  In this paper, a new framework for crossover of scaling law is proposed: a
crossover of scaling law can be described by a self-similar solution. A
crossover emerges as a result of the interference from similarity parameters of
the higher class of the self-similarity. This framework was verified for the
dynamical impact of solid sphere onto a viscoelastic board. All the physical
factors including the size of spheres and the impact of velocity are
successfully summarized using primal dimensionless numbers which construct a
self-similar solution of the second kind, which represents the balance between
dynamical elements involved in the problem. The self-similar solution gives two
different scaling laws by the perturbation method describing the crossover.
These theoretical predictions are compared with experimental results to show
good agreement. It was suggested that a hierarchical structure of similarity
plays a fundamental role on crossover, which offers a fundamental insight to
self-similarity in general.
