W. Han, K. L. Kuttler, M. Shillor and M. Sofonea,

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 $\beta$, 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 K. L. Kuttler and
M. Shillor (Commun. Contemp. Math. 1999). 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.