For practical applications of the elastic metamaterials, dynamic behavior of finite structures made of elastic metamaterials with frequency dependent properties are analyzed theoretically and numerically. First, based on a frequency-dependent mass density and Young's modulus of the effective continuum, the global dynamic response of a finite rod made of elastic metamaterials is studied. It is found that due to the variation of the effective density and Young's modulus, the natural frequency distribution of the finite structure is altered. Furthermore, based on the spectral approach, the general wave amplitude transfer function is derived before the final transmitted wave amplitude for the finite-layered metamaterial structure with decreasing density is obtained using the mathematical induction method. The analytical analysis and finite element solutions indicate that the increased transmission wave displacement amplitude and reduced stress amplitude can be controlled by the impedance mismatch of the adjacent layers of the layered structure.

## Introduction

Liu et al. fabricated the first artificial metamaterial capable of generating a negative effective mass near the resonant frequency of substructures made of rubber-coated lead (Pb) spheres [1]. Since then, researchers have focused on elaborating on the physical meaning of the negative effective mass phenomenon [2–6], as well as designing new types of acoustic/elastic metamaterials with negative effective mass and/or negative effective modulus [7,8]. Recently, the realization of metamaterials with a near-zero-index has become the subject of increasing interest [9–12]. This shift to non-negative material properties opens a new chapter of metamaterial research, which expands the definition of metamaterials to a material with “on-demand” effective properties without the constraints imposed by what nature provides [13–15].

In the past, layered structures have demonstrated the ability to achieve vibration isolation and reduce the overall stress amplitude of dynamic disturbances [16,17]. The earliest theoretical discussions for wave propagation in media consisting of a number of elastic layers can be traced back to Thomson's derivation in 1950 [18], which introduced the transfer matrix approach in the time domain to relate the displacements and stresses at the bottom of a layer to those at its top. Due to the complexity of wave propagation in finite layered structures, frequency domain approaches have recently been proposed to perform analysis and determine optimal designs of layered structures [16,17]. However, it is extremely difficult to find suitable natural materials for the multilayered structures to actually control the displacement and stress transmission amplitudes quantitatively, due to the limitation of their inherent properties. On the other hand, elastic metamaterials can deliver any desired dynamic mass density and/or Young's modulus by controlling the wave frequency. This is much more efficient when constructing layered structures capable of controlling wave propagation characteristics. Recently, Srivastava and Willis studied out-of-plane wave propagation toward the interface between a finite layered medium and its homogeneous medium [19]. It was found that the evanescent waves from boundary layers should be considered to meet continuous conditions of the displacement and stress fields.

Due to the catastrophic effect, vibration and mechanical resonance is a significant problem in industry. Within the band gap, resonant-based metamaterials could provide a solution for subwavelength vibration isolation and/or absorbing unwanted noise, which can be interpreted by the frequency-dependent properties. It is well known that the unnatural mechanical properties inherent to elastic/acoustic metamaterial, including negative effective mass, negative effective modulus, and near-zero mass density, originate from its internal structure. Therefore, elastic metamaterials can be designed to have any mass density or modulus by simply changing its internal microstructure. However, much less attention has been paid on the dynamic behavior of the metamaterials within the passing band especially when the structure is in a finite length. It can be anticipated that the global dynamic spectrum behavior of the finite metamaterial-based structure will be altered due to the embedded microstructures and cannot simply captured by using the simple Cauchy medium. To fill this gap and elucidate the influence of the frequency-dependent properties on the natural frequency of the finite structure, the dynamic response of the finite metamaterial-based structure with frequency dependent properties is analyzed analytically and numerically.

In this paper, spectrum and vibration analysis of finite elastic metamaterial-based rod is performed analytical and numerically to illustrate the effects of the frequency-dependent properties on the resonate frequency of the finite structure. Then, the spectral approach [16,17,20,21] is utilized and the transfer function for the general wave amplitudes is derived for wave propagation in the finite metamaterial-based layered structure. Finally, the transmitted wave amplitude and its range for one special case in which a layered structure comprised of continuum media with decreasing density is studied.

