Time and rate effects may have a crucial role in many engineering applications. In this work, an innovative viscoelastic and viscoplastic framework is presented within the framework of the overstress theory. The peculiarity of this approach is the use of two elastic and two plastic, rate-dependent mechanisms, having very different viscosity rates. This implies the use of two Perzyna-type, yield surfaces and the need of splitting the total strain increments in four fractions. Model simulations, obtained with a multiaxial formulation, are presented and compared with laboratory test results.