A linear complementarity problem(LCP) formulation combined with an arc-length method is developed for post-buckling analysis of geometrically nonlinear structures with frictional contact constraints. The arc-length method with updated normal plane constraint is used to trace the equilibrium paths of the structures after limit points. The unknown contact variables such as contact status and contact forces can be directly solved in this formulation without any ad-hoc technique. For numerical implementation, a path-following algorithm is generated as a predictor-corrector procedure. The initial load increment is determined in the predictor phase, considering the change of contact status. In the corrector phase, the unknown load scale parameter used in the arc-length method is expressed in terms of contact forces, and eliminated to formulate as a linear complementarity problem. This can be solved using a standard LCP solution technique such as Lemke's algorithm.
Several post-buckling problems with contact constraints are solved to test the algorithm and to illustrate the detail complex behaviors. The results are compared with those of commercial package, ABAQUS. The first example is a post-buckling problem of a clamped shallow circular arch forced with a frictionless flat punch. The result shows that mesh pattern of the structure has a considerable effect on numerical solution in the post-buckling range. The second example is a frictional contact between a rigid surface and a frame with two equal legs under large displacement. It is seen that the present solution by LCP and the ABAQUS solution by the Lagrange multiplier method are in good agreement. In this example, the effect of the friction is increased with diminution of the frame angle. The third example is a frictional contact problem of a clamped shallow circular arch buckled by a circular punch. It is found that the present method is well applicable to the post-buckling analysis of a frictional contact problem with this snap-through phenomenon. In case of ABAQUS, stable solutions were obtained using the displacement control method with a constant increment in the downward rigid body displacement, but no solution was possible when an arc-length method is used. As the last example, a complicated snap-buckling problem of a clamped shallow circular arch with two circular punches is investigated. In this example, ABAQUS even using the displacement control method is unable to generate a solution possibly due to a snap-back buckling phenomenon unlike the previous examples. The present method is successful in dealing with this type of complex contact buckling problems. From the results of several numerical examples, it is shown that the present method is stable and has good convergence even when a complicated snapbuckling occurs in contact problems. By including the detail phenomenon of contact and friction, it has been possible to study not only the local state at an interface but also its effect on the global structural behavior.
