Highly ecient and reusable energy-absorbing metamaterials exploiting soft rate-dependent frictional interfaces

1 Energy-absorbing materials with both high absorption eﬃ- 2 ciency and good reusability are ideal candidates of impact 3 protection products. Despite the prosperous needs, the 4 current designs are either eﬃcient but one-time-use, or 5 reusable but low capacity. Here, we show that metama- 6 terials with unprecedentedly high energy-absorbing eﬃ- 7 ciency and good reusability can be designed, reaching an 8 energy-absorbing capacity of >2000 kJ/kg per lifetime. 9 The extraordinary performance is achieved by exploiting 10 rate-dependent frictional dissipation between soft elastomer 11 and hard constituents in a porous structure. Particularly, 12 the compliant elastomer in the metamaterials ensures a 13 large real contact area, while the stiﬀ porous supporting 14 frame oﬀers high and robust compressive pre-stress for 15 the sliding interfaces, both of which are essential for vast 16 frictional dissipation. Owing to the rate-dependent friction 17 of elastomer interface, the metamaterials also exhibit a self- 18 adapting feature such that more energy can be absorbed 19 when subjected to higher impact rates. We believe this de- 20 sign opens an avenue to develop high-performance reusable 21 energy-absorbing metamaterials that enable completely 22 novel designs of machines or structures. 23


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Energy-absorbing materials and structures 1 , such as foams 25 in helmets and car bumpers, play an important role in keep- 26 ing humans and objects safe from unexpected impacts 2-4 . 27 This protective functionality demands two iconic features 28 of their force-displacement relations: hysteresis and limited 29 peak force 5 . They ensure impact energy is dissipated with-30 out imposing high stress onto the protected targets. Hence, 31 an ideal force-displacement curve of an energy-absorbing 32 material is rectangular-shaped and has a long and flat force 33 * corresponding author, qunyang@tsinghua.edu.cn † corresponding author, lihuajin@seas.ucla.edu ‡ corresponding author, zhaozh@tsinghua.edu.cn plateau. Achieving such curves is crucial for creating excep-34 tional energy-absorbing materials, and has attracted great 35 research interest in exploring original design strategies 6,7 . 36 The most well-known energy-absorbing mechanism is 37 damaging constituent materials, such as ductile metals 8,9 , 38 brittle foams 10-14 and ceramics 15,16 . Besides, in order to 39 maintain a long yielding force plateau, curved shapes 17 or 40 auxetic materials 18 are introduced to prevent structures 41 from immediately losing their load-carrying capabilities 42 due to localization. The mechanisms of damage and plas-43 tic flow can dissipate a huge amount of energy, benefited 44 from bond breakage or dislocations motion at the molecule 45 level. Taking commercial aluminum foams as an exam-46 ple 19,20 , their energy-absorbing capacities are as high as 47 30 kJ/kg or 30 MJ/m 3 . The excellent performance has en-48 dowed them with broad applications in engineering, such 49 as protecting cargo 21 and preventing collapse of rocks in 50 mining 22 . However, they are usually for one-time usage, af-51 ter which the constituents are permanently damaged. This 52 shortcoming can be partially overcome by incorporating 53 damage-tolerant micro-lattices [23][24][25][26][27] or phase-transforming 54 constituents 28-30 that allow the materials to undergo cyclic 55 loadings, although the performance decreases along cycles. 56 To completely remove the one-time-usage limitation, a 57 promising way is designing the microstructures of metama-58 terials, which provide a vast space to gain outstanding me-59 chanical properties that are otherwise hard to achieve [31][32][33][34] . 60 By introducing non-damage energy dissipation mechanisms 61 into microstructures, researchers have developed reusable 62 energy-absorbing metamaterials. A well-investigated mech-63 anism is mechanical instability of micro-cells, such as buck-64 ling of flexible beams [35][36][37][38][39] and shells 40 , and nonlinear forces 65 between magnets 41-43 . The assembled structures, obtained 66 by connecting a series of these micro-cells, often produce 67 hysteric saw-tooth force-displacement curves 44,45 . The 68 metamaterials constructed in this way are reusable since the 69 deformation is elastic. Nevertheless, their energy-absorbing 70 capabilities are typically several orders of magnitude lower 71 than those of the non-reusable ones, which significantly 72 limits their potential applications. For example, one kind 1 of micro-beam based metamaterial 46 only has an energy-2 absorption capacity around 0.15 kJ/kg or 0.015 MJ/m 3 .

