# Thermo-Electro-Mechanical Analysis of a Curved Functionally Graded Piezoelectric Actuator with Sandwich Structure

^{*}

## Abstract

**:**

_{31}varying continuously along the radial direction of the curved actuator. Based on the theory of linear piezoelectricity, analytical solutions are obtained by using Airy stress function to examine the effects of material gradient and heat conduction on the performance of the curved actuator. It is found that the material gradient and thermal load have significant influence on the electroelastic fields and the mechanical response of the curved FGP actuator. Without the sacrifice of actuation deflection, smaller internal stresses are generated by using the sandwich actuator with functionally graded piezoelectric layer instead of the conventional bimorph actuator. This work is very helpful for the design and application of curved piezoelectric actuators under thermal environment.

## 1. Introduction

_{31}or the elastic parameter s

_{33}under different loading conditions.

_{31}varying linearly along the thickness direction under different electrical and heat conduction conditions. Based on the theory of piezoelectricity, an FGP sandwich cantilever under electrical and thermal loads were studied in [23], in which all material parameters are assumed to vary in the direction of the thickness according to a power law distribution. Yang and Xiang [24] used the Timoshenko beam theory to investigate the static bending and dynamic response of FGP actuators under combined thermal-electro-mechanical loading.

_{31}varies along the radial direction of the circularly curved beam. However, there is very limited work of studying the thermal load effect in the curved FGP actuator configuration. The bending behavior of a circularly curved FGP cantilever actuator under an applied electrical load and heat conduction was investigated in our previous work [29]. It was found that thermal effect was significant on the electroelastic field of the curved actuator. To the authors’ best knowledge, there is no investigation of the thermal effect on the performance of sandwiched FGP structures thus far. It is, therefore, the objective of the current study to investigate the thermal-electro-elastic fields of a curved FGP actuator with sandwich structure under electrical and thermal loads. By using Airy stress function, analytical solutions are derived and numerical results are presented to show the effects of material gradients and thermal loads on the stresses, displacements, electric displacements and electric potential of the curved actuator. These results can also demonstrate the advantages of using the sandwiched FGP actuator over the traditional piezoelectric bimorph actuator.

## 2. Formulation of the Problem

_{31}in the middle FGP layer is assumed to vary along the radial direction while approaches to the corresponding values of the homogeneous piezoelectric layers at the upper and lower surfaces, respectively. It is assumed that all the piezoelectric layers are poled in radial direction. For the analysis of this device, a polar coordinate system (r, θ) is used, and the thickness of the k th layer is determined by ${h}_{k}={R}_{k+1}-{R}_{k}$ (k = 1~3), where k = 1, 2 and 3 refers to the lower layer, the FGP layer and the upper layer of the actuator. The actuator is subjected to an electric potential V

_{0}between the outer surface of layer 3 and the inner surface of layer 1, and a thermal conduction occurs along the radial direction due to the temperature rise difference, i.e., T

_{o}on the outer surface of layer 3 and T

_{i}on the inner surface of layer 1.

_{31}[9,28,30]. Therefore, to make the analysis mathematically tractable, only the piezoelectric coefficient ${g}_{31}$ of the piezoelectric media is assumed to vary along the radial direction in the sandwich structure, while all the other material coefficients are assumed as constant. A Taylor series expansion is used to describe the arbitrary function g

_{31}(r) for the FGP layer in terms of the following N th-order polynomial function as [28]:

_{31}(r) could be of arbitrary format. For the case that the FGP is exponentially graded along the radial direction, g

_{31}can be expressed as:

_{31}(r) using Taylor’s series expansion;

_{0}is taken as the average radius of the actuator, i.e., ${R}_{0}=\frac{{R}_{1}+{R}_{4}}{2}$.

## 3. Solution of the Problem

## 4. Results and Discussion

_{31}are assumed as constants in the current work. The elastic, piezoelectric (except g

