  Viscoelasticity and rate-dependent adhesion of soft matter lead to
difficulties in modeling the 'relatively simple' problem of a rigid sphere in
contact with a viscoelastic half-space. For this reason, approximations in
describing surface interactions and viscous dissipation processes are usually
adopted in the literature. Here, we develop a fully deterministic model in
which adhesive interactions are described by Lennard-Jones potential and the
material behaviour with the standard linear solid model. Normal
loading-unloading cycles are carried out under different driving conditions.
When loading is performed in quasi-static conditions and, hence, unloading
starts from a completely relaxed state of the material, the effective surface
energy is found to monotonically increase with the contact line velocity up to
an asymptotic value reached at high unloading rates. Such result agrees with
existing theories on viscoelastic crack propagation. If loading and unloading
are performed at the same non-zero driving velocity and, hence, unloading
starts from an unrelaxed state of the material, the trend of the effective
surface energy {\Delta}{\gamma}eff with the contact line velocity is described
by a bell-shaped function in a double-logarithmic plot. The peak of
{\Delta}{\gamma}eff is found at a contact line velocity smaller than that makes
maximum the tangent loss of the viscoelastic modulus. Furthermore, we show
Gent\&Schultz assumption partly works in this case as viscous dissipation is no
longer localized along the contact perimeter but it also occurs in the bulk
material.
