Non-integer-dimensional architected materials enabling synergistic acoustic, mechanical, and fluid coupling

Abstract

Architected materials have long struggled to achieve true multifunctionality, as attempts to combine acoustic insulation, mechanical robustness, and ventilation often rely on hybridized or modular designs that compromise scalability. Here we introduce a dimension-driven strategy that exploits non-integer-dimensional architected materials (NDAMs) to achieve multifunctional integration within a single topological framework. As a proof of concept, Menger sponge-inspired NDAMs were fabricated by high-resolution additive manufacturing, demonstrating three capabilities: broadband acoustic insulation through self-similarity induced scattering and resonance, tunable mechanical energy absorption via stress redistribution, and enhanced airflow efficiency enabled by drag-reducing multiscale channels. These functionalities arise intrinsically from fractal hierarchy, without reliance on material heterogeneity or external hybridization. Crucially, the dimensional parameter serves as a scalable and fabrication-accessible handle, bridging abstract fractional geometry with real-world engineering. This work establishes NDAMs as a powerful design axis for next-generation multifunctional metamaterials, with potential applications in aerospace, transport, and biomedical systems.

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New concepts

Multifunctionality in architected metamaterials is typically pursued through hybridization or modular assembly, approaches that inherently introduce property trade-offs, geometric redundancy, and scalability barriers. A genuinely intrinsic route has remained elusive. The central challenge is that most design strategies are restricted to integer-dimensional frameworks, which fundamentally limit how energy pathways can be organized across scales. Here, we argue that non-integer dimensionality offers a missing degree of freedom: it elevates fractal hierarchy from a descriptive mathematical construct to a prescriptive design variable. This perspective is novel in that it shifts multifunctional integration from “adding discrete modules” to “embedding coupling directly into geometry”, thereby allowing acoustic scattering, mechanical stress redistribution, and fluid transport to arise within a single topological principle. Unlike prior approaches, this paradigm does not rely on material heterogeneity or structural add-ons, but instead exploits dimension itself as a tunable parameter. By doing so, it establishes a scalable and generalizable foundation for architected metamaterials, opening pathways to multifunctional systems that are both fabrication-accessible and theoretically rigorous.

1. Introduction

Dimension control has emerged as a central paradigm in advanced material design due to its decisive role in tailoring functionality.^1–3 By precisely manipulating geometric parameters across different dimensional spaces, scientists and engineers have developed a new class of rationally designed structures—metamaterials. These materials exhibit tunable and evolving functionalities, enabling diverse applications across a range of fields.^4–18 However, many natural phenomena exhibit irregularity and complexity that cannot be fully captured by conventional integer-based dimension frameworks. To address this, the mathematical concepts of fractal geometry were proposed, enabling the representation of non-integer dimensional spaces and self-similar forms. Fig. 1(A) illustrates the two foundational mathematical definitions underlying this geometric framework: non-integer dimension and self-similar set, both of which serve as cornerstones for encoding hierarchical complexity into physical structures. Non-integer dimensionality quantitatively characterizes how structural detail evolves with the observation scale, as captured by the scaling law , where N(ε) is the number of subunits, and 1/ε is the scaling factor. This formulation leads to non-integer dimensional values associated with fractals, reflecting their inherent scaling behavior and recursive self-similarity. In parallel, Hutchinson¹⁹ formalized that recursive affine transformations, which involve contraction mappings achieved by scaling, rotation, and translation, provide a deterministic framework for generating self-similar sets. According to the iterated function system (IFS) formalism, any set invariant under such transformations converges to a unique compact fractal set K, satisfying , where S_i are affine contraction maps acting on a metric space. Together, these two definitions provide the theoretical basis for constructing a wide variety of fractal geometries with controllable complexity and self-similar topology. The translation of this mathematical framework into practical geometric forms is exemplified by classical fractal constructions, such as the Koch snowflake and Sierpiński triangle, as shown in Section S1, SI. These fractals visually demonstrate how simple recursive rules can generate complex and scale-invariant patterns.

Fig. 1 Bridging mathematical formulation and multiphysics properties through non-integer dimensional geometry. (A) The foundational mathematical framework of fractals, including the definition of dimension based on scaling and the recursive construction of self-similar sets through affine mappings. (B) Fractal geometries preserve identical structural features across scales such as scale hierarchy, void topology, and connectivity. (C) Multiphysics functionality including sound insulation, energy absorption, and air ventilation without relying on material heterogeneity, addressing core application demands. (D) Multiphysical responses of fractal geometry, showing how scale-sensitive mechanisms manifest in acoustic wave propagation, mechanical stress redistribution, and fluid transport pathways.

Non-integer dimensionality functions not only as a mathematical concept but also as a guiding principle that drives the development and practical applications of numerous scientific disciplines, much like many other advanced theoretical frameworks.^20–24 Functional integration design is a key trend in the development of architected materials (AMs).^25–33 The coupled interplay among sound insulation, energy absorption, and air ventilation has emerged as a key research focus in both engineering and physics, owing to its vital role in multifunctional applications and the intrinsic physical synergy among these processes. Fig. 1(B) and (D) illustrate the conceptual transformation from mathematics to structure and functional integration, anchored in non-integer dimensional fractals and guided by multiphysics, illustrating how these principles translate into functional physical responses. Due to their self-similarity and scale hierarchy, fractal structures exhibit identical geometric properties across different length scales, such as consistent void topology, connectivity, and dimensional profile. Beyond geometric self-similarity, the non-integer dimensionality itself plays a fundamental role in determining physical performance. Structures with non-integer dimensions between classical integers, for example, between 2D and 3D, possess unique spatial complexity and hierarchical porosity that cannot be captured by conventional geometries. When interacting with different physical fields, such as acoustic waves, mechanical stress, or fluid flow, these same geometries can yield distinct, scale-sensitive responses. In acoustics, fractal self-similarity leads to an ultra-high refractive index and supports efficient manipulation of both short- and long-wavelength (λ) acoustic waves, enabling broadband subwavelength sound control. Mechanically, fractal structures embody intrinsic multi-scale characteristics, where distinct topological morphologies orchestrate stress flow (S) pathways and deformation modes, giving rise to diverse mechanical performance orientations across stiffness, strength, cushioning, and energy absorption. In fluid dynamics, the hierarchical porosity of fractal architectures facilitates scale-dependent flow pathways, allowing for precise control of fluid flux (I) and suppression of turbulence and flow separation. A common feature among these domains is that the desired functionalities are realized solely through the intrinsic geometric characteristics of the fractal architecture, without requiring any additional structural components or hybrid designs. Overall, this generalized mathematical framework encodes scale hierarchy and self-similarity directly into geometry, thereby enabling fractal structures to act as a conceptual bridge linking abstract mathematical theory with real-world multiphysical integration.

