History-dependent quasi-variational inequalities arising in contact mechanics

2011 ◽  
Vol 22 (5) ◽  
pp. 471-491 ◽  
Author(s):  
MIRCEA SOFONEA ◽  
ANDALUZIA MATEI
Keyword(s):  
Variational Inequalities ◽  
Frictional Contact ◽  
Deformable Body ◽  
Regularity Result ◽  
Contact Problems ◽  
Uniqueness Result ◽  
Constitutive Law ◽  
Signorini Condition ◽  
Normal Damped Response ◽  
Weak Solvability

We consider a class of quasi-variational inequalities arising in a large number of mathematical models, which describe quasi-static processes of contact between a deformable body and an obstacle, the so-called foundation. The novelty lies in the special structure of these inequalities that involve a history-dependent term as well as in the fact that the inequalities are formulated on the unbounded interval of time [0, +∞). We prove an existence and uniqueness result of the solution, then we complete it with a regularity result. The proofs are based on arguments of monotonicity and convexity, combined with a fixed point result obtained in [22]. We also describe a number of quasi-static frictional contact problems in which we model the material's behaviour with an elastic or viscoelastic constitutive law. The contact is modelled with normal compliance, with normal damped response or with the Signorini condition, as well, associated to versions of Coulomb's law of dry friction or to the frictionless condition. We prove that all these models cast in the abstract setting of history-dependent quasi-variational inequalities, with a convenient choice of spaces and operators. Then, we apply the abstract results in order to prove the unique weak solvability of each contact problem.

A CLASS OF EVOLUTIONARY VARIATIONAL INEQUALITIES WITH APPLICATIONS IN VISCOELASTICITY

2005 ◽  
Vol 15 (10) ◽  
pp. 1595-1617 ◽  
Author(s):  
M. BARBOTEU ◽  
J. R. FERNÁNDEZ ◽  
T.-V. HOARAU-MANTEL
Keyword(s):  
Variational Inequalities ◽  
Error Estimates ◽  
Contact Problems ◽  
Approximate Solutions ◽  
Uniqueness Result ◽  
Viscoelastic Materials ◽  
Evolutionary Equations ◽  
Fully Discrete ◽  
Evolutionary Variational Inequalities ◽  
Weak Solvability

We consider a class of abstract evolutionary variational inequalities arising in the study of contact problems for viscoelastic materials. We prove an existence and uniqueness result, using arguments of nonlinear evolutionary equations and Banach's fixed point theorem. We then consider the numerical approximation of the problem. By introducing a fully discrete scheme we show the existence of a unique solution and derive error estimates on the approximate solutions, which represent the main results of the work. We apply the abstract results in the study of two contact problems involving viscoelastic materials. The first one is a Signorini frictionless problem for Kelvin–Voigt materials and the second one is a frictional contact problem for linear Maxwell materials. We provide the variational and numerical analysis of the first problem including error estimates and numerical simulations and we prove the unique weak solvability of the second problem.

INTEGRODIFFERENTIAL HEMIVARIATIONAL INEQUALITIES WITH APPLICATIONS TO VISCOELASTIC FRICTIONAL CONTACT

2008 ◽  
Vol 18 (02) ◽  
pp. 271-290 ◽  
Author(s):  
STANISŁAW MIGÓRSKI ◽  
ANNA OCHAL ◽  
MIRCEA SOFONEA
Keyword(s):  
Frictional Contact ◽  
Deformable Body ◽  
Monotone Operators ◽  
Uniqueness Result ◽  
Second Order ◽  
Material Behavior ◽  
Constitutive Law ◽  
Memory Term ◽  
Hemivariational Inequalities ◽  
Banach Fixed Point Theorem

We consider a class of abstract second-order evolutionary inclusions involving a Volterra-type integral term, for which we prove an existence and uniqueness result. The proof is based on arguments of evolutionary inclusions with monotone operators and the Banach fixed point theorem. Next, we apply this result to prove the solvability of a class of second-order integrodifferential hemivariational inequalities and, under an additional assumption, its unique solvability. Then we consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is dynamic, the material behavior is described with a viscoelastic constitutive law involving a long memory term and the contact is modelled with subdifferential boundary conditions. We derive the variational formulation of the problem which is of the form of an integrodifferential hemivariational inequality for the displacement field. Then we use our abstract results to prove the existence of a unique weak solution to the frictional contact model.

