A Single-Phase Aluminum-Based Chiral Metamaterial with Simultaneous Negative Mass Density and Bulk Modulus

Abstract

We propose a single-phase chiral elastic metamaterial capable of simultaneously exhibiting negative effective mass density and negative bulk modulus in the ultrasonic frequency range. The unit cell consists of a regular hexagonal frame connected to a central circular mass through six obliquely oriented, slender aluminum beams. The design avoids the manufacturing complexity of multi-phase systems by relying solely on geometric topology and chirality to induce dipolar and rotational resonances. Dispersion analysis and effective parameter retrieval confirm a double-negative frequency region from 30.9 kHz to 34 kHz. Finite element simulations further demonstrate negative refraction behavior when the metamaterial is immersed in water and subjected to 32 kHz and 32.7 kHz incident plane wave. Equifrequency curves (EFCs) analysis shows excellent agreement with simulated refraction angles, validating the material’s double-negative performance. This study provides a robust, manufacturable platform for elastic wave manipulation using a single-phase metallic metamaterial design.

1. Introduction

In recent years, metamaterials—man-made structures designed to exhibit unconventional properties absent in natural materials—have garnered extensive interest for their remarkable capacity to control wave propagation [1,2,3,4]. Particularly noteworthy are double-negative (DNG) metamaterials, which simultaneously exhibit negative effective mass density and negative effective modulus, enabling phenomena such as negative refraction [5,6], acoustic cloaking, and imaging beyond the diffraction limit [7,8].

Double-negative (DNG) behavior in metamaterials is commonly realized by inducing localized resonances within periodic architectures. In acoustic and elastic metamaterials, negative effective mass density typically emerges near dipolar resonance frequencies, whereas negative bulk modulus arises around monopolar or rotational resonances [9,10,11,12]. Traditionally, these effects have been achieved through composite designs, incorporating multi-material inclusions, soft components, or intricate internal mechanisms [13,14,15,16,17,18,19,20]. However, such multi-phase configurations often suffer from manufacturing complexity, reduced mechanical reliability, and interfacial losses, which hinder their scalability and practical deployment [21,22].

To address these challenges, research has increasingly focused on single-material metamaterials, where the desired resonant behaviors are obtained purely through architectural design rather than material heterogeneity [23,24]. Among these, chiral metamaterials—which inherently lack mirror symmetry—have proven effective in simultaneously activating rotational and translational modes. This coupling facilitates the concurrent emergence of negative density and modulus [25,26]. Integrating chirality into lattice-based systems has further enabled the development of novel mechanisms for negative stiffness, auxetic behavior, and customized dynamic responses [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43].

In this study, we present a chiral elastic metamaterial constructed entirely from aluminum—a single-phase metallic material—designed to exhibit simultaneous negative effective density and modulus. The unit cell consists of a regular hexagonal outer ring connected to a central circular mass by six obliquely oriented slender beams with rectangular cross-sections. These angled connectors not only serve as elastic linkages but also introduce chirality into the structure. Functionally, the configuration resembles a mechanical mass-spring system, where the rigid outer frame and central inclusion act as masses, and the slender beams serve as springs. Dipolar resonances arising from this setup lead to negative effective mass density, while the rotational motion induced under compression causes outward expansion of the hexagonal frame, resulting in a negative effective bulk modulus.

To quantitatively evaluate the dynamic response of the metamaterial, we perform dispersion analysis along the longitudinal direction and extract its effective acoustic parameters through homogenization. Within the frequency window of 30.9–34 kHz, the third dispersion branch demonstrates a negative group velocity, aligning with the simultaneously negative values of retrieved mass density and modulus-confirming the presence of a double-negative (DNG) regime.

To verify the associated wave manipulation capabilities, we construct a triangular prism-shaped metamaterial composed of the proposed unit cells and immerse it in a fluid domain. Finite element simulations indicate that when a 32.7 kHz plane wave impinges perpendicularly on the wedge base, the transmitted wave refracts anomalously toward the same side of the normal as the incident wave—a distinctive signature of negative refraction. This behavior is corroborated by equifrequency curves (EFCs) analysis, with the theoretical refraction angle exhibiting excellent agreement with numerical simulation results.

