Anisotropic bistable auxetic material

Abstract

An anisotropic auxetic bistable metamaterial (ABAM), has: building blocks having a quadrilateral shape and having: two unit cells conjointly defining four sides and four corners, the two unit cells disposed on opposite sides of a bisecting line, a unit cell having: three sides interconnected and including two of the four sides and the bisecting line, and three slits each extending from a respective one of the three sides towards a respective opposite one of the three sides; a rotating member bounded by the three slits; translating members disposed around the rotating member; and hinges at ends of the three slits, the rotating member being rotatable relative to the translating members about the hinges, wherein the assembly is non-plastically and reversibly deformable from a collapsed position to an expanded position and reversibly deformable from the expanded position to the collapsed position.

Description

ANISOTROPIC BISTABLE AUXETIC MATERIALCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This disclosure claims benefit from United States patent application No. 63 / 557,809 filed on February 26, 2024, the entire contents of which are incorporated by reference herein.TECHNICAL FIELD

[0002] The disclosure relates generally to mechanical metamaterials and, more particularly, to auxetics materials.BACKGROUND

[0003] Shape morphing in kirigami may enable functionalities appealing to a diverse range of applications. At the core of such shape shifting lies the architecture of the repeating unit, where multiple symmetries may confer deployment obeying uniform scaling transformation. Existing bistable materials may be suitable for their intended purposes, but improvements are sought.SUMMARY

[0004] In one aspect, there is provided an anisotropic auxetic bistable metamaterial (ABAM), comprising: an assembly of building blocks, a building block of the building blocks having a quadrilateral shape and having: two unit cells conjointly defining four sides and four corners, the two unit cells disposed on opposite sides of a bisecting line extending across two opposite ones of the corners, a unit cell of the two unit cells having: three sides interconnected to one another and including two of the four sides and the bisecting line, and three slits each extending from a respective one of the three sides towards a respective opposite one of the three sides, the three slits free from intersection between each other; a rotating member bounded by the three slits; translating members disposed around the rotating member; and hinges at ends of the three slits, the rotating member being rotatable relative to the translating members about the hinges, wherein the assembly is non-plastically and reversibly deformable from a collapsed position to an expanded position upon application of a first load, and the assembly being non-plastically and reversibly deformable from the expanded position to the collapsed position upon application of a second load, wherein the translating members are geometrically frustrated in the expanded position.

[0005] The ABAM defined above and described herein includes, in certain embodiment, one or more of the following features, in whole or in part, and in any combination.

[0006] In some embodiments, the assembly is bistable and thereby defining a first stable state in the collapsed position and a second stable state in the expanded position, and wherein in the first stable state the assembly maintains the collapsed position after removal of the second load applied thereon and, in the second stable state, the assembly maintains the expanded position after removal of the first load applied thereon.

[0007] In some embodiments, between the expanded position and the collapsed position, the building blocks move relative to each other such that the assembly deforms toward a closest one of the expanded position and the collapsed position after removal of either of the first load and the second load applied on the assembly.

[0008] In some embodiments, a dimension of the assembly of the building blocks taken in a direction normal to a direction of application of the first load being greater in the expanded position than that in the collapsed position.

[0009] In some embodiments, a ratio of a slit length (a1) of a slit of the three slits to a length (11) of a side of the three sides being parallel to the slit ranges from 0.4 to 0.8.

[0010] In some embodiments, angles defined two adjacent ones of the three sides range from 10 degrees and 160 degrees.

[0011] In some embodiments, an end of a slit of the slits is spaced apart from an adjacent slit by a gap having a thickness (t), a ratio of the thickness to a length of a side of the three sides being parallel to the slit being from 0.03 to 0.08.

[0012] In some embodiments, the ratio of the thickness to the length is about 0.04.

[0013] In some embodiments, a ratio of a thickness of the building block to a length of a side of the three sides is about 0.3.

[0014] In some embodiments, the three sides are of equal length, the ABAM featuring a threefold rotational symmetry and reflectional symmetry in both of the expanded position and the collapsed position.

[0015] In some embodiments, two of the three sides are of equal length and a third one of the three sides has a different length, the ABAM featuring reflections and glide reflections with parallel axes.

[0016] In some embodiments, the three sides are all of different lengths, the ABAM being devoid of both rotational and reflectional symmetry in both of the expanded position and the collapsed position.

[0017] In another aspect, there is provided a building block for an anisotropic auxetic bistable metamaterial (ABAM), comprising: two unit cells conjointly defining four sides and four corners, the two unit cells disposed on opposite sides of a bisecting line extending across two opposite ones of the corners, a unit cell of the two unit cells having a triangular rotating member surrounded by three translating members rotatably connected to the triangular rotating member by hinges, wherein the building block is non-plastically and reversibly deformable from a collapsed position to an expanded position upon application of a first load, and the building block being non- plastically and reversibly deformable from the expanded position to the collapsed position upon application of a second load.

[0018] The building block for an ABAM defined above and described herein includes, in certain embodiment, one or more of the following features, in whole or in part, and in any combination.

[0019] In some embodiments, three slits are located between adjacent sides of the triangular rotating member and the three translating members.

[0020] In some embodiments, a ratio of a slit length (a1) of a slit of the three slits to a length (11) of a side of the unit cell being parallel to the slit ranges from 0.4 to 0.8.

[0021] In some embodiments, an end of a slit of the slits is spaced apart from an adjacent slit by a gap having a thickness (t), a ratio of the thickness to a length of a side of the unit cell being parallel to the slit being from 0.03 to 0.08.

[0022] In some embodiments, the ratio of the thickness to the length is about 0.04.

[0023] In some embodiments, angles defined two adjacent ones of sides of the unit cell range from 10 degrees and 160 degrees.

[0024] In some embodiments, a ratio of a thickness of the building block to a length of a side of the unit cell is about 0.3.

[0025] In some embodiments, the three sides are of equal length, the ABAM featuring a threefold rotational symmetry and reflectional symmetry in both of the expanded position and the collapsed position; or two of the three sides are of equal length and a third one of the three sides has a different length, the ABAM featuring reflections and glide reflections with parallel axes; or wherein the three sides are all of different lengths, the ABAM being devoid of both rotational and reflectional symmetry in both of the expanded position and the collapsed position.

[0026] In accordance with one aspect, there is provided an anisotropic auxetic bistable metamaterial (ABAM), comprising: an assembly of building blocks, a building block of the building blocks having a quadrilateral shape and having: four sides interconnected to one another at four corners; a bisecting line extending between two opposite ones of the four corners; two unit cells each disposed on a respective opposite side of the bisecting line, the two unit cells each having a triangular shape and having: three sides interconnected to one another, the three sides including two of the four sides and the bisecting line, and three slits each extending from a respective one of the three sides towards a respective opposite one of the three sides, the three slits free from intersection between each other; triangularly-shaped rotating members each bounded by the three slits of a respective one of the two unit cells; and translating members disposed around the two triangularly-shaped rotating members, wherein junctions between the triangularly-shaped rotating members and the translating members are defined at ends of the three slits, the junctions being hinges about which the triangularly-shaped rotating members rotate relative to the translating members, wherein the assembly is non-plastically and reversibly deformable from a collapsed position to an expanded position upon application of a first load, and the assembly being non-plastically and reversibly deformable from the expanded position to the collapsed position upon application of a second load, wherein the translating members are geometrically frustrated in the expanded position.

