Abstract Symplectic analytical singular element (SASE) is a special crack-tip element in the framework of finite element method (FEM) for modelling cracks. A group of SASEs, which have been developed by the authors, for various crack problems almost in two-dimensional (2D), but not touched yet more complex ones, i.e., in three-dimensional (3D) domain. The underlying reason lies in the fact that analytical symplectic eigen solution for 3D elastic crack problem is not yet available, a 3D SASE thus cannot be constructed through a simple generalization from the 2D SASEs. This study aims to fill out this 3D gap, we thus use a trail displacement field to construct a 3D SASE. The derived strain and stress fields still possess singularity in the vicinity of crack-tip. Finite element formulation is derived based on the minimum total potential energy principle. It is found that some of key features of a 2D SASE still exist in the developed 3D one, e.g., the SIFs can be calculated accurately without any post-processing. Numerical examples are considered to show the accuracy and performance of the proposed element.

中文翻译：

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三维弹性体裂纹的一种新的裂纹尖端奇异单元

摘要 辛解析奇异单元(SASE)是在有限元法(FEM)框架下用于裂纹建模的特殊裂纹尖端单元。一组由作者开发的 SASE，用于解决几乎二维 (2D) 中的各种裂纹问题，但尚未涉及更复杂的问题，即三维 (3D) 域。根本原因在于，3D 弹性裂纹问题的解析辛本征解尚不可用，因此无法通过对 2D SASE 的简单概括来构建 3D SASE。本研究旨在填补这个 3D 空白，因此我们使用轨迹位移场来构建 3D SASE。导出的应变和应力场在裂纹尖端附近仍然具有奇异性。有限元公式是基于最小总势能原理推导出来的。发现2D SASE的一些关键特征仍然存在于开发的3D SASE中，例如无需任何后处理即可准确计算SIF。数值例子被认为是显示所提出的元素的准确性和性能。