For years, the design of lightweight, strong porous materials and effective cushioning has emphasized geometric symmetry and regularity. Metamaterials are a prime example: by arranging basic units (unit cells) such as square or hexagonal honeycombs and trusses in an orderly pattern, they produce mechanical properties the base material does not have. Uniform lattices have the advantage that their deformation is easy to predict, and their stiffness and light weight can be designed theoretically.
However, this regularity can also be a weakness under heavy loads or impacts. Because every internal cell has the same shape and dimensions, members along a given cross-section tend to deform all at once. When some beams buckle, the deformation spreads in a chain reaction to identical neighboring structures, forming a localized crushing band. As a result, the load cannot be distributed across the whole material, and deformation concentrates in specific regions.
A joint research team led by Associate Professor Carlos Portela of MIT (Massachusetts Institute of Technology) and Professor Albert-László Barabási of Northeastern University proposed a design approach that departs from regular lattices. Inspired by plant leaf veins, dragonfly wing veins, and complex network theory, they created "scale-rich network-based metamaterials," in which beam length, thickness, and connectivity are deliberately made non-uniform. The findings were published open-access on November 11, 2026, in the peer-reviewed journal Nature in the paper "Scale-rich network-based metamaterials" (DOI: 10.1038/s41586-026-11191-2). The journal's peer review file also documents the review process, including two named reviewers and an additional anonymous reviewer.
What distinguishes this study is that it does not simply place members at random. Using mathematically controlled parameters, it lets beams of different dimensions coexist within a single structure. By combining thick, long beams with thin, short ones, the design aims to distribute buckling across the entire structure under load and suppress localized collapse.
A multiscale microscopic skeleton designed with two parameters
Attempts to introduce irregularity into metamaterials have been made before, such as randomly deforming lattice cells or generating irregular polygons through Voronoi tessellation. In conventional approaches, however, the dimensions of the beams making up the structure often fall within a relatively narrow range. In this study, members with widely differing lengths and thicknesses are incorporated into the same structure, designing a network that spans multiple scales.
The team developed an algorithm that generates a beam network according to the desired mechanical properties. First, a random position and angle are set within the target region, and a straight line through that point is extended in both directions. The extension stops where the line meets the boundary of the region or an existing beam, forming a junction. Another random position and angle are then chosen and the same operation is repeated, adding new beams in the gaps of the existing skeleton. During this process, beam thickness is also changed step by step, forming a network in which members of different dimensions are interwoven.
The beam thickness is controlled by the following power law with respect to the generation step .
Here, is the initial beam thickness, and is an exponent that determines the rate at which thickness decreases as generation steps progress. By adjusting these two parameters, a wide range of networks can be generated, from structures in which extremely thin beams are woven through the gaps of a thick skeleton to relatively uniform ones. Leaf veins and complex networks were sources of design inspiration, but the work does not directly reproduce the mechanical behavior of biological tissue or social networks. It was developed as a method of geometrically controlling the non-uniformity of member dimensions while taking the material's physical constraints into account.
The algorithm showed it may be possible to design structures with a wider range of mechanical properties than before. The team analyzed the relationship between the elastic anisotropy index , which expresses how much stiffness differs by direction, and relative density, which indicates the volume fraction of the structure relative to solid material. The closer is to 1, the smaller the directional difference in stiffness; lower values mean stronger anisotropy. The analysis showed that scale-rich networks covered 77% of the combinations of relative density and anisotropy that were investigated.
This 77% figure does not represent an improvement in strength or impact absorption over conventional structures; it represents the range of designable property parameters. In simulations, structures with strong anisotropy () could be generated at relative densities of 17–64%, while nearly isotropic structures () could be realized at relative densities of 33–72%. In other words, by varying and , it is possible to design anything from structures that deform easily in a particular direction to those that support loads relatively uniformly regardless of direction.
