Study Of Contact Problems Research Articles

A differential variational inequality in the study of contact problems with wear

We start with a mathematical model which describes the sliding contact of a viscoelastic body with a moving foundation. The contact is frictional and the wear of the contact surfaces is taken into account. We prove that this model leads to a differential variational inequality in which the unknowns are the displacement field and the wear function. Then, inspired by this model, we consider a general differential variational inequality in reflexive Banach spaces, governed by four parameters. We prove the unique solvability of the inequality as well as the continuous dependence of its solution with respect to the parameters. The proofs are based on arguments of monotonicity, compactness, convex analysis and lower semicontinuity. Then, we apply these abstract results to the mathematical model of contact for which we deduce the existence of a unique solution as well as the existence of optimal control for an associate optimal control problem. We also present the corresponding mechanical interpretations.

The evaluation of shear deformation for contact analysis with large displacement

A common problem encountered in the study of contact problem is the failure to obtain stable and accurate convergence result when the contact node is close to the element edge, which is referred as “critical area”. In previous studies, the modification of the element force equation to apply it to a node-element contact problem using the Euler-Bernoulli beam theory [1]. A simple single-element consists two edges and a contact point was used to simulate contact phenomenon of a plane frame. The modification was proven to be effective by the converge-ability of the unbalanced force at the tip of element edge, which enabled the contact node to “pass-through”, resulting in precise results. However, in another recent study, we discover that, if shear deformation based on Timoshenko beam theory is taken into consideration, a basic simply supported beam coordinate afforded a much simpler and more efficient technique for avoiding the divergence of the unbalanced force in the “critical area”. Using our unique and robust Tangent Stiffness Method, the improved equation can be used to overcome any geometrically nonlinear analyses, including those involving extremely large displacements.

On the Solution of the Eigenvalue Complementarity Problem by a Line Search Filter-SQP Algorithm

In this paper, the Eigenvalue Complementarity Problem (EiCP) with real symmetric matrices is addressed, which appears in the study of contact problem in mechanics. We discuss a quadratic programming formulation to the problem. The resulting problems are nonlinear programs that can be solved by a line search filter-SQP algorithm.

Nano-Contact Problem with Surface Effects on Triangle Distribution Loading

This work presents a theoretical study of contact problem. The Fourier integral transform method based on the surface elasticity theory is adopted to derive the fundamental solution for the contact problem with surface effects, in which both the surface tension and the surface elasticity are considered. As a special case, the deformation induced by a triangle distribution force is discussed in detail. The results are compared with those of the classical contact problem. At nano-scale, the contributions of the surface tension and the surface elasticity to the stress and displacement are not equal at the contact surface. The surface tension plays a major role to the normal stress, whereas the shear stress is mainly affected by the surface elasticity. In addition, the hardness of material depends strongly on the surface effects. This study is helpful to characterize and measure the mechanical properties of soft materials through nanoindentation.

Tribocorrosion d’un acier inoxydable en milieu marin artificiel

The friction and wear of stainless steels in sea-water environment is of prime interest for naval applications. The behaviour of the tribosystem alumina pin / 316L stainless steel disc in a 30g/l NaCl solution buffered at pH 8 exhibited the coupling between mechanical and electrochemical parameters. One objective of this study is to determine methods for the comprehension and interpretation of tribocorrosion tests. The friction coefficient is quite sensitive to the applied potential and it is particularly true around the free potential under friction. In our experimental conditions, sliding friction shifts the value of the free potential towards more cathodic value. This behaviour shows the sensibility of potential to frictional perturbation of the surface. In passive domain of materials, an increase of intensity is observed under friction. This evolution is due to depassivation / repassivation phenomena. The overall wear volume shows a dependence to the applied normal load and to the imposed electrochemical potential. Except for the beginning of the test, the evolution of wear is linear in time during a 5 hours test. The total wear, addition of mechanical and electrochemical wear, is investigated and an explanation of its global behaviour is proposed on the basis of an electrochemical and mechanical framework. This eroded volume can be principally attributed to corrosive mechanisms. A mechanical modelling related to evaluation of elasto-plastic pressure distribution is proposed and gives the stabilised wear regime. The experimental wear regime is in agreement with this model. The simplicity of this tribocorrosion system allows an automatic analysis and interpretation of tests results. Interactions between electrochemical and mechanical behaviour are studied. We can explain the mechanical wear by study of contact problem and electrochemical wear by analysis with Faraday law. Despite the great complexity of industrial problems (wear and corrosion of two antagonists,...), this method could be adapted to interpret various tribotest systems.

Stressed state of a transversally isotropic half-space in the case of a nonplane elliptical die under forces and bending moments

UDC 539.3 An exact solution is constructed for the problem of the stressed state of an elastic tramversally isotropic half-space affected by a nonplane elliptical die when the die is subjected to compressive forces and bending moments. The problem is solved in the absence of frictional forces. Expressions are given for determining the compressive load, the transition of points in the die bottom, projections of the rotation vector, the semimajor axis, and the eccentricity of the contact area. A large number of studies of contact problems for elliptical dies were maae by Lur'e [2], Galin [1, 4], and Mossakovskii [3]. Those studies, however, dealt mainly with the stresses in an isotropic solid. Such problems for a transversally isotropic half-space were studied little. In this work we construct an exact solution for problems of the stressed state of an elastic transversally isotropic halfspace under the action of a nonplane elliptical die. The bottom of the die is assumed to be absolutely smooth. The die is considered as an absolutely hard solid subjected to compressive forces and bending moments. The surface of the die bottom in the region of contact with the bourdary of the half-space is approximated by an elliptical paraboioid. The equation of the surface is represented by a power series expansion. The plane bounding the half-space is taken to be the xy plane. Suppose that the z axis is directed on the outside of the solid normal to the plane of isotropy and the x and y axes are oriented arbitrarily in that plane. For a nonplane elliptical die the contact area S is an elliptical region in the plane Z = 0,

Friction laws of fractal type and the corresponding contact problems

The aim of the present paper is the study of contact problems allowing for debonding, i.e. unilateral contact problems, with fractal type friction laws. The iterative method proposed here is based on the description of a fractal as the attractor of a contractive operator T: C 0 → C 0. Using this method, the fractal friction law is approximated by a sequence of nonmonotone classical C 0-curves. The numerical treatment of each arizing nonmonotone problem is accomplished by an advanced solution method which approximates the nonmonotone stress—strain problem by a sequence of monotone problems.

Ultrasonic and laser measurements in structural contact problems

This paper deals with two experimental techniques recently set up by the authors which enable the study of contact problems between elastic bodies. The more widely used experimental techniques for determining the response of elastic bodies in unilateral contact include moire fringes, holography and photoelasticity. The experimental techniques proposed here utilize either the ultrasonic pulse transit time method or the laser beam intensity method. Both techniques are applied to measure the relative displacements and the contact area of beam and plate models in unilateral contact with an elastic half-space model. The experimental results, compared with those obtained by numerical analysis, show the efficiency and validity of the measurement techniques proposed.

Numerical analysis of certain contact problems in elasticity with non-classical friction laws

An algorithm for the numerical analysis of a highly nonlinear variational inequality encountered in the study of contact problems with non-classical friction laws is described. Numerical results obtained for a representative example problem are given.