On Numerical Analysis of Thermomechanical Contact Problems

2009 ◽  
Author(s):  
G. Zavarise ◽  
B.A. Schrefler ◽  
P. Wriggers
Keyword(s):  
Numerical Analysis ◽  
Contact Problems ◽  
Thermomechanical Contact

Numerical Analysis of Microscopically Elastic Contact Problems

1995 ◽  
pp. 305-312 ◽  
Author(s):  
G. Zavarise ◽  
B. A. Schrefler
Keyword(s):  
Numerical Analysis ◽  
Contact Problems ◽  
Elastic Contact

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.

Numerical analysis of hemivariational inequalities in contact mechanics

Acta Numerica ◽  
2019 ◽  
Vol 28 ◽  
pp. 175-286 ◽  
Author(s):  
Weimin Han ◽  
Mircea Sofonea
Keyword(s):  
Numerical Analysis ◽  
Mathematical Models ◽  
Contact Mechanics ◽  
Numerical Solutions ◽  
Deformable Body ◽  
Industrial Process ◽  
Contact Problems ◽  
Hemivariational Inequalities ◽  
Uniqueness Results ◽  
Simulation Results

Contact phenomena arise in a variety of industrial process and engineering applications. For this reason, contact mechanics has attracted substantial attention from research communities. Mathematical problems from contact mechanics have been studied extensively for over half a century. Effort was initially focused on variational inequality formulations, and in the past ten years considerable effort has been devoted to contact problems in the form of hemivariational inequalities. This article surveys recent development in studies of hemivariational inequalities arising in contact mechanics. We focus on contact problems with elastic and viscoelastic materials, in the framework of linearized strain theory, with a particular emphasis on their numerical analysis. We begin by introducing three representative mathematical models which describe the contact between a deformable body in contact with a foundation, in static, history-dependent and dynamic cases. In weak formulations, the models we consider lead to various forms of hemivariational inequalities in which the unknown is either the displacement or the velocity field. Based on these examples, we introduce and study three abstract hemivariational inequalities for which we present existence and uniqueness results, together with convergence analysis and error estimates for numerical solutions. The results on the abstract hemivariational inequalities are general and can be applied to the study of a variety of problems in contact mechanics; in particular, they are applied to the three representative mathematical models. We present numerical simulation results giving numerical evidence on the theoretically predicted optimal convergence order; we also provide mechanical interpretations of simulation results.

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