A new analysis method of dealing with frictional dynamic contact is developed for two-dimensional deformable bodies. It is based on the complementarity formulation of the contact compatibility and the Coulomb friction law, and implemented for a typical time step. A direct time integration using a second order polynomial approximation of the displacement field is adopted and found to be suitable for accommodating velocity discontinuities and complementarity relations imposed for discretized contact pairs in the potential contact region. It has the stability property same as the Newmark method. For spatial discretization, the finite element method is used. To describe contact gap and relative slip of a potential contact pair, a general node-to-edge contact is used and implemented for linear and quadratic element segments of the target body. The resulting linear complementarity problem (LCP) for each time step is solved using Lemke's algorithm.
Five examples are solved to show the applicability and efficiency of the proposed method. In the first simple example of a longitudinal impact between two elastic rods, the feature of dynamic contact such as the velocity discontinuities is well represented. Good agreement is seen compared to the results of one-dimensional theoretical solution, ABAQUS solution and other references. The contact status on the contact surface such as stick, slip, and separation is well represented which cannot be considered in one-dimensional wave theory. In the second example, an oblique impact of a rectangular plate against a rigid wall is considered. It is shown that the friction has a significant effect on the motion of the body after impact. The body tends to rebound backward with high friction. The example of an oblique impact of a plate with a round boundary has clearly shown large efficiency when compared to other iterative methods. In the fourth example, an axisymmetric problem of a steel sphere and a spherical glass shell is modeled by axisymmetric elements. The abrupt separation of the bodies at the end of the impact represents a velocity discontinuity and is well described by the present method. This has not been possible in the other existing methods when based on the Hertzian contact theory. The detailed local stress distribution is rather easily reproducible. As the last example, an oblique impact between two elastic rectangular plates with friction is considered. Eight-noded quadratic elements are employed. The quadratic element is found to be very well usable for contact analysis.
In summary, the proposed LCP formulation provides a computationally efficient procedure for the dynamic analysis of two-dimensional frictional contact. The local contact-impact phenomenon is shown well simulated. To solve more realistic and complicated problems, however, an extension to three-dimensional geometry must be included.