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Combinatorial Design of Textured Mechanical Metamaterials

Corentin Coulais, Eial Teomy, Koen de Reus, Yair Shokef, Martin van Hecke

arXiv:1608.00625v1cond-mat.softcond-mat.mtrl-scicond-mat.stat-mech

TL;DR

The paper addresses how to characterize compatible configurations in structured metamaterial patterns. It develops combinatorial classifications and recursion relations, yielding exact values and asymptotic bounds for the number of compatible metacube configurations.

  • Problem

    The paper studies how patterned brick arrangements constrain the number and structure of compatible configurations.

  • Method

    The approach classifies z-brick patterns using binary row and column vectors, counts compatible layer configurations, and combines symmetry-based recursion relations.

  • Results

    Exact values of the number of compatible L × L × L metacube configurations are obtained, alongside asymptotic lower and upper bounds.

  • Takeaways & Limitations

    The combinatorial classification provides a systematic basis for enumerating compatible configurations across metacube sizes.

Abstract

from arXiv · show

The structural complexity of metamaterials is limitless, although in practice, most designs comprise periodic architectures which lead to materials with spatially homogeneous features. More advanced tasks, arising in e.g. soft robotics, prosthetics and wearable tech, involve spatially textured mechanical functionality which require aperiodic architectures. However, a naïve implementation of such structural complexity invariably leads to frustration, which prevents coherent operation and impedes functionality. Here we introduce a combinatorial strategy for the design of aperiodic yet frustration-free mechanical metamaterials, whom we show to exhibit spatially textured functionalities. We implement this strategy using cubic building blocks - voxels - which deform anisotropically, a local stacking rule which allows cooperative shape changes by guaranteeing that deformed building blocks fit as in a 3D jigsaw puzzle, and 3D printing. We show that, first, these aperiodic metamaterials exhibit long-range holographic order, where the 2D pixelated surface texture dictates the 3D interior voxel arrangement. Second, they act as programmable shape shifters, morphing into spatially complex but predictable and designable shapes when uniaxially compressed. Third, their mechanical response to compression by a textured surface reveals their ability to perform sensing and pattern analysis. Combinatorial design thus opens a new avenue towards mechanical metamaterials with unusual order and machine-like functionalities.

III. AUTHOR CONTRIBUTIONS

The authors’ contributions were divided across conceptualization, formulation and solution of the spin problem, experiments, simulations, and manuscript preparation.

  • C.C. and M.v.H. conceived the main concepts.
  • C.C., E.T., Y.S. and M.v.H. formulated the spin problem, while E.T. and Y.S. solved it.
  • C.C. and K.d.R. performed experiments and simulations with input from E.T., Y.S. and M.v.H.
  • C.C. and M.v.H. wrote the manuscript with contributions from all authors.

V. METHODS

The methods combine spin-based combinatorial design, bounded configuration counting, voxel fabrication, mechanical testing, simulation, and lock-and-key analysis.

  • Combinatorial design: Binary vectors c_i and r_j determine z-brick placement, while remaining sites are filled with x- or y-bricks.The construction restricts z-bricks to patterns of columns, rows, and their intersections.
  • Combinatorial design: Each prescribed texture has at least two compatible L × L × 1 motifs, which can be stacked in arbitrary order to form metacubes.For a 5 × 5 smiley texture, six compatible motifs are identified.
  • Configuration counting: Compatible configuration counts are bounded and recursively evaluated through the number Q of in-plane solutions for each texture.The exact evaluation reaches L = 14, while asymptotic lower and upper bounds are also derived.
  • Materials and fabrication: Specimens were fabricated from silicone rubber or sintered polyurethane using 3D printing and moulding.The 5 × 5 × 5 specimens used silicone rubber, while the 10 × 10 × 10 sample used sintered polyurethane.
  • Mechanical testing: Mechanical tests compressed metacubes uniaxially while controlling displacement and measuring force under flat or textured boundaries.Textured boundaries were used for the shape-morphing and lock-and-key experiments.
  • Simulation and analysis: Nonlinear Abaqus simulations and quadratic force-displacement fits were used to estimate stiffness and compare numerical with experimental responses.The fitted form was F(u) = ku + ηu^2.
  • Lock-and-key analysis: In lock-and-key experiments, incompatible key textures localized frustration along internal domain walls, whose frustrated-side count equals A + C.A single misplaced pixel frustrates four x/y sides and one opposing z side, while touching defects share an unfrustrated interface.

