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Development of accurate piezoelectric beam models based on Boley's method

机译：基于Boley方法的精确压电梁模型的开发

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The solution of the bi-potential equation gives analytical formulas for two-dimensional isotropic problems. In 1954, Boley published a fast-converging, iterative method for rectangular cross-section elastic beams, in order to calculate in-plane stresses and deflections. In this contribution, Boley's iterative method is extended to piezoceramic materials. After defining potential functions for the in-plane stresses and electric fields, two coupled partial differential equations for these two potential functions are obtained. Extending the Boley's iterative method, by including the piezoelectric effect, it is shown that the solution of the first iteration exactly yields the analytical results for thin piezoelectric beams that are already known in the open literature. The first iteration also shows that the distribution of the electric potential through the beam thickness is a second-order polynomial with respect to the thickness coordinate. The second and third iterations are found to be higher-order terms, which depend on the thickness-to-length ratio of the beam. They can be interpreted as correction terms to obtain more accurate results for moderately thick beams. In this work, analytical and numerical results for a hinged-hinged bimorph under a sinusoidal force (sensing problem) or under a sinusoidal voltage load (actuation problem) are compared to analytical 2D plane stress results. It is shown that for higher values of the thickness-to-length ratio (i.e. moderately thick or very thick beams), where the Bernoulli-Euler beam bending theory yields unsatisfactory results, the latter are modified by the higher order Boley's method iterations such that the results converge to analytical 2D results.
机译：双势方程的溶液给出了二维各向同性问题的分析公式。 1954年，Boley发布了一种用于矩形横截面弹性束的快速迭代方法，以计算面内应力和偏转。在这一贡献中，Boley的迭代方法延伸到压电陶瓷材料。在限定面内应力和电场的潜在功能之后，获得了这两个潜在功能的两个耦合的部分微分方程。通过包括压电效应来扩展Boley的迭代方法，示出了第一迭代的解决方案精确地产生了在开放文献中已知的薄压电束的分析结果。第一迭代还示出了通过光束厚度的电位分布是相对于厚度坐标的二阶多项式。发现第二和第三迭代是高阶项，这取决于光束的厚度到长度比。它们可以被解释为校正术语，以获得适度厚梁的更准确的结果。在这项工作中，与分析2D平面应力结果相比，在正弦力（传感问题）下或在正弦电压（致动问题）下进行铰接铰接的双芯片的分析和数值结果。结果表明，对于厚度到长度比的较高值（即，适度厚或非常厚的光束），其中Bernoulli-euler光束弯曲理论产生不令人满意的结果，后者由更高阶Boley的方法迭代修改结果会聚到分析2D结果。

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