Design of Stretch-Dominated Metamaterials Avoiding Bandgap Resonance

Abstract

Mechanical metamaterials subjected to dynamic loads within their bandgaps can still experience significant undesired structural responses, which are referred to as bandgap resonances. This paper proposes a design method for stretch-dominated metamaterials to avoid such a phenomenon. The metamaterials are modeled as pin-jointed bar structures, and the sufficient condition for preventing their bandgap resonances is derived: they must exhibit spatial inversion symmetry and satisfy certain boundary conditions. A matrix-form perturbation expression of the bandgap is then provided to generate expected bandgaps by adjusting the node coordinates and element cross-sectional areas of the unit cells under symmetry constraint. As an example, a two-dimensional metamaterial is designed to achieve an expected bandgap of 600–1000 Hz. The frequency-response analyses show that the sufficient condition ensures the suppression of bandgap resonances.

1. Introduction

Bandgap is a fundamental property of materials in solid-state physics governing their electrical conductivity characteristics. In the context of mechanical metamaterials, the presence of a bandgap in the dispersion function signifies that mechanical waves within a specific frequency range cannot propagate, constituting the basis for vibration isolation materials and topological insulators. Mechanical wave bandgaps are commonly categorized into Bragg bandgaps [1] and local resonant bandgaps [2], originating from the Bragg scattering and local resonator–wave interaction, respectively. Owing to the ability to isolate mechanical waves, bandgap mechanical metamaterials are broadly applicable in engineering vibration and sound insulation. Vasconcelos et al. [3] proposed a metamaterial-based interface to attenuate pressure waves induced by the impact hammer acting on the top of an offshore monopile. Zuo et al. [4] applied a star-shaped metamaterial to the shell of an underwater vehicle to reduce engine noise emission. A lattice-structured metamaterial capable of both sound insulation and ventilation was developed by Li et al. [5] and can be used as a roadside noise barrier. Seismic metamaterials composed of periodically arranged building foundations have been validated for shielding seismic waves and train-induced vibrations [6]. Replacing conventional aggregates with local resonant masses [7] enables concrete structural members to function as metamaterials capable of dissipating elastic waves induced by dynamic loads.

Dynamic loads within the bandgaps may still induce significant structural responses in the metamaterials, which are known as bandgap resonances [8]. For example, Zhang et al. [9] designed a negative stiffness metamaterial that exhibits a peak in its frequency-response curve within the bandgap range. Jiang et al. [10] designed negative stiffness metamaterials whose frequency-response curves do not exactly match their bandgaps. The chiral and hexagonal lattices proposed by Li et al. [11,12] also experience large frequency responses within the bandgap ranges. Bandgap resonances can similarly be observed in the frequency-response curves of the metamaterials studied in Refs. [13,14,15,16]. The presence of bandgap resonances suggests that certain eigenfrequencies of the metamaterial lie within the bandgap range, thereby compromising its vibration isolation performance. Therefore, it is necessary to prevent this phenomenon in the engineering design of bandgap metamaterials.

Bandgap resonance attracted early attention in the field of solid-state physics. In calculating the specific heat capacity of crystals, periodically arranged atoms are modeled as vibrating one-dimensional atomic chains. Thus, some early studies focused on the correspondence between the eigenfrequency spectra and the dispersion functions of finite atomic chains. For example, Wallis [17] identified an eigenfrequency within the bandgap while analyzing the spectrum of a finite diatomic chain. Recent studies continue to concentrate on the bandgap resonances in simple metamaterials. Based on topology energy band theory, Rosa et al. [18] proposed a method to analyze and manipulate the bandgap resonance of an elastic beam. Ba’ba’a et al. derived closed-form expressions for the characteristic equations governing the natural frequencies of a finite one-dimensional rod [19] and a mass-spring chain [20] using the transfer matrix method and analyzed the factors contributing to bandgap resonances. Park et al. [21] numerically and experimentally investigated how unit cell symmetry, boundary conditions, material/geometric properties, and the number of unit cells affect the existence of bandgap resonances in bilayer beams. The modified perturbed tridiagonal n-Toeplitz method is used by Ramakrishnan et al. [22] to estimate the bandgap resonance characteristics in 1D and 2D monoatomic phononic crystal lattices.

Calculating the dispersion function via Bloch’s theorem assumes that the metamaterials exhibit translational symmetry [23], necessitating that they be either infinitely extended or subjected to periodic boundary conditions [24]. However, finite metamaterials in engineering possess non-periodic boundaries, so their spectra deviate from the dispersion functions. This suggests that the existence of bandgap resonances is determined by the boundary conditions. Bastawrous et al. [8] introduced boundary conditions as perturbations to the dynamic stiffness matrix and analytically derived a closed-form condition for the existence of bandgap resonance in finite diatomic chains. Guo et al. [25] and Sugino et al. [26] solved dynamic differential equations with boundary conditions substituted to obtain the spectra of finite two-phase plates. Jin et al. [27] incorporated boundary conditions in the form of a Fourier series into the dynamic stiffness matrix and solved the spectrum of a two-phase metamaterial plate. By comparing the obtained spectra with the dispersion functions of the metamaterials, the existence of bandgap resonances can be identified.

Analytical calculations of the spectra of metamaterials require explicit expressions for the eigenfrequencies, which pose a substantial mathematical challenge. As a result, existing studies have primarily focused on simple systems [8,17,18,19,20,21,22,23,24,25,26,27], such as two-phase plates or diatomic chains. However, metamaterials designed for specific bandgaps are usually complex, making it difficult to analytically solve their spectra. Therefore, to ensure the vibration isolation performance of a designed metamaterial, bandgap resonance should preferably be avoided without solving the spectrum, which is one of the topics of this paper.

