Tunable metamaterials for impact absorption

Abstract

The present disclosure provides energy absorbing materials and methods of fabricating the same. The energy absorbing materials of the present disclosure comprise a plurality of repeating cells, wherein each of the plurality of cells comprises a three-dimensional plate lattice, wherein a least a portion of plate lattice is prebuckled.

Description

[0001] TUNABLE METAMATERIALS FOR IMPACT ABSORPTION

[0002] CROSS-REFERENCE TO RELATED APPLICATION

[0003] This application claims the benefit of priority to U.S. Provisional Patent Application Serial No. 63 / 546,832 entitled "TUNABLE METAMATERIALS FOR IMPACT ABSORPTION," filed November 1, 2023, the disclosure of which is incorporated herein by reference in its entirety.

[0004] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0005] This invention was made with government support under DE-NA0003525 awarded by the Department of Energy and AWD-23-03-0079 awarded by the Department of Defense. The government has certain rights in the invention.

[0006] BACKGROUND OF THE DISCLOSURE

[0007] Traditional methods of shielding fragile goods and human tissues from impact energy rely on isotropic foam materials, which are inexpensive and offer adequate energy absorption performance. However, the mechanical properties of these foams are inferior to an emerging class of metamaterials, which have predominantly been fabricated in simple 2.5 -dimensional geometries using conventional fabrication. There is a need in the art to develop new technologies that enable a better control over the geometry of engineering metamaterials across multiple length scales. The present disclosure addresses this need.

[0008] BRIEF SUMMARY OF THE INVENTION

[0009] In one aspect, an energy absorbing material is provided. The energy absorbing material includes a plurality of repeating cells, wherein each of the plurality of repeating cells that include a three-dimensional plate lattice that includes: a network of a plurality of plate members; a plurality of void regions; a plurality of edges, wherein each edge from the plurality of edges includes a set of nodal points, and is formed by connecting edges of at least two plates from the plurality' of plate members, and wherein at least a portion of the plurality of edges is geometrically prebuckled, whereby each edge from the plurality of edges is positionally skewed relative to a cell’s vertical centerline at a local and spatially varying magnitude.

[0010] In one aspect, a method of fabricating an energy absorbing material is provided. The method includes: i. simulating an energy absorbing material comprising a set of nodal points, wherein the node position of each of the plurality of nodal points is independently defined and / or adjusted without losing its connectivity to the material; ii. performing numerical simulation(s) to determine a response of the material when subjected to a load impact test; and iii. printing the material, wherein the energy' absorbing material includes a plurality of repeating cells, wherein each of the plurality' of repeating cells comprises a three-dimensional plate lattice comprising: a network of a plurality of plate members; a plurality' of void regions; a plurality' of edges, wherein each edge from the plurality' of edges comprises a set of nodal points, and is formed by connecting edges of at least two plates from the plurality of plate members, and wherein at least a portion of the plurality' of edges is geometrically prebuckled, whereby each edge from the plurality' of edges is positionally skewed relative to a cell’s vertical centerline at a local and spatially vary ing magnitude.

[0011] In one aspect, a non-transitory computer readable medium is provided. The medium includes a computer readable program code that, when executed, causes a processor to: i. simulate an energy absorbing material, ii. perform numerical simulation(s) to determine a response of the material when subj ected to a load impact test; and iii. communicate with a printer and print the matenal. wherein the energy absorbing material includes a plurality' of repeating cells, wherein each of the plurality' of repeating cells comprises a three-dimensional plate lattice comprising: a network of a plurality of plate members; a plurality’ of void regions; a plurality' of edges, wherein each edge from the plurality of edges comprises a set of nodal points, and is formed by connecting edges of at least two plates from the plurality of plate members, and wherein at least a portion of the plurality of edges is geometrically prebuckled, whereby each edge from the plurality of edges is positionally skewed relative to a cell’s vertical centerline at a local and spatially varying magnitude.

[0012] BRIEF DESCRIPTION OF THE DRAWINGS

[0013] The following detailed description of exemplary embodiments of this disclosure will be better understood when read in conjunction with the appended drawings. For the purpose of illustrating, non-limiting embodiments are shown in the drawings. It should be understood, however, that the description in this disclosure is not limited to the precise arrangements and instrumentalities of the embodiments show n in the drawings.

[0014] FIGS. 1A-1E depict designers of stochastic foam energy absorbers (e.g., polyurethane foam micrograph (FIG. 1 A) exert only rudimentary control over stress strain response via adjustment of relative density and choice of base material (FIG. IB). These foamed materials tend to collapse at a near-constant stress over a wide range of strains, allowing high energy absorption without transmitting high forces. Beyond a critical specific energy' absorbed, the materials undergo densification (FIG. 1C, at ea) and force transmission is extremely sensitive to additional strain. Dynamic simulations of impact samples with increasing density (FIG. ID) reveal the importance of matching specific energy absorbed before reaching densification strain sa to specific impact energy'. Soft materials absorb insufficient energy' before densifying, and subsequently (t>20ms in this example) transfer large forces, w hile stiff materials absorb impact energy with increasingly higher forces and lower deformation. Optimal performance (lowest transferred impact force) is achieved by an intermediate density material. By sacrificing the isotropy' of stochastic materials, architectured materials with outstanding properties in a single direction can be achieved (FIG. IE). Foam micrograph in FIG. 1A was taken using a scanning electron microscope (SEM) with an acceleration voltage of 15 kV.

[0015] FIGS. 2A-2F depict typical samples fabricated via additive manufacturing for empirical testing in quasistatic and dynamic compression. FIG. 2A shows plate lattice sample and equivalent-density TMPS gyroid foam used as reference, while (FIG. 2B) shows effects of adjusting geometric prebuckling parameter e* on a global (upper) and local (lower) scale. Plate lattices fabricated via fused deposition modeling additive manufacturing technology closely resemble designed geometry, achieving sub-1 mm wall thickness throughout a sample with exterior dimensions 64 * 64 x 48mm. Plate lattices are printed at the lower limit of resolution of the available commodity hardware, with the wall thickness created by a single pass of the 0.4 mm orifice nozzle. A Keyence VK X-l 100 profilometer is used to scan a section cut (FIG. 2C) through a manufactured sample, producing a high-resolution stitched image (FIG. 2F). A custom algorithm is used to compensate for wall eccentricity and gather 2500 measurements of wall thickness, which shows periodic variation in the z direction due to print layer lines (FIG. 2D). Mean wall thickness is 587 ± 40 pm at one standard deviation, slightly below the design thickness of 600 pm (FIG. 2E). In subsequent finite element simulations of plate lattice behavior, this geometry is approximated as uniform thickness equal to the mean value of 587 pm.

[0016] FIGS. 3A-3F depict plate lattice materials exhibit pronounced differences in macroscale stress / strain response relative to isotropic foam materials of similar relative densities, resulting in improved impact absorption performance. Adjustment of geometric prebuckling parameter e* affords density’ -independent control of peak stress and stress plateau magnitude (FIG. 3A), while not affecting densification strain. All empirical tests are performed twice on each of four replicates; centerline show s mean response while shaded region shows 2o confidence bounds. Not only do plate lattice samples show’ a 28% average increase in efficiency (FIG. 3B, black points mark peak efficiency), the efficiency peaks are less pronounced, indicating less dramatic falloff in efficiency when subjected to variations in impact stress. Simulations across a wide range of plate lattice geometries (FIG. 3C) show that high-efficiency designs are obtained across a wide range of impact stresses (these efficiency peaks are indicated by points colored by density in FIG. 3B; full responses are given in FIG. 3D). Plate lattice stress response character remains consistent across the range of tested strain rates (2 orders of magnitude, FIG. 3E), including a sharp increase and local maximum followed by a long and flat plateau. Foam materials of various densities exhibit similar, but less pronounced changes (97% increase in stress plateau versus 140% change for PL), though being made of the same material and having similar specific density (FIG. 3F).

[0017] FIGS. 4A-4E depict empirical results from impact testing (FIG. 4 A) show how the underlying stress-strain response (FIG. 3A-3F) of plate lattices drives favorable impact performance. In tests on a plate lattice sample and reference foam sample w ith identical mass, volume, base material, impact mass, and impact velocity, the PL transmits higher initial forces, allowing it to fully absorb the energy of impact without densifying (FIG. 4A, upper right inset images), leading to 3.5x reduction in peak transmitted load in this test. All empirical tests are performed twice on each of four replicates; centerline shows mean response while shaded region shows 2o confidence bounds. This peak reaction force transmitted by plate lattices is insensitive to increasing impact energy (compared to reference foams) across the range tested in experiment (FIG. 4C, square marks), and simulated impacts across a wider range of energies simulation (FIG. 4B, and insets in FIG. 4A) indicate that plate lattices offer robust performance to even higher specific impact energies (FIG. 4C, circular marks) before densification occurs (knee point at 1500 J / kg). Simulations of each of the 100 impacts in FIG. 4B across a slew of plate lattices with p* E [0.020.22] indicate that this trend holds across a wider range of relative densities than was accessible in experiment: a relatively constant stress is transmitted over several orders of magnitude of impact energy (envelope bounded by the post-densification response in FIG. 4D). An analogous envelope of peak stress transmitted before densification for foams with p* S [0.010.60] is derived analytically (dashed line in FIG. 4D), with the response of a representative foam sample (p* = 0. 11) indicated by a solid black line. Plate lattice of equivalent density absorbs 10x higher energy before densifying (indicated by red arrow). Allowing p* to vary freely and comparing the maximum useful work absorbed at a specific stress transmitted shows (each are normalized by solid material stiffness and plotted in FIG. 4E) shows that plate lattices can manage the relationship between these two across a ~10x wider range, and that samples with graded geometric prebuckling provide higher peak useful absorbed energy than all other samples tested.

[0018] FIGS. 5A-5B depict plate lattice materials dissipate energy (e.g., included area in cyclic load curve, (FIG. 5A) effectively at higher specific energies than equivalent density isotropic foams. Repeated loading induces significant softening of the structure, and normalized dissipated energy drops to 25% after 1000 compressions to densification at 0.5 Hz (FIG. 5A). In dynamic testing, low energy impacts excite only linear elastic deformations in plate lattice samples (FIG. 5B), resulting in relatively low energy dissipation (high CoR). Augmenting test data with 100 simulated impacts reveals that as deformation increases at higher impact energies, a larger fraction of that energy is dissipated; this trend reverses beyond a critical impact energy that results in full densification, after which bulk compression in the material leads to increasing CoR.

