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(Fig. protocol ation doi:10.1073/pnas.2018610118/-/DCSupplemental at online information supporting contains article This 2 1 BY-NC-ND) (CC 4.0 NoDerivatives License distributed under is article access open This ulse a 7 2021. 17, May Published Editorial the by invited editor guest a is K.B. Submission. Direct PNAS Board. a is article This interest.y competing no D.M.J.D., declare authors A.B., The and data; analyzed paper.y the C.C. J.v.d.L., wrote and C.C. D.M.J.D., D.M.J.D., and A.B., A.B., research; research; performed designed C.C. 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APPLIED PHYSICAL SCIENCES Oligomodal Tilings In contrast to the highly localized line modes, the additional zero-energy modes present in oligomodal tilings are markedly different: They typically exhibit deformations covering the whole ABC sample (Fig. 2I). By definition, their number remains constant with increasing system size. There is, in principle, no upper bound on the number of modes in an oligomodal tiling, but, by construction, this number of modes in the tiling cannot exceed the number of modes in the supercell. These spatially extended deformations, which, to the best of our knowledge, have not been reported before, offer the prospect of unconventional bulk ... properties. In the following, we focus on the simplest oligomodal config- monomodal oligomodal plurimodal uration that we uncovered: the bimodal tiling of Fig. 3J, which hosts two bulk zero-energy modes irrespective of system size. In order to understand the nature of these zero-energy modes (and, controllable in fact, also to interpret and classify all three classes of tilings; see SI Appendix), we introduce a vertex representation that maps out Fig. 1. Oligomodal materials. (A) Monomodal materials have a single zero-energy mode, hence a single property, that can be obtained via a simple actuation protocol. (B) Oligomodal materials have a small but fixed number of zero-energy modes larger than one, hence a few dis- A D E tinct properties, that can be selected with a simple actuation proto- col, for example, uniaxial compression. (C) Plurimodal materials have a Unit cell Supercell Metamaterial large number of zero-energy modes that grows with system size, and hence are kinematically able to host a large number of properties, but they often require complex actuation protocols, for example, multiaxial loading.
2 that is, for an n × n tiling, there are 4n possible configurations. B C It is thus impossible to sample all of the possible configura- tions, even numerically. Therefore, we restrict our attention to a subset of configurations. Namely, we will focus on periodic tilings of 2 × 2 supercells (Fig. 2E), which is a very small frac- tion of the design space; we discuss an example of quasiperiodic F G tilings in SI Appendix. Any choice of four orientations defines a 2 × 2 supercell and thus a tiling; hence there are 256 pos- sible configurations. By using the isometries of the square, we can reduce these to only 10 mechanically nonequivalent config- urations (Fig. 2F). A numerical study of the kinematics (Mate- rials and Methods) then reveals a rich spectrum of zero-energy modes. This spectrum contains the three classes of metamateri- als defined in Fig. 1. For system sizes n > 4, monomodal tilings (Fig. 2 G and H, red) have a single zero-energy mode; oligo- modal tilings have a constant zero-energy mode number larger than one (Fig. 2 G and I, green); and plurimodal tilings have a number of zero-energy modes that increases linearly with n (Fig. 2 G and J, blue). HI J All three classes of tilings share a common zero-energy mode, which is the rotating-squares mechanism (13), which spans the size of the system, and where the unit cells have the same defor- mation in an alternating manner with a lattice spacing periodicity of two (Fig. 2H). Monomodal tilings only have this zero-energy mode, while oligomodal and plurimodal tilings have additional zero-energy modes. In particular, plurimodal tilings include most periodic tilings and feature modes that are typically localized along lines (Fig. 2J and SI Appendix), which we term line modes. The existence of line modes in a periodic tiling implies that such a Fig. 2. Combinatorial design space. (A) Bimodal unit cell that has two tiling is plurimodal, and explains why the number of modes grows zero-energy deformation modes. (B) Mode 1, actuating four angles, with with the size of the sample n. Such lines modes have, in fact, pre- constant square angle indicated. (C) Mode 2, actuating all five angles, with viously been identified in highly symmetric lattices (14, 15), and dashed symmetry line. (D) The 2 × 2 supercell made up from stacking the highly localized deformations related to line modes have been unit cell in four possible orientations. (E) Periodic 6 × 6 metamaterial made observed in foams under compression (16–22). We discuss a par- from stacking a 2 × 2 supercell. (F) All mechanically nonequivalent 2 × 2 supercells, featuring monomodal (red), oligomodal (green), and plurimodal ticularly intriguing example of nonperiodic plurimodal tilings, in (blue) mode scalings. (G) Mode scaling, NM, with the tiling’s side length, SI Appendix, namely, a quasi-crystalline tiling that gains a single n, for all 10 tilings in D and the average of random tilings (red triangular zero-energy line mode when its side length is doubled, thus lead- crosses). (H) Rotating-squares mechanism of monomodal tilings. (I) Spatially ing to a number of modes that scales logarithmically with the size extended mode typical of oligomodal tilings. (J) Line localized mode typical of the sample n. of plurimodal tilings.
