Deformation and stability of initially stressed hyperelastic plates

Abstract

Initial/residual stress is inherent in nearly all natural and engineered structures. This paper presents a comprehensive theory for modelling residually stressed, growing plates. By constructing a two-dimensional representation of three-dimensional solid mechanics, we avoid any need for prior assumptions about deformation fields. This approach reformulates both the initial stress fields and deformation gradients in three-dimensional space through planar quantities, yielding a set of plate equations that govern their interactions. This framework enables modelling of various naturally and artificially generated planar structures with residual stress and growth, such as plant leaves and additively manufactured plates.

To explore the wrinkling instabilities that often arise in such structures, we derive a principal solution for an initially stressed, growing plate supported by Winkler foundations. We then apply linear perturbation to examine bifurcation phenomena, solving the resulting governing equations analytically and computationally. The numerical scheme is validated with analytical results and shows promise for solving more geometrically complex instability problems.