Abstract
Flat sheets patterned with folds, cuts or swelling regions can deform into complex three-dimensional shapes under external stimuli1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24. However, current strategies require prepatterning and lack intrinsic shape selection5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24. Moreover, they either rely on permanent deformations6,12,13,14,17,18, preventing corrections or erasure of a shape, or sustained stimulation5,7,8,9,10,11,25, thus yielding shapes that are unstable. Here we show that shape-morphing strategies based on mechanical multistability can overcome these limitations. We focus on undulating metasheets that store memories of mechanical stimuli in patterns of self-stabilizing scars. After removing external stimuli, scars persist and force the sheet to switch to sharply selected curved, curled and twisted shapes. These stable shapes can be erased by appropriate forcing, allowing rewritable patterns and repeated and robust actuation. Our strategy is material agnostic, extendable to other undulation patterns and instabilities, and scale-free, allowing applications from miniature to architectural scales.
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Data availability
The data that support the findings of this study are available in the online version of this paper. Source data are provided with this paper. All other data are available on request from the corresponding author.
Code availability
The code that supports the findings of this study is available on request from the corresponding author.
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Acknowledgements
We thank A. Jensenius, O. Jensenius and L. Bisiot for their discovery of the reshaping behaviour of groovy metasheets. We thank H. Bense, C. Coulais, L. Domino, T. Jules, E. Filipov, M. Labousse, B. Overvelde, M. Serra-Garcia, Y. Shokef and A. Schenning for discussions. We are grateful to the precision manufacturing departments at AMOLF and Leiden University’s Institute of Physics, and to J. Mesman-Vergeer, D. Ursem and J. Verlinden for technical support.
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M.v.H. conceived and guided the work, and performed exploratory experiments and simulations. A.S.M. performed experiments, simulations and analytical calculations. M.v.H. and A.S.M. jointly developed the conceptual framework and wrote the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Groovy sheet fabrication.
a, Flat plastic film (orange) is placed on an aluminium mould (grey) with machined grooves. Screw-fastened rods push the film into each groove. b, Mould geometry is set by amplitude A, wavelength λ, mean radius R, curve angle θ, facet length l and mould spacing d. The film’s mid-surface is indicated (orange line). c, Top: cross-section photograph of a 75 μm-thick groovy sheet after forming in the mould (grey line). Bottom: image analysis yields an average amplitude and pitch A = 5.4 ± 0.3 mm and λ = 10.9 ± 0.3 mm. d, Groove height as a function of arc length s (left) is used to calculate the local curvature κ (right). Averaged values (black lines) yield radius of curvature R = 2.0 ± 0.2 mm and facet length l = 4.0 ± 0.1 mm, close to the mould’s shape (dashed lines, R = 1.5 mm, l = 4.4 mm).
Extended Data Fig. 2 Popping energy of a single defect.
a, Experimental setup. An indenter (grey) is attached by a ball magnet to a sheet (orange, N grooves, width W = 100 ± 5 mm) and held by two supports at w = 50 mm. Forces F are measured while moving the indenter over displacement u. b, Force-displacement curve measured while popping (upper curve) and un-popping (lower curve) a sheet with N = 5 grooves. Defect formation corresponds to 1.7 mNm of work (area under curve); un-popping to 0.1 mNm. c, Popping of sheets with N ∈ [0.5, 1, 2, 3, 5]. Two snap-through events i and ii (grey areas) and stable defects are seen for N = 1 to 5. Defect at N = 0.5 (black curve) is unstable. d, Work done during indentation (see c) reaches a plateau value around N = 5.
Extended Data Fig. 3 3D imaging of groovy sheets.
a, A sinusoidal fringe pattern is projected over distance l1 on a reference plane. The projector’s optical axis (grey dashed line) is orthogonal to the reference. A CCD camera (optical axis: grey dashed line) at distance l2 away from the projector captures the fringe pattern, which may be distorted by a sample (orange lines). b, Typical recorded images with and without sample (bottom, top). c, Sample height profile (colour bar) is calculated from an image set (left; see main text) and filtered (right). d, Typical filtered height profile (colour bar) of a scarred groovy sheet.