Overall, this work demonstrates a novel and structurally simple route to realizing double-negative acoustic metamaterials using a monolithic aluminum design. The integration of chirality within a single-phase framework offers a promising avenue for developing robust, scalable, and manufacturable platforms for advanced control of elastic and acoustic waves.

2. Methods

2.1. Metamaterial Design

The designed elastic metamaterial is composed of a periodically arranged unit cell with a regular hexagonal topology, entirely constructed from single-phase aluminum, characterized by Young’s modulus E = 70 GPa, Poisson’s ratio v = 0.33, and density ρ = 2700 kg/m³, as shown in Figure 1a. Each unit cell comprises an outer hexagonal ring of side length L = 10 mm and thickness t = 1.2 mm, enclosing a central circular inclusion with radius r = 5 mm. The two components are linked by six narrow rectangular beams of width d = 0.15 mm, which are evenly distributed and inclined at an angle θ = 80° relative to the corresponding side of the hexagonal outer frame, introducing a chiral geometric feature. These slender beams are intentionally engineered with reduced cross-sectional dimensions to serve as flexible connectors.

To better understand the physical mechanisms responsible for the emergence of negative effective parameters, the proposed chiral metamaterial can be interpreted using an equivalent mass-spring model, as introduced by Liu et al. [25]. In this framework, the metamaterial unit cell functions analogously to a coupled oscillator system, where the outer hexagonal frame and central circular inclusion represent discrete mass elements, and the obliquely oriented slender beams act as elastic springs.

Specifically, the dipolar resonance arises from the out-of-phase oscillation between the central mass and the surrounding frame under incident acoustic excitation. This motion corresponds to a translational mode where the inertial response of the inner and outer masses leads to a frequency band with negative dynamic mass density [11]. Simultaneously, due to the chiral beam configuration, rotational motion is induced when the structure is subjected to omnidirectional (hydrostatic) compression from all directions. This rotation couples with lateral expansion of the frame, mimicking a rotational spring mechanism, which under specific conditions gives rise to a negative bulk modulus.

The coupling between these dipolar and rotational resonances is geometrically governed by the beam orientation angle (θ), beam stiffness (determined by cross-sectional dimensions), and the mass distribution. This dual-resonance mechanism aligns with the theoretical insights from Ref. [25], where both mass density and modulus become negative due to localized structural resonance rather than material composition.

2.2. Dispersion Analysis

The dispersion curves of the metamaterial are calculated using the “Solid Mechanics (Elastic Waves)” module in COMSOL Multiphysics 6.0. We consider metamaterials to have an infinite extent in the direction perpendicular to the plane, which corresponds to a plane-strain problem. The dispersion curve is obtained by applying Floquet periodic boundary conditions on the outer boundaries of the unit cell. In this study, dispersion analysis was performed under plane-strain conditions, focusing on in-plane vibrational modes. Out-of-plane modes, which would be relevant for finite-thickness plates, are not considered in this two-dimensional model. The dispersion curve, as shown in Figure 2a, depicts the relationship between frequency and wave vector within the first Brillouin zone (Figure 1b) for the designed double-negative (DNG) metamaterial. The curve demonstrates the typical behavior of a metamaterial with negative effective properties, particularly in the frequency range 30.9 kHz to 34 kHz. The negative slope observed in the curve around the Γ point indicates the presence of a negative group velocity. This phenomenon is a direct consequence of the simultaneous negative effective mass density and negative bulk modulus, which leads to a negative group velocity—a condition where the phase of the wave propagates in the opposite direction to the energy flow. This unique characteristic causes wavefronts to refract on the same side of the normal as the incident beam, thus enabling negative refraction [1,5]. Such behavior is unattainable in conventional materials and underlies advanced functionalities including subwavelength focusing, acoustic cloaking, and wave redirection [2,15]. The physical origin of these negative effective parameters stems from local resonances, which reverse the sign of the constitutive parameters in specific frequency bands, leading to a backward-wave regime characteristic of double-negative metamaterials [15]. The double-negative (DNG) behavior is a hallmark of the metamaterial’s ability to manipulate acoustic waves in unconventional ways, such as in wave focusing or deflection.

