Boundary element formulations for contact problems in elasticity

The present work is concerned with the study of contact between deformable bodies. As a principal mechanism of load transfer between mechanical components, the contact effects may play a fundamental role in the analysis of structural behavior. It is thus essential to be able to perform contact stress analysis of a component accurately and efficiently. The overall objective of this study was to develop formulations for an effective analysis of contact problems using a combination of the Boundary Element Method (BEM) and Mathematical Programming (MP) techniques. The assumptions of small displacement and linear elasticity are made. A detailed study of both frictionless and frictional problems is presented. The contact problems are written in the form of MP problems whose solution corresponds to the solution of the original mechanical problem.;It is shown that the presence of rigid body displacements in contact configuration may be resolved using linear programming. Two new formulations corresponding to frictionless and frictional contact problems, respectively, where contacting bodies may possess rigid body displacements along a certain direction, are presented. The present methods also allow to determine if whether the contact effect can provide sufficient kinematic restrictions for an otherwise underconstrained body. The problem of optimum shape design of the contour of contacting surfaces is addressed with a simple linear programming formulation. A sequence of design changes leading to an optimal configuration is driven by the objective to reduce peak contact tractions. The above developments are validated through a series of carefully selected example problems. The results are compared with analytical results available in the literature or with results obtained using alternative solution methods to demonstrate the effectiveness of the proposed formulations.;Two original approaches based on quadratic and linear programming techniques were developed. The Quadratic Programming (QP) formulations are based on minimum principles defined on the contact boundary for the frictionless contact and the reduced friction contact, respectively. The complex contact problem is decomposed into a sequence of simplified contact subproblems and an incremental loading sequence. These subproblems are: (a) contact between the elastic bodies under prescribed tangential frictional tractions and (b) contact between the elastic bodies under prescribed normal contact tractions along the contact surface. The Linear Programming (LP) formulations treat contact conditions by optimizing the size of the applied load increments in such a way that only one nodal contact condition changes in any load increment. The resulting sequence of load increments guarantee an accurate approximation of the deformation path of the structure and provides the most accurate results for a given mesh distribution.