_{31}), dielectric constants, thermal expansion and pyroelectric coefficients for the different layers of the sandwiched piezoelectric actuator are taken as those for the PZT-4 [28,31] in the numerical calculation, and the typical value of thermal conductivity is taken as $\kappa =2.1{\text{Wm}}^{-\text{1}}{\text{K}}^{-\text{1}}$. All these material constants are listed in Table 1. It should be mentioned that the derived solutions in the previous section are applicable for any arbitrary format of g31 with Taylor series expansion. For case study of the actuator configuration in Figure 1, the middle FGM layer is assumed as exponentially graded along the radial direction as shown in Equation (10), and the coefficients of Taylor expansion of g31 can be determined from its values at the boundaries of the FGM layer. The upper layer of the sandwich structure is taken as PZT-4 and its piezoelectric constant is ${g}_{31}({R}_{3})={g}_{31}^{(3)}=-17.8\times {10}^{-3}{\text{m}}^{2}{\text{C}}^{-1}$, while the piezoelectric constant for the lower layer is assumed as ${g}_{31}({R}_{2})={g}_{31}^{(1)}=-9.35\times {10}^{-3}{\text{m}}^{2}{\text{C}}^{-1}$. The geometry of the curved FGP actuator with sandwich structure is fixed with R

_{1}= 15.5 mm, R

_{2}= 16.0 mm, R

_{3}= 17.0 mm and R

_{4}= 17.5 mm.

Elastic constant (10^{−12}m^{−2}N^{−1}) | Piezoelectric constant (10^{−3}m^{2}C^{−1}) | Dielectric constant (10^{6}mF^{−1}) | Thermal expansion (10^{−6}K^{−1}) | Pyroelectric constant (10^{3}NC^{−1}K^{−1}) | Thermal conductivity (Wm^{−1}K^{−1}) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${s}_{11}^{D}$ | ${s}_{13}^{D}$ | ${s}_{33}^{D}$ | ${s}_{44}^{D}$ | ${g}_{33}$ | ${g}_{15}$ | ${\zeta}_{11}$ | ${\zeta}_{33}$ | ${\alpha}_{\theta}$ | ${\alpha}_{r}$ | $\rho $ | $\kappa $ |

7.95 | −3.03 | 7.91 | 17.91 | 23.91 | 40.36 | 76.87 | 99.65 | 2.89 | 1.29 | 5.56 | 2.1 |

_{31}, the distribution of radial stress ${\sigma}_{rr}$ and hoop stress ${\sigma}_{\theta \theta}$ along the radial direction of the actuator is plotted in Figure 2 when the actuator is subjected to an electric potential V

_{0}= 100 V and the temperature rise at the upper surface of layer 3 and the lower surface of layer 1 is T

_{o}= 10 °C and T

_{i}= 0 °C, respectively. From these two figures, it is clearly illustrated that the curves are almost identical for N = 20 and 30. Therefore, it can be concluded that convergence is obtained when using the Taylor series up to 20 terms to expand the piezoelectric coefficient g

_{31}(r) in the current case study. Taking different values of N is equivalent to the change of material gradient for the FGP layer, therefore, the discrepancy among the curves with different N indicates the stress distribution is significantly affected by the material gradient. It is seen from Figure 2(b) that the hoop stress will be continuous at the interfaces when N = 20 and 30, unlike an abrupt change for a smaller N, indicating the discontinuity of hoop stresses at the interfaces of dissimilar media can be eliminated by using graded material, which may prevent the possible failure of the structures at the interfaces. In addition, non-zero stresses $\left({\sigma}_{rr},{\sigma}_{\theta \theta}\right)$ are always observed in this sandwich actuator with thermal conduction. These internal stresses should be considered in the design of curved FGP actuator with sandwich structure. The distribution of electric displacement D

_{r}and electric potential Φ along the radial direction are shown in Figure 3 with different N. It is observed that the material gradient has relatively large influence on the electric displacement but no significant influence on electric potential distribution as Φ is almost identical for different values of N.

**Figure 2.**The distribution of stresses along radial direction for the curved FGP actuator when V

_{0}= 100 V and T

_{o}= 10 °C (

**a**) radial stress; (

**b**) hoop stress.

**Figure 3.**The distribution of (

**a**) electric displacement D

_{r}and (

**b**) electric potential Φ along the radial direction of the curved FGP actuator when V

_{0}= 100 V.

_{o}= (0 °C, 2 °C, 5 °C, 10 °C), while the temperature on the inner surface of layer 1 is kept constant. It is clearly indicated that the thermal loading has a significant effect on the stress distribution in the curved actuator. With the increase of the temperature, the magnitude of the stresses decreases. Under the same loading condition, the distribution of the electric field in the actuator is presented in Figure 5. It is seen that the thermal conduction significantly changes the distribution of electric field as expected, i.e., the electric field increases with the increase of the thermal loading. The influence of thermal loading can also be observed from the distribution of radial displacement ${u}_{r}$ and hoop displacement ${u}_{\theta}$ along the circumferential direction of the curved actuator as shown in Figure 6. The significant effect of thermal loading on the electroelastic fields of the curved FGP actuator observed from these figures indicate that it is necessary to consider the thermal effect in the design and optimization of the curved FGP sandwich actuator.

