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

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

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

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

Model and Simulations of a Wood Frog Population

This work presents and simulates a mathematical model  for the dynamics of a population of Wood Frogs. The model consists of a system of five coupled impulsive differential equations for the larvae, juveniles (early, middle, and late) and the mature adult populations. A simulation result depicts possible dynamics of the frogs' population when during one year the larvae population dies out. This provides a tool for the study of the resilience of the population and the conditions that may lead to its survival and flourishing or extinction

Mathematical model and simulations of MERS outbreak: Predictions and implications for control measures

The Middle East Respiratory Syndrome (MERS) has been identified in 2012 and since then outbreaks have been reported in various localities in the Middle East and in other parts of the world. To help predict the possible dynamics of MERS, as well as ways to contain it, this paper develops a mathematical model for the disease. It has a compartmental structure similar to SARS models and is in the form of a coupled system of nonlinear ordinary differential equations (ODEs). The model predictions are fitted to data from the outbreaks in Riyadh (Saudi Arabia) during 2013-2016. The results reveal that MERS will eventually be contained in the city. However, the containment time and the severity of the outbreaks depend crucially on the contact coefficients and the isolation rate constant. When randomness is added to the model coefficients, the simulations show that the model is sensitive to the scaled contact rate among people and to the isolation rate. The model is analyzed using stability theory for ODEs and indicates that when using only isolation, the endemic steady state is locally stable and attracting. Numerical simulations with parameters estimated from the city of Riyadh illustrate the analytical results and the model behavior, which may have important implications for the disease containment in the city. Indeed, the model highlights the importance of isolation of infected individuals and may be used to assess other control measures. The model is general and may be used to analyze outbreaks in other parts of the Middle East and other areas