# Cusp-shaped Elastic Creases and Furrows

## TL;DR

This paper investigates the formation and morphology of cusp-shaped creases and furrows on soft surfaces, revealing universal scaling laws and providing a theoretical framework for understanding their self-folding bifurcation.

## Contribution

It introduces a controlled experimental setup to study surface folding, demonstrating universal cusp-shaped profiles and developing a similarity theory for their formation and bifurcation.

## Key findings

- Furrows and creases exhibit a universal cusp shape with width scaling as y^{3/2}.
- A critical deformation triggers bifurcation from furrow to crease.
- Theoretical models accurately predict fold length and profile singularities.

## Abstract

The surfaces of growing biological tissues, swelling gels, and compressed rubbers do not remain smooth, but frequently exhibit highly localized inward folds. We reveal the morphology of this surface folding in a novel experimental setup, which permits to deform the surface of a soft gel in a controlled fashion. The interface first forms a sharp furrow, whose tip size decreases rapidly with deformation. Above a critical deformation, the furrow bifurcates to an inward folded crease of vanishing tip size. We show experimentally and numerically that both creases and furrows exhibit a universal cusp-shape, whose width scales like $y^{3/2}$ at a distance $y$ from the tip. We provide a similarity theory that captures the singular profiles before and after the self-folding bifurcation, and derive the length of the fold from large deformation elasticity.