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1 This article was downloaded by:[youssef, Hamdy M.] On: 22 February 2008 Access Details: [subscription number ] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of Thermal Stresses Publication details, including instructions for authors and subscription information: Two-Temperature Generalized Thermopiezoelasticity of Finite Rod Subjected to Different Types of Thermal Loading E. Bassiouny a ; Hamdy M. Youssef b a Mathematical Department, Faculty of Science, Fayoum University, Fayoum, Egypt b Mathematical Department, Faculty of Science, King Saud University, Al-Kharj, Saudi Arabia Online Publication Date: 01 March 2008 To cite this Article: Bassiouny, E. and Youssef, Hamdy M. (2008) 'Two-Temperature Generalized Thermopiezoelasticity of Finite Rod Subjected to Different Types of Thermal Loading', Journal of Thermal Stresses, 31:3, To link to this article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Journal of Thermal Stresses, 31: , 2008 Copyright Taylor & Francis Group, LLC ISSN: print/ x online DOI: / TWO-TEMPERATURE GENERALIZED THERMOPIEZOELASTICITY OF FINITE ROD SUBJECTED TO DIFFERENT TYPES OF THERMAL LOADING E. Bassiouny 1 and Hamdy M. Youssef 2 1 Mathematical Department, Faculty of Science, Fayoum University, Fayoum, Egypt 2 Mathematical Department, Faculty of Science, King Saud University, Al-Kharj, Saudi Arabia In this work the theory of two-temperature generalized thermoelasticity, based on the theory of Youssef is used to solve boundary value problems of one-dimensional finite piezoelectric rod with loading on its boundary with different types of heating. The governing equations are solved in the Laplace transform domain by using a direct approach. The general solution obtained is applied to specific problems of a finite piezoelectric rod subjected to two types of heating: a thermal shock type, and a ramp type. The inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. The conductive temperature, the dynamical temperature, the stress, the strain and the displacement distributions are shown graphically. Keywords: Finite rod; Generalized thermoelasticity; Piezothermoelasticity; Piezoelectric materials; Two-temperature INTRODUCTION It is very well known now that in the classical coupled and uncoupled theories of thermoelasticity, the heat conduction equations are of diffusion type. This means that the heat wave will propagate with infinite speed, which contradicts physical observations. Many theories attempt to generalize and amend the classical theories of thermoelasticity and aim to eliminate the paradox accompanying the infinite speed of heat propagation in the classical theories. Among all these theories, there are two different famous theories of generalized thermoelasticity. The first theory is based on the modified Fourier s Law of heat conduction and allows for one relaxation time. This is the Lord and Schulman theory [1]. The second theory was developed by Green and Lindsay [2]. It modifies both the energy equation and the Duhamel Neuman relation. It admits two relaxation times. There are few investigations of propagation of heat wave in piezoelectric materials especially the propagation of a heat wave in a finite piezoelectric rod. The case of finite piezoelectric rod totally differs from the Received 30 January 2007; accepted 6 August Address correspondence to Hamdy M. Youssef, Faculty of Engineering, Umm Al-Qura University, P.O. Box 5555, Makkah, Saudi Arabia. yousefanne@yahoo.com 233

3 234 E. BASSIOUNY AND H. M. YOUSSEF semi-infinite case, since in the latter elastic waves travel in one direction only, while for the finite rod, there will be reflected elastic waves from both ends. GOVERNING EQUATIONS In the absence of body force, free charge and inner heat sources, the generalized thermo-piezoelectric governing differential equations as in Bassiouny and Ghaleb [3] and Tianhu et al. [4]: Equations of motion: ijj = ü i (1) Equation of entropy production (in the absence of inner heat source): Stress-strain-temperature: Gauss equation and electric field relation: q ii = T 0 (2) ij = c ijkl e kl h kij D k ij (3) D ii = 0 (4) E i = v i (5) E i = h ikl e kl + ik D k d i T (6) Equation of entropy density: = ij e ij + d i D i + ct (7) Strain-displacement relations: e ij = 1 2 u ij + u ji (8) We will assume the following new form of the heat conduction equation which includes the conductive temperature instead of the thermo-dynamical temperature [5]: q i + A ij q j = k ij j (9) is the conductive temperature and it satisfies the relation T = a ii (10) in which a>0 is the two-temperature parameter and k ij are the components of thermal conductivity tensor. In these equations, a comma followed by a suffix denotes material derivatives and a superposed dot denotes the derivatives with respect to time.