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The relatively low performance of reusable energy-4 absorbing metamaterials is mainly caused by two reasons: 5 the constituent materials can only sustain limited forces, 6 and only a small portion of the microstructures contribute 7 to energy dissipation. A strategy to improve the capacity 8 is increasing the peak forces of the saw-teeth 47-49 through 9 increasing the maximum stresses and using stiffer materi- simultaneously achieve high-reusability and high-capacity 19 metamaterials for energy absorption (Fig. 1a). 20 Here, we survey the frictional mechanism in depth, 21 and realize a kind of high-performance reusable energy- between two parts increase with their interface's real con-28 tact area at micro-scale rather than the nominal contact 29 area at macro-scale. Real contact area of two hard ma-30 terials is much smaller than nominal contact area, while 31 that of a soft material and a hard material is almost the 32 same as nominal contact area. Hence, we inserted stiff 33 rods/ropes into smaller diameter holes of a soft porous 34 elastomer to achieve large real contact area. Meanwhile, 35 another stiff porous frame is interwoven with the elastomer 36 to significantly enlarge the prestress between the sliding 37 parts (Fig. 1b, c). The obtained load-displacement curve 38 has a long plateau, whose height and length can be easily 39 tuned by tailoring the geometry (Fig. 1d, Fig. 1: Design of the proposed energy-absorbing metamaterials with both high capacity and high reusability. a A typical energy-absorbing material shows a trade-off between reusability and energy-absorbing capacity. b-c The proposed metamaterial is composed of stiff rods/ropes and a porous reinforced elastomer, which can slide between each other. The porous reinforced elastomer has a stiff frame interwoven with a soft silicone elastomer, inspired by tendon-bone interface shown in upper right corner of (b). d The metamaterial can be subjected to compression or tension. Here, H 0 is the total height, including the supporting platform, and A is the crosssectional area of the reinforced elastomer. e The measured force balances the friction forces between the rods/ropes and reinforced elastomer. The force F is expected to first sharply increase with the displacement u in the static friction region, and then reaches a plateau in the dynamic friction region, which shows an ideal rectangular shape for energy absorption.

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Interwoven structure of the proposed metamateri-53 als. Figure 1b  Cell i contains only an elastomer, ii and iii involve stiff frames with sharp and smooth corners, respectively, to reinforce the elastomer. Each cell has a height 5 mm and flat-to-flat distance 10 mm. The diameter of the rods 2.0 mm is slightly larger than that of the holes 1.8 mm to create pressure between the elastomer and the rods. The lower images show the corresponding manufactured seven-column samples. b Measured load-displacement curves for structures in (a) with a compression loading rate v = 500 mm/min. All the curves show nearly flat force plateaus F p when the displacement is high. The two metamaterials with interwoven structures exhibit higher F p , 3 times as many as that of the one with only an elastomer. Inset: images showing the elastomer without a supporting frame is more compliant and undergoes a larger deformation. c-d Stress-strain curves of structure ii and iii under 200 loading-unloading cycles in reusability test. After all cycles, the capacity of the former decreases by about 14% , while that of the latter varies only about 4%.
tomer structure (gray) with a stiff porous polylactic acid 1 (PLA) frame (yellow) to obtain a reinforced elastomer that 2 has a better load-bearing capability. This interwoven struc-  The porous stiff frame was designed to have a number of 7 hollow columns in a hexagonal packing, since a hexagonal 8 packing has a better compression strength than a square or 9 triangular one 56 due to higher connectivity. Meanwhile, its 10 surface was modified from a minimal surface to guarantee 11 smoothness everywhere 57 , and avoid unnecessary stress

