Two Temperature Heat Flux of Semi Infinite Piezoelectric Ceramic Rod

The theory of two-temperature generalized thermoelasticity is used to solve the problem of heating a semi-infinite rod made of a piezoelectric ceramic material within the framework of generalized thermopiezoelasticity theory by supplying the rod a certain amount of heat uniformly distributed over a finite time period to the finite end of the rod. The Laplace transform formalism is used to solve the proposed model. Inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. The physical parameters (i.e., conductive temperature, dynamical temperature, stress, strain, and displacement distributions) are investigated graphically.


Introduction
Piezoelastic ceramics have become now prominent and extensively used in many engineering applications especially in mechatronics, structronic systems, sensors, actuators and intelligent structures [1][2][3].
Piezoelectric ceramic have been extensively used in many engineering and industrial applications.So, the theory of generalized thermo-piezoelectricity has been the object of numerous investigations in the last decades or so, concerning both its theoretical foundations and the applications.
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 the physical observations.Widespread attention to eliminate this paradox has been given to thermoelasticity theories which admit a finite speed for the propagation of thermal waves.Many authors have formulated generalized theories involve a hyperbolic-type heat equation and are referred to as gen-eralized thermoelasticity.
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 [4].The second theory was developed by Green and Lindsay [5].It modifies both the energy equation and the Duhamel-Neuman relation.It admits two relaxation times.In this work we solve the problem of heating of a semi-infinite piezoelectric rod by supplying a certain amount of heating uniformly distributed over a finite time period to the finite end.The rod is assumed to be at rest at zero temperature initially.

Governing Equations
In the absence of body force, free charge and inner heat sources, we consider generalized thermo-piezoelectric governing differential equations [6,7] as follows: Equations of motion: Equation of entropy increment (in the absence of inner heat source): Stress-strain-temperature: Gauss equation and electric field relation: , ,i Equation of entropy density: Strain-displacement relations: Fourier's law for heat conduction where  is the conductive temperature and it satisfies the relation in which is the two-temperature parameter and are the components of thermal conductivity tensor.
In the above equations, a comma followed by a suffix denotes material derivatives and a superposed dot denotes the derivatives with respect to time

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.We assume the following form for the displacement component [6]: We consider the following forms of the linearized basic equations in one-dimensional formulation:   where  is the coefficient of the linear thermal expansion, k is the coefficient of thermal conductivity and x is the coordinate taken along the rod.
It is convenient now to introduce the following dimensionless variables: From Gauss's law, since there is no free charge inside the piezoelectric rod we have and the following relation between the conductive temperature and the thermo dynamical one: where The rod is supplying a certain amount of heat uniformly distributed over a finite period of time to the near end and is traction free.The boundary conditions are: where o F is the heat flux intensity and   o t t  the well-known impulse function: while the initial conditions are assumed to be: Applying the Laplace transform defined by: to both sides of Equations ( 23)-( 27), we obtain: where s denotes the complex argument related to the Laplace transform.
The transformed boundary conditions take the forms Thus, the transformed component of the heat flux vector becomes while the Equations (30) become and the corresponding transformed initial conditions of the Equations (31) assume the form: Eliminating  between Equations ( 35) and (37), we get: where Substituting from Equation (42) into Equation (37), we obtain Using Equation ( 42) we can easily eliminate  be- tween Equations ( 33) and (44) to obtain where Solving Equations ( 42) and (45) together we get the following fourth order equation where, and . It is worth mentioning here that the roots of Equation (47) are functions of s and assume the forms: where Thus the solutions of the Equations ( 42) and (45) satisfying the boundary conditions at infinity are: respectively, where 1 2 are the roots of the Equation (47) and , k k , A B are parameters depending on s to be determined using the boundary conditions while the relations between the coefficients , , , Using the boundary conditions (38) and (40) it is easy to verify that: where Substituting form Equations (54) into the Equation (53) we can easily evaluate the coefficients C and D: Therefore, the transformed heat conduction and the strain given by the Equations ( 51) and ( 52) respectively become: where where Substituting the expressions of  and of e from Equations (57) and (59) into the Equation (44) we get the following form of the thermodynamical function  : where Using Equations ( 59) and (61) it is easy to put  given by the Equation (34) in the form: Equations ( 57), (59), ( 61) and (63) are the complete solutions of the , , and e

 
In order to invert the Laplace transform, we adopt a numerical inversion method based on a Fourier series expansion [8].
Using this method, the inverse   where N is a sufficiently large integer representing the number of terms in the truncated Fourier series, chosen such that  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 singularities of   f s .The optimal choice of c was obtained according to the criteria described in

Numerical Results and Discussion
Distributions of unknown functions along the rod for three different time moments, t = 0.1, 0. Figures 1-4 illustrate the effects of the presence of the two temperature parameter on the conductive temperature  , thermodynamical temperature  , stress  and strain distribution curves.We note that the two temperature parameter has significant effects on the unknown function fields especially on the strain distribution and the conductive temperature at the nearest end of the rod, it increases the amplitude of the unknowns at the nearest end of the rod and it has no effects far from the nearest end as seen on Figures 1 and 4.

e
The presence of the two temperature parameter leads to removing decreases the number of peaks as shown in

Conclusion
We considered a two temperature generalized thermppiezoelectric model to study the effects of different pa-rameters on the thermomechanical behavior of a semi infinite rod made of piezoelectric material.We can conclude that the presence of the two temperature parameter leads to a direct relation between the time and the amplitude of the thermodynamical temperature and an inverse relation with the heat conduction.The absolute values of the amplitude of the stress and the strain are in direct relation with the time for all values of the two temperature parameter.The values of the stress and the strain resemble the same sign for all values of time and the two temperature parameter.

Acknowledgements
The authors would like to thank the support of the Deanhip of Scientific Research in Salman Bin Abdul-Aziz s The components of thermal conductivity.
2 and 0.3 and for different values of the relaxation time: numerically where the effects of the presence of the two temperature parameter on the unknown functions are illustrated in Figures 1-28.

i
The electric potential function.


The principal stress component.