2.3. Effective Parameter Extraction and Modal Analysis

The effective dynamic parameters are obtained by scanning in the frequency domain through the effective medium theory based on homogenization and the theoretical calculation formula proposed in Refs [25]. The signs of the effective properties are assigned according to the direction of the group velocity associated with the dispersion branches.

Under the assumption of long-wavelength behavior, the metamaterial unit cell is approximated as a homogeneous medium, where the detailed internal microstructure is averaged out. In this approximation, the unit cell’s macroscopic response is represented through the averaged boundary conditions such as stress, displacement, and acceleration, effectively neglecting the finer details of the internal structure. The governing equations for the system are formulated as follows:

$$F_{\alpha }={\displaystyle \frac{1}{V}}{\displaystyle {\int }_{\partial V}{\sigma }_{\alpha \beta }d{s}_{\beta }}$$

(1)

$${\ddot{U}}_{\alpha }={\displaystyle \frac{1}{S}}{\displaystyle {\int }_{\partial V}{\ddot{u}}_{\alpha }ds}$$

(2)

$${\sigma }_{\alpha \beta }={\displaystyle \frac{1}{V}}{\displaystyle {\int }_{\partial V}{\sigma }_{\alpha \gamma }{x}_{\beta }d{s}_{\gamma }}$$

(3)

$${\epsilon }_{\alpha \beta }={\displaystyle \frac{1}{2V}}{\displaystyle {\int }_{\partial V}(u_{\alpha }d{s}_{\beta }+u_{\beta }d{s}_{\alpha }})$$

(4)

where $\alpha ,\beta ,\gamma =1,2,S,V,\partial V$ represent the surface area, volume, and external boundary of the cell, respectively, ${\sigma }_{\alpha \beta }$, ${\ddot{u}}_{\alpha }$, $u_{\alpha }$ represent the stress, acceleration, and displacement, respectively, and $d{s}_{\alpha }=n_{\alpha }ds$, where $n_{\alpha }$ represents the unit external normal vector of the boundary.

The effective mass density, effective bulk modulus and effective shear modulus of two-dimensional isotropic solid metamaterials can be expressed as

$${\rho }_{\mathrm{eff}}=F_{\alpha }/{\ddot{U}}_{\alpha },K_{\mathrm{eff}}={\displaystyle \frac{1}{2}}{\sigma }_{\alpha \alpha }/{\epsilon }_{\alpha \alpha },{\mu }_{\mathrm{eff}}={\displaystyle \frac{1}{2}}{\sigma }_{\alpha \beta }^{\prime }/{\epsilon }_{\alpha \beta }^{\prime }$$

(5)

where ${\sigma }_{\alpha \alpha }$ and ${\epsilon }_{\alpha \alpha }$ represent the principal diagonal components of stress and strain, and ${\sigma }_{\alpha \beta }^{\prime }$ and ${\epsilon }_{\alpha \beta }^{\prime }$ represent the non-principal diagonal components of stress and strain.

To retrieve the effective parameters shown in Figure 2, we employed a homogenization-based approach under the long-wavelength assumption, as detailed in Appendix A. In this method, periodic boundary conditions were applied to the unit cell, and effective dynamic properties were extracted by averaging the stress, strain, and acceleration fields. The explicit expressions and numerical implementation steps are provided to ensure reproducibility and clarity.

