scholarly journals A New Class of History–Dependent Evolutionary Variational–Hemivariational Inequalities with Unilateral Constraints

2020 ◽  
Author(s):  
Stanisław Migórski ◽  
Biao Zeng
Keyword(s):  
Frictional Contact ◽  
Hemivariational Inequalities ◽  
Normal Velocity ◽  
Unilateral Constraints ◽  
Response Condition ◽  
New Class ◽  
Fixed Point Principle ◽  
Selected Functions ◽  
Normal Damped Response ◽  
Weak Solvability

Abstract In this paper we study a new abstract evolutionary variational–hemivariational inequality which involves unilateral constraints and history–dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected functions together with a fixed-point principle for history–dependent operators. Then, we apply the abstract result to show the unique weak solvability to a dynamic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity combined with the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.

Hemivariational inequalities modeling electro-elastic unilateral frictional contact problem

2017 ◽  
Vol 23 (3) ◽  
pp. 329-347 ◽  
Author(s):  
Piotr Gamorski ◽  
Stanisław Migórski
Keyword(s):  
Contact Problem ◽  
Frictional Contact ◽  
Coupled System ◽  
Contact Friction ◽  
Hemivariational Inequalities ◽  
Unilateral Constraints ◽  
Frictional Contact Problem ◽  
Uniqueness Of The Solution ◽  
Pseudomonotone Mappings ◽  
Weak Solvability

We study a class of abstract hemivariational inequalities in a reflexive Banach space. For this class, using the theory of multivalued pseudomonotone mappings and a fixed-point argument, we provide a result on the existence and uniqueness of the solution. Next, we investigate a static frictional contact problem with unilateral constraints between a piezoelastic body and a conductive foundation. The contact, friction and electrical conductivity condition on the contact surface are described with the Clarke generalized subgradient multivalued boundary relations. We derive the variational formulation of the contact problem which is a coupled system of two hemivariational inequalities. Finally, for such system we apply our abstract result and prove its unique weak solvability.

A dynamic elastic-visco-plastic unilateral contact problem with normal damped response and Coulomb friction

2010 ◽  
Vol 21 (3) ◽  
pp. 229-251 ◽  
Author(s):  
CHRISTOF ECK ◽  
JIŘÍ JARUŠEK ◽  
MIRCEA SOFONEA
Keyword(s):  
Contact Problem ◽  
Variational Equation ◽  
Auxiliary Problem ◽  
Regularity Of Solutions ◽  
Normal Velocity ◽  
Response Condition ◽  
The Coefficient Of Friction ◽  
The Galerkin Method ◽  
System Coupling ◽  
Normal Damped Response

We consider a dynamic frictional contact problem between an elastic-visco-plastic body and a foundation. The contact is modelled with a normal damped response condition of such a type that the normal velocity is restricted with unilateral constraint, associated with the Coulomb law in which the coefficient of friction may depend on the velocity. We derive a variational formulation of the problem which has the form of a system coupling an integro–differential equation for the stress field with an evolutionary variational inequality for the displacement field. This inequality is approximated by a variational equation using a smoothing of the friction and the penalty approximation of the unilateral condition. The existence of a weak solution to the variational equation is proved by the Galerkin method for an auxiliary problem with given viscoplastic part of the stress and a fixed point argument. The solvability of the original problem is proved by passing to the limit of the penalty parameter and the smoothing parameter. This convergence is based on a certain regularity of solutions which is verified with the use of a local rectification of the boundary and a translation method.

scholarly journals A new class of hyperbolic variational–hemivariational inequalities driven by non-linear evolution equations

2020 ◽  
Vol 32 (1) ◽  
pp. 59-88
Author(s):  
STANISŁAW MIGÓRSKI ◽  
WEIMIN HAN ◽  
SHENGDA ZENG
Keyword(s):  
Dynamical System ◽  
Evolution Equations ◽  
Frictional Contact ◽  
Maximal Monotone ◽  
Contact Boundary ◽  
Hemivariational Inequalities ◽  
Response Condition ◽  
Linear Evolution ◽  
Linear Evolution Equation ◽  
Non Linear

