1 Introduction

Acoustic periodic structures, due to their extraordinary merit of prohibiting wave propagation in the acoustic band gaps, are being paid wider and wider academic and engineering attention during the past few decades^[1-3]. Based on this unique stop-band property, many novel pass-band functionalities have been realized, e.g., the waveguiding through designated paths^[4-5], the linear acoustic diode where waves propagate in one direction and are forbidden in other directions^[6-7], and the surface wave propagation without leaking into the bulk^[8-9]. These pass-band functionalities not only give birth to various extraordinary wave propagation characteristics, which cannot be acquired in normal materials, but also make wave propagation behaviors easier to be controlled and manipulated. These outstanding performances for acoustic periodic structures should be attributed to two aspects. One is the strict periodic arrangement, and the other is the material or geometric difference between the neighboring unit cells. It should be pointed out that, in most of the recent studies, only the interaction or interface between the neighboring unit cells has been considered. Therefore, it has become an intriguing question whether an acoustic periodic structure with nonlocal interactions bears special functionalities.

In the nonlocal elastic (NLE) theory for continuum media, the long range interatomic forces are considered by assuming that, in the constitutive equations, the stress at one point depends on the strains at all points in the entire body. This theory can be traced back to the work in the middle of 1960s, and has been systematically summarized by Eringen^[10]. The nonlocal effect is significant when the structural size becomes small enough and is comparable to the atomic constant. The NLE theory has already been used to successfully describe systems such as nanoscaled structures exhibiting size dependent phenomena^[11-15]. Wave propagation behaviors in nanoscaled periodic structures with the size effect have also been investigated with the NLE theory^[16-18]. Nevertheless, the nonlocal effect mentioned above stems from the natural atomic forces, and does not contribute a lot in the macroscopic structure likely to support the wave propagation in a low acoustic frequency range. Therefore, it is very interesting to see if an artificial enhancement of the nonlocal effect could lead to special wave propagation behaviors. Therefore, we use the concept of nonlocal effect originated from micro/nanoscaled structures into the macro periodic lattice chain to investigate whether the abnormal but interesting phenomena will occur.

In this paper, a simple monoatomic lattice chain is investigated with the weakly nonlinear springs connecting the masses. Without loss of generality, the nonlocal effect is assumed to be confined only to the second-nearest masses^[19], and is realized by the linear springs shown in Fig. 1. An active control action is also exerted on every mass. The main focus of this work would be on how the nonlocal effect affects the property of the dispersion curve and how different the energy transferring manners would be at different points of the dispersion curve. The effects of nonlinearity^[20-25] and active control^[26-29] on the dispersion curve are also carefully investigated. Wang et al. developed the perturbation approach to the nonlinear elastic wave phononic crystals containing both monoatomic^[27] and diatomic^[30] lattice chain models. These valuable results demonstrate that the wave amplitude and hard and soft nonlinearities can affect the band gap structures.

2 Modeling: the theoretical approach

As shown in Fig. 1, the periodic system we are considering is a one-dimensional monoatomic lattice chain with the lattice constant a, which is connected by the weakly nonlinear springs between the neighboring masses. We just give 5 masses in Fig. 1 for example, where j is an arbitrary integer representing the mass number. The linear springs between the second-nearest masses are used to model the nonlocal interaction. The stiffness of the nonlocal linear spring is k₂^*. Numerical results will show that it is this consideration that gives birth to the anomalous zero and negative group velocities. The nonlinear internal force between two adjacent masses is

where k₁^* and Γ are the linear and nonlinear constants of the nonlinear spring, respectively, δ is the relative displacement between neighboring masses, and ε is a small value indicating the weak nonlinear effect. The asymptotic solution of this nonlocal monoatomic lattice chain is achieved based on the Lindstedt-Poincaré perturbation technique^[28]. In addition, the stiffness for the active piezoelectric spring is denoted by k_p^[27].

The wave motion in the above-mentioned monoatomic lattice chain is

(1)

where u^(j) represents the displacement of the jth mass.

Here, we introduce the dimensionless variables as follows:

where τ, , and μ are the dimensionless time, the circular frequency, and the wave number, respectively, α, η, and are the dimensionless strength of the nonlocality, the active control, and the nonlinearity, respectively. Then, Eq.(1) can be normalized as follows:

(2)

3 Asymptotic solution

Based on the Lindstedt-Poincaré perturbation technique, the asymptotic solutions corrected up to the second order can be expressed as follows:

(3)

Substituting Eq.(3) into Eq.(2), we have

(4)

The terms corresponding to different orders of ε can be identified as follows:

(5a)

(5b)

Without loss of generality, the solution of Eq.(5a) can be expressed as follows:

(6)

where A is the wave amplitude, and c.c. denotes the complex conjugate parts.

