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REVIEW 3 major objections 5 minor 41 references

Rigid kirigami sheets can be inverse-designed so anisotropic snap-through locks them into prescribed 3D shapes without continuous actuation.

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2026-07-30 16:19 UTC pith:FT6FZZY4

load-bearing objection Solid methods paper: anisotropic bistable unit library plus inverse assignment for rigid kirigami, with real FEM/experiment support; the out-of-plane omission is a real but already-flagged scope limit, not a collapse of the claim. the 3 major comments →

arxiv 2607.26941 v1 pith:FT6FZZY4 submitted 2026-07-29 cond-mat.soft cond-mat.mtrl-sciphysics.class-ph

Instability-induced bistable shape-morphing kirigami structures

Xiaoyuan Ying , Marcelo A. Dias This is my paper
classification cond-mat.soft cond-mat.mtrl-sciphysics.class-ph
keywords bistable kirigamianisotropic shape morphinggeometric frustrationinstability-induced deploymentinverse designHencky bar-chainauxetic metamaterial
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Most kirigami morphing either stays monostable, needs soft materials, or assumes every unit expands the same way in every direction. This paper shows that is not enough: bistability depends on the specific anisotropic stretch path each triangular unit follows, not just overall area change. The authors build a semi-analytical energy model of the slender ligaments, map how tilting angle and edge-wise stretch set the energy well, then use conformal flattening of a target surface to assign heterogeneous units from a precomputed library. The flat laser-cut pattern deploys by instability into a self-locked 3D shape. Experiments on rigid Delrin dome and double-dome specimens stay free-standing and match the target contours. A sympathetic reader cares because the method brings programmable, load-bearing shape change to stiff sheet materials instead of only soft gels or continuously powered actuators.

Core claim

An inverse design pipeline that couples Hencky bar-chain ligament energetics to anisotropic edge scale factors and a bistable unit library can turn a flat rigid kirigami tessellation into a prescribed three-dimensional surface that is a stable energy minimum after snap-through, without external constraints or soft substrates.

What carries the argument

The Hencky bar-chain model of each ligament, constrained by end-angle and positional closure, which yields the full energy landscape along an anisotropic deployment path parametrized by edge scale factors (λ1, λ2, λ3) and tilting angle β; that landscape supplies the bistable-strain and bistability metrics used to select units after conformal surface flattening.

Load-bearing premise

Design energy is computed along a straight-line interpolation between flat and target shapes and counts only in-plane ligament stretch and bend, treating out-of-plane folding costs as irrelevant once the target geometry is fixed.

What would settle it

Fabricate a designed negative-curvature (saddle) specimen from the same rigid sheet; if the deployed state fails to remain free-standing or the measured energy well disappears relative to the in-plane prediction, the neglect of dihedral bending has invalidated the selection.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Heterogeneous rigid bistable kirigami can be cut from a single stiff sheet and still self-lock into curved load-bearing shapes.
  • Tilting angle β becomes a practical handle to place the bistable strain where geometric compatibility demands it.
  • The same unit library can be paired with thermal, magnetic, or other actuators once the passive energy wells are designed.
  • Surfaces whose local stretch stays inside the mapped bistable region of the ternary diagram are fabricable by this route; those outside are not without new unit topologies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extending the library to other cut topologies would enlarge the admissible stretch range and reduce the positive-curvature bias the authors already flag.
  • Because boundary units are deliberately left undeployed by global rescaling, large free edges may systematically under-constrain interior units on open or multiply connected domains.
  • Coupling the present in-plane selector to a lightweight dihedral energy term at design time is a direct next experiment suggested by the paper’s own limitation discussion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript presents an inverse-design framework for rigid, anisotropic bistable kirigami tessellations that deploy from flat cut sheets into prescribed 3D surfaces via instability-induced snap-through of slender ligaments. A Hencky bar-chain (HBM) model supplies unit-level energy landscapes under prescribed end kinematics; anisotropic deployment is parametrized by edge scale factors (λ1,λ2,λ3) and internal angles, and a precomputed (β,t) library is used to match conformal edge-wise stretches from Boundary First Flattening while maximizing a bistability index η. Finite-element comparisons support the HBM metrics, uniaxial tests show snap-through and open-state stability, and laser-cut Delrin dome and double-dome prototypes free-stand with reported front-view RMSE of 0.025 and 0.071.

