Shear Jamming in Frictionless Particulate Media

We numerically study two-dimensional packings of frictionless bidisperse disks created using compresive and simple shearing protocols. To create jammed packings by compression, we start $N$ particles from random positions and grow their diameters followed by relaxation of particle overlaps using energy minimization. These compressed packings exist over a range of packing fractions $\phi$. As a result, during compression the system may reach a $\phi$ above the minimum value before jamming. If this unjammed packing is then sheared by a strain $\gamma$, it can jam. Using a combination of compression and shearing, we can define jamming protocols as trajectories in the $(\phi, \gamma)$ plane that yield jammed packings. In this plane, we can reach a particular point $(\phi_n, \gamma_n)$ in many ways. We will focus on two protocols: (1) shearing to $\gamma_n$ at $\phi=0$ followed by compression to $\phi_n$ at $\gamma= gamma_n$ and (2) compression to $\phi_n$ at $\gamma=0$ followed by shearing to $\gamma_n$ at $\phi=\phi_n$. For protocol 1, we find that the probability of finding a jammed packing at $\phi$ and $\gamma$, $P(\phi,\gamma)=Q(\phi)$ is indepependent of $\gamma$. For protocol 2, we use a simple theory to deduce $P(\phi,\gamma)$ from $Q(\phi)$.