Tunable multi-metamaterials intergrated with auxiliary magnetorheological resonators

In recent years, there has been a surge in interest in utilizing multi-metamaterials for various purposes, such as vibration control, noise reduction, and wave manipulation. To enhance their performance and tunability, auxiliary resonators and magnetorheological elastomers (MREs) can be effectively integrated into these structures. This research aims to formulate the wave propagation analysis of periodic architected structures integrated with MRE-based auxiliary resonators. For this purpose, cantilever MRE beams are embedded into conventional unit cells of square and hexagonal shapes. Integrating MREs into multi-metamaterial structures allows for real-time tuning of the material properties, which enables the multi-metamaterial to adapt dynamically to changing conditions. The wave propagation in the proposed architected structures is analyzed using the finite element method and Bloch’s theorem. The studied low-frequency region is significant, and the addition of MRE resonators leads to the formation of a mixture of locally resonant and Bragg-type stop bands, whereas the basic structures (pure square and hexagonal) do not exhibit any specific band gaps in the considered region. The effect of different volume fractions and applied magnetic fields on the wave-attenuation performance is also analyzed. It is shown that band gaps depend on the material parameters of the resonators as well as the applied magnetic flux stimuli. Moreover, the area of band gaps changes, and their operating frequency increases by increasing the magnetic flux around the periodic structure, allowing for the tuning of wave propagation areas and filtering regions using external magnetic fields. The findings of this study could serve as a foundation for designing tunable elastic/acoustic metamaterials using MRE resonators that can filter waves in predefined frequency ranges.


Introduction
In recent years, architected materials have become a popular area of research due to their ability to exhibit unprecedented properties, such as a high strength-to-weight ratio, unusual mechanical properties, and multiple functions.Architected materials can be classified into two types: metamaterials and lattice materials.Metamaterials are artificial materials that have properties not found in natural materials, and they are created by arranging sub-wavelength structures in a periodic pattern.Lattice materials, on the other hand, are materials that consist of a repeated unit cell structure with a well-defined geometry.One of the key advantages of architected materials is their ability to exhibit properties that are not available in natural materials.For example, researchers have developed architected materials that can bend light in unusual ways [1], leading to the development of novel optical devices.Similarly, architected materials can be designed to have high strength and stiffness while remaining lightweight, making them ideal for use in aerospace applications where weight reduction is critical.Another area where architected materials find applications is in energy absorption and dissipation.The ability of these materials to absorb and dissipate energy during impact has led to their use in protective gear and automotive crash systems.The design of architected materials relies heavily on computational modelling and simulation tools, such as finite element analysis and topology optimization.These tune band gap areas.Sepehri et al [25] tuned the stop band area with piezoelectric actuators and electrical stimulation.These methods help tune the band gap area; however, they all have limitations like low time response, wires, and required external medium.As an unconventional method, magnetic field excitation with features of fast response and intelligent adjustability can effectively apply to the development of adaptive metamaterials.
In presence of the magnetic-field activation, metamaterials integrating with magnetorheological elastomers (MREs) can achieve the adaptive regulation of the band gaps and propagation characteristics to rapidly control the low-frequency noise and vibration.Magnetorheological elastomers (MREs) are a type of smart material which has features of magnetic-induced modulus stiffening (MR effect) and magnetostriction.These characteristics can be altered rapidly and reversibly their mechanical properties in response to magnetic fields using low power.The first contribution, magnetic-induced modulus stiffening, refers to the change in the mechanical properties of MREs when subjected to an external magnetic field .Typically composed of a soft elastomeric matrix embedded with magnetically responsive particles, MREs exhibit a remarkable increase in their elastic modulus upon applying a magnetic field.This stiffening effect arises from the alignment and reorganization of the embedded particles in response to the magnetic field, leading to enhanced load-bearing capabilities and improved mechanical performance.Magnetostriction involves the deformation of MRE-based structures due to the magnetic field.When an external magnetic field is applied to an MRE, the embedded particles experience a change in their magnetic orientation, resulting in the rearrangement of the elastomer matrix.This rearrangement induces a strain within the material, causing a change in its overall dimensions.This Magnetostrictive behaviour enables MREs to undergo reversible and controllable changes in shape and size in response to varying magnetic field strengths, making them suitable for applications requiring shape morphing or adaptive structures.Understanding the mechanism behind these effects is crucial for harnessing the potential of MREs in various applications, such as vibration control [26], energy absorption [27], and shape morphing [28].
These unique properties, especially magnetic-induced modulus stiffening, have garnered the attention of researchers in various fields, including acoustics.In the area of acoustics, MREs have been used to design sound absorbers, acoustic metamaterials, and low-frequency wave filters.A few studies have explored the design and performance of tunable multi-metamaterials based on MREs.Some studies have focused on optimizing the resonator parameters to achieve the desired performance [29][30][31], while others have investigated specific applications, such as noise reduction in different industries [32][33][34].
Cao et al [35] introduced an inventive sound absorber concept that incorporated the use of Magnetorheological Elastomers (MREs).In their design, a panel was constructed using micro-perforated sheets and coated with a layer of MREs.Notably, their research demonstrated the ability to dynamically adjust the sound absorption capabilities of this absorber by controlling the mechanical properties of the MRE layer through the application of external magnetic fields.This innovative approach aligns with the broader context of tunable sound absorbers, as highlighted in previous research utilizing Magnetic Particles-Incorporated Magnetorheological Elastomers (MP-MREs) and offers promising potential for enhancing sound absorption applications by manipulating resonance frequencies and acoustic properties.Gorshkov et al [29] conducted research in the field of intelligent acoustic metamaterials with the ability to actively adjust band gaps in real time using magnetic fields.They focused on the idea of creating core-shell 2D-phononic materials, resulting in acoustic metamaterials that can be tuned in real time.These materials have simple structures and offer multiple adjustable band gaps across different frequency ranges.Their approach involves manipulating the interplay between various modes of vibration (known as multimode eigenoscillations) in both the cores and shells of the material, thus allowing for precise control of the band gaps.They employed a multi-resonator phononic metamaterial mechanism within a system of multiple vibrators to create these band gaps.Additionally, they modified the elastic properties of magnetoelastic materials, which contributed to the tunability of the band gaps.Wang et al [36] extended the use of Magnetorheological Elastomers (MREs) and Gradient Resonators (MGRs) to create a novel metamaterial plate, focusing on low-frequency and broadband flexural wave manipulation.Their innovation leveraged adaptive magnetic field adjustments to achieve low-frequency operability, broadband capabilities, and tunable band gap characteristics.The study introduced a bandgap prediction model based on the Plane Wave Expansion (PWE) method, incorporating equivalent stiffness terms to estimate the magneto controlled MRE modulus.Numerical analyses validated the advantages of the gradient resonator design and effective MRE tuning, resulting in significant low-frequency bandgap broadening.Zhang et al [37] introduced an innovative magnetic field-induced asymmetric mechanical metamaterial employing Hard Magnetic Active Elastomers (hMAEs).This remotely controllable metamaterial demonstrated tunable band gaps spanning a wide low-frequency spectrum.
Their metamaterial design incorporated resonating units comprised of hMAE, supported by highly deformable curved beams within an elastomeric matrix.When exposed to a magnetic field, these resonating units entered an unstable regime, undergoing significant configuration and stiffness variations.These controlled transformations had a profound impact on elastic wave propagation.The magnetoactive metamaterial exhibited the remarkable capability of tuning bandgaps over a broad low-frequency range, enabling remote and reversible control of its performance.
Studies on magnetorheological elastomers (MREs) have underscored their vast potential in sound control device development, notably in the creation of adaptable sound absorbers and acoustic metamaterials.MREbased solutions have emerged as efficient and versatile means of addressing diverse sound control requirements.
Research investigating the interplay between structural shapes and band-gap frequencies in magnetorheological (MR) acoustic metamaterials has revealed significant insights.This exploration demonstrates that manipulating the geometry of structures within MR metamaterials provides precise control over band-gap frequencies, emphasizing shape as a crucial tunability factor [38,39].Resonators, ranging from simple to complex shapes, have been extensively studied, showing that diverse shapes can generate unique bandgap features, offering enhanced control over acoustic wave propagation.Additionally, researchers have explored the impact of nonlinear effects on shape and band-gap frequencies, potentially leading to adaptive materials adjusting their band gaps in response to external conditions [40].The practical applications of this research, particularly in noise reduction, vibration control, and acoustic cloaking, aim to create materials with superior performance for effective noise mitigation, improved vibration damping, and adaptive acoustic devices [41].
Xu et al [48] delved into the application of magnetorheological elastomers (MREs) in 3D locally resonant acoustic metamaterial (LRAM) cores for vibration isolation.Through the integration of MRE cladding, the study showcased the controllable manipulation of elastic band structures and bandgap characteristics in cubic and spherical LRAM units using external magnetic fields and variations in MRE cladding thickness.Employing the finite element method, the research analyzed elastic band structures, transmission, and vibration modes.The results demonstrated the feasibility of using an external magnetic field to control the LRAM, allowing the manipulation of the MRE cladding's elastic modulus or thickness to alter the LRAM elastic bandgap's central position and width, showcasing practical implications for MRE isolation bearings.Furthermore, the study considered lattice-based metamaterials for adaptive designs, focusing on square and hexagonal lattices.The researchers strategically introduced MRE-based resonators at the joints of tunable unit cells to enhance lowfrequency vibration band gap coverage.The study explored the adaptability of these architected structures in modulating band gaps under varying applied magnetic fields, scrutinizing the dynamic evolution of the band gap and filtering area.Investigating critical facets, the research contributes to the expanding field of acoustic metamaterials and advances the innovative frontier of MRE-based sound control technology [42].In addition, Jafari et al [43] considered a totally MRE-created square lattice, aiming to understand the effect of changing the magnetic field around the MRE-based lattice without resonators.While they provided a specific design for square lattice metamaterials, their unit cell was entirely made of MRE material, making it challenging to adapt easily.In contrast, the present manuscript proposes non-magnetic materials as the main unit cells and MREbased materials as resonators, aiming for easier adaptability and providing more characteristics about tunable materials.
In an effort to push the boundaries of innovative applications of MREs, this research focuses on crafting tunable multi-metamaterials for acoustic wave manipulation.Specifically, it explores the application of square and hexagonal lattices, addressing their deficiency in low-frequency vibration band gap coverage.The introduction of MRE-based resonators strategically enhances band gap regions, elevating the overall sound control efficiency of these structures.The study examines the adaptability of these architected structures in modulating band gaps under varying applied magnetic fields, scrutinizing the dynamic evolution of the band gap and filtering area.This comprehensive exploration establishes a nuanced relationship between the band gap area, iron volume fraction, and the applied magnetic flux density, contributing not only to the field of acoustic metamaterials but also advancing the innovative frontier of MRE-based sound control technology.

