A Design of Tunable Band Gaps in Anti-tetrachiral Structures Based on Shape Memory Alloy

Benefitted from the properties of band gaps, elastic metamaterials (EMs) have attracted extensive attention in vibration and noise reduction. However, the width and position of band gaps are fixed once the traditional structures are manufactured. It is difficult to adapt to complex and changeable service conditions. Therefore, research on intelligent tunable band gaps is of great importance and has become a hot issue in EMs. To achieve smart control of band gaps, a design of tunable band gaps in anti-tetrachiral structures based on shape memory alloy (SMA) is proposed in this paper. By governing the phase transition process of SMA, the geometric configuration and material properties of structures can be changed, resulting in tunable band gaps. Therein, the energy band structures and generation mechanism of tunable band gaps in different states are studied, realizing intelligent manipulation of elastic waves. In addition, the influence of different geometric parameters on band gaps is investigated, and the desired bandgap position can be customized, making bandgap control more flexible. In summary, the proposed SMA-based anti-tetrachiral metamaterial provides valuable reference for the application of SMA materials and the development of EMs.


Introduction
Metamaterials are composite materials/structures with artificial periodic properties, and they have special physical properties and functions that natural materials cannot achieve [1][2][3].Many metamaterials have been widely applied in the fields of electromagnetics, thermodynamics and acoustics [4][5][6][7][8].Elastic metamaterials (EMs) create conditions for vibration rejection and elastic wave control, which can be used for subwavelength focusing through negative refraction [9], energy harvesting [10,11] and sensing [12].However, a common drawback of most EMs is that they are only effective in the designed frequency range.When exposed to complex or variable service environments, their functionality is at risk of failure.Therefore, the band gaps should be adjusted repeatedly during the whole service process.However, for conventional EMs, the adjustment of band gaps during service is challenging because their bandgap properties are severely restricted once they are manufactured.Hence, it is crucial to design EMs with tunable band gaps.Wang [13] has reported a comprehensive review of tunable metamaterials, and there also exist other tunable mechanisms, such as using liquids [14] and infrared radiation [15].In general, the adjustment of band gaps mainly depends on changes in geometric configurations or material properties, which can be achieved using additional external stimuli or smart materials, such as mechanical loading [16], piezoelectric [17][18][19] or magnetic and electromechanical coupling [20,21] and shape memory polymer/alloy [22][23][24].Therein, mechanical loading is a more convenient method in engineering.However, to maintain the deformed state, the metamaterials need to be under loading during the whole process.Meanwhile, to achieve significant changes in structural geometry, the traditional materials should be subjected to large strains, which may result in failure to recover the deformation.In the current design of tunable band gaps, there is a tough balance between broadband tunability and structural stability.Therefore, the purpose of designing tunable EMs is to increase the effective bandwidth, reduce costs and decrease size.It is necessary to establish a tunable structure with a simple form and easy manufacturing process.
In this paper, an SMA-based anti-tetrachiral structure is proposed.SMA is an intelligent alloy with shape memory effect.It can transition between high-temperature austenite and low-temperature martensite.Within the range of maximum phase strain, any deformed state after the phase change can be restored to the initial shape.Meanwhile, it can maintain the phase change state without additional external forces [25].In addition, chiral structures possess the advantages of lightweight and easy manufacturing.They deform through the rotation of the central ring, which drives the connected ligaments to bend [26].By combining different deformation mechanisms and various physical behaviors, it provides more potentiality for tunable band gaps.By controlling the temperature and strain of SMA, the geometric configuration and material properties can be changed simultaneously, and these dual changes increase the ability to tune band gaps.Benefitted from the rapid development of high-temperature SMA, large phase transitions can be reverted [27].Compared with ordinary metamaterials, SMA can realize remarkable changes in configurations without damaging the structure itself.It breaks through the limitation of traditional structures that cannot recover from mechanical loading.The adjustment of band gaps is more convenient and flexible, and the control range of band gaps can also be improved.The combination of anti-tetrachiral structure and SMA provides more possibilities for adjusting band gaps, which will greatly improve the elastic wave manipulation capability.

