Conical wave propagation and diffraction in two-dimensional hexagonally packed granular lattices

Linear and nonlinear mechanisms for conical wave propagation in two-dimensional lattices are explored in the realm of phononic crystals. As a prototypical example, a statically compressed granular lattice of spherical particles arranged in a hexagonal packing configuration is analyzed. Upon identifying the dispersion relation of the underlying linear problem, the resulting diffraction properties are considered. Analysis both via a heuristic argument for the linear propagation of a wave packet and via asymptotic analysis leading to the derivation of a Dirac system suggests the occurrence of conical diffraction. This analysis is valid for strong precompression, i.e., near the linear regime. For weak precompression, conical wave propagation is still possible, but the resulting expanding circular wave front is of a nonoscillatory nature, resulting from the complex interplay among the discreteness, nonlinearity, and geometry of the packing. The transition between these two types of propagation is explored.

Figure 1

(a) Orientation of the index convention. The $m$ axis represents the horizontal direction. The $m$ axis and the $n$ axis meet at an angle of $\theta =\pi /3$. In the absence of precompression, the equilibrium distance between the centers of adjacent beads is the bead diameter $d$. (b) The hexagonal lattice is compressed uniformly on all boundaries, which induces a static overlap $\delta$ such that the equilibrium distance between the centers of adjacent beads is $d-\delta$. Under these compression conditions, the static equilibrium configuration has the hexagonal symmetry. The amount of static force $F_0$ required to induce the static overlap will depend on the number of beads in contact with the boundary. The compression amount of the boundary is greatly exaggerated here for clarity.

Figure 2

The dispersion surface $\omega (k,\ell )$ as given by Eq. (11). Parameter values are $d=M=\gamma =1$ and $\delta =0.1$. (a) Contour plot of the bottom dispersion surface. The Brillouin zone is shown as the black line, and the Dirac points as black points. (b) Same as (a), but for the top dispersion surface. (c) Both dispersion surfaces in $\left(k,\ell ,\omega \right)$ space.

Figure 3

(a) Zoom-in of the dispersion surface near the Dirac point $(k_d,{\ell }_d)=(4\pi /3,0)$. The transparent layers are the first-order approximations given by ${\omega }_{cr}\pm \alpha \sqrt{{(k-k_d)}^2+{(l-l_d)}^2}$, where the slope is $\alpha =3b_2/2{\omega }_{\mathrm{cr}}.$ (b) Zoom-in of the dispersion surface near the Dirac point $(k_0,{\ell }_0)=(0,0)$. Here, there are four cones meeting at the Dirac point rather than two, as in (a). Note that no first-order approximation is shown in (b).

Figure 4

Conical-like diffraction in a hexagonal lattice. Images from left to right show the dynamical evolution. The color intensity corresponds to the magnitude of the displacement. The position of the bead in each panel corresponds to the position in physical space (see also Fig. 1). Parameter values are the same as in Fig. 2. The initial conditions are a superposition of Bloch modes at the Dirac points $(k_d,{\ell }_d)=(0,4\pi /3)$ and $(k_d,{\ell }_d)=(2\pi \sqrt{3},2\pi /3)$; see Eq. (29). Images from top to bottom correspond to increasing initial intensity. (a)–(c) Evolution with a low-amplitude excitation (the maximum strain is $0.6\%$ of the static overlap $\delta )$, thus inducing near-linear dynamics. (d)–(f) Weakly nonlinear evolution (the maximum strain is $10\%$ of the static overlap $\delta )$. (g)–(i) Strongly nonlinear evolution (the maximum strain is $53\%$ of the static overlap $\delta )$.

Figure 5

Exciting the quadruple Dirac point $(k_0,{\ell }_0)=(0,0)$. Parameter values are the same as in Fig. 2. Note that the stability condition $\delta /d<1/3$ is satisfied. The color intensity corresponds to the magnitude of the displacement. (a) The initial condition with a low-amplitude (i.e., near-linear) excitation (the maximum strain is $0.6\%$ of the static overlap $\delta )$. (b) Given sufficient evolution time, two rings propagating at different speeds will form. (c) Surface map of the magnitude of the displacement along the $n=0$ axis. Solid and dashed lines are the approximations of the group speeds ${\alpha }_{-}$ and ${\alpha }_{+}$, respectively.

Figure 6

Exciting the quadruple Dirac point $(k_0,{\ell }_0)=(0,0)$, with parameters as in Fig. 2 with the exception of the precompression, which has the value $\delta =0.4$. The stability condition $\delta /d<1/3$ is violated (i.e., ${\alpha }_{-}$ is purely imaginary). (a) The initial condition with a low-amplitude (i.e., near-linear) excitation (the maximum strain is $0.6\%$ of the static overlap $\delta )$. The color intensity corresponds to the magnitude of the displacement. (b), (c) The amplitude grows exponentially since the group speed ${\alpha }_{-}$ is purely imaginary. The condition for validity of the model [which for these parameter values is $(\sqrt{3}(d-\delta )-d)/2\approx 0.02]$ is exceeded at $t\approx 9.7$.

Figure 7

(a) Evolution of a low-amplitude Gaussian modulating a Bloch function at an arbitrary wave number. The characteristic features of conical diffraction are absent, while hyperbolic structures can be discerned. (b) Evolution after exciting four particles at the center of a purely nonlinear chain.