Materials and methods
Figure 1 illustrates the proposed magnetoactive metamaterial design.The systems consist of in-plane (2D) arrays of conventional periodic non-magneto active beam configurations (in the form of square and hexagonal patterns as shown in black color) with added magnetoactive resonators (red beams added to the joints).The proposed resonators are made of magnetorheological elastomers with specific iron volume fractions.In order to study the effect of magnetic flux (  B ) on wave propagation of the tunable periodic structure, it is exposed to the magnetic flux, as shown in figure 1(c).The magnetic flux alters the stiffness of the magnetoactive beams and, consequently, the natural frequency of the resonators, which helps to tune wave propagation properties.In both unit cells, the length of passive non-magnetoactive beams is considered to be L = 5 mm, and the width and thickness of these beams are equally calculated using the relation presented by Phani et al [44], as for slenderness ratio of l = 50 in which d is considered to be width and thickness.Length, width and thickness Vectors of ( ) , are the unit vectors along the (X, Y) directions that are shown in figure 1(c).
Due to the periodic and lattice nature of the proposed configurations and based on the concept of Bravais lattice, one can effectively choose and study a reference unit cell and then generalize the results to all its close and distant neighboring cells using some periodicity-based methods, such as the Bloch theorem [45].
The displacement of all points in the periodic structure can be subsequently defined based on the points in the reference unit cell.Thus, it is practically possible to formulate the governing equations of motion for the reference unit cell and then generalize it for the entire architected metamaterial using periodic boundary conditions.
The governing equations of motion for the unit cell can be generally presented in finite element form as [43]: where M , nr C , nr and K nr represent the non-reduced (subscript of nr) mass, damping, and stiffness matrices of the unit cell, respectively.Also,  F T describes the total external forces applied to the unit cell.In order to study wave propagation analysis, the displacement vector of the unit cell (  u ) under harmonic excitation can be considered as harmonic in the form of   = w u qe , i t in which  q is the nodal displacement amplitude vector, and w is the excitation frequency.Thus, by considering the harmonic excitation, the equation of motion may be described as: Based on the Bloch theorem, the displacement of any point in a specified unit cell, (  q ), can be described with an equation related to the non-reduced displacement of the reference unit cell (  q nr ) as: where m and n represent the position vector of a point inside the periodic metamaterial and k 1 and k 2 represent the components of wave vector (κ).By substituting equation (3) in equation (2), the governing equations of motion for the periodic metastructure can be considered as follows: In order to consider Bloch-Floquet periodic boundary conditions and solve equation (4) for the periodic metamaterial, the unit cell should be transferred to the reciprocal space (k-space) shown in figure 2. This transformation is applied to the direct vectors (   a a , 1 2 ) to evaluate the reciprocal vectors ( ).For the square lattice, the reciprocal vectors are and for the hexagonal lattice, these are ( ) Based on these reciprocal vectors, the reciprocal space for each periodic structure is presented in figure 2: After applying the periodic boundary condition in the reciprocal space, one can conduct wave propagation analysis in reciprocal (k-space) using the following governing equation: where M , r C , r and K r are reduced periodic mass, damping and stiffness matrixes, respectively, and  q r is the reduced displacement vector in the periodic boundary condition.
Magnetorheological elastomers with silicone rubber (Mold Max, Smooth-on) as the host matrix impregnated with different volume fractions of magnetic iron particles are considered to develop beam-like resonators integrated into the joints of the periodic structure (red beam in figure 1).Under the application of the magnetic flux, the elastic properties of these magnetomotive resonators can be adaptively altered due to their field-dependent material properties.The magnetic flux also induces an external magnetic force (Maxwell force).The Maxwell force vector (  F T ), applied over the volume (V ) can be defined as [46]: where T is the strain-displacement matrix [47] and s M is the Maxwell stress tensor, which can be represented as [48]: M in which  B and  H are the magnetic flux density and applied magnetic field strength in the magnetoactive elastomer, respectively, and I is the identity tensor.It is noted that the magnetic flux density is related to the magnetic field strength as: 0 where  M is the magnetization vector field which expresses the density of permanent or induced magnetic dipole moments in a magnetic material.By substituting the relation between the magnetization vector and applied magnetic field strength ( , one can write: where c is the magnetic susceptibility and m 0 is the permeability in the vacuum (m p = ´--Hm 4 10 0 7 1 ).The governing equations of motion for the periodic structure in the presence of applied magnetic flux density,  B, are solved considering equations ( 6)-( 9) in equation (5), and also taking into account the Neo-Hookean hyperelastic material model for a nearly incompressible magnetoelastic medium [49,50], linear elastic wave motion [51,52], and field-dependent material properties of magnetorheological elastomers [26].Equation (5) can thus be finally cast into the following form: in which the reduced magnetic field-dependent mass matrix (M r B , ) and stiffness matrix (K r B , ) of the periodic structure and resonators depend on various factors, such as the external magnetic flux (  B ), elastic modulus (E), reduced wave vector (κ), unit cell geometry, density, and magnetic susceptibility (c) of the elastomer, which are all influenced by the specific amount of iron volume fraction present in the material.It is noted that equation (10) has been solved in the Reciprocal space (figure 2) for both hexagonal and square lattices.

