Two approaches for computational contact mechanics are explored to provide an accu rate and robust numerical simulation of frictional contact. First, a novel quasi-static contact algorithm (Method of First Violation) is developed, which provides a solution over the load history that is admissible in the sense that all equality and inequality constraints are satisfied at each load step and every point between load steps. The central idea of this algorithm is to calculate each load increment to be exactly which is necessary to cause the first nodal inequality constraint to be violated. At the end ofthat load increment, the status of the associated node is switched to a different status (such as from contact to non-contact) where the inequality constraint is replaced with an equality constraint, and a new inequality constraint is introduced. Importantly, the constraints of the previous and the new node status are simultaneously satisfied at that point in the loading cycle.The Method of First Violation (MFV) algorithm is illustrated using a compliance formulation for degrees of freedom on the surface of a half-plane loaded by a rigid indenter with friction. The compliance formulation permits considering only the degrees of freedom in the vicinity of contact, therefore extremely fine surface discretization is possible using only a modest number of degrees of freedom. Because this contact problem has been well studied using analytic techniques and because of the extremely fine mesh resolution, it is possible to perform a critical comparison with analyticalsolutions.An added benefit of this algorithm is its greatly improved robustness. This indenter/half-plane discretization was also used to test MFV where friction was represented by a two-parameter (static and kinematic) model. Though such problems are notoriously ill-conditioned, the MFV remained robust. It was also used to explore a problem involving the Dahl friction model.Second, a compliance matrix formulation is derived for a two-dimensional elastic disk and for a hole in an infinite elastic plane. The circular disk/hole surface is discretized finely with a uniform distribution of nodes and a set of basis functionsfor traction are defined so that each basis function has a value of 1 in the vicinity of the corresponding node and a value of zero elsewhere. The surface displacement field associated with each traction basis function is obtained through a laborious derivation involving Fourier series and terms of the Michell’s Airy stress function. The compliance matrix is constructed using the calculated displacement fields evaluated at each node. Fortunately, one may restrict attention to only the tractions and displacements in the vicinity of contact because all surface tractions outside that region are zero so the resulting compliance matrix is of tractable size. This semianalytical formulation provides a direct construction of a compliance matrix with an extremely fine surface discretization, easy implementation of friction models, and accommodation of elastic coupling.The Method of First Violation and the compliance formulation of two-dimensional circular bodies are implemented together to numerically simulate multiple contact problems. Five different geometries are designed using cylinders, holes, and half-plane, where two contacting bodies are pressed against each other and then sheared cyclically for four cycles. Surface tractions, stick ratios, steady-state dissipation, and error in surface tractions are calculated using Goodman decoupling and full decoupling approximations. The results are compared with calculations employing full elastic coupling to assess the qualitative and quantitative ramifications of the decoupling assumptions. The surface tractions calculated using Goodman decoupling show qualitative similarity with those of full elastic coupling but showed some severe errors inpredicting the dissipation. The full decoupling approximation simulations predictions are significantly different from the predictions of full coupling calculations. However, they are no worse than the Goodman decoupling approximations in predicting dissipation. This ability to introduce extremely fine mesh resolution in a contact region with a robust contact algorithm allows us to explore many contact characteristics through numerical simulation, which were previously intractable.
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