4 THERMOPIEZOELASTICITY OF FINITE ROD WITH LOADING 235 ONE-DIMENSIONAL FORMULATION Consider a piezoelectric rod of finite length h. Let one end be raised to a given temperature by a thermal effect to a prescribed temperature of known function, which is then kept constant all the time, while the other end of the rod is fixed and maintained at the initial temperature. At the initial time the rod is at rest, in an undeformed state. For the one-dimensional problem the rod coincides with x-axis, and we assume the following form for the displacement component: u x = ux t u y = u z = 0 (11) We will consider the following forms of the linearized basic equations in onedimensional formulation: u x2 x = u 2 t 2 (12) = + 2 u hd (13) ( k 2 x = 2 t + 0 x 2 t 2 ) C E + T 0 e (14) T = a 2 (15) x 2 e = u (16) x D x = 0 (17) E = v (18) x = t t is the coefficient of the linear thermal expansion and x is the coordinate taken along the rod. It is convenient now to introduce the following dimensionless variables: u = c 0 u t = c 2 0 t = D = h + 2 D + 2 = C E k c 2 0 = T T 0 T 0 = c 2 0 = T 0 T 0 = + 2 t 0 = c2 0 t 0 x = c 0 x (19) From Gauss s law, since there is no free charge inside the piezoelectric rod, we have which gives D x = 0 (20) D = const (21)

5 236 E. BASSIOUNY AND H. M. YOUSSEF Substituting from Eq. (19) into Eqs. (12) (18) and dropping the primes for convenience, we obtain the following set of non-dimensional equations [6]: 2 x 2 = 2 e x 2 2 x = 2 e 2 t 2 (22) = e D (23) ( t t 2 ) + e (24) and the following relation between the conductive temperature and the thermodynamical one: = T = 2 x 2 (25) = C E and = ac We raise one end of the rod by a known function to a prescribed temperature, while the other end is fixed and maintained at the initial temperature such that the initial conditions are assumed to be: ux 0 = 0 0 x h ux 0 = 0 t 0 x h (26) x 0 = 0 0 x h (27) x 0 = x 0 = 0 0 x h (28) We assume that the rod is subjected to heating of a general function Ft and is traction free, so that the boundary conditions take the following forms: Applying the Laplace transform defined by: 0 0t= Ft (29) e0t= 0 0t= 0 (30) Lft = fs = to both sides of Eqs. (22) (25), we obtain: 0 e st ftdt (31) d 2 ē dx 2 d2 dx 2 = s2 ē (32) =ē D s (33) d 2 dx 2 = s + 0s 2 + s + 0 s 2 ē (34)

6 THERMOPIEZOELASTICITY OF FINITE ROD WITH LOADING 237 = d2 (35) dx 2 s denotes the complex argument related to the Laplace transform. The boundary conditions (29) (30) now take the form: 0s= Fs h s = 0 (36) ē0s=ēh s = 0 (37) Eliminating between Eqs. (34) and (35), we are left with: d 2 dx2 = L + Lē (38) L = Ls = s + 0 s s + 0 s 2 Substituting from Eq. (38) into Eq. (35), we obtain = 1 L Lē (39) Using Eq. (38) we can easily eliminate between Eq. (32) and Eq. (39) to obtain d 2 ē = M + N ē (40) dx2 M = Ms = L1 L 1 + L and N = Ns = s2 + L1 L 1 + L (41) Solving Eqs. (38) and (40) together we get the following fourth-order equation k 4 ak 2 + b = 0 (42), a = L + N and b = LN LM. It is worth mentioning here that the roots of Eq. (42) are functions of s. Thus the solutions of Eqs. (38) and (40) will take the following forms: = ē = 2 A i e xk i + B i e xk i (43) i=1 2 C i e xk i + D i e xk i (44) i=1 respectively, ±k 1 ±k 2 are the roots of Eq. (42) and A i, B i, C i, D i are parameters depending on s.

