1. Introduction
Impact phenomena often occur in nature and engineering applications [
1,
2,
3], such as earthquakes, explosions, vehicle collisions, etc. These impact phenomena will not only lead to the destruction of buildings but also damage human life and health [
4,
5]. Therefore, how to carry out impact protection is an important content in the research of impact problems [
6,
7,
8]. According to the mechanism of action, there are two common ways of impact protection: the first way is to confine the impact energy to the acceptable range of the impacted object [
9]; the second way is to reduce the speed of impact wave propagation and extend the time of impact wave reaching the impacted object so that the impacted object has enough time to respond to the impacts [
10].
Granular material consists of closely ordered particles in contact with each other, which has unique wave propagation characteristics and has potential applications in impact energy harvesting and mitigation [
11], impact damper [
12], non-destructive test [
13], switches [
14] and actuating devices [
15]. In 1983, Nesterenko first presented the concept of granular materials and showed the highly nonlinear solitary waves in granular materials [
16]. From then on, more and more attention has been paid to granular materials. As the waves in granular materials propagate through the contact of neighboring granules, the properties of the granules and the contact interactions are widely studied.
The elastic interaction without plastic deformations is widely studied in 1D granular chains [
17]. Li [
18] assembled chains of elastic cylindrical granules oriented at a variable angle with respect to each other and analyzed the effect of static precompression, alignment angles, and granules’ eccentricity on the signal’s transmissibility. The results showed that cylindrical granular chain has excellent tunability and can be used as passive, tunable acoustic filtering devices and vibration absorbers. Ngo et al. studied the dynamic response of elastic 1D granular chains of uniform hollow spheres [
19] and ellipsoidal granules [
20]. They found that the geometry of the granules could affect wave propagation and dissipation. Khatri studied the waves in chains of elastic cylindrical granules and found that the orientation angle between the granules could be used to tune the wave propagation [
21].
For influences of materials on wave propagations in elastic granular chains, Boechler et al. [
22] studied the granular chain of the steel and aluminum alternating granules and found the modulational instability and discrete breathers. Daraio et al. [
23] studied the wave propagations in PTFE granular chains and found that the waves propagate at a very low speed. Daraio [
24] also found that stainless steel granules under the same static precompression force demonstrate a higher absolute increase of the solitary wave speed compared to the PTFE system through experiments and numerical calculations. This feature can be used to make tunable acoustic focusing lenses.
Sen [
25] and Rosas [
26] used theoretical, numerical, and experimental methods to investigate the regulation of solitary waves in 1D composite elastic granular chains from granule size, density, Young’s modulus, and arrangement, respectively. These methods could give granular chains that can be engineered with impact wave amplitudes and custom wave velocities. Huang [
27] adopted the method of molecular dynamics and formed an energy trap by changing the mass ratio of granules in the heavy-light composite granular chain so as to realize the attenuation of the propagation velocity of the solitary wave and achieve the effect of impact delay. Chen [
28] et al. studied the periodic binary granular chain using the binary collision approximation theory, and the impact delay can be realized by reducing the radius of the granule at the periodic position.
The applications of granular chains are not only in low-velocity impact but also in high-velocity impact, where plastic deformation or even structural damage could occur. For these inelastic cases, plastic deformations happen once the stress reaches the yield strength, and the residual deformations remain after the impact. Therefore, the energy dissipation due to plastic deformations cannot be neglected. Investigations have shown that wave propagations and contact behaviors in elastic–plastic granular chains are different from elastic ones [
29,
30]. The crucial interactions between the elastic–plastic granules have been widely studied by using analytical methods and finite element analysis. Stronge [
31] analyzed the contact process of the elastic–plastic granules and proposed a widely used contact model, which divided the loading process into the elastic loading phase, the elastic–plastic loading phase, and the fully plastic loading phase. Pal [
29] presented the force-displacement in detail by using a nonlinear finite element analysis to simulate the wave propagations in elastic–plastic chains.
On [
32] studied the elastic–plastic loading conditions in the homogeneous brass granular chain through the split Hopkinson pressure bar (SHPB) experiment and found that the plastic deformation of granules under high-speed impact leads to the formation of plastic waves with slower wave speeds in the granular chains. Feng [
33] studied the fluctuation characteristics caused by multiple impacts and compression processes between single contact points using the SHPB experiment and numerical simulation based on the elastic–plastic contact model [
34], where the results revealed the adjustability of the impact energy dissipation. By adding individually adjustable granules at different locations in a homogeneous elastic–plastic granular chain, Pal [
35] and On [
36] tuned the ratios of pressure peaks, leading wave velocities, local contacts, and total dissipation at each contact point along the chain direction [
29]. Burgoyne [
37] used the discrete element method as well as SHPB experiments to study the homogeneous and binary elastic–plastic long chains. The results showed that the active control of impact wave propagation speed under high-speed impact could be realized by changing granule material properties, and the wave speed gradually decreased with the decrease of contact stiffness, which improves the impact delay effect.
Compared with the conventional granule arrangement, the impact buffering effect of the segmented composite granule chain is more significant. Daraio [
38] experimentally captured weak separation pulses in the soft granule segments of the low elastic modulus in the multi-segment granular chain. Zhang and Xu [
39,
40] used experimental and numerical simulation methods to create staggered composite granular chains of materials to attenuate impact wave amplitudes and design equivalent waves [
41]. Wang [
42] studied the energy decay caused by the transitional behavior in the light granule region of a one-dimensional three-segment composite granular chain by means of the momentum conservation principle. Through experiments and numerical simulation, Wang [
43] and Wu [
44] found that increasing the length of the light chain of the three-segment granular chain can cause more energy dissipation of the granule chain as a whole.
Compared with 1D granular chains, more complex problems such as rotation, friction, tangential force, boundary configurations, and so on should be considered in 2D and 3D granular systems [
45,
46]. Therefore, the study of 2D and 3D granular systems is complicated. However, the properties of 1D granular chains can be scaled to predict the behaviors of 2D and 3D granular systems [
47] and simplify the computational cost [
48]. Therefore, the segmented composite 1D granular chains are selected as the study emphasis in this study, especially the propagation characteristics and energy dissipation mechanism under the influence of elastic–plastic deformations. By changing the material properties of adjustable-segment granules, the influences of yield strength on the energy dissipation and impact wave propagation process in the elastic–plastic granular chain are obtained, and the impact buffering characteristics are analyzed. The research results in this paper could provide a theoretical basis for energy buffering under high-speed impact conditions.
The rest of the paper is organized as follows. The finite element model for the 1D elastic–plastic granular chains is presented in
Section 2. The results of the wave propagations and the influences of impact buffering are discussed in
Section 3. Finally, the conclusions are drawn in
Section 4.