We consider a mathematical model which describes the quasistatic
contact between two deformable bodies. The bodies are assumed to
have a viscoelastic behavior that we model with Kelvin-Voigt
constitutive law. The contact is frictionless and is modeled with
the classical Signorini condition with zero-gap
function. We derive a variational formulation of the problem and
prove the existence of a unique weak solution to the model by
using arguments of evolution equations with maximal monotone
operators. We also prove that the solution converges to the
solution of the corresponding elastic problem, as the viscosity
tensors converge to zero. We then consider a fully discrete
approximation of the problem, based on the augmented Lagrangian
approach, and present numerical results of two-dimensional test
problems.