International Journal of Heat and Mass Transfer 66 (2013) 451–460

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Fractional order heat conduction law in micropolarthermo-viscoelasticity with two temperatures

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.07.047

⇑ Corresponding author. Tel.: +91 1662263367; fax: +91 1662276240.E-mail addresses: spannu_gju@yahoo.com (S. Deswal), kapilkalkal_gju@rediff-

mail.com (K.K. Kalkal).

Sunita Deswal, Kapil Kumar Kalkal ⇑Department of Mathematics, G. J. University of Science and Technology, Hisar 125001, Haryana, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 January 2013Received in revised form 13 June 2013Accepted 14 July 2013Available online 15 August 2013

Keywords:Fractional order theoryMicropolar thermo-viscoelasticityTwo temperature parameterLaplace–Fourier transformsDistributed thermal source

Present work is concerned with the transient solution of a half-space problem in the context of fractionalorder micropolar thermo-viscoelasticity involving two temperatures whose surface is acted upon by auniformly distributed thermal source. Medium is assumed initially quiescent. The formulation is appliedto the fractional generalization of the Lord–Shulman theory with microstructure effects and the non-dimensional equations are handled by employing an analytical–numerical technique based on Laplaceand Fourier transforms. The numerical estimates of the displacement, stresses and temperatures are com-puted for magnesium crystal like material and corresponding graphs are plotted to illustrate and com-pare theoretical results. All the fields are found to be significantly affected by the fractional parameter,viscosity and two temperature parameter. The phenomenon of finite speed of propagation is observedgraphically for each field. Some particular cases have also been inferred from the present study.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Under the assumption of continuum hypothesis of an elasticbody, the classical theory of elasticity is based on linear stress–strain law (Hooke’s law). In this theory, the transmission of loadacross a surface element of an elastic body is described by a forcestress (force per unit area) and the motion is characterized bytranslational degrees of freedom only. For materials possessinggranular structure, it is found that the classical theory of elasticityis inadequate to represent complete deformation. Certain discrep-ancies are observed between the results obtained experimentallyand theoretically, particularly, in dynamical problems involvingelastic vibrations of high frequencies and short wavelengths, i.e.,vibrations due to the generation of ultrasonic waves. The reasonfor these discrepancies lies in the microstructure of the material,which exerts special influence at high frequencies and short wave-lengths [1]. This influence of microstructure results in the develop-ment of new type of waves, not found in the classical theory ofelasticity. Metals, polymers, composites, soils, rocks, concrete aretypical media with microstructures. More generally, most of thenatural and manmade materials including engineering, geologicaland biological media possess a microstructure.

Cosserat and Cosserat [2] were the first who rendered impor-tance to the microstructure of a granular body and incorporateda local rotation of points in addition to the translation assumed

in classical theory of elasticity. Consequently, there exists couplestress (a torque per unit area) in addition to the force stress. Thistheory is known as ‘Cosserat theory’ after their names or the theoryof elasticity with couple stress. This theory was in dormant for somany years and did not get sufficient attention. In the 1960s, Erin-gen and Suhubi [3,4] and Eringen [5] gave modern formulations ofCosserat medium equations, which became known as the equa-tions of the micropolar theory of elasticity or the theory of asym-metric elasticity. Within such a theory, solids can undergomacro-deformations and micro-rotations. The motion in this kindof solids is completely characterized by the displacement vector~uð~X; tÞ and the micro-rotation vector ~/ð~X; tÞ while in the case ofclassical elasticity, the motion is characterized by the displacementvector only. A historical development of the theory of micropolarelasticity is given in a recent monograph of Eringen [6]. This theoryis expected to find applications in the treatment of mechanics ofgranular materials, composite fibrous materials and particularlymicrocracks and microfractures.

Recent years have seen an evergrowing interest in the investi-gation of dynamical interaction between thermal and mechanicalfields in solids due to their manifold applications in variousbranches of engineering, science and technology. While in service,structural elements are frequently subjected to not only force loadsbut also non-uniform heating causing thermal stresses. Thesestresses themselves or in combination with mechanical stressesdue to external loads may cause the material to fracture. Therefore,to perform a complete strength analysis of structures, it is neces-sary to know the magnitude and distribution of thermal stresses.In this connection, issues associated with the determination of

Notations

rij components of force stress tensormij components of couple stress tensor~/ microrotation vectork� ke 1þ a0

@@t

� �l⁄ le 1þ a1

@@t

� �b�1 b1e 1þ b1

@@t

� �b1e ð3ke þ 2le þ kÞat

b1 ð3kea0 þ 2lea1Þ atb1e

h = T � T0 thermodynamical temperature/ = / � T0 conductive temperatureke; le Lame’s constantsa0,a1 viscoelastic relaxation timesa,b,c,k micropolar material constantsat coefficient of linear thermal expansion

a two-temperature parameterj microinertiaT absolute temperatureT0 temperature of the medium in its natural state as-

sumed to be |h/T0|� 1ui components of the displacement vectorq density of the mediumeij components of the strain tensore cubical dilatationcE specific heat at constant strainK⁄ thermal conductivitys0 thermal relaxation timeC Gamma functionm fractional order parameter such that 0 < m 6 1

452 S. Deswal, K.K. Kalkal / International Journal of Heat and Mass Transfer 66 (2013) 451–460

temperature fields and thermal stresses are of importance anddraw the attention of experts of different professions. Keepingthe above applications in view, the micropolar elasticity theorywas further extended to include the thermal effects by Nowacki[7–9] and Eringen [10]. One can refer to Dhaliwal and Singh [11]for a review on the micropolar thermoelasticity and a historicalsurvey of the subject as well as to Eringen and Kafadar [12] inthe continuum physics series, in which the general theory ofmicromorphic media has been summed.

