A numerical procedure is developed for the solution of elasto-plastic contact problems under large deformation. The contact surface is assumed unbonded and frictionless. The problem is formulated as a sequence of partial differential equations with inequalities. To consider large strains and large displacements, the measure of stress and strain must be of a form that is independent of the current rate of rigid body motion. Therefore, the 2nd PiolaKirchhoff stress and the Green-Lagrange strain is used. Since plastic deformation is a path-dependent phenomenon, an incremental method is employed. A flow theory of plasticity has been used to describe the plastic deformation. By the formulation derived, it is possible to handle unloading with possible separation.
A minimization problem has been proposed, which is equivalent to the incremental form posed as partial differential equations with inequalities. The equivalence is shown by considering the necessary conditions of the minimization problem. In order to solve this minimization problem, a sequential quadratic programming method has been adopted. The finite element method is utilized as a numerical approximation technique. However, since the contact surface is discretized, the presumed contact pairs on the mating surface may not match after a load increment. For this reason, the matrices in the equation describing the contact gap are derived to treat this situation. The dual problem is found computationally more efficient than the primal one. A modified simplex method is employed.
A computer program has been written to solve two dimensional or axisymmetric elastoplastic contact problems based on the proposed formulation. The constitutive coefficients in the elastoplastic range has been calculated 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.
Several examples have been solved to test the method. The first example is a ball indentation problem. The ball is assumed rigid and the material properties of the indented body are taken from an experiment that was carried out in an earlier study for Al 2024. Meyor hardness obtained from the numerical calculation is in close agreement with the experimental results. The second problem is a bending problem of a plate against an elastic hemicylinder. Under the plane strain condition, concentrated loads that remain perpendicular to the plate plane for all load steps are applied at both ends of the plate. Unloading is also considered. The material is assumed to be elastic-perpectly plastic. The numerical results appear to be in agreement with the observed behavior.
As an application of the method, a three-point bending problem is selected. This problem is a multi-body contact problem. The punch and the supporters are assumed to be rigid and the material properties of the deformable body are taken from a tensile test. Experiments are carried out for specimens that have various widths. The load-displacement curve from the numerical results for plane strain condition is in good agreement with that from the experimental results for a width-to-thickness ratio of 5. The numerical results of the contact problem are compared with those that are obtained for a model where the contact force is substituted by a concentrated force. It is shown that this simplified model is not adequate for an elaborate analysis for the three-point bending problem. A detailed discussion on the local phenomena in the contacting zone is given. The developed numerical algorithm in the present form is only applicable up to the point of unstability, as shown by the example.
In summary, the method is shown to be a valuable tool for detailed analysis of complicated phenomena which involve solid contact. The analysis results can supply important informations such as residual stress and permanent deformation in addition to usual stress and displacement histories.