データを表で見る
| 設計可能領域の網羅率 (%) | |
|---|---|
| スケール混在型ネットワーク | 77 |
Compression tests confirm distributed deformation and suppressed local buckling
To verify the mechanical properties of the designed structures, the team manufactured actual specimens and ran compression tests. The specimens were centimeter-scale resin structures made with a vat photopolymerization 3D printer, with a printing resolution of about 100 \mu\text. To reduce the effect of the specimen's outer edge on the measurements and evaluate the average mechanical properties of the whole structure, the total number of beams was set to .
The specimens had a shape in which a two-dimensional planar network was extruded a fixed distance in the thickness direction, so that the mechanical response under plane strain could be evaluated. The relative density of specimens made this way was 20–75%. For materials, they used Formlabs Clear v4, which is stiff and brittle, for evaluating the linear elastic range, and Anycubic Tough Resin 2.0, which is highly tough, for large-deformation and impact tests. The layer pitch was 0.025 mm, and specimens were measured 72 hours after printing.
In uniaxial compression tests involving large deformation, the performance of the scale-rich network was compared with a regular square lattice, a hexagonal lattice, and a Voronoi lattice, which is an irregular single-scale structure. Three specimens of each structure were fabricated at a nominal relative density of 42%, and for the scale-rich type the condition was adopted to suppress the effect of anisotropy. Tests were displacement-controlled at a strain rate of 5 \times 10^-3\ \text^-1. This is a quasi-static test in which the specimen is compressed relatively slowly rather than struck.
The main results and verification conditions are summarized below.
| 構造・実証手法 | 構造設計の規則 | 証拠の性質 | 基準および測定条件 | 測定された主要指標 | 適用範囲・制約事項 |
|---|---|---|---|---|---|
| 構造設計空間の探索 | べき乗則に基づく梁厚制御（\lambda_t = \lambda_0 t^） | 計算機シミュレーション | 相対密度17〜75%の範囲で有効弾性テンソルを解析 | 密度対異方性プロット領域の77%を網羅（ から まで） | 強度や衝撃吸収率の向上ではなく、設計可能な物性パラメータの範囲を示す |
| 擬似静的圧縮試験 | 多重スケールネットワーク（） | 実験室での変位制御圧縮試験 | 公称相対密度42%、ひずみ速度 5 \times 10^-3\ \text^-1、各構造3体 | 靭性は対照構造と同等以上。変形局所化指標はボロノイ格子より低い値を記録 | 2次元押出形状、単一密度での比較であり、すべての条件での優位性を示すものではない |
| 落錘衝撃試験 | 段階的に梁寸法を変えた多重スケールネットワーク | 実験室での衝撃試験（9,000 Hzの高速度撮影） | 質量6.2 kg、落下高さ0.8020 m、衝突速度約4 m/s、ひずみ速度 100\ \text^-1 | 比較対象の中で最大荷重 F_ が最小、比エネルギー散逸 W_} が最大 | 相対密度40%の特定樹脂試験片による単一衝撃試験。人体保護性能や繰り返し耐久性は未検証 |
| 脆弱コアの保護 | ザボンの果皮に着想を得た密度勾配構造 | 光弾性観察と有限要素解析 | 同等密度の正方形・六角形格子と比較。中央にPMMA製コアを配置 | 実験では光弾性縞の発生を抑制。シミュレーションではコア内部の応力が大幅に低下 | コア内部応力の低減量は数値解析によるもので、直接測定値ではない |
| 弾性波制御 | 屈折率勾配を持つネットワークレンズ | ブロッホ波解析と動的有限要素シミュレーション | 相対密度20〜75%、リューネブルク型屈折率分布 | 有効屈折率1.5〜8.3を達成（対照格子は1.0〜1.9） | 数値シミュレーションによる実証であり、実際の弾性波レンズは製造していない |
In the compression tests, the stress-strain curve was integrated up to 40% engineering strain, before the material densifies, to evaluate "toughness," a measure of energy absorption capacity. The scale-rich network showed toughness equal to or greater than the three control structures. Furthermore, its "deformation localization metric," which shows how strongly deformation is concentrated in particular regions, stayed at a low value of 0.2 or below until macroscopic strain reached 25%. Deformation concentration was smaller than in the Voronoi lattice under the same conditions, confirming that load-induced deformation was distributed across the whole structure.
