Highly nonlinear solitary wave propagation in Y-shaped granular crystals with variable branch angles

We study the propagation of highly nonlinear waves in a branched (Y-shaped) granular crystal composed of chains of spherical particles of different materials, arranged at variable branch angles. We experimentally test the dynamic behavior of a solitary pulse, or of a train of solitary waves, crossing the Y-junction interface, and splitting between the two branches. We describe the dependence of the split pulses’ speed and amplitude on the branch angles. Analytic predictions based on the quasiparticle model and numerical simulations based on Hertzian interactions between the particles are found to be in excellent agreement with the experimental data.

Figure 1

(a) Experimental setup showing the Y-shaped guide and indicating the number of beads composing each portion of the system. (b) Schematic of the composition of a particle with an embedded piezogauge [12, 13, 14].

Figure 2

(a) Schematic of the Y-shaped system with the notation and direction of the solitary wave speeds in the system. (b) Schematic of the quasiparticles representation of the Y-shaped system and its corresponding effective speeds.

Figure 3

Experimental results for single solitary waves propagating in uniform Y-shaped systems composed of stainless steel beads. (a) $\alpha$ $=$ $\beta$ $=$ 30${}^{\circ }$. (b) $\alpha$ $=$ $\beta$ $=$ 45${}^{\circ }$. (c) $\alpha$ $=$ 30${}^{\circ }$ and $\beta$ $=$ 60${}^{\circ }$. The $y$-axes’ scale for curves (a) and (b) is 10 N per division. The $y$-axes’ scale for curve (c) is 15 N per division. The striker was dropped from the same height in all cases, and the striker velocity was 0.56 m/s.

Figure 4

Experimental results for trains of solitary waves propagating in uniform Y-shaped systems composed of stainless steel beads. (a) $\alpha$ $=$ $\beta$ $=$ 30${}^{\circ }$. (b) $\alpha$ $=$ 30${}^{\circ }$ and $\beta$ $=$ 60${}^{\circ }$. The $y$-axes’ scale for all the curves is 15 N per division. The striker was dropped from the same height in all cases, and the striker velocity was 0.45 m/s.

Figure 5

Experimental results for a single solitary wave propagating in nonuniform symmetric Y-shaped systems. (a) shows results obtained in a Y-shaped system in which the main stem was composed of aluminum beads and the two branches were composed of stainless steel beads for $\alpha$ $=$ $\beta$ $=$ 30${}^{\circ }$. The $y$-axes’ scale for all the curves is 5 N per division, and the striker velocity was 0.71 m/s. (b) reports results obtained in a Y-shaped system in which the main stem and one branch are composed of stainless steel beads, whereas, the other branch is composed of aluminum beads for $\alpha$ $=$ $\beta$ $=$ 30${}^{\circ }$. The $y$-axes’ scale for all the curves is 15 N per division, and the striker velocity was 0.67 m/s.

Figure 6

Comparison of the solitary wave shapes in uniform symmetric Y-shaped systems composed of stainless steel beads ($\alpha$ $=$ $\beta$ $=$ 30${}^{\circ }$) for (a) incident and reflected solitary waves in the main stem, (b) transmitted solitary wave in the branch. The solid (blue) lines represent the experimental data. The dashed (red) lines are the discrete element simulation results. The dotted (black) lines are the theoretical wave shapes. The striker was dropped from the same height in all cases, and the striker velocity was 0.56 m/s.

Figure 7

(a) Dependence of ratios of solitary wave speeds on the branch angles in a symmetric uniform Y-shaped system ($\alpha$ $=$ $\beta$). (b) Dependence of ratios of solitary wave amplitudes on the branch angles in an asymmetric uniform Y-shaped system. The solid lines represent the results obtained from the quasiparticle model; the dashed lines represent results from the discrete element simulations. The ratios of the reflected wave to the incident wave are in red (bottom curves), the ratios of the transmitted wave to the incident wave are in black (top curves). Error bars denote the standard deviations obtained from five experimental measurements repeated with the same striker velocity for a fixed configuration of the system.

Figure 8

(a) Dependence of ratios of solitary wave speeds on the branch angle $\beta$ in a symmetric uniform Y-shaped system. (b) Dependence of ratios of solitary wave amplitudes on branch angle $\beta$ in an asymmetric uniform Y-shaped system. The solid lines represent the results from the quasiparticle model; the dashed lines represent the results from the discrete element simulations. The ratios of the reflected wave to the incident wave are in red (bottom curves), the ratios of the transmitted wave in branch $\alpha$ to the incident wave are in black (top curves), and the ratios of the transmitted wave in branch $\alpha$ to the incident wave are in blue (middle curves). Angle $\alpha$ was fixed at 30${}^{\circ }$. Error bars denote the standard deviations obtained from five experimental measurements repeated with the same striker velocity for a fixed configuration of the system.

Figure 9

(a) and (b) Energy density plots for single solitary wave propagation in symmetric uniform Y-shaped system $\alpha$ $=$ $\beta$ $=$ 45${}^{\circ }$. (c) and (d) Energy density plots for single solitary propagation in asymmetric uniform Y-shaped systems $\alpha$ $=$ 30${}^{\circ }$ and $\beta$ $=$ 60${}^{\circ }$.

Figure 10

Dependence of the wave speed ratios on the branch angle in a symmetric ($\alpha$ $=$ $\beta$) uniform Y-shaped system traversed by trains of solitary waves. (a) The wave speed ratios correspond to the speed of the first solitary wave in the transmitted train to the corresponding one in the incident train. (b) The wave speed ratios correspond to the speed of the second solitary wave in the train to the corresponding one in the incident train. The solid lines represent the results from the quasiparticle model; the dashed lines correspond to the results obtained with the discrete element simulations. Error bars denote the standard deviations obtained from five experimental measurements repeated with the same striker velocity for a fixed configuration of the system.

Figure 11

Dependence of the wave speed ratios on the branch angle in an asymmetric uniform Y-shaped system traversed by trains of solitary waves. (a) The wave speed ratios correspond to the speed of the first solitary wave in the transmitted train to the corresponding one in the incident train. (b) The wave speed ratio corresponds to the speed of the second solitary wave in the transmitted train to the corresponding one in the incident train. The solid lines represent the results from the quasiparticle model; the dashed lines are the discrete element simulation results. Error bars denote the standard deviations obtained from five experimental measurements repeated with the same striker velocity for a fixed configuration of the system.

Figure 12

Dependence of the wave speed ratios on the branch angle in a symmetric nonuniform Y-shaped system. (a) System composed of a main stem filled with aluminum particles and branches filled with stainless steel particles. (b) System composed of a main stem and a branch filled with stainless steel particles and the other branch filled with aluminum particles. The dashed lines report results obtained with the discrete element simulations. Error bars denote the standard deviations obtained from five experimental measurements repeated with the same striker velocity for a fixed configuration of the system.