Modular origami approach for rigid foldable steel load-bearing plate lattices in arbitrary sizes

Abstract

An approach for fabricating rigid foldable steel load-bearing structures is disclosed. The invention employs unit cells with an expanded-truncated rectangular pyramid geometry, formed from metallic sheet stock through a progressive folding process. The modular design reduces strain concentrations, eliminates edge-edge connections, and enables robust mechanical joining methods such as riveting. The manufacturing process uses press-forming molds to ensure precision and accommodate springback in metallic materials. The resulting structures exhibit customizable relative densities, enhanced mechanical performance, and improved energy absorption, making them suitable for static and dynamic loading scenarios.

Description

CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of priority of U.S. provisional application No. 63 / 671,675, filed Jul. 15, 2024, the contents of which are herein incorporated by reference.BACKGROUND OF THE INVENTION

[0002] The present invention relates to novel approaches for fabricating rigid load-bearing structures.

[0003] In cellular solids nature found ways to enhance material properties via geometric configurations. A distinctive aspect of these materials is that the geometry of their unit cells influences their properties to the same extent as their constitutive materials [6] (Gibson and Ashby 88). All cellular materials can be classified in two principal families. If the unit cell is confined to its edges, it is classified as an open cell. On the other hand, if the unit cell is contained in both edges and facets, it is identified as a closed cell material.

[0004] Architected materials are an endeavor to mimic nature's approach to cellular solids. These topology-oriented materials are an interest in current research because they offer methods for producing solids with engineered properties.

[19] (Liu 20) demonstrated that given identical relative densities, constituent materials, and topologies, closed-cell architected materials outperform their open-cell counterparts in mechanical properties. [1] (Berger et al. 17)'s pioneering work introduced the term plate lattice, showcasing architectures that achieve the Hashin-Shtrikman bound for isotropic stiffness.

[0005] However, there exists a gap between the proven properties of plate lattices and their manufacturability. Most existing research on lattice materials has depended on additive manufacturing (AM) techniques, specifically powder bed fusion (PBF), which are capable of producing complex geometries using load bearing materials across various industries [4] (Fidan et al. 24). Yet, for employing lattices as bulk materials in architectural and engineering applications, AM currently lacks material flexibility, operates at high energy consumption levels relative to production rates [8] (Gutowski et al. 17), and demonstrates a direct correlation between part quality and energy density for structural components

[14] (Liu et al. 18).

[0006] By contrast, origami methods offer a unique capability since they transform existing materials rather than adding or subtracting. By inherently encoding 3D spatial information within a 2D domain, origami has emerged as a potent design and manufacturing tool for metamaterials and cellular materials. [2] (Cheung et al. 14) revealed highly anisotropic, elastic, and rigid metamaterials based on volumetric tiling of the Miura-Ori pattern. [5] (Filipov et al. 15) showed that origami tube-like structures could be hierarchically assembled to create cellular materials with varied mechanical properties.

[15] (Miyazawa et al. 21) introduced configurable multicellular origami materials with customized static responses within the same material. Furthermore, [9] (Jamalimehr et al. 22) developed metamaterials with a self-locking unit cell geometry that bears loads along the deployment axis, among many other studies.

[0007] This aforementioned research path exploits origami as a design method increasing complexity in the unit cell geometry, which presents significant manufacturing challenges if attempting manufacturing with structural materials.Structural Origami Cellular Materials.

[0008] The most successful attempt at origami engineering to manufacture large scale folded structural elements from load-bearing materials was in 2005 with the establishment of Foldcore GmbH. Work by its founder introduced origami-based open cells for the aerospace industry and related sandwich panels, however these primarily utilized composites as constitutive materials in the form of sandwich panels.

[12] (Klett et al. 07) [Klett and Drechsler 11]. In the domain of metallic origami structures, research by

[13] (Li et al. 11) presented folded sandwich construction approaches with open facets, studying various unit cells for energy absorption.

[0009] While continuum sandwich panel manufacturing has matured as an industry, its technology has not heretofore exploited the benefits of folding and assembling for generating larger lattice materials.