In this work, a representative fractal architecture, the Menger sponge, is utilized as a conceptual demonstration to validate the multifunctional potential of non-integer dimensional geometry in achieving acoustic, mechanical, and fluid performance concurrently. Leveraging its multi-scale properties, the Menger sponge induces diverse local resonance modes across a broad frequency range, resulting in broadband and robust sound insulation performance. The mechanical properties of the designed fractals are intrinsically governed by fractal hierarchy, exhibiting a functional transition from high stiffness and energy absorption to enhanced buffering with increasing fractal order. Additionally, the aerodynamic performance was experimentally investigated, revealing a significant increase in ventilation efficiency and heat dissipation with increasing fractal order. This proof-of-concept sets the stage to address key limitations in current multifunctional material design paradigms.

Despite growing interest in multifunctional materials, conventional integer-dimensional AMs, such as honeycombs and lattice frameworks, typically rely on structural hybridization or material heterogeneity to achieve diverse functionalities. These approaches introduce design complexity, increase manufacturing costs, and limit scalability. In contrast, the proposed design strategy demonstrates that non-integer dimensional fractal architectures can intrinsically integrate acoustic, mechanical, and fluid functionalities within a single, dimensionally driven framework. This work introduces a dimension-driven design philosophy, wherein non-integer dimensionality serves as a fundamental organizing principle to couple multiphysics functionalities through a shared geometric backbone. By elevating fractal dimension from a descriptive metric to a prescriptive design variable, this study validates the feasibility of using fractal structures to integrate acoustics, mechanics, and fluid dynamics, and further establishes a scalable “one-to-many” design framework based on intrinsic geometric integration. This intrinsic integration eliminates the need for hybridization and provides a structurally elegant, fabrication-efficient, and generalizable solution to meet the growing demand for high-performance multifunctional systems.

2. Non-integer dimensional mechanisms for multifunctional integration

Leveraging geometric transformation to tailor physical properties has emerged as a powerful design approach. Non-integer-dimensional architected materials (NDAMs) exhibit hierarchical self-similarity across scales, enabling coordinated improvement in acoustic attenuation, ventilation efficiency, and mechanical crashworthiness. As a proof of concept, we begin with a simple cube as the geometric starting point for fractal construction, as shown in Fig. 2(A). The cube is divided into smaller segments using a scaling factor of 3, and the central cube along with the six cubes centered on each face is removed. This recursive process generates a self-similar, sponge-like geometry, with each iteration producing an exponentially increasing number of voids and channels. The resulting structure resembles a Menger sponge fractal. Mathematically, this fractal architecture follows the definition of fractal dimension, where the effective dimensionality decreases from 3 to approximately 2.7268 (log 20/log 3). The structure also satisfies the definition of a self-similar set, formed by affine mappings involving scaling and rotation. The relative density (volume) and total surface area are calculated in mathematics^34,35where i represents the fractal order. It is noteworthy that, as illustrated in Fig. 2(B), such non-integer dimensional configurations exhibit a counterintuitive mathematical property. After numerous iterations, the volume of the shape approaches zero, while its surface area tends toward infinity. This paradoxical property is the defining characteristic of fractals: they exist in a state between dimensions, occupying space without fully occupying it. The fractal nature enables efficient material distribution and multiphysics properties, making them highly suitable for advanced materials design. Moreover, this geometrical evolution process continues to reduce materials, making them ideal for lightweight structural applications.

Fig. 2 Structural design and fabrication. (A) Generation of a fractal Menger sponge by iterative scaling and translation from a 3D cube, resulting in a structure with fractional dimension (D = 2.73). (B) Counterintuitive mathematical properties of the fractal, characterized by a fractional dimension, which trends toward zero volume and infinite surface area. (C) Schematic illustration of the DLP printing technology used to fabricate the polymer AMs. (D) Fabricated first-, second-, and third-order fractal samples, with SEM images of surface and pore vicinity, confirming well-preserved multiscale topology with minor irregularities.

Moreover, additive manufacturing, also known as 3D printing, provides a viable approach for fabricating multiscale fractal architectures. As shown in Fig. 2(C), the NDAMs were fabricated via digital light processing (DLP), which offers high precision and resolution across scales, with the printed samples reaching a characteristic size of 27 mm (see Section S2 in the SI for fabrication details). This technique enables the realization of intricate hierarchical structures with both macroscopic geometry and microscopic features.^25,36,37 In particular, scanning electron microscopy (SEM) observations of the fabricated samples (as shown in Fig. 2(D)) confirm that the hierarchical pore topology and multiscale features were well preserved, with only minor surface irregularities. With continued advances in AM technologies, fractal geometries with higher orders and enhanced structural complexity can be achieved, offering a scalable pathway for integrating multiscale functionalities into a single architected material system.

Overall, the paradoxical geometry of non-integer dimensional fractals, together with their scalable fabrication through additive manufacturing, establishes a robust platform for lightweight yet multifunctional architectures. Building on this structural foundation, the following sections explore how such geometries translate into enhanced acoustic, mechanical, and fluid–thermal performances.

3. Results and discussion

3.1. Sound insulation and multi-bandgap performance

The multifunctional potential of NDAMs is first examined through their capability for acoustic wave attenuation. Phononic crystals (PnCs) with acoustic bandgaps are highly effective in blocking sound.^38–40 The primary objective of sound insulation is to achieve multiple bandgaps through the integration of acoustic physics and innovative geometric designs. Leveraging Bloch's theory, the eigenfrequencies of the acoustic band structures and the transmission characteristics of these systems were calculated using finite element methods. Fig. 3(A) illustrates the air phase of first- to third-order NDAMs. It is worth noting that the acoustic impedance of solid materials is significantly higher than that of air, therefore, the solid phase is treated as a rigid boundary. Additionally, among various PnCs, only three-dimensional ones with full phononic bandgaps hold promise for achieving full manipulation of acoustic or elastic waves. Consequently, the irreducible Brillouin zone for the cubic lattice is also depicted in Fig. 3(A). The band structure diagrams of first- to third-order NDAMs were compared, where the gray-shaded regions in Fig. 3(B) represent acoustic bandgaps. It is evident that the number and proportion of bandgaps across the entire frequency range are widely distributed. The fractal design inherently facilitates the formation of multi-bandgap properties. The multi-scale configuration generates scalable pores and voids, providing diverse transmission attenuation pathways across the frequency spectrum.^39,41 Specifically, the self-similar fractal geometry significantly broadens the total bandgap regions where sound wave propagation is prohibited. Additionally, multi-bandgaps emerge on a subwavelength scale, effectively demonstrating low-frequency sound insulation with the proposed NDAMs. Two quantitative metrics were employed to compare the bandgap properties among NDAMs of different orders,⁴² as shown in Fig. 3(C). For the first bandgap, the second-order NDAM exhibits the lowest normalized starting frequency of 0.322. In terms of the widest bandgap, the first-order NDAM achieves the largest normalized bandwidth of 0.24, corresponding to enhancements of 107% and 167% compared with the second- and third-order NDAMs, respectively.