Results and discussion
The proposed 2D periodic architected metamaterial with embedded magnetoactive resonators is subjected to a magnetic field, as shown in figure 1(c).The slenderness ratio of the beam elements in the passive conventional configurations is considered to be λ = 50, with each beam's length being L = 5 mm, considering accotrding to [44].Thus, the thickness (d) and depth of each beam are selected to be = = t d 0.346 mm.Additionally, the conventional passive periodic configurations are made of Ecoflex 00-30, which has a density of 1070 kg m −13 , an elastic modulus of 225 kPa, and a Poisson's ratio of 0.45.
The resonators in this study are made of magnetorheological elastomers (MREs), and their properties are provided in table 1.These properties were obtained through experimental measurements conducted by Lee et al [53] on MREs produced using a combination of silicone rubber (Mold Max, Smooth-on) and varying iron volume fractions of magnetic particles.Considering that the working frequency of the designed filter is below 1000 Hz, the variation of the elastic modulus in MREs, as indicated by Lopez et al [54] and Clayton et al [55], is not found to be significant within this frequency range.Based on this information, it is assumed that the elastic modulus of the MREs is frequency-independent for the purpose of the present study, which investigates the magnetic field-dependent behaviour of MREs and its influence on the metamaterial's band gaps.The effective magnetic susceptibility difference (∆ 2) for the different iron volume fractions (f) of MREs (ranging from 3.14% to 18.48%) as well as their elastic modulus (E), density (ρ) and Poisson's ratio (ν) are presented in table 1.The Multiphysics model of the proposed adaptive MRE-based metastructure with periodic boundary governed by equation (10) has been developed in COMSOL Multiphysics.It is important to note that in the analysis, the MRE material is treated as a hyperelastic material.Consequently, the influence of damping is considered, using a damping ratio (ζ) calculated as ( ) w abs represents the absolute part of the frequency, respectively.

