WO2023092232A1 - Tessellations of unit chains having folded and unfolded configurations - Google Patents

Abstract

A unit chain for a tessellation for a metamaterial, has: a plurality of panels distributed about a central polygonal aperture having an even number of edges, the plurality of panels including: quadrilateral panels each extending from a respective one of the edges and away from the central polygonal aperture, and triangular panels each secured to two adjacent quadrilateral panels via hinges, the triangular panels interspaced between the quadrilateral panels, the quadrilateral panels pivotable relative to the triangular panels via the hinges, the unit chain having a flat configuration in which the plurality of panels are parallel to one another and a folded configuration in which a first group of the triangular panels are contained in a first plane and a second group of the triangular panels are contained in a second plane offset from the first plane, and in which the triangular panels are non-parallel to the quadrilateral panels.

Description

TESSELLATIONS OF UNIT CHAINS HAVING FOLDED AND UNFOLDED CONFIGURATIONS

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims benefit from United States provisional application number 63/283,337 filed on November 26, 2021 , the entire contents of which are incorporated by reference herein.

TECHNICAL FIELD

[0002] This disclosure generally relates to the field of foldable metamaterial and, more particularly, to reconfigurable materials based on origami and kirigami.

BACKGROUND

[0003] Origami and kirigami, the arts of folding and cutting paper, have inspired the development of a plethora of scale-invariant reconfigurable materials and structures that can deploy either spatially or in-plane. Their application spans a multitude of sectors across- disciplines, from mechanical memories, robotic actuators, thermally tunable structures, multistable devices, complex 3D geometries and programmable surfaces to flexible electronics. Origami crease and kirigami cut patterns have been also used to design mechanical metamaterials with distinct geometric and mechanical properties, such as reconfigurability, flatfoldability and bistable auxeticity among others.

[0004] Existing origami-inspired metamaterials have been proposed to offer a certain level of programmability, yet they are unable to attain concurrently rigid-foldability, flat-foldability and loadbearing capacity along the deployment direction. For example, rigid-foldable material systems with multiple degrees of freedom have been shown to be either floppy or in need of precise control of the folding sequence, a characteristic that severely limits their capacity to withstand loads acting along multiple directions. Non-flat-foldable concepts, on the other hand, have limited reconfigurability, making their size and volume large, as opposed to flat-foldable materials which reduce both. In addition, most of the existing concepts that use structural instability or the Kreseling pattern to achieve reconfigurability, are essentially non-rigid-foldable. To fold, they must overcome a large energy barrier that bends and stretches their panels; their reliance on panel compliance therefore sacrifices load-bearing capacity. On the other hand, foldable patterns that offer some load resistance, can do so in certain directions only; they collapse along others, and loose stiffness in the direction of deployment. This aspect can have a problematic outcome in applications where the load direction is uncertain during the entire service life, as a slight change in the direction of the acting forces, for example, can spoil structural performance, and unexpectedly turn a stiff configuration into a floppy one. Improvements are therefore sought.

SUMMARY

[0005] This disclosures presents a framework for designing a topological class of rigidly flat- foldable metamaterials that may be reprogrammed in-situ to reconfigure along multiple directions, some flat-foldable and others lockable, the latter may be multi-directionally stiff even along the deployment direction. The underpinning concept combines notions of origami and kirigami to introduce a crease pattern that is built cellular in its flat configuration, and then stacked with the minimum number of layers to steer folding along one trajectory only. The concept introduces excisions in the form of shaped voids in an origami crease with a twofold benefit. First, the voids may relax the deformation constraints enacted by the rigidity of the faces of the parent origami; second, the intracellular spaces formed by the emergent voids may enable face contact. Besides load-bearing capacity, the disclosed concepts may bring about additional hallmarks including topology and symmetry switch, which altogether may enlarge the degree of in-situ programmability of stiffness, permeability, yield strength and other properties. Finally, a simple yet effective fabrication process that may be automated to impart three-dimensionality in the flat configuration.

[0006] In one aspect, there is provided a unit chain, comprising: a plurality of plates circumferentially distributed about a central polygonal aperture, the central polygonal aperture having an even number of edges, the plurality of plates including: quadrilateral plates each extending from a respective one of the edges and away from the central polygonal aperture, and triangular plates each secured to two adjacent quadrilateral plates via hinges such that the triangular plates are interspaced between the quadrilateral plates, the unit chain having a flat configuration in which all of the plurality of plates are co-planar and a folded configuration in which a first group of the triangular plates are contained in a first plane and a second group of the triangular plates are contained in a second plane offset from the first plane.

[0007] In some embodiments, in the folded configuration, dihedral angles defined between successive triangular plates and quadrilateral plates are less than 90 degrees and in which at least two of the triangular plates of the first group are in abutment against one another and/or in which at least two of the triangular plates of the second group are in abutment against one another. [0008] In some embodiments, in the folded configuration, external pressure exerted on the unit chain to move the first plane toward the second plane is opposed by a cooperation of the triangular plates and/or the quadrilateral plates being in abutment against one another.

[0009] In another aspect, there is provided a tessellation comprising: unit chains each having: a plurality of plates circumferentially distributed about a central polygonal aperture, the central polygonal aperture having an even number of edges, the plurality of plates including: quadrilateral plates each extending from a respective one of the edges and away from the central polygonal aperture, and triangular plates each secured to two adjacent ones of the quadrilateral plates via hinges such tht the triangular plates are interspaced between the quadrilateral plates, each two adjacent ones of the unit chains secured to one another via two respective triangular plates, the tessellation having a flat configuration in which all of the plurality of plates are co-planar and a folded configuration in which a first group of the triangular plates are contained in a first plane and a second group of the triangular plates are contained in a second plane offset from the first plane.

[0010] In some embodiments, in the folded configuration, dihedral angles defined between successive triangular plates and quadrilateral plates is less than 90 degrees and in which at least two of the triangular plates of the first group are in abutment against one another and/or in which at least two of the triangular plates of the second group are in abutment against one another.

[0011] In some embodiments, the two adjacent ones of the unit chains are indirect neighbors, a first unit chain of the unit chains having at least one direct neighbor, the first unit chain and the at least one direct neighbor sharing a quadrilateral panel.

[0012] In some embodiments, the unit chains are first unit chains of a first layer of unit chains, the tessellation comprising a second layer of unit chains disposed below the first layer of unit chains, the second layer of unit chains having second unit chains, the second unit chains having triangular plates and quadrilateral plates interspaced between the triangular plates, a first subset of the triangular plates of the first unit chains secured to a first subset of the triangular plates of the second unit chains.

[0013] In some embodiments, the tessellation includes a third layer of unit chains disposed below the second layer of unit chains, the third layer of unit chains having third unit chains including triangular plates and quadrilateral plates interspaced between the triangular plates, a first subset of the triangular plates of the third unit chains secured to a second subset of the triangular plates of the second unit chains. [0014] In yet another aspect, there is provided a tessellation comprising: unit chains each having: a plurality of plates circumferentially distributed about a central polygonal aperture, the central polygonal aperture having an even number of edges, the plurality of plates including: quadrilateral plates each extending from a respective one of the edges and away from the central polygonal aperture, and triangular plates each secured to two adjacent ones of the quadrilateral plates via hinges such that the triangular plates are interspaced between the quadrilateral plates, the unit chains including a first unit chain and a second unit chain disposed below the first unit chain, a first subset of the triangular plates of the first unit chain secured to a first subset of the triangular plates of the second unit chain.

[0015] In some embodiments, the unit chains include a third unit chain disposed below the second unit chain, a first subset of the triangular plates of the third unit chain secured to a second subset of the triangular plates of the second unit chain, the second subset different from the first subset.

[0016] In another aspect, there is provided a tessellation comprising: a plurality of layers of unit chains stacked above one another, each unit chains of the plurality of layers of unit chains having: a plurality of plates circumferentially distributed about a central polygonal aperture, the central polygonal aperture having an even number of edges, the plurality of plates including: quadrilateral plates each extending from a respective one of the edges and away from the central polygonal aperture, and triangular plates each secured to two adjacent ones of the quadrilateral plates via hinges such that the triangular plates are interspaced between the quadrilateral plates, each two adjacent ones of the unit chains secured to one another via two respective triangular plates, the plurality of layers of unit chains including at least a first layer and a second layer, a first subset of the triangular plates of the unit chains of the first layer secured to a first subset of the triangular plates of the unit chains of the second layer, the tessellation having a flat configuration in which all of the plurality of plates are co-planar and a folded configuration in which the first subset of the triangular plates of the unit chains of the first layer and the first subset of the triangular plates of the unit chains of the second layer are in a first plane, a second subset of the triangular plates of the unit chains of the first layer are in a second plane offset from the first plane, and a second subset of the triangular plates of the unit chains of the second layer are in a third plane offset from both of the first plane and the second plane. [0017] In some embodiments, the tessellation includes a third layer of unit chains, a first subset of the triangular plates of the unit chains of the third layer secured to the second subset of the triangular plates of the second layer.