Significance. If the design pipeline is mechanically reliable, the work meaningfully extends bistable kirigami beyond soft/isotropic systems toward monolithic rigid-sheet fabrication with programmable anisotropic deployment and self-locking equilibria—relevant to deployable structures, soft robotics, and adaptive architecture. Strengths include a tractable semi-analytical ligament model with documented HBM–FEM agreement on ε_bist and η (~1–2%), an explicit anisotropic bistable region in (α1,α2,β) space, and free-standing experimental prototypes rather than constrained holds. The combination of conformal surface parametrisation with a mechanics-based unit library is a concrete, reusable design strategy.

major comments (3)
  1. [§2.3.3; Discussion] §2.3.3 states that excluding out-of-plane dihedral bending from the optimisation “involves no loss of generality” because dihedral angles are fixed by the target and therefore “do not affect the optimal unit selection.” That does not follow: dihedral angles evolve from the flat state along the deployment path, so E_out(ξ) is path-dependent and can raise or eliminate the second minimum even when the in-plane η used for library lookup is positive. The Discussion correctly flags this risk for negative K, but the Results claim should be retracted or replaced by evidence (e.g., full-shell FEM energy vs ξ for the designed dome/double dome) that the selected units retain a bistable well once dihedral costs are included—especially for the mixed-K double dome.
  2. [§2.2 Eq. (7); Fig. 7; Suppl. S3] §2.2–2.3 and Fig. 7 evaluate deployment energy along the linear nodal interpolation q(ξ)=(1−ξ)q(0)+ξq(1). Library metrics (Fig. 5) and global E/Emax curves are therefore path-constrained, not proven stationary paths of the unconstrained assembled sheet. Please justify this kinematic family (or compare against quasi-static FEM/experimental deployment paths) and state clearly that reported bistability is conditional on that path class; otherwise the inverse-design guarantee is overstated relative to what is computed.
  3. [§2.4; Fig. 8] Experimental validation (Fig. 8) reports free-standing front-view contour RMSE only. For the central claim of programmable bistable 3D equilibria, please add at least (i) a measure of out-of-plane/shape error beyond a single silhouette and (ii) evidence that the deployed state is an energy minimum of the physical specimen (e.g., small perturbation recovery, or measured force–displacement / snap-through of the full tessellation), not only geometric resemblance after manual deployment.
minor comments (5)
  1. [Fig. 2; Fig. 5] Fig. 2 caption and panel labels: β ranges are written inconsistently (0°–12° in text/curves vs axes); unify units (degrees vs radians) with Fig. 5 (β=0.13 rad).
  2. [§2.2 Eq. (8)] Eq. (8) writes (λ1,λ2,λ3) with a trailing factor (1+ε_bist) and angle sines; briefly state the edge-length convention (which edge is reference length 1) to avoid ambiguity when mapping from BFF edge scales.
  3. [Introduction] “quarigid” in the Introduction appears to be a typo for “quasi-rigid.”
  4. [Abstract] Keywords and abstract repeat “instability-induced” / bistable claims; consider tightening abstract quantitative claims (e.g., cite RMSE or HBM–FEM error) once revised.
  5. [§2.3.1; Suppl. S4] Suppl. S4–S5 are helpful; a short forward reference in §2.3.1 to the discrete λij definition (Eq. S21) would help readers who skip the supplement.

