Weak solutions for contact problems involving viscoelastic materials with long memory

Matei, Andaluzia; Ciurcea, Raluca

doi:10.1177/1081286511400515

Cited by 12 publications

(14 citation statements)

References 21 publications

(25 reference statements)

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“…The space M is a Hilbert space, see, e.g., [11]. Furthermore, Y is a Hilbert space being the dual of the Hilbert space M. Everywhere below ·, · Y,M will denote the dual pairing between Y and M. For a regular enough function u which verifies Problem 1, a Lagrange multiplier λ ∈ Y can be introduced as follows:…”

Section: Second Variational Formulationmentioning

confidence: 99%

On the relationship between alternative variational formulations of a frictional contact model

Matei

2019

Journal of Mathematical Analysis and Applications

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We consider a frictional contact model, mathematically described by means of a nonlinear boundary value problem in terms of PDE. We draw the attention to three possible variational formulations of it. One of the variational formulations is a variational inequality of the second kind and the other two are mixed variational formulations with Lagrange multipliers in dual spaces. As main novelty, we establish the relationship between these three variational formulations. We also pay attention to the recovery of the formulation in terms of PDE starting from the mixed variational formulations.Key words: boundary value problem, frictional contact, variational inequality of the second kind, mixed variational formulation, dual Lagrange multipliers, weak solutions.

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Section: Second Variational Formulationmentioning

confidence: 99%

On the relationship between alternative variational formulations of a frictional contact model

Matei

2019

Journal of Mathematical Analysis and Applications

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“…Also, by (55) the bilinear form e (·, ·) defined in (63) verifies Assumption 3 with M e = ‖ μ ‖ L ∞ ( Ω ) and m e = μ *. To prove that the bilinear form b (·, ·) verifies Assumption 4, arguments similar to those used in [6, 24] can be used; for the convenience of the reader, we shall indicate below the justification. First,…”

Section: A Frictional Contact Problemmentioning

confidence: 99%

“…For a variational approach via dual Lagrange multipliers in viscoplasticity we refer to [5], where a frictionless unilateral contact problem was analyzed. For a variational approach via dual Lagrange multipliers in viscoelasticity with long memory we refer to [6], where the weak solvability of a nonlinear antiplane frictional contact problem was studied.…”

Section: Introductionmentioning

confidence: 99%

An evolutionary mixed variational problem arising from frictional contact mechanics

Matei

2012

Mathematics and Mechanics of Solids

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We consider an abstract mixed variational problem which consists of a system of an evolutionary variational equation in a Hilbert space X and an evolutionary inequality in a subset of a second Hilbert space Y , associated with an initial condition. The existence and the uniqueness of the solution is proved based on a fixed point technique. The continuous dependence on the data was also investigated. The abstract results we obtain can be applied to the mathematical treatment of a class of frictional contact problems for viscoelastic materials with short memory. In this paper we consider an antiplane model for which we deliver a mixed variational formulation with friction bound dependent set of Lagrange multipliers. After proving the existence and the uniqueness of the weak solution, we study the continuous dependence on the initial data, on the densities of the volume forces and surface tractions. Moreover, we prove the continuous dependence of the solution on the friction bound. KeywordsEvolutionary mixed variational problem, saddle point, fixed point, friction bound dependent set, dual Lagrange multipliers, weak solutions, continuous dependence on the data, viscoelasticity, frictional contact problems

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“…Reference in the field are [23,24,25]. The analysis of various mixed variational problems associated to mathematical models which describe the contact between a deformable body and a foundation can be found in [9,10,11,12,15,16,17] and, more recently, in [1,2,3,27].…”

Section: Introductionmentioning

confidence: 99%

Convergence and Optimization Results for a History-Dependent Variational Problem

Sofonea¹,

Matei²

2019

Acta Appl Math

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We consider a mixed variational problem in real Hilbert spaces, defined on on the unbounded interval of time [0, +∞) and governed by a history-dependent operator. We state the unique solvability of the problem, which follows from a general existence and uniqueness result obtained in [26]. Then, we state and prove a general convergence result. The proof is based on arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Finally, we consider a general optimization problem for which we prove the existence of minimizers. The mathematical tools developed in this paper are useful in the analysis of a large class of nonlinear boundary value problems which, in a weak formulation, lead to history-dependent mixed variational problems. To provide an example, we illustrate our abstract results in the study of a frictional contact problem for viscoelastic materials with long memory.

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Weak solutions for contact problems involving viscoelastic materials with long memory

Cited by 12 publications

References 21 publications

On the relationship between alternative variational formulations of a frictional contact model

On the relationship between alternative variational formulations of a frictional contact model

An evolutionary mixed variational problem arising from frictional contact mechanics

Convergence and Optimization Results for a History-Dependent Variational Problem

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