Fig. 3 Sound insulation properties of NDAMs. (A) Air phase diagram of first- to third-order NDAMs and the first Brillouin zone of the 3D structure with a lattice constant a = 27 mm. The band structure is obtained by sweeping the wave vector k along the edges of the irreducible Brillouin zone (highlighted by the red track). (B) Band structure of the corresponding unit cell, with the shaded area indicating the acoustic band gap. The frequency f is normalized as f = c/a, where c represents the velocity of sound in air. (C) Comparison of two key metrics: the first and widest acoustic band gaps of NDAMs, alongside a one-dimensional mass-spring lattice representation of first- to third-order NDAMs. In this model, m_i represents the masses, and k_ij denotes the stiffness between neighboring local-resonance channels. (D) Sound pressure distributions of the corresponding NDAMs under normal incidence at normalized frequencies of 0.874, 0.606, and 0.740, respectively. (E) Experimental setup for measuring the sound pressure level of the proposed design. (F) Measured SPL results of first- to third-order NDAMs, compared with the case without a structure. (G) SPL-based evaluation of stability and noise attenuation performance for different fractal configurations. Stars indicate optimal performance per metric.

To further investigate the acoustic wave attenuation, multi-bandgap properties, and underlying physical mechanisms of NDAMs, the fractal design was first simplified into a one-dimensional mass-spring lattice, as illustrated in Fig. 3(C). As the fractal order increases, self-similar cubes are progressively removed, leading to higher-order NDAMs introducing additional local resonance modes. The interactions among different oscillators, coupled through spring connections, give rise to new and multiple bandgaps. To elucidate the behavioral patterns of acoustic insulation motions, especially at bandgaps, some typical sound pressure field distributions of corresponding NDAMs are predicted by numerical simulations, as shown in Fig. 3(D). The primary function of soundproof structures is to suppress the propagation of sound waves effectively. When sound waves pass through such a design, they encounter porous materials or complex geometries that induce phenomena such as reflection, refraction, and scattering.⁴³ Specifically, the fractal characteristics of the NDAMs generate intricate geometries and multiple voids, forcing sound waves to undergo repeated reflection and scattering within the structure, thereby substantially diminishing their propagation energy. Each instance of reflection and scattering disperses the sound wave energy, leading to gradual attenuation. Furthermore, the internal pores and surfaces of the NDAMs interact with the sound waves, facilitating the conversion of acoustic energy into heat or other dissipative forms. As the sound waves penetrate into the structure, the multi-layered pores further enhance energy dissipation through amplified reflection and scattering processes. Numerical analysis at normal incidence of Fig. 3(D) reveals significant pressure differentials at specific frequencies, indicating the substantial acoustic energy dissipation. This phenomenon, driven by multiple scattering and reflection effects, leads to the formation of bandgap zones where acoustic wave propagation is effectively suppressed. Meanwhile, owing to the energy dissipation behavior enabled by the fractal configuration, the sound insulation properties are greatly enhanced when three NDAMs are aligned in series, as detailed in Section S3 of the SI.

Finally, 3 × 3 planar arrays of first- to third-order NDAM unit cells were fabricated to experimentally validate their sound reduction properties.^44,45 A photograph of the experimental setup is shown in Fig. 3(E). The structures were placed inside a rigid square tube made of 5 mm-thick acrylic plates, with an overall NDAM sample size of 81 mm × 81 mm × 27 mm. A loudspeaker and a sound level meter were positioned on either side of the structure to measure the A-weighted curves of sound pressure level (SPL), which approximates the response of the human auditory system. Compared with the reference case without the structure, the SPL was significantly reduced, as shown in Fig. 3(F). To quantitatively assess the acoustic performance under different fractal orders, four SPL evaluation metrics were extracted and categorized into stability and noise attenuation, as shown in Fig. 3(G). In terms of stability, the third-order structure achieved the lowest standard deviation, indicating the most consistent insulation ability. The first-order structure exhibited the smallest peak-to-valley value, reflecting minimal SPL fluctuation amplitude. Regarding noise attenuation, the first-order configuration also provided the highest energy dissipation ratio (14.6%) and exhibited the maximum SPL reduction (6 dB(A)), demonstrating superior sound insulation capability. The experimental findings confirm the effectiveness and robustness of fractal-based designs for sound insulation, highlighting their potential for practical applications in acoustic engineering.

3.2. Mechanical and energy absorption performance

Through compression tests and numerical simulations, this study reveals how the geometric evolution of NDAMs drives the evolution of their mechanical characteristics, including elastic and plastic responses, energy absorption capacity, and deformation mechanisms. Fig. 4(A) illustrates the stress–strain curves for NDAMs across different fractal orders, derived from quasi-static compression experiments. The initial linear segment of each curve corresponds to the elastic deformation regime, characterized by a proportional relationship between stress and strain. In this regime, the first-order NDAM shows a higher elastic modulus than its higher-order counterparts, indicating greater initial stiffness and resistance to elastic deformation. Fig. 4(B) illustrates the distribution of Young's modulus, with isotropy measured using the Zener anisotropy index A (see Section S4, SI).²⁵ An A value closer to 1 indicates better isotropy. The first-order NDAM exhibits a value of 0.7024, indicating relatively uniform elastic performance across different directions.⁴⁶ However, with increasing structural order, the modulus surfaces exhibit pronounced fluctuations, indicating a higher degree of anisotropy and enhanced directional dependence of the elastic modulus in higher-order NDAMs.

Fig. 4 Mechanical properties of NDAMs. (A) Stress–strain curves of first- to third-order NDAMs under quasi-static compression experiment with a lattice constant a = 27 mm. (B) Young's modulus surfaces and Zener anisotropy index (A) for NDAMs of different orders, where values closer to A = 1 indicate better isotropy. (C) Experimental and simulated plastic failure modes of NDAMs. (D) Evolution of stress distribution characteristics in NDAMs with increasing hierarchy, detailing the transition of tensile, compressive, and shear stress patterns. σ_x represents lateral tensile stress, σ_z corresponds to vertical compressive stress, and τ_xz denotes shear stress in the x–z plane. (E) Analysis of the intrinsic relationship between hierarchical levels and mechanical properties. (F) Radar charts of normalized key mechanical performance parameters for NDAMs.

