Multi-physics coupling of piezoelectric laminated beam with temperature-dependent material property

doi:10.1007/s10483-026-3418-8

Abstract

Abstract:

Piezoelectric laminated beams (PLBs) have attracted increasing attention because of their high energy density and ease of integration. However, the temperature sensitivity of their material parameters poses a critical challenge for accurately predicting their response under thermal and vibration excitations. To bridge this gap, we construct a coupled thermo-electro-elastic forced vibration model of the PLB configuration with a tip mass, characterized by a temperature-dependent material property, and derive its closed-form solutions. Furthermore, the dynamic responses induced by thermal and vibration excitations are decoupled and analyzed. The decoupling analysis indicates that the global displacement is predominantly governed by the displacement induced by the thermo-electric coupling effect under static and low-frequency thermal-vibration excitations, with the displacement amplitude on the order of micrometers. Due to the insignificant displacement, the thermal strain is excluded from the multi-physics coupling model, allowing for individual investigation of the effect of temperature-dependent material property on the dynamic response. The results show that the structural resonant frequency decreases with the increasing thermal source intensity, which is attributed to the deterioration of the flexural rigidity. The corresponding maximum voltage and power decrease by 4.5% and 9.7% with the increasing surface thermal source intensity from 0 kW/m² to 2.0 kW/m², respectively. Overall, this study is promising to promote the investigation of multi-physics coupling and provides critical insights into the performance degradation of piezoelectric structures under the combination of thermal and vibration excitations.

Key words: multi-physics coupling, temperature-dependent material property, thermo-electric coupling effect, energy harvesting, Green’s function

2010 MSC Number:

O327

Huirong ZHANG, Gantong CHEN, Bohao DUAN, Shengxi ZHOU. Multi-physics coupling of piezoelectric laminated beam with temperature-dependent material property. Applied Mathematics and Mechanics (English Edition), 2026, 47(8): 1811-1834.

TrendMD

Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks

URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3418-8

https://www.amm.shu.edu.cn/EN/Y2026/V47/I8/1811

Figures/Tables 14

Fig. 1

Fig. 2

Fig. 3

Table 1

Calculation results at different accumulation orders and FEM result"

r_m	$T (y 0) / ° C$	e₁/%	r_m	$T (y 0) / ° C$	e₁/%
1	64.000 1		30	64.000 0	$< 0.5$
5	64.000 0	$< 0.5$	40	64.000 0	$< 0.5$
10	64.000 0	$< 0.5$	50	64.000 0	$< 0.5$
20	64.000 0	$< 0.5$	FEM result	64.000 0	$< 0.5$
Note that the precision of the numerical results is determined as $1 × 10 − 4 .$ $e 1 = \| T i (y 0) − T j (y 0) \| / T i (y 0);$ $e 2 = \| T FEM (y 0) − T 50 (y 0) \| / T FEM (y 0) < 0.5 % .$ The superscripts i and j indicate the orders, and $j < i,$ where $j = 1,$ 5, 10, 20, 30, 40; $i = 5,$ 10, 20, 30, 40, 50. $T FEM (y 0)$ denotes the FEM result

