Investigation on bandgap modulation of locally resonant metamaterials with shape memory alloy resonators under temperature and multi-mode pre-deformation

Abstract

Elastic and acoustic metamaterials with locally resonant (LR) arrays can generate bandgaps that attenuate or block elastic waves. Although extensive research has explored thermally induced tuning or single-mode mechanical pre-deformation, systematic comparative analyses of different pre-deformation strategies remain insufficient. To address this research gap, this study incorporates shape memory alloys (SMAs) into metamaterial architectures to realize tunable bandgaps via pre-tension, pre-bending, pre-twisting, and composite pre-deformations strategies, while also exploring the feasibility of achieving lower-frequency bandgaps. Experiments demonstrate that distinct pre-deformation modes induce differentiated shifts in the bandgap, and systematic trends throughout heating are presented. As temperature increases, these shifting effects progressively diminish due to the shape memory effect of the SMA, thereby establishing a temperature-dependent tuning range. To further elucidate the underlying tuning mechanism, numerical simulations investigate symmetric pre-bending and pre-twisting at various angles, revealing the quantitative relationship between pre-deformation magnitude, temperature, and bandgap tunability. The contributions of bending and twisting in hybrid configurations are also examined, providing insights into the optimal structural design of such systems. Finally, the study extends to two-dimensional metamaterial plate structures incorporating SMA resonators. Simulation show broader and more flexible bandgap tunability under thermal actuation, underscoring their potential for adaptive low-frequency vibration isolation and control.

Keywords

metamaterials local resonance shape memory alloy bandgap pre-deformation

Introduction

In recent years, artificially engineered metamaterials have attracted considerable attention due to their ability to exhibit frequency bandgaps that inhibit wave propagation, offering new possibilities for wave manipulation beyond conventional materials. Owing to these unique properties, metamaterials have been explored in a wide range of fields, including vibration and noise suppression, wave filtering, energy localization, and signal control (Almirall et al., 2022; Li et al., 2024a; Ma et al., 2020; Tamma et al., 2024). These applications collectively underscore the transformative role of metamaterials in engineering and physical sciences.

Among various types of metamaterials, locally resonant (LR) metamaterials are particularly attractive because they can generate low-frequency bandgaps through subwavelength resonators (Casalotti et al., 2018; Chen et al., 2013; Ho et al., 2003; Hu et al., 2017; Liu et al., 2000; Matlack et al., 2016; Nouh et al., 2015; Peng and Frank Pai, 2014; Sugino et al., 2017; Xiao et al., 2012; Zhu et al., 2014). However, for conventional LR metamaterials, the bandgap characteristics are generally fixed once the structure is fabricated, which significantly limits their adaptability to varying operational environments.

To overcome this limitation and achieve precisely control low-frequency narrow-band bandgaps, extensive efforts have been devoted to the development of tunable metamaterials. One of the most mature and widely adopted approaches relies on piezoelectric-based tuning strategies, including surface-bonded piezoelectric patches, shunted piezoelectric circuits, and hybrid piezoelectric–mechanical resonators. Numerous studies have demonstrated that piezoelectric actuation enables active or semi-active modulation of bandgaps, offering advantages such as fast response speed, high control precision, wide actuation bandwidth, and compatibility with low-power electronic systems. These features make piezoelectric tuning particularly suitable for real-time and adaptive vibration control applications (Chen et al., 2016; Gao and Wang, 2024; Li et al., 2024b; Ma et al., 2021; Sun et al., 2024; Wenjing and Yize, 2022).

In contrast, shape memory alloy (SMA)-based tuning strategies rely on the intrinsic thermo-mechanical phase transformation behavior of SMAs, which is associated with the shape memory effect (SME). Specifically, during heating, SMAs undergo a reversible phase transformation from the martensitic phase to the austenitic phase, accompanied by a pronounced increase in elastic modulus and internal stress. By exploiting this temperature or deformation-induced modulation of material stiffness, SMA-integrated metamaterial structures can achieve large and reversible variations in effective stiffness, resulting in substantial bandgap shifts.

Although SMA-based tuning generally exhibits a slower response speed compared with piezoelectric actuation, it offers several distinct advantages, including a large tunable range, relatively simple structural implementation, and the capability of temperature-driven passive or semi-active tuning. These features make SMA-based metamaterials particularly suitable for applications where rapid actuation is not the primary requirement but wide-range bandgap tunability is desired. Leveraging this property, Sousa et al. developed a locally resonant metamaterial beam structure in which the thermally induced stiffness variation of SMA resonators allows dynamic tuning of the bandgap frequency range (Candido de Sousa et al., 2018b). They further integrated the transverse vibration analysis model with an SMA spring to systematically investigate the coupled effects of thermoelastic phase transformation and stress hysteresis behavior on bandgap regulation and damping characteristics, thereby establishing the pivotal role of SMA resonators in tunable metamaterial design (Candido de Sousa et al., 2018a).

Beyond the temperature-driven mechanism, Chuang et al. utilized the strain-releasing behavior of bi-directionally bent SMA structures upon heating, integrating planar bent SMA elements into metamaterial structures (Chuang et al., 2019). Their study demonstrated that introducing pre-deformation into SMA resonant cavities enables reversible tuning of bandgap characteristics between Bragg-type and local resonance mechanisms in planar metamaterial beams, offering a more flexible approach for bandgap control. Subsequently, Wang et al. expanded this line of research through numerical simulations and experimental validation (Yin et al., 2022). They revealed the bandgap evolution behavior of SMA-resonator-based metamaterial beams under thermal cycling and quantitatively analyzed the effects of pre-stretched SMA resonators on the position and width of the bandgap.

Existing studies have primarily focused on bandgap tuning in SMA-resonator-based metamaterials under single-mode pre-deformations; however, comprehensive investigations into bandgap evolution under diverse pre-deformation modes remain limited. In practical applications, the bandwidth, onset frequency, and tunability range of the bandgap play a pivotal role in determining metamaterial performance, particularly for low-frequency vibration attenuation. Such low-frequency bandgaps hold significant promise for mitigating vibrations and noise in large-scale rotating machinery, transportation systems, civil engineering structures, and precision instruments.

Therefore, this study aims to investigate the feasibility of achieving low-frequency bandgap tuning through diverse pre-deformation strategies. By integrating numerical simulations with experimental verification, we systematically analyze the dynamic bandgap evolution induced by thermally driven phase transformation in SMA resonators under pre-stretching, pre-bending, pre-twisting, and combined pre-deformation conditions. Furthermore, the specific effects of varying bending and twisting angles on bandgap characteristics are explored. Through a quantitative analysis, the nonlinear relationship between pre-deformation magnitude and bandgap response is elucidated. Additionally, the synergistic mechanism within compound pre-deformation configurations is examined, focusing on the coupled effects and modulation capacity of bending and twisting during bandgap formation. Finally, building upon the model in Wang et al. (2021), the investigation is extended to two-dimensional metamaterial plates incorporating SMA resonators, analyzing bandgap variations through simulation. These findings not only enrich the theoretical framework of multi-deformation modulation in SMA-based metamaterials but also provide fundamental support and practical guidance for the design of next-generation programmable and environmentally responsive metamaterial devices.

Mechanism of bandgap formation for metamaterial beam structures with resonators

Based on the uniform Euler-Bernoulli beam model with arbitrary boundary conditions, as shown in Figure 1, $EI$ is defined as bending stiffness, $m$ as the linear mass density (i.e. mass per unit length), and $L$ as the length of the beam. The transverse base motion of the beam is denoted as $w_{b}$ and its relative transverse vibration is represented by $w_{(x, t)}$ , thus the absolute displacement $w_{abs}$ is given by (1), according to the formulation presented in Sugino et al. (2016):

w_{abs} = w_{b} + w_{(x, t)}

(1)

Figure 1.

Beam model with undamped resonators.

