The effect of diffusion on two-dimensional problem of generalized thermoelasticity with Green–Naghdi theory

International Communications in Heat and Mass Transfer 36 (2009) 857–864

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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t

The effect of diffusion on two-dimensional problem of generalized thermoelasticity with Green–Naghdi theory☆ Mohamed I.A. Othman ⁎, Sarhan Y. Atwa, R.M. Farouk Faculty of Science, Department of Mathematics, Zagazig University, P.O. Box 44519, Zagazig, Egypt

a r t i c l e

i n f o

Available online 30 May 2009 Keywords: Thermoelastic diffusion Generalized thermoelasticity Green–Naghdi theory (of types II, III)

a b s t r a c t The present paper is concerned with the investigation of disturbances in a homogeneous, isotropic elastic medium with generalized thermoelastic diffusion. The formulation is applied to the generalized thermoelasticity based on the Green and Naghdi (GN) theory under the effect of diffusion. The analytical expressions for displacement components, stresses, temperature field, concentration and chemical potential are obtained in the physical domain by using the normal mode analysis. These expressions are calculated numerically for a copper-like material and depicted graphically. Effect of presence of diffusion is analyzed theoretically and numerically. Comparisons are made with the results predicted by the type II and type III in the presence and absence of diffusion. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction The governing equations for the displacement and temperature fields as given by the linear dynamical theory of thermoelasticity, introduced by Biot [1] consist of the two coupled partial differential equations. The displacement field is governed by a wave-type equation and the temperature field is governed by a diffusion type equation. The properties of the later are such that a portion of the solution extends to infinity. That is, if an isotropic, homogeneous elastic medium is subjected to a mechanical or thermal disturbance, the effect of disturbance will be felt instantaneously at distances infinitely far from its source. Moreover, this effect will be felt in both the temperature and the displacement fields, since the governing equations are coupled. Physically, this means that a portion of the disturbance has an infinite velocity of propagation. Such behavior is physically inadmissible and contradicts existing theories of heat transport mechanisms. Lord and Shulman [2] derived the theory of generalized thermoelasticity, which is also referred to as extended thermoelasticity theory, by modifying the Fourier's law of heat conduction with the introduction of a thermal relaxation time parameter. The modified heat conduction equation in this theory is of the wave type and it ensures the finite speeds of propagation of heat and elastic waves. This theory was extended by Dhaliwal and Sherief [3] to general anisotropic media in the presence of heat sources. Othman [4] applied the normal mode analysis to study the effect of rotation on plane

☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail addresses: [email protected] (M.I.A. Othman), [email protected] (S.Y. Atwa), [email protected] (R.M. Farouk). 0735-1933/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2009.04.014

waves in generalized thermoviscoelasticity with one relaxation time. Othman [5], used the same method to study two-dimensional problems of generalized thermoelasticity with one relaxation time with the modulus of elasticity dependent on the reference temperature for non-rotating and rotating medium, respectively. Green and Lindsay's [6] theory contains two constants that act as relaxation times and modifies all the equations of the coupled theory not the heat conduction equation only. The two theories both ensure finite speeds of propagation for heat wave. Using the Green–Lindsay's theory, Othman and Song [7] studied the effect of rotation on plane waves of generalized electro-magneto-thermoviscoelasticity with two relaxation times. Othman and Song [8] studied the reflection of magneto-thermoelastic waves from a rotating elastic half-space in generalized thermoelasticity under three theories. The theory of thermoelasticity without energy dissipation is another generalized theory and was formulated by Green–Naghdi [9]. It includes the “thermal-displacement gradient” among its independent constitutive variables and differs from the previous theories in that it does not accommodate dissipation of thermal energy. Diffusion can be defined as the movement of particles from an area of high concentration to an area of lower concentration until equilibrium is reached. It occurs as a result of second law of thermodynamics which states that the entropy or disorder of any system must always increase with time. Diffusion is important in many life processes. There is now a great deal of interest in the study of this phenomenon, due to its many applications in geophysics and industrial applications. In integrated circuit fabrication, diffusion is used to introduce “dopants” in controlled amounts into the semiconductor substrate. In particular, diffusion is used to form the base and emitter in bipolar transistors, form integrated resistors, form the source/drain regions in MOS transistors and dope poly-silicon gates in MOS transistors. In most of these applications, the