[0018] In a first aspect, there is provided a unit chain for a tessellation for a metamaterial, comprising: a plurality of panels circumferentially distributed about a central polygonal aperture, the central polygonal aperture having an even number of edges, the plurality of panels including: quadrilateral panels each extending from a respective one of the edges and away from the central polygonal aperture, and triangular panels each secured to two adjacent quadrilateral panels via hinges, the triangular panels interspaced between the quadrilateral panels, the quadrilateral panels pivotable relative to the triangular panels via the hinges, the unit chain having a flat configuration in which the plurality of panels are parallel to one another and a folded configuration in which a first group of the triangular panels are contained in a first plane and a second group of the triangular panels are contained in a second plane offset from the first plane, and in which the triangular panels are non-parallel to the quadrilateral panels.

[0019] The unit chain described above may include any of the following features, in any combinations.

[0020] In some embodiments, in the folded configuration, one or more of: a first triangular panel of the triangular panels abuts a second triangular panel of the triangular panels; and a first quadrilateral panel of the quadrilateral panels abuts a second quadrilateral panel of the quadrilateral panels.

[0021] In some embodiments, in the folded configuration, dihedral angles defined between successive triangular panels and quadrilateral panels are less than 90 degrees.

[0022] In some embodiments, in the folded configuration, at least two of the triangular panels of the first group are in abutment against one another and/or in which at least two of the triangular panels of the second group are in abutment against one another.

[0023] In some embodiments, in the folded configuration, external pressure exerted on the unit chain to move the first plane toward the second plane is opposed by a cooperation of one or more of the triangular panels and the quadrilateral panels being in abutment against one another.

[0024] In some embodiments, the plurality of panels are parts of a single monolithic body. [0025] In some embodiments, the hinges are living hinges.

[0026] In some embodiments, the hinges are defined by fold lines.

[0027] In a second aspect, there is provided a tessellation comprising: unit chains interconnected to one another, a unit chain of the unit chains having: a plurality of panels circumferentially distributed about a central polygonal aperture, the central polygonal aperture having an even number of edges, the plurality of panels including quadrilateral panels each extending from a respective one of the edges and away from the central polygonal aperture, and triangular panels each secured to two adjacent ones of the quadrilateral panels via hinges, the triangular panels are interspaced between the quadrilateral panels, the tessellation having a flat configuration in which all of the plurality of panels are parallel and a folded configuration in which a first group of the triangular panels are contained in a first plane and a second group of the triangular panels are contained in a second plane offset from the first plane and in which the triangular panels are non-parallel to the quadrilateral panels.

[0028] The tessellation described above may include any of the following features, in any combinations.

[0029] In some embodiments, each pair two unit chains of the unit chains disposed adjacent one another are interconnected by one or more of a triangular panel of the triangular panels and a quadrilateral panel of the quadrilateral panels.

[0030] In some embodiments, the pair of the two unit chains share a common quadrilateral panel, the central aperture having at most six edges.

[0031] In some embodiments, a triangular panel of a first unit chain of the pair of the two unit chains is connected to a triangular panel of a second unit chain of the pair of the two unit chains, the central aperture having more than six edges.

[0032] In some embodiments, in the folded configuration, for the unit chain of the unit chains, one or more of: a first triangular panel of the triangular panels abuts a second triangular panel of the triangular panels; and a first quadrilateral panel of the quadrilateral panels abuts a second quadrilateral panel of the quadrilateral panels.

[0033] In some embodiments, in the folded configuration, dihedral angles defined between successive triangular panels and quadrilateral panels is less than 90 degrees. [0034] In some embodiments, in the folded configuration, at least two of the triangular panels of the first group are in abutment against one another and/or in which at least two of the triangular panels of the second group are in abutment against one another.

[0035] In some embodiments, the unit chains are first unit chains of a first layer of unit chains, the tessellation comprising a second layer of unit chains disposed below the first layer of unit chains, the second layer of unit chains having second unit chains, the second unit chains having triangular panels and quadrilateral panels interspaced between the triangular panels, a first subset of the triangular panels of the first unit chains secured to a first subset of the triangular panels of the second unit chains.

[0036] In some embodiments, the tessellation includes a third layer of unit chains disposed below the second layer of unit chains, the third layer of unit chains having third unit chains including triangular panels and quadrilateral panels interspaced between the triangular panels, a first subset of the triangular panels of the third unit chains secured to a second subset of the triangular panels of the second unit chains.

[0037] In some embodiments, the plurality of panels are parts of a single monolithic body.

[0038] In some embodiments, the hinges are living hinges.

[0039] In some embodiments, the hinges are defined by fold lines.

[0040] Many further features and combinations thereof concerning the present improvements will appear to those skilled in the art following a reading of the instant disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figs. 1A to 1C illustrate how a unit chain in accordance with one embodiment is built;

[0042] Figs. 1 D and 1 E show the unit chain of Fig. 1 C in a partially folded configuration and in a folded configuration;

[0043] Figs. 2A to 2E show planar views of unit chains in accordance with a plurality of possible embodiments, the unit chains being shown in their flat configuration;

[0044] Figs. 3A to 3E show three dimensional views of the unit chains of Figs. 2A to 2E in a partially folded configuration; [0045] Figs. 4A to 4E show three dimensional views of the unit chains of Figs. 2A to 2E in their folded configuration;

[0046] Figs. 5A to 5E are planar views of tessellations generated with the unit chains of Figs. 2A to 2E, the tessellations shown in their flat configuration;

[0047] Figs. 6A to 6E are three dimensional views of the tessellations of Figs. 5A to 5E in their folded configuration;

[0048] Fig. 7A illustrates an assembly process to generate a stacking of three of the unit chains of Fig. 2B and illustrates many possible folded and flat configurations;

[0049] Fig. 7B is a graph illustrating a variation of a number of degrees of freedom (m) as a function of a number of stacked layers (n) for a given N even-sided primitives (N= 4, 6, 8, 10, 12);

[0050] Fig. 7C is a three dimensional view of the unit chain of Fig. 2B shown in a partially folded configuration;

[0051] Fig. 7D is a graph illustrating a variation of a number of post-bifurcation modes as a function of a number of sides of the N even-sided primitives;

[0052] Figs. 8A to 8C are top views of a tessellation of the unit chains of Fig. 2A shown in a plurality of configurations;

[0053] Figs. 8D to 8G are top views of a tessellation of the unit chains of Fig. 2B shown in a plurality of configurations;

[0054] Figs. 8H to 8M are top views of a tessellation of the unit chains of Fig. 2C shown in a plurality of configurations;

[0055] Figs. 8N to 8U are top views of a tessellation of the unit chains of Fig. 2D shown in a plurality of configurations;

[0056] Fig. 9A is a graph illustrating a dimensionless total energy landscape subjected to two representative in-plane biaxial forces as a function of the dihedral angles;

[0057] Figs. 9B and 9C are mode phase diagrams; [0058] Fig. 9D is a graph illustrating a total energy landscape of a unit subjected to uniformly applied out-of-plane loads;

[0059] Fig. 10A is a top view of a tessellation made with the unit chains of Fig. 2A;

[0060] Fig. 10B is a graph illustrating a stress-strain curve of the tessellation of Fig. 10A measured in two locked configurations;

[0061] Fig. 10C is a graph illustrating a stress-strain curve of a tessellation of the unit chains of Fig. 2B measured in two locked configurations;

[0062] Fig. 10D is a graph illustrating a normalized compressive Young’s modulus and normalized yield strength as a function of the density of the tessellations;

[0063] Fig. 10E illustrates directions of the applied compressive loads;

[0064] Fig. 10F illustrates stress-strain responses of the tessellations subjected to the loads illustrated in Fig. 10E;

[0065] Fig. 10G shows top views of the tessellations of Fig. 10A in mixed-mode configurations;

[0066] Fig. 10H is a graph illustrating an out-of-plane normalized compressive Young’s modulus for the seven (1 to 7) configurations of Fig. 10G;

[0067] Fig. 101 is a graph illustrating the normalized open channel area in the out-of-plane direction for the seven configurations of Fig. 10G;

[0068] Fig. 11 A is a planar view of a unit chain in accordance with another embodiment;

[0069] Fig. 11 B is a planar view of a tessellation of the unit chains of Fig. 11 A shown in a flat configuration; and

[0070] Fig. 11C is a top view of the tessellation of Fig. 11 B shown in a folded configuration. DETAILED DESCRIPTION

[0071] Introduction

[0072] Referring to Figs. 1 A to 1 E, to generate a rigidly-foldable unit that is flat-foldable and can lock into a stiff state upon panel self-contact, a primitive network of bars 1 connected in a planar loop is obtained. The network is a planar N-bar linkage that forms a regular N even-sided polygon (e.g., square, hexagon, etc). Each of the four bars 1 has length a and enclosing a square void. Extruding each bar outward to the length b (Fig. 1 B) in the x-y plane and at a given angle Φ as shown in Fig. 1 B yields four quadrilateral panels, herein parallelograms, which

may be connected using triangles, herein isosceles triangles, with a vertex angle of By

prescribing the folding profiles (dash lines) at each boundary between interfacing panels, a planar assembly of rigid surfaces that altogether can spatially fold along their connecting valley (V) and mountain (M) folding lines is obtained. The conceptual process can be thought as complementary to the fabrication steps, where the unit void (e.g., square aperture) is first excised from a planar sheet of paper (kirigami cuts), and then folded along prescribed dashed lines (origami folds), thereby generating a hybrid architecture.

