Quasistatic Frictional Contact and Wear of a Beam

A problem of frictional contact between an elastic beam and a moving foundation and the resulting wear of the beam is considered. The process is assumed to be quasistatic, the contact is modeled with normal compliance, and the wear is described by the Archard law. Existence and uniqueness of the weak solution for the problem is proved using the theory of strongly monotone operators and the Cauchy-Lipschitz theorem. It is also shown that growth of the the wear function is at most linear. Finally, a numerical approach to the problem is considered using a time semi-discrete scheme. The existence of the unique solution for the discretized scheme is established and error estimates on the approximate solutions are derived

A frictional contact problem with wear diffusion

This paper constructs and analyzes a model for the dynamic frictional contact between a viscoelastic body and a moving foundation. The contact involves wear of the contacting surface and the diffusion of the wear debris. The relationships between the stresses and displacements on the contact boundary are modeled by the normal compliance law and a version of the Coulomb law of dry friction. The rate of wear of the contact surface is described by the differential form of the Archard law. The effects of the diffusion of the wear particles that cannot leave the contact surface on the surface are taken into account. The novelty of this work is that the contact surface is a manifold and, consequently, the diffusion of the debris takes place on a curved surface. The interest in the model is related to the wear of mechanical joints and orthopedic biomechanics where the wear debris are trapped, they diffuse and often cause the degradation of the properties of joint prosthesis and various implants. The model is in the form of a differential inclusion for the mechanical contact and the diffusion equation for the wear debris on the contacting surface. The existence of a weak solution is proved by using a truncation argument and the Kakutani--Ky Fan--Glicksberg fixed point theorem.Comment: 22 page

Dual formulation of a quasistatic viscoelastic contact problem with tresca's friction law

International audienceWe consider quasistatic evolution of a viscoelastic body which is in bilateral frictional contact with a rigid foundation, We derive two variational formulations for the problem: the primal formulation in terms of the displacements and the dual formulation in terms of the stress field. We prove the existence of a unique solution to each one and establish the equivalence between the two variational formulations. We also prove the continuous dependence of the solution on the friction yield limit

Models and simulations of dynamic frictional contact of a thermoelastic beam

International audienceWe investigatea mathematicalmodel for the dynamic thermomechanical behavior of a viscoelasticbeam that is in frictional contact with a rigid moving surface. Friction is modeled by a version of Coulomb's law with slip dependent coefficientof friction,taking into account the frictional heat generation.We prove the existence and uniqueness of the weak solution, describe an algorithm for the numerical solutions and present results of numerical simulations, including the frequency distribution of the noise generated by the stick/slip motion. We also show that when the surface moves too fast there are no steady solutions and therefore the system is thermally unstable

Thermal analysis of the grinding process

A two-dimensional mathematical model for the thermal aspects of a grinding process is presented. The model includes heat conduction in the grinding wheel, workpiece, and coolant. The heat generation through friction, heat loss to the environment as well as debris, and the interaction among the three components are described in detail. A finite-element algorithm is implemented to solve the nonlinear problem. Numerical results, such as temperatures in the grinding wheel and workpiece, are presented

Dynamic thermoviscoelastic thermistor problem with contact and nonmonotone friction

The paper studies the evolution of the thermomechanical and electric state of a thermoviscoelastic thermistor that is in frictional contact with a reactive foundation. The mechanical process is dynamic, while the electric process is quasistatic. Friction is modeled with a nonmonotone relation between the tangential traction and tangential velocity. Frictional heat generation is taken into account and so is the strong dependence of the electric conductivity on the temperature. The mathematical model for the process is in the form of a system that consists of dynamic hyperbolic subdifferential inclusion for the mechanical state coupled with a nonlinear parabolic equation for the temperature and an elliptic equation for the electric potential. The paper establishes the existence of a weak solution to the problem by using time delays, a priori estimates and a convergence method.Comment: 25 page

A Quasistatic Contact Problem for an Elastoplastic Rod

AbstractWe consider a mathematical model which describes the quasistatic contact of an elastoplastic rod with an obstacle. It is based on the Prandtl–Reuss flow law and unilateral conditions imposed on the velocity. Two weak formulations are presented and existence and uniqueness results established. The proofs are based on approximate problems with viscous regularization, which have merit on their own and may be used as the basis for convergent numerical algorithms for the problem

A Mathematical Model for Outgassing and Contamination

A model for the mathematical description of the processes of outgassing and contamination in a vacuum system is proposed. The underlying assumptions are diffusion in the source, convection and diffusion in the cavity, mass transfer across the source-cavity interface, and a generalization of the Langmuir isotherm for the sorption kinetics on the target. Three approximations are considered where the asymptotic behavior of the model for large time is shown as well as the dependence and sensitivity of the model on some of the parameters. Some numerical examples of the full model are then presented together with a proof of the uniqueness of the solution