In contrast, the square lattice showed continuous buckling of beams in the same horizontal row, forming a clear crushing band. In the hexagonal lattice, shear deformation concentrated along a diagonal of about 45 degrees. In the scale-rich network, on the other hand, concentration of deformation in a specific cross-section was suppressed, and beams deformed in multiple regions. However, while identical specimens of the regular lattices showed almost the same stress-strain curves, the scale-rich type showed a spread in response due to the random generation conditions. The team interprets this difference as showing that stiffness and yield behavior can be tuned through internal arrangement even at the same relative density.
As the reason deformation is distributed, the team points to differences in the "critical buckling load" of individual beams. Thick, short beams and thin, long beams begin to buckle at different loads, so not all members deform at the same time. Even if thin beams buckle locally, the surrounding thick beams continue to carry the load, which suppresses concentration of deformation into a single crushing band. This explanation is consistent with numerical simulation analysis of buckling sensitivity and with the result that load-bearing performance dropped when beams within a specific size range were deliberately removed.
Impact tests show lower peak force and greater energy dissipation
When evaluating the performance of cushioning materials, it matters not only how much impact energy they can absorb but also whether they can limit the peak load transmitted to the structures inside. Even with large energy absorption, an extremely large reaction force at the instant of impact could damage the protected equipment or human body. Impact-absorbing materials are therefore required to dissipate as much energy as possible while avoiding sharp rises in load.
The team conducted dynamic drop-weight impact tests using specimens with relative density aligned at 40%. For comparison, they used a variety of structures, including square and hexagonal lattices as well as Voronoi lattices, extended Voronoi (E-Voronoi) lattices, k-nearest-neighbor networks, and Gilbert networks. In the experiment, a rigid impactor with a mass of 6.2 kg was dropped from a height of 0.8020 m and struck the specimen at a speed of about 4 m/s. The strain rate was about 100\ \text^-1, and the deformation process was filmed with a high-speed camera at 9,000 frames per second while a load cell measured the load over time.
The footage and load measurements showed that the way crushing progressed differed greatly by structure. In the square lattice, buckling cascaded layer by layer immediately after impact, and sharp peaks were observed in which the load rose abruptly each time. In the hexagonal lattice, a distinct failure plane formed along a diagonal, and in the Voronoi lattice too, localized crushing bands formed, though irregularly. All of these show a tendency for deformation to concentrate in some regions.
In the scale-rich network, by contrast, beams deformed at various positions within the structure with time lags, and the load was distributed. As a result, the load-versus-time curve showed a relatively flat region (plateau) with few prominent peaks. The peak load F_ at impact was the lowest among the compared structures, and the "specific energy dissipation (W_})," defined as the impactor's loss of kinetic energy divided by the specimen's mass, recorded the highest value. In other words, under these experimental conditions, more energy per unit mass was dissipated while suppressing the instantaneously transmitted load.
That said, this result does not necessarily hold under all impact conditions. The advantage was confirmed only under specific conditions: a relative density of 40%, specimens made of Anycubic Tough Resin 2.0, and an impact speed of about 4 m/s. Whether similar performance is maintained when impact speed or temperature changes, or when impacts are applied repeatedly to the same structure, has not been verified.
Applications: protecting fragile interiors and controlling elastic waves
To explore the potential applications of scale-rich networks, the team also conducted an experiment on protecting fragile internal structures and numerical simulations of controlling elastic waves.
For the protection experiment, they designed a structure inspired by the thick rind of the pomelo, a citrus fruit. A pomelo's rind has a non-uniform internal density and serves to protect the fruit from impacts such as drops. Drawing on this feature, the team varied the density of nucleation points radially when generating the network, producing a structure with a dense skeleton at the periphery. At its center they placed a brittle square core made of polymethyl methacrylate (PMMA) to simulate equipment to be protected.
