A second-order cone linear complementarity approach for three-dimensional frictional contact problems

Frictional contact problems frequently arise in variousengineering applications, but their solutions, especially the solutions ofthree dimensional (3D) frictional contact problems, are challenging sincethe conditions for contact and friction are highly nonlinear and non-smooth.The 3D frictional contact problem is nonlinear and non-differentiable atleast in three aspects: 1) The unilateral contact law, combining a geometriccondition of impenetrability, a static condition of no-tension and an energycondition of complementarity, is represented by a multi-valuedforce-displacement relation. 2) The friction law, governed by a relationbetween reaction force and local relative velocity, is also multi-valued. 3)The Coulomb friction law in 3D space is expressed as a nonlinear inequalitythat is non-differentiable in the ordinary sense. In this paper, we proposea new linear second-order cone complementarity formulation for the numericalfinite element analysis of 3D frictional contact problem by using theparametric variational principle. Specifically, we develop a regularizationtechnique to resolve the multi-valued difficulty involved in the unilateralcontact law, and utilize a second-order cone complementarity condition tohandle the regularized Coulomb friction law in contact analysis. Wereformulate the governing equations of the 3D frictional contact problem asa linear second-order cone complementarity problem (SOCCP) via theparametric variational principle and the finite element method. Comparedwith the linear complementarity formulation of 3D frictional contactproblems, the proposal SOCCP formulation avoids the polyhedral approximationto the Coulomb friction cone so that the problem to be solved has muchsmaller size and the solution has better accuracy. A semismooth Newtonmethod is used to solve the obtained linear SOCCP. Numerical examples arecomputed and the results confirm the effectiveness and robustness of theSOCCP formulation developed.