## Effective Mass and Young's Modulus of Metamaterials in Lattice Form

A common approach to model an infinite discrete metamaterial is to use a lattice system of mass-spring units [2–6]. An example of this modeling approach is shown in Fig. 1, which depicts a one-dimensional (1D) mass-in-mass lattice system where each unit cell in the lattice is arranged at a distance, *d*, from its neighboring cell [5].

*j*th unit are obtained as

*γ*” (= 1, 2) in the

*j*th unit cell of the lattice. Similarly, for the (

*j*+

*n*)th unit cell, the harmonic displacement is expressed as

where *A _{γ}* is the wave amplitude,

*q*is the wavenumber, and

*ω*is the angular frequency.

*A*

_{1}and

*A*

_{2}, from which the dispersion relation is obtained as the determinant of the coefficient matrix

The mass-in-mass system could be further represented as a monatomic lattice with a single effective mass as shown in Fig. 2.

*j*th unit cell is written as

*j*+

*n*)th unit cell is written as

The next step is to homogenize the effective monatomic mass lattice model into its corresponding continuum representation [5,6,22]. The representative unit cell in Fig. 2 can be represented as two masses connected by a spring shown in Fig. 3.

*d*is the unit cell size,

*E*

_{eff}is the effective Young's modulus of the 1D continuum model, and

*A*is the cross-sectional area of the continuum model. Substituting Eq. (9) into Eq. (10) and normalizing with respect to the static Young's modulus $Est=k1d/A$, the ratio of effective and static moduli can be obtained as

Based on Eqs. (4), (7), and (11), Fig. 4 shows the dispersion curves, effective mass, and effective Young's modulus for the material constants *θ* = *m*_{2}/*m*_{1} = 0.5, *δ* = *k*_{2}/*k*_{1} = 0.8, and local resonance frequency $\omega 0=k2/m2=20$ (rad/s). The resulting band gap corresponds to both the negative effective mass and negative effective Young's modulus frequency regions. It is clear that the local resonance frequency, *ω*_{0}, is the cause of the negative effective material properties which is expected.

## Influence of Frequency-Dependent Properties on Rod Vibrations

Unlike the constant physical properties of materials used in conventional structures, the effective mass density and Young's modulus of a structure made of metamaterials are highly dependent on the excitation frequency applied. Although metamaterial substructures (unit cells) have been designed to tune the overall structural stiffness [14] or to isolate dynamic disturbances [15], the dynamic response of a structure using metamaterials as structural building blocks is still an unelucidated problem to the best of our knowledge. To study this problem, the vibration response of a metamaterial rod is analyzed in this section.

### Analytical Analysis of Dynamic Behavior in Finite Elastic Metamaterial Rod.

*A*, and its total length as

*L*. Based on Hamilton's principle and the assumption that no damping is present in the structure, the governing equation of motion of the continuum rod can be expressed as

where ρ_{eff} = *m*_{eff}/*Ad*.

*α*is a real number. Thus, Eq. (14) can be rewritten as Eq. (15), and the solution from Eq. (15) can be interpreted as either the left-to-right ($C1e\u2212\alpha x$) or right-to-left ($C2e\alpha x$) directions of the traveling incident or reflected wave [20], respectively. When the incident wave is superimposed on the reflected wave traveling in the opposite direction, both waves' displacement amplitudes monotonically decrease along the propagation direction exponentially. It is apparent that for any boundary conditions, such as free-free end and fixed-free end, only a trivial solution can be obtained from Eq. (15), which indicates that there is no natural resonance frequency in the band gap frequency region

*n*, there are two sets of unique resonance frequencies. For the first mode in each set, the two solutions can be obtained from Eq. (19) below. Compared with Eq. (7), it is easy to see that the effective mass is zero at $\omega 2$, which corresponds to the upper frequency cutoff for the band gap shown in Fig. 4

For free vibrations of the finite rod with a fixed-free boundary condition as shown in Fig. 6, the parameter *n* in Eqs. (17) and (18) needs to be replaced by $n+(1/2)$, but for the high-order modes ($n\u2192\u221e$), the form of the resonance frequencies remains the same as in Eq. (20).

