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EN
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 normal damped response and unilateral constraint for the velocity field, associated to a version of Coulomb’s law of dry friction. Our aim is to present a detailed description of the numerical modelling of the problem. To this end, we use a penalty method to approximate the constraint. Then, we provide numerical simulations in the study of a two-dimensional example and compare the penalty model with the original one.
EN
This work considers a mathematical model that describes quasistatic evolution of an elastic 2D bar that may come in frictional contact with a deformable foundation. We present the model and some of its underlying assumptions. In particular, the novelty in the model is that both vertical and horizontal motions are taken into account, which makes it especially useful when frictional contact is concerned. Contact is described with the normal compliance condition and friction with the Coulomb law of dry friction. We introduce a hybrid variational formulation of the problem and a numerical discretization based on a uniform time step and the finite element method in space. The numerical algorithm has been implemented, and we present computer simulations that illustrate the mechanical behavior of the system with emphasis on frictional aspects of the problem.
EN
We construct a mathematical model which describes the contact between an elastic body and an obstacle, the so-called foundation. The contact is frictional and is modelled with normal compliance and unilateral constraint, associated to a slip- dependent version of Coulomb’s law of dry friction. We present a detailed description Of the model, then we provide numerical simulations in the study of a two-dimensional Example. Our aim is to underline the influence of the parameters involved in the boundary conditions, which could give rise to different status of the material points c>n the contact surface.
4
A Damageable Spring
EN
The evolution of material damage in a nonlinear spring is modeled, analyzed, and numerically simulated. The material damage is described by a damage function whose evolution depends on the mechanical energy in the system and the damage threshold. The model is in the form of two coupled nonlinear ordinary differential equations. The existence of the unique solution is proved using arguments for evolutionary equations with maximal monotone operators, differential equations, and fixed points. The scaling properties of the model are discussed. A numerical algorithm for the problem is presented and four simulations of the system behavior depicted. In particular, the changes in the oscillations of the system as damage progresses are shown.
EN
We consider a quasistatic problem which describes the contact between a viscoplastic body and an obstacle, the so-called foundation. The contact is frictionless and is modelled with a version of the normal compliance condition in which the penetration is restricted with unilateral constraint. The mathematical analysis of the problem, including, existence, uniqueness and convergence results, was provided by Barboteu et al. (2011). Here we present numerical simulations in the study of an academic two-dimensional contact example.
6
Regularity of Solutions to Dynamic Contact Problems
EN
We investigate the regularity in time of solutions of two dynamic frictionless contact problems. The first problem describes the vibration of a mass-spring system and the second one concerns the contact of an elastic rod. In both models the contact is described with normal compliance. We show that the regularity of the normal compliance function determines the regularity of the solution.
7
A Dynamic Piezoelectric Contact Problem
EN
We consider a mathematical model, which describes the dynamic process of contact between a piezoelectric body and an electrically conductive foundation. The material's behavior is modeled with a nonlinear electro-viscoelastic constitutive law; the contact is frictionless and is described with the normal compliance condition and a regularized electrical conductivity condition. We state the variational formulation for the problem, and then we introduce a fully discrete scheme, based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. We implement this scheme in a numerical code and, in order to verify its accuracy, we present numerical simulations in the study of a two-dimensional test problem.
EN
In this paper detailed description of temperature influence on MR damper working parameters has been described. Prototype of such a device and its mathematical description taking into considerations a thermal balance, have been proposed. Basing on experimental results, a connection between temperature and operating parameters of tested MR damper has been revealed.
PL
W pracy szczegółowo omówiono wpływ temperatury na charakterystyki dyssypacji energii amortyzatora wypełnionego cieczą magnetoreologiczną. Zaproponowano pewne rozwiązanie konstrukcyjne prototypowego amortyzatora MR oraz jego model matematyczny, uwzględniający bilans cieplny. Na podstawie wyników badań eksperymentalnych wykazano ścisłe powiązanie parametrów eksploatacyjnych amortyzatora MR z jego temperaturą pracy.
9
A Model for Adhesive Frictional Contact
EN
The aim of this paper is to present a mathematical model which describes the quasistatic process of adhesive frictional contact between a deformable body and an obstacle, the so-called foundation. The material's behavior is assumed to be elastic, with a nonlinear constitutive law; the adhesive contact is modelled with a surface variable, the bonding field, associated to the normal compliance condition and the static version of Coulomb's law of dry friction. We describe the assumptions which lead to the mathematical model of the process and derive a variational formulation of the problem; then, under a smallness assumption on the coefficient of friction, we prove the uniqueness of the solution for the model.
EN
We study a mathematical problem describing the friction- less adhesive contact between an elastic body and a foundation. The adhesion process is modelled by a surface variable, the bonding field, and the contact is modelled with a normal compliance condition; the tangential shear due to the bonding field is included; the elastic consti- tutive law is assumed to be nonlinear and the process is quasistatic. The problem is formulated as a nonlinear system in which the unknowns are the displacement, the stress and the bonding field. The existence of a unique weak solution for the problem is established by using arguments for differential equations followed by the construction of an appropriate contraction mapping.
EN
We consider a mathematical model which describes the frictional contact between a deformable body and an obstacle, say a foundation. The body is assumed to be linear elastic and the contact is modeled with a version of Coulomb's law of dry friction in which the normal stress is prescribed on the contact surface. The novelty consists here in the fact that we consider a slip dependent coefficient of friction and a quasistatic process. We present two alternative yet equivalent formulations of the problem and establish existence and uniqueness results. The proofs are based on a new result obtained in [10] in the study of evolutionary variational inequalities.
12
Numerical Analysis and Simulations of Quasistatic Frictionless Contact Problems
EN
A summary of recent results concerning the modelling as well as the variational and numerical analysis of frictionless contact problems for viscoplastic materials are presented. The contact is modelled with the Signorini or normal compliance conditions. Error estimates for the fully discrete numerical scheme are described, and numerical simulations based on these schemes are reported.
13
Variational Analysis of a Frictional Contact Problem for the Bingham Fluid
EN
We consider a mathematical model which describes the flow of a Bingham fluid with friction. We assume a stationary flow and we model the contact with damped response and a local version of Coulomb's law of friction.The problem leads to a quasi-variational inequality for the velocity field. We establish the existence of a weak solution and, under additional assumptions, its uniqueness. The proofs are based on a new result obtained in (Motreanu and Sofonea, 1999). We also establish the continuous dependence of the solution with respect to the contact conditions.
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