**Figure 4.**The distribution of stresses along the radial direction for the curved FGP actuator with different T

_{o}when V

_{0}= 100 V (

**a**) radial stress; (

**b**) hoop stress.

**Figure 5.**The distribution of the electric field along the radial direction of the curved FGP actuator when V

_{0}=100 V.

**Figure 6.**The variation of displacements with the angle θ in the middle of the actuator when V

_{0}= 100 V (

**a**) radial displacement; (

**b**) hoop displacement.

_{o}= 10 °C. It is clearly indicated that the magnitude of both stresses reduces drastically compared with those of the bimorph actuator. The curves obtained are smoothly changing along the radial direction of the actuator and no sharp peaks are observed at the interface for the FGP actuator. The distribution of the displacement fields in the middle of both piezoelectric bimorph and FGP sandwich actuator is depicted in Figure 8 for comparison. It is observed that the FGP sandwich actuator provides relative larger displacements compared to piezoelectric bimorph actuator for the same loading conditions. Tabular results are also provided to supplement the graphic presentation for stress and displacement distribution as shown in Table 2. For example, the radial stress at r = 17 mm (i.e., the interface of the bimorph) for both the bimorph and FGP actuators under different thermal loads are quantitatively shown in this table. It is seen that, by using the FGP sandwich actuator, the magnitude of the radial stress in the bimorph actuator has decreased significantly as illustrated by percentage. Also larger displacement at the free end of the FGP sandwich actuator is always observed compared to that of the bimorph actuator under different thermal loads. Moreover, with the decrease of the temperature rise T

_{o}, the difference of the free end displacements of these two type actuators increases. Based on these graphical displays and tabular data, the advantages of using FGP sandwich actuator over bimorph actuator are clearly demonstrated. Therefore, it is concluded that FGPMs are very important for the design and optimization of an actuator by generating less internal stresses while providing larger deflections for actuation. In addition, the thermal conduction has a significant effect on the electroelastic fields of the piezoelectric structure, which should also be considered for the design purpose of curved FGP actuator.

**Figure 7.**Comparison of the distribution of stresses along the radial direction for bimorph and FGP sandwich actuator when V

_{0}= 100 V (

**a**) radial stress; (

**b**) hoop stress.

**Figure 8.**Comparison of the variation of displacements with θ for bimorph and FGP sandwich actuator when V

_{0}= 100 V (

**a**) radial displacement; (

**b**) hoop displacement.

**Table 2.**Comparison of radial stress and free end displacements of the bimorph and FGP sandwich actuators under different thermal loads when V

_{0}= 100 V.

Thermal loads T_{o} (°C) | ${{\sigma}_{rr}|}_{r=17\text{mm}}$ (KPa) | Difference (%) | ${{u}_{r}|}_{\theta ={0}^{\circ}}$ (μm) | Difference (%) | ${{u}_{\theta}|}_{\theta ={0}^{\circ}}$ (μm) | Difference (%) | |||
---|---|---|---|---|---|---|---|---|---|

Bimorph | FGP | Bimorph | FGP | Bimorph | FGP | ||||

0 | 4.983 | 0.184 | −96.307 | −0.731 | −0.877 | +19.973 | −0.565 | −0.673 | +19.115 |

2 | 4.462 | 0.207 | −95.361 | −1.410 | −1.540 | +9.220 | −1.014 | −1.110 | +9.467 |

5 | 3.680 | 0.243 | −93.397 | −2.429 | −2.534 | +4.323 | −1.687 | −1.764 | +4.564 |

10 | 2.377 | 0.302 | −87.295 | −4.128 | −4.191 | +1.526 | −2.808 | −2.855 | +1.674 |

## 5. Conclusions

_{31}of the FGP layer is assumed to vary exponentially along the radial direction. By using Airy stress function, the electroelastic fields of the actuator are obtained analytically. Simulation results are presented to show the influence of material gradient and thermal conduction on the curved actuator configuration. It is found that the material gradient has a significant influence on the stresses and electric displacement, but not on the electric potential. However, thermal conduction has a significant effect on all the electroelastic fields of the curved FGP actuator with sandwich structure. By comparing the FGP sandwich actuator with piezoelectric bimorph, it is clearly indicated that much smaller internal stresses with no compromise of deflections could be achieved by the FGP sandwich actuator. This work is expected to provide helpful guidelines for the design and optimization of curved piezoelectric actuator with the consideration of thermal effect.

