O. Chau

Variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion

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 field on the contact surface, and contact is described by a modified normal compliance condition. The problem is formulated as a coupled system of a variational equality for the displacements and a differential equation for the bonding field. 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 field.

A Dynamic Frictional Contact Problem with Normal Damped Response

We consider a mathematical model which describes the frictional contact between a viscoelastic body and a reactive foundation. The process is assumed to be dynamic and the contact is modeled with a general normal damped response condition and a local friction law. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution, using results on evolution equations with monotone operators and a fixed point argument. We then introduce and study a fully discrete numerical approximation scheme of the variational problem, in terms of the velocity variable. The numerical scheme has a unique solution. We derive error estimates under additional regularity assumptions on the data and the solution.

Quasistatic frictional problems for elastic and viscoelastic materials

We consider two quasistatic problems which describe the frictional contact between a deformable body and an obstacle, the so-called foundation. In the first problem the body is assumed to have a viscoelastic behavior, while in the other it is assumed to be elastic. The frictional contact is modeled by a general velocity dependent dissipation functional. We derive weak formulations for the models and prove existence and uniqueness results. The proofs are based on the theory of evolution variational inequalities and fixed-point arguments. We also prove that the solution of the viscoelastic problem converges to the solution of the corresponding elastic problem, as the viscosity tensor converges to zero. Finally, we describe a number of concrete contact and friction conditions to which our results apply.

A convergence result in elastic-viscoplastic contact problems with damage

This work deals with the approximation of the contact problem for a viscoplastic material with the Signorini contact conditions by the problem with normal compliance, when the surface deformability coefficient converges to zero, i.e., when the surface stiffness tends to infinity, which represents a perfectly rigid obstacle. The possible damage of the material caused by compression or tension is taken into account. The approximate problem is formulated as a variational inequality and its convergence to the Signorin problem is proved. Then, the fully discrete scheme for the two problems is described and its convergence established. Results of numerical simulations, based on these schemes, are presented in one and two dimensions which show the convergence.

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

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 effect, whose evolution is described by an ordinary differential 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.

Quasistatic Thermoviscoelastic Frictional Contact Problem with Damped Response

We analyze a problem which describes the frictional contact between a thermoviscoelastic body and a rigid foundation. The process is assumed to be quasistatic and the contact is modeled by a general normal damped response condition with friction law and heat exchange. Then we present a variational formulation of the problem, which is set in an abstract form as a system of evolution equations for the displacements and temperature. We establish the existence and uniqueness of the weak solution, using general results on evolution equations with monotone operators and fixed point arguments. Finally, we study the continuous dependence of the solution with respect to the initial data and contact conditions.

Numerical analysis of a contact problem for elastic-visco-plastic materials with damage

We study a quasistatic frictionless viscoplastic contact problem with normal compliance and damage for elastic-visco-plastic bodies. The mechanical damage of the material, caused by excessive stress or strain, is described by a damage function whose evolution is modelled by a parabolic inclusion. We provide a variational formulation for the mechanical problem that has a unique solution. We then study a fully discrete scheme for the numerical solution of the problem and derive error estimates for the approximate solutions. Finally, we present some numerical results.

Analysis of a quasistatic displacement-traction problem for viscoelastic materials

We consider an initial and boundary value problem which describes the evolution of a viscoelastic body submitted to body forces and surface tractions. The viscoelastic constitutive law is assumed to be nonlinear and the process is quasistatic. We prove the existence and the uniqueness of the solution using arguments of monotone operators theory and a version of Cauchy-Lypchitz theorem. We establish the continuous dependence of the solution on the elasticity operator. Finally, we study the behavior of the solution when the viscosity operator converges to zero.