There has been very much written in recent years concerningthe problem of propagation of thermal waves at finite speed. Thearticles of Dreyer and Struchtrup [13] and Caviglia et al. [14] pro-vide an extensive survey of work on experiments involving thepropagation of heat as a thermal wave. They reported instanceswhere the phenomenon of second sound has been observed in sev-eral kinds of materials. Extensive reviews on the second sound the-ories can be found in the works of Chandrasekharaiah [15,16]. Ageneralized theory of linear micropolar thermoelasticity that ad-mits the possibility of ‘‘second sound’’ effects was established in[17]. Using the Green and Lindsay theory [18], Dost and Tabarrok[19] propounded another new model of generalized micropolarthermoelasticity that permits the propagation of thermal wavesat a finite speed. Chandrasekharaiah [20,21] obtained the equa-tions for a generalization of micropolar thermoelasticity equations,which is called the heat flux dependent micropolar thermoelastic-ity and proved variational and reciprocal principles for his equa-tions. Ciarletta [22] used the procedure proposed by Green andNaghdi [23] to derive a new linear theory of micropolar thermo-elasticity. In this theory, in contrast to the theories developed in[17,19,20], the heat flow does not involve energy dissipation. Sher-ief et al. [24] proposed the generalized equations for the linear the-ory of micropolar thermoelasticity based on Lord–Shulman theory[25]. A uniqueness theorem is also provided in the same article. Asan illustrative example, they have solved a half-space problemusing Laplace and Hankel transforms whose boundary is rigidlyfixed and subjected to an axisymmetric thermal shock. Severalresearchers in past including Kumar and his co-workers [26–28],Othman and Singh [29], Zakaria [30] among several others havestudied many interesting problems based on micropolar general-ized thermoelasticity theories.

Effect of internal friction on the propagation of plane waves inan elastic medium may be attributed to the fact that dissipationaccompanies vibrations in solid media due to the conversion ofelastic energy to heat energy. Several mathematical models havebeen used by authors [31,32] to accommodate the energy

dissipation in vibrating solids where it is observed that internalfriction produces attenuation and dispersion; hence, the effect ofthe viscoelastic nature of material medium in the process of wavepropagation cannot be neglected. Also with the rapid developmentof polymer science and plastic industry as well as wide use ofmaterials under high temperature in modern technology andapplication of biology and geology in engineering, the theoreticalstudy and applications in viscoelastic materials have become animportant task for solid mechanics. Further, as pointed out byFreudenthal [33], most of the solids, when subjected to dynamicloading, exhibit viscous effects. Keeping these facts in mind, sev-eral problems on wave propagation in a linear viscoelastic solidhave been explored by many research workers. One can refer toIlioushin and Pobedria [34] for a formulation of a mathematicaltheory of thermal viscoelasticity and the solutions of some bound-ary value problems.

Gurtin and Williams [35,36] have suggested that there are noa priori grounds to assume that the second law of thermodynam-ics for continuous bodies involve only a single temperature, i.e., itis more logical to assume a second law in which the entropy con-tribution due to heat conduction is governed by one temperature,that of the heat supply by another. Chen and Gurtin [37] andChen et al. [38,39] have formulated a theory of heat conductionin deformable bodies which depends on two distinct tempera-tures – the conductive temperature / and the thermodynamicaltemperature h. The conductive temperature is due to the thermalprocesses and the heat exchange with the external world and thethermodynamical temperature is due to the mechanical processesinherent between the particles of elastic materials. Chen et al.[38] have pointed out that the difference between these two tem-peratures is proportional to heat supply and these temperaturesbecome equal for time-independent situation in the absence ofheat supply. However, for time-dependant cases the two temper-atures are in general different, regardless of the heat supply. Thekey element that sets the two-temperature thermoelasticity apartfrom the classical theory of thermoelasticity is the materialparameter a ðP 0Þ, called the temperature discrepancy. Specifi-cally if a = 0, then / = h and the field equations of two-tempera-ture thermoelasticity reduce to those of classical theory ofthermoelasticity. Youssef [40] extended this theory in the frameof generalized theory of heat conduction by introducing thermalrelaxation parameters in the constitutive relations and proposeda two-temperature theory of generalized thermoelasticity.Uniqueness of the solution for this theory has also been derivedby Youssef in the same article.

S. Deswal, K.K. Kalkal / International Journal of Heat and Mass Transfer 66 (2013) 451–460 453

Several interesting models have been developed by using frac-tional calculus to modify many existing models of physical pro-cesses, particularly in the fields of heat conduction, diffusion,viscoelasticity, control theory, mechanics of solids etc. One canstate that the whole theory of fractional derivatives and integralswas established in the second half of the 19th century. The firstapplication of fractional calculus was given by Abel in the solutionof an integral equation that arises in the formulation of the taut-ochrone problem. An account of the historical developments ofthe theory of fractional calculus as well as references to variouscontributions may be found in the books by Oldham and Spanier[41] and Miller and Ross [42]. One can refer to Podlubny [43] fora survey of many applications of fractional calculus in the area ofscience and engineering.