[0010] The present disclosure is directed to the manufacture of metallic folded plate lattices. By employing discretely assembled unit cells, embodiments of the present invention decrease the folding pattern complexity, facilitating simple manufacturing. By having fewer creases to fold, these embodiments simplify the use of load-bearing materials such as steel to create load bearing plate lattices.SUMMARY OF THE INVENTION

[0011] In one aspect of the present invention, there is disclosed an approach for fabricating rigid load-bearing structures in arbitrary sizes. This method addresses longstanding challenges in manufacturing plate lattices, particularly the limitations of additive manufacturing (AM) techniques, such as scalability, energy inefficiency, and high costs. By utilizing sheet stock and progressive folding techniques, the invention simplifies the design and assembly process, enabling the use of structural materials like steel in a scalable and cost-effective manner. The sheet stock can be made of any bendable material including, but not limited to, metal, carboard and plastic. In a preferred embodiment, the sheet stock is made of metal, such as steel, aluminum, titanium or copper, to name a few.

[0012] In another aspect of the present invention, there is disclosed discretely assembled unit cells based on an expanded-truncated rectangular pyramid geometry, which reduces the complexity of folding patterns and minimizes strain concentrations in metallic sheets. The modular design facilitates robust mechanical connections, such as riveting or welding, and eliminates reliance on edge-edge connections, thereby improving manufacturability and structural integrity. Additionally, the invention introduces a scalable manufacturing process using press-forming molds, which ensures precision and repeatability while accommodating the springback phenomenon inherent in metallic materials.

[0013] This approach offers several key benefits, including enhanced mechanical performance, customizable relative densities, and improved energy absorption characteristics, making the lattices suitable for both static and dynamic loading scenarios. The modular assembly method further enables the creation of large-scale structures with high strength-to-weight ratios, while reducing material waste and production costs. The invention is particularly advantageous for applications in aerospace, automotive, and architecture, where lightweight, strong, and cost-effective load-bearing structures are critical.

[0014] By overcoming the limitations of existing fabrication methods and leveraging the principles of origami, this invention provides a transformative solution for manufacturing steel plate lattices, unlocking their potential for widespread use in engineering and architectural applications.

[0015] These and other features, aspects and advantages of the present invention will become better understood with reference to the following drawings, description and claims.BRIEF DESCRIPTION OF THE DRAWINGS

[0016] FIG. 1 illustrates a metallic folded plate lattice in accordance with a preferred embodiment of the present invention, wherein FIG. 1A shows partial folding states of the unit cells, FIG. 1B shows stacked unit cells in respectively reversed configurations (one stack upside down compared to the other), and FIG. 1C shows a 3 by 3 by 3 lattice showing cells fastened together in a sturdy construction.

[0017] FIG. 2 illustrates plate lattices formed by expanded-truncated rectangular pyramid unit cells in accordance with a preferred embodiment of the present invention, wherein FIG. 2A shows progressive folding from the flat state to the final configuration, FIG. 2B shows a semi regular cuboctahedron lattice with cuboctahedron and octahedron voids, FIG. 2C shows a mirrored expanded-truncated rectangular pyramid that realizes the principal planes of the octahedron, and (FIG. 2D) shows vertex connected octahedra forming a semi regular lattice.

[0018] FIGS. 3A, 3B and 3C illustrate unit cell parameters in accordance with a preferred embodiment of the present invention, wherein the height (h), width (w) and expanded flaps (b) are denoted, as well as the folded and unfolded degree 5 vertex of the unit cell.

[0019] FIG. 4 illustrates an expansion process in accordance with a preferred embodiment of the present invention, wherein FIG. 4A shows an intermediate cell state used to calculate folding parameters, FIG. 4B shows expansion directions, and FIG. 4C shows expanded-truncated rectangular pyramid unit cells.

[0020] FIG. 5 illustrates manufacturing molds and partially folded unit cells in accordance with a preferred embodiment of the present invention.

[0021] FIG. 6 illustrates an assembly process for a plate lattice size 3 by 3 by 3 in accordance with a preferred embodiment of the present invention.

[0022] FIG. 7 illustrates a periodic boundary conditions diagram in accordance with a preferred embodiment of the present invention, with FIG. 7A showing RVE selected,

[0023] FIG. 7B showing lattice vector components, and FIG. 7C showing meshed cell with na and nb being paired nodes as they satisfy equation 12.

[0024] FIG. 8 illustrates simulations for ultimate compressive strength vs. relative density for all samples.

[0025] FIG. 9 illustrates simulations for elastic modulus vs. relative density.

[0026] FIG. 10 illustrates qualitative comparison for simulated (left) and testing (right) for (1) local shell buckling, (2) unit cell shearing, and (3) shear propagation.

