Buckling and wrinkling from geometric and energetic viewpoints

TL;DR

This paper investigates buckling and wrinkling in inhomogeneously growing tissues using geometric embedding and energetic methods, revealing consistent results and self-similar patterns with implications for biological tissue organization.

Contribution

It introduces a dual approach combining conformal geometry and energy minimization to analyze buckling in growing tissues, highlighting self-similar behavior and potential biological significance.

Findings

Both approaches yield consistent buckling profiles.

Self-similar, fractal-like patterns emerge in the structures.

Implications for understanding cell proliferation in tissues.

Abstract

We discuss shape profiles emerging in inhomogeneous growth of squeezed tissues. Two approaches are used simultaneously: i) conformal embedding of two-dimensional domain with hyperbolic metrics into the plane, and ii) a pure energetic consideration based on the minimization of the total energy functional. In the latter case the non-uniformly pre-stressed plate, which models the inhomogeneous two-dimensional growth, is analyzed in linear regime under small stochastic perturbations. It is explicitly demonstrated that both approaches give consistent results for buckling profiles and reveal self-similar behavior. We speculate that fractal-like organization of growing squeezed structure has a far-reaching impact on understanding cell proliferation in various biological tissues.