Axi-symmetric generalized thermoelastic diffusion problem with two-temperature and initial stress under fractional order heat conduction

Author’s Accepted Manuscript Axi-symmetric generalized thermoelastic diffusion problem with two-temperature and initial stress under fractional order heat conduction Sunita Deswal, Kapil Kumar Kalkal, Sandeep Singh Sheoran www.elsevier.com/locate/physb

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S0921-4526(16)30186-7 http://dx.doi.org/10.1016/j.physb.2016.05.008 PHYSB309474

To appear in: Physica B: Physics of Condensed Matter Received date: 23 October 2015 Revised date: 11 February 2016 Accepted date: 10 May 2016 Cite this article as: Sunita Deswal, Kapil Kumar Kalkal and Sandeep Singh Sheoran, Axi-symmetric generalized thermoelastic diffusion problem with twotemperature and initial stress under fractional order heat conduction, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2016.05.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Axi-symmetric generalized thermoelastic diffusion problem with two-temperature and initial stress under fractional order heat conduction Sunita Deswal, Kapil Kumar Kalkal, Sandeep Singh Sheoran∗ Department of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar-125001, Haryana (India)

Abstract A mathematical model of fractional order two-temperature generalized thermoelasticity with diffusion and initial stress is proposed to analyze the transient wave phenomenon in an infinite thermoelastic half-space. The governing equations are derived in cylindrical coordinates for a two dimensional axi-symmetric problem. The analytical solution is procured by employing the Laplace and Hankel transforms for time and space variables respectively. The solutions are investigated in detail for a time dependent heat source. By using numerical inversion method of integral transforms, we obtain the solutions for displacement, stress, temperature and diffusion fields in physical domain. Computations are carried out for copper material and displayed graphically. The effect of fractional order parameter, two-temperature parameter, diffusion, initial stress and time on the different thermoelastic and diffusion fields is analyzed on the basis of analytical and numerical results. Some special cases have also been deduced from the present investigation. Keywords: Axi-symmetric, Fractional parameter, Two-temperature, Initial stress, Diffusion, Laplace and Hankel transforms. 2000 MSC: 74A15, 80A20

∗

corresponding author Email addresses: spannu [email protected] (Sunita Deswal), kapilkalkal [email protected] (Kapil Kumar Kalkal), sandeep [email protected] (Sandeep Singh Sheoran)

Preprint submitted to Elsevier

May 12, 2016

1. Introduction Diffusion is one of the mechanism in the transport phenomenon that occurs in nature. The notion of diffusion can be described by two different approaches: (i) phenomenological approach, (ii) atomistic approach. The phenomenological approach is based on the Fick’s laws, according to which diffusion is the passive movement of particles from regions of higher concentration to the regions of lower concentration. Also, it occurs in response to a concentration gradient expressed as the change in the concentration due to change in position. According to the atomistic approach, diffusion is considered as a result of the random walk of the diffusing particles. Nowacki [1, 2, 3] put forward the theory of thermoelastic diffusion, in which the coupled thermoelastic model was formulated to study the dynamical problems of diffusion in solids. Later, Sherief et al. [4] developed the theory of generalized thermoelastic diffusion with one relaxation time that predicts finite speeds of propagation for thermoelastic and diffusive waves. Sharma et al. [5] presented an analysis of a two-dimensional problem concerning with plane strain deformation in the generalized thermoelastic diffusive medium. Xia et al. [6] studied the effect of diffusion on the dynamical response of thermoelastic and diffusive fields in an infinite body with a cylindrical cavity. Deswal and Choudhary [7] solved a two-dimensional generalized thermodiffusion problem with thermal relaxation using Laplace and Fourier transforms. Deswal and Kalkal [8] examined the thermo-mechanical interactions for generalized electro-magneto-thermoviscoelasticity based on G-L theory in a half-space with diffusion. Using Laplace and Hankel transforms technique, Sherief and Hussein [9] solved two dimensional axi-symmetric problem of generalized thermoelastic diffusion with one relaxation time for a thick plate. Gurtin and Williams [10, 11], Chen and Gurtin [12] and Chen et al. [13] introduced the two temperature theory of thermoelasticity by modifying the classical Clausius-Duhem inequality and proposed that the heat conduction on a deformable body depends upon two different temperatures: the conductive temperature Φ and the thermodynamical temperature θ, the first is caused by the thermal processes and the second is caused by the mechanical processes inherent between the particles and the layers of elastic material. In the two-temperature theory a material parameter a∗ (> 0) appears and for limiting value a∗ → 0, one can obtain Φ → θ, which is the case of the classical theory of thermoelasticity. Quintanilla [14] discussed the existence, 2

structural stability, convergence and spatial behaviour of two-temperature thermoelasticity. Youssef [15] established the two-temperature theory of generalized thermoelasticity by introducing thermal relaxation parameters in the constitutive relations and obtained the uniqueness theorem for a homogeneous and isotropic body. Magana and Quintanilla [16] studied the uniqueness and growth of solutions in the context of the two-temperature generalized thermoelasticity. Kumar and Mukhopadhyay [17] proposed an analytical model of two-temperature generalized thermoelasticity theory to investigate the effect of thermal relaxation parameter on the wave fields. Kumar et al. [18] investigated the thermoelastic interactions for the theory of two-temperature thermoelasticity with two relaxation parameters by deriving the short-time approximate solutions for the fields in an unbounded medium with cylindrical cavity. The problems related to initially stressed elastic medium have found numerous applications in various fields, such as earthquake engineering, seismology and geophysics. The elastodynamics of a body under initial stress is exposed in the treatise of Biot [19]. The effect of pre-stress on wave propagation through homogeneous anisotropic media has interested many authors. It has stimulated a great deal of research and in case of initial finite elastic deformations, a classical textbook on the subject has been written by Ogden [20]. Montanaro [21] developed the linear theory of thermoelasticity with initial stress for an isotropic medium. Othman and Atwa [22] applied normal mode analysis to investigate the thermoelastic plane waves for an elastic solid half-space under hydrostatic initial stress. Deswal et al. [23] studied the effects of magnetic field and initial stress on the dynamical interaction of a fractional order generalized thermoelastic half-space. During last few decades, fractional order differential equations have been successfully employed for modeling many different processes and systems, specifically in the area of physics, chemistry, engineering, astrophysics, mechanics, quantum mechanics, nuclear physics and quantum field theory etc. Models for anomalous transport process in the form of time and/or space fractional convection diffusion wave equation attained much attention and have been considered by many researchers. Moreover, the global dependency and non local property of fractional derivative is one of the main reasons for its growing popularity. The historical development of the subject fractional The calculus can be investigated in Ross [24] and Miller and Ross [25]. review article by Rossikhin and Shitikova [26] is devoted to the analysis of 3

new trends and results in the field of fractional calculus and its application to dynamic problems of solid mechanics. The results obtained are critically estimated in the view of the role of the fractional calculus in engineering problems. Povstenko [27] developed a quasi-static uncoupled theory of thermoelasticity based on the heat conduction equation with a Caputo time-derivative of fractional order and found the associated thermal stresses. Sherief et al. [28] introduced the fractional order theory of thermoelasticity by using the methodology of fractional calculus in combination with L-S theory of thermoelasticity and proved uniqueness theorem and derived variational principle and reciprocity theorem. Youssef [29] formulated the theory of fractional order generalized thermoelasticity by introducing the Riemann-Liouville fractional integral operator into the generalized heat conduction equation, proved uniqueness theorem and solved one dimensional problem to discuss the effect of fractional parameter on the fields. Ezzat [30] constructed a new mathematical model of fractional heat conduction law in which the generalized Fourier’s law of heat conduction is modified by using the new Taylor’s series expansion of time fractional order (Jumarie [31]). In this model, the fractional heat conduction equation takes the form: ∂ α q τ0α = −k∇θ, 0 < α ≤ 1. q + Γ(α + 1) ∂tα where q is heat flux vector, τ0 is the thermal relaxation time, k is thermal conductivity, Γ is the Gamma function, θ = T − T0 , T is the absolute temperature, T0 is reference temperature assumed to obey the inequality |θ/T0 |  1 and α is fractional order parameter. El-Karamany and Ezzat [32] introduced the two-temperature fractional thermoelasticity theory for non-homogeneous anisotropic elastic solid, proved uniqueness and reciprocal theorems and established the convolutional variational principle. In this theory, the Fourier’s law of heat conduction is replaced by: q +

