Failure mode engineered high-energy-absorption metamaterials with biomimetic hierarchical microstructures and articial grain boundaries

For numerous engineering applications, there is a high demand for protective lightweight structures with outstanding energy absorption performance and the ability to prevent catastrophic structural failures. In nature, most species have evolved with hierarchical biological structures that possess novel mechanical properties, including ultrahigh specific energy absorption, progressive laminated failure modes, and ability for crack arrestment, in order to defend themselves from hostile environments. In this study, a novel protective metamaterial having spherical hollow structures (SHSs) was developed with different hierarchical microstructures. An artificial failure mode engineering strategy was proposed by tailoring the microstructures of SHS unit cells. To demonstrate the effectiveness of the proposed method, a composite hierarchical SHS lattice structure was developed using a biomimetic laminated failure mode and through a hardening mechanism, mimicking crystal grain boundaries. The quasi-static compressive results indicated a significant improvement in the specific energy absorption, an enhanced plateau stress magnitude, and an obvious delay in the densification stage for the composite hierarchical SHS lattice owing to the constraining effect of its mesoscale grain boundaries and an increased number of intensively engineered laminated failure levels. This novel type of metamaterial was shown to be immensely beneficial in designing lightweight protective aerospace components such as turbine blade lattice infills. experimental stress – strain curve for BHC-SHS and SHC-SHS unit cells; (e-f) experimental and simulational failure analysis of BHC-SHS and SHC-SHS unit cells; (g-h) EAC with different strain for BHC-SHS and SHC-SHS unit cells; and (i-l) relationship among the compressive relative density, density, EAC, and topological density for BHC-SHS and SHC-SHS unit cells.


Introduction
Metamaterials are artificially engineered materials with complicated microstructures designed to possess novel properties such as a high stiffness-to-weight ratio [1] , a negative Poisson's ratio [2] , improved fracture resistance [3,4] , improved damage tolerance [5] , vibration migration [6] , and enhanced specific energy absorption [7,8] . Among the different metamaterial structural designs, the spherical hollow structure (SHS) exhibits tremendous energy absorption and impact resistance capabilities and is lightweight because of the high porosity and highly plastic behavior of its microstructure. Numerous studies have been conducted to demonstrate the energy absorption advantages and potential applications of SHSs. Gao et al. [9] experimentally characterized the mechanical behavior of metallic hollow sphere (MHS) materials. In their work, the stress-strain curve of a typical MHS foam showed three phases: (1) the elastic phase, (2) the plateau phase, and (3) the densification phase, wherein the wide range of the plateau phase was found to be the main source of its superior energy absorption capability. Song et al. [10] discussed the energy absorption properties of an MHS material under impact loading.
Their research indicates that the MHS is an effective protective material to shield bridge piers against vehicle impact. To further explore the relationship between the energy absorption behavior and the design parameters of SHS materials, the mechanical properties of SHS were studied with different sphere sizes, packing patterns, and relative densities. To investigate the compressive energy absorption responses of the adhesively bonded hollow steel spheres, steel foams with two different sphere sizes were fabricated by Yiatros et al. [11] . The experimental results revealed that the steel foam with smaller spheres possessed a higher energy absorption value and plateau stress. Yu et al. [12] studied the fundamental mechanical characteristics and energy absorption capacity (EAC) of MHSs through experimental, numerical, and analytical approaches. The quasi-static and dynamic compression experimental results indicated significant strain hardening phenomena and an obvious dynamic effect. Empirical functions have been established between the elastic modulus, yield strength, plateau stress, and relative density for the face-centered-cubic-(FCC)-packed MHSs, while analytical relations were derived for the yield strength, EAC, and strain hardening behavior at different relative densities.
Nevertheless, the SHS structures proposed in these studies were manufactured with identical hollow spheres, wherein the structure of the SHS was not optimized to further improve its energy absorption potential. Thus, several innovative composite design features have been recently proposed to enhance the energy absorption and impact performance of SHS structures.