In the last few years, fractional calculus has also been intro-duced in the theory of thermoelasticity. A quasi-static uncoupledtheory of thermoelasticity based on fractional heat conductionequation has been put forward by Povstenko [44]. The theory ofthermal stresses based on the heat conduction equation with Cap-uto time-fractional derivative is used by Povstenko [45] to investi-gate thermal stresses in an infinite body with a circular cylindricalhole. Sherief et al. [46] have proposed a new model of thermoelas-ticity using fractional calculus with second sound effects, proved auniqueness theorem and derived reciprocity relation and varia-tional principle. Youssef [47] has also constructed a new theoryof thermoelasticity using the methodology of fractional calculusand discussed one-dimensional application. In this theory, the dif-ferent values of the fractional parameter m with wide rangeð0 < m 6 2Þ cover different cases of conductivity; (0 < m < 1)corre-sponds to weak conductivity, m = 1 for normal conductivity andð1 < m 6 2Þ corresponds to strong conductivity. However, for themodel [46], fractional parameter ranges between 0 and 1. An an-other new theory of fractional order generalized thermoelasticityusing the new Taylor series expansion of time-fractional orderhas been developed by Ezzat [48] with one relaxation time. Veryrecently, Shaw and Mukhopadhyay [49] have suggested a frac-tional order micropolar generalized thermoelasticity theory withtwo temperatures using the fractional order theory derived bySherief et al. [46]. The uniqueness theorem, reciprocity theoremand a variational principle on this theory are also provided in thesame article.

In the current work, a new model of time-fractional derivativeof order m has been considered in the context of micropolar gener-alized thermo-viscoelasticity theory with two temperatures. Thework is articulated as follows: Section 2 is devoted to the basicequations and constitutive relations of the linear theory of frac-tional order micropolar generalized thermo-viscoelasticity for iso-tropic and homogeneous material with two temperatures. InSection 3, we have formulated a two-dimensional problem inCartesian coordinates having the surface of the half-space as theplane z = 0. Section 4 deals with the solution methodology of theformulated problem. Sections 5, 6 and 7 are concerned with thethermal shock application of the problem, deduction of some par-ticular cases of interest and inversion of the transforms respec-tively. To illustrate the theoretical results obtained in thepreceding sections, a numerical example is given in Section 8. Inlast section, we have presented some conclusions which emergednaturally from the present investigation. The application of thepresent model cannot be ruled out in the physical world. This mod-el may be simulated to some geophysical situations. The crude flu-ids and the granular rocks may be best approximated withmicropolar theory of thermoelasticity. It also finds application inthe problem of raptures and cracks and in various problems ofengineering interest. Considering the multifarious applications ofthe above problem and the non-existence of systematic investiga-tion in the context of fractional order micropolar generalized

thermo-viscoelastic medium with two temperatures has moti-vated the authors to study the propagation of waves in such aninteresting medium.

2. Field equations and relations

In the context of fractional order generalized micropolar ther-mo-viscoelasticity theory with two temperatures proposed byShaw and Mukhopadhyay [49], the energy equation and the equa-tions of motion as well as the constitutive relations for a linearhomogeneous and isotropic medium in the absence of body forcesand heat sources are in the form:

K�r2/ ¼ qcE 1þ s0@m

@tm

� �@h@tþ b�1T0 1þ s0

@m

@tm

� �@

@tr �~u; ð1Þ

ðk� þ l�Þrðr:~uÞ þ ðl� þ kÞr2~uþ kr�~/� b�1rh ¼ q@2~u@t2 ; ð2Þ

ðaþ bþ cÞrðr �~/Þ � c r� ðr�~/Þ þ kr�~u� 2k~/

¼ qj@2~/

@t2 ; ð3Þ

rij ¼ k�ur;rdij þ l�ðui;j þ uj;iÞ þ kðuj;i � eijr/rÞ � b�1hdij; ð4Þ

mij ¼ a/r;rdij þ b/i;j þ c/j;i: ð5Þ

The relation between conductive and thermodynamical tempera-tures is expressed as

/� h ¼ ar2/ ð6Þ

Eqs. (1)–(6) reduce to fractional order micropolar thermo-visco-elasticity theory with one temperature when a = 0. Puttinga = b = c = k = 0, the equations reduce to the fractional order ther-mo-viscoelasticity theory with two temperatures, while for m = 1and a = 0, the above equations are same as established by Sheriefet al. [24]. In this particular case, Eq. (1) is modified to the form

K�r2h ¼ qcE 1þ s0@

@t

� �@h@tþ b�1T0 1þ s0

@

@t

� �@

@tr:~u : ð7Þ

In the above fractional order heat conduction Eq. (1), we havetaken into consideration the following definition of fractional orderderivative

@m

@tm f ðx; tÞ ¼f ðx; tÞ � f ðx;0Þ m! 0

I1�m @@t f ðx; tÞ 0 < m < 1

@f ðx;tÞ@t m ¼ 1

8><>: ; ð8Þ

where Im is the Riemann–Liouville integral operator of order m,which commutes i.e.

