Elasto-buoyant heavy spheres: a unique way to test non-linear elasticity

TL;DR

This study introduces a novel experimental approach using heavy spheres to probe large deformations in ultra-soft elastic materials, revealing a specific scaling law that challenges traditional models and advances understanding of non-linear elasticity.

Contribution

The paper presents the first systematic experimental and theoretical analysis of large deformations in ultra-soft elastic media using heavy spheres, establishing a new scaling law and a validated asymptotic model.

Findings

Penetration depth scales as δ ∼ a^{3/2}

Ideal neo-Hookean models are inconsistent with observed data

An asymptotic analytic model explains the experimental results

Abstract

Extra-large deformations in ultra-soft elastic materials are ubiquitous, yet systematic studies and methods to understand the mechanics of such huge strains are lacking. Here we investigate this complex problem systematically with a simple experiment: by introducing a heavy bead of radius $a$ in an incompressible ultra-soft elastic medium. We find a scaling law for the penetration depth ($\delta$) of the bead inside the softest gels as $\delta \sim a^{3/2}$. While this result is inconsistent with an ideal neo-Hookean model of elastic deformation, according to which the displacement fields must diverge, it is vindicated by an original asymptotic analytic model developed in this article. This model demonstrates that the observed relationship is precisely at the demarcating boundary of what would be required for the field variables to either diverge or converge. This correspondence between a unique mathematical prediction and the experimental observation ushers in new insights into the behavior of the deformations of strongly non-linear materials.