Extended Data Fig. 4 Computational model of a groovy sheet.
a, Unit cell: a square lattice (grey bars) of Hookean edge springs (length l = 1, stiffness ks) cross-braced by diagonal springs (length \(\sqrt{2}l\), stiffness ksd) and connected by nodes (circles) with torsional hinges (stiffness kt, resting angle ϕ0 along \(\widehat{{\boldsymbol{x}}}\) and π along \(\widehat{{\boldsymbol{y}}}\)). b, A structure of Nx by Ny nodes mimics a groovy sheet of \(N=\frac{{N}_{x}-1}{2}\) grooves. Indices n show node numbering convention.
Extended Data Fig. 5 Experimental and simulated defect interactions.
a, Experimental differential mean curvature ΔKm (calculated from difference between surface with and without defects) and Gaussian curvature Kg (colour bar) of a 75 μm-thick groovy sheet with two distant defects in adjacent grooves. ξ, ν are local sheet coordinates. Right: 3D-scan showing region of interest (dashed box). Greyscale indicates z-height. b, ΔKm and Kg for defects at intermediate distance. Zoom-in of Kg shows typical check pattern. c, ΔKm and Kg for bound defects. Zoom-in of Kg shows distinct diamond pattern. d, Left: top view of two simulated grooves of length W = 20l where l is the square unit cell size. Right: two defects (plus signs) at mutual distance d, resulting in material strain (colour bar). e, Elastic energy \({\mathcal{E}}\) due to material strain is shown as a function of defect spacing d. A small dip at d = 0 is seen (arrow), indicating short-range attraction.
Extended Data Fig. 6 Scar formation.
a, A groovy sheet (14 cm wide with 12 grooves, thickness 50 μm, and design wavelength, pitch, and curvature radius λ = 6.2 mm, A = 6.6 mm, and R = 1 mm) is attached to an in Instron UTM under point forcing (white arrows). Scale bar: 2 cm. b, Applied extension d of the sheet over time t. c, Extensional force F as a function of time t (black line). Snapping events are tracked via changes of the video signal (orange line, arbitrary units) within the area shown in a (dashed box). Large changes correspond to snapping events (dashed grey lines). d, Mimicking the experimental groovy sheet, a numerical model of size Nx = Ny = 21 (see Methods) is sequentially extended under point forcing (black arrows). e, Numerical force-displacement curve. Minor force drops due to defect nucleation and a major force drop due to sudden scar formation are observed. f, Snapshots of the numerical model; corresponding simulation steps are indicated by coloured stars in e. Linear spring strain ϵ is shown (colour bar). g, Force-displacement curve fi(di) of a theoretical bistable unit i. The unit switches from low-stiffness state si = 0 (no defect) to high-stiffness state si = 1 (defect) at critical force \({f}_{i}^{c}\). h, A model of 12 chained bistable units as in g is explored (parameters are fc(0) = 1, λ = 0.1, c = 0.004, and p = 2). Under applied extension, the force F (left axis) and state of each unit is calculated (right axis, legend). Minor force drops correspond to single units switching state, starting at the forcing locations and propagating inward; a major force drop is seen when remaining units avalanche into the defect state.
Extended Data Fig. 7 Shape metrics of 75 μm-thick groovy sheets with one or two scars.
