Nonlinear nonlocal phononic crystals with roton-like behavior

Abstract

Thanks to their extraordinary dynamic characteristics, nonlinear phononic crystals (PCs) have found invaluable academic and industrial significance, in recent years. The dependence of wave propagation properties in phononic structures on material and geometrical specifications has led to innovative approaches to design them for specific needs. Along with the ongoing efforts to design and analyze more effective PCs, in this paper, we investigate the roton-like behavior of PCs by introducing a nonlinear monoatomic chain with the third-neighbor interactions able to exhibit backward energy flow in the Brillouin zone. We study the amplitude-dependent roton-like behavior of such crystals in detail, here. Furthermore, we analyze the effects of amplitude and nonlinear coefficients on the dispersion diagrams and backscattering regions of the introduced chain. We then expand the study to the nonlinear two-dimensional lattices to investigate the roton-like behavior in such structures. The results reveal that the roton-like behavior can lead to extraordinary characteristics in PCs and including nonlinearity can provide an extra level of control over their behavior.

Appendices

Appendix 1

The three terms on the right-hand side of Eq. (6) are obtained as:

$$2{D}_{0}{D}_{1}{u}_{j}^{\left(0\right)}= \mathrm{i}{\omega }_{0}{D}_{1}A{e}^{\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{\mathrm{i}\mu j}-\mathrm{i}{\omega }_{0}{D}_{1}\overline{A}{e }^{-\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{-\mathrm{i}\mu j}= \mathrm{i}{\omega }_{0}{D}_{1}A{e}^{\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{\mathrm{i}\mu j}+c.c.,$$

(23)

$${{\alpha }_{1}\left({u}_{j}^{\left(0\right)}-{u}_{j-1}^{\left(0\right)}\right)}^{3}+{\alpha }_{1}{\left({u}_{j}^{\left(0\right)}-{u}_{j+1}^{\left(0\right)}\right)}^{3}= {\alpha }_{1} \left\{{\left[\frac{A}{2}\left(1-{e}^{-\mathrm{i}\mu }\right){e}^{\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{\mathrm{i}\mu j}+ \frac{\overline{A}}{2 }\left(1-{e}^{\mathrm{i}\mu }\right){e}^{-\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{-\mathrm{i}\mu j}\right]}^{3}+ {\left[\frac{A}{2}\left(1-{e}^{\mathrm{i}\mu }\right){e}^{\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{\mathrm{i}\mu j}+ \frac{\overline{A}}{2 }\left(1-{e}^{-\mathrm{i}\mu }\right){e}^{-\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{-\mathrm{i}\mu j}\right]}^{3}\right\}= \frac{3}{8}{\alpha }_{1}{A}^{2}\overline{A }\left(6-4{e}^{\mathrm{i}\mu }-4{e}^{-\mathrm{i}\mu }+{e}^{-2\mathrm{i}\mu }+{e}^{2\mathrm{i}\mu }\right){e}^{\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{\mathrm{i}\mu j}+\frac{{\alpha }_{1}{A}^{3}}{8}\left(1-3{e}^{-\mathrm{i}\mu }+3{e}^{-2\mathrm{i}\mu }-{e}^{-3\mathrm{i}\mu }\right){e}^{3\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{3\mathrm{i}\mu j}+c.c.$$

(24)

$${{\alpha }_{3}\left({u}_{j}^{\left(0\right)}-{u}_{j-3}^{\left(0\right)}\right)}^{3}+{\alpha }_{3}{\left({u}_{j}^{\left(0\right)}-{u}_{j+3}^{\left(0\right)}\right)}^{3}= {\alpha }_{3} \left\{{\left[\frac{A}{2}\left(1-{e}^{-3\mu }\right){e}^{\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{\mathrm{i}\mu j}+ \frac{\overline{A}}{2 }\left(1-{e}^{3\mathrm{i}\mu }\right){e}^{-\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{-\mathrm{i}\mu j}\right]}^{3}+ {\left[\frac{A}{2}\left(1-{e}^{3\mathrm{i}\mu }\right){e}^{\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{\mathrm{i}\mu j}+ \frac{\overline{A}}{2 }\left(1-{e}^{-3\mathrm{i}\mu }\right){e}^{-\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{-\mathrm{i}\mu j}\right]}^{3}\right\}= \frac{3}{8}{\alpha }_{3}{A}^{2}\overline{A }\left(6-4{e}^{3\mathrm{i}\mu }-4{e}^{-3\mathrm{i}\mu }+{e}^{-6\mathrm{i}\mu }+{e}^{6\mathrm{i}\mu }\right){e}^{\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{\mathrm{i}\mu j}+\frac{{\alpha }_{3}{A}^{3}}{8}\left(1-3{e}^{-3\mathrm{i}\mu }+3{e}^{-6\mathrm{i}\mu }-{e}^{-9\mathrm{i}\mu }\right){e}^{3\mathrm{i}{\omega }_{0}{T}_{0}}{e}^{3\mathrm{i}\mu j}+c.c.$$

(25)

Substituting Eqs. (23)–(25) in Eq. (6) and following the routine mathematical procedures, Eq. (9) can be reached.