2 of 9 | PNAS Bossart et al. https://doi.org/10.1073/pnas.2018610118 Oligomodal metamaterials with multifunctional mechanics Downloaded by guest on October 6, 2021 Downloaded by guest on October 6, 2021 lgmdlmtmtraswt utfntoa mechanics multifunctional with metamaterials defor- Oligomodal of al. gradient et Bossart constant a with system, the of size the spans ,cmiigbt ntcl oe fFg 3 Fig. of (Fig. modes domains southwest cell two from unit into both going deformation combining axis the 3L), symmetry splitting northeast, a to features which mode, 3 rotating-squares the 3 find (Fig. we mode First, 3J modes. Fig. the zero-energy in on spanning analysis shown an possi- tiling such the Performing oligomodal of modes. shape deformation spatial soft the ble into insight 2G direct zero-energy Fig. provides of model in number through the shown predict edges, analysis to modes structural the used standard via we the that neighbors computation Unlike 3 its search. Fig. to combinatorial in vertex a shown one vertices from gated two the follows: as of D formulated combinations be linear can Only constraints geometric quantifies 3 edge the each (Fig. tation, on deformation arrowheads hinge of to corresponds number the edge the each deformations. and cell, hinge, unit single-cell a a of to corresponds representation vertex Each for graphical allows simple representation This a physics Methods). statistical and (2D) (Materials of two-dimensional (23–25) reminiscent in is found that models graph, vertex directed a onto deformations angle also See Boundaries. Textured by Actuation Selective (M rest at sample (J design terial (A–C undeformed 3. Fig. J MN BDE AB D–F and lgmdlmtmtras (A–I metamaterials. Oligomodal ny eod efidanwbl ioanzero-energy bidomain bulk new a find we Second, only. G r loe.Teegoerccntansaepropa- are constraints geometric These allowed. are ,wt oaigsursmd (K mode rotating-squares with ), K sepce,uigtefis ntcl oeo Fig. of mode cell unit first the using expected, as ) ,adsbetdt etrdbudr odtos1(N 1 conditions boundary textured to subjected and ), ,dfre nisfis oe(D–F mode first its in deformed ), 2 cm ,tevertex the Methods), and (Materials C 5 cm A, ntcl fFg 2A Fig. of cell Unit ) and D, eel w itntsystem- distinct two reveals iulzdi etxrpeetto.(M representation. vertex in visualized (L) mode bidomain and ) KL .I hsrepresen- this In G). ,addfre nisscn oe(G–I mode second its in deformed and ), oi S1. Movie D–I hsmode This . nvre (A, vertex in wiearw) ooe lissidct oepolarization hole indicate ellipses Colored arrows). (white (O) 2 and ) and D, el aetosf eomto oe Fg 3 (Fig. unit modes 3D-printed deformation the soft Crucially, two 3C). have (Fig. cells cell unit idealized see 3B; the (Fig. hinges Methods compliant and and als struts curved rigid 3J of Fig. of a tiling create oligomodal we printing, of 3D made textured frus- multimaterial metamaterial while Using chosen choice others. of allow the mode suitably trating compression zero-energy a of uniaxial of exciting simple sets preferential under that conditions demonstrate boundary we rate-dependent First, Boundaries strain Textured by Actuation or Selective texture boundary properties— either mechanical viscoelasticity. effective using distinct modes— two deformation by soft obtain these thus actuate demonstrate and selectively We can modes. we deformation soft that distinct metamate- geometry two flexible axis. oligomodal with create the to rials northwest modes use zero-energy to we distinct paper, southeast two the with the of would remainder in deformation the oriented In of same, gradient If the large. the as remain rectangular, deformation twice were of be would gradient tiling deformation the the a the large, and for as same, Therefore, the twice would axis. be northwest would to that tiling southeast the along mation ,3-rne (B, 3D-printed G), F ofiuain.(J configurations. ) O G o eal) uhta hi hpscicd with coincide shapes their that such details), for and E, 16 H ,adshmtcrpeetto (C, representation schematic and ), × h 8 The –L) H h Dpitdui el r made are cells unit 3D-printed The . 16 h 16 The –O) ntcls(i.3M (Fig. cells unit https://doi.org/10.1073/pnas.2018610118 × ceai lgmdlmetama- oligomodal schematic 8 × 6oiooa metamaterial oligomodal 16 I ,floigthe following ), E and PNAS Ω and F, endin defined H Materi- | that ) f9 of 3 I in )