The application of mechanical metamaterials requires the design of material microstructures tailored to specific bandgap properties. Recent studies have employed topology optimization or deep learning to design metamaterials with expected bandgaps. Cool et al. [28] presented a topology optimization framework to obtain bandgaps simultaneously insolating acoustic and structural waves. Zhang et al. [29] employed a deep learning network to predict curved rod lattices with expected bandgap characteristics. Wan et al. [30] predicted two-dimensional phononic crystal lattices that satisfy expected bandgaps with a deep learning network. Metamaterials designed via these methods can exhibit expected bandgaps, but they do not necessarily avoid bandgap resonances. It is essential to conduct a theoretical analysis on the conditions for avoiding bandgap resonance, which can then be incorporated into the design of metamaterials.

This paper proposes a design method for stretch-dominated metamaterials that achieve expected bandgaps while avoiding bandgap resonance. The paper is organized as follows: Section 2 derives the sufficient condition for avoiding the bandgap resonance in stretch-dominated metamaterials. Section 3 presents the equations for designing the unit cell parameters under symmetry constraint. A design example is illustrated in Section 4, which is used to validate both the sufficient condition and the corresponding design method. Finally, Section 5 draws some conclusions.

2. Sufficient Condition for Avoiding Bandgap Resonance

A two-dimensional stretch-dominated metamaterial is used as an example to demonstrate the derivation of the sufficient condition for avoiding bandgap resonance. As shown in Figure 1, a 3 × 3 metamaterial, denoted by m, is modeled as a pin-jointed bar structure and subjected to a boundary condition that can be practically realized in engineering. Figure 2a and Figure 3a show the distributions of modal forces and displacements on the boundaries corresponding to any eigenfrequency ω_j of m. Except for the corner nodes, the horizontal components of the node forces are zero on the top and bottom edges, while the vertical components are zero on the left and right edges. Similarly, the vertical components of the node displacements are zero on the top and bottom edges, and the horizontal components are zero on the left and right edges.

According to Figure 2a and Figure 3a, the boundary node displacement and force vectors of m can be written as $d_{\mathrm{t}/\mathrm{b}}$ = [0, 0, $d_{\mathrm{t}/\mathrm{b}}^{2x}$, 0, …, $d_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}}-1)x}$, 0, 0, 0]^T, $d_{\mathrm{l}/\mathrm{r}}$ = [0, 0, 0, $d_{\mathrm{l}/\mathrm{r}}^{2y}$, …, 0, $d_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}}-1)y}$, 0, 0]^T, $f_{\mathrm{t}/\mathrm{b}}$ = [$f_{\mathrm{t}/\mathrm{b}}^{1x}$, $f_{\mathrm{t}/\mathrm{b}}^{1y}$ 0, $f_{\mathrm{t}/\mathrm{b}}^{2y}$, …, 0, $f_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}}-1)y}$, $f_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}})x}$, $f_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}})y}$]^T, $f_{\mathrm{l}/\mathrm{r}}$ = [$f_{\mathrm{l}/\mathrm{r}}^{1x}$, $f_{\mathrm{l}/\mathrm{r}}^{1y}$, $f_{\mathrm{l}/\mathrm{r}}^{2x}$, 0, …, $f_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}}-1)x}$, 0, $f_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}})x}$, $f_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}})y}$]^T, where $d_{\mathrm{t}/\mathrm{b}}$ and $f_{\mathrm{t}/\mathrm{b}}$ are the displacement and force vectors of the nodes on the top or bottom edge of m; $d_{\mathrm{l}/\mathrm{r}}$ and $f_{\mathrm{l}/\mathrm{r}}$ are the displacement and force vectors of the nodes on the left or right edge of m; $d_{\mathrm{t}/\mathrm{b}}^{ux}$ and $d_{\mathrm{t}/\mathrm{b}}^{uy}$ (u = 1, …, n_t/b) are the x and y components of the displacement of the u-th node on the top or bottom edge, respectively, and $f_{\mathrm{t}/\mathrm{b}}^{ux}$ and $f_{\mathrm{t}/\mathrm{b}}^{uy}$ (u = 1, …, n_t/b) are the corresponding force components; $d_{\mathrm{l}/\mathrm{r}}^{ux}$ and $d_{\mathrm{l}/\mathrm{r}}^{uy}$ (u = 1, …, n_l/r) are the x and y components of the displacement of the u-th node on the left or right edge, and $f_{\mathrm{l}/\mathrm{r}}^{ux}$ and $f_{\mathrm{l}/\mathrm{r}}^{uy}$ (u = 1, …, n_l/r) are the corresponding force components; n_t/b is the number of nodes on the top or bottom edge, and n_l/r is that on the left or right edge; and the superscript T denotes the transpose of a vector or matrix.

