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SUMMARYIlh this paper, a formulation is presented for the finite element treatment of multibody, large deformation f~ctional contact problems. The term multibody is used to mean that when two bodies mechanically contact, both may be deformable. A novel aspect of the approach advocated is that the equations governing contact are developed in the continuum setting jirst, before deriving the corresponding finite element eQUatiOnS. This feature distinguishes the current work from many earlier treatments of contact problems and renders it considerably more general. In particular, the approach yields a characterization of the frictional constraint (assuming a Coulomb law) suitable for arbitrary discretizations in either two or three dimensions. A geometric framework is constructed within which both frictionless and frictional response are naturally described, making subsequent finite element discretization a straightforward substitution of finite-dimensional solution spaces for their continuum counterparts. To our knowledge, this general formulation and implementation of the frictional contact problem in a finite element setting has not been reported previously in the literature.The development includes exact linearization of the statement of virtual work, which enables optimal convergence properties for Newton--Raphson solution strategies, and which appears to be highly desirable (if not essential) for the general robustness of implicit finite element techniques. Since the theory and subsequent linearization require no limitations on the amount of deformation or relative sliding that can oucur, the resulting treatment of frictional contact is suitable for a wide range of examples displaying significant non-linear behaviour. This assertion is substantiated through presentation of a variety of examples in both two and three dimensions.

Work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.

SUMMARYThis paper presents a mortar-based formulation for the solution of two dimensional frictional contact problems involving finite deformation and large sliding. As is widely recognized, traditional node-tosurface contact formulations have several drawbacks in solution of deformable-to-deformable contact problems, including lack of general patch test passage, degradation of spatial convergence rates, and robustness issues associated with the faceted representation of contacting surfaces. The mortar finite element method, initially proposed as a technique to join dissimilarly meshed domains, has been shown to preserve optimal convergence rates in tied contact problems (see (Discretization Methods and Iterative Solvers Based on Domain Decomposition, Springer-Verlag, Heidelberg, 2001) for a recent review), and is examined here as an alternative spatial discretization method for large sliding contact. In particular, a novel description for frictional sliding conditions in large deformation mortar formulations is proposed in this work.In recent years, the mortar element method has already been successfully implemented to solve frictional contact problems with linearized kinematics (see (Int. J. Numer. Meth. Engng 1993; 36: 3451)). However, in the presence of large deformations and finite sliding, one must face difficulties associated with the definition and linearization of contact virtual work in the case where the mortar projection has a direct dependence on the tangential relative motion along the interface. In this paper, such a formulation is presented, with particular emphasis on key aspects of the linearization procedure and on the robust description of the friction kinematics. Some novel techniques are proposed to treat the non-smoothness in the contact geometry and the searching required to define mortar segments. A number of numerical examples illustrate the performance and accuracy of the proposed formulation.

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SUMMARYThis paper proposes a formulation of dynamic contact problems which enables exact algorithmic conservation of linear momentum, angular momentum, and energy in ÿnite element simulations. It is seen that a Lagrange multiplier enforcement of an appropriate contact rate constraint produces these conservation properties. A related method is presented in which a penalty regularization of the aforementioned rate constraint is utilized. This penalty method sacriÿces the energy conservation property, but is dissipative under all conditions of changing contact so that the global algorithm remains stable. Notably, it is also shown that augmented Lagrangian iteration utilizing this penalty kernel reproduces the energy conserving (i.e. Lagrange multiplier) solution to any desired degree of accuracy. The result is a robust, stable method even in the context of large deformations, as is shown by some representative numerical examples. In particular, the ability of the formulation to produce accurate results where more traditional integration schemes fail is emphasized by the numerical simulations.