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 Numerical analysis and simulations of a frictional contact problem with damage and memory

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hailing Xuan ◽  
Xiaoliang Cheng
Keyword(s):  
Contact Problem ◽  
Frictional Contact ◽  
Numerical Solutions ◽  
Deformable Body ◽  
Mechanical Damage ◽  
Coupled System ◽  
Constitutive Law ◽  
Optimal Order ◽  
Order Error ◽  
Nonlinear Parabolic

<p style='text-indent:20px;'>In this paper, we study a frictional contact model which takes into account the damage and the memory. The deformable body consists of a viscoelastic material and the process is assumed to be quasistatic. The mechanical damage of the material which caused by the tension or the compression is included in the constitutive law and the damage function is modelled by a nonlinear parabolic inclusion. Then the variational formulation of the model is governed by a coupled system consisting of a history-dependent hemivariational inequality and a nonlinear parabolic variational inequality. We introduce and study a fully discrete scheme of the problem and derive error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived for the linear finite element method. Several numerical experiments for the contact problem are given for providing numerical evidence of the theoretical results.</p>

scholarly journals Variational and numerical analysis of the Signorini′s contact problem in viscoplasticity with damage

2003 ◽  
Vol 2003 (2) ◽  
pp. 87-114 ◽  
Author(s):  
J. R. Fernández ◽  
M. Sofonea
Keyword(s):  
Contact Problem ◽  
Numerical Solutions ◽  
Mechanical Damage ◽  
Parabolic Type ◽  
Approximate Solutions ◽  
Damage Function ◽  
Optimal Order ◽  
Order Error ◽  
Fully Discrete ◽  
Solution Regularity

We consider the quasistatic Signorini′s contact problem with damage for elastic-viscoplastic bodies. The mechanical damage of the material, caused by excessive stress or strain, is described by a damage function whose evolution is modeled by an inclusion of parabolic type. We provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution of the model. We then introduce and study a fully discrete scheme for the numerical solutions of the problem. An optimal order error estimate is derived for the approximate solutions under suitable solution regularity. Numerical examples are presented to show the performance of the method.

scholarly journals Analysis of a Dynamic Viscoelastic Contact Problem with Normal Compliance, Normal Damped Response, and Nonmonotone Slip Rate Dependent Friction

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Mikaël Barboteu ◽  
David Danan
Keyword(s):  
Contact Problem ◽  
Frictional Contact ◽  
Slip Rate ◽  
Approximation Schemes ◽  
Discrete Approximation ◽  
Normal Compliance ◽  
Fully Discrete ◽  
Rate Dependent ◽  
Fully Discrete Approximation ◽  
Normal Damped Response

We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with a combination of a normal compliance and a normal damped response law associated with a slip rate-dependent version of Coulomb’s law of dry friction. We derive a variational formulation and an existence and uniqueness result of the weak solution of the problem is presented. Next, we introduce a fully discrete approximation of the variational problem based on a finite element method and on an implicit time integration scheme. We study this fully discrete approximation schemes and bound the errors of the approximate solutions. Under regularity assumptions imposed on the exact solution, optimal order error estimates are derived for the fully discrete solution. Finally, after recalling the solution of the frictional contact problem, some numerical simulations are provided in order to illustrate both the behavior of the solution related to the frictional contact conditions and the theoretical error estimate result.

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