 Journal of Applied Mathematics and Physics, 2013, 1, 110-120 Published Online November 2013 (http://www.scirp.org/journal/jamp) http://dx.doi.org/10.4236/jamp.2013.15017 Open Access JAMP Fractional Order Two Temperatur e Thermo-Elastic Behavior of Piezoelectric Materials Essam Bassiouny1,2, Refaat Sabry1,3,4 1Department of Mathematics, Faculty of Science and Humanitarian Studies, Salman Bin AbdulAziz University, Al Kharj, KSA 2Department of Mathematics, Faculty of Science, Fayoum University, Fayoum, Egypt 3Theoretical Physics Group, Department of Physics, Faculty of Science, Mansoura University, Damietta Branch, New Damietta, Egypt 4International Centre for Advanced Studies in Physical Sciences, Faculty of Physics and Astronomy, Ruhr University Bochum, Bochum, Germany Email: esambassiouny@yahoo.com, Sabry_nonlinear@yahoo.com Received July 13, 2013; revised August 21, 2013; accepted August 27, 2013 Copyright © 2013 Essam Bassiouny, Refaat Sabry. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT A new mathematical model of time fractional order heat equation and fractional order boundary condition have been constructed in the context of the generalized theory of thermo piezoelasticity. The governing equations have been ap-plied to a semi infinite piezoelectric slab. The Laplace transform technique is used to remove the time-dependent terms in the governing differential equations and the boundary condition. The solution of the problem is first obtained in the Laplace transform domain. Furthermore, a complex inversion formula of the transform based on a Fourier expansion is used to get the numerical solutions of the field equations which are represented graphically. Keywords: Fractional Order; Two Temperature; Generalized Thermoelectricity; Weak Diffusion; Thermal Loading; Piezoelectric Materials; Ceramics 1. Introduction Chen and Gurtin [1-3] have formulated a theory of heat conduction in deformable bodies, which depends upon two distinct temperatures, the conductive temperature  and the dynamical temperature T. Regarding time inde-pendent situations, the difference between these two temperatures is proportional to the heat supply. However, in the absence of any heat supply, the two temperatures are identical [1,2]. On the other hand in time dependent problems, particularly for wave propagation problems, the two temperatures are generally different regardless of the presence of heat supply. The two temperatures T, and the strain are found to have representations in the form of a traveling wave plus a response, which occur instantaneously throughout the body . Warren and Chen  investigated the wave propagation in the two- temperature theory of thermoelasticity, but Youssef  investigated this theory in the context of generalized thermoelasticity. Because the non-local property of fractional order of differential equations (FODE), FODE becomes prominent and extensively used in many applications in fluid me- chanics, physics, engineering, viscoelasticity and many other fields. The presence of the fractional order operator in the differential equations affects the history of the system, which means that the next states of the system will depend, on the current state and also upon all of its previous states, making it more realistic: Caputo , Mainardi  and Podlubny . FODE has been used successfully in modeling of various physical phenomena and in many applications such as chemistry, biology, electronic, wave propagation and viscoelasticity Hilfer , Caputo and Mainardi , Caputo , Bagley and Torvik , Koeller  and Rossikhin and Shitikova . In the second half of the 19th century, both of the the-ory of fractional derivatives and integrals were estab-lished. The first application of fractional derivatives was applied by Abel to solve an integral equation that arises in the formulation of the Tautochrone problem [16,17]. Various definitions and approaches of fractional de-rivatives have become the main purpose of many studies E. BASSIOUNY, R. SABRY 111[18,19]. Kimmich  study time—fractional diffusion—wave equation and use the Riemann-Liouville fractional inte- gral as follows:   101d,0 2,,ttfIftft0 (1) where  is the Gamma function. Fujita [20,21] considered a fractional order heat wave equation for the case 12 obtained from the non local constitutive equation for the heat flux components in the form iq1, 12iiqIT (2) Povestenko [22,23] used the heat Equation (2) to study the theories of thermal stresses based on space-time frac-tional telegraph equations. To eliminate the paradox of the instantaneous propa-gation of heat, Cattaneo  introduced a law of heat conduction to replace the classical Fourier law of heat conduction. The propagation of