Additionally, the beam is subjected to a distributed external load $f_{(x, t)}$ per unit length. To simplify the model, it is assumed to be undamped. A number of undamped spring-mass resonators are attached to the beam, with their relative positions denoted as $k_{j}$ , masses as $m_{j}$ , relative displacements as $u_{j}$ , and natural frequencies as $w_{a, j}$ , where $j = 1 \cdot \cdot \cdot \cdot \cdot \cdot S$ . Consequently, equation (2) can be obtained:

w_{a, j} = \sqrt{k_{j} / m_{j}}

(2)

Based on the above conditions, the dynamic equation of the beam is derived as shown in equation (3; Sugino et al., 2016):

EI \frac{\partial^{4} w}{\partial x^{4}} + m \frac{\partial^{2} w}{\partial x^{2}} - \sum_{j = 1}^{s} m_{j} w_{a, j}^{2} u_{j} (t) δ (x - x_{j}) = f (x, t) - m w_{b}^{″} (t)

(3)

Where $δ (x)$ is the Dirac delta function, and the governing equation for each resonator is given by equation (4; Sugino et al., 2016):

u_{j}^{″} (t) + w_{a, j}^{2} u_{j} (t) + \frac{\partial^{2} w}{\partial t^{2}} (x_{j}, t) = - w_{b}^{″} (t)

(4)

Assuming that the natural frequencies $w_{i}$ and mode shapes $ϕ_{i}$ of the beam model are known, the normalization of the mode shapes is given by equation (5):

\int_{0}^{L} ϕ_{i} (x) ϕ_{j} (x) dx = L δ_{ij}

(5)

Where $δ_{i j}$ is the Kronecker delta.

By employing an assumed mode expansion with $N$ terms and adopting the corresponding mode shapes of a simply supported beam as basis functions, we assume to obtain equation (6):

w (x, t) = \sum_{k = 1}^{N} ϕ_{k} (x) η_{k} (t)

(6)

Multiplying both sides of equation (3) by $ϕ_{i} (x)$ and integrating both sides from 0 to $L$ yields equation (7):

\begin{matrix} \int_{0}^{L} ϕ_{i} (x) [EI \frac{\partial^{4} w}{\partial x^{4}} + m \frac{\partial^{2} w}{\partial x^{2}} - \sum_{j = 1}^{S} m_{j} w_{a, j}^{2} u_{j} (t) δ (x - x_{j})] dx \\ = \int_{0}^{L} ϕ_{i} (x) [f (x, t) - m w_{b}^{″} (t)] dx \end{matrix}

(7)

Based on the fundamental linear dynamic equation (8):

EI \frac{\partial^{4} w (x, t)}{\partial x^{4}} + m \frac{\partial^{2} w (x, t)}{\partial t^{2}} = f (x, t)

(8)

Under free vibration conditions, when $f (x, t) = 0$ , equation (9) is obtained:

EI \frac{\partial^{4} w (x, t)}{\partial x^{4}} + m \frac{\partial^{2} w (x, t)}{\partial t^{2}} = 0

(9)

Substituting equation (6) into equation (9) yields equation (10):

\sum_{k = 1}^{N} [E I ϕ_{k}^{″ ″} (x) η_{k} (t) + m ϕ_{k} (x) η_{k}^{″} (t)] = 0

(10)

To simplify the equation, each modal amplitude $η_{k} (t)$ is assumed to oscillate in a harmonic form, as expressed in equation (11):

η_{k} (t) = Φ_{k} e^{i w_{k} t}

(11)

Substituting equation (11) into equation (10) and simplifying yields equation (12):

\sum_{k = 1}^{N} [EI ϕ_{k}^{" "} (x) - {mw}_{k}^{2} ϕ_{k} (x)] Φ_{k} e^{i w_{k} t} = 0

(12)

Since $e^{i w_{k} t}$ is linearly independent for different $k$ , the equation must hold for each $k$ , which requires that the corresponding coefficients be zero, leading to equation (13):

EI \frac{d^{4} ϕ_{k} (x)}{d x^{4}} η_{k} (t) = {mw}_{k}^{2} ϕ_{k} (x) η_{k} (t)

(13)

Substituting equation (13) into one term of equation (7) and combining it with equation (5) yields equation (14):

\begin{array}{l} \int_{0}^{L} ϕ_{i} (x) E I \frac{\partial^{4} w}{\partial x^{4}} d x = \int_{0}^{L} ϕ_{i} (x) \sum_{k = 1}^{N} m w_{k}^{2} ϕ_{k} (x) η_{k} (t) d x \\ = \sum_{k = 1}^{N} m w_{k}^{2} η_{k} (t) \int_{0}^{L} ϕ_{i} (x) ϕ_{k} (x) d x = \sum_{k = 1}^{N} m w_{k}^{2} η_{k} (t) • L δ_{i k} \\ = m L w_{i}^{2} η_{i} (t) \end{array}

(14)

Simplifying the other terms in equation (7) results in equations (15)–(17):

\begin{matrix} \int_{0}^{L} ϕ_{i} (x) δ (x - x_{j}) dx = {\begin{matrix} ϕ_{i} (x_{j}), x_{j} \in [0, L] \\ 0, other \end{matrix} \\ - \sum_{j = 1}^{S} m_{j} w_{a, j}^{2} u_{j} (t) \int_{0}^{L} ϕ_{i} (x) δ (x - x_{j}) dx \\ = - \sum_{j = 1}^{S} m_{j} w_{a, j}^{2} u_{j} (t) ϕ_{i} (x_{j}) \end{matrix}

(15)

m \sum_{k = 1}^{N} η_{k}^{"} (t) \int_{0}^{L} ϕ_{i} (x) ϕ_{k} (x) dx = m \sum_{k = 1}^{N} η_{k}^{"} (t) L δ_{ik}

(16)

\begin{array}{l} m L w_{i}^{2} η_{i} (t) + m L \sum_{k = 1}^{N} η_{k}^{″} (t) δ_{i k} \\ \sum_{j = 1}^{S} m_{j} [- μ_{j}^{″} (t) - ϕ_{k} (x_{j}) η_{k}^{″} (t) - w_{b}^{″} (t)] ϕ_{i} (x_{j}) \\ = \int_{0}^{L} ϕ_{i} (x) f (x, t) d x - \int_{0}^{L} m_{j} ϕ_{i} (x) w_{b}^{″} (t) d x \end{array}

(17)

Substituting the simplified equations (14)–(16) back into equation (7) yields equation (17). Dividing both sides by $mL$ results in equations (18) and (19):

\begin{array}{l} \sum_{i = 1}^{N} [δ_{i k} + \sum_{j = 1}^{S} {\hat{m}}_{j} ϕ_{i} (x_{j}) ϕ_{k} (x_{j})] η_{i}^{″} (t) + \sum_{j = 1}^{S} {\hat{m}}_{j} ϕ_{k} (x_{j}) μ_{j}^{″} (t) + w_{k}^{2} η_{k} (t) \\ = - w_{b}^{″} (t) (\frac{1}{L} \int_{0}^{L} ϕ_{k} (x) d x + \sum_{j = 1}^{S} {\hat{m}}_{j} ϕ_{k} (x_{j})) \end{array}

(18)

u_{j}^{″} (t) + w_{a, j}^{2} u_{j} (t) + \sum_{i = 1}^{N} η_{i}^{″} (t) ϕ_{i} (x_{j}) = - w_{b}^{″} (t)

(19)

After modal analysis, the bandgap exists within the frequency range given by equation (20; Sugino et al., 2016):

w_{t} < w < w_{t} \sqrt{1 + μ}

(20)

Where $w_{t}$ is the added natural frequency, and $μ$ is the ratio of the absorber’s added mass to the plain beam’s mass. This frequency range defines the bandgap limits and provides a simple method for locating the bandgap within the desired frequency range. This assumed mode analysis can be easily extended to any typical continuous vibration structure with dampers. The bandwidth of the bandgap is determined by the mass ratio and the attached natural frequency, and can be expressed as equation (21; Sugino et al., 2016):

Δ w = w_{t} (\sqrt{1 + μ} - 1)

(21)

Based on the aforementioned modal analysis, the bandgap range of the resonator-integrated metamaterial structure can be systematically determined. The primary determinant influencing this bandgap range is identified as the supplementary natural frequency of the resonator, as mathematically formulated in equation (22):

w_{t} = \sqrt{k_{j} / m_{j}} = \sqrt{E_{j} I_{j} / m_{j}}

(22)

where $E_{j}$ denotes the elastic modulus of the resonator, and $I_{j}$ represents its second moment of area.