VI. EXTENDED DATA

The extended data documents motif construction, implementation dimensions, experimental setups, compressed metacube faces, sensory measurements, and exact configuration counts through L = 14.

  • Motif design: Six compatible motifs can realize a 5 × 5 smiley texture, and varying stacking order yields 65 smiley metacubes.
  • Implementation: The printed unit cell uses a = 11.46 mm, D = 10.92 mm, and w = 3.6 mm, with outer walls thickened by 0.27 mm.
  • Experimental setup: The lock-and-key experiment is shown through a textured clamp and a side view of the setup.
  • Compression response: A 10 × 10 × 10 metacube under compression shows motif stacking, an inverted opposite-face pattern, and checkerboard transverse faces.
  • Sensory properties: Five force-compression experiments are fitted quadratically, with numerical results and a stiffness comparison between simulations and experiments.The fit region is 0.8 mm ≤ u ≤ 2.5 mm.
  • Configuration counts: Exact values of Ω are tabulated for L × L × L metacubes up to L = 14.

I. MOVIES

The accompanying movies show periodic and aperiodic metacubes under compression, while supplementary combinatorics derive and visualize their compatible-configuration counts.

  • Movies: The periodic 5 × 5 × 5 movie shows compression triggering a transformation into alternating elongated and flattened bricks.
  • Movies: The 10 × 10 × 10 smiley movie shows a checkerboard clamp producing a designed smiley surface texture.
  • Movies: The opposite face morphs into the inverted smiley pattern, while a side face morphs into a checkerboard pattern.
  • Combinatorics: For a given texture, Q counts compatible L × L × 1 configurations, which can be stacked in any order to generate Q^L metacubes.

1. Maximal number of compatible configurations

For a k × L layer, the uniform textures maximize compatible solutions, with the maximum established inductively as Q = 2k + 2L −1.

  • 1. Maximal number of compatible configurations: The textures ˜σz ≡+1 and ˜σz ≡−1 have the maximal number of solutions, Q = 2k + 2L −1.The result is proved by induction on L.
  • 1. Maximal number of compatible configurations: For L = 1, a texture with p positive spins has Q = 2p + 2k−p solutions, maximized when p = 0 or p = k.These cases give Q = 2k + 1, matching the general formula.
  • 1. Maximal number of compatible configurations: At the induction step, the solution bound is maximal when the top row is uniform, with p = 0 or p = k.The resulting value is the required Q = 2k + 2L −1.

2. Second maximal number of compatible configurations

Among nonuniform L × L textures, the second-highest solution count occurs for an otherwise uniform texture with one exceptional row or column.

  • 2. Second maximal number of compatible configurations: The second-maximal textures are all +1 or all −1 except for one oppositely signed row or column.This classification excludes the two uniform textures with the absolute maximum.
  • 2. Second maximal number of compatible configurations: For these textures, the number of solutions is Q = 2L−1 + 2L.The same value follows from the extremal nonuniform case p = 1 or p = L −1.
  • 2. Second maximal number of compatible configurations: For a nonuniform top row containing p positive and L −p negative spins, the maximal solution count occurs when every column is uniform.The count is Q = 2p −1+2L−p −1+2L before optimizing over p.
  • 2. Second maximal number of compatible configurations: Because p cannot equal 0 or L, the maximum is attained at p = 1 or p = L −1.This yields Q = 2L−1 + 2L.

