Invariants of nonlinearity in the phononic characteristics of granular chains

In this work, we analyze the phononic characteristics of wave motion in precompressed monoatomic and diatomic granular chains, with emphasis on the evolving spatial features of wave packets. A Taylor series expansion of the governing equations is considered to approximate the granular chain with the Fermi-Pasta-Ulam chain. Within this approximation, the envelope modulation of the first-order features of the wave profile is monitored and the characteristics of this modulation are determined by studying the evolution of one of the distinctive features of the spatial profile. A set of constants that describe the quantitative effects of nonlinearity on the response are determined for monoatomic and diatomic chains and interpreted as invariants of quadratic nonlinearity. The universality of these invariants is verified by constructing inverse problems to estimate the contact power law from the wave response of granular chains with arbitrary nonlinear force interaction. The imposed power law is recovered exactly from numerical simulations for a number of considered scenarios, paving the way for inverse characterization of nonlinearity from experimental data.

Figure 1

1D monoatomic chain of spheres.

Figure 2

Displacement profile of the monoatomic granular chain at the end of simulation for two different excitation frequencies. The excitation frequencies are chosen to lie in the dispersive region of the Brillouin zone and the time of simulation is chosen as an integral multiple of the time taken to realize the entire burst in the system. $n$ corresponds to the index of the sphere in the chain.

Figure 3

Wave packet at the end of simulation in a spring-mass system with quadratic nonlinearity. The filtered long- and short-wavelength spectral components of this wave packet and the wave profile in the corresponding linear system are also shown. $n$ is the mass index in the chain.

Figure 4

Variation of tail amplitude ($u_t$) of the wave profile with respect to the excitation frequency ($\omega$) in a spring-mass chain with quadratic nonlinearity. The left figure shows the variation of the tail amplitude $\left(u_t\right)$ while the one on the right shows the variation of the normalized tail amplitude $[u_t/\left(\overline{G}{u}_0^2\right)]$.

Figure 5

Variation of tail amplitude with nondimensional excitation frequency in monoatomic granular chains for different power-law interactions.

Figure 6

Diatomic chain of spheres.

Figure 7

Displacement profile at the end of simulation for three different diatomic granular chains. The displacement of the heavier mass is shown in the figure. The nondimensional excitation frequency is the same for all three different diatomic chains and the time of simulation is chosen as an integral multiple of the time taken to realize the entire burst in the system. $n$ corresponds to the index of the unit cell.

Figure 8

Parametrization of tail amplitude of the heavier mass in a nonlinear diatomic spring-mass chain for $\alpha =0.3$.

Figure 9

Parametrization of tail amplitude of the heavier mass in a nonlinear diatomic spring-mass chain for $\alpha =0.6$.

Figure 10

Parametrization of tail amplitude of the heavier mass in a nonlinear diatomic spring-mass chain for different mass ratio ($\alpha$).

Figure 11

Variation of normalized maximum tail amplitude in the nonlinear diatomic spring-mass chain as a function of mass ratio ($\alpha$).

Figure 12

Variation of normalized tail amplitude as a function of nondimensional excitation frequency in the steel-Al diatomic granular chain for different power-law interactions.