discontinuities of solu-tions in this theory was investigated by Ezzatt and Kara-many . Ezzatt and Karamany [25-28] established a new model of fractional heat equation based on a Taylor expansion of time-fractional order. They studied the non-homoge- neous anisotropic elastic solid using two new models and they considered the uniqueness theorem in linear frac-tional two temperatures thermoelasticity and the theory of a perfect conducting thermoelastic medium. Also, they constructed a new model of electro-thermoelasticity in the context of a new consideration of heat conduction with fractional order. Sherief et al.  used the following form of the heat conduction law , 01iioqTqtt 1 (3) and derived the governing equations of the fractional order theory of thermoelasticity using Caputo  defini-tion of fractional derivatives of order 0 of ab-solutely continuous function ft given by:  1ddftIftt (4)   10dttsIftfs s (5) where I is the fractional integral of the function ft of order  defined by . In the limit as 1, Equation (3) can be reduced to Cattaneo law . A new formula of heat conduction has been considered in the context of fractional integral operator defined by Youssef  who introduces the following form of heat conduction law 1, 02ioi iqq IT  (6) Taking into consideration the works of Fujita [20,21] and Povestenko [22,23], Youssef proved the uniqueness of the solutions in this case. In the present work a model for generalized ther-mopiezoelasticity has been constructed in the context of the fractional heat equation where 020 to describe different types of diffusion where 1 corresponds to weak diffusion, 1 corresponds to normal diffu-sion, 12 corresponds to strong diffusion and 2 corresponds to ballistic diffusion. This is used to investigate the propagation of thermal wave through a semi infinite slab subjected to thermal loading of frac-tional order of exponential type applied for finite period of time. 2. Governing Equations In the absence of body force, free charge and inner heat sources, we consider generalized thermo-piezoelectric governing differential equations Youssef  and Yous- sef and Bassiouny  follows: Equations of motion: 2,2iij jut (7) Equation of entropy increment (in the absence of inner heat source): ,,ii oqTt  (8) Stress-strain-temperature: ,ijijkl klkijkijce hD (9) Gauss equation and electric field relation: ,0iiD (10) ,iEvi (11) iikl klikkiEhe Dd (12) Equation of entropy density: ij ijiiedDcT (13) Strain-displacement relations: ,,12ijijj ieuu (14) The heat conduction 212iioE oICTtt   e (15) Open Access JAMP E. BASSIOUNY, R. SABRY 112 where   101d, 02,, 0ttfIftft (16) The thermodynamical temperature  relates with the conductive temperature  by the relation ,iia (17) in which is the two-temperature parameter. 0aIn the above equations, a comma followed by a suffix denotes material derivatives and a superposed dot de-notes the derivatives with respect to time. 3. One Dimension Formulation Consider a semi-infinite piezoelectric rod occupying the region . At the near end a uniform flow of heat is supplied to the rod during a finite period of time. All the state functions field will depend only on the dimension x and the time t. We assume the following form for the displacement component: 0x,, 0xyuuxtuuz (18) We consider the following forms of the linearized ba-sic equations in one-dimensional formulation: 2222ux2uxt  (19) 2uhDx  (20) 22122oEoICTtxt  e (21) 22ax (22) ,uxtex (23) 0Dx (24) ,vEx (25) where 32,t  t is the coefficient of the linear thermal expansion, is the coefficient of ther-mal conductivity and x is the coordinate taken along the rod. It is convenient now to introduce the following dimen-sionless variables: 2222, , , ,2, , , ,2, , , 2ooooooooo ooEoooTTucutct TTtctxcx cTCqhqDD ckc Tk      (26) From Gauss’s law, since there is no free charge inside the piezoelectric rod we have 0,Dx (27) which gives constD (28) Substituting from Equation (26) into Equations (19)- (25) and dropping the primes for convenience, we obtain the following set of non-dimensional equations Youssef  and Youssef and Bassiouny  follows: 2222e22exxt (29) eD (30) 22122oIetxt  (31) and the following relation between the conductive tem-perature and the thermodynamical one: 22x (32) where 2222,,and oEacCx   (33) The boundary conditions are: 10,toett (34)  ,0,0, ,0,0,0,0,0,0tetettttt 2 (35) where 0, while the initial conditions are as-sumed to be: ,0 0,,0 0,,0 0,0xex xx (36) Applying the Laplace transform defined by:  0dstLftfse ft t (37) to both sides of Equations (29)-(32), we obtain: Open Access JAMP E. BASSIOUNY, R. SABRY 11322222ddddesexx (38) Des (39) 22121d ,doo2sssssx e (40) 22d.dx (41) Using Equations (29) and (30) with the definitions (23) and (37) we can obtain 2ddusx (42) where s denotes the complex argument related to the Laplace transform. The transformed boundary conditions take the forms 10,toesFst (43) where  11oFsss (44) while the Equations (35) become    ,0, 0,,0,0, 0,0,sesesss  , (45) and the corresponding transformed initial conditions of the Equations (36) assume the form: ,0,0,00, 0xex xx (46) Eliminating  between Equations (40) and (41), we get: 2DLLe (47) where 222212 dand doossLLs Dxsss  Substituting from Equation (47) into Equation (41) we obtain 1LLe (48) Using Equation (47) we can easily eliminate  be-tween Equation (38) and (48) to obtain 2De MNe (49) where  2111and 1LLMMs LsLLNNs L  (50) Solving Equations (47) and (49) together we get the following fourth order equation 42 0kakb (51) where 21111oossass (52) 221111 1oosss sbss   (53) It is worth mentioning here that the roots of equation (51) are functions of s and assume the forms: 221244, 22aa baa bkk  (54) Thus the solutions of the Equations (47) and (49) sat-isfying the boundary conditions at infinity are: 1122xkAeA exk (55) 1122xkeBe Bexk (56) where 121,,AAB and 2 are coefficients depending on s to be determined using the boundary conditions (43) and (45). B21222121oFsANksNs kks (57) 22122121oFsANksNs kks (58) While the constants 12 are related to the con-stants ,BB12,AA according to the following relations:  211,1iiioooABkssss 1,2i (59) Substituting from Equations (57) and (58) into the Equations (55) and (56) the heat conduction and the strain field in the Laplace domain take the forms: 1222122221,11xk xkooFsxs NskkssNkessNke  (60) 122112221,11111oxkooxkooexs sAk ssseAk ssse  (61) Using the expressions of  and of e from Equa-tions (60) and (61) to find the thermodynamical function Open Access JAMP E. BASSIOUNY, R. SABRY 114  and the stress in the Laplace transformed domain, thus Equations (39) and (48) become: 1212,xkxk Dxs ees (62) 112,2xkxkxse e (63) 112,2xkuxs ueuexk (64) where 21111iii ooAks ss (65) 21ii iAk (66) 2iuskii (67) Equations (60)-(64) are the complete solutions of the ,, ,e and u, respectively, in the Laplace trans-formed domain. In order to invert the Laplace transform, we adopt a numerical inversion method based on a Fourier series expansion Honig . Using this method, the inverse ft of the Laplace transform fs is approximated by  11111ππReexp, 202,ct Nkeikftfcf ctttt1iktt(68) where N is a sufficiently large integer representing the number of terms in the truncated Fourier series, chosen such that 111ππexp ReexpiNiN tctf ctt (69) where 1 is a prescribed small positive number that corresponds to the degree of accuracy required and Re is the real part. The parameter c is a positive free parameter that must be greater than the real part of all the singulari-ties of fs. The optimal choice of c was obtained ac-cording to the criteria described in . 4. Numerical Results and Discussion To investigate the role of various physical parameters involved in the current problem, we have investigated the role of varying the angular frequency of thermal vibra-tion on different system parameters, where it is ob-served that increasing increases the heat conduction, as depicted in Figure 1(a). Such behavior is in accor- dance with the fact that increasing the thermal vibrations will increase the kinetic energy of the ceramic slab molecules and results in increasing the amount of heat transferred by conduction mechanism. Variation of the  Figure 1. Variation of (a) Heat conduc tion  (b) Thermo- dynamical temperature and (c) Displacement against x for various values of angular frequency of thermal vibration  at 0=0.3 =0.4=0.30.75t. thermodynamical temperature for different , shows a peculiar behavior, as illustrated in Figure 1(b) and re- flected through the variation of the displacement distri-bution shown in Figure 1(c). It is found that increasing the thermal vibrations will increase the amplitude of the thermodynamic temperature. The effect of increasing the thermal vibrations frequency on the stress and strain shows the same qualitative behavior, as illustrated in Open Access JAMP E. BASSIOUNY, R. SABRY Open Access JAMP 115Figure 2. Both of the stress and strain decreases initially by increasing x, but at a certain critical point further in-crease in x increases the stress and strain. It is observed in Figure 2 that the thermal vibration increases the stress and strain in a symmetrical way with respect to a critical point. The effect of increasing time is shown to increase the heat conduction as well as the thermodynamic tempera- ture, as reflected in Figures 3(a) and (b), respectively. In fact it shows the same qualitative behavior of increasing Figure 2. The role of varying angular frequency  on the (a) Stress and (b) Strain at 0=0.3 =0.4=0.30.75t. Figure 3. Variation of (a) Heat conduction  (b) Thermodynamical temperature and (c) Displacement against x for differ-nt value of time at 0=0.2 =0.30.