∂ α q τ0α = −k∇Φ, Γ(α + 1) ∂tα

0 < α ≤ 1,

where Φ is the conductive temperature. Ezzat and Fayik [33] derived the fractional order theory of thermoelastic diffusion in elastic solids and established variational principle, uniqueness 4

theorem and reciprocity theorem. In this theory, the equation of mass flux vector takes the form   ∂α τα ηj = −DP, i, 0 < α ≤ 1, 1+ Γ(α + 1) ∂tα where ηj denotes the flow of the diffusing mass vector, D denotes the diffusion constant and P = p − P0 , p is the chemical potential per unit mass at non-equilibrium, P0 is the chemical potential per unit mass at natural state. Ezzat and El-Karamany [34] solved one dimensional problem on twotemperature magneto-thermoelasticity with fractional order heat conduction using state space approach and Laplace transform. Recently, Mashat et al. [35] analyzed the thermoelastic interactions in a traction free thermoelastic half-space with axi-symmetric temperature distribution under fractional order heat conduction. The present study is motivated by the importance of thermoelastic diffusion process in the field of oil extraction, the fields associated with the advent of semiconductor devices and the advancement of microelectronics. This model may be simulated to some geophysical situations. The assumption of initially stressed elastic medium finds great significance in the fields of seismology, geomechanics, earthquake engineering and soil dynamics etc. In fact, the earth is an initially stressed medium, therefore it is of great interest to study the phenomenon of wave propagation in the presence of initial stress. It is realized that the models of two-temperature thermoelasticity may be of more relevance to real situations. The proposed concept of non-locality utilizing fractional differential operator in this model opens new perspective in the study of thermoelastic deformations in solid mechanics. Moreover, non-standard formulations are essential for a constant progress in the development of new materials. Due to the multifarious applications as discussed above and non-existence of systematic investigations in the context of the fractional order two-temperature generalized thermoelasticity theory with diffusion and initial stress in the axi-symmetric thermoelastic medium, has inspired the authors to study the current problem. In this paper, we study the dynamical interactions of the thermal, elastic and diffusion fields under the fractional order generalized thermoelasticity theory with diffusion, two-temperature and initial stress. The governing equations are derived in cylindrical coordinates for a two dimensional axisymmetric problem. By adopting a direct approach, an exact solution of the problem is first obtained in the Laplace and Hankel transforms space 5

and then in the physical domain by using a numerical inversion technique of integral transforms. The derived expressions are computed numerically for copper material and the results are presented in graphical form. The results are validated by comparing them with those cited in the literature as special cases. 2. Basic governing equations We investigate the dynamic interactions in an axi-symmetric generalized thermoelastic half-space subjected to a time dependent heat source under the fractional order two-temperature generalized thermoelasticity theory with diffusion and initial stress. The basic field equations and constitutive relations for the proposed model of generalized thermoelasticity are given as: (i) the stress-strain-temperature-diffusion relations (Sherief et al. [4], Montanaro [21]): ¯ kk − β1 θ − β2 C)δij − S(δij + ωij ), μeij + (λe σij = 2¯

(1)

1 1 eij = (ui,j + uj,i ), ωij = (ui,j − uj,i ), 2 2

(2)

where

(ii) equation of motion: σji,j = ρu¨i , (iii) heat conduction equation (Ezzat and Fayik [33]):   τ0α ∂ α+1 ∂ (ρCE θ + β1 T0 e + aT0 C) , + kΦ,ii = ∂t Γ(α + 1) ∂tα+1

(3)

(4)

(iv) modified law of mass diffusion with the time-fractional derivative is established as (Ezzat and Fayik [33]):   τα ∂ α+1 ∂ DbC,ii = Dβ2 e,ii + Daθ,ii + C, (5) + ∂t Γ(α + 1) ∂tα+1 (v) the equation of chemical potential is given as (Sherief et al. [4]): P = −β2 ekk + bC − aθ, 6

(6)

(vi) the relation between conductive temperature Φ and thermodynamical temperature θ is given as: Φ − θ = a∗ Φ,ii ,

(7)

where ui are the components of displacement vector u, σij are the components of the stress tensor, eij are the components of strain tensor, δij is the Kronecker delta function, e is the cubical dilatation, ρ is the density of the ¯ + 2¯ μ)αt , αt is the coefficient of linmedium, CE is the specific heat, β1 = (3λ ¯ + 2¯ ¯ μ)αc , ear thermal expansion, λ and μ ¯ are the Lame’s constants, β2 = (3λ αc is the coefficient of linear diffusion expansion, S is the initial pressure, τ0 is the thermal relaxation time, τ is the diffusion relaxation time, C = c − C0 , c is the non-equilibrium concentration, C0 is the mass concentration at natural state, a is the measure of thermodiffusive effect, b is a measure of diffusive effect, a∗ is the two-temperature parameter, a comma followed by a suffix denotes material derivative and a superposed dot denotes the derivative with respect to time t. 3. Statement of the problem A thermoelastic homogeneous isotropic thermally conducting half-space with diffusion and initial stress is considered as the domain of the problem with a cylindrical coordinate system (r, θ, z) and z−axis is assumed to be the axis of symmetry pointing vertically into the medium so that half-space occupies the region z ≥ 0. According to the characteristics of axi-symmetric problem, all the field variables are independent of θ and will be the functions of r, z and t. A time harmonic thermal source is assumed to be acting at the origin of the cylindrical polar coordinate system. Now, we restrict our analysis to a two dimensional problem in rz−plane. The components of displacement vector u = (ur , uθ , uz ) assume the form u = ur = u(r, z, t), v = uθ = 0, w = uz = w(r, z, t). The initial and regularity conditions for the thermo-diffusive half-space

7

with two-temperature are given as (Sharma et al. [5]): u(r, z, 0) = u(r, ˙ z, 0) = 0, w(r, z, 0) = w(r, ˙ z, 0) = 0, ˙ θ(r, z, 0) = θ(r, z, 0) = 0, ˙ z, 0) = 0, Φ(r, z, 0) = Φ(r, ˙ z, 0) = 0, C(r, z, 0) = C(r, P (r, z, 0) = P˙ (r, z, 0) = 0, for z ≥ 0, −∞ < r < ∞,

(8)

u(r, z, t) = w(r, z, t) = θ(r, z, t) = 0, Φ(r, z, t) = C(r, z, t) = P (r, z, t) = 0, for t > 0 when z → ∞.

