A numerical procedure is developed for the solution of elastoplastic contact problems. The contacting surface was assumed unbonded and frictionless. Under the assumption of small deformations and displacements, the contact problem was formulated as a sequence of partial differential equations with inequalities. Since plastic deformation is a path dependent phenomenon, an incremental method was employed. In this study the flow theory of plasticity was used to describe plastic deformations. An equivalent minimization problem was proposed for the incrementalform of contact problem given as partial differential equations with inequalities. The equivalence of these two problems was shown by considering the necessary conditions of the minimization problem.
The finite element method was adopted as a numerical approximation technique. The resulting finite element discretization of the equivalent minimization problem becomes a quadratic programming problem. The dual was found computationally more efficient than the original problem, and a computer program was developed to solve two dimensional or axisymmetric elastoplastic contact problem under proportional loading. The constitutive coefficients in the elastoplastic range were obtained from the Prandtl-Reuss equation, the von Mises yield criterion and the isotropic hardening rule.
Several examples were solved to test the method for elastic perfectly plastic material. Also a computer simulation of a ball indentation test was performed. The first example is an indentation problem by a flat punch. Under the plane stress condition the limit load obtained is found 4.6% less than the theoretical value. The distribution of contact pressure is in good agreement with that of the theory. For plane strain assumption it is found about 2.6% less than that obtained by the slip line method. The second example, a contact problem between a piston rod and a pin under the plane stress condition shows good agreement with the literatuer available only for elastic range. As an application of the method, a ball indentation problem was selected. The ball was assumed rigid. Compression tests were performed to find the stress-strain property of indented material. The hardness obtained was compared with the computational results. Meyer hardness obtained from the calculation was in good agreement with the test result. The simulation indicates that Meyer's equation is fairly good for representation of the indentation force in terms of the indented radius. Tabor's relationship for representative strains was studied obtaining 0.1 for the proportional constant, but this constant was found dependent on the material involved, contrary to the previous report in the literature.