Circularity Check

0 steps flagged

No significant circularity: inverse design selects units so in-plane HBM minima match targets by construction, then checks that claim externally with FEM and free-standing experiments.

full rationale

The load-bearing chain is standard mechanics-plus-geometry inverse design, not a tautology. Ligament energy (Eqs. 2–5, S1–S8) is obtained by constrained minimization of discrete bending/stretch springs under prescribed end kinematics; bistable strain ε_bist and depth η are read off those landscapes, not fitted to the target surfaces. Anisotropy is parametrized by independent geometric inputs (λ1,λ2,λ3) or (α1,α2,α3), scanned to build a precomputed (β,t) library (Fig. 5). Surface design pulls edge-wise scale factors from BFF conformal flattening (external method [41]) and assigns library units that match those factors while maximizing η—so the in-plane second minimum sits at the prescribed deployed shape by design intent, which is the definition of inverse design rather than a circular “prediction.” Global E/Emax curves (Fig. 7) and free-standing Delrin dome/double-dome contours (Fig. 8, RMSE 0.025/0.071) plus FEM comparisons (Fig. 2) are external checks of the model and assembly, not re-use of fitted targets. Self-citations ([24], [27], [28]) supply prior kirigami context and are not uniqueness theorems that force the present result. Linear path interpolation q(ξ) and omission of out-of-plane dihedral energy are modeling assumptions that may affect correctness (especially for negative K), but they are stated openly and do not make the reported energy equal to its inputs by definition. No self-definitional loop, fitted-input-as-prediction, or load-bearing self-citation chain is present.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 1 invented entities

The claim rests on classical beam/spring discretization, standard conformal surface parametrization, and several modeling closures (rigid panels, joint idealization, linear deployment path, in-plane-only optimization) plus geometric design handles (β, t, library matching). No new physical entities are postulated; free parameters are geometric/design choices and discretization resolution rather than fitted universal constants.

free parameters (5)
  • tilting angle β (per unit / library) = design-selected per element from library
    Primary geometric handle chosen so bistable equilibrium matches target (α1,α2,α3,ε_bist) while maximizing η; swept in [0, π/20].
  • ligament thickness t (spatial field) = spatially varying; sheet 1.5 mm Delrin
    Independent handle; ΔE∝Et³ used to assign higher t where perturbation resistance is needed, with t/l≤1/12.
  • HBM segment count N and end-clustered spacing = N=12 end-clustered
    Discretization chosen for ~5% energy error vs FEM at N=12; affects quantitative energy but is numerical, not physics.
  • gap w, in-radius a, cell length L = process- and design-dependent
    Unit geometry parameters fixed by design/fabrication constraints; example isotropic case uses w/L=0.1, t/L=0.015, a/L=0.6.
  • global boundary rescaling of conformal λ field = global rescaling so rim λ=1
    Rim units forced to λ=1 for boundary compatibility; relative interior λ preserved to keep target shape.
axioms (6)
  • domain assumption Flanks and triangular core are rigid relative to slender ligaments; interfaces act as perfect rotational joints.
    Stated in §2.1 to close the HBM unit model; neglects panel compliance and local stress concentrations.
  • domain assumption Ligament mechanics are Euler–Bernoulli bending plus axial stretch, discretized as Hencky bar-chain energy with rotational and extensional springs.
    Eqs. (2)–(5) and Supplementary S1; standard slender-beam modeling choice.
  • ad hoc to paper Intermediate deployment configurations follow linear interpolation of nodal coordinates between closed and target open states.
    Eq. (7) / S10; motivated by smooth monotonic rigid-body rotations but not derived from dynamics or quasi-static path continuation.
  • domain assumption Conformal (BFF) flattening plus edge-wise λi adequately represents the metric needed for unit selection on the triangular grid.
    §2.3.1 and S4–S5; standard discrete differential geometry practice, with acknowledged residual triangle anisotropy.
  • ad hoc to paper Out-of-plane dihedral bending can be omitted from unit selection without changing the optimal assignment once the target surface is fixed.
    §2.3.3 and Discussion; authors note bias against negative-K surfaces where this fails.
  • standard math Constrained energy minimization (interior-point) yields the relevant equilibrium branch for bistability metrics η and ε_bist.
    Standard nonlinear programming applied to Eqs. (3)–(5); branch-tracking subtleties not fully formalized.
invented entities (1)
  • Precomputed anisotropic bistable unit library indexed by (β,t) over (α1,α2,α3,ε_bist) independent evidence
    purpose: Maps required edge scale factors to a fabricable unit that maximizes bistability depth at the target strain.
    Engineering construct built from the paper’s parametric sweep; not a new physical particle/force, but the key invented design object.

