Variational and numerical analysis of a dynamic frictionless contact problem with adhesion (original) (raw)
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Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion
Journal of Computational and Applied Mathematics, 2003
A model for the adhesive, quasistatic and frictionless contact between a viscoelastic body and a deformable foundation is described. The adhesion process is modelled by a bonding ÿeld on the contact surface, and contact is described by a modiÿed normal compliance condition. The problem is formulated as a coupled system of a variational equality for the displacements and a di erential equation for the bonding ÿeld. The existence of a unique weak solution for the problem and its continuous dependence on the adhesion parameters are established. Then, the numerical analysis of the problem is conducted for the fully discrete approximation. The convergence of the scheme is established and error estimates derived. Finally, representative numerical simulations are presented, depicting the evolution of the state of the system and, in particular, the evolution of the bonding ÿeld.
Analysis of a frictional contact problem with adhesion
We consider a mathematical model which describes the contact between a deformable body and an obstacle, the so-called foundation. The contact is frictional and is modelled with a version of normal compliance condition and the associated Coulomb's law of dry friction in which the adhesion of contact surfaces is taken into account. The evolution of the bonding field is described by a first order differential equation and the the material's behavior is modelled with a nonlinear elastic constitutive law. We derive a variational formulation of the problem then, under a smallness assumption on the coefficient of friction, we prove the existence of a unique weak solution for the model. The proof is based on arguments of timedependent variational inequalities, differential equations and Banach fixed point theorem. Finally, we extend our results in the case when the piezoelectric effect is taken into account, i.e. in the case when the material's behavior is modelled with a nonlinear electro-elastic constitutive law.
Variational Analysis of a Elastic-Viscoplastic Contact Problem with Friction and Adhesion
2009
The aim of this paper is to study the process of frictional contact with adhesion between a body and an obstacle. The material's behavior is assumed to be elastic-viscoplastic, the process is quasistatic, the contact is modeled by the Signorini condition and the friction is described by a non local Coulomb law coupled with adhesion. The adhesion process is modelled by a bonding field on the contact surface. We derive a variational formulation of the problem, then, under a smallness assumption on the coefficient of friction, we prove an existence and uniqueness result of a weak solution for the model. The proof is based on arguments of timedependent variational inequalities, differential equations and Banach fixed point theorem.
A viscoelastic frictionless contact problem with adhesion
Applicable Analysis, 2001
We consider two quasistatic frictionless contact problems for viscoelastic bodies with long memory. In the first problem the contact is modelled with Signorini's conditions and in the second one is modelled with normal compliance. In both problems the adhesion of the contact surfaces is taken into account and is modelled with a surface variable, the bonding field. We provide variational formulations for the mechanical problems and prove the existence of a unique weak solution to each model. The proofs are based on arguments of time-dependent variational inequalities, differential equations, and a fixed point theorem. Moreover, we prove that the solution of the Signorini contact problem can be obtained as the limit of the solutions of the contact problem with normal compliance as the stiffness coefficient of the foundation converges to infinity.
Analysis of a Bilateral Contact Problem with Adhesion and Friction for Elastic Materials
2010
We consider a mathematical model which describes a contact problem between a deformable body and a foundation. The contact is bilateral and is modelled with Tresca's friction law in which adhesion is taken into account. The evolution of the bonding field is discribed by a first order differential equation and the material's behavior is modelled with a nonlinear elastic constitutive law. We derive a variational formulation of the mechanical problem and prove the existence and uniqueness result of the weak solution. Moreover, we prove that the solution of the contact problem can be obtained as the limit of the solution of a regularized problem as the regularizaton parameter converges to 0. The proof is based on arguments of time-dependent variational inequalities, differential equations and Banach fixed point theorem.
Applicable Analysis, 2013
We consider a model for quasistatic frictional contact between a viscoelastic body and a foundation. The material constitutive relation is assumed to be nonlinear. The mechanical damage of the material, caused by excessive stress or strain, is described by the damage function, the evolution of which is determined by a parabolic inclusion. The contact is modeled with the normal compliance condition and the associated version of Coulomb's law of dry friction. We derive a variational formulation for the problem and prove the existence of its unique weak solution. We then study a fully discrete scheme for the numerical solutions of the problem and obtain error estimates on the approximate solutions.
A frictionless contact problem for viscoelastic materials
Journal of Applied Mathematics, 2002
We consider a mathematical model which describes the contact between a deformable body and an obstacle, the so-called foundation. The body is assumed to have a viscoelastic behavior that we model with the Kelvin-Voigt constitutive law. The contact is frictionless and is modeled with the well-known Signorini condition in a form with a zero gap function. We present two alternative yet equivalent weak formulations of the problem and establish existence and uniqueness results for both formulations. The proofs are based on a general result on evolution equations with maximal monotone operators. We then study a semi-discrete numerical scheme for the problem, in terms of displacements. The numerical scheme has a unique solution. We show the convergence of the scheme under the basic solution regularity. Under appropriate regularity assumptions on the solution, we also provide optimal order error estimates.
Variational Analysis of an Electro-Viscoelastic Contact Problem with Friction and Adhesion
Journal of the Korean Mathematical Society, 2016
We consider a mathematical model which describes the quasistatic frictional contact between a piezoelectric body and an electrically conductive obstacle, the so-called foundation. A nonlinear electroviscoelastic constitutive law is used to model the piezoelectric material. Contact is described with Signorini's conditions and a version of Coulomb's law of dry friction in which the adhesion of contact surfaces is taken into account. The evolution of the bonding field is described by a first order differential equation. We derive a variational formulation for the model, in the form of a system for the displacements, the electric potential and the adhesion. Under a smallness assumption which involves only the electrical data of the problem, we prove the existence of a unique weak solution of the model. The proof is based on arguments of time-dependent quasi-variational inequalities, differential equations and Banach's fixed point theorem.
Analysis and numerical simulations of a dynamic contact problem with adhesion
Mathematical and Computer Modelling, 2003
The dynamic process of frictionless contact between a viscoelastic body and a reactive foundation is modelled, analyzed, and simulated. The contact is adhesive and it is described by introducing an internal variable, the bonding field p, which messures the fractional density of active bonds. The evolution of fl is described by an ordinary differential equation that depends on the process history, taking into account possible adhesive degradation during cycles of debonding and rebonding.
Viscoelastic frictionless contact problems with adhesion
Journal of Inequalities and Applications, 2006
We consider two quasistatic frictionless contact problems for viscoelastic bodies with long memory. In the first problem the contact is modelled with Signorini's conditions and in the second one is modelled with normal compliance. In both problems the adhesion of the contact surfaces is taken into account and is modelled with a surface variable, the bonding field. We provide variational formulations for the mechanical problems and prove the existence of a unique weak solution to each model. The proofs are based on arguments of time-dependent variational inequalities, differential equations, and a fixed point theorem. Moreover, we prove that the solution of the Signorini contact problem can be obtained as the limit of the solutions of the contact problem with normal compliance as the stiffness coefficient of the foundation converges to infinity.