[0019] FIG. 5C shows similar behaviors in scans across impact energy at various relative densities: as p* increases, PLs dissipate energy effectively (low CoR) across a wider range of impact energies. FIG. 6 is a typical response of energy-absorbing materials consists of a linear elastic regime at small deformations, a near-constant stress plateau oPfollowing cell collapse at oc, and a stiff regime that follows densification strain ea. Specific energy absorbed by the material is the area under the loading curve which is divided into elastic energy stored and returned to the environment Ue and dissipated energy Ua.

[0020] FIG. 7 is a custom test apparatus consists of an impact carriage A which slides freely along a vertical rail of extruded aluminum. Sample under test B rests on a stationary platen positioned below an impact arm on the carriage. Upon activation of the release latch C, the impact carriage falls under the acceleration of gravity and strikes the sample. The resulting im-pact forces are reacted to load cells Di, D2 while high-speed camera E records the vertical position of the impact carriage. Impact energy is controlled by varying drop height and impact mass; test matrix used in empirical and numerical experiments indicated in upper left. Integration of load cell data and differentiation of camera position data (right) produces consistent velocity signals; this indicates correct calibration of camera and load cell subfixtures.

[0021] FIG. 8 is a mechanical characterization of TPU material used in this work is conducted via uniaxial tensile testing (n=8 samples) according to ASTM D412. A nonlinear curve fitting procedure is used to extract coefficients for a 2ndorder Ogden hyperelasticity model. The fit shows excellent agreement with the empirical data when assumptions of perfect incompressibility and a uniaxial strain field are applied to the model. A 3D nonlinear finite element analysis of the uniaxial tensile test is used to verify the hyperelastic model implementation in Abaqus, which confirms the uniaxial assumption and agreement with empirical data.

[0022] FIG. 9 shows that numerical models of dynamic impact experiments closely match empirical results on a qualitative and quantitative basis. Upper two rows show results from finite element simulations on hexahedral volumetric (C3D8R) and quadrilateral shell (S4R) computational meshes, respectively; bottom row shows four still images from a high-speed video capture of an impact with identical mass and impact velocity. Both simulations qualitatively match empirical deformation modes, namely the onset of local buckling shortly after impact, followed by progressive collapse until densification and reversal of momentum.

[0023] FIG. 10 depicts computational meshes using shell finite elements are capable of capturing geometric detail of plate lattice structures with fewer degrees of freedom while retaining high element quality and the general advantages of the finite element method: straightforward implementation of nonlinear constitutive material models, accurate application of boundary loads, and robust contact handling. In the explicit time integration scheme used for simulations in this work, stable time increment increases by a factor of 10 for simulations on shell meshes relative to volumetric meshes; this compounds the time savings offered by reducing the nodal degrees of freedom and the number of contact surfaces, resulting in an average 18x reduction in solve time. Subplot at left shows quasistatic stress / strain results for plate lattices with increasing unit cell size;

[0446] lattices were chosen to be sufficiently large to capture homogenized response while balancing computational cost and fabrication time.

[0024] FIG. 11 shows kinematics of input and output nodes for candidate deceleration function and various values of ar.

[0025] FIG. 12 shows kinematics of input and output nodes for candidate deceleration function and various values of a .

[0026] FIG. 13 shows kinematics of input and output nodes for candidate deceleration function and various values of ar.

[0027] DETAILED DESCRIPTION OF THE DISCLOSURE

[0028] Stochastic cellular materials are ubiquitous in energy absorption, vibration, isolation, and shock mitigation applications. Introducing voids into a solid material reduces the relative density and can adjust the mechanical properties by a factor of 1000 or more from those of the constituent material, yielding improved energy absorption performance. By carefully choosing a base material and controlling process parameters that govern the average pore size, designers exert rudimentary control over the macroscale mechanical properties of the resulting metamaterial, which tend to scale with relative density p* (n e [1, 4]):

[0029] Yet the popularity of foams in impact absorption applications is pragmatic (they are inexpensive, volume-filling, and isotropic), not due to optimal performance. To date, no foam material has been demonstrated which outperforms architectured (non-stochastic) materials on a specific modulus, toughness, or energy absorption basis. Designers cannot directly prescribe the microscale geometry of foam materials; stochastic physical processes like bubble nucleation, cell growth, and phase change separate their design intentions from resulting microscale geometry. Although extensive experimental and analytical work has been dedicated to understanding the scaling behavior and performance bounds on foamed materials, their performance remains sensitive to manufacturing details.

[0030] Unlike the design of other metamaterials (e.g., minimally compliant materials), energy absorbing materials must simultaneously balance several performance criteria to provide useful protection. High performance designs absorb the kinetic energy of a particular impact scenario while limiting peak loads transferred to protected objects. This cannot be achieved by materials at upper or lower limits of specific stiffness or strength. In fact, the stress / strain response of a theoretical ideal absorber is necessarily linked to a particular impact energy and compression distance at hand and compresses that entire distance at a constant force such that the integral f Fdx exactly totals the input energy. The practical design and use of foams remains an empirical exploration of base material selection on the one hand, and process parameters that underdetermine microscale geometry on the other.

[0031] Additive manufacturing (AM) offers a promising alternative to this paradigm. By controlling the deposition of build material at resolutions separated by four orders of magnitude from the build volume, it is possible to create metamaterials with deterministic, fully controlled geometry across multiple length scales. Additionally, microstructure geometry can be locally or directionally tailored, enabling region-specific and functionally graded mechanical properties. This level of control over local mechanical properties is not a hallmark of working with foam materials. Finally, advanced AM technologies can co-print with multiple distinct base materials of widely varying mechanical properties. Taken together, AM enhances design freedom and flexibility, and enables searches over microstructure geometry and material constitution in pursuit of higher performance metamaterials.

[0032] In one aspect, the present disclosure is related to investigating the quasistatic and dynamic mechanical response of locally tunable, additively manufactured elastomeric plate lattice materials (FIG. 2A). In contrast to any previous work, the designs with graded geometric prebuckling in the out-of-plane direction is evaluated. An efficient, fully scripted geometry generation and simulation pipeline was used to explore more widely into the plate lattice design space and impact test gamut than is accessible via empirical testing.

[0033] The methods described herein allow for the broadest characterization of plate lattice impact performance to date, demonstrating that these materials offer exceptional properties relative to industry-standard stochastic foams, for example, the plate lattice material described herein absorb 6 times more energy relative to industry-standard stochastic, foams at equivalent density, and up to 10 times more energy while transmitting equivalent peak forces. The methods presented herein enable the development of high performance, geometryspecific metamaterials which protect fragile goods from a wide bandwidth of impact energies.

[0034] Reference will now be made in detail to certain embodiments of the disclosed subject matter, examples of which are illustrated in part in the accompanying drawings. While the disclosed subject matter will be described in conjunction with the enumerated claims, it will be understood that the exemplified subject matter is not intended to limit the claims to the disclosed subject matter.

[0035] Throughout this document, values expressed in a range format should be interpreted in a flexible manner to include not only the numerical values explicitly recited as the limits of the range, but also to include all the individual numerical values or sub-ranges encompassed within that range as if each numerical value and sub-range is explicitly recited. For example, a range of "about 0.1% to about 5%" or "about 0.1% to 5%" should be interpreted to include notjust about 0.1% to about 5%, but also the individual values (e.g., 1%, 2%, 3%, and 4%) and the sub-ranges (e.g., 0.1% to 0.5%, 1.1% to 2.2%, 3.3% to 4.4%) within the indicated range. The statement "about X to Y" has the same meaning as "about X to about Y," unless indicated otherwise. Likewise, the statement "about X, Y, or about Z" has the same meaning as "about X, about Y, or about Z," unless indicated otherwise.

[0036] In the methods described herein, the acts can be carried out in any order, except when a temporal or operational sequence is explicitly recited. Furthermore, specified acts can be carried out concurrently unless explicit claim language recites that they be carried out separately. For example, a claimed act of doing X and a claimed act of doing Y can be conducted simultaneously within a single operation, and the resulting process will fall within the literal scope of the claimed process.

[0037] Definitions

[0038] The term "about" as used herein can allow for a degree of variability in a value or range, for example, within 10%, within 5%, or within 1% of a stated value or of a stated limit of a range, and includes the exact stated value or range.

[0039] In this document, the terms "a," "an," or "the" are used to include one or more than one unless the context clearly dictates otherwise. The term "or" is used to refer to a nonexclusive "or" unless otherwise indicated. The statement "at least one of A and B" or "at least one of A or B" has the same meaning as "A, B, or A and B." In addition, it is to be understood that the phraseology or terminology employed herein, and not otherwise defined. is for the purpose of description only and not of limitation. Any use of section headings is intended to aid reading of the document and is not to be interpreted as limiting: information that is relevant to a section heading may occur within or outside of that particular section. All publications, patents, and patent documents referred to in this document are incorporated by reference herein in their entirety, as though individually incorporated by reference.

[0040] As used herein the term "prebuckling" refers to the intentional inclusion of small geometric deviations from perfectly orthogonal honeycomb materials, to control the nature of the relationship between a design’s stress and strain.

[0041] As used herein the terms "prebuckling parameter" or "prebuckling factor" refers to the degree of prebuckling included in a design, with higher prebuckling factor indicating that deviations of larger magnitude have been added.

[0042] As used herein the term "nodal point" refers to a point in a three-dimensional lattice material where multiple structural members join together.

[0043] As used herein the term "out-of-plane" refers to the direction in which honeycomb lattice structures are extruded, which exhibits better / superior properties than the other directions.

[0044] The term "substantially" as used herein refers to a majority of, or mostly, as in at least about 50%, 60%, 70%, 80%, 90%, 95%, 96%, 97%, 98%, 99%, 99.5%, 99.9%, 99.99%, or at least about 99.999% or more, or 100%. The term "substantially free of' as used herein can mean having none or having a trivial amount of, such that the amount of material present does not affect the material properties of the composition including the material, such that the composition is about 0 wt% to about 5 wt% of the material, or about 0 wt% to about 1 wt %, or about 5 wt% or less, or less than, equal to, or greater than about 4.5 wt%, 4, 3.5, 3, 2.5, 2, 1.5, 1, 0.9, 0.8. 0.7, 0.6, 0.5, 0.4. 0.3, 0.2, 0.1, 0.01, or about 0.001 wt% or less. The term "substantially free of can mean having a trivial amount of, such that a composition is about 0 wt% to about 5 wt% of the material, or about 0 wt% to about 1 wt%, or about 5 wt% or less, or less than, equal to, or greater than about 4.5 wt%, 4, 3.5, 3, 2.5, 2, 1.5, 1, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1, 0.01, or about 0.001 wt% or less, or about 0 wt%.

