## Yale Graduate School of Arts and Sciences Dissertations

Spring 2021

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mechanical Engineering & Materials Science (ENAS)

Dry granular materials are collections of macroscopic-sized grainsthat interact via purely repulsive and dissipative contact forces. In this thesis, we describe discrete element method (DEM) simulations to characterize the structural and mechanical properties of jammed packings of spherical particles. Many previous studies have shown that the bulk modulus of jammed sphere packings depends only weakly on the pressure during isotropic compression, whereas the ensemble-averaged shear modulus $\langle G \rangle$ increases as a power-law in pressure $P$ at large pressures. However, the origin of the power-law scaling of the shear modulus with pressure is not well-understood. In particular, why is the power-law exponent for $\langle G\rangle$ versus $P$ close to $0.5$ for packings of both frictionless and frictional spherical particles with repulsive linear spring interactions and how does the exponent vary with the form of the repulsive interaction potential? In the first project, we focus on the mechanical response of jammed packings of $N$ frictionless spherical particles during isotropic compression. We show that $\langle G \rangle$ hastwo key contributions: 1) continuous variations of the shear modulus as a function of pressure along geometrical families, for which the interparticle contact network does not change and 2) discontinuous jumps during compression that arise from changes in the contact network. We find that the form of the shear modulus $G^f$ for jammed packings within near-isostatic geometrical families is largely determined by the affine response. We further show that the ensemble-averaged shear modulus, $\langle G(P) \rangle$, is not simply a sum of two power-laws, but $\langle G(P) \rangle \sim (P/P_c)^a$, where $a \approx (\alpha-2)/(\alpha-1)$ in the $P \rightarrow 0$ limit, $\alpha$ is the exponent that controls the form of the purely repulsive interparticle potential, and $\langle G(P) \rangle \sim (P/P_c)^b$, where $b \gtrsim (\alpha-3/2)/(\alpha-1)$ above a characteristic pressure that scales as $P_c \sim N^{-2(\alpha-1)}$. In the second project, we investigate whether jammming preparation protocol of frictional spherical particles gives rise to differences in their mechanical properties. We find that the average contact number and packing fraction at jamming onset are similar (with relative deviations $< 0.5\%$) for packings generated via isotropic compression and simple shear. In contrast, the average stress anisotropy $\langle \hat{\Sigma}_{xy} \rangle= 0$ for packings generated via isotropic compression, whereas $\langle \hat{\Sigma}_{xy} \rangle > 0$ for packings generated via simple shear. To investigate the difference in the stress state of jammed packings, we develop two additional packing-generation protocols: 1) we shear unjam compression jammed packings and then re-jam them use simple shear and 2) we decompression unjam shear-jammed packings and then re-jam them using isotropic compression. Comparing stress anisotropy distributions of the original jammed packings and the re-jammed packings, we find that there are nonzero stress anisotropy deviations $\Delta \hat{\Sigma}_{xy}$ between the jammed and re-jammed packings, but the deviations are smaller than the fluctuations in the stress anisotropy obtained by enumerating the force solutions within the null space of the contact networks of the jammed packings. These results emphasize that even though the compression and shear jamming protocols generate jammed packings with the same contact networks, there can be residual differences in the normal and tangential forces at each contact, and thus differences in the stress anisotropy of the packings. We conclude the thesis with suggested directions for future research. Topics include studies of geometrical families in packings of frictional, spherical particles, studies of the pressure-dependence of the shear modulus for non-spherical particles, and studies of the mechanical properties of frictional, non-spherical particles.