This work is aimed at establishing a unified fractional thermoelastic model for piezoelectric structures, and shedding light on the influence of different definitions on the transient responses. Theoretically, based upon Cattaneo-type equation, a unified form of fractional heat conduction law is proposed by adopting Caputo Fabrizio fractional derivative, Atangana Baleanu fractional derivative and Tempered Caputo fractional derivative. Then, thermoelastic model of fractional order is formulated for piezoelastic materials by combining the unified heat conduction law and the governing equations of elastic and electric fields. Numerically, the present theoretical model is applied to study the transient responses of piezoelectric medium that is subjected to a thermal shock. The governing equations are analytically derived and numerically solved with the aids of Laplace transform method. The obtained results are graphically illustrated, and the influences of different definitions of fractional derivative and different fractional order are revealed. This work may be helpful for understanding the multi-coupling effect of elastic, thermal and electric fields, and for inspiring further developments of fractional calculus.

中文翻译：

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压电材料的分数阶热弹性

这项工作旨在为压电结构建立统一的分数热弹性模型，并阐明不同定义对瞬态响应的影响。在理论上，基于Cattaneo型方程，采用Caputo Fabrizio分数导数、Atangana Baleanu分数导数和Tempered Caputo分数导数，提出了分数热传导定律的统一形式。然后，结合统一热传导定律和弹性电场控制方程，建立了压电弹性材料的分数阶热弹性模型。在数值上，本理论模型被应用于研究压电介质受到热冲击的瞬态响应。借助拉普拉斯变换方法对控制方程进行解析推导和数值求解。得到的结果以图形方式说明，揭示了不同分数导数定义和不同分数阶的影响。这项工作可能有助于理解弹性场、热场和电场的多重耦合效应，并有助于启发分数微积分的进一步发展。