7 238 E. BASSIOUNY AND H. M. YOUSSEF Using the boundary conditions (36) and (37) we get: A 1 = 0L k2 2e2hk 1 k1 2 B 1 = 0 k2 2 L k2 2 e2hk 1 1 k1 2 (45) k2 2 e2hk 1 1 A 2 = 0k1 2 Le2hk 2 k1 2 B 2 = 0 L k1 2 k2 2 e2hk 2 1 k1 2 (46) k2 2 e2hk 2 1 while the relation between the coefficients A i B i and C i D i assume the forms: C i = k2 i LA i L and Substituting from Eqs. (45) (47) into Eqs. (43) and (44) we get 1 = D i = k2 i LB i (47) L = 1 sinhh xk sinhh xk 2 (48) 0 L k 2 2 k 2 1 k2 2 sinhhk 1 and 2 = 0 k 2 1 L k 2 1 k2 2 sinhhk 2 (49) ē = e 1 sinhh xk 1 + e 2 sinhh xk 2 (50) e 1 = 0L k 2 2 k2 1 L Lk 2 1 k2 2 sinhhk 1 and e 2 = 0k 2 1 Lk2 2 L Lk 2 1 k2 2 sinhhk 2 (51) Substituting from Eqs. (48) and (50) into Eq. (39) we get ū = u 1 coshh xk 1 + u 2 coshh xk 2 (52) 1 = 1 1 Le 1 and 2 = 1 2 Le 2 (53) Substituting from Eqs. (50) and (52) into Eq. (33) we get = 1 sinhh xk sinhh xk 2 D s (54) 1 = e = e 2 2 (55) Substituting from Eq. (50) into Eq. (16) after using the Laplace transform, we get ū = u 1 coshh xk 1 + u 2 coshh xk 2 (56)

8 THERMOPIEZOELASTICITY OF FINITE ROD WITH LOADING 239 u 1 = e 1 /k 1 and u 2 = e 2 /k 2 (57) Application I (Thermal shock problem). We consider the finite rod 0 x h at a uniform temperature T 0 with its boundary x = 0 subjected to thermal shock [7]: Ft = F 0 Ht (58) F 0 is a constant representing the strength of the shock on the boundary, and Ht is the Heavyside unit step function. After using the Laplace transform, we have 0 = Fs = F 0 (59) s Thus, we get the complete solution for this application in the Laplace transform domain by substituting Eq. (59) into Eqs. (48) (56). Application II (Ramp-type heating). We consider the finite rod 0 x h at a uniform temperature T 0 with its boundary x = 0 subjected to thermal shock [8]: 0 t 0 F F0t= 0 t 0 <t t t 0 0 F 0 t>t 0 (60) F 0 is constant and t 0 is the ramp-type parameter. After using the dimensionless variables (19) and the Laplace transform defined previously, we have 0 = Fs = F 01 e st 0 t 0 s 2 (61) Thus, we get the complete solution for this type of heating on the Laplace transform domain by substituting Eq. (61) into Eqs. (48) (56). Numerical Inversion of the Laplace Transform In order to invert the Laplace transform, we adopt a numerical inversion method based on a Fourier series expansion, [9] and [10]. By this method the inverse ft of the Laplace transform fs is approximated by [ ( ft = ect 1 t 1 2 fc N + R1 f c + ik ) ( )] ikt exp t 1 t 1 0 <t 1 < 2t (62) k=1

9 240 E. BASSIOUNY AND H. M. YOUSSEF N is a sufficiently large integer representing the number of terms in the truncated Fourier series, chosen such that [ ( expctr1 f c + in ) ( )] int exp t 1 t 1 (63) 1 1 is a prescribed small positive number that corresponds to the degree of accuracy required. The parameter c is a positive free parameter that must be greater than the real part of all the singularities of fs. The optimal choice of c was obtained according to the criteria described in [9]. Numerical Results and Discussion The numerical values of the thermal temperature, the dynamical temperature, stress and strain have been calculated for small time t = 025 and for a wide range of x = 00 uptox = 10. In the calculation process, the following constants are necessary to be known including F 0 = 10= = = 01D= 10 7, and o = 002. The numerical results are displayed graphically [7] and [8]. In Figure 1, we display the conductive temperature of the thermal shock heating application for the values of = 00 and = 01, and we notice that the parameter makes the speed of the wave propagation of the conductive temperature vanish at long distance of x. In Figure 2, we display the dynamical temperature of the thermal shock heating application for the values of = 00 and = 01, and we observe that the parameter makes the speed of the wave propagation of the dynamical temperature vanish at long distance of x. In Figures 3 5, we display the stress, the strain and the displacement respectively of the thermal shock heating application with the different values of = 00 and = 01, and we notice that the parameter has a significant effect. In Figures 6 and 7, we display the conductive temperature and the dynamical temperature respectively of the ramp type heating application with different values of t o = 00, t o = 02, and t o = 03, and we observe that the parameter t o makes the conductive temperature and the dynamical temperature decrease when it increases. Figure 1 The conductive heat distribution with thermal shock heating.

