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Kinematics of sliding can initiate microstructural and frictional changes in the surface and near-surface material of some polymers [1] and in layer-lattice materials as graphite and molybdenum disulphide [4]. In these cases, microstructures of the sliding surfaces reorient themselves, e.g. in the direction of sliding. It may reduce (or increase) friction and wear of the materials. From the mathematical point of view, the induced friction and wear depend on the sliding direction and on a shape of the sliding path, i.e. they depend on a sliding path curvature. First- second- and higher-order equations describe evolutions of friction and wear induced by the sliding path curvature, see [2, 3, 4].

Composition duality methods in the context of optimization problems in mechanics have been the basis for analysis and approximation of minimization and related minimax problems, as studied by Ekeland and Temam [1]. Abstract convex functionals of the type () = ()+(), ∈ , have been considered on a reflexive Banach space . Associated two-field Lagrangians have been of the type (, ) = () – ()+ < (), >, (, ) ∈ × , denoting the topological dual of another reflexive Banach space . Then,corresponding duality principles establish conditions for the solvability equivalence of the primal and mixed optimality condition problems.

The model problem is considered for a contact interaction between plate and massive body under adhesive joint, which simulates the work of the friction lining and its metal base. The conditions are sought by way of the choice of favorable angles of the joint geometry depending on the mechanical properties of joining materials, which promote development of the optimal structure of the friction lining joint with a metal base under external thermal and mechanical action.

The method (NSCD) deals with frictional unilateral contact between rigid or deformable bodies. It was originated around 1984 by J.J. Moreau as the method (). It was extended as the () method by M. Jean to deal with more general applications, such as nite element modelling. Numerical applications of the method for granular materials are currently customized in C language by J.J. Moreau. The method has been implemented by M. Jean in a now obsolete fortran77 general purpose software. F. Dubois is the author of a completely remodelled new version written in fortran90, open to research scientists for further developments and applications. This is an open source software governed by a (i.e. GPL). In the NSCD method the basic laws such as Coulomb’s law and the inelastic shock law are described as non smooth laws in terms of multimappings. The dynamical equation is discretized according to a low order implicit algorithm. The main unknowns are the relative velocities between contactors at some overlapping moments with the time steps (leap frog technique) and the mean reaction impulses during the time step. Assuming provisional values for contacts neighbouring of a given contact, values of the reactions for this given contact are obtained discussing the intersection of the graphs of affine mappings. Values of the reactions are updated, and all contacts are processed successively as long as necessary to obtain a satisfactory convergence. This may be described as a non linear block Gauss Seidel method. The software is dedicated to applications with a large number of 2D or 3D contactors. It is a modular software written in fortran90 with an object oriented organisation:

The frictional behaviour of elastomer materials still raises a lot of questions. The sliding of rubber on a rough surface cannot be described by using a constant frictioncoefficient. An appropriate constitutive law has to represent the complex dependancies on sliding velocity or normal stress. As rough tracks have a fractal character it is necessary to use a staggered procedure over the relevant length scales.

A recently proposed micro-mechanical model for the stress-strain behavior of filled elastomers has been formulated on the basis of an extended tube model of rubber elasticity that permits a sound molecular-statistic description of bulk polymer networks up to large strains [1]-[4].

A general anisotropic model of unilateral contact with adhesion and friction is proposed. The model is applied to the numerical analysis of contact between pelvis and acetabulum after arthroplasty.

There are two approaches to the description of impact phase in impact mechanics, namely, [1]. Both approaches give the same restitution law if external control is absent. However, the situation changes dramatically if one can apply an impulsive control during the phase of contact. Indeed, a real interaction force acts when a contact takes place only. The difference in these two models is as follows. In the case of visco-elastic media the mathematical approach implies an existence of interaction force acting during a whole period of an existence of body-obstacle deformation, which leads to appearance of a drawing force pulling the body into obstacle. In reality (in the physical model) the interaction force vanishes when a contact is lost, even if the deformation still presents. In this paper a control problem based on the physical model (which is more realistic) is considered.

Tribometers can be used for preselecting tribo-materials in an early development stage. The test set-ups range from simple model tests to complex component tests [1]. All measured tribological variables are not only materialparameters but they also depend on the used tribosystem. Therefore the results of tribometer measurement often cannot be transferred to complex tribological systems or to components. The different test setups can be classified according to the -Curve. Typical hydrodynamic running components like camshafts operate in the fully lubricated region of the curve. Tribological damage (like seizing) only occurs if the lubricant film fails, for example due to severe overload. These conditions can be reproducibly obtained by choosing the appropriate tribological model, which operates under boundary lubrication conditions. Two different metallic tribo-materials (Al-based and Cu-based) were investigated by using two different configurations (ball-ondisc, ring-on-disc).

We consider stone bridges with unilateral mortar joints and elastic or rigid stones [1], [2]. Opening or sliding indicates crack initiation and propagation. The ultimate load has been calculated by using a path - following technique. Lack of a solution at a certain level of loading indicates onset of failure, which can be checked by the solvability conditions of the corresponding variational or hemivariational inequality [1]. For validation and comparison the ultimate load is recalculated by the classical collapse mechanism method of Heyman, in a modern implementation with linear programming [3]. The theoretical di.culty of using upper bound style techniques, especially in the presence of friction (in general, nonassociated models), can be followed in [4].