APPLIED PHYSICAL SCIENCES closely mimic the zero-energy modes of the idealized unit cell (Fig. 3 F and I). Following the deformation prescribed by the A B vertex representation (Fig. 3 K and L), we then compress our metamaterial using two sets of eight indenters on each side, that apply the load on every other unit cell and where the bottom set of indenters is either antialigned (Fig. 3N) or aligned (Fig. 3O) with the top set of indenters. To quantify the deformations of each unit cell, we use particle tracking (OPENCV [a computer vision library] and Python) and custom-made tracking algorithms C D to quantify the flattening f and orientation φ with respect to the horizontal of each pore and calculate the polarization (26) nx +ny Ω := (−1) f cos 2φ, where nx (ny ) is the unit cell’s column (row). Using this protocol, we observe that we can actuate either the rotating-squares mode (Fig. 3N) or the bidomain mode (Fig. 3O) by changing the location of load application (see also Movie S1). This example demonstrates that the zero-energy modes of oligomodal metamaterials can be selectively actuated by tuning the boundary texture. In the following, we build on this result to achieve mode selection under a simpler actuation protocol that does not resort to boundary texture. E F Selective Actuation by Dissipation To achieve this goal, we leave the realm of static responses and harness viscoelastic dissipation to tune the dynamical response of our metamaterial (26–28), previously applied to alter the direc- tion of a mechanical mode (26, 28). Instead, we now harness viscoelasticity to switch between modes altogether. To this end, we use the oligomodal metamaterial design of Fig. 3J, which we rotate by 45◦ (Fig. 4 A–D). With this rotation of the lat- tice, we expect the domain wall seen in Fig. 3O to rotate by G H 45◦ (Fig. 4 B and D), and thus to become perpendicular to the loading axis as well as to provide mirror symmetry about the transverse axis. The selective actuation of modes via viscoelasticity is possible because the two modes—the rotating-square mechanism (Fig. 4 A and C) and the bidomain mode (Fig. 4 B and D)—involve dis- tinct sets of hinges. While, in the rotating-square mechanism, all hinges but the one at the center of the supercell are actuated (white hinges in Fig 4A), in the bidomain mode, all hinges includ- ing the one at the center of the supercell are actuated (white and black hinges in Fig 4B). Therefore, in order to favor the actuation of the rotating-squares mechanism (respectively, the bidomain I mode), the central hinge should be stiffer (respectively, softer) than the other hinges. Conversely, in order to favor the actua- tion of the bidomain mode, the central hinge should be softer than the other hinges. To obtain a selective actuation of the modes, we tune the stiffness of hinges by making them viscoelas- tic. Specifically, we use 1) viscoelastic hinges—whose stiffness increases with the strain rate at which they are actuated—for the hinges that are actuated in the rotating-squares mechanism (white hinges in Fig 4 A and B) and 2) elastic hinges—whose stiffness does not depend on the strain rate at which they are actuated—for the hinge that is only actuated in the bidomain mode (black hinge in Fig 4B). Fig. 4. Mode selection using strain rate. (A–D) Schematic supercell represen- Next, we prove the feasibility of the concept described above tation (A and B) and schematic and vertex model representation (C and D) of with finite element simulations (Abaqus; see Materials and Meth- the rotating-squares mode expected under slow actuation (A and C) and of ods and SI Appendix). In order to selectively actuate one mode the bidomain mode expected under fast actuation (B and D). The tiling design or the other, we use a combination of stiff material for the rigid follows Fig. 3, but is rotated by 45◦.