(9)

and

It is more convenient to introduce non-dimensional variables as follows: (r , z  , u , w ) = c0 η0 (r, z, u, w), (t , τ0 , τ  ) = c20 η0 (t, τ0 , τ ), 1 β2 β1 1 σij = ¯ σij , C  = ¯ C, θ = ¯ θ, P  = P, (10) β2 λ + 2¯ μ λ + 2¯ μ λ + 2¯ μ β1 1 Φ = ¯ Φ, S  = ¯ S, λ + 2¯ μ λ + 2¯ μ ¯ + 2¯ λ μ ρCE , η0 = . ρ k  with ψ  = (0, −ψ, 0) Using the Helmholtz decomposition u = ∇φ + ∇ × ψ  = 0, we seek the displacement components u and w in the form and ∇.ψ   ∂ 2ψ ∂φ ∂2 ∂φ 2 (11) + , w= − ∇ − 2 ψ, u= ∂r ∂r∂z ∂z ∂z

where c20 =

∂2 ∂2 1 ∂  where ∇2 = + + and the potential functions φ and ψ ∂r2 r ∂r ∂z 2 represents the dilatational and rotational parts respectively of displacement vector u. Introducing the dimensionless parameters and potential functions from Eqs. (10) and (11) into Eqs. (1)-(7), we get the following relations (dropping

8

the prime notation from now)    ∂ψ ∂2 2 2 2 φ+ − θ − C − S, (12) σzz = ∇ φ − 2β ∇ − 2 ∂z ∂z     2  ∂2ψ  2  ∂  2 ∂2 2 ∂ φ 2 + σzr = 2β − β + S/2 ∇ − 2 ψ ,(13) β − S/2 ∂r∂z ∂r ∂z 2 ∂z  2  ∂ ψ 1 ∇2 ψ = , (14) 2 β − S/2 ∂t2 ∂ 2φ (15) ∇2 φ − θ − C = 2 , ∂t    τ0 ∂ α+1  ∂ 2 + ∇ φ + ε C , (16) ∇2 Φ = θ + ε 1 2 ∂t Γ(α + 1) ∂tα+1   τ0 ∂ α+1 ∂ 4 2 2 C = 0, (17) + ∇ φ − α1 ∇ C + α2 ∇ θ + α3 ∂t Γ(α + 1) ∂tα+1 α4 ∇2 Φ = Φ − θ, (18) 2 (19) P = −∇ φ + α1 C − α2 θ, ¯ + 2¯ μ ¯ β12 T0 β1 T0 b(λ μ) = , α = , where β 2 = ¯ , ε , ε1 = 2 1 2 ¯ ρCE β2 β2 ρCE (λ + 2¯ μ) λ + 2¯ μ ¯ + 2¯ ¯ + 2¯ a(λ μ) (λ μ) α2 = , α3 = , α4 = ηc20 η02 . 2 β1 β2 Dβ2 η0 4. Solution of the problem In this section we obtain the analytical solution in Laplace-Hankel transform domain of the formulation presented in previous section. The Laplace transform of the function f (r, z, t) is defined as  ∞ ¯ f (r, z, t)e−st dt, (20) L[f (r, z, t)] = f (r, z, s) = 0

and the Hankel transform of the function f (r, z, s) is defined as  ∞ ∗ f (r, z, s)rJ0 (ξr)dr, H[f (r, z, s)] = f (ξ, z, s) =

(21)

0

where J0 is Bessel function of first kind of order zero, s and ξ are the Laplace and Hankel transform parameters respectively. 9

Using the definition of Hankel transform, we can find the relation   2 ∂ f (r, z, s) 1 ∂f (r, z, s) = −ξ 2 f ∗ (ξ, z, s). + H ∂r2 r ∂r

(22)

Applying the Laplace and Hankel transforms to Eqs. (14)-(19) and using the relation (22) appropriately, we get the relations in the joint transform domain as   2 d 2 − λ4 ψ¯∗ = 0, (23) dz 2  2  d 2 2 − (ξ − s ) φ¯∗ − θ¯∗ − C¯ ∗ = 0, (24) 2 dz  2    2   d d 2 ∗ ∗ 2 ∗ ∗ ¯ ¯ ¯ ¯ (25) − ξ Φ − a2 θ + ε 1 − ξ φ + ε2 C = 0, dz 2 dz 2     4 2 d2 d 2 ∗ ∗ 2 d 4 ¯ α1 2 − (α1 ξ + α3 a2 ) C − − 2ξ + ξ φ¯∗ dz dz 4 dz 2  2  d 2 −α2 − ξ θ¯∗ = 0, (26) dz 2   d2 2 ¯ ∗ + θ¯∗ = 0, α4 2 − (α4 ξ + 1) Φ (27) dz  2  d 2 ¯∗ − ξ φ + P¯ ∗ − α1 C¯ ∗ + α2 θ¯∗ = 0, (28) 2 dz       s2 τ0α sα τ α sα 2 2 ∗ where λ4 = ξ + 2 , a2 = s 1 + , a2 = s 1 + . β − S/2 Γ(α + 1) Γ(α + 1) ¯ ∗ , θ¯∗ , φ¯∗ and C¯ ∗ . It can be seen that Eqs. (24)-(27) are coupled in Φ By applying elimination procedure, we can achieve the following six order differential equation 

where Q =

  ∗ ∗ ∗ ∗ d4 d2 d6 ¯ , C¯ , θ¯ = 0, + Q + N + I φ¯ , Φ dz 6 dz 4 dz 2

 1 1  (F − 3Eξ 2 ), N = G − 2F ξ 2 + 3Eξ 4 , E E

10

(29)

I= and

 1  H − Gξ 2 + F ξ 4 − Eξ 6 , E

E = a2 α4 (α1 + α2 )(1 + 2 ) + [a2 α4 (1 − ε2 ) + 1] (α1 − 1), F = −[a2 (α1 + α2 ){α4 ε2 s2 + ε1 + ε2 }] + a2 a∗2 α3 α4 (ε1 + ε2 ) +(a2 α4 (1 − ε2 ) + 1)(α1 s2 + α3 + a∗2 ) + a2 (α1 − 1)(1 − ε2 ), G = a2 a∗2 α3 (α4 ε2 s2 + ε1 + ε2 ) + α3 a∗2 s2 (a2 α4 (1 − ε2 ) + 1) +a2 (1 − ε2 )(α1 s2 + α3 a∗2 ), H = −a2 α3 a∗2 s2 (1 − ε2 ). Now, Eq. (29) can be factorized as  2  2   2 d d d 2 2 2 ¯ ∗ , φ¯∗ , θ¯∗ , C¯ ∗ ) = 0, − λ1 − λ2 − λ3 (Φ 2 2 2 dz dz dz

(30)

where λ1 , λ2 and λ3 are the roots with positive real part of the characteristic equation λ6 + Qλ4 + N λ2 + I = 0,

(31)

which after solved by the Cardan’s method, gives

 √ 1 1 (2l sin m − Q), λ2 = −Q − l( 3 cos m + sin m) , λ1 = 3 3

 √ 1 λ3 = −Q + l( 3 cos m − sin m) , 3 2Q3 − 9QN + 27I sin−1 n ,n= where l = Q2 − 3N , m = . 3 2l3 The solutions of Eqs. (23) and (30) can be expressed in terms of characteristic values λ1 , λ2 , λ3 and λ4 . Assuming the regularity condition as in Eq. (9) that all the fields are bounded at infinity (Sharma et al. [5]), we can obtain φ¯∗ = A1 e−λ1 z + A2 e−λ2 z + A3 e−λ3 z , ¯ ∗ = B1 e−λ1 z + B2 e−λ2 z + B3 e−λ3 z , Φ C¯ ∗ = C1 e−λ1 z + C2 e−λ2 z + C3 e−λ3 z , θ¯∗ = D1 e−λ1 z + D2 e−λ2 z + D3 e−λ3 z , ψ¯∗ = A4 e−λ4 z , 11