Given a constant resonator mass, the bandgap shift is primarily governed by the resonator’s stiffness, which depends critically on both the material modulus and the geometric configuration. Shape memory alloys, as smart materials capable of recovering pre-strains and exhibiting significant thermally induced modulus variations, demonstrate exceptional potential for adaptive local resonator applications.

As indicated by equation (22), geometric reconfiguration of the SMA resonator alters its moment of inertia, thereby shifting the metamaterial bandgap. Consequently, in addition to thermal modulus tuning, mechanical pre-deformation serves as an effective mechanism for bandgap regulation. In this study, pre-stretching, pre-bending, pre-twisting, and combined pre-deformations are introduced into the SMA resonators to systematically investigate their effects on the bandgap tuning characteristics of the metamaterial beams.

Simulation and experiments

Setup

The SMA utilized in this study is a NiTi alloy manufactured by XHE Corporation (China). Given that the elastic modulus of SMA varies with temperature, a Discovery DMA850 instrument from TA Instruments (New Castle, DE) was employed to characterize the relationship between the elastic modulus of SMA (also referred to as the storage modulus in DMA testing) and temperature. To ensure measurement accuracy, a constant heating and cooling rate of 1°C/min was maintained, and the resulting curve is presented in Figure 2.

Figure 2.

Relationship between storage modulus of SMA and temperature.

As shown in Figure 2, the elastic modulus of SMA varies with temperature during both heating and cooling processes. Given the similar trends observed in both thermal cycles, this analysis primarily focuses on the heating phase. When the SMA is heated to approximately 44°C, its elastic modulus reaches a minimum of around 50 GPa, marking the onset of the martensite-to-austenite transition (Austenite start temperature, $A_{s}$ ). As the temperature rise to approximately 60°C, the phase transition is fully completed, and the SMA entirely transforms into austenite (Austenite finish temperature, $A_{f}$ ), where the elastic modulus reaches approximately 83 GPa. Conversely, during the cooling process from 80°C, the austenite-to-martensite transition initiates at 57°C (Martensite start temperature, $M_{s}$ ), where the modulus remains near 83 GPa. As the temperature further decreases to approximately 41°C, the transformation is complete (Martensite finish temperature, $M_{f}$ ), and the modulus returns to approximately 50 GPa. Consequently, when utilizing these SMA elements as local resonators in the metamaterial beam model (Figure 1), the bandgap range can be dynamically adjusted by controlling the temperature of the SMA. By fitting the DMA test data of SMA, a numerical relationship between the elastic modulus and temperature can be obtained. As indicated by equation (22), these temperature-induced modulus variations directly alter the resonator stiffness, thereby shifting the bandgap frequency and determining the tunable range of the metamaterial structure

In this study, a steel main beam with dimensions of 290 mm × 15 mm × 2 mm, an elastic modulus of 200 GPa, and a density of 7850 kg/m³ was utilized. Five SMA resonators, each measuring 185 mm × 10mm × 0.8 mm with a density of 6360 kg/m³, were uniformly attached to the steel beam using high-temperature adhesive. This metamaterial beam model was adapted from our previously published structural design (Yin et al., 2022), with several modifications made to better suit the present study: the SMA resonator length was increased from 139 to 185 mm, while their thickness was reduced from 1.6 to 0.8 mm. Additionally, the length of the steel beam was extended from 250 to 290 mm. to ensure a fully symmetric distribution of the SMA resonators along the beam. Furthermore, the boundary conditions were adjusted from the previously used free–free configuration (Yin et al., 2022) to a clamped–free configuration. These structural optimizations serve to establish a robust platform for systematically investigating the bandgap tuning mechanisms under coupled pre-deformation and thermal excitations.

In this study, finite element simulations were conducted using ANSYS, where the metamaterial model with the specified geometric dimensions was imported and analyzed using hexahedral elements with modal and harmonic response solvers; the steel beam was assigned material properties corresponding to Q235 steel, while the SMA resonators were defined using density and Poisson’s ratio provided by the manufacturer together with a temperature-dependent elastic modulus extracted from Figure 2; the boundary conditions were implemented by fixing one end of the beam and leaving the other end free, a point mass was applied at the accelerometer location, and a harmonic force together with an elastic support was assigned at the exciter location. This approach facilitates the numerical simulation of acceleration frequency response functions (FRFs) at locations corresponding to the experimental nodes, enabling direct comparison with experimental data. Given that the variation in the elastic modulus of SMA during heating and cooling is generally consistent, the analysis focuses solely on the heating process until the pre-deformed structures fully recover their original shape. Specifically, bandgap shifts were evaluated at four distinct stages: room temperature (initial state), the austenite start point ( $A_{s}$ ), the austenite finish point ( $A_{f}$ ), and the fully recovered state at 80°C. These critical points were analyzed to quantify bandgap tunability and derive relevant conclusions.

As illustrated in Figure 3(a), the experimental setup was established to characterize the vibration transmission frequency response functions (FRFs) of the locally resonant metamaterial structure. This experimental approach has been widely employed in previous studies on locally resonant (LR) structures (Xiao et al., 2013). In the experiment, one end of the steel beam was fixed using a clamp to establish a clamped-free boundary condition. The exciter was installed at the front end of the extended section beyond the clamped end of the beam and was driven by a power amplifier and a signal generator to produce a swept-frequency excitation signal.

Figure 3.

Actual (a) and schematic (b) setup of the experiments.

To capture the vibration response at both ends of the main beam, two PCB352C34 accelerometers, each weighing 5.8 g, were mounted at the input (fixed end) and output (free end) of the main beam to capture the corresponding acceleration signals. These signals were acquired using a DEWESOFT SIRIUS modal analyzer to compute the frequency response functions. The acceleration responses at the input and output ends were denoted as $u_{i}^{″} (w)$ and $u_{o}^{″} (w)$ , respectively. The vibration transmissibility $T (w)$ is then given by equation (23):

T (w) = | \frac{u_{o} (w)}{u_{i} (w)} | = | \frac{- w^{2} u_{o} (w)}{- w^{2} u_{i} (w)} | = | \frac{μ_{o}^{″} (w)}{μ_{i}^{″} (w)} |

(23)

Here, $u_{o} (w)$ and $u_{i} (w)$ represent the displacement at the output and input ends, respectively.

The schematic diagram of the experimental setup is shown in Figure 3(b). In this study, the entire structure was placed inside a temperature-controlled chamber (Figure 4) to regulate the temperature of the SMA resonators. As the temperature increased, the pre-deformed SMA resonators gradually recovered, and their elastic modulus varied in accordance with the trend illustrated in Figure 2. By adjusting the temperature, tests were conducted to investigate the effects of different pre-deformation modes as well as non-deformed SMA resonators on the bandgap characteristics of the metamaterial structure. Concurrently, the experimental process also provided data support for the ANSYS simulations, particularly regarding structural dimensional variations and material parameter selection.

Figure 4.

Experimental setup of the temperature-controlled chamber.

Results

Thermally tunable bandgap control

Given the similarity in the variation of the elastic modulus of the SMA used in this study during heating and cooling, this research primarily focuses on analyzing key temperature nodes during the heating process (as listed in Table 1). The selected temperatures are based on previously reported NiTi SMA phase-transformation characteristics (Bhattacharya, 2003; Brinson, 1993; Kumar and Lagoudas, 2008), to obtain the vibration transmission frequency response function of the beam. The experimental and simulation results are presented in Figure 5, where we use $M %$ and $A %$ to represent the percentages of martensite and austenite.

Table 1.

The selected experimental temperatures during heating processes (Bhattacharya, 2003; Brinson, 1993; Kumar and Lagoudas, 2008).

Heating process
25°C	Room temperature
44°C	Starting temperature of austenite transition ( $A_{s}$ )
60°C	Finishing temperature of austenite transition ( $A_{f}$ )
80°C	The temperature at which the SMA completely recovers after deformation

Figure 5.

Comparison of the simulated and experimental bandgaps of the SMA-based metamaterial structure at different temperatures (a).