The midsection of the stress–strain curve in Fig. 4(A), corresponding to the plastic deformation stage, constitutes the primary energy absorption phase. While the first-order NDAM exhibits a limited plastic region, the second- and third-order counterparts maintain load-bearing capacity over a broader strain range, indicating longer effective energy absorption strokes. Beyond a certain strain, localized failures arise, consistent with the intrinsic quasi-brittle fracture of the printed resin^13,27 (further details are provided in Section S5 of the SI), resulting in a marked drop in the stress curves. Moreover, the stress level remains modest throughout compression because it is bounded by the intrinsic strength of the photopolymer matrix. Fig. 4(C) shows the plastic collapse modes of the NDAMs. Plastic deformation in the first-order NDAM is concentrated around the main pore edges, whereas in higher-order NDAMs, it extends into the secondary pores, revealing the regulatory role of structural hierarchy in deformation behavior. To further elucidate the intrinsic relationship between structural order and plastic deformation mechanisms, Fig. 4(D) depicts the stress distribution characteristics of the NDAMs. The stress response of the first-order NDAM is dominated by tensile-compressive coupling: vertical columns endure the majority of compressive stress while simultaneously transmitting tensile stress to horizontal columns. Additionally, geometric discontinuities induce pronounced stress concentrations at pore edges and corners. In the second-order NDAM, the introduction of secondary pores facilitates the increase in the transmission paths of stress flow. This leads to a reduction in the magnitude of tensile stress and a more uniform stress distribution. Although shear stress concentration is alleviated, localized concentrations remain at the edges of secondary pores. For the third-order NDAM, stress concentration is further alleviated, with additional stress transmission paths along vertical columns. As shown in Fig. 4(E), increasing the structural order of NDAMs results in enhanced porosity, multi-scaling, and geometric uniformity. The significant rise in cell numbers and the nested configuration of diverse pores form a multiscale hierarchical geometric network within the structure. This network establishes numerous stress transmission pathways, effectively reducing localized stress concentrations. Moreover, shear bands are formed along the connecting lines between the vertices of the pores, amplifying the contribution of shear stress in the structural response. In summary, the progressive geometric refinement and stress redistribution mechanisms contribute to the superior buffering of higher-order NDAMs.

Fig. 4(F) quantifies the key crashworthiness metrics of NDAMs across different orders using radar charts. The elastic modulus (E), specific energy absorption (SEA), and mean crushing force (MCF) represent the structural capacity for load-bearing and energy absorption,^47–50 while crushing force efficiency (CFE) and peak force (PF) correspond to buffering performance and impact protection capabilities.^51,52 For intuitive comparison, all metrics are normalized: higher E, SEA, CFE, and MCF values (closer to 1 after normalization) indicate better performance, whereas a lower PF is preferable. The raw data prior to normalization is provided in Section S6, SI. The first-order NDAM demonstrates superior performance in E, SEA, and MCF, highlighting its high stiffness and energy absorption capacity. However, its PF value is the highest, suggesting that it may cause greater damage to flexible impactors during collisions. The second-order NDAM excels in CFE, reflecting a more uniform impact load distribution and a balanced capability for energy absorption and buffering. The third-order NDAM achieves the lowest PF value alongside a relatively high CFE, showcasing significant advantages in buffering and impact protection. This transition highlights the inherent multifunctionality and adaptive reconfigurability of NDAMs.

3.3. Ventilation and thermal dissipation performance

Ventilation and thermal dissipation refer to the fundamental processes of air exchange and heat removal, governed by natural or forced convection.^53–55 Here, the ventilation functionality of fractal design is further explored. As shown in Fig. 5(A), experiments were conducted to measure the ventilation efficiency of first- to third-order NDAMs within a duct system. The experimental setup consists of a straight tube, an electric fan positioned at the inlet to provide airflow (with an air velocity v_in of 5.6 m s⁻¹), and an anemometer installed at the outlet to record the velocity rate.^56–58 For each NDAM sample placed at the center of the airflow tube, the outlet velocity v_out was recorded, and the ventilation efficiency was calculated as the ratio v_out/v_in, as shown in Fig. 5(B). A substantial improvement in ventilation efficiency was observed with increasing fractal order, reaching a maximum of 67% for the third-order NDAM. Although porosity at the windward surface also increases with fractal order, the rise in outlet velocity shows a greater growth rate. This indicates that the improvement in ventilation efficiency cannot be attributed solely to porosity, but also to the hierarchical channel configuration.

Fig. 5 Ventilation and thermal dissipation enabled by non-integer dimensional configurations. (A) Experimental configuration for airflow evaluation of fractal structures from first to third order. (B) Comparison of outlet velocity and porosity, showing ventilation enhancement with fractal order. (C) Air flow field distributions and (D) drag force reduction trends. (E) Schematic of the enhancement mechanism illustrating multi-scale drag distribution and Bernoulli effects. (F) Schematic of the structure with a fixed-temperature heat source (308 K) applied to the top boundary of the air domain, consistently used across all fractal orders. (G) Temperature field evolution indicating improved thermal dissipation with increasing fractal complexity.

The superior ventilation performance observed in higher-order NDAMs is attributed not only to increased porosity but also to the fractal geometry, which guides airflow more efficiently. To elucidate the physical mechanisms behind this enhancement, a numerical analysis was conducted using computational fluid dynamics (CFD). The isotropic nature of the structure allows a vertical cut-plane to represent the airflow distribution within the NDAMs, as illustrated in Fig. 5(C). When air flows through the structure, a significant increase in airflow velocity is observed. Physically, according to Bernoulli's principle, when airflow passes through small apertures, a reduction in cross-sectional area as it transitions from a wide channel to a narrow region leads to an increase in velocity accompanied by a corresponding pressure drop. Detailed descriptions of the pressure distribution are provided in Section S7, SI. This pressure differential drives the flow from high- to low-pressure regions, releasing pressure potential energy, which further accelerates the fluid and enhances the outlet velocity. The airflow velocity v_i in the i-th pore can be estimated using Bernoulli's principle, , where Δp_i represents the pressure difference across the corresponding fractal channel. As the fractal order increases, the number of channels grows exponentially. This multi-scale configuration provided by the fractional dimension effectively distributes fluid drag across multiple scales, significantly reducing aerodynamic drag per unit area, as shown in Fig. 5(D). By mitigating localized pressure concentrations caused by single large-scale features and reducing pressure fluctuations in fluid impact regions, the fractal design enhances overall flow efficiency. The presence of pores at varying scales within the fractal structure optimizes flow distribution, enhances channel utilization, and disperses airflow across multiple pathways. This reduces localized high drag that could otherwise result from the concentration of large pores, thereby improving overall ventilation efficiency. For example, the third-order NDAM exhibits a more uniform velocity distribution, lower aerodynamic drag, and higher ventilation efficiency compared with the first- and second-order designs. Further analyses under varying inlet velocities v_in consistently show a decreasing trend in drag with increasing fractal order. Detailed results are provided in Section S8 of the SI. Furthermore, the fractal structure's inherent propensity for infinite surface area growth expands its contact interface with airflow, thereby improving air exchange efficiency. This results in increased outlet velocity and a substantial enhancement in ventilation performance.

