ≤ x ≤ 1.6 and then finally diminishes to zero. The effect of fractional parameter α on displacement component u is significant in the range 0 ≤ x ≤ 1. In figure 2, we have compared the values of temperature field θ for α = 0.5 and α = 1.0. The value of θ is -0.04605 initially, it gradually increases in the range 0 ≤ x ≤ 0.6 attaining maximum value at x = 0.6 and then ultimately approaches to zero for both the values of α. In the initial range 0 ≤ x ≤ 0.2, θ has almost same values for both cases and the difference is significantly pronounced in the range 0.2 ≤ x ≤ 1.4. Figure 3 displays the distribution of the stress component σ with distance x for α = 0.5 and α = 1.0. We notice from the figure that σ has zero value at x = 0 which agrees with the boundary conditions, then it decreases to zero with increase in x for both the values of α. However, the rate of decay is much


ACCEPTED MANUSCRIPT faster in case of α = 1.0. Figures 4-6 depict the space variation of the displacement u, temperature θ and stress σ under three different theories, namely (i) Fractional order generalized visco-thermoelasticity with temperature dependent modulus of elasticity (FGVTT) (ii) Fractional order generalized thermoelasticity with temperature dependent modulus of elasticity (FGTT) (iii) Fractional order generalized viscothermoelasticity (FGVT). Figure 4 shows the variation of displacement u with distance x under


the theories mentioned above. It is clearly depicted in figure that u starts with maximum value at x = 0 and then gradually decreases to zero with increase in distance x for all theories. We can see from the figure that the value of u for FGTT and FGVT theories is less as compared to FGVTT theory which implies that as we neglect the effect of viscosity or temperature dependent modulus of elasticity from considered theory, it acts to decrease the value of displacement field. The difference is much pronounced for FGTT theory which indicates that effect of viscosity is more than temperature dependent modulus of elasticity on displacement field. In figure 5, we have plotted the variation of temperature θ with distance x for the theories FGVTT, FGTT and FGVT. We observe that the behaviour for temperature field is similar in nature for all the theories i.e. it begins with negative values on the boundary of half space, then increases to maximum and thereafter tends to zero. We also observe that the values of θ are negative in the range 0 ≤ x ≤ 0.4 and positive in the range 0.4 ≤ x ≤ 1.1. It can be seen that the magnitudes of θ for FGTT and FGVT theories are greater than the magnitude for FGVTT theory which means that the magnitude of θ increases if we neglect the effect of viscosity or temperature dependent modulus of elasticity and the difference is significant for the range 0 ≤ x ≤ 0.4 and it is more pronounced for FGVT theory than FGTT theory. Figure 6 presents the variation of stress in the context of FGVTT, FGTT and FGVT theories. It is noticed from figure that the magnitude of σ is less for FGVT theory and much greater for FGTT theory as compared to FGVTT theory which clearly indicates the effect of temperature dependent modulus of elasticity and viscosity on stress field. We see that the stress distribution in case of FGVTT and FGVT theories tends to zero much earlier than FGTT theory. The difference is found


ACCEPTED MANUSCRIPT to be much significant for FGTT theory.

8

CONCLUSIONS

The paper analyzes the effects of fractional parameter, viscosity and temperature dependent modulus of elasticity on displacement (u), temperature (θ) and stress (σ) fields in a homogeneous


thermoelastic half space whose surface is subjected to thermal shock. State space approach is used to obtain the expressions for the physical quantities. The theoretical and numerical results reveal that all the considered parameters, namely, fractional parameter, viscosity and temperature dependent modulus of elasticity have significant effects on the field variables. The results can be summarized as follows: 1. All the physical variables tend to zero after some distance x which make us to conclude that the phenomenon of finite speed of wave propagation is manifested for all the considered field variables. Hence, it is possible to generalize results for practical applications. 2. It is observed from the figures that the fields variables u, θ and σ are strongly affected by the fractional parameter α. 3. From figures 1-3, we deduce that the pattern for u, θ and σ is observed to be smoother for α = 0.5. 4. As we can see from figures 4-6 that if we neglect the effect of the temperature dependent modulus of elasticity parameter from the considered theory (i.e. FGVTT) than it decreases the magnitude of displacement and stress fields but increases the magnitude of temperature. The difference is significantly pronounced in temperature but silently pronounced in displacement and stress fields.


ACCEPTED MANUSCRIPT 5. Figures 4-6 also reveal that the displacement and stress fields are highly influenced by viscosity rather than temperature field. It is also concluded that pattern for displacement and stress fields for FGTT theory take much large distance to approach zero value as compared to FGVTT and FGVT theories but in contrary the temperature field diminishes to zero at same point for FGVTT, FGTT and FGVT theories.


6. The displacement and stress fields are highly influenced by viscosity while temperature dependent modulus of elasticity has small effect on these field quantities but in the contrary, for temperature field the effect of temperature dependent modulus of elasticity is more pronounced than viscosity. 7. For all the theories FGVTT, FGTT and FGVT the behavior for displacement u, temperature θ and stress field σ is similar.

References [1] H. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Solids, vol. 15, pp. 299-309, 1967. [2] A.E. Green and K.A. Lindsay, Thermoelasticity, J. Elast., vol. 2, pp.1-7, 1972. [3] R. Dhaliwal and H. Sherief, Generalized thermoelasticity for anisotropic media, Quart. Appl. Math., vol. 33, pp. 1-8, 1980. [4] R.B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Therm. Stress., vol. 22, pp. 451-476, 1999. [5] H. Sherief, State space formulation for generalized thermoelasticity with one relaxation time including heat sources, J. Therm. Stress., vol. 16, pp. 163-180, 1993.