[0027] FIG. 11 illustrates force-displacement results for the simulated lattice and the tested lattice.DETAILED DESCRIPTION OF THE INVENTION

[0028] The following detailed description, together with the figures, is of the best currently contemplated modes of carrying out exemplary embodiments of the invention. The description is not to be taken in a limiting sense but is made merely for the purpose of illustrating the general principles of the invention, since the scope of the invention is best defined by the appended claims.

[0029] Broadly, an embodiment of the present invention provides an approach for fabricating rigid load-bearing plate structures in arbitrary sizes. The invention employs discretely assembled unit cells based on an expanded-truncated rectangular pyramid geometry, which, in a preferred embodiment, are folded from sheet stock and assembled into larger regularly repeating structures, sometimes referred to as “lattices” or “plate lattices” herein and in the accompanying Figures. The modular design and manufacturing process overcome the limitations of existing fabrication methods, enabling scalable, energy-efficient, and cost-effective production of steel plate lattices with superior mechanical properties.

[0030] The design of this architected plate lattice begins with the topology of the unit cell, which, in this context, is essentially a folded geometry. This section outlines the design process and shows the geometric transformations necessary to satisfy manufacturing and mechanical constraints.

[0031] Embodiments of the present invention focus on plate-based cellular materials inspired by semi-regular octahedral lattices. This configuration has gained attention for its capability to be constructed from either a face-connected cuboctahedron unit cell or a vertex-connected octahedron unit cell. This architecture, interpreted as a beam-lattice, satisfies Maxwell's rigidity criteria with the lowest vertex connectivity [3] (Cheung 12).

[0032] For present purposes, this implies a reduction in the number of creases that must be folded to form a unit cell.

[0033] Referring now to FIGS. 1A, 1B, 1C, 2A, 2B, 2C, 2D, 3A, 3B, 3C, 4A, 4B, 4C, 5, 6, 7A, 7B, 7C, 8, 9, 10, and 11, FIGS. 2A-2C illustrate the present approach to utilizing the vertex-connected octahedron unit cell. Embodiments of the present invention employ an expanded-truncated rectangular pyramid as the base shape to form an approximately octahedral unit cell. When these folded geometries are mirrored at their bases, their lateral planes realize the principal facets of an octahedron. In the plane of mirroring, a steel layer is inserted where the two rectangular pyramids will be riveted together to form an octahedron and, thereby, restrain its degrees of freedom (DOFs).

[0034] The rationale for the truncation and expansion is to satisfy assembly and mechanical constraints. It is important to note that these modifications compromise the lattice's isotropy. As shown in (Parra Rubio et al. 23), the engineering of boundary regions is crucial for discretely assembled cellular materials that will be mechanically joined. By avoiding edge-edge connections and instead providing facet-facet contacts, the assembly process is simplified and can rely on robust and accessible mechanical connections such as riveting, welding or screwing for modular assembly.

[0035] The unit cell has 5 DOFs: h, b, w, e, θ2 and ρ1. The process that is followed, and the parameters, can be seen in FIGS. 3 and 4. First, fold an octahedron-like shape from a flat preform by adding inside-reverse folds at each corner. Second, truncate the top of this geometry as seen in FIG. 4A. Next, take the corner vertex as shown in FIG. 3 and calculate the folding angles θ2 and ρ1. The final step is to apply the expansion as shown in FIG. 4C.

[0036] Analytical form and DOFs of the expanded-truncated rectangular pyramid. In this section one assumes that h, w, b, e, and θ2 are fixed, since this uniquely determines the geometry of the flat state. The goal is to describe the geometry of any intermediate or final folded state as a function of ρ1. In the general case, determining the kinematics of such a rigid origami structure is complicated and usually solved numerically

[18] (Tachi 09). But this design has a number of symmetries that enable a relatively simple analytical solution.

[0037] As depicted in FIG. 3, each of the four vertices is identical in the flat state. It is required that they evolve identically through all intermediate states. Thus, one only needs to consider the kinematics of a single vertex.