[0045] Throughout this disclosure, various aspects of the disclosure can be presented in a range format. It should be understood that the description in range format is merely for convenience and brevity and should not be construed as an inflexible limitation on the scope of the disclosure. Accordingly , the description of a range should be considered to have specifically disclosed all the possible subranges as well as individual numerical values within that range. For example, description of a range such as from 1 to 6 should be considered to have specifically disclosed subranges such as from 1 to 3, from 1 to 4, from 1 to 5, from 2 to 4, from 2 to 6, from 3 to 6 etc., as well as individual numbers within that range, for example. 1, 2, 2.7, 3, 4, 5, 5.3, and 6. This applies regardless of the breadth of the range.

[0046] Non-limiting Material Selection for Energy Absorbers

[0047] Soft elastomeric materials exhibit glass transition temperatures far below ambient conditions and support high strains without fracture or yield, meaning that deformations are highly reversible and structures may absorb many repeated impacts. Additionally, empirical investigation of shock propagation through buckling elastomeric meta-structures has revealed that high-strain rate impacts can activate deformation modes not induced by quasistatic loading, which reduce force transmission through a sample under test.

[0048] AM (additive manufacturing) allows designers to exercise even more design freedom, enabling higher tunability in material response than is possible in stochastic materials. Liquid crystal elastomers (LCEs), which undergo mechanically-induced, energy-dissipating phase transition under large strains, have recently been explored as AM base materials for energy’ absorption. Empirical tests show that beam lattice networks of these materials dissipate up to 4 times more energy than identical geometries printed from elastomeric base materials.

[0049] As described herein, the quasistatic and dynamic mechanical response of locally tunable additively manufactured elastomeric plate lattice materials can be explored as shown in FIG. 2A. An efficient, fully scripted geometry generation and simulation pipeline can be used to explore more widely into the plate lattice design space and impact test gamut than is accessible via empirical testing. The methods described herein can allow’ for the broadest characterization of plate lattice impact performance to date, demonstrating that these materials offer exceptional properties relative to industry -standard stochastic foams: they absorb 6 times more energy at equivalent density and up to 10 times more energy' yvhile transmitting equivalent peak forces. The computational design workflow’ presented here enables the development of high performance, geometry -specific metamaterials which protect fragile goods from a wide bandwidth of impact energies.

[0050] Energy Absorbing Materials

[0051] In one aspect, described herein is an energy absorbing material comprising: a plurality of repeating cells, wherein each of the plurality of cells comprises a three-dimensional plate lattice having: a network of plurality of plate members, a plurality of void regions, and a plurality of edges. Higher performance can be achieved by plate lattices, which are conceptually similar to their strut counterparts: a set of nodal points is connected by high aspect ratio structural members, sometimes in a repeating pattern. Plate lattices can provide substantially higher stiffness and strength than optimal truss-lattices of equal mass using analytical techniques and several hundred finite element simulations, a result that holds under high strain rate testing. Lattices derived from the shear planes of cr stal systems can achieve the theoretical Hashin- Schtrickman bounds on modulus and the Suquet bounds on strength for isotropic cellular solids. Comparisons between analytical, numerical, and experimental results for energy absorption and strength in quasistatic compression of ultralight (p* = 0.01) metamaterials based on hollow truss lattices and minimal surfaces have shown similarly high specific strength. Simulations in this work can identify optimal geometries with nonuniform wall thickness. This points to another design principle emerging from various works on AM metamaterials: nonuniform, locally tuned, or graded geometric properties are generally advantageous in energy absorption. A detailed examination of the progressive collapse behavior of similar functionally graded lattices shows density grading can prevent undesirable diagonal shear banding behavior, leading to more repeatable energy absorption than constant-density counterparts. If the direction of an impact load is known a priori, the optimization of mechanical properties in that direction can be prioritzied- an approach exemplified in structural honeycomb metamaterials widely used in aerospace, automobile, railway, and packaging applications.

[0052] In certain embodiments, each edge from the plurality of edges comprises a set of nodal points. In certain embodiments, each edge from the plurality of edges is formed by connecting edges of at least two plates from the plurality' of plates.

[0053] In certain embodiments, at least a portion of the plurality of edges is geometrically prebuckled, whereby each edge from the plurality of edges is positionally skew ed relative to a cell’s vertical centerline. A prebuckling factor e“ is defined to control the local eccentricity of the lattice and increasing e* skews nodal positions / edges of the plate lattice about the cell vertical centerline at a local and spatially varying magnitude.

[0054] In certain embodiments, each edge from the plurality of edges is positionally skewed along its entire length.

[0055] In certain embodiments, the plate members comprise regular geometric shape(s). In certain embodiments, the regular geometric shape is a polygon selected from the group consisting of is a hexagon, square, and quadrilateral. In certain embodiments, the cell comprises at least one wall having graded thickness.

[0056] In certain embodiments, the cell has graded density along at least one dimension.

[0057] In certain embodiments, material absorbs about 2 to 20 times more energy before densifying compared to a reference stochastic foam at equivalent density. In certain embodiments, material described herein absorbs about at least about 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, to about 20 times more energy’ before densifying compared to a reference stochastic foam at equivalent density. Although integer values are listed herein, the amount of energy the material can absorb is on a continuum such that all fractional values between the indicated integer values are also contemplated and included as if written expressly, such as 2.5, 2.6, 3.2, 3.3, and so on.

[0058] In certain embodiments, the critical impact energy value for the material is higher than the critical impact energy value for the stochastic reference foams at equivalent density. In certain embodiments, the critical impact energy value for the material is at least two times higher than the critical impact energy’ value for the stochastic reference foams at equivalent density. In certain embodiments, the critical impact energy value for the material is higher, at least by an order of magnitude, than the critical impact energy' value for the stochastic reference foams at equivalent density.

[0059] In certain embodiments, the energy’ absorbing material is composed of at least one base material. Non-limiting examples of the base materials include, for example, thermoplastic polyurethane (TPU), liquid crystal elastomers, nylon, urethane methacrylates, and the like. In certain embodiments, the energy’ absorbing material is composed of metals and / or alloys such as, for example, steel, nickel alloys, cobalt chrome, copper, titanium, aluminum, refractory metals such as tungsten. In certain non-limiting embodiments, the energy absorbing material is composed of composite materials, such as any of the base materials described herein combined with any of the metals described herein. The energy absorbing material can also include carbon fiber, graphite, carbon nanotubes, and the like.

[0060] Methods:

[0061] In another aspect, this disclosure provides a method for fabricating an energy absorbing material as described herein, wherein the method comprises: printing the material (wherein the material is a s described elsewhere herein), wherein the printing comprises defining the position of each nodal point from the set of nodal points.

[0062] In certain embodiments, the printing comprises additively mixing a plurality of base materials by prescribing the amounts of each of the plurality of base materials to fabricate the material.

[0063] In certain embodiments, the position of each of the plurality of nodal points is independently defined by prescribing a value for continuously adjustable prebuckling parameter (e*).

[0064] In certain embodiments, the method comprises independently prescribing the size of each cell from the plurality of cells.

[0065] In certain embodiments, the method comprises prescribing the wall thickness for each cell from the plurality of cells.

[0066] In certain embodiments, wherein the wall has non-uniform / graded thickness. Non- uniform or graded thickness as used herein refers to a thickness that can vary along a gradient, for example a linear gradient (thinner to thicker), an exponential or other power law gradient, and the like.

[0067] In yet another aspect, the disclosure provides a method for identifying an optimal geometry for fabricating an energy' absorbing meta-material , wherein the method comprises the steps of: simulating the energy absorbing material(wherein the material is as described elsewhere herein) having a set of nodal points, wherein node position of each of the plurality of nodal points is independently defined / adjusted without losing its connectivity to the material, performing numerical simulation(s) to determine a response of the material when subjected to a load impact test; and printing the structure.

[0068] In certain embodiments, the simulating comprises generating a visual rendering of the material.

[0069] In certain embodiments, the simulating comprises displaying the visual rendering of the material on a display device.

[0070] In certain embodiments, wherein the numerical simulation(s) comprise(s) quasistatic compression simulations, dynamic impact simulations, and / or dimensional reduction of plate geometry'.

[0071] In certain embodiments, the response includes macroscale stress / strain response.

[0072] In certain embodiments, the method further comprises the steps of empirically determining a response of the printed structure when subjected to a load impact test and comparing the empirically determined response with the response determined by' performing the numerical simulation (s).

[0073] In certain embodiments, a non-transitory computer readable medium is provided. The non-transitory computer readable medium includes readable program code that, when executed, causes a processor to: i. simulate an energy absorbing material as described herein. ii. perform numerical simulation(s) to determine a response of the material when subjected to a load impact test: and iii. communicate with a printer and print the material.

[0074] It is to be understood that wherever values and ranges are provided herein, all values and ranges encompassed by these values and ranges, are meant to be encompassed within the scope of the present disclosure. Moreover, all values that fall within these ranges, as well as the upper or lower limits of a range of values, are also contemplated by the present application.

[0075] It should be understood that the methods and compositions that would be useful in the present disclosure are not limited to the particular formulations set forth in the examples. The following examples are put forth so as to provide those of ordinary skill in the art with a complete disclosure and description of how to make and use the compositions and methods of the disclosure and are not intended to limit the scope of what the inventors regard as their disclosure.

[0076] EXPERIMENTAL EXAMPLES

[0077] The disclosure is further described in detail by reference to the following experimental examples. These examples are provided for purposes of illustration only and are not intended to be limiting unless so specified. Thus, the disclosure should in no way be construed as being limited to the following examples, but rather, should be construed to encompass any and all variations which become evident as a result of the teaching provided herein.