Table 1

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

[1]	RINDI, A., BALDASSARRE, L., PANARA, D., MELI, E., RIDOLFI, A., FRILLI, A., NOCCIOLINI, D., and PANCONI, S. An efficient iterative approach for the analysis of thermal instabilities in rotating machines. Journal of Vibration and Acoustics, 139(6), 061019 (2017)
[2]	MATNEY, A., MURTHY, R., SONG, P. C., WANG, X. Q., and MIGNOLET, M. P. Nonlinear reduced order modeling of heated structures with temperature-dependent properties. AIAA Journal, 63(1), 244–259 (2025)
[3]	NESARHOSSEINI, S., ANSARI, R., OSKOUIE, M. F., and ROUHI, H. Thermally induced vibration analysis of Timoshenko beams based on the micropolar thermoelasticity. Acta Mechanica, 234(5), 1957–1971 (2023)
[4]	ZHAO, X., ZHU, W. D., and LI, Y. H. Analytical solutions of nonlocal coupled thermoelastic forced vibrations of micro-/nano-beams by means of Green’s functions. Journal of Sound and Vibration, 481, 115407 (2020)
[5]	KONG, D. J., HE, Z. Z., LIU, C. B., and ZHANG, C. L. Analysis of multi-field coupling behaviors of sandwich piezoelectric semiconductor beams under thermal loadings. Applied Mathematics and Mechanics (English Edition), 46(8), 1571–1590 (2025) https://doi.org/10.1007/s10483-025-3284-7
[6]	LOTFAN, S. and CIGEROGLU, E. Coupled thermo-elastic dynamics of rotating pre-twisted blades exposed to thermal shock including nonlinear rotational effects. Journal of Sound and Vibration, 594, 118669 (2025)
[7]	YAMAMOTO, R., HEGENDÖRFER, A., MERGHEIM, J., and KAKIMOTO, K. I. Temperature-dependent vibration energy harvesting performance of polyimide/(Na, K)NbO₃ piezoelectric composites. Japanese Journal of Applied Physics, 61, SN1028 (2022)
[8]	NODA, N. Thermal stresses in materials with temperature-dependent properties. Applied Mechanics Reviews, 44(9), 383–397 (1991)
[9]	YANG, W. and YOU, P. B. Thermal stress analysis and fatigue life assessment of bridge structures under multi-physical field coupling. International Journal of Heat and Technology, 41(6), 1561–1572 (2023)
[10]	BOWEN, C. R., KIM, H. A., WEAVER, P. M., and DUNN, S. Piezoelectric and ferroelectric materials and structures for energy harvesting applications. Energy & Environmental Science, 7(1), 25–44 (2014)
[11]	IQBAL, M., NAUMAN, M. M., KHAN, F. U., ABAS, P. E., CHEOK, Q., IQBAL, A., and AISSA, B. Vibration-based piezoelectric, electromagnetic, and hybrid energy harvesters for microsystems applications: a contributed review. International Journal of Energy Research, 45(1), 65–102 (2021)
[12]	TOPRAK, A. and TIGLI, O. Piezoelectric energy harvesting: state-of-the-art and challenges. Applied Physics Reviews, 1(3), 031104 (2014)
[13]	MAO, J. J., GAO, W., LIU, C. R., CAO, D. X., and LAI, S. Transition analysis of meta-stable and bi-stable nonlinear behavior in piezoelectric vibration energy harvesting through a pre-shaped curved beam model. Applied Mathematics and Mechanics (English Edition), 46(11), 2017–2034 (2025) https://doi.org/10.1007/s10483-025-3319-6
[14]	MANOACH, E. and RIBEIRO, P. Coupled, thermoelastic, large amplitude vibrations of Timoshenko beams. International Journal of Mechanical Sciences, 46(11), 1589–1606 (2004)
[15]	MOHAMMADI, R. and HOSSEINI, M. Modeling and free vibration analysis of a rotating functionally graded thin-walled hub-blade system under aerothermoelastic loading. Aerospace Science and Technology, 146, 108935 (2024)
[16]	PRIYA, S. and INMAN, D. J. Energy Harvesting Technologies, Springer, New York (2009)
[17]	TSUSHIMA, N. and SU, W. H. A study on adaptive vibration control and energy conversion of highly flexible multifunctional wings. Aerospace Science and Technology, 79, 297–309 (2018)
[18]	DAI, H. L., ABDELKEFI, A., and WANG, L. Theoretical modeling and nonlinear analysis of piezoelectric energy harvesting from vortex-induced vibrations. Journal of Intelligent Material Systems and Structures, 25(14), 1861–1874 (2014)
[19]	RAZZAQ, S., ASGHAR, A., and IQBAL, S. Advances in Hybrid Conducting Polymer Technology, Springer, Switzerland (2021)
[20]	HANISZEWSKI, T., BUCKI, S., MARGIELEWICZ, J., GĄSKA, D., KUANG, Y., and LITAK, G. Energy harvesting system with a hyperelastic mechanical vibration amplifier. Mechanical Systems and Signal Processing, 224, 112038 (2025)
[21]	LI, H. T., ZHENG, T. Y., QIN, W. Y., TIAN, R. L., DING, H., JI, J. C., and CHEN, L. Q. Theoretical and experimental study of a bi-stable piezoelectric energy harvester under hybrid galloping and band-limited random excitations. Applied Mathematics and Mechanics (English Edition), 45(3), 461–478 (2024) https://doi.org/10.1007/s10483-024-3098-5
[22]	ERTURK, A. and INMAN, D. J. A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters. Journal of Vibration and Acoustics, 130(4), 041002 (2008)
[23]	DAQAQ, M. F. Transduction of a bistable inductive generator driven by white and exponentially correlated Gaussian noise. Journal of Sound and Vibration, 330(11), 2554–2564 (2011)