[0038] Within each vertex, by construction θ1=π / 2 radians. Set 02=05 and 03=04 to obtain symmetry about crease 3 in the flat state. Ensure that this symmetry is preserved during folding by keeping ρ1=ρ5 and ρ2=ρ4. With these constraints, the vertex has a single kinematic degree of freedom. The sum of the face angles is 2π, so one may express θ3 in terms of θ2.θ3=3⁢π4-θ2(1)

[0039] Since θ1 is a right angle, it is convenient to work in a coordinate system centered at the vertex where x{circumflex over ( )} lies along crease 5, y{circumflex over ( )} along crease 1, and z{circumflex over ( )}=x{circumflex over ( )}×y{circumflex over ( )}. Define vectors x1, . . . , x5 lying on the five creases, each one with unit magnitude. The positions of x1 and x5 are fixed.x1=y^(2)x5=x^(3)

[0040] The positions of x2 and x4 in the flat state are readily found via rotation about z{circumflex over ( )}. Their positions in any intermediate state can be found by applying subsequent rotations about y{circumflex over ( )} and x{circumflex over ( )}, respectively.x2=-sin⁡(θ2)⁢cos⁡(ρ1)⁢x^+cos⁡(θ2)⁢y^-sin⁡(θ2)⁢sin⁡(ρ1)⁢z^(4)x4=cos⁡(θ2)⁢x^-sin⁡(θ2)⁢cos⁡(ρ1)⁢y^-sin⁡(θ2)⁢sin⁡(ρ1)⁢z^(5)

[0041] By construction x3 has a magnitude of one, and by symmetry it lies in the plane characterized by x=y. As such, it is convenient to parameterize x3 with respect to a single degree of freedom a.x3=a⁢x^+a⁢y^-p1-2⁢a2⁢z^(6)

[0042] The angle between x2 and x3 is θ3. Since these are both unit vectors, one may write this as x2·x3=cos (θ3). This produces an equation that is quadratic in a. Its two solutions are given below in terms of temporary values u1, u2, u3, and u4.u1=sin⁡(θ2)⁢sin⁡(ρ1)(7)u2=u1⁢u12+2⁢cos⁡(θ2)⁢sin⁡(θ2)⁢(1-cos⁡(ρ1))_(8)u3=(sin⁡(θ2)⁢cos⁡(ρ1)-cos⁡(θ2))⁢cos⁢θ2+π4(9)u4=1+u12-2⁢cos⁡(θ2)⁢sin⁡(θ2)⁢cos⁡(ρ1)(10)a=(u3±u2)u4(11)

[0043] For all physically realizable configurations both roots are real, indicating that faces three and four can be folded toward the inside or outside of the pyramid. To prepare to fold these faces inward, select the maximum of the two solutions. Since u1, u2, u3, and u4 depend only on θ2 and ρ1, this gives us a closed form solution for x3.

[0044] Given x1, . . . , x5, it is easy to compute the crease angles. The cross product of two adjacent crease vectors gives the normal vector of the face they enclose. Each crease angle can be computed as the angle between the normal vectors of the crease's two adjacent faces.Manufacturing.

[0045] Embodiments of the present invention introduce a simple and scalable manufacturing method for fabricating metallic plate lattices via progressive folding and modular assembly. These embodiments employ stainless steel sheet stock with a thickness of 180 microns. In a preferred embodiment, the manufacturing process includes progressive folding, other manufacturing techniques can be used, such as stamping, molding, casting, die casting, thermoforming, roll forming, hydroforming, explosive forming, electromagnetic forming, additive manufacturing, incremental sheet forming, machining, and combinations thereof, or other automated or non-automated methods.

[0046] The process of folding metallic sheets necessitates overcoming plastic deformation. Creases form local areas of high strain, which necessitates the selection of a material with high toughness with a high ultimate stress. However, the inherent trade-off between toughness and strength in materials

[17] (Ritchie 11) requires a compromise. Martensitic steels are ideal for their strength, whereas austenitic steel offers better formability. Embodiments of the present invention use cold worked austenitic 301 stainless steel, as this material is widely available, recyclable and economically affordable.Unit Cell Folding.

[0047] After calculating the unit cell and all its intermediate folding states, progress to cutting the stainless steel sheets in its unfolded configuration. Employ the Fab-Light 3000 laser cutter equipped with a 3KW laser. To mitigate local malformations at the vertices of the truncated facet, FIG. 5 shows how to introduce small holes at each vertex, where multiple creasing lines converge.

[0048] The next step involves progressive molding. Create three distinct molds, each corresponding to a different folding state of the unit cell. Mold A corresponds to a 20% total fold, mold B to 60%, and mold C to 105%. All molds are designed to serve as press forming molds with concave and convex shapes for the corresponding partial folding configurations. 3D print each mold using PLA. A 2-ton manual arbor press is utilized to shape the metal into the desired mold forms.

[0049] Mold A's role is to imprint the crease map onto the stainless steel preform. Mountains and valleys are precisely stamped onto the material to guarantee successful assembly and overall lattice precision. Repetitive and accurate creasing is achieved by employing pins to prevent translations and aligning features to avoid undesired rotations.