As illustrated in Figure 5, the experimental data exhibit good agreement with the simulation results. Figure 5(a1)–(a4) depict the bandgap evolution of the SMA-based metamaterial beam during the heating process. Consistent with the temperature nodes listed in Table 1, significant shifts in the bandgap frequency range are clearly observable. Specifically, Figure 5(a1)–(a2) indicate that as the metamaterial beam is heated from 25°C to 44°C, the bandgap shifts toward lower frequencies. Conversely, as the temperature further increases to 60°C and 80°C (as shown in Figure 5(a3)–(a4)), the bandgap gradually shifts toward higher frequencies. This phenomenon corroborates the theoretical relationship defined in equation (22). As the temperature rises from 25°C to approximately 44°C, the elastic modulus of the SMA exhibits a pronounced decrease (as shown in Figure 2). This reduction in modulus directly leads to a decrease in the natural angular frequency of the local resonator $w_{t}$ , thereby driving the entire bandgap toward lower frequencies. In contrast, as the temperature continues to rise from 44°C to 60°C and even up to 80°C, the elastic modulus of the SMA increases significantly. This recovery enhances $w_{t}$ , consequently shifting the bandgap back toward higher frequencies.

These results demonstrate that the temperature-dependent variation in the elastic modulus of SMA plays a pivotal role in determining the bandgap tunability.

Throughout the heating process, the metamaterial structure exhibited a significant evolution in its bandgap frequency range. Specifically, as the temperature increased from 25°C to 44°C, the onset frequency of the bandgap shifted from 332 to 312 Hz, as shown in Figure 5(a1) and (a2). With a further temperature rise to 60°C, the onset frequency surged to 404 Hz, as depicted in Figure 5(a3). Finally, as illustrated in Figure 5(a4), when the temperature reached 80°C, the onset frequency further shifted to 414 Hz. Throughout the heating process, the variation between the minimum (312 Hz) and maximum (414 Hz) onset frequencies indicates a tunable range of approximately 32.7% for the metamaterial structure.

Pre-Deformation Enabled Thermally Tunable Bandgap Control

To investigate the influence of specific deformation modes on the bandgap of the main beam, the SMA resonators were subjected to axial pre-stretching, pre-twisting, and pre-bending at room temperature

Pre-tensioning enabled thermally tunable bandgap control

Axial pre-stretching of the SMA resonator was performed using an ETM504C universal testing machine. The resonators were stretched from an initial length of 185–195 mm. However, due to the superelastic properties of SMA, the actual length after unloading was 191 mm, corresponding to a tensile strain of approximately 3.24%.

Driven by the Shape Memory Effect (SME) of SMA, the resonator gradually recovers its deformation once the temperature reaches the point $A_{s}$ . Therefore, bandgap experiments were conducted on the metamaterial structure incorporating pre-stretched SMA resonators at four temperature conditions: room temperature (25°C), $A_{s}$ (44°C), $A_{f}$ (60°C) and the fully recovered state (80°C), as shown in Figure 6(a1)–(a4). With increasing temperature, the bandgap onset frequency shifted from 300 Hz at 25°C to 280 Hz at 44°C, subsequently rising to 378 Hz at 60°C, and finally reaching 414 Hz upon full shape recovery at 80°C. Compared to the undeformed structure, the pre-stretching induced a downward frequency shift of approximately 10.7% at room temperature. Throughout the heating process, the minimum and maximum onset frequencies were observed at 280 and 414 Hz, respectively, yielding a total tunable range of approximately 47.9%. Notably, the experimental and simulation results for the pre-stretched configuration demonstrate strong agreement with the theoretical trend described by equation (22). The lower onset frequency observed relative to the undeformed structure is attributed to the reduction in the resonator’s cross-sectional moment of inertia ( $I_{j}$ ) caused by stretching, which drives the bandgap toward lower frequencies under the same temperature.

Figure 6.

Comparison of simulation and experimental results for the bandgap of stretched SMA metamaterial structures at different temperatures (a).

Pre-bending and pre-twisting enabled thermally tunable bandgap control

Subsequently, pre-bending deformation was induced in the SMA resonators using a mold, targeting initial bending angles of 120° and 80°. Due to the superelasticity of the SMA material, the actual angles after unloading were approximately 75° and 30°, as shown in Figure 7. Similarly, the SMA resonators were subjected to pre-twisting using a torsion testing machine, with initial torsion angles of 210° and 150°, and the actual angles after unloading were approximately 65° and 30°, as shown in Figure 8. These pre-deformed resonators were uniformly distributed along the main beam, as illustrated in Figures 7(b) and 9. The bandgap characteristics of the metamaterial structures were subsequently characterized throughout the heating process, spanning from room temperature (25°C) to the state of full shape recovery. Representative results are summarized in Figures 10 and 11.

Figure 7.

(a) Schematic diagram of bending and (b) metamaterial structure with bent SMA resonators.

Figure 8.

(a) Torsion schematic diagram and (b) torsion testing machine torsion diagram.

Figure 9.

Metamaterial structure with twisted SMA resonators.

Figure 10.

Simulation and experimental results of bandgaps in SMA metamaterial structures with different bending angles (a₁)–(a₄) represent bandgap diagrams for a 30° bent structure heated to different temperature points, while (b₁)–(b₄) correspond to a 75° bent structure heated to different temperature points.

Figure 11.

Simulation and experimental results of bandgaps in SMA metamaterial structures with different twist angles (a₁)–(a₄) represent the bandgap diagrams for the structure with a 30° twisted SMA resonator heated to different temperature points, while (b₁)–(b₄) correspond to the bandgap diagrams for the structure with a 60° twisted SMA resonator heated to different temperature points.

The experimental results indicate that for the 30° pre-bent resonator, the bandgap onset frequency shifted downward by 12 Hz at 25°C relative to the undeformed structure. It further advanced by 20 Hz at 44°C, shifted backward by 96 Hz at 60°C, and continued to shift backward by an additional 16 Hz at 80°C, as illustrated in Figure 10(a). For the 75° pre-bent resonator, the tunable bandgap range was significantly expanded. It exhibited a downward frequency shift of approximately 17% relative to the undeformed structure (Figure 5 (a1)), with a maximum variation of about 55.6% throughout the heating process, as shown in Figure 10(b).

For the 30° pre-twisted resonator, compared to the bandgap of the undeformed structure at 25°C (Figure 5(a1)), the bandgap onset frequency shifted downward by 70 Hz at 25°C. It decreased by an additional 10 Hz at 44°C, before shifting backward by 128 Hz at 60°C, and continued to shift backward by an additional 36 Hz at 80°C, as illustrated in Figure 11(a). In the case of the 65° pre-twisted resonator, the downward frequency shift was even more pronounced. It exhibited a reduction of approximately 69.4% relative to the undeformed structure (Figure 5(a1)), achieving a maximum tunable variation of about 125% throughout the heating process, as shown in Figure 11(b). These results demonstrate that increasing the pre-deformation magnitude—particularly via pre-twisting—can substantially enhance the bandgap tunability of the metamaterial structures

It is concluded that, under pre-bending and pre-twisting conditions, the experimental results and numerical simulations demonstrate strong agreement with the trend predicted by equation (22), thereby validating the effectiveness of the theoretical model. Specifically, SMA resonators subjected to pre-bending and pre-twisting undergo geometric distortion, which reduces their effective moment of inertia ( $I_{j}$ ). Under the combined influence of temperature-dependent variations in elastic modulus and moment of inertia, the bandgap exhibits a pronounced shift compared with the undeformed structure. As the pre-deformation angle increases, this geometric distortion becomes more significant, leading to a further reduction in the moment of inertia and, consequently, enhanced bandgap tunability. These effects induced by pre-bending and pre-twisting can be well interpreted within the theoretical framework of equation (22), providing a comprehensive understanding of the intrinsic mechanisms governing bandgap evolution.

Compound pre-deformation enabled thermally tunable bandgap control

This study primarily investigates the bandgap shift of the same type of SMA under single symmetric pre-deformation conditions. To broaden the research scope and uncover further tuning potentials, this section introduces asymmetric composite configurations. These structures are utilized to elucidate the more complex bandgap tuning characteristics inherent in SMA-based metamaterials.

Building upon the symmetric configurations of pre-bending, pre-twisting, and pre-stretching, a series of combined experimental investigations were conducted. Specifically, five types of composite structures were analyzed through experimental and simulation comparisons: (1) Combined 30° pre-bending and 30° pre-twisting; (2) Combined 75° pre-bending and 30° pre-twisting; (3) Combined 65° pre-twisting and 30° pre-bending; (4) Combined 30° pre-bending and 3 mm pre-stretching; (5) Combined 75° pre-bending and 3 mm pre-stretching. The experimental structures are shown in Figure 12, while the corresponding experimental and simulation results are presented in Figure 13 and Figure S1 in Supplemental Materials.