Figure 2b illustrates the relationship between the normalized bulk modulus and frequency. The plot reveals a pronounced negative bulk modulus around the frequency range of 30.6–34.3 kHz, supporting the design of the metamaterial as a double-negative material. The negative bulk modulus indicates the material’s ability to reverse the usual behavior of wave propagation, which is essential for negative refraction. This phenomenon arises from the material’s structural configuration, in which the coexistence of negative effective modulus and negative effective mass density results in acoustic wave propagation characterized by opposite group and phase velocity directions. This behavior enables a range of unconventional acoustic effects with significant application potential. For instance, the negative refraction observed in such double-negative media allows for the realization of flat acoustic lenses, capable of focusing sound at subwavelength scales—commonly referred to as acoustic superlensing [1,15,26]. These devices overcome the diffraction limit and have important implications in high-resolution medical imaging and nondestructive evaluation. Additionally, the reversed energy flow characteristic of these materials can be exploited in directional sound control, enabling compact beam-steering or wavefront reshaping devices without the need for bulky curved components [7,8]. Such functionalities have also been utilized in the design of acoustic cloaks, which guide sound around an object to render it undetectable [22].

The sharp transitions observed in the plot suggest that the metamaterial operates most effectively within a narrow frequency range, further confirming that this frequency band is critical for the desired negative refraction effects. The blue shaded region in the figure marks the frequency range where the bulk modulus is negative, highlighting the material’s operational window for negative refraction applications.

The results presented in Figure 2c, which shows the relationship between the normalized mass density and frequency, demonstrate a similarly significant shift in material properties at the targeted frequency. As the mass density approaches negative values from 29.5 to 44.5 kHz, the metamaterial continues to exhibit its double-negative characteristics. The negative mass density is integral to the negative refraction effect, as it contributes to the reversal of wave propagation direction when combined with the negative bulk modulus. The red-shaded region indicates the frequency range where the mass density becomes negative.

To elucidate the physical mechanism behind the double-negative properties of the metamaterial, we selected the frequencies of four points in Figure 2b and c, respectively, for modal analysis, as illustrated in Figure 3, where arrows indicate the local displacement fields of the structure. It should be noted that the modal analyses presented in Figure 3 were performed on a single isolated unit cell subjected to specific harmonic displacement excitations rather than Bloch–Floquet periodic boundary conditions. In each case, different displacement configurations were imposed on the outer boundary to excite and visualize characteristic local deformation modes that are responsible for the emergence of negative mass density and bulk modulus. The observed differences between Figure 3a,d (and between Figure 3b,e) at the same frequency arise from these distinct boundary conditions and excitation modes. Detailed descriptions of the boundary conditions, displacement setups, and numerical procedures are provided in Appendix B.

In Figure 3a, the mode shape at a low frequency of 500 Hz (below the resonant frequency associated with the effective bulk modulus) is presented. It can be observed that the outer frame undergoes outward expansion, which in turn drives a clockwise rotation of the central circular inclusion through the chiral beams. This behavior corresponds to a positive bulk modulus. At 32.7 kHz, as shown in Figure 3b, the central inclusion exhibits a significantly enhanced clockwise rotational displacement, as indicated by the color scale. This strong rotational motion induces further outward expansion of the outer frame via the chiral beams, resulting in a negative effective bulk modulus. Conversely, at 34.5 kHz, shown in Figure 3c, the inclusion rotates counterclockwise, thereby opposing the expansion of the outer frame. This leads to increased stiffness and corresponds to a large positive bulk modulus near the peak of the dynamic modulus response. The effective mass density behavior is clarified in Figure 3d–f. At 500 Hz (Figure 3d), the displacement directions of both the outer frame and central inclusion are identical, indicating positive effective mass density. At 32.7 kHz (Figure 3e), the inclusion and outer frame move in opposite directions, with the inclusion exhibiting a larger displacement, which leads to negative effective mass density. In contrast, at 29 kHz (Figure 3f), although there is a slight difference in displacement magnitude, both components move in the same direction, indicating the material still exhibits positive effective mass density. These modal observations confirm that the emergence of negative effective parameters is closely linked to the coupled rotational and translational resonances, which are in turn governed by the geometry of the chiral beams.