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M.I.A. Othman et al. / International Communications in Heat and Mass Transfer 36 (2009) 857–864

Nomenclature λ, μ ρ CE t T To θ σij eij ekk ui k k⁎ δij P C d a b ν β αt αc u φ ψ

Lame's constants density specific heat at constant strain time absolute temperature reference temperature chosen so that j T temperature increment components of stress tensor components of strain tensor dilatation components of displacement vector thermal conductivity matrial characteristic of the theory Kronecker delta chemical potential per unit mass mass concentration thermodiffusion constant measure of thermodiffusion effect measure of diffusive effect = (3λ + 2μ)αt = (3λ + 2μ)αc coefficient of linear thermal expansion coefficient of linear diffusion expansion displacement vector scalar potential vector potential

− To To

jb1

permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The present study is motivated by the importance of thermoelastic diffusion process in the field of oil extraction. The theory of thermodiffusion is also applied in the description of thermomechanical treatment of porous media of sintered powder metals. Thermodiffusion methods have been successfully applied in the last few years in improving the mechanical properties of product made of powder metals. 2. Basic equations and problem formulation Following Sherief et al. [16], the governing equations for an isotropic, homogeneous elastic solid with generalized thermodiffusion at uniform temperature T0 in the undisturbed state, in the absence of body forces and heat loads are: (i) the equation of motion :: ρu i = μui;jj + ðλ + μ Þuj;ij − m θ;i − β C;i ;

ð1Þ

(ii) the heat conduction equation under GN theory (of type II and III) : : :: :: 4 kθ;ii +k θii = ρCE θ + rT0 ui;i + aT0 C;

ð2Þ

(iii) the generalized diffusion equation : dβekk;ii + daθii + C − dbCii = 0;

ð3Þ

(iv) the constitutive equations concentration is calculated using what is known as Fick's law. This is a simple law that does not take into consideration the mutual interaction between the introduced substance and the medium into which it is introduced or the effect of the temperature on this interaction. The phenomenon of diffusion is used to improve the conditions of oil extractions (seeking ways of more efficiently recovering oil from oil deposits). These days, oil companies are interested in the process of thermoelastic diffusion for more efficient extraction of oil from oil deposits. The thermodiffusion process also helps the investigation in the field associated with the advent of semiconductor devices and the advancement of microelectronics. Thermodiffusion in the solids is one of the transport processes that has great practical importance. Most of the research associated with the presence of concentration and temperature gradients has been made with metals and alloys. The first critical review was published in the work of Oriani [10]. With the advancement of a nuclear energetics the interest in thermodiffusion has returned to metallic oxides that often heats up in inhomogeneous temperature field [11] in connection with technological conditions. Thermodiffusion in an elastic solid is due to the coupling of the fields of temperature, mass diffusion and that of strain. Heat and mass exchange with the environment during the process of thermodiffusion in an elastic solid. The concept of thermodiffusion is used to describe the process of thermomechanical treatment of metals (carbonizing, nitriding steel, etc.), these processes are thermally activated, their diffusing substances being, e.g, nitrogen, carbon etc. They are accompanied by deformations of the solid. Nowacki [12–15] developed the theory of thermoelastic diffusion. In this theory, the coupled thermoelastic model is used. This implies infinite speeds of propagation of thermoelastic waves. Sherief et al. [16] developed the theory of generalized thermoelastic diffusion that predicts finite speeds of propagation for thermoelastic and diffusive waves. The reflection phenomena of P and SV waves from free surface of an elastic solid with thermodiffusion was considered by Singh [17]. Sherief and Saleh [18] worked on a problem of a thermoelastic half-space with a

σ ij = 2μeij + ðλekk − mθ − βC Þδij ;

ð4Þ

P = − βekk + bC − aθ:

ð5Þ

The diffusion relaxation time ensures that the equation satisfied by the concentration C also predicts finite speed of propagation of matter from one medium to the other. We use a fixed Cartesian coordinate system (x,y,z) with origin on the surface z = 0, which is stress free and with z-axis directed vertically into the medium. The region z N 0 is occupied by the elastic solid with generalized thermodiffusion. We restrict our analysis parallel to xz plane. The boundary of the medium is assumed to be thermally insulated. The chemical potential is also assumed to be a known function of time. We assume that all quantities are functions of the coordinates x, z and time t and independent of coordinate y. So the displacement vector will have the components ux = uðx; z; t Þ; uy = 0; uz = wðx; z; t Þ and e =

