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Multi-step self-guided pathways for shape-changing metamaterials
Corentin Coulais, Alberico Sabbadini, Fré Vink, Martin van Hecke
TL;DR
Macroscopic mechanical systems generally require external control for multi-step reconfiguration, unlike autonomous molecular and self-assembly processes. This paper introduces mechanically designed metamaterials that execute self-guided topological reconfigurations under uniform compression, with pathways controlled through nonlinear elements and hierarchical architecture. The authors demonstrate alternative two-step pathways, pathway-error suppression through self-contacts, and increased numbers of reconfiguration steps through hierarchy.
Problem
Macroscopic mechanical systems require external control to execute multi-step pathways, motivating self-guided mechanical reconfiguration.
Method
The authors combine strongly nonlinear links with multimodal hierarchical architectures and rationally designed link parameters to produce sequential reconfigurations under compression.
Results
The metamaterials exhibit controlled sequential pathways, including two-step folding in alternative geometries under uniaxial compression.
Takeaways & Limitations
Hierarchical architectures extend the number of distinct reconfiguration steps, while self-contacts suppress pathway errors.
Takeaways & Limitations
Boundary friction and frustration cause distortions and misfoldings, so the reported bending-angle analysis is restricted to central regions of the samples.
Abstract
from arXiv · showhide
Multi-step pathways, constituted of a sequence of reconfigurations, are central to a wide variety of natural and man-made systems. Such pathways autonomously execute in self-guided processes such as protein folding and self-assembly, but require external control in macroscopic mechanical systems, provided by, e.g., actuators in robotics or manual folding in origami. Here we introduce shape-changing mechanical metamaterials, that exhibit self-guided multi-step pathways in response to global uniform compression. Their design combines strongly nonlinear mechanical elements with a multimodal architecture that allows for a sequence of topological reconfigurations, i.e., modifications of the topology caused by the formation of internal self-contacts. We realized such metamaterials by digital manufacturing, and show that the pathway and final configuration can be controlled by rational design of the nonlinear mechanical elements. We furthermore demonstrate that self-contacts suppress pathway errors. Finally, we demonstrate how hierarchical architectures allow to extend the number of distinct reconfiguration steps. Our work establishes general principles for designing mechanical pathways, opening new avenues for self-folding media, pluripotent materials, and pliable devices in, e.g., stretchable electronics and soft robotics.
METHODS
The hierarchical architecture replaces square building blocks with recursively generated cross-shaped subpatterns, increasing internal links and enabling additional reconfiguration steps.
- Hierarchical Design: The metamaterial hierarchy starts from an n × n rotating square mechanism and replaces each square with five smaller squares at each higher rank.For rank m, the structure contains n2 × 5^(m−1) squares.
- Hierarchical Design: Each increase from rank m−1 to rank m adds internal links connecting the recursively inserted subpatterns.The construction includes α-links between unit cells, β-links for m ≥2, and γ-links for m ≥3.
Hinged Tessellations
Hinged tessellations are freely hinging structures that can fold into fully area-filling configurations. The paper identifies rotating-square, unequal-square, and cross-based architectures as examples that form multiple self-contacts.
- Hinged Tessellations: Hinged tessellations are freely hinging structures that can be folded into a fully area-filling structure.The rotating square mechanism is one example.
- Hinged Tessellations: Unequal-square and cross-based variations can also form area-filling structures with multiple self-contacts.These variations arise among the hierarchical metamaterial pathways.
Zero Modes
The number of internal zero modes follows from Maxwell counting of square degrees of freedom and link constraints. Higher hierarchy increases the mode count, while stiffening one link family changes the constraints and available floppy modes.
- Zero Modes: Maxwell counting starts with three degrees of freedom per square, subtracts two constraints per connection, and removes three global degrees of freedom.The resulting quantity is nz, the number of internal degrees of freedom.
- Zero Modes: For hierarchy m = 2, nz = 3n2 + 4n −3, yielding 61 internal zero modes for n = 4 structures.The count applies to the hierarchy-2 structures shown in Figs. 1 and 2.
- Zero Modes: For hierarchy m = 3, nz = 23n2 + 4n −3, yielding 216 internal zero modes for the n = 3 structure.This is the hierarchy-3 structure shown in Fig. 3.
