A framework for crossover of scaling law as a self-similar solution : dynamical impact of viscoelastic board

TL;DR

This paper introduces a new framework using self-similar solutions to describe the crossover of scaling laws, validated through experiments on the impact of a sphere on a viscoelastic board, revealing hierarchical similarity structures.

Contribution

It proposes a novel self-similar solution framework for scaling law crossover, verified through experiments involving viscoelastic impact dynamics.

Findings

Successful summarization of physical factors using dimensionless numbers

Prediction of two different scaling laws for crossover

Good agreement between theoretical predictions and experimental results

Abstract

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.