10 THERMOPIEZOELASTICITY OF FINITE ROD WITH LOADING 241 Figure 2 The thermodynamical heat distribution with thermal shock heating. Figure 3 The stress distribution with thermal shock heating. Figure 4 The strain distribution with thermal shock heating.

11 242 E. BASSIOUNY AND H. M. YOUSSEF Figure 5 The displacement distribution with thermal shock heating. Figure 6 The conductive heat distribution with ramp-type heating. Figure 7 The thermodynamical heat distribution with ramp-type heating.

12 THERMOPIEZOELASTICITY OF FINITE ROD WITH LOADING 243 Figure 8 The stress distribution with ramp-type heating. Figure 9 The strain distribution with ramp-type heating. Figure 10 The displacement distribution with ramp-type heating.

13 244 E. BASSIOUNY AND H. M. YOUSSEF In Figures 8 10, we display the stress, the strain and the displacement respectively of the ramp type heating application with different values of t o = 00, t o = 02, and t o = 03 it is evident that the parameter t o has a significant effect it makes the absolute value of the mention fields decreases when it increases. CONCLUSION In the framework of this article, one finds that the theory of twotemperature generalized thermoelasticity has reclaimed the discontinuity points for the temperature, the stress and the strain. It means that the response to the thermal shock or ramp-type heating is not felt instantly at very large distances from the bounding plane of the medium. This is not the case for coupled and uncoupled theories of thermoelasticity, the response to the shock is felt instantly at infinitely far points from the source of disturbances. NOMENCLATURE A ij The components of relaxation time a The two-temperature parameter C E Specific heat at constant strain The elastic constants c ijkl c o D i d i E i e ijkl h ijk k ij q i T T o t t o u i v i T ik ij ij c +2 Longitudinal wave speed The components of electric displacement The pyroelectric constants The components of electric field vector The components of strain tensor The piezoelectric coefficients The components of thermal conductivity The components of the heat flux vector Absolute temperature Reference temperature Time Ramping time parameter Components of displacement vector The electric potential function T o = Dimensionless thermoelastic coupling constant +2 Coefficient of linear thermal expansion The components of dielectric tensor The thermal modulus T The angular frequency of thermal vibration Kronecker delta function Dimensionless mechanical coupling constant C E The entropy density C E The thermal viscosity k C E T o

14 THERMOPIEZOELASTICITY OF FINITE ROD WITH LOADING 245 ij o T T o The dynamical temperature increment such that T T o 1 T o Lame s constants Mass density Components of stress tensor xx The principal stress component One relaxation time parameter The conductive temperature ac0 22 Dimensionless two-temperature parameter REFERENCES 1. H. W. Lord and Y. Shulman, A Generalized Dynamical Theory of Thermoelasticity, J. Mech. Phys. Solids, vol. 15, pp , A. E. Green and K. E. Lindsay, Thermoelasticity, J. Elasticity, vol. 2, pp. 1 7, E. Bassiouny and A. F. Ghaleb, A One Dimensional Problem in the Generalized Theory of Thermopiezoelasticity, in G. A. Maugin (ed.), The Mechanical Behavior of Electromagnetic Solid Continua, pp , Elsevier Science Publisher B. V. (North Holland), IUTAM-IUPAP, H. Tianhu, T. Xiaogeneg, and S. Yapeng, State Space Approach to One-Dimensional Shock Problem for a Semi-Infinite Piezoelectric Rod, Int. J. Eng. Sci., vol. 40, pp , H. M. Youssef, Theory of Two-Temperature Generalized Thermoelasticity, IMA J. Appl. Math., vol. 71, no. 3, pp , N. M. El Maghraby and H. Youssef, State Space Approach to Generalized Thermoelastic Problem with Thermomechanical Shock, Appl. Math. Compt., vol. 156, pp , H. Tianhu, T. Xiaogeneg, and S. Yapeng, Two-Dimensional Generalized Thermal Shock a Thick Piezoelectric Plate of Infinite Extent, Int. J. Eng. Sci., vol. 40, pp , H. M. Youssef, Problem of Generalized Thermoelastic Infinite Medium with Cylindrical Cavity Subjected to a Ramp-Type Heating and Loading, Archive Appl. Mech., vol. 75, pp , G. Hanig and U. Hirdes, A Method for the Numerical Inversion of Laplace Transform, J. Comp. Appl. Math., vol. 10, pp , H. Chu, C. Chen, and C. Weng, Applications of Fourier Series Technique to Transient Heat Transfer, Chem. Eng. Commun., vol. 16, pp , 1984.