ImInf ðtÞ ¼ InImf ðtÞ ¼ Imþnf ðtÞ;

and is defined as

Imf ðtÞ ¼ 1CðmÞ

Z t

0

1

ðt � sÞ1�m f ðsÞds

3. Problem formulation

A homogeneous, isotropic, fractional order micropolar general-ized thermoelastic solid with two temperatures is considered in anundisturbed state at uniform temperature T0. We shall use the rect-angular Cartesian coordinate system (x,y,z) having the surface ofthe half-space as the plane z = 0, with z-axis pointing inward. Weobtain the solution of the half-space problem whose surface z = 0

454 S. Deswal, K.K. Kalkal / International Journal of Heat and Mass Transfer 66 (2013) 451–460

is traction free and subjected to a uniformly distributed thermalshock. It is assumed that there are no body forces, body couplesor heat sources affecting the medium. The deformation of the med-ium is thus due solely to the thermal shock. We restrict our anal-ysis to x–z plane. Thus all the quantities appearing in Eqs. (1)–(6)are independent of the variable y. So the displacement vector ~uand micro-rotation vector ~/ will have the components

u ¼ ux ¼ uðx; z; tÞ; v ¼ uy ¼ 0; w ¼ uz ¼ wðx; z; tÞ;

~/ ¼ ð0;/2;0Þ: ð9Þ

Substituting Eq. (9) into Eqs. (4), (5), the arising stresses can be ex-pressed as

rxx ¼ ðk� þ 2l� þ kÞ @u@xþ k�

@w@z� b�1h; ð10Þ

rzz ¼ ðk� þ 2l� þ kÞ @w@zþ k�

@u@x� b�1h; ð11Þ

rzx ¼ l� @w@xþ ðl� þ kÞ @u

@z� k/2; ð12Þ

mxy ¼ c@/2

@x; mzy ¼ c

@/2

@z: ð13Þ

Combination of the Eqs. (2), (3), and (9), yields the followingequations of motion

ðk� þ l�Þ @@x

@u@xþ @w@z

� �þ ðl� þ kÞr2u� k

@/2

@z� b�1

@h@x

¼ q@2u@t2 ; ð14Þ

ðk� þ l�Þ @@z

@u@xþ @w@z

� �þ ðl� þ kÞr2wþ k

@/2

@x� b�1

@h@z

¼ q@2w@t2 ; ð15Þ

c r2/2 þ k@u@z� @w@x

� �� 2k/2 ¼ qj

@2/2

@t2 : ð16Þ

Introducing the expressions defined in Eq. (9) into the govern-ing equation of the temperature field and the relation betweenconductive and thermodynamical temperatures, we obtain

K�r2/ ¼ @

@tþ s0

@mþ1

@tmþ1

!ðqcEhþ b�1T0eÞ; ð17Þ

/� h ¼ ar2/; ð18Þ

where e is the cubical dilatation given as

e ¼ div ~u ¼ @u@xþ @w@z

;

and r2 ¼ @2

@x2 þ @2

@z2 is the Laplacian operator.To transform the above equations into non-dimensional forms,

we define the following non-dimensional variables

ðx0; z0Þ ¼ x�

c1ðx; zÞ; ðt0; s00;a00;a01;b

01Þ ¼ x�ðt; s0;a0;a1;b1Þ;

ðu0;w0Þ ¼ qx�c1

b1eT0ðu;wÞ; ðh0;/0Þ ¼ 1

T0ðh;/Þ;

r0ij ¼rij

b1eT0; /02 ¼

qc21

b1eT0/2; m0ij ¼

x�

c1b1eT0mij; ð19Þ

where

x� ¼ qc21cE

K�; c2

1 ¼ðke þ 2le þ kÞ

q:

In order to solve the governing equations, we introduce the po-tential functions q and w. Displacement components u and w areexpressed in terms of these potentials by the following relations:

u ¼ @q@xþ @w@z

; w ¼ @q@z� @w@x

and w¼ð�~UÞy; ð20Þ

where ~Uðx; z; tÞ is the vector potential function.With the aid of expressions (19) and (20), the governing equa-

tions can be put in more convenient form as (dropping the primesfor simplicity)

1þ d0@

@t

� �r2 � @2

@t2

" #q� 1þ b1

@

@t

� �h ¼ 0; ð21Þ

a0 þ a1 1þ a1@

@t

� �� �r2 � @2

@t2

" #w� a0/2 ¼ 0; ð22Þ

r2 � 2a2 � a3@2

@t2

!/2 þ a2r2w ¼ 0; ð23Þ

r2/� @

@tþs0ðx�Þm�1 @

mþ1

@tmþ1

" #h�a4

@

@tþs0ðx�Þm�1 @

mþ1

@tmþ1

" #r2q¼0;

ð24Þ

/� h ¼ g1r2/; ð25Þ

where a0 ¼ kqc2

1; a1 ¼ le

qc21; a2 ¼ kc2

1

cx�2; a3 ¼ qjc2

1c ; a4 ¼

b21eT0

qK�x�, d0 ¼kea0þ2lea1

qc21

and g1 ¼ ax�2

c21: Elimination of h in Eqs. (21) and (25)

results in

1þ d0@

@t

� �r2 � @2

@t2

" #qþ 1þ b1

@

@t

� �ðg1r2 � 1Þ/ ¼ 0: ð26Þ

Also, substituting the value of h from Eq. (25) into Eq. (24), one canobtain

r2 þ ðg1r2 � 1Þ @

@tþ s0ðx�Þm�1 @

mþ1

@tmþ1

( )" #/

� a4@

@tþ s0ðx�Þm�1 @

mþ1

@tmþ1

( )r2q ¼ 0: ð27Þ

4. Solution of the problem

Following the solution methodology through integral transformtechnique, we now operate Laplace and Fourier transforms on Eqs.(22), (23), (26), and (27). The Laplace and Fourier transforms of afunction f(x,z, t) with parameters s and n are defined by therelations