Comparison of frequencies between lattices without and with integrating resonators and the mechanism of the band gap generation
In order to demonstrate how resonators enhance the wave propagation properties of conventional lattices, the band diagrams of the real part of the natural frequencies of square and hexagonal lattices with and without resonators are shown in figures 3 and 4, respectively.After evaluating the damping ratio for square and hexagonal lattice structures, both with and without resonators, it is determined that the resulting damping ratio (z ) is lower than 0.003.This implies that the damping has a negligible influence on the analysis of free wave motion and the dispersion relation [56].Therefore, all subsequent results are derived solely from the real part of the frequencies.
Resonators are made of magnetorheological elastomer with an iron volume fraction of 3.14%.As shown in figure 3(a), the conventional unit cell (without a resonator) of the square lattice is ineffective in filtering noise Table 1.Material properties of magnetorheological elastomers with different volume fractions [53].In this context, figure 3(d) serves as a pivotal visualization, illustrating the intricate process behind the generation of band gaps, specifically targeting the first and second band gap areas.These two band gap regions manifest themselves between the 3rd and 4th branches (w) as well as between the 8th and 9th branches.Notably, the vibrational modes of the conventional square lattice in the Multi-Metamaterial lattice exhibit no discernible displacement.In contrast, the resonators' mode shapes and vibrations become instrumental in defining these specific band gap areas at the respective frequencies.What makes these band gaps particularly noteworthy is their origin-they stem from the local resonance effect.This characteristic offers the unique advantage of adjustability, enabling the creation of adaptive wave filters tailored to specific needs and applications.
The behavior of the hexagonal lattice differs significantly from that of the square lattice in the absence of a resonator.Notably, the hexagonal lattice demonstrates the presence of a distinct band gap region nestled between the 6th and 7th vibrational branches, a feature visually represented as a shaded gray area in figure 4(a).This intriguing phenomenon aligns with findings reported in selected references [15,57] and is further elucidated through the mode shapes showcased in figure 4(b) (specifically denoted as B and C), which unmistakably signify a Bragg-type bandgap between the 6th and 7th branches.Upon the incorporation of resonators into this lattice structure, a remarkable transformation occurs, giving rise to new and noteworthy band gap regions within the band diagram.These newly observed band gaps are prominently highlighted in a vivid shade of orange in figure 4(c).To comprehensively understand the origin of these emergent band gaps, we must consider various underlying mechanisms, including locally resonance, Bragg-type phenomena, and their complex interplay.Notably, within the lattice structure, points ¢ A and ¢ C point to locally resonant bandgaps, which, as we demonstrate in this manuscript, are subject to precise control using the methodologies outlined.Examining figure 4(d), we observe that the conventional hexagonal lattice exhibits no discernible displacement in these specific mode shapes under consideration.In contrast, the resonators within the lattice exhibit pronounced vibrations in their natural modes.Mode ¢ B , however, stands out as it manifests a distinct behavior, reflecting a mode shape that is a fusion of the fundamental hexagonal lattice structure and the resonators.This intriguing observation prompts us to question whether the initially identified gray band gap can be unequivocally attributed to a locally resonant bandgap.While the original band gap region undergoes size reduction and shifts to lower frequencies as a result of this modification, the introduction of resonators introduces an advantageous feature.It bestows the lattice with adaptive stop bands at higher frequencies, and notably, these stop bands can be precisely manipulated by altering the material properties of the resonators through magnetic flux adjustments.This adaptability opens promising avenues for tailoring the lattice's performance to specific applications and needs.
When incorporating resonators into square and hexagonal lattice unit cells, two key distinctions significantly influence the lattice's behavior and band-gap frequencies.Firstly, the arrangement of resonators differs: square lattices accommodate four resonators per joint in accordance with their four-fold rotational symmetry, while hexagonal lattices feature three resonators per joint to match their inherent six-fold rotational symmetry.This dissimilarity in resonator placement directly affects how resonator modes interact with lattice modes, impacting the formation of band gaps.Secondly, the coupling strength between resonator modes and lattice modes varies due to the differing number of resonators per joint.This disparity influences the hybridization between localized resonator modes and extended lattice modes, leading to distinct modifications in band-gap frequencies.These nuances highlight the critical role of lattice symmetry and resonator coupling in shaping band-gap characteristics and the importance of considering them in the design of lattice structures for specific applications.

Magnetoactive metamaterials in different iron volume fractions 3.2.1. The influence of the magnetic field on material properties of the MRE-based materials
To investigate the influence of the magnetic field on material properties of the MRE-based materials, the relationship between Young's Modulus and the magnetic field is examined in figure 5, considering various iron volume fractions.The findings indicate that the disparity in Young's Modulus between lower and higher magnetic fields becomes more pronounced with increasing iron volume fraction.This implies that the dependency of Young's Modulus on the magnetic field is more significant in higher volume fractions.

