What is the Best Viscous Approximation to a Rate-Independent Process?

Viscous approximations of a rate-independent process with regulated inputs are considered with a general viscosity operator. It is shown that the limit as the viscosity coefficient tends to zero defines a continuous rate-independent input-output mapping with respect to the uniform topology in the space of regulated functions, the limits are, however, in general different for different viscosity operators. Examples show that if the viscosity operator is chosen independently of the energy potential, the limit jump trajectories may violate both the normality rule and the maximal dissipation principle.