�f ðx; z; sÞ ¼Z 1

0f ðx; z; tÞe�stdt; ð28Þ

and f ðn; z; sÞ ¼Z 1

�1

�f ðx; z; sÞeinxdx; ð29Þ

where over-bar and over-cap denote the Laplace and Fouriertransforms respectively. Applying the above transforms underhomogeneous initial conditions on Eqs. (22), (23), (26), and(27) and then eliminating / and /2 from the resulting expres-sions, we obtain the following system of ordinary differentialequations:

S. Deswal, K.K. Kalkal / International Journal of Heat and Mass Transfer 66 (2013) 451–460 455

ðD4 þ L1D2 þM1Þq ¼ 0; ð30Þ

ðD4 þ L2D2 þM2Þw ¼ 0; ð31Þ

where

L1 ¼� g1 g2ð2g1n

2 þ 1Þ þ s2g1

� þ n2ð1þ d0sÞ þ g3

�g1g1g2 þ ð1þ d0sÞ ;

M1 ¼g1ð1þ g1n

2Þðs2 þ n2g2Þ þ n2g3

g1g1g2 þ ð1þ d0sÞ ;

L2 ¼ � 2ðn2 þ a2Þ þ a3s2 þ s2 � a0a2

fa0 þ a1ð1þ a1sÞg

� ;

M2 ¼ n2ðn2 þ 2a2 þ a3s2Þ þ s2ð2a2 þ a3s2Þ þ n2ðs2 � a0a2Þfa0 þ a1ð1þ a1sÞg ;

and

g1 ¼ s 1þ s0

x�ðx�sÞm

n o; g2 ¼ fð1þ d0sÞ þ a4ð1þ b1sÞg;

g3 ¼ ð1þ d0sÞn2 þ s2: ð32Þ

The solutions of Eqs. (30) and (31) can be expressed as

q ¼ a�1 expð�k1zÞ þ a�2 expð�k2zÞ ¼X2

n¼1

a�ne�knz; ð33Þ

/ ¼ b�1 expð�k1zÞ þ b�2 expð�k2zÞ ¼X2

n¼1

b�ne�knz; ð34Þ

w ¼ a�3 expð�k3zÞ þ a�4 expð�k4zÞ ¼X4

p¼3

a�pe�kpz; ð35Þ

/2 ¼ b�3 expð�k3zÞ þ b�4 expð�k4zÞ ¼X4

p¼3

b�pe�kpz; ð36Þ

where the terms containing exponentials of growing nature in thespace variable z have been discarded due to the regularity conditionof the solution at infinity and k2

1;2 and k23;4 are the roots of the char-

acteristic equations

k4n þ L1k

2n þM1 ¼ 0; ðn ¼ 1;2Þ ð37Þ

k4p þ L2k

2p þM2 ¼ 0; ðp ¼ 3;4Þ ð38Þ

respectively satisfying the relations

k21 þ k2

2 ¼ �L1; k21k

22 ¼ M1; ð39Þ

k23 þ k2

4 ¼ �L2; k23k

24 ¼ M2: ð40Þ

Here a�1; a�2; a�3 and a�4 are constants to be determined by applyingthe appropriate boundary conditions and the coefficientsb�1; b�2; b�3 and b�4 are defined by the relations

b�n ¼a4g1ðk2

n � n2Þa�nð1þ g1g1Þðk

2n � n2Þ � g1

¼ Hna�n; ðn ¼ 1;2Þ

b�p ¼1a0fa0 þ a1ð1þ a1sÞgðk2

p � n2Þ � s2h i

a�p ¼ Hpa�p: ðp ¼ 3;4Þ

Now to achieve the solution of thermodynamical temperaturedistribution, we will make use of Eq. (25). Operating Laplace andFourier transforms on this equation, one can obtain

h ¼ f1� g1ðD2 � n2Þg/: ð41Þ

Substitution of Eq. (34) into above equation yields

h ¼X2

n¼1

c�ne�knz; ðn ¼ 1;2Þ ð42Þ

where

c�n ¼ f1� g1ðk2n � n2Þgb�n:

System of Eqs. (33)–(36) together with (42) constitutes the generalsolution in the Laplace–Fourier transform domain of two-dimen-sional problems in Cartesian coordinates. This solution dependson the parameters a�1; a�2; a�3 and a�4. In order to apply this formu-lation to the solution of a specific problem, the boundary conditionsof the problem are used to evaluate the above four quantities. Thisenables us to find the solution for many specific problems in the La-place–Fourier transform domain.

5. Thermal shock application

We consider the problem of a half-space X, which is defined as

X ¼ fðx; y; zÞ : �1 < x <1; �1 < y <1; 0 6 z <1g:

The boundary of the half-space (z = 0) has no traction anywhere andis acted upon by a uniformly distributed thermal source of magni-tude /0, which depends on the coordinate x and the time t. Mathe-matically, these boundary conditions can be expressed as

rzzðx; 0; tÞ ¼ rzxðx;0; tÞ ¼ mzyðx;0; tÞ ¼ 0; ð43Þ

/ðx;0; tÞ ¼ /0HðjLj � xÞe�bt; ð44Þ

where H(.) is the Heaviside unit step function. This means that heatis applied on the surface of the half-space on a narrow band ofwidth 2L surrounding the x-axis to keep it at temperature /0, whilethe rest of the surface is kept at zero temperature.