Appendix 2: Range of wave amplitude

To investigate the allowable amplitude range for the weakly nonlinear chain of the present analysis, a deeper look at the perturbation assumptions is necessary. According to Eq. (11), the nonlinear frequency is calculated as $\Omega = {\omega }_{0}+\varepsilon {\omega }_{1}$, which leads to the following equation:

$${\Omega }^{2}={\omega }_{0}^{2}+2\varepsilon {\omega }_{0}{\omega }_{1}$$

(26)

To make sure that the ${\varepsilon }^{0}$-and ${\varepsilon }^{1}$-order equations are described correctly, the inequality $2\varepsilon {\omega }_{0}{\omega }_{1}\ll {\omega }_{0}^{2}$ should be satisfied. Hence, the following equation is obtained:

$$\frac{3\varepsilon \left[{\alpha }_{1}{A}^{2}\left(\mathrm{cos}2\mu - 4\mathrm{cos}\mu +3\right)+ {\alpha }_{3}{A}^{2}\left(\mathrm{cos}6\mu - 4\mathrm{cos}3\mu +3\right)\right]}{2{\omega }_{0}^{2}}\ll 1$$

(27)

Since the perturbation parameter $\varepsilon$ is assumed small, it is taken to be in the same order as $\ll 1$. Hence, the relationship between the amplitude, stiffness, and wavenumber becomes:

$$A<\sqrt{\frac{2{\omega }_{0}^{2}}{3\left({\alpha }_{1}{A}^{2}\left(\mathrm{cos}2\mu - 4\mathrm{cos}\mu +3\right)+ {\alpha }_{3}{A}^{2}\left(\mathrm{cos}6\mu - 4\mathrm{cos}3\mu +3\right)\right)}}$$

(28)

A similar approach for the 2D nonlinear lattice leads to the following equation for the amplitude:

$$A<\sqrt{\frac{2{\omega }_{0}^{2}}{3\beta }}$$

(29)

where $\beta$ is defined by Eq. (21). Assuming constant values for linear and nonlinear stiffness values ($k=\alpha =1$), the wavenumber-dependent amplitudes can be determined for 1D and 2D nonlinear PCs. Figure 

Fig. 13

The allowable amplitude range for a a 1D PC with roton-like behavior, and b a 2D nonlinear lattice with roton-like behavior ($\Gamma -X$ path only)

13 illustrates the maximum values of amplitude, for which, the asymptotic expansion remains valid. Figure 13a illustrates the variation of the amplitude along with the first Brillouin zone in a 1D nonlinear PC. Based on the illustrated results, the maximum allowed amplitude is denoted by the minimum value of the amplitude in the Brillouin zone, i.e., ${A}_{\mathrm{max}}=1.78$. As a result, the asymptotic expansion remains valid for amplitudes below $A=1.78$. Similarly, the maximum allowed amplitude for the 2D lattice with roton-like behavior is ${A}_{\mathrm{m}}=1.88$, as depicted in Fig. 13b. Note that for the sake of convenience, the variation of the amplitude is illustrated only in the $\Gamma -X$ path along the irreducible Brillouin zone. More information regarding the details of the procedure can be found in [41].

Appendix 3: Propagation of large-amplitude waves in roton-like structures

To provide a better numerical perception of the propagation of a wave having a large amplitude in a nonlinear periodic structure with roton-like behavior, a numerical approach similar to the one presented by Chen et al. [46] is adopted, here. In this approach, a single mass in the middle of the chain is excited by a temporal Gaussian pulse, ${u}_{0}\left(t\right)={A}_{0}\mathrm{cos}\left(\omega t\right)\mathrm{exp}\left(-{\left(t/\tau \right)}^{2}\right)$, where $\tau =100/\omega$ and the carrier frequency $\omega$ lies in the region for which we have three distinct wavenumbers for each frequency. The schematic representation of the pulse excitation on the middle mass is illustrated in Fig. 

Fig. 14

The schematic representation of applying the Gaussian pulse to a nonlinear chain with roton-like behavior

14.

The numerical response of the nonlinear chain with roton-like behavior for different excitation amplitudes of ${A}_{0}=2$, ${A}_{0}=5$ and ${A}_{0}=10$ is illustrated in Fig. 

Fig. 15

The time-dependent response of the 700-mass nonlinear chain with roton-like behavior for different amplitudes of a ${A}_{0}=2$, b ${A}_{0}=5$ and c ${A}_{0}=10$. (${k}_{1}={\alpha }_{1}={\alpha }_{3}=1$, ${k}_{3}=3$, $\varepsilon =0.01$)

15a–c, respectively. In the case of low-amplitude excitation, the propagation is similar to the linear chain [46], i.e., exciting the center mass of the chain leads to the generation of two triplets of wave packets at each side. These three packets are consistent with Fig. 4 and correspond to positive and negative energy flows, as previously discussed. However, as the amplitude is increased, chaotic behavior tends to emerge and the propagation patterns turn from a more focused pattern to a more distributed one. This is observed in Fig. 15b, where the three distinct wave packets can still be noticed. However, unlike the low-amplitude propagation, in the case with ${A}_{0}=5$, a distributed propagation pattern is further noticed in another direction. Furthermore, as the amplitude is increased to ${A}_{0}=10$, the three distinct patterns fade away and a more chaotic propagation pattern is observed.