Au Aw + : Ax Az

ð6Þ

Then the field Eqs. (1)–(4) become ðλ + μ Þ

:: Ae Aθ AC 2 + μj u − m −β = ρu ; Ax Ax Ax

ð7Þ

ðλ + μ Þ

:: Ae Aθ AC 2 + μj w − m −β = ρw ; Az Az Az

ð8Þ

: : :: :: 4 2 2 k j θ + k j θ = ρCE θ + mT0 e + aT0C;

: 2 2 2 dβj e + daj θ + C − dbj C = 0;

ð9Þ

ð10Þ

M.I.A. Othman et al. / International Communications in Heat and Mass Transfer 36 (2009) 857–864

σ zz = ðλ + 2μ Þ

σ zx = μ

Aw Au +λ − mθ − βC; Az Ax


 Aw Au + : Ax Az 2

ð11Þ

subsequent analysis we are taking into consideration the case of low speed so that centrifugal effects can be ignored.

ð12Þ

3. Normal mode analysis

2

A A is a two-dimensional Laplace operator. Where j2 = Ax 2 + Az2 The governing equations can be put into a more convenient form by using the following non-dimensional variables

˜ ˜ ρc ω ω ˜ fuV; wVg = 1 fu; wg; fx; zg; t V= ωt; c1 mT0 σ ij C θ P n V o ρc2 ; C V= ; θ V= ; P V= ; / ; ψV = 1 f/; ψg: σijV = mT0 mT0 ρ T0 β

fx V; z Vg =

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

i h i 4 4 4 4 4 4 4 4 C; u; w; e; φ; ψ; θ; σ ij ðx; z; t Þ = C ; u ; w ; e ; / ; ψ ; θ ; σ ij ðzÞ expðωt + imxÞ:

ð22Þ

ð13Þ

˜ = ρCE c21 = k: Where c21 = ðλ + 2μ Þ = ρ and ω With the aid of the expressions relating displacement components u, w to the scalar potential φ and vector potential ψ in dimensionless form given by u = φx − ψz ; w = φz + ψx :

859

Where ω is the (complex) frequency, i is the imaginary unit, m is the wave number in the x-direction and C⁎, u⁎, w⁎, e⁎, φ⁎, ψ⁎, θ⁎, and σij⁎ are the amplitudes of the functions. Using Eq. (22), we can obtain the following equations from Eqs. (15)–(18) h

i 2 2 2 4 4 4 D − m − ω φ ðzÞ − θ ðzÞ − a1 C ðzÞ = 0;

ð23Þ

h

i 2 2 2 4 D − m − a2 ω ψ ðzÞ = 0;

ð24Þ

ð14Þ

Using the quantities given by Eqs. (11) and (12) in Eqs. (7)–(12), we obtain the equations in dimensionless form (dropping the dashes for convenience) as "

# A2 j − 2 φ − θ − a1 C = 0; At 2

ð15Þ

h  i   2 2 2 4 2 2 2 4 4 e D − m − ω θ ðzÞ − e1 ω D − m φ ðzÞ − a3 ωC ðzÞ = 0; ð25Þ

"

# A2 j − a2 2 ψ = 0; At 2

ð16Þ

h

i   h  i 4 2 2 4 4 2 2 4 2 2 4 C ðzÞ = 0: D − 2m D + m φ ðzÞ − a4 D − m θ ðzÞ + a5 ω − a6 D − m

ð26Þ

:: : :: 2 2: e2 j θ + e3 j θ − θ = e1 e + a3C;

ð17Þ

: 4 2 2 j / + a4 j θ + a5C − a6 j C = 0:

ð18Þ

where, D = d : dz Eliminating θ⁎(z), φ⁎(z) and C⁎(z) between Eqs. (23), (25) and (26) we get the following ordinary differential equation satisfied by φ⁎(z)   6 4 2 4 D − AD + BD − E φ ðzÞ = 0;

