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Contact between bodies is most commonly analyzed using quadrilateral contact elements that are based on 8-node brick (hexahedral) continuum finite elements. As a quadrilateral contact surface, in comparison to a triangular contact surface (tetrahedral continuum elements), is not necessarily flat, or it deforms as deformable body deforms, contact formulation turns to be a complex problem. Recent developments in contact routines based on the Moving Friction Cone (MFC) approach for flat triangular contact elements enabled significant simplifications in the element formulation, what is used herein. The MFC formulation of contact is based on the single gap vector, instead of two vectors (slip and stick one). The curved contact surface is defined in a parametric form, thus enabling finite deformations and a Lagrangian definition of contact.

In this paper a 3D beam-to-beam contact element is presented, to deal with contact problems in the coupled electric - mechanical fields. The beams are supposed to get in contact in a pointwise manner, the detection of the contact points and the computation of all contributions are carried out using a fully symmetric treatment of the two beams. Concerning the mechanical field, Hertz theory of contact for elastic bodies is considered. The contact area is varying according to the beamto- beam angle, being circular only in the case of perpendicular beams. This variation of the shape is taken into account too. The problem is semi-coupled: the mechanical field influences the electric one because of the dependence of the voltage distribution on the contact area.

Complex 3D contact formulations, including various types of smoothing, advanced friction laws and sensitivity analysis have become computationally progressively expensive and, for moderate size problems, can be the main obstacle for practical application of these formulations. The aim of the paper is to present a general symbolic description of contact problems and to study the efficiency and the accuracy of several formulations of contact finite elements. For the study to be comprehensive, an approach is needed that enables derivation of the required formulas (e.g. element residual and tangent), their finite element coding, and running of some benchmark problems in an objective way. This has been achieved by using a symbolic approach to derivation of formulas and automatic code generation.

For static or incremental contact problems with Coulomb friction there are satisfactory and well known existence results for the coercive case, , when the elastic body is anchored so that rigid body motions are not possible, see [3, 1, 6, 7, 2]. The articles by Jaru and Cocu, [7, 2] indeed contain results for the noncoercive case, , when rigid body motions are possible. However, the compatibility conditions which are used to ensure the existence of a solution, are the same that guarantee that the corresponding contact problem without friction has a solution. The condition is essentially that the applied force field should push the elastic body towards the obstacle. One of few previous articles containing friction-dependent compatibility conditions is [1].

In this paper we consider the frictional contact problem involving the Signorini contact model and the Coulomb friction law in elastostatics. We focus on the behavior of the set of solutions in the finite-dimensional case where the friction coefficient is a parameter. We study local uniqueness and Lipschitz or continuation of solutions. We come to the conclusion that for any contact status there exists a generalized eigenvalue problem and that the solutions are locally unique if the friction coefficient is not an eigenvalue.

In this paper, we extend the existence results obtained in [1], for a dynamic unilateral contact problem with nonlocal friction between a Kelvin-Voigt viscoelastic body and a rigid obstacle, to the contact between two viscoelastic bodies and to a cracked viscoelastic body. We give classical and primal variational formulations for the problems. Penalized formulations are investigated by using an abstract existence and uniqueness result. Several estimates on the penalized solutions are established which allow to pass to the limit and to prove the existence of solutions for these dynamic unilateral contact problems with friction.

When a rigid body slides with friction on a surface, hopping motion is observed: this is an everyday phenomenon. In rigid bodies mechanics, this phenomenon appears when it is no longer possible to compute the reaction contact forces. The difficulty is overcome by a motion theory involving velocity discontinuities. Velocity discontinuities may result either from an obstacle which makes impossible to compute the acceleration: this is a or from the impossibility to compute the reaction forces: this is a . We describe two examples: the Klein and Painlevé sthenic incompatibilities.

We consider two quasistatic contact problems for viscoplastic materials. The contact is modeled with Signorini’s condition in the first problem, and with normal compliance in the second one. In both problems the adhesion of the contact surfaces, caused by glue, is taken into account and the evolution of the bonding field is described by a first order differential equation. For each model, we provide the variational formulation, state a result on the existence of a unique weak solution, and indicate that the solution of the Signorini problem can be obtained as the limit of the solutions of the problem with normal compliance as the stiffness coefficient of the foundation tends to infinity. We also introduce and discuss a fully discrete scheme for solving the Signorini problem; under certain solution regularity assumptions, an optimal order error estimate holds.

Some optimal a estimates are given for the solutions to the Signorini problem with Coulomb friction (the so-called Coulomb problem) and a uniqueness criterion is exhibited. Recently, nonuniqueness examples have been presented in the continuous framework. It is proven, here, that if a solutions satisfies an hypothesis on the tangential displacement and if the friction coefficient is small enough, it is the unique solution to the problem.

We consider two-body contact problems in elastoplasticity (plasticity with isotropic hardening) with and without friction and present solution procedures based on the coupling of finite elements and boundary elements. One solution method consists in rewriting the problem with penalty terms taking care of the frictional contact conditions [4], see also [8]. Then, its discretized version is solved by applying the radial return algorithm for both friction and plastification. We perform a segment-to-segment contact discretization which allows also to treat friction. Another solution procedure uses mortar projections [2] together with a Dirichlet-to- Neumann (DtN) algorithm for the frictional contact part [6]; here we still use radial return for the plasticity part. Furthermore, extending the approach in [7] we can rewrite the contact problems with friction (given as variational inequalities without regularization) as saddle point problems and directly apply Uzawa’s algorithm. Comments are given for adaptive procedures [5]. Numerical benchmarks are given for small deformations and demonstrate the wide applicability of the given methods.