Models for elastic shells with incompatible strains

TL;DR

This paper rigorously derives asymptotic theories for the shapes of residually strained elastic shells with nontrivial curvature, generalizing classical plate theories to curved geometries and different growth regimes.

Contribution

It extends existing models by providing a unified derivation for shells with incompatible strains, covering both weakly and strongly curved regimes, and generalizes the F"oppl-von Kármán energy.

Findings

Derived asymptotic theories for residually strained shells

Unified treatment of weakly and strongly curved regimes

Generalized classical shell energy models

Abstract

The three-dimensional shapes of thin lamina such as leaves, flowers, feathers, wings etc, are driven by the differential strain induced by the relative growth. The growth takes place through variations in the Riemannian metric, given on the thin sheet as a function of location in the central plane and also across its thickness. The shape is then a consequence of elastic energy minimization on the frustrated geometrical object. Here we provide a rigorous derivation of the asymptotic theories for shapes of residually strained thin lamina with nontrivial curvatures, i.e. growing elastic shells in both the weakly and strongly curved regimes, generalizing earlier results for the growth of nominally flat plates. The different theories are distinguished by the scaling of the mid-surface curvature relative to the inverse thickness and growth strain, and also allow us to generalize the classical F\"oppl-von K\'arm\'an energy to theories of prestrained shallow shells.