=0.175. e E. BASSIOUNY, R. SABRY 116 the frequency of the thermal vibrations on the heat con- duction, illustrated in Figure 1(a). Such behavior can be strain will be an increasing function in x, as illustrated in Figure 4. The amount of energy delivered to the ceramic explained on the basis that increasing the time of heating the slab will increase the amount of energy delivered to the slab. The amount of energy delivered to the ceramic slab increases the entropy of the thermodynamic tem- perature. The minimum points of the stress and strain curves are shown to be an increasing function in x as time increases. Far from the near end of the slab the ef- fect of time damped as x increases and the stress and slab is a factor of heating time, which is the key answer to such behavior. An inverse proportion is noticed be- tween the value of the fractional order (i.e., the measure of the system memory) and the heat conduction at the near end of the slab, whereas a slight change in the dis- tribution curves is noticed for large values of x, as de- picted in Figure 5(a). The thermodynamic temperature is a decreasing function in the fractional order as shown in Figure 4. The role of varying time on the (a) Stress and (b) Strain at 0=0.2 =0.30.75 =0.1 . Figure 5. Variation of (a) Heat c onduction , (b) Thermodynamical temperature and (c) Displacement against x for differ-ent fractional order parameter at 00.40.2=0.3= 0.1t =. Open Access JAMP E. BASSIOUNY, R. SABRY 117 Figure 5(b). It is apparent that increasing the system ractional order leads to reverse tdisplacement distribution. The amount of energy deliv-erhe effect of heating as in ed to the ceramic slab is affected by the weak conduc- fFigure 5. The qualitative behavior of the displacement distribution is shown to resemble the same behavior as in the case of the effects of thermal angular vibration and time but here; the fractional has a slight effect on the tivity imposed on the material by the system memory retained through the fractional parameter. Such behavior is confirmed through the role of the fractional parameter on the stress and strain curves as displayed in Figure 6. 00.4= 0.2= 0.3= 0.1t.Figure 6. The role of varying fractional order parameter on the (a) Stress and (b) Strain at Figure 7. Variation of (a) Heat c onduction ; (b) Thermodynamic temperature ; (c) Displacement U against x for differ- ent value of the two-temperature parameter at 00.25= 0.75= 0.3= 0.1t. Open Access JAMP E. BASSIOUNY, R. SABRY 118 Figure 8. The role of varying the two-temperature parameter  on the (a) Stress and (b) Strain at 0.25t 0= 0.75= 0.3= 0.1. pThe two temperaturearameter ω, which depends on o distinct temperatures, the conductive temperature antwd the thermodynamic temperature where the difference between these two temperatures is proportional to the heat supply, is found to increase the heat conduction as shown in Figure 7(a). The thermodynamic temperature increases by increasing ω as illustrated in Figure 7(b). The displacement is found to have a critical behavior at 0.4x, as depicted in Figure 7(c) where the displace-ment distribution curves for all values of ω pass by this point and the distribution becomes positive after it. The strain distribution confirms similar qualitative be- havior as the stress distribution as illustrated in Figure 8. Both distributions of the stress and strain are decreased by increasing ω for 0.4critical , as shown in Figure 8. However, for 0.4 the stress and strain distributions remain decreasing funcn the two temperature pa- rameter ω. tions iREFERENCES  P. J. Chen andy of Heat Con- duction InvolZeitschrift M. E. Gurtin, “On a Theorving Two Temperatures,” für Angewandte Mathematik und Physik ZAMP, Vol. 19, No. 4, 1968, pp. 614-627. http://dx.doi.org/10.1007/BF01594969  P. J. Chen, M. E. GurtinNon-Simple Heat Conduction,” Zeits and W. O. 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Time. t:iu: Components of displaceectric potential function. i: The elv2oT: Dimensionless thermoelastiT: The components of dielectric tensor. ikij: The thermal modulus. 32 T . c coupling con-stant. : The angular frequency of thermal vibration. : Kronecker delta function. ijECnstant : Dimensionless mechanical coupling co: The entropy density. ECk: The thermal viscosity. EoCCToTTthat : The dynamical temperature increment such 1ooTTT. ,: Lamé’s constants. : Mass density. ij: Components of stress tensor. xx: The principal stress component. o: One relaxation time parameter. Open Access JAMP