13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- The eXtended Finited Element Method for frictional contact problem Zhiqiang Hu1,*, Guogang Fan1, Gao Lin1 1 Faculty of Infrastructure Engineering, School of Hydraulic Engineering, Dalian University of Technology, No.2 Linggong Road, Ganjingzi District, Dalian, 116024, China * Corresponding author: huzhq@dlut.edu.cn Abstract Frictional contact is often observed in the problems with the presence of crack surface. In order to take effects of contact of crack surfaces on the structural response, in the framework of mesh-based approaches, e.g. Finite Element Method (FEM) or Boundary Element Method (BEM), usually the contact surfaces need to be discretized and nodes are placed on the contact surfaces, although the meshes for both contact surfaces are not necessary to be matched with each other. However, the crack surface will evolve under loading, so remeshing is needed to make the meshes consistent with the geometry of crack surfaces. In this paper, we deal with the frictional contact problems resulting from presence of crack surfaces by combining the eXtended Finite Element Method (XFEM) and B-Differential Equation Method (BDEM). XFEM is used to model the discontinuities of displacement fields in the interior of the elements without the need for the remeshing of the domain. In BDEM, the normal and tangential contact conditions are formulated as B-differentiable equations and satisfied accurately. The B-differentiable Newton solution strategy with the good convergence performance is employed to solve with system equations. The Numerical examples including 2D and 3D frictional contact problems are given to demonstrate the effectiveness and accuracy of the presented approach. Keywords Frictional contact, eXtended Finite Element Method, B-differential Equation 1. Introduction Separation, stick and frictional sliding are often observed in the problems with the presence of crack surface or crack propagation. In order to take effects of contact on the behavior of cracks and structure, in the framework of mesh-based approaches, e.g. traditional Finite Element Method (FEM) or Boundary Element Method (BEM), remeshing is needed to trace the crack surfaces and make the meshes consistent with the geometry of crack surfaces. Although the meshes for contact surfaces in the framework of FEM and BEM are not necessary to be matched with each other, remeshing will result in the increase of the computational cost and additional mapping of the computational results from the original meshes to updated meshes. The invention of embedded discontinuity method, in which the discontinuity surfaces can be embedded in the element and traced effectively without remeshing, e.g., eXtended Finite Element Method[1], provides an alternative method to deal with contact problem, especially for the case in which contact surfaces evolve in the structure subjected to the complicated cyclic loads. The key feature of XFEM is that the discontinuity across crack surface can be resolved by additional enrichment functions and additional nodal degrees of freedom. The contact problem which is solved in the content of XFEM is firstly proposed by J. Dolbow Möes et al. [2]. The contact condition were enforced by penalty method and the LArge Time INcrement method (LATIN) was employed to solve the system equations. Recently, FS. Liu and R.I. Borja [3] proposed the Petrov-Galerkin variational formation for the frictional contact problem, and the augmented Lagrangian technique was used for the enforcement of contact conditions. I. Nistor et al. [4] developed a hybrid X-FEM contact element for frictionless large sliding contact problem, in which the augmented Lagrangian method was also employed. Due to the good performance and convergence property of B-differentiable Equations method to solve the frictional contact problem, so we extended this method to model the two and
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