[0073] The fold lines, which are shown in dashed lines, of both the valleys V and the mountains M panels may enable the system to act as a kinematic chain. Here, its configurational changes is defined using m independent dihedral angles θ₁, θ₂, ..., θ_m. Each dihedral angle specifies the angle between the triangular panel and its adjacent quad panel as shown in Fig. 1 D, and m denotes the mobility or nontrivial degrees of freedom (DoF) of the kinematic chain that exclude rigid-body motions. Since two dihedral angles, one concave and the other convex, can be always identified between connected panels, a dihedral angle is considered as convex. In addition, it has been assumed the mountain and valley fold lines are constrained to remain on two parallel planes during folding. This strategy, as explained later, may be enforced by using unit chain stacking, and it may enable the unit to engage motion along a single DoF. In this case, the out-of-plane rise h can be expressed as a function of three geometric parameters by: h = a sin Φ sin θ₁. (1)

[0074] During folding, a process that decreases θ₁ , the “unit kinematic chain”, in short

“unit chain”, can reach a lock state, denoted with the superscript “L”, where self-contact between panels forbids any further motion (Fig. 1 E). In a lock state, the acute dihedral angles are given by θ^L = cos^{- 1} (cot Φ tan (2)

[0075] The generative process illustrated in Figs. 1A to 1 C for a unit chain with square primitive may be abstracted to other primitives, i.e., regular N even-sided polygons, by merely varying N. The outcome is a class of planar unit chains that spatially reconfigure within the voids and lock upon self-contact of their panels, thus behaving stiff under compression.

[0076] Unit Chains

[0077] Referring to Fig. 2A, a unit chain 10 is described in more detail. The unit chain

10 includes a plurality of plates or panels that are circumferentially distributed about a central polygonal aperture 11. The central polygonal aperture 11 is bounded by edges 12; a number of the edges being even (e.g., 4, 6, 8, ...). The unit chain 10 includes quadrilateral panels 13 each extending from a respective one of the edges 12 and away from the central aperture 1 1 . The unit chain 10 includes triangular panels 14 that are each secured to two adjacent ones of the quadrilateral panels 13 via hinges 15 (dashed lines). The triangular panels 14 are interspaced between the quadrilateral panels. The quadrilateral and triangular panels 13, 14 may be part of a single monolithic body of the unit chain. The hinges 15 may be fold lines, living hinges, or mechanical hinges in some embodiments. The expression “living hinges”, also called “integral hinges” are created by a thin flexible hinge made from the same material as the two pieces it connects. It may be at thinned or cut to allow the rigid pieces to bend along the line of the hinge. The quadrilateral panels 13 are pivotable relative to the triangular panels 14 via the hinges 15.

[0078] The unit chain 10 has a flat or unfolded configuration in which all of the plurality of panels are parallel to one another and a folded configuration in which a first group of the triangular panels 14 are contained in a first plane (e.g., valleys V, Fig. 1 C) and a second group of the triangular panels 14 are contained in a second plane (e.g., mountains M, Fig. 1 C) offset from the first plane. In the folded configuration, the triangular panels 14 are non-parallel to the quadrilateral panels 13. In this folded configuration, dihedral angles defined between successive triangular panels and quadrilateral panels may be less than 90 degrees. In the folded configuration, at least two of the triangular panels 14 of the first group are in abutment against one another and/or at least two of the triangular panels 14 of the second group are in abutment against one another. This creates a locked configuration in which a compressive force exerted on the unit chain 10 to move the first plane towards the second plane is resisted by the different panels being in abutment against one another. In otherwords, in the folded configuration, external pressure exerted on the unit chain to move the first plane toward the second plane is opposed by a cooperation of the triangular panels and/or the quadrilateral panels being in abutment against one another. In other words, in the folded configuration, one or more of a first triangular panel of the triangular panels abuts a second triangular panel of the triangular panels, and a first quadrilateral panel of the quadrilateral panels abuts a second quadrilateral panel of the quadrilateral panels. In some embodiments, in the folded configuration, one or more of an edge and/or a corner of a first triangular panel of the triangular panels abuts an edge and/or a corner of a second triangular panel of the triangular panels; and an edge and/or a corner of a first quadrilateral panel of the quadrilateral panels abuts an edge and/or a corner of a second quadrilateral panel of the quadrilateral panels.

[0079] Figs. 2B to 2E illustrate different embodiments of unit chains. For instance, a unit chain having a hexagonal aperture is shown at 20, a unit chain having an octagonal aperture is shown at 30, a unit chain having a decagonal aperture is shown at 40, and a unit chain having a dodecagonal aperture is shown at 50. Each of those unit chains are built similarly to the unit chain 10 described above with reference to Fig. 2A and includes quadrilateral panels joined to triangular panels via hinges, or fold lines 15. Figs. 3A to 3E illustrate the same unit chains 10, 20, 30, 40, 50 as they are being folded toward their folded configuration. Figs. 4A to 4E illustrate the same unit chains 10, 20, 30, 40, 50 in their folded configuration. As explained above, self-contact between the different panels may allow the unit chains to resist compression in the folding direction.

[0080] In all embodiments, the unit chain has a flat configuration in which the plurality of panels are parallel to one another and a folded configuration in which a first group of the triangular panels are contained in a first plane and a second group of the triangular panels are contained in a second plane offset from the first plane and in which the triangular panels are nonparallel to the quadrilateral panels.

[0081] Tessellations

[0082] Referring now to Figs. 5A to 5E, tessellations of the above-described unit chains

10, 20, 30, 40, 50 are shown in their flat or unfolded configurations. In other words, each of these unit chains may be grouped together to form an assembly of unit chains of the same kind. Figs. 6A to 6E illustrate each of the tessellations 110, 120, 130, 140, 150 described above with reference to Figs. 5A to 5E, but in their folded configurations. [0083] Referring to Fig. 5A, a tessellation of the unit chains 10 of Fig. 2A is shown at

1 10. One of the unit chains 10 is highlighted in Fig. 5A. The unit chains 10 of the tessellation 110 may be either a peripheral unit chain or a central unit chain. A peripheral unit chain is a unit chain located on the edges of the tessellation whereas a central unit chain is surrounded all around by neighboring unit chains 10. In the embodiment shown, a unit chain of the tessellation 1 10 may have four direct neighbors and four indirect neighbors. Direct neighbors correspond to two neighboring unit cells 10 sharing a common quadrilateral panel. Indirect neighbors correspond to two neighboring unit cells 10 secured via two of their triangular panels. These two triangular panels are therefore joined together to define a quadrilateral panel. In the present case, the two triangular panels of two indirect neighbors may have a shape corresponding to that of the central apertures of the unit cell 10. One of the direct neighbors is referred to with reference numeral 10A and one of the indirect neighbors is referred to with reference numeral 10B.

[0084] Referring now to Fig. 5B, a tessellation of the unit chains 20 of Fig. 2B is shown at 120. One of the unit chains 20 is highlighted in Fig. 5B. In the embodiment shown, a central unit chain may have six direct neighbors that share a common quadrilateral panel and six indirect neighbors. One of the direct neighbors is referred to with reference numeral 20A and one of the indirect neighbors is referred to with reference numeral 20B. The unit chain and one of its indirect neighbors 20B are connected to one another via the quadrilateral panel of a corresponding direct neighbor 20A. In the present embodiment, each of the triangular panels of a central unit chain is secured to a triangular panel of a respective indirect neighbor via a quadrilateral panel of a respective direct neighbor.

[0085] Referring now to Fig. 5C, a tessellation of the unit chains 30 of Fig. 2C is shown at 130. One of the unit chains 30 is highlighted in Fig. 5C. In the embodiment shown, a unit chain 30 may have up to four indirect neighbors. None of the unit chain has direct neighbors. In other words, for each of the unit chains 30, the quadrilateral panels are not shared with any other one of the unit chains of the tessellation 130. In the present embodiment, each other one of the triangular panels of a central unit chain is secured to a triangular panel of a neighboring unit chain. In other words, each of the triangular panels that is secured to a corresponding triangular panel of an indirect neighbor is disposed circumferentially between two triangular panels that are free of direct connection with the other unit chains 30. The triangular panels may therefore include connecting triangular panels and non-connecting triangular panels that are circumferentially interspaced around the central aperture. It will be appreciated that the two connecting triangular panels of two neighboring unit chains 30 that are secured together form a secondary quadrilateral panel that has a shape that may differ from that of the quadrilateral panel of the unit chains 30. Here, the tessellation 130 is provided in a matrix form of four unit chains 30 by four unit chains 30. However, the tessellation 130 may include any suitable number of unit chains 30 (e.g., 3x5, 10x10, etc).

