Arc-Length Method for Frictional Contact with a Criterion of Maximum Dissipation of Energy

In this paper, we propose an arc-length equilibrium path-following method for quasi-static frictional contact problems incorporating a criterion of maximum dissipation of energy, which is applicable to cases in which there exist critical points along the equilibrium path. The Coulomb friction law and the unilateral contact condition are considered.

It is well known that the frictional contact problems may have limit points and successive stable and/or unstable bifurcation points [

1

], even if small rotations and small strains are assumed. This implies that the corresponding incremental problem does not have unique solution in general. Moreover, this problem often has bifurcated paths such that most sliding contact nodes become stuck and the loading parameter decreases, which are referred to as

trivial unloading paths

. Our aim is to propose a path-following method that can automatically avoid tracing trivial unloading paths, which seem to be uninteresting from the practical point of view. To this end, we attempt to follow the path with the maximum dissipation of energy when the corresponding incremental problem has some solutions. At each loading stage, the incremental displacements and the reactions are obtained by solving a

mathematical program with complementarity constraints

(MPEC). Algorithms that do not have any criterion to select among multiple solutions may compute trivial unloading solutions.

A regularization scheme of the MPEC is also proposed. In contrast with the fact that the original MPEC fails to satisfy any standard constraint qualification, it is shown that the regularized problem satisfies the linear independence constraint qualification at a feasible solution. This implies that, in the inner iteration of the arc-length method, we can solve the proposed regularized problem by using the conventional nonlinear programming approach.

It has been shown in the numerical examples that the proposed method can automatically avoid tracing trivial unloading paths, even when there exist successive bifurcation points due to friction and/or some limit points.