Frictionless contact-detachment analysis: iterative linear complementarity and quadratic programming approaches

The object of the paper concerns a consistent formulation of the classical Signorini’s theory regarding the frictionless contact problem between two elastic bodies in the hypothesis of small displacements and strains. The employment of the symmetric Galerkin boundary element method, based on boundary discrete quantities, makes it possible to distinguish two different boundary types, one in contact as the zone of potential detachment, called the real boundary, the other detached as the zone of potential contact, called the virtual boundary. The contact-detachment problem is decomposed into two sub-problems: one is purely elastic, the other regards the contact condition. Following this methodology, the contact problem, dealt with using the symmetric boundary element method, is characterized by symmetry and in sign definiteness of the matrix coefficients, thus admitting a unique solution. The solution of the frictionless contact-detachment problem can be obtained: (i) through an iterative analysis by a strategy based on a linear complementarity problem by using boundary nodal quantities as check quantities in the zones of potential contact or detachment; (ii) through a quadratic programming problem, based on a boundary min-max principle for elastic solids, expressed in terms of nodal relative displacements of the virtual boundary and nodal forces of the real one.