Influence of iron volume fraction on adaptive wave propagation properties for square-lattice and hexagonallattice metamaterials under application of different magnetic fields
In this section, wave propagation analysis was conducted on the proposed square and hexagonal unit cells with embedded MRE-based resonators to investigate the impact of changing iron volume fraction of added resonators, f, and external magnetic flux density, B, in the range of 0 to 200 mT, on their wave propagation properties and band gap areas.Firstly, the square periodic unit cell was considered, and branch diagrams for four different iron volume fractions and two magnetic fields (B = 75 mT and B = 200 mT) were obtained.The results are displayed in figure 6.Specifically, two band gap areas were selected for discussion: the first band gap area is colored in yellow between the 3rd and 4th branches, and the second band gap area is colored in crimson between the 8th and 9th branches.This helps to observe the variation of these band gaps under the different iron volume fractions and applied magnetic flux densities.It can be realized that the branches related to these areas are constant lines representing local resonance, irrespective of the amount of volume fraction of magnetic particles and applied magnetic flux densities.The first band gap area is crucial because it is in the range of low frequencies  The analysis of the results indicates that the band gap frequencies have undergone substantial adjustments, resulting in an approximately two-fold increase.However, these changes are not uniform across all the selected band gap areas.For instance, the second band gap (crimson region) exhibits distinct modifications that are dissimilar to those observed in the first band gap.This suggests that the external magnetic flux affecting the band gap properties may have different impacts on different parts of the band structures.
When the magnetic field is increased, it exerts a stronger force on the magnetic particles, causing them to align more closely with the field direction.This increased alignment of the magnetic moments within the particles enhances their cooperative interactions, leading to a higher elastic modulus of the resonators.The changes in the elastic modulus of the MRE resonators have a direct impact on the material's band-gap frequency.This, in turn, influences the phononic band structure, leading to shifts in the band-gap frequencies.Essentially, the enhanced magnetic interactions result in alterations in the material's ability to control the propagation of acoustic or mechanical waves, leading to shifts in the band-gap characteristics.
As shown in figure 6 (a/c/e/g), altering the volume fraction of resonators results in noticeable changes in the number of bandgap regions within the 0-1000 Hz frequency range.The observed variation in the quantity of these band gaps across different iron volume fractions is a crucial observation, indicating intricate material behavior.According to literature [58][59][60][61], this phenomenon is likely attributed to the evolving mechanical and magnetic characteristics of the material as the iron volume fraction undergoes changes.Siebert et al [59] demonstrated that applying a magnetic field increases the induced yield stress in MRE-based materials.Additionally, Shah et al [58] showed that different percentages of magnetic particles inside MR elastomers cause changes in magneto-viscous properties of ferrofluid.Therefore, comparing the effects of different magnetic particles on bandgap areas would have extensive value in research.For example, at an iron volume fraction of 3.14%, the conditions may not favor the formation of additional band gaps.However, as the iron volume fraction increases within the range of 8.58% to 16.27%, significant transformations occur in the material's stiffness and magnetic responsiveness.These transformations have the potential to facilitate the emergence of additional band gaps.The noticeable increase in the width of both the first and second band gaps as the iron volume fraction rises can be attributed to shifts in the material's effective stiffness and magnetic properties.Higher iron volume fractions may induce increased stiffness, leading to modifications in resonant frequencies and consequently influencing the width of these band gaps.Furthermore, the presence of a greater number of iron particles may intensify the material's magnetic response, which, in turn, impacts the magnetic fielddependent properties of the metamaterial.As In figure 3, the formation of bandgaps within the first mode (between the 3rd and 4th branches) and the second mode (between the 8th and 9th branches) is a result of a locally resonant bandgap mechanism.To understand the mechanisms responsible for altering the frequencies of these bandgaps, it is essential to examine the manipulation of material properties within MRE-based resonators.This manipulation leads to changes in the inherent natural frequencies of the resonators due to variations in magnetic fields.For a more comprehensive grasp of this intriguing phenomenon, figure 7 presents the mode shapes of a multi-metamaterial square lattice at a specific volume fraction of f = 13.94%.We investigate the impact of two distinct magnetic field strengths, denoted as B = 60 mT and B = 160 mT, within this context.To facilitate understanding of the related bandgaps in the first and second bandgap areas, namely the 3rd, 4th, 8th, and 9th branches, specific branches are color-coded with blue, red, green, and yellow dots, respectively.Figure 7(b) reveals that the mode shapes related to points A and Dare purely locally resonant and depend on the vibrational modes of the resonators (first and second mode shapes).In contrast, points B and C exhibit combinations of two mode shapes of the base square lattice and the resonators.When the magnetic field is increased to B = 160 mT, figure 7(d) demonstrates that the mode shapes of the specific branches remain unchanged, but their magnitudes increase.To facilitate a detailed analysis of these specific points, as shown in two dispersion curves (figure 7(a) and (c)), their values are presented in table 2.
Analysis of table 2 reveals that increasing the magnetic field around the multi-metamaterial square lattice results in an overall increase in all frequencies due to the rising elastic moduli of the resonators.An intriguing observation in table 2 and figure 7 is that, with an increasing magnetic field, the bandgap areas become wider.This phenomenon can be attributed to the effect of the magnetic field on the natural frequencies of MRE-based materials.As explained by Eloy et al [62], increasing the magnetic field has a greater impact on higher mode shapes compared to lower natural modes, affecting the thickness of the bandgap area under varying magnetic field strengths.
To better realize the variation in the band gap areas, the evolution of the first and second band gap areas of the proposed multi-metamaterial square lattice with respect to magnetic flux densities and different volume fractions of iron particles in the MRE resonators are shown in figure 8.These diagrams depict all the band gap areas within the 0-1000 Hz range.However, the focus is on the first and second band gap areas, and their upper and lower bounds, together with mean values (i.e., the mean of the upper and lower bonds of the band gaps), are shown in all Figures.These lines are helpful for precisely identifying the presence of stop band regions and their  The results show that increasing the magnetic flux density not only raises the frequency of this specific band gap but also expands the thickness (i.e., the difference between the upper and lower bonds) of the band gap.For instance, the thickness of the band gap for the hexagonal lattice with an iron volume fraction of f = 16.27%changes from 19.3 Hz under 75 mT to 57.2 Hz under 200 mT.The increase in the band gap frequency with the enhancement of magnetic field strength can be attributed to the phenomenon where higher mode frequencies are influenced more significantly than lower frequencies by the magnetic field.This differential effect results in the band gap shifting to higher frequencies as the magnetic flux density increases.Additionally, this alteration in frequency has a direct impact on the thickness of the band gap, causing it to expand.This phenomenon aligns with findings in a research paper by Eloy et al [62] , which also studied MRE-based resonators.In both cases, the magnetic field's influence on the material's mechanical properties and resonance characteristics results in similar outcomes.This represents an astonishing increase of almost 196% in the first band gap area by just increasing magnetic flux density in the amount of 125 mT (from 75 to 200 mT).
The mechanism responsible for generating bandgaps within a hexagonal metamaterial, specifically within the first bandgap area (between the 6th and 7th branches, colored blue and red in figures 10(a), (c)) and the second bandgap area (between the 15th and 16th branches, colored green and yellow in figures 10(a), (c)), is primarily attributed to locally resonant phenomena.To gain insight into the dynamic behavior of these bandgaps concerning variations in magnetic field strength, we investigate the mode shapes of these specific band structures within a multi-metamaterial hexagonal lattice, with a fixed volume fraction of f = 13.94%,under two distinct magnetic field conditions, B = 60 mT and B = 160 mT.resonators, specifically the first and second mode shapes.In contrast, points B and C reveal a combination of two mode shapes originating from the underlying square lattice and the resonators.As the magnetic field is elevated to B = 160 mT, figure 10(d) demonstrates that the mode shapes of the specific branches remain unaltered but experience an increase in magnitude.Notably, an essential distinction between square and hexagonal lattices emerges from this analysis, shedding light on the mechanism of bandgap generation and alteration.The 19th branch of the two specific band diagrams in figures 10(a), (c) is marked with black dots.The starting points of these specific branches are denoted as E and ¢ E for B = 60 mT and B = 160 mT, respectively.It is noteworthy that the mode shape of this specific branch undergoes changes with increasing magnetic field strength, leading to the expansion of the bandgap between the 13th and 19th branches in higher magnetic fields.For an in-depth analysis of these specific points, as depicted in the two dispersion curves (figure 10(a) and (c)), their corresponding values are presented in table 3. Examination of table 3 reveals that augmenting the magnetic field around the multi-metamaterial hexagonal lattice results in a collective increase in all frequencies, primarily attributed to the heightened elastic moduli of the resonators.
A striking observation in table 3 and figure 10 is that, as the magnetic field strength increases, the bandgap areas become more expansive.This intriguing phenomenon can be ascribed to the magnetic field's influence on  the natural frequencies of Magnetostrictive Resonance Elasticity (MRE)-based materials, as previously discussed by Eloy et al [62].
The significance of comprehending the correlation between magnetic flux density and wave propagation characteristics may be better understood using figure 11, which shows the results for the variation of the band gap areas for hexagonal lattice with respect to applied magnetic flux density and various iron volume fractions.Results clearly show that as the magnetic flux density increases, there is a corresponding increase in the band gaps of materials with higher iron volume fractions.This trend is not limited to any specific band gap region, as similar behavior has been observed across a range of different iron volume fractions within the frequency range of 0-1000 Hz.It is evident that the periodic hexagonal structure with MRE-based resonators and iron volume fractions exceeding f = 11.47%exhibits a remarkable filtering behavior when subjected to an increase in external magnetic flux.This finding suggests that materials with high iron volume fractions could be advantageous for a range of applications that require effective filtering capabilities.This understanding can facilitate the development and optimization of materials with superior functionality for diverse applications.

