# Intermediate asymptotics on dynamical impact of solid sphere on mili-textured surface

## TL;DR

This paper investigates the scale-dependent impact dynamics of a solid sphere on an elastic surface, revealing how inertial and elastic forces compete across different scales through a theoretical model validated by experiments.

## Contribution

It introduces a new dimensionless framework capturing intermediate asymptotics in impact behavior, combining dimensional analysis with energy conservation.

## Key findings

- Theoretical power-law behaviors match experimental data.
- Two distinct asymptotic regimes identified based on scale.
- Dimensionless parameters effectively describe impact dynamics.

## Abstract

Complex phenomena incorporating several physical properties are abundant while they are occasionally revealing the variation of power-law behavior depending on the scale. In this present work, the global scaling-behavior of dynamical impact of solid sphere onto elastic surface is described. Its fundamental dimensionless function was successfully obtained by applying the dimensional analysis combined with the solution by energy conservation complementally. It demonstrates that its power-law behavior is given by the competition between two power-law relations representing inertial and elastic property respectively which is strengthened by scale size of sphere. These factors are successfully summarized by the newly defined dimensionless parameters which gives two intermediate asymptotics in different scale range. These power-law behaviors given by the theoretical model were compared with experimental results, showing good agreement. This study supplies the insights to dimensional analysis and self-similarity in general.