Figure 12.

Experimental structural diagram of bandgaps in SMA metamaterials with different composite pre-deformations (a₁)–(a₂), (b₁)–(b₂), (c₁)–(c₂), (d₁)–(d₂), and (e₁)–(e₂) correspond to the 30° bending–30° twisting composite structure, 75° bending–30° twisting composite structure, 30° bending–65° twisting composite structure, 30° bending–3 mm stretching composite structure, and 75° bending–3 mm stretching composite structure, respectively.

Figure 13.

Simulation and experimental results of bandgaps in metamaterial structures based on SMA resonators with 75°bending–30° twisting composite structure at various temperatures (a).

Based on the experimental and simulation data in Figure 13 and Figure S1 in Supplemental Materials, the starting frequency, ending frequency, and bandwidth of the bandgap for different configurations were analyzed and summarized in Table 2. The results indicate that, compared to torsional pre-twisting, pre-bending exerts a more significant influence on the bandgap tuning of composite structures. For instance, in the 75° pre-bending and 30° pre-twisting composite structure with SMA resonators, the bandgap starting frequency is shifted forward by approximately 50 Hz compared to the undeformed structure at both 25°C and 44°C. Even as the temperature rises to 60°C, accompanied by the gradual recovery of the SMA deformation, the starting frequency remains about 30 Hz ahead of the original structure. Similarly, in the 75° pre-bending and 3 mm pre-stretching composite structure, the bandgap starting frequency is advanced by about 30 Hz compared to the undeformed structure. These findings demonstrate that composite pre-deformations of SMA can effectively regulate the bandgap characteristics of metamaterial structures, with pre-bending deformation playing a particularly prominent role.

Table 2.

Bandgap characteristics of metamaterial structures with different composite pre-deformations.

SMA composite configurations	Original structure (Hz)	30° Bending–30° twisting (Hz)	75° Bending–30° twisting (Hz)	30° Bending-65° twisting (Hz)	30° Bending–3 mm stretching (Hz)	75° Bending–3 mm stretching (Hz)
Bandgap starting frequency–bandgap ending frequency (bandwidth) at 25°C	332–358 (26)	324–350 (26)	284–306 (22)	326–350 (24)	322–348 (26)	302–326 (24)
Bandgap starting frequency–bandgap ending frequency (bandwidth) at 44°C)	312–336 (24)	302–326 (24)	266–286 (20)	306–330 (24)	304–326 (24)	280–302 (22)
Bandgap starting frequency–bandgap ending frequency (bandwidth) at 60°C	404–435 (31)	396–427 (31)	374–403 (29)	382–411 (29)	396–426 (30)	384–413 (29)
Bandgap starting frequency–bandgap ending frequency (bandwidth) at 80°C	414–446 (32)	414–446 (32)	414–446 (32)	414–446 (32)	414–446 (32)	414–446 (32)

According to the data in Table 2, it can be clearly observed that the experimental results and numerical simulations remain in strong agreement with the theoretical prediction of equation (22). For SMA resonators structures with composite pre-deformations, the coupled variations of elastic modulus ( $E_{j}$ ) and moment of inertia ( $I_{j}$ ) under thermal loading jointly lead to a significant bandgap shift of up to 50 Hz compared with the undeformed structure.

Discussion

Synergistic regulation and contribution analysis of multi-parameter coupling

Based on the experimental and simulation data in Section 3.2.2, this study compares the performance of metamaterial structures with non-deformed SMA against those with various pre-deformed SMA (bending, twisting, stretching, and their composite configurations) under four temperature conditions. Following the methodology described in Bement (1989) and Montgomery (2017), Analysis of Variance was conducted to quantify the contribution rates of temperature, pre-deformation, and their synergistic effect term to bandgap regulation were quantified, and the specific calculation results are listed in Table 3 below.

Table 3.

$ρ$ of temperature, pre-deformation, and their synergistic effects to bandgap regulation.

Pre-deformation mode	Temperature contribution (%)	Deformation contribution (%)	Synergistic contribution (%)	Total (%)
Pre-stretching 6 mm	93.49	4.84	1.67	100
Pre-bending 30°	98.98	0.74	0.28	100
Pre-bending 75°	88.48	8.44	3.08	100
Pre-twisting 30°	83.27	12.49	4.24	100
Pre-twisting 65°	58.40	29.10	12.50	100
75° Bending–30° twisting	89.30	7.73	2.97	100
75° Bending–3 mm stretching	94.47	4.00	1.53	100

To facilitate this comparison, the bandgap data of the non-deformed SMA metamaterial structure was used as a baseline and compared with those of metamaterial structures with bending, twisting, stretching, and composite pre-deformed SMA configurations, respectively. Within the analytical framework, two independent variables are defined: temperature (T) and pre-deformation mode (D). Specifically, $N_{T} = 4$ represents the four characteristic temperature levels, while $N_{D} = 2$ denotes the number of deformation modes in each comparison. The variable $y_{i j}$ represents the initial bandgap frequency for the i-th deformation mode at the j-th temperature level, and $\bar{\bar{y}}$ signifies the grand mean of all observations.

As shown in equation (24), the total sum of squares ( $S S_{T o t a l}$ ) represents the overall deviation of all experimental observations from the grand mean, reflecting the total variation of the frequency distribution during the bandgap tuning process:

S S_{Total} = \sum_{i = 1}^{N_{D}} \sum_{j = 1}^{N_{T}} {(y_{ij} - \bar{\bar{y}})}^{2}

(24)

As shown in equation (25), the sum of squares for the temperature main effect ( $S S_{T}$ ) quantifies the frequency fluctuation induced solely by thermal changes and the resulting phase transformation of the SMA:

S S_{T} = N_{D} \cdot \sum_{j = 1}^{N_{T}} {({\bar{y}}_{T_{j}} - \bar{\bar{y}})}^{2}

(25)

In this formula, ${\bar{y}}_{T_{j}}$ denotes the average frequency at the j-th temperature level across all deformation modes.

As shown in equation (26), the sum of squares for the pre-deformation main effect ( $S S_{D}$ ) accounts for the frequency fluctuations arising from structural configuration changes and geometric nonlinearity:

S S_{D} = N_{T} \cdot \sum_{i = 1}^{N_{D}} {({\bar{y}}_{D_{i}} - \bar{\bar{y}})}^{2}

(26)

In this equation, ${\bar{y}}_{D_{I}}$ is the average frequency for the i-th deformation mode across all temperature levels.

As shown in equation (27), the synergistic interaction sum of squares ( $S S_{T \times D}$ ) is used to extract the non-linear coupling effect between temperature and pre-deformation, thereby accurately capturing the multi-physical synergy between the thermal phase transformation of the SMA and structural configuration evolution:

S S_{T \times D} = S S_{Total} - S S_{T} - S S_{D}

(27)

As shown in equation (28), the percentage contribution rate ( $ρ$ ) for each parameter is quantified through normalization to objectively reflect the influence weight of each factor on the bandgap characteristics:

ρ_{i} = \frac{S S_{i}}{S S_{T o t a l}} \times 100 %

(28)

Where $ρ_{i}$ is the percentage contribution of factor i, and $S S_{i}$ is the corresponding sum of squares for temperature (T), pre-deformation (D), or their interaction ( $T \times D$ ).

Quantitative analysis confirms that temperature remains the dominant factor driving the bandgap evolution of metamaterial structures. For example, under the 30° pre-bending condition of SMA, the temperature contribution rate is as high as 98.98%, which indicates that the phase transformation process of SMA constitutes the core physical basis for frequency regulation. However, this dominance is significantly modulated by the geometric configuration. In a completely symmetric pre-stretching SMA metamaterial structure, a 6 mm axial stretch produced a 1.67% temperature-pre-deformation synergistic bandgap regulation contribution; as the magnitude of deformation increases, this temperature-pre-deformation synergistic effect shows a significant nonlinear intensification trend. Specifically, the synergistic bandgap regulation contribution rate of twisting and temperature surged from 4.24% at the time of SMA 30° symmetric twisting to 12.50% at 65° twisting; the synergistic bandgap regulation contribution rate of bending and temperature increased from 0.28% at the time of SMA 30° symmetric bending to 3.08% at 75° bending. This trend profoundly reveals that large-scale geometric deformation strengthens the deep coupling between the SMA thermal-induced phase transformation (material nonlinearity) and configuration evolution (geometric nonlinearity), thereby significantly enhancing the regulation sensitivity of the metamaterial structures.

