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Nonlinear Growing Caps

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Sobota, PMH 
Seffen, Keith A 


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.



Actuation, Growth, Buckling, Morphing Structures, Active Control, Adaptive Structures, Bistability, Analytical Solution, Post-buckling

Journal Title

Proceedings of the IASS Symposium 2018

Conference Name

Annual Symposium of the International Association for Shell and Spatial Structures 2018 - "Creativity in Structural Design"

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International Association for Shell and Spatial Structures (IASS)