Viscosity solutions for dynamic problems with slip-rate dependent friction

Ionescu, Ioan R.

doi:10.1090/qam/1914436

Cited by 13 publications

(11 citation statements)

References 19 publications

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“…The other data were: the wear constant k = 0.5 · 10 −6 M P a −1 and viscoelasticity constant in (1) ε = 0.001 [7]. Functionsū 0 andū 1 in (3) were equal to 0.…”

Section: Numerical Resultsmentioning

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 other data were: the wear constant k = 0.5 · 10 −6 M P a −1 and viscoelasticity constant in (1) ε = 0.001 [7]. Functionsū 0 andū 1 in (3) were equal to 0.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…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). …”

Section: Problem Formulationmentioning

confidence: 99%

Rolling Contact Problem with the Generalized Coulomb Friction

Chudzikiewicz

Myśliński

2006

Proc Appl Math and Mech

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“…The proof is based on penalization of the inequality (11), friction regularization and employment of localization and shifting technique due to Lions and Magenes. For details of the proof see [3].. o REMARK 2 Assuming that f satisfies the conditions (9), (10) and using the same arguments as in proof of Theorem 2.1 in [5] one can prove that the solution to (11) is unique.…”

Section: (Iv) the Friction Coefficient T Is Bounded Than There Eximentioning

confidence: 99%

“…Usually [2,3] this contact problem is considered either with a constant or suitable small functional friction coefficient depending on spatial variables. Numerous experiments indicate [5,7], that in the study of many frictional processes (stick -sleep motions, earthquake modelling, etc.) the friction coefficient has to be considered variable during the slip.…”

Section: Introductionmentioning

confidence: 99%

“…The evolution of the body in the unilateral contact is described by a variational inequality of the second order [2,4]. Under the assumption that the friction coefficient is bounded and Lipschitz continuous with respect to the sliding velocity, the existence of solutions to the viscoelastic dynamic contact problems is shown in [3,5]. For regularity result with respect to time variable see [7].…”

Section: Introductionmentioning

confidence: 99%

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Shape Optimization of Contact Problems with Slip Rate Dependent Friction

Myśliński¹

IFIP International Federation for Information Processing

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This paper deals with the formulation of a necessary optimality condition for a shape optimization problem of a viscoelastic body in unilateral dynamic contact with a rigid foundation. The contact with Coulomb friction is assumed to occur at a portion of the boundary of the body. The contact condition is described in velocities. The friction coefficient is assumed to be bounded and Lipschitz continuous with respect to a slip velocity. The evolution of the displacement of the viscoelastic body in unilateral contact is governed by a hemivariational inequality of the second order. The shape optimization problem for a viscoelastic body in contact consists in finding, in a contact region, such shape of the boundary of the domain occupied by the body that the normal contact stress is minimized. It is assumed, that the volume of the body is constant. Using material derivative method, we calculate the directional derivative of the cost functional and we formulate a necessary optimality condition for this problem.

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Interaction of faults under slip‐dependent friction. Non‐linear eigenvalue analysis

Ionescu

Wolf

2004

Math Methods in App Sciences

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SUMMARYWe analyse the evolution of a system of ÿnite faults by considering the non-linear eigenvalue problems associated to static and dynamic solutions on unbounded domains. We restrict our investigation to the ÿrst eigenvalue (Rayleigh quotient). We point out its physical signiÿcance through a stability analysis and we give an e cient numerical algorithm able to compute it together with the corresponding eigenfunction.We consider the anti-plane shearing on a system of ÿnite faults under a slip-dependent friction in a linear elastic domain, not necessarily bounded. The static problem is formulated in terms of local minima of the energy functional. We introduce the non-linear (static) eigenvalue problem and we prove the existence of a ÿrst eigenvalue=eigenfunction characterizing the isolated local minima. For the dynamic problem, we discuss the existence of solutions with an exponential growth, to deduce a (dynamic) non-linear eigenvalue problem. We prove the existence of a ÿrst dynamic eigenvalue and we analyse its behaviour with respect to the friction parameter. We deduce a mixed ÿnite element discretization of the non-linear spectral problem and we give a numerical algorithm to approach the ÿrst eigenvalue=eigenfunction. Finally we give some numerical results which include convergence tests, on a single fault and a two-faults system, and a comparison between the non-linear spectral results and the time evolution results.

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Viscosity solutions for dynamic problems with slip-rate dependent friction

Cited by 13 publications

References 19 publications

Rolling Contact Problem with the Generalized Coulomb Friction

Rolling Contact Problem with the Generalized Coulomb Friction

Shape Optimization of Contact Problems with Slip Rate Dependent Friction

Interaction of faults under slip‐dependent friction. Non‐linear eigenvalue analysis

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