Design of SMA-Based Anti-tetrachiral Structures
In order to achieve significantly tunable band gaps, the designed structures are required to possess both geometric configurations and material changes.Considering the specified mechanical characteristics under working conditions, a tunable bandgap design of anti-tetrachiral lattice with SMA is proposed in this paper.The anti-tetrachiral lattice structure is composed of slender SMA ligament beams and rings, whose geometry is displayed in Fig. 1b.Therein, the parameters of length S and thickness t of the ligament and radius r of the node circle can be changed to adjust the properties of the structures.The band gaps can be analyzed based on the unit cell with Floquet periodic boundary conditions.The first Brillouin zone is shown in Fig. 1c, and the wave vector SMA has unique shape memory effects.When the temperature is lower than the start temperature of austenite (A s ), but higher than the start temperature of martensite (M s ), SMA is loaded and later unloaded, the elastic strain is recovered and the transformation strain is retained.The austenite phase completely transforms to martensite.Therefore, without external mechanical load, the cell geometry can be fixed in a stable state.When heated to the austenitic transformation end temperature (A f ), the transformation strain can be completely recovered.After SMA is trained by loading-unloading-heating, the structure will switch between the initial state and the phase state, and the configuration and material properties will be changed during the training process.The whole deformation process is shown in Fig. 2.
Benefitted from the deformation mechanism of chiral structures and the self-tuning capability of SMA, the combined design offers more possibilities for tunable band gaps.Specifically, a transition analysis of the proposed SMA-based anti-tetrachiral structure is conducted, and the geometric configurations are displayed in Fig. 3.The whole training process includes three stages: loading, unloading and heating (the detailed process is explained in Sect.3).By the self-tuning shape memory effect, the bandgap characteristics and transmission properties can be controlled through phase transition-induced changes in geometric configuration and material properties.

Phase Transition Analysis
In this section, after briefly describing the constitutive equations for the phase transition of SMA, the phase transformation of the anti-tetrachiral structures with an initial configuration is analyzed using the finite element method, and a new configuration, i.e., the transformed configuration, is obtained including the material properties and geometries.

Constitutive Equation of SMA Phase Transition
This paper adopts the SMA constitutive equation proposed by Lagoudas [28,29].A phenomenological constitutive model of SMA is established using the Gibbs energy function, in which the total Gibbs free energy is decomposed into the following two phases.
where A and M represent the austenite and martensite, respectively.
For the mixed free energy, Lagoudas provided the following form where h(ξ ) is the phase transformation strengthening function where b M , b A , μ 1 and μ 2 are the phase transformation strengthening parameters.
The phase transition of SMA is a dissipative process involving entropy change.According to the second law of thermodynamics, the Clausius-Planck dissipation inequality can be obtained: For any σ and Ṫ , if the inequality in Eq. ( 4) holds, the following constitutive equation can be derived: where ε is the strain tensor and S is the specific entropy.Therefore, the inherent dissipation can be expressed as Reviewing the free energy in Eq. ( 1), the thermodynamic stress conjugated with the internal variable can be determined as follows In Eq. ( 8), represents the difference between the amounts of austenite and martensite.
It can be assumed that the rate change of the transformation strain is proportional to the rate change of the martensite volume fraction, and the direction is along the direction of the equivalent deviatoric stress tensor.The evolution of the transformation strain is given by the flow law below.
∂Φ ∂σ (11) where the phase transition function Φ can be modeled as where Y is a material parameter.Benefitted from the simple form of the macroscopic phenomenological constitutive model, it can efficiently predict various mechanical effects.For more details, please refer to [28,29].