Benefiting from the enhanced ventilation efficiency, the NDAMs also demonstrate superior thermal dissipation capability, as revealed by coupled fluid–thermal analyses. A fixed-temperature heat source (308 K) was applied at the top boundary of the air phase region, as indicated by the red-highlighted sides.²⁹ This thermal condition was uniformly used across all fractal orders. As shown in Fig. 5(G), the spatial distribution of temperature exhibits a significant evolution with increasing fractal order. Specifically, the enhanced diffusion of heat toward cooler regions and the suppression of hotspot intensity in higher-order configurations indicate a marked improvement in thermal dissipation performance. This enhancement stems from the non-integer dimensional design strategy. By recursively embedding self-similar features across scales, the fractal topology enables an expansion of thermal pathways, effectively increasing the available surface area and forming hierarchical channels for multidirectional heat flow. The increased number of branching thermal channels reduces thermal resistance, facilitating efficient heat spreading in both lateral and vertical directions, while the enhanced airflow channels support convective cooling. Together, these effects demonstrate that the non-integer dimensional framework, particularly at higher fractal orders, provides markedly improved thermal dissipation performance.

3.4. Integration mechanisms for all-in-one multifunctionality

The preceding sections demonstrated the acoustic, mechanical, and fluid–thermal performances of fractal architectures with non-integer dimensionality. To extend beyond individual functionalities, this section elucidates the underlying integrated mechanism that unifies these responses within a single geometric framework. As illustrated in Fig. 6, the multifunctional performance can be rationalized through three cooperative mechanisms of energy flow manipulation: acoustic energy shielding, airflow energy transport, and mechanical energy absorption. These categories highlight how the coexistence of solid and air phases, governed by hierarchical self-similarity, enables the direct control of multiple energy transport pathways within a unified geometric framework, without relying on material heterogeneity or modular assembly.

Fig. 6 Integration mechanisms of energy flow manipulation in NDAMs. (A) Acoustic energy shielding enabled by multipath scattering and localized resonances in hierarchical cavities, facilitating broadband sound attenuation. (B) Airflow energy transport through interconnected multiscale pore networks, which facilitate ventilation and convective heat transfer. (C) Mechanical energy absorption realized by hierarchical stress redistribution in the fractal backbone, enabling progressive load transfer, impact buffering, and improved energy dissipation.

For the acoustic domain, as shown in Fig. 6(A), the self-similar cavity networks induce multipath scattering and localized resonances, as evidenced by the distributed particle-velocity fields and spatially varied acoustic pressure. These multiscale interactions enable broadband sound attenuation across multiple wavelength ranges, functioning as an intrinsic mechanism of acoustic energy shielding. For the fluid–thermal domain, as shown in Fig. 6(B), the interconnected pore channels provide hierarchical pathways for airflow and heat transfer. By redistributing fluid flux across multiple scales, the structure reduces global flow resistance and promotes smoother flow transitions. Meanwhile, the extended diffusion paths enhance convective cooling, enlarging low-temperature regions and suppressing thermal hotspots. These effects constitute an effective mechanism of airflow energy transport, combining ventilation with thermal dissipation. For the mechanical domain, as shown in Fig. 6(C), the fractal backbone forms a nested framework that guides stress redistribution across scales. Concentrated stresses are redirected along multiple hierarchical paths, facilitating adaptive load transfer from local features to global framework. This mechanical energy absorption mechanism enables a functional transition of the structure, from high stiffness and strength at lower orders to progressive energy absorption and impact buffering at higher orders.

Overall, these three mechanisms demonstrate how non-integer dimensionality intrinsically encodes multifunctionality purely through geometric design. By coupling acoustic, fluid-thermal, and mechanical energy flows within a single fractal framework, this strategy provides a scalable and integrative paradigm for all-in-one multifunctional materials.

4. Scientific advances and perspectives

The concept of non-integer dimensionality provides not only a mathematical foundation for embedding hierarchical complexity and self-similarity into geometry, but also a physical mechanism for coupling geometry with function across multiple domains. Fractal architectures with non-integer dimensionality inherently generate multiscale features that simultaneously manipulate wave propagation, stress redistribution, and fluid transport. As summarized in Fig. 7, this dimension-guided strategy reveals how a single geometric framework can encode intrinsic multifunctionality, offering a scalable route toward architected materials where diverse physical responses are governed by the same set of dimensional parameters.

Fig. 7 Comparison between integer- and non-integer dimensional design strategies for multifunctional metamaterials. (A) Contour plots show the performance distributions of four target functionalities: sound insulation, ventilation efficiency, load-bearing performance, and buffering capacity. These maps collectively establish a unified design space for tuning multiphysics functionalities using two governing topological descriptors, non-integer dimension D and fractal order i. (B) Integer-dimensional architected structures (e.g., lattice geometries) achieve multifunctionality by adding discrete modules, resulting in complex “many-to-many” hybrid designs. (C) NDAMs enable intrinsic multifunctionality through hierarchical topology governed by controllable fractal dimension, fractal order, and scaling factor, representing a “one-to-many” design paradigm. (D) Radar chart comparing the proposed fractal-based design with conventional architected structures.^44,60–62 (E) Demonstration of potential applications in architecture, highspeed railway, and automobile systems, where simultaneous sound insulation, ventilation, and impact resistance are required.

To quantify this principle, we systematically analyzed multifunctional responses across a range of non-integer dimensions, thereby revealing the inherent tunability and robustness of fractal architectures (see the SI, Section S9). As shown in Fig. 7(A), the multifunctional behavior can be consistently parameterized by two topological descriptors: the non-integer dimension D and the fractal order i. Within the explored range (2.35 ≤ D ≤ 2.73), these descriptors govern acoustic, mechanical, and fluid responses in a coherent and physically interpretable manner. Although the dataset includes only three discrete fractal orders per dimension, we introduced low-degree empirical scoring functions to interpolate performance trends in the (D,i) design space.⁵⁹ Rather than aiming for precise quantitative prediction, these functions capture essential dependencies, enabling interpretable design maps that highlight high-performance regions. This mapping demonstrates that by tuning only two governing parameters, targeted multifunctional optimization can be achieved without structural add-ons or compositional heterogeneity.

The resulting contour plots identify distinct high-performance regions, supporting targeted structure selection by modulating only two governing parameters: the non-integer dimension and fractal order. This dimensional mapping demonstrates that multifunctionality can be compactly encoded within a low-dimensional design space rather than accumulated through multiple structural modules. Such a perspective naturally raises the question of how this NDAM strategy compares with existing AMs in enabling multiphysics coupling. As shown in Fig. 7(B), conventional designs such as lattices or honeycombs are rooted in integer dimensions, where each structural element is defined within fixed Euclidean coordinates. To realize multiple functions such as noise control, energy absorption, or airflow enhancement, they typically adopt additive strategies, layering or combining separate modules to provide performance. This approach exemplifies the “many-to-many” mode, in which each physical function requires a distinct structural feature. The accumulation of discrete elements increases geometric redundancy, reduces integration efficiency, and elevates fabrication costs. More critically, it introduces incompatibilities across domains, underscoring the lack of an intrinsic, unified principle for multiphysics integration.