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Steel or carbon fiber rods were inserted to the cylindrical 18 holes in the reinforced elastomer. The diameter of the rods 19 is slightly larger than that of the holes, which creates 20 pressure between the elastomer and the rods along the 21 frictional surfaces. When the rods are pushed downward 22 (Fig. 1d), a compression force F balances the friction forces. 23 When F is low, there is no relative displacement between 24 the rods and the elastomer due to static friction, and F 25 linearly increases with the deformation of the elastomer. 26 When F is high, relative sliding between the rods and the 27 elastomer occurs, and F remains at a plateau corresponding 28 to the dynamic friction force (Supplementary Movie 1). 29 For the convenience of quantifying the energy-absorbing 30 capacity, strain of the proposed metamaterials was defined 31 as displacement u divided by the initial height H 0 (Fig. 1d), 32 and stress as force F divided by the cross-sectional area A. 33 Then, the stress-strain curve is expected as Fig. 1e, and 34 the area of the stress-strain loop is the absorbed energy 35 per unit volume. In addition, replacing the rods with 36 ropes (right part of Fig. 1d) should yield similar frictional 1 behavior against tensile loads.  Load-displacement performance of a single col-48 umn. Next, we would like to understand how physical pa-49 rameters affect energy-absorbing capacity, which is mainly 50 featured by F p of a single column. Obviously, F p can be 51 modified by tailoring the parameters, e.g., the height h of 52 the elastomer, or the diameter d of the steel rods (Fig. 3a). 53 Theoretically, frictional dynamics says where µ is the coefficient of dynamic friction between the 55 elastomer and the rods, and N is the total normal contact 56 force acted on the rods. Intuitively, increasing the column 57 height h enlarges the contact area, and increasing the di-58 ameter difference between rods and holes d − d h enlarges 59 the prestress. As a result, N , and therefore F p , should be 60 proportional to both h and d − d h , which was validated by 61 varying a single geometric parameter in controlled experi-62 ments (Fig. 3b-c). To simplify the experiments, only four 63 steel rods were inserted into the central column for each 64 seven-column module (right in Fig. 3a); thus the obtained 65 plateau values F p are about 1/7 of that in Fig. 2b. More-66 over, we observed F p increases notably with the loading 67 rate v (Fig. 3d), which indicates that energy-absorbing 68 capability gets higher for a higher impact velocity. This 69 attractive feature will be quantified and understood in the 70 next few paragraphs.

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Mechanisms of rate-dependent energy-absorbing 72 behavior. The plateau force F p = µN relies on the load-73 ing rate v, indicating that at least one of µ and N changes 74 with v. Further quantitative studies reveal they both are, 75 and only in that case can the velocity curve in Fig. 3d be 76 understood. More specifically, we attribute the metama-77 terials' rate-dependent energy-absorbing behavior to the 78 synergy of two iconic characteristics of elastomers: rate-79 dependent frictional coefficient, and nonlinear hyperelastic 80 property in large deformation.