(32) (33) (34) (35) (36)

where Ai , Bi , Ci , Di (i = 1, 2, 3) and A4 are the cofficients, dependent on the variables s and ξ. Now, compatibility among the Eqs. (32)-(35) along with the Eqs. (24)(27) allows us to deduce the following relations: Bi = bi Ai , Ci = ci Ai and Di = di Ai (i = 1, 2, 3), where bi = − di =

α4 λ2i e1 λ4i −

di , ci = λ2i − di − (ξ 2 + s2 ), − (α4 ξ 2 + 1) [2e1 ξ 2 + e2 ]λ2i + e1 ξ 4 + e2 ξ 2 + α3 a∗2 s2 , e3 λ2i − [e3 ξ 2 + α3 a∗2 ]

e1 = α1 − 1, e2 = α1 s2 + α3 a∗2 , e3 = α1 + α2 . 5. Application: Heat source We have considered a fractional order two-temperature generalized thermoelastic half-space with diffusion and initial stress. The bounding plane to the surface z = 0 is subjected to a time dependent heat source. Therefore, at the surface of the half-space, the sum of normal stress and initial stress must be equal to zero and shear stress and diffusion concentration must also vanish. Mathematically, the boundary conditions of the problem can be expressed as: σzz (r, z, t) + S σzr (r, z, t) Φ(r, z, t) C

= = = =

0, 0, f (r, t), 0.

(37) (38) (39) (40)

The function representing the applied heat source is assumed as: f (r, t) =

F0 δ(r)eιωt , 2πr

(41)

where δ(.) is the Dirac-delta function and F0 is a constant representing the strength of the source. In the Laplace-Hankel transform domain, Eq. (41) can be expressed as f¯∗ (ξ, s) =

F0 . 2π(s + ιω) 12

(42)

Applying Laplace and Hankel transforms to Eqs. (37)-(40) and using Eqs. (12), (13), (32)-(36) and (42) in the resulting equations, we arrive at the following set of linear equations b1 A1 + b2 A2 + b3 A3 = P ∗ , Q1 A1 + Q2 A2 + Q3 A3 + Q4 A4 = 0, R1 A1 + R2 A2 + R3 A3 + R4 A4 = 0, c1 A1 + c2 A2 + c3 A3 = 0,

(43) (44) (45) (46)

where Qi = λ2i − (1 − 2β 2 )ξ 2 − ci − di , Ri = −2β 2 λi , (i = 1, 2, 3), Q4 = −2β 2 ξ 2 λ4 , R4 = (β 2 − S/2)λ24 + (β 2 + S/2)ξ 2 , P ∗ =

F0 . 2π(s + ιω)

Solution of system of linear Eqs. (43)-(46) is given by A1 =

Δ1 Δ2 Δ3 Δ4 , A2 = , A3 = , A4 = , Δ Δ Δ Δ

(47)

where Δ = b 1 Δ 1 + b2 Δ 2 + b3 Δ 3 , Δ1 = P ∗ [−Q2 R4 c3 + Q3 R4 c2 + Q4 (R2 c3 − R3 c2 )] , Δ2 = −P ∗ [−Q1 R4 c3 + Q3 R4 c1 + Q4 (R1 c3 − R3 c1 )] , Δ3 = P ∗ [−Q1 R4 c2 + Q2 R4 c1 + Q4 (R1 c2 − R2 c1 )] , Δ4 = −P ∗ [Q1 (R2 c3 − R3 c2 ) − Q2 (R1 c3 − R3 c1 ) + Q3 (R1 c2 − R2 c1 )] . Now, using the results obtained in Eq. (47) along with the Eqs. (11)-(13), (19) and (32)-(36) simultaneously, we obtain the expressions of thermoelastic and diffusion fields in the Laplace-Hankel transform domain as  1  Δ1 e−λ1 z + Δ2 e−λ2 z + Δ3 e−λ3 z − λ4 Δ4 e−λ4 z , Δ   1 ¯∗ = b1 Δ1 e−λ1 z + b2 Δ2 e−λ2 z + b3 Δ3 e−λ3 z , Φ Δ   1 θ¯∗ = d1 Δ1 e−λ1 z + d2 Δ2 e−λ2 z + d3 Δ3 e−λ3 z , Δ  1  c1 Δ1 e−λ1 z + c2 Δ2 e−λ2 z + c3 Δ3 e−λ3 z , C¯ ∗ = Δ  1  E1 Δ1 e−λ1 z + E2 Δ2 e−λ2 z + E3 Δ3 e−λ3 z , P¯ ∗ = Δ u¯∗ = −ξ

13

(48) (49) (50) (51) (52)

 1   S1 Δ1 e−λ1 z + S2 Δ2 e−λ2 z + S3 Δ3 e−λ3 z + S4 Δ4 e−λ4 z − S/s, (53) Δ  1   (54) S1 Δ1 e−λ1 z + S2 Δ2 e−λ2 z + S3 Δ3 e−λ3 z + S4 Δ4 e−λ4 z , = Δ

∗ σ ¯zz = ∗ σ ¯zr

where Ei = −(λ2i − ξ 2 ) + α1 ci − α2 di , Si = λ2i − (1 − 2β 2 )ξ 2 − ci − di , Si = 2β 2 ξλi , S4 = −2β 2 ξλ4 , S4 = − [(β 2 − S/2) λ24 + (β 2 + S/2) ξ 2 ], (i = 1, 2, 3). 6. Special cases 6.1. Neglecting the diffusion effect: If the diffusion effect is removed from the thermoelastic medium, then the problem will reduce to axi-symmetric fractional order two-temperature generalized thermoelasticity with initial stress. In this special case, the diffusive wave will disappear from the medium. Hence, by modifying Eqs. (12)-(19), we can obtain the governing equations corresponding to the relevant problem as:    ∂ψ ∂2 2 2 2 φ+ − θ − S, (55) σzz = ∇ φ − 2β ∇ − 2 ∂z ∂z     2  ∂2ψ  2  ∂  2 ∂2 2 ∂ φ 2 + σzr = 2β − β + S/2 ∇ − 2 ψ ,(56) β − S/2 ∂r∂z ∂r ∂z 2 ∂z  2  ∂ ψ 1 ∇2 ψ = , (57) 2 β − S/2 ∂t2 ∂ 2φ (58) ∇2 φ − θ = 2 , ∂t    τ0 ∂ α+1  ∂ 2 2 + ∇ φ , (59) ∇ Φ= θ + ε 1 ∂t Γ(α + 1) ∂tα+1 α4 ∇2 Φ − Φ + θ = 0. (60) Solving the above system of Eqs. (58)-(60) in Laplace-Hankel transform domain simultaneously, we obtain a fourth-order differential equation satis¯ ∗ and θ¯∗ as fied by φ¯∗ , Φ   4 2  ∗ ∗ ∗ ∗ d ∗ d ∗ ¯ , θ¯ = 0, φ¯ , Φ +B +C A (61) 4 2 dz dz 14

where A∗ = 1+a2 α4 a3 , B ∗ = − [2ξ 2 + s2 + a2 {(α4 a3 ξ 2 + s2 ) + a3 (α4 ξ 2 + 1)}], C ∗ = ξ 2 (ξ 2 + s2 ) + a2 (α4 a3 ξ 2 + s2 )(a3 ξ 2 + s2 ), a3 = 1 + ε1 . The solution of Eqs. (57) and (61) can be expressed as (under the assumption of the regularity condition that all the fields are bounded at infinity)   (62) φ¯∗ = A∗1 e−λ1 z + A∗2 e−λ2 z , ∗ ∗ −λ1 z ∗ −λ2 z ¯ + B2 e , (63) Φ = B1 e z z ∗ ∗ −λ ∗ −λ (64) θ¯ = D1 e 1 + D2 e 2 , ∗ ∗ −λ4 z ¯ ψ = A3 e , (65) ∗ −B ± B ∗ 2 − 4A∗ C ∗ where Bi∗ = ei A∗i , Di∗ = ei A∗i , λi = , (i = 1, 2), 2A∗

