An augmented lagrangian treatment of contact problems involving friction

Finite element formulation of large deformation impact-contact problems with friction

In this paper, a formulation is presented for the finite element treatment of multibody, large deformation frictional 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 first, 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 occur, 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.

A continuum-based finite element formulation for the implicit solution of multibody, large deformation-frictional contact problems

The numerical treatment of contact problems involves the formulation of the geometry, the statement of interface laws, the variational formulation and the development of algorithms. In this paper we give an overview with regard to the different topics which are involved when contact problems have to be simulated. To be most general we will derive a geometrical model for contact which is valid for large deformations. Furthermore interface laws will be discussed for the normal and tangential stress components in the contact area. Different variational formulations can be applied to treat the variational inequalities due to contact. Several of these different techniques will be presented. Furthermore the discretization of a contact problem in time and space is of great importance and has to be chosen with regard to the nature of the contact problem. Thus the standard discretization schemes will be discussed as well as techiques to search for contact in case of large deformations.

Finite Element Algorithms for Contact Problems

Finite element analysis of nonsmooth contact

Algorithms and numerical results for finite element approximations of contact problems with non-classical friction laws

Numerical analysis of certain contact problems in elasticity with non-classical friction laws

In this paper we prove a general approximation theorem useful in obtaining order of convergence estimates for the approximation of the solution of a class of variational inequalities that characterize the Signorini problem with non-local friction Several special cases that can be derived from the general error estimate are also presented

Error estimates for the approximation of a class of variational inequalities arising in unilateral problems with friction

A number of finite element methods for the analysis of nonlinear problems in rubber elasticity are outlined. Several different finite element schemes are discussed. These include the augmented Lagrangian method, continuation or incremental loading methods, and associated Riks-type methods which have the capability of incorporating limit point behavior and bifurcations. Algorithms for the analysis of limit point behavior and bifurcations are described and the results of several numerical experiments are presented. In addition, a brief survey of some recent work on modelling contact and friction in elasticity problems is given. These results pertain to the use of new nonlocal and nonlinear friction laws.

Finite element methods for nonlinear elastostatic problems in rubber elasticity

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Papers

Algorithms and numerical results for finite element approximations of contact problems with non-classical friction laws

Numerical analysis of certain contact problems in elasticity with non-classical friction laws

Error estimates for the approximation of a class of variational inequalities arising in unilateral problems with friction

Finite element methods for nonlinear elastostatic problems in rubber elasticity