Effects of the temperature difference at two surfaces

Effects of the temperature difference at two surfaces of the spherical shell on the dimensionless stresses, radial displacement and electric potential in the radial direction for shell 1 are presented in Figure 4(a)–(d). For the sake of simplicity, numerical calculations are carried out with the temperature of the outer surface assumed to be fixed, i.e. , and the temperature of the inner surface varying as a parameter. Material properties are assumed to vary linearly in the radial direction, i.e. . It is revealed from Figure 4(a) that the absolute values of dimensionless radial stress increase as the temperature difference increases. It is shown in Figure 4(b) that the rate of change of the dimensionless circumferential stress increases as the temperature difference increases. It is also observed that the response curves of the dimensionless circumferential stress intersect at a radial position of about , indicating an invariant value of the circumferential stress for different values of temperature differences. Variation of the dimensionless radial displacement is plotted in Figure 4(c), which shows an increase as the temperature difference increases. It is observed from Figure 4(d) that the dimensionless electric potential sensed via the wst-1 layer increases as the temperature difference increases.
Effects of the ratio of the thickness of the FGM layer to the thickness of the piezoelectric layer (i.e. ) for shell type 1 are shown in Figure 5(a)–(e). It is observed from Figure 5(a) that the effect of on the dimensionless radial stress is less than the other parameters. It is shown in Figure 5(b) and (c) that the dimensionless circumferential stress and radial displacement increase as increases. However, the behavior for the temperature distribution is the opposite, as depicted in Figure 5(d). The effect of the geometric parameter, , on the distribution of the electric potential is found to be the highest among all other parameters, as shown in Figure 5(e), indicating a decrease in the dimensionless electric potential as increases.
Next, the effect of the grading parameter, , for shell type 2 is studied, and the results are depicted in Figure 6(a)–(d). As expected, the interface continuity conditions for the radial stress, radial displacement and temperature distribution, and also boundary conditions, are fully satisfied. The sensor and actuator layers are assumed to be orthotropic with constant material properties. Hence, the distribution of parameters through the thickness is expected to be almost linear. However, distributions of parameters in the FGM shell are nonlinear due to the power law grading of material properties through the thickness. It is shown in Figure 6(a) that the absolute value of the dimensionless radial stress across the thickness increases in magnitude as the grading parameter, , increases. It is also observed from Figure 6(b) that the dimensionless circumferential stress increases in the actuator and sensor layers as increases. It is interesting to note that for , the circumferential stress in the host layer is almost constant. Another interesting point is the discontinuity of the circumferential stresses between layers, as shown in Figure 6(b). It is also revealed from Figure 6(c) and (d) that the dimensionless radial displacement increases and the temperature distribution decreases as increases. Comparison of results presented in Figures 3 and 6 indicates that the effect of the actuator layer (i.e. electric excitation) on the distribution of parameters is high.
Effects of the temperature difference at two surfaces of the spherical shell on the distribution of dimensionless parameters through the thickness of shell 2 are presented in Figure 7(a)–(d). Numerical results are calculated with the temperature of the outer surface assumed to be zero and that of the inner surface to vary as a parameter. It is shown in Figure 7(a) that the dimensionless radial stress is almost constant, irrespective of the variation in temperature difference. It is observed from Figure 7(b) that dimensionless circumferential stresses decrease in the actuator layer but are constant in the FGM and sensor layers as the temperature difference increases. The dimensionless electric potential sensed via the sensor layer increases as the thermal excitation increases, as shown in Figure 7(d).