In the first part of this two-paper series, a new Fragile Points Method (FPM), in both primal and mixed formulations, is presented for analyzing flexoelectric effects in 2D dielectric materials. In the present paper, a number of numerical results are provided as validations, including linear and quadratic patch tests, flexoelectric effects in continuous domains, and analyses of stationary cracks in dielectric materials. A discussion of the influence of the electroelastic stress is also given, showing that Maxwell stress could be significant and thus the full flexoelectric theory is recommended to be employed for nanoscale structures. The present primal as well as mixed FPMs also show their suitability and effectiveness in simulating crack initiation and propagation with flexoelectric effect. Flexoelectricity, coupled with piezoelectric effect, can help, hinder, or deflect the crack propagation paths and should not be neglected in nanoscale crack analysis. In FPM, no remeshing or trial function enhancement are required in modeling crack propagation. A new Bonding-Energy-Rate (BER)-based crack criterion as well as classic stress-based criterion are used for crack development simulations.

在这个由两篇论文组成的系列的第一部分中，介绍了一种新的脆点法 (FPM)，包括原始公式和混合公式，用于分析 2D 介电材料中的挠曲电效应。在本文中，提供了许多数值结果作为验证，包括线性和二次补丁测试、连续域中的挠曲电效应以及介电材料中的固定裂纹分析。还讨论了电弹性应力的影响，表明麦克斯韦应力可能很大，因此建议将完整的挠曲电理论用于纳米级结构。目前的原始和混合 FPM 也显示了它们在模拟具有挠曲电效应的裂纹萌生和扩展方面的适用性和有效性。柔性电，加上压电效应，可以帮助、阻碍或偏转裂纹扩展路径，在纳米级裂纹分析中不应被忽视。在 FPM 中，在模拟裂纹扩展时不需要重新网格划分或试验函数增强。新的基于键合能量速率 (BER) 的裂纹准则和经典的基于应力的准则用于裂纹发展模拟。