Existence of variational solutions to the problem of contact of a beam with a DNC obstacle

This work establishes the existence of weak solutions to a mathematical model for the vibrations of an Euler-Bernoulli beam that may come in contact with a reactive obstacle situated below it. The obstacle reaction is described by the Damped Normal Compliance (DNC) contact condition, which extends both the normal compliance and the damped normal response contact conditions. The model consists of the Euler-Bernoulli beam, clamped at the left end and free at the right and, with the added DNC contact resistance. The classical and variational formulations of the model are presented. A Galerkin method is applied to the version of the beam with added viscosity. Then, the necessary estimates derived, and the existence of the unique solution to the problem with viscosity proved. Then, passing to the limit of vanishing viscosity, a solution to the problem without viscosity is obtained and its uniqueness is established. This work is an addition to the currently expanding Mathematical Theory of Contact Mechanics (MTCM).