[0086] Referring now to Fig. 5D, a tessellation of the unit chains 40 of Fig. 2D is shown at 140. One of the unit chains 40 is highlighted in Fig. 5D. In the embodiment shown, a unit chain 40 may have up to four indirect neighbors. None of the unit chains 40 has direct neighbors. In other words, for each of the unit chains 40, the quadrilateral panels are not shared with any other one of the unit chains 40 of the tessellation 140. In the present embodiment, the triangular panels includes connecting triangular panels and non-connecting triangular panels that are circumferentially interspaced around the central aperture. It will be appreciated that the two connecting triangular panels of two neighboring unit chains 40 that are secured together form a secondary quadrilateral panel that has a shape that may differ from that of the quadrilateral panels of the unit chains 40. In the present case, out of ten triangular panels, four are connecting triangular panels and six are non-connecting triangular panels that are free of connection with adjacent unit chains 40. The non-connecting triangular panels are interspaced with the connecting triangular panels. In the embodiment shown, each connecting triangular panel is located between a single non-connecting triangular panel and a pair of non-connecting triangular panels. Here, the tessellation is provided in a matrix form of four unit chains 40 by four unit chains 40. However, the tessellation 140 may include any suitable number of unit chains 40 (e.g., 3x5, 10x10, etc).

[0087] Referring now to Fig. 5E, a tessellation of the unit chains 50 of Fig. 2E is shown at 150. One of the unit chains 50 is highlighted in Fig. 5E. In the embodiment shown, a unit chain 50 may have up to six indirect neighbors. None of the unit chains 50 has direct neighbors. In other words, for each of the unit chains 50, the quadrilateral panels are not shared with any other one of the unit chains 50 of the tessellation 150. In the present embodiment, the triangular panels includes connecting triangular panels and non-connecting triangular panels that are circumferentially interspaced around the central aperture. It will be appreciated that the two connecting triangular panels of two neighboring unit chains 50 that are secured together form a secondary quadrilateral panel that has a shape that may differ from that of the quadrilateral panels of the unit chains 50. In the present case, out of twelve triangular panels, six are connecting triangular panels and six are non-connecting triangular panels that are free of connection with adjacent unit chains 50. The non-connecting triangular panels are interspaced with the connecting triangular panels. In the embodiment shown, each connecting triangular panel is located between two non-connecting triangular panels. Here, the tessellation is provided in a matrix form of four unit chains 50 by four unit chains 50. The rows are staggered. The tessellation 150 may include any suitable number of unit chains 50.

[0088] Tessellations may be generated from unit chains having any suitable number of triangular and quadrilateral panels without departing from the scope of the present disclosure.

[0089] Unit stacking

[0090] Referring now to Fig. 7A, a stacking of the unit chains 20 described above with reference to Fig. 2B is shown at 220. The assembly 220 is built by vertically stacking a plurality of the unit chains 20 of Fig. 2B above one another (vertical stacking). In the present embodiment, the assembly 220 includes three unit chains 20, but more or less unit chains 20 may be used. Understandably, a similar assembly may be constructed with any of the unit chains 10, 20, 30, 40, 50 described above with reference to Figs. 2A to 2E.

[0091] In the embodiment shown, the stacking 220 is built by securing triangular panels

14 of one unit chain 20 to triangular panels 14 of an adjacent unit chain 20. As described above, when folding the unit chain 20, some triangular panels 14 become valleys V (Fig. 1 C) and some triangular panels 14 become mountains M (Fig. 1 C). In the assembly 220, the valleys of a first unit chain 20 are secured to the mountains of a second unit chain 20 located below the first unit chain 20. Then, the valleys of the second unit chain 20 are secured to the mountains of a third unit chain 20 located below the second unit chain 20, and so on.

[0092] The tessellations disclosed in Figs. 5A and 5E may be vertically stacked upon one another. Such tessellations includes a plurality of layers (e.g., tessellations 110, 120, 130, 140, 150) of unit chains 10, 20, 30, 40, 50 stacked above one another. The plurality of layers of unit chains may include at least a first layer and a second layer. A first subset of the triangular panels of the unit chains of the first layer may be secured to a first subset of the triangular panels of the unit chains of the second layer. Such tessellations may have a flat configuration in which all of the plurality of panels are co-planar and a folded configuration in which the first subset of the triangular panels of the unit chains of the first layer and the first subset of the triangular panels of the unit chains of the second layer are in a first plane, a second subset of the triangular panels of the unit chains of the first layer are in a second plane offset from the first plane, and a second subset of the triangular panels of the unit chains of the second layer are in a third plane offset from both of the first plane and the second plane. In Figs. 6A to 6E, two to four layers are provided, although any suitable number of layers may be used. In some embodiments, the tessellation includes a third layer of unit chains, a first subset of the triangular panels of the unit chains of the third layer secured to the second subset of the triangular panels of the second layer.

[0093] Kinematic

[0094] Below the distinctive kinematics of the unit chains 10, 20, 30, 40, 50 is investigated. To do so, the dimensionless extrusion

is introduced and the notation

to discriminate between unit chains is used. N_N refers to the generic class of unit chains, where and Φ can assume any values. For demonstrative purpose, the focus is brought on a subset (Figs. 2A to 2E), namely the subclass defined by

and Φ = π/3, i.e., N_W(1, π/3) denoted as for simplicity, with N = 4, 6, 8, 10 and 12.

[0095] Upon folding, the panels of the unit chains 10, 20, 30, 40, 50 are in relative motion before coming into contact. To study the kinematics, assumptions are made. First, it is assumed the panels 13, 14 as infinitely rigid and the fold lines 15 act as rotational hinges. Second, to ease the formulation of the kinematic constraints, the unit chain is replaced with a triangulated network of inextensible elements connected through pin joints. Third, the edges of the triangular panels are modelled as bars, whereas the quad panels are replaced with two triangles satisfying the planarity condition of their interplanar angles overthe entire folding process. With the assumptions above, a generic pin-jointed network of inextensible bars is examined and the number of mechanisms i.e., DoFs, and types of motion, i.e., kinematic paths, the disclosed unit chain can attain in space is determined.

[0096] To determine the mobility of the unit chains 10, 20, 30, 40, 50, the rigidity matrix

R pertinent to its structural assembly is formulated. In general, the number of degrees of freedom, m, for a pin-jointed triangulated network is given by m = 3j - n_K - r, where j is the total number of joints, n_K the number of external kinematic constraints, and rthe rank of R. For the disclosed unit chains, m also represents the number of independent dihedral angles. For the unit chain 10 in Fig. 2A, m = 5 represents the number of mechanisms it can attain, i.e., its DoFs, m can be reduced to 1 if the mountain and valley M, V (Fig. 1 D) fold lines are constrained to remain on two parallel planes during the folding process. The specific values and the relationships these dihedral angles assume define distinct types of motion, as described below. [0097] There are specific trajectories the disclosed unit chains 10, 20, 30, 40, 50, 60 can follow during motion. Each of them can be uniquely defined by a relation between the independent parameters, in the present case the dihedral angles. For example, the equality of dihedral angles, i.e., θ₁ = ft, represents one type of motion, whereas θ₁ = π - θ₂ describes another type of motion. Each relationship between angles defines a given trajectory, namely a kinematic path. For example, the kinematic path can be defined by the relation of equality between its dihedral angles.

[0098] Still referring to Fig. 7A, ifthe triangular panels 14 shown in Fig. 2A do not remain parallel during folding, the unit chains 10, 20, 30, 40, 50 may be endowed with manifold DoFs, which for are five. Having too many DoFs, m, may be problematic, as the unit chains act as a multi-degree-of-freedom mechanism; in this case, the unit tends to be floppy, and folding may not be unequivocally and easily controlled for use as a reconfigurable load-bearing material. One way to prune m is to act along the third direction (z), and stack layers of unit chains one on the top of the other. Fig. 7A shows this strategy applied to the unit chain 20 (Fig. 2B) having

Here, the mountain facets of the triangular panels 14 of the top unit are bonded to the valley facets of the adjacent unit (below). By stacking and bonding three unit chains 20, the mountain and valley triangular panels 14 are set to lie in parallel planes. If made out of a real material with sufficiently high elastic modulus, the non-negligible thickness of the triangular panels 14 may have the effect of restricting the relative rotation range of two connected quad panels 13, allowing only equal and opposite rotation to satisfy kinematic compatibility and to avoid encountering the energy barrier caused by the bending or stretching panels during deformation.