Li et al. [13] developed a novel hybrid structure by assembling the MHSs into the tetrahedra and octahedra of the inner spaces of the wire-woven bulk kagome (WBK). It was found that the buckling effect of the WBK structure was resisted by including MHSs, where improved stiffness, strength, and energy absorption values were observed. The high-strain-rate compressive responses of Al380-Al2O3 hollow spheres with different sphere sizes and size distributions were studied by Maria et al. [14] . In their study, a higher peak strength, plateau strength, and toughness of the hollow sphere foams were observed for foams containing spheres with a larger thickness-to-diameter ratio. Liu et al. [15] proposed a density grade thin-walled design for MHS arrays to improve the dynamic properties of cellular materials. Consequently, the transformation of the MHS packing pattern was observed. It was also found that the gradient profile of the MHS arrays generated higher energy absorption value against high impact velocities. However, these hollow spheres are usually formed through traditional powder metallurgy techniques and are connected based on conventional bonding methods, such as sintering [16] and adhesive bonding [17] . Typically, these conventional manufacturing methods lead to a poorly controlled microstructure, limited design potential, and reduced SHS mechanical response owing to their fabrication limits.
Nevertheless, the maturing additive manufacturing (AM) technology now allows researchers to develop materials with more precisely controlled microstructures without a significant increase in the manufacturing cost [18] . For instance, a spherical body-centeredcubic-(BCC)-shell composite lattice structure were 3D-printed by Yuan et al. [19] with carbon nanotube reinforced polyimide (CNT/PA12). Compared with traditional elastomer foams, a higher energy absorption value and a larger energy absorption scale factor with different lattice densities were obtained. Dai et al. [20] proposed and fabricated hollow sphere structures with perforations (PHSSs) in simple cubic (SC), BCC, and FCC packing arrangements through selective laser melting (SLM). The results of the uniaxial compressive tests revealed that the specific energy absorption and strength of the additively manufactured PHSSs outperformed the traditional MHSs. It was also observed that the large deformation and EAC of PHSSs are sensitive to the geometric design parameters, including the wall thickness, hole diameter, and sphere arrangement. To improve the damage tolerance of additively manufactured ceramic materials, Sajadi et al. [21] 3D-printed spherical architected structures with silica-filled preceramic polymers and coated the surface of the spherical lattice with a flexible epoxy polymer. The proposed surface modification technique has proved to be effective in improving the damage tolerance of ceramic-based composite SHS lattices compared to their counterparts.
Some researchers have further utilized the design complexity offered by AM to develop lattice structures with artificially distributed non-uniform microstructures to further promote their functional performance. For example, Alonso et al. [5] proposed an innovative distribution of lattice microstructures mimicking the hardening strategies of crystals to improve the overall damage tolerance of lattice materials. In their research, the artificially designed grain boundary effectively prevented cracks from propagating across it in the overall lattice structures. Inspired by the crack deflection of wood sap channels, Manno et al. pre-engineered multiple vertical crack paths in 2D honeycomb lattice structures by varying the relative density of the microstructural honeycomb cells [22] . Based on the finite element analysis (FEA) results, the 2D honeycomb lattice with a bio-inspired crack path significantly increased its fracture energy.
Gao et al. [3] engineered the crack path of octet-truss lattice structures by optically programming the distribution of the mechanical properties of resin materials. As a result, a 152% increase in fracture energy was observed for the optically programmed octet-truss lattice structures compared to that achieved by their conventionally manufactured counterparts without preengineered crack paths. For SHS lattice structures, however, limited research efforts have been devoted toward the design of an effective energy-absorbing SHS lattice structure with an artificially engineered improvement of its mechanical performance by varying its microstructures. To achieve this novel energy absorbing SHS design, however, the development of a spherical lattice surface that is smoothly connected to the lattice structure, with varying mechanical properties in different lattice regions, remains a challenge.
Moreover, most biological structures have evolved with interesting macroscopic distributions of hierarchical microstructures to achieve excellent mechanical performance.
These features effectively increase survivability in severe natural environments and thus act as a valuable design database for material scientists. For example, Sadeghzade et al. [4] mimicked the hierarchical microstructure of spicule structures to develop novel crack-deflecting circular structures. Their results indicated that the spicule-inspired design strategy improves both the strength and flexibility of the cylindrical structure. Gu et al. [23] investigated the fracture toughness of brick-and-mortar composite structures with respect to the volume fraction of their stiff and soft components and the number of mineral bridges. To improve the fracture toughness of the material, it is recommended by their work to increase the fraction of the soft components until the maximum value of the fracture toughness is reached, while the presence of mineral bridges is shown to be effective in deflecting cracks in the material. A crack-arresting material was also proposed by Gu et al. [24] . This material mimics the hierarchical structures of the conchshell, which itself possesses alternating sheets of mineralized calcium carbonate structures separated by organic layers. Their results revealed that the catastrophic failure mode of the structure can be prevented by increasing the number of hierarchical levels. Meza et al. [25] studied the mechanical robustness and damage tolerance of hierarchically designed nanolattices with three orders of octahedron hierarchical structures, wherein the combinations of solid polymers, hollow ceramics, and polymer/ceramic composites were compared. It was found that an appropriate hierarchical design of the microstructures can efficiently distribute the load over each region of the entire structure. Therefore, designing a bio-inspired SHS (BH-SHS) metamaterial with hierarchical topologies and proposing an effective failure mode engineering method for BH-SHS materials based on the use of artificially alternated lattice microstructures is a promising approach for the development of next-generation protective SHS metamaterials.