ð27Þ

where, σ zz =

Aw Au + a8 − θ − a1 C; Az Ax


σ zx = a7

 Aw Au + : Ax Az

g1 = 2m + ω +

ð20Þ

g2 = m + m ω +

2

2

4

2

2

g3 = a1 m +

Where, 2

2

ω2 ð1 + e1 Þ ; e

ð19Þ

2 4

a1 =

βρ c ρ a aρ c1 ρ c1 ; a5 = ; a = 1 ; a3 = ; ; a4 = ˜ 0 ˜ βm mT0 2 μ CE ω d βm ωT

a6 =

bρ2 c21 μ λ m2 T ; a7 = ; a8 = ; e1 = 2 2 0 ; 2 2 βmT0 ρc1 ρc1 ρ c1 CE

ð21Þ

2

2 2 4 m ω ð1 + e1 Þ + ω ; e

a1 ω 2 − a3 ω ; e

2

2

g4 = 2m ð1 + a4 Þ + a4 ω ;

4

2

ð28Þ

ð29Þ

ð30Þ

ð31Þ

2

g5 = m ð1 + a4 Þ + a4 m ω ;

ð32Þ

g6 = a6 + a1 a4 ;

ð33Þ

˜ 4 k ωk e2 = 2 ; e3 = 2 : ρc1 CE ρc1 CE Where ε1 is usually the thermoelastic-coupling factor, ε2 is the characteristic parameter of the GN theory (of type II) and ε3 is the characteristic parameter of the GN theory (of type III), in the

2

g7 = a5 ω + m g6 ;

ð34Þ

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M.I.A. Othman et al. / International Communications in Heat and Mass Transfer 36 (2009) 857–864

g8 = a1 ð1 + a4 Þ − g6 ;

ð35Þ

where   2 2 2 H1n = Kn − m − ω − H2n = a1 ;

g9 = − g1 g6 − g7 + a1 g4 + g3 ð1 + a4 Þ;

ð50Þ

n = 1; 2; 3;

ð36Þ h

g10 = − g2 g6 − g1 g7 + a1 g5 + g3 g4 ;

ð37Þ

H2n

   i 2 2 2 2 2 2 a3 ω Kn − m − ω + a1 e1 ω Kn − m    ; = a3 ω + a1 e Kn2 − m2 − a1 ω2

n = 1; 2; 3: ð51Þ

g11 = g2 g7 − g3 g5 :

ð38Þ

And

4

g A = 9; g8

g B = 10 ; g8

−g11 E= ; e = e2 + ωe3 : g8

ψ ðzÞ = M4 ðm; ωÞe ð39Þ

In a similar manner we can show that θ⁎(z), φ⁎(z) and C⁎(z) satisfy the equations h

The solution of Eq. (24) is

6

4

2

D − AD + BD − E

in o 4 4 4 φ ðzÞ; C ðzÞ; θ ðzÞ = 0;

− K4 z

;

ð52Þ

where

K4 =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 + a2 ω2 :

ð53Þ

ð40Þ In order to obtain the displacement components u and w, using Eq. (22), Eq. (14), becomes

Eq. (40) can be factorized as

4

6

4

2

ð42Þ

The solution of Eq. (41) which is bounded as z → ∞, is given by 4

φ ðzÞ =

3 X

Mn ðm; ωÞ e

− Kn z

:

4

ð54Þ

41

where, K2n (n = 1,2,3) are the roots of the following characteristic equation K − AK + BK − E = 0:

4

u ðzÞ = imφ ðzÞ − Dψ ðzÞ;

    2 2 2 2 2 2 4 D − K2 D − K3 φ ðzÞ = 0: D − K1

ð43Þ

4

4

4

w ðzÞ = Dφ ðzÞ + imψ ðzÞ;

ð55Þ

Substitution of Eqs. (54), (55), (43), and (52) into Eqs. (19), and (20), we get 4

σ zz ðzÞ =

3 X

H3n Mn ðm; ωÞe

− Kn z

− H1 M4 ðm; ωÞe

− K4 z

;

ð56Þ

n=1

n=1

Similarly 4

C ð zÞ =

3 X

MnVðm; ωÞe

− Kn z

4

;

ð44Þ

σ zx ðzÞ =

3 X

−H4n Mn ðm; ωÞe

− Kn z

− H2 M4 ðm; ωÞe

− K4 z

;