[0050] Once the creasing map is imprinted, use Mold B to apply a significantly larger strain on the creases to prepare it for the final mold.

[0051] Mold C intentionally overfolds the cell, taking into account springback phenomenon. The springback angle has been empirically calculated.Discrete Lattice Assembly.

[0052] Embodiments of the present invention introduce a modular approach for assembling pre-folded unit cells into lattices. Most origami tessellations used to create cellular structures are based on parallel origami consisting of multiple vertices from which creases emerge to connect other vertices. Materials like paper, which easily deform out of plane, compliantly deform to accommodate the folding process, achieving the desired final shape. However, folding monolithic tessellations from load-bearing materials that resist out-of-plane deformation poses substantial challenges. The present method involves discretizing the tessellation and folding individual unit cells for subsequent assembly. This technique facilitates the manufacturing of metallic plate lattices in an arbitrary array of n by m by i unit cells. Additionally, the geometry streamlines assembly since all connections are aligned along the Z-axis, presenting significant automation potential through gantry systems

[11] (Jenett 20) or robot swarms [7] (Gregg et al. 24)

[10] (Jenett et al. 19). For this iteration embodiments of the present invention use stainless steel blind rivets as the mechanical connectors. Other connectors are contemplated as a matter of application specific design choice.

[0053] FIG. 6 illustrates the assembly strategy for a lattice composed of 3×3×3 unit cells. The construction progresses from the bottom up, requiring coordination only with elements in the immediately adjacent layer. This 2.5D assembly method alternates between two modes, ensuring row-by-row assembly by coordinating between the upper and lower layers. The first mode involves aligning the truncated face with its corresponding mirror cell, shown in FIG. 6 step 1 for the coordination motion and step 2 for the outcome. The second mode involves the coordination and assembly of expanded base flaps, enclosing a base plate, demonstrated in step 3. This procedure is repeated until the desired structure is achieved.Modeling.

[0054] To analyze the mechanical response of the proposed plate lattices under uniaxial compression, nonlinear finite element simulations were conducted with ABAQUS CAE. This involved a stochastic search across numerous unit cell candidates to select one optimized for performance and manufacturability, followed by detailed simulation of a compression test on a 3×3×3 lattice.Periodic Boundary Conditions.

[0055] Simulating plate lattices becomes computationally intensive with increasing mesh size, especially in lattices with numerous unit cells. Lattices with fewer elements face boundary node dominance, while those with extensive unit cells achieve optimal performance through internal node predominance.

[0056] To simulate infinitely large cellular solids with minimal computational cost, Periodic Boundary Conditions (PBC) was employed. This approach involves selecting a Representative Volume Element (RVE)—the unit cell—and adjusting the DOFs of boundary nodes and the constitutive equation, significantly reducing element count, convergence time, and computational power.

[0057] As the RVE is spatially periodic, FIG. 7 illustrates the definition of RVE as the bounding box of our unit cell, delineated by three lattice vectors LVx, LVy, and LVz.

[0058] The unit cell is meshed with symmetric seeding across the XY, XZ, and YZ planes. Nodes are selected if they satisfy the following equation:nb-na=n1⁢LVx+n2⁢LVv+n3⁢LVz(12)where nb and na are the node coordinates and n1, n2 and n3 are the three components of any possible linear combination of the 3D Lattice Vector:[n1n2n3]=

