A class of dynamic contact problems with Coulomb friction in viscoelasticity

Cocou, Marius

doi:10.1016/j.nonrwa.2014.08.012

Cited by 22 publications

(16 citation statements)

References 28 publications

(41 reference statements)

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“…This work extends the results in [7] to the case of a coefficient of friction depending on the sliding velocity. Using new three and four-field variational formulations, expressed as an evolution variational equation coupled with pointwise constraints, existence and improved regularity results are established.…”

Section: Marius Cocousupporting

confidence: 72%

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A dynamic viscoelastic problem with friction and rate-depending contact interactions

Cocou¹

2020

Evolution Equations &Amp; Control Theory

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The aim of this work is to study a dynamic problem that constitutes a unified approach to describe some rate-depending interactions between the boundaries of two viscoelastic bodies, including relaxed unilateral contact, pointwise friction or adhesion conditions. The classical formulation of the problem is presented and two variational formulations are given as three and four-field evolution implicit equations. Based on some approximation results and an equivalent fixed point problem for a multivalued function, we prove the existence of solutions to these variational evolution problems.

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Section: Marius Cocousupporting

confidence: 72%

“…Based on Ky Fan's fixed point theorem, an elastic contact problem with relaxed unilateral conditions and pointwise Coulomb friction in the static case was studied in [26], the extension to an elastic quasistatic contact problem was investigated in [8] and the corresponding viscoelastic dynamic case was analyzed in [7].…”

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confidence: 99%

A dynamic viscoelastic problem with friction and rate-depending contact interactions

Cocou¹

2020

Evolution Equations &Amp; Control Theory

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“…1 is a particular case of problem Q. As it is straightforward to verify the assumptions (1 -5), (7 -15), and also (16) if F M is sufficiently small, by Theorem 2.3 we obtain the following existence result.…”

Section: Applications To Two Quasistatic Contact Problemsmentioning

confidence: 69%

“…A static contact problem with relaxed unilateral conditions and pointwise Coulomb friction was studied in [15], based on abstract formulations and Ky Fan's fixed point theorem. Recently, an extension of these results to dynamic contact problems in viscoelasticity was treated in [16,17].…”

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confidence: 99%

A variational inequality and applications to quasistatic problems with Coulomb friction

Cocou

2017

Applicable Analysis

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The aim of this paper is to study an evolution variational inequality that generalizes some contact problems with Coulomb friction in small deformation elasticity. Using an incremental procedure, appropriate estimates and convergence properties of the discrete solutions, the existence of a continuous solution is proved. This abstract result is applied to quasistatic contact problems with a local Coulomb friction law for nonlinear Hencky and also for linearly elastic materials.

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“…Recently, the existence and uniqueness of the weak solutions for dynamic contact problems have been established in numerous publications (see e.g., []). The motion equation for the models of these problems is

u^{''} - div σ = f

, where u denotes the displacement vector, σ is the stress tensor and the mass density is taken to be one.…”

Section: Introductionmentioning

confidence: 99%

Existence of a class of variational inequalities modelling quasi‐static viscoelastic contact problems

Peng

2019

Z Angew Math Mech

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This work concerns the existence for a class of variational inequalities arising in quasi-static frictional contact problems for viscoelastic materials. We first consider dynamic contact problems described by evolutionary variational inequalities with a small parameter in the inertial term. Then we study the asymptotic behavior of their solutions when the parameter tends to zero. Based on a time-discretization technique and monotone operators theory, the inequalities are solved in the form of evolutionary inclusions in the framework of evolution triples. We show that the limit functions of dynamic problems are solutions to the quasi-static variational inequalities. Applications of the abstract result to frictional contact problems with multivalued constitutive law and normal damped response are given. Moreover, some comments on nonmonotone boundary problem are presented. K E Y W O R D S hemivariational inequality, nonlinear inclusion, quasi-static problem, variational inequality, viscoelastic contact A M S S U B J E C T C L A S S I F I C A T I O N S (( 1.1) andHere, is a Banach space of admissible displacements, and are operators related to the viscoelastic constitutive law, is a convex functional related to contact boundary conditions, and is a linear operator on . The parameter is positive, the function represents the given body forces and surface traction, and 0 , 1 represents the initial displacement and velocity, PENG respectively. Moreover, the derivative of displacement ′ is with respect to the time variable , the interval of interest is [0, ], and the symbol ⟨⋅, ⋅⟩ denotes the duality pairing between the Banach space and its dual * . Recently, the existence and uniqueness of the weak solutions for dynamic contact problems have been established in numerous publications (see e.g., [4][5][6][7][8]). The motion equation for the models of these problems is ′′ − div = , where denotes the displacement vector, is the stress tensor and the mass density is taken to be one. This equation represents the balance of the momentum in the system and governs the evolution of the state of the body. The variational formulations the equation are expressed by variational (resp. hemivariational) inequalities when the multivalued monotone (resp. nonmonotone) boundary conditions are considered.The present work is motivated by the quasi-static model of viscoelastic contact problems. This model comes from the situations in which the system configuration and the external forces and traction evolve slowly in time in such a way that the accelerations in the system are rather small and negligible so that the inertial term can be neglected. Thus the equation of motion is approximated by the equilibrium equation −div = . In this case, at each time instant the system is very close to equilibrium and the external forces are balanced by the internal stresses. The variational formulations of these problems can be described by variational inequalities of the form: find ∶ [0, ] → such that for almost all ∈ [0, ]

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A class of dynamic contact problems with Coulomb friction in viscoelasticity

Cited by 22 publications

References 28 publications

A dynamic viscoelastic problem with friction and rate-depending contact interactions

A dynamic viscoelastic problem with friction and rate-depending contact interactions

A variational inequality and applications to quasistatic problems with Coulomb friction

Existence of a class of variational inequalities modelling quasi‐static viscoelastic contact problems

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