We study the contact interaction between rigid punches and viscoelastic foundations with thin coatings for the cases in which the punch and coating surfaces are conformal (mutually repeating). Such problems can arise, for example, when the punch immerses into a solidificating coating before its complete solidification; as a result, the surface takes the shape of the punch base. Examples of such coatings can be a layer of glue, concrete at its young age, many polymeric materials. We consider plane contact problems for inhomogeneous aging viscoelastic basements in the case of their conformal contact with rigid punches. We present the statements of the problems and derive their basic mixed integral equation. The solution of this equation is constructed by using the generalized projection method. We present numerical computations of model problems, including the problem in which the shape of the punch base is described by a rapidly oscillating function.