[0078] Materials and Methods

[0079] Sample Design and Fabrication:

[0080] The custom constructive geometry script allows designers to exert unprecedented control over PL geometry and realize designs not previously explored, including those with nonuniform wall thickness, prebuckling, and cell sizes. In this work fused material deposition (FDM) 3D printing and a soft thermoplastic polyurethane (TPU) elastomer were utilized to fabricate all samples under test. The TPU material chosen for this study has a nominal Shore A hardness of 92, which strikes a balance between being soft enough to enable relatively low- force characterization experiments, and stiff enough to allow robust, repeatable fabrication of many samples with high yield. Samples were fabricated using a commodity fused filament fabrication (FFF) 3D printer (Prusa MK3s, Prusa Research) fitted with an upgraded direct drive filament extruder designed for higher torque (Bondtech Prusa i3 Upgrade, Bondtech, AB) and a high-flowrate nickel-coated brass nozzle with 0.6mm orifice diameter (Bondtech CHT, Bondtech, AB). Fabrication files (geode) were generated for all samples using the open-source slicing program PrusaSlicer, with 100% infill and linear extrusion rate of 30 mm / s.

[0081] Empirical Testing:

[0082] A low-force, high-stroke load frame was used for quasistatic mechanical characterization (810E5 All-Electric Dynamic Test Machine, Test Resources). In compression tests, a sample was loaded between two parallel aluminum plates with 100 mm diameter, and the upper plate was lowered at a rate of 0.5 mm / s (s' = 1 's ') up to siatt = 0.75 while vertical displacement of the upper plate and compression force was logged at 1 kHz (Table 2). All samples were compressed at ambient conditions and allowed to recover for a minimum of 1000 s between tests to mitigate temperature-dependent stiffness changes and allow sufficient recover}' time for viscoelastic effects.

[0083] A custom impact fixture was designed and fabricated to characterize dynamic PL behavior, capable of delivering prescribed impact energy and measuring the force reacted by the fixture as well as the sample compression. The fixture consists of an aluminum carriage with a cantilever impact arm that rides freely on a vertical rail 2 m in height, and a test plate placed directly below the impact arm. The user prescribes the impact mass by loading the arm with up to 20 kg in cast steel plates, and the impact velocity by adjusting the carriage drop height (Table 2).

[0084] Immediately before impact, a microprocessor simultaneously triggers the start of load cell signal collection on a data acquisition card (Labjack U6 Pro, ©Labjack), and the capture of high-speed video on a suitable camera (Phantom v710, ©Vision Research). Position data is extracted from high-speed video using the common visual fiducial system AprilTag. In post processing, in-house code performs temporal alignment of these signals, yielding synchronized force-displacement data which allows the extraction of dynamic stress-strain curves. All testing is performed at ambient conditions and samples are allowed to recover for a minimum of 1000 s between tests to mitigate temperature-dependent stiffness changes and allow recovery from viscoelastic effects. Numerical Modeling:

[0085] Simulations were conducted using the nonlinear finite element package Abaqus (©Dassault Systemes), highly capable in solving complex continuum mechanics problems. All simulations, regardless of loading rate, were performed using the Abaqus dynamic explicit solver, in order to access the general contact algorithm, as in (S Townsend et al, Mater. Des. 195, 108930 (2020), R Adams, Mater. Des. 213, 110368 (2022)). Coulomb friction is defined at all contact surfaces with a sliding coefficient of friction equal p = 0.75, imitating real-world conditions where lattice materials may contact plastics, other elastomers, or flesh. A parameter sweep over this coefficient of friction value shows simulations are relatively insensitive to p so long as it is high enough to prevent nonphysical large-scale sliding on the impact platens. A hyper-viscoelastic constitutive model is implemented to capture the TPU’s nonlinear stress-strain relationship and rate-dependent elastic effects, yielding a material model which adequately captures material behavior across a wide range of loading rates (Table 3).

[0086] To prevent over-estimation of critical buckling load in numerical analysis of plate lattices with zero geometric prebucklmg (designs with perfectly vertical walls), these designs were perturbed in the undeformed configuration by the lowest-energy buckling mode. The eigenmode was extracted using a linear perturbation analysis step and scaled so that the peak displacement magnitude was equal to 10% of the wall thickness. Predictably, designs which include geometric prebucklmg (e* > 0) are insensitive to geometric perfections, so this step was omited.

[0087] Numerical and empirical experiments are aimed at obtaining a homogenized macroscale response for various geometries tested. In order to determine the number of unit cells required to adequately recover the macroscale response of plate latices, a scaling study w as performed in which the lattice size was incrementally increased until the quasistatic compression response converged. Latice aspect ratio was controlled via geometric parameters wc and he to ensure a progressive buckling collapse mechanism, rather than macroscale buckling.

[0088] Table 1. Design Parameterization & Typical Values

[0089] Table 2. Quasistatic and Dynamic Test Fixtures

[0090] DMA Impact Fixture

[0091] Crosshead Velocity (m / s) 5e-5-5e-l 1.0 - 3.5

[0092] Impact Mass (kg) N / A 3-17

[0093] Load Capacity7(N) 720 9800

[0094] Load Collection (kHz) 1 12.5

[0095] Position Collection (kHz) 1 25

[0096] Table 3. Ogden-Prony hyper-viscoelastic material model

[0097] Ogden Model Prony Model

[0098] 1 0.028 4.247 0 0.503 0.001 0

[0099] 2 7.812 -2.079 0 0.186 0.01 0

[0100] 3 0.018 0.1 0

[0101] ASTM dogbone testing; estimated from impact tests, assumed

[0102] Definitions for Energy Absorption Performance

[0103] To distinguish the macroscale stress-strain response of a cellular solid from the local stress and strain fields, the lattice strain eiatt is defined as the change in height of a sample normalized by its undeformed height, and lattice stress oiatt as the force applied to one face of a lattice sample divided by the proj ected area of that face. Integration of the area under the stress-strain curve during a loading cycle yields the spe-cific energy absorption on a per- volume basis, UA = crdc. In keeping with previous results, “useful absorbed energy” up to densification was intergrated, after which bulk modulus dominates the macroscale response, and the material is extremely sensitive to variations in impact energy. Normalizing absorbed energy by the effective density7p*ps(where psdenotes solid density) gives specific energy absorption on a per-mass basis SEAm (Equation 1)

[0104] Several nondimensional measures of a structure's energy absorbing efficiency have been suggested earlier, including the Janssen factor J, the cushion factor C, the Rusch curve, energy absorption efficiency E, ideality I, and energy -absorption diagrams; the nonconsensus stems from the high level of empiricism in energy absorbing material development.

[0105] Normalizing absorbed energy UA at a given strain by the peak stress exhibited up to that strain produces a nondimensionalized measure of energy absorption efficiency E* (Equation 2)

[0106] E* = 1 across all strains for theoretical ideal energy absorbers. In real materials E* drops rapidly following densification, leading to a mathematical definition of ea as the lattice strain associated with maximum energy absorption efficiency. Normalizing UA by peak stress rather than instantaneous o(s) accounts for possible stress peaks early in the response not considered in early analysis of monotonically stiffening cellular solids. Upon unloading, energy is either returned to the environment (shaded area UE) or dissipated (included area UD). The ratio of dissipated energy’ to total absorbed energy is equivalent to the coefficient of restitution CoR ty pically computed for a constant-mass impact:

[0107] (Equation 3)

[0108] Quantifying Impact Energy Absorption Performance

[0109] Investigation into energy absorptive materials can be conducted via distinct experimental methods, depending on the strain rates of interest. Load frames are capable of prescribing quasistatic to moderately dynamic ( 105- 10's1) strain rates s' with high precision, and are used to study energy' absorption in compression for metallic, elastomeric, and polymeric materials. For higher (101— 104s-1) strain rates, researchers use impact tests; impact energy is prescribed by controlling the mass and impact velocity of the impacting object. Typical outputs from these experiments include force / accel eration profiles and measurements of dissipated energy.

[0110] Empirical Experimentation

[0111] Quasistatic Test Fixture

[0112] A low-force, high-stroke load frame was used for quasistatic mechanical characterization (810E5 All-Electric Dy namic Test Machine, Test Resources). This load frame was used in both uniaxial tensile testing of raw material samples for constitutive model extraction (discussed elsewhere herein), and for quasistatic compression testing. In these compression tests, a sample was loaded between two parallel aluminum plates with 100mm diameter, and the upper plate was lowered at a rate of 0.5 mm / s (s' = 101s ') up to siat= 0.75. During each test the time, vertical displacement of the upper plate, and compression force was monitored and logged at 1 kHz. All samples were compressed at ambient conditions and allowed to recover for a minimum of 1000 s between tests to mitigate temperature-dependent stiffness changes and allow sufficient recovery time for viscoelastic effects.

[0113] Dynamic Impact Fixture

[0114] For exploring dynamic behavior, a custom impact fixture was designed and fabricated, capable of delivering prescribed impact energy’ and measuring the force reacted by the fixture as well as the sample compression at high frequencies. This text fixture is very useful for extracting the response of candidate materials, as it profiles realistic impact scenarios with application-specific impact energy and strain rate. The fixture consists of an aluminum carriage with a cantilever impact arm that rides freely on a vertical rail 2 m in height, and a test plate placed directly below the impact arm (FIG. 7, center). The user prescribes the impact mass by loading the arm with up to 20 kg in cast steel plates, and the impact velocity by adjusting the carriage drop height. When the user activates the release mechanism, a latch opens and allows the carriage to fall down the vertical rail. At a certain (user-controllable) distance above the impact site, a magnet on the impact carriage closes a reed switch mounted on the rail, sending a signal to a microprocessor (Arduino Nano, ©Arduino). This microprocessor simultaneously triggers the start of load cell signal collection on a data acquisition card (Labjack U6 Pro, ©Labjack), and the capture of high speed video on a suitable camera (Phantom v710, ©Vision Research). Position data is extracted from high-speed video using the common visual fiducial system AprilTag. In post processing, in-house code performs temporal alignment of these signals, yielding synchronized force-displacement data which allows the extraction of dynamic stress-strain curves. All testing is performed at ambient conditions and samples are allowed to recover for a minimum of 1000 s between tests to mitigate temperature-dependent stiffness changes and allow recovery from viscoelastic effects.