Spatial inversion of m in the x-direction (i.e., mirroring across the y-axis) yields the metamaterial m^x. The distributions of modal forces and displacements on the boundaries are shown in Figure 2b and Figure 3b, and the corresponding frequency remains constant at ω_j. According to Figure 2b and Figure 3b, the boundary node displacement and force vectors of m^x can be written as $d_{\mathrm{t}/\mathrm{b}}^{\mathrm{x}}$ = [0, 0, $-d_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}}-1)x}$, 0, …, $-d_{\mathrm{t}/\mathrm{b}}^{2x}$, 0, 0, 0]^T, $d_{\mathrm{r}/\mathrm{l}}^{\mathrm{x}}$ = [0, 0, 0, $d_{\mathrm{l}/\mathrm{r}}^{2y}$, …, 0, $d_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}}-1)y}$, 0, 0]^T, $f_{\mathrm{t}/\mathrm{b}}^{\mathrm{x}}$ = [$-f_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}})x}$, $f_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}})y}$, 0, $f_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}}-1)y}$, …, 0, $f_{\mathrm{t}/\mathrm{b}}^{2y}$, $-f_{\mathrm{t}/\mathrm{b}}^{1x}$, $f_{\mathrm{t}/\mathrm{b}}^{1y}$]^T, $f_{\mathrm{r}/\mathrm{l}}^{\mathrm{x}}$ = [$-f_{\mathrm{l}/\mathrm{r}}^{1x}$, $f_{\mathrm{l}/\mathrm{r}}^{1y}$, $-f_{\mathrm{l}/\mathrm{r}}^{2x}$, 0, …, $-f_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}}-1)x}$, 0, $-f_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}})x}$, $f_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}})y}$]^T, where $d_{\mathrm{t}/\mathrm{b}}^{\mathrm{x}}$ and $f_{\mathrm{t}/\mathrm{b}}^{\mathrm{x}}$ are the displacement and force vectors of the nodes on the top or bottom edge of m^x; and $d_{\mathrm{r}/\mathrm{l}}^{\mathrm{x}}$ and $f_{\mathrm{r}/\mathrm{l}}^{\mathrm{x}}$ are the displacement and force vectors of the nodes on the right or left edge of m^x.

Spatial inversion of m in the y-direction (i.e., mirroring across the x-axis) yields the metamaterial m^y. The distributions of modal forces and displacements on the boundaries are shown in Figure 2c and Figure 3c, and the corresponding frequency remains constant at ω_j. According to Figure 2c and Figure 3c, the boundary node displacement and force vectors of m^y can be written as $d_{\mathrm{b}/\mathrm{t}}^{\mathrm{y}}$ = [0, 0, $d_{\mathrm{t}/\mathrm{b}}^{2x}$, 0, …, $d_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}}-1)x}$, 0, 0, 0]^T, $d_{\mathrm{l}/\mathrm{r}}^{\mathrm{y}}$ = [0, 0, 0, $-d_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}}-1)y}$, …, 0, $d_{\mathrm{l}/\mathrm{r}}^{2y}$, 0, 0]^T, $f_{\mathrm{b}/\mathrm{t}}^{\mathrm{y}}$ = [$f_{\mathrm{t}/\mathrm{b}}^{1x}$, $-f_{\mathrm{t}/\mathrm{b}}^{1y}$, 0, $-f_{\mathrm{t}/\mathrm{b}}^{2y}$, …, 0, $-f_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}}-1)y}$, $f_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}})x}$, $-f_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}})y}$]^T, $f_{\mathrm{l}/\mathrm{r}}^{\mathrm{y}}$ = [ $f_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}})x}$, $-f_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}})y}$, $f_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}}-1)x}$, 0, …, $f_{\mathrm{l}/\mathrm{r}}^{2x}$, 0, $f_{\mathrm{l}/\mathrm{r}}^{1x}$, $-f_{\mathrm{l}/\mathrm{r}}^{1y}$]^T, where $d_{\mathrm{b}/\mathrm{t}}^{\mathrm{y}}$ and $f_{\mathrm{b}/\mathrm{t}}^{\mathrm{y}}$ are the displacement and force vectors of the nodes on the bottom or top edge of m^y; and $d_{\mathrm{l}/\mathrm{r}}^{\mathrm{y}}$ and $f_{\mathrm{l}/\mathrm{r}}^{\mathrm{y}}$ are the displacement and force vectors of the nodes on the left or right edge of m^y.

Spatial inversion of m^x in the y-direction yields the metamaterial m^xy. The distributions of modal forces and displacements on the boundaries are shown in Figure 2d and Figure 3d, and the corresponding frequency remains constant at ω_j. According to Figure 2d and Figure 3d, the boundary node displacement and force vectors of m^xy can be written as $d_{\mathrm{b}/\mathrm{t}}^{\mathrm{xy}}$ = [0, 0, $-d_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}}-1)x}$, 0, …, $-d_{\mathrm{t}/\mathrm{b}}^{2x}$, 0, 0, 0]^T, $d_{\mathrm{r}/\mathrm{l}}^{\mathrm{xy}}$ = [0, 0, 0, $-d_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}}-1)y}$, …, 0, $d_{\mathrm{l}/\mathrm{r}}^{2y}$, 0, 0]^T, $f_{\mathrm{b}/\mathrm{t}}^{\mathrm{xy}}$ = [$-f_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}})x}$, $-f_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}})y}$, 0, $-f_{\mathrm{t}/\mathrm{b}}^{(n_{\mathrm{t}/\mathrm{b}}-1)y}$, …, 0, $-f_{\mathrm{t}/\mathrm{b}}^{2y}$, $-f_{\mathrm{t}/\mathrm{b}}^{1x}$, $-f_{\mathrm{t}/\mathrm{b}}^{1y}$]^T, $f_{\mathrm{r}/\mathrm{l}}^{\mathrm{xy}}$ = [ $-f_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}})x}$, $-f_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}})y}$, $-f_{\mathrm{l}/\mathrm{r}}^{(n_{\mathrm{l}/\mathrm{r}}-1)x}$, 0, …, $-f_{\mathrm{l}/\mathrm{r}}^{2x}$, 0, $-f_{\mathrm{l}/\mathrm{r}}^{1x}$, $-f_{\mathrm{l}/\mathrm{r}}^{1y}$]^T, where $d_{\mathrm{b}/\mathrm{t}}^{\mathrm{xy}}$ and $f_{\mathrm{b}/\mathrm{t}}^{\mathrm{xy}}$ are the displacement and force vectors of the nodes on the bottom or top edge of m^xy; and $d_{\mathrm{r}/\mathrm{l}}^{\mathrm{xy}}$ and $f_{\mathrm{r}/\mathrm{l}}^{\mathrm{xy}}$ are the displacement and force vectors of the nodes on the right or left edge of m^xy.

