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Dynamic analysis of flexoelectric systems in the frequency domain with isogeometric analysis

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Abstract

A numerical procedure based on isogeometric analysis (IGA) is developed to analyze the dynamic response of flexoelectric systems in the frequency domain. In materials or composites with an effective flexoelectric response, a polarization can be induced by local strain gradients. In general, these effects are small in the static regime. However, larger effects may be induced by dynamic loads, and can be used in energy harvesters converting mechanical vibrations into electrical energy. In this work, the equations describing frequency response of flexoelectric systems under dynamic loads are first described. Then, an IGA discretization procedure is employed to handle the \(C^1\) continuity of the displacement fields. The conditions of both open and close-circuits are formulated. The numerical methodology is used to evaluate the sensitivity of different parameters such as load resistors, dynamic scale parameter, and the use of flexoelectric or non-flexoelectric materials on the frequency response of output voltage, power and displacements of beam-like structures, possibly incorporating structural geometrical features. The potential of IGA with respect to mesh refinement (h-refinement) and higher order approximation (p-refinement) for modeling complex geometries within the present framework is invetigated.

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Acknowledgements

Xing Chen acknowledges the support from China Scholarship Council (CSC No. 202106370116)

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Authors and Affiliations

  1. Key Laboratory of Traffic Safety on Track, Ministry of Education, School of Traffic and Transportation Engineering, Central South University, Changsha, 410075, China

    Xing Chen & Song Yao

  2. Université Gustave Eiffel, CNRS, MSME UMR 8208, 77454, Marne-la-Vallée, France

    Xing Chen & Julien Yvonnet

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Correspondence to Julien Yvonnet.

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Appendix: Numerical values of material matrices

Appendix: Numerical values of material matrices

$$\begin{aligned}&{\mathbf {C}}_1=\begin{bmatrix} 132.1&{} 84&{} 0\\ 84&{} 155.6&{} 0\\ 0&{} 0 &{} 35.8 \end{bmatrix}(GPa)&\end{aligned}$$
(76)
$$\begin{aligned}&\varvec{\mu }=\begin{bmatrix} 1.365&{}1.365&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}2.641&{}2.641&{}0 \end{bmatrix} (\times 10^{-4}\,\text {C}\,\mathrm {m}^{-1})&\end{aligned}$$
(77)
$$\begin{aligned}&\varvec{e}=\begin{bmatrix} -0.5217&{} -0.5217&{} 0\\ 0&{} 0&{} 0 \end{bmatrix} (\text {C}\,\mathrm {m}^{-2})&\end{aligned}$$
(78)
$$\begin{aligned}&\varvec{\alpha }_1=\begin{bmatrix} 2.102&{} 0\\ 0&{} 4.065 \end{bmatrix}(\text {nC}^2\, \mathrm {N}^{-1}\,\text {m}^{-2})&\end{aligned}$$
(79)
$$\begin{aligned}&{\mathbf {g}}=\ell ^2\begin{bmatrix} 132.1&{}0&{}0&{}84&{}0&{}0\\ 0&{}132.1&{}84&{}0&{}0&{}0\\ 0&{}84&{}132.1&{}0&{}0&{}0\\ 84&{}0&{}0&{}132.1&{}0&{}0\\ 0&{}0&{}0&{}0&{}35.8&{}0\\ 0&{}0&{}0&{}0&{}0&{}35.8 \end{bmatrix}\times 10^9&\end{aligned}$$
(80)

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Chen, X., Yao, S. & Yvonnet, J. Dynamic analysis of flexoelectric systems in the frequency domain with isogeometric analysis. Comput Mech 71, 353–366 (2023). https://doi.org/10.1007/s00466-022-02244-0

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