ACCEPTED MANUSCRIPT [6] H. Sherief and F. Hamza, Generalized thermoelastic problem of a thick plate under axisymmetric temperature distribution, J. Therm. Stress., vol. 17, pp. 435-453, 1994. [7] N.M. El-Maghraby and H. Youssef, State space approach to generalized thermoelastic problem with thermomechanical shock, J. Appl. Math. Comput., vol. 156, pp. 577-586, 2004. [8] A.A. Ilioushin and B.E. Pobedria, Fundamentals of the Mathematical Theories of Thermal


Viscoelasticity, Nauka, Moscow, 1970. [9] R.I. Tanner, Engineering Rheology, Oxford University Press, 1988. [10] R. Huilgol and N. Phan-Thien, Fluid Mechanics of Viscoelasticity, Elsevier, Amsterdam, 1997. [11] M.I.A. Othman, M.A. Ezzat, S.A. Zaki and A.S. El-Karamany, Generalized thermoviscoelasticity plane waves with two relaxation times, Int. J. Engg. Sci., vol. 40, pp.13291347, 2002. [12] M.A. Ezzat, M.I.A. Othman and A.S. El-Karamany, State space approach to generalized thermo-viscoelasticity with two relaxation times, Int. J. Engg. Sci., vol. 40, pp. 283-302, 2002. [13] M.A. Ezzat and A.S. El-Karamany, The uniqueness and reciprocity theorem for generalized thermo-viscoelasticity plane wave with two relaxation times, J. Therm. Stress., vol. 25, pp. 507-522, 2002. [14] M.I.A. Othman, Generalized electro-magneto-thermo-viscoelastic in case of 2-D thermal shock problem in a finite conducting medium with one relaxation time, Acta Mech., vol.169, pp. 37-51, 2004. [15] J.N. Sharma and R. Chand, Transient thermoviscoelastic waves in a half-space due to thermal loads, J. Therm. Stress., vol. 28, pp. 233-252, 2005.


ACCEPTED MANUSCRIPT [16] M.A. Ezzat, The relaxation effect on volume properties of electrically conducting viscoelastic material, Mater. Sci. Engg., vol. 130, pp. 11-23, 2006. [17] M.A. Ezzat and A.S. El-Karamany, State space approach of two- temperature magnetoviscoelasticity theory with thermal relaxation in a medium of perfect conductivity, J. Therm. Stress., vol. 32, pp. 819-838, 2009.


[18] M.A. Ezzat, M. Zakaria and A. Abdel-Barry, Generalized thermoelasticity with temperature dependent modulus of elasticity under three theories, J. Appl. Math. Comput., vol. 14, pp. 193-212, 2004. [19] M.I.A. Othman, State space approach to generalized thermoelasticity plane waves with two relaxation times under dependence of modulus of elasticity on the reference temperature, Can. J. Phys., vol. 81, pp. 1403-1418, 2003. [20] M.I.A. Othman and Kh. Lotfy, Two dimensional problem of generalized magneto thermoelasticity under the effect of temperature dependent properties for different theories, M M M S, vol. 5, pp. 235-242, 2009. [21] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. [22] K.S. Miller and B. Ross, An Introduction to the Fractional Integrals and Derivatives-Theory and Applications, Wiley, New York, 1993. [23] Y.Z. Povstenko, Fractional heat conduction equation and associated thermal stress, J. Therm. Stress., vol. 28, pp. 83-102, 2005. [24] H. Sherief, A.M.A. El-Sayed and A.M. Abd El-Latief, Fractional order theory of thermoelasticity, Int. J. Solids Struct., vol. 47, pp. 269-275, 2010. [25] H. Youssef, Theory of fractional order generalized thermoelasticity, J. Heat Trans., vol. 132, pp. 1-7, 2010.


ACCEPTED MANUSCRIPT [26] M.A. Ezzat, Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer, Physica B, vol. 405, pp. 4188-4194, 2010. [27] M.A. Ezzat, Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer, Physica B, vol. 406, pp. 30-35, 2011. [28] G. Jumarie, Derivation and solutions of some fractional Black-Scholes equations in coarse-


grained space and time. Application to Merton’s optimal portfolio, Comput. Math. Applic., vol. 59, pp. 1142-1164, 2010. [29] H. Youssef, State-space approach to fractional order two-temperature generalized thermoelastic medium subjected to moving heat source. Mech. Adv. Mat. Struct., vol. 20, pp. 47-60, 2013. [30] H. Sherief and A.M. Abd El-Latief, Application of fractional order theory of thermoelasticity to a 1D problem for a half space, Z. Angew. Math. Mech., vol. 94, pp. 509-515, 2014. [31] L.Y. Bahar and R.B. Hetnarski, State space approach to thermoelasticity, J. Therm. Stress., vol. 1, pp. 135-145, 1978. [32] G. Honig and U. Hirdes, A method for the numerical inversion of Laplace transforms, J. Comput. Appl. Math., vol. 10, pp. 113-132, 1984.



ACCEPTED MANUSCRIPT

Figure 1: Displacement distribution u for different values of α at t = 0.1



ACCEPTED MANUSCRIPT

Figure 2: Temperature distribution θ for different values of α at t = 0.1



ACCEPTED MANUSCRIPT

Figure 3: Stress distribution σ for different values of α at t = 0.1



ACCEPTED MANUSCRIPT

Figure 4: Displacement distribution u at α = 0.5, t = 0.1



ACCEPTED MANUSCRIPT

Figure 5: Temperature distribution θ at α = 0.5, t = 0.1



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Figure 6: Stress distribution σ at α = 0.5, t = 0.1


Mechanics of Advanced Materials and Structures ...

Mechanics of Advanced Materials and Structures ...

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