## Acknowledgments

## References

- Yao, K.; Koc, B.; Uchino, K. Longitudinal-bending mode micromotor using multilayer piezoelectric actuator. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2001**, 48, 1066–1071. [Google Scholar] [CrossRef] [PubMed] - Li, L.; Kumar, P.; Kanakraju, S.; Devoe, D.L. Piezoelectric AlGaAs bimorph microactuators. J. Micromech. Microeng.
**2006**, 16, 1062–1066. [Google Scholar] [CrossRef] - Luo, Y.; Lu, M.; Cui, T. A polymer-based bidirectional micropump driven by a PZT bimorph. Microsyst. Technol.
**2011**, 17, 403–409. [Google Scholar] - Zhao, H.; Stanley, K.; Wu, Q.M.J.; Czyzewska, E. Structure and characterization of a planar normally closed bulk-micromachined piezoelectric valve for fuel cell applications. Sens. Actuator A Phys.
**2005**, 120, 134–141. [Google Scholar] [CrossRef] - Zou, Q.; Tan, W.; Kim, E.S.; Loeb, G.E. Single-and triaxis piezoelectric-bimorph accelerometers. J. Microelectromech. Syst.
**2008**, 17, 45–57. [Google Scholar] [CrossRef] - Zhu, X.H.; Meng, Z.Y. Operational principle, fabrication and displacement characteristics of a functionally gradient piezoelectric ceramic actuator. Sens. Actuator A Phys.
**1995**, 48, 169–176. [Google Scholar] [CrossRef] - Wu, C.C.M.; Kahn, M.; Moy, W. Piezoelectric ceramics with functional gradients: A new application in material design. J. Am. Ceram. Soc.
**1996**, 79, 809–812. [Google Scholar] [CrossRef] - Takagi, K.; Li, J.F.; Yokoyama, S.; Watanabe, R.; Almajid, A.; Taya, M. Design and fabrication of functionally graded PZT/Pt piezoelectric bimorph actuator. Sci. Technol. Adv. Mater.
**2002**, 3, 217–224. [Google Scholar] [CrossRef] - Hauke, T.; Kouvatov, A.; Steinhausen, R.; Seifert, W.; Beigea, H.; Langhammer, H.T.; Abicht, H.P. Bending behavior of functionally gradient materials. Ferroelectrics
**2000**, 238, 759–766. [Google Scholar] [CrossRef] - Kruusing, A. Analysis and optimization of loaded cantilever beam microactuators. Smart Mater. Struct.
**2000**, 9, 186–196. [Google Scholar] [CrossRef] - Almajid, A.; Taya, M.; Hudnut, S. Analysis of out-of-plane displacement and stress field in a piezocomposite plate with functionally graded microstructure. Int. J. Solids Struct.
**2001**, 38, 3377–3391. [Google Scholar] [CrossRef] - Taya, M.; Almajid, A.A.; Dunn, M.; Takahashi, H. Design of bimorph piezo-composite actuators with functionally graded microstructure. Sens. Actuator A Phys.
**2003**, 107, 248–260. [Google Scholar] [CrossRef] - Huang, D.J.; Ding, H.J.; Chen, W.Q. Piezoelasticity solutions for functionally graded piezoelectric beams. Smart Mater. Struct.
**2007**, 16, 687–695. [Google Scholar] [CrossRef] - Huang, D.J.; Ding, H.J.; Chen, W.Q. Analysis of functionally graded and laminated piezoelectric cantilever actuators subjected to constant voltage. Smart Mater. Struct.
**2008**, 17, 065002. [Google Scholar] [CrossRef] - Shi, Z.F. General solution of a density functionally gradient piezoelectric cantilever and its applications. Smart Mater. Struct.
**2002**, 11, 122–129. [Google Scholar] [CrossRef] - Liu, T.T.; Shi, Z.F. Bending behavior of functionally gradient piezoelectric cantilever. Ferroelectrics
**2004**, 308, 43–51. [Google Scholar] [CrossRef] - Xiang, H.; Shi, Z. Electrostatic analysis of functionally graded piezoelectric cantilevers. J. Intell. Mater. Syst. Struct.
