APPLICATION OF THE LINEARLY INDEPENDENT HIGH-ORDER NUMERICAL MANIFOLD METHOD IN FRACTURE MECHANICS
XU Dongdong1,YANG Yongtao2,ZHENG Hong2,WU Aiqing1
(1. Key Laboratory of Geotechnical Mechanics and Engineering of Ministry of Water Resources,Yangtze River Scientific Research Institute,Hubei,Wuhan 430010,China;2. State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences,Hubei,Wuhan 430071,China)
Abstract:The numerical manifold method(NMM) has succeeded in providing a unified solution to continuum and discontinuum problems and therefore it is highly suitable for solving fracture mechanics problems. However,the conventional high-order NMM using the first order polynomial as the local displacement function has the problem of linear dependence,which restricts to a certain degree its further development and application. A new NMM framework was established in this research by introducing a new localized displacement function,as well as a special displacement function for modeling the stress singularity around crack tips. A new paradigm that eliminates the problem of linear dependence is then derived to solve linear elastic fracture mechanics problems. The numerical examples show that:(1) The proposed method successfully eliminates the problem of linear dependence;(2) For classic linear elastic fracture problems,the stress intensity factors at the crack tip can be calculated accurately even if the mesh is relatively sparse;(3) The stress function at interpolation points inside the physical domain is continuous;(4) All the degrees of freedom defined on non-singular physical patches are physically meaningful,with the third to the fifth being the strain components at the interpolation point of the patch. As a result,the stress components at the interpolation point can be directly obtained. Finally,the proposed approach can be easily extended to other methods based on the theory of the partition of unity.
张慧华,祝晶晶. 复杂裂纹问题的多边形数值流形方法求解[J]. 固体力学学报,2013,34(1):38-46.(ZHANG Huihua,ZHU Jingjing. Numerical manifold analysis of complex crack problems on polygonal elements[J]. Chinese Journal of Solid Mechanics,2013,34(1):38-46.(in Chinese))
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