[0027] The ABAM as defined above and described herein may also be a bistable assembly and thereby defining a first stable state in the collapsed position and a second stable state in the expanded position, and wherein in the first stable state the assembly maintains the collapsed position after removal of the second load applied thereon and, in the second stable state, the assembly maintains the expanded position after removal of the first load applied thereon.

[0028] In the ABAM as defined above and described herein, between the expanded position and the collapsed position, the building blocks move relative to each other such that the assembly deforms toward a closest one of the expanded position and the collapsed position after removalof either of the first load and the second load applied on the assembly, a dimension of the assembly of the plurality of interconnected building blocks taken in a direction normal to a direction of application of the first load being greater in the expanded position than that in the collapsed position.

[0029] In the present disclosure, symmetry breaking in bistable kirigami permits geometric frustration and anisotropic morphing, enabling arbitrarily scaled deployment in planar and spatial bistable domains. With an analysis on their symmetry properties complemented by a systematic investigation integrating semi-analytical derivations, numerical simulations, and experiments on elastic kirigami sheets, the present disclosure presents relations between slit symmetry, geometric frustration, and anisotropic bistable deployment. Furthermore, asymmetric kirigami units may be leveraged in planar and flat-to-3D demonstrations to showcase the pivotal role of shear deformation in achieving target shapes and functions so far unattainable with uniformly stretchable kirigami. The role of slit symmetry breaking in controlling the anisotropic bistable deployment of soft kirigami metamaterials is disclosed, enriching the range of achievable functionalities for applications spanning deployable space structures, wearable technologies, and soft machines.BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Reference is now made to the accompanying figures in which:

[0031] Fig. 1 is a plan view of a building block of an anisotropic bistable auxetic material (ABAM) in accordance with one embodiment;

[0032] Fig. 2 illustrates a tessellation or assembly of a plurality of the building blocks of Fig. 1 in a collapsed configuration;

[0033] Fig. 3 illustrates the tessellation of Fig. 2 in an expanded configuration;

[0034] Fig. 4 illustrates a design space for the tessellation of Fig. 2 visualized in ternary plot obtained by varying symmetry leading to programmable anisotropic expansion;

[0035] Fig. 5A illustrates a cut pattern for the unit cell of Fig. 1 and shows the unit cell in both the collapsed and expanded configurations;

[0036] Fig. 5B illustrates a polar plot of engineering normal strain s and shear strain y of unit cell;

[0037] Fig. 5C presents graphs illustrating evolution of stress and energy density as a function of strain obtained by numerical simulation under periodic boundary conditions;

[0038] Figs. 5D to 5F illustrate contour plots characterizing energy (bistability Emin / Emax), anisotropy (shear strain y), and expansion (volumetric strain eV) of ABAM for varying geometric parameters (A1 , A2, A3) and given inner triangle size (a1 / £1 = 0.6);

[0039] Fig. 5G illustrates trade-offs between bistability and anisotropy of the anisotropic auxetic bistable metamaterial;

[0040] Figs. 5H to 5J illustrate cut view of 3D surfaces characterizing energy (bistability Emin / Emax), anisotropy (maximum shear strain ymax), and expansion (volumetric strain eV) of bistable auxetics for varying geometric parameters (A1 , A2, A3) and given values of a1 Ztl (t / £1 = 0.04);

[0041] Figs. 6A to 6D present graphs illustrating uniaxial tensile response of isotropic and anisotropic bistable auxetic material (experimental (solid line) and FE (dashed lines, shifted vertically by 3N for ease of reading) force-displacement curves during loading (tension) and unloading (compression) for BAM and ABAM. (a1 / £1 = 0.6 and t / tl = 0.04);

[0042] Figs. 7A to 7E illustrate planar deployment of anisotropic auxetic bistable material (shear band composed of “p31 m” and “cm” unit cells on Fig. 7A; Letter “M” composed of “p31 m” and “cm” unit cells in Fig. 7B; mechanical switch leveraging ABAM shear strain to align rigid hooks attached to translational Y-strut motifs and delivering load bearing capacity in Figs. 7C-7E); and

[0043] Figs. 8A to 8E illustrate three-dimensional bistable deployment of anisotropic bistable auxetic material (3D target shape composed of three semi-cylindrical shells featuring a peak point and a saddle point in Fig. 8A; 2D precursor of 3D target in Fig. 8B; unit cell selection and 3D target decomposed into three sections and flattened, scaling metrics between flattened target and 2D precursor, including volumetric strain eV and shear strain y, are extracted to compare flattened target and 2D precursor. "p31 m” BAM unit cells (A1 = A2 = A3 = 60°) are selected for isotropic scaling, whereas “cm” ABAM unit cells (A1 = A3 = 45°, A2 = 90°) are chosen for anisotropic scaling in Fig. 8C; physical realization of 2D precursor and its 3D deployment in Figs. 8E and 8E).DETAILED DESCRIPTIONIntroduction

[0044] Perforating a thin sheet with a distinct motif of slits has been leveraged in kirigami metamaterials to attain auxeticity, shape shifting, multistable deployment, and other unusual responses. In contrast to other shape-morphing metamaterials, such as folded tessellations (origami) and bilayer films, kirigami patterns may achieve fairly complex in-plane and out-of-plane deformations through the opening of rationally designed cuts, making them appealing in a range of sectors, from aerospace deployable structures, solar cells, biomedical devices, and robotics.

[0045] The diverse set of properties and functionality that a planar kirigami pattern can deliver stems mainly from the tessellation of its repeating shape, e.g., triangular (kagome pattern), square, hexagonal and other polygonal tiles, each embedding an intrinsic symmetry of their constituent slits, e.g., rotational, reflectional, and glide reflectional symmetry.

[0046] Symmetry may be a useful tool in creating complex architected materials for kirigami metamaterials applied to rigid deployable patterns with kinematics described by non-deformable panels and pure rotational hinges. The deployment of all these patterns does not involve geometric frustration, which denotes local disordered deployments caused by geometric incompatibility, such as local deformation of panels due to slit geometry that violates rigid- deployable constraints and out-of-plane morphing due to non-uniform strains in non-periodic patterns. This focus has so far excluded the investigation of exotic modes of deformation, e.g., anisotropic and inhomogeneous deployment, pattern formation of complex order, and historydependent mechanical computation among other functionalities, that - on the other hand - have been studied in frustrated soft metamaterials. The concept of geometric frustration in soft kirigami and its interplay with slit symmetry remains elusive and calls for further investigation due to its promise to unlock shape transformations and properties thus far unattainable in kirigami metamaterials.