As can be seen from Figure 2, the boundary node displacement vectors satisfy

$$d_{\mathrm{r}}=d_{\mathrm{l}}^{\mathrm{x}},d_{\mathrm{r}}^{\mathrm{y}}=d_{\mathrm{l}}^{\mathrm{xy}},d_{\mathrm{b}}=d_{\mathrm{t}}^{\mathrm{y}},d_{\mathrm{b}}^{\mathrm{x}}=d_{\mathrm{t}}^{\mathrm{xy}}$$

(1)

Moreover, Figure 3 indicates that the boundary node force vectors satisfy the following relationships except for corner nodes:

$$f_{\mathrm{r}}+f_{\mathrm{l}}^{\mathrm{x}}=0,f_{\mathrm{r}}^{\mathrm{y}}+f_{\mathrm{l}}^{\mathrm{xy}}=0,f_{\mathrm{b}}+f_{\mathrm{t}}^{\mathrm{y}}=0,f_{\mathrm{b}}^{\mathrm{x}}+f_{\mathrm{t}}^{\mathrm{xy}}=0$$

(2)

Let nodes 1, 2, 3, and 4 denote the lower right corner node of m, the lower left corner node of m^x, the upper right corner node of m^y, and the upper left corner node of m^xy, respectively. Then, their node forces satisfy

$$f_1+f_2+f_3+f_4=\left\{\begin{array}{c}{f}_b^{x{n}_{\mathrm{b}}}\\ f_b^{y{n}_b}\end{array}\right\}+\left\{\begin{array}{c}-f_b^{x{n}_{\mathrm{b}}}\\ f_b^{y{n}_b}\end{array}\right\}+\left\{\begin{array}{c}{f}_b^{x{n}_{\mathrm{b}}}\\ -f_b^{y{n}_b}\end{array}\right\}+\left\{\begin{array}{c}-f_b^{x{n}_{\mathrm{b}}}\\ -f_b^{y{n}_b}\end{array}\right\}=0$$

(3)

Equations (1)–(3) indicate that the corresponding nodes on the bottom edge of m and the top edge of m^y have identical displacements, and their node forces are in equilibrium. The same relationship holds between the right edge of m and the left edge of m^x, the top edge of m^xy and the bottom edge of m^x, and the right edge of m^y and the left edge of m^xy. This suggests that m, m^x, m^y, and m^xy can be merged into a new structure M. Figure 4a,b show the distributions of its modal forces and displacements on the boundaries, and the corresponding frequency is still ω_j.

The boundary node displacement and force vectors of M can be written as

$$d_{\mathrm{t}}^{\mathrm{M}}={[0,0,d_{\mathrm{t}}^{2x},0,\dots ,d_{\mathrm{t}}^{(n_{\mathrm{t}}-1)x},0,0,0,-d_{\mathrm{t}}^{(n_{\mathrm{t}}-1)x},0,\dots ,-d_{\mathrm{t}}^{2x},0,0,0]}^{\mathrm{T}}$$

(4)

$$d_{\mathrm{b}}^{\mathrm{M}}={[0,0,d_{\mathrm{t}}^{2x},0,\dots ,d_{\mathrm{t}}^{(n_{\mathrm{t}}-1)x},0,0,0,-d_{\mathrm{t}}^{(n_{\mathrm{t}}-1)x},0,\dots ,-d_{\mathrm{t}}^{2x},0,0,0]}^{\mathrm{T}}$$

(5)

$$d_{\mathrm{l}}^{\mathrm{M}}={[0,0,0,d_{\mathrm{l}}^{2y},\dots ,0,d_{\mathrm{l}}^{(n_{\mathrm{l}}-1)y},0,0,0,-d_{\mathrm{l}}^{(n_{\mathrm{l}}-1)y},\dots ,0,-d_{\mathrm{l}}^{2y},0,0]}^{\mathrm{T}}$$

(6)

$$d_{\mathrm{r}}^{\mathrm{M}}={[0,0,0,d_{\mathrm{l}}^{2y},\dots ,0,d_{\mathrm{l}}^{(n_{\mathrm{l}}-1)y},0,0,0,-d_{\mathrm{l}}^{(n_{\mathrm{l}}-1)y},\dots ,0,-d_{\mathrm{l}}^{2y},0,0]}^{\mathrm{T}}$$

(7)

$$f_{\mathrm{t}}^{\mathrm{M}}={[f_{\mathrm{t}}^{1x},f_{\mathrm{t}}^{1y},0,f_{\mathrm{t}}^{2y},\dots ,,0,f_{\mathrm{t}}^{(n_{\mathrm{t}})y},\dots ,0,f_{\mathrm{t}}^{2y},-f_{\mathrm{t}}^{1x},f_{\mathrm{t}}^{1y}]}^{\mathrm{T}}$$

(8)

$$f_{\mathrm{b}}^{\mathrm{M}}=-{[-f_{\mathrm{t}}^{1x},f_{\mathrm{t}}^{1y},0,f_{\mathrm{t}}^{2y},\dots ,,0,f_{\mathrm{t}}^{(n_{\mathrm{t}})y},\dots ,0,f_{\mathrm{t}}^{2y},f_{\mathrm{t}}^{1x},f_{\mathrm{t}}^{1y}]}^{\mathrm{T}}$$

(9)

$$f_{\mathrm{l}}^{\mathrm{M}}={[f_{\mathrm{l}}^{1x},f_{\mathrm{l}}^{1y},f_{\mathrm{l}}^{2x},0,\dots ,f_{\mathrm{l}}^{(n_{\mathrm{l}/\mathrm{r}})x},0,\dots ,f_{\mathrm{l}}^{2x},0,f_{\mathrm{l}}^{1x},-f_{\mathrm{l}}^{1y}]}^{\mathrm{T}}$$