[0115] Material Characterization

[0116] In this work fused material deposition (FDM) 3D printing and a soft thermoplastic polyurethane (TPU) elastomer (©SainSmart) were used to fabricate all samples under test. This material was chosen for its mechanical properties (high elongation to fracture, low elastic modulus) and for its availability, low cost, and ease of use. Samples can be fabricated quickly on commodity hardware for 0.1% the cost of metal structures manufactured for related works. The TPU material chosen for this study has a nominal Shore A hardness of 92, which strikes a balance between being soft enough to enable relatively low-force characterization experiments, and stiff enough to allow robust, repeatable fabrication of many samples with high yield. Similar to many commodity 3D printing filaments, mechanical characterization is left to the end user, with the manufacturer supplying only nominal hardness and density.

[0117] In order to build an accurate numerical model of this base material, specimens were designed and tested according to ASTM standard D412 (Die C. 33x6* 1.6 mm test region). Samples were stretched until failure or by 600% engineering strain, which was dictated by the maximum travel available on the load frame used for this characterization. Uniaxial tensile testing was carried out on eight samples of each material, which were divided into two groups of four (A and B). For group A, the infill direction was specified to alternate between +45 deg and -45 deg offset from the pull direction across print layers. For group B. the infill direction was specified to alternate between parallel to and orthogonal to the pull direction across print layers. Samples were fabricated using a commodity fused filament fabrication (FFF) 3D printer (Prusa MK3s, Prusa Research) fitted with an upgraded direct drive filament extruder designed for higher torque (Bondtech Prusa i3 Upgrade. Bondtech, AB) and a high- flowrate nickel-coated brass nozzle with 0.6mm orifice diameter (Bondtech CHT, Bondtech, AB). Fabrication files (geode) were generated for all samples using the open-source slicing program PrusaSlicer, with 100% infill and linear extrusion rate of 30 mm / s.

[0118] Raw data measured during each tension test (uniaxial extension distance and tensile force) is converted into quantities relevant to hyperelastic material models by accounting for as-fabricated cross-sectional area and test region length. Data is averaged across samples and decimated to 20 datapoints in order to adequately capture the shape of the stress-stretch curve then fit to a hyperelasticity model using a general nonlinear fitting function fitnlm() (©Mathworks). It was chosen to fit the data extracted from this testing to the 2nd order (N = 2) Ogden model for hyperelasticity, which quantifies the strain energy density IF of a material point as a function of its principal stretches i.

[0119] Under the assumptions of isotropic incompressibility' and uniaxial strain state, principal Cauchy stress on is expressed as a function of principal stretch X. producing the equation utilized during curve fitting (though the plot is against engineering strain e in FIG.8 for visual purposes):

[0120] (Equation 4)

[0121] The Ogden model is a general and widely used model for rubber-like solids and is appropriate for the large deformations involved in collapse problems studied here. Optimized model coefficients (Table 4) satisfy the Drucker stability condition for all magnitudes of uniaxial tension, biaxial tension, uniaxial compression, and pure shear.

[0122] The TPU material used in this study exhibits viscoelastic (rate-dependent) behavior, expressed as a higher apparent elastic modulus at increased rates of deformation. The Prony series model is the most common method of capturing these time-dependent elastic effects, in which copies of the time-invariant elasticity model are scaled by dimensionless relaxation constants gi, exponentially decayed at rates governed by time constants xi, and superimposed. However, incorporating this effect into a simulation material model typically requires extensive testing across a range of loading rates and temperatures, and poses a barrier in industry and research. As the load frame used for quasistatic hyperelasticity model calibration was not able to achieve strain rates relevant to the impact loading of interest in this work, posed a 3rd order Prony series model was posed with xi assumed to be powers of 10. An inverse search was then performed for gi using a gradient-free optimization routine which sought to minimize differences between reference empirical force-displacement curves and simulated analogues. Given the relatively small number of design variables, the optimization routine (Matlab’s built in fminsearchbnd()) converged quickly at a minimum value, yielding the gi given in Table 4.

[0123] Table 4: Coefficients used for Ogden-Prony model for hyper-viscoelastic material model coefficients in bold typeface are determined via nonlinear fit to uniaxial test data gathered on dogbone samples, coefficients in italics are determined by inverse design using dynamic simulations of impact experiments. Remaining coefficients are assumed values based on incompressibility condition and impact loading rates.

[0124] _ Ogden Model _ Prony Model ., (MPa) «, Di (MPa) gi (s) Ki (MPa)

[0125] 1 0.0282 4.2479 0 0.5039 0.001 0

[0126] 2 7.8125 -2.0790 0 0.1863 0.01 0

[0127] 3 - - - 0.0181 0.1 0 Table 5: Design parameterization for plate lattices analyzed in this work, and typical parameter values

[0128] Parameter Variable Typical Value

[0129] Number of Unit Cells [nxny z]

[0446] Unit Cell Height he 8 mm Unit Cell Length wc12 mm Wall Thickness twall 0.6 mm Geometric Prebuckling e* 0.05-0.25

[0130] Sample Design and Fabrication

[0131] For large or geometrically complex lattices, which can consist of hundreds of thousands of facets, boundary representation (BREP) based modeling (e.g., Solidworks ©Dassault Systemes, Fusion360 ©Autodesk, Inc.) presents a significant computational overhead and memory footprint. Additionally, these tools were designed with traditional (non- AM) fabrication processes in mind and can struggle to create locally varying geometries (e.g., graded wall thickness) that are facile to realize using 3D printing. Finally, design representations produced by these modeling approaches are separated from analysis-ready computational meshes by a tedious, often manual mesh generation step. To avoid these computational bottlenecks and manual processes, a constructive geometry script capable of generating plate lattices of given geometrical properties was developed.

[0132] Plate lattice geometries, like more widely known strut lattices, can be fully defined by the position and connectivity of nodes. In this work, a previously introduced, modified version of a plate lattice was adopted, which features repeated quadrilateral plates and a global, continuously adjustable prebuckling parameter used to control macroscale compression behavior. In this instantiation, node locations can be freely adjusted on a local basis while preserving node connectivity. This change allows for a wealth of plate lattice geometries not previously explored, including those with nonuniform wall thickness, prebuckling parameter, and cell sizes. A prebuckling factor e* is defined to control the local eccentricity of the lattice; increasing e* skews nodal positions about the cell vertical centerline, and the lattice approaches a honeycomb material as e* 0. The design space parameterization used in the remainder of this work, along with typical parameter values, is given in Table 5. The meshing capabilities of the Geometry and Image-Based Bioengineering Add-On (GIBBON) w ere incorporated to perform face offsetting and quadrilateral subdivision. The script that was developed exports finite-thickness meshes of plate lattice geometries to the opensource slicing software PrusaSlicer for conversion into fabrication files. This slicer was chosen because of recent improvements in printing thin features, as it can dynamically control extrusion width to + / - 50% of a printer’s nozzle diameter.

[0133] Reference Geometry

[0134] In order to establish a reference point for the performance of plate lattice materials, several control samples were fabricated for empirical testing. The geometry of these samples is derived from gyroid triply periodic minimal surfaces (TPMS), well-known periodic shell structures with zero mean curvature. TPMS are defined by compact analytical functions in three dimensions, which can be tailored to control the length scale of the periodic cells and the shell thickness. They are also particularly amenable to fabrication via fused filament AM, as they are self-supporting and formed by long, continuous extrusion paths, unlike beam lattices. Finally, TPMS structures respond to compression in an extremely similar fashion to industry -standard polymer foams: they are isotropic, and exhibit linear elastic deformation at small strains, a broad, constant stress plateau, and rapid geometric stiffening following densification. Owing to their ease of design and fabrication, these structures are well- characterized in literature, and have been shown to mimic the performance of foamed materials when fabricated from metals, plastics, and elastomers.

[0135] In this work, gyroid structures are fabricated using the same base material and print settings as plate lattices and are printed to identical external dimensions and volume density, ensuring that any differences in impact absorption performance can be attributed to the geometry' of the samples. Three varieties are fabricated, w ith cell size and shell thickness adjusted to achieve 9, 11, and 13% relative density'.

[0136] Numerical Experimentation

[0137] A closer look at simulation outputs shows that numerical models of dynamic impact experiments closely match empirical results on a qualitative and quantitative basis. The upper two rows of FIG. 9 show results from finite element simulations on hexahedral volumetric (C3D8R) and quadrilateral shell (S4R) computational meshes, respectively; lower row shows four still images from a high-speed video capture of an impact with identical mass and impact velocity'. Both simulations qualitatively match empirical deformation modes, namely the onset of local buckling shortly after impact, followed by progressive collapse until densification and reversal of momentum. Local stress fields are more finely’ resolved in the volumetric simulations compared to shell simulations, as expected. The lowest row of FIG. 9 plots evolution of force and displacement for two separate drop tests and their simulation analogues. Both simulations match the initial portion of the empirically measured collapse force very well, but volumetric elements more closely match empirical data for the remainder of the impact. For this work, both simulations are judged to be sufficiently accurate in the prediction of plate lattice behavior when subjected to impact loads with strain rates roughly equal to 50 s ' (the strain rate of the empirical data used for Prony series fitting described in elsewhere herein). Typical simulations on shell finite elements execute 18* faster than their volumetric counterparts, which enables sweeping investigations of plate lattice performance; shell elements are used exclusively for the remainder of this disclosure.

[0138] Numerical simulation is most valuable when placed inside the design cycle of a mechanical component, enabling iterative evaluation of candidate designs and automated / programmatic characterization of a design space. In this work, a numerical simulation pipeline is constructed to link high level design parameterizations to predicted mechanical performance as seamlessly as possible. Samples which are specified by combinations of parameters of Table 5 are automatically converted to computational meshes using an in-house script, avoiding human interaction with third-party CAD software. The mesh data, boundary' conditions, material models, contact interactions, solver settings and output data requests are automatically written to simulation input files and dispatched via the command line, avoiding human interaction with FEA GUIs. The open-source software package Abaqus2Matlab was used to automatically retrieve key results and port them the computing platform Matlab, which was used for data processing and visualization.

[0139] Table 6: Simulation details for quasistatic and dynamic simulations used in this work.

[0140] Simulations were conducted using the nonlinear finite element package Abaqus (©Dassault Systemes), highly capable in solving complex continuum mechanics problems. All simulations, regardless of loading rate, were performed using the Abaqus dynamic explicit solver, in order to access the general contact algorithm. Coulomb friction is defined at all contact surfaces with a sliding coefficient of friction equal p = 0.75, imitating real- world conditions where lattice materials may contact plastics, other elastomers, or flesh. A parameter sweep over this coefficient of friction value shows simulations are relatively insensitive to p so long as it is high enough to prevent large-scale sliding on the impact platens.