ð57Þ

n=1

n=1

where 4

θ ð zÞ =

3 X

MWn ðm; ωÞe

− Kn z

;

ð45Þ

H1 = imð1 − a8 ÞK4 ;

ð58Þ

n=1

where Mn(m,ω), Mn′(m,ω), and Mn″(m,ω) are parameters depending on m and ω. Substituting Eqs. (43)–(45) into Eqs. (23)–(25) we get MnVðm; ωÞ = H1n Mn ðm; ωÞ;

n = 1; 2; 3;

ð46Þ

MWn ðm; ωÞ = H2n Mn ðm; ωÞ;

n = 1; 2; 3:

ð47Þ

We thus have C ð zÞ =

3 X

3 X n=1

H1n Mn ðm; ωÞe

H2n Mn ðm; ωÞe

2

ð59Þ

  2 2 H3n = Kn − a8 m − H2n − a9 H1n ; H4n = − 2imKn ;

n = 1; 2; 3:

n = 1; 2; 3;

ð60Þ

ð61Þ

4. Application − Kn z

;

ð48Þ

;

ð49Þ

n=1

θ ð zÞ =

2

H2 = K4 + m ;

− Kn z

In this section, the general solutions for displacement, stresses, temperature field, concentration and chemical potential, will be used to yield the response of a half-space subject to a fixed load acting with uniform constant F(x). The final solution in the original domain (x,z,t) is obtained numerically.

M.I.A. Othman et al. / International Communications in Heat and Mass Transfer 36 (2009) 857–864

In the half-space, the load F(x) is fixed in normal direction. The surface z = 0 is assumed to be thermally insulated that there is no variation of temperature and concentration on it. Therefore, for this loading case the boundary conditions are

σ zz = − F ðxÞH ðt Þ;

σ zx = 0;

Aθ = 0; Az

AC = 0; Az

at

where F(x) = F0δ(x). Substituting the expressions of the variables considered into the above boundary conditions, we can obtain the following equations satisfied by the parameters 3 X

H3n Mn − H1 M4 = − F0 ;

where, S2n (n = 1,2) are the roots of the following characteristic equation 4

ð63Þ

2

S − g1 S + g2 = 0:

z = 0; ð62Þ

ð72Þ

The solution of Eq. (71) is given by 2 X

4

φ ðzÞ =

Gn ðm; ωÞe

− Sn z

:

ð73Þ

;

ð74Þ

n=1

Similarly 2 X

4

θ ð zÞ =

GV n ðm; ωÞe

− Sn z

n=1

n=1

3 X

861

where Gn(m,ω) and Gn′(m,ω), are parameters depending on “m” and ω. Substituting Eqs. (73) and (74) into Eq. (76) we get H4n Mn − H2 M4 = 0;

ð64Þ

GV n ðm; ωÞ = H5n Gn ðm; ωÞ;

ð75Þ

n = 1; 2:

n=1

We thus have 3 X

ð65Þ

H2n Kn Mn = 0;

n=1

2 X

4

θ ð zÞ =

H5n Gn ðm; ωÞe

− Sn z

;

ð76Þ

n=1

3 X

where ð66Þ

H1n Kn Mn = 0:

n=1

  2 2 2 H5n = Sn − m − ω ;

ð77Þ

n = 1; 2;

The solution of Eq. (68) is the same in Eq. (52) and Solving Eqs. (63)–(66), we get the parameters Mn (n=1,2,3), are defined as follows: M1 = ΔΔ1 ; M2 = ΔΔ2 ; M3 = ΔΔ3 and M4 = ΔΔ4 ; where, Δ, Δj, j=1, 2, 3, 4 are defined in Appendix A. 5. Particular case

4

σ zz ðzÞ =

2 X

H3 Gn ðm; ωÞe

− Sn z

− H1 G3 ðm; ωÞe

− K4 z

;

ð78Þ

n=1

4

σ zx ðzÞ =

2 X

−H6n Sn Gn ðm; ωÞe

− Sn z

− H2 G3 ðm; ωÞe

− K4 z

;

ð79Þ

n=1

Neglecting diffusion effect, by taking C = a = b = β = 0, we obtain the expressions for displacement components, stresses and temperature field in the generalized thermoelastic medium are obtained as

where

h

H3 = m ð1 − a8 Þ + ω ;