(10)

$$f_{\mathrm{l}}^{\mathrm{M}}=-{[f_{\mathrm{l}}^{1x},-f_{\mathrm{l}}^{1y},f_{\mathrm{l}}^{2x},0,\dots ,f_{\mathrm{l}}^{(n_{\mathrm{l}/\mathrm{r}})x},0,\dots ,f_{\mathrm{l}}^{2x},0,f_{\mathrm{l}}^{1x},f_{\mathrm{l}}^{1y}]}^{\mathrm{T}}$$

(11)

where $d_{\mathrm{t}/\mathrm{b}/\mathrm{l}/\mathrm{r}}^{\mathrm{M}}$ and $f_{\mathrm{t}/\mathrm{b}/\mathrm{l}/\mathrm{r}}^{\mathrm{M}}$ are the displacement and force vectors of the nodes on the top, bottom, left, or right edge of M.

According to Equations (4)–(11), the node displacements and forces on the boundaries of M satisfy the periodic boundary condition [28]. However, M may not be a translationally symmetric metamaterial because m is not necessarily identical to m^x, m^y, and m^xy. Conversely, if m remains unchanged after the spatial inversions in both the x- and y-directions, M is a metamaterial satisfying the periodic boundary condition. This requires that the unit cell of m exhibits spatial inversion symmetry along the x- and y-axes. In this case, the spectrum of M must coincide with the dispersion function obtained via Bloch’s theorem, indicating that none of the eigenfrequencies of M, including ω_j, lie within the bandgap. Since any eigenfrequency ω_j of m lies outside the bandgap, the metamaterial m can avoid bandgap resonance.

The above derivation can be generalized to three-dimensional metamaterials and some other boundary conditions. For two- and three-dimensional cases, the applicable boundary conditions are listed in

Table 1. Boundary conditions of two-dimensional metamaterials for avoiding bandgap resonance.

Case	Boundary Condition
	x	y
1	c1	c1
2	c2	c1
3	c1	c2
4	c2	c2

and

Table 2. Boundary conditions of three-dimensional metamaterials for avoiding bandgap resonance.

Case	Boundary Condition
	x	y	z
1	c2, c3	c1, c3	c1, c2
2	c1	c1, c3	c1, c2
3	c2, c3	c2	c1, c2
4	c1	c2	c1, c2
5	c2, c3	c1, c3	c3
6	c1	c1, c3	c3
7	c2, c3	c2	c3
8	c1	c2	c3

, respectively.

In Table 1 and Table 2, x, y, and z represent nodes on the boundaries normal to the x-, y- and z-directions, respectively; c₁, c₂ and c₃ represent constraints applied in the x-, y- and z- directions, respectively.

In summary, the sufficient condition for avoiding bandgap resonance can be expressed as

Theorem 1.

A d-dimensional metamaterial can avoid bandgap resonance if both of the following conditions are satisfied: 1. the unit cell exhibits spatial inversion symmetry along all d-coordinate axes; 2. one of the boundary conditions listed in Table 1 or Table 2 is imposed.

The mechanism by which the above sufficient condition suppresses bandgap resonance is explained as follows. Bandgap resonances are in fact defect-induced modes localized at symmetry-breaking defects [22]. The spatial inversion symmetry of the metamaterial ensures the absence of bulk defects and their associated defect modes. The boundaries of a finite metamaterial also act as defects, giving rise to vibrational modes localized at the edges. Imposing the boundary conditions listed in Table 1 or Table 2 constrains the displacements of the boundary nodes, which eliminates these modes and suppresses bandgap resonances.

3. Perturbation of Bandgap

The following generalized characteristic equation [31,32,33,34] can be used to solve the dispersion function of the metamaterial consisting of unit cells with spatial inversion symmetry:

$$\widehat{D}(k)\widehat{d}(k)=0$$

(12)

where $\widehat{D}(k)=T_{\mathrm{d}}{(k)}^{\mathrm{H}}[K-\omega {(k)}^{2}M]T_{\mathrm{d}}(k)$; $K$ and $M$ are the d·n × d·n stiffness and mass matrices of the unit cell, respectively, and their explicit forms can be found in Ref. [34]; n is the number of nodes in the unit cell; d is the dimension of the unit cell; the superscript H denotes the Hermite transpose; $T_{\mathrm{d}}(k)$ is the d·n × d·r reduction matrix introducing Bloch’s theorem [34], and r is the number of inequivalent nodes [34]; ω(k) is the circular frequency, and k is the d × 1 wave vector [35]; $\widehat{d}(k)=T_{\mathrm{d}}(k)d$ and d is the d·n × 1 node displacement vector of the unit cell. Once k is given, the generalized eigenvalues ω_j(k)² (j = 1, 2, …, d·r) and their corresponding eigenvectors ${\widehat{d}}_j(k)$ (d·r × 1) can be solved via Equation (12). Thus, the dispersion function of the metamaterial, i.e., the k‒ω relation, is established with a total of d·r branches of k ‒ ω_j.