[100] ⁢ or

[010] ⁢ or

[001] ⁢ or

[110] ⁢ or

[011] ⁢ or

[101] ⁢ or

[111] (13)One now pairs their displacements with the following equation:ub-ua=H⁡(Xb-Xa)(14)where H is the displacement gradient matrix and Xa and Xb are the coordinate of the selected nodes at the undeformed mesh state. The matrix X can be now designed with three virtual nodes that will impose a macroscopic deformation. In this case, as sz,a constant strain value is desired that corresponds to uniaxial compression in the Z-axis:H=[ux,VirtualNode⁢1⁢uy,VirtualNode⁢1⁢uz,VirtualNode⁢1ux,VirtualNode⁢2⁢uy,VirtualNode⁢2⁢uz,VirtualNode⁢2ux,VirtualNode⁢3⁢uy,VirtualNode⁢3⁢uz,VirtualNode⁢3]=[00000000sz](15)PBC may be implemented using Abaqus scripting, and a batch of simulations may be run to obtain the unit cell stress response to the strain. Simulations for different heights, different ρ1 values and same θ 2 are possible.FIGS. 8 and 9 present simulation results employing Periodic Boundary Conditions (PBC) on steel folded unit cells with a thickness of 180 microns. The samples are labeled with part numbers, which are detailed in Table 1. The simulations account for the elastoplastic behavior of stainless steel 301, parameters of which were derived from dogbone tests conducted using an Instron 5985. The geometry is meshed using SR4 elements with 5 Gauss integration points. The unit cell was subjected to a strain of 0.1, following which the results were processed to calculate the stress and stiffness values of the unit cell. Notably, instead of calculating Cauchy stresses, one obtains first Piola-Kirchhoff stresses, as the analysis is based on the initial volume state.TABLE 1Part names and unit cell parameters.Part NumberHeight (mm)ρ1(deg)θ2(deg)A12051106A22053106A32054106A42056106A52057106A62058106A72059106A82060106A92061106B12351106B22353106B32354106B42356106B52357106B62359106B72360106B82361106B92362106C12552106C22554106C32556106C42557106C52558106C62559106C72560106C82561106D13051106D23052106D33054106D43056106D53057106D63058106D73059106D83060106Discussed here is manufacture of a lattice with the sample A8, as it combines both good performance in the simulations and manufacturability. Performance-wise it maximizes stresses and stiffness. This model is more easily manufactured as it tiles well in the unfolded state, making efficient use of the sheet feedstock and the folding angles are suitable for good formability.Modelling of a 3×3×3 Lattice.A 3×3×3 lattice is simulated using sample 8. The unit cell parameters can be seen in Table 1. The simulation parameters are the same as the one defined in the PBC subsection, but in this case the lattice is uniaxially compressed by 20 mm.ResultsDescribed here is a 3×3×3 lattice fabricated using the selected A8 topology. The cellular structure weighs 560 grams, measuring 150 mm×150 mm×123 mm, resulting in a relative density of 0.023, exceeding the anticipated 0.019. This additional weight is attributed to the parasitic mass of steel rivets utilized in assembly, which was not accounted for in the performance projections of the unit cell. FIG. 10 depicts the steel lattice subjected to uniaxial compression testing, using an Instron 5985, conducted at a strain rate of 10 mm / min.

[0065] FIG. 11 compares the mechanical responses of both the simulated and actual lattices. The simulation indicates a stiffer response under load, with a maximum load of 37 kN before plastic deformation begins. Conversely, the test sample displayed a slightly less stiff behavior but closely matched maximum load value around 33 kN. This stress level corresponds to 1.47 MPa, closely aligning with the projected 1.37 MPa from PBC Simulations for that Unit Cell.

[0066] While the stress projections and simulations were accurate, two factors may explain the observed discrepancy in stiffness: First, the model does not account for radii in the folds, assuming them to be infinitely sharp. Besides, some dome curvature was noted in the manufactured truncated facets, potentially increasing the unit cell's compliance. Second, the effect of the rivet assembly was not simulated, although no rivets were observed to fail by shear during compression testing.CONCLUSIONS

[0067] Embodiments of the present invention introduced a modular origami design and manufacturing approach for producing meso-scale steel plate lattices. The approach relies on progressive folding through press forming molding, a base technology that can be scaled to high throughput while demanding less energy consumption than competing processes. Also described here is a method to evaluate mechanical responses of parametrized topologies, fabricated the optimal configuration, and conducted mechanical testing. This method presents a straightforward avenue for the automated production of meso-scale metallic lattices for structural applications at a considerably lower cost.

[0068] Regarding industrial utilization and commercialization, the architected structures of embodiments of the present invention demonstrate promising size scalability owing to their custom relative density. This makes them suitable candidates for on-site deployed structures in aerospace and architecture. Additionally, their exceptional energy absorption characteristics and strength-to-cost ratio position them as excellent candidates for applications within the automotive sector.

[0069] It should be understood, of course, that the foregoing relates to exemplary embodiments of the invention and that modifications may be made without departing from the spirit and scope of the invention as set forth in the following claims.REFERENCES

[0070] [1] (Berger et al. 17) J. Berger, Wadley, H., and R. McMeeking. “Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness.” Nature.

[0071] [2] (Cheung et al. 14) Kenneth C. Cheung, Tomohiro Tachi, Sam Calisch, and Koryo Miura. “Origami interleaved tube cellular materials.” Smart Materials and Structures.