[0141] To prevent over-estimation of critical buckling load in numerical analysis of plate lattices with zero geometric prebuckling (designs with perfectly vertical walls), these designs were perturbed in the undeformed configuration by the lowest-energy buckling mode. The eigenmode was extracted using a linear perturbation analysis step and scaled so that the peak displacement magnitude was equal to 10% of the wall thickness. Predictably, designs which include geometric prebuckling (e* > 0) are insensitive to geometric perfections, so this step was omitted. Numerical and empirical experiments are aimed at obtaining a homogenized macroscale response for various geometries tested. In order to determine the number of unit cells required to adequately recover the macroscale response of plate lattices, a scaling study was performed in which the lattice size was incrementally increased until the quasistatic compression response converged (Figure 10, left). Lattice aspect ratio was controlled via geometric parameters wcand he to ensure a progressive buckling collapse mechanism, rather than macroscale buckling.

[0142] Quasistatic Compression Simulations

[0143] To simulate the baseline response of elastomeric metamaterials, simulations to 75% uniaxial compression (siatt = -0.75) were conducted. Geometries with identical exterior dimensions were simulated as fabricated samples consisting of

[0446] unit cells, sufficiently many to reduce the impacts of frictional effects at contact surfaces on homogenized results while remaining tractable for fabrication and empirical testing. Analytical rigid surfaces are defined at the upper and lower extents of the lattice, general contact is enabled to prevent penetration of these surfaces by the lattice. Each surface is linked each to a reference node, and boundary conditions are applied to these nodes: an encastre boundary condition to the lower plate’s reference node, and a constant (vertical downward) velocity to the upper plate’s reference node, matched to the crosshead velocity' in quasistatic empirical testing. Follow ing previous work in numerical characterization of energy absorbing unit cells which undergo elastic buckling, boundary conditions on the ±x and ±y faces of the samples are left free, implying that constraining effects of neighboring unit cells are minor. Variable mass scaling was implemented to increase the stable time increment for the explicit solver, reducing the estimated solution increments to 500,000. Throughout the simulation, kinetic energy was monitored to ensure that it remained less than 0. 1% of the total energy of the problem, ensuring that inertial effects did not impact measured reaction forces. Key outputs from quasistatic compression simulations include the reaction force at the lower plate, which can be converted to lattice stress using the projected area in the out-of-plane direction. These results are found to closely match empirical data gathered on a variety of plate lattice designs.

[0144] Dynamic Impact Simulations

[0145] To better understand the real-world performance of elastomeric plate lattice metamaterials, dynamic simulations of impacts using physically relevant specific impact energies were conducted. The geometry for these simulations was identical to that of the quasistatic simulations, but the boundary conditions were adjusted to model an impact scenario. The upper contact surface was linked to a reference node with a user-specified point mass and was given a user-specified initial velocity (vertically downward). In these simulations, displacement of the upper plate and reaction force at the lower plate are nonconstant solution variables and are monitored throughout the simulation. In these experiments, no variable mass scaling was enabled, as inertial forces strongly interact with internal and external forces.

[0146] Dimensional Reduction of Plate Geometry.

[0147] The use of dimensionally reduced shell finite elements was investigated to enable high-throughput characterization of plate lattice geometries. When discretizing thin-wall, large-span structures, many small volumetric finite elements must be used in order to preserve element quality (FIG. 10. right). Shell meshes can represent the same geometry using fewer elements while maintaining excellent element quality. Shell finite elements consider solution quantities constant through the thickness dimension and integrate material properties to produce zero-thickness representations of shell structures with stiffness in tension, compression, shear, and bending. Due to their outstanding utility in structural analysis, many variations of shell finite elements have been implemented in both commercial (e.g., Abaqus) and open-source (e.g. FEBio) software, including variants specialized for small and large deformations, multiscale and multiphysics problems, buckling, composites, and higher-order formulation. In this work, general purpose, large strain formulation, four-node quadrilateral shell elements with reduced integration (Abaqus type S4R) are utilized. These are ideal for analyses of thin, large-span structures and have previously been used to model complex progressive buckling and self-contact of hierarchical honeycomb structures in out-of-plane compression. In the explicit time integration scheme used for simulations in this work, stable time increment increases by a factor of 10 for simulations on shell meshes relative to volumetric meshes; this compounds the time savings offered by reducing the nodal degrees of freedom and the number of contact surfaces, resulting in an average 18x reduction in solve time. Based on both qualitative and quantitative agreement to experimental data and to dynamic simulations using hexahedral elements, shell finite elements are utilized in the large explorations of plate lattice geometry and impact load cases presented in the following section.

[0148] Derivation of Analytical Envelope for Isotropic Foams

[0149] Ashby derives the following for predicting the collapse stress ocof isotropic cellular solids with an elastic buckling collapse mechanism

[0150] (Equation 5) with Esindicating the stiffness of the solid material, p* the specific density of the cellular solid and A a proportionality constant determined to be 0.05.

[0151] Making the conservative assumption that densification strain sa is related to relative density by (Equation 6) and that useful absorbed energy (here ’‘useful’’ indicates that densification has not yet occurred, and hence transmitted stresses are controlled) is simply (Equation 7) it can be algebraically rearranged to express both useful energy absorbed as a function of relative density: (Equation 8)

[0152] Equations 5 and 8 are normalized my solid material stiffness Esand plotted against each other. Kinematic Bounds and Ideal Energy Absorbers

[0153] Consider a model system consisting of a protected object of mass m, shielded from an input acceleration pulse by a passive elastic-dissipative material. The haversine function is frequently used to model the deceleration profile experienced by body during an impact scenario (e.g. a vehicle during frontal collision): (Equation 9) where a defines the haversine amplitude and p defines the period.

[0154] By integration of the haversine pulse, the velocity and position of the input node can be described as a function of time:

[0155] The desired acceleration profile for the protected object features the minimum possible peak acceleration magnitude. A promising candidate acceleration profile can be defined by the piecewise function:

[0156] 0 for t>tf where arrepresents a fraction of the peak acceleration (arG (0, 1]), tr is the time at which this fraction of peak acceleration is reached, and tf is the time at which all energy is absorbed (zero relative velocity between input and output).

[0157] This function prescribes the acceleration profile of the protected object such that it matches the input acceleration up to a threshold arm and decelerates at a constant rate thereafter.

[0158] The state of the protected object at the onset of constant deceleration can be determined by integration of the haversine pulse through tr:

[0159] The time tr at which all energy is absorbed (i.e. Uhsv= Pout, tr > p / 2) occurs before the pulse ends for high ar, after the pulse ends for low ar, or simultaneously with the conclusion of the pulse at ar.crit. This critical ar value can be determined by setting the time required to reach constant deceleration tr and the time required to absorb the remaining energy at constant acceleration nr -niout (ti) / arm equal to the period p:

[0160] (Equation 11)

[0161] Without loss of generality. ? and m can be set to unity and Equation 11 can be solved for ar numerically, yielding: (Equation 12)

[0162] The kinematics of the input and output nodes can be plotted for various arusing the equations above. The difference in position between the input and output nodes at the moment when all energy is absorbed is defined as the clearance needed to absorb the kinetic energy of the input pulse. Intuitively, this required clearance increases as the protected object decelerates more gently. Plotting these two quantities against each other (for arbitrary p, m) yields a curve describing the expected reduction in peak acceleration for a given clearance.

[0163] The required material response of the elastic-dissipative material shielding the protected object can be plotted by inferring force applied to the protected object from its a(t) profile and mass (assumed to be constant m = 1). Assuming full compression of the media and unit cross sectional area, the intrinsic material response can be plotted for media generating various reductions in peak acceleration. Interestingly, this material response, derived making an assumption about a candidate deceleration profile, recovers the constantstress compression profile of the “Ashby foam”: a perfectly rigid response up to a specified collapse stress, after which the material compresses at a constant stress up until full densification.

[0164] Example 1:

[0165] Plate Lattice Metamaterials: Plate lattices (PL), like more widely known strut lattices, can be fully defined by the position and connectivity of nodes. In this work a modified version of a plate lattice is adopted, which features repeated quadrilateral plates and a global, continuously adjustable prebuckling parameter used to control macroscale compression behavior. To avoid bottlenecks and limitations of traditional Computer Aided Design (CAD) tools, a custom constructive geometry code which produces plate lattice geometries to designers’ specifications was implemented. With a single click the tool automatically produces a computational mesh of the design suitable for finite element analysis and a fabrication file ready for additive manufacturing; the generation of these representations is typically a time- and labor-intensive manual process.

[0166] Using this tool node locations can be freely adjusted on a local basis while preserving node connectivity7, unlocking a wealth of plate lattice geometries not previously explored, including those with nonuniform wall thickness, prebuckling parameter, and cell sizes. A prebuckling factor e* is defined to control the local eccentricity of the lattice: increasing e* skews nodal positions about the cell vertical centerline, and the lattice approaches a honeycomb material as e* — > 0 (FIG. 2B). The design space parameterization used in the remainder of this work, along with typical parameter values, is given in Table 1.

[0167] Plate latices are printed with wall thicknesses at the lower resolution limit of the printer hardware, so cross-section analysis was performed to quantify deviations from the design dimensions. Samples were sectioned through the vertical wall with a razor blade (FIG. 2C) and were imaged using a Keyence VK X-l 100 optical profilometer (FIG. 2D, FIG. 2F). Taking 2500 optical measurements over half the height of the plate lattice, a mean wall thickness of 587± 40 pm was determined, slightly below the design wall thickness of 600 pm. Thickness measurements appear normally distributed, and local measurements clearly mirror the “rectified sine wave” wall profile expected in an extruded-filament based AM (FIGS. 2D. 2E).

[0168] In order to establish a reference point for the performance of plate lattice materials, control samples with gyroid triply periodic minimal surface (TPMS) geometry were fabricated for empirical testing. Owing to their ease of design and fabrication, these structures are well-characterized in literature, and have been shown to mimic the performance of foamed materials when fabricated from metals, plastics, and elastomers. In this work, gyroid structures are fabricated using the same base material and print settings as plate lattices and are printed to identical external dimensions and volume density, ensuring that any differences in impact absorption performance can be attributed to the geometry of the samples.

