一种新的无网格脆弱点法（FPM），在每一点上的最小未知数，在两个理论下的挠曲电分析与裂纹扩展。第II部分：确认和讨论
A New Meshless Fragile Points Method (FPM) With Minimum Unknowns at Each Point, For Flexoelectric Analysis Under Two Theories with Crack Propagation. Part II: Validation and discussion
摘要：
在这两篇论文系列的第一部分中，提出了一种新的脆弱点方法（FPM），在原始和混合配方，用于分析二维介电材料中的挠曲电效应。在本文中，提供了一些数值结果作为验证，包括线性和二次补丁测试，在连续域的挠曲电效应，和分析的固定裂纹在介电材料。的电弹性应力的影响的讨论也给出了，麦克斯韦应力可能是显着的，因此建议采用全挠曲电理论的纳米尺度结构。目前的原始以及混合FPMs也显示出其适用性和有效性，在模拟裂纹的萌生和扩展与挠曲电效应。挠曲电效应与压电效应相结合，可以帮助、阻碍或偏转裂纹扩展路径，在纳米尺度裂纹分析中不应被忽略。在FPM中，不需要重新网格划分或测试函数增强来模拟裂纹扩展。一个新的键合能量率（BER）为基础的裂纹准则，以及经典的基于应力的准则用于裂纹发展模拟。其他复杂的问题，如动态裂纹的发展，断裂，破碎和三维挠曲电分析将在我们未来的研究。
英文摘要：
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 nano-scale 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 nano-scale 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. Other complex problems such as dynamic crack developments, fracture, fragmentation and 3D flexoelectric analyses will be given in our future studies.
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