Buckling by disordered growth

TL;DR

This work investigates how spatially disordered growth influences buckling in an elastic rod, motivated by developmental processes like brain folding. By combining analytic results for a uniformly growing rod with numerical sampling of random growth fields, the authors show that growth variability can raise or lower the buckling threshold, and that the shift correlates with spatial moments of the growth field. The study demonstrates that the arrangement of growth variability—such as distributed disorder or localized growth islands—can be tuned to trigger or avoid buckling, providing a potential design principle for controlling morphogenesis in biological tissues. The approach blends morphoelastic theory, asymptotic analysis, and finite-element simulations to quantify how micro-scale growth variability propagates to a macro-scale mechanical instability.

Abstract

Buckling instabilities driven by tissue growth underpin key developmental events such as the folding of the brain. Tissue growth is disordered due to cell-to-cell variability, but the effects of this variability on buckling are unknown. Here, we analyse what is perhaps the simplest setup of this problem: the buckling of an elastic rod with fixed ends driven by spatially varying growth. Combining analytical calculations for simple growth fields and numerical sampling of random growth fields, we show that variability can increase as well as decrease the growth threshold for buckling, even when growth variability does not cause any residual stresses. For random growth, we find that the shift of the buckling threshold correlates with spatial moments of the growth field. Our results imply that biological systems can either trigger or avoid buckling by exploiting the spatial arrangement of growth variability.

Figures (1)

• Figure S1: Finite element mesh for the quarter rod $0\leqslant x\leqslant 1/2$, $-\uppi/2\leqslant\phi\leqslant\uppi/2$ plotted with respect to the Cartesian basis $(x,y,z)$). Parameter value: $h=1/20$.