Comparing Eqs. (17) and (20), it can be found that for a conventional continuum rod, the high-order resonance frequency approaches infinity. However, for the rod made of elastic metamaterials, the resonance frequency converges to a finite value. When conducting vibrational analysis, this characteristic will influence the natural frequency distribution of structures made of elastic metamaterials, which is demonstrated in Sec. 3.2.

### Numerical Simulation of Dynamic Behavior in Finite Elastic Metamaterial Rod.

Based on Eq. (15), there should be no natural resonance frequency located in the band gap frequency region. To verify this analytical solution, Fig. 7 compares the dispersion curve calculated with Eq. (3) with the natural resonance frequency calculated with Eq. (18), for the specific set of material constants: *θ* = *m*_{2}/*m*_{1} = 0.5, *δ* = *k*_{2}/*k*_{1} = 0.8, *d* = 1 m, *L* = 100 m and $\omega 0=k2/m2=20$ (rad/s). It is obvious that there is no resonance frequency (Fig. 7(b)) within the band gap frequency region (Fig. 7(a)), and as predicted in Sec. 3.1, $\omega 2$ and $\omega \u221e$ define the upper and lower bounds of the band gap, respectively.

Figure 8 shows the free-free boundary condition for the mass-in-mass lattice model with the same set of material constants used in Fig. 7.

Using the analytical solution obtained from Eq. (18) and the numerical solution obtained using abaqus, the two sets of natural resonance frequency of metamaterial rod can be compared as shown in Fig. 9. It should be noted that the natural frequencies obtained from the two approaches start and end (dashed line) at the same frequencies, which verifies that no resonant frequency is present within the band gap region for either solution. In addition, it is also noted that the analytical and finite element method solutions are in good agreement for the frequency regions with a small mode number but begin to diverge for higher modes. This is caused by the finite number of resonators considered in the numerical model, which cannot obtain the infinite number of modes obtained analytically with Eq. (18). Comparing the resonance frequencies obtained for the finite elastic metamaterial rod with 100 and 20 resonators shown in Fig. 9, it is clear that if more resonators are considered in the finite elastic metamaterial rod, more modes can be observed.

In Fig. 10, the mass-in-mass lattice model and its effective continuum rod with a prescribed displacement as input are created in abaqus. The material constants are the same as those used in Fig. 7, except now *L* = 20 m.

To show the difference in the natural frequency distribution, Fig. 11 compares the steady-state displacement of the remote end of the conventional material rod with the last unit cell's outer mass, $m1$, of the elastic metamaterial rod. The mass density and Young's modulus of the conventional material rod are *ρ* = 63 kg/m^{3} and *E* = 10,000 Pa, respectively. The obvious band gap in Fig. 11(a) indicates that the resonators inside the rod can significantly isolate the vibration. Furthermore, as shown in Fig. 11(b), due to the variation of the effective mass density and Young's modulus, the rod's resonance frequency distribution is altered, especially in the frequency region close to the local resonance frequency of the unit cells' internal resonator, $m2$. This interesting phenomenon requires special attention because the resonance frequency concentration below and above the band gap will amplify the input force/displacement instead of attenuating it.

## Wave Propagation in Finite Layered Structures

Due to the wide technological applications, layered structures have been studied extensively [16,17]. The Earth's strata and general composite materials are two typical examples of layered structures. Although the earliest theoretical discussions can be traced back to Thomson's derivation in 1950, conventional materials with their inherent properties limit their applications in multilayered structures.

As discussed in Sec. 2, elastic metamaterials can exhibit arbitrary mass densities or Young's moduli depending on the wave excitation frequency. Thus, elastic metamaterials have an advantage as candidates in layered structures capable of unique wave manipulation properties. In Sec. 4.1, the transfer function is derived for a propagating wave in a layered structure. Then, the resulting transmitted wave amplitude range for the case of a layered structure comprised of elastic metamaterials with a decreasing effective density is studied.

