Frctionless contact: step by step analysis and mathematical programming technique

The object of the paper concerns a consistent formulation of the classical Signorini's theory regarding the frictionless unilateral contact problem between two elastic bodies in the hypothesis of small displacements and strains. A variational approach employed in conjunction with the Symmetric Boundary Element Method (SBEM) leads to an algebraic formulation based on generalized quantities [1]. The contact problem is decomposed into two sub-problems: one is purely elastic, the other pertains to the unilateral contact conditions alone [2,3]. Following this methodology, the contact problem, by symmetric BEM, is characterized by symmetry and sign definiteness of the coefficient matrix, thus admitting a unique solution. The solution of the frictionless unilateral contact problem has been obtained: • by means of a quadratic programming problem [2], as optimization problem developed in terms of discrete variables, by using Karnak.sGbem code [4] coupled with MatLab. • through a step by step analysis by using nodal quantities as the check elements. Indeed the detachment or contact phenomenon occurs when the traction or the displacement is greater than the cohesion or reference gap, respectively [3]. The innovative approach is given meanly by the only boundary discretization by using the SBEM approach, by the elastic relation written for each bem-e involving the only quantities of the contact zone. In the examples some comparisons of the two strategies will be shown.