By contrast, this work adopts a dimensionally driven “one-to-many” design paradigm, in which NDAM architectures inherently integrate multiple functionalities without the need for additional specialized design features. As shown in Fig. 7(C), the fractal architecture derives its performance advantages from the tunability of non-integer dimensions, which intrinsically govern hierarchical porosity, mass distribution, and scaling behavior. This adjustable dimensionality enables precise control over physical responses across scales, offering a systematic approach to tailoring multifunctional performance within a unified topological framework. Unlike integer-dimensional designs that rely on adding separate functional elements, the physical behaviors here are directly governed by dimensional parameters, reflecting a shift from shape-based strategies to dimensionally driven material design. Importantly, this principle is not restricted to the Menger sponge fractal. Other non-integer dimensional geometries, such as Hilbert curves and Sierpinski-type architectures, follow the same fundamental logic: dimensionality acts as the governing variable that organizes connectivity, void topology, and hierarchical scaling. Despite their distinct morphologies, these structures share the capacity to translate fractional dimensionality into tunable pathways for wave, stress, and flow transport. This universality highlights that the NDAM paradigm provides more than individual structural prototypes; it establishes a transferable design framework that unifies diverse fractal families under a common multifunctional principle.

To validate this conceptual shift from “many-to-many” to “one-to-many,” we benchmark the proposed NDAM strategy against state-of-the-art designs. This difference in design philosophy is further quantified using the radar plot (as shown in Fig. 7(D)), which compares the overall multifunctional performance of the proposed design with representative state-of-the-art AMs across five key evaluation metrics: sound insulation, mechanical energy absorption, ventilation capability, fabrication feasibility, and intrinsic multifunctionality.^44,60–62 These metrics were selected to comprehensively reflect both core functional attributes (acoustic, mechanical, and fluid performance) and practical design considerations (manufacturing ease and integration level), thereby providing a balanced perspective on applicability and scalability. Detailed scoring and evaluation criteria for each category are provided in Section S10 of the SI. The proposed non-integer dimensional design achieves consistently high performance across all dimensions, highlighting its capacity to simultaneously satisfy functional demands and engineering constraints through a geometry-unified, intrinsically multifunctional strategy. From an application perspective, as shown in Fig. 7(E), the intrinsic multifunctionality of NDAM lends itself well to diverse real-world applications, such as architectural partition systems combining soundproofing and airflow, high-speed railway barriers mitigating aerodynamic impact while absorbing noise, and automotive components integrating impact absorption with thermal ventilation.

It is also worth noting that while the proposed non-integer dimensional design paradigm demonstrates remarkable intrinsic multifunctionality and scalability, several limitations should be acknowledged. The fabrication of high-order fractals remains challenging due to their complex multi-scale topology and fine geometric features. Additionally, the theoretical modeling and performance prediction of such fractal structures are often complicated by a geometrical complexity that transcends classical Euclidean frameworks and requires advanced computational strategies. Nevertheless, these challenges are not fundamental roadblocks but rather reflect the nascent stage of dimension-driven material design. Continued advances in high-resolution manufacturing, multi-physics simulation, and topological optimization are expected to mitigate these constraints. Overall, this work establishes non-integer dimensional geometry as a powerful, dimension-driven design paradigm that intrinsically couples multiple physical functionalities through a unified geometric framework. By shifting the basis of multifunctionality from additive complexity to scalable, dimensionally driven control, this paradigm offers a promising and fabrication-accessible pathway toward the next generation of high-performance materials, bridging mathematical theory and real-world engineering needs.

5. Conclusions

This study presents a dimensionally-driven strategy for multifunctional material design, leveraging the mathematical concept of non-integer fractal dimensions to intrinsically couple sound insulation, mechanical energy absorption, and airflow ventilation within a unified structure. Taking the Menger sponge as a representative example, we demonstrate how fractal architectures inherently generate multiscale features that govern physical responses across multiple domains. In acoustics, the self-similar hierarchy induces multi-bandgap behavior, enabling broadband sound attenuation via local resonance and wave scattering. In mechanics, stress redistribution and hierarchical deformation pathways enhance energy dissipation and impact resilience. For fluid dynamics, the fractal pore network improves ventilation and thermal dissipation by mitigating flow resistance and promoting uniform airflow via hierarchical multiscale channels. Rather than adding discrete modules to achieve different functions, the proposed design harnesses dimensional tunability as the underlying mechanism, shifting the paradigm from geometry-driven assembly to dimensionally governed integration. This mathematical-to-physical mapping expands the design space beyond traditional integer dimensional frameworks, offering a scalable and fabrication-efficient approach to multifunctional integration.

While promising, this dimension-driven paradigm also presents emerging challenges, including fabrication difficulties at high fractal orders and the need for advanced modeling tools to capture geometrical complexities that transcend classical Euclidean frameworks. Nonetheless, these limitations reflect the early stage of this approach rather than inherent constraints. The potential of non-integer dimensionality as a generalizable design framework will further unfold and bridge mathematical principles and real-world engineering needs.

6. Experimental work

Design and fabrication: detailed computer-aided design (CAD) models were developed in SolidWorks and then converted into STL files for additive manufacturing. The fractal designs consist of various pores at millimeter or submillimeter scales. Therefore, digital light processing 3D printing technology is employed, as it is often used to produce fine components for such applications due to its high precision and efficiency. The curable resin used in this study is standard resin from Nova 3D, China. After the specimens are printed, the raw samples are first thoroughly washed with isopropyl alcohol (IPA) for approximately 20 minutes to remove excess resin and then dried using a high-power blower. Once fully dried, the samples are placed in an ultraviolet (UV) box and post-cured for 30 minutes under exposure to light with a wavelength of 405 nm emitted by an LED lamp.

Quasi-static compression tests: these were performed using an MTS Insight 30 universal testing machine. The samples were placed on a rigid horizontal platform and compressed vertically at a constant displacement rate of 1 mm min⁻¹. The reaction force and displacement were recorded in real time via built-in sensors. Meanwhile, a professional camera was employed to capture the deformation process, enabling detailed analysis of the deformation modes and energy absorption behavior of the structures.

Acoustic insulation testing: to evaluate sound insulation performance, 3 × 3 arrays of first- to third-order samples were fabricated and tested. The experimental setup consisted of a rigid square acoustic tube (81 mm × 81 mm cross-section, constructed from 5 mm-thick acrylic plates), within which the samples were tightly fitted. A loudspeaker was placed at the inlet of the tube, while a sound level meter was positioned at the outlet to measure the transmitted SPL. The A-weighted SPL, which closely reflects human auditory perception, was used to evaluate acoustic performance across frequency ranges.

Air ventilation testing: ventilation performance was assessed using a duct-based airflow measurement setup. First- to third-order samples were inserted into a straight test tube, with an electric fan installed upstream to generate airflow at a constant inlet velocity of 5.6 m s⁻¹. An anemometer was mounted at the downstream outlet to measure the exit airflow velocity, enabling quantitative evaluation of ventilation efficiency and flow resistance associated with each structure.

