Nonlinear Growing Caps

Abstract

Engineers capture growth strains in two ways, reflecting the inherent bending-stretching nature of shells: by a strain gradient through the thickness or by an average in-plane value. We analyse their interaction by assuming a uniform displacement and growth-strain field in shells with elastic spring supports and a radial force applied to their outer boundary. The increased degree of statical indeterminancy enriches the variety of existing solutions and we distinguish two in-plane actuation modes which can induce Gaussian curvature via radially varying quadratic expansions in either the circumferential or radial direction. Using a Rayleigh-Ritz approach, we find closed-form solutions of the Föppl-von Kármán shell equations for the buckling thresholds, bistable limits and the post-buckled shape, which show good agreement with finite element reference solutions and available results from the literature. Moreover, we show that ‘natural’ growth modes, which evoke a change of shape without incurring elastic strain energy, can be achieved by employing quadratic radial expansions only. Additionally, we study unsupported shells subjected to higher-order actuation distributions, which give rise to natural growth modes with varying wavenumbers. Our approach dramatically simplifies an otherwise non-trivial general solution, and may be applied in novel generations of smart materials with actively tunable material properties.