  a2 ei + ε1 (λi 2 − ξ 2 ) ei = , ei = λi 2 − (ξ 2 + s2 ). 2 2 λi − ξ In the absence of diffusion field, the boundary conditions for the fractional order two-temperature generalized thermoelastic half-space with initial stress are given by Eqs. (37)-(39). Now, following the similar procedure as described in previous section, we obtain a system of linear equations as

e1 A∗1 + e2 A∗2 = P ∗ , Q∗1 A∗1 + Q∗2 A∗2 + Q∗3 A∗3 = 0, R1∗ A∗1 + R2∗ A∗2 + R3∗ A∗3 = 0,

(66) (67) (68)

where Q∗i = λi 2 − (1 − 2β 2 )ξ 2 − ei , Ri∗ = −2β 2 λi , (i = 1, 2), Q∗3 = −2β 2 ξ 2 λ4 , R3∗ = (β 2 − S/2) λ24 + (β 2 + S/2) ξ 2 . The solution of the above system of equations leads to the following expressions A∗1 =

Δ∗1 Δ∗2 Δ∗3 ∗ ∗ , A = , A = , 2 3 Δ∗ Δ∗ Δ∗

(69)

where Δ∗1 = P ∗ (Q∗2 R3∗ − R2∗ Q∗3 ) , Δ∗2 = P ∗ (Q∗3 R1∗ − R3∗ Q∗1 ) , Δ∗3 = P ∗ (Q∗1 R2∗ − R1∗ Q∗2 ) , Δ∗ = e1 (Q∗2 R3∗ − R2∗ Q∗3 ) − e2 (Q∗1 R3∗ − R1∗ Q∗3 ). Finally, the requisite expressions of displacement component u, conductive temperature Φ, thermodynamical temperature θ, normal stress σzz and

15

shear stress σzr are obtained in Laplace-Hankel transform domain as  1 ∗ −λ1 z ∗ ∗ −λ2 z ∗ −λ4 z + Δ2 e − λ 4 Δ3 e , u¯ = −ξ ∗ Δ1 e Δ

 ¯ ∗ = 1 e1 Δ∗1 e−λ1 z + e2 Δ∗2 e−λ2 z , Φ Δ∗  1  ∗ −λ1 z ∗  ∗ −λ2 z ¯ θ = ∗ e1 Δ1 e + e2 Δ 2 e , Δ

 1 ∗ ∗ ∗ −λ1 z ∗ ∗ −λ2 z ∗ ∗ −λ4 z σ ¯zz = ∗ S1 Δ1 e + S2 Δ2 e + S3 Δ3 e − S/s, Δ

 1 ∗ ∗∗ ∗ −λ1 z ∗∗ ∗ −λ2 z ∗∗ ∗ −λ4 z + S2 Δ2 e + S3 Δ3 e . σ ¯zr = ∗ S1 Δ1 e Δ

(70) (71) (72) (73) (74)

where Si∗ = λi 2 − (1 − 2β 2 )ξ 2 − ei , Si∗∗ = 2β 2 ξλi , (i = 1, , 2), S3∗ = −2β 2 ξ 2 λ4 . S3∗∗ = − [(β 2 − S/2) λ24 + (β 2 + S/2) ξ 2 ]. 6.2. Neglecting fractional effect: The wave propagation phenomena for two-temperature generalized theory of thermoelasticity with diffusion and initial stress can be obtained if we remove the fractional order parameter α appearing in the heat conduction equation and mass diffusion equations. In addition, if we omit the twotemperature and initial stress effects from the medium (i.e. a∗ = 0, S = 0), then the analytical solutions obtained for the resulting formulation coincide with those obtained by Sherief and Hussein [9] with appropriate changes in the boundary conditions. 6.3. Neglecting two-temperature and initial stress effects: Under the assumption that the generalized thermoelastic medium is without initial stress and the heat conduction is independent of two-temperature, the problem is reduced to the fractional order generalized thermoelasticity with diffusion for a homogeneous isotropic thermoelastic medium. In this special case, we shall substitute a∗ = 0, S = 0 and omit conductive temperature Φ from the field equations and disregard the two-temperature relation. If we further neglect the diffusion effect from the medium, then the results of the relevant problem coincide with Mashat et al. [35].

16

7. Inversion of the integral transforms We shall now outline the numerical inversion method used to find the solution in the physical domain. The fields in the Laplace-Hankel transform domain are the functions of the form f¯∗ (ξ, z, s). To find the functions in the form f (r, z, t), we first invert the Hankel transform by using

H

−1

[f¯∗ (ξ, z, s)] = f¯(r, z, s) =



∞

0

ξ f¯∗ (ξ, z, s)J0 (rξ)dξ.

(75)

Now, for fixed values of ξ, r and z, the function f¯(r, z, s) in Eq. (75) can be considered as the Laplace transform f¯(s) of the function f (t). In order to find Laplace inverse transform of the function f¯(s) to get the function f (t), we apply a numerical inversion method based on Fourier series expansion explained by Honig and Hirdes [36]. The integral of Eq. (75) is evaluated by following the procedure given in Rakshit and Mukhopadhyay [37]. 8. Numerical results and discussions In recent years, the thermal diffusion method has been successfully used in the production of powder metallurgy and in improving the mechanical properties. The purpose of the present study is to promote wide applications of the thermoelastic diffusion process. To understand the interaction phenomena, we have evaluated the numerical results of the non-dimensional displacement component u, conductive temperature Φ, thermodynamical temperature θ, concentration C, chemical potential P and normal stress σzz and displayed graphically along z−axis. For numerical computation, we consider material properties of copper metal, whose physical data is given below k = 386 W m−1 K −1 , T0 = 293 K, CE = 383.1 J kg −1 K −1 , τ = 0.2 s, αt = 1.78 × 10−5 K −1 , αc = 1.98 × 10−4 kg −1 m3 , ρ = 8954 kg m−3 , b = 0.9 × 106 m5 kg −1 s−2 , D = 0.85 × 10−8 kg s m−3 , τ0 = 0.02 s, E = 36.9 × 1010 kg m−1 s−2 , a = 1.2 × 104 m2 s−2 K −1 , α4 = 0.1. ¯ and μ The generalized Lame’s constants λ ¯ are given as ¯= λ