pith-pipeline@v1.2.0-daily-grok45 · 20296 in / 3923 out tokens · 70936 ms · 2026-07-30T16:19:05.172070+00:00 · methodology

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read the original abstract

Deployable shape-morphing structures that transform from flat sheets into stable three-dimensional configurations are highly desirable for applications ranging from soft robotics and biomedical devices to adaptive architecture and aerospace systems. Existing kirigami-based morphing systems primarily rely on isotropic deployment, compliant soft materials, or external constraints to maintain deployed shapes, which limits geometric programmability, structural integrity, and applicability in rigid-material systems. Here, we present an inverse design framework for anisotropic bistable kirigami structures that enables programmable shape morphing through controlled geometric frustration and instability-induced deployment. The framework combines a semi-analytical mechanical model with geometry to establish a direct connection between geometric transformation and the underlying energy landscape. We show that instability-induced shape morphing leads to tunable bistability and directional deployment in anisotropic kirigami structures. The results are validated through finite element simulations and experiments, demonstrating stable deployed configurations and programmable anisotropic morphing. The proposed framework further provides a general design strategy that can be integrated with various active actuation systems, enabling broader engineering applications.

Figures

Figures reproduced from arXiv: 2607.26941 by Marcelo A. Dias, Xiaoyuan Ying.

Figure 1
Figure 1. Figure 1: Kirigami unit cell and tessellation. (a) Triangular auxetic unit cell parametrised by ligament thickness t, gap width w, in-radius a, tilting angle β, and unit cell length L. (b) Undeployed 4 × 5 tessellation. (c) Deployed unit cell with discrete beam representation of one ligament: nodes (circles) connected by extensible bars with segment angles θi , relative rotations φi , and axial extensions ei ; Θ1, Θ… view at source ↗
Figure 2
Figure 2. Figure 2: Bistability curves and Comparison between HBM and FEM. (a) Strain energy density versus strain for units with different tilt￾ing angle β computed with the semi-analytical model. (b) Bistable strain ϵbist(top) and bistability η(bottom) versus β from HBM and FEM.Definitions of ϵbist and η are given in Supplementary Fig. S4. strain at which the deployed-state energy minimum occurs, and the bista￾bility η = ∆U… view at source ↗
Figure 3
Figure 3. Figure 3: Uniaxial tensile test of uniform and nonuniform tessel￾lations. (a) force-displacement curve during loading(tension). (b) Pho￾tographs of uniform tessellation(top) and nonuniform tessellation(bottom) in the closed state and open state with side views. and pronounced out-of-plane deformation during stretching, whereas the uni￾form tessellation deploys essentially in-plane. 2.2 Effect of anisotropic deployme… view at source ↗
Figure 4
Figure 4. Figure 4: Effect of deployment path on bistability. (a) Initial closed state. (b) Isotropically scaled open configuration with uniform edge scale factors (λ1, λ2, λ3) = (1.54, 1.54, 1.54). (c) Energy–displacement curve for isotropic scaling, exhibiting a well-defined bistable response. (d) Anisotropi￾cally scaled open configuration with unequal edge scale factors (λ1, λ2, λ3) = (1.46, 1.56, 1.63). (e) Energy–displac… view at source ↗
Figure 5
Figure 5. Figure 5: Sensitivity of bistability to anisotropic deformation and tilting angle β. (a) Each unit deforms from an equilateral initial shape to a target shape parametrised by edge scale factors (λ1, λ2, λ3) and internal angles (α1, α2, α3); intermediate states follow linear interpolation of nodal coordinates. (b) Bistable region in (α1, α2, α3, β) space; highlighted planes correspond to the slices in (c–f). (c,d) Te… view at source ↗
Figure 6
Figure 6. Figure 6: Inverse design framework for bistable kirigami tessella￾tions. The target surface is conformally flattened via BFF, yielding a spa￾tially varying scale-factor field [λmin, λmax]. A regular triangular grid is over￾laid on the planar domain; barycentric interpolation converts vertex-based scale factors into edge-wise values (λ1, λ2, λ3) for each element. Each element is then assigned a bistable unit from the… view at source ↗
Figure 7
Figure 7. Figure 7: Inverse design applied to two target surfaces. (a) Dome (K > 0) and (b) double dome (mixed K): target geometry, conformal scale￾factor field, and the deployed tessellation configuration. (c) Normalised global in-plane deployment energy E/Emax versus deployment parameter ξ for the dome (left) and double dome (right). 14 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Fabrication and experimental validation. (a) Dome and (b) double dome: flat cut pattern (left), free-standing deployed specimen (centre), and front-view contour comparison between target and experiment (right). RMSE and maximum deviation are reported as fractions of the normalised span. 3 Discussion and Conclusion To conclude, this work bridges the gap between rigid-material fabrication and programmable sh… view at source ↗