### Analytical Analysis of Wave Propagation in Finite Layered Structures.

where $tj=(xj+1\u2212xj)/cj,\u2009\alpha j=Zj/Zj+1$, and $Zj=Ej\rho j$.

*T*, is a complex matrix, the real amplitude of

*A*can be obtained by taking the absolute value as

_{N}*A*. For the numerator of Eq. (30), if

_{N}*N*= 2, it can be found that $t11t22\u2212t12t21=\alpha $. Supposing $N=m\u22652$, $t11\u2032t22\u2032\u2212t12\u2032t21\u2032=\alpha m\u22121$ and $\alpha 1=\alpha 2=\u2026=\alpha $, then for $N=m+1$, the transmission amplitude transfer function can be expressed as

Based on Eq. (32), it is clear that for the numerator $t11t22\u2212t12t21=\alpha N\u22121$.

where the maximum value is reached when, $\omega t1=((2m+1)\pi )/2$, $(m=0,1,\u2026)$ and $\omega tj=p\pi $, ($p=1,2,\u2026,j=2,3,\u2026,n$); while the minimum value ($1/\alpha N\u22121$) is reached when $\omega t1=m\pi $, (*m* = 0, 1,…) and $\omega tj=p\pi $, ($p=1,2,\u2026,j=2,3,\u2026,n$).

*A*can be obtained as

_{N}*N*layers, the stress input is written in the form

*N*th layer to the unit input stress can then be expressed as

*A*/

_{N}*A*

_{1}can be obtained as

The minimum value, $1/\alpha N\u22121$, is reached when $\omega t1=((2m+1)\pi )/2$, ($m=0,1,\u2026$) and $\omega tj=p\pi $, ($p=1,2,\u2026,j=2,3,\u2026,n$), while the maximum value is reached when $\omega t1=m\pi $, ($m=0,1,\u2026$) and $\omega tj=p\pi $, ($p=1,2,\u2026,j=2,3,\u2026,n$).

Comparing the maximum and minimum conditions with those of the transmitted wave displacement amplitude, when the increased final transmitted wave displacement amplitude reaches its maximum value, the reduced stress amplitude reaches a minimum value. Furthermore, both values are controlled by the constant adjacent impedance rate “$\alpha $”.

### Numerical Simulation of Wave Propagation in a Finite-Layered Metamaterial Structure.

*A*) obtained from Eq. (30), the mass-in-mass lattice model and its effective continuum model are established with the finite element method software abaqus (shown in Fig. 13). The model consists of three sections where the third section is designed to be sufficiently long to avoid the wave reflection from the right end boundary. According to Eqs. (7) and (11), the effective density and effective Young's modulus of each section can be obtained from Eq. (39) as

_{N}Considering the unit cell cross-sectional area (*S* = 1 m^{2}) and the periodical spacing between each unit cell to be *d* = 1 m, to make the adjacent impedance rate $\alpha =2$ at the frequency $f=5\u2009Hz$, the effective density and effective Young's modulus are chosen as: $\rho eff1$ = 16 kg/m^{3}, $Eeff1$ = 640,000 Pa, for the first section; $\rho eff2$ = 4 kg/m^{3}, $Eeff2$ = 640,000 Pa, for the second section; $\rho eff3$ = 1 kg/m^{3}, $Eeff3$ = 640,000 Pa, for the third section. Based on Eq. (40), $k1$ and $k2$ can be obtained as: *k*^{1}_{1} = 643,947 N/m, *k*^{1}_{2} = 6579.7 N/m, for the first section; *k*^{2}_{1} = 640,987 N/m, *k*^{2}_{2} = 13,159.5 N/m, for the second section; and *k*^{3}_{1} = 640,247 N/m, *k*^{3}_{2} = 14,412.8 N/m, for the third section. In addition, to satisfy the condition of the maximum value of Amp(*A _{N}*), we set $\omega t1=\pi /2$ and $\omega t2=\pi $, and the lengths of first two sections can be calculated as

*L*

_{1}= 10 m and

*L*

_{2}= 40 m.