In this work, the SHS lattice structures were tailored with artificially distributed biomimetic hierarchical topologies, and a significant improvement in the specific energy absorption was observed for the composite SHS lattice structures with the pre-engineered biometric laminated failure mode and crystal-inspired mesoscale grain boundaries. Specifically, the design and modeling methods proposed in Section 2 were based on simulation and experimental efforts to

Results
In this study, the SHC-SHS, BHC-SHS, and CHC-SHS lattice structures with pre-engineered failure modes were manufactured by the SLM technique using AlMgScZr powders, and quasistatic compression tests were performed to examine the mechanical responses of these samples at a compressive rate of 2 mm/min. The compositions of the powders are shown in Figure S1 and summarized in Table S2. The compressive stress-strain curves and failure propagation diagrams of the printed lattice samples are presented in Figure 1, where the experimental recodings together with progressing stress-strain curves are delivered in Video S3 to Video S7.
The quasi-static compressive behaviors and failure mechanisms were first derived for the BHC-SHS and SHC-SHS lattice structures with two distinct levels of the σ (N) gradients.
Subsequently, a CHC-SHS lattice structure was developed through a crystal-inspired hardening distribution strategy of different unit cell topologies and a biometric laminated failure mode design with an optimal gradient of σ (N), on the basis of the experimental results of the BHC-SHS and SHC-SHS gradient lattice structures. The compressive test results of the gradient BHC-SHS (BHC-G1/2) and SHC-SHS (SHC-G1/2) lattice structures are summarized in Figure   1a-b, wherein G1 has a larger level of σ (N) gradient than G2. Similar to the compressive behavior of the BH-SHS unit cells, the stress-strain curves of the BHC-G1/2 and SHC-G1/2 lattice structures could be characterized into three different stages: • I: The wavy stage was composed of the first (I1) and secondary (I2) sub-wavy stages, wherein a significant fluctuation of stress was observed.
• II: The climbing plateau stage, wherein a steady development of the stress was observed.
• III: The densification stage, wherein the structures of the lattice were forced into contact, and an exponential increase in the stress resulted.
where is the weight of the lattice structure. For instance, the , , and SEA for the BHC-G1 lattice structure are 2.1 MPa/MPa, 73%, and 13.3 J/g, respectively, which are 55%, 9%, and 35% higher, respectively, than those of the BHC-G2 lattice structure. For the SHC-G1/2 lattice structure, the values of = 1.4 MPa/MPa, = 69%, and SEA = 13.81 J/g were observed for the SHC-G1 lattice structure, which were 40%, 2%, and 1% higher, respectively, than those for the SHC-G2 lattice structure. To explore the detailed mechanisms that confer Based on these analyses, the CHC-SHS lattice structure was engineered to possess solidhardening layer-wise composition of the unit cells from the BHC-G1 and SHC-G1 lattice structures for an optimal ( ) gradient; a detailed design diagram of this is shown in Figure   5c. According to the compressive test results shown in Figure 1c, a significant improvement in the stress magnitude for the plateau stage and a noticeable delay in the densification strain are obtained. Table S6 lists the corresponding compressive stages for each type of unit cell of the CHC-SHS lattice structure during compression. The results indicated that an increased number of failure levels was observed for the CHC-SHS lattice structure because of the combination of two unit-cell topologies with different failure mechanisms. Moreover, it was observed that the constraining effects between the boundaries of different cell topologies shifted the failure sequence of the SHC-SHS unit cells earlier such that a greater stress magnitude occurred during the plateau stage of the CHC-SHS lattice structure. Quantitively, the CHC-SHS lattice entered 10 sub-stages before densification, wherein = 2 MPa/MPa and = 0.75 were obtained.
