On the elastodynamic solution of frictional contact problems using variational inequalities

Abstract

This article is concerned with the development, implementation and application of variational inequalities to treat the general elastodynamic contact problem. The solution strategy is based upon the iterative use of two subproblems. Quadratic programming and Lagrange multipliers are used to solve the respective ÿrst and second subproblems and to identify the candidate contact surface and contact stresses. This approach guarantees the imposition of the active kinematic contact constraints, avoids the use of special contact elements and the interference of the user in dictating the accuracy of the solution. A modiÿed Newmark formulation is developed to integrate the resulting time-dependent variational inequality. This newly devised implicit time integration scheme is unconditionally stable, second-order accurate, avoids numerical oscillations present in the traditional Newmark method, and does not cause numerical dissipation. To demonstrate the versatility and accuracy of the newly proposed algorithm, several examples are examined and compared with existing solutions where the penalty method has been employed.

The variational method in contact problems. The present state of the problem and trends in its development

“…Formally, these problems can be solved using Ostrogradskii's method but problems arise in its practical implementation, which are caused both by the variability of the contact, adhesion and slippage zones as well as by wave processes. The first problem was solved by Czekanski et al 48 using an iterative process (without proof of convergence) and the Chang and Hulbert scheme was used for the integration with respect to time. An implicit difference scheme was proposed for solving a similar problem and its convergence was proved.…”

Simulation based optimization of the NC-shape grinding process with toroid grinding wheels

“…The discretization of the dynamic Signorini problem can be carried out either by finite difference schemes such as the Newmark method in time and finite elements in space [17] or by finite elements in space and time [18]. Both methods are implemented in the finite-element library SOFAR [19], which is the platform that was used for the presented calculations.…”

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

“…(28)) has a non-differentiable frictional term j( Á ). In order to overcome this difficulty, we adopt the following regularization approximation [6]:…”

“…In earlier studies, Meguid and his collaborators treated elastic [4,5], and elasto-dynamic [6] contact problems using appropriate variational inequalities formulations, contact search and solution algorithms. In this work, we devote our attention to the treatment of dynamic elasto-plastic frictional contact problems.…”