- Zero Modes: When β-links are stiff and α-links floppy, the structure has one soft mode; when β-links are floppy and α-links stiff, the m = 2 case has n2 + 6n −3 floppy modes.The latter count follows from subtracting additional constraints imposed by the stiff α-links.
Rational Design of Linkages
The paper rationally designs link thicknesses and boundary-compatible super-cells to select deformation pathways. Linear and nonlinear analyses show that the thickness ratio controls mode competition and pathway robustness.
- Rational Design of Linkages: Relative beam thickness tα/tβ controls the critical buckling strains and modes that initiate the deformation pathway.The buckling load scales as approximately t3, so tuning link thicknesses selects the required instability.
- Rational Design of Linkages: The design analysis uses a 2 × 2 super-cell in a uniformly compressed square box with periodic boundary conditions.Additional constraints enforce aligned edges and equal horizontal and vertical dimensions.
- Rational Design of Linkages: For an m = 2, n = 2 super-cell with square periodic boundary conditions, the flexible-hinge limit has six zero modes.The modes are represented using an orthogonal basis of motions a–f.
- Rational Design of Linkages: When tα/tβ is small, one mode is significantly softer than the others; when tα/tβ is larger, several modes compete.For tα ≪ tβ, only motion a is soft, whereas for tβ ≪ tα, motions b–e are soft.
- Rational Design of Linkages: The soft-mode crossover is centered near tα/tβ ≃1.5, with motion a dominating at small ratios and motions b and c contributing at large ratios.The analysis associates an isolated lowest mode at small ratios with robust pathway I and mode competition at large ratios with greater complexity.
- Rational Design of Linkages: Nonlinear buckling analysis confirms the crossover and finds that buckling modes are nearly combinations of motions a and b.Other motions are suppressed and essentially irrelevant in this analysis.
Symmetry broken links
A lateral offset in the β-links breaks symmetry and helps select pathway II despite competing deformation modes. The offset magnitude is fixed at half the β-link thickness.
- Pathway II is harder to execute because competing modes and β-link deformations interfere with the intended motion.
- The design introduces a lateral β-link offset with magnitude |δβ| = 0.5tβ and alternating chirality.The offset pattern is chosen to be consistent with motion b.
- For tα/tβ = 0.4 and tα/tβ = 2.3, simulations show no instability and deformation behavior that illustrates the crossover scenario.
Design guidelines and experimental validation
The authors translate numerical mode-selection results into practical thickness-ratio and offset guidelines, then test pathway selection experimentally. Pathway I is robust at low ratios, whereas pathway II requires offsetting and remains vulnerable without it.
- Design guidelines: Pathway I is expected without offsets for tα/tβ ≲1, with increasing fidelity at smaller ratios.At tα/tβ = 0.5, the buckling mode overlaps motion a by more than 90%.
- Design guidelines: With |δβ| = 0.5tβ and alternating chirality, pathway II is expected for tα/tβ ≳2.At tα/tβ = 2.3, the nonlinear mode overlaps motion b by 75%.
- Experimental validation: Experiments without offsets produced clear pathway I at tα/tβ = 0.5 and 0.4, but a disordered pathway at 1.4.Earlier less accurately manufactured 3D-printed samples also showed clear pathway I.
- Experimental validation: Without offsets, experiments at tα/tβ = 4 showed substantial disorder that prevented successful pathway II execution.Disorder appeared to nucleate near the boundaries.
Numerical simulations
Numerical simulations model representative metamaterial super cells with hyperelastic material behavior and periodic square boundary conditions. Linear, nonlinear stability, and imperfect nonlinear compression analyses probe their modes and reconfiguration behavior.
- Finite-element simulations of representative metamaterial elements use Abaqus/Standard.
- Model definition: The model varies tα, tβ, and δβ using a neo-Hookean material with E = 8.0 MPa and ν = 0.49999 in plane stress.
- Boundary conditions: Periodic square boundary conditions constrain boundary-node displacements while preserving a square-shaped periodic cell.
- Analysis types: Linear eigenmode analysis calculates the lowest eigenmodes and their eigenfrequencies.
- Analysis types: Nonlinear bifurcation analysis determines the bifurcation point with relative accuracy 5×10^-2.
- Analysis types: Imperfect nonlinear compression studies offset β-links by compressing the structure to a strain of 6.6%.