Furthermore, the research further reveals the “regulatory equivalence” phenomenon in composite pre-deformed SMA metamaterial structures, opening a new path for asymmetric bandgap design. Experimental data shows that the synergistic bandgap regulation contribution rate of 75° bending–30° twisting composite pre-deformed SMA with temperature is 2.97%, which is almost equal to the 3.08% bandgap contribution rate of the fully symmetric 75° pre-bending SMA metamaterial structure. This indicates that by strategically introducing a twisting angle, the bandgap regulation capacity of symmetric bending SMA configurations can be effectively replicated or even compensated for in asymmetric configurations. A similar synergistic effect was also verified in the stretch-bend composite pre-deformation coupling: introducing only a 3 mm pre-stretching into the SMA’s bending structure achieved a 1.53% synergistic bandgap contribution rate for the structure, which is equivalent to the 1.67% synergistic contribution rate of the 6 mm completely symmetric pre-stretching SMA metamaterial structure mentioned above.

In summary, by quantitatively deconstructing the contribution rates, the independent influences of temperature and pre-deformation on SMA performance and their interactions can be more clearly identified. The interaction between these parameters is essentially a multi-physical synergy rather than a simple linear superposition. Even in the case of decomposing symmetric structures into composite deformations, the system still exhibits extremely high regulation efficiency. This not only highlights the superior flexibility of the proposed SMA-based metamaterial structure in multi-parameter coupled regulation, but also proves its innovative application value for achieving precise, low-frequency vibration suppression in complex mechanical environments.

Bandgap evolution across the heating process

Based on the experimental and numerical results presented in Section 3.2, it is evident that the bandgap characteristics of SMA-resonator-based metamaterial structures—encompassing undeformed, pre-stretched, pre-bent, pre-twisted, and combined pre-deformations—consistently align with the trends predicted by equation (22). To further validate the applicability of equation (22), we not only conducted experimental observations at selected temperature points, but also performed more systematic numerical simulations covering the entire heating process, with a wider temperature range and smaller temperature increments. As illustrated in Figure 14 and Figure S2, the bandgap evolution of SMA resonators under various pre-deformation modes during the heating process shows excellent agreement with the theoretical prediction of equation (22) and the trend illustrated in Figure 2. These results clearly demonstrate that under thermal loading, SMA resonators undergo not only modulus variations but also geometrical dimensional changes, which consequently alter their moment of inertia and lead to more pronounced bandgap shifts relative to the undeformed structure. By integrating experimental evidence, numerical simulations, and theoretical modeling, this study elucidates the intrinsic mechanisms governing bandgap evolution under the coupled effects of temperature and pre-deformation.

Figure 14.

Temperature-driven bandgap evolution of SMA-based metamaterial structures with different pre-deformation modes compared to the original structure (The dash-dotted lines represent the onset frequencies of the bandgaps, while the solid lines denote the cutoff frequencies. (a) Pre-twisting compared with the original structure, (b) pre-bending compared with the original structure, (c) bending–twisting composite pre-deformation compared with the original structure, and (d) bending–stretching composite pre-deformation compared with the original structure.).

Bandgap variation under symmetrical pre-deformation angles

Effect of twisting angle on bandgap shift and bandwidth reduction

By measuring the bandgap of SMA resonator-based metamaterial structures with pre-twisting angles of 30° and 65°, it can be observed that the bandgap starting frequency shifts to lower frequencies as the twisting angle increases. To further investigate this trend, simulations were conducted over a broader range of twisting angles. The results, presented in Figure S3 in Supplemental Materials and Figure 15.

Figure 15.

(a) Bandgap curves of metamaterial structures based on SMA resonators with different twisting angles and (b) bar chart of bandwidths of metamaterial structures based on SMA resonators with different twisting angles.

The simulation results indicate that as the twisting angle increases from 30° to 180°, the bandgap starting frequency decreases significantly from 274 to 102 Hz. To quantitatively describe the regulation mechanism of the twisting angle on the bandgap, the relationships between the twisting angle ( $θ_{twist}$ ) and the starting bound frequency ( $f_{start}$ ), ending bound frequency ( $f_{end}$ ), and bandwidth ( $Δ f$ ) were fitted using polynomial functions. The corresponding functional relationships are expressed in equation (29):

{\begin{matrix} f_{start} = - 4.31 e^{- 5} θ_{twist}^{3} + 1.92 e^{- 2} θ_{twist}^{2} - 3.42 θ_{twist} + 332 \\ f_{end} = - 5.46 e^{- 5} θ_{twist}^{3} + 2.31 e^{- 2} θ_{twist}^{2} - 3.89 θ_{twist} + 358 \\ Δ f = 6 e^{- 4} θ_{twist}^{2} - 0.208 θ_{twist} + 26 \end{matrix}

(29)

The scientific mechanism of frequency regulation is fundamentally based on the reduction of the equivalent cross-sectional moment of inertia in SMA resonators due to twisting deformation, as detailed in equation (22). This structural change causes a downward shift in bandgap boundaries, while simultaneously narrowing the bandwidth ( $Δ w$ ) by decreasing the parameter $w_{t}$ according to equation (21). Specifically, as evidenced in Figure 15 and Figure S3, the twisting deformation exerts a significant nonlinear influence, causing the bandwidth to compress from 20 Hz at 30° to 8 Hz at 180°, alongside the emergence of a new second-order bandgap at 466–494 Hz when the twisting angle reaches 180°.

To quantitatively evaluate the reliability of the transfer function in equation (29), a comprehensive error analysis was performed, yielding a Coefficient of Determination (R²; Draper and Smith, 1998) as high as 0.9998 and a Root Mean Square Error (RMSE; Hyndman and Koehler, 2006) of 0.67–0.86 Hz. Furthermore, the p-values (Montgomery, 2017) for all key parameters are less than 0.001, indicating that the observed correlation between the twisting angle and frequency offsets is statistically inevitable rather than accidental. In this model, the R² demonstrates that the formula explains 99.98% of the data variance, while the low RMSE reflects exceptional local prediction accuracy and strong consistency with the simulation data. Consequently, an increase in the SMA resonator’s twisting angle shifts the bandgap to lower frequencies while simultaneously compressing its width.

Effects of bending angle on bandgap frequency shift and bandwidth variation

Similarly, simulations were conducted over a broader range of bending angles, with results shown in Figure S4 in Supplemental Materials and Figure 16. As the bending angle increases, the bandgap starting frequency shifts from 322 Hz at a bending angle of 30° to 202 Hz at 180°. To quantitatively describe the regulation mechanism of the bending angle on the bandgap, polynomial fitting was applied to the simulation data. The functional relationships between the bending angle ( $θ_{bend}$ ) and the starting bound frequency ( $f_{start}$ ), ending bound frequency ( $f_{end}$ ), and bandwidth ( $Δ f$ ) are expressed in equation (30):

{\begin{matrix} f_{start} = - 3.83 e^{- 7} θ_{bend}^{4} + 1.63 e^{- 4} θ_{bend}^{3} - 2.5 e^{- 2} θ_{bend}^{2} \\ - 4.5 e^{- 2} θ_{bend} + 332 \\ f_{end} = - 2.71 e^{- 7} θ_{bend}^{4} + 1.15 e^{- 4} θ_{bend}^{3} - 1.84 e^{- 2} θ_{bend}^{2} \\ - 1.25 e^{- 1} θ_{bend} + 358 \\ Δ f = 1.11 e^{- 7} θ_{bend}^{4} - 4.8 e^{- 5} θ_{bend}^{3} + 6.64 e^{- 2} θ_{bend}^{2} \\ - 7.96 e^{- 1} θ_{bend} + 26 \end{matrix}

(30)

Figure 16.

(a) Bandgap curves of metamaterial structures based on SMA resonators with different bending angles and (b) bar chart of bandwidths of metamaterial structures based on SMA resonators with different bending angles.

