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

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