Penalization of history-dependent variational inequalities

2013 ◽  
Vol 25 (2) ◽  
pp. 155-176 ◽  
Author(s):  
M. SOFONEA ◽  
F. PĂTRULESCU
Keyword(s):  
Variational Inequalities ◽  
Viscoelastic Body ◽  
Regularity Result ◽  
Uniqueness Result ◽  
Constitutive Law ◽  
Quasivariational Inequalities ◽  
Penalization Method ◽  
New Class ◽  
Mechanical Interpretation ◽  
Appl Math

The present paper represents a continuation of Sofonea and Matei's paper (Sofonea, M. and Matei, A. (2011) History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22, 471–491). There a new class of variational inequalities involving history-dependent operators was considered, an abstract existence and uniqueness result was proved and it was completed with a regularity result. Moreover, these results were used in the analysis of various frictional and frictionless models of contact. In this current paper we present a penalization method in the study of such inequalities. We start with an example which motivates our study; it concerns a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation; the material's behaviour is modelled with a constitutive law with long memory, the contact is frictionless and is modelled with a multivalued normal compliance condition and unilateral constraint. Then we introduce the abstract variational inequalities together with their penalizations. We prove the unique solvability of the penalized problems and the convergence of their solutions to the solution of the original problem, as the penalization parameter converges to zero. Finally, we turn back to our contact model, apply our abstract results in the study of this problem and provide their mechanical interpretation.

Numerical analysis of history-dependent variational–hemivariational inequalities with applications to contact problems

2015 ◽  
Vol 26 (4) ◽  
pp. 427-452 ◽  
Author(s):  
MIRCEA SOFONEA ◽  
WEIMIN HAN ◽  
STANISŁAW MIGÓRSKI
Keyword(s):  
Numerical Analysis ◽  
Real World ◽  
Viscoelastic Body ◽  
Contact Problems ◽  
Uniqueness Result ◽  
Constitutive Law ◽  
Hemivariational Inequalities ◽  
New Class ◽  
Nonlinear Anal ◽  
Dependence Result

A new class of history-dependent variational–hemivariational inequalities was recently studied in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618). There, an existence and uniqueness result was proved and used in the study of a mathematical model which describes the contact between a viscoelastic body and an obstacle. The aim of this paper is to continue the analysis of the inequalities introduced in Migórski et al. (2015Nonlinear Anal. Ser. B: Real World Appl.22, 604–618) and to provide their numerical analysis. We start with a continuous dependence result. Then we introduce numerical schemes to solve the inequalities and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modelled with a viscoelastic constitutive law, the contact is given in the form of normal compliance, and friction is described with a total slip-dependent version of Coulomb's law.

Weak solvability of a piezoelectric contact problem

2009 ◽  
Vol 20 (2) ◽  
pp. 145-167 ◽  
Author(s):  
STANISŁAW MIGÓRSKI ◽  
ANNA OCHAL ◽  
MIRCEA SOFONEA
Keyword(s):  
Mathematical Model ◽  
Variational Formulation ◽  
Frictional Contact ◽  
Constitutive Law ◽  
Potential Fields ◽  
Material Behaviour ◽  
Abstract Result ◽  
Electrically Conductive ◽  
Piezoelectric Body ◽  
Weak Solvability

We consider a mathematical model which describes the frictional contact between a piezoelectric body and a foundation. The material behaviour is modelled with a non-linear electro-elastic constitutive law, the contact is bilateral, the process is static and the foundation is assumed to be electrically conductive. Both the friction law and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system of two coupled hemi-variational inequalities for the displacement and the electric potential fields, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proof is based on an abstract result on operator inclusions in Banach spaces.

A CLASS OF EVOLUTIONARY VARIATIONAL INEQUALITIES WITH VOLTERRA-TYPE TERM

2004 ◽  
Vol 14 (04) ◽  
pp. 557-577 ◽  
Author(s):  
A. RODRÍGUEZ-ARÓS ◽  
J. M. VIAÑO ◽  
M. SOFONEA
Keyword(s):  
Variational Inequalities ◽  
Numerical Approximation ◽  
Contact Problems ◽  
Uniqueness Result ◽  
Long Term Memory ◽  
Term Memory ◽  
Frictionless Contact ◽  
Fully Discrete ◽  
Evolutionary Variational Inequalities

We consider a class of abstract evolutionary variational inequalities arising in the study of frictionless contact problems for linear viscoelastic materials with long-term memory. We prove an existence and uniqueness result, by using arguments of time-dependent elliptic variational inequalities and Banach's fixed point theorem. We then consider numerical approximation of the problem by introducing spatially semi-discrete, time semi-discrete and fully discrete schemes. For both schemes, we show the existence of a unique solution and derive error estimates. Finally, we apply the abstract results to the analysis and numerical approximation of the Signorini frictionless contact problem between two viscoelastic bodies with long-term memory.

Sign in / Sign up

Export Citation Format

Share Document