Formulations and computational methods for contact problems in solid mechanics

A study of existing formulations and computational methods for contact problems is conducted. The purpose is to gain insights into the solution procedures and pinpoint their limitations so that alternate procedures can be developed. Three such procedures based on the augmented Lagrangian method (ALM) are proposed. Small-scale benchmark problems are solved analytically as well as numerically to study the existing and proposed methods.; The variational inequality formulation for frictionless contact is studied using the two bar truss-wall problem in a closed form. Sub-differential formulation is investigated using the spring-wall contact and the truss-wall friction problems. A two-phase analytical procedure is developed for solving the truss-wall frictional contact benchmark problem.; The variational equality formulation for contact problems is studied using the penalty method along with the Newton-Raphson procedure. Limitations of such procedures, mainly due to their dependence on the user defined parameters (i.e., the penalty values and the number of time steps), are identified. Based on the study it is concluded that alternate formulations need to be developed.; Frictionless contact formulation is developed using the basic concepts of ALM from optimization theory. A new frictional contact formulation (ALM1) is then developed employing ALM. Automatic penalty update procedure is used to eliminate dependence of the solution on the penalty values. Dependence of the solution on the number of time steps in the existing as well as ALM1 formulations is attributed to a flaw in the return mapping procedure for friction. Another new frictional contact formulation (ALM2) is developed to eliminate the dependence of solution on the number of time steps along with the penalty values. Effectiveness of ALM2 is demonstrated by solving the two bar and five bar truss-wall problems. The solutions are compared with the analytical and existing formulations. Design sensitivity analysis of frictional contact problems is also studied and potential advantages of ALM2 over the existing formulations to obtain the sensitivity coefficients are identified. Finally, future directions of the research and conclusions are given.