K.L. Kuttler

SET-VALUED PSEUDOMONOTONE MAPS AND DEGENERATE EVOLUTION INCLUSIONS

We develop the theory of evolution inclusions for set-valued pseudomonotone maps. The problems we investigate are where B=B(t) is a linear operator that may vanish and A is a set-valued pseudomonotone operator. We prove the existence of unique solutions of such, possibly degenerate, problems. We apply the theory to the problem of dynamic frictional contact with a slip dependent friction coefficient and prove the existence of its unique weak solution. This theory opens the way for the investigation of sophisticated dynamical models in mechanics and frictional contact problems.

Elastic beam in adhesive contact

Abstract Dynamic and quasistatic processes of contact with adhesion between an elastic or viscoelastic beam and a foundation are considered. The contact is modeled with the Signorini condition when the foundation is rigid, and with normal compliance when it is deformable. The adhesion is modeled by introducing the bonding function β, the evolution of which is described by an ordinary differential equation. The existence and uniqueness of the weak solution for each of the problems is established using the theory of variational inequalities, fixed point arguments and the existence and uniqueness result in Commun. Contemp. Math. 1(1) (1999) 87–123. The numerical approximations of the quasistatic problem with normal compliance are considered, based on semi-discrete and fully discrete schemes. The convergence of the solutions of the discretized schemes is proved and error estimates for these approximate solutions are derived.

Dynamic contact with normal compliance wear and discontinuous friction coefficient

We apply the recent theory of evolution inclusions for set-valued pseudomonotone maps, developed in Kuttler and Shillor [Commun. Contemp. Math., 1 (1999), pp. 87--123] to the problem of dynamic frictional contact with normal compliance and wear. The friction coefficient is assumed to be slip rate dependent, and may be continuous, or discontinuous in the form of a graph with a vertical segment at the origin, representing the transition from the static to the dynamic value. The wear of the contacting surfaces is modeled by the Archard law. We prove the existence of a weak solution for the problem. We establish the uniqueness of the weak solution in the case when the friction coefficient is continuous. We also show that the problem with prescribed wear depends continuously on the wear.

Unilateral dynamic contact of two beams

Dynamic unilateral contact between two beams is investigated. The contact is modeled with the Signorini or normal compliance conditions. The model is in the form of a variational inequality for which existence theorems are established. A numerical algorithm for the problem is obtained by using the method of lines, in which the problem is approximated by a system of ordinary differential equations. Simulation results when one of the beams is driven periodically are presented. The characteristics of vibrations transmission across the joint between the beams are numerically investigated.

Quasilinear evolution equations in nonclassical diffusion

After describing the motivation leading to some nonclassical diffusion equations, we formulate a general abstract nonlinear evolution equation and establish existence of solutions. Then we return to the original equation and discuss particular initial-boundary value problems.

Dynamic Adhesive Contact of a Membrane

This work presents a dynamic model for adhesive contact between a stretched viscoelastic membrane and a reactive obstacle that lies beneath it. The adhesion is described by a bonding field, and the model allows for failure, that is complete debonding in finite time. It is two-dimensional, but retains the essential mathematical structure of the full three-dimensional model. It is set as a hyperbolic equation for the vibrations of the membrane coupled with a nonlinear ordinary differential equation for the evolution of the bonding field. Existence and uniqueness of regular solutions are established in the case of positive viscosity, and in the case of no viscosity, existence of weak solutions is obtained, while the uniqueness of the solutions remains unresolved.

Models and simulations of dynamic frictional contact of a beam

We investigate a mathematical model for the dynamic thermomechanical behavior of a viscoelastic beam that is in frictional contact with a rigid moving surface. Friction is modeled by a version of Coulombs law with slip dependent coefficient of friction, taking into account the frictional heat generation. We prove the existence and uniqueness of the weak solution, describe an algorithm for the numerical solutions and present results of numerical simulations, including the frequency distribution of the noise generated by the stick/slip motion. We also show that when the surface moves too fast there are no steady solutions and therefore the system is thermally unstable.

Quasi-static thermoviscoelastic contact problem with slip dependent friction coefficient

We prove existence and uniqueness of the weak solution for a quasi-static thermoviscoelastic problem which describes bilateral frictional contact between a deformable body and a moving rigid foundation. Friction is modeled with slip rate dependent friction coefficient, and it may depend either on the current slip rate or on the accumulated slip over the contact history. The frictional heat generated in the process is taken into account. The proof is based on the existence of solutions for a regularized problem, a priori estimates and a fixed-point argument, which provides the solution when the friction coefficient is sufficiently small.

Analysis and simulations of vibrations of a beam with a slider

A model for vibrations of a beam with a slider is derived, analysed and numerically simulated. It describes a viscoelastic beam that is clamped at one end to a vibrating device, while the other end moves between two stops attached to a slider. The contact is described by the normal compliance or by the Signorini conditions. The existence of weak solutions is established using the theory of set-valued pseudomonotone operators. The model is discretized using fourth-order spatial discretization, the solutions are numerically simulated and their results presented. The dynamics of the vibrations are depicted and so are the noise characteristics of the system.

The Galerkin method and degenerate evolution equations

Abstract Existence, uniqueness and regularity results are obtained for an abstract equation of the form By ′ + Ay = f , where B is not 1-1 and may vanish and A is a nonlinear operator. The desired solution is obtained as a limit of solutions of finite dimensional ordinary differential equations. The method used here generalizes the usual Galerkin method by allowing B to be degenerate and by making weaker assumptions of coercivity than are customary in most applications of the Galerkin method to evolution equations.