ð80Þ

H6n = 2imSn ;

ð81Þ

h

2

2

2

i

2

4

4

D − m − ω φ ðzÞ − θ ðzÞ = 0; i 2 2 2 4 D − m − a2 ω ψ ðzÞ = 0;

h  i   2 2 2 4 2 2 2 4 e D − m − ω θ ðzÞ − e1 ω D − m φ ðzÞ = 0:

2

ð67Þ

ð68Þ

n = 1; 2:

In this case the boundary conditions are σ zz = − F ðxÞH ðt Þ;

ð69Þ

Eliminating θ⁎(z) and φ⁎(z) between Eqs. (67) and (69) we get the following ordinary differential equation satisfied by

σ zx = 0;

Aθ = 0; Az

at z = 0:

ð82Þ

Substituting from the expressions of the variables considered into the above boundary conditions, we can obtain the following equations satisfied by the parameters 2 X

H3 Gn − H1 G3 = − F0 ;

ð83Þ

H6n Gn − H2 G3 = 0;

ð84Þ

H5n Sn Gn = 0:

ð85Þ

n=1

h

4

2

D − g1 D + g2

o in T 4 / ðzÞ; θ ðzÞ = 0:

ð70Þ

2 X n=1

Eq. (70) can be factorized as    2 2 2 2 4 D − S1 D − S2 φ ðzÞ = 0;

ð71Þ

2 X n=1

862

M.I.A. Othman et al. / International Communications in Heat and Mass Transfer 36 (2009) 857–864

Fig. 1. Variation of displacement distribution w with distance x (with diffusion).

Fig. 2. Variation of temperature distribution θ with distance x (with diffusion).

Fig. 3. Variation of stress distribution σzx with distance x (with diffusion).

Fig. 4. Variation of stress distribution σzz with distance x (with diffusion).

Fig. 5. Variation of concentration C with distance x (with diffusion).

Fig. 6. Variation of chemical potential P with distance x (with diffusion).

M.I.A. Othman et al. / International Communications in Heat and Mass Transfer 36 (2009) 857–864

863

Fig. 9. Stress distribution σzx with distance x (without diffusion). Fig. 7. Variation of displacement distribution w with distance x (without diffusion).

The variations of the vertical component of displacement w, the temperature distribution θ, component σzx, normal force stress σzz, concentration C, chemical potential P with distance x at z = 0 for Green–Naghdi theory of type II and type III at t = 0.1 and t = 0.3 have been shown for (i) thermoelastic solid with diffusion (TESD) by solid line and dot line (type II), and (ii) thermoelastic solid with diffusion (TESD) by dashed line and dashed with dot line (type III). These

variations are shown in Figs. 1–6. The values of the physical quantities for (TESD) have an oscillatory behavior in the range 0 ≤ x ≤ 4 for both types II and III. Fig. 1 shows the variation of the vertical component of displacement w with distance x for types II and III. It is clear that the time has an increasing effect for both types II and III in the ranges 0 ≤ x ≤ 0.9 and 2.4 ≤ x ≤ 4 while has a decreasing effect in the range 0.9 ≤ x ≤ 2.4. Fig. 2 depicts the variation of temperature distribution θ with distance x for types II and III. It is clear that the time has a decreasing effect for both types II and III in the ranges 0 ≤ x ≤ 0.75 and 2.5 ≤ x ≤ 4, while has an increasing effect in the range 0.75 ≤ x ≤ 2.5. Fig. 3 shows the variation of stress component σzx with distance x for types II and III. It is clear that the time has a decreasing effect for both types II and III in the ranges 0 ≤ x ≤ 1.65 and x ≥ 3.25, while has an increasing effect in the range 1.65 ≤ x ≤ 3.25. Fig. 4 shows the variation of normal force stress σzz with distance x for types II and III. It is clear that the time has a decreasing effect for both types II and III in the ranges 0 ≤ x ≤ 0.5 and 2.1 ≤ x ≤ 3.7, while has an increasing effect in the range 0.5 ≤ x ≤ 2.1. Figs. 5 and 6 show the variation of the concentration C and the chemical potential P with distance x for types II and III. It is clear that the time has an increasing effect for both types II and III in the ranges 0 ≤ x ≤ 0.9 and 2.4 ≤ x ≤ 4 while has a decreasing effect in the range 0.9 ≤ x ≤ 2.4. The variations of the vertical component of displacement w, the temperature distribution θ, stress component σzx, and normal force stress σzz, with distance x at z = 0 for (GN) theory of type II and type III at t = 0.1 and t = 0.3 have been shown for (i) thermoelastic solid without diffusion (TES) by solid line and dot line (type II) and (ii)

Fig. 8. Temperature distribution θ with distance x (without diffusion).