(E and F) Numerical realization of the parts and soft material for the compliant hinges. In particular, schematic sample of C and D, numerically compressed in vertical direction to we dress our metamaterial with two types of elastic and vis- 7% strain with an average strain rate ε˙ = 9.3 · 10−6 s−1 (E) and ε˙ = 0.11 s−1 coelastic hinges (see Fig. 9B) whose properties correspond to (F). The legend indicates the Von Mises stress (megapascals). Insets contain (visco)elastic elastomers we will use in the experiments (Fig. 5 enlarged views of an individual hinge. (G and H) For the cases of E and F, A and B; see Materials and Methods for details about the simula- respectively, the average hole row polarization Ω as function of strain ε, up to 10% strain, and normalized vertical coordinate yn. The polarization is defined tions). For slow actuation rates, both types of compliant hinges ny as Ω := (−1) f cos 2φ, where ny is the unit cell’s row, starting at zero at the have a very similar torsional stiffness (Materials and Methods). bottom; f is the flattening of the ellipse; and φ is the orientation of the ellipse When the actuation rate increases, the viscoelastic hinges stiffen with respect to the first diagonal. (I) Absolute value of the average sample dramatically: They become more than 5 times stiffer than the polarization Ω as function of strain ε and strain rate ε˙. The legend indicates elastic hinges (Materials and Methods). As expected, we find that, the average row hole polarization Ω for G–I.
4 of 9 | PNAS Bossart et al. https://doi.org/10.1073/pnas.2018610118 Oligomodal metamaterials with multifunctional mechanics Downloaded by guest on October 6, 2021 Downloaded by guest on October 6, 2021 lgmdlmtmtraswt utfntoa mechanics multifunctional with metamaterials Oligomodal al. et Bossart material viscoelastic the with of rate timescales strain of relaxation large the a to at respect metamaterial the compressing 4 hand, (Fig. mechanism rotating-squares the l cosawd ag fsris nldn tlrestrains large at including strains, of range wide a across ble above. described selection mode 4 of 4 (Fig. mechanism (Fig. rates rates strain strain large low for actuated 4 (Fig. supercell 4 (Fig. mode viscoelastic the strain of timescales low relaxation a of the at material to metamaterial respect the with compressing rate hand, one the on orange. in 4, Fig. of results the following strain results, of function as spread also polarization See sample pores. average deformed Corresponding and polarized While highly deformation, to 4. no spond Fig. hence in and polarization defined no as to legend, the with line oe fe compression and a after pores (C (C rates. slow under metamaterial modal olsi tasaet aeila et olwn h apedsg fFig. of design sample the vis- following and 4 rest, (green) at elastic material (white), (transparent) rigid coelastic of made metamaterial Oligomodal (A) 5. 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APPLIED PHYSICAL SCIENCES Strain Rate-Dependent Poisson’s Ratio rates, there is a range of strains ε ∈ [0.3, 0.45] for which the In this last section, we ask whether the selection of distinct modes Poisson’s ratio can switch from positive to negative by tuning can also lead to selection of distinct mechanical properties. the strain rate. Therefore, there is a range of strains for which Specifically, we probe whether the rotating-squares mechanism our metamaterial is multifunctional, hosting several functions and the bidomain modes lead to distinct values of the Poisson’s that can be selected with a robust actuation protocol: auxetic ratio, and, therefore, whether one can obtain distinct values of under slow compression rate and nonauxetic at fast compression the Poisson’s ratio using strain rate. Intuitively, we expect the rate. presence of the domain wall to frustrate the overall lateral defor- mation and therefore to induce a larger Poisson’s ratio for the Discussion bidomain mode than for the rotating-squares mechanism—see To conclude, we have introduced a class of materials, oligomodal Materials and Methods. metamaterials, that strike a delicate kinematic balance that We measure, both experimentally and numerically, the Pois- allows them to exhibit a limited number of zero-energy modes. son’s ratio as a function of the strain ε for the same different Based on these zero modes, we have created flexible metama- strain rates as before. While we always find a positive Poisson’s terials, whose deformations can then be selectively and robustly ratio in the prebuckling phase, we find that compressing at a actuated using relatively simple loading protocols such as uni- small strain rate leads to a response with a negative nonlinear axial compression, thus providing an example of multifunctional Poisson’s ratio at strains ε > 0.04 (Fig. 6A, blue), while compress- mechanics. ing at large strain rate induces larger values of the Poisson’s Our approach is fundamentally different from and is com- ratio (Fig. 6B, blue). To