On quasi-static contact problem with generalized Coulomb friction, normal compliance and damage

Gasiński, Leszek; Kalita, Piotr

doi:10.1017/s0956792515000583

Cited by 5 publications

(14 citation statements)

References 18 publications

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“…[25,26,33] and the references therein). Other aspects including evolution inclusions or boundary conditions which are multivalued, nonmonotone, and of subdifferential form can be found in [4,5,[16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Multivalued nonmonotone dynamic boundary condition

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In this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities, reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the Laplace operator with Wentzell (dynamic) boundary conditions perturbed by a multivalued nonmonotone operator expressed in terms of Clarke subdifferentials. We show that one can reformulate the problem so that standard techniques can be applied. We use the well-established theory of boundary hemivariational inequalities to prove that under growth and general sign conditions, the dynamic boundary hemivariational inequality admits a weak solution. Moreover, in the situation where the functionals are expressed in terms of locally bounded integrands, a “filling in the gaps” procedure at the discontinuity points is used to characterize the subdifferential on the product space. Finally, we prove that, under a growth condition and eventually smallness conditions, the Faedo–Galerkin approximation sequence converges to a desired solution.

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Section: Introductionmentioning

confidence: 99%

Multivalued nonmonotone dynamic boundary condition

et al. 2021

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“…Introduction. This paper is a follow up of [15], [14], and [12]. We study an evolution of the displacement of a Kelvin-Voigt viscoelastic body which can come into contact with a foundation.…”

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“…According to this model, the damage is represented by a function β : Ω × [0, T ] → [0, 1], where the value 0 means that the body is fully damaged, and the value 1 means that it is not damaged at all. Such approach has been recently extensively studied for springs (see [2,6]), beams (see [1,18]) and for three-dimensional deformable solids (see [4,12,14,19,20,25,27]).…”

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“…The contact conditions assumed here are the same as in [12]. The foundation is penetrable and the reaction force density depends on the penetration depth 1010 LESZEK GASIŃSKI AND PIOTR KALITA through so called normal compliance law.…”

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“…For more information about the multivalued contact conditions and subdifferentials see the monographs [7,8,21,22,24,26]. The novelty of this article with respect to [12] lies in the fact that while in [12] the model was assumed to be quasistatic, here we assume that it is dynamic, i.e., instead of the equilibrium equation −Div σ(t) = f 0 (t), studied in [12], we consider the momentum equation u(t) − Div σ(t) = f 0 (t).…”

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On dynamic contact problem with generalized Coulomb friction, normal compliance and damage

Gasiński¹,

Kalita²

2020

Evolution Equations &Amp; Control Theory

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A frictional contact problem with wear diffusion

Kalita

Szafraniec

Shillor

2019

Z. Angew. Math. Phys.

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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.

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On quasi-static contact problem with generalized Coulomb friction, normal compliance and damage

Cited by 5 publications

References 18 publications

Multivalued nonmonotone dynamic boundary condition

Multivalued nonmonotone dynamic boundary condition

On dynamic contact problem with generalized Coulomb friction, normal compliance and damage

A frictional contact problem with wear diffusion

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