Influence of magnetic fields on the location of two particular band gaps for square-lattice and hexagonal-lattice metamaterials in different iron volume fractions
The mean lines (average of upper and lower bounds) for two specific band gaps (first and second) are also shown in figure 12 with dashed black and blue lines and distinctive markers.Subsequently, cubic equations are utilized to fit these mean lines using least-square minimization in a Matlab environment, and the resulting data for coefficients of the polynomials are tabulated in tables 4 and 5 for the first band gap regions in various resonators with differing iron volume fractions for both square and hexagonal configurations, respectively.
To facilitate a comparison of bandgap frequencies between square and hexagonal lattice structures, the specific mean bandgap frequencies are plotted in figures 12(c)-(e).The outcomes clearly demonstrate that the bandgap frequency within the square lattice surpasses that of the hexagonal lattice.This disparity can be attributed to the inherent characteristics, overall stiffness, and mass density of these structural configurations.Our choice of beam dimensions aligns with the model put forth by Phani et al [32], as evidenced in their research.They have illustrated that, under identical properties and dimensions, the mode shapes and natural frequencies of the square lattice consistently outperform those of the hexagonal lattice.Indeed, it is evident that variations exist in certain specifications, which highlight the influence of resonators on different bandgaps.Additionally, the number of branches associated with bandgap occurrences may contribute to the disparities observed in bandgap behavior between square and hexagonal lattice structures.
Results in tables 4 and 5 have also been shown in figures 10(a) and (b) to gain a better understanding of the relationship between first band gap mean line frequency, applied magnetic field and iron volume fractions.The results clearly demonstrate that the influence of iron volume fractions becomes increasingly significant at higher  magnetic flux densities, particularly evident when examining the zoomed regions in both lower and higher magnetic flux areas.Further examination of the results reveals a saturation phenomenon occurring as the iron volume fraction exceeds 11.47%.Notably, it is observed that the maximum frequency tends to occur consistently at an iron volume fraction of 11.47%, regardless of the magnetic flux density.At lower iron volume fractions (ranging from 3.14% to 11.47%), there is a positive correlation between iron volume fraction and bandgap frequency.This positive correlation is attributed to the changes in the material's mechanical properties, such as the elastic modulus, which are influenced by the presence of iron particles.However, as the iron volume fraction increases beyond 11.47%, it eventually reaches a point of magnetic saturation.At this saturation point, the material's magnetic properties no longer respond significantly to additional iron content, leading to a plateau in its behavior.In practical terms, as the iron volume fraction continues to increase above 11.47%, it results in an increase in the total mass of the resonators due to the higher iron content.However, this increase in mass does not translate into a proportional increase in the material's elastic modulus.Instead, the material's magnetic properties remain relatively constant beyond this point, causing a decline in the bandgap frequency.This phenomenon occurs because the material reaches a state where additional iron does not significantly impact its magnetic response, and the increase in mass starts to outweigh any potential benefits in terms of resonance behavior.The bandgap frequency then appears to decline with increasing iron volume fraction due to this magnetic saturation effect.This can be better realized in figures 13(a) and (b), which show the variation of the band gap mean line frequency with respect to iron volume fraction at magnetic flux densities of 75, 125 and 200 mT.
As show in figure 13 and table 6 maximum frequency value for the different iron volume fraction under application of different magnetic fields is different for square and hexagonal lattices which shows the different effect of magnetic field on MRE-based metamaterials.
Results in figure 13 confirm that the optimum band gap area occurs at an iron volume fraction of approximately 12%, regardless of the applied magnetic flux density.As expected, increasing the magnetic flux density substantially increases the band gap mean value.Results also show that both square and hexagonal lattices generally exhibit similar behaviour across different magnetic flux densities, irrespective of the number of resonators in each joint and the basic structures.
Cubic polynomials representing the mean frequency associated with the second band gap areas are formulated using least square error minimization for both square and hexagonal lattices, and results are provided in tables 7 and 8 as well as figures 14(a) and (b), respectively.Results suggest that second band gap areas  exhibit slightly different trends compared to those for the first band gap areas, which may be attributed to the existence of the Bragg-type band gap at a higher frequency range.Unlike square lattice, results for hexagonal lattice in figure 14(b) show that the relationship between iron volume fractions and bandgap frequencies varies depending on the magnetic flux density.For magnetic flux densities below 50 mT, it is observed that iron volume fractions in the range of 3.14% to 13.94% exhibit a negative correlation with bandgap frequency.The apparent negative correlation between iron volume fraction and bandgap frequency within a specific range is influenced by the complex interplay of these two mechanisms, leading to non-linear behaviors.It's crucial to recognize that the behavior of bandgaps in such hybrid materials is inherently non-linear.The simultaneous presence of local resonance and periodicity introduces intricate interference patterns and resonant interactions that are highly sensitive to changes in material properties.However, it's noteworthy that when the iron volume fraction surpasses 13.94%, a positive correlation emerges with bandgap frequency.As the magnetic flux densities increase beyond 50 mT, this trend is reversed, and the band gap frequencies increase with increasing iron volume fractions up to around 12% and then remain almost unchanged by any further increase.These findings suggest that the band gap behaviour of the structures is influenced by the interplay between magnetic field effects on resonators and the existence of Bragg-type band gap areas, which result in complex and non-linear relationships between the applied magnetic flux density, iron volume fraction, and band gap frequency.The results presented in figure 14 also reveal that the behaviour of the second band gap in both square and hexagonal lattices changes significantly around a magnetic flux density of B = 75 mT.To further elucidate the results in this critical magnetic flux density region, results for variation of the second band gap mean line frequency versus various iron volume fractions under magnetic fields of B = 75 mT, 125 mT, and 200 mT are presented in figure 15. Results in figure 15 indicate that the second band gap is highly sensitive to the magnetic flux density.Specifically, as the magnetic flux density increases from B = 75 mT to B = 200 mT, the bandgap frequency exhibits varying trends across different iron volume fractions.While there is a sharp increase in bandgap frequency for iron volume fractions below 12%, it is important to note that at a magnetic flux density of 75 mT, there is a slight decrease in bandgap frequency within the range of 3.14% to 6%, rather than a sharp increase, as depicted in figure 15(a).Moreover, beyond f = 12%, the band gap frequency remains nearly constant, confirming that a further increase in magnetic flux density does not affect the second band gap.
In figures 13 and 15, the initial increase followed by a decrease in the band gap with the iron volume fraction can be attributed to the interplay of two key factors: magnetic-induced modulus stiffening (MR effect) and density of the resonators.Initially, as the iron volume fraction increases, the magnetic-induced modulus stiffening (MR effect) becomes more prominent.This effect happens because the stronger magnetic field encourages stronger interaction among the magnetic particles.As a result, the band gap initially increases with the iron volume fraction.However, as the iron volume fraction continues to increase, another factor comes into play, which is the density of the resonators.At higher iron volume fractions, the density of the resonators increases, leading to a decrease in the natural frequency of the resonators.This decrease in natural frequency affects the overall band gap behaviour, causing it to decrease despite the MR effect.Therefore, the observed trend in figures 13 and 15, where the bandgap initially increases and then decreases with the iron volume fraction, can be attributed to the intricate interplay between the magnetic-induced modulus stiffening (MR effect) and the frequency analysis suggests that the magnetoactive periodic structures can be used as effective wave filters in a variety of applications.
The difference between hexagonal and square lattice configurations can be attributed the underlying lattice symmetries, which fundamentally shape how waves interact within these structures.A hexagonal lattice boasts a six-fold rotational symmetry.On the other hand, a square lattice features four-fold rotational symmetry along with mirror symmetry.These distinct symmetries influence the dispersion of waves and the permissible propagation directions, leading to notable differences in the iso-frequency diagrams.Importantly, given the magnetostrictive nature of MRE-based resonators, the arrangement of unit cells significantly affects the coupling between mechanical strain and magnetic fields.As hexagonal unit cell has three resonators in each node and the square unit cell has four resonators in each node, their behavier has significanly diferance in isofrequencies.

