A new algorithm for numerical solution of 3D elastoplastic contact problems with orthotropic friction law H. W. Zhang, S. Y. He, X. S. Li, P. Wriggers Abstract 3D elastoplastic frictional contact problems with orthotropic friction law belong to the unspecified bound- ary problems with nonlinearities in both material and geometric forms. One of the difficulties in solving the problem lies in the determination of the tangential slip states at the contact points. A great amount of computa- tional efforts is needed so as to obtain high accuracy numerical results. Based on a combination of the well known mathematical programming method and iterative method, a finite element model is put forward in this paper. The problems are finally reduced to linear com- plementarity problems. A specially designed smoothing algorithm based on NCP-function is then applied for solving the problems. Numerical results are given to demonstrate the validity of the model and the algorithm proposed. Keywords 3D contact problem, Orthotropic friction law, Elastoplasticity, Mathematical programming method, Iterative method, Linear complementarity problem, Smoothing algorithm 1 Introduction Contact problems are of particular importance in various engineering applications such as structure-structure interaction, machine design and metal forming. They are hard to solve because of their geometric and material nonlinearities. It is a general phenomenon that the non- linear properties in both material form and contact form will occur at the same time. The similar properties between the material nonlinearity and contact problems have been demonstrated by laboratory test. Due to the practical importance of the elastoplastic contact analysis, the problems have been receiving extensive research work over the years. There are much more literatures on elastic contact problems than on elastoplastic contact problems, for the former, see the papers (Mijar and Arora, 2000; Stavrou- lakis and Antes, 2000; Christensen et al., 1998; Wriggers, 1995) and the references therein. The recent research work on elastoplastic contact problem is available in (Chris- tensen, 2002; Tin-Loi and Xia, 2001). In Christensen (2002), the frictional contact problem for elastoplastic bodies is reformulated as a set of unconstrained, non- smooth equations, which is solved by Pang’s Newton method for B-differentiable equations. In Tin-Loi (2001), elastoplastic frictionless contact problem is considered, which is formulated as a mixed complementarity problem and solved by the industry-standard complementarity solver named PATH. Zhong et al. (1997) developed a parametric variational principle for the analyses of contact problems and elas- toplastic structures. In Zhong’s method (Zhong and Sun, 1988; Zhong et al., 1997; Zhang et al., 1998), both contact and elastoplastic problems can be formulated as the same form of parametric programming problems after finite element discretization. Then they are finally reduced to linear complementarity problems. Parametric variational principle and parametric pro- gramming method is still used in this paper. But there will be two differences: firstly, instead of using the usual isotropic friction law as used in most reports on contact problems, an orthotropic friction law is considered; secondly, a new method for solving linear complemen- tarity problems will be adopted. It is well known that the friction force is always opposite to the slip direction for isotropic friction. But in many cases, isotropic friction can not give practical solutions because the contacting surfaces or the mechanical properties of materials might be anisotropic. 1 Computational Mechanics 34 (2004) 1–14 Ó Springer-Verlag 2004 DOI 10.1007/s00466-004-0548-2 Received: 22 August 2003 / Accepted: 22 December 2003 Published online: 18 February 2004 H. W. Zhang (&), S. Y. He, X. S. Li Department of Engineering Mechanics, State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116024, China E-mail: zhanghw@dlut.edu.cn P. Wriggers Institut fu ¨r Baumechanik und Numerische Mechanik, Universita ¨t Hannover, Appelstr. 9A, D-30167 Hannover, Germany The project is jointly supported by the National Natural Science Foundation (10225212, 50178016, 10302007), the National Key Basic Research Special Foundation (G1999032805), the Special Funds for Major State Basic Research Projects and the Foundation for University Key Teacher by the Ministry of Education of China. The authors are also grateful to the referees for their careful reading and detailed remarks on an earlier version of the paper. 1