The aim of the paper is to introduce and investigate a dynamical system which consists of a variational–hemivariational inequality of hyperbolic type combined with a non-linear evolution equation. Such a dynamical system arises in studies of complicated contact problems in mechanics. Existence, uniqueness and regularity of a global solution to the system are established. The approach is based on a new semi-discrete approximation with an application of a surjectivity result for a pseudomonotone perturbation of a maximal monotone operator. A new dynamic viscoelastic frictional contact model with adhesion is studied as an application, in which the contact boundary condition is described by a generalised normal damped response condition with unilateral constraint and a multivalued frictional contact law.

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 QUASISTATIC FRICTIONAL CONTACT PROBLEM WITH NORMAL DAMPED RESPONSE FOR THERMO-ELECTRO-ELASTIC-VISCOPLASTIC BODIES

2021 ◽  
Vol 10 (12) ◽  
pp. 3549-3568
Author(s):  
A. Hamidat ◽  
A. Aissaoui
Keyword(s):  
Existence And Uniqueness ◽  
Variational Formulation ◽  
Contraction Mapping ◽  
Original Problem ◽  
Frictional Contact ◽  
Unique Fixed Point ◽  
Mathematical Problem ◽  
Response Condition ◽  
Electric Potential Field ◽  
Normal Damped Response

We consider a mathematical problem for quasistatic contact between a thermo-electro--elastic-viscoplastic body and an obstacle. The contact is modeled by a general normal damped response condition with friction law and heat exchange. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution. The proof is based on the formulation of four intermediate problems for the displacement field, the electric potential field and the temperature field, respectively. We prove the unique solvability of the intermediate problems, then we construct a contraction mapping whose unique fixed point is the solution of the original problem.

Variational-hemivariational inverse problem for electro-elastic unilateral frictional contact problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Othmane Baiz ◽  
Hicham Benaissa ◽  
Zakaria Faiz ◽  
Driss El Moutawakil
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Contact Problem ◽  
Existence And Uniqueness ◽  
Frictional Contact ◽  
Hemivariational Inequalities ◽  
Frictional Contact Problem ◽  
Uniqueness Of A Solution

AbstractIn the present paper, we study inverse problems for a class of nonlinear hemivariational inequalities. We prove the existence and uniqueness of a solution to inverse problems. Finally, we introduce an inverse problem for an electro-elastic frictional contact problem to illustrate our results.

PREDICATION FOR RELATIVE MOTION OF THE COLONOSCOPE IN COLONOSCOPY

2013 ◽  
Vol 13 (03) ◽  
pp. 1350023 ◽  
Author(s):  
WU BIN CHENG ◽  
MICHAEL A. J. MOSER ◽  
SIVARUBAN KANAGARATNAM ◽  
WEN JUN ZHANG
Keyword(s):  
Frictional Force ◽  
Frictional Contact ◽  
Linear Complementarity ◽  
Intestinal Wall ◽  
Deformable Objects ◽  
Dynamic Friction ◽  
Unilateral Constraints ◽  
Common Procedure ◽  
Mechanical Compliance ◽  
Signorini's Problem

Colonoscopy is common procedure frequently carried out. It is not without its problems, which include looping formation. Looping formation prevents the tip of the colonoscope itself from advancing, thus further probing induces a risk of perforation, significant patient discomfort, and failure of colonoscopy. During colonoscopy, the manipulated colonoscope for intubation in the colon goes through the friction between the colonoscope and the colon. Due to major frictional force, the sigmoidal colon forms looping with the scope during intubation. The interactive frictional force between the colon and the colonoscope is highly complex because of frictional contact between two deformable objects. In this paper, contact force computation was formulated into a linear complementarity problem (LCP) by linearizing Signorini's problem, which was adapted into non-interpenetration with unilateral constraints. Frictional force was computed by the mechanical compliance of finite element method (FEM) models with the consideration of dynamic friction between the colonoscope and the intestinal wall. Furthermore, we presented a mathematical model of the elongation of the colon that predicts the motion of scope relative to the intestinal wall in colonoscopy.

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.

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