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Reference graph

Works this paper leans on

41 extracted references

  1. [1]

    Shape morphing mechanical metamaterials through reversible plasticity.Science robotics, 7(63):eabg2171, 2022

    Dohgyu Hwang, Edward J Barron III, ABM Tahidul Haque, and Michael D Bartlett. Shape morphing mechanical metamaterials through reversible plasticity.Science robotics, 7(63):eabg2171, 2022. 17

    2022

  2. [2]

    Snap-induced morphing: From a single bistable shell to the origin of shape bifurcation in interacting shells.Journal of the Mechanics and Physics of Solids, 170:105116, 2023

    Mingchao Liu, Lucie Domino, Iris Dupont de Dinechin, Matteo Taffe- tani, and Dominic Vella. Snap-induced morphing: From a single bistable shell to the origin of shape bifurcation in interacting shells.Journal of the Mechanics and Physics of Solids, 170:105116, 2023

    2023

  3. [3]

    Static bistability of spherical caps.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 474(2213):20170910, 2018

    Matteo Taffetani, Xin Jiang, Douglas P Holmes, and Dominic Vella. Static bistability of spherical caps.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 474(2213):20170910, 2018

    2018

  4. [4]

    Bistable auxetic mechanical metamaterials inspired by ancient geometric motifs.Extreme Mechanics Letters, 9:291–296, 2016

    Ahmad Rafsanjani and Damiano Pasini. Bistable auxetic mechanical metamaterials inspired by ancient geometric motifs.Extreme Mechanics Letters, 9:291–296, 2016

    2016

  5. [5]

    One string to pull them all: Fast assembly of curved structures from flat auxetic linkages—supplemental information.ACM Trans

    Akib Zaman, Jacqueline Aslarus, Jiaji Li, and Stefanie Mueller. One string to pull them all: Fast assembly of curved structures from flat auxetic linkages—supplemental information.ACM Trans. Graph, 44(6), 2025

    2025

  6. [6]

    Computational inverse design of surface-based inflatables.ACM Transactions on Graphics (TOG), 40(4):1–14, 2021

    Julian Panetta, Florin Isvoranu, Tian Chen, Emmanuel Si´ efert, Beno ˆ ıt Roman, and Mark Pauly. Computational inverse design of surface-based inflatables.ACM Transactions on Graphics (TOG), 40(4):1–14, 2021

    2021

  7. [7]

    Multimaterial 4d printing with tailorable shape memory polymers.Scientific reports, 6(1):31110, 2016

    Qi Ge, Amir Hosein Sakhaei, Howon Lee, Conner K Dunn, Nicholas X Fang, and Martin L Dunn. Multimaterial 4d printing with tailorable shape memory polymers.Scientific reports, 6(1):31110, 2016

    2016

  8. [8]

    4d printing of shape-morphing liquid crystal elastomers.Chem & Bio Engineering, 1(6):488–515, 2024

    Tongzhi Zang, Shuang Fu, Junpeng Cheng, Chun Zhang, Xili Lu, Jian- she Hu, Hesheng Xia, and Yue Zhao. 4d printing of shape-morphing liquid crystal elastomers.Chem & Bio Engineering, 1(6):488–515, 2024

    2024

  9. [9]

    Pro- grammable active kirigami metasheets with more freedom of actuation

    Yichao Tang, Yanbin Li, Yaoye Hong, Shu Yang, and Jie Yin. Pro- grammable active kirigami metasheets with more freedom of actuation. Proceedings of the National Academy of Sciences, 116(52):26407–26413, 2019