Fig. 10. Stress distribution σzz with distance x (without diffusion).

Solving Eqs. (83)–(85), we get the parameters Gn (n = 1,2), are defined as follows: G1 = ΔΔ65 ; G2 = ΔΔ75 and G3 = ΔΔ85 ; where, Δ5, Δj, j = 6, 7, 8 are defined in Appendix B. 6. Numerical results With an aim to illustrate the problem, we will present some numerical results. The material chosen for the purpose of numerical computation is copper, the physical data for which is given by Thomas [19] in SI units: T0 = 293 K; ρ = 8954 kg = m3 ; CE = 383:1 J = ðkg KÞ; α t = 1:78 × 10 − 5 K − 1 ;     2 10 2 kg = m s ; μ = 3:86 × 10 kg = m s ;   −4 3 −8 3 4 2 2 kg s = m ; a = 1:2 × 10 m = s K ; α c = 1:98 × 10 m = kg; d = 0:85 × 10   6 5 2 4 b = 0:9 × 10 m = kg s ; z = 3; k = 1:2; m = 2; Hðt Þ = 1; F0 = 10; δðxÞ = 1: 10

K = 386 W = ðm KÞ; λ = 7:76 × 10

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thermoelastic solid without diffusion (TES) by dashed line and dashed with dot line (type III). These variations are shown in Figs. 7–10. The values of the physical quantities for (TES) have an oscillatory behavior in the range 0 ≤ x ≤ 4 for both types II and III. Fig. 7 shows the variation of the vertical component of displacement w with distance x for types II and III. It is clear that the time has an increasing effect for both types II and III in the ranges 0.3 ≤ x ≤ 1.9 and x ≥ 3.4 while has a decreasing effect in the range 1.9 ≤ x ≤ 3.4. Fig. 8 depicts the variation of temperature distribution θ with distance x for types II and III. It is clear that the time has an increasing effect for type II in the ranges 0 ≤ x ≤ 1.6 and 3.1 ≤ x ≤ 4 and for type III in the ranges 0 ≤ x ≤ 1 and 2.5 ≤ x ≤ 4 while has a decreasing effect in the range 1.6 ≤ x ≤ 3.1 for type II and in the range 1 ≤ x ≤ 2.5 for type III. Fig. 9 shows the variation of stress component σzx with distance x for types II and III. It is clear that the time has an increasing effect for both types II and III in the ranges 0 ≤ x ≤ 1 and 2.5 ≤ x ≤ 4, while has a decreasing effect in the range 1 ≤ x ≤ 2.5. The magnitude of these oscillations increases with increase in time for the two cases of type II and type III. Fig. 10 shows the variation of normal force stress σzz with distance x for types II and III. It is clear that the time has a decreasing effect for both types II and III in the ranges 0 ≤ x ≤ 1.6 and x ≥ 3.25, while has an increasing effect in the range 1.6 ≤ x ≤ 3.25. The value of variation of change in all quantities in the presence of diffusion is a long range comparing with the absence of diffusion for different values of time (with and without energy dissipation). 7. Concluding remarks According to the analysis above we can conclude that the presence of diffusion plays a significant role in all the quantities and has an important effect on the vertical and normal components of displacement, the temperature, the stress components, concentration, and the chemical potential. Diffusion and thermal parameters also have a significant effect on the quantities. The method used in the present article is applicable to a wide range of problems in thermodynamics. Appendix A Δ = H13 K3 ½−K1 H21 ðH2 H32 + H1 H42 Þ + K2 H22 ðH2 H31 + H1 H41 Þ + H12 K2 ½K1 H21 ðH2 H33 + H1 H43 Þ − K3 H23 ðH2 H31 + H1 H41 Þ + H11 K1 ½−K2 H22 ðH2 H33 + H1 H43 Þ + K3 H23 ðH2 H32 + H1 H42 Þ; Δ1 = F0 H2 K2 K3 ½−H13 H22 + H12 H23 ; Δ2 = F0 H2 K1 K3 ½H13 H21 − H11 H23 ; Δ3 = F0 H2 K1 K2 ½−H12 H21 + H11 H22 ; Δ4 = F0 ½K1 K2 H43 ðH12 H21 − H11 H22 Þ + K1 K3 H42 ð−H13 H21 + H11 H23 Þ + K3 K2 H41 ðH13 H22 − H12 H23 Þ:

Appendix B Δ5 = S1 H51 ðH2 H3 + H1 H62 Þ − S2 H52 ðH2 H3 + H1 H61 Þ; Δ6 = F0 S2 H2 H52 ; Δ7 = − F0 S1 H2 H51 ; Δ8 = F0 ðS1 H51 H62 − S2 H52 H61 Þ: References [1] M. Biot, Thermoelasticity and irreversible thermodynamics, J. Appl. Phys. 27 (1956) 240–253. [2] H.W. Lord, Y. Shulman, A generalized thermodynamical theory of thermoelasticity. elasticity, J. Mech. Phys. Solids 15 (1967) 299–309. [3] R. Dhaliwal, H. Sherief, Generalized thermoelasticity for an isotropic media, J. Quat. Appl. Math. 33 (1980) 1–8. [4] M.I.A. Othman, Effect of rotation and relaxation time on a thermal shock problem for a half-space in generalized thermoviscoelasticity, Acta Mech. 174 (2005) 129–143. [5] M.I.A. Othman, Lord–Shulman theory under the dependence of the modulus of elasticity on the reference temperature in two-dimensional generalized thermo elasticity, J. Therm. Stress. 25 (2002) 1027–1045. [6] A.E. Green, K.A. Lindsay, Thermoelasticity, J. Elasticity 2 (1972) 1–7. [7] M.I.A. Othman, Y.Q. Song, Effect of rotation on plane waves of the generalized electromagneto-thermo-viscoelasticity with two relaxation times, Applied Mathematical Modelling 32 (2008) 811–825. [8] M.I.A. Othman, Y.Q. Song, Reflection of magneto-thermo-elastic waves from a rotating elastic half-space, Int. J. Eng. Science 46 (2008) 349–364. [9] A.E. Green, P.M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity 31 (1993) 189–208. [10] R.A. Oriani, Thermomigration in solid metals, J. Phys. Chem. 30 (1969) 339–351. [11] R.E. Fryxel, E.A. Aitken, High temperature studies of urania in a thermal gradient, J. Nucl. Mater. 30 (1969) 50–56. [12] W. Nowacki, Dynamic problems of thermoelastic diffusion in solids, J. I Bulletin de l' Acadermic Polonaise des Sciences Serie des Sciences Techniques vol. 22 (1974) 55–64. [13] W. Nowacki, Dynamic problems of thermoelastic diffusion in solids, J. II Bulletin de l' Acadermic Polonaise des Sciences Serie des Sciences Techniques vol. 22 (1974) 129–135. [14] W. Nowacki, Dynamic problems of thermoelastic diffusion in solids, J. III Bulletin de l' Acadermic Polonaise des Sciences Serie des Sciences Techniques vol. 22 (1974) 266–275. [15] W. Nowacki, Dynamic problems of diffusion in solids, J. Eng. Frac. Mech. 8 (1976) 261–266. [16] H. Sherief, F. Hamza, H. Saleh, The theory of generalized thermoelastic diffusion, Int. J. Eng. Sci. 42 (2004) 591–608. [17] B. Singh, Reflection of P and SV waves from free surface of an elastic solid with generalized thermodiffusion, J. Earth. Syst. Sci. 114 (2005) 159–168. [18] H. Sherief, H. Saleh, A half-space problem in the theory of generalized thermoelastic diffusion, Int. J. Solids Struct. 42 (2005) 4484–4493. [19] L. Thomas, Fundamentals of Heat Transfer [M], Prentice-Hall Inc, Englewood Cliffs, New Jersey, 1980.