Ev , η(1 + v)(1 − 2v) 17

μ ¯=

E , 2η(1 + v)

where η is initial stress parameter, E is Young’s modulus and v is Poisson ratio. The numerical data for other parameters related to the initially stressed medium are given as: η = 2.5, ν = 0.33. and S = 1. For isotropic elastic medium with no initial stress, we take η = 1 and S = 0. Also, we have taken ω = 1.0, F0 = 10 for computation purpose. We analyze the effect of fractional parameter α, diffusion, time t, initial stress and two-temperature on the fields, by dividing the graphical representations into three categories. In Category-I (Figures: 1-6), the profiles of different fields are illustrated to depict the effect of fractional order parameter α for values 0.1, 0.5, 1.0 at t = 0.1 and r = 1.0. In Category-II (Figures: 7-12), the responses of the field variables are analyzed to show the effect of diffusion for two different times (t = 0.1, 0.2) at α = 0.5 and r = 1.0. In Category-III (Figures: 13-18), we have investigated the dynamical response of the field variables at α = 0.5, t = 0.1 and r = 1.0 for the cases: (i) Fractional order two-temperature generalized thermoelsticity with diffusion and initial stress (FTTDIS), (ii) Fractional order two-temperature generalized thermoelsticity with diffusion (FTTD), (iii) Fractional order generalized thermoelsticity with diffusion and initial stress (FDIS). 8.1. Category-I Figure 1 is plotted to analyze the transient phenomenon of displacement component u for three different values of fractional order parameter α (0.1, 0.5, 1.0). It can be observed that the displacement field undergoes expansion as well as compression deformation, which is due to the fact that as thermal shock is exposed, the parts of the half-space near the boundary surface expand towards the unconstrained direction yielding negative displacement initially and then shift from negative to positive gradually, thereafter reach to the steady state and finally become zero at the heat wave front. Moreover, the fractional parameter α shows its dominance effectively on the displacement field. The dynamic interactions of the conductive temperature Φ and the thermodynamical temperature θ are presented in Figures 2 and 3 by considering three different values of fractional order parameter α (0.1, 0.5, 1.0). As seen from these figures, the temperature fields attain maximum positive values at the point of the application of thermal source, then decrease continuously to reach to the steady state and finally diminish to zero at the thermal wave front. Physically, this phenomenon can be interpreted as: when the thermal 18

source acts on the surface of the half-space, it increases intrinsic energy of particles in contact with the bounding surface which causes to increase the temperature initially. On the other hand, thermal energy propagates in to the medium due to temperature gradient in the form of heat conduction. Moreover, fractional order parameter α has prominent effect on these fields. Figure 4 is drawn to analyze the variations of the concentration C for different values of fractional parameter α (0.1, 0.5, 1.0). As illustrated, concentration field follows similar pattern for all the values and is also consistent with the boundary condition assumed (i.e. zero initial value). In addition, for all the cases the propagation of the diffusive wave for concentration field is restricted to a bounded region and this region enlarges with increase in the value of the fractional order parameter α which ensures that the diffusive wave for concentration field predicts finite speed. In Figure 5, the diffusion wave phenomenon for chemical potential P is depicted for different values of fractional parameter α taken as 0.1, 0.5, 1.0. The chemical potential P begins with maximum positive value and decays gradually to reach to the steady state and finally rests to zero at diffusive wave front. Moreover, increase in the values of fractional parameter α acts to increase chemical potential field significantly. Figure 6 describes the variations of the normal stress σzz for three different values of fractional parameter α considered as 0.1, 0.5, 1.0. As we have noticed, the stress field occupies some initial positive value at the boundary of half-space, suddenly jumps upward to attain stationary point, then decreases to a minimum value and finally reaches to the steady state with a constant value. This kind of phenomenon has happened due to the strength of initial stress present in the medium. It is also clear from the figure that the fractional order parameter α significantly affects the stress field. 8.2. Category-II It is clear from Figure 7 that the curves of the displacement field u corresponding to different times (t = 0.1, 0.2) with and without diffusion, follow similar pattern and experience both expansion and compression deformations. Moreover, time t has decreasing effect on the displacement field u while diffusion shows both increasing as well as decreasing effects on this field. From Figures 8 and 9, it is evident that the variations of conductive temperature Φ and thermodynamical temperature θ begin with positive values, then decrease gradually and finally diminish at the thermal wave front. We 19

have also observed that time t increases the values of both the fields. But, in the absence of diffusion, the conductive temperature increases while thermodynamical temperature decreases. Figures 10 and 11 show the variations of concentration C and chemical potential P for diffenent values of time t (0.1, 0.2) with and without diffusion. As can be seen from these figures, the waves corresponding to concentration C and chemical potential P disappear from the medium in the absence of diffusion, which is a realistic phenomenon. However, time t causes to decrease the values of these fields significantly. Moreover, the distributions of these fields have non-zero values only in a bounded region of the half-space. Outside this region, the values vanish identically. Figure 12 displays the variations of stress field σzz for different values of time t (0.1, 0.2) with and without diffusion. It is observed that the curves corresponding to different times follow almost similar pattern with difference in magnitudes. Figure indicates that the effects of time t and diffusion are soundly pronounced. Although, the values of stress field decrease with time t and increase in the absence of diffusion. 8.3. Category-III Figure 13 shows that the profiles of displacement field u for FTTDIS, FTTD and FDIS begin with negative values, which is due to the fact that as the surface of the half space (z = 0) is exposed to thermal source then it undergoes thermal expansion deformation and move towards the unconstrained direction. With the passage of time, the expansion part enlarges and moves inside the medium dynamically, causing the negative to positive region transform. In addition, the effect of two-temperature and initial stress is quite significant on the displacement field. Figure 14 reveals that the nature of conductive temperature Φ for FTTDIS, FTTD and FDIS is similar with coincident initial point with maximum positive value 0.2566 at the boundary z = 0 of half-space, which is in complete justification to the heat source applied. Furthermore, the absence of initial stress causes to increase the values of Φ, while Φ decreases in the absence of two-temperature. Figure 15 represents the plots of thermodynamical temperature θ for FTTDIS, FTTD and FDIS. It is apparent from the figure that θ begins with positive values 0.209, 0.2297, 0.2566 for the cases for FTTDIS, FTTD and FDIS respectively, afterward decreases gradually with increase in distance z until reaches to the steady state. The difference of the distribution 20

is much prominent about the close proximity of the source and it becomes indistinct along the passage of time. However, the absence of initial stress and two-temperature causes to push the vibrations in upward direction. It is evident from Figure 16 that the profiles of concentration field C for FTTDIS, FTTD and FDIS have coincident initial point with zero value, which is consistent with the boundary conditions applied, afterward jump to attain maximum values and finally decrease gradually to diminish to zero value at the diffusive wave front. It is also concluded that all the variations show similar pattern and as the medium is made free from initial stress and two-temperature, the vibrations of concentration field increase consequently. Moreover, the phenomena of finite wave speed is also predicted by this field. The effect of initial stress and two-temperature is analyzed on the distribution of the chemical potential P in Figure 17 and it is found that initial stress is very much dominant on this field than two-temperature (however significant). Also, the patterns of the variations of P for FTTDIS, FTTD and FDIS are alike and restricted to a bounded region, which make us sure to conclude that the wave phenomena with second sound has occurred for all the cases. Figure 18 represents the variations of normal stress σzz for FTTDIS, FTTD and FDIS. The figure reveals that the profiles of stress distribution σzz for FTTDIS and FDIS are tensile in nature, while for FTTD, it changes from tensile to compressive. Also, the variations of stress field for FTTDIS and FDIS have large difference in magnitude to that of FTTD, which is due to the presence of initial stress in the medium for FTTDIS and FDIS. However, the effect of two-temperature on this field is quite observable. 9. Conclusions In this paper, we have presented an axi-symmetric model of generalized thermodiffusion with two temperature and initial stress based on the methodology of the fractional calculus. The proposed theoretical framework has been applied to a time harmonic thermal source and the results are presented. The fractional order theory of thermoelasticity discussed within the article is non-local, thus, all fields examined (displacement, temperature, diffusion fields and stress) at the specific point of interest depend on the information from its surrounding - contrary to generalized thermoelasticity theories with integer order in which the results are local. From the analysis of the illustrations, we can arrive at the following conclusions: 21