[0047] With a focus on soft kirigami metamaterials, the present disclosure uses the intrinsic relation between slit symmetry, geometric frustration, and anisotropic scaling in a representative class of uniformly scaled bistable auxetic kirigami. By breaking slit symmetry and allowing sheet deformability, it is demonstrated how to program their anisotropic bistable deployment, departing from uniform scaling transformations that are frustration-free to non-uniform scaling transformations that are geometrically frustrated. Through a combined approach of semi-analytical derivations, numerical simulations, and experiments, the symmetry relations of bistable kirigami is mapped and their role on bistability, anisotropy, deployment magnitude as well as geometrically frustrated deployed state is investigated. The present disclosure describes a set of anisotropic in-plane deployments of bistable kirigami that leverage shearing deformations to achieve target shapes and functions. The present disclosure also employs a shape target that simultaneously contains positive, negative, and zero Gaussian curvatures to show the potential of arbitrarily scaled kirigami for generic flat-to-3D deployment, hence proposing a design strategy for anisotropic shape-morphing of kirigami sheets.

[0048] An “auxetic” may be defined as a structure or material having a negative Poisson’s ratio. That is, when stretched, such material or structure become thicker in a direction being perpendicular to a direction of the applied force. This occurs due to its internal structure and the way it deforms when uniaxially loaded.Anisotropic Bistable Auxetic Material (ABAM)

[0049] Referring now to Fig. 1 , a unit cell of a metamaterial is shown at 10. The unit cell 10 is first described than a metamaterial, which is built by tessellating a plurality of the unit cells 10 will be described below. The unit cell 10 may also be referred to as building block.

[0050] In the present embodiment, Fig. 1 represents two unit cells 10 secured to one another. Each of the unit cells 10 has a triangular shape including three sides 11 . The unit cell 10 includes three slits 12. The three sides 11 may include a first side 11A, a second side 1 1 B, and a third side 1 1 C. The second side 11 B is disposed circumferentially between the first side 11A and the third side 1 1 C relative to a central axis of the unit cell 10. Thus, a slit of the slits 12 extends from the first side towards the third side, which is opposite to the first side 11 A, and parallel to the second side 11 B. Stated differently, the slits 12 includes a first slit 12A extending parallel to the first side 1 1 A, a second slit 12B extending parallel to the second side 11 B, and a third slit 12C extending parallel to the third side 11 C. The first slit 12A extends from the second side 11 B towards the third side 11 C; the second slit 12B extends from the third side 11 C towards the first side 1 1A; and the third slit 12C extends from the first side 11 A towards the second side 1 1 B. Each of the slits has an end that is offset from the other slits. In other words, Gaps G remain between ends of the slits and adjacent slits. A dimension of this gap 13 is denoted by reference numeral “t” in Fig. 1 .

[0051] The unit cell 10 may be varied in shape by varying some parameters. These parameters include a first length 11 , a second length I2, and a third length I3 of respectively thefirst, second, and three sides 11 A, 11 B, 11 C. The parameters further includes a first angle A1 between the first side 11A and the third side 11 C, a second angle A2 between the second side 1 1 B and the third side 11 C, and a third angle A3 between the second side 1 1 B and the first side 1 1A. The parameters may also include a length t of the gap between the end of one of the slits 12 and an adjacent slit. The parameters may also include lengths a1 , a2, a3 of sides of a triangle defined by the slits 12. For instance, the length a1 is associated to the second slit 12B and extends from an intersection between a projection of the third slit 12C to the second slit 12B to a projection of the second slit 12B to the first slit 12A. The lengths of the slits may be defined as the lengths of the three sides of an internal triangle defined by the three slits 12.

[0052] In Fig. 1 , two unit cells 10 are secured to one another to form a quadrilateral building block 20. In this case, the second side of one of the unit cells is secured to the second side of an adjacent one of the unit cells. The two unit cells are assembled such as to form the quadrilateral building block. The two unit cells 10 are monolithically connected to one another at their respective second sides.

[0053] The unit cell 10 may include a rotating member 13 and translating members 14. In this case, the unit cell 10 includes a single rotating member 13, which is triangularly shaped and defined by the three slits 12, and includes three translating members 14 distributed around the rotating member 13. The translating members 14 are defined between the sides 11 and the slits 12. For instance, one of the translating members 14 is defined between the first side 1 1A and the first slit 12A.

[0054] Referring now to Figs. 2-3, an assembly, referred to herein as a tessellation 100, of four building blocks 20 is shown. The tessellation is shown as a matrix of 2x2, but any suitable configuration (e.g., 3x3, 2x3, etc) may be envisioned. The tessellation 100 is shown in a collapsed configuration in Fig. 2 and in an expanded configuration in Fig. 3. In this configuration, there are 8 unit cells 10, or four building blocks 20 interconnected to one another to form the tessellation 100. When applying a force along direction D1 on the tessellation 100, it expands and increases in width in a direction perpendicular to the direction D1.

[0055] As shown in Fig. 1 , the building block 20 has a quadrilateral shape and has four sides interconnected to one another at four corners; a bisecting line (shown in dashed line in Fig. 1) extending between two opposite ones of the four corners; and two unit cells 10 each disposed ona respective opposite side of the bisecting line, the two unit cells 10 each having a triangular shape.

[0056] Each of the unit cells 10 has three sides interconnected to one another, the three sides including two of the four sides and the bisecting line, and three slits each extending from a respective one of the three sides towards a respective opposite one of the three sides, the three slits free from intersection between each other. The building block 20 includes triangularly-shaped rotating members 13 (one by unit cell 10) each bounded by the three slits of a respective one of the two unit cells 10; and translating members 14 disposed around the two triangularly-shaped rotating members 13.

[0057] In the embodiment shown, junctions between the triangularly-shaped rotating members 13 and the translating members 14 are defined at ends of the slits 12. The junctions act as hinges 15 about which the triangularly-shaped rotating members 13 rotate relative to the translating members 14.

[0058] As shown in Figs. 2-3, the assembly is non-plastically and reversibly deformable from a collapsed position to an expanded position upon application of a first load, and the assembly is non-plastically and reversibly deformable from the expanded position to the collapsed position upon application of a second load. The translating members are geometrically frustrated in the expanded position. In the context of the present disclosure, the expression “geometrically frustrated” (or simply “frustrated”) implies that these members are not stress free. Put differently, a geometrically frustrated member may be subjected to a force, such as a flexion force, when the assembly is in the expanded position, even if said assembly is stable and holds itself in the expanded position.

