Three-Dimensional Thermoelasticity Analysis of Viscoelastic FGM Plate Embedded in Piezoelectric Layers under Thermal Load
Abstract
1. Introduction
2. Governing Equations
2.1. Temperature Field
2.2. FGM Layer
3. Solution Procedure
3.1. Temperature Gradient
3.2. FGM Layer
3.3. Piezoelectric Layer
4. Numerical Results and Discussion
5. Conclusions
- Increasing the length-to-thickness ratio leads to decrease in deflections and increase in stresses.
- In the absence of an applied voltage when , the effect of the piezoelectric layer thickness on the thermo-elastic behavior becomes negligible.
- Stiffness of the plate decreases by increasing and, accordingly, stresses and the deflection decrease.
- The effect of a temperature difference in the lower region is more significant than in the upper region due to the thermal barrier behavior of the FGM core at the upper surface.
- Increasing the relaxation time constant causes the stiffness of the viscoelastic plate and, accordingly, stress components to increase and the displacement to decrease.
- Increasing the relaxation time constant causes the rate of convergence to the elastic behaviour to decrease.
- The effect of the applied voltage near the outer region of the FGM layer is more significant due to the actuator layer’s effect.
- Deflection of the plate increases by increasing the applied voltage.
- Through-thickness distribution of deflection is linear in piezoelectric and FGM layers with different slope.
- Increasing the time constant causes delay in the steady state condition for stresses and displacement.
- Increasing the time constant causes a decrease in the transverse displacement .
- The maximum values of transverse normal stress are not at the mid-thickness of the plate, which is due to the FGM property.
Author Contributions
Funding
Conflicts of Interest
Nomenclature
a, b, h | Plate dimensions in x-, y-, and z-directions |
Subscripts designating FGM, actuator, and sensor layers, respectively | |
Temperature distribution | |
, | Temperature at the bottom and top surfaces, respectively |
, | Temperature at the bottom and top surface of FGM layer, respectively |
Thermal conductivity coefficient for FGM layer | |
Thermal expansion coefficient | |
Stress–temperature coefficients in x-, y-, and z-directions | |
Pyroelectric constant | |
Relaxation moduli coefficients | |
Electric displacement | |
Elasticity constant | |
Electric field in x-, y-, and z-directions | |
Young’s modulus | |
Piezoelectric coefficient | |
Dielectric constants | |
d1 | Piezoelectric modulus |
Thermal conductivity coefficient for piezoelectric layer in the x-, y-, and z-directions | |
Thicknesses of the FGM and piezoelectric layers | |
Half-wave numbers in the x- and y-directions | |
Displacement components in the x-, y- and z-directions | |
Normal stresses | |
Shear stresses | |
Normal strains | |
Shear strains | |
Relaxation time constant | |
State vectors of the FGM and piezoelectric layers | |
Electric voltage | |
Poisson’s ratio |
Appendix A
References
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Elasticity Constant [109 Nm−2] | |||||||||
---|---|---|---|---|---|---|---|---|---|
Sensor (PZT-4) | 139 | 78 | 74 | 139 | 74 | 115 | 25.6 | 25.6 | 30.5 |
Actuator (Ba2 NaNb5 O15) | 239 | 104 | 50 | 274 | 52 | 135 | 65 | 66 | 76 |
