Wave manipulation using a bistable chain with reversible impurities

We systematically study linear and nonlinear wave propagation in a chain composed of piecewise-linear bistable springs. Such bistable systems are ideal test beds for supporting nonlinear wave dynamical features including transition and (supersonic) solitary waves. We show that bistable chains can support the propagation of subsonic wave packets which in turn can be trapped by a low-energy phase to induce energy localization. The spatial distribution of these energy foci strongly affects the propagation of linear waves, typically causing scattering, but, in special cases, leading to a reflectionless mode analogous to the Ramsauer-Townsend effect. Furthermore, we show that the propagation of nonlinear waves can spontaneously generate or remove additional foci, which act as effective “impurities.” This behavior serves as a new mechanism for reversibly programming the dynamic response of bistable chains.

Figure 1

(a) Schematic illustration of the system composed of bistable springs. The impact is applied to the leftmost unit ($n=1$) by setting an initial velocity. The rightmost mass ($n=N$) is connected to a dashpot. (b) The relationship between force and strain of a bistable spring is expressed by a trilinear function [cf. Eq. (1)]. The stiffness parameters are $(K_I,K_S,K_{II})=(1,-1,1.5)$ and the strain parameters are $(r_S,r_{II})=(0.4,1)$. (c) The bistable energy landscape obtained from the trilinear force-strain relationship of panel (b).

Figure 2

Amplitude-dependent behavior for a bistable chain composed of strain-hardening springs ($K_{II}/K_I=1.5$). All unit cells are initially in Phase I. We apply an impact velocity to the first unit cell ($n=1$) by setting $v_1(t=0)\ne 0$. We consider four different values of the impact velocities: (a) $v_1(t=0)=0.1$, (b) 0.552, (c) 2, and (d) 15. The color indicates the strain $r_n$ as a function of the unit index $n$ and time $t$.

Figure 3

Trapping behavior. (a) The spatial distribution of the subsonic wave packet generated by the impact velocity $v_1(t=0)=0.552$ is shown at different instants of time $t$. The waveforms for $t=585$ and 660 correspond to the black vertical lines in Fig. 2, respectively. Each strain profile is shifted in the space domain so that the peak of the strain profile is located at $n=0$ to ease visualization. (b) The strain change of $n=70$ is shown as a function of time $t$. The shaded gray area indicates the spinodal regime [see also Fig. 1]. To quantify the wave shape, we measure time widths in which the strain is in the spinodal region, $t_{SpL}$ and $t_{SpR}$, bounded by the red vertical lines. (c) Surface plot of the energy $E_n\left(t\right)=\frac{1}{2}m{\dot{u}}_n^2\left(t\right)+U_n\left(t\right)$ where $U_n$ is the bistable potential-energy function of Fig. 1. Note that the colormap is adjusted for clarity of illustration of the energy leakage so that energies greater than $4\times {10}^{-3}$ all appear with a dark color. (d) The peak energy of the propagating wave packet is shown as a function of $t$. $\mathrm{\Delta }U$ is the energy barrier measured from the energy minimum in Phase II [see Fig. 1]. The red vertical line indicates the trapping point at which the peak energy becomes lower than that of the energy barrier $\mathrm{\Delta }U$. (e) We calculate the time difference $t_{SpR}-t_{SpL}$, which indicates the symmetric (if $t_{SpR}-t_{SpL}=0$) or asymmetric waveform (otherwise).

Figure 4

Schematic illustrations for a chain with a (a) single and (b) double impurity. Analytical transmission and reflection coefficients for (c) single- and (d) double-impurity cases. The two different stiffness ratio cases (left) $\gamma =0.5$ and (right) $\gamma =2.5$ are shown. The reflection and transmission coefficients for (e) single- and (f) double-impurity cases are plotted as a function of stiffness ratio $\gamma$ and wave number $k$. (e), (f) The gray dashed lines indicate $\left|R\right|=0$ (or $\left|T\right|=1$), and the reflectionless mode, which is expressed by Eq. (12), is denoted by the white dashed lines.