[0099] The strategy above not only turns off DoFs, but may also provide the potency for a robust reconfiguration along a single route. Having now layers of unit chains, a generic multilayered N even-sided unit chain with n stacking layers may be denoted by N_Nn_n. Through a rigidity analysis of N_Nn_n, the role of layer stacking, n, on the DoFs may be assessed. The outcome of this investigation is shown in Fig 7B., where θ₁ / π/2 is assumed, and θ₁ = π/2 refers to the kinematic bifurcation. From the plot, the DoFs of the single layer unit chains increases linearly with N through the relation m = 2N - 3 which can be reduced to 1 if multiple layers are stacked upon them. Stacking layers can thus enact reconfigurability along one single path. The minimum number of layers, n, necessary to trim m to 1 is not unique, rather it depends on the unit chain primitive. Fig. 7B shows that for unit chains N₄ and N₆,

= 3, whereas for N_N>6,

= 5. The focus is now on multilayered units with a single DoF prior to bifurcation, denoted by

and investigate their behavior at the instant of kinematic bifurcation and post-bifurcation.

[0100] In the early stage of folding, the

chains 10, 20, 30, 40, 50 may travel along one route governed by

However, as soon as it reaches a configuration where all its dihedral angles are π/2, i.e., kinematic bifurcation, the DoFs instantaneously grow, and multiple kinematic paths become active. The post-bifurcation modes may be classified into either two-dimensional (2D) flat-foldable, or three-dimensional (3D) lockable. The first class collects modes where the multilayered chain N_Nn_fl can access a set of fully flat patterns that are distinct from the initial flat pattern. The second describes 3D lock states, where self-contact between adjacent panels at the end of their folding path imparts compressive stiffness.

[0101] The relations that define the post-bifurcation modes belonging to a given kinematic path for a single-DoF stacked unit chain

is studied below. As illustrated in Fig. 7A, in the pre- and post-bifurcation stages, the triangular panels 14 (six in dark green in the middle layer) in each set of mountains and valleys may remain parallel as

= 3. Three of them are mountain panels that lay in a mountain plane, and the others rest in a valley plane. The distance between them is h which can be calculated through relation (1). In a given plane, there are six fold lines (for the unit chain 20 of Fig. 2B) 15 (two per triangle) which form a total of six dihedral angles. The fold lines 15 that are parallel form a pair of dihedral angles. This pair may contain either equal or supplementary angles, a condition that defines the type of post-bifurcation mode. Equality of dihedral angles in a given pair may define regular modes, which in Fig. 7A belong to paths 1 and 2; this implies that the reconfiguration of the unit chain 20 may lead to a folded pattern that is compatible with its original tessellation. In contrast, supplementarity of dihedral angles in all pairs, i.e., the dihedral angles sum up to 180 in each pair, may give rises to irregular modes, and path 3 in Fig. 7A shows an example. Irregular modes may be attained only in a single unit chain but not in a tessellated pattern, as they forego folding congruence between the initial and the final pattern, revealing that the flat foldable tessellation cannot be unpacked to its initial pattern. Only one irregular mode may exist for with N > 4 and for N/2 equal to an odd

number, e.g., modes of shown in Fig. 7B.

[0102] Given the folding incompatibility of irregular modes, the focus is now brought on the regular counterparts and study the conditions that can be used for a given kinematic path to count the number of existing modes and define the characteristics of each of them. The goal is to demonstrate the existence of relations between distinct pairs of dihedral angles, which may in turn govern the kinematic paths

can access post-bifurcation. In the denomination of dihedral angles, acute angles are specified with

and obtuse angles with

The geometry of the units enforces the condition

= π during their entire range of motion. As an example, three pairs of dihedral angles are illustrated in Fig. 7C, two

and one

for a total of six dihedral angles. Since in

which has 1 DoF, each pair contains equal angles, all are equal as are all

A given sequence with angle pairs, e.g., two

and one

depicted in Fig. 7C, can be simply denoted by the series of angle pairs, e.g.,

and in compact form with the power indicating the repeated pairs, e.g.,

. This notation allows discriminating between kinematic modes that emerge at bifurcation. For example, (Fig. 7A) can travel along four regular modes:

obtuse angles are engaged only), acute angles are engaged only)

(2 acute angles

and 1 obtuse angle),

(1 acute and 2 obtuse), and one irregular, shown with

Modes containing identical pairs of dihedral angles belong to the same kinematic path, and have been designated by swapping Jis and Os, i.e.,

and belong to path 2, and and to path

1 .

[0103] With the notions above, the regular modes of a generic

may be characterized and the total number of possible reconfiguration modes may be determined. To do so, Polya enumeration theorem of combinatorics is used. This theory may enable to (i) count the regular modes and then (ii) define their kind, hence enabling differentiation among them. The problem of finding all independent regular modes of a generic unit can now be treated as

the classical necklace problem. Here the goal is to reconstruct the colored pattern of a necklace of a number of beads, each colored either in white or black, from the knowledge of a limited set of information. In the present case, the equivalent necklace is the disclosed unit chain with

N/2 coloured beads, and each color represents a type of dihedral angles, either

. By applying the Polya enumeration theory, all reconfiguration modes the disclosed

chain can attain from knowledge of N/2 numbers of

and

are determined. It is also assumed that the beads can be rotated around the necklace, and that the necklace can be flipped over. By applying this theory to for instance, all the possible modes can be collected in a generating function of the form

, which describes the four regular modes illustrated in Fig. 7 A, and where the sum of the powers of

in each mode is N/2. Similar results can be obtained for other unit chains

By using a predictor-corrector type incremental method implemented in an in-house Matlab routine, it is possible to visualize the post-bifurcation kinematic paths of

[0104] Referring now to Fig. 7D, to understand how the total number of post-bifurcation modes varies with N , the total number of lockable modes c^L and flat-foldable modes c^F versus the sides of the unit chain primitive, N is plotted. The best curve fits are included to provide phenomenological closed-form relations that characterize folding paths and differentiate lockable modes (c^L = 0.21exp(0.3N )) from flat-foldable modes (c^F = 0.54exp(0.16N )) as a function of N for a generic

The results show that the number of regular modes grows exponentially, hence providing a rich platform to program and design the disclosed class of lockable metamaterials.

[0105] Symmetry and Topology

[0106] Referring now to Figs. 8A to 8U, from a single multilayered unit chain the attention is turned to their periodic tessellations forming multilayered material systems. Multilayered unit chains are orderly connected to follow a tessellation pattern that replicates a given crystallographic arrangement of atoms. Figs. 8A, 8D, 8H, and 8N show top-views of representative patterns for N = 4, 6, 8 and 10. While several others also exist, here the focus is now brought on tessellations with the most compact pattern. The goal is to show that upon folding along a given path the disclosed material systems undergo switches in symmetry and apparent topology, both hallmarks of their in-situ programmability.

[0107] To study changes in symmetry, classical notions of crystallography are used.

Each pattern is formed by tessellating in plane a representative unit (darker region enclosed by a red boundary in Figs. 8A, 8D, 8H, and 8N) that is defined by the periodic vectors that preserve translational symmetry. Upon reconfiguration, the material system can access a new kinematic path at bifurcation that may cause the smooth transition of certain dihedral angle pairs from

to the result is a break in symmetry that takes place after bifurcation. This phenomenon is visualized by the patterns of Figs. 8B to 8C, 8E to 8G, 8I to 8M, and 8O to 8U, each denoted by their characteristic crystallographic point group and Schoenflies symbol. A variation in the lattice point groups (symmetry) translates into a change of the elastic constants defining the elastic tensor; the symmetry shift has thus endowed the material system with another set of elastic properties. For example, the lattice made of

(Figs. 8A, 8D, 8H, and 8N) is shown in one of its lock modes,

in the second column. In this mode

has C_2h symmetry, i.e., a two-fold rotational symmetry, and its elastic compliance matrix contains 13 elastic constants which corresponds to a monoclinic behavior. Upon switching to lock mode

(third column), its symmetry changes to C_4h , a four-fold rotational symmetry, resulting in an elastic compliance matrix with only 7 independent constants.

[0108] Besides symmetry, the other aspect that can be harnessed to enable in-situ programmability is the apparent change of topology the disclosed periodic systems undergo during reconfiguration after bifurcation. Here to define the topology of a given lattice configuration, the connectivity: Z_f is referred to, the number of faces that meet at an edge, which is equivalent to a fold-line as opposed to a cut-edge, and Z_e, the number of edges that meet at a vertex. The values of Z_e and Z_f are calculated for the corresponding spatial lattice upon the assumption that in the lock state coincident edges and vertices form a single edge and a single vertex. These values are shown on top of each configuration in Figs. 8A to 8U, separated by a vertical line after the respective Schoenflies symbol; the use of parentheses indicates that more than one connectivity value exist. A change in Z_e and Z_f denotes topology differentiation, whereas topology is invariant to other geometric parameters, such as the length of the primitive void and the panel thickness. Results show that transitioning from the partially folded unit to the lock modes enhances the connectivity. An increase in connectivity values from the partially folded state to the lock states attests a break in topology, an outcome that impacts the deformation mode of the panels and leads to a transition from bending to stretching as shown later by compression data from out-of-plane experiments. The versatility to impart topological changes when travelling across a kinematic bifurcation further contributes to an additional degree of in-situ tunability. This suggests that not only mechanical properties, such as Young’s modulus and yield strength, but also the mechanism of deformation can be switched on demand to address application requirements.