Phase Transition Analysis of Anti-tetrachiral Structures
The basic material properties of SMA used in the antitetrachiral structures are shown in Table 1.Therein, the initial parameters of length S and thickness t of the SMA ligament Austenite heat capacity at constant pressure Martensite heat capacity at constant pressure and radius r of the node circle are set as 10 mm, 1 mm and 4 mm, respectively, as displayed in Fig. 1b.According to Lagoudas' constitutive relationship for SMA under compression/tension, a static analysis is implemented using the finite element method based on COMSOL.First, the geometry of the chiral structure is modeled, and the material properties of SMA are assigned.Then, the finite element model of SMA is established based on the platform provided by COMSOL.The transformed configuration is obtained by applying compressive/tensile force (SMA undergoes phase change in this loading period) and unloading.The transformed configuration can be recovered to the initial configuration through heating.This process is implemented by applying force and temperature loads described in Fig. 4 on the finite element model.The geometric configuration and stress distribution are displayed in Fig. 5. Initially, the structure (Fig. 5a) is in a pure austenite state at a temperature of 260 K (below A s ).A uniaxial compressive load is applied, causing deformation (Fig. 5b).And then, the load is removed while maintaining the temperature, resulting in partial elastic strain recovery and transformation strain retention, as the structure transforms from austenite to martensite.In this case, the transformed configuration is with the geometry shown in Fig. 5c, and the material properties are in the martensite state.To recover the transformed state back to the initial state, the temperature is raised to 350 K, which is higher than A f , as shown in Fig. 5d.
The stress-strain curve (Fig. 6a) shows a maximum strain of 2.5% during the loading process.After unloading, the elastic strain of 0.4% is recovered, and the transformation strain of 2.1% is retained.All transformation strain is recovered after heating.From the displacement curve (Fig. 6b), the chiral structure reaches a maximum displacement of 3.76 mm after loading.The elastic deformation of 0.58 mm is recovered, and the phase change deformation of 3.18 mm is retained after unloading.In Fig. 6c, the martensitic transformation of 50.5% is achieved, which is induced by stress.As shown in Fig. 5, the stress mainly exists in the beam structures, while the ring structures almost have no stress.Therefore, the phase transition mainly occurs in the beam structures, resulting in a change of Young's modulus from 90 to 45.3 GPa (the blue line in Fig. 6d).And Young's modulus of the rings remains at 90 GPa throughout the whole training process, as shown by the green line in Fig. 6d.Finally, the structure is heated to 350 K, and the material properties and geometric configuration return to the initial state.
Under uniaxial tensile load at the same temperature as the uniaxial compressive load, the initial structure undergoes geometric configuration changes and stress distribution, as depicted in Fig. 7.The stress-strain curve for the entire process is shown in Fig. 8a.Loading generates a maximum strain of 4.17%, followed by the elastic strain recovery of 0.57% upon unloading, while retaining a transformation strain of 3.6%.After heating, all transformation strains are restored.The displacement curve in Fig. 8b illustrates a deformation displacement of 5.69 mm during loading, with the elastic deformation of 0.8 mm recovered at the end of unloading, resulting in a phase deformation of 4.89 mm.This process achieves a 79.3% martensitic transformation (Fig. 8c), resulting in a change in Young's modulus from 90 to 35.4 GPa (the blue line in Fig. 8d), while Young's modulus of the rings remains at 90 GPa.Subsequently, upon heating, the material properties and geometric configuration are restored to their initial state.The SMA-based anti-tetrachiral structure enables multiple morphological changes under different loads, offering enhanced possibilities for applications in elastic wave control.

Analysis Model for Tunable Band Gaps
The proposed SMA-based anti-tetrachiral metamaterial provides a simple and effective unit design, which can maintain a stable phase transition state and return to its original structure by controlling the external load and temperature.Therefore, the proposed structure can realize intelligent manipulation of elastic waves.In this section, using the configurations and materials detailed in Sect.3, the bandgap distribution, generation mechanism and transmission are mainly discussed.

Elastic Wave Propagation Theory
The finite element method is commonly used for calculating metamaterials.It has the advantages of strong compatibility, good convergence, a small amount of computation required and high efficiency in calculations.
The governing equation [30] for the elastic wave propagation can be written as where u is the displacement vector; ρ, E and μ are the material density, Young's modulus and Poisson's ratio, respectively; ω denotes the angular frequency; and ∇ represents the differential operator.
Based on Bloch's theorem [31], the solution of phononic crystal can be calculated as where r represents the position vector, k is the Bloch wave vector, i is the imaginary number and a represents the lattice constant.The Bloch's theorem described in Eq. ( 14) is applied to the unit cell boundary along the periodic direction, therein, the displacement field u has to satisfy the following condition Combining the governing equation in the unit cell (Eq.( 13)) and the boundary condition (Eq.( 15)), the eigenvalue equation can be obtained as where K is the global stiffness matrix of the structure, U denotes the displacement vector and M represents the global mass matrix.