Substituting Eq.(6) into Eq.(5a), we have

(7)

Since A≠0, it can be derived that

(8)

Substituting Eq.(6) into Eq.(5b) leads to

(9)

where N_ST means the terms that do not produce the secular terms. To make the solution of Eq.(9) free of the secular terms, the coefficient of e^i(μj+τ) should equal zero. Then, we have

(10)

(11)

(12)

As a result, the normalized frequency can be derived as follows:

(13)

(14)

4 Dispersion relation analysis

In this part, we will discuss how the dispersion relation varies with the parameters α, η, and , which present the nonlocal effect, the active control, and the nonlinear degree, respectively. We will finally come to the conclusion on how to actively and efficiently manipulate the wave propagation based on different combinations of these tunable parameters.

Figure 2 shows the dispersion curves for different values of the nonlocal parameter α. It can be seen that the nonlocal effect does not change the frequencies at the boundaries of the Brillouin zone. However, it only enhances the frequencies in the middle part of the dispersion curve when α increases. Especially, when α exceeds 0.25+, the slope of the curve, which can be calculated from Eqs.(13)-(14), will not always be positive. For example, the dispersion curve for α =0.8 reaches its maximum at about μ =1.8. Therefore, the left part with μ < 1.8 has a positive slope, which means that the group velocity is positive; while the right part with μ>1.8 has a negative slope, which means that the group velocity is negative. The energy transferring properties corresponding to different group velocities will be numerically illustrated and discussed in Section 5.

The effect of the nonlinear parameter on the dispersion curve is demonstrated in Fig. 3. It can be seen that a larger nonlinear parameter corresponds to a higher frequency except at μ =0. This frequency enhancement mechanism of the nonlinear effect differs from that of the nonlocal effect by the fact that the frequency at the right boundary of the Brillouin zone is also lifted up (in Fig. 2, this point remains unchanged). Moreover, one can observe in Fig. 3 that the nonlinear effect does not give rise to the negative group velocity, and the slope of the dispersion curve is always positive. It is also noticed that the nonlinear effect is unable to change the frequency at the left boundary of the Brillouin zone. This task will be accomplished by the active control.

Figure 4 presents the effects of the active control on the dispersion relation, where the active control parameter η changes from 0.0 to 0.6. It can be clearly observed that the nonzero η introduces a cutoff frequency at the left boundary of the Brillouin zone, and this cutoff frequency increases as η increases. Therefore, a band gap emerges in the low frequency range from 0.0 to the cutoff frequency due to the existence of the active control. One can also observe that the active control lifts up the frequency not only at the left boundary of the Brillouin zone but also at the right boundary of the Brillouin zone. In other words, it enhances the frequency by lifting up the entire dispersion curve, which agrees with the conclusion in Ref.[27]. Nevertheless, although the active control provides an effective controlling method on the wave propagation behavior, it still cannot give rise to a zero or negative group velocity.

Since the zero and negative group velocities seem to be uniquely possessed by the nonlocal effects, the mechanism is intriguing. Here, we deserve a short discussion to get a better understanding for it. As we can see in Fig. 1, the chain system actually has three correlated parts. One is the masses denoted by j-2, j, j+2, ... connected by nonlocal springs, one is the masses denoted by j-1, j+1, ... connected by nonlocal springs, and the third one is the normal monoatomic lattice chain connected by the nonlinear springs. These three parts may have different propagation modes and different modes, which couple with each other to exchange their energy. In this way, the chain system is likely to perform different group velocities at specific wave numbers. This situation is somewhat similar to the case of wave propagation in plates. In plates, bulk waves are successively reflected by the surfaces, and mode conversion occurs when different propagation modes are coupled. The energy exchange between different modes makes it possible for researchers to observe zero or negative group velocities^[31-35]. What is new in this study is that the mode coupling mechanism occurs in a purely one-dimentional system.

After separately analyzing the nonlocal, nonlinear, and active control effects, let us combine all these three aspects together to see how flexible and arbitrary the controlling method could be. Figure 5 shows that an increasing active control parameter η lifts up the frequency at the left boundary of the Brillouin zone, and a decreasing nonlinear parameter lowers the frequency at the right boundary of the Brillouin zone. Therefore, the dispersion curve of larger η and smaller may have a crossover point with the dispersion curve of smaller η and larger . Based on this feature, we can get different group velocities at the same dispersion point by adjusting corresponding parameters.

Figure 6 shows the effects of all these three parameters on the dispersion curve. One can observe several phenomena simultaneously from the figure, i.e., the low frequency band gap caused by the active control parameter η, the frequency enhancement caused by the nonlinear parameter and the active control parameter η, and the zero and negative group velocities caused by the nonlocal parameter α. In summary, the adjustment of these parameters provides rich ways to take the control of various wave propagation behaviors, and may have potential values in practical applications.