obtain more insight on these results, plementary to recent developments based on poroelasticity (29) we measure the Poisson’s ratio as a function of the strain ε and magnetoelasticity (30). In these works, multifunctionality is and the strain rate ε˙ from the numerical simulations (Fig. 6 A achieved through elaborate actuation multiphysics couplings and and B, orange and Fig. 6C) and find that the Poisson’s ratio via loading in the bulk: The loading is exerted on the bulk by remains positive in the prebuckled region, while it becomes neg- external fields, such as a chemical potential or a magnetic field. ative in the buckled region. The rotating-squares mechanism and This is different from our case, where the deformation modes are the bidomain mode both exhibit negative Poisson’s ratios, yet selected in a purely mechanical fashion, and where the loading is the Poisson’s ratio is larger and therefore less negative in the exerted at the boundaries. These approaches could, in fact, ben- case of the bidomain mode, which is consistent with the experi- efit from employing oligomodal architectures, in order to reduce ments. Notably, the Poisson’s ratio in the rotating-square mode the number of involved degrees of freedom to the minimum shows larger spatial dispersion; this is a signature of the snap- required by the desired functions. ping transition that occurs when the deformation snaps from the Our approach is broadly applicable for the design of flexi- bidomain mode into the rotating-square mode. Interestingly, as ble metamaterials with low-energy strain pathways. In particular, the buckling strain is pushed to larger values for larger strain our approach could be further generalized to other types of unit cells, to other lattices, and to 3D tilings and applied to more types of functionalities such as chiral responses (31–33), shape morphing (4, 11, 34), enhanced energy absorption (26, 35) and nonlinear wave propagation (36, 37). In particular, we expect the switchable nature of oligomodal metamaterials to enable the control of deformation pathways, which could be potentially useful for dissipation of energy. Moreover, there is no reason to assume that a zero-energy mode-based method as we use is the only way to design flexible oligomodal metamate- rials. Alternative analysis and design methods that can induce clear modal separation, such as topology optimization, may be used instead. Also, our design space, mostly restricted to peri- odic tilings of 2 × 2 supercells, could be dramatically augmented, for instance, by considering periodic tilings of larger supercells, by using alternative nonperiodic tiling strategies constructed via fractal substitution rules and mirroring (see SI Appendix for an example), or by harnessing defects (12, 38). Finally, data-driven approaches such as machine learning could provide a powerful alternative to better understand the structure–property relation- ship (33) and to rationally design oligomodal tilings. Additional open questions that remain are how to effectively design large deformation responses, which variety of multifunctionality can be achieved, and how to produce these materials on a large scale effectively. Materials and Methods Kinematics of Multimode Combinatorial Metamaterials. Repeating our bimode cell with arbitrary orientations on a square lattice yields a large variety of configurations. To understand how the balance between con- Fig. 6. Strain rate-dependent Poisson’s ratio. (A and B) The average Pois- straints and degrees of freedom gives rise to zero-energy modes and states son’s ratio ν, as function of ε for the experimental sample of Fig. 5 C and of self-stress, we use the Maxwell–Calladine index theorem (14). It provides D (blue, with error bars) and numerical sample Fig. 4 E and F (orange, with a relation between the number of states of self-stress NSS, the number of −6 −1 −1 spatial spread) at a strain rate of ε˙ = 9.3 · 10 s (A) and of ε˙ = 0.11 s zero-energy modes NM, the dimension d, the number of hinges N, and the (B). (C) The absolute value of the average sample Poisson’s ratio ν as func- number of constraints c, tion of strain ε and strain rate ε˙ for the numerical sample of Fig. 4 E and F. The hashed area indicates the range of ε and ε˙, where positive–negative d(d + 1) Poisson’s ratio switching can be observed. dN − c = + NM − N , [1] 2 SS