Figure 14 illustrates the time history of the transmitted wave amplitude for the mass-in-mass lattice and the effective continuum model. It is easily observed that the final transmitted wave amplitude of the effective continuum model is amplified as much as 400%, which is the exact maximum value, $\alpha 2$, predicted by Eq. (34). It is also noted that, for the mass-in-mass model (Fig. 14(b)), the transmitted wave amplitude fluctuates between +4 and −4, which is due to the inner mass-spring system. Furthermore, compared to the effective continuum model, the wave amplitude in the mass-in-mass model takes a much longer time to reach a steady-state. This phenomenon is caused by the dispersive characteristics in the mass-in-mass lattice model, which reduces the velocity of the wave-front of the imperfect harmonic input wave. Since the effective continuum model can reach the steady-state wave amplitude of the mass-in-mass lattice model quickly and accurately, the effective continuum model is adopted to verify the analytical solution in Sec. 4.1.

To verify the minimum transmitted stress value obtained from Eq. (38), the unit displacement input is replaced by a unit force input (shown in Fig. 15(a)) with the same three sections in the effective continuum model. Figure 15(b) compares the time history of the input stress with the final transmitted stress where it is clear that the input stress amplitude has been reduced by roughly 75%, which matches the minimum value, $1/\alpha 2$, predicted by Eq. (38).

To verify the solution given by Eq. (34), Fig. 16 shows a five-section conventional continuum model with unit displacement input. The mass densities and Young's Moduli for the conventional solids are *ρ*_{1} = 64 kg/m^{3}, *ρ*_{2} = 16 kg/m^{3}, *ρ*_{3} = 4 kg/m^{3}, *ρ*_{4} = 1 kg/m^{3}, *ρ*_{5} = 0.25 kg/m^{3}, *E*_{1} = *E*_{2} = *E*_{3} = *E*_{4} = *E*_{5} = 640,000 Pa. To satisfy the condition for reaching the maximum or minimum value of Amp (*A _{N}*), the lengths of the first four sections can be calculated as:

*L*

_{1}= 5 m,

*L*

_{2}= 20 m,

*L*

_{3}= 40 m,

*L*

_{4}= 80 m (maximum design);

*L*

_{1}= 10 m,

*L*

_{2}= 20 m,

*L*

_{3}= 40 m,

*L*

_{4}= 80 m (minimum design).

Figure 17 shows the transmitted wave time history for the maximum and minimum transmission amplitude designs, respectively. Although the transient wave time histories are different for the initial time domain, the two steady-state wave amplitudes (Figs. 17(a) and 17(b)) match the maximum and minimum values as predicted by Eq. (34). Physically, this phenomenon is due to the mode control in each section by elastic metamaterials.

## Conclusion

This paper is focused on the analysis of finite elastic metamaterial structures with frequency-dependent mechanical properties, and main attention is paid on the frequency application from band gap to the passing band. Based on a simplified mass-in-mass metamaterial lattice model, the equations for the frequency-dependent mass density and Young's modulus for its equivalent continuum model are derived. By utilizing the frequency-dependent mechanical properties, the vibration response of the finite elastic metamaterial rod is studied. It is found that no natural resonance frequency can exist within the band gap frequency region of the corresponding metamaterial. In addition, due to the variation of the effective mass density and Young's modulus, the natural resonance frequency distribution of the rod is changed, especially in the frequency region close to the local resonance frequency of the unit cells in the structure. Finally, wave propagation behavior in layered structures made of elastic metamaterials is investigated. Based on the spectral approach, the general wave amplitude transfer function is derived for wave propagation in the layered structures. The final transmitted wave amplitude range of the layered elastic metamaterial structure with decreasing density is obtained using the mathematical induction method. It is found that when the final transmitted wave displacement amplitude reaches its maximum value, the reduced stress amplitude reaches its minimum value. Subsequently, the amplitudes can be controlled by the impedance mismatch of the adjacent layers in the layered elastic metamaterial structure.

## Funding Data

Air Force Office of Scientific Research (Grant No. FA9550-15-1-0016).

Defense Threat Reduction Agency (Grant No. DTRA1-12-1-0047).