非均匀平面外弹性力学中的内摆线夹杂：应力奇异性与应力折减
Hypocycloidal Inclusions in Nonuniform Out-of-Plane Elasticity: Stress Singularity vs. Stress Reduction
摘要：
利用复势函数和保角映射方法，得到了无限大线弹性平面中$n$尖点内摆线形孔洞和刚性夹杂在非均匀远反平面载荷作用下的应力场解和应力强度因子.结果表明，内摆线形状的空隙可以导致更高的应力强度因子比相应的星形裂纹引起的，这是违反直觉的，因为后者通常会产生更严重的应力场的材料。此外，据观察，当控制远程加载的多项式的阶数$m$增长时，由内摆线形空隙和星形裂纹产生的应力场趋于重合，从而从失效分析的观点来看，它们变得等同。最后，特殊的几何形状和加载条件被发现，有没有应力奇异性的夹杂物尖点和应力甚至减少的情况下的夹杂物。因此，在存在尖锐的楔形体的情况下的应力降低因子（SRF）的概念被引入，与已知的存在具有光滑边界的夹杂物的情况下的应力集中因子（SCF）的定义进行对比。本文提出的结果提供了标准，这将有助于在超强复合材料的设计，应力奇异性总是促进失败。此外，他们将有助于找到特殊的条件下，电阻可以优化存在的夹杂物与非光滑边界。
英文摘要：
Stress field solutions and Stress Intensity Factors (SIFs) are found for $n$-cusped hypocycloidal shaped voids and rigid inclusions in an infinite linear elastic plane subject to nonuniform remote antiplane loading, using complex potential and conformal mapping. It is shown that a void with hypocycloidal shape can lead to a higher SIF than that induced by a corresponding star-shaped crack; this is counter intuitive as the latter usually produces a more severe stress field in the material. Moreover, it is observed that when the order $m$ of the polynomial governing the remote loading grows, the stress fields generated by the hypocycloidal-shaped void and the star-shaped crack tend to coincide, so that they become equivalent from the point of view of a failure analysis. Finally, special geometries and loading conditions are discovered for which there is no stress singularity at the inclusion cusps and where the stress is even reduced with respect to the case of the absence of the inclusion. The concept of Stress Reduction Factor (SRF) in the presence of a sharp wedge is therefore introduced, contrasting with the well-known definition of Stress Concentration Factor (SCF) in the presence of inclusions with smooth boundary. The results presented in this paper provide criteria that will help in the design of ultra strong composite materials, where stress singularities always promote failure. Furthermore, they will facilitate finding the special conditions where resistance can be optimized in the presence of inclusions with non-smooth boundary.
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