According to matrix perturbation theory [33,34], the perturbation of branch j caused by the increments of unit cell parameters, including node coordinates and element cross-sectional areas, can be calculated by

$$\Delta {\omega }_j(k)=O_j(k)\Delta \epsilon$$

(13)

where $\Delta \epsilon ={\{\Delta A_1,\Delta A_2,\dots ,\Delta A_b,\Delta x_1^1,\dots ,\Delta x_1^d,\dots ,\Delta x_n^1,\dots ,\Delta x_n^d\}}^{\mathrm{T}}$ is the (b + d·n) × 1 incremental unit cell parameter vector; b is the number of elements in the unit cell; A_i (i = 1,…, b) is the cross-sectional area of the i-th element; and $x_p^k$(p = 1, …, n, k = 1, …, d) is the k-th coordinate of the p-th node. O_j(k) is a 1 × (b + d·n) row vector whose v-th component is

$${(O_j(k))}_v=\left\{\begin{array}{c}{\displaystyle \frac{1}{2{\omega }_j(k)}}{\widehat{d}}_{\overline{j}}{(k)}^{\mathrm{H}}{\displaystyle \frac{\partial \widehat{D}(k)}{\partial A_v}}{\widehat{d}}_{\overline{j}}(k),v\le b\\ {\displaystyle \frac{1}{2{\omega }_j(k)}}{\widehat{d}}_{\overline{j}}{(k)}^{\mathrm{H}}{\displaystyle \frac{\partial \widehat{D}(k)}{\partial x_p^k}}{\widehat{d}}_{\overline{j}}(k),v\text{>}b\end{array}\right.$$

(14)

where v = b + d·(p−1) + k for the second term; ${\widehat{d}}_{\overline{j}}(k)$ is the eigenvector normalized by ${\widehat{d}}_{\overline{j}}{(k)}^{\mathrm{H}}{T}_{\mathrm{d}}{(k)}^{\mathrm{H}}M{T}_{\mathrm{d}}(k){\widehat{d}}_{\overline{j}}(k)=1$; ${\displaystyle \frac{\partial \widehat{D}(k)}{\partial A_v}}=T_{\mathrm{d}}{(k)}^{\mathrm{H}}[{\displaystyle \frac{\partial K}{\partial A_v}}-{\omega }_j{(k)}^2{\displaystyle \frac{\partial M}{\partial A_v}}]T_{\mathrm{d}}(k)$, ${\displaystyle \frac{\partial \widehat{D}(k)}{\partial x_p^k}}=T_{\mathrm{d}}{(k)}^{\mathrm{H}}[{\displaystyle \frac{\partial K}{\partial x_p^k}}-{\omega }_j{(k)}^2{\displaystyle \frac{\partial M}{\partial x_p^k}}]T_{\mathrm{d}}(k)$. The explicit forms of ${\displaystyle \frac{\partial K}{\partial A_v}}$,${\displaystyle \frac{\partial K}{\partial x_p^k}}$, ${\displaystyle \frac{\partial M}{\partial A_v}}$, and ${\displaystyle \frac{\partial M}{\partial x_p^k}}$ are given in Ref. [34].

An illustrative bandgap between the lower branch ω_l(k) and the upper branch ω_u(k) is shown in Figure 5. The range of the bandgap is [ω_l(k_l), ω_u(k_u)], where k_l and k_u are the wave vectors corresponding to the maximum value of ω_l(k) and the minimum value of ω_u(k), respectively. Substituting k_l and k_u into Equation (13), the perturbation of ω_l(k_l) and ω_u(k_u) can be expressed as

$$\Delta \omega =\left\{\begin{array}{c}\Delta {\omega }_{\mathrm{u}}(k_{\mathrm{u}})\\ \Delta {\omega }_{\mathrm{l}}(k_{\mathrm{l}})\end{array}\right\}=\left\{\begin{array}{c}{O}_{\mathrm{u}}(k_{\mathrm{u}})\\ O_{\mathrm{l}}(k_{\mathrm{l}})\end{array}\right\}\Delta \epsilon =O\Delta \epsilon$$

(15)

The difference between the expected bandgap [ω_l, ω_u] shown in Figure 5 and the current bandgap is Δω_t = {ω_u − ω_u(k_u), ω_l − ω_l(k_l)}^T. By substituting Δω_t into Equation (15), Δε for adjusting unit cell parameters and tuning bandgap can be solved. However, Δε must satisfy spatial inversion symmetry, or it will break the symmetry of the unit cell. Figure 6 shows a unit cell with eight nodes and 12 elements. An increment vector adjusting the coordinates of nodes 5, 6, 7, and 8 with $\Delta x_5^1=1$, $\Delta x_6^1=-1$, $\Delta x_7^1=-1$, and $\Delta x_8^1=1$ satisfies spatial inversion symmetry and is denoted as Δε_g. Similarly, ΔA₅ = 1, ΔA₆ = 1, ΔA₇ = 1, and ΔA₈ = 1 can be assembled into another Δε_g satisfying spatial inversion symmetry. If there are at most τ linearly independent Δε_g, then any incremental unit cell parameter vector satisfying spatial inversion symmetry can be expressed as

$$\Delta {\epsilon }_{\mathrm{sym}}=\{\Delta {\epsilon }_{g1},\Delta {\epsilon }_{g1},\dots \Delta {\epsilon }_{g\tau }\}{\{{\kappa }_1,{\kappa }_2,\dots ,{\kappa }_{\tau }\}}^{\mathrm{T}}={{\rm E}}_g\Delta \kappa$$

(16)

where Δκ is an arbitrary τ × 1 coefficient vector.

Substituting Equation (16) and Δω_t into Equation (15), it can be obtained that

$$\Delta {\omega }_{\mathrm{t}}=O{{\rm E}}_g\Delta \kappa$$

(17)

The least squares solution [36] of Equation (17) is

$$\Delta \kappa ={(O{{\rm E}}_g)}^{\mathrm{T}}{[(O{{\rm E}}_g){(O{{\rm E}}_g)}^{\mathrm{T}}]}^{-1}\Delta {\omega }_{\mathrm{t}}$$

(18)

By substituting Equation (18) into Equation (16), the resulting Δε_sym can be used to adjust the unit cell parameters and tune the bandgap under the constraint of spatial inversion symmetry.

