Contact problems for nonlinearly elastic materials: weak solvability involving dual Lagrange multipliers

Matei, Andaluzia; Ciurcea, Raluca

doi:10.21914/anziamj.v52i0.2212

Cited by 8 publications

(17 citation statements)

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“…Using definition (5.20) inequalities (2.4), (2.6) and assumptions (3.2), (3.20) we deduce that operator A verifies (5.11) with constants m A = m ε and M A = d E Q ∞ + c 2 0 L ν , respectively. Next, as it was shown in [16], definition (4.6) implies that the bilinear form b(·, ·) satisfies (5.13), i.e. there exist M b > 0 and α b > 0 such that…”

Section: Proof Of Theorem 41mentioning

confidence: 88%

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A mixed variational formulation of a contact problem with adhesion

Pătrulescu

2016

Applicable Analysis

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A mathematical model describing the contact between a viscoplastic body and a deformable foundation is analyzed under small deformation hypotheses. The process is quasistatic and in normal direction the contact is with adhesion, normal compliance, memory effects and unilateral constraint. We derive a mixed-variational formulation of the problem using Lagrange multipliers. Finally, we prove the unique weak solvability of the contact problem. ARTICLE HISTORY

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Section: Proof Of Theorem 41mentioning

confidence: 88%

“…We recall that the use of Lagrange multipliers represents a mathematical tool to remove the unilateral constraints. Concerning the literature in the field, see for instance [12][13][14][15] and recent papers [16,17].…”

Section: Introductionmentioning

confidence: 99%

A mixed variational formulation of a contact problem with adhesion

Pătrulescu

2016

Applicable Analysis

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“…(24) show that the operator R satisfies condition (4) and, obviously, the functional ' verifies (5). Next, as showed, e.g., in [9], the bilinear form b. ; / is continuous and satisfies the "inf-sup" condition. We conclude from here that condition (6) holds.…”

Section: A Viscoelastic Contact Modelmentioning

confidence: 90%

“…Existence and uniqueness results in the study of stationary mixed variational problems with Lagrange multipliers, together with various applications in Solid Mechanics, can be found in [3-5, 8, 10] and the references therein. Reference concerning the analysis of mixed variational problems associated with contact problems include [6,7,9], for instance.…”

Section: Introductionmentioning

confidence: 99%

A Class of Mixed Variational Problems with Applications in Contact Mechanics

Sofonea

2014

Springer Proceedings in Mathematics &Amp; Statistics

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We provide an existence result in the study of a new class of mixed variational problems. The problems are formulated on unbounded interval of time and involve history-dependent operators. The proof is based on generalized saddle point theory and various estimates, combined with fixed point arguments. Then, we consider a new mathematical model which describes the frictionless contact between a viscoelastic body and an obstacle. The process is quasistatic and the contact is modelled with a version of the normal compliance condition with unilateral constraint, which describes both the hardness and the softness of the foundation. We list the assumption on the data, derive a variational formulation of the problem, then we use our abstract result to prove its weak solvability.

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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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Contact problems for nonlinearly elastic materials: weak solvability involving dual Lagrange multipliers

Cited by 8 publications

References 0 publications

A mixed variational formulation of a contact problem with adhesion

A mixed variational formulation of a contact problem with adhesion

A Class of Mixed Variational Problems with Applications in Contact Mechanics

Convergence and Optimization Results for a History-Dependent Variational Problem

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