4.4 Coupled Thermopiezoelectricity in One-Dimensional
4.4.4 Results and Discussion
/p2¼ C4 n22þa1n2
Lcgn22þa1Lcgn2
en2x ð4:81cÞ
/p3¼up1 r21þa1r1
þC1a4ðr1þa1Þer1x Lcgr21þa1Lcgr1
ð4:81dÞ
/p4¼up2 r22þa1r2
þC2a4ðr2þa1Þer2x
Lcgr22þa1Lcgr2 ð4:81eÞ
/p5 ¼up3
sv 2a1 s
v
þhpa4 vsþa1
Lcg sv 2a1Lcg sv ð4:81fÞ
4.4.3.4 Numerical Inversion of the Laplace Transform
Having obtained the coupled and uncoupled solutions in the Laplace domain, we now employ the so-called fast Laplace inverse transform [14]. Performing this numerical inversion will transform the results into the time domain for later anal- ysis. The form of this inversion process have been previously stated in Sect.4.2.3 and is omitted here. The only change concerns the values of the constraint parameters, which are stated below:
aT¼100
15 ;Ln¼5;N¼900
and are therefore indiscernible in thefigures. Intuitively, the maximum temperature occurs at the location of the heat source at this timeðx¼tt¼0:1Þ. On the other hand, the maxima of displacement, stress and electric potential all occur ahead of this point as shown in Figs.4.17,4.18and4.19. These results are in agreement with those found by Babei and Chen [33] analytically.
The effects of non-Fourier heat conduction can be seen in Figs.4.16and4.18.
For the classical coupled and classical uncoupled solutions, the temperature dis- tribution is diffusive, and as such thermal wave characteristics are not observed due to the parabolic nature of Fourier heat conduction. Contrarily, in the generalized distributions, the presence of thermal wavefronts is obvious due to thefinite thermal wave speed and hyperbolic heat conduction. Therefore, there are distinguishable undisturbed portions of the rod in the distributions of temperature and stress.
Table 4.2 Material properties of the left end of the rod [3,31]
Properties Cadmium Selenide
c0 (GPa) 74.1
e0 C
m2 0.347
b0106 N Km2
0.621
201011 C2 Nm2
9.03
p0106 C Km2
–2.94 q0103 kg
m3 7.60
CE103 J kg K
0.42
K0 W mK
12.9
Fig. 4.16 Comparison of the temperature distribution based on different thermoelasticity theories at non-dimensional timet¼0:2. [Reproduced from [3] with permission from World Scientific Publishing Co., Inc.]
4.4 Coupled Thermopiezoelectricity in One-Dimensional… 157
The time history of the difference between the temperature solutions of the coupled and uncoupled theories is depicted in Fig.4.20. The amplitude offluctu- ation remains constant after the heat source leaves the rod attexit¼2.
Figures4.21,4.22and4.23display the effect of the non-homogeneity indexkon the histories of displacement, temperature and electric displacement, respectively.
This analysis is performed at a non-dimensional location ofx¼0:5 based on the classical coupled thermoelasticity theory. The absolute mean value offluctuation for the displacement distribution increases as the value ofk increases. It is shown in Fig.4.22that before the heat source exits the rod, the temperature at each location increases monotonically. After this point, the temperature will reach its constant value while exhibiting smallfluctuations.
Fig. 4.17 Comparison of the displacement distribution based on different thermoelasticity theories at non-dimensional timet¼0:2.
[Reproduced from [3] with permission from World Scientific Publishing Co., Inc.]
Fig. 4.18 Comparison of the stress distribution based on different thermoelasticity theories at non-dimensional timet¼0:2. [Reproduced from [3] with permission from World Scientific Publishing Co., Inc.]
In Fig.4.23the history of electric displacement with different non-homogeneity indices is depicted. It is clear to see that the absolute value of electric displacement increases when k becomes larger. The distribution smoothly increases until the thermal disturbance reaches the end of the rod, at which point it remains constant.
This phenomenon is consistent with the results obtained by Babei and Chen for a homogeneous rod under L-S theory [33].
The effect of non-homogeneity index on the distribution of stress is depicted in Fig.4.24. The results are again analyzed for classical coupled thermoelasticity at non-dimensional timet¼0:2. Before it reaches its maximum, the absolute value of stress decreases whenkincreases. This relationship is completely reversed after the maximum of stress occurs. Once again, thesefindings can also be observed for the coupled thermoelasticity analysis based on L-S theory.
Fig. 4.19 Comparison of the electric potential distribution based on different
thermoelasticity theories at non-dimensional timet¼0:2.
[Reproduced from [3] with permission from World Scientific Publishing Co., Inc.]
Fig. 4.20 Time history of the difference of the temperature distribution for coupled and uncoupled thermoelasticity at non-dimensional timet¼0:2.
[Reproduced from [3] with permission from World Scientific Publishing Co., Inc.]
4.4 Coupled Thermopiezoelectricity in One-Dimensional… 159
In general, certain phenomena can be noted based on the results found in this section. In classical coupled thermoelasticity, there are no wave fronts in the dis- tributions of temperature or stress, but they exist for the generalized L-S solutions.
Additionally, the extrema of temperature and stress based on classical coupled and classical uncoupled thermoelasticity are lower compared to those based on the generalized theory. For any thermoelasticity theory discussed, an increase in k results in an increase of the absolute value of electric displacement after the thermal disturbance has left the rod. For classical coupled thermoelasticity, an increase ink diminishes the dynamic response of displacement, temperature and electric potential. Nonetheless, variations in non-homogeneity have no effect on the con- stant temperature reached once the heat source exits the rod [3].
Fig. 4.21 Effect of the non-homogeneity index on the displacement history.
[Reproduced from [3] with permission from World Scientific Publishing Co., Inc.]
Fig. 4.22 Effect of the non-homogeneity index on the temperature history.
[Reproduced from [3] with permission from World Scientific Publishing Co., Inc.]