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SIFs of interfacial crack using generalized extended finite element method
SU Yi , WANG Shengnan , LU Longkun
College of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China
Received: 2015-06-09; Accepted: 2015-07-03; Published online: 2016-06-20 12: 00
Foundation item: Aeronautical Science Foundation of China (2010ZF56016)
Corresponding author. E-mail:wangshna@nwpu.edu.cn
Abstract: Generalized extended finite element method (GXFEM) is a new numerical simulation method which combines both the generalized finite element and the extended finite element. The principle of the generalized extended finite element method for analyzing the stress intensity factor (SIF) of bi-material interfacial cracks is proposed. A new enriched function for bi-material interfacial crack tip is proposed, and the twelve crack tip enriched functions are reduced to the six ones. Because of the discontinuity of bi-material interface, enrichment functions based on level set are added in the displacement mode of the conventional finite element method. And the node freedom of the crack element and crack tip element are also generalized. Besides, the calculation precision is improved. A comparison of literature method and GXFEM calculations of numerical examples shows the the accuracy and reliability of the proposed method.
Key words: generalized extended finite element method (GXFEM)     interfacial crack     crack tip enriched function     level set     stress intensity factor (SIF)

1 GXFEM的基本原理 1.1 位移逼近方程的建立

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 图 1 双材料界面裂纹的加强结点 Fig. 1 Enriched nodes for a bi-material interfacial crack
1.2 GXFEM离散方程的建立

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BiuBiaBibBic分别为常规结点、裂纹贯穿单元结点附加、裂纹尖端单元结点附加和材料界面单元附加的应变转换矩阵,其定义如式 (26)～式(29)所示：

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2 双材料界面断裂力学

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3 双材料界面裂纹的区域积分

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σij ui,1由广义扩展有限元算得,ui,1auxεikauxσijaux由辅助位移场求出：

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4 数值算例 4.1 中心界面裂纹

 图 2 双材料板内的界面裂纹 Fig. 2 Interfacial crack in center of a bi-material plate
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 E1/E2 α β 本文GXFEM(L/a=30) Sukumar等[13](L/a=30) 准确结果(L/a=∞) F1 F1误差/% F2 F2误差/% F1 F1误差/% F2 F2误差/% F1 F1误差/% 2 0.333 0.095 1.0026 0.1 -0.0408 2.8 1.002 0.1 -0.0411 3.5 1.0011 -0.0397 4 0.600 0.171 1.0051 0.2 -0.0735 2.1 1.004 0.1 -0.0743 3.2 1.0035 -0.0720 8 0.778 0.222 1.0076 0.2 -0.0955 1.7 1.007 0.2 -0.0967 3.0 1.0059 -0.0939 20 0.905 0.259 1.0097 0.2 -0.1115 1.5 1.009 0.1 -0.1127 2.6 1.0081 -0.1098 40 0.951 0.272 1.0106 0.2 -0.1173 1.4 1.010 0.1 -0.1185 2.4 1.0090 -0.1157 100 0.980 0.280 1.0110 0.1 -0.1209 1.3 1.010 0 -0.1220 2.2 1.0096 -0.1194 1000 0.998 0.285 1.0108 0.1 -1.0229 1.0 1.010 0 -1.2396 1.9 1.0100 -0.1217

4.2 内部含副裂纹的单边界面裂纹

 图 3 单边界面裂纹和副裂纹板 Fig. 3 Edge interfacial cracked with minor crack plate

 图 4 不同E2/E1下,无量纲应力强度因子随裂纹之间相互作用域的变化 Fig. 4 Normalized stress intensity factors changing with crack interaction domain for different E2/E1
5 结论

1) GXFEM结合了XFEM和GFEM的优点,不需要在裂纹尖端设置过密的网格,同时能够保证较高的计算精度。

2) 本文在运用GXFEM处理界面裂纹时,使用三角变换将裂纹尖端位移场逼近函数由12个减少到6个,在不损失精度的情况下大大减少了计算量；采用偏移富集函数对混合单元的问题进行了处理,提高了计算的收敛性；应用GFEM对裂纹尖端和裂纹面的元素通过增加结点广义位移参数,提高了计算精度。算例表明,采用本文的GXFEM计算双材料界面裂纹的应力强度因子是成功和有效的。

3) FEM、GFEM及XFEM均可以看作GXFEM在特定条件下的简化。在处理不连续问题时,只需要对不连续的区域增加富集函数、进行结点自由度广义化,对其他区域则采用FEM进行处理,从而可以减少建模的复杂性和工作量。

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#### 文章信息

SU Yi, WANG Shengnan, LU Longkun

SIFs of interfacial crack using generalized extended finite element method

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(6): 1162-1168
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0376