Conclusions
Magneto-adaptive wave propagation has numerous potential applications in the industry due to its ability to provide real-time wireless tunability to material properties, enabling dynamic adaptation to changing environmental conditions.In the field of vibration control, noise reduction, and wave manipulation, there has been a growing interest in the use of multi-metamaterials with auxiliary resonators and magnetorheological elastomers (MREs) to enhance their performance and tunability.In this study, the integration of MRE resonators into multi-metamaterial structures was investigated to advance and fine-tune wave propagation analysis in magnetoactive periodic architected structures.Cantilever MRE beams were added to conventional unit cells of square and hexagonal shapes, and their wave-attenuation performance was analyzed under different volume fractions and magnetic fields.The studied low-frequency region is significant, and the addition of MRE resonators led to the formation of a mixture of locally resonant and Bragg-type stop bands, whereas the basic passive conventional structures (pure square and hexagonal) did not exhibit any specific band gaps in the considered region.Results can provide important guidance on the design of tunable elastic/acoustic metamaterials using MRE resonators to filter waves in predefined frequency ranges.In addition to vibration control and noise reduction, magneto-adaptive wave propagation has potential applications in fields such as acoustic insulation, sensing, and communication.The use of external magnetic fields to tune wave propagation areas and filtering regions in real-time can lead to the more efficient and effective performance of these materials in a variety of industrial applications.The finite element method and Bloch's theorem were employed to analyze wave propagation in the considered architected structures, and the results showed that the area of band gaps changes, and their operating frequency increases by increasing the magnetic flux around the periodic structure.This allows for tuning wave propagation areas and filtering regions with external magnetic fields.Iso-frequency diagrams are also provided under various magnetic flux densities.These diagrams play an important role in the analysis of vibration in periodic structures, enabling researchers to identify the vibrational modes and frequencies of the system.This information is also crucial for designing and optimizing structures for vibration isolation, damping, and control.The findings of this study could serve as a foundation for designing tunable elastic/acoustic metamaterials using MRE resonators that can filter waves in predefined frequency ranges.

2 )
of each configuration are presented in blue shaded areas as shown in figures 1(a) and (b).for the representative vectors for square lattice are 

Figure 1 .
Figure 1.Schematic of 2D periodic architected metamaterial with Auxiliary Resonators (Red resonators) and their specific unit cells areas (shaded part) and direct vectors (green arrows): (a) Square lattice (b) Hexagonal lattice (c) periodic structure subjected to the magnetic flux.

Figure 2 .
Figure 2. Reciprocal space for (a) square lattice and (b) hexagonal lattice with their first Brillouin zone (blue shaded area), irreducible Brillouin zone (orange shaded area) and high symmetry points (red dotted).