To further evaluate the mechanical performance of the BHC-G1/2, SHC-G1/2, and CHC-SHS lattice structures, the normalized average plateau stress, densification strain, specific strength, and SEA were derived (Figure 2a-b), whose specific strength (SS) is calculated using Equation (17): where is the density of the lattice structure. It was found that the CHC-SHS lattice had an increased SEA value of 15.25 J/g, which was 10.4% higher than that of the SHC-G1 lattice and 14.3% higher than that of the BHC-G1 lattice. This was due to the fact that the CHC-SHS where , 1 , , 2 , and , 2 are the effective moduli of the 1, 2, and 3 lattice layers, respectively; 132 , 212 , 282 , 180 , 320 , and 420 are the moduli of the BHC132, BHC212, BHC282, SHC180, SHC320, and SHC420 unit cells calculated by Equation (9)  (c-e) Ashby charts reflecting the SEA, density-normalized plateau stress, densification strain, and density of lightweight structures including aluminum alloy foams [26][27][28][29] , aluminum alloy lattice structures [30] , different designs of SHS [11,13,20] , and the CHC-SHS cell designed in this work.

Discussion
In summary, this study provided a biometric hierarchical SHS design and a multilevel failure  Figure 3. In this study, the honeycomb structure was applied as the hierarchical design of the surface topology of BH-SHS owing to its enhanced mechanical energy absorption properties [8,31,32] , wherein a bending-dominated honeycomb SHS (BHC-SHS) and a stretch-dominated honeycomb SHS (SHC-SHS) were selected so that different compressive failure mechanisms could be achieved.
where is the radius of the circle 3 tangent to the cross-sectional curve of the bonding surface, represents the tangential points between the bonding surface and the base sphere, ( ) describes the cross-sectional curve of the bonding surface, and calculates the curvature of ( ) at .
To develop the cutting wireframe for pure SHS unit cells, three subprocesses were  Table S1 summarizes the design parameters of the BH-SHS unit cells, wherein the beam thickness, the shell thickness, and the lower limits of the topological densities were determined on the basis of the balance achieved by offering the smallest relative density while maintaining a good geometrical agreement between the CAD design and fabrication results of the BH-SHS unit cells.

Mechanical modeling of BH-SHS unit cells.
In this study, the BH-SHS unit cells were manufactured using the SLM technique; the detailed information on the building platform and the process parameters are listed in Table 1. The AlMgScZr powder, as shown in Figure 4a, was selected as the building material. The material composition of the powder is summarized in Figure S1. The scanning electron microscope (SEM) images of the elementary distributions of the powder are shown in Figure S1.  Table 2, wherein good consistency of the mechanical performance is observed among the printed samples. The detailed dimensions of the specimens are provided in Table S3. To study the compressive characteristics of the BH-SHS unit cells and to model their mechanical properties based on the topological types and densities, compressive experiments and simulations were performed. The detailed recordings of the simulations are given in Video S1 and Video S2. During the compressive process, the compression rate was set to 2 mm/min to satisfy the quasi-static condition of the compression, wherein each unit cell was compressed beyond its densification stage. To improve the accuracy of the simulation, the structural dimensions of the as-built BH-SHS unit cells were measured using an optical microscope and compared with the original designs. For each BH-SHS unit cell, measurements were performed for a representative unit area with a full range of beam angles, as shown in Figure S2a, with the measurement results summarized in Figure S2b. This indicated that the actual beam thicknesses of the BH-SHS unit cells ranged from 887 μm to 1055 μm, which was offset −113 μm to +55 μm from the designed value, while the shell thicknesses of BH-SHS unit cells was offset approximately −100 μm to −200 μm from the original designs. As reported in previous studies [33][34][35] ,    adhesion [36,37] , which also resulted in the observation of a rough surface finish, as shown in Figure S3 and Figure S4. These defects caused a higher stress concentration and, hence, an earlier plastic deformation of the structures and reduced mechanical performance. Combining the experimental stress-strain curves with the simulated failure sequence and the stress distribution results, the compressive process of the BHC-SHS and SHC180 unit cells could be characterized into four stages: • I: The first wavy stage, wherein one of the top/bottom areas (A) collapsed.
• II: The secondary wavy stage, which corresponded to the collapse of the top/bottom area (A) that persisted in stage I, and the post-collapse deformation of the regions that were deformed at stage I.
• III: The plateau stage, wherein plastic deformation occurred in the middle area (B).
• IV: The densification stage, wherein the entire structure of the unit cells was forced to contract.
For the SHC260 to SHC420 unit cells, different characterization of the failure sequence was observed: • I-II: The wavy stage, wherein the middle area (B) collapsed.
