Analysis of a contact problem with normal damped response and unilateral constraint

Barboteu, Mikaël; Danan, David; Sofonea, Mircea

doi:10.1002/zamm.201400304

Cited by 12 publications

(9 citation statements)

References 35 publications

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“…The unique solvability of the inclusion is proved by a standard fixed point argument similar to those used in many papers, for instance in [15,[19][20][21] and [23]. Furthermore, we note that the abstract convergence result of Theorem 13 is based on arguments and assumptions similar to those exploited, for instance in [2,15,25] and [27]. In the second part of the paper we analyze a mathematical model of a contact problem for viscoelastic materials with history-dependent operators and a slipdependent friction.…”

Section: Introductionmentioning

confidence: 89%

A Dynamic Contact Problem with History-Dependent Operators

Ogorzaly

2016

J Elast

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In this paper we present results on existence, uniqueness and convergence of solutions to the Cauchy problem for abstract first order evolutionary inclusion which contains two operators depending on the history of the solution. These results are applicable to a dynamic contact problem for viscoelastic materials with a normal compliance contact condition with memory and a friction law in which the friction bound depends on the magnitude of the tangential displacement. The proofs are based on recent results for hemivariational inequalities and a fixed point argument.

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Section: Introductionmentioning

confidence: 89%

A Dynamic Contact Problem with History-Dependent Operators

Ogorzaly

2016

J Elast

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“…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

u : [0, T] \to V

such that for almost all

t \in [0, T]

\begin{matrix} left {false⟨ ξ (t) + B u (t) - f (t), v - u^{'} (t) false⟩}_{V} + J (t, L v) - J (t, L u^{'} false(t false)) \geq 0 \forall v \in V with ξ (t) \in A (t, u^{'} false(t false)), \end{matrix}

and

u false(0 false) = u_{0} .

There is considerable literature on the mathematical theory and applications of quasi‐static models, see, e.g., [] and the references therein. Especially, Han and Sofonea studied a class of variational inequalities like in a Hilbert space by assuming that A is single‐valued, strongly monotone and and both A and B are Lipschitz continuous, and therefore Banach fixed theorem and some standard arguments on elliptic variational inequality can be exploited to the existence and uniqueness of solutions.…”

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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“…The process is supposed to be the subject of thermal effects, friction and wear of contacting surfaces. Mathematical models in Contact Mechanics can be found in [1,2,7,8,9,10,12,13,14,17,18,19,21,22,23,24] .…”

Section: Introductionmentioning

confidence: 99%

A History-Dependent Frictional Contact Problem With Wear for Thermoviscoelastic Materials

Chouchane

Selmani

2019

Mathematical Modelling and Analysis

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In this manuscript we study a contact problem between a deformable viscoelastic body and a rigid foundation. Thermal effects, wear and friction between surfaces are taken into account. A variational formulation of the problem is supplied and an existence and uniqueness result is proved. The idea of the proof rested on a recent result on history-dependent quasivariational inequalities. Finally, a perturbation of the data is initiated and a convergence result is demonstrated when the perturbation parameter converges to zero.

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Analysis of a contact problem with normal damped response and unilateral constraint

Cited by 12 publications

References 35 publications

A Dynamic Contact Problem with History-Dependent Operators

A Dynamic Contact Problem with History-Dependent Operators

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

A History-Dependent Frictional Contact Problem With Wear for Thermoviscoelastic Materials

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