Keywords: boundary element method, contact problem, non-linear formulation.

In several structural systems the external load is transferred among the structural elements by contact. Thus, the mechanical efficiency depends on the interaction between theses components and its surfaces. The knowledge of the structural behavior of the surfaces on contact is very important mainly in automobile and aeronautic industries. This work deals with analysis of contact problems using the boundary element method (BEM). The BEM is known to be a robust and accurate technique for this type of problem, because the contact occurs on the boundaries. The proposed formulation for contact is based on the use of singular and hyper singular integral equations by the BEM. When the contact occurs between crack surfaces, the formulation adopted is the dual version of the BEM, in which singular and hyper singular integral equations are employed along the opposite sides of the contact boundaries. This technique avoids singularities of the resulting algebraic system of equations, in spite of the fact that the material points coincide for the two opposite sides of the contact region. When the contact occurs among the external body surfaces, the sub-region technique is used. In this case we can adopt other combinations of integral equations for the solid discretization. The first combination is named singular sub-region technique (SST), where only a singular integral equation is adopted. The second scheme uses only a hyper singular integral equation. We named this scheme the total hyper-singular sub-region technique (THST). Finally, we can use a singular integral equation along the external boundary and a hyper singular integral equation on the contact surfaces. This scheme is named the hyper singular sub-region technique (HST).

For the non linear contact formulation we adopted the Coulomb's law. A BEM formulation using a tangent operator will be presented. Using a tangent operator to solve the non linear system of algebraic equations in the context of BEMs has shown to be a more accurate and stable procedure in which convergence is more easily achieved, Leonel and Venturini [1]. This formulation uses the derivative of the set of equations to construct the corrections on the non-linear process. This formulation was shown to bge as accurate as the classical approach, however it is very fast in terms of computational work. Examples of simple and multi-region contact problems are shown to illustrate the applicability of the proposed scheme.

1

E.D. Leonel, W.S. Venturini, "Non-linear boundary element formulation with tangent operator to analyse crack propagation in quasi-brittle materials", Engineering Analysis with Boundary Elements, 34, 122-129, 2010. doi:10.1016/j.enganabound.2009.08.005

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