Wave tailoring by precompression in confined granular systems

We present a granular system whose response under an impact load can be varied from rapidly decaying to almost constant amplitude waves by an external regulator. The system consists of a granular chain of larger spheres surrounded by small spheres, confined in a hollow cylindrical tube and supporting wave propagation along the axis of the cylinder. We demonstrate using numerical simulations that the response can be controlled by applying radial precompression. These observations are then complemented by an asymptotic analysis, which shows that the decay in the leading wave is due to energy leakage to the oscillating small beads in the tail of the wave. This system has potential applications in systems requiring tuning of elastic waves.

Figure 1

Top (a) and side (b) views of the granular chain confined between rigid walls. By applying a radial precompression $\delta$, the wave propagation characteristics can be controlled.

Figure 2

Peak contact forces between the contacting big spheres. (a) Response of the $n=8$ system, showing that the peak forces decays rapidly with zero precompression but remains almost constant with increasing precompression. (b) Response of $n=9$ system, showing that the peak force remains almost constant for all values of precompression $a$.

Figure 3

Peak contact forces between the contacting big spheres. (a) Response for $n=6$ showing a near solitary wavelike behavior, while the $n=7$ system (b) has a decaying leading wave for low levels of precompression.

Figure 4

Wave speeds with precompression $a$ for the four confined systems indicating that the leading wave's speed increases both with precompression and with number of small spheres.

Figure 5

Bead velocities for the $n=8$ system. (a) At two lattice sites showing decay of leading wave and large oscillations behind it for $a=0$ and almost no decay and small oscillations behind it for $a=1.9\times {10}^{-3}$. The curves corresponding to $a=1.9\times {10}^{-3}$ are shifted by 1200 time units for clarity. (b) The velocity profiles of the large and small beads for two precompression levels: $a=0$ (solid curves) and $a=1.9\times {10}^{-3}$ (dashed curves).

Figure 6

Variation of the problem parameters with $n$, along with the normalized force decay rate for fractional $n$ in a fictitious system, showing that systems with $n=7,8$ have rapid decay compared to systems with $n=6,9$.

Figure 7

Peak force decay rate with density ratio between small and big beads. The peaks correspond to near solitary waves, while the troughs are resonances [8] where the decay is rapid.

Figure 8

Normalized peak contact force along the chain for different impact velocities and precompression levels, validating the scaling law (11).

Figure 9

Numerical and asymptotic zeroth-order solution for the big spheres, showing a good agreement. As precompression increases, the speed of the wave increases and its amplitude decreases.

Figure 10

Asymptotic solution for first-order axial (a) and radial (b) velocities of the small beads for three precompression levels for $n=8$ system. As the precompression increases, the magnitude of oscillations in the tail decreases, thereby indicating a reduction in the energy leaking from the primary wave.

Figure 11

Numerical force decay rate for the $n=8$ system with precompression $a$ along with the normalized kinetic energy in the tail obtained from the asymptotic solution. The slopes are equal, demonstrating that the force decay is due to the energy leakage into the oscillating small beads behind the leading wave.