Numerical Analysis of Contact Problems with Friction on Nano-indentation by a Modified Augmented Lagrangian Optimization Approach

In this paper a general finite element formulation, using a modified augmented Lagrangian (AL) optimization method, has been developed to analyze the quasi-static frictional contact behavior of nano-indentation procedure.Based on the geometrical kinematic conditions of contacting parts, a novel improved AL approach has been used.This novel solution procedure for contact formulation overcomes the difficulty of dependence of the solution process on two user-specified parameters, i.e. the penalty parameter for contact constraints and the number of load steps. For overcoming the numerical difficulties of variational equality formulations, a new non-classical friction law has been used, in which the stick and slip conditions of classical law have been changed to a modified elastic friction law.Finally, the indentation procedure of rigid nano-indenters into an elastic layer has been evaluated to verify the results. The numerical algorithm in an applied example is accurately converged for stick and slip friction conditions and solution convergence is analyzed using automatic load stepping and contact load predicting.