scholarly journals A HISTORY-DEPENDENT FRICTIONAL CONTACT PROBLEM WITH WEAR FOR THERMOVISCOELASTIC MATERIALS

2019 ◽  
Vol 24 (3) ◽  
pp. 351-371
Author(s):  
Lamia Chouchane ◽  
Lynda Selmani
Keyword(s):  
Contact Problem ◽  
Thermal Effects ◽  
Variational Formulation ◽  
Frictional Contact ◽  
Viscoelastic Body ◽  
Perturbation Parameter ◽  
Convergence Result ◽  
Uniqueness Result ◽  
Quasivariational Inequalities ◽  
Wear And Friction

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.

Analysis of a sliding frictional contact problem with unilateral constraint

2016 ◽  
Vol 22 (3) ◽  
pp. 324-342 ◽  
Author(s):  
Mircea Sofonea ◽  
Yahyeh Souleiman
Keyword(s):  
Contact Problem ◽  
Dry Friction ◽  
Variational Formulation ◽  
Frictional Contact ◽  
Convergence Result ◽  
Uniqueness Result ◽  
Unilateral Constraint ◽  
Penalization Method ◽  
Result Theorem ◽  
Equivalence Result

We consider a mathematical model that describes the equilibrium of an elastic body in frictional contact with a moving foundation. The contact is modeled with a multivalued normal compliance condition with unilateral constraint, associated with a sliding version of Coulomb’s law of dry friction. We present a description of the model, list the assumptions on the data and derive its primal variational formulation, in terms of displacement. Then we prove an existence and uniqueness result, Theorem 3.1. We proceed with a penalization method in the study of the contact problem for which we present a convergence result, Theorem 4.1. Finally, under additional hypotheses, we consider a variational formulation of the problem in terms of the stress, the so-called dual variational formulation, and prove an equivalence result, Theorem 5.3. The proofs of the theorems are based on arguments of monotonicity, compactness, convexity and lower semicontinuity.

scholarly journals Numerical Analysis of a Sliding frictional contact problem with Normal Compliance and Unilateral Contact

2020 ◽  
Author(s):  
Yahyeh Souleiman ◽  
Mikael Barboteu
Keyword(s):  
Numerical Analysis ◽  
Contact Problem ◽  
Frictional Contact ◽  
Nonlinear Partial Differential Equations ◽  
Viscoelastic Body ◽  
Sliding Contact ◽  
Memory Term ◽  
Optimal Order ◽  
Normal Compliance ◽  
Order Error

Abstract This paper represents a continuation of [15] and [18]. Here, we consider the numerical analysis of a non trivial frictional contact problen in a form of a system of evolution nonlinear partial differential equations. The model describes the equilibrium of a viscoelastic body in sliding contact with a moving foundation. The contact is modeled with a multivalued normal compliance condition with memory term restricted by a unilateral constraint, and is associated to a sliding version of Coulomb's law of dry friction. After a description of the model and some assumptions, we derive a variational formulation of the problem, which consists of a system coupling a variational inequality for the displacement field and a nonlinear equation for the stress field. Then, we introduce a fully discrete scheme for the numerical approximation of the sliding contact problem. Under certain solution regularity assumptions, we derive an optimal order error estimate and we provide numerical validation of this result by considering some numerical simulations in the study of a two-dimensional problem.

scholarly journals A VISCOELASTIC CONTACT PROBLEM WITH ADHESION AND SURFACE MEMORY EFFECTS

2014 ◽  
Vol 19 (5) ◽  
pp. 607-626 ◽  
Author(s):  
Mircea Sofonea ◽  
Flavius Patrulescu
Keyword(s):  
Long Memory ◽  
Existence And Uniqueness ◽  
Variational Formulation ◽  
Viscoelastic Body ◽  
Uniqueness Result ◽  
Unilateral Constraint ◽  
Memory Effects ◽  
Constitutive Law ◽  
Classical Formulation ◽  
Viscoelastic Contact

We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and an obstacle, the so-called foundation. The material's behavior is modelled with a constitutive law with long memory. The contact is with normal compliance, unilateral constraint, memory effects and adhesion. We present the classical formulation of the problem, then we derive its variational formulation and prove an existence and uniqueness result. The proof is based on arguments of variational inequalities and fixed point.

scholarly journals Variational and Numerical Analysis for Frictional Contact Problem with Normal Compliance in Thermo-Electro-Viscoelasticity

