A numerical scheme is developed for the analysis of three-dimensional incipient sliding contact problems with an orthotropic friction law of the Coulomb type. The friction law is approximated as a polyhedral friction law. In this law, the slip equations are defined in the principal directions. In the general sense, the directions of slip are normal to the polyhedral friction domain. Using the polyhedral friction law, a linear complementarity problem (LCP) formulation in an incremental form is obtained. The LCP has the same form as the Kuhn-Tucker conditions of a quadratic programming. Accordingly, an equivalent minimization problem is derived. Lemke's complementary pivoting algorithm is used for solving the LCP.
As a simple application, a two-dimensional incipient sliding contact with bulk tension is solved. It is found that the stick region shifts in the direction of the tangential loading, and frictional tractions are repeating after an initial cycle of loading and unloading.
For the numerical verification of the proposed three-dimensional LCP formulation, the problem with an isotropic friction condition is solved. The LCP results are in good agreement with the analytical solutions. By analyzing the problem with an orthotropic friction condition, it is found that oscillating tangential loading produces a cyclic response after the initial loading and unloading. It is also found that as the ratio of the minor friction coefficient to the major decreases, the deviation between the directions of frictional traction and tangential loading becomes larger.
The incipient sliding contact subject to a combined tangential loading and twisting moment is also formulated as a LCP and solved. The numerical results agree well with physical reasoning.
In summary, the proposed LCP formulation provides computationally efficient procedures for determining frictional traction distribution and slip on contact area, and it is easily applicable to the problem with arbitrary contact geometries once the influence matrix is obtained. Furthermore, the existence of a solution is ensured in this numerical scheme.