• The variations of the thermoelastic and diffusive fields in the fractional order generalized thermodiffusion medium with two temperature and initial stress are restricted to a limited region and outside this region, these variations rest to zero on the thermal and diffusive wave fronts accordingly (except for normal stress field σzz in Figures 6, 12 and 18 for FTTDIS and FDIS, which is due to the presence of initial stress in the medium). • The fractional order theory of thermoelasticity presented in this paper describes the behaviour of particles of an elastic body more realistically then the theory of generalized thermoelasticity with integer order. • The fractional order parameter α shows significant dominance on all the fields. However, the magnitude of all the fields increases according to increase in the values of fractional parameter α except for displacenent component u and stress distribution σzz , for which it shows both increasing and decreasing effects. • All the physical fields show qualitatively similar pattern for different values of time t with a significant difference in magnitude. Although, the numerical values of all the fields decrease with increase in the values of time t. • The effect of the initial stress is found to be very much prominent on the thermal and diffusive fields. It is observed that, initial stress lessens the magnitude of the fields u, P and σzz , while it acts to increase the the values of temperature fields and concentration field. • All the fields show significant sensitivity towards the two temperature parameter. In the limiting case a∗ → 0, we get Φ → θ. We have also obtained this fact numerically as well as graphically. • Diffusion plays an important role in the variations of all the field quantities. In the absence of diffusion, the distributions of conductive temperature Φ and stress σzz increase significantly and thermodynamical temperature θ decreases. However, displacement component u is experiencing both increasing and decreasing effects of diffusion. • From the profiles of temperature distribution, it is observed that the heat wave front moves forward with a finite speed in the medium with 22

the passage of time which proves that the the fractional order generalized thermodiffusive theory is very close to the behaviour of the elastic materials. Theory of thermoelasticity with fractional order time derivatives is a new branch of research. In literature, there are a few number of investigations with fractional order thermoelasticity. The results presented in this paper should prove useful for researchers in material science, designers of new materials, low temperature physicists, as well as for those working on the development of a theory of hyperbolic thermoelasticity with fractional order. The introduction of fractional order time derivatives, diffusion, two temperature and initial stress to the axi-symmetric generalized thermoelastic half-space provides a more realistic model for these studies. 10. Acknowledgement One of the authors, Sandeep Singh Sheoran, is thankful to University Grants Commission, New Delhi, for the financial support as SRF Vide Letter no. F.17-11/2008 (SA-1). [1] W. Nowacki, Dynamical problems of thermoelastic diffusion in solids-I, Bull. L’Aca. Pol. Sci. 22 (1974) 55-64. [2] W. Nowacki, Dynamical problems of thermoelastic diffusion in solidsII, Bull. L’Aca. Pol. Sci. 22 (1974) 129-135. [3] W. Nowacki, Dynamical problems of thermoelastic diffusion in solidsIII, Bull. L’Aca. Pol. Sci. 22 (1974) 257-266. [4] H. Sherief, F. Hamza, H. Saleh, The theory of generalized thermoelastic diffusion, Int. J. Eng. Sci. 42 (2004) 591-608. [5] N. Sharma, R. Kumar, P. Ram, Plane strain deformation in generalized thermoelastic diffusion, Int. J. Thermophys. 29 (2009) 1503-1522. [6] R. Xia, X. Tian, Y. Xen, The influence of diffusion on generalized thermoelastic problems of infinite body with a cylindrical cavity, Int. J. Eng. Sci. 47 (2009) 669-679.

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[7] S. Deswal, S. Choudhary, Thermally induced vibrations in a generalized thermoelastic medium with diffusion, J. Vib. Cont. 17 (2011) 16011610. [8] S. Deswal, K. Kalkal, A two-dimensional generalized electro-magentothermoviscoelastic problem for a half-space with diffusion, Int. J. Therm. Sci. 50 (2011) 749-759. [9] H. Sherief, E.M. Hussein, Two-dimensional problem for a thick plate with axi-symmetric distribution in the theory of generalized thermoelastic diffusion, Math. Mech. Solid. (2014) doi: 10.1177/1081286514524759. [10] M.E. Gurtin, W.O. Williams, On the Clausius-Duhem inequality, ZAMP 17 (1966) 626-633. [11] M.E. Gurtin, W.O. Williams, An axiom foundation for continuum thermodynamics, Arch. Rat. Mech. Anal. 26 (1967) 83-117. [12] P.J. Chen, M.E. Gurtin, On a theory of heat conduction involving two temperatures, ZAMP 19 (1968) 614-627. [13] P.J. Chen, M.E. Gurtin, W.O. Williams, On the thermodynamics of non-simple elastic materials with two temperatures, ZAMP 20 (1969) 107-112. [14] R. Quintanilla, On existence, structural stability, convergence and spatial behaviour in thermoelasticity with two temperatures, Acta. Mech. 168 (2004) 61-73. [15] H.M. Youssef, Theory of two-temperature generalized thermoelasticity, IMA J. Appl. Math. 71 (2006) 383-390. [16] A. Magana, R. Quintanilla, Uniqueness and growth of solutions in twotemperature generalized thermoelastic theories, Math. Mech. Solid. 14 (2009) 622-634. [17] R. Kumar, S. Mukhopadhyay, Effect of the relaxation time on plane wave propagation under two-temperature thermoelasticity, Int. J. Eng. Sci. 48 (2010) 128-139.

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[18] R. Kumar, A. Kumar, S. Mukhopadhyay, An investigation on thermoelastic interactions under two-temperature thermoelasticity with two relaxation parameters, Math. Mech. Solid. (2014) doi:10.1177/1081286514536429. [19] M. Biot, Mechanics of Incremental Deformation, Wiley, New York, 1965. [20] R.W. Ogden, Non-Linear Elastic Deformations, Dover Publications, Inc., New York, 1984. [21] A. Montanaro, On singular surface in isotropic linear thermoelasticity with initial stress, J. Acous. Soc. Am. 106 (1999) 1586-1588. [22] M.I.A. Othman, Y. Atwa, Thermoelastic plane waves for an elastic solid half-space under hydrostatic initial stress of type III, Meccanica 47 (2012) 1337-1347. [23] S. Deswal, S.S. Sheoran, K.K. Kalkal, The effect of magnetic field and initial stress on fractional order generalized thermoelastic half-space, J. Math. 489863 (2013) 1-11. [24] B. Ross, The development of fractional calculus 1695-1900, His. Math. 4 (1977) 75-89. [25] K.S. Miller, B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equation, Wiley, New York, 1993. [26] Y.A. Rossikhin, M.V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results, Appl. Mech. Rev. 63 (2010) (010801) 1-52. [27] Y.Z. Povstenko, Fractional heat conduction equation and associated thermal stress, J. Therm. Stress. 28 (2004) 83-102. [28] H. Sherief, A.M.A. El-Sayed, A.M.A. El-Latief, Fractional order theory of thermoelasticity, Int. J. Solid. Struct. 47 (2010) 269-275. [29] H.M. Youssef, Theory of fractional order generalized thermoelasticity, J. Heat. Trans. 132 (2010) (061301) 1-7.