[0072] [3] (Cheung 12) Kenneth C. Cheung. “Digital Cellular Solids: reconfigurable composite mate-rials.” Ph.D. thesis, Massachusetts Institute of Technology, 2012.

[0073] [4] (Fidan et al. 24) Fidan, Ismail, Vivekan, Naikwadi, Suhas Alkunte, Roshan Mishra, and Khalid Tantawi. “Energy Efficiency in Additive Manufacturing: Condensed Review.” Technologies.

[0074] [5] (Filipov et al. 15) Evgueni T. Filipov, Tomohiro Tachi, and Glaucio H. Paulino. “Origami tubes assembled into stiff, yet reconfigurable structures, metamaterials.” PNAS.

[0075] [6] (Gibson and Ashby 88) Lorna J. Gibson and Michael F. Ashby. Cellular Solids and Structure and Properties. Cambridge University Press, 1988.

[0076] [7] (Gregg et al. 24) Christine E. Gregg, Damiana Catanoso, Olivia Irene B. Formoso, Irina Kostitsyna, Megan E. Ochalek, Taiwo J. Olatunde, In Won Park, Frank M. Sebastianelli, Elizabeth M. Taylor, Greenfield T. Trinh, and Kenneth C. Cheung. “Ultralight, strong, and self-reprogrammable mechanical metamaterials.” Science Robotics.

[0077] [8] (Gutowski et al. 17) Timothy Gutowski, Sheng Jiang, Daniel Cooper, Gero Corman, and Jan-Anders; Schudeleit Timo; Wegener Konrad; Sabelle Matias; Ramos-Grez Jorge; Sekulic Dusan P Hausmann, Michael; Manson. “Note on the Rate and Energy Efficiency Limits for Additive Manufacturing.” Journal of Industrial Ecology.

[0078] [9] (Jamalimehr et al. 22) Amin Jamalimehr, Morad Mirzajanzadeh, Abdolhamid Akbarzadeh, and Damiano Pasini. “Rigidly flat-foldable class of lockable origami-inspired meta-materials with topological stiff states.” Nature Communications.

[0079]

[10] (Jenett et al. 19) Benjamin Jenett, Amira Abdel-Rahman, Kenneth Cheung, and Neil Gershenfeld. “Material-Robot System for Assembly of Discrete Cellular Structures.” IEEE Robotics and Automation Letters.

[0080]

[11] (Jenett 20) Ben Jenett. “Discrete Mechanical Metamaterials.” Ph.D. thesis, MIT, 2020. [Klett and Drechsler 11] Yves Klett and Klaus Drechsler. “Designing Technical Tessellations.” Origami 5: International Meeting of Origami Science, Mathematics and Education.

[0081]

[12] (Klett et al. 07) Y Klett, K Drechsler, M Kolax, H Wentzel, and R Kehrle. “Design of multifunctional folded core structures for aerospace sandwich applications.” CEAS European Air and Space Conference.

[0082]

[13] (Li et al. 11) Zhejian Li, Wensu Chen, and Hong Hao. “Quasi-static-crushing-behaviours-of-folded-open-top-truncated-pyramid-structures-with-interconnected-side-walls.pdf.”Origami 7: International Meeting of Origami Science, Mathematics and Education.

[0083]

[14] (Liu et al. 18) Z. Y. Liu, C. Li, X. Y. Fang, and Y. B. Guo. “Energy Consumption in Additive Manufacturing of Metal Parts.” Procedia Manufacturing.

[0084]

[15] (Miyazawa et al. 21) Yasuhiro Miyazawa, Hiromi Yasuda, Hyungkyu Kim, James H. Lynch, Kosei Tsujikawa, Takahiro Kunimine, and Jordan R. Raney Jinkyu Yang. “Heterogeneous origami-architected materials with variable stiffness.” Communications Materials.

[0085]

[16] (Parra Rubio et al. 23) Alfonso Parra Rubio, Klara Mundilova, David Preiss, Erik Demaine, and Neil Gershenfeld. “Kirigami Corrugations: Strong and Modular and Programmable Plate Lattices.” Proceedings of the ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 8: 47th Mechanisms and Robotics Conference (MR).

[0086]

[17] (Ritchie 11) R. Ritchie. “The conflicts between strength and toughness.” Nature.

[0087]

[18] (Tachi 09) Tomohiro Tachi. “Simulation of Rigid Origami.” Origami 4: Fourth International Meeting of Origami Science, Mathematics, and Education 4. doi: 10.1201 / b10653-20.

