Geometry and mechanics of thin growing bilayers

Abstract

We investigate how thin sheets of arbitrary shapes morph under the isotropic in-plane expansion of their top surface, which may represent several stimuli such as nonuniform heating, local swelling and differential growth. Inspired by geometry, an analytical model is presented that rationalizes how the shape of the disk influences morphing, from the initial spherical bending to the final isometric limit. We introduce a new measure of slenderness that describes a sheet in terms of both thickness and plate shape. We find that the mean curvature of the isometric state is three fourths the natural curvature, which we verify by numerics and experiments. We finally investigate the emergence of a preferred direction of bending in the isometric state, guided by numerical analyses. The scalability of our model suggests that it is suitable to describe the morphing of sheets spanning several orders of magnitude.

Curvature-Induced Instabilities of Shells

“…We note that the critical stimuli in open and closed shells could also be determined via the method for nonlinear deformations presented in [13], and a formal comparison should be investigated in future work. We believe our study is a valuable contribution towards the generic understanding of curvature-driven instabilities in thin curved shells, as it generalizes previous experiments on plates [17] and elastica with a natural curvature [37]. Due to current limitations of the coating setup [18], we hope that our study will motivate experiments on more general surfaces, e.g.…”

“…Forθ → 0, shells tend to plates. Flat plates of radius r under curvature stimuli bifurcate atκ p h = ±a(h/r) 2 with a = 10 + 7 √ 2 [17]. Then, for large R and small θ, but r = Rθ finite, shells are expected to behave like plates if we identifyκ p R = κ p R − 1, i.e.…”

“…For θ → 0 we derived κ b R = ±a/θ 2 + 1 , from the flat case presented in [17], assuming the radius of the plate to be equal to Rθ. When θ → π, we estimated a divergence of the critical curvature as 1/(θ − π).…”

“…Thus, κ =κ+1/R measures the curvature stimulus ( Fig. 1 (a)) [17,19]. Specifically, for a bilayer with VPS 8 on the outside, we find κ > 0, and by switching the order of the layers, we can induce a negative natural curvature (κ < 0).…”

“…To describe the mechanics of slender structures with arbitrary stimuli, classical shell mechanics was extended recently to model bodies that do not possess a stressfree configuration [12][13][14], leading to the non-Euclidean shell theory [15]. Despite recent progress [16,17], the role of curvature-altering stimuli, and their interplay with geometric frustration and instabilities in thin, initially curved shells, remains poorly understood.In this Letter, we combine precision experiments with non-Euclidean shell theory to reveal how curvature stimuli induce mechanical instabilities in spherical shells. Our experiments demonstrate symmetry-breaking as well as snap-through shape transitions depending on the amount of stimulus and the deepness of the shell.…”