Unprecedented mechanical wave energy absorption observed in multifunctional bioinspired architected metamaterials

Abstract

In practical engineering, noise and impact hazards are pervasive, indicating the pressing demand for materials that can absorb both sound and stress wave energy simultaneously. However, the rational design of such multifunctional materials remains a challenge. Herein, inspired by cuttlebone, we present bioinspired architected metamaterials with unprecedented sound-absorbing and mechanical properties engineered via a weakly-coupled design. The acoustic elements feature heterogeneous multilayered resonators, whereas the mechanical responses are based on asymmetric cambered cell walls. These metamaterials experimentally demonstrated an average absorption coefficient of 0.80 from 1.0 to 6.0 kHz, with 77% of the data points exceeding the desired 0.75 threshold, all with a compact 21 mm thickness. An absorptance-thickness map is devised for assessing the sound-absorption efficiency. The high-fidelity microstructure-based model reveals the air friction damping mechanism, with broadband behavior attributed to multimodal hybrid resonance. Empowered by the cambered design of cell walls, metamaterials shift catastrophic failure toward a progressive deformation mode characterized by stable stress plateaus and ultrahigh specific energy absorption of 50.7 J/g—a 558.4% increase over the straight-wall design. After the deformation mechanisms are elucidated, a comprehensive research framework for burgeoning acousto-mechanical metamaterials is proposed. Overall, our study broadens the horizon for multifunctional material design.

Introduction

Acoustic and stress waves serve as carriers of mechanical wave energy^1,2. The prevalence of noise and the potential hazards posed by impacts in practical engineering necessitate the pursuit of materials adept at absorbing such energy. The crux lies in the simultaneous optimization of the sound-absorbing and mechanical properties of these materials^3,4,5. For example, in aerospace engineering, sound-absorbing liners must not only dampen noise but also provide structural resilience against potential local impacts^6,7. The need for lightweight multifunctional materials drives innovation. Unlike traditional methods that use separate materials, multifunctional materials integrate functions, reducing the system size and weight for increased efficiency. However, creating ideal multifunctional materials that are lightweight yet strong, energy-absorbing, and effective in sound attenuation remains a major challenge in materials science. In simpler terms, achieving this requires a new approach to material design principles.

In the realm of advanced materials design, mimicking natural biological structures has emerged as a prevailing concept⁸. Additive manufacturing techniques have led to the rise of bioinspired metamaterials, a category of engineered materials with unique properties that are not typically found in nature⁹. In recent decades, researchers have diligently pursued innovative approaches aimed at increasing the acoustic absorption efficacy of conventional materials, including perforated panels, foams, and fabrics. Consequently, a plethora of novel sound-absorbing metamaterials^10,11, alongside advanced porous materials exemplified by aerogels¹² and graphene foams¹³, have emerged as promising avenues for exploration. More recently, microlattices have emerged as a focal point of research interest because of their potential to mitigate these challenges. However, researchers find themselves ensnared in a conundrum, striving to strike a delicate balance between high strength and broad-spectrum sound absorption^4,14. Therefore, the attainment of materials that embody the desired characteristics without compromising either acoustic or mechanical integrity necessitates a new design approach.

Henceforth, the pursuit of an ideal absorber for mechanical wave energy is initiated from a direct concept—leveraging the inherent mechanical robustness of bioinspired materials and introducing sound-absorption capabilities via architectural design. In this context, the microstructure of cuttlebone emerges as a compelling candidate because of its inherent strength, toughness, and adaptable cellular features^15,16. While prior investigations into cuttlebone-inspired materials have focused predominantly on enhancing mechanical strength through the utilization of sinusoidal undulating walls^9,17, two critical aspects have been overlooked: 1) the multilayered “wall-septa” structure, which presents untapped potential for the design of sound absorbers, and 2) the undervalued asymmetric cambering pattern of cell walls, which is important in mechanics. Structurally, cuttlebone is composed of undulating cell walls encased by plates arranged in layers. The introduction of surface porosity to the plates facilitates air exchange between layers, thereby enabling dissipation mechanisms through frictional loss at the pores. Furthermore, the internal partitions within the cuttlebone serve as inspiration for integrating heterogeneous cells into resonators, thus enhancing the sound-absorbing performance via multimodal resonance^11,18. Concurrently, to improve the mechanical properties, the overlooked cambering structural attribute is meticulously analyzed in our study. The aim of this analysis is to harness the innate biological advantages of this feature in mechanics. Consequently, the proposed metamaterials are anticipated to exhibit broadband sound absorption, high strength, and damage tolerance.

The multifunctional bioinspired architected metamaterials, termed MBAMs, are fabricated via selective laser melting (SLM) additive manufacturing with high-strength Ti6Al4V alloy. This material is chosen for its practical engineering benefits: it provides high strength, corrosion resistance, and high-temperature performance, contributing to stable sound absorption. Experimental tests revealed a notably high average absorption coefficient $\alpha$ of 0.80 from 1.0 to 6.0 kHz, with 77% of the data points surpassing a threshold of 0.75, despite its remarkably compact thickness of 21 mm. This milestone marks a departure from the trade-off between absorptance and thickness observed in all sound-absorbing materials. Moreover, these metamaterials exhibit exceptional mechanical properties, with an average modulus of 4.93 GPa, strength of 211 MPa, and a specific energy absorption of 50.7 J/g at a low density of 1.53 g/cm³. Overall, the metamaterial epitomizes a lightweight multifunctional wave energy absorbing material, exhibiting thin yet remarkable sound-absorptive capabilities, high strength, highly efficient energy absorption, and damage tolerance. This study introduces a compelling design paradigm to engineer multifunctional materials with desirable acoustic and mechanical properties.