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(Fig. boundary the from 3π , 3π = 4 1. + δ antb eaie Eq. negative, be cannot sin(C 3π , and , 4 and 3 cos(C 2 + √ + β 2 + ε. hslinearization This ε. B) c = + , ,wt naddi- an with 7, D), oconstruct To . n 2 − 4n N c + = = [6] [4] [5] [3] [2] 1. 2 7 2 inigseradtninisedo uebnigtrin a affect can bending/torsion, expe- pure hinges, of compliant instead Imperfect cells: tension unit and the shear of details riencing the of design emtyo i.5 h ih ry(akga)mtra srgd(compliant). rigid is specified. material otherwise gray) unless millimeters (dark in gray are light units The All 5. Fig. of geometry 3J Fig. 180 in shown 3J tiling Fig. same in shown 192 are tilings are such, the following as Fabrication. designed and and, of Design regime change Sample nonlinear length the require into modes extended zero-energy be other infinitesimal. to the rods of the All S1 finite. 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APPLIED PHYSICAL SCIENCES Table 1. Mechanical properties of compliant hinges of Fig. 5 εtransverse. We use these quantities to compute the nonlinear Poisson’s ratio ν := − ∂ε /∂ε averaged over applied strain deformation steps Stiffness Elite double 32 Agilus 30 transverse axial of δεaxial = 0.005. Fast Slow Fast Slow Hinge Material Properties. We measure the mechanical properties of a Bending (Nmm/rad) 24 20 120 20 single compliant hinge of the sample in Fig. 5 using the same uni- ±1 ±2 ±25 ±2 axial testing device. We perform three different experiments: bend- Tension (N/mm) 31 24 630 44 ing, tension, and shear. For bending, we pull on the hinge up to ±2 ±1 ±150 ±10 0.30-rad deflection at rates of 0.002 and 20 mm/s. For tension and Shear (N/mm) 10 8.1 190 18 shear, we pull on the hinge up to 1.0-mm deflection at rates of ±1 ±0.4 ±30 ±3 0.002 and 20 mm/s. We report the average of the measured stiffnesses in Table 1. Bending, tensile, and shear stiffness under fast (20 mm/s) and slow (0.002 We observe that the stiffnesses at long timescales are similar for both mm/s) loading rate. types of hinges. However, at short timescales, the stiffnesses of the elas- tic (Elite Double 32) hinges increases only by approximately 20 to 30% compared to long timescales, while the stiffnesses of the viscoelastic (Agilus 30) hinges can increase by more than 1,300%. This difference the linear and nonlinear deformation of mechanisms significantly (39). For in loading-rate dependency allows us to obtain the switchable response the samples of Figs. 3 and 5, we negate these effects in two ways. First observed in Fig. 5. of all, the central unit cells are made from a much more rigid material than the elastic hinges, such that the central unit cells do not deform Simulating Oligomodal Metamaterials. We use dynamic explicit nonlinear during loading. Next, we use tapered rigid parts as seen in Fig. 8. A finite element methods (Abaqus 2018, Dassault Systemes)` to predict and central narrow section between the rigid sections greatly increases resis- design the dynamic response of oligomodal metamaterials, as well as to tance to shear and tension while offering only a negligible resistance to study the effects of changing material and geometry parameters. We model bending. the geometry of the sample of Figs. 4 and 5 as seen in Fig. 9 using 2D tri- When the actuation is aligned with the primitive lattice, the bidomain angular quadratic elements (CPE6). Plane strain elements are used as the mode leads to shear–compression coupling. However, such a mode in a 45◦ rigid material to prevent the hinges from expanding and contracting in real- orientation can lead to a positive Poisson’s ratio, which is markedly differ- ity (as in Fig. 5). A mesh density featuring at least four elements through ent from the rotating-squares mode, that has a negative Poisson’s ratio (SI the hinge thickness and at least eight elements through the hinge thick- Appendix). Therefore, we choose to rotate the sample by 45◦ in Fig. 5. ness at the center is used, after performing a mesh convergence study Without imperfection, the configuration in Fig. 5 would have an inherent showing that mesh refinement leads to little to no differences in global buckling preference toward the rotating-squares mode of Fig. 5C. How- deformation. ever, this preference is tuned by designing an imperfection (40) favoring We model the stiff plastic components as a linear elastic material the bidomain