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Mustapha Bouallala ◽  
El Hassan Essoufi ◽  
Mohamed Alaoui
Keyword(s):  
Contact Problem ◽  
Variational Formulation ◽  
Frictional Contact ◽  
Approximate Solutions ◽  
Numerical Approach ◽  
Normal Compliance ◽  
Fully Discrete ◽  
Tresca’S Friction ◽  
Discrete Finite Element ◽  
Finite Element Schemes

In this paper, we consider a mathematical model of a contact problem in thermo-electro-viscoelasticity with the normal compliance conditions and Tresca’s friction law. We present a variational formulation of the problem, and we prove the existence and uniqueness of the weak solution. We also study the numerical approach using spatially semidiscrete and fully discrete finite element schemes with Euler’s backward scheme. Finally, we derive error estimates on the approximate solutions.

scholarly journals OPTIMIZATION PROBLEMS FOR A THERMOELASTIC FRICTIONAL CONTACT PROBLEM

2021 ◽  
Vol 26 (3) ◽  
pp. 444-468
Author(s):  
Othmane Baiz ◽  
Hicham Benaissa ◽  
Rachid Bouchantouf ◽  
Driss El Moutawakil
Keyword(s):  
Contact Problem ◽  
Frictional Contact ◽  
Optimization Problems ◽  
Variational Equation ◽  
Coupled System ◽  
Convergence Result ◽  
Elliptic Type ◽  
Frictional Contact Problem ◽  
Quasi Variational Inequality ◽  
Thermoelastic Body

In the present paper, we analyze and study the control of a static thermoelastic contact problem. We consider a model which describes a frictional contact problem between a thermoelastic body and a deformable heat conductor obstacle. We derive a variational formulation of the model which is in the form of a coupled system of the quasi-variational inequality of elliptic type for the displacement and the nonlinear variational equation for the temperature. Then, under a smallness assumption, we prove the existence of a unique weak solution to the problem. Moreover, we establish the dependence of the solution with respect to the data and prove a convergence result. Finally, we introduce an optimization problem related to the contact model for which we prove the existence of a minimizer and provide a convergence result.

A PIEZOELECTRIC CONTACT PROBLEM WITH SLIP DEPENDENT COEFFICIENT OF FRICTION

2004 ◽  
Vol 9 (3) ◽  
pp. 229-242 ◽  
Author(s):  
M. Sofonea
Keyword(s):  
Mathematical Model ◽  
Weak Solution ◽  
Contact Problem ◽  
Coefficient Of Friction ◽  
Dry Friction ◽  
Variational Formulation ◽  
Frictional Contact ◽  
Coupled System ◽  
Piezoelectric Body ◽  
The Coefficient Of Friction

We consider a mathematical model which describes the static frictional contact between a piezoelectric body and an obstacle. The constitutive relation of the material is assumed to be electroelastic and involves a nonlinear elasticity operator. The contact is modelled with a version of Coulomb's law of dry friction in which the coefficient of friction depends on the slip. We derive a variational formulation for the model which is in form of a coupled system involving as unknowns the displacement field and the electric potential. Then we provide the existence of a weak solution to the model and, under a smallness assumption, we provide its uniqueness. The proof is based on a result obtained in [14] in the study of elliptic quasi‐variational inequalities.

Convergence results for primal and dual history-dependent quasivariational inequalities

2018 ◽  
Vol 149 (2) ◽  
pp. 471-494
Author(s):  
Mircea Sofonea ◽  
Ahlem Benraouda
Keyword(s):  
Fixed Point ◽  
Variational Formulation ◽  
Continuous Dependence ◽  
Viscoelastic Body ◽  
Rigid Foundation ◽  
Quasivariational Inequalities ◽  
Class A ◽  
Convergence Results ◽  
Mechanical Interpretation ◽  
Primal And Dual

AbstractWe consider a class of history-dependent quasivariational inequalities for which we prove the continuous dependence of the solution with respect to the set of constraints. Then, under additional assumptions, we associate with each inequality in the class a new inequality, the so-called dual variational inequality, for which we state and prove existence, uniqueness, equivalence and convergence results. The proofs are based on various estimates, monotonicity and fixed-point arguments for history-dependent operators. Our abstract results are useful in the study of various mathematical models of contact. To provide an example, we consider a boundary value problem which describes the equilibrium of a viscoelastic body in contact with an elastic-rigid foundation. We list the assumptions on the data and derive both the primal and the dual variational formulation of the problem. Then, we state and prove existence, uniqueness and convergence results. We also provide the link between the two formulations, together with their mechanical interpretation.

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