2.4. Finite Element Validation of Negative Refraction

The verification of negative refraction was carried out through frequency-domain analysis in the acoustic-solid interaction module of COMSOL Multiphysics 6.0. We assembled 256 unit cells into a triangular prism-shaped metamaterial wedge with a base angle of 30°, embedded in water as the background medium with density ρ = 1000 kg/m³, sound speed c = 1500 m/s. The surrounding boundaries were treated with Perfectly Matched Layers (PML) to eliminate reflection artifacts, as shown in Figure 4. A 32.7 kHz Gaussian beam was normally incident on the wedge boundary. At this frequency, the effective parameters are: effective density ρ_eff = −2259 kg/m³, effective bulk modulus K_eff = −7.98 GPa, and effective shear modulus u_eff = 3.01 GPa. Simulation results indicate that the transmitted wave inside water bends toward the same side of the normal as the incident beam, confirming negative refraction. The wave enters the metamaterial from the water medium normally and is refracted by the metamaterial wedge structure. As shown by the white arrows, the refracted wave and the incident wave are on the same side of the normal, indicating a negative refraction effect. This unusual behavior is characteristic of metamaterials with negative effective mass density and negative effective modulus, as the waves are guided along paths opposite to the typical refraction expected in conventional materials.

Figure 5 presents the EFCs of the metamaterial at 32.7 kHz, which is crucial for understanding the propagation characteristics of the acoustic waves. The EFCs depict the relationship between the frequency and wave vector components in the periodic medium. The red circle represents the sound line of water, indicating the propagation of sound at the speed of sound in water. v_g and k are the group velocity and wave vector in metamaterial, respectively. v_gw and k_w are the group velocity and wave vector in water, respectively. The color gradient illustrates the frequency distribution, where the red regions correspond to higher frequencies and blue regions to lower frequencies.

The dashed line within the EFCs indicates the direction of the normal, with the black arrows showing the expected direction of propagation. As seen, the wave experiences a significant bending, which is a hallmark of negative refraction. The wave vectors in the metamaterial are positioned such that they correspond to a wave propagation direction opposite to that of conventional materials. According to the conservation of the tangential wave vectors component and Snell’s law, the refracted angle is calculated to be −17.2°, consistent with the simulated result.

The sound pressure field at 32 kHz is presented in Figure 6. At this frequency, the effective mass density, bulk modulus, and shear modulus of the metamaterial are −2922 kg/m³, −3.71 GPa, and 3.03 GPa, respectively. It is observed that the refracted wave follows the same direction as the incident wave, remaining on the same side of the normal. Notably, there is a slight horizontal deviation in the position of the refracted wave relative to the wave source. This deviation occurs because the EFCs at 32 kHz does not form a perfect circle, causing the acoustic wave at the base of the wedge to shift from the vertical direction as it transitions from water into the metamaterial. To accurately determine the refracted wave’s direction, we plotted the EFCs of the negative passband for the single-phase solid metamaterial, as shown in Figure 7. Although the EFCs at 32 kHz is no longer circular, the direction of the refracted wave can still be determined from its geometric shape. The comparison reveals that the refracted wave’s direction predicted by the EFCs matches the simulation results.

To examine the deflection of refracted waves when an acoustic wave is normally incident on the metamaterial, we transformed the wedge-shaped region into a rectangular region to enhance experimental observation, as shown in Figure 8. The results show that the acoustic wave exits the surface of the metamaterial in a vertical direction, while a lateral displacement occurs in the horizontal plane. When compared to the negative refraction diagram, it is evident that the propagation direction of the acoustic wave remains the same in both the wedge-shaped and rectangular regions. This direction is dictated by the group velocity vector at 32 kHz, which corresponds to the tangent direction of the EFCs at this frequency. The comparison between theoretical predictions based on the EFCs and the numerical simulation results demonstrates excellent agreement, confirming the consistency of the model.

