4.4 Coupled Thermopiezoelectricity in One-Dimensional
4.4.6 Results of Dual Phase Lag Model Analysis
The problem analyzed in this section is identical to that of Sect.4.4.4. The only modification is the addition of the temperature gradient phase lag, which has a non-dimensional value of 0.04. Since the solution in this section was based on a general approximation of C-T thermoelasticity theory, it is possible to study the response of the rod based on L-S theory by settingt1 andt2 equal to zero. In this case, the problem truly is identical to the preceding one and so these results are used in this section for comparison purposes [12]. The analysis based on C-T theory, however, is performed for botht2¼0 andt2¼ sqffiffi
p2in order to investigate the effects of this parameter on the results.
In Figs.4.25,4.26,4.27and4.28, the distributions of displacement, temperature change, stress, and electric potential are depicted based on two approximations of Fig. 4.25 Comparison of the
displacement distribution based on C-T and L-S theories. [Reproduced from [40] with permission from SAGE Publications Ltd.]
Fig. 4.26 Comparison of the temperature distribution based on C-T and L-S theories.
[Reproduced from [40] with permission from SAGE Publications Ltd.]
4.4 Coupled Thermopiezoelectricity in One-Dimensional… 163
C-T theory as well as L-S theory. The results are shown for non-dimensional time t¼0:1333, thus the non-dimensional location of the heat source is x¼tt¼0:0667. As seen previously, the maximum temperature in the rod occurs at this point for L-S theory, but the same is not true for C-T. Nonetheless, it will be seen in Fig.4.29 that as the temperature gradient phase lag decreases, the tem- perature maximum tends to occur at the location of the thermal disturbance.
Figures4.26and4.27clearly show that the thermal wavefront based on C-T theory witht26¼0 is located farther ahead than the wavefront based on the L-S theory, and furthermore that whent2¼0, no wavefront is observed.
In the following two figures, the effect of the temperature gradient phase lag sh¼t1 on temperature is studied for the C-T theory witht26¼0 and t2¼0. The Fig. 4.27 Comparison of the
stress distribution based on C-T and L-S theories.
[Reproduced from [40] with permission from SAGE Publications Ltd.]
Fig. 4.28 Comparison of the electric potential distribution based on C-T and L-S theories. [Reproduced from [40] with permission from SAGE Publications Ltd.]
material is still considered homogeneous at this point and the results are shown at the same non-dimensional timet¼0:1333. It can be concluded that as the phase lag increases, the wave propagation speed increases, thus forcing the wave fronts to move ahead farther. Moreover, the wavefronts weaken as sh decreases and they eventually disappear att1¼0:01, as seen in Fig.4.29. In Fig.4.30, no wavefronts are observed at all due to the heatflux phase lag being equal to zero. We can also conclude that a decrease int1increases the absolute values of the extrema for the temperature distribution in the C-T theory whether or nott2¼0.
We will now study this problem on the basis of functionally graded media.
As seen below in Fig.4.31for the elastic constant, the non-homogeneity index k holds an exponential relationship with the material properties of the thermopiezo- electric rod.
Fig. 4.29 Effect of the phase-lag of temperature gradient on the temperature distribution for C-T theory witht26¼0. [Reproduced from [40] with permission from SAGE Publications Ltd.]
Fig. 4.30 Effect of the phase-lag of temperature gradient on the temperature distribution for C-T theory witht2¼0. [Reproduced from [40] with permission from SAGE Publications Ltd.]
4.4 Coupled Thermopiezoelectricity in One-Dimensional… 165
In Figs.4.32, 4.33,4.34 and 4.35, the distributions of displacement, tempera- ture, electric potential, and stress are depicted with varying non-homogeneity indices for C-T theory with sh¼t1 ¼0:04 and t26¼0 at non-dimensional time t¼0:1333. An increase inklowers the absolute value of displacement, temperature and electric potential and also reduces the heights of the wavefronts seen in the distributions. The locations of the wavefronts, however, remains the same for varying values of non-homogeneity. Finally, whenkis increased, the absolute value of stress decreases before it reaches its maximum and increases after its maximum.
For C-T theory witht2 ¼0, as well as L-S theory, similar results can be observed.
Through the dual phase lag results presented in this section, it is possible to study C-T theory with different values for phase lags, as well as generalized L-S Fig. 4.31 Effect of
non-homogeneity index on the distribution of elastic constant. [Reproduced from [40] with permission from SAGE Publications Ltd.]
Fig. 4.32 Effect of non-homogeneity index on the displacement distribution.
[Reproduced from [40] with permission from SAGE Publications Ltd.]
theory. In the C-T theory, sq and sh can be interpreted as two relaxation times, whereas in L-S theory we only account for one relaxation time [40]. The results found are reduced to those for coupled L-S theory when t1¼t2¼0. However, using C-T theory with two phase lagsshandsqallows the consideration of the fact that the heat flux vector may precede the temperature gradient or vice versa. In addition, non-equilibrium thermodynamic transitions and microscope effects of energy exchange in high-rate heating applications are significant setbacks which are addressed by the dual phase lag C-T theory [40]. This approach to thermopiezo- electric problems provides a multiphysical description of functionally graded materials on microscopic and macroscopic scales while including other more generalized thermoelasticity theories.
Fig. 4.33 Effect of non-homogeneity index on the temperature distribution.
[Reproduced from [40] with permission from SAGE Publications Ltd.]
Fig. 4.34 Effect of non-homogeneity index on the electric potential distribution. [Reproduced from [40] with permission from SAGE Publications Ltd.]
4.4 Coupled Thermopiezoelectricity in One-Dimensional… 167