According to equation (30), bending deformation exerts a significant influence on the bandgap characteristics of the metamaterial; however, the observed trend contrasts sharply with that of twisting deformation. This phenomenon suggests that the mechanical properties of the SMA resonator are fundamentally altered during the bending process. Specifically, as the bending angle increases, the resonator’s configuration transitions from a flat extension to a concentrated “mass block” effect, which significantly increases the effective participating mass in the resonance. Based on the relationship established in equation (21), this increase in the mass ratio μ enhances the coupling strength between the resonator and the host beam, directly causing the overall bandwidth ( $Δ w$ ) to expand from 24 Hz at 30° to 50 Hz at 180°. To quantitatively evaluate the reliability of the transfer function in equation (30), a comprehensive error analysis was performed. The fitting formula demonstrates exceptional precision, with a Coefficient of Determination (R²) as high as 0.9999 and a p-value < 0.001, indicating that the correlation between bending angles and frequency responses is statistically inevitable. Additionally, the RMSE (0.46–0.72 Hz) quantifies the minimal average deviation between the simulated and predicted values, ensuring strong consistency between the fitting formula and the simulation data.

In summary, as the bending angle increases, the bandgap starting frequency shifts to lower frequencies, albeit to a lesser extent compared to twisting deformation. Crucially, bending deformation significantly broadens the bandwidth, thereby enhancing the structure’s tunability.

Bandgap variation patterns under different combined pre-deformation modes

Based on the bandgap characteristics of the metamaterial structure with composite-deformed SMA at room temperature, as shown in Table 2, it is observed that under constant twisting angle and tensile strain, an increase in bending angle induces a significant low-frequency shift of the bandgap. For instance, the bandgap of the metamaterial structure with symmetrically bent SMA has an onset frequency of 288 Hz and an ending frequency of 317 Hz, whereas the structure with a composite deformation of 75° bending and 30° twisting exhibits an onset frequency of 286 Hz and an ending frequency of 306 Hz. Although the bandwidth of the composite structure is slightly narrower than that of the symmetric structure, the overall bandgap shift remains comparable. Therefore, in scenarios where an SMA elements of the ideal dimensions for symmetric deformation are unavailable, a composite deformation strategy can be employed by combining two half-sized SMA pieces, thereby effectively tuning the bandgap to the desired frequency range.

Conversely, when the bending angle is held constant, increasing the twisting angle yields a trend distinct from that observed in purely symmetric twisting. Rather than shifting significantly toward lower frequencies, the bandgap moves toward higher frequencies, and the bandwidth gradually decreases. As shown in Figure 17(a1), simulation results of the 75° bending–65° twisting composite structure indicate that when the twisting angle becomes excessive, the bandgap vanishes. Similarly, as shown in Figure 17(a2)–(a3), no bandgap is observed in the metamaterial structures with 3 mm stretching–30° twisting and 3 mm stretching–65° twisting composite deformations.

Figure 17.

The bandgap simulation results of metamaterial structures with different composite deformation modes are as follows: (a₁)–(a₃) correspond to the 75° bending–65° torsion composite structure, the 3 mm stretching–30° torsion composite structure, and the 3 mm stretching–65° torsion composite structure, respectively. (b₁)–(d₃) represent the bandgap simulation results of metamaterial structures under the composite deformation conditions of 120°, 150°, and 180° bending combined with 45°, 65°, and 90° torsion, respectively.

To further explore the interplay of twisting and bending in the process of composite deformation, extensive simulation analyses were conducted, as illustrated in Figure 17(b1)–(d3). It is evident that when a small twisting angle is combined with a large bending angle, the bandgap persists and shifts progressively toward lower frequencies as the bending angle increases. However, when the twisting angle increases to 65° or 90°—even when coupled with substantial bending angles of 120°, 150°, or 180°—the bandgap vanishes. This indicates that bending deformation plays the dominant role in regulating the bandgap of SMA-based metamaterial structures with composite deformation. While sufficient bending combined with minor twisting enables effective modulation, excessive twisting may completely suppress bandgap formation within the 0–500 Hz range. Therefore, in the design of metamaterial structures with composite deformation, bending deformation should serve as the primary tuning mechanism, supplemented by auxiliary twisting and stretching deformations, to achieve the desired bandgap tuning effect.

Bandgap characteristics of two-dimensional metamaterial structures based on SMA resonators

Compared with one-dimensional metamaterial structures, the application of SMA to the bandgap tuning of two-dimensional metamaterial plate structures holds significant research value. According to the bandgap theory of two-dimensional plate structures proposed by Wang et al. (2021). in conjunction with Bloch theory, the lower bound frequency of the bandgap $f$ , as expressed in equation (26), is jointly determined by the stiffness $k$ of the SMA beams and the attached mass m, with the schematic of the overall structure shown in Figure 18(a) and (b).

Figure 18.

Two-dimensional metamaterial structure with SMA resonators: (a) resonant unit structure with SMA resonators and(b) two-dimensional metamaterial structure with SMA resonators.

On this basis, a two-dimensional metamaterial plate model incorporating SMA resonators was established, Specifically, the model comprises a 350 mm × 70 mm × 2 mm steel substrate plate, 30 mm × 6 mm × 1 mm SMA beams, 6 mm × 5 mm × 2 mm resin connection blocks, and 10 mm × 6 mm × 40 mm lead attached masses. Numerical simulations were performed under the corresponding boundary conditions to obtain its bandgap characteristics, as illustrated in Figure 19(a)–(d). The simulation results reveal that as the temperature increases from 25°C to 80°C, the onset frequencies of both the first and second-order bandgaps exhibit a trend of advancing followed by retreating. This behavior is consistent with the variation in SMA elastic modulus shown in Figure 2 and aligns with the theoretical prediction of equation (26), which states that under constant attached mass, the lower bound frequency of the bandgap depends solely on the stiffness of the SMA beams. Notably, compared with conventional one-dimensional metamaterial structures, the two-dimensional plate structure demonstrates a broader bandgap range, highlighting its greater potential for vibration attenuation applications.

f = \frac{1}{2 π} \sqrt{\frac{k}{m}}

(31)

Figure 19.

Simulated bandgap diagrams of the two-dimensional metamaterial plate structure with SMA resonators (a–d correspond to temperatures of 25°C, 44°C, 60°C, and 80°C, respectively).

Conclusion

This study systematically investigates the influence of temperature coupled with various pre-deformation forms on the bandgap characteristics of SMA resonator-based metamaterial beams. Initially, the relationship between the elastic modulus of SMA and temperature was characterized through testing SMA-based metamaterial structures under different thermal conditions. Based on these properties, the effects of pre-stretching, pre-bending, pre-twisting, and composite pre-deformations on bandgap tunability were examined. Experimental results demonstrate that the coupling of temperature and pre-deformation enables effective regulation of the bandgap across different frequency ranges, particularly facilitating low-frequency bandgap tuning. Analysis of Variance was employed to quantify the contribution rates of temperature, pre-deformation, and their synergistic effect. The dominance of temperature is significantly modulated by geometric configurations. Notably, the synergistic contribution exhibits a “nonlinear intensification” as deformation increases, revealing a deep coupling between the SMA’s thermal phase transformation and structural geometric nonlinearity. Moreover, the research identifies a “regulatory equivalence” in composite configurations, where asymmetric combinations of twisting, bending, and stretching achieve tuning efficiencies comparable to fully symmetric structures, thereby highlighting the superior flexibility of multi-parameter coupled regulation. To more intuitively present the bandgap evolution of SMA-resonator-based metamaterial structures under various pre-deformation modes throughout the entire heating process, numerical simulations were conducted, accounting for the dynamic evolution of SMA geometry and elastic modulus during heating. The simulation results indicate that the bandgap evolution trends under different pre-deformation conditions are in excellent agreement with theoretical predictions, thereby revealing the intrinsic mechanism of bandgap variation under the coupled effects of temperature and pre-deformation. Subsequently, the bandgap evolution of metamaterial structures incorporating symmetrically pre-bending and pre-twisting SMA resonators was analyzed. By simulating the bandgap curves under different pre-deformation angles, the advantages and limitations of SMA metamaterials based on pre-bending and pre-twisting were revealed, providing a theoretical basis for optimal selection of pre-deformation parameters in practical applications. Additionally, the effects of composite pre-deformations were investigated by evaluating the relative contributions of pre-bending and pre-twisting angles to bandgap regulation, thus offering guidance for optimizing composite pre-deformation strategies for more efficient bandgap control. Finally, the investigation was extended from one-dimensional SMA metamaterial beam structure to two-dimensional SMA-based metamaterial plate. The results indicate that the two-dimensional structure exhibits a broader bandgap range and demonstrates higher sensitivity and flexibility under temperature variations, highlighting its significant potential for low-frequency vibration suppression applications.

