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We use simulations to probe the flow properties of dense two-dimensional magnetorheological fluids. Prior results from both experiments and simulations report that the shear stress $\sigma$ scales with strain rate $\dot \gamma$ as $\sigma \sim \dot \gamma^{1-\Delta}$, with values of the exponent ranging between $2/3 <\Delta \le 1$. However it remains unclear what properties of the system select the value of $\Delta$, and in particular under what conditions the system displays a yield stress ($\Delta = 1$). To address these questions, we perform simulations of a minimalistic model system in which particles interact via long ranged magnetic dipole forces, finite ranged elastic repulsion, and viscous damping. We find a surprising dependence of the apparent exponent $\Delta$ on the form of the viscous force law. For experimentally relevant values of the volume fraction $\phi$ and the dimensionless Mason number (which quantifies the competition between viscous and magnetic stresses), models using a Stokes-like drag force show $\Delta \approx 0.75$ and no apparent yield stress. When dissipation occurs at the contact, however, a clear yield stress plateau is evident in the steady state flow curves. In either case, increasing $\phi$ towards the jamming transition suffices to induce a yield stress. We relate these qualitatively distinct flow curves to clustering mechanisms at the particle scale. For Stokes-like drag, the system builds up anisotropic, chain-like clusters as the Mason number tends to zero (vanishing strain rate and/or high field strength). For contact damping, by contrast, there is a second clustering mechanism due to inelastic collisions., Comment: 16 pages, 7 figures (including appendices)

We report on numerical simulations of simple models of athermal, bidisperse, soft-core, massive disks in two dimensions, as a function of packing fraction $\phi$, inelasticity of collisions as measured by a parameter $Q$, and applied uniform shear strain rate $\dot\gamma$. Our particles have contact interactions consisting of normally directed elastic repulsion and viscous dissipation, as well as tangentially directed viscous dissipation, but no inter-particle Coulombic friction. Mapping the phase diagram in the $(\phi,Q)$ plane for small $\dot\gamma$, we find a sharp first-order rheological phase transition from a region with Bagnoldian rheology to a region with Newtonian rheology, and show that the system is always Newtonian at jamming. We consider the rotational motion of particles and demonstrate the crucial importance that the coupling between rotational and translational degrees of freedom has on the phase structure at small $Q$ (strongly inelastic collisions). At small $Q$ we show that, upon increasing $\dot\gamma$, the sharp Bagnoldian-to-Newtonian transition becomes a coexistence region of finite width in the $(\phi,\dot\gamma)$ plane, with coexisting Bagnoldian and Newtonian shear bands. Crossing this coexistence region by increasing $\dot\gamma$ at fixed $\phi$, we find that discontinuous shear thickening can result if $\dot\gamma$ is varied too rapidly for the system to relax to the shear-banded steady state corresponding to the instantaneous value of $\dot\gamma$., Comment: 27 pages, 36 figures, new version corresponds to published version

We carry out constant volume simulations of steady-state, shear driven flow in a simple model of athermal, bidisperse, soft-core, frictionless disks in two dimensions, using a dissipation law that gives rise to Bagnoldian rheology. Focusing on the small strain rate limit, we map out the rheological behavior as a function of particle packing fraction $\phi$ and a parameter $Q$ that measures the elasticity of binary particle collisions. We find a $Q^*(\phi)$ that marks the clear crossover from a region characteristic of strongly inelastic collisions, $QQ^*$, and give evidence that $Q^*(\phi)$ diverges as $\phi\to\phi_J$, the shear driven jamming transition. We thus conclude that the jamming transition at any value of $Q$ behaves the same as the strongly inelastic case, provide one is sufficiently close to $\phi_J$. We further characterize the differing nature of collisions in the strongly inelastic vs weakly inelastic regions, and recast our results into the constituent equation form commonly used in discussions of hard granular matter., Comment: 14 pages, 17 figures, revised to published version; dissipative part of stress computed with higher accuracy, anisotropy in pressure considered in new Appendix

We demonstrate that the elasticity of jammed solids is nonlocal. By forcing frictionless soft sphere packings at varying wavelength, we directly access their transverse and longitudinal compliances without resorting to curve fitting. The observed wavelength dependence of the compliances is incompatible with classical (local) elasticity, and hence quantifies the amplitude of nonlocal effects. Three distinct length scales, two of which diverge, control the amplitude of both nonlocal effects and fluctuations about the mean response. Our results identify new, more accurate constitutive relations for weakly jammed solids, including emulsions, foams, and granulates., Comment: 5 pages, 4 figures

The shear response of soft solids can be modeled with linear elasticity, provided the forcing is slow and weak. Both of these approximations must break down when the material loses rigidity, such as in foams and emulsions at their (un)jamming point -- suggesting that the window of linear elastic response near jamming is exceedingly narrow. Yet precisely when and how this breakdown occurs remains unclear. To answer these questions, we perform computer simulations of stress relaxation and shear startup experiments in athermal soft sphere packings, the canonical model for jamming. By systematically varying the strain amplitude, strain rate, distance to jamming, and system size, we identify characteristic strain and time scales that quantify how and when the window of linear elasticity closes, and relate these scales to changes in the microscopic contact network. Our findings indicate that the mechanical response of jammed solids are generically nonlinear and rate-dependent on experimentally accessible strain and time scales., Comment: 10 pages, 9 figures