Design principle

Natural microscopic features

As depicted in Fig. 1Ai, the cuttlebone is situated dorsally within the cuttlefish anatomy. This structure features a sturdy dorsal shield enveloping its dorsal aspect, beneath which lies a porous chambered arrangement. The characteristic “wall–septa” microstructure of cuttlebone was revealed by scanning electron microscopy (SEM), as shown in Fig. 1Aii¹⁶. Each chamber is partitioned into individual chambers by horizontal septa, and each chamber is supported by vertical walls. These walls exhibit an undulating pattern reminiscent of sinusoidal waves. As such, this intricate multilayered microstructure is mimicked in the design of high-strength mechanical metamaterials^9,17. Nevertheless, there is an underappreciated microscopic characteristic of cuttlebone: the growing camber of vertical walls from the bottom to the top, as illustrated in an SEM image of a sectioned cuttlebone from the side view (Fig. 1Bi). Disregarding this nonlinear camber feature in metamaterial design may simplify the modeling and analysis, but doing so would compromise the biomechanical advantages conferred by this feature. For clarity, this biological feature in Fig. 1Bi is reconstructed in Fig. 1Bii. The curvature of the vertical walls undergoes gradual abnormal alteration. The evolving camber morphology leads to a distinctive cross-sectional profile for each wall, which essentially conforms to a specific nonlinear relationship derived from the cuttlefish itself, as described by Eq. (1)¹⁹

$$P=\left(0.998+0.002{e}^{6.55\left(\frac{H}{{H}_{0}}-0.11\right)}\right){P}_{0}$$

(1)

where ${H}_{0}$ and ${P}_{0}$ represent the chamber height and the cross-sectional length close to the bottom profile, respectively, and where $P$ is the cross-sectional length at a height $H$.

Fig. 1: Design principle.

Ai A cuttlefish with a cuttlebone. The “S” direction points from the dorsal to the ventral side. Aii A widely explored microstructure feature involving a multilevel wall-septum with horizontally undulating walls. Reproduced with permission¹⁶. Copyright 2022, Elsevier. Bi SEM image of the side view of the cuttlebone, showing the growth of the cell walls from the bottom to the top. Bii Reconstructed profiles of cell walls, further revealing the cambered pattern. Biii The wall length $P/{P}_{0}$ against the height $H/{H}_{0}$ according to Eq. (1), which mathematically characterizes the nonlinear increase in wall length. Ci Cascaded resonant plate design inspired by the multilayered wall-septum microstructure, with dissipative pores introduced. Cii Heterogeneous design based on cascaded resonance, dividing the metamaterial into two parallel parts (Part 1 and Part 2). Ciii The air phase of the unit cell, with the geometries governed by the solid cell walls. Di Schematic of the metamaterial and the cambered cell walls. Dii Morphology of cell walls with four camber levels: straight-wall and A₀ = 0.5, 1.0, and 1.5. E Weakly-coupled design recipe: elements/units significant for absorbing sound wave energy but negligible for stress wave energy, and vice versa.

In Fig. 1Biii, the plot displays the relationship between the $P/{P}_{0}$ and $H/{H}_{0}$ coordinates based on Eq. (1), depicted by the black dotted curve. Notably, the length of the cell wall profile increases gradually without apparent changes until $H/{H}_{0}$ reaches approximately 0.5. Once beyond this point, the growth trend becomes sharp, shaping a curve reminiscent of a parabola and reaching its maximum curvature.

Heterogeneous acoustic absorbers

Upon revisiting the local resonance theory, the multilayered septum and chambers in Fig. 1Aii are found to offer favorable conditions for designing resonant structures. Thus, we designed three cascaded panels by introducing sound-dissipative pores into the panels (see Fig. 1Ci). Owing to the critical effect of dissipative pores, we conceptualized a heterogeneous arrangement for highly efficient sound absorption (Fig. 1Cii) based on the coherent coupling theory. Each resonant panel within every layer is divided into two parts, labeled Part 1 and Part 2, with a consistent pore size and arrangement maintained within each segment across layers. Consequently, this heterogeneous metamaterial is acoustically divided into six segments (Supplementary Fig. S1A). The pore diameter is labeled ${d}_{p,l}$, with the subscripts $p$ and $l$ representing the specific parallel part and cascaded layer, respectively ($p$ ≤ 2; $l$ ≤ 3). By utilizing this heterogeneity, we create a multimodal hybrid resonance system comprising two parallel circuits, each containing three series circuits. This configuration is visually represented in an electroacoustic analogy map (Supplementary Fig. S1B).

The air phase within a metamaterial cell is illustrated in Fig. 1Ciii, with its geometry determined by the cell walls shown in Fig. 1Di. These cell walls are strictly designed to mimic the natural curvature of the cuttlebone, adhering to a nonlinear relationship described by Eq. (1), and they also conform horizontally (in the x‒y plane) to a sinusoidal pattern defined by the function in Eq. (2).

$$y={A}_{0}\sin \frac{2}{3}x$$

(2)

where ${A}_{0}$ represents the bottom profile length of the unit cell. Therefore, the length of a metamaterial unit corresponds to one period ($T$) of the function, which measures 3 mm in the x direction (Supplementary Fig. S1C). The resonant panels of each unit cell contain four pores, delineated by “sinusoidal wave” patterns, positioned alternately in adjacent cell walls (Fig. 1Ciii). The effective sound-absorbing characteristics of Helmholtz resonators are governed by their acoustic impedance, a property inherently reliant on geometry. Specifically, in addition to pore diameter, pore thickness ${t}_{p}$, cavity depth $D$, and surface porosity $\sigma$ are three other significant factors (Supplementary Fig. S1D).