Tunable Band Gaps and Transmission Analysis
The energy band structure is studied by solving the eigenvalue equations, and the bandgap distributions of the initial state, uniaxial compression state, uniaxial tension state and recovery state are analyzed, as shown in Fig. 9 When the uniaxial tensile load is unloaded, the lattice size is 49.53 mm.Compared to the initial structure, a new band gap is opened at the low-frequency position, as illustrated in Fig. 9c.The new band gap [8190 − 13,394 Hz, 5204 Hz] is located between the 12th and 13th orders with a large width, which overcomes the problem that the band gap of the initial anti-tetrachiral structure has a narrow width and high position.The second band gap [15,060 Hz,4354 Hz] exists between the 16th and 17th orders.The upper and lower boundaries of the band gap move to low frequency, and the bandgap width is nearly doubled compared to the initial structure.The last band gap appears at the 20th and 21st orders, and the bandgap position is [20,112 Hz,3726 Hz].When the temperature increases above A f , the geometric configuration and material properties of the whole structure change to the initial structure, so the energy band structure diagram is the same as that of the initial structure, as shown in Fig. 9d.
The geometric configuration and material properties strongly influence the width and position of the lattice band gaps.Structural reconfiguration can lead to the emergence of new band gaps or the closure of existing ones, enabling the "on" or "off" control of elastic wave transmission.This impressive tuning capability primarily relies on the specially designed bending ligaments.By manipulating the temperature and deformation of SMA, the bandgap or passband region of a permanent structure can be transformed into a passband or band gap of a temporary structure.SMA materials overcome the limitations of conventional materials, which require energy to maintain their shape, thus expanding the range of bandgap control.Furthermore, the two control methods (geometric reconstruction and material property change) can be coupled together to adjust band gaps, and thus achieve extensive tuning functionality.
To understand the bandgap formation mechanism, Fig. 10 illustrates the vibration modes at the upper and lower edges (high symmetrical point Γ ) of the band gaps.The initial bandgap generation mechanism is observed first.The lower edge mode (Fig. 10a) shows inward deformation of one diagonal ring and bending movement of the surrounding straight beam, while the other diagonal ring remains stationary.The upper edge mode (Fig. 10b) exhibits simultaneous inward deformation of all four rings, while the individual beams remain stationary.
Upon unloading the uniaxial compressive load, the lower edge mode of the first band gap (Fig. 10c) displays the same inward motion of the diagonal rings and bending of the curved beam.The upper edge mode (Fig. 10d) shows inward deformation of the circular ring, with the curved beams remaining stationary in fourth-order rotational symmetry.The lower edge mode of the second band gap (Fig. 10e) demonstrates inward motion of the diagonal rings, while the curved beams remain stationary with low-order rotational symmetry.The upper edge mode (Fig. 10f) primarily exhibits bending motion of the curved beam, while the four rings remain stationary.
Upon unloading the uniaxial tensile load, the vibration modes at the band gaps are shown in Fig. 10g-i.The lower edge mode of the first band gap (Fig. 10g) shows downward translational motion of the transverse curved beam.The upper edge mode (Fig. 10h) consists of compressive deformation of the diagonal ring and bending motion of the beam, similar to the vibration mode in Fig. 10c.The lower edge mode of the second band gap (Fig. 10i) demonstrates inward motion of the diagonal rings in a different direction (while the other diagonal remains stationary) and bending motion of the curved beam.The upper edge mode (Fig. 10j) displays flattened inward motion of the diagonal ring, while the curved beam remains stationary, similar to the vibration mode in Fig. 10e.The lower edge mode of the third band gap (Fig. 10k) is axisymmetric, with inward motion of the rings and stationary beams.The mode in Fig. 10l is characterized by bending motion of the longitudinal curved beam, while the ring remains stationary.When the structure is heated, it returns to the initial configuration, and the corresponding vibration modes at the bandgap boundary are the same as the initial structure.
By scanning the frequency domain of the finite structure, the transmission response is obtained.Theoretically, the transmission rate should decay rapidly in the frequency range corresponding to the band gap, which indicates that the transmission coefficient provides a useful representation of the bandgap location.The tunable transmission characteristics are depicted in Fig. 11.An acceleration signal is applied at the left end of the plate structure, and the right end is used to receive the signal.From the transmission response of the initial structure (Fig. 11a), the first band gap corresponds to the position where transmission response decays.Interestingly, it can be seen that it decays significantly in the frequency range of 13,710-17,100 Hz, but this band gap does not exist in Fig. 9a, which proves the existence of the deaf band [32].The structure after the phase transition is shown in Fig. 11b   and c.Compared to the initial structure, the position of the wave decay is significantly different, which proves the shift of the bandgap position as well as the opening of a new band gap after the phase transition.From the above analysis, by the self-tuning shape memory effect, the transmission properties of the chiral structure with SMA can be controlled, and the geometric configuration and effective stiffness have significant effects on wave propagation, which provides a strong proof for the proposed scheme.
To further visualize the wave propagation in the proposed chiral structure, Fig. 12 shows the displacement fields of the 5 × 2 cells under excitation frequencies of 10,000 Hz and 20,000 Hz.Theoretically, the incident wave within the bandgap frequency decays rapidly, while the wave can pass through the metamaterial when the excitation frequency is located outside the band gap.For both the initial structure and the structure after compression (Fig. 12a and Fig. 12b), it can be clearly seen that the 10,000-Hz excitation lies in the passband range and the elastic wave can propagate in the plate structure.However, the elastic wave cannot propagate after compression, as shown in Fig. 12c.Under 20,000-Hz excitation, the elastic wave decays rapidly after the first cell at the left end and has almost no response at the right end for the chiral structure after compression and tension (Fig. 12e  and f), which prevents the propagation of the elastic wave, while the 20,000-Hz excitation causes the whole initial structure to deform, as shown in Fig. 12d.Therefore, the proposed structures can stably modulate the elastic wave, which can achieve intelligent manipulation without additional continuous energy input.

