Mechanics of invagination and folding: hybridized instabilities when one soft tissue grows on another

TL;DR

This paper models the folding behavior in soft layered solids due to differential growth, revealing complex morphologies and transitions that explain biological structures like intestinal villi and brain cortex folds.

Contribution

It introduces a minimal elastic model capturing the transition from smooth to cusped folds and diverse morphologies in layered tissues, unifying biological folding phenomena.

Findings

Identifies a transition from supercritical to subcritical folding regimes.

High amplitude fold spacing converges to about four layer thicknesses.

Diverse morphologies including zigzag and triple-junction patterns are explained.

Abstract

We address the folding induced by differential growth in soft layered solids via an elementary model that consists of a soft growing neo-Hookean elastic layer adhered to a deep elastic substrate. As the layer/substrate modulus ratio is varied from above unity towards zero we find a first transition from supercritical smooth folding followed by cusping of the valleys to direct subcritical cusped folding, then another to supercritical cusped folding. Beyond threshold the high amplitude fold spacing converges to about four layer thicknesses for many modulus ratios. In three dimensions the instability gives rise to a wide variety of morphologies, including almost degenerate zigzag and triple-junction patterns that can coexist when the layer and substrate are of comparable softness. Our study unifies these results providing understanding for the complex and diverse fold morphologies found in biology, including the zigzag precursors to intestinal villi, and disordered zigzags and triple-junctions in mammalian cortex.