[0088]

[19] (Liu 20) Y. Liu. “Mechanical properties of a new type of plate-lattice structures.” Journal of Mechanical Sciences.

Claims

1. A regular repeated structure, comprising:a plurality of unit cells, each unit cell having an expanded-truncated rectangular pyramid geometry, wherein:the unit cell includes a height (h), width (w), expanded flaps (b), truncation (e), and angles (θ2 and ρ1);the expanded flaps (b) provide facet-facet contact points for assembly; andthe unit cells are discretely assembled into a structure using mechanical connectors.

2. The structure of claim 1, wherein the unit cells are formed of sheet stock.

3. The structure of claim 2, wherein the sheet stock is made of metal.

4. The structure of claim 1, wherein the mechanical connectors are rivets.

5. The structure of claim 1, wherein the unit cells are assembled into a lattice structure configured in an arbitrary array of n×m×i unit cells.

6. The structure of claim 2, wherein the unit cells are folded using a progressive folding process comprising:cutting the sheet stock into a flat preform configuration;imprinting a crease map onto the flat preform using a press-forming mold;applying intermediate strain to the flat preform using a second press-forming mold; andoverfolding the unit cell using a third press-forming mold to account for springback.

7. The structure of claim 1, wherein the sheet stock is stainless steel.

8. The structure of claim 1, wherein the unit cells are formed via a process selected from the group of processes including essentially folding, stamping, molding, casting, die casting, thermoforming, roll forming, hydroforming, explosive forming, electromagnetic forming, additive manufacturing, incremental sheet forming, machining, and combinations thereof.

9. A method of manufacturing a regular repeated structure, comprising:cutting sheet stock into a flat preform configuration;progressively folding the flat preform into a unit cell having an expanded-truncated rectangular pyramid geometry, wherein the folding process includes imprinting a crease map, applying intermediate strain, and overfolding to account for springback;assembling a plurality of unit cells into a structure using mechanical connectors; andconfiguring the structure into an arbitrary array of n×m×i unit cells.

10. The method of claim 9, wherein the sheet stock is steel.

11. The method of claim 9, wherein the mechanical connectors are rivets.

12. The method of claim 9, further comprising introducing small holes at vertices of the flat preform where multiple creasing lines converge to mitigate local malformations.

13. The method of claim 9, further comprising modifying the geometry of the unit cells to include triangular truncated pyramids or other configurations to customize properties.

14. The method of claim 9, wherein the progressively folding step further comprises placing the flat preform into a mold and pressing the flat preform to form a fold to further shape the preform.

15. A method of manufacturing a regular repeated structure, comprising:forming the flat preform into a unit cell having an expanded-truncated rectangular pyramid geometry, wherein the forming process includes imprinting a crease map, applying intermediate strain, and accounting for springback;assembling a plurality of unit cells into a structure using mechanical connectors; andconfiguring the structure into an arbitrary array of n×m×i unit cells.

16. The method of claim 15, wherein the unit cells are formed via a process selected from the group of processes including essentially folding, stamping, molding, casting, die casting, thermoforming, roll forming, hydroforming, explosive forming, electromagnetic forming, additive manufacturing, incremental sheet forming, machining, and combinations thereof.

17. A method for simulating the mechanical response of a regular repeated structure using periodic boundary conditions, comprising:selecting a representative volume element (RVE) corresponding to a unit cell of the plate lattice, the RVE defined by a bounding box delineated by three lattice vectors LVx, LVy, and LVz;meshing the unit cell with symmetric seeding across the XY, XZ, and YZ planes;identifying pairs of boundary nodes (na,nb) such that the difference in their coordinates satisfies:nb-n⁢a=n⁢1⁢L⁢V⁢x+n⁢2⁢LVy+n⁢3⁢LVz,where n1, n2, n3 are components of any possible linear combination of the three lattice vectors;pairing the displacements of the identified node pairs according toub-ua=H⁡(Xb-Xa),where H is a displacement gradient matrix and Xa, Xb are the coordinates of the selected nodes in the undeformed mesh state;designing the displacement gradient matrix H with virtual nodes to impose a macroscopic deformation, including a constant strain value corresponding to uniaxial compression in the Z-axis;and running a batch of finite element simulations to obtain the unit cell stress response to the applied strain.

18. The method of claim 17, wherein the finite element simulations account for the elastoplastic behavior of the metallic material forming the plate lattice structure.

19. The method of claim 17, wherein the unit cell parameters, including height, folding angles, and relative density, are varied across simulations to optimize mechanical performance characteristics.

Images