To achieve optimal resonance, an impedance model of the metamaterial is established, and the pore diameter is optimized via a genetic algorithm. Considerations for determining other acoustic parameters are provided in Supplementary Section S1, Supplementary Material. The sound absorption coefficient ($\alpha$) under an incident sound wave is calculated by

$$\alpha =1-{\left|\frac{{Z}_{s}/{Z}_{0}-1}{{Z}_{s}/{Z}_{0}+1}\right|}^{2}$$

(3)

where ${Z}_{0}$ represents the characteristic impedance of air, which is calculated as ${Z}_{0}={\rho }_{0}{c}_{0}$; ${Z}_{s}$ represents the total surface impedance in this multimodal resonance system.

Cambered cell walls

The mechanical properties of the proposed metamaterials are intricately tied to the design of their cell walls. Given a specific ${A}_{0}$ in Eq. (2), a profile length corresponds to a particular height. Combining this with Eq. (1) results in a horizontally undulating and vertically cambered pattern (Fig. 1Di). Here,$\,{t}_{{total}}$ represents the thickness of the present compact metamaterial (21 mm). Other parameters include the unit height ${H}_{0}$, which also serves as the resonant cavity depth $D$; the original wall thickness ${{t}}_{0}$; and the “period” of the cell walls $T$. The detailed geometric values are given in Supplementary Table S1. ${A}_{0}$ and ${A}_{T}$ denote the initial bottom and top amplitudes of the asymmetric walls, respectively. Herein, four camber-level cases are examined: straight-wall and A₀ = 0.5, 1.0, and 1.5 (Fig. 1Dii). The wall thickness of these configurations is varied to maintain a constant relative density of 33%.

Weakly-coupled design recipe

As discussed above, we conceptualized a “weakly-coupled” design approach to attain the desired multifunctionality, by which the acoustic and mechanical units could be almost independently designed (Fig. 1E). Thus, this design approach allows for a high degree of customization in terms of sound and stress wave energy absorption performance. Specifically, the geometries and arrangement patterns of dissipative pores in resonant panels govern the sound-absorptive performance^10,11. However, their influence on axial mechanical properties remains negligible: that is, they have minimal impact on stress wave absorption. Furthermore, the cell walls act as the basic units for absorbing stress wave energy, and their shapes, as determined by Eqs. (1) and (2), govern the shapes of the air phases; however, in such an isovolumetric case, the shapes of the air phases have a minimal impact on sound absorption^18,20,21. We term this a “weakly-coupled” design rather than an “independent” design based on the degree of coupling intensity, as the functionalities are not entirely independent. For example, if the resonant plate, which is crucial for sound absorption, is significantly thicker than the cell walls are, it can affect the mechanical response.

With the assistance of advanced additive manufacturing technology, the MBAM samples were successfully fabricated. Figure 2A offers a concise overview of the selective laser melting (SLM) fabrication method utilized to manufacture these metamaterials. In Fig. 2B, the untested MBAM samples are displayed, showing varying levels of camber, with each group being fabricated in triplicate. The geometries and densities of the as-printed samples are given in the Supplementary Material. All the samples consisted of 3 × 3 × 3 cells measuring 21 mm in the x, y, and z directions.

Fig. 2: Experimental sound-absorptive performance.

A Schematic illustration of the SLM 3D printing technique. B As-printed samples and Ci sound absorption curves of one representative set of straight-wall structures and MBAMs. Cii Comparison of the average sound absorption and Ciii bandwidth with $\alpha$ greater than 0.75 within 1–6 kHz for all the samples. D A benchmark map showing the tradeoff between the average sound absorption ${\alpha }_{a{ve}}$ and total thickness, revealing the unprecedented absorptance-thickness overall performance of the MBAMs compared with that of structured aerogels³³, flexible aerogels³⁴, phononic crystals³⁵, hybrid lattices⁵, interpenetrating lattices³, hierarchical plate lattices³⁶, ABH lattices³⁷, multiorder microperforated panels (MPP)²⁶, non-local absorbers¹¹, truss absorbers³⁸, TPMS absorbers³⁹, recoverable absorbers¹⁴, and heterogeneous lattices¹⁷.

Materials and methods

Fabrication of the samples

The samples are manufactured via the SLM technique with Ti6Al4V as the base material by an XDM 750 machine (China). SLM processing is carried out in a vacuum environment using particles smaller than 53 μm. The disk laser is operated at 500 W, while the printing layer thickness is set at 60 μm, with a scanning speed of 1200 mm/s. The materials are then wire-cut from the substrate, with any residual powders removed. The material parameters are determined via tensile tests conducted on an Instron 8501 apparatus (Supplementary Fig. S3A), with the stress‒strain curves detailed in Supplementary Fig. S3B. The material parameters include a Young’s modulus ${E}_{s}$ of 104 GPa, initial yield strength ${\sigma }_{0}$ of 864 MPa, Poisson’s ratio $\nu$ of 0.3, and density ${\rho }_{s}$ of 4.43 g/cm³.

Sound absorption measurements

A standard two-microphone setup (SKC ZT13) in accordance with ISO 10534-2 standards is utilized for sound absorption coefficient measurements (Supplementary Fig. S4). The samples are rigidly backed in a sample holder with a diameter of 30 mm. The small tube provides an effective test frequency ranging from 0.8 kHz to 6.3 kHz. The cubic samples are wrapped by four rigid polymeric adapters to cater to the circular tube. The effects of the solid adapters on the measurements are neglected. All reported absorption coefficients are averaged from three datasets.