[0059] The assembly is bistable and thereby defines a first stable state in the collapsed position and a second stable state in the expanded position. In the first stable state, the assembly maintains the collapsed position after removal of the second load applied thereon and, in the second stable state, the assembly maintains the expanded position after removal of the first load applied thereon. Between the expanded position and the collapsed position, the building blocks move relative to each other such that the assembly deforms toward a closest one of the expanded position and the collapsed position after removal of either of the first load and the second load applied on the assembly. A dimension of the assembly of the plurality of interconnected buildingblocks taken in a direction normal to a direction of application of the first load is greater in the expanded position than that in the collapsed position.

[0060] The present assembly may be bistable in the manner described in United States Patent No. 10,767,032, the entire contents of which are incorporated herein by reference.

[0061] Although not shown in Figs. 1 -3, it is appreciated that the deformation between the collapsed and expanded configurations is done in a plane. That is, the deformation is not out of plane in this configuration. The unit cells therefore have a thickness selected to ensure that, when geometrically frustrated, the different members do not buckle out of plane when in the expanded configuration.

[0062] The ABAM may be characterized by geometrical ratios. For instance, a ratio of a slit length a1 , a2, a3 of the slits to a length 11 , I2, I3 of a side of the unit cell may range from 0.4 to 0.8. Angles defined two adjacent ones of the three sides may range from 10 degrees and 160 degrees. An end of a slit of the slits is spaced apart from an adjacent slit by the gap G. The gap G has a thickness t. A ratio of the thickness t to a length 11 , 12, 13 of the sides being parallel to the slit may range from 0.03 to 0.08. In some embodiments, the ratio of the thickness to the length is about 0.04. Herein, the expression “about” implies variations of plus or minus 10%. A ratio of a thickness of the building block, extending transversally to the building block, to the length 11 , I2, I3 of the sides may be about 0.3.

[0063] As will be discussed further below, many configurations may be obtained by selecting the lengths 11 , 12, I3 of the sides of the unit cells. For instance, when the three sides are of equal length, the ABAM may feature a threefold rotational symmetry and reflectional symmetry in both of the expanded position and the collapsed position. When two of the three sides are of equal length and a third one of the three sides has a different length, the ABAM may feature reflections and glide reflections with parallel axes. Also, when the three sides are all of different lengths, the ABAM may be devoid of both rotational and reflectional symmetry in both of the expanded position and the collapsed position.Results

[0064] Referring to Fig. 4, the tessellation 100 may be designed such that its expanded state vary in shape. This may be done by selecting the parameters of the unit cell 10 described abovewith reference to Fig. 1. Hence, the tessellation 100 may be anisotropic. This implies that the tessellation 100 may expand differently in one direction compared to in other direction.

[0065] The geometry consists of rotating equilateral triangles that are surrounded by translational Y-strut motifs, which are periodically tessellated with three-fold rotational symmetry. In bistable auxetic material, geometric frustration is encountered only when transitioning from the collapsed configuration to the expanded configuration, where the rotating units uniformly push away their neighboring translational Y-strut motifs in all directions and impart isotropic scaling to their deployment.

[0066] In the present disclosure, interplay between slit symmetry and geometric frustration results in an arbitrary kirigami unit where the three-fold symmetry of their slits is broken. The objective is to investigate the physical mechanisms underpinning non-uniform deployment and its frustrated equilibrium states, the signature of the presently disclosed Anisotropic Bistable Auxetic Metamaterials (ABAM).

[0067] The periodic pattern of ABAM includes of six mutually intersecting slits nesting two triangle units each within a larger triangular module. To describe ABAM in a compact and insightful form, the present disclosure focuses on a subset of geometric parameters, i.e., the internal angles A1 , A2, A3, that govern the symmetry of the kirigami pattern, and present a ternary plot where the internal angles lie on its axes (Fig. 4). Each point in the parameter space is defined by a set of Ai (i = 1 , 2, 3) values that can describe the geometry of any slit pattern applied to a unit cell 10 with prescribed base length 11 , internal cut ratio al / 11 , and normalized hinge thickness t / 11 . For example, the intersection of the three bisectors (A1 = A2 = A3) denotes a kirigami pattern with an equilateral elementary triangle that belongs to the “p31 m” wallpaper group, i.e., its geometry in its two stable states features threefold rotational symmetry and reflectional symmetry. As described above, this is BAM exhibiting geometrical frustration at the intermediate states of deployment whereas its closed and deployed states are frustration-free. By altering the geometry of the elementary triangle, ABAM can access other wallpaper groups with reduced symmetry. For example, if the elementary triangle is isosceles (A1 = A2, A2 = A3, or A1 = A3), ABAM that lie along any of the bisectors of the ternary plot belong to the “cm” class, i.e., there exist reflections and glide reflections with parallel axes. Fig. 4 shows two types of “cm” ABAM, one describing a triangle with two long sides and one short side (pattern drawn in blue and ternary plot bisectors with solid lines), or the other with two equal sides shorter than the third side (pattern drawn in orange and ternary plot bisectors with dashed lines). Lastly, the most general case is for ABAMbelonging to the “p1 ” class (A1 f 2. t A3), represented by any point that is not on the bisectors, e.g., lower right side of Fig. 4, where there exists neither rotational nor reflectional symmetry.

[0068] While the location in the ternary plot determines the symmetry group of ABAM, the symmetry the ternary plot itself conveys important insights on the existence of dissimilar sets of geometric parameters that describe the same ABAM. With the shape of an equilateral triangle, the ternary plot naturally exhibits three-fold rotational symmetry around its center and reflection symmetry with respect to its three bisectors. If we take an arbitrary ABAM and denote it as B(A1 , A2, A3 ), symmetry operations applied to the ternary plot enable to obtain at most 6 distinct ABAM patterns that are all obtained by the six possible permutations of the angles (e.g., B(A1 , A3, A2 ) and B(A3, A1 , A2 )). Since all these six ABAM patterns can be simply obtained through rotational and reflectional operations applied to one of them, this notion proves that the symmetry of the ternary plot, i.e., the geometric parameter space, correlates with that of their ABAM patterns, the physical space, and that these kirigami patterns are all identical. As a result, the study of the entire parametric space of our ternary plot can be reduced to the model of one of its subsets, e.g., one describing shapes with angles A1 < A2 < A3 (or equivalently 11 < I2 < 3), i.e., the lower-right region bounded by the red triangle.

[0069] Following Neumann’s principle, the symmetry of physical properties in a crystal can only be higher or equal to the symmetry of the crystal lattice. To enable non-uniform scaling of deployment, the symmetry of the uniformly scalable baseline, i.e., BAM should be reduced. To assess the scaling mode, unit cells are analyzed in their deployed state under periodic boundary conditions while ignoring finite size effects. For the “p31 m” BAM, it is observed that the deployed unit cell also belongs to “p31 m” class, and the pertinent strain tensor values must remain unchanged from rotational symmetry operations of angles 120° and -120°. This in turn constrains the strain tensor in the deployed state to carry an isotropic symmetry, i.e., it only permits a uniform change of the scaling factor; here the internal cut parameters are uniformly scaled during deployment. In ABAM, the unit cells with a lower order of symmetry undergo anisotropic scaling in the deployed state, and cause residual geometric frustration in the Y-strut beams in the open stable state.