Invoking the previously defined transforms to the boundary

conditions (43) and (44) alongwith /00 ¼/0T0

and plugging the

expressions (33)–(36) of the variables considered, we obtain thefollowing expressions for the displacement components, stressesand temperature fields as

u ¼ � 1D

inðD1e�k1z þ D2e�k2zÞ þ k3D3e�k3z þ k4D4e�k4z �

; ð45Þ

w ¼ � 1D

k1D1e�k1z þ k2D2e�k2z � inðD3e�k3z þ D4e�k4zÞ �

; ð46Þ

rzx ¼1D

inb5ðk1D1e�k1z þ k2D2e�k2zÞ þ f3D3e�k3z þ f4D4e�k4z �

; ð47Þ

rzz ¼1D

f1D1e�k1z þ f2D2e�k2z � inb5ðk3D3e�k3z þ k4D4e�k4zÞ �

; ð48Þ

mzy ¼ �a5

Dðk3H3D3e�k3z þ k4H4D4e�k4zÞ; ð49Þ

/ ¼ 1D½H1D1e�k1z þ H2D2e�k2z�; ð50Þ

h ¼ 1Df1� g1ðk

21 � n2ÞgH1D1e�k1z þ f1� g1ðk

22 � nÞgH2D2e�k2z

�;

ð51Þ

where

D¼ðf3k4H4� f4k3H3ÞðH2f1�H1f2Þþn2k3k4b5ðb1�b2ÞðH4�H3ÞðH1k2�H&2k1Þ;

D1 ¼ �f ðn; sÞ½f2ðf3k4H4 � f4k3H3Þ � n2k2k3k4b5ðb1 � b2ÞðH4 � H3Þ�;

456 S. Deswal, K.K. Kalkal / International Journal of Heat and Mass Transfer 66 (2013) 451–460

D2 ¼ f ðn; sÞ½f1ðf3k4H4 � f4k3H3Þ � n2k1k3k4b5ðb1 � b2ÞðH4 � H3Þ�;

D3 ¼ �if ðn; sÞnk4H4b5ðf1k2 � f2k1Þ;

D4 ¼ if ðn; sÞnk3H3b5ðf1k2 � f2k1Þ;

fn ¼ ðb1 þ b0g1HnÞk2n � ðb2 þ b0g1HnÞn2 � b0Hn; ðn ¼ 1;2Þ

fp ¼ b4ðk2p � HpÞ þ b3ðn2 þ HpÞ; ðp ¼ 3;4Þ

f ðn; sÞ ¼ 2/0 sinðnLÞnðsþ bÞ ; b0 ¼ ð1þ b1sÞ; b1 ¼ ð1þ d0sÞ;

b2; ¼ ð1� a0 � 2a1Þð1þ a0sÞ; b3 ¼ a1ð1þ a1sÞ; b4 ¼ a0 þ b3;

b5 ¼ b3 þ b4; a5 ¼cx�2

qc41

: ð52Þ

6. Particular cases

6.1. Case I: without viscous effect

In order to discuss the problem of wave propagation in the frac-tional order micropolar generalized thermoelasticity with twotemperatures, it is sufficient to set zero value to the viscous relax-ation times, i.e., a0 = a1 = 0. Accordingly, we have k� ¼ ke;

l� ¼ le; b�1 ¼ b1e; b1 ¼ d0 ¼ 0. Taking into consideration the abovementioned modifications, the corresponding expressions of thephysical fields can be obtained from the Eqs. (45)–(52).

6.2. Case II: with one temperature

By setting a = 0 and consequently g1 = 0 in the governing equa-tions, we get the required expressions for different distributionsfrom Eqs. (45)–(52) alongwith the expressions defined in (52) forone temperature case, i.e., the case when conductive temperaturecoincides with the thermodynamical temperature.

-1

0

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

men

t

zm=0.1m=0.75m=1.0

6.3. Case III: without fractional effect

To reduce the problem to that of micropolar generalized ther-mo-viscoelasticity with two temperatures, we just disregard thefractional values of m which is the fractional order parameter inthe generalized heat conduction equation. The correspondingexpressions for displacements, stresses and temperature fieldscan be procured from the Eqs. (45)–(52). If we also ignore the ef-fects of viscosity and two temperature parameter in this limitingcase, then our results are in quite good agreement with thoseachieved by Kumar and Deswal [26] by making slight modifica-tions in the boundary conditions.

-6

-5

-4

-3

-2

norm

al d

ispl

ace

Fig. 1. Distribution of displacement for various values of m.

7. Inversion of the transforms

The transformed components of displacements, temperaturesand stresses can be formally expressed as function of z and theparameters of Laplace and Fourier transforms s and n respectivelyand hence are of the form f ðn; z; sÞ. In order to obtain the solution ofthe problem in the physical domain, we invert the double trans-forms in Eqs. (45)–(52) by adopting the methodology of Youssef[50].