Mechanical property measurements

Compression experiments are executed with a Shimadzu AG25-TB universal testing machine. The samples are subjected to axial quasistatic loading at a strain rate of 0.002 s^–1. Compression is directed along the parallel pair of faces aligned with the SLM building direction. The materials are centrally positioned on the plate and compressed unidirectionally by a mobile rigid plate until densification is achieved. Deformation modalities are documented via a digital camera. The experimental findings are methodically presented as three sets of averaged results supplemented with their corresponding standard deviations.

Finite element method

Concerning the sound-absorbing characteristics, we develop a finite element (FE) model via COMSOL Multiphysics. We utilize the Thermo-viscous Acoustics Module incorporating built-in boundary layer theory, as depicted in Supplementary Fig. S5. The primary physical equations built in the Thermo-viscous Module are presented in Supplementary Eqs. (S5)–(S7). The incident sound waves are propagated via the pressure-acoustics module, whereas a perfectly matched layer (PML) is implemented to simulate nonreflecting infinite fields above the model domain. For the compression model, we adopt the same geometries of the presented multicell structures to simulate their plastic large-strain deformation. The validation of this model via stress‒strain curves is illustrated in Supplementary Fig. S6, with modeling details given in the Supplementary Material.

Results and discussion

Ultrahigh, broadband, and efficient sound-absorptive properties

Figure 2Ci shows the measured coefficients of a representative group of samples. Evidently, MBAMs demonstrate superior ability in absorbing sound wave energy, characterized by a high $\alpha$ with two resonant peaks. The average sound absorption coefficients ${\alpha }_{a{ve}}$ are all greater than 0.77 within 1.0–6.0 kHz, and the maximum ${\alpha }_{a{ve}}$ reaches 0.80 (A₀ = 0.5), with 77% of the data points surpassing 0.75. The absorption measurements of all the samples are compared in Supplementary Fig. S7; despite differences in initial amplitude, they exhibit similar values and trends. Remarkably, each group of samples not only exhibits excellent test repeatability but also demonstrates very similar absorptance levels despite variations in the camber level of the cell walls. As illustrated in Fig. 2Cii, the average $\alpha$ values among these groups are closely aligned, with a relative error ranging within 1.7%. The bandwidth with $\alpha$ exceeding 0.75 (${B}_{0.75}$) for the MBAMs also remains consistent across different camber levels (Fig. 2Ciii). This finding suggests that, for these metamaterials with identical cavity volumes of air phases (Fig. 1Ciii), variations in their configurations have negligible influence on the acoustic absorption, which conforms to resonance theories^4,22. What truly matters are the factors within the theoretical model. For example, the thickness of the resonant plate ${t}_{p}$ and the air cavity height $D$ noticeably influence the absorption spectrum, particularly the bandwidth (Supplementary Fig. S8).

With respect to the sound-absorption efficiency, two critical factors should be considered: 1) absorptance, generally characterized by ${\alpha }_{a{ve}}$²³, and 2) the total thickness of the sample^24,25. However, past studies have not considered these factors simultaneously. Here, we introduce an innovative approach by creating a map that depicts the relationship between absorptance and material thickness. This map helps illustrate the overall efficiency of sound wave energy absorption in metamaterials. In Fig. 2D, the horizontal axis represents the total thickness of the samples, whereas the vertical axis represents the ${\alpha }_{a{ve}}$ calculated within the range of 1.0–6.0 kHz. Although there is no linear relationship between the average $\alpha$ and total thickness, a dotted line representing the general trend can be derived, as shown in the map.

Notably, the presented metamaterials exhibit an ultrahigh ${\alpha }_{a{ve}}$ ranging from 0.77 to 0.8, achieved with a remarkably compact thickness of 21 mm, far surpassing the general trend. In contrast, most reported metamaterial absorbers cluster around the 30-mm-thick region, with many cellular structures that are not specifically designed for sound absorption, resulting in a lower ${\alpha }_{a{ve}}$ below 0.6. We henceforth refer to these structures as qualified absorbers. The nonlocal¹¹ and multiorder MPP²⁶ absorbers represent two prominent examples of thick absorbers known for their ultrabroadband high absorption, which extends even into the low-frequency region. However, their large thicknesses somewhat limit their engineering applicability. Advanced materials such as structured aerogels cover a relatively large area on the map, ranging from approximately 10 to 50 mm in thickness. These materials offer an ${\alpha }_{a{ve}}$ of approximately 0.75. Similarly, advanced lattice absorbers, including hybrid and interpenetrating lattice structures, also surpass the general trend. Despite their small thicknesses, typically approximately 20 mm, they can also achieve an ${\alpha }_{a{ve}}$ of 0.75. Overall, our groundbreaking MBAMs exhibit unprecedented sound wave energy absorptance, achieving remarkable efficacy with a thickness of a mere 21 mm. The remarkable sound-absorption performance of these materials surpasses that of existing advanced sponges and metamaterial absorbers.