Compressive load was applied to this specimen from the outside, and the stress distribution inside the core was observed using photoelasticity. Photoelasticity uses the birefringence that arises in transparent materials under load, visualizing stress concentrations as fringe patterns through polarizing plates. When the core was surrounded by square or hexagonal lattices of the same relative density, external load was transmitted to the core and clear photoelastic fringes appeared inside the PMMA. With the scale-rich network, in contrast, the appearance of fringes was greatly suppressed, showing that the surrounding skeleton distributed the load and reduced stress concentration on the core.
The team further analyzed the stress inside the core in detail with nonlinear finite element simulation. Under the same macroscopic strain, the internal stress of the core surrounded by the scale-rich network was substantially lower than with square or hexagonal lattices. This analysis is consistent with the photoelastic observations, but the stress reduction itself is a value derived from simulation, not measured directly by sensors inside the core.
Another application is controlling "elastic waves," such as sound waves and vibrations that travel through solids. The team used Bloch wave analysis and the finite element method to investigate how elastic waves propagate within scale-rich networks. They found that at relative densities of 20–75%, whereas the effective refractive index of square and hexagonal lattices was 1.0–1.9, that of the scale-rich network could be tuned across a wide range of 1.5–8.3. This suggests that geometric design of the structure could allow the direction and propagation speed of elastic waves to be controlled over a wide range.
Using this property, the team designed an elastic-wave focusing lens of the "Luneburg lens" type, in which the refractive index varies continuously from the center toward the periphery. In steady-state dynamic finite element simulations, elastic waves entering from one end were seen to refract inside the network and converge on a specific focal point on the opposite side. However, this is a proof of concept by numerical analysis; the lens was not actually manufactured, and its elastic-wave focusing performance was not measured.
Challenges remaining before practical use
This study presents a new method that, unlike conventional designs repeating regular lattices, mathematically combines members of multiple length scales. Rather than simply making a structure irregular, controlling beam dimensions and arrangement makes it possible to tune a material's anisotropy and how it deforms. In particular, distributing buckling and crushing that tend to concentrate locally in conventional lattices, and achieving both reduced peak load and improved specific energy dissipation under certain impact conditions, is a significant result.
At the same time, these results cannot be directly linked to improved safety in helmets, bulletproof vests, or cushioning for aerospace equipment. What was confirmed is mechanical behavior under specific materials, geometries, and boundary conditions, and challenges remain to be solved before practical products can be made.
The first challenge is extending the structures to three dimensions. The physical tests used centimeter-scale specimens in which a two-dimensional network was extruded in the thickness direction, mainly evaluating deformation under plane strain. Actual protective gear and structural members, however, are subject to combined bending, torsion, and shear in addition to compression. Whether non-uniform member dimensions can suppress cascading buckling under such complex three-dimensional stress states will require further verification.
The second challenge is the effect of the base material. The experiments used photopolymer resins such as Formlabs Clear v4 and Anycubic Tough Resin 2.0, but metals and carbon-fiber composites differ greatly in elastic, plastic deformation, and fracture properties. For example, if similar fine networks were manufactured from aluminum or titanium alloys, stress concentrations at beam joints could lead to fatigue failure or crack initiation. The advantages of the geometric design are not necessarily maintained with other materials.
Third, durability and manufacturing reproducibility must also be verified. The drop-weight tests applied a single impact to each specimen, and residual deformation and accumulated damage after repeated impacts were not evaluated. In addition, the spread in stress-strain response caused by random generation conditions indicates a high degree of design freedom but is also a quality-control challenge for keeping the performance of each product within a fixed range. Practical use will require manufacturing technology that controls unit-to-unit variation and reliably reproduces the required mechanical properties.
The results show that uniform, regular structures are not necessarily optimal for improving a material's strength or impact absorption. Deliberately combining members of different dimensions opens the possibility of designing how loads are transmitted and how deformation proceeds inside a material. The next important steps toward practical use will be extending the approach to three-dimensional bulk structures and testing it in practical materials such as metals and ceramics, clarifying performance under complex loading conditions and repeated impacts.