8. Numerical results and discussion

In order to illustrate the contribution of fractional parameter,viscosity effect and two temperature parameter on the field vari-ables, a numerical analysis is carried out. Material chosen for thispurpose is magnesium crystal, the physical data for which is givenas [26,51,52]

q ¼ 1:74� 103 kg m�3; ke ¼ 9:4� 1010 kg m�1s�2;

le ¼ 4:0� 1010 kg m�1s�2;

k ¼ 1:0� 1010 kg m�1s�2; c ¼ 0:779� 10�9 kg ms�2;

j ¼ 0:2� 10�19 m2;

a ¼ 0:074� 10�15 m2; at ¼ 2:36� 10�5K�1;

K� ¼ 2:510 W m�1K�1;

cE ¼ 9:623� 102 J kg�1K�1; T0 ¼ 293 K; s0 ¼ 0:02 s;a0 ¼ 0:06 s; a1 ¼ 0:09 s:

The values of other non-dimensional parameters arising in the pres-ent analysis are taken as /0 = 10, L = 1, b = 1.

Considering the above physical data, non-dimensional fieldvariables have been evaluated and results are presented in theform of graphs at different positions of z at t = 0.1 and x = 1.0. Fromapplication point of view, we have divided the graphs into two cat-egories. First category (Figs. 1–5) depicts the influence of fractionalparameter m (0.1,0.75,1.0) on the distribution of field variables.The investigation into the effect of viscosity and two temperatureparameter has been carried out in the second category (Figs. 6–10). In this category, all the field quantities have been examinedfor three different cases: (i) fractional order micropolar thermo-viscoelasticity with two temperatures (FMTVT, solid line), (ii) frac-tional order micropolar thermoelasticity with two temperatures(FMTT, dashed line) and (iii) fractional order micropolar thermo-viscoelasticity (FMTV, dotted line). All the figures in second cate-gory are displayed at m = 0.1.

8.1. Category I

Fig. 1 represents the spatial variation of displacement compo-nent w with distance z for different values of fractional orderparameter taking the distance axis 0 6 z 6 1:7. Displacement fieldstarts with the maximum value (in magnitude) in all the threecases, then it shifts from negative into positive gradually and at last

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ther

mod

ynam

ical

tem

pera

ture

z

m=0.1m=0.75m=1.0

Fig. 2. Distribution of thermodynamical temperature for various values of m.

0

1

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4

cond

uctiv

e te

mpe

ratu

re

z

m=0.1m=0.75m=1.0

Fig. 3. Distribution of conductive temperature for various values of m.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

norm

al f

orce

str

ess

z

m=0.1m=0.75m=1.0

Fig. 4. Distribution of force stress for various values of m.

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

tang

entia

l co

uple

str

ess

z

m=0.1m=0.75m=1.0

Fig. 5. Distribution of couple stress for various values of m.

-7

-6

-5

-4

-3

-2

-1

0

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

norm

al d

ispl

acem

ent

z FMTVTFMTTFMTV

Fig. 6. Viscosity and two temperature parameter effects on displacement field.

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ther

mod

ynam

ical

tem

pera

ture

z

FMTVTFMTTFMTV

Fig. 7. Viscosity and two temperature parameter effects on thermodynamicaltemperature.

S. Deswal, K.K. Kalkal / International Journal of Heat and Mass Transfer 66 (2013) 451–460 457

becomes zero at the heat wave front. Effect of fractional parameterm is quite pertinent on displacement field and can be easily noticedfrom the figure. Increment in the value of fractional parameter hascaused decrement in the numerical values of displacement field.Hence, it has a decreasing effect. Figs. 2 and 3 have been plottedto observe the variations of thermodynamical and conductive tem-peratures. Both the fields have similar behavior qualitatively hav-ing some difference in magnitudes. However, magnitude ofvariation of temperature fields at small values of m is larger as

compared to the case when m = 1 which clearly indicates thatthe fractional parameter m has a decreasing effect. When the ther-mal source irradiates the surface of the half-space, the temperaturerise velocity becomes very fast because of the high energy intensityin a very short time, causing the temperature gradient to increasedramatically. As a result, temperature fields attain their peak val-ues in the beginning. As time passes, the temperature rise velocityand the temperature gradient decrease, so the temperature fields

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2 1.4

cond

uctiv

e te

mpe

ratu

re

z

FMTVTFMTTFMTV

Fig. 8. Viscosity and two temperature parameter effects on conductivetemperature.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

norm

al f

orce

str

ess

z

FMTVTFMTTFMTV

Fig. 9. Viscosity and two temperature parameter effects on force stress.

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

tang

entia

l cou

ple

stre

ss

z

FMTVTFMTTFMTV

Fig. 10. Viscosity and two temperature parameter effects on couple stress.

458 S. Deswal, K.K. Kalkal / International Journal of Heat and Mass Transfer 66 (2013) 451–460

weaken to a comparatively small value and ultimately reach tosteady state after some distance z P 1:4. Thus, it can be seen thatthe thermodynamics is a short time effect.

Variation in normal force stress with spatial coordinate z hasbeen displayed in Fig. 4. It starts with a zero value which is com-pletely in agreement with the boundary conditions. At the

beginning, the vibration amplitude quickly rises to its maximumvalue, which is called the peak deflection. Then the amplitudedamps quickly to near zero. After that, the vibration weakensslowly. It can also be noticed from the plot that the stress distribu-tion for fractional order theory (m = 0.1, m = 0.75) have large valuesin comparison to Lord–Shulman theory (m = 1.0) which illuminatesthat the fractional parameter is having a significant decreasing ef-fect on the profile of force stress field. Another important phenom-enon observed is that solution vanishes outside a bounded regionof space which shows the existence of wave front. This is the differ-ence between the hyperbolic heat conduction model and Fourierheat conduction model. Fig. 5 shows the variation of tangentialcouple stress with location z for three values of fractional parame-ter. In this figure all the curves have coincident starting point withvalue zero which leads to satisfy the boundary conditions. It can beseen that the maximum impact zone of fractional parameter m isaround z = 0.1 and this impact dies out as we move away fromthe boundary.