Piezoelectric coefficients [coul·m−2] | |||||||||
Sensor | −5.2 | −5.2 | 15.1 | 12.7 | 12.7 | ||||
Actuator | −0.4 | −0.3 | 4.3 | 3.4 | 2.8 | ||||
Dielectric constants [10−9 farads·m−1] | η1 | η2 | η3 | ||||||
Sensor | 6.5 | 6.5 | 5.6 | ||||||
Actuator | 1.96 | 2.01 | 0.28 | ||||||
Thermal conductivity | |||||||||
Sensor | 2.1 | 3.15 | |||||||
Actuator | 8.6 | 12.9 | |||||||
Thermal expansion [] | |||||||||
Sensor | 1.97 | 2.62 | |||||||
Actuator | 4.39 | 2.45 | |||||||
Piezoelectric modulus pyroelectric constant: | |||||||||
Sensor | −3.92 | 5.4 | |||||||
Actuator | −3.92 | 5.4 |
[41] | [42] | [38] | Present | [41] | [42] | [38] | Present | |
---|---|---|---|---|---|---|---|---|
3.043 | 3.043 | 3.0431 | 3.043 | 28.54 | 28.53 | 28.53 | 28.53 | |
2.144 | 2.143 | 2.1443 | 2.143 | 28.46 | 28.45 | 28.448 | 28.45 | |
1.901 | 1.901 | 1.9012 | 1.900 | 28.44 | 28.43 | 28.432 | 28.43 | |
−1.681 | −1.681 | −1.681 | −1.681 | −1.703 | −1.703 | −1.7027 | −1.703 | |
−0.6822 | −0.6822 | −0.6823 | −0.6860 | −0.8080 | −0.8081 | −0.8081 | −0.808 | |
0.08266 | 0.08240 | 0.08242 | 0.08241 | 0.08553 | 0.08528 | 0.08527 | −0.08552 | |
−1018 | −1018 | −1018 | −1018 | −1003 | −1003 | −1003 | −1003 | |
−204.7 | −204.8 | −204.82 | −204.821 | −251.2 | −251.2 | −251.208 | −251.2084 | |
−74.03 | −73.53 | −73.525 | −73.525 | −76.59 | −76.10 | −76.12 | −76.1239 | |
4.203 | 4.186 | 4.1875 | 4.1875 | 0.3135 | 0.3122 | 0.3123 | 0.3123 | |
6.300 | 6.217 | 6.23 | 6.2342 | 0.1178 | 0.04067 | 0.04051 | 0.4051 |
0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 | ||
---|---|---|---|---|---|---|---|
600 | −20.620 | −20.850 | −21.100 | −21.392 | −21.706 | −22.040 | |
800 | −15.547 | −16.507 | −17.509 | −18.574 | −19.576 | −20.704 | |
900 | −13.856 | −15.088 | −16.403 | −17.635 | −18.929 | −20.265 | |
600 | 5.6785 | 6.0438 | 6.1482 | 6.3048 | 6.4092 | 6.5658 | |
800 | 22.015 | 22.276 | 22.380 | 22.484 | 22.589 | 22.797 | |
900 | 27.443 | 27.704 | 27.756 | 27.860 | 28.017 | 28.173 | |
600 | 0 | 0.4945 | 0.7732 | 0.7438 | 0.4281 | 0 | |
800 | 0 | 0.4155 | 0.7031 | 0.7167 | 0.4301 | 0 | |
900 | 0 | 0.3892 | 0.6798 | 0.7076 | 0.4307 | 0 | |
600 | −199.79 | −181.63 | −239.87 | −257.41 | −258.66 | −202.92 | |
800 | −137.79 | −124.63 | −187.27 | −225.47 | −248.64 | −199.16 | |
900 | −117.12 | −105.22 | −169.73 | −214.82 | −246.14 | −197.91 |
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Feri, M.; Krommer, M.; Alibeigloo, A. Three-Dimensional Thermoelasticity Analysis of Viscoelastic FGM Plate Embedded in Piezoelectric Layers under Thermal Load. Appl. Sci. 2023, 13, 353. https://doi.org/10.3390/app13010353
Feri M, Krommer M, Alibeigloo A. Three-Dimensional Thermoelasticity Analysis of Viscoelastic FGM Plate Embedded in Piezoelectric Layers under Thermal Load. Applied Sciences. 2023; 13(1):353. https://doi.org/10.3390/app13010353
Chicago/Turabian StyleFeri, Maziyar, Michael Krommer, and Akbar Alibeigloo. 2023. "Three-Dimensional Thermoelasticity Analysis of Viscoelastic FGM Plate Embedded in Piezoelectric Layers under Thermal Load" Applied Sciences 13, no. 1: 353. https://doi.org/10.3390/app13010353
APA StyleFeri, M., Krommer, M., & Alibeigloo, A. (2023). Three-Dimensional Thermoelasticity Analysis of Viscoelastic FGM Plate Embedded in Piezoelectric Layers under Thermal Load. Applied Sciences, 13(1), 353. https://doi.org/10.3390/app13010353