We carry out constant volume simulations of steady-state, shear driven, rheology in a simple model of bidisperse, soft-core, frictionless disks in two dimensions, using a dissipation law that gives rise to Bagnoldian rheology. We carry out a detailed critical scaling analysis of our resulting data for pressure $p$ and shear stress $\sigma$, in order to determine the critical exponent $\beta$ that describes the algebraic divergence of the Bagnold transport coefficients, as the jamming transition is approached from below. We show that it is necessary, for the strain rates considered in this work, to consider the leading correction-to-scaling term in order to achieve a self-consistent analysis of our data. Our resulting value $\beta\approx 5.0\pm 0.4$ is clearly larger than the theoretical prediction by Otsuki and Hayakawa, and is consistent with earlier numerical results by Peyneau and Roux, and recent theoretical predictions by DeGiuli et al. We have also considered the macroscopic friction $\mu\equiv \sigma/p$ and similarly find results consistent with Peyneau and Roux, and with DeGiuli et al. Our results confirm that the shear driven jamming transition in Bagnoldian systems is well described by a critical scaling theory (as was found previously for Newtonian systems), and we relate this scaling theory to the phenomenological constituent laws for dilatancy and friction., Comment: 20 pages, 21 figures; revised manuscript according to published version

In a recent paper [S. Mandal et al., Phys. Rev. E 88, 022129 (2013)] the nature of spatial correlations of plasticity in hard sphere glasses was addressed both via computer simulations and in experiments. It was found that the experimentally obtained correlations obey a power law whereas the correlations from simulations are better fitted by an exponential decay. We here provide direct evidence--- via simulations of a hard sphere glass in 2D---that this discrepancy is a consequence of the finite system size in the 3D simulations. By extending the study to a 2D soft disk model at zero temperature, the robustness of the power-law decay in sheared amorphous solids is underlined. Deviations from a power law occur when either reducing the packing fraction towards the supercooled regime in the case of hard spheres or changing the dissipation mechanism from contact dissipation to a mean-field type drag for the case of soft disks., Comment: 5 pages, 5 figures

We investigate the criticality of the jamming transition for overdamped shear-driven frictionless disks in two dimensions for two different models of energy dissipation: (i) Durian's bubble model with dissipation proportional to the velocity difference of particles in contact, and (ii) Durian's "mean-field" approximation to (i), with dissipation due to the velocity difference between the particle and the average uniform shear flow velocity. By considering velocity correlations, finite-size behavior of pressure, and the pressure analog of viscosity, we argue that these two models share the same critical behavior., Comment: 4 pages, 4 figures. Also includes Supplemental material: 3 pages, 5 figures

We use numerical simulations to investigate the effect of different dissipative models on the shearing rheology of massive soft-core frictionless disks in two dimensions. We show that the presence of Newtonian (overdamped) vs Bagnoldian (inertial) rheology is related to the formation of large connected clusters of disks, and that sharp transitions may exist between the two as system parameters vary. In the limit of strongly inelastic collisions, we find that rheological curves collapse to a well-defined limit when plotted against an appropriate dimensionless strain rate., Comment: 6 pages, 5 figures, revised to published version

We numerically study the distributions of global pressure that are found in ensembles of statically jammed and quasistatically sheared systems of bidisperse, frictionless, disks at fixed packing fraction $\phi$ in two dimensions. We use these distributions to address the question of how pressure increases as $\phi$ increases above the jamming point $\phi_J$, $p\sim |\phi - \phi_J|^y$. For statically jammed ensembles, our results are consistent with the exponent $y$ being simply related to the power law of the interparticle soft-core interaction. For sheared systems, however, the value of $y$ is consistent with a non-trivial value, as found previously in rheological simulations., Comment: 6 pages, 6 figures

We carry out a finite size scaling analysis of the jamming transition in frictionless bi-disperse soft core disks in two dimensions. We consider two different jamming protocols: (i) quench from random initial positions, and (ii) quasistatic shearing. By considering the fraction of jammed states as a function of packing fraction for systems with different numbers of particles, we determine the spatial correlation length critical exponent $\nu\approx 1$, and show that corrections to scaling are crucial for analyzing the data. We show that earlier numerical results yielding $\nu<1$ are due to the improper neglect of these corrections., Comment: 5 pages, 4 figures -- slightly revised version as accepted for Phys. Rev. E Rapid Communications

The jamming of bi-disperse soft core disks is considered, using a variety of different protocols to produce the jammed state. In agreement with other works, we find that cooling and compression can lead to a broad range of jamming packing fractions $\phi_J$, depending on cooling rate and initial configuration; the larger the degree of big particle clustering in the initial configuration, the larger will be the value of $\phi_J$. In contrast, we find that shearing disrupts particle clustering, leading to a much narrower range of $\phi_J$ as the shear strain rate varies. In the limit of vanishingly small shear strain rate, we find a unique non-trivial value for the jamming density that is independent of the initial system configuration. We conclude that shear driven jamming is a unique and well defined critical point in the space of shear driven steady states. We clarify the relation between glassy behavior, rigidity and jamming in such systems and relate our results to recent experiments., Comment: 10 pages, 11 figures, significantly expanded version as accepted for publication in PRE