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<p>The piezoelectric cantilever beam energy harvester incorporating simultaneously the strain gradient effects, piezoelectric and flexoelectric effects was studied in this paper. The flexoelectric effect describes the electric polarization induced by the strain gradient in dielectric materials. This effect represents a more pronounced force-electric coupling compared to the piezoelectric effect at micro-nano scale. A new mechanical model for a piezoelectric cantilever beam energy harvester incorporating strain gradient and flexoelectric effects was developed in this paper. Based on the Euler–Bernoulli beam theory, the boundary conditions and governing equations were derived using Gibbs free energy and Hamilton’s variational principle. This model effectively captures the mechano-electronic response characteristics at micro-nano scale. The multi-order modes of the cantilever beam were obtained by solving the governing equations. The frequency response of the harvester’s output voltage and power were then determined through modal analysis. Epoxy resin and BaTiO<sub>3</sub> were used as the elastic and piezoelectric layer materials, respectively, and the parameter study is performed to systematically examine the effects of various parameters such as the strain gradient parameter, flexoelectric parameter, the concentrated mass at the free end, and the external resistance on the energy harvester’s performance. The numerical analysis shows that the enhancements in the strain gradient parameter, the concentrated mass, the flexoelectric parameter, and the external resistance significantly increase the voltage output across each resonant frequency mode, although these parameters affect the voltage frequency response curves in distinct ways. Furthermore, the flexoelectric effect contributes prominently to mechano-electronic coupling behaviors at micro-nano scales. The power frequency response reaches its maximum at specific values of the concentrated mass and the external resistance. Notably, power output from higher-order modes is significantly greater than that of the fundamental mode, particularly near the short-circuit state. These results provide a theoretical guidance for the design and optimization of piezoelectric energy harvester that leverage flexoelectric and strain gradient effects simultaneously.</p>

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Energy harvesting of nano-cantilever beam with simultaneous consideration of strain gradient, piezoelectric and flexoelectric effects

  • 期刊论文

  • 发表日期：

    2026年7月26日

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  • Ziwei Wang,
  • Peijun Wei,
  • Lina Zhao,
  • Xiao Guo

摘要

The piezoelectric cantilever beam energy harvester incorporating simultaneously the strain gradient effects, piezoelectric and flexoelectric effects was studied in this paper. The flexoelectric effect describes the electric polarization induced by the strain gradient in dielectric materials. This effect represents a more pronounced force-electric coupling compared to the piezoelectric effect at micro-nano scale. A new mechanical model for a piezoelectric cantilever beam energy harvester incorporating strain gradient and flexoelectric effects was developed in this paper. Based on the Euler–Bernoulli beam theory, the boundary conditions and governing equations were derived using Gibbs free energy and Hamilton’s variational principle. This model effectively captures the mechano-electronic response characteristics at micro-nano scale. The multi-order modes of the cantilever beam were obtained by solving the governing equations. The frequency response of the harvester’s output voltage and power were then determined through modal analysis. Epoxy resin and BaTiO3 were used as the elastic and piezoelectric layer materials, respectively, and the parameter study is performed to systematically examine the effects of various parameters such as the strain gradient parameter, flexoelectric parameter, the concentrated mass at the free end, and the external resistance on the energy harvester’s performance. The numerical analysis shows that the enhancements in the strain gradient parameter, the concentrated mass, the flexoelectric parameter, and the external resistance significantly increase the voltage output across each resonant frequency mode, although these parameters affect the voltage frequency response curves in distinct ways. Furthermore, the flexoelectric effect contributes prominently to mechano-electronic coupling behaviors at micro-nano scales. The power frequency response reaches its maximum at specific values of the concentrated mass and the external resistance. Notably, power output from higher-order modes is significantly greater than that of the fundamental mode, particularly near the short-circuit state. These results provide a theoretical guidance for the design and optimization of piezoelectric energy harvester that leverage flexoelectric and strain gradient effects simultaneously.