The Time Domain Solutions For Generalized Piezoelectric-thermoelastic Problems

In the classical thermoelastic theory, the governing equation of temperature field on the basis of Fourier heat conduction law is of diffusion type, predicting infinite propagation speed of heat , which is contrary to experimental observations. To solve such paradox, the generalized thermoelastic theories were developed in attempt to amend the classical thermoelasticity. In1967, Lord and Shulman (L-S) modified the Fourier heat conduction law by introducing one thermal relaxation time factor and heat flux rate in the Fourier heat conduction equation. In 1972, Green and Lindsay (G-L) postulated the G-L type generalzied themoelastic theory by introducing two thermal relaxation times, and one of them was introduced in the energy equation and the other in constitutive equations. Both theories can characterize heat disturbance propagating with finite velocity in media, which is so-called second sound effect.Due to the direct and converse piezoelectric effect of piezoelectric materials, they can be widely used as actuators and sensors in various fields such as aerospace, instruments, mechanical engineering etc.. With the wide applications of piezoelectric materials, they have attracted intense attentions in investigating both the static and dynamic thermoelastic responses of piezoelectric materials. The devices made of piezoelectric materials usually suffer from heat disturbances when they are in service, and the devices undergo thermal expansion deformation due to heat, resulting in piezoelectric effect in the piezoelectric devices. It is necessary to understand the law of heat propagating in piezoelectric media, especially in the case of extreme heat environment. The investigation on generalized piezoelectric-thermoelastic problem is less, and even less on rotation problem.Due to the mathematical complexity in the governing equations caused by the coupling of different variables and by the derivatives with respect to not only spatial coordinates but also time, integral-transformation technique is commonly applied to solving the generalized piezoelectric-thermoelastic problem. The procedure of integral-transformation technique is as follows: first, transform the considered problem form time domain to Laplace or Fourier transformation domain, then solve the problem to get the solutions in transformation domain, and lastly transform the solutions numerically from transformation domain to time domain by Laplace or Fourier numericla inversion to get the solutions in time domain. This method unavoidably causes loss of precision due to the truncation error and discretization error introdued in the process of numerical inversion. To avoid loss of precision, the so-called direct finite element method was proposed to solve generzlied thermoelstic problems directly in time domain.In this paper, a one-dimensional problem of a piezoelectric slim strip and a two-dimensional problem of a roating infinite piezoelectric plate with finite thickness are considered in the frame of generalized thermolastic theories. The governing equations of the problems are formulated, and the corresponding finite element equations are derived from the vitural displacment principle established in the context of the generalzied theories. Then the finite element equations are solved directly in time domain by means of the so-called direct finite element method to guarrantee a high calculation precision. The related variables such as temperature, displacement, stress, and induced potential distribution in the plate are numerically obtained and presented graphically. From the distribution of temperate,the position of heat wave front and temperature jump can be clearly observed, which implies that the unique characterics of heat wave, that is, a temperature jump occuring in the position of heat front wave, can be prcisly captured by means of direct finite element method. Also can be found from the distritubions of the considered variables is the coupling effects as well as the fact that heat propagates as a wave with finite speed. The coupling effects show that stress jump occures at the same position where the temperature jump occures, and the induced electric potential jump occures at the same position where the displacement jump occures. The results also show that the rotation effect acts to significantly decrease the magnitude of the real part of the dimensionless displacement and the induced potential distribution and insignificantly affect the magnitude of temperature.