    2019

  10. [10]

    Designing responsive buckled surfaces by halftone gel lithography.science, 335(6073):1201–1205, 2012

    Jungwook Kim, James A Hanna, Myunghwan Byun, Christian D San- tangelo, and Ryan C Hayward. Designing responsive buckled surfaces by halftone gel lithography.science, 335(6073):1201–1205, 2012. 18

    2012

  11. [11]

    Pro- grammed buckling by controlled lateral swelling in a thin elastic sheet

    Marcelo A Dias, James A Hanna, and Christian D Santangelo. Pro- grammed buckling by controlled lateral swelling in a thin elastic sheet. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 84(3):036603, 2011

    2011

  12. [12]

    Xiaohu Zhou, Tianzhen Li, Jiahui Wang, Fan Chen, Dan Zhou, Qi Liu, Liyun Zhang, Jiayan Shen, and Xuechang Zhou. Shape morphing of anisotropy-encoded tough hydrogels enabled by asymmetrically-induced swelling and site-specific mechanical strengthening.Journal of Materials Chemistry B, 6(29):4731–4737, 2018

    2018

  13. [13]

    4d printing of a self-morphing polymer driven by a swellable guest medium.Soft Matter, 14(5):765–772, 2018

    Jheng-Wun Su, Xiang Tao, Heng Deng, Cheng Zhang, Shan Jiang, Yuyi Lin, and Jian Lin. 4d printing of a self-morphing polymer driven by a swellable guest medium.Soft Matter, 14(5):765–772, 2018

    2018

  14. [14]

    Physics-aware differentiable design of magnetically actuated kirigami for shape morphing.Nature communications, 14(1):8516, 2023

    Liwei Wang, Yilong Chang, Shuai Wu, Ruike Renee Zhao, and Wei Chen. Physics-aware differentiable design of magnetically actuated kirigami for shape morphing.Nature communications, 14(1):8516, 2023

    2023

  15. [15]

    Reprogrammable shape morphing of magnetic soft ma- chines.Science advances, 6(38):eabc6414, 2020

    Yunus Alapan, Alp C Karacakol, Seyda N Guzelhan, Irem Isik, and Metin Sitti. Reprogrammable shape morphing of magnetic soft ma- chines.Science advances, 6(38):eabc6414, 2020

    2020

  16. [16]

    Magnetic kirigami dome metasheet with high deformability and stiffness for adaptive dynamic shape-shifting and multimodal manipula- tion.Science Advances, 10(49):eadr8421, 2024

    Yinding Chi, Emily E Evans, Matthew R Clary, Fangjie Qi, Haoze Sun, Saarah Niesha Cant´ u, Catherine M Capodanno, Joseph B Tracy, and Jie Yin. Magnetic kirigami dome metasheet with high deformability and stiffness for adaptive dynamic shape-shifting and multimodal manipula- tion.Science Advances, 10(49):eadr8421, 2024

    2024

  17. [17]

    Super- adaptive electroactive programmable adhesive materials to challenging surfaces: From intelligent soft robotics to xr haptic interfaces.InfoMat, 7(2):e12640, 2025

    Seung Hwan Jeon, Gui Won Hwang, Jinhyung Kim, Dohyun Lim, Yong Son, Tae-Heon Yang, Da Wan Kim, and Changhyun Pang. Super- adaptive electroactive programmable adhesive materials to challenging surfaces: From intelligent soft robotics to xr haptic interfaces.InfoMat, 7(2):e12640, 2025

    2025

  18. [18]

    Electric field driven soft morphing matter.Advanced Materials, page 2419077, 2025

    Ciqun Xu, Charl FJ Faul, Majid Taghavi, and Jonathan Rossiter. Electric field driven soft morphing matter.Advanced Materials, page 2419077, 2025

    2025

  19. [19]