Experimental techniques
The metamaterial samples use cast silicone rubber mounted on plywood and are fabricated by waterjet cutting. Equi-biaxial compression is measured mechanically while high-resolution imaging records deformation.
- Materials and fabrication: Samples use 10 mm-thick Shore 80A silicone rubber with Young’s modulus E = 8 MPa, selected to limit gravity, viscous, and creeping effects.
- Materials and fabrication: Waterjet cutting and plywood backing provide precise fabrication, while ellipses marked on the squares enable deformation detection.The marked ellipses measure 2.5 mm × 4 mm on 4.5 mm-square elements.
- Link geometry: Hinges are rectangular beams with thicknesses tα, tβ, or tγ ranging from 0.8 mm to 4.0 mm.Their height matches the 10 mm sheet thickness.
- Mechanical testing: Equi-biaxial compression uses a custom aluminium V-shaped press with displacement accuracy of 10 µm and force accuracy of 0.1 N.Fine powder reduces boundary friction.
- Imaging: A high-resolution CMOS camera with 3858 px × 2764 px resolution records the samples under LED front and back lighting.The press windows are coated to reduce front-light reflections.
Image tessellation techniques
Image tessellation and tracking techniques were used to extract square positions and link-bending angles, while boundary regions were excluded from angle analysis because of distortions and misfoldings.
- Image tessellation techniques: A semi-automatic tracking algorithm extracted space-time trajectories for nearly all squares and bending angles.Detection errors occurred near compression-cell edges and when squares came into contact.
- Image tessellation techniques: 28 central squares for rank II and 29 for rank three were used to compute bending angles in the main figures.Wider analyses over 64 and 165 squares showed larger scatter but preserved the main trends and multi-step nature.
Sequential pathways with alternative topologies
The authors construct two alternative metamaterial geometries that execute clear two-step folding pathways under uniaxial compression. These designs use link groups with different buckling thresholds so sequential buckling is triggered by self-contact.
- Sequential pathways with alternative topologies: Two alternative geometries demonstrate that sequential folding pathways are not limited to hierarchical structures under biaxial compression.Both use the same rubber and waterjetting fabrication technique as the main-text samples.
- Sequential pathways with alternative topologies: Different buckling thresholds cause two link groups to buckle sequentially under compression.The designs couple the link groups in series so that one group destabilizes before the other.
- Sequential pathways with alternative topologies: In the diluted square lattice, thinner links buckle first, fold until self-contact, and trigger buckling of thicker-link columns.Because this structure is soft to lateral shear, the experiment uses lateral sliding boundaries.
- Sequential pathways with alternative topologies: A second geometry removes the soft shear modes, requires no lateral boundaries, and also exhibits a clear two-step sequence.Together, the alternative designs broaden the topology options for sequential pathways under uniaxial compression.
EXTENDED DATA
The extended data document the metamaterial architectures, hinged motions, numerical mode analyses, compression setup, boundary effects, and alternative two-step topologies. Together, these figures provide structural, computational, experimental, and geometric context for the reported pathways.
- Hierarchical Design: The hierarchical metamaterial uses rotating-square units and replaces each square with five smaller squares at successive generations.The architecture includes α-links between units and internal β- and γ-links at higher ranks.
- Hinged Tessellations: The rotating-square mechanism, unequal-square mechanism, and linked crosses are shown as freely hinging tessellations.These structures can fold into area-filling configurations with multiple self-contacts.
- Mode analysis: Extended Data Figures ED3 and ED4 analyze zero-energy motions and eigenmodes for a 2 × 2 super cell with periodic square boundaries.The mode analysis relates bending-angle vectors and their projections onto the system’s motions.
- Nonlinear analysis: Symmetry-broken links produce distinct deformed states and strain-dependent projections for varying tα/tβ ratios.The analyzed cases use tα + tβ = 3 mm and a β-link offset of 0.5tβ.
- Experimental and boundary context: Compression experiments and a custom fixture document biaxial loading, while boundary analyses compare angle trends across central regions of different sizes.The boundary study reports larger scatter over wider regions but preserves the main trends and multi-step behavior.
- Alternative topologies: Extended Data Figure ED10 presents alternative linked-square and slanted-bar geometries with two-step folding under uniaxial compression.The accompanying passages specify link dimensions and thicknesses for these alternative designs.