Numerical simulation: the acoustic simulations were conducted in COMSOL Multiphysics to analyze both the band structure and sound insulation performance of the proposed design. For bandgap calculations, the pressure acoustics module was employed to compute the sound pressure field and eigenfrequencies under steady-state harmonic excitation, following the linear acoustic equation

where p is the acoustic pressure, ρ₀ is the air density, c is the speed of sound, ω is the angular frequency, and r = (x, y, z) is the position vector. Periodic boundary conditions based on Bloch's theorem were applied as:

p(r + a) = p(r)e^i(k·a) (4)

and the corresponding eigenfrequencies were extracted to obtain the acoustic band structures. For sound transmission loss analysis, both the pressure acoustics and thermoviscous acoustics modules were employed to account for the thermoviscous losses and boundary-layer damping within the narrow multiscale channels. The detailed simulation procedures and results are provided in Section S3, SI. To analyze the mechanical performance, the specimen was modeled using 8-node solid elements to account for large deformation behavior. Material properties were defined with an ideal elastic–plastic constitutive model, characterized by a density of 1270 kg m⁻³, a Young's modulus of 1008 MPa, a Poisson's ratio of 0.33, and a yield strength of 40 MPa. The specimen was supported by a rigid plate at the bottom, while a rigid plate at the top applied a uniform downward velocity of 0.05 m s⁻¹ to simulate quasi-static compression. Normal contact was defined as ‘hard contact’, while tangential contact was modeled using the ‘penalty’ method with a friction coefficient of 0.2. To evaluate the ventilation performance, steady-state airflow simulations were performed based on the Reynolds-averaged Navier–Stokes (RANS) equations, incorporating the Shear stress transport (SST) k–ω turbulence model.⁶³ This formulation inherently includes the effects of air viscosity and viscous drag, enabling accurate prediction of pressure loss and flow resistance within the narrow multiscale channels. The SST k–ω model effectively captures both near-wall behavior and free-stream turbulence, making it suitable for resolving multi-scale flow features in the proposed design. The computational domain was discretized using a structured Cartesian mesh, with local refinement applied in regions exhibiting strong velocity gradients, particularly within the hierarchical pores of the fractal architectures. This mesh adaptation strategy ensured accurate resolution of airflow behavior while maintaining computational efficiency.

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Supplementary information (SI) is available. See DOI: https://doi.org/10.1039/d5mh01768h.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (52505299), the Hunan Provincial Natural Science Foundation of China (2023JJ10074), and the science and technology innovation Program of Hunan Province (2023RC1011).