    Topology and dynamics of active nematic vesicles

    Felix C Keber, Etienne Loiseau, Tim Sanchez, Stephen J DeCamp, Luca Giomi, Mark J Bowick, M Cristina Marchetti, Zvonimir Dogic, and 19 Andreas R Bausch. Topology and dynamics of active nematic vesicles. Science, 345(6201):1135–1139, 2014

    2014

  20. [20]

    Mechanics of active surfaces

    Guillaume Salbreux and Frank J¨ ulicher. Mechanics of active surfaces. Physical Review E, 96(3):032404, 2017

    2017

  21. [21]

    Flexible mechanical metamaterials.Nature Reviews Ma- terials, 2(11):1–11, 2017

    Katia Bertoldi, Vincenzo Vitelli, Johan Christensen, and Martin Van Hecke. Flexible mechanical metamaterials.Nature Reviews Ma- terials, 2(11):1–11, 2017

    2017

  22. [22]

    Mechanical metamaterials and their engineering applications.Advanced Engineering Materials, 21(3):1800864, 2019

    James Utama Surjadi, Libo Gao, Huifeng Du, Xiang Li, Xiang Xiong, Nicholas Xuanlai Fang, and Yang Lu. Mechanical metamaterials and their engineering applications.Advanced Engineering Materials, 21(3):1800864, 2019

    2019

  23. [23]

    Boundary curvature guided programmable shape-morphing kirigami sheets.Nature communications, 13(1):530, 2022

    Yaoye Hong, Yinding Chi, Shuang Wu, Yanbin Li, Yong Zhu, and Jie Yin. Boundary curvature guided programmable shape-morphing kirigami sheets.Nature communications, 13(1):530, 2022

    2022

  24. [24]

    Inverse design of programmable shape-morphing kirigami structures.International Jour- nal of Mechanical Sciences, 286:109840, 2025

    Xiaoyuan Ying, Dilum Fernando, and Marcelo A Dias. Inverse design of programmable shape-morphing kirigami structures.International Jour- nal of Mechanical Sciences, 286:109840, 2025

    2025

  25. [25]

    Origami spring-inspired shape morphing for flexible robotics.Soft Robotics, 9(4):798–806, 2022

    Qianying Chen, Fan Feng, Pengyu Lv, and Huiling Duan. Origami spring-inspired shape morphing for flexible robotics.Soft Robotics, 9(4):798–806, 2022

    2022

  26. [26]

    Spherical image analysis of origami and kirigami.Proceedings of the Royal Society A Mathematical Physical and Engineering Science, 482(2337):20250820, 2026

    Martin G Walker and Marcelo A Dias. Spherical image analysis of origami and kirigami.Proceedings of the Royal Society A Mathematical Physical and Engineering Science, 482(2337):20250820, 2026

    2026

  27. [27]

    On local kirigami mechanics i: Isometric conical solutions.Journal of the Mechanics and Physics of Solids, 151:104370, 2021

    Souhayl Sadik and Marcelo A Dias. On local kirigami mechanics i: Isometric conical solutions.Journal of the Mechanics and Physics of Solids, 151:104370, 2021

    2021

  28. [28]

    On local kirigami mechanics ii: Stretchable creased solutions.Journal of the Mechanics and Physics of Solids, 161:104812, 2022

    Souhayl Sadik, Martin G Walker, and Marcelo A Dias. On local kirigami mechanics ii: Stretchable creased solutions.Journal of the Mechanics and Physics of Solids, 161:104812, 2022

    2022

  29. [29]

    A perspective on the revival of structural (in) stability with novel opportunities for function: From buckliphobia to buckliphilia

    Pedro M Reis. A perspective on the revival of structural (in) stability with novel opportunities for function: From buckliphobia to buckliphilia. Journal of Applied Mechanics, 82(11):111001, 2015. 20

    2015

  30. [30]

    Buckling-induced smart applica- tions: recent advances and trends.Smart Materials and Structures, 24(6):063001, 2015

    Nan Hu and Rigoberto Burgue˜ no. Buckling-induced smart applica- tions: recent advances and trends.Smart Materials and Structures, 24(6):063001, 2015

    2015

  31. [31]

