Eric DeGiuli

Breakdown of continuum elasticity in amorphous solids

We show numerically that the response of simple amorphous solids (elastic networks and particle packings) to a local force dipole is characterized by a lengthscale lc that diverges as unjamming is approached as lc ∼ (z - 2d)(-1/2), where z ≥ 2d is the mean coordination, and d is the spatial dimension, at odds with previous numerical claims. We also show how the magnitude of the lengthscale lc is amplified by the presence of internal stresses in the disordered solid. Our data suggests a divergence of lc ∼ (pc - p)(-1/4) with proximity to a critical internal stress pc at which soft elastic modes become unstable.

Force distribution affects vibrational properties in hard-sphere glasses

Significance How a liquid becomes rigid at the glass transition is a central problem in condensed matter physics. In many scenarios of the glass transition, liquids go through a critical temperature below which minima of free energy appear. However, even in the simplest glass, hard spheres, what confers mechanical stability at large density is highly debated. In this work we show that to quantitatively understand stability at a microscopic level, the presence of weakly interacting pairs of particles must be included. This approach allows us to predict various nontrivial scaling behavior of the elasticity and vibrational properties of colloidal glasses that can be tested experimentally. It also gives a spatial interpretation to recent, exact calculations in infinite dimensions. We theoretically and numerically study the elastic properties of hard-sphere glasses and provide a real-space description of their mechanical stability. In contrast to repulsive particles at zero temperature, we argue that the presence of certain pairs of particles interacting with a small force f soften elastic properties. This softening affects the exponents characterizing elasticity at high pressure, leading to experimentally testable predictions. Denoting ℙ(f)∼fθe, the force distribution of such pairs and ϕc the packing fraction at which pressure diverges, we predict that (i) the density of states has a low-frequency peak at a scale ω*, rising up to it as D(ω)∼ω2+a, and decaying above ω* as D(ω)∼ω−a where a=(1−θe)/(3+θe) and ω is the frequency, (ii) shear modulus and mean-squared displacement are inversely proportional with 〈δR2〉∼1/μ∼(ϕc−ϕ)κ, where κ=2−2/(3+θe), and (iii) continuum elasticity breaks down on a scale ℓc∼1/δz∼(ϕc−ϕ)−b, where b=(1+θe)/(6+2θe) and δz=z−2d, where z is the coordination and d the spatial dimension. We numerically test (i) and provide data supporting that θe≈0.41 in our bidisperse system, independently of system preparation in two and three dimensions, leading to κ≈1.41, a≈0.17, and b≈0.21. Our results for the mean-square displacement are consistent with a recent exact replica computation for d=∞, whereas some observations differ, as rationalized by the present approach.

Unified theory of inertial granular flows and non-Brownian suspensions

Rheological properties of dense flows of hard particles are singular as one approaches the jamming threshold where flow ceases both for aerial granular flows dominated by inertia and for over-damped suspensions. Concomitantly, the length scale characterizing velocity correlations appears to diverge at jamming. Here we introduce a theoretical framework that proposes a tentative, but potentially complete, scaling description of stationary flows. Our analysis, which focuses on frictionless particles, applies both to suspensions and inertial flows of hard particles. We compare our predictions with the empirical literature, as well as with novel numerical data. Overall, we find a very good agreement between theory and observations, except for frictional inertial flows whose scaling properties clearly differ from frictionless systems. For overdamped flows, more observations are needed to decide if friction is a relevant perturbation. Our analysis makes several new predictions on microscopic dynamical quantities that should be accessible experimentally.

On variational arguments for vibrational modes near jamming

Amorphous solids tend to present an abundance of soft elastic modes, which diminish their transport properties, generate heterogeneities in their elastic response, and affect non-linear processes like thermal activation of plasticity. This is especially true in packings of particles near their jamming transition, for which effective medium theory and variational arguments can both predict the density of vibrational modes. However, recent numerics support that one hypothesis of the variational argument does not hold. We provide a novel variational argument which overcomes this problem, and correctly predicts the scaling properties of soft modes near the jamming transition. Soft modes are shown to be related to the response to a local strain in more connected networks, and to be characterized by a volume

Theory of the jamming transition at finite temperature.

1/\delta z

Laws of granular solids: geometry and topology.

, where

Effect of friction on dense suspension flows of hard particles

\delta z

Phase diagram for inertial granular flows.

is the excess coordination above the Maxwell threshold. These predictions are verified numerically.

On the granular stress-geometry equation

A theory for the microscopic structure and the vibrational properties of soft sphere glass at finite temperature is presented. With an effective potential, derived here, the phase diagram and vibrational properties are worked out around the Maxwell critical point at zero temperature T and pressure p. Variational arguments and effective medium theory identically predict a non-trivial temperature scale T(∗) ∼ p((2-a)/(1-a)) with a ≈ 0.17 such that low-energy vibrational properties are hard-sphere like for T ≳ T(∗) and zero-temperature soft-sphere like otherwise. However, due to crossovers in the equation of state relating T, p, and the packing fraction ϕ, these two regimes lead to four regions where scaling behaviors differ when expressed in terms of T and ϕ. Scaling predictions are presented for the mean-squared displacement, characteristic frequency, shear modulus, and characteristic elastic length in all regions of the phase diagram.

Friction law and hysteresis in granular materials

In a granular solid, mechanical equilibrium requires a delicate balance of forces at the disordered grain scale. To understand how macroscopic rigidity can emerge in this amorphous solid, it is crucial that we understand how Newtons laws pass from the disordered grain scale to the laboratory scale. In this work, we introduce an exact discrete calculus, in which Newtons laws appear as differential relations at the scale of a single grain. Using this calculus, we introduce gauge variables that describe identically force- and torque-balanced configurations. In a first, intrinsic formulation, we use the topology of the contact network, but not its geometry. In a second, extrinsic formulation, we introduce geometry with the Delaunay triangulation. These formulations show, with exact methods, how topology and geometry in a disordered medium are related by constraints. In particular, we derive Airys expression for a divergence-free, symmetric stress tensor in two and three dimensions.