5 Spatial wave analysis

To simulate a nonreflecting wave propagation, at each end, a perfectly matched layer (PML) is attached to the lattice chain, which has 500 cells. The PML consists of a section of a damped (linear viscous) chain. The PML damping profile is chosen as follows (see Fig. 7):

(15)

where C(j) is the damping coefficient of the jth mass. N_pml is the number of the PML masses, and here we take N_pml =30.

The PML modifies Eq.(1) by adding a damping term as follows:

(16)

The initial displacement and velocity are

(17a)

(17b)

where H(x) is the Heaviside function, and

(18)

Note that Eq.(17) represents the initial wavepacket with its center at j=0, we take N_cy =7. A replacement of j by j-m will thus represent the input wavepacket with the center at m.

The numerical integration routine of MATLAB, ODE45 is used to calculate the nonlinear system in Eq.(16). To monitor the spatial evolution of the input wavepacket, we consider the transient wave propagation in the finite monoatomic chain system. The initial condition is described by Eqs.(17)-(18). In the simulation process, we take ω₀ =1 rad·s^-1. Based on the relation

we have

From the asymptotic solution of described by Eqs.(13) and (14), we can get the asymptotic dimensionless time τ.

One can observe in Fig. 8(a) that, the typical points at P₁, P₂, and P₃ have positive, zero, and negative group velocities, respectively. As has been analyzed in Section 4, this unique property of one dispersion curve, which possesses three types of group velocities, attributes to the nonlocal interaction. To get a deeper insight into how energy is transported with different group velocities, the wavepacket evolutions at these three typical points are investigated. Figures 8(b), 8(c), and 8(d) depict the wavepacket evolutions at the points P₁, P₂, and P₃, respectively. In Fig. 8(b), the wavepacket has a positive group velocity, and propagates from the left to the right, which is also the direction of the corresponding phase velocity. More interesting phenomena happen at P₂ and P₃. P₂ has a zero group velocity. As expected, the center of the wavepacket in Fig. 8(c) keeps stationary. With time goes by, its amplitude decreases, and its width extends. This indicates a pure diffusion process without any drift. Actually, this phenomenon can be understood as a standing wave mode, which makes possible the energy concentration in a limited region. At P₃, however, the group velocity becomes negative, and the wavepacket center moves backward from the right to the left, which is the direction opposite to the phase velocity. This unusual wave propagation behavior, combined with the above two propagation manners, can be used in practical circumstances, which calls for different energy transferring needs.

In Fig. 9(a), it can be seen that, the larger the nonlocal coefficient α is, the higher the dispersion curve is. This can be easily understood because a stiffer system corresponds to a higher frequency, although these three curves meet at the same points under the boundaries of the Brillouin zone. As a result, the absolute value of the slope of the dispersion curve increases with α, which indicates different group velocity magnitudes at the same wave number. Comparing Figs. 9(b), 9(c), and 9(d), we can find that, a larger magnitude of the group velocity exactly corresponds to a faster propagation of the wavepacket in the backward direction since the group velocity is negative. The same propagation property can also be obtained at the points with positive group velocities, and will not be presented here for brevity.

Therefore, with the nonlocal interaction effects of the second-nearest masses, we can find a way to control both the direction and the speed of the transporting energy. Moreover, we can even let the energy be localized in a designated area. These novel phenomena can be used to arbitrarily harness energy transferring and to effectively realize different wave propagation behaviors. Moreover, with these merits of the nonlocal effect, one can also go deep into the investigations on the nonlocal effect of several nearest masses, in a two-dimensional periodic network, in a continuous system, etc.

6 Conclusions

In this paper, the Lindstedt-Poincaré method is used to calculate the dispersion relation of a one-dimensional monoatomic lattice chain connected by nonlinear springs. Both nonlocal and active control effects are considered. The dispersion relation shows that the nonlocal effect can effectively enhance the frequency except at the boundaries of the Brillouin zone. Moreover, a sufficiently large nonlocal effect introduces zero and negative group velocities into the dispersion curve, and provides different ways of transporting energy including forward-propagation, localization, and backward-propagation of wavepackets. Apart from the controllable direction of energy transferring, the speed of energy transferring can also be enhanced by enlarging the nonlocal effect.

The nonlinear effect can also enhance the frequencies at the right boundary of the Brillouin zone, but it is unable to produce zero or negative group velocities. The active control effect is also unable to produce zero or negative group velocities. However, the active control effect can enhance the frequencies at the boundaries of the Brillouin zone, and therefore gives birth to the nonzero cutoff frequency and band gap in the low frequency range. With the combination of adjusting all these effects, the geometry of the dispersion curve can be effectively modified, which shows its potential functionality in manipulating wave propagation and energy transferring. In the further investigation, the analysis and design will also be extended to new types of periodic structures to see if interesting phenomena and functionalities can be realized.