[0169] Example 2:

[0170] Metamaterial Constitutive Response. The mechanical response to compression loading in plate lattice materials is characterized by mapping the stress-strain response of various geometries using empirical and numerical methods. Relative to reference foams, plate lattice materials exhibit pronounced differences in response to compression, including higher initial stiffness and higher stress plateaus at equivalent density (FIG. 3A), resulting in up to 6x increase in useful energy absorbed and higher energy' absorption efficiency E":

[0171] (Equation 13) where o and E indicate compressive lattice stress and strain, respectively. As indicated by three reference foams plotted in FIG. 3A, increasing the relative density of a foam sample is the only means by which designers can exert control over the materiaFs mechanical response, which raises, shortens, and pitches upwards the stress plateau, resulting in lower peak efficiency.

[0172] Excellent agreement between simulated and empirical results is achieved in quasistatic compression tests across various geometries, allowing to leverage numerical models to explore a wider range of wall thicknesses and prebuckling parameters than was possible to test in experiment (FIG. 3C). It was found that an average error between mean empirical response and simulated response is below 10% across the entire range of strains in all quasistatic simulations, which feature complex geometric and material nonlinearities, selfcontact, and frictional interaction.

[0173] Plate lattices with minimal prebuckling (e* —> 0) exhibit an initial peak in the stress strain response (FIG. 3A) associated with the onset of localized buckling and subsequent geometric softening, as plate members are loaded in bending rather than stretching. This effect can be partially mitigated by increasing the geometric prebuckling factor, which makes this peak less pronounced at the cost of simultaneously lowering the plateau stress. This density-independent, continuously tunable parameter enables precise matching of plateau stress of a plate lattice material to a high-level requirement on acceleration for a particular impact scenario without impacting the densification strain.

[0174] Increasing geometric prebuckling e" lowers the stress at which the lattice material is most efficient (highest E* but the value of that efficiency remains higher than for the reference foams tested (FIG. 3B). Efficiencies predicted by finite simulations are near to but uniformly lower than those computed from quasistatic experimental data. This can likely be attributed to over-estimation of collapse stress ocor plateau stress oPby finite element simulations. Simulations of plate lattice compression with a variety of prebuckling level and wall thicknesses indicate that designs with E* > 0.4 can be found across all relative densities tested, p* 6 [0.02, 0.22],

[0175] Quasistatic simulations of plate lattices with various wall thicknesses and prebuckling level reveal favorable scaling of fundamental specific properties with relative density' (FIG. IE). Relative stiffness E / Esscales with p*1 18, slightly worse than the linear scaling predicted by Ashby, with less-eccentric designs exhibiting higher specific stiffness at a given density’. In simulation, linearized elastic modulus E is defined as the secant modulus at 1% compressive strain, and the linearized elastic modulus of the solid material Esis taken as the sum of the hyperelasticity coefficients pi in the Ogden model (Table 3), a reasonable approximation for small-deformation modulus.

[0176] Example 3:

[0177] Impact Performance: Impact testing remains most reliable means of understanding the performance of impact absorbing materials in conditions that closely mimic practical use cases. Empirical and numerical results support the favorable performance of plate lattice materials relative to reference foam materials on the basis of specific energy absorption, prevention of impact load transmission, useful energy bandwidth, and energy' dissipation. FIG. 4A shows a representative empirical impact test conducted on PL and reference foam samples with identical impact mass, impact velocity, sample volume, sample density, and constitutive material. Energy absorbed early in the PL compression (blue) prevents the sample from reaching full densifi cation and limits peak forces reacted by a factor of 3.5 relative to foam sample (orange).

[0178] Consistent with previous results, fabricated samples show rate dependent macroscale response over two orders of magnitude variation on loading rate (FIGS. 3E,3F). Characteristic stress / strain response remains consistent across the range of tested strain rates, including a sharp increase and local maximum followed by a long and flat stress plateau. Foam materials of various densities exhibit similar, but less pronounced changes (97% increase in stress plateau versus 140% change for PL), though being made of the same material and having similar specific density. Simulations indicate a possible explanation: plate lattices feature highly nonuniform strain fields (and therefore strain rate fields). Thus, peak local strain rates for plate lattices exceed those of foams even for identical siatt and excite higher frequency components of the materials viscoelastic (strain rate-stiffening) response. Simulations of impact produce stress / strain responses very similar to those measured in experiments (FIG. 3E), lending credibility7to the dynamic finite element model - in particular, the viscoelasticity component of the material model.

[0179] An important consideration in the design of any mechanical system for real-world use is robustness. This term refers to a mechanical system’s sensitivity to disturbances that arise in real-world manufacturing and operating conditions. Robust systems operate far from performance cliffs and can tolerate dimensional variations and slight changes in loading conditions without dramatic changes in performance metrics.

[0180] FIG. 4C shows the peak impact force reacted by a reference foam material and a plate lattice material with identical specific density7. The foam material densifies across all impact energies tested, so the peak force transmitted is dominated by the material’s bulk modulus and scales approximately linearly with increasing specific impact energy. In contrast, plate lattice material shows remarkable robustness to increasing impact energy, and transmit a relatively constant peak load across the range of energies tested. Analytical treatment of stretching- and bending-dominated foams suggests that an analogous range of impact energies over which foams offer robust performance exists (125 J / kg for this material and relative density), but this is below the lower limit of specific impact energy attainable by test apparatus.

[0181] Simulations of 100 impact tests with energies that extend beyond the limits of the physical test apparatus closely agree with empirical results (FIG. 4C). Beyond a critical impact energy Ucrit.any real-world energy absorber fully densifies and becomes highly sensitive to variations in impact energy-simulations indicate that this critical value for the plate lattices tested is around 1500 J / kg - an order of magnitude higher than reference foams at equivalent density.

[0182] To explore the performance bandwidth of plate lattice materials more thoroughly, an additional 800 simulated impact tests are performed on plate lattice materials with relative density p* ranging from 0.02 to 0.22 (FIG. 4D). These simulations indicate a trade-off between plate lattice density, the peak stress reacted over the high-performance bandwidth of impact energies, and the bandwidth itself. First, the peak stress reacted during each impact less energetic than Ucritscales linearly with increasing density7- this is a confirmation of quasistatic simulation results showing near-linear scaling of collapse stress with p*. Second, as p* increases, densification strain decreases - less compression distance is available for robust operation. This manifests as a reduction in bandwidth at higher relative densities. Third, simulated plate lattice performance bandwidth is much higher than that of reference foams (4 - 10x higher, FIG. 4D) when compared across equivalent peak stress transmitted. A relationship between specific impact energy and peak force transmitted is derived for stretch- dominated foams that collapse via elastic buckling, allowing us to identify Ucritfor relative densities ranging from 1% to 50% (FIG. 4C). The response of one such foam with specific density p* = 0.20 is predicted analytically and indicated by a solid line in FIG. 4C. Equivalent density plate lattices exhibit an order of magnitude increase in bandwidth of useful energy absorbed at equivalent peak stress.

[0183] As an example, if a design specification required a peak stress transmission of 140 kPa for specific impact energies up to 100 J / kg, a designer using the reference foams tested here might reasonably choose a foam with relative density 20%, which would offer robust performance up to 145 J / kg. Simulations presented here indicate that a plate lattice material designed to transmit the same peak stress would have a relative density of 4% (5 times lighter) and offer robust performance up to 650 J / kg (4 times higher). More broadly, this example demonstrates the power of tightly7coupled empirical and numerical test environments, as numerical simulations of impacts augment empirical data and reveal trends beyond the limits of what is feasible or tractable to investigate experimentally.

[0184] Naturally, increasing the relative density of any cellular solid will heighten the collapse and plateau stresses which dictate the per-strain absorbed energy up to the point of densification ea. It will also decrease this sa, indicating that a peak in useful absorbed energy oda (energy absorbed after sa is not useful) exists for any cellular material at a particular relative density. This peak useful absorbed energy and the maximum stress transmitted during compression to Ed at the relative density at which it occurs are plotted against each other in FIG. 4E (they are normalized by the linearized stiffness of the base material Es). This yields a density- and material-independent response curve which is linked to the microstructure geometry of the sample only, showing the envelope of normalized impact energies that can be absorbed and peak stresses which are transmitted during the absorption. Simulations indicate that plate lattice samples operate inside an envelope which is ~10x wider in each dimension relative to foams (foam response is predicted analytically). This highlights the intrinsic advantages of PL geometry when compressed in the out of plane direction, across all values of global geometric prebuckling e* tested.

[0185] Additionally, e* is a continuously adjustable, density-independent parameter, meaning that this envelope can be shifted along the stress axis (not possible for stochastic foam materials). Critically, because the geometry of PL samples was defined using a home-built constructive geometry script, e* can also be locally adjusted, enabling grading or other functional variation in the out of plane direction. Results from one such graded design, with linear variation in e* between 0.0 and 0.2, exhibits the highest peak in useful w ork of any design explored in this w ork (FIG. 4E).

[0186] Example 4

[0187] Energy Dissipation: An additional consideration in the performance of energy absorbing materials is the fraction of impact energy that is dissipated during an impact. Common dissipation mechanisms include material viscoelasticity, plasticity, phase change, and incidental or deliberate frictional interactions. The dissipation is quantified using coefficient of restitution CoR. which can be computed either from the included area in a load / unload curve (FIGS. 1C and 5A)or by comparing the incident and exit velocity in a dynamic impact scenario: where Uedenotes the energy7stored and elastically returned to the environment. Ua denotes the dissipated energy7, and vin and vout indicate the incident and exit velocity, respectively. While proper selection of base material and microscale geometry can effectively limit the transfer of high forces by increasing the duration of the impact even in a perfectly elastic scenario, in practice energy7dissipation can be advantageous in limiting the rebound of protected objects following impact. The plate lathee dissipation is first and response to repeated loading via quasistatic cyclic compression tests, which indicate significant softening (FIG. 5 A) over 1000 load / unload cycles. The normalized dissipated energy drops to only 25% after 1000 load cycles (compression to densification at 0.5 Hz). This can be attributed to a combination of inherent temperature-dependent stiffness of the base material, which was warm to the touch by the end of testing, and possible permanent damage to the sample via local yield in regions of high strain.