[24]	STANTON, S. C., OWENS, B. A. M., and MANN, B. P. Harmonic balance analysis of the bistable piezoelectric inertial generator. Journal of Sound and Vibration, 331(15), 3617–3627 (2012)
[25]	ZHOU, S. X. and ZUO, L. Nonlinear dynamic analysis of asymmetric tristable energy harvesters for enhanced energy harvesting. Communications in Nonlinear Science and Numerical Simulation, 61, 271–284 (2018)
[26]	HUANG, D. M., ZHOU, S. X., and LITAK, G. Theoretical analysis of multi-stable energy harvesters with high-order stiffness terms. Communications in Nonlinear Science and Numerical Simulation, 69, 270–286 (2019)
[27]	ZHAO, L., HU, G. B., ZHOU, S. X., PENG, Y., XIE, S. R., and LI, Z. J. Magnetic coupling and amplitude truncation based bistable energy harvester. International Journal of Mechanical Sciences, 273, 109228 (2024)
[28]	ZHANG, B., YANG, G., HU, B. X., XIONG, Y. P., and ZHOU, S. X. Optimized design of self-powered SSHI interface circuit for enhanced vibration energy harvesting. Smart Materials and Structures, 34(2), 025025 (2025)
[29]	ZHANG, H., ZHOU, S. X., and LIM, C. W. Flutter suppression of a heated panel in supersonic airflow using energy harvesters. AIAA Journal (2026) https://doi.org/10.2514/1.J066769
[30]	ZHAO, X., IEGAINK, F. J. N., ZHU, W. D., and LI, Y. H. Coupled thermo-electro-elastic forced vibrations of piezoelectric laminated beams by means of Green’s functions. International Journal of Mechanical Sciences, 156, 355–369 (2019)
[31]	ZHANG, H. R., INMAN, D. J., and ZHOU, S. X. Coupled thermo-electric-elastic piezoelectric vibration energy harvester with axial movement: modeling, verification, and analysis. Journal of Vibration and Acoustics, 146(5), 051002 (2024)
[32]	ARROYO, E., JIA, Y., DU, S., CHEN, S., and SESHIA, A. High temperature performance of a piezoelectric micro cantilever for vibration energy harvesting. Journal of Physics: Conference Series, 773(1), 012001 (2016)
[33]	ARROYO, E., JIA, Y., DU, S. J., CHEN, S. T., and SESHIA, A. A. Experimental and theoretical study of a piezoelectric vibration energy harvester under high temperature. Journal of Microelectromechanical Systems, 26(6), 1216–1225 (2017)
[34]	BARKER, S., VASSILEVSKI, K. V., WRIGHT, N. G., and HORSFALL, A. B. High temperature vibration energy harvester system. IEEE Sensors, IEEE, 300–303 (2011)
[35]	KIM, S. B., PARK, J. H., AHN, H., LIU, D., and KIM, D. J. Temperature effects on output power of piezoelectric vibration energy harvesters. Microelectronics Journal, 42(8), 988–991 (2011)
[36]	WACHTMAN, J. B., TEFFT, W. E., LAM, D. G., and APSTEIN, C. S. Exponential temperature dependence of Young’s modulus for several oxides. Physical Review, 122(6), 1754–1759 (1961)
[37]	CHO, C. H. Characterization of Young’s modulus of silicon versus temperature using a “beam deflection” method with a four-point bending fixture. Current Applied Physics, 9(2), 538–545 (2009)
[38]	LI, W. G., KOU, H. B., ZHANG, X. Y., MA, J. Z., LI, Y., GENG, P. J., WU, X. Z., CHEN, L. M., and FANG, D. N. Temperature-dependent elastic modulus model for metallic bulk materials. Mechanics of Materials, 139, 103194 (2019)
[39]	EN 1999-1-2: 2007. Eurocode 9: Design of Aluminium Structures — Part 1-2: Structural Fire Design, European Committee for Standardization (2007)
[40]	SU, M. N. and YOUNG, B. Material properties of normal and high strength aluminium alloys at elevated temperatures. Thin-Walled Structures, 137, 463–471 (2019)
[41]	LI, P., LI, C. L., and CONG, B. L. Piezoelectric-thermo-elastic coupling effect analysis for piezoelectric vibration energy harvester. Microsystem Technologies, 24(9), 3823–3832 (2018)
[42]	XIAO, Y. S. and WU, Z. Multi-objective optimization on thermomechanical behaviors of temperature-dependent graphene platelet reinforced sandwich plates. Chinese Journal of Aeronautics, 38(5), 103363 (2025)
[43]	YANG, Z. Y., ZHU, H., YU, F., WU, P., and FANG, H. Thermo-mechanical coupled behavior of laminated beams with temperature-dependent viscoelastic interlayers. European Journal of Mechanics A, 100, 105000 (2023)
[44]	GUO, X. X., WANG, Z. M., WANG, Y., and ZHOU, Y. F. Analysis of the coupled thermoelastic vibration for axially moving beam. Journal of Sound and Vibration, 325(3), 597–608 (2009)
[45]	ZHANG, H. R., SUI, W. T., YANG, C. Q., ZHANG, L. A., SONG, R. J., and WANG, J. L. An asymmetric magnetic-coupled bending-torsion piezoelectric energy harvester: modeling and experimental investigation. Smart Materials and Structures, 31(1), 015037 (2022)
[46]	GROVER, D. and SHARMA, J. N. Transverse vibrations in piezothermoelastic beam resonators. Journal of Intelligent Material Systems and Structures, 23(1), 77–84 (2012)
[47]	ERTURK, A. and INMAN, D. J. Piezoelectric Energy Harvesting, Wiley, Chichester, U. K. (2011)
[48]	ÖZISIK, M. N. Heat Conduction, Wiley, New York (1993)
[49]	ASMAR, N. H. Partial Differential Equations with Fourier Series and Boundary Value Problems, Pearson Education, Upper Saddle River, N. J. (2005)