Pathways, Scaling Laws and Analytical Solutions for Crease Formations in a Gel Layer

TL;DR

This paper provides an analytical framework for understanding crease formations in swelling gel layers, revealing pathways, bifurcation types, and scaling laws, supported by experimental validation and invariance of critical quantities.

Contribution

It introduces a novel analytical approach using coupled series-asymptotic expansions to solve crease formation problems in gels, offering new insights into pathways and scaling laws.

Findings

Three pathways to crease depending on layer thickness

Analytical solutions for post-bifurcation branches

Critical quantities invariant with thickness at crease formation

Abstract

An analytical study on crease formations in a swelling gel layer is conducted. By exploring the smallness of the layer thickness and using a method of coupled series-asymptotic expansions, the original nonlinear eigenvalue problem of partial differential equations is reduced to one of ordinary differential equations. The latter problem is then solved analytically to obtain closed-form solutions for all the post-bifurcation branches. With the available analytical results, a number of deep insights on crease formations are provided, including the unveiling of three pathways to crease (depending on the layer thickness), determination of the bifurcation type, establishment of a lower bound for mode numbers and two scaling laws. Also, a number of experimental results are captured, which are then nicely interpreted based on the analytical solutions. In particular, it is shown that some critical physical quantities are invariant with respect to the thickness at the moment of crease formation. It appears that the present work offers a comprehensive understanding on crease formation, a widely-spread phenomenon.