13th International Conference on Fracture June 16–21, 2013, Beijing, China -10- 6. Conclusions In this paper, a combined eXtend Finite Element method and B-differentiable equations method for solving contact problems in which the contact surfaces are embedded in the elements. There are some salient features for the presented methods. Firstly, in the framework of XFEM, it is very convenient to construct the finite element meshes with minor consideration of the position of crack or contact surfaces. Due to the assumption of small deformation, the contact pairs are easily defined at the intersection points between the contact surfaces and the edges of elements. Secondly, the contact conditions are formulated as B-differentiable equations by the quantities at the contact pairs and satisfied exactly. Thirdly, the B-differential Newton method with guaranteed convergence property is utilized to solve the system equations consisting of equilibrium equations and contact conditions. The 2D and 3D frictional contact examples show the high accuracy and good convergence property of the presented method. Acknowledgements The first author greatly appreciates the financial support of State Scholarship fund provided by Chinese Scholarship Council and Professor Ted Belytschko to offer him a visiting scholarship position in Northwestern University. He also thanks postdoctors Qinglin Duan, Jeong-Hoon Song, Thomas Menouillard at Northwestern University, Dr. Fushen Liu at Stanford University for their kindly help in studying the XFEM and constructive discussion and suggestions. The support of the China National Science Foundation under grant 51178069 is gratefully acknowledged. References [1] N. Möes, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46 (1999) 131–150. [2] J. Dolbow, N. Möes, T. Belytschko, An extended finite element method for modeling crack growth with frictional contact, Computer Methods in Applied Mechanics and Engineering, 190 (2001) 6825–6846. [3] F.S. Liu, R. I. Borja, A contact algorithm for frictional crack propagation with the extended finite element method, International Journal for Numerical Methods in Engineering, 76 (2008) 1489–1512. [4] I. Nistor, M. L. E. Guiton, P. Massin, N. Möes, S. Géniaut, An X-FEM approach for large sliding contact along discontinuities, International Journal for Numerical Methods in Engineering, 78 (2009) 1407–1431. [5] P. Christensen, A. Klarbring, J.S. Pang, N. Stromberg, Formulation and comparison of algorithms for frictional contact problems, International Journal for Numerical Methods in Engineering, 42(1998), 145-173 [6] J.S. Pang, Newton's method for B-differentiable equations, Mathematics of Operations Research, (15) 1990 311-341 14. [7] N. Sukumar, N. Möes, B. Moran, T. Belytschko, Extended finite element method for three-dimensional crack modeling, International Journal for Numerical Methods in Engineering, 48 (2000) 1549-1570
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