[0109] As a natural outcome of the changes described above, the disclosed systems may undertake variations in other mechanical properties, such as relative density and Poisson’s ratio. Changes in void size also appear in a lattice that folds into a given mode. For example, for a lattice made of

units, voids can be fully closed upon folding from

mode to

or

hence attesting a switch in permeability from a non-zero value to 0. Similarly, for the lattice made of units, a reduction of the voids area occurs when switching from

mode to or

and the voids completely close if the system flat-folds into mode While not quantified here,

this result qualitatively attests the versatility of the disclosed systems to tune on-the-fly flow permeability.

[0110] Rigidity and Load-bearing capacity [0111] Once folded into the lock state, the multi-layered unit chain N_Nn„ may inherit compressive load-bearing capacity as panels reach self-contact and prevent from further motion. The load-bearing capacity of the disclosed class of foldable material systems is now studied in relation to their layer stacking. The condition that guarantees their structural rigidity in one of their lock state can be determined by studying their pin-jointed counterpart made of a triangulated network. By doing so, it is possible to formulate the general problem that predicts the rigidity of a structure by theoretical analysis. While the units are subjected to compressive loads, it is assumed coincident bars as a single bar and multiple coincident joints as a single joint. The results can be expressed for a single unit chain as a function of the number of bars and joints at its lock and partially folded configurations along with the conditions of rigidity.

[0112] Carried out for the general lock state

where all dihedral angles are acute, the disclosed rigidity analysis reveals that N4 becomes rigid with a single layer, whereas at least two layers must be stacked for N₆ and three layers for N_N>6. Knowing the minimum number of stacked layers provides an essential guideline to make the disclosed unit chain stiff in a lock state under compression. The conditions to access lockable as well as flat-foldable paths become crucial for design purposes as investigated in the next section.

[0113] Energy Landscape

[0114] Prior to bifurcation, the disclosed multi-layered system

folds with 1 DoF through the application of in-plane forces. At the point of kinematic bifurcation, however, multiple kinematic paths become accessible. The entry into a specific path depends on the interplay between the components of the compressive forces, applied in its plane at the bifurcation instant. To study their role, a generic N_Nn_n (with n ≥

bi-axially compressed uniformly is examined and the energy landscape of each kinematic path is studied. The goal is to find the path and pertinent mode with the lowest energy level.

[0115] The formulation of the total energy of N_Nn_n is underpinned by three main assumptions. 1) The system consists of infinitely rigid panels; 2) frictionless rotational springs (hinges) of uniform stiffness per unit length, k, connect the panels; and 3) when unloaded, the system is in the configuration mode

where all dihedral angle pairs are obtuse. This is the zero-energy state of the system denoted by the dihedral angle θ⁰; it specifies a configuration that is either flat or partially folded due to the likely presence of residual stresses induced by a given manufacturing process. θ^L, on the other hand, is the acute dihedral angle once the unit is locked. In the disclosed analysis, if θ⁰ > a condition implying that upon the application of an out-of-

plane load (z-direction in Fig. 1 E), the system may only fold from its zero-energy state (specified by θ⁰) to its fully developed (flat) configuration.

[0116] Mode-phase diagram: the role of in-plane confinement

[0117] Referring now to Figs. 9A to 9D, a pair of compressive in-plane forces, f_x and f_y, applied uniformly in a quasi-static mannerto the disclosed system and oriented along the principal directions x and y (Fig. 1A) is considered. r_B = f_y/f_x with f_x,f_y e [0, ∞ ] and r_B ∈ [0,1] are defined to discriminate between the relative magnitude of the applied forces, and derive an expression of the total energy as a function of the applied forces and dihedral angle, i.e., or

Here, two representative systems are examined for

demonstrative purposes, and their energy landscapes are mapped into mode-phase diagrams.

[0118] We start with

_{4 n} described by the representative set of parameters: k = 1/3 N, n = 3, θ⁰ = π and a = b = 15 mm. Fig. 9A illustrates two typical curves of the dimensionless total energy, each for a given value of (f_x,r_B). Upon bifurcation, when θ = π/2, the total energy curve splits into two branches, each representing a reconfiguration mode

and

Between these two, the system chooses the mode which has the lowest energy level. This outcome can be determined by examining /) the magnitude of the energy of all branches immediately before and after the bifurcation, and //) the gradient of the energy of all branches at bifurcation, for example, The magnitude and the ratio of the in-plane biaxial loads, i.e., (f_x,r_B),

govern the relative energy level of each energy branch, dictating the configuration mode the disclosed system would travel after bifurcation. For example, Fig. 9A shows the role of r_B in entering a given post-bifurcation mode. For a load case (f_x,r_B) = (1,1/3), the system at bifurcation chooses the

energy branch until reaching the lock state in this mode; in contrast once subjected to (f_x,r_B) = (1,1), the system follows the

energy branch to reach the lock state.

[0119] The simple example above suggests the prospect to generate a mode-phase diagram that maps the activation of a given mode with respect to the relative magnitude of the in- plane confinement forces f_x and f_y . Fig. 4B visualizes such a map for

with a = b = 15 mm and θ⁰ = π . Each color is assigned to a region that describes a given configuration mode. The boundaries separating modes

and

are obtained by equating the gradient of the total energy of each branch at bifurcation

[0120] The application of the mode-phase diagram is showcased in demonstrative scenarios defined by families of in-plane applied forces. Each family is specified by (f_x, r_B) where r_B is maintained constant over the entire loading history. Two load families are considered. The first is (f_x,r_B = 1/3), where f_x and f_y can be proportionally scaled to respect the one third ratio, and the (dimensionless) total energy landscape of one specific case (f_x = 1, r_B = 1/3) of that family is shown in Fig. 9A by the upper curve. In Fig. 9B , (f_x,r_B = 1/3) is shown by the load-path ABCD crossing three domains. In the triangular region, an increase of the force magnitudes from A to B is not sufficient to reconfigure the disclosed system, which remains flat (θ = π ) in its fully developed state. Once the load magnitude reaches the first domain boundary (black dash line), point B, the system starts to reconfigure and fold along the only kinematic path up to bifurcation, point C. Immediately after bifurcation, the system can in principle access two modes (

and branches in Fig. 9A), yet it enters <40 due to the lower energy it requires for activation (Fig. 9A). After entering in mode

the system momentarily continues to reconfigure until it locks at point D (Figs. 9A and 9B). At this stage regardless of the magnitude of f_x and f_y satisfying r_B = 1/3, no further folding is possible as the panels have come into self-contact, thus imparting stiffness.

[0121] The second family of in-plane forces is (f_x, r_B = 1) is examined, shown in Fig.

9B by the green load-path AEFG. The total energy landscape of one specific load case (f_x = 1, r_B = 1) for that family is illustrated in Fig. 9A lower curve. Similar to ABCD, the system remains in its flat configuration for a load increase from A to E. A further increase of the in-plane confinement makes the system exit the initial region (black dash line) and gradually deform up to the bifurcation point F (second boundary threshold). Thereafter, it enters the lowest energy branch ( in Fig. 9A and light blue region in Fig. 9B) with an energetically stable state that occurs slightly prior to its lock state, point G. An additional load increase makes the system fold until it reaches its lock state.

[0122] The map in Fig. 9B shows that the only way for to access mode

is with

r_B = 1; in contrast, any other values of r_B would bring the system to lock in mode

This is an outcome that is distinctive to

and not necessarily to other systems. For instance, for

there are more kinematic paths available, i.e., multiple ranges of r_B exist to switch between modes. Fig. 4C shows the mode-phase diagram of

which is to be interpreted as explained above fo (Fig. 4B). As it can be observed here, r_B = 3/4 (diagonal in the light green region) defines a condition for which the system, initially in

mode, enters

mode immediately after bifurcation. Above that value, r_B > 3/4, the unit enters its lockable mode

(dark green), whereas below, r_B < 3/4, it accesses its flat-foldable mode

(brown region). Once entered in one of these post-bifurcation modes, one with the minimum energy, the system remains in that region until it locks due to panel self-contact. Once rigidity under compression is attained, the system can no longer reconfigure, and no other regions beyond the red boundaries (see Fig. 9B and C) are accessible through in-plane compressive forces.

[0123] The mode-phase diagrams here developed for

showcase the potency of tuning the kinematics and properties of the disclosed systems by controlling in-plane forces. By harnessing in-plane confinement, it may be possible to encode desired post-bifurcation modes of reconfiguration. A natural question now arises on the third front, the out-of-plane confinement as described below.

[0124] Lock state domains governed by out-of-plane confinement

[0125] If a compressive force acts in the third direction (z axis in Fig. 1A), how much compression is required to be applied to the disclosed system after bifurcation to bring it to its lock state, where it attains structural rigidity? To address this question, the role of an out-of-plane compressive force f_o uniformly applied to a generic N_Nn_n is studied and the magnitude of f_o necessary to drive and keep N_Nn_n in its lock state without the need of in-plane confinement is assessed.

