Analysis of a dynamic viscoelastic unilateral contact problem with normal damped response

2016 ◽  
Vol 28 ◽  
pp. 229-250 ◽  
Author(s):  
Jiangfeng Han ◽  
Stanisław Migórski ◽  
Huidan Zeng
Keyword(s):  
Contact Problem ◽  
Unilateral Contact ◽  
Normal Damped Response

Tykhonov well-posedness of a frictionless unilateral contact problem

2019 ◽  
Vol 25 (6) ◽  
pp. 1294-1311 ◽  
Author(s):  
Zhenhai Liu ◽  
Mircea Sofonea ◽  
Yi-bin Xiao
Keyword(s):  
Contact Problem ◽  
Variational Formulation ◽  
Unilateral Contact ◽  
Well Posedness ◽  
Unilateral Constraints ◽  
The Family ◽  
Contact Models ◽  
Mechanical Interpretation ◽  
Well Posed ◽  
Natural Way

We consider a frictionless contact problem, Problem [Formula: see text], for elastic materials. The process is assumed to be static and the contact is modelled with unilateral constraints. We list the assumptions on the data and derive a variational formulation of the problem, Problem [Formula: see text]. Then we consider a perturbation of Problem [Formula: see text], which could be frictional, governed by a small parameter [Formula: see text]. This perturbation leads in a natural way to a family of sets [Formula: see text]. We prove that Problem [Formula: see text] is well-posed in the sense of Tykhonov with respect to the family [Formula: see text]. The proof is based on arguments of monotonicity, pseudomonotonicity and various estimates. We extend these results to a time-dependent version of Problem [Formula: see text]. Finally, we provide examples and mechanical interpretation of our well-posedness results, which, in particular, allow us to establish the link between the weak solutions of different contact models.

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 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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