    Har- nessing bistability for directional propulsion of soft, untethered robots

    Tian Chen, Osama R Bilal, Kristina Shea, and Chiara Daraio. Har- nessing bistability for directional propulsion of soft, untethered robots. Proceedings of the National Academy of Sciences, 115(22):5698–5702, 2018

    2018

  32. [32]

    Beyond developable: computational design and fabrication with auxetic materials.ACM Transactions On Graphics (TOG), 35(4):1–11, 2016

    Mina Konakovi´ c, Keenan Crane, Bailin Deng, Sofien Bouaziz, Daniel Piker, and Mark Pauly. Beyond developable: computational design and fabrication with auxetic materials.ACM Transactions On Graphics (TOG), 35(4):1–11, 2016

    2016

  33. [33]

    Compu- tational design of deployable auxetic shells.Advances in architectural geometry, pages 94–111, 2018

    Mina Konakovi´ c-Lukovi´ c, Pavle Konakovi´ c, and Mark Pauly. Compu- tational design of deployable auxetic shells.Advances in architectural geometry, pages 94–111, 2018

    2018

  34. [34]

    Bistable auxetic surface structures.ACM Transactions on Graphics (TOG), 40(4):1–9, 2021

    Tian Chen, Julian Panetta, Max Schnaubelt, and Mark Pauly. Bistable auxetic surface structures.ACM Transactions on Graphics (TOG), 40(4):1–9, 2021

    2021

  35. [35]

    Durable bistable auxetics made of rigid solids.Journal of Materials Research, 33(3):300–308, 2018

    Xiao Shang, Lu Liu, Ahmad Rafsanjani, and Damiano Pasini. Durable bistable auxetics made of rigid solids.Journal of Materials Research, 33(3):300–308, 2018

    2018

  36. [36]

    Rapid deployment of curved surfaces via programmable auxetics

    Mina Konakovi´ c-Lukovi´ c, Julian Panetta, Keenan Crane, and Mark Pauly. Rapid deployment of curved surfaces via programmable auxetics. ACM Transactions on Graphics (TOG), 37(4):1–13, 2018

    2018

  37. [37]

    Locally var- ied auxetic structures for doubly-curved shapes

    Jan Friedrich, Sven Pfeiffer, and Christoph Gengnagel. Locally var- ied auxetic structures for doubly-curved shapes. InHumanizing Dig- ital Reality: Design Modelling Symposium Paris 2017, pages 323–336. Springer, 2017

    2017

  38. [38]

    Design optimisation of kirigami-based auxetic metamaterials with multistability and shape-morphing capabil- ity.Virtual and Physical Prototyping, 20(1):e2450286, 2025

    Eui-Hyun Kim and Keun Park. Design optimisation of kirigami-based auxetic metamaterials with multistability and shape-morphing capabil- ity.Virtual and Physical Prototyping, 20(1):e2450286, 2025. 21

    2025

  39. [39]

    ¨Uber die angen¨ aherte L¨ osung von Stabilit¨ atsproblemen im Raum mittels der elastischen Gelenkkette

    Heinrich Hencky. ¨Uber die angen¨ aherte L¨ osung von Stabilit¨ atsproblemen im Raum mittels der elastischen Gelenkkette. PhD thesis, Verlag nicht ermittelbar, 1921

    1921

  40. [40]

    From kirigami to hydrogels: a tutorial on designing conformally transformable surfaces.Journal of Applied Mechanics, 90(4):044801, 2023

    Yue Wang, Yingying Ren, and Tian Chen. From kirigami to hydrogels: a tutorial on designing conformally transformable surfaces.Journal of Applied Mechanics, 90(4):044801, 2023

    2023

  41. [41]

    Boundary first flattening.ACM Transactions on Graphics (ToG), 37(1):1–14, 2017

    Rohan Sawhney and Keenan Crane. Boundary first flattening.ACM Transactions on Graphics (ToG), 37(1):1–14, 2017. 22 Supplementary Material S1 Discrete Beam Model Kb,i ei φix0 xN-1 θ(s) a b k(s) Figure S1: (a) Diagram of Euler beam in continuous case, and (b) Diagram of Euler beam in discrete case We model each ligament using the Hencky bar-chain model (HBM...

    2017