Figure 3 .
Figure 3.Comparison of the band diagram of square lattice: (a) Conventional lattice (without resonator); (b) Eigenmodes shapes of conventional lattice in specific points of A, B and C; (c) multi-Metamaterial lattice (with resonators); (d) Eigenmodes shapes of Multi-Metamaterial lattice in specific points of ¢ A , ¢ B and ¢ C .
and vibration within the 0-1000 Hz range, which is potentially harmful to both biological and industrial components.The inclusion of magnetoactive beams as resonators in this structure (as illustrated in figure1) led to the emergence of band gap regions in the branch diagram, which is dedicated in orange in figure3(c).To gain a deeper insight into the underlying mechanisms responsible for generating these band gaps, the three primary vibrational modes of the conventional square lattice at specific high-symmetry points denoted as X are scrutinized in figure3(b).For a direct comparison and correlation, corresponding points in the form of ¢ A , ¢ B , and ¢ C within figure 3(d) are established to align with those identified in the conventional square lattice.

Figure 4 .
Figure 4. Comparison of the band diagram of hexagonal lattice: (a) Conventional lattice (without resonator); (b) Eigenmodes shapes of conventional lattice in specific points of A, B and C; (c) multi-Metamaterial lattice (with resonators); (d) Eigenmodes shapes of Multi-Metamaterial lattice in specific points of ¢ A , ¢ B and ¢ C .
Hz).For example, the first stop band of the square lattice with resonators with the properties of iron volume fraction f = 3.14% in the magnetic flux density of B = 75 mT is around 132.4-142.4Hz.This band gap area is changed to 236.6-258.4Hz by increasing the magnetic flux density to B = 200 mT.Confirming the

Figure 5 .
Figure 5. Dependence of material properties (Young's Modulus) to the magnetic field for various iron volume fractions.

Figure 7 .
Figure 7.Comparison of the band diagrams of Multi-Metamaterial Square lattice with iron volume fraction of (f = 13.94%): (a) Dispersion curve in B = 60 mT; (b) Eigenmodes shapes of specific points of A, B, C and D in B = 60 mT; (c) Dispersion curve in B = 160 mT; (d) Eigenmodes shapes of specific points of ¢ A , ¢ B , ¢ C and ¢ D in B = 160 mT.

Table 2 .
Frequency of natural modes for square lattice in volume fraction of f = 13.94% in points of A, B, C and D in magnetic field of B = 60 mT and in points of A', B', C' and D' in magnetic field of B = 160 mT.B = 60 mT A = 127.29 Hz B = 142.56Hz C = 505.36Hz D = 640.62Hz B = 160 mT A' = 241.92Hz B' = 269.73Hz C' = 722.69Hz D' = 859.20 Hz relationship with variations in external magnetic flux density.Examination of results reveals that the second band gap areas (represented by the crimson area) exhibit similar behavior for all iron volume fractions.The differences between the lower and upper bounds (thickness) of the band gap areas and start points of these band gap areas are mainly attributed to field-dependent material properties of MRE and the effect of magnetic flux on different effective magnetic susceptibility.For instance, the maximum thickness of the second band gap (i.e., the difference between the upper and lower bonds of the second band gap) for the iron volume fraction of f = 3.14% is 102.46Hz occurring at 125 mT, which is less than the maximum thickness of 139.48 Hz (occurring at 200 mT) for the iron volume fraction of f = 16.14%.The findings suggest that increasing the iron volume fraction in the square lattice with MRE-based resonators can significantly broaden the stopband area, a desirable outcome in wave propagation analysis.These findings can provide important guidance for the design and optimization of periodic structures to enhance their band gap areas for various applications.Same as the square lattice, the band diagrams for the hexagonal lattice with MRE-based resonators are presented in figure 9.These band diagrams are presented for four different iron volume fractions (i.e., f = 3.14%, 8.85%, 11.47%, and 16.27%) in two magnetic flux densities (i.e., B = 75 mT, and 200mT).The yellow and crimson regions, representing the first and second band gap, are distinguished from other regions due to their significant contribution to the locally resonant band gap.The first and second band gaps under different magnetic fields are between the fifth and sixth and 14th and 15th mode shapes, respectively.For instance, the first band gap frequencies for the hexagonal structure with MRE resonators having f = 3.14%, 8.85%, 11.47%, and 16.27% under the magnetic flux of B = 75 mT are 120.9-133.3Hz, 137.59-156.6Hz, 142.5-163.4Hz and 140.6-159.9Hz respectively, which increases to 223.4-248.6Hz, 259.1-305.6Hz, 270.2-324.1 Hz and 261.5-318.7 Hz, respectively, when the magnetic flux density increases to B = 200 mT.
Figure 10(b) elucidates that mode shapes associated with points A and D predominantly exhibit local resonance and rely on the vibrational modes of the

Figure 8 .
Figure 8. Band gap evolution of multi-Metamaterial square lattice with respect to magnetic flux density for different iron volume fractions: (a) f = 3.14%; (b) f = 8.85%; (c) f = 11.47%; (d) f = 16.27% (First band gap: yellow area with the mean line passing through diamonds; Second Band gap: crimson area with the mean line passing through circles).

Figure
Figure Comparison of the band diagrams of Multi-Metamaterial Hexagonal lattice with iron volume fraction of (f = 13.94%):(a) Dispersion curve in B = 60 mT; (b) Eigenmodes shapes of specific points of A, B, C, D and E in B = 60 mT; (c) Dispersion curve in B = 160 mT; (d) Eigenmodes shapes of specific points of ¢ A , ¢ B , ¢ C , ¢ D and ¢ E in B = 160 mT.

Figure 11 .Figure 12 .
Figure 11.Band gap evolution of multi-Metamaterial hexagonal lattice with respect to magnetic flux density for different iron volume fractions: (a) f = 3.14%; (b) f = 8.85%; (c) f = 11.47%; (d) f = 16.27% (First band gap: yellow area with the mean line passing through diamonds; Second Band gap: crimson area with the mean line passing through circles).

Figure 13 .Figure 14 .
Figure 13.Variation of first band gap mean line frequency in various iron volume fractions for magnetic fields of B = 75 mT, 125 mT, and 200 mT: (a) Square lattice ; (b) Hexagonal lattice.

Figure 15 .
Figure 15.Variation of second band gap frequency versus various iron volume fractions under magnetic fields of = B 75mT, 125mT, and 200mT: (a) lattice; (b) Hexagonal lattice.

Table 4 .
Equation of the first band gap mean frequency

Table 5 .
Equation of the first band gap mean frequency