Flexoelectricity is the development of an electric field under an applied strain gradient. This effect is highly size-dependent and is significant at nanoscales. Unsymmetric magnetoelectric (ME) composites undergo bending deformations under a magnetic field that results in non-zero strain gradients. Hence, it is important to give due consideration to flexoelectricity, along with direct piezoelectricity in these composites. This article presents a finite-element model that predicts the static and dynamic behavior of ME nanobeams subjected to magnetic fields. The effect of material nonlinearity of the ferromagnetic phase and flexoelectricity in the ferroelectric phase on the ME coupling are determined, and the obtained results show agreement with the literature. It is found that flexoelectricity results in substantially higher static ME coefficients. However, the resonant ME response is weakened by flexoelectricity. The effect of surface elasticity on the ME coupling coefficient is also investigated by making use of the Gurtin–Murdoch theory. The surface effects are found to improve the ME coefficient at bending resonance, while its effect is trivial at static and axial resonant conditions. Furthermore, the effect of load resistance on the ME voltage and charge coefficients is determined. The results indicate clear changes in the trend between short-circuit and open-circuit conditions.

中文翻译：

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柔性电和表面弹性对非对称复合材料中非线性磁电耦合的影响

柔性电是在施加的应变梯度下电场的发展。这种效应高度依赖于尺寸，并且在纳米尺度上很显着。不对称磁电 (ME) 复合材料在磁场下发生弯曲变形，导致非零应变梯度。因此，重要的是要适当考虑挠曲电性以及这些复合材料中的直接压电性。本文提出了一个有限元模型，该模型可预测 ME 纳米梁在磁场作用下的静态和动态行为。确定了铁磁相的材料非线性和铁电相中的挠曲电对 ME 耦合的影响，所得结果与文献一致。发现挠曲电导致显着更高的静态ME系数。然而，共振ME响应被挠曲电减弱。还利用 Gurtin-Murdoch 理论研究了表面弹性对 ME 耦合系数的影响。发现表面效应可提高弯曲共振时的 ME 系数，而其在静态和轴向共振条件下的影响微不足道。此外，还确定了负载电阻对 ME 电压和电荷系数的影响。结果表明短路和开路条件之间的趋势明显变化。而它的影响在静态和轴向共振条件下是微不足道的。此外，还确定了负载电阻对 ME 电压和电荷系数的影响。结果表明短路和开路条件之间的趋势明显变化。而它的影响在静态和轴向共振条件下是微不足道的。此外，还确定了负载电阻对 ME 电压和电荷系数的影响。结果表明短路和开路条件之间的趋势明显变化。