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Frictional coefficient µ, especially for elastomers 58 , can 82 be rate-dependent 59-61 ; therefore, we first conducted fric-83 tion experiments on a Tribometer (Rtec MFT-V) to quan-84 tify µ between our silicone elastomer and steel (see details 85 in Supplementary Fig. 3). In the tribometer experiments, 86 a steel sphere was pressed by a downward force F z onto 87 an elastomer disc, placed on a rigid flat plate (inset of 88 Fig. 3e). The plate was rotated to control the sliding 89 speed, meanwhile, F z was maintained to be an almost con-90 stant. Considering that µ relies on normal contact stress 62 91 and elastomer thickness, we prepared two elastomer discs 92 of thicknesses t = 0.4 and 1.3 mm, since the thickness of 93 the elastomer surrounding a single rod varies along the rod; 94 and applied normal force F z = 1.1 and 3.0 N respectively, 95 ensuring the nominal normal contact stress acted on the 96 steel sphere approaches the average normal stress applied 97 to the rods, which is about 0.5 MPa according to the finite 98 element method (FEM) simulations (for details see Sup-99 plementary Fig. 4), in which the constitutive behavior of 100 silicone elastomer is assumed to follow the third order Og-101 den hyperelastic model 63 (see Methods and Supplementary 102 Fig. 5). Taking the ratio of the lateral force to the normal 103 force at different loading rates v, the frictional coefficient 104 µ versus v can be obtained. Averaging the measured data 105 of the two elastomer samples yields µ-v curve in Fig. 3e. 106 When v rises from 57 to 471 mm/min, µ increases by 30%, 107 while F p = µN increases by 60%, greater than µ. shear forces F from the rods, its large deformation pat-5 tern tends to shrink the holes, which in turn enlarges N . 6 This was validated via a FEM model of a column of the 7 metamaterial, as shown in Fig. 3f (for details see Meth-8 ods and Supplementary Fig. 4). Rods and the elastomer 9 were assumed to bond together. In quasi-static numer-10 ical simulations, each rod has a diameter of 1.8 mm in 11 the stress-free state, and then was expanded to 2.0 mm to 12 introduce prestress, after which the rods were forced to 13 move downward by displacement u. Integrating normal 14 stress over the interface Ω gives the normal force N . The 15 obtained N − F curve, black solid line in Fig. 3g, confirms 16 that N increases with F .
where α h/L is the ratio of the elastomer height to 49 the core length. For a given core, E m monotonically in-50 creases with L. When L is fixed, we define a dimension- which only depends on two dimensionless quantities α 53 and ρ c /ρ r . WhileĒ m monotonically decreases with ρ c /ρ r , 54 it non-monotonically increases and then decreases as α 55 increases, reaching the maximal value at an optimal α 56 (Fig. 4b).
57 Two samples were made to demonstrate the practical 58 performance of the design. The first sample (Fig. 4c) used 59 2mm-diameter carbon fiber rods as cores. The length of 60 the rods is 10 cm and the height of the elastomer is 3 cm, 61 so that the length ratio is α = 0.3. The corresponding 62 dimensionless energy-absorbing quantityĒ m approaches 63 the maximal value (see the red dashed line in Fig. 4b). This 64 metamaterial only weights 33 g, but can dissipate energy 65 as large as 80 ∼ 140 J, depending on the loading rate 66 (Fig. 4d) The second sample adopted Kevlar ropes as cores 3 (Fig. 4e). In detail, we used a needle to thread two strands 4 of 1.2mm-diameter Kevlar ropes into a 1.8mm-diameter 5 elastomer hole. Since the ropes are more compliant, the 6 prestress in this sample is lower than the first one. We  Fig. 4b). The metamaterial weights 12 121 g, but can dissipate energy as large as 820 ∼ 1190 J, 13 depending on the loading rate (Fig. 4e). In other words, 14 the energy-absorbing capacity is 7 ∼ 10 kJ/kg, which is 15 even better than the first sample. with relatively large capacity decreases notably in less than 29 10 cycles (dotted lines in Fig. 5). If the energy-absorbing 30 capacity per cycle is multiplied by the number of repeated 31 cycles to calculate the total energy-absorbing capacity in 32 the entire life of a material, then a conservative estimate 33 of our design is 10 × 2000 = 2000 kJ/kg per life, since the 34 metamaterials are still intact and reusable after the tested 35 cycles; and this value is at least 40 times as many as the 36 others ( Supplementary Fig. 6). 37 We attribute the extraordinary performance of the pro-38 posed metamaterials into two strategies of our design. 39 Firstly, we utilized friction between soft elastomer and hard 40 constituents rather than between hard particles, which is 41 the strategy of the previous frictional metamaterials. In-42 volving elastomer helps to achieve tightly contacted fric-43 tional interfaces that have a large amount of real contact 44 area. Secondly, the interwoven structure strongly anchors 45 the soft elastomer on the stiff supporting frame, which em-46 powers the reinforced elastomer with robust load-bearing 47 capability and applies larger prestress at the frictional 48 interface. These interfaces notably improve the capac-49 ity of energy-absorbing metamaterials based on friction 50 as shown in Fig. 5. In addition, the rate-dependent fric-51 tional behavior of elastomer interfaces enables the reported 52 metamaterials passive adaptability to fit impacts with dif-53 ferent velocities. We believe our design strategy opens a 54 new technical path to obtain high-performance reusable 55 energy-absorbing metamaterials.

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Fabrication of the reinforced elastomer. The rein-58 forced elastomer samples were manufactured by combing 59 fused deposition modeling (FDM) 3D printing technology 60 (Hori Z300 3D Printer) and mold-casting process. Materi-61 als adopted were commercial 3D printing PLA filaments 62 (eSUN poly lactic acid), bright-finished 304 stainless steel 63 rods of different diameters and a commercially available 64 silicone elastomer (Dongguan ShinBon New Material Co., 65 Ltd., China). The elastomer consists of two liquid con-66 stituent parts. They were mixed in a ratio of 1:1, then 67 poured into 3D printed molds assembled with PLA frame 68 and 1.8mm-diameter steel rods, and finally cured for 6 69 hours at room temperature. These steel rods were then 70 removed after curing, and the corresponding holes left 71 in the elastomer were subsequently inserted with thicker 72 rods/ropes.

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Measurements of the stress-strain curve of the 74 silicone elastomer. To quantify the mechanical prop-75 erties of the elastomer, uniaxial tension and compression 76 tests were conducted using the single-axis MTS testing 77 machine (E44.104). Due to the large deformation and low 78 stiffness of the elastomer, video extensometry, instead of 79 conventional clip-on extensometry, was used to measure 80 the tensile strain. Specifically, a ruler was placed vertically 81 on one side of the tension specimen, and the coordinates 82 of the markers on the specimen were video recorded (see 83 Supplementary Fig. 5). (C3D8RH) were adopted for the silicone elastomer due to 29 its incompressibility, and eight-node linear elements with 30 reduced integration (C3D8R) were adopted for the PLA 31 and steel rod. All geometric parameters of the model were 32 consistent with those of the experimental samples.