a, One scar causes rolling. Left: schematic showing central scar sign and location at distance Wf from sheet edge. Middle: 3D-scanned image of deformed sheet. Shape metrics for twisting (groove angle α) and rolling (scar radius Rscar) are indicated. Right: shape metrics measured for increasing scar-edge distance Wf. Saturation of scar radius and a sudden symmetry-breaking twisting bifurcation are observed for increasing Wf. b, Two equal-signed scars cause rolling and twisting. Left to right: scar signs and locations at mutual distance D are shown. 3D-scanned image shows groove angle α and central curvature radius Rc. Radius Rc increases weakly with D and decreases weakly with Wf (colors). Groove angle α decreases weakly with D. Note that this configuration is not stable for D < 90 mm. c, Two opposite-signed scars produce a helical shape. Left to right: scar signs and locations are shown schematically. 3D-scanned image shown shape metrics of twist per groove β and central groove wavelength λc. The latter shows a weak decrease with scar-scar distance D and varies between the resting wavelength λ and groove arc length s (dashed lines). Prediction of λc for a purely helicoidal shape is shown (solid line, see Eq. (5)). Twist angle β initially increases, then decreases with D. Only left-twisting states are reported; equivalent right-twisting states exist as well.
Extended Data Fig. 8 Ruled surfaces approximate scarred sheet shapes.
a, A helicoidal base curve of radius R and pitch 2πT with local coordinate frame \(\{\widehat{{\boldsymbol{t}}},\widehat{{\boldsymbol{n}}},\widehat{{\boldsymbol{b}}}\}\) (pink line) approximates a scar line. b, Straight rules of length Wf extending at an orthogonal angle β away from the base curve (grey lines) approximate the sheet’s grooves.
Extended Data Fig. 9 Admissible sheet shapes for parallel scars in the ruled-surface model.
a, State diagram of double-scarred shapes. The rescaled scar radius R* as function of rescaled pitch T* and groove angle β is shown (colour bar). Contours of constant R* are shown for clarity (dashed grey lines). b, Zoom-in of experimentally relevant portion of the state diagram. Real sheets with parallel scars exhibit \(R* \lesssim \frac{1}{2}\). c, Examples of double-scarred shapes throughout the phase diagram, corresponding to coloured dots in b; scar lines (blue/green) and grooves (grey) are indicated. Constant Wf = 1 and scar arc length of π/40Wf between adjacent grooves were chosen for visualization. Coils (left), cylinders (bottom left), cones (bottom) and helices can be distinguished. Point T* = β = 0 produces cylindrical surfaces with arbitrary value of R* (purple star, dashed box); there are no other surfaces with β = 0. Solid (dashed) black boxes indicate qualitative experimentally observed shapes for equal (opposite)-parity scars.
Extended Data Fig. 10 Twisting transition in a simple elastic model.
a, Left: sheets are modelled as ruled surfaces. The scar line forms a helical directrix (orange line) with radius of curvature Rscar and pitch 2πT. Grooves (grey lines) are approximated as straight lines of length Wf at angle β to the scar, resulting in a cylindrical shape twisted over angle α. Right: elastic deformation modes of the sheet cost energy (see Eq. (19)) and are resisted with stiffness Kk (scar rolling), Kgs (scar torsion), Kβ (wing angle changes), Ks (sheet splay), and Kr (sheet rolling). b, Energy-minimizing sheet shapes as function of Wf for given deformation resistance Kk, Kgs, Kβ = 1, Ks, Kr = 0.1. Consistent with experiments (see Extended Data Fig. 7a), Rscar grows from its initial value R0 by amount ΔR. A twisting bifurcation in T and α is observed at critical \({W}_{f}^{\ast }\). Post-bifurcation values T* and α* at the end of the studied width range Wf = 150 mm are indicated. Sheet shapes shown left correspond to black circles in each graph. c, The overall sheet shape and existence of the twisting bifurcation, quantified by ΔR, \({W}_{f}^{* }\), T*, and α*, are robust against variations of deformation resistances over several orders of magnitude (legend).
Supplementary information
Supplementary Video 1
Defects interact in a groovy sheet. Thin corrugated sheets support snap-through defects in each groove. Defect pairs interact attractively or repulsively, depending on their separation distance.
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Meeussen, A.S., van Hecke, M. Multistable sheets with rewritable patterns for switchable shape-morphing. Nature 621, 516–520 (2023). https://doi.org/10.1038/s41586-023-06353-5
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DOI: https://doi.org/10.1038/s41586-023-06353-5
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