The unit cell parameters should be modified in small steps due to the strong nonlinearity in their relationship with the dispersion function. Thus, Δε_sym is scaled by

$$\Delta {\epsilon }_{\mathrm{s}}=s{\displaystyle \frac{\Delta {\epsilon }_{\mathrm{sym}}}{\left|\right|\Delta {\epsilon }_{\mathrm{sym}}\left|\right|}}$$

(19)

where s is a specified scaling factor; ||•|| denotes the two-norm of the vector. The expected bandgap can be obtained by iteratively adjusting the unit cell parameters with Δε_s until ||Δω_t|| falls below a specified tolerance c.

4. Example

Figure 7 shows a square unit cell with a side length of 10 mm that exhibits spatial inversion symmetry along the x- and y-axes. The origin of the coordinate system is at the center of the unit cell with the axes shown in Figure 7. The node coordinates are listed in

Table 3. Node coordinates of initial unit cell (mm).

Node	Coordinate		Node	Coordinate		Node	Coordinate
	x-	y-		x-	y-		x-	y-
1	−5.00	5.00	11	−2.00	1.00	21	−2.00	−1.00
2	5.00	5.00	12	0.00	1.00	22	0.00	−1.00
3	5.00	−5.00	13	2.00	1.00	23	2.00	−1.00
4	−5.00	−5.00	14	4.00	1.00	24	4.00	−1.00
5	−4.00	3.00	15	−4.00	0.00	25	−4.00	−3.00
6	−2.00	3.00	16	−2.00	0.00	26	−2.00	−3.00
7	0.00	3.00	17	0.00	0.00	27	0.00	−3.00
8	2.00	3.00	18	2.00	0.00	28	2.00	−3.00
9	4.00	3.00	19	4.00	0.00	29	4.00	−3.00
10	−4.00	1.00	20	−4.00	−1.00

. All the elements of the unit cell have a circular cross-section with a radius of 0.3 mm (area A = 0.283 mm²) and are made of plastic with a Young’s modulus of E = 1.2 GPa and a density of ρ = 0.9 g/cm³.

For this unit cell, the dispersion function obtained using Equation (12) consists of 52 branches, whose significant overlap makes it difficult to depict their surfaces on a plane. To illustrate the band structure, a frequency distribution function for branch j can be constructed as follows:

$$D_j(\omega )={\displaystyle \underset{\Pi }{\iint }{\left.\mathrm{d}A\right|}_{{\omega }_j(k)=\omega }}(j=1,2,\dots ,d\cdot r)$$

(20)

where Π denotes the first Brillouin zone [35]; dA = dk_xdk_y is the area of the infinitesimal element of Π; and k_x and k_y are the x and y components of k, respectively. According to Equation (20), D_j(ω) represents the total area of the infinitesimal elements for which ω_j = ω. If D_j(ω) ≠ 0, there must exist a wave vector k such that ω_j(k) = ω. Therefore, ω_j(k) lies within the frequency domain with nonzero D_j(ω). The frequency distribution of the dispersion function can be visualized by plotting D_j(ω) for all branches, with ω as the horizontal axis and area as the vertical axis. Figure 8 shows 27 lower-frequency branches distributed between 0 and 3100 Hz.

This example aims to achieve a bandgap of 600–1000 Hz by designing the coordinates of nodes 5–29 and the cross-sectional areas of all elements. According to Figure 8, the expected bandgap is most likely generated between branches 3 and 4. The unit cell parameters are iteratively adjusted using Equations (15)–(19) with a tolerance of c = 2 and a scaling factor of s = 0.01. The resulting unit cell is shown in Figure 9, whose element cross-sectional areas and node coordinates are listed in

Table 4. Element cross-sectional areas of the unit cell satisfying expected bandgap (mm²).

Element	Area	Element	Area	Element	Area	Element	Area
1-7	0.107	1-2	0.001	13-14	0.220	23-18	0.276
1-6	0.229	3-4	0.001	10-15	0.233	24-18	0.229
1-5	0.159	2-3	0.000	10-16	0.229	24-19	0.233
1-10	0.105	1-4	0.000	11-16	0.276	20-21	0.220
1-15	0.038	5-6	0.231	11-17	0.622	21-22	0.458
2-7	0.107	6-7	0.305	12-17	0.321	22-23	0.458
2-8	0.229	7-8	0.305	13-17	0.622	23-24	0.220
2-9	0.159	8-9	0.231	13-18	0.276	25-20	0.157
2-14	0.105	5-10	0.157	14-18	0.229	25-21	0.387
2-19	0.038	5-11	0.387	14-19	0.233	26-21	0.250
3-27	0.107	6-11	0.250	15-16	0.202	26-22	0.509
3-28	0.229	6-12	0.509	16-17	0.261	27-22	0.246
3-29	0.159	7-12	0.246	17-18	0.261	28-22	0.509
3-24	0.105	8-12	0.509	18-19	0.202	28-23	0.250
3-19	0.038	8-13	0.250	20-15	0.233	29-23	0.387
4-27	0.107	9-13	0.387	20-16	0.229	29-24	0.157
4-26	0.229	9-14	0.157	21-16	0.276	25-26	0.231
4-25	0.159	10-11	0.220	21-17	0.622	26-27	0.305
4-20	0.105	11-12	0.458	22-17	0.321	27-28	0.305
4-15	0.038	12-13	0.458	23-17	0.622	28-29	0.231

and

Table 5. Node coordinates of the unit cell satisfying expected bandgap (mm).