Physical mechanisms

The presented MBAMs comprise two parallel resonant systems, each intricately woven with three cascaded resonant cells (Fig. 3A). Remarkably, each cavity type within this ensemble contributes to a distinct surface impedance, lending a multifaceted dimension to the overall design. Leveraging the principles of coupled-mode theory, the reflection coefficient ($R$) of a resonator within a one-port system is given as follows:

$$R=1-\frac{2{Q}_{{\rm{rad}}}^{-1}}{2i\left(\frac{\omega }{{\omega }_{r}}-1\right)+{Q}_{{\rm{rad}}}^{-1}+{Q}_{{\rm{loss}}}^{-1}}$$

(4)

where $\omega$ and ${\omega }_{r}$ symbolize the angular frequency and the angular resonant frequency, respectively. ${Q}_{{\rm{rad}}}^{-1}$ and ${Q}_{{\rm{loss}}}^{-1}$ represent the radiation and intrinsic-loss factors, respectively. ${Q}_{{\rm{rad}}}^{-1}=2\gamma /{\omega }_{r}$ and ${Q}_{{\rm{loss}}}^{-1}=2\Gamma /{\omega }_{r}$, with $\gamma$ and $\Gamma$ representing the radiative and dissipative decay rates, respectively. The intricate interplay between these decay rates is pivotal in determining the absorption coefficients of the system. When the condition of “critical coupling” is met, characterized by γ = Γ, perfect absorption is attained.

Fig. 3: Sound-absorptive behaviors and physical mechanisms.

A Schematic of three-layered cascaded air cavities arranged in parallel. B Multimodal resonance system revealing the coupling effect between parallel resonators. Ci Comparative analysis of the absorptive performance among the measured data and the analytical and FEM results. Cii Relative impedance demonstrated by the acoustic resistance and reactance. Ciii Representation of the reflection coefficient in the complex frequency plane. Di Sound absorption performance and Dii relative impedance of Part 1 and Part 2. The reflection coefficient in the complex frequency plane is depicted in Diii for Part 1 and Div for Part 2. Energy dissipation of Part 1 and Part 2 resonators: Ei viscous and thermal dissipation and Eii energy density at resonant frequencies (f₁, f₂, and f₃). Energy conversion distributions of Fi Part 1 and Cii Part 2, with the velocity of the air particles indicated by red arrows.

Figure 3B schematically plots the multimodal resonant system of MBAMs, showing the coupling behavior between the Part 1 and Part 2 absorbers. The individual component absorbers are designed in inherent weak-resonance mode, collectively realizing a remarkable enhancement in overall absorption. A numerical model of the metamaterials is established via COMSOL Multiphysics (Supplementary Fig. S5). We undertake a comparison between the high-fidelity numerical models and experimental data, exemplified by the straight-wall case (Fig. 3Ci); this reveals substantial agreement between the predicted and tested data both in terms of trends and resonant frequencies. This alignment notably validates the reliability of the numerical models. From Eq. (3), the resonance peak corresponds to a set of impedance-matching conditions. When excellent absorption is attained, near-coincidence between the structural surface impedance and the impedance of air is achieved, delineated by the real part Re (${Z}_{s}/{Z}_{0}$) = 1 and the imaginary part Im (${Z}_{s}/{Z}_{0}$) = 0. In Fig. 3Cii, a well-matched impedance is discerned at approximately 4.0–5.0 kHz. From approximately 2.0–3.0 kHz, an undesirable mismatching state is evident, characterized by a resistance peak and a resonance dip. This incongruous matching state elucidates the absorption valley observed in Fig. 3Ci. The damping state according to the distributions of ${{\log }_{10}\left|R\right|}^{2}$ in the complex frequency plane in physics is plotted in Fig. 3Ciii. In the lossless case, the reflection coefficient encompasses a complex conjugate zero and a pole. Herein, the first and third zeros are situated below the real frequency axis, indicating a slightly overdamped state, which is a relatively favorable damping state¹¹. The second zero closely aligns with the real frequency axis, indicating a delicate balance between energy loss and leakage within the system, which accounts for the ultrahigh absorption peak in Fig. 3Ci. The black contour line delineates the frequency domain where $\alpha$ > 0.5. Remarkably, the curve encompasses the real frequency axis across nearly the entire frequency range, nearing the total half-absorption.

In physics, unprecedented sound-absorptive properties are realized by the coupling effect among individual resonators within radiation fields. To gain a deeper understanding, we compare the sound absorption of these resonators (Part 1 and Part 2). In Fig. 3Di, a notable increase in absorption is observed for Part 1 within 1.5–5.0 kHz. For Part 2, the absorption bandwidth significantly increases beyond 3.5 kHz. Fig. 3Dii elucidates the suboptimal absorption of both Part 1 and Part 2 through impedance-matching state analysis. Part 1 exhibits an overdamped state within 1.0–3.0 kHz, but this state improves at approximately 4.5 kHz, achieving quasitotal absorption (Fig. 3Diii). For Part 2, although the zeros closely approach the real frequency axis, none are present above 4 kHz (Fig. 3Div), consequently yielding a narrow absorption band. Briefly, the resonators in both Part 1 and Part 2 distinctly exhibit weak resonance. This characteristic weak resonance, when combined, yields an overall synergistic enhancement through the coupling effect.