As described earlier, the emergence of negative bulk modulus and negative mass density in the metamaterial is achieved through geometric optimization, rather than being a consequence of the intrinsic properties of the base material. Among the various parameters, the dimensional tuning of the chiral beams is particularly critical in enabling the desired dynamic response. Moreover, the frequency range over which the metamaterial exhibits double-negative properties is also highly dependent on its geometric configuration. For example, when the inclination angle of the chiral beams is adjusted to 84° (A 4° difference in beam inclination angle compared to the configuration in Figure 1), the double-negative branch in the dispersion curve spans from 33.8 kHz to 38.2 kHz, as shown in Figure 9a, indicating that the metamaterial supports negative refraction within this band. Specifically, at 36.0 kHz, a refraction angle of 15° is observed in Figure 9b, confirming the structure’s effectiveness in manipulating wave propagation at higher frequencies.

Distinct from prior research, which employs a combination of soft matrix and embedded inclusions to achieve double-negative behavior in a limited frequency range, our monolithic aluminum-based metamaterial exhibits a significantly broader double-negative frequency band from 30.9 to 34.0 kHz. The engineered chiral hexagonal unit cell induces coupled translational and rotational resonances, giving rise to a broad double-negative band, as confirmed by effective medium theory, dispersion analysis, and modal characterization. The structure also achieves accurate negative refraction (−17.2° at 32.7 kHz), well supported by Snell’s law and equifrequency contour analysis, enabling effective anisotropic wave control. Compared to conventional multi-material or hybrid metamaterials, the proposed single-phase aluminum structure offers notable advantages, including fabrication simplicity, enhanced mechanical robustness, and elimination of interfacial failure. This comparison underscores the novelty and practical advantages of our approach in advancing manufacturable, double-negative acoustic metamaterials.

Despite these strengths, the current study is based solely on numerical simulations and lacks experimental validation. Potential limitations include structural anisotropy, minor deviations from effective medium assumptions at low frequencies, and limited tunability. Nevertheless, this work introduces a novel paradigm for achieving double negativity purely through geometric chirality in a single-phase solid, offering a scalable and robust platform for subwavelength acoustic lenses, beam-steering devices, and underwater acoustic systems.

3. Conclusions

We have designed and analyzed a chiral single-phase elastic metamaterial based on aluminum that simultaneously exhibits negative effective mass density and negative effective modulus within a frequency range of 30.9–34 kHz, which employs obliquely oriented slender beams to induce local resonance and chiral coupling. Dispersion analysis and effective medium theory confirm the presence of a double-negative frequency band, while finite element simulations demonstrate the negative refraction behavior when the material is embedded in a water background. The agreement between the simulated results and theoretical EFCs further substantiates the material’s double-negative characteristics.

This innovative design eliminates the need for multi-phase materials and simplifying the fabrication process while endowing the metamaterial with exceptional durability and scalability. This study offers a robust, scalable, and practical platform for the realization of metallic acoustic metamaterials with negative refraction, providing a foundation for applications in wave steering, subwavelength imaging, and acoustic cloaking, all using manufacturable, single-material designs. Future work will focus on extending the double-negative bandwidth and experimental validation.

The proposed metamaterial design is fully compatible with current manufacturing techniques. All structural elements, including the slender beams (minimum width: 0.15 mm) and outer frame (maximum side length: 10 mm), can be fabricated from aluminum using CNC micromachining or wire EDM, both of which offer the high precision required for maintaining consistent resonance behavior. For larger-scale applications or complex geometries, metal additive manufacturing techniques, such as selective laser melting (SLM), provide an efficient and flexible alternative. The use of a single-phase material simplifies the process by eliminating interfacial bonding challenges common in multi-material systems. Moreover, the periodic tiling of unit cells enables scalable construction, either as 2D surfaces or as volumetric metamaterial blocks, depending on the application need. This manufacturability, combined with geometric tunability, makes the proposed metamaterial a promising candidate for industrial implementation in acoustic wave manipulation devices.