C. Exact derivation of ZQ

The exact derivation classifies textures by full rows and columns, then recursively counts textures with each solution number Q to obtain Ω(L).

  • C. Exact derivation of ZQ: An exact recursion for ZQ(L) can, in principle, be solved numerically for any finite L.Together with Eq. (SI1), it gives the exact number of compatible spin configurations Ω(L).
  • C. Exact derivation of ZQ: Textures are divided into types according to whether they contain full rows, full columns, or both, with signs and counts recorded explicitly.The categories include type 0, C, R, and CR families.
  • C. Exact derivation of ZQ: For each texture type, recurrence equations relate counts with Q solutions to smaller textures with q < Q and reduced dimensions.The recurrences are summarized in Section II C 7.
  • C. Exact derivation of ZQ: Uniform textures have Q = 2Lx + 2Ly −1 solutions and form the maximal-solution case in the recursion.The two textures are ˜σz all +1 and ˜σz all −1.

2. Types CR±(px, py), 1 ≤px ≤Lx −1, 1 ≤py ≤Ly −1

For CR± textures with interior positive-spin rows and columns, solution counts are obtained by decomposing the texture into quadrants or a smaller block and combining the resulting cases recursively.

  • 2. Types CR±(px, py), 1 ≤px ≤Lx −1, 1 ≤py ≤Ly −1: A CR+(px, py) texture places px positive columns on the left and py positive rows on top, leaving a bottom-right block of reduced size.The block has q solutions and belongs to type 0, C−, R−, or CR−.
  • 2. Types CR±(px, py), 1 ≤px ≤Lx −1, 1 ≤py ≤Ly −1: If at least one leftmost ˜σx spin is +1, the remaining constraints produce 2px −1 solutions.The rightmost ˜σx spins are fixed to +1 and ˜σy is fixed to −1.
  • 2. Types CR±(px, py), 1 ≤px ≤Lx −1, 1 ≤py ≤Ly −1: If all leftmost ˜σx spins are −1 and at least one topmost ˜σy spin is +1, the case contributes 2py −1 solutions.The rightmost ˜σx spins are −1 and the bottom ˜σy spins are +1.
  • 2. Types CR±(px, py), 1 ≤px ≤Lx −1, 1 ≤py ≤Ly −1: If the selected leftmost ˜σx and topmost ˜σy spins are all −1, the remaining constraints come only from the block, contributing q solutions.This is the residual case in the block decomposition.
  • 2. Types CR±(px, py), 1 ≤px ≤Lx −1, 1 ≤py ≤Ly −1: Combining the cases gives Q = 2px + 2py −2 + q solutions for the full texture.The same decomposition underlies the recursion relations for the CR family.

7. Final result

The recursion relations, together with symmetry relations and numerical solutions for finite L, determine exact values of the compatible-configuration count Ω(L).

  • The recursion relations can be combined with symmetry relations to determine Z0^2(Lx, Ly).
  • Numerical solutions of the recursion equations were obtained for L ≤ 14.
  • These solutions were substituted into Eq. (SI1) to obtain exact values of Ω(L).

D. Estimates

The estimates characterize which Q values contribute to Ω(L), validate an approximation using exact finite-size data, and produce tighter bounds on the configuration count.

  • Many Q values have ZQ(L) = 0 and therefore do not contribute to Ω(L).
  • For 9 ≤ L ≤ 14, L values near Q = 2n + 1 have ZQ(L) ≈ A(L)Q^-L and dominate the contribution to Ω.
  • For L = 14, retaining only the 14 dominant Q points gives about 0.36 × Ω(14).
  • The fraction of relevant Q values follows f ≈ 1.8 × e^-0.27L for L ≥ 4.
  • Using exact f(L) values for L ≤ 14 yields a good approximation for Ω(L), while the resulting approximate upper bound is close to the asymptotic lower bound.
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