In general, we identify two synchronous mechanisms contributing to effective absorption: i) viscous friction dissipation and ii) thermal dissipation. In Fig. 3Ei, the intrinsic energy dissipations by Part 1 and Part 2 exhibit distinct characteristics, as depicted. In physics, as the incident wave traverses the microperforations, viscous boundary layers are generated due to the vibration and friction of air molecules, thereby facilitating dissipation ${E}_{v}=1/2(\eta {{v}_{t}}^{2}/{\phi }_{v})$²⁷, where ${v}_{t}$ denotes the tangential velocity with $t$ of time evolution and where ${\phi }_{v}$ denotes the viscous boundary layer. Thermal dissipation is primarily induced by the disruption of adiabatic sound wave propagation, resulting in energy conversion and loss within thermal boundary layers. The constituent thermal flux ${E}_{t}$ is calculated as ${E}_{t}=[\left({\rm{\gamma }}-1\right){p}^{2}\sqrt{w\chi /(2{\rho }_{0}{C}_{p})}]/(2{\rho }_{0}{{c}_{0}}^{2})$²⁷, where $\gamma$, $p$, $\chi$, and ${C}_{p}$ refer to the specific heat ratio, the pressure outside of the boundary layer, the thermal diffusivity, and the specific heat capacity of air, respectively. The energy density of the resonator components at the three resonant frequencies (1.86 kHz, 3.90 kHz, and 4.70 kHz, denoted as f₁, f₂, and f₃, respectively) is compared in Fig. 3Eii. Part 2 predominantly contributes to sound wave energy dissipation at the first two absorption peaks, whereas Part 1 results in slightly greater dissipation. Figure 3Fi, ii illustrates the sound wave energy dissipation within the air phases of Parts 1 and 2, respectively. As shown, the cavities contribute minimally to energy dissipation; conversely, the narrow air necks, particularly within the second and third layers, dissipate the majority of incident energy. At frequency f₁, both Part 1 and Part 2 demonstrate that the second- and third-layer air necks play dominant roles, whereas the first layer dominates f₂ and f₃. This observation highlights the synergistic interaction among the resonators. The red arrows, symbolizing air particle velocities, exhibit a nonuniform distribution, with higher velocities concentrated around the narrow regions. Particularly near the narrow sections, air molecules display severe vibrations adhering to the walls. The well-matched impedance encourages air particles to congregate around the microperforations, facilitated by a siphonic effect. These particles subsequently pass through the pores, with the incident sound energy being converted into heat.

Superior stiffness, strength, damage tolerance, and energy absorption

The mechanical responses of the MBAMs are explored through axial compression tests. Figure 4A presents the stress–strain curves of one representative specimen set. Notably, in instances where the walls are straight or A₀ = 0.5, the stress clearly sharply decreases, nearly reaching zero. This strikingly unstable stress decrease occurs three times, corresponding to the catastrophic collapse of the three-layered cell walls. When A₀ = 0.5, it has a greater compressive stress than the straight-wall structure does, but it fails to disrupt the commonly observed failure modes inherent in straight-wall structures^9,28, resulting in catastrophic failure. With increasing camber level, the overall stress‒strain response presents a more moderate trajectory, effectively mitigating severe failure. Intriguingly, for A₀ = 1.5, instead of exhibiting a sharp decline in stress, a serrated stress curve emerges, reminiscent of deformation modes in multilayered energy absorbers. However, the mechanical advantages of our design do not follow a linear progression with increasing camber level. The desired stress‒strain response is observed with A₀ = 1.0, with an unprecedented level of stability, characterized by a long and smooth stress plateau, while consistently high stress levels are maintained across these configurations.

Fig. 4: Measured mechanical properties.

A Stress‒strain curves, B SEA, specific modulus, and specific strength, and C fluctuation coefficient of both the straight-wall structure and our MBAMs (A₀ = 0.5, 1.0, and 1.5). Ashby maps of Di modulus against density, Dii compressive strength against density, and Diii SEA against relative density, compared with advanced mechanical metamaterials: Ti6Al4V octet lattice⁴⁰, Ti6Al4V gyroid⁴¹, Ti6Al4V hierarchical lattice⁴², 316 L steel RD lattice⁴³, 316 L steel plate lattice³⁶, 316 L steel TPMS⁴⁴, Inconel 718 hybrid lattice⁵, Maraging steel TPMS and gyroid⁴⁵, 316 L steel shell-FCC⁴⁶, 316 L steel BCC^47,48, Ti6Al4V diamond and cuboctahedron⁴⁹, 316 L steel TPMS⁵⁰, and 316 L steel hybrid⁵¹.

Figure 4B presents the measured specific energy absorption (SEA), specific strength, and specific modulus of our MBAMs. The SEA is calculated via the formula ${SEA}={\int }_{0}^{{\varepsilon }_{{den}}}\sigma d\varepsilon /{\rho }_{c}$, where ${\varepsilon }_{{den}}$ denotes the densification strain, i.e., the integral termination point, and where ${\rho }_{c}$ denotes the density. Notably, the SEA tends to increase followed by a decrease, with A₀ = 1.0 resulting in the highest SEA value. This trend can be attributed to the stress‒strain curves, where the area under the curve along the x-axis represents the energy absorption capacity. In contrast, both the specific strength and modulus follow a similar pattern, reaching their peaks when A₀ = 0.5, with the straight-wall structure exhibiting the lowest values. An index, the fluctuation coefficient $\tau$, is introduced here to quantitatively describe the stability of the crushing response, which is calculated as

$$\tau =\frac{{\sigma }_{\max }-{\sigma }_{\min }}{{\sigma }_{{ave}}}$$

(5)

where ${\sigma }_{\max }$, ${\sigma }_{\min }$, and ${\sigma }_{{ave}}$ refer to the maximum, minimum, and average stresses, respectively, with strains ranging from 0.1–0.5.

Based on the measured data, we generally divide the stress fluctuations into three phases: catastrophic failure (τ > 1.5), serrated phase(0.75 < τ < 1.5), and stable phase(0.1 < τ < 0.75). Remarkably, with A₀ of 1.0, the value of τ falls within the stable phase with the lowest fluctuation (Fig. 4C). The mechanical superiority of the MBAMs is elucidated through a comprehensive comparison with mainstream reported AM-enabled microlattice metamaterials, which incorporate a variety of commonly used materials, such as Ti6Al4V and 316 L steel, via Ashby maps. In this analysis, three critical indices are considered: the modulus, strength, and SEA.