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[30] M.A. Ezzat, Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer, Phys. B 405 (2010) 4188-4194. [31] G. Jumarie, Derivation and solutions of some fractional Black-Scholes equations in coarse-grained space and time. Application to Merton’s optimal portfolio, Comp. Math. Appl. 59 (2010) 1142-1164. [32] A.S. El-Karamany, M.A. Ezzat, Convolutional variational principle, reciprocal and uniqueness theorems in linear fractional two-temperature thermoelasticity, J. Therm. Stress. 34 (2011) 264-284. [33] M.A. Ezzat, M.A. Fayik, Fractional order theory of thermoelastic diffusion, J. Therm. Stress. 34 (2011) 851-872. [34] M.A. Ezzat, A.S. El-Karamany, Fractional order heat conduction law in magneto-thermoelasticity involving two temperatures, ZAMP 62 (2011) 937-952. [35] D.S. Mashat, A.M. Zenkour, A.E. Abouelregal, Fractional order thermoelasticity theory for a half-space subjected to an axisymmetric heat distribution, Mech. Adv. Mat. Struct. 22 (2015) 925-932. [36] G. Honig, U. Hirdes, A method for the numerical inversion of Laplace transforms, J. Comp. Appl. Math. 10 (1984) 113-132. [37] M. Rakshit, B. Mukhopadhyay, A two dimensional thermoviscoelastic problem due to instantaneous point heat source, Math. Comp. Model. 46 (2007) 1388-1397.

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Figure Captions: Figure 1: Effect of fractional parameter α on the distribution of displacement component u at t = 0.1. Figure 2: Effect of fractional parameter α on the distribution of conductive temperature Φ at t = 0.1. Figure 3: Effect of fractional parameter α on the distribution of thermodynamical temperature θ at t = 0.1. Figure 4: Effect of fractional parameter α on the distribution of concentration C at t = 0.1. Figure 5: Effect of fractional parameter α on the distribution of chemical potential P at t = 0.1. Figure 6: Effect of fractional parameter α on the distribution of stress σzz at t = 0.1. Figure 7: Effect of diffusion on the variations of displacement field u for different times (t=0.1, 0.2) at α = 0.5. Figure 8: Effect of diffusion on the variations of conductive temperature Φ for different times (t=0.1, 0.2) at α = 0.5. Figure 9: Effect of diffusion on the variations of thermodynamical temperature θ for different times (t=0.1, 0.2) at α = 0.5. Figure 10: Effect of diffusion on the variations of concentration C for different times (t=0.1, 0.2) at α = 0.5. Figure 11: Effect of diffusion on the variations of chemical potential P for different times (t=0.1, 0.2) at α = 0.5. Figure 12: Effect of diffusion on the variations of stress σzz for different times (t=0.1, 0.2) at α = 0.5.

27

Figure 13: Dependence of displacement component u on two-temperature and initial stress at t = 0.1 and α = 0.5. Figure 14: Dependence of conductive temperature Φ on two-temperature and initial stress at t = 0.1 and α = 0.5. Figure 15: Dependence of thermodynamical temperature θ on two-temperature and initial stress at t = 0.1 and α = 0.5. Figure 16: Dependence of concentration C on two-temperature and initial stress at t = 0.1 and α = 0.5. Figure 17: Dependence of chemical potential P on two-temperature and initial stress at t = 0.1 and α = 0.5. Figure 18: Dependence of stress σzz on two-temperature and initial stress at t = 0.1 and α = 0.5.

28




Displacement component u

0.02



     

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Distance z

-0.02

-0.04

-0.06

-0.08

-0.1

-0.12



Figure 1: 0.26

 

0.24

 

0.22

Conductive temperature




 

0.2

0.18 0.16 0.14 0.12 0.1

0.08 0.06 0.04 0.02

0 0

0.1

0.2

0.3

0.4

0.5

0.6

Distance z Figure 2:

29

0.7

0.8

0.9

1 

0.24

0.22

Thermodynamical temperature



0.2
















0.18 0.16 0.14 0.12

0.1 0.08 0.06 0.04 0.02 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance z



Figure 3: 0.024






  

Concentration C

0.02

  

0.016

0.012

0.008

0.004

0

0

0.4

0.8

1.2

1.6

2

2.4

Distance z Figure 4:

30

2.8

3.2

3.6

4

4.4 

3.5

Chemical Potential P

3
















2.5

2

1.5

1

0.5

0 0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

Distance z



Figure 5: 0.8

    

0.7

zz

 

Stress distribution




0.6

0.5

0.4

0.3

0.2

0.1

0 0

0.4

0.8

1.2

1.6

2

Distance z

Figure 6:

31

2.4

2.8

3.2

3.6

4 

Displacement component u

0.03

0.01

-0.01

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

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1.4

1.5

Distance z With Diffusion (t = 0.1)

-0.03

With diffusion (t = 0.2) Without diffusion (t = 0.1)

-0.05

Without diffusion (t = 0.2) -0.07

-0.09

-0.11



Figure 7: 0.26 With Diffusion (t = 0.1)

0.24

With Diffusion (t = 0.2)

0.22

Without Diffusion (t = 0.1)

0.2

Conductive temperature




Without Diffusion (t = 0.2)

0.18 0.16 0.14 0.12 0.1 0.08 0.06

0.04 0.02 0 0

0.1

0.2

0.3

0.4

0.5

Distance z Figure 8:

32

0.6

0.7

0.8

0.9

1 

0.22 With Diffusion (t = 0.1) 0.2

With Diffusion (t = 0.2)

Thermodynamical temperature



0.18

Without Diffusion (t = 0.1)

0.16

Without Diffusion (t = 0.2)

0.14 0.12 0.1 0.08

0.06 0.04 0.02 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance z



Figure 9: 0.024 With Diffusion (t = 0.1) With Diffusion (t = 0.2)

Concentration C

0.02

0.016

0.012

0.008

0.004

0

0

0.4

0.8

1.2

1.6

2

2.4

Distance z Figure 10:

33

2.8

3.2

3.6

4 

3.5

With Diffusion (t = 0.1)

Chemical potential P

3

With Diffusion (t = 0.2)

2.5

2

1.5

1

0.5

0

0

0.4

0.8

1.2

1.6

2

2.4

Distance z

2.8

3.2

3.6

4 

Figure 11: 0.8

With Diffusion (t = 0.1) 0.7

With Diffusion (t = 0.2)

zz

Without Diffusion (t = 0.1) 0.6

Without Diffusion (t = 0.2)

Stress distribution




0.5

0.4

0.3

0.2

0.1

0 0

0.4

0.8

1.2

1.6

2

Distance z Figure 12:

34

2.4

2.8

3.2

3.6

4



0.03

Displacement component u

0.01 0

-0.01

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Distance z FTTDIS

-0.03

FTTD

-0.05

FDIS

-0.07

-0.09

-0.11

-0.13 

Figure 13: 0.26

FTTDIS

0.24

FTTD

0.22

Conductive temperature




FDIS

0.2 0.18 0.16 0.14 0.12 0.1 0.08

0.06 0.04 0.02 0 0

0.1

0.2

0.3

0.4

0.5

0.6

Distance z Figure 14:

35

0.7

0.8

0.9

1 

0.26 FTTDIS

0.24

FTTD

0.22

FDIS

Thermodynamical temperature



0.2 0.18 0.16

0.14 0.12

0.1 0.08

0.06 0.04 0.02 0 0

0.1

0.2

0.3

0.4

0.5

Distance z

0.6

0.7

0.8

0.9

1 

Figure 15: 0.028 FTTDIS FTTD

0.024

Concentration C

FDIS 0.02

0.016

0.012

0.008

0.004

0

0

0.4

0.8

1.2

1.6

2

Distance z

Figure 16:

36

2.4

2.8

3.2

3.6

4 

4

FTTDIS 3.5

FTTD

Chemical potential P

FDIS 3

2.5

2

1.5

1

0.5

0

0

0.4

0.8

1.2

1.6

2

2.4

Distance z

2.8

3.2

3.6

4 

Figure 17: 0.75 FTTDIS 0.65

FTTD

zz

FDIS 0.55

Stress distribution




0.45

0.35

0.25

0.15

0.05 -0.05 0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

2.75

3

3.25

3.5

3.75

4

Distance z

-0.15



Figure 18:

37