Modulus estimates and cavitation in higher dimensions

TL;DR

This paper investigates cavitation in higher-dimensional elasticity, establishing refined modulus bounds and introducing a new directional dilatation to better understand when cavitation occurs or is prevented.

Contribution

It introduces a novel directional dilatation combined with angular dilatation to improve modulus estimates for detecting cavitation in higher dimensions.

Findings

Refined modulus bounds for cavitation detection

Introduction of directional dilatation method

Examples illustrating cavitation occurrence and prevention

Abstract

We explore the phenomenon of cavitation in higher-dimensional elasticity, defining it as the mapping of a punctured ball onto a non-degenerate ring domain. Crucially, for the class of locally quasiconformal mappings (or more general mappings) defined on the punctured ball $0<|x|<1$ in $\mathbb R^n$ that we examine, cavitation is equivalent to a failure of continuous extension to the origin. While existing modulus estimates prove insufficient for reliably detecting cavitation in this setting, our study establishes refined modulus bounds. This is achieved by introducing a novel directional dilatation which, in conjunction with the known angular dilatation, overcomes the limitations of previous methods. We illustrate our theoretical findings with several examples that demonstrate both cavitation occurrence and its absence.