Higher-order two-way electromechanical coupling between strain gradients and electric field, known as flexoelectricity, has a strong influence on the micro- and nanoelectromechanical devices characterization as it is highly pronounced on smaller scales. Flexoelectricity in dielectrics and piezoelectrics has been well analyzed recently, however, its influence on the behavior of ferroelectric materials has not been explored yet. On the one hand, the current study can be seen as an expansion of ferroelectric material behavior towards higher-order electromechanical coupling. On the other hand, it can be treated as an advancement of flexoelectricity to non-linear material behavior, including inherent residual stresses and polarization. Either way of thinking is equally justified and leads to a substantial advancement in material modeling that incorporates small scales and strong electromechanical loading that drives the ceramics out of the linear response domain. For the numerical implementation, the mixed FEM is successfully employed. This involves the formulation from scratch of a three-dimensional mixed FE for flexoelectricity in dielectrics/piezoelectrics. After its successful verification, the micromechanical ferroelectric switching model is additionally implemented into the mixed FE. Initial investigations involve single and polycrystalline representative volume element simulations, which enable the analysis of the altered hysteresis loops. Following this, novel simulations of the truncated pyramid problem for flexoelectricity + ferroelectricity are performed. The introduced fully coupled, multi-scale, mixed FEM-based electromechanical model brings insights into the combined impact of the first-order nonlinear (ferroelectricity) and higher-order (flexoelectricity) responses.

中文翻译：

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存在挠曲电的铁电材料的数值模拟

应变梯度和电场之间的高阶双向机电耦合（称为挠曲电）对微米和纳米机电器件的表征具有重大影响，因为它在较小的尺度上非常明显。最近，电介质和压电体中的挠曲电性已得到很好的分析，但其对铁电材料行为的影响尚未得到探索。一方面，当前的研究可以被视为铁电材料行为向高阶机电耦合的扩展。另一方面，它可以被视为挠曲电对非线性材料行为（包括固有残余应力和极化）的进步。任何一种思考方式都是同样合理的，并且会导致材料建模方面的重大进步，该材料建模结合了小尺度和强大的机电负载，可将陶瓷驱动出线性响应域。对于数值实现，成功地采用了混合有限元法。这涉及从头开始制定用于电介质/压电体挠曲电的三维混合有限元。成功验证后，微机械铁电开关模型又被应用到混合有限元中。初步研究涉及单晶和多晶代表性体积元模拟，这使得能够分析改变的磁滞回线。在此之后，对柔性电+铁电的截棱锥问题进行了新颖的模拟。引入的全耦合、多尺度、基于混合有限元的机电模型可以深入了解一阶非线性（铁电性）和高阶（挠性电性）响应的综合影响。