A numerical analysis method of large displacement nonlinear dynamic contact problem with friction is developed for two-dimensional deformable multi bodies. A complementarity formulation, derived from a contact compatibility condition and Coulomb's friction law, and the linearization of nonlinear structural equations are combined for each time step. The total Lagrangian formulation is used. For numerical integration, a predictor and corrector algorithm with a single step time integration scheme which uses a second order approximation of displacements is used. The stability property of this time integration method is known as the same as the Newmark method. For spatial discretization , the finite element method is used. A hierarchical scheme with a circular territory is adopted to handle node to segment contact, which consists of the pre-contact searching and post-contact searching. Reliable detection of impact time and positions are presented using a second order approximation of displacements. Velocity discontinuities during impact are considered using an impact condition. The resulting linear complementarity problem is solved using Lemke's algorithm iteratively for each time step to enforce dynamic equilibrium at the end of the time step.
Three numerical examples are solved to show the applicability and the validity of the proposed method and the results compared with analytical solutions and where available and with those of commercial finite element code. The first simple example is an impact of a block and an inclined plane. The results are compared with the theoretical solutions for a simplified model with the assumption of perfectly elastic impact. Some difference is seen which reflects the significance of the elastic deformation during contact. The contact status on the contact surface such as stick, slip, and separation is well represented. In the second example, an impact of a block against a cantilever beam is considered. The results are compared with those by ABAQUS, a commercial FE code. The displacements by the present method tend to show relatively small tangential motion which seem to come from the differences of the time integration scheme and the treatment of tangential impulse in impact condition. For both ours and ABAQUS, the numerical results are rather sensitive to the time steps. The results show more abrupt corners in the displacements, which indicates that the proposed method can depict velocity discontinuities well. The third example is an impact of two cantilever beams. The results show nonlinear dynamic motion well. As the last example, an impact of a ring and a block is considered. The results show the propagation of motion along the beam and a mode of deformation of the ring during impact.
In summary, the proposed method provides an effective procedure for the large displacement nonlinear dynamic analysis of a two-dimensional frictional contact. The local phenomenon of impact is shown well simulated. It is, however, also seen that in contact problems the detail behaviors are very sensitive to the size of time steps. It is therefore very difficult to tell that a solution is better than the other. To solve more realistic and complicated problems an extension to three dimensional geometry and material nonlinearity should be implemented.
