Granular Impact Model as an Energy-Depth Relation

Velocity-squared drag forces are common in describing an object moving through a granular material. The resulting force law is a nonlinear differential equation, and closed-form solutions of the dynamics are typically obtained by making simplifying assumptions. Here, we consider a generalized version of such a force law which has been used in many studies of granular impact. We show that recasting the force law into an equation for the kinetic energy versus depth, K(z), yields a linear differential equation, and thus general closed-form solutions for the velocity versus depth. This approach also has several advantages in fitting such models to experimental data, which we demonstrate by applying it to data from 2D impact experiments. We also present new experimental results for this model, including shape and depth dependence of the velocity-squared drag force

Topology of Force Networks in Granular Media under Impact

We investigate the evolution of the force network in experimental systems of two-dimensional granular materials under impact. We use the first Betti number, $\beta_1$, and persistence diagrams, as measures of the topological properties of the force network. We show that the structure of the network has a complex, hysteretic dependence on both the intruder acceleration and the total force response of the granular material. $\beta_1$ can also distinguish between the nonlinear formation and relaxation of the force network. In addition, using the persistence diagram of the force network, we show that the size of the loops in the force network has a Poisson-like distribution, the characteristic size of which changes over the course of the impact

Reynolds Pressure and Relaxation in a Sheared Granular System

We describe experiments that probe the evolution of shear jammed states, occurring for packing fractions $\phi_S \leq \phi \leq \phi_J$, for frictional granular disks, where above $\phi_J$ there are no stress-free static states. We use a novel shear apparatus that avoids the formation of inhomogeneities known as shear bands. This fixed $\phi$ system exhibits coupling between the shear strain, $\gamma$, and the pressure, $P$, which we characterize by the `Reynolds pressure', and a `Reynolds coefficient', $R(\phi) = (\partial ^2 P/\partial \gamma ^2)/2$. $R$ depends only on $\phi$, and diverges as $R \sim (\phi_c - \phi)^{\alpha}$, where $\phi_c \simeq \phi_J$, and $\alpha \simeq -3.3$. Under cyclic shear, this system evolves logarithmically slowly towards limit cycle dynamics, which we characterize in terms of pressure relaxation at cycle $n$: $\Delta P \simeq -\beta \ln(n/n_0)$. $\beta$ depends only on the shear cycle amplitude, suggesting an activated process where $\beta$ plays a temperature-like role.Comment: 4 pages, 4 figure