Mesoscopic approach to granular crystal dynamics

We present a mesoscopic approach to granular crystal dynamics, which comprises a three-dimensional finite-element model and a one-dimensional regularized contact model. The approach investigates the role of vibrational-energy trapping effects in the dynamic behavior of one-dimensional chains of particles in contact (i.e., granular crystals), under small to moderate impact velocities. The only inputs of the models are the geometry and the elastic material properties of the individual particles that form the system. We present detailed verification results and validate the model comparing its predictions with experimental data. This approach provides a physically sound, first-principles description of dissipative losses in granular systems.

Figure 1

(a) Finite-element mesh of a single spherical particle. (b) Coefficient of restitution $e\left(v_i\right)=1-0.0247v_i^{0.61}$ obtained from best fitting (solid curve) a numerical campaign of head-on impacts of $9.525$-mm stainless steel beads (symbols). Five $v_i$ are employed and $e$ is computed from $\frac{T_f}{T_0}=\frac{1}{2}(1+e^2)$, where $T_0$ and $T_f$ are the initial and final kinetic energies of the center-of-mass motion obtained from the three-dimensional finite-element simulations. Two uniform tetrahedral meshes are employed to assess convergence: (i) 53$k$ nodes, 275$k$ elements ($\circ$) and (ii) 410$k$ nodes, 2.2$M$ elements ($\diamond$), per spherical particle. The mesh referred to as (i) is depicted in (a).

Figure 2

Dynamics of (a)–(c) one bead and (d)–(f) a one-dimensional chain impacted by an identical $9.525$-mm stainless steel bead. Predictions of the one-dimensional (three-dimensional) model are depicted by dashed (solid) curves. Energies of the three-dimensional model are effectively compared by subtracting the vibrational components explicitly computed with the model in (b) and (e). Exact conservation of linear momentum in both models in (a) and (d) and total energy in the three-dimensional model in (b) and (e) are depicted by dash-dotted curves.

Figure 3

One-dimensional regularized contact model for $v_{\text{imp}}=0.63$ m/s. (a) Time history of measured forces (dashed curves) and numerical predictions with and without gravitational forces. (b) Conservation of total linear momentum and time history of individual $J_k$. (c) Time history of total $E$, kinetic $T$, and potential $V$ energies.

Figure 4

Comparison of predictions obtained with our one-dimensional numerical model, with and without gravitational forces, with measured values (solid dots) for all five values of $v_{\text{imp}}$. (a) Evolution of the solitary-wave full width at half maximum as a function of the particle's position in the chain. (b) Decay of the maximum force. (c) Speed of the solitary wave versus the maximum force and its best fit $V_s\propto F_m^{0.168}$. Five different valus of $v_{\text{imp}}$ are used: $0.31$, $0.44$, $0.63$, $0.89$, and $1.25$ m/s. Error bars represent the standard deviation from five samples of each data point.