mode of Fig. 5D, which has been designed using nonlinear vis- with a Young’s modulus of E = 1 GPa and Poisson’s ratio ν = 0.3. The coelastic finite element methods (Abaqus; see SI Appendix). Furthermore, as hinges are modeled as flexible nearly incompressible hyperelastic mate- the samples are tested horizontally, the bottom of the sample features small rials [Abaqus specific implementation of Arruda–Boyce material (41)], pockets with 5-mm steel ball bearings to reduce the effects of friction with with a material large strain stiffening parameter λm = 2 and a Pois- the bottom surface. son’s ratio ν = 0.47, corresponding to a ratio of the bulk and shear We produce the samples in Fig. 3 using additive manufacturing using modulus K/µ = 15.7. The elastic hinges (black in Fig. 9B) have E = 1 a Stratasys Objet500 Connex3 3D printer. The hinges are produced from a MPa. The viscoelasticity of the viscoelastic hinges (gray in Fig. 9B) is highly viscoelastic rubber-like PolyJet Photopolymer (Stratasys Agilus 30), accounted for using a three-term Prony series, with relaxation strengths −1 which is cocured with the central parts, which are made from a much η123 = (0.45, 0.26, 0.10) and timescales τ123 = (0.047, 0.97, 18) s (26). To stiffer Photopolymer (Stratasys VeroWhitePlus). The square hinges and cen- study the effects of changing material parameters, we vary the instan- tral parts of the sample of Fig. 5 are produced in the same way. However, taneous Young’s modulus between E0 = 0.25 MPa to 4.0 MPa, spanning the triangular hinges of the sample of Fig. 5 are cast from an elastic two- over the real E0 = 3.25 MPa (26), while the results of Figs. 4–6 apply component silicon-based rubber (Zhermack Elite Double 32), and are then E0 = 1 MPa. glued to the sample using Wacker Elastosil E43 glue. We anticipate that Reference points are created in the middle of the top and bottom stiff advances in additive manufacturing, using, for instance, silicone resins, will square segments (yellow in Fig. 9A), which are rigidly coupled to all nodes allow single-step manufacturing of the elastic–viscoelastic samples of Fig. 5 on these square elements, with the boundary conditions visualized in white. in the near future. The bottom reference nodes are restricted from moving in a vertical direc- tion, and one is also constrained horizontally. The top reference nodes are Experimental Methods. We compress the periodic bimode sample in Fig. 3 homogeneously compressed vertically using displacement control. All nodes M–O using a uniaxial testing device (Instron 5943 with a 500-N load cell), are free to rotate. with laser-cut Plexiglas indenters, denoted by the white arrows in the figure, at a strain rate of ε˙ = 1.30 · 10−3s−1 to a strain of ε = 0.026. The sample is positioned horizontally to enhance overall mechanical stability, with gravity preventing out-of-plane buckling. The quasiperiodic sample in SI Appendix, A B Fig. S1 is produced and tested in a similar way. However, the sample is compressed, instead, in the direction from the upper left to the bottom right. The sample in Fig. 5 is compressed using the same uniaxial testing machine with straight surfaces on the top and bottom, at various strain rates. The sample is also positioned horizontally to enhance overall mechan- ical stability, with gravity also preventing out-of-plane buckling. For Fig. 3 (Fig. 5), the tests are recorded using a high-resolution 3,858 × 2,748 (2,048 × 2,048) monochrome CMOS camera Basler acA3800-14um with Fuji- non 75-mm lens (acA2040-90um with Fujinon 50-mm lens), inducing a spatial resolution of 0.07 mm (0.1 mm). Using particle tracking (OPENCV and Python), we extract and track the four central holes of the stiff sections, which we use to calculate the position and ellipticity of all of the pores, whose reconstruction has been overlaid in Fig. 3 (Fig. 5). For Fig. 5, we track the average horizontal (vertical) distance between the centers of the ellipses on the sides (top and bottom) to compute Fig. 9. Dynamic oligomodal finite element model: (A) modeled geometry the width (height) of the sample, from which we extract the compres- with boundary conditions and nodes used for computing polarization with
sive axial strain εaxial and the strain transverse to the loading direction, (B) enlarged supercell with material correspondence.
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