Dynamic frictionless contact with adhesion (original) (raw)

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.

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.

An Elastic Contact Problem with Adhesion and Normal Compliance

Journal of Applied Analysis, 2006

We study a mathematical problem describing the frictionless adhesive contact between an elastic body and a foundation. The adhesion process is modelled by a surface variable, the bonding field, and the contact is modelled with a normal compliance condition; the tangential shear due to the bonding field is included; the elastic constitutive law is assumed to be nonlinear and the process is quasistatic. The problem is formulated as a nonlinear system in which the unknowns are the displacement, the stress and the bonding field. The existence of a unique weak solution for the problem is established by using arguments for differential equations followed by the construction of an appropriate contraction mapping.

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.

Existence and Regularity for Dynamic Viscoelastic Adhesive Contact with Damage

Applied Mathematics and Optimization, 2006

A model for the dynamic process of frictionless adhesive contact between a viscoelastic body and a reactive foundation, which takes into account the damage of the material resulting from tension or compression, is presented. Contact is described by the normal compliance condition. Material damage is modelled by the damage field, which measures the pointwise fractional decrease in the loadcarrying capacity of the material, and its evolution is described by a differential inclusion. The model allows for different damage rates caused by tension or compression. The adhesion is modelled by the bonding field, which measures the fraction of active bonds on the contact surface. The existence of the unique weak solution is established using the theory of set-valued pseudomonotone operators introduced by Kuttler and Shillor (1999). Additional regularity of the solution is obtained when the problem data is more regular and satisfies appropriate compatibility conditions.

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.

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.

Variational and numerical analysis of a dynamic frictionless contact problem with adhesion

Journal of Computational and Applied Mathematics, 2003

We study a dynamic frictionless contact problem between a viscoelastic body and an obstacle, the so-called foundation. The contact is subjected to an adhesion e ect, whose evolution is described by an ordinary di erential equation. For the variational formulation of the contact problem, we present and prove an existence and uniqueness result. A fully discrete scheme is introduced to solve the problem. Under certain solution regularity assumptions, we derive an optimal order error estimate. Some numerical examples are included to show the performance of the method.

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.

Analysis of a unilateral contact problem taking into account adhesion and friction

Journal of Differential Equations, 2012

In this paper, we investigate a contact problem between a viscoelastic body and a rigid foundation, when both the effects of the (irreversible) adhesion and of the friction are taken into account. We describe the adhesion phenomenon in terms of a damage surface parameter according to Frémond's theory, and we model unilateral contact by Signorini conditions, and friction by a nonlocal Coulomb law. All the constraints on the internal variables as well as the contact and the friction conditions are rendered by means of subdifferential operators, whence the highly nonlinear character of the resulting PDE system. Our main result states the existence of a global-in-time solution (to a suitable variational formulation) of the related Cauchy problem. It is proved by an approximation procedure combined with time discretization.