• III: The plateau stage, which corresponded to the continuous collapse and fracture of the middle area (B). • In addition, the compressive wavy strength at stages I and II was defined as the maximum compressive stress within the range of those stages. The total EAC was calculated as the energy absorbed per unit volume of the structure until its densification strain as shown in Equation (4): where is the densification strain and is the total EAC. Based on these theoretical calculations and experimental results, the following mechanical models were established: where the topological density of a unit cell is represented by number of individual topological surfaces , represents the type of the surface topology, is the relative density of the unit  Multilevel failure mode engineering. Based on the derived mechanical models of the BH-SHS unit cells, a multilevel failure mode engineering strategy was proposed. To illustrate this, failure propagation is artificially tailored through the multilevel design of the failure sequences, as shown in Figure 5a. Specifically, the provided design space was first pixelized with size identical to the cubic dimensions of the BH-SHS unit cells, wherein the failure sequence of each pixel was represented by failure factor ( , , ) ranging from 1 to . This failure factor was calculated as the safety level that indicated the extent to which each BH-SHS unit cell pixel could move prior to its collapse, and the condition in Equation (10) needed to be met to guarantee the correct failure sequence: ( , , ) = ( , , ) ( , , ) where ( , , ) is the pre-designed failure factor, ( , , ) represents the safety factor of the secondary wavy stage, and ( , , ) is the estimated stress located at ( , , ). For adjacent failure levels, Equation (13) should be satisfied to avoid the simultaneous collapse of the wavy stages of their unit cells: ( , , ) < ( +1 , +1 , +1 ), = 1, … , − 1 (13) where ( , , ) represents the coordinates of the pixel located at failure level with failure factor . Combining the results from Equations (7), (8), and (13), the relationship between the wavy strength and topological density of the BH-SHS unit cells at adjacent failure levels is established in Equations (14) and (15): ( , , ) ( +1 , +1 , +1 ) < , +1 ( ( , , )) ( , , ) − , +1 ( ( +1 , +1 , +1 )) ( +1 , +1 , +1 ) < , +1 ( ( +1 , +1 , +1 )) − ( ( , , )) where , = ( , , ) ( , , ) represents the ratio of the estimated stress located in regions A and B. With these conditions satisfied, the structures within the design space would deform and collapse with a pre-engineered failure sequence. Moreover, typical failure modes, including "V" shaped, "X" shaped, and layer-wise failure shapes, as shown in Figure 5b, can be easily developed by the proposed method. According to previous research [3,[38][39][40][41] , composite structures with laminated combinations of mechanical properties are widely found in nature, and many promoted mechanical properties, such as crack deflection, high energy absorption, and high impact resistance. Furthermore, a layer-wise gradient design of the mechanical properties allowed the structure to exhibit moderate failure and a long stress plateau without undergoing a catastrophic structural collapse while also avoiding a decrease in stress during the compressive process, thereby producing a higher SEA. Therefore, a multilevel laminated failure mode was proposed to develop a composite BH-SHS lattice structure with improved SEA, shown in Figure 5c. To study the effect of the gradients for stage I strength ( ) of both SHC-SHS and BHC-SHS lattice structures with an artificially engineered layer-wise failure mode on their compressive energy absorption behaviors, two levels of ( ) gradients were designed on the basis of the proposed mechanical models of the BH-SHS unit cells. To further improve the energy absorption performance of the BH-SHS lattice structure, a composite honeycomb SHS (CHC-SHS) lattice with multi-failure modes was developed using a combination of the crystal-inspired hardening mechanism and the layer-wise failure mode.
For the composition of the CHC-SHS lattice structure, both SHC-SHS and BHC-SHS unit cells were layer-wise distributed with an optimal ( ) gradient, wherein a hardening grain boundary design pattern was included within each layer of the lattice. This multilevel failure mode design was expected to offer a longer plateau to the CHC-SHS lattice structure with an increased number of failure levels, which were enabled by the distinctive failure mechanisms of different topological designs of the BH-SHS unit cells. In addition, the artificial grain boundaries could potentially improve the relative magnitude of the stress plateau of the lattice structure by constraining the effect of the unit cells with different topologies. Figure 5d shows a case study of applying the BH-SHS unit cells as protective infills for an aerospace turbine blade. In this case study, the failure mode of the turbine blade was artificially engineered to resist the potential impact and protect its shaft from the disintegrated turbine fragments. To illustrate the generality of the proposed method, additional application examples, including a damage-deflective wheel for the space rover and a protective structural infill for a quadradrone, are provided in Figure S5.