On the use of variational inequalities to model impact problems of elasto–plastic media

Abstract

A new variational inequality-based formulation is presented for treating dynamic elasto–plastic contact problems. The incremental variational inequality representing this class of problems is developed in an updated Lagrangian framework. A new technique for representing the kinematic contact conditions, based on C 1-continuous spline interpolation and an intermediate surface with uniquely defined normal, is developed. The solution algorithm is based upon the iterative use of mathematical programming and Lagrange multipliers to identify the candidate contact surface and contact stresses. This approach guarantees the accurate imposition of the active kinematic contact constraints and avoids the use of special contact elements. The dynamic variational inequalities formulations for the two sub-problems are solved using the generalized- α time integration scheme. The solution strategy accounts for the effect of friction through the use of an appropriate regularization technique in the virtual work expressions. This newly developed approach leads to a significant reduction in numerical oscillations in impact and dynamic frictional problems, and is less sensitive to variations in the time increment. It also reduces the number of iterations needed to achieve convergence. The robustness and accuracy of the proposed FE algorithm are demonstrated by application to a number of case studies.