8.2. Category II

Fig. 6 has been plotted to examine the effects of viscosity andtwo temperature parameter on the pattern of displacement field.Displacement field starts with negative values having magnitudes�5.3138, �4.7018 and �6.0557 for the theories FMTVT, FMTT andFMTV respectively, which confirms the significant impacts of vis-cosity and two temperature parameter on displacement. For thenon-viscous solid, absolute variation of this function is found tobe smaller than that of a viscous solid and this difference is quitepertinent in the range z 6 0:8. This is attributed to the fact thatthere is a large internal friction among the molecules of the solidat the time of application of load and the internal friction decreaseswith the passage of time. Two temperature parameter has also af-fected displacement field significantly and it acts to increase themagnitude.

The dynamic effects of viscosity and two temperature parame-ter on temperature fields, namely, thermodynamical temperatureand conductive temperature are well expressed in the Figs. 7 and8. It is noted that the values of temperature fields are maximumfor all the three cases in the vicinity of source which is physicallyreasonable and then diminish to zero gradually. From the physicalstandpoint, when the upper surface of the half-space is exposed tothe thermal source, heat flux passes each infinitesimal element ofthe half-space sequentially. According to the energy conservationlaw, one portion of heat flux is absorbed by the element and makesan increase in its intrinsic energy, which is embodied as the stepincrease of temperature. The other portion of the heat flux contin-ues to propagate due to the temperature gradient in the form ofheat conduction. From the standpoint of mathematics, the non-Fourier thermal conduction equation is a damped wave equation,with the coefficient of @h

@t representing the amount of damping. Thisis the difference between the Fourier and non-Fourier heat conduc-tion. Here the rate of decay is faster in the case of FMTVT theory ascompared to FMTV theory which is again faster in FMTT theory. Itis also observed that due to the presence of viscosity term, themagnitude of temperature fields decreases.

Fig. 9 is drawn to observe the effects of viscosity and two tem-perature on force stress against distance z by taking m = 0.1. As ex-pected, stress field is having a coincident starting point of zeromagnitude for all the three cases which agrees with the boundaryconditions. It is noticed that the mechanical relaxation times, i.e.,time dependence of thermoelastic parameters has a significant ef-fect on force stress distribution. Stress field attains significantlylarge values in case of viscous solid as compared to non-viscous.However, amplitude of vibrations gets suppressed due to the pres-ence of two temperature parameter and the rate of decay becomes

S. Deswal, K.K. Kalkal / International Journal of Heat and Mass Transfer 66 (2013) 451–460 459

slower. Hence, it has a decreasing effect and the maximum effectzone is around 0:2 6 z 6 1:2. It is further observed in figure thatthe time to reach steady state is greater for viscous case which sup-ports the physical facts. Fig. 10 depicts the variation of couplestress versus distance z by taking distance axis 0 6 z 6 1:0. Ascan be seen from the plot, the couple stress is compressive in nat-ure and it shows negligible impact of both the factors near theboundary of the half-space. For all the three models, couple stressfield shows the same nature. Figure also indicates that the effect oftwo temperature parameter is more significant than the effect ofviscosity term and couple stress distribution has non-zero valuesonly in a bounded region of the half-space.

9. Conclusion

The main goal of this work is to introduce a new mathematicalmodel of heat conduction with time fractional order m for isotropicmaterial as an improvement and progress in the field of micropolarthermoelasticity. The reason of this development is that a frac-tional model can describe simply and elegantly the complex char-acteristics of a thermoelastic material. According to the aboveanalysis, we can conclude the following points:

(i) All the fields are restricted in a limited region which is inaccordance with the notion of generalized thermoelasticitytheory and supports the physical facts.

(ii) The effect of fractional parameter on all the studied fields isvery much significant.

(iii) New classification of materials must be constructed accord-ing to the fractional parameter that describes the ability ofthe material to conduct heat.

(iv) Presence of viscosity term has strongly affected all the phys-ical fields.

(v) It is also observed that the theories of coupled thermoelas-ticity, Lord–Shulman theory [25] and Sherief et al. [46] frac-tional order theory can be obtained as limiting cases.

(vi) From the distributions of temperature, we have found awave type heat propagation in the medium. With the pas-sage of time, the heat wave front moves forward with a finitespeed, which proves that the generalized micropolar ther-moelasticity theory with fractional heat transfer is very closeto the behavior of elastic materials.

(vii) Two temperature parameter has a salient effect on all thefields.

The results presented in this paper will prove useful forresearchers in material science, designers of new materials, lowtemperature physicists, as well as for those working on the devel-opment of the hyperbolic thermoelasticity theory with fractionalderivative heat transfer. Applications of this problem are found inthe fields of seismology, geomechanics, earthquake engineeringand soil dynamics etc., where the interest is about various phe-nomena occurring in earthquakes and measuring of displacement,stresses and temperature field due to the presence of certainsources. Problem assumes great significance in an earthquakepreparation region when we think of the variation in particle mo-tion as a possible precursor for earthquake prediction. The intro-duction of viscosity term and two temperature parameter to themicropolar generalized thermoelastic medium with fractional heatconduction provides a more realistic model for these studies.

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