We consider a mathematical model which describes the contact
between a deformable body and an obstacle, the so-called
foundation. The body is assumed to have a viscoelastic behavior
that we model with the Kelvin-Voigt constitutive law. The contact
is frictionless and is modeled with the well-known Signorini
condition in a form with a zero gap function. We present
two alternative yet equivalent weak formulations of the problem
and establish existence and uniqueness results for both
formulations. The proofs are based on a general result on
evolution equations with maximal monotone operators. We then
study a semi-discrete numerical scheme for the problem, in terms
of displacements. The numerical scheme has a unique solution. We
show the convergence of the scheme under the basic solution
regularity. Under appropriate regularity assumptions on the
solution, we also provide optimal order error estimates.