Dense granular materials display a complicated set of flow properties, which differentiate them from ordinary fluids. Despite their ubiquity, no model has been developed that captures or predicts the complexities of granular flow, posing an obstacle in industrial and geophysical applications. Here we propose a 3D constitutive model for well-developed, dense granular flows aimed at filling this need. The key ingredient of the theory is a grain-size-dependent nonlocal rheology-inspired by efforts for emulsions-in which flow at a point is affected by the local stress as well as the flow in neighboring material. The microscopic physical basis for this approach borrows from recent principles in soft glassy rheology. The sizedependence is captured using a single material parameter, and the resulting model is able to quantitatively describe dense granular flows in an array of different geometries. Of particular importance, it passes the stringent test of capturing all aspects of the highly nontrivial flows observed in split-bottom cells