Formation of desired three-dimensional (3D) shapes from flat thin sheets with programmed non-uniform deformation profiles is an effective strategy to create functional 3D structures. Liquid crystal elastomers (LCEs) are of particular use in programmable shape morphing due to their ability to undergo large, reversible, and anisotropic deformation in response to a stimulus. Here we consider a rectangular monodomain LCE thin sheet divided into one high- and one low-temperature strip, which we dub a ‘bistrip’. Upon activation, a discontinuously patterned, anisotropic in-plane stretch profile is generated, and induces buckling of the bistrip into a rolled shape with a transitional bottle neck. Based on the non-Euclidean plate theory, we derive an analytical model to quantitatively capture the formation of the rolled shapes from a flat bistrip with finite thickness by minimizing the total elastic energy involving both stretching and bending energies. Using this analytical model, we identify the critical thickness at which the transition from the unbuckled to buckled configuration occurs. We further study the influence of the anisotropy of the stretch profile on the rolled shapes by first converting prescribed metric tensors with different anisotropy to a unified metric tensor embedded in a bistrip of modified geometry, and then investigating the effect of each parameter in this unified metric tensor on the rolled shapes. Our analysis sheds light on designing shape morphing of LCE thin sheets, and provides quantitative predictions on the 3D shapes that programmed LCE sheets can form upon activation for various applications.
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Shapes of Fe nanocrystals encapsulated at the graphite surface
We describe and analyze in detail the shapes of Fe islands encapsulated under the top graphene layers in graphite. Shapes are interrogated using scanning tunneling microscopy. The main outputs of the shape analysis are the slope of the graphene membrane around the perimeter of the island, and the aspect ratio of the central metal cluster. Modeling primarily uses a continuum elasticity (CE) model. As input to the CE model, we use density functional theory to calculate the surface energy of Fe, and the adhesion energies between Fe and graphene or graphite. We use the shaft-loaded blister test (SLBT) model to provide independent stretching and bending strain energies in the graphene membrane. We also introduce a model for the elastic strain in which stretching and bending are treated simultaneously. Measured side slopes agree very well with the CE model, both qualitatively and quantitatively. The fit is optimal for a graphene membrane consisting of 2–3 graphene monolayers, in agreement with experiment. Analysis of contributions to total energy shows that the side slope depends only on the properties of graphene/graphite. This reflects delamination of the graphene membrane from the underlying graphite, caused by upward pressure from the growing metal cluster. This insight leads us to evaluate the delamination geometry in the context of two related, classic models that give analytic results for the slope of a delaminated membrane. One of these, the point-loaded circular blister test model, reasonably predicts the delamination geometry at the edge of an Fe island. The aspect ratio also agrees well with the CE model in the limit of large island size, but not for small islands. Previously, we had speculated that this discrepancy was due to lack of coupling between bending and stretching in the SLBT model, but the new modeling shows that this explanation is not viable.
more » « less- NSF-PAR ID:
- 10302748
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- New Journal of Physics
- Volume:
- 22
- Issue:
- 2
- ISSN:
- 1367-2630
- Page Range / eLocation ID:
- Article No. 023016
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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