Dynamic Contact Problems With Small Coulomb Friction for Viscoelastic Bodies: Existence of Solutions

Jarušek, Jiřı́; Eck, Christof

doi:10.1142/s0218202599000038

Cited by 58 publications

(49 citation statements)

References 8 publications

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“…First, we comment on the existence result for (8). The important step is to derive the following uniform estimate.…”

Section: Definition 43mentioning

confidence: 99%

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Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

Migórski

2012

Open Mathematics

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Abstract:We survey recent results on the mathematical modeling of nonconvex and nonsmooth contact problems arising in mechanics and engineering. The approach to such problems is based on the notions of an operator subdifferential inclusion and a hemivariational inequality, and focuses on three aspects. First we report on results on the existence and uniqueness of solutions to subdifferential inclusions. Then we discuss two classes of quasi-static hemivariational ineqaulities, and finally, we present ideas leading to inequality problems with multivalued and nonmonotone boundary conditions encountered in mechanics.

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“…First, we comment on the existence result for (8). The important step is to derive the following uniform estimate.…”

Section: Definition 43mentioning

confidence: 99%

“…[9-12, 15, 16]. More information on modeling and analysis of contact phenomena can be found in [5,7,8,20].…”

Section: Introductionmentioning

confidence: 99%

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

Migórski

2012

Open Mathematics

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“…and c 1 ijkl (x) are components of Hook's tensor satysfiying usual symmetry, boundedness and ellipticity conditions [8]. We use here and throughout the paper the summation convention over repeated indices [3].…”

Section: Problem Formulationmentioning

confidence: 99%

“…Denote by u = (u 1 , u 2 ), u = u(t, x), t ∈ [0, T ], T > 0 given, x ∈ Ω, a displacement of the strip and by σ(u(t, x)) = {σ ij (u(t, x))}, i, j = 1, 2, a stress tensor of the strip. We consider viscoelastic bodies obeying KelvinVoigt law [8]:…”

Section: Problem Formulationmentioning

confidence: 99%

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Rolling Contact Problem with the Generalized Coulomb Friction

Chudzikiewicz

Myśliński

2006

Proc Appl Math and Mech

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This paper deals with the numerical solution of the wheel -rail rolling contact problems. The unilateral dynamic contact problem between a rigid wheel and a viscoelastic rail lying on a rigid foundation is considered. The contact with the generalized Coulomb friction law occurs at a portion of the boundary of the contacting bodies. The Coulomb friction model where the friction coefficient is assumed to be Lipschitz continuous function of the sliding velocity is assumed. Moreover Archard's law of wear in the contact zone is assumed. This contact problem is governed by the evolutionary variational inequality of the second order. Finite difference and finite element methods are used to discretize this dynamic contact problem. Numerical examples are provided. Problem FormulationConsider deformations of a viscoelastic strip Ω of height h lying on a rigid foundation due to the wheel rolling with the linear velocity V (see Fig. 1). Denote by, a stress tensor of the strip. We consider viscoelastic bodies obeying KelvinVoigt law [8]:where the strainand c 1 ijkl (x) are components of Hook's tensor satysfiying usual symmetry, boundedness and ellipticity conditions [8]. We use here and throughout the paper the summation convention over repeated indices [3]. The contact problem consists in finding the displacement field u satisfying [10]whereThe following initial and boundary conditions are imposedwhere u 0 , u 1 are given functions. On the boundary (0, T ) × Γ 2 a surface traction vector F = σ ij (u), i, j = 1, 2, is a priori unknown and is given by the conditions of contact and friction. Under the assumptions, that the strip displacement is small, the contact conditions on the boundary (0, T ) × Γ 2 take a form [3]:where r 0 is the wheel radii, h 0 is a distance between wheel axis and the rail, u N and F N = σ N denote normal components [3] of the displacement u an stress σ on the boundary (0, T ) × Γ 2 respectively. z + N denotes positive part of z N . Moreoverwhere u T and F T = σ T denote tangential components [3] of the displacement u and stress σ on the boundary (0, T ) × Γ 2 respectively, F = (F N , F T ). The choice of functions h N and h T allows to formulate different contact models. Function µ denotes a friction coefficient. Usually this coefficient is assumed to be constant. Numeruous experiments indicate [4,9] that this coefficient is dependent on sliding velocity or temperature. Therefore in this paper the friction coefficient is assumed to depend on the sliding velocityu T . Function w = w(t, x) in (5) denotes the distance between the bodies due to wear and satisfies the Archard law [2],This function is an internal state variable to model the wear process taking place at the contact interface [2]. Assuming that there is no wear and the friction coefficient µ is Lipschitz continuous function of the sliding velocityu T there exists [5,7,8,10] the solution to the contact problem (1) -(6).

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The solvability of a coupled thermoviscoelastic contact problem with small Coulomb friction and linearized growth of frictional heat

Eck

Jarušek²

1999

Math. Meth. Appl. Sci.

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Dynamic Contact Problems With Small Coulomb Friction for Viscoelastic Bodies: Existence of Solutions

Cited by 58 publications

References 8 publications

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

Rolling Contact Problem with the Generalized Coulomb Friction

The solvability of a coupled thermoviscoelastic contact problem with small Coulomb friction and linearized growth of frictional heat

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