Elastic beam in adhesive contact

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.