Figure 5

Analysis of transmission. (a) Comparison between analytics (black line) and numerics (red markers) for a stiffness ratio of $\gamma =0.5$. (b) Spatiotemporal plots of velocity profiles for excitation frequencies of (left) $\omega =0.6$ and (right) $\omega =1.0$, respectively. The inset shows the temporal dependence of the velocity $v_n$ for (gray) $n=-269$ and (red) $n=31$. (c), (d) Same as the top panels but for the case corresponding to a double-impurity chain.

Figure 6

(a) Schematic illustration of a collision of a propagating wave packet with an impurity. (b), (c) Spatiotemporal plots of strain wave profiles for the (b) single- and (c) double-impurity cases. We apply the initial velocity $[v_1(t=0)=0.55]$ to the first particle to generate the subsonic propagating wave packet.

Figure 7

Collision of a (subsonic) propagating wave packet with a single impurity. (a) Surface plot of the energy $E_n\left(t\right)=\frac{1}{2}m{\dot{u}}_n^2\left(t\right)+U_n\left(t\right)$ where $U_n$ is the bistable potential-energy function as shown in Fig. 1. The red vertical line indicates the collision point. (b) The peak energy of the propagating wave packet is shown as a function of time.

Figure 8

Scattering by multiple double impurities. (a) The schematic illustration highlights the application of a harmonic excitation (with excitation frequency ${\omega }_{ex}$) to the first unit ($n=1$) of the chain with multiple double impurities. The stiffness parameters for the host and impurities are $K_{\mathrm{host}}=1$ and $K_{\mathrm{imp}}=1.5$, respectively. (b) The transmission and reflection coefficients for the double impurity with $\gamma =1.5$ are shown as a function of frequency $\omega$. Surface plots of strain wave propagation are presented for (c) ${\omega }_{ex}=1.10$ and (d) ${\omega }_{ex}=1.73$.

Figure 9

Amplitude-dependent phase-transition behavior. (a) Schematic illustration for examining the steady-state configuration after an impact is applied to the chain. We extract the final configurations of the chain at $t=3000$ and present them as a function of the impact velocity $v_1$ for (b) the softening ($K_{II}/K_I=0.5$) and (c) hardening ($K_{II}/K_I=1.5$) chains.

Figure 10

Wave form analysis of the supersonic solitary wave. The blue solid line indicates the strain profile of the supersonic solitary wave with the wave speed of 1.076 obtained from the simulation [see Fig. 2]. The red dashed line indicates the analytical solution based on the Padé approximation [Eq. (B5)].

Figure 11

Numerical simulations to obtain the transmission coefficient. (a) Spatiotemporal plot of velocity. We embed a single impurity (stiffness ratio $\gamma =0.5$) in a chain composed of 601 units and apply a harmonic excitation ($\omega =0.4$) to the chain. (b) Temporal distribution of the velocity profiles $v_n$ for (gray) $n=-269$ and (red) $n=31$.

Figure 12

Analytical transmission compared with numerical simulations for (a), (b) single-impurity and (c), (d) double-impurity cases. We calculate transmission coefficients for stiffness ratios (left) $\gamma =1.5$ and (right) $\gamma =2.5$.

Figure 13

Effect of multiple double impurities on phase transitions. Spatiotemporal plots for (a) a homogeneous chain (Phase I) and (b) a chain with double impurities ($K_{\mathrm{imp}}=K_{II}=1.5$). We control the displacement of the first unit at constant speed $v_1=1.2$. The gray (black) dashed lines indicate the phase boundary (linear wavefront). (c) We plot the speed of the phase boundary ($V_{PB}$) as a function of the applied speed $v_1$. The inset shows the difference between the phase boundary speeds for the homogeneous chain without impurities and the chain with multiple impurities, which is defined as $\mathrm{\Delta }{V}_{PB}=V_{PB}^i-V_{PB}^h$.

Figure 14

A chain with multiple impurities under tension. To apply the tension, we control the displacement of the first particle at constant speed $v_1$. Spatiotemporal plots of the strain wave profiles for (a) $v_1=-0.1$ and (b) $v_1=-0.5$.