As illustrated, the MBAMs demonstrate exceptional modulus values (Fig. 4Di) and strength (Fig. 4Dii), despite having a low density of 1.53 g/cm³. This performance surpasses the typical density-dependent trend (blue region), where a higher density usually correlates with higher performance, which is observed in most microlattice or porous metallic foams. Additionally, at similar densities, our metamaterial outperforms other advanced mechanical metamaterials, exhibiting higher specific moduli and strengths. The relatively comparable competitor is the Inconel 718 hybrid lattice, owing to its favorable mechanical properties and density of approximately twice that of our Ti6Al4V MBAMs, which also underscores the mechanical merits of our metamaterials. From the SEA Ashby chart (Fig. 4Diii), the varying cambered levels enable the MBAMs to exhibit a large divergence in SEA. This disparity is elucidated by examining the stress‒strain curves presented in Fig. 4A. Notably, the lowest SEA is obtained with A₀ of 0.5, whereas the highest SEA is observed with A₀ of 1.0. Regardless of the cambered level, the measured SEA of the MBAMs consistently surpasses that of mainstream microlattice energy absorbers. Particularly noteworthy are the cases of A₀ = 1.0 (SEA of 36.3 J/g) and A₀ = 1.5 (SEA of 50.7 J/g), notably exceeding those of mainstream microlattices and exhibiting a gap above the corresponding general trend. This heightened SEA can be attributed to the smooth, flat, and elongated plateau stress curves observed in Fig. 4A, thereby demonstrating their ability to absorb stress wave energy.

Deformation and enhancement mechanisms

Figure 5A compares the deformation modes of the straight-wall structure and the MBAM (A₀ = 1.0) at compressive strains of 0.1, 0.3, and 0.5. Experimental displacement contour plots generated via digital image correlation (DIC) for A₀ of 1.0 are compared with those obtained via the FEM with a displacement of 0.6 mm (Fig. 5B), revealing great agreement between them and thus suggesting the high reliability of the FEM results. Specifically, as shown in Fig. 5A, one layer of the straight-wall structure experiences severe buckling when the strain reaches 0.1. Severe buckling is then associated with the release of strain energy, and hence a large dip in the stress–strain curve (Fig. 4A). At a strain of 0.3, initial fracture occurs in the stress-concentrated region, particularly near the junction between the plate and walls. This fracture is evident in both the sample and the FEM results, leading to catastrophic failure of this layer, accompanied by severe buckling in another layer. With continued loading, almost all the walls fracture, ultimately losing their load-bearing ability until being compressed to a dense state.

Fig. 5: Mechanical responses and deformation mechanisms at the structural and wall levels.

A Experimental and numerical deformation modes of both the straight-wall and the MBAM (A₀ = 1.0) with compressive strains of 0.1, 0.3, and 0.5, revealing the transition from catastrophic failure to the desired damage-tolerant layer-by-layer deformation. B DIC and FEM displacement contours at 0.6 mm. Ci Mises stress and PEEQ distributions of the straight-wall structure and MBAMs (one row of cell walls). Cii Numerically calculated principal stress distributions of one row of the cell wall of the MBAM. Ciii Schematic illustrating the microscopic postfracture performance, depicting asymmetric fracture accompanied by effective stress transfer and enhanced wall interactions.

Conversely, the proposed metamaterials initially fracture around the top of the walls adjacent to the plates—the area with the highest curvature—enabling severe buckling or failure. As loading progresses, one layer is gradually moderately crushed, followed by the other two layers, accompanied by intense wall interactions between crushed and uncrushed wall parts; this leads to high strength and high damage tolerance. Furthermore, the present MBAMs increase the stress in the large-curvature region. This stress distribution shift offers several advantages: (1) decreased stress at the plate‒wall junction reduces the likelihood of severe buckling; (2) higher stress levels at high-curvature regions facilitate crack initiation, controlling the initial strain energy release and enhancing mechanical robustness; and (3) elevated stress distribution at the middle parts, which is crucial for directional damage propagation and enhanced postfracture contact performance (Fig. 5A). Briefly, the present metamaterials feature the absence of abrupt macroscopic shear bands or severe buckling, ushering in a graceful damage-tolerant layer-by-layer deformation mode. The interconnected and compacted parts within the metamaterial act as robust load-bearing components, facilitating a more uniform dispersion of applied strain across the structure.

In Fig. 5Ci, the stress distribution of one row of the cell wall of our metamaterial (A₀ = 1.0) is displayed, and that of the other camber-level cases is illustrated in Supplementary Fig. S9. The stress of the straight-wall structure disperses uniformly, resulting in the accumulation and convergence of strain energy. In contrast, the cambered pattern design enables the high-stress regions to gather around the maximum curvature area, forming a long arc-shaped region of elevated stress (marked by dotted curves). Thus, cracks occur in this region with a moderate release of strain energy. Furthermore, the equivalent plastic strain distribution (PEEQ) in Fig. 5Ci corroborates this deformation behavior. In the straight-wall case, the highest PEEQ is observed near the bottom of the walls. However, our MBAMs are concentrated inside the arc-shaped region, suggesting a more controlled deformation mechanism and potentially enhanced mechanical performance while mitigating structural failure. To delve deeper into the mechanics, the numerical principal stress distributions within the cell walls are depicted in Fig. 5Cii. The tensile and compressive stresses represent the maximum and minimum principal stresses, respectively. The red arrows denote tensile stress, whereas the blue arrows signify compressive stresses, with the arrow length indicating the relative stress magnitude. As depicted, tensile stresses are primarily concentrated at the maximum-curvature region near the top, with generally horizontal orientations. As concentrated tensile stress can lead to unit cracking, horizontally distributed stress inevitably initiates subtle cracks. Consequently, the metamaterials exhibit graceful progressive failure (Fig. 5A), offering superior damage tolerance. A representative microscopic deformation response of the cuttlebone under compression is schematically summarized in Fig. 5Ciii. As depicted, the initial fracture manifests at the arc-shaped corner, strategically directing maximum stress to well-defined locations, thus enabling manipulation of the fracture path. With subsequent loading, the cambered walls undergo progressive fracture, with certain portions deposited onto the plate. This process facilitates efficient stress transfer and heightened wall interactions, ultimately achieving an optimal balance between strength and damage tolerance²⁹.

