Wrinkling as a mechanical instability in growing annular hyperelastic plates

TL;DR

This paper develops a mathematical framework to analyze growth-induced wrinkling in annular hyperelastic plates, revealing how boundary conditions and geometry influence instability modes, with applications to biological tissues and soft materials.

Contribution

It introduces a variational approach to derive governing equations without kinematic assumptions and performs a bifurcation analysis to identify critical growth factors for wrinkling.

Findings

Asymmetric bifurcation is often the preferred mode of instability.

Boundary constraints and geometry significantly affect the critical growth factor.

The model applies to biological tissues, hydrogels, and growing elastic films.

Abstract

Growth-induced instabilities are ubiquitous in biological systems and lead to diverse morphologies in the form of wrinkling, folding, and creasing. The current work focusses on the mechanics behind growth-induced wrinkling instabilities in an incompressible annular hyperelastic plate. The governing differential equations for a two-dimensional plate system are derived using a variational principle with no apriori kinematic assumptions in the thickness direction. A linear bifurcation analysis is performed to investigate the stability behaviour of the growing hyperelastic annular plate by considering both axisymmetric and asymmetric perturbations. The resulting differential equations are then solved numerically using the compound matrix method to evaluate the critical growth factor that leads to wrinkling. The effect of boundary constraints, thickness, and radius ratio of the annular plate on the critical growth factor is studied. For most of the considered cases, an asymmetric bifurcation is the preferred mode of instability for an annular plate. Our results are useful to model the physics of wrinkling phenomena in growing planar soft tissues, swelling hydrogels, and pattern transition in two-dimensional films growing on an elastic substrate.