scholarly journals Variational analysis for some frictional contact problems

2019 ◽  
Vol 38 (7) ◽  
pp. 21-36
Author(s):  
Leila Ait Kaki ◽  
M. Denche
Keyword(s):  
Weak Solution ◽  
Variational Inequalities ◽  
Variational Analysis ◽  
Frictional Contact ◽  
Coupled System ◽  
Contact Problems ◽  
Variational Problems ◽  
Unique Weak Solution ◽  
Piezoelectric Body ◽  
Evolutionary Variational Inequalities

We consider a class of evolutionary variational problems which describes the static frictional contact between a piezoelectric body and a conductive obstacle. The formulation is in a form of coupled system involving the displacement and electric potentiel fieelds. We provide the existence of unique weak solution of the problems. The proof is based on the evolutionary variational inequalities and Banach's xed point theorem.

scholarly journals Evolutionary variational inequalities arising in quasistatic frictional contact problems for elastic materials

1999 ◽  
Vol 4 (4) ◽  
pp. 255-279 ◽  
Author(s):  
Dumitru Motreanu ◽  
Mircea Sofonea
Keyword(s):  
Weak Solution ◽  
Variational Inequalities ◽  
Continuous Dependence ◽  
Frictional Contact ◽  
Contact Problems ◽  
Elastic Materials ◽  
Friction Law ◽  
Lipschitz Continuous ◽  
Linear Elastic ◽  
Evolutionary Variational Inequalities

We consider a class of evolutionary variational inequalities arising in quasistatic frictional contact problems for linear elastic materials. We indicate sufficient conditions in order to have the existence, the uniqueness and the Lipschitz continuous dependence of the solution with respect to the data, respectively. The existence of the solution is obtained using a time-discretization method, compactness and lower semicontinuity arguments. In the study of the discrete problems we use a recent result obtained by the authors (2000). Further, we apply the abstract results in the study of a number of mechanical problems modeling the frictional contact between a deformable body and a foundation. The material is assumed to have linear elastic behavior and the processes are quasistatic. The first problem concerns a model with normal compliance and a version of Coulomb's law of dry friction, for which we prove the existence of a weak solution. We then consider a problem of bilateral contact with Tresca's friction law and a problem involving a simplified version of Coulomb's friction law. For these two problems we prove the existence, the uniqueness and the Lipschitz continuous dependence of the weak solution with respect to the data.

SET-VALUED PSEUDOMONOTONE MAPS AND DEGENERATE EVOLUTION INCLUSIONS

1999 ◽  
Vol 01 (01) ◽  
pp. 87-123 ◽  
Author(s):  
KENNETH L. KUTTLER ◽  
MEIR SHILLOR
Keyword(s):  
Weak Solution ◽  
Friction Coefficient ◽  
Frictional Contact ◽  
Contact Problems ◽  
Pseudomonotone Operator ◽  
Unique Weak Solution ◽  
Dynamical Models ◽  
Evolution Inclusions ◽  
Theory Of Evolution ◽  
Pseudomonotone Maps

We develop the theory of evolution inclusions for set-valued pseudomonotone maps. The problems we investigate are [Formula: see text] where B=B(t) is a linear operator that may vanish and A is a set-valued pseudomonotone operator. We prove the existence of unique solutions of such, possibly degenerate, problems.We apply the theory to the problem of dynamic frictional contact with a slip dependent friction coefficient and prove the existence of its unique weak solution.This theory opens the way for the investigation of sophisticated dynamical models in mechanics and frictional contact problems.

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.

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.

A New Finite Element Analysis of Frictional Contact Using Nondifferential Optimization

1996 ◽  
Author(s):  
M. H. Refaat ◽  
S. A. Meguid
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Variational Inequality ◽  
Mathematical Programming ◽  
Variational Inequalities ◽  
Ad Hoc ◽  
Frictional Contact ◽  
Contact Problems ◽  
Element Analysis ◽  
Regularization Techniques

Abstract Current solution schemes of variational inequalities arising in frictional contact problems adopt penalty and regularization techniques. The convergence and accuracy of these schemes are governed by user-defined parameters. To overcome the difficulties associated with the ad hoc use of such parameters, the variational inequality of the general frictional contact problem is treated in this paper using mathematical programming. A new non-differential optimization (NDO) technique, in association with quadratic programming, is used to treat the resulting variational inequalities.

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.

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