References

1. W. Liu, S. Janbaz, D. Dykstra, B. Ennis and C. Coulais, Nature, 2024, 634, 842–847.
2. P. Jiao, J. Mueller, J. R. Raney, X. Zheng and A. H. Alavi, Nat. Commun., 2023, 14, 6004.
3. L. R. Mezaa, A. J. Zelhofera, N. Clarkeb, A. J. Mateosa, D. M. Kochmanna and J. R. Greera, Proc. Natl. Acad. Sci. U. S. A., 2015, 112(37), 11502–11507.
4. J. Chen, J. Xiao, D. Lisevych, A. Shakouri and Z. Fan, Nat. Commun., 2018, 9, 4920.
5. L. Wu, Y. Wang, Z. Zhai, Y. Yang, D. Krishnaraju, J. Lu, F. Wu, Q. Wang and H. Jiang, Appl. Mater. Today, 2020, 20, 100671.
6. G. Lee, W. Choi, H. M. Seung, S. Kim and J. Rho, Proc. Natl. Acad. Sci. U. S. A., 2025, 122(18), e2425407122.
7. P. Fu, Z. Xu, T. Zhou, H. Li, J. Wu, Q. Dai and Y. Li, Nat. Commun., 2024, 15, 6258.
8. L. Huang, S. Huang, C. Shen, S. Yves, A. S. Pilipchuk, X. Ni, S. Kim, Y. K. Chiang, D. A. Powell, J. Zhu, Y. Cheng, Y. Li, A. F. Sadreev, A. Alù and A. E. Miroshnichenko, Nat. Rev. Phys., 2024, 6, 11–27.
9. J. Park, G. Lee, H. Kwon, M. Kim and J. Rho, Adv. Funct. Mater., 2024, 34(39), 2403550.
10. G. Lee, W. Choi, B. Ji, M. Kim and J. Rho, Adv. Sci., 2024, 11(19), 240009.
11. R. Xu, C. Chen, J. Sun, Y. He, X. Li, M. Lu and Y. Chen, Int. J. Extreme Manuf., 2023, 5, 042013.
12. J. Chen, B. Liu, G. Peng, L. Zhou, C. Tan, J. Qin, J. Li, Z. Hong, Y. Wu, M. Lu, F. Cai and Y. Huang, Adv. Sci., 2025, 12, 2500170.
13. Y. Tian, X. Zhang, B. Hou, A. Jarlöv, C. Du and K. Zhou, Proc. Natl. Acad. Sci. U. S. A., 2024, 121(43), e2407362121.
14. J. Guo, Y. Fang, R. Qu and X. Zhang, Mater. Today, 2025, 66(4), 321–338.
15. Z. Guo, Z. Li, K. Zeng, J. Ye, X. Lu, Z. Lei and Z. Wang, Adv. Mater. Technol., 2024, 2400934.
16. Z. Li, X. Wang, S. Ramakrishna, Y. Lu, Z. Wang and L. Cheng, Mater. Today, 2025, 89, 151–171.
17. C. Liu, J. Shi, W. Zhao, X. Zhou, C. Ma, R. Peng, M. Wang, Z. H. Hang, X. Liu, J. Christensen, N. X. Fang and Y. Lai, Phys. Rev. Lett., 2021, 127, 084301.
18. Y. Mi, W. Zhai, L. Cheng, C. Xi and X. Yu, Appl. Phys. Lett., 2021, 118, 114101.
19. J. E. Hutchinson, Fractals and Self Similarity, Indiana Univ. Math. J., 1981, 30, 713–747.
20. Y. Liu, W. Xu, M. Chen, D. Pei, T. Yang, H. Jiang and Y. Wang, Mater. Res. Express, 2019, 6, 116211.
21. H. Zhong, V. O. Kompanets, Y. Zhang, Y. V. Kartashov, M. Cao, Y. Li, S. A. Zhuravitskii, N. N. Skryabin, I. V. Dyakonov, A. A. Kalinkin, S. P. Kulik, S. V. Chekalin and V. N. Zadkov, Light: Sci. Appl., 2024, 13, 264.
22. Z. Guo, Z. Li, K. Zeng, X. Lu, J. Ye and Z. Wang, Phys. Rev. B: Condens. Matter Mater. Phys., 2024, 109, 104113.
23. Z. Guo, Z. Li, K. Zeng, X. Lu, J. Ye and Z. Wang, Mater. Des., 2024, 241, 112943.
24. B. Kushwaha, K. Dwivedi, R. S. Ambekar, V. Pal, D. P. Jena, D. R. Mahapatra and C. S. Tiwary, Adv. Eng. Mater., 2021, 23, 2001471.
25. Z. Li, K. Zeng, Z. Guo, Z. Wang, X. Yu, X. Li and L. Cheng, Adv. Funct. Mater., 2024, 2420207.
26. Z. Li, X. Wang, K. Zeng, Z. Guo, C. Li, X. Yu, S. Ramakrishna, Z. Wang and Y. Lu, NPG Asia Mater., 2024, 16, 45.
27. Z. Guo, L. Du, Z. Yang, K. Zeng, Z. Lei, Z. Wang, Z. Li and Z. Fan, Adv. Sci., 2025, e08951.
28. X. Li, X. Yu, M. Zhao, Z. Li, Z. Wang and W. Zhai, Adv. Funct. Mater., 2022, 2210160.
29. X. Li, Y. Cheng, Y. Zhou, L. Shi, J. Sun, G. W. Ho and R. Wang, Adv. Mater., 2024, 2406714.
30. Y. Liu, Y. Wang, H. Ren, Z. Meng, X. Chen, Z. Li, L. Wang, W. Chen, Y. Wang and J. Du, Nat. Commun., 2024, 15, 2984.
31. G. Lee, H. Kang, J. Yun, D. Chae, M. Jeong, M. Jeong, D. Lee, M. Kim, H. Lee and J. Rho, Nat. Commun., 2024, 15, 6537.
32. G. Lee, S.-J. Lee, J. Rho and M. Kim, Mater. Today Energy, 2023, 37, 101387.
33. G. Lee, J. Park, W. Choi, B. Ji, M. Kim and J. Rho, Mech. Syst. Signal Process, 2023, 200, 110593.
34. Y. Zhang, W. Z. Jiang, Y. Pan, X. C. Teng, H. H. Xu, H. Yan, X. H. Ni, J. Dong, D. Han, W. Q. Chen, J. Yang, Y. M. Xie, Y. Lu and X. Ren, Sci. Adv., 2024, 10, eads0892.
35. R. Herrmann, Chaos, Solitons Fractals, 2015, 70, 38–41.
36. V. Karthikeyan, J. U. Surjadi, X. Li, R. Fan, V. C. S. Theja, W. J. Li, Y. Lu and V. A. L. Roy, Nat. Commun., 2023, 14, 2069.
37. C. Zhu, H. B. Gemeda, E. B. Duoss and C. M. Spadaccini, Adv. Mater., 2024, 2314204.
38. T. Gao, K. Liu, Q. Ma, J. Ding, Z. Hu, K. Wei, X. Song, Z. Li and Z. Wang, Virtual. Phys. Prototyping, 2025, 20, 1.
39. T. Ma, X. Li, X. Tang, X. Su, C. Zhang and Y. Wang, J. Sound Vib., 2022, 536, 117115.
40. X. Man, T. Liu, B. Xia, Z. Luo, L. Xie and J. Liu, J. Sound Vib., 2018, 423, 322–339.
41. H. Ruan and D. Li, Eng. Struct., 2024, 308, 118003.
42. J. Liu, L. Li, B. Xia and X. Man, Int. J. Solids Struct., 2018, 132–133, 20–30.
43. S. Cheng, X. Li, Q. Zhang, Y. Sun, Y. Xin, Q. Yan, Q. Ding and H. Yan, Nanostruct., 2024, 61, 101289.
44. D. P. Elford, L. Chalmers, F. V. Kusmartsev and G. M. Swallowe, J. Acoust. Soc. Am., 2011, 130, 2746–2755.
45. D. Zong, W. Bai, M. Geng, X. Yin, J. Yu, S. Zhang and B. Ding, ACS Nano, 2022, 16, 13740–13749.
46. C. Wu, X. Yuan, Y. Ye, L. Wang, H. Li, C. Gao, Y. Huang and X. Wu, J. Phys. D: Appl. Phys., 2024, 57, 295501.
47. N. Viet, S. Ilyas and W. Zaki, Int. J. Solids Struct., 2024, 288, 112591.
48. H. Yin, W. Zhang, L. Zhu, F. Meng, J. Liu and G. Wen, Compos. Struct., 2023, 304(1), 1–40.
49. V. Rai, H. Ghasemnejad, J. Watson, J. Gonzalez-Domingo and P. Webb, Eng. Struct., 2019, 199, 109620.
50. Z. Lei, J. Liu, Z. Guo, J. Ye, H. Di, F. Gao and Z. Wang, Eng. Struct., 2025, 341, 120827.
51. M. Zhao, H. Qing, Y. Wang, J. Liang, M. Zhao, Y. Geng, J. Liang and B. Lu, Mater. Des., 2021, 200, 109448.
52. Z. Wang, Z. Lei, Z. Li, K. Yuan and X. Wang, Thin-Walled Struct., 2021, 167, 108235.
53. M. Bogahawaththa, D. Mohotti, P. Hazell, H. Wang, K. Wijesooriya and C. Lee, Thin-Walled Struct., 2025, 206, 112704.
54. J. Wang, C. Du and Y. Wang, Case. Stud. Therm. Eng., 2021, 28, 101387.
55. P. Raphe, H. Fellouah, S. Poncet and M. Ameur, Build. Environ., 2021, 204, 108118.
56. X. Wang, Q. Wang, S. Yoo, T. Gomyo, H. Sotokawa, J. Chung, K. Kuga and K. Ito, Build. Environ., 2024, 262, 111773.
57. X. Yuan, Q. Li, C. Wu, Y. Huang and X. Wu, Extreme Mech. Lett., 2024, 66, 102119.
58. Y. Zhu, R. Dong, D. Mao, X. Wang and Y. Li, Phys. Rev. Appl., 2022, 19, 014067.
59. A. Bucciarelli, S. Chiera, A. Quaranta, V. K. Yadavalli, A. Motta and D. Maniglio, Adv. Funct. Mater., 2019, 29, 1901134.
60. X. Li, M. Zhao, X. Yu, J. W. Chua, Y. Yang, K. M. Lim and W. Zhai, Mater. Des., 2023, 234, 112354.
61. Z. Wang, Composites, Part B, 2019, 166, 731–741.
62. Z. Xu, H. Gao, Y. Ding, J. Yang, B. Liang and J. Cheng, Phys. Rev. Appl., 2020, 14, 054016.
63. Y. Zhang, D. Zhou, J. Feng, R. Xue and G. Chen, Phys. Fluids, 2024, 36, 105194.

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Footnote

† Z. C. Guo and Z. P. Lei contributed equally to this work.

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