Abstract The linear elastic and viscoelastic problems with V-shaped notches under in-plane loading are investigated in this paper. We consider two different sets of boundary conditions on the radial edges: Free-Free case and Clamped-Clamped case. A novel singular finite element (SASE) with arbitrary high-order accuracy for the notched problems is proposed. Firstly, via Laplace transform, the original viscoelastic problem is transformed into a corresponding elastic one. Then we construct the SASE by using elastic symplectic eigen solutions with higher order expanding terms. The SASE can depict the characteristics of displacement fields and singular stress fields in the vicinity of the notch vertex having arbitrary opening angle. By taking advantage of the symplectic eigen solutions in the SASE, Mode I and/or Mode II notch stress intensity factors can be determined directly without any post-processing. Fine finite element meshes are not required. Numerical examples are provided to illustrate the validity and accuracy of the present method.

中文翻译：

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用于弹性和粘弹性平面问题中 V 型缺口分析的辛解析奇异元

摘要 本文研究了面内载荷作用下具有V形缺口的线弹性和粘弹性问题。我们在径向边缘上考虑两组不同的边界条件：Free-Free 情况和 Clamped-Clamped 情况。针对缺口问题提出了一种具有任意高阶精度的新型奇异有限元（SASE）。首先，通过拉普拉斯变换，将原来的粘弹性问题转化为相应的弹性问题。然后我们通过使用具有高阶扩展项的弹性辛本征解来构造 SASE。SASE可以刻画具有任意张角的缺口顶点附近的位移场和奇异应力场的特征。通过利用 SASE 中的辛本征解，模式 I 和/或模式 II 缺口应力强度因子可以直接确定，无需任何后处理。不需要精细的有限元网格。提供了数值例子来说明本方法的有效性和准确性。