Supplemental Material

sj-docx-1-jim-10.1177_1045389X261441424 – Supplemental material for Investigation on bandgap modulation of locally resonant metamaterials with shape memory alloy resonators under temperature and multi-mode pre-deformation

Supplemental material, sj-docx-1-jim-10.1177_1045389X261441424 for Investigation on bandgap modulation of locally resonant metamaterials with shape memory alloy resonators under temperature and multi-mode pre-deformation by Ming Jun Gao, Chao Yin, Zong Jun Li, Li Jia Deng, Chen Wang, Tao Xi Wang, Xing Shen and Fu Jian in Journal of Intelligent Material Systems and Structures

Footnotes

ORCID iDs

Ming Jun Gao

Chen Wang

Author contributions

Ming Jun Gao: Conceptualization, Investigation, Formal analysis, Writing-original draft, Data Curation, Validation, Resources. Chao Yin: Conceptualization, Investigation, Validation. Zong Jun Li: Resources, Data Curation. Li Jia Deng: Investigation, Data Curation. Chen Wang: Investigation, Funding acquisition, Project administration. Tao Xi Wang: Supervision, Writing–Review & Editing, Methodology, Investigation, Conceptualization, Project administration. Xing Shen: Writing–Review & Editing. Fu Jian: Writing–Review & Editing.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Aeronautical Science Foundation of China (Grant No. 20230015052002).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

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

Supplemental material

Supplemental material for this article is available online.

References

Almirall

Fernández-García

Gil

(2022) Wearable metamaterial for electromagnetic radiation shielding. Journal of the Textile Institute 113(8): 1586–1594.

Bement

(1989) Taguchi Techniques for Quality Engineering. Taylor & Francis.

Bhattacharya

(2003) Microstructure of Martensite: Why It Forms and How It Gives Rise to the Shape-Memory Effect. Oxford University Press.

Brinson

(1993) One-dimensional constitutive behavior of shape memory alloys: Thermomechanical derivation with non-constant material functions and redefined martensite internal variable. Journal of Intelligent Material Systems and Structures 4(2): 229–242.

Candido de Sousa

Sugino

De Marqui Junior

, et al. (2018a) Adaptive locally resonant metamaterials leveraging shape memory alloys. Journal of Applied Physics 124(6): article no. 064505.

Candido de Sousa

Tan

De Marqui

, et al. (2018b) Tunable metamaterial beam with shape memory alloy resonators: Theory and experiment. Applied Physics Letters 113(14): article no. 143502.

Casalotti

El-Borgi

Lacarbonara

(2018) Metamaterial beam with embedded nonlinear vibration absorbers. International Journal of Non-Linear Mechanics 98: 32–42.

Chen

Wang

Wen

, et al. (2013) Wave propagation and attenuation in plates with periodic arrays of shunted piezo-patches. Journal of Sound and Vibration 332(6): 1520–1532.

Chen

Huang

(2016) An adaptive metamaterial beam with hybrid shunting circuits for extremely broadband control of flexural waves. Smart Materials and Structures 25(10): 105036.

10.

Chuang

K-C

Wang

D-F

(2019) A tunable elastic metamaterial beam with flat-curved shape memory alloy resonators. Applied Physics Letters 114(5): article no. 051903.

11.

Draper

Smith

(1998) Applied Regression Analysis. John Wiley & Sons.

12.

Gao

Wang

(2024) An active tunable piezoelectric metamaterial beam for broadband vibration suppression by optimization. Acta Mechanica Sinica 40(3): 523235.

13.

Cheng

Yang

, et al. (2003) Broadband locally resonant sonic shields. Applied Physics Letters 83(26): 5566–5568.

14.

Tang

Banerjee

, et al. (2017) Metastructure with piezoelectric element for simultaneous vibration suppression and energy harvesting. Journal of Vibration and Acoustics 139(1): 011012.

15.

Hyndman

Koehler

(2006) Another look at measures of forecast accuracy. International Journal of Forecasting 22(4): 679–688.

16.

Kumar

Lagoudas

(2008) Introduction to Shape Memory Alloys. Shape Memory Alloys: Modeling and Engineering Applications. Springer. pp.1–51.

17.

Yang

Zhao

, et al. (2024a) A new seismic metamaterial design with ultra-wide low-frequency wave suppression band utilizing negative Poisson’s ratio material. Engineering Structures 319: 118821.

18.

Liu

Zhang

Mao

, et al. (2000) Locally resonant sonic materials. Science 289(5485): 1734–1736.

19.

Liu

Zhou

, et al. (2024b) Broadening bandgaps in a multi-resonant piezoelectric metamaterial plate via bandgap merging phenomena. Scientific Reports 14(1): 16127.

20.

Chen

, et al. (2021) Tuning characteristics of a metamaterial beam with lateral-electric-field piezoelectric shuntings. Journal of Sound and Vibration 491: 115738.

21.

Matlack

Bauhofer

Krödel

, et al. (2016) Composite 3D-printed metastructures for low-frequency and broadband vibration absorption. Proceedings of the National Academy of Sciences 113(30): 8386–8390.

22.

T-X

Fan

Q-S

Z-Y

, et al. (2020) Flexural wave energy harvesting by multi-mode elastic metamaterial cavities. Extreme Mechanics Letters 41: 101073.

23.

Montgomery

(2017) Design and Analysis of Experiments. John Wiley & Sons.

24.

Nouh

Aldraihem

Baz

(2015) Wave propagation in metamaterial plates with periodic local resonances. Journal of Sound and Vibration 341: 53–73.

25.

Peng

Frank Pai

(2014) Acoustic metamaterial plates for elastic wave absorption and structural vibration suppression. International Journal of Mechanical Sciences 89: 350–361.

26.

Sugino

Leadenham

Ruzzene

, et al. (2016) On the mechanism of bandgap formation in locally resonant finite elastic metamaterials. Journal of Applied Physics 120(13): article no. 134501.

27.

Sugino

Xia

Leadenham

, et al. (2017) A general theory for bandgap estimation in locally resonant metastructures. Journal of Sound and Vibration 406: 104–123.

28.

Sun

Han

(2024) Novel hybrid-controlled graded metamaterial beam for bandgap tuning and wave attenuation. European Journal of Mechanics-A/Solids 103: 105178.

29.

Tamma

Boonjue

Wiboonjaroen

, et al. (2024) Performance improvement of slot antenna with metamaterial for modern wireless communication. Results in Engineering 23: 102686.

30.

Wang

Zhang

, et al. (2021) Bandgap properties in metamaterial sandwich plate with periodically embedded plate-type resonators. Mechanical Systems and Signal Processing 151: 107375.

31.

Wenjing

Yize

(2022) Active control on band gap properties and interface transmission of elastic waves in piezoelectric metamaterial beams. Applied Mathematics and Mechanics 43(1): 14–25.

32.

Xiao

Wen

Wang

, et al. (2013) Theoretical and experimental study of locally resonant and Bragg band gaps in flexural beams carrying periodic arrays of beam-like resonators. Journal of Vibration and Acoustics 135(4): 041006.

33.

Xiao

Wen

(2012) Flexural wave band gaps in locally resonant thin plates with periodically attached spring–mass resonators. Journal of Physics D Applied Physics 45(19): 195401.

34.

Yin

Geng

Shen

, et al. (2022) Bandgap variation of a locally resonant metamaterial induced by temperature variation and pre-tension in the shape memory alloy resonators. Smart Materials and Structures 31(5): 055012.

35.

Zhu

Liu

, et al. (2014) A chiral elastic metamaterial beam for broadband vibration suppression. Journal of Sound and Vibration 333(10): 2759–2773.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

2.58 MB