Creases and cusps in growing soft matter

TL;DR

This paper explores the formation of creases and cusps in growing soft materials, using complex analysis and nonlinear bifurcation theory to understand various buckling patterns like wrinkles and folds.

Contribution

It provides a modern analytical framework for understanding elastic instabilities in soft matter growth, emphasizing complex analysis and pattern diversity.

Findings

Rich variety of buckling profiles including wrinkles, folds, and cusps

Role of physical parameters in pattern formation below and above Biot's instability threshold

Application of complex analysis to nonlinear elastic instability analysis

Abstract

The buckling of a soft elastic sample under growth or swelling has highlighted a new interest in materials science, morphogenesis, and biology or physiology. Indeed, the change of mass or volume is a common fact of any living species, and on a scale larger than the cell size, a macroscopic view can help to explain many features of common observation. Many morphologies of soft materials result from the accumulation of elastic compressive stress due to growth, and thus from the minimization of a nonlinear elastic energy. The similarity between growth and compression of a piece of rubber has revived the instability formalism of nonlinear elastic samples under compression, and in particular Biot's instability. Here we present a modern treatment of this instability in the light of complex analysis and demonstrate the richness of possible profiles that an interface can present under buckling, even if one restricts oneself to the two spatial dimensions. Special attention is given to wrinkles, folds and cusps, a surprising observation in swelling gels or clays. The standard techniques of complex analysis, nonlinear bifurcation theory and path-independent integrals are revisited to highlight the role of physical parameters at the origin of the observed patterns below and above the Biot threshold.