[0126] The potential energy of the system hinges is expressed as V = V(θ) = nNkb(θ - θ⁰)² and the work of a uniformly applied external force f_o as W_o = nf_oasin Φ (sin θ⁰ - sin θ), where n is the number of stacked layers and the other parameters are defined in Fig. 1A. The potential energy due to gravity is here neglected since the panels of the disclosed system are made of cellulose paperboard, a lightweight material with gravitational potential energy of few orders of magnitude lower than that of the hinges and the work of the external forces. If V' = V(π - θ) is introduced, and a generic mode with

is denoted, where ζ is the total number of pairs for the acute dihedral angles

and ζ ' counts the total number of pairs for the obtuse dihedral angles

with ζ + ζ ' the total energy of the

mode may be expressed as (3)

[0127] The first expression in Eq. 3 describes the total energy of the system in mode

prior to bifurcation, while the second gives the total energy after bifurcation.

[0128] Under an out-of-plane force, a generic system N_Nn_n is in equilibrium at the lock state when the total energy has a stationary value. Given N_Nn_n has one DoF, and the total energy and its derivative are continuous functions, the minimum out-of-plane force f^L at the lock state may be determined by solving which yields:

(4)

[0129] Equations (3) and (4) can be used to map the total energy landscape of a system under an uniform out-of-plane compression as a function of the supplementary of the dihedral angle θ, i.e., π - θ (Fig. 1A ). For demonstrative purpose,

folding in mode

is examined. Fig 4D shows its energy curves (Eq. 3) for three representing values of the out-of-plane load normalized by the lock load, i.e.,

Setting

yields the boundary (thick curve) between two energy domains, one (below) satisfying

and the other (above) The red point,

which all curves (three shown) pass through, represents the zero-energy state, described above as the state of the system immediately after manufacturing, either flat (ideal case) or marginally folded due to residual stress from fabrication.

[0130] Subject to uniform out-of-plane compression, the material system can lock into its self-contact state under two conditions. First, the magnitude of the uniformly applied force f_o should be greater than the minimum out-of-plane force, f^L, required to lock up the unit. Second, the dihedral angle of the disclosed unit should be larger than a threshold value defined by the maximum energy barrier of the system. The interplay between f_o and f^L described by these conditions gives rise to three domains:

[0131] Region I (light brown) Here fall configurations defined by supplementary

angles for which the disclosed system can reach an equilibrium that is either stable or

unstable if Since f_o <f^L, the system cannot access the lock state from a given

configuration, e.g., lower orange point, and it tends to fold back to its equilibrium point along the “flat-fold” direction towards the zero-energy point (red).

[0132] Region II:

In this domain, the disclosed system can potentially reach the lock state, but a difference in the outcome exists as determined by the stability of equilibrium. Region II splits into two subdomains (Ila and llb), each defined by the slope of the energy curve, i.e., the sign of where is expressed as a function of ( π - θ).

[0133] Region Ila: The condition of equilibrium here is unstable

despite the out-of-plane force being largerthan the minimum locking force. In this region, a system partially folded at a given dihedral angle by the applied in-plane forces is prone to fold back to its fully developed (flat) state.

[0134] Region lib: This is the lockable domain, bounded by the

locus of points (dot line), which satisfies the condition Upon imposing the

condition in Eq. (3), the dihedral angle may be expressed as a function of the load f_o

when the energy is maximum from which, it is possible to obtain:

[0135] Substituting Eq. (5) into (3) yields the lockable domain boundary of maximum energy (dot bound in Fig. 9D) given as a function of the dihedral angle

(6)

[0136] Equation (6) traces points of the dot boundary that are unstable configurations of equilibrium, where the total energy attains maximum values, one of which is shown by the blue point of the representative energy curve If in-plane forces lead the system to reach one

unstable dihedral angle, a small perturbation prompt the system to naturally abandon it. Thus, once the in-plane forces succeed in generating dihedral angles larger than those described by Eq. (4), the disclosed system accesses the descending path in the lockable domain. Here spontaneous folding towards the lock state occurs, and the in-plane forces are no longer needed. The magnitude of the out-of-plane action (f_o >f^L) enables lifting the in-plane confinement. Once in the lockable domain, e.g., orange point, the system is drawn to fold towards a stable configuration of equilibrium until it has to arrest due to panel self-contact. [0137] The analysis above has revealed the interplay between in-plane and out-of-plane confinement. The former can be imparted through the bi-axiality ratio to program and steer the reconfiguration mode (either lockable or flat-foldable) during the folding process. The latter, in particular its magnitude (f_o >f^L) and the threshold value of the dihedral angle, i.e., the lockable domain boundary, set the conditions for spontaneous folding into the lock state without need of in-plane compression. Once self-contact is attained, the disclosed system becomes a stiff structure, and it is ready to sustain compressive loads exerted in all three directions, as described below.

[0138] Assessment of mechanical performance and in-situ property programmability

[0139] A set of representative proof-of-concept specimens made of cellulose paperboard in their lock states under compression is examined. In particular, samples with

and unit chains are considered as the disclosed representative material systems. The purpose is to demonstrate their capacity to achieve on demand distinct mechanical properties, meaning a given set of properties can be in-situ programmed post-fabrication through uniform and non-uniform application of the applied forces. Their Young’s modulus E* (the tangent of the compressive stress-strain curve assessing panel engagement as opposed to initial slippage) and the yield strength σ* (the peak stress before densification) have been experimentally investigated.

[0140] Referring now to Figs. 10A to 101, prior to carrying a campaign of experiments on the disclosed material system, the properties of finite size specimens is studied to ensure they are representative of those of their periodic counterparts. In a convergence analysis, the minimum number of primitive unit cells required for in-plane tessellation along the basis vectors {e₁, e₂} (Fig. 10A) is experimentally determined. The results show a minimum of 7 unit-cells is required along the in-plane (diagonal) directions, i.e with I being the number of cells in a given

diagonal direction. In addition, in the out-of-plane direction, n = 6 layers need to be stacked to parallel the response of a periodic material system. Given the role of tessellation along the three directions, the disclosed material system is here denoted by and two representative

systems are now studied. The aim is to assess the following

characteristics in their lock configurations: Load-bearing capacity at different lock modes; Mechanical properties scaling laws; Multi-directional stiffness; and Properties attained in hybridmode configurations. [0141] Fig. 10A shows the top view of

along with an inset of a representative primitive unit cell (yellow) in the lock mode

Fig. 10B reports its engineering stress-strain curves obtained in two lock configurations

and and in Fig. 10C the corresponding

response of

in its two lock configurations

The shaded domain describes the dispersion of the results obtained from testing three samples for each material system in a given mode. From the plots, three regions are identified: (i) An initial nonlinear regime, describing panels not yet engaged under compression, hence unable to establish a proper contact. (ii) A linear material response for both locked states of each system. (iii) The stress-peak and the following regime is indicative of progressive panel buckling and creasing. The distinct curves show that the disclosed system can attain distinct responses upon switching between its lock modes. The disclosed measurements reveal that the Young’s modulus and yield strength of

are approximately reduced of an order of magnitude when switching from , and the

Young’s modulus of is reduced approximately 3 times while its yield strength about 2

times when switching from

Minor deviations have been observed in regions (i) and (ii), as opposed to large values attained in region (iii) far from the peak. The stiffness and strength values measured for the disclosed specimens are comparable with those reported in the literature of kirigami-based concepts made of similar paperboard material.

[0142] Fig. 10D shows the normalized Young’s modulus, and normalized yield

strength (where E_s and σ_f are the Young’s modulus and failure stress of the base material in the machine direction, MD) fo measured at three values of relative density. The results

are obtained for samples designed with three geometric parameters a = 10 mm, 15 mm and 20 mm; for each of them, five identical samples are fabricated, and the testing is repeated five times. What has been observe is that the normalized Young’s modulus scales almost linearly with the relative density which obeys the classical scaling law for stretching-dominated

structures. The normalized yield strength, on the other hand, scales almost quadratically with the relative density for the given set of the disclosed measured data, hence not obeying

the classical laws of stretch or bending dominated 3D cellular materials. This deviation can be attributed to the presence of additional deformation and failure mechanisms, which are governed by hinge stiffness, panel self-contact and the presence of geometrical imperfections. [0143]

in the lock configuration is tested under in-plane and out-of-plane compression (Fig. 10E) along two in plane directions (A and B) at 45° and 90° with respect to the x-axis, and along the z-direction. Representative curves of their engineering stress-strain responses in Fig. 10F attest a stiffness and load-bearing capacity comparable in all main three directions. The largest strength (A) and stiffness (B) observed during in-plane testing are attributed to the presence of double-layered panels, i.e., quad panels bonded to stacked layers, an aspect that confers additional anisotropy and larger strength to bear the compressive load beyond the elastic regime. The maximum load-bearing capacity and stiffness measured in both the in-plane experiments are similar, if the initial - more compliant - response of the unit loaded at 45° is not considered. This is due to the occurrence of a shear deformation that is dominant at the start of the compression test. Overall, the disclosed

paperboard specimens locked in mode and realized with a = 15 mm weighs ~ 40 gr, and can withstand up to 850 N under compression along the z-direction (Movie S11), while up to 1450 N when compressed in the inplane directions. In the lock mode

with identical geometric parameters and weight can resist 1000 N under out-of-plane compression.

