A numerical procedure is developed for the solution of contact problems with the Coulomb friction based on a complementarity problem formulation. It is shown to be very natural for the description of frictional contact. The equations are presented in an incremental form to accommodate the loading-path dependency friction phenomena.
The basis of the formulation is the complementarities existing between pairs of variables, that is, contact pressure and gap, slackness in the friction equation and slippage, and constraint force and rigid body displacement. The detailed derivation is limited to two dimensional problems.
Two different classes of problems are tested in detail. The linear elastic contact problem is one, where boundary element method is found very efficient. Another problem is the treatment of elasto-plastic, large deformation. The updated Lagrangian method is adopted. The constitutive coefficients in the plastic range have been obtained from the Prandtl-Reuss equation, the von Mises yield criterion and the isotropic hardening rule. They have been adjusted to accommodate the large deformation analysis. Through several numerical examples, the friction effect has been clearly shown, expecially the loading-path dependency. The state at the loading stage is different from that at the unloading stage even when the externally applied load is the same. This fact means that the numerical solution can be dependent on the size of an incremental step, which is the case in the numerical experimentation. The proper selection of the size which can quarantee uniqueness within an incremental step is important but this is left remained unsolved at the present time.
The numerical examples in an eleasto-plastic large deformation cases are solved using finite element discretization. The solutions seem reasonable as intuitivelly judged. The developed program does not include a treatment of buckling or instability. This puts a limit on the amount of applied load and it has not been possible to get into a relatively large deformation zone.
