A frictional contact problem with wear diffusion (original) (raw)
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Analysis of viscoelastic contact with normal compliance, friction and wear diffusion
Comptes Rendus De L Academie Des Sciences Series Iib Mechanics, 2003
We consider a quasistatic problem of frictional contact between a viscoelastic body and a moving foundation. The contact is with wear and is modeled by normal compliance and a law of dry friction. The novelty in the model is that it allows for the diffusion of the wear debris over the potential contact surface. Such kind of phenomena arise in orthopaedic biomechanics and influence the properties of joint prosthesis. We derive a weak formulation of the problem and state that, under a smallness assumption on the problem data, there exists a unique weak solution for the model. To cite this article: M.
Quasistatic viscoelastic contact with friction and wear diffusion
Quarterly of Applied Mathematics, 2004
We consider a quasistatic problem of frictional contact between a deformable body and a moving foundation. The material is assumed to have nonlinear viscoelastic behavior. The contact is modeled with normal compliance and the associated law of dry friction. The wear takes place on a part of the contact surface and its rate is described by the Archard differential condition. The main novelty in the model is the diffusion of the wear particles over the potential contact surface. Such phenomena arise in orthopaedic biomechanics where the wear debris diffuse and influence the properties of joint prosthesis and implants. We derive a weak formulation of the model which is given by a coupled system with an evolutionary variational inequality and a nonlinear evolutionary variational equation. We prove that, under a smallness assumption on some of the data, there exists a unique weak solution for the model.
Viscoelastic sliding contact problems with wear
We consider a mathematical model which describes the sliding contact with wear between a viscoelastic body and a rigid moving foundation We consider both the dynamic and quasi-static cases and we model the wear with a version of Archard's law. We derive the variational formulation of the model and prove existence and uniqueness results. The proofs are based on arguments of evolution equations with monotone operators and Banach's fixed-point theorem, in the case of the dynamical model, and on Cauchy-Lipschitz theorem in the case of the quasi-static model. We also establish the continuous dependence of the solution with respect to parameters related to the velocity of the moving foundation.
Variational Analysis of a Frictional Contact Problem with Wear and Damage
Mathematical Modelling and Analysis
We study a quasistatic problem describing the contact with friction and wear between a piezoelectric body and a moving foundation. The material is modeled by an electro-viscoelastic constitutive law with long memory and damage. The wear of the contact surface due to friction is taken into account and is described by the differential Archard condition. The contact is modeled with the normal compliance condition and the associated law of dry friction. We present a variational formulation of the problem and establish, under a smallness assumption on the data, the existence and uniqueness of the weak solution. The proof is based on arguments of parabolic evolutionary inequations, elliptic variational inequalities and Banach fixed point.
A Mixed Variational Formulation of a Contact Problem with Wear
Acta Applicandae Mathematicae, 2017
We consider a mathematical model which describes the sliding frictional contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic, the material's behavior is described with a viscoplastic constitutive law with internal state variable and the contact is modelled with normal compliance and unilateral constraint. The wear of the contact surfaces is taken into account, and is modelled with a version of Archard's law. We derive a mixed variational formulation of the problem which involve implicit history-dependent operators. Then, we prove the unique weak solvability of the contact model. The proof is based on a fixed point argument proved in Sofonea et al. (Commun.
Analytical Solution of the Contact Problem for a System of Bodies under Collective Wear
Mechanics of Solids, 2017
The contact problem is considered for a system of bodies subject to wear on a common base. The deformation properties of the bodies and the base are described by the Winkler model. The problem is reduced to a system of ordinary differential equations and an integral equation of hereditary type with difference kernel. The solution of the problem is constructed with the use of the Laplace transform. The asymptotics of the solution at large times is studied. The continuity conditions for the contact of each of the bodies with the base are derived.
A dynamic thermoviscoelastic contact problem with friction and wear
International Journal of Engineering Science, 1997
This paper deals with a contact problem describing the mechanical and thermal evolution of a damped extensible thermoviscoelastic beam under the Cattaneo law, relating the heat flux to the gradient of the temperature. The beam is rigidly clamped at its left end whereas the right end of the beam moves vertically between reactive stops like a nonlinear spring. Existence and uniqueness of the solution is proved, as well as the exponential decay of the related energy. Then, fully discrete approximations are introduced by using the classical finite element method and the implicit Euler scheme to approximate the spatial variable and to discretize the time derivatives, respectively. An a priori error estimates result is proved, from which the linear convergence of the algorithm is deduced. The case where the two stops are rigid is also studied from the point of view of the existence and longtime behavior of the solutions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximation and the behavior of the solution.
Analysis of steady state wear processes for inhomogeneous materials and varying contact loads
Journal of Computational and Applied Mechanics, 2020
The transient wear process on a frictional interface of two elastic bodies in relative steady sliding motion induces shape evolution of contact interface and tends to a steady state. Then wear growth develops at constant contact stress and strain distributions. In previous papers these cases were analyzed for the fixed sliding velocity between the bodies and for fixed loads [1-4]. The cases of periodic sliding under fixed normal loads were treated in [5]. The cases of periodic loads for fixed or varying sliding velocity were investigated in [6]. The variational procedure and minimization of the response functional corresponding to the wear dissipation power were applied. The modified Archard wear rule was assumed. The specific examples were solved assuming fixed values of wear parameters in the contact domain. In the present paper the previous analyses are extended to cases when the wear parameters can vary along the sliding path and similarly, the sliding velocity and normal load can vary periodically. The cases of a ring segment-on rotating disk and translating punch-on strip are considered, providing wear analysis accounting for temperature effect.
Advances in Mathematical Physics, 2016
We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with a combination of a normal compliance and a normal damped response law associated with a slip rate-dependent version of Coulomb’s law of dry friction. We derive a variational formulation and an existence and uniqueness result of the weak solution of the problem is presented. Next, we introduce a fully discrete approximation of the variational problem based on a finite element method and on an implicit time integration scheme. We study this fully discrete approximation schemes and bound the errors of the approximate solutions. Under regularity assumptions imposed on the exact solution, optimal order error estimates are derived for the fully discrete solution. Finally, after recalling the solution of the frictional contact problem, some numerical simulations are provided in order to illustrate both the behavior of the solutio...