Effects of Geometric Parameters on Tunable Band Gaps
The geometric properties can be defined in terms of two dimensionless coefficients (μ and λ, where μ S/r and λ t/r ).Eight sets of parameters are modeled separately for the finite element analysis, where the values of dimensionless parameters are different (μ 2, 3, 4, 5; λ 0.05, 0.1, 0.2, 0.4).The initial geometric configuration and the geometric configuration after the phase transition are illustrated in Fig. 13, and the boundary conditions and external loads are set the same as those in Sect.3.2.The initial configurations with different parameters produce various martensitic phase Figure 14b displays the effect of parameter λ on the bandgap distribution by varying λ from 0.05 to 0.4 while fixing μ 2. When λ 0.05, the band gaps become In summary, the SMA chiral structure with different parameters can shift the position of band gaps, and the band gaps can be customized to satisfy engineering requirements.It is worth emphasizing that the band gap can be restored to its initial state through temperature regulation, which is not achievable with conventional materials.The introduction of SMA overcomes the challenge of narrow band gaps in common chiral structures and expands the application of chiral structures.

Conclusion
This paper proposes an SMA-based anti-tetrachiral structure to achieve tunable band gaps.The study includes a static mechanical analysis of the structure and explores the effects of temperature and stress on its geometric configuration and material properties after the phase transition.By examining energy band structures and vibration modes, the researchers analyze the tunable bandgap characteristics, demonstrating a wide adjustable range from the "on" to "off" states.The bending ligaments, manually trained , are crucial

Fig. 1 Fig. 2
Fig. 1 Anti-tetrachiral structure: a architected materials with 3 × 3 unit cells; b the geometric configuration of unit cell design; c the first Brillouin zone

Fig. 3
Fig. 3 Deformation mechanism of anti-tetrachiral structures based on SMA

Fig. 4 Fig. 5
Fig. 4 Conditions of phase change process: a the boundary conditions; b the load conditions; c the temperature conditions

Fig. 6 Fig. 7 Fig. 8
Fig. 6 Whole training process under uniaxial compressive load: a the stress-strain curve; b the displacement curve; c the martensite volume fraction curve; d Young's modulus (the blue and green lines represent Young's moduli of the beams and rings, respectively)

Fig. 9
Fig. 9 Bandgap diagrams in four states: a the initial state; b the uniaxial compression state; c the uniaxial tension state; d the state after recovery

Fig. 10
Fig. 10 Vibration mode at the high symmetrical point Γ of the bandgap edge: a, b the initial structure bandgap edge; c-f the bandgap edge under uniaxial compression; g-l the bandgap edge under uniaxial tension

Fig. 11
Fig. 11 Transmission characteristic curves: a the initial state; b the uniaxial compression state; c the uniaxial tension state; d the anti-tetrachiral structure after recovery

Fig. 12
Fig. 12 Displacement field of the plate structure under 10,000-Hz and 20,000-Hz excitations: a and d the initial structure; b and e the structure after uniaxial compression; c and f the structure after uniaxial tension

Table 1
Material properties of SMA Compared to the initial structure, SMA undergoes phase transformation, increasing the tunability of the band gap due to the two material properties.The upper and lower boundaries of the band gap move to the low frequency at the same time, and the bandgap width is twice of that of the initial structure.At the 20th and 21st orders, the second band gap appears at[21,941 − 26,790 Hz, 4849 Hz].Since band gaps below 25,000 Hz are of interest, only part of the band gap is demonstrated.