[0188] Dynamic tests indicate that reference foam materials are much more dissipative (2x lower CoR at equivalent p*) than plate lattices for low-energy impacts (Figure 5B, square marks). At increasing impact energies (which are beyond Ucritfor these foams), foam materials respond more elastically. By augmenting the experimental dataset with an additional 100 simulations of impact tests (Figure 5B, blue circular marks), a more detailed picture of the plate lattice dissipative response emerges. At low specific impact energies, the lattice responds relatively elastically (CoR > 0.5). Above a critical energy, which is the energy required to initiate local elastic buckling of the plate lattice walls, the material becomes more dissipative with increasing impact energy, eventually falling to 50% of its initial value. However beyond a critical impact energy’ Ucrit, the material behaves more elastically and CoR begins to increase with additional impact energy. This can be attributed to the material’s bulk response being much more elastic than the macroscale lattice response. As impact energy increases and drives the material beyond the point of densification, bulk effects increasingly and eventually overwhelmingly influence the macroscale lattice response. This trend holds across 800 simulations of impacts on plate lattices with relative density p* ranging from 0.02 to 0.22 (FIG. 5C). Denser lattices require more energy to initiate buckling, but also display highly dissipative behavior over a wider range of impact energies - this is directly linked to the impact energy bandwidth increase shown in FIG. 4D.

[0189] Example 5:

[0190] This w ork focuses on the mechanical characterization of additively manufactured plate lattice metamaterials with exceptional specific stiffness and specific energy absorption in out-of-plane compression. By leveraging additive fabrication technologies capable of directly writing the microscale geometry of these materials, scaling relationships established for stochastic materials were exceeded. By sacrificing isotropy, near-linear specific stiffness scaling with relative density and increased specific energy absorption relative to isotropic plate lattices is demonstrated. The design, geometry construction, and simulation pipeline described herein enables continuous and local control over geometric attributes, allowing practitioners to exert unprecedented control over metamaterial microstructure and predict the behavior of those materials before fabricating. Extensive empirical and numerical testing of plate lattice designs across several orders of magnitude of strain rate is conducted, in both prescribed-velocity and prescribed impact energy tests and it is demonstrated that plate lattice materials uniformly outperform reference foams in the out-of-plane direction.

[0191] Enumerated Embodiments

[0192] The following enumerated embodiments are provided, the numbering of which is not to be construed as designating levels of importance:

[0193] Embodiment 1 provides an energy absorbing material comprising: a plurality' of repeating cells, wherein each of the plurality of repeating cells comprises a three-dimensional plate lattice comprising: a network of a plurality of plate members; a plurality' of void regions; a plurality of edges, wherein each edge from the plurality of edges comprises a set of nodal points, and is formed by connecting edges of at least two plates from the plurality of plate members, and wherein at least a portion of the plurality of edges is geometrically prebuckled, whereby each edge from the plurality of edges is positionally skewed relative to a cell's vertical centerline at a local and spatially varying magnitude.

[0194] Embodiment 2 provides the energy absorbing material of embodiment 1 , wherein at least a portion of the plurality of edges is positionally skewed along its entire length.

[0195] Embodiment 3 provides the energy absorbing material of any one of embodiments 1-2, wherein each plate member from the plurality of plate members comprise a regular geometric shape.

[0196] Embodiment 4 provides the energy absorbing material of any one of embodiments 1-3, wherein the regular geometric shape is a polygon selected from the group consisting of hexagon, square, and quadrilateral.

[0197] Embodiment 5 provides the energy absorbing material of any one of embodiments 1-4, wherein the cell comprises at least one wall having graded wall thickness.

[0198] Embodiment 6 provides the energy' absorbing material of any one of embodiments 1-5, wherein the cell has graded density along at least one dimension. Embodiment 7 provides the energy absorbing material of any one of embodiments 1-6, wherein the material absorbs at least two times more energy before densifying compared to a reference stochastic foam having equivalent density.

[0199] Embodiment 8 provides the energy absorbing material of any one of embodiments 1-7, wherein the plate lattices have a critical impact energy value that is at least two times higher than the critical impact energy value for a stochastic reference foam at equivalent density.

[0200] Embodiment 9 provides the energy absorbing material of any one of embodiments 1-8, wherein the material comprises at least one selected from the group consisting of thermoplastic polyurethane(TPU), a liquid crystal elastomer, nylon, a urethane methacrylate, steel, a nickel alloy, cobalt chrome, copper, titanium, aluminum, and tungsten.

[0201] Embodiment 10 provides method for fabricating an energy absorbing material, wherein the method comprises: printing the material of any one of embodiments 1-9, wherein the printing comprises defining a position of each nodal point from the set of nodal points.

[0202] Embodiment 1 1 provides the method of embodiment 10, wherein the printing comprises additively mixing a plurality7of base materials by prescribing an amount of each of the plurality of base materials to fabricate the material.

[0203] Embodiment 12 provides the method of any one of embodiments 10-1 1, wherein the position of each of the plurality7of nodal points is independently defined by prescribing a value for continuously adjustable prebuckling parameter (e*)-

[0204] Embodiment 13 provides the method of any one of embodiments 10-12, wherein the method comprises independently controlling the size of each cell from the plurality of cells.

[0205] Embodiment 14 provides the method of any one of embodiments 10-13, wherein the method comprises prescribing a wall thickness for each cell from the plurality of cells.

[0206] Embodiment 15 provides the method of any one of embodiments 10-14, wherein the wall has non-uniform thickness.

[0207] Embodiment 1 provides a method of fabricating an energy absorbing material, wherein the method comprises: i. simulating the energy absorbing material of claim 1 comprising the set of nodal points, wherein node position of each of the plurality of nodal points is independently defined and / or adjusted without losing its connectivity to the material; ii. performing numerical simulation(s) to determine a response of the material when subjected to a load impact test; and iii. printing the material.

[0208] Embodiment 17 provides the method of embodiment 16, wherein the numerical simulation(s) comprise(s) quasistatic compression simulations, dynamic impact simulations, and / or dimensional reduction of plate geometry.

[0209] Embodiment 18 provides the method of any one of embodiments 16-17, wherein the response includes macroscale stress / strain response.

[0210] Embodiment 19 provides the method of any one of embodiments 16, wherein the method further comprises the steps of empirically determining a response of the printed material when subjected to a load impact test and comparing the empirically determined response with the response determined by performing the numerical simulation (s).

[0211] Embodiment 20 provides a non-transitory computer readable medium comprising computer readable program code that, when executed, causes a processor to: i. simulate an energy absorbing material of any one of embodiments 1-9, ii. perform numerical simulation(s) to determine a response of the material when subjected to a load impact test; and iii. communicate with a printer and print the material.

[0212] Embodiment 21 provides the non-transitory computer readable medium of embodiment 20, wherein the printing of the material is according to any one of embodiments 10-19.

[0213] Other Embodiments

[0214] The disclosures of each and every patent, patent application, and publication cited herein are hereby incorporated herein by reference in their entirety7. While this disclosure has been disclosed with reference to specific embodiments, it is apparent that other embodiments and variations of this disclosure may be devised by others skilled in the art without departing from the true spirit and scope of the disclosure. The appended claims are intended to be construed to include all such embodiments and equivalent variations.

Claims

CLAIMSWhat is claimed is:

1. An energy absorbing material comprising: a plurality' of repeating cells, wherein each of the plurality of repeating cells comprises a three-dimensional plate lattice comprising: a network of a plurality of plate members; a plurality of void regions; a plurality of edges, wherein each edge from the plurality of edges comprises a set of nodal points, and is formed by connecting edges of at least two plates from the plurality of plate members, and wherein at least a portion of the plurality of edges is geometrically prebuckled, whereby each edge from the plurality of edges is positionally skewed relative to a cell’s vertical centerline at a local and spatially varying magnitude.

2. The energy absorbing material of claim 1, wherein at least a portion of the plurality' of edges is positionally skewed along its entire length.

3. The energy absorbing material of claim 1, wherein each plate member from the plurality of plate members comprise a regular geometric shape.

4. The energy absorbing material of claim 3, wherein the regular geometric shape is a polygon selected from the group consisting of hexagon, square, and quadrilateral.

5. The energy absorbing material of claim 1, wherein the cell comprises at least one wall having graded wall thickness.

6. The energy absorbing material of claim 1. wherein the cell has graded density along at least one dimension.

7. The energy absorbing material of claim 1, wherein the material absorbs at least two times more energy before densifying compared to a reference stochastic foam having equivalent density.

8. The energy absorbing material of claim 1, wherein the plate lattices have a critical impact energy value that is at least two times higher than the critical impact energy value for a stochastic reference foam at equivalent density.

9. The energy absorbing material of claim 1, wherein the material comprises at least one selected from the group consisting of thermoplastic polyurethane(TPU), a liquid crystal elastomer, nylon, a urethane methacrylate, steel, a nickel alloy, cobalt chrome, copper, titanium, aluminum, and tungsten.

10. A method for fabricating an energy absorbing material, wherein the method comprises: printing the material of claim 1, wherein the printing comprises defining a position of each nodal point from the set of nodal points.

11. The method of claim 10, wherein the printing comprises additively mixing a plurality of base materials by prescribing an amount of each of the plurality of base materials to fabricate the material.

12. The method of claim 10, wherein the position of each of the plurality of nodal points is independently defined by prescribing a value for continuously adjustable prebuckling parameter (e*).

13. The method of claim 10, wherein the method comprises independently controlling the size of each cell from the plurality of cells.

14. The method of claim 10, wherein the method comprises prescribing a wall thickness for each cell from the plurality of cells.

15. The method of claim 14, wherein the wall has non-uniform thickness.

16. A method of fabricating an energy7absorbing material, wherein the method comprises: i. simulating the energy absorbing material of claim 1 comprising the set of nodal points, wherein node position of each of the plurality of nodal points isindependently defined and / or adjusted without losing its connectivity to the material; ii. performing numerical simulation(s) to determine a response of the material when subjected to a load impact test; and iii. printing the material.

17. The method of claim 16. wherein the numerical simulation(s) comprise(s) quasistatic compression simulations, dynamic impact simulations, and / or dimensional reduction of plate geometry.

18. The method of claim 16, wherein the response includes macroscale stress / strain response.

19. The method of claim 16, wherein the method further comprises the steps of empirically determining a response of the printed material when subjected to a load impact test and comparing the empirically determined response with the response determined by performing the numerical simulation (s).

20. A non-transitory computer readable medium comprising computer readable program code that, when executed, causes a processor to: i. simulate an energy absorbing material of claim 1, ii. perform numerical simulation(s) to determine a response of the material when subjected to a load impact test; and iii. communicate with a printer and print the material.

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