**2007**, 18, 719–726. [Google Scholar] [CrossRef] - Shi, Z.F.; Chen, Y. Functionally graded piezoelectric cantilever beam under load. Arch. Appl. Mech.
**2004**, 74, 237–247. [Google Scholar] [CrossRef] - Wang, B.L.; Noda, N. Thermally induced fracture of a smart functionally graded composite structure. Theor. Appl. Fract. Mech.
**2001**, 35, 93–109. [Google Scholar] [CrossRef] - Joshi, S.; Mukherjee, A.; Schmauder, S. Numerical characterization of functionally graded active materials under electrical and thermal fields. Smart Mater. Struct.
**2003**, 12, 571–579. [Google Scholar] [CrossRef] - Lee, H.J. Layerwise laminate analysis of functionally graded piezoelectric bimorph beams. J. Intell. Mater. Syst. Struct.
**2005**, 16, 365–371. [Google Scholar] [CrossRef] - Chen, Y.; Shi, Z.F. Exact solutions of functionally gradient piezothermoelastic cantilevers and parameter identification. J. Intell. Mater. Syst. Struct.
**2005**, 16, 531–539. [Google Scholar] [CrossRef] - Xiang, H.J.; Shi, Z.F. Static analysis for functionally graded piezoelectric actuators or sensors under a combined electro-thermal load. Eur. J. Mech. A Solids
**2009**, 28, 338–346. [Google Scholar] [CrossRef] - Yang, J.; Xiang, H.J. Thermo-electro-mechanical characteristics of functionally graded piezoelectric actuators. Smart Mater. Struct.
**2007**, 16, 784–797. [Google Scholar] [CrossRef] - Rye, D.H.; Wang, K.W. Characterization of surface-bonded piezoelectric actuators on curved beams. Smart Mater. Struct.
**2002**, 11, 377–388. [Google Scholar] [CrossRef] - Shi, Z.F. Bending behavior of piezoelectric curved actuator. Smart Mater. Struct.
**2005**, 14, 835–842. [Google Scholar] [CrossRef] - Zhang, T.T.; Shi, Z.F. Two-dimensional exact analysis for piezoelectric curved actuators. J. Micromech. Microeng.
**2006**, 16, 640–647. [Google Scholar] [CrossRef] - Shi, Z.F.; Zhang, T.T. Bending analysis of a piezoelectric curved actuator with a generally graded property for the piezoelectric parameter. Smart Mater. Struct.
**2008**, 17, 045018. [Google Scholar] [CrossRef] - Zaman, M.; Yan, Z.; Jiang, L.Y. Thermal effect on the bending behavior of curved functionally graded piezoelectric actuators. Int. J. Appl. Mech.
**2010**, 2, 787–805. [Google Scholar] [CrossRef] - Kouvatov, A.; Steinhausen, R.; Seifert, W.; Hauke, T.; Langhammer, H.T.; Beige, H.; Abicht, H. Comparison between bimorphic and polymorphic bending devices. J. Eur. Ceram. Soc.
**1999**, 19, 1153–1156. [Google Scholar] [CrossRef] - Chen, C.D. On the singularities of the thermo-electro-elastic fields near the apex of a piezoelectric bonded wedge. Int. J. Solids Struct.
**2006**, 43, 957–981. [Google Scholar] [CrossRef]

## Appendix A

© 2011 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Yan, Z.; Zaman, M.; Jiang, L.
Thermo-Electro-Mechanical Analysis of a Curved Functionally Graded Piezoelectric Actuator with Sandwich Structure. *Materials* **2011**, *4*, 2151-2170.
https://doi.org/10.3390/ma4122151

**AMA Style**

Yan Z, Zaman M, Jiang L.
Thermo-Electro-Mechanical Analysis of a Curved Functionally Graded Piezoelectric Actuator with Sandwich Structure. *Materials*. 2011; 4(12):2151-2170.
https://doi.org/10.3390/ma4122151

**Chicago/Turabian Style**

Yan, Zhi, Mostafa Zaman, and Liying Jiang.
2011. "Thermo-Electro-Mechanical Analysis of a Curved Functionally Graded Piezoelectric Actuator with Sandwich Structure" *Materials* 4, no. 12: 2151-2170.
https://doi.org/10.3390/ma4122151