Node	Coordinate		Node	Coordinate		Node	Coordinate
	x-	y-		x-	y-		x-	y-
1	−5.00	5.00	11	−1.87	1.01	21	−1.87	−1.01
2	5.00	5.00	12	0.00	1.21	22	0.00	−1.21
3	5.00	−5.00	13	1.87	1.01	23	1.87	−1.01
4	−5.00	−5.00	14	3.95	0.98	24	3.95	−0.98
5	−3.92	2.87	15	−4.03	0.00	25	−3.92	−2.87
6	−1.85	3.01	16	−2.09	0.00	26	−1.85	−3.01
7	0.00	3.14	17	0.00	0.00	27	0.00	−3.14
8	1.85	3.01	18	2.09	0.00	28	1.85	−2.99
9	3.92	2.87	19	4.03	0.00	29	3.92	−2.87
10	−3.95	0.98	20	−3.95	−0.98

, respectively. The frequency distribution of the updated dispersion function is shown in Figure 10, where a bandgap of 600.0–998.6 Hz can be found.

A metamaterial consisting of 4 × 4 designed unit cells is shown in Figure 11a. The metamaterial is denoted as PBC, FBC, or SBC when the periodic, free, or specific boundary condition is imposed. The specific boundary condition is illustrated in Figure 11b and corresponds to case 3 in Table 1. In addition, the symmetry of the designed unit cell can be slightly broken by adjusting the coordinate of node 15 to (−4.03, 1.00), resulting in a bandgap of 601.0–904.7 Hz. The 4 × 4 metamaterial, consisting of this asymmetric unit cell and subjected to the specific boundary condition, is denoted as SBCA.

A frequency-response function is defined as follows for PBC, FBC, SBC, or SBCA:

$$y(\omega )=20{\log }_{10}({\displaystyle \frac{\left|\right|d_{\mathrm{out}}(\omega )\left|\right|}{\left|\right|d_{\mathrm{in}}(\omega )\left|\right|}})$$

(21)

where ||d_in(ω)|| is the displacement amplitude at input node 1, and ||d_out(ω)|| is that at output node 2 or 3. These nodes are labeled in Figure 11a. With a modal damping of 0.02, ||d_in(ω)|| and ||d_out(ω)|| are calculated using the modal superposition method (MS) [35] and the finite element software ANSYS 2021 R2, respectively.

The MS is implemented in MATLAB 2024a, and the procedure includes the following steps:

• Set up the dynamic equation for the metamaterial, with its specific form given in Ref. [34].
• Calculate the eigenfrequencies and eigenmodes of the metamaterial.
• Calculate the displacements at the input and output nodes, i.e., d_in and d_out, under a simple harmonic load using the displacement equation [37] of the modal superposition method.

The metamaterials are also modeled in ANSYS with all the elements simulated by Link180. A harmonic response analysis is conducted using the command HROPT to obtain the displacements d_in and d_out, with the solution method [38] set to FULL.

The frequency-response curves and eigenfrequencies of PBC, FBC, SBC, and SBCA are shown in Figure 12, Figure 13, Figure 14 and Figure 15, where the dashed lines indicate that the d_out obtained by ANSYS is zero. As illustrated in Figure 12, the eigenfrequencies and frequency-response curves of PBC are consistent with the bandgap, confirming that the metamaterial with the expected bandgap has been successfully designed. However, when the metamaterial is subjected to the free boundary condition, eigenfrequencies emerge within the bandgap, as illustrated in Figure 13. Figure 14 shows that the bandgap resonances in SBC are effectively suppressed by imposing the specific boundary condition, thereby validating the second condition of Theorem 1. For the asymmetric metamaterial, even when the specific boundary condition is imposed, bandgap resonances still emerge in the frequency-response curves of Figure 15. This validates the first condition of Theorem 1.

5. Conclusions and Outlook

This paper proposes a design method for stretch-dominated metamaterials that exhibit expected bandgaps while avoiding bandgap resonance. The conclusions are summarized as follows:

First, the sufficient condition for avoiding bandgap resonance is derived, specifically including spatial inversion symmetry and the boundary conditions listed in either Table 1 or Table 2. The symmetry constraint should be incorporated into the design process of bandgap metamaterials to ensure their vibration isolation performance.

Second, equations for adjusting unit cell parameters are formulated to obtain expected bandgaps. The matrix-form perturbation expression of the bandgap is established, which represents the sensitivity of the bandgap with respect to node coordinates and element cross-sectional areas. It is then used to design the unit cell under the symmetry constraint to generate the expected bandgap.

Third, a two-dimensional metamaterial example is presented to validate the proposed sufficient condition and design method. The designed metamaterial, satisfying spatial inversion symmetry and the boundary condition in Table 1, achieves both the expected bandgap and the suppression of bandgap resonance, thus validating the proposed design method. However, bandgap resonances emerge once either of these two conditions is not satisfied. This validates the sufficient condition for avoiding bandgap resonance.

Although the derived condition is sufficient for avoiding bandgap resonance, the symmetry constraint is overly restrictive, thus significantly reducing the number of design parameters. In addition, although the boundary conditions listed in Table 1 and Table 2 are feasible in engineering practice, they substantially increase the complexity of the metamaterials. Future work should further investigate more readily implementable conditions for avoiding bandgap resonance, including weaker symmetry constraints and simpler boundary conditions.

Stretch-dominated metamaterials represent one class of mechanical metamaterials with significant application potential, alongside bend-dominated, origami, kirigami, honeycomb, and sandwich structures. The sufficient condition for avoiding bandgap resonance and the corresponding design method proposed in this study cannot be directly extended to these metamaterials. For instance, the perturbation expression for designing the bandgap of bend-dominated metamaterials must be re-derived based on beam elements. Future work will further investigate the bandgap resonances and design methods for metamaterials of other structural forms.