[0144] The application of uniformly applied forces at the instant of kinematic bifurcation causes the material system to fold into one of its lock modes, each with its own set of properties (e.g., Fig. 10B and 10C) as discussed above. Here, the outcome of a non-uniform set of ferees locally applied in given zones has been observed, hence bringing the system into a mixed-mode configuration, i.e., a state that is partially folded and encompasses a combination of lockable and flat-foldable modes. Fig. 10G depicts 7 hybrid-mode configurations for the disclosed

specimen with square primitive side a = 10 mm. The top row shows the emergence of voids increasingly appearing from configuration 4 to 7. In the second row, the distribution of the attainable modes, i.e.,

which can concurrently form in a given hybrid-mode, is highlighted with a given color. Here the number of the hybrid-mode configurations examined is 7, although several other modes are possible.

[0145] We now examine the effective compressive Young’s modulus and permeability of the configurations shown in Fig. 10G, being these properties representative of an additional degree of in-situ programmability. Fig. 10H reports the normalized Young’s modulus (normalized with respeetto that of the configuration (1), i.e., E*⁽¹⁾) measured in each mixed-mode configuration. Measurements of the Young’s modulus E* for shows only a 1.2-fold decrease occurs

by increasing the tessellation level from (3,3) to (7,7). This difference is small compared to the decrease observed for a switch from mode

to mode

(see Fig. 5B) or

(e.g. see config (1) with (7) in Fig. 10H). This result suggests that the stiffness values measured for mixed-mode configurations are quite insensitive to finite-size effects. The stiffness value of the material system made of only mode

(config (1) Fig. 10G) drops by adding regions made of mode

For example, a drop to half can be registered upon switching from configuration (1) to (3). Configuration (4) with all three mode-regions

shows slightly higher stiffness than configuration (2), a result that might be attributed to the distribution of regions with a two-fold symmetry mode, as opposed to the other region possessing only mirror symmetry. In this mode, only a single channel (an out-of-plane void) is formed as opposed to configurations (1), (2) and (3), which have no open channels or voids. The change of open channels through sequence of reconfigurations is shown in the bar-chart of Fig. 10I. Here plotted is the normalized open channel area, where the area of the open channels in the out-of-plane direction, A_ch, is normalized

with respect to that of the configuration (4),

having only a single channel. The change in open channel area can be considered as a descriptor of the system permeability, which scales linearly with the conduit area as in a porous medium. In addition, the trends show that the compressive Young’s modulus and permeability are antagonist, and that for a given value of permeability, the pattern with higher degree of symmetry emerging in the mode region distribution (second row Fig. 10G) provides higher stiffness. The addition of mode regions

weakens the disclosed material system while the existence of mode region

increases the permeability of the material in the out-of-plane direction.

[0146] Referring now to Figs. 11A to 1 1 C, another embodiment of a chain unit and its associated tessellation is shown at 60 and 160, respectively. In the embodiment shown, the central aperture 1 1 has six sides. The hexagonal shape of the central aperture 11 is skewed. This implies that angles between two successive edges 12 are not constant all around the central aperture 11 . Any of the unit chain 10, 20, 30, 40, 50, 60 described above with reference to Figs. 2A to 2E may be modified such that their central apertures 11 is skewed. A tessellation of the unit chain 60 is shown at 160 and depicted in its flat configuration on Fig. 11 B. The same tessellation 160 is shown on Fig. 1 1 C in one of its folded configurations.

[0147] In some embodiments, length of all of the edges 12 bounding the central apertures 11 may be equal. In some other embodiments, the central apertures 1 1 may have irregular shapes such that the lengths of all of the edges 12 may differ from one edge to the other. [0148] In some embodiments, the bistable snapping of the panels adjacent to the emergent voids may be harnessed to store and dissipate energy, hence working as a damper.

[0149] As can be seen therefore, the examples described above and illustrated are intended to be exemplary only. The scope is indicated by the appended claims.

Claims

WHAT IS CLAIMED IS:

1 . A unit chain for a tessellation for a metamaterial, comprising: a plurality of panels circumferentially distributed about a central polygonal aperture, the central polygonal aperture having an even number of edges, the plurality of panels including: quadrilateral panels each extending from a respective one of the edges and away from the central polygonal aperture, and triangular panels each secured to two adjacent quadrilateral panels via hinges, the triangular panels interspaced between the quadrilateral panels, the quadrilateral panels pivotable relative to the triangular panels via the hinges, the unit chain having a flat configuration in which the plurality of panels are parallel to one another and a folded configuration in which a first group ofthe triangular panels are contained in a first plane and a second group of the triangular panels are contained in a second plane offset from the first plane, and in which the triangular panels are non-parallel to the quadrilateral panels.

2. The unit chain of claim 1 , wherein, in the folded configuration, one or more of: a first triangular panel of the triangular panels abuts a second triangular panel of the triangular panels; and a first quadrilateral panel of the quadrilateral panels abuts a second quadrilateral panel of the quadrilateral panels.

3. The unit chain of claim 1 , wherein, in the folded configuration, dihedral angles defined between successive triangular panels and quadrilateral panels are less than 90 degrees.

4. The unit chain of claim 1 , wherein, in the folded configuration, at least two ofthe triangular panels of the first group are in abutment against one another and/or in which at least two of the triangular panels of the second group are in abutment against one another.

5. The unit chain of claim 1 , wherein, in the folded configuration, external pressure exerted on the unit chain to move the first plane toward the second plane is opposed by a cooperation of one or more of the triangular panels and the quadrilateral panels being in abutment against one another. 6 The unit chain of claim 1 , the plurality of panels are parts of a single monolithic body. 7 The unit chain of claim 1 , wherein the hinges are living hinges. 8 The unit chain of claim 1 , wherein the hinges are defined by fold lines. 9 A tessellation comprising: unit chains interconnected to one another, a unit chain of the unit chains having: a plurality of panels circumferentially distributed about a central polygonal aperture, the central polygonal aperture having an even number of edges, the plurality of panels including quadrilateral panels each extending from a respective one of the edges and away from the central polygonal aperture, and triangular panels each secured to two adjacent ones of the quadrilateral panels via hinges, the triangular panels are interspaced between the quadrilateral panels, the tessellation having a flat configuration in which all of the plurality of panels are parallel and a folded configuration in which a first group of the triangular panels are contained in a first plane and a second group of the triangular panels are contained in a second plane offset from the first plane and in which the triangular panels are non-parallel to the quadrilateral panels. 10 The tessellation of claim 9, wherein each pair two unit chains of the unit chains disposed adjacent one another are interconnected by one or more of a triangular panel of the triangular panels and a quadrilateral panel of the quadrilateral panels. 11 The tessellation of claim 10, wherein the pair of the two unit chains share a common quadrilateral panel, the central aperture having at most six edges. 12 The tessellation of claim 10, wherein a triangular panel of a first unit chain of the pair of the two unit chains is connected to a triangular panel of a second unit chain of the pair of the two unit chains, the central aperture having more than six edges. 13 The tessellation of claim 9, wherein, in the folded configuration, for the unit chain of the unit chains, one or more of: a first triangular panel of the triangular panels abuts a second triangular panel of the triangular panels; and a first quadrilateral panel of the quadrilateral panels abuts a second quadrilateral panel of the quadrilateral panels. 14 The tessellation of claim 9, wherein, in the folded configuration, dihedral angles defined between successive triangular panels and quadrilateral panels is less than 90 degrees. 15 The tessellation of claim 9, wherein, in the folded configuration, at least two of the triangular panels of the first group are in abutment against one another and/or in which at least two of the triangular panels of the second group are in abutment against one another. 16 The tessellation of claim 9, wherein the unit chains are first unit chains of a first layer of unit chains, the tessellation comprising a second layer of unit chains disposed below the first layer of unit chains, the second layer of unit chains having second unit chains, the second unit chains having triangular panels and quadrilateral panels interspaced between the triangular panels, a first subset of the triangular panels of the first unit chains secured to a first subset of the triangular panels of the second unit chains. 17 The tessellation of claim 16, comprising a third layer of unit chains disposed below the second layer of unit chains, the third layer of unit chains having third unit chains including triangular panels and quadrilateral panels interspaced between the triangular panels, a first subset of the triangular panels of the third unit chains secured to a second subset of the triangular panels of the second unit chains. 18 The tessellation claim 9, wherein the plurality of panels are parts of a single monolithic body. 19 The tessellation of claim 9 , wherein the hinges are living hinges. 20 The tessellation of claim 9, wherein the hinges are defined by fold lines.