Research framework and prospects

While multifunctional sound-absorbing and mechanical metamaterials have gained attention in recent years, the key to their material design remains elusive. Here, we pioneer a research framework for these metamaterials based on the coupled modes between the sound-absorbing and mechanical properties. We delineate two distinct mechanisms: (1) weakly-coupled, and (2) strongly-coupled, as shown in Fig. 6A. As shown, the initial stage is the unit design. Under the weakly-coupled principle, a common approach involves pairing resonant plates with axial cells. Resonant plates influence sound absorption, whereas vertical cells control stress‒strain responses. Our MBAMs embody this strategy by allowing customization for specific needs, such as achieving broad absorption and maintaining a stable stress plateau. Generally, sound absorption and mechanical responses are mutually exclusive rather than mutually influencing. However, in the strongly-coupled design principle, the fundamental units consist of periodic cells with three-directional symmetry, often seen as simple cubic or octet lattice structures. Within these cells, elements such as rods and plates act as load-bearing components, serving crucial roles in determining both mechanical properties and sound-absorbing capabilities. Addressing these two types of issues requires distinct strategies: in weakly-coupled scenarios, designing independent units enables functional separation and manipulation. Conversely, in strongly-coupled cases, establishing a structure‒property relationship is advisable. The use of a multiobjective optimization method to handle trade-offs and identify optimal geometries that achieve the Pareto front is recommended. The coupling performance could be assessed by examining the overall rating distribution of each functionality via the selected criteria of the measured data (Supplementary Fig. S10). In our work, the data depicted in the map are discrete and lack apparent correlation, which is indicative of the weakly-coupled performance among these metrics. Furthermore, with a sufficiently large dataset, we can evaluate the correlation performance via models such as the Pearson coefficient (Supplementary Eq. (S8)). Figure 6B shows representative examples of the two design approaches. In the weakly-coupled paradigm, typical examples are MPP-like structures^30,31,32. Resonant plates and interior cells function independently, governing distinct aspects of sound absorption and mechanical behavior. In contrast, in the strongly-coupled case, lattice structures exhibit a prevalent archetype characterized by an enhanced interaction between functionalities, where structural elements concurrently impact the acoustic and mechanical properties.

Fig. 6: Research framework of acousto-mechanical metamaterials.

A Summarizing the weakly-coupled and strongly-coupled mechanism design approaches for manipulating sound-absorbing and mechanical properties, revealing their distinctions in terms of unit design, research approach, and final performance outcomes. B Specimen images of typical structures representing the weakly-coupled and strongly-coupled paradigms.

With respect to this emerging research direction, we see numerous avenues worth exploring, including novel structural designs, materials, and functionalities. Examples include bioinspired designs, nonuniform unit cell arrangements, and innovative energy-dissipating mechanisms, all of which show considerable promise. These multifunctional materials hold substantial practical potential, particularly in the construction and transportation industries. While many are currently in the lab-scale phase, advancements in additive manufacturing could expedite their integration into diverse large-scale engineering projects.

Conclusion

In summary, this study introduces MBAMs, a new class of multifunctional bioinspired architected metamaterials. Inspired by the cuttlebone microstructure and following a weakly-coupled design approach, these materials incorporate dissipative pores into multilayer plates. This design enables efficient sound absorption through air friction damping mechanisms. Furthermore, a high-fidelity acoustic impedance model is developed to fine-tune the pore diameter, facilitating comprehensive analyses of impedance matching and damping state evaluations. The experimental data reveal the exceptional sound absorption of the MBAMs across a wide frequency range (1.0–6.0 kHz), with an average absorption coefficient of 0.80 and an impressive 77% of the data points surpassing 0.75, reaching the quasitotal half-absorption. Remarkably, this achievement is accomplished within a compact 21 mm thickness; this suggests a breakthrough in the absorptance‒thickness trade-off observed in other reported absorbers. Delving deeper into the underlying physics unveils the cornerstone of such performance: the multimodal hybrid resonance, fueled by the far-field coupling effect between the heterogeneously designed resonators.

Moreover, the advantages of our MBAMs extend beyond sound absorption. Empowered by the cambered-wall design, these materials demonstrate ultrahigh strength, exceptional energy absorption, and high damage tolerance. A low density of 1.53 g/cm³ with an average modulus of 4.93 GPa, strength of 211 MPa, and a specific energy absorption of 50.7 J/g is achieved. Future applications may bid farewell to catastrophic failure and severe buckling, as a progressive, graceful deformation pattern emerges in a damage-tolerant manner. The stress distribution is concentrated around the maximum-curvature regions, which guides directional stress propagation, enhances effective stress transfer, and facilitates wall interactions, thus enhancing the structural strength. In brief, these multifunctional materials bridge a pivotal gap within the realms of materials science and engineering.