# Frontiers of Mechanical Engineering

 Front. Mech. Eng.    2019, Vol. 14 Issue (3) : 332-341     https://doi.org/10.1007/s11465-019-0540-3
 RESEARCH ARTICLE
A family of non-conforming crack front elements of quadrilateral and triangular types for 3D crack problems using the boundary element method
Guizhong XIE1, Fenglin ZHOU2(), Hao LI1, Xiaoyu WEN1, Fannian MENG1
1. Henan Provincial Key Laboratory of Intelligent Manufacturing of Mechanical, Mechanical and Electrical Engineering Institute, Zhengzhou University of Light Industry, Zhengzhou 450002, China
2. College of Mechanical Engineering, Hunan University of Technology, Zhuzhou 412007, China
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 Abstract This study focuses on establishing non- conforming crack front elements of quadrilateral and triangular types for 3D crack problems when the dual boundary element method is applied. The asymptotic behavior of the physical variables in the area near the crack front is fully considered in the construction of the shape function. In the developed quadrilateral and triangular crack front elements, the asymptotic term, which captures the asymptotic behavior of the physical variable, is multiplied directly by the conventional Lagrange shape function to form a new crack front shape function. Several benchmark numerical examples that consider penny-shaped cracks and straight-edge crack problems are presented to illustrate the validity and efficiency of the developed crack front elements. Corresponding Authors: Fenglin ZHOU Just Accepted Date: 24 May 2019   Online First Date: 08 July 2019    Issue Date: 24 July 2019
 Cite this article: Guizhong XIE,Fenglin ZHOU,Hao LI, et al. A family of non-conforming crack front elements of quadrilateral and triangular types for 3D crack problems using the boundary element method[J]. Front. Mech. Eng., 2019, 14(3): 332-341. URL: http://journal.hep.com.cn/fme/EN/10.1007/s11465-019-0540-3 http://journal.hep.com.cn/fme/EN/Y2019/V14/I3/332
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 Fig.1  Three-node, non-conforming quadratic element Fig.2  Nine-node non-conforming quadratic element Fig.3  Six-node non-conforming quadratic element Fig.4  Local coordinate system at Point O Fig.5  Mesh of the crack surface. (a) Non-conforming quadrilateral element; (b) non-conforming triangular element Fig.6  Results of the normalized SIF $KI$ calculated by non-conforming crack front elements of the quadrilateral type Fig.7  Results of the normalized SIF $KI$ calculated by non-conforming crack front elements of the triangular type Fig.8  Results of $Δu$ over the crack surface obtained using our method Fig.9  Influence of position parameter k Fig.10  Penny-shaped crack under uniform inclined traction Fig.11  Results of three normalized SIFs $KI$, $KII$, and $KIII$ calculated by non-conforming crack front elements of the quadrilateral type Fig.12  Results of three normalized SIFs $KI$, $KII$, and $KIII$ calculated by non-conforming crack front elements of the triangular type Fig.13  Geometry model of the singular edge crack and mesh Fig.14  Normalized stress intensity factors along the crack front for the single edge crack