[0070] In the context of the present disclosure, the expression “geometric frustration” implies that, in the expanded configuration, even if the ABAM is stable in its expanded configuration, some of its member may be subjected to forces. This is better illustrated in Fig. 3 that shows some of the members slightly deviating from a straight line. Thus, in this state, the members are stillsubjected to a force in flexion even if the ABAM is stable and remains by itself in the expanded configuration without any external force applied thereto.

[0071] The inner triangles appear almost undeformed because the frustration is mitigated by the compliance of slender Y-strut units. For “cm” ABAM (Figure 4), the two beams of equal length exhibit deflections in opposite directions, while the last beam is undeformed, a phenomenon that preserves the “cm” class in the deployed stable state. On the other hand, in “p1 ” ABAM, all beams of the Y-strut unit are deformed, and the deployed unit cell can exhibit only “p1 ” symmetry.Bistability, anisotropy and non-uniform scaling

[0072] To investigate the role of anisotropic scaling during ABAM deployment, we track the evolution of three quantitative measures, each expressed as a function of the geometrical parameters A2, A3, a1 / £1 , and t / £1 : bistability, described by the valley over the peak ratio of the strain energy density Emin / Emax, anisotropy, measured by the maximum value of engineering shear strain ymax for all choices of direction a in plane , which is obtained from the right Cauchy- Green deformation tensor Cij following finite strain theory:

[0073]

[0074] magnitude of scaling, defined here by the engineering volumetric strain eV, computed from the principal in-plane strains s1 and s2:

[0076] These strain measures are here selected for simplicity. While other strain measures, such as true strain, could be considered, they would not affect the overall findings in this work.

[0077] All the metrics above are obtained through a set of numerical simulations of various unit cells under periodic boundary conditions. The unit cell is initially loaded under uniaxial tension along the horizontal direction beyond its bistable state, and then is released to return the bistable state. Moreover, the deployed state of any generic ABAM can be determined with a good level of accuracy through a semi-analytical model that assumes rigid triangles, Euler-Bernoulli beams, and pure rotational hinges. In our model described in, the open state should satisfy the balanceof in-plane bending moments at the junction of three beams without any external work. This translates to:

[0078]

[0079] where Wi is the is the width of the ith beam, Ai is the length of the ith beam, expressed as a function of A1 , A2, and A3 and ut® are the unknown transverse displacement at the tip of the ith beam. The geometrical constraints imposed by the rigid blocks combined with the periodicity of the pattern provide the two additional equations required to determine ut®, that ultimately enable to describe the ABAM configuration in the deployed state.

[0080] Fig. 5A shows an ABAM unit cell with “cm” symmetry (A1 = 40°, A2 = A3 = 70°) in its equilibrium states, which are obtained both from our semi-analytical model and numerical simulations with periodic boundary conditions. This unit cell design is representative of the designs shown in Figure 4. The results are overall in good agreement, although the former does not account for the stretching and compression in the living hinges, hence slightly overestimating the strain level (both normal and shear strains) as shown in the polar plot of Figure 5B. Here, the hourglass-shaped curve of the engineering normal strain s is observed, indicating the highly anisotropic scaling of the deployed unit cell with a horizontal expansion 200% above its vertical counterpart. The engineering shear strain y, which describes the distortion of the material through the measurement of the local changes of angles, is also depicted in the polar plot to quantify the magnitude of the anisotropic distortion of the deployed unit cell along all directions, ymax = 0.38 is the maximum value that appears at the direction of 45°. Figure 5C shows the evolution of the stress and strain energy density as a function of an effective engineering strain applied to the periodic unit cell along the horizontal directions. The stress curve exhibits a negative stiffness, i.e., the negative slope in the stress strain plot, around s = 0.5, until the stress finally transits from negative to positive values. The existence of negative stress leads to a local minimum of the energy density curve Emin that is far from the initially undeformed state, where ABAM reaches its bistable deployed state. Additionally, we note that ABAM in the bistable state can maintain its shape without external forces (Figure 5C).

[0081] Next, a parametric analysis with numerical simulations is performed to unveil the relation between slit symmetry breaking and anisotropic bistable scaling. To manipulate the symmetry properties of ABAM, we sweep the internal angles 1, 2, and A3 between 10° and 160°with an increment of 5°. We first investigate the global parameters by prescribing the hinge thickness (f / / i = 0.04) and the size of the internal triangle (a-i / / ) = 0.6), values that ensure bistability for a sizeable range of ABAM patterns. In a second step, we also explore the role of the internal parameters, i.e., the internal triangle size a1 / £1 varying from 0.4 to 0.8, to capture the influence of slit symmetry breaking in a broader design space. Although in this work we focus on the role of slit symmetry breaking in an elastomer kirigami metamaterials, the choice of other material types could also affect their mechanical response.

[0082] Figure 5D shows the first set of results of the parametric study, where the bistability index Emin / Emax is plotted on the ternary plot. What we observe here is the role of symmetry breaking in ABAM, indicating that a departure from symmetry, with the central point being the BAM that has the highest degree of symmetry (p31 m), diminishes the energy barrier. The map shows that excessive distortion in the slit pattern hinders bistability and can eventually downgrade ABAM to a monostable kirigami for internal angles Ai (i = 1 , 2, 3) above 105°. While ABAM lying on the dashed bisectors (thickset “cm” ABAM) of the ternary plot mark the direction where bistability weakens most rapidly, the maximum engineering shear strain ymax, i.e., anisotropy (Figure 5E), and the volumetric strain eV, i.e., magnitude of scaling (Figure 5F) measured, also show a strong dependency on the symmetry group. The slender “cm” ABAM, marked by solid bisectors (e.g., lower left side in Figure 4), exhibit the highest levels of anisotropy and normal expansion with large values of shear and volumetric strains, whereas the thickset “cm” ABAM, marked by dashed bisectors (e.g., upper right side in Figure 4), tend to exhibit low values of anisotropy and normal expansion. On the other hand, an ABAM of the “p1 ” symmetry group shows moderate responses bounded by those of the slender and thickset “cm” ABAM.

[0083] The most regular “p31 m” BAM provides the highest bistability, while “p1 ” and “cm” ABAM exhibit a reduced value, but they offer a sizeable tunability for anisotropic scaling. This observation suggests that slit symmetry breaking controls the trade-off between bistability and anisotropy of scaling and Figure 5G shows their antagonistic nature. The “p31 m” BAM unit cells with varying size of the inner triangle (a1 / £1 from 0.4 to 0.8) show no shear strains (red dots) and are unable to attain anisotropic deployment. For “p1 ” ABAM, domains with various shear strains can be accessed for all values of a1 / £1 from 0.4 to 0.8. Lastly, slender “cm” ABAM (solid lines) form the lower bound of the domains of “p1 ” ABAM in terms of bistability index Emin / Emax, indicating that the slender “cm” ABAM is the most effective in achieving anisotropic deployment while preserving bistability. In contrast, the thickset “cm” ABAM (dashed lines), is the least bistable among ABAM for a given shear strain.

[0084] Figure 5H to 5J further explore the role of slit symmetry breaking in a broader design space of ABAM. In particular “cm” ABAM (A1 = A2), taken as representative, is characterized by the three metrics (Emin / Emax, ymax, eV) in a 3D plot for three values of a1 / £1 , each visualized by a color in its corresponding section view. As can be observed in Figure 5H, the design space of ABAM expands with a1 / £1 , while the bistability index Emin / Emax still increases monotonically for ABAM that departs from the most symmetric BAM, “p31 m” (A1 = A2 = A3 = 60°). Figure 5I and 5J show that both the shear strain and the volumetric strain are higher for larger values of a1 / £1 , whereas the “cm” ABAM trace the directions in which the strains vary either most rapidly or slowly. These results further demonstrate that slit symmetry governs the anisotropic bistable deployment. In some kirigami patterns, a unique and extreme case of strain can be achieved by using the slender "cm" ABAM design. This design features a combination of angles and ratios that result in a maximum engineering shear strain of 1.12 and a maximum volumetric strain of 3.37. This showcases the potential of symmetry-breaking in kirigami patterns for creating anisotropic shape morphing.From repeating unit to ABAM tessellation: experimental characterization of tensile response

[0085] Referring to Fig. 6, through a set of experiments and numerical simulations, the anisotropic bistable response in ABAM specimens undergoing a cycle of tension (loading) and compression (unloading) is characterized. Four specimens, designed with given geometry and symmetry group, were laser cut from 6.35 mm-thick natural rubber sheets and pulled at a strain rate of 10 mm / min (Video S1 and S2, Supporting Information). The periodic array for each sample comprises 5 x 5 unit cells.

[0086] Figure 6 shows the bistable responses of all fabricated specimens, able to retain their deployed state even without external load. The deformation observed during tensile test is typical of a mechanism featuring rotational hinges, where the Y-strut motifs in the “cm” and “p1” ABAM specimens are frustrated, hence being deformed unlike those in the “p31 m” BAM specimen (Figure 6). During the tension process, all specimens go through various metastable states before they reach the fully open stable state. In the unloading stage, the negative values in the loaddisplacement curves demonstrate that the specimens can resist the action of the external compression and maintain their fully open shapes until the compressive load overcomes their energy barrier and guides them back to their initial state. Being monolithic with the rest of the structure, neighboring units undergo relative rotation by hinge bending. As a result, geometric frustrations occur with the relative rotation between the triangles and the Y-strut motifs, aphenomenon that in turn results in structural bistability. The unit cells in the corner may not deploy uniformly as the others do in the middle. This phenomenon occurs because their motion is not confined by adjacent unit cells, and can be avoided by pulling the kirigami with distributed loading.

[0087] An additional observable feature, which is specific to the ABAM, is the programmable anisotropic deformation of its unit cells, enabling the tailoring of both the magnitude and the direction of non-uniform morphing. In contrast to the “p31 m” BAM specimen that expands uniformly in all directions and maintains its macroscopic outline in the deployed state, the “cm” and “p1 ” ABAM specimens display distinct and tunable expansion in response to the vertical pull, showing an envelope that is dissimilar from its initial undeformed state. To better illustrate the anisotropic deformation of ABAM, a square area was initially painted on the surface of the specimens. In the deployed state, the two “cm” ABAM specimens elongate mainly in the pulled direction but with distinct stretch ratios, whereas the “p1 ” ABAM specimen mainly expands in an oblique direction.

[0088] As perthe load-displacement curves, all specimens start with a short linear stage, then enter a saw-toothed plateau that alternates between positive and negative values of stiffness, and finally increase monotonically with the displacement. A large hysteresis loop appears between the loading and unloading curves mainly due to the negative stiffness behavior of the unit cells that dissipate energy via snap-through buckling. In the presented experiments, the energy loss may also be partially attributed to the friction between the specimen and the clamps introduced to suppress out-of-plane buckling as well as the material viscoelasticity of rubber.

[0089] The results of these experiments are supported by numerical analyses that capture the overall behavior of the specimens. The load-displacement curves show three major peaks under tension and a negative portion during the compression stage, indicating structural bistability. The first two peaks are caused by the snapping of unit cells around the clamps, and the final peak is caused by the opening of the remaining unit cells. The experimental curves in Figure 6 also show minor peaks in addition to the major peaks in the numerical curves. This factor can be attributed to tiny misalignments and manufacturing defects that emerged during the fabrication of our specimens, which mildly affected the local snap-through events. The numerical model does not account for these defects.Programming planar bistable deployment: anisotropic snapping

[0090] After uncovering the physical mechanism underpinning anisotropic bistable scaling of ABAM, its potential for programming planar arbitrary deployment is showcased. The first simple deployment target is a “shear band”, which we aim to achieve with an initially rectangular-shaped kirigami. To do so, we start by combining “p31 m” and “cm” ABAM unit cells in a rectangular domain (Figure 7 A). In the initial state, 4 x 4 “cm” ABAM unit cells (A1 = A3 = 45° and A2 = 90°) are sandwiched between eight layers of “p31 m” BAM unit cells (A1 = A2 = A3 = 60°). Once deployed, the “p31 m” unit cells expand uniformly in all directions with no shearing, whereas the “cm” unit cells exhibit sizeable shear strains that skew the material by about 25.5°. The outcome is the transformation of the initially rectangular sheet into a “shear band”. As a second example, we add complexity to the planar shape target, aiming to deploy a rectangular-shaped kirigami into a letter “M”. Figure 7B illustrates the result where the oblique strokes are formed by the shearing of “cm” unit cells and the other strokes consist of isotropic scaling “p31 m” unit cells (Video S4, Supporting Information). In these examples, while “p31 m” BAM maintains its initial shape with isotropic expansion in the deployed state, ABAM with reduced symmetry undergoes anisotropic scaling with considerable shear strains. The collective shearing of the symmetry-broken kirigami units provide the driving force to distort the material from its initial rectangular shape to a more complex target configuration.

[0091] With the anisotropy in shape-morphing, ABAM can leverage additional components to achieve new functions that go beyond shape-shifting. Figure 7C shows a demonstrative example where we leverage the shearing of “cm” unit cells (A1 = A3 = 71.6°, A2 = 36.8°) to control the assembly of a load bearing mechanism. The “cm” unit cells are initially tessellated in a parallelogram manner, hence enabling them to deploy into a bistable rectangle profile. In this demonstration, the effective shear strain of ABAM is harnessed to align functional components (blue). In the physical realization, a set of acrylic hooks are attached on the surfaces of the translational Y-strut motifs in the “cm” unit cells to act as load-bearing components. When the “cm” unit cells switch to the deployed state, the hooks interlock with each other and form a chain that can resist tensile forces. For example, although the ABAM made from 2-mm-thick rubber sheet is extremely soft and floppy, the augmented ABAM structure can leverage the hooks as an exoskeleton substrate that altogether can withstand a weight of 3 kg (Video S5, Supporting Information). Along with interlock hooks for load-bearing demonstration, ABAM can also be combined with other functional components, such as electrical circuits, imaging devices, and magnetic materials, to achieve a wide range of applications beyond shape morphing.Programming three-dimensionally bistable anisotropic deployment

[0092] Besides planar deployment, ABAM can anisotropically deploy from a flat state to a 3D stable shape. To demonstrate this potential, we choose a simple yet comprehensive 3D shape target (Figure 8A) that simultaneously contains positive (dome-like surfaces), zero (ruled surfaces), and negative (saddle-like surfaces) Gaussian curvatures. The 3D target comprises three semi-cylindrical shells, including two perpendicular-cut segments (i) and (iii) at the ends (red), and one oblique-cut segment (ii) in the middle (blue). This shape target ensures that the Gaussian curvature is positive at the peak between segments (i) and (ii), zero within each segment, and negative at the saddle between segments (ii) and (iii).

[0093] For simplicity, we assume that the 2d precursors of both the whole 3D target and its constituent segments all have rectangle shapes (Figure 8B). To determine the scaling ratio between the precursor and the target, we first divide the 3D target into three segments (Figure 5C). Segments (i) and (iii) of the target can be flattened as rectangles, but segment (ii), being an oblique-cut semi-cylindrical shell, can only be flattened as a chevron shape. Comparing the geometry of the flattened targets and their precursors, all segments exhibit changes in their sizes with non-zero volumetric strains eV 0. In addition, segment (ii) also displays a nontrivial shear strain y 0, which determines the oblique angle 0 of the 3D target (Figure 8G) as:

[0094] To attain isotropic scaling segments (i) and (iii) (eV 0 and y = 0), we select the “p31 m” unit cell (A1 = A2 = A3 = 60°), while for the anisotropic scaling segment (ii) (eV 0 and y 0), we chose a square-shaped “cm” unit cell (A1 = A3 = 45°, A2 = 90°) to ease the tessellation.

[0095] Figure 8D shows the fabricated sample of the 2d precursor, assembled with 4 columns of “cm” unit cells that are sandwiched between eight columns of “p31 m” unit cells. Due to the symmetry of the flattened chevron target, the segment (ii) is tessellated with two mirrored packs of 4x4 “cm” unit cells. In Figure 8E-8G, the deployed state of the sample can tightly match the 3D printed target surface, which proves the dominant role of shear strain in achieving the target (Equation 4). This demonstrates the potential of ABAM and the key role of non-uniform scaling for three-dimensional bistable deployment which could not be otherwise obtained with existing isotropically scalable kirigami. We also note that the 3D target surface here is GO continuous, and it contains only two unit cell designs. For 3D surfaces with more complex shapes, one potentialapproach is to improve the design of the unit cells with inverse design strategies and optimization algorithms. These methods have been successfully applied in the field of shape-morphing kirigami. Although the ABAM specimen in this work cannot resist gravity because they are fabricated with soft rubber sheets, it is possible to create free-standing 3D surfaces by increasing the stiffness of ABAM. For example, we can fabricate ABAM with stiffer materials, while the stress concentration at the hinges can be reduced by tailoring the geometry of the kirigami constituents, such as the living hinges. Another way is to optimize the unit cell design for high stiffness at the bistable state or incorporate exoskeletons, as shown in our demonstration with interlock hooks.Conclusions

[0096] To conclude, our combined semi-analytical, experimental, and numerical investigations have shown that symmetry breaking of slit pattern induces geometric frustration and anisotropic shape morphing in bistable kirigami and controls the trade-offs between bistability and anisotropy of scaling. While symmetry breaking has been used to generate anisotropic response in other metamaterial architectures, this work unlocks anisotropic morphing in kirigami metamaterials and further unveils how symmetry groups affects geometric frustration and their anisotropic bistable shape shifting. For example, "p31 m” BAM with threefold rotational symmetry features frustration-free bistable isotropic expansion, “cm” ABAM with lower symmetry undergoes frustrated anisotropic deployment with reduced bistability, and “p1” ABAM with least symmetry exhibits a moderate response that is frustrated and bounded by “cm” ABAM. The shape morphing capability of symmetry broken kirigami is demonstrated via a set of in-plane and out-of-plane deployments with various target shapes that cannot be achieved without shear strains, the key indicator of anisotropic deformation. This work has thus established the link between symmetry, frustration, and anisotropic bistable deployment in soft kirigami metamaterials, which can offer a new design route for non-uniform shape morphing.

[0097] The symmetry-broken kirigami in this work is entirely made of soft elastic materials to accommodate geometric frustration, which limits the stiffness of the kirigami especially in the deployed state. A potential approach to overcome this limitation is to take a multimaterial design strategy with both soft and rigid components, which for example could increase the load bearing capacity of kirigami metamaterials for engineering applications such as impact energy absorption. Moreover, symmetry-broken kirigami can be made responsive to multiphysical stimuli through the integration of active materials, such as shape memory polymers, hydrogels, and dielectric elastomers.

[0098] While the concluding demonstrations of our work show how to harness slit symmetry breaking to tailor anisotropic deployment in bistable kirigami, our strategy holds generality and can also be extended to other kirigami patterns as well as other types of metamaterials and metasurfaces. For example, the four-fold symmetry of the square-cut motifs can be reduced by changing the square cuts into rectangular cuts, rhombic cuts, parallelogram cuts, trapezoidal cuts, and even arbitrary quadrilateral cuts. Besides symmetry breaking of slit pattern and geometric design, the asymmetric distribution of multiple materials within a structure, e.g., materials with distinct viscoelasticity and coefficient of thermal expansion, could also be exploited to attain uncommon yet highly desired anisotropy in properties and functionality for a diverse range of applications in aerospace, robotics, and other sectors.

[0099] It is noted that various connections are set forth between elements in the preceding description and in the drawings. It is noted that these connections are general and, unless specified otherwise, may be direct or indirect and that this specification is not intended to be limiting in this respect. A coupling between two or more entities may refer to a direct connection or an indirect connection. An indirect connection may incorporate one or more intervening entities. The term “connected” or "coupled to" may therefore include both direct coupling (in which two elements that are coupled to each other contact each other) and indirect coupling (in which at least one additional element is located between the two elements).

[0100] It is further noted that various method or process steps for embodiments of the present disclosure are described in the preceding description and drawings. The description may present the method and / or process steps as a particular sequence. However, to the extent that the method or process does not rely on the particular order of steps set forth herein, the method or process should not be limited to the particular sequence of steps described. As one of ordinary skill in the art would appreciate, other sequences of steps may be possible. Therefore, the particular order of the steps set forth in the description should not be construed as a limitation.

[0101] Furthermore, no element, component, or method step in the present disclosure is intended to be dedicated to the public regardless of whether the element, component, or method step is explicitly recited in the claims. As used herein, the terms “comprises”, “comprising”, or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus.

[0102] While various aspects of the present disclosure have been disclosed, it will be apparent to those of ordinary skill in the art that many more embodiments and implementations are possible within the scope of the present disclosure. For example, the present disclosure as described herein includes several aspects and embodiments that include particular features. Although these particular features may be described individually, it is within the scope of the present disclosure that some or all of these features may be combined with any one of the aspects and remain within the scope of the present disclosure. References to “various embodiments,” “one embodiment,” “an embodiment,” “an example embodiment,” etc., indicate that the embodiment described may include a particular feature, structure, or characteristic, but every embodiment may not necessarily include the particular feature, structure, or characteristic. Moreover, such phrases are not necessarily referring to the same embodiment. The use of the indefinite article “a” as used herein with reference to a particular element is intended to encompass “one or more” such elements, and similarly the use of the definite article “the” in reference to a particular element is not intended to exclude the possibility that multiple of such elements may be present.

[0103] The embodiments described in this document provide non-limiting examples of possible implementations of the present technology. Upon review of the present disclosure, a person of ordinary skill in the art will recognize that changes may be made to the embodiments described herein without departing from the scope of the present technology. Yet further modifications could be implemented by a person of ordinary skill in the art in view of the present disclosure, which modifications would be within the scope of the present technology.

Claims

CLAIMS1 . An anisotropic auxetic bistable metamaterial (ABAM), comprising: an assembly of building blocks, a building block of the building blocks having a quadrilateral shape and having: two unit cells conjointly defining four sides and four corners, the two unit cells disposed on opposite sides of a bisecting line extending across two opposite ones of the corners, a unit cell of the two unit cells having: three sides interconnected to one another and including two of the four sides and the bisecting line, and three slits each extending from a respective one of the three sides towards a respective opposite one of the three sides, the three slits free from intersection between each other; a rotating member bounded by the three slits; translating members disposed around the rotating member; and hinges at ends of the three slits, the rotating member being rotatable relative to the translating members about the hinges, wherein the assembly is non-plastically and reversibly deformable from a collapsed position to an expanded position upon application of a first load, and the assembly being non-plastically and reversibly deformable from the expanded position to the collapsed position upon application of a second load, wherein the translating members are geometrically frustrated in the expanded position.

2. The ABAM of claim 1 , wherein the assembly is bistable and thereby defining a first stable state in the collapsed position and a second stable state in the expanded position, and wherein in the first stable state the assembly maintains the collapsed position after removal of the second load applied thereon and, in the second stable state, the assembly maintains the expanded position after removal of the first load applied thereon.

3. The ABAM of claim 2, wherein between the expanded position and the collapsed position, the building blocks move relative to each other such that the assembly deformstoward a closest one of the expanded position and the collapsed position after removal of either of the first load and the second load applied on the assembly.

4. The ABAM of claim 3, wherein a dimension of the assembly of the building blocks taken in a direction normal to a direction of application of the first load being greater in the expanded position than that in the collapsed position.

5. The ABAM of any one of claims 1 to 4, wherein a ratio of a slit length (a1) of a slit of the three slits to a length (11) of a side of the three sides being parallel to the slit ranges from 0.4 to 0.8.

6. The ABAM of any one of claims 1 to 5, wherein angles defined two adjacent ones of the three sides range from 10 degrees and 160 degrees.

7. The ABAM of any one of claims 1 to 6, wherein an end of a slit of the slits is spaced apart from an adjacent slit by a gap having a thickness (t), a ratio of the thickness to a length of a side of the three sides being parallel to the slit being from 0.03 to 0.08.

8. The ABAM of claim 7, wherein the ratio of the thickness to the length is about 0.04.

9. The ABAM of any one of claims 1 to 8, wherein a ratio of a thickness of the building block to a length of a side of the three sides is about 0.3.

10. The ABAM of any one of claims 1 to 9, wherein the three sides are of equal length, theABAM featuring a threefold rotational symmetry and reflectional symmetry in both of the expanded position and the collapsed position.1 1 . The ABAM of any one of claims 1 to 9, wherein two of the three sides are of equal length and a third one of the three sides has a different length, the ABAM featuring reflections and glide reflections with parallel axes.

12. The ABAM of any one of claims 1 to 9, wherein the three sides are all of different lengths, the ABAM being devoid of both rotational and reflectional symmetry in both of the expanded position and the collapsed position.

13. A building block for an anisotropic auxetic bistable metamaterial (ABAM), comprising:two unit cells conjointly defining four sides and four corners, the two unit cells disposed on opposite sides of a bisecting line extending across two opposite ones of the corners, a unit cell of the two unit cells having a triangular rotating member surrounded by three translating members rotatably connected to the triangular rotating member by hinges, wherein the building block is non-plastically and reversibly deformable from a collapsed position to an expanded position upon application of a first load, and the building block being non-plastically and reversibly deformable from the expanded position to the collapsed position upon application of a second load.

14. The building block of claim 13, wherein three slits are located between adjacent sides of the triangular rotating member and the three translating members.

15. The building block of claim 14, where a ratio of a slit length (a1) of a slit of the three slits to a length (11) of a side of the unit cell being parallel to the slit ranges from 0.4 to 0.8.

16. The building block of claim 14 or 15, wherein an end of a slit of the slits is spaced apart from an adjacent slit by a gap having a thickness (t), a ratio of the thickness to a length of a side of the unit cell being parallel to the slit being from 0.03 to 0.08.

17. The building block of claim 16, wherein the ratio of the thickness to the length is about 0.04.

18. The building block of any one of claims 13 to 17, wherein angles defined two adjacent ones of sides of the unit cell range from 10 degrees and 160 degrees.

19. The building block of any one of claims 13 to 18, wherein a ratio of a thickness of the building block to a length of a side of the unit cell is about 0.3.

20. The building block of any one of claims 13 to 19, wherein: the three sides are of equal length, the ABAM featuring a threefold rotational symmetry and reflectional symmetry in both of the expanded position and the collapsed position; or two of the three sides are of equal length and a third one of the three sides has a different length, the ABAM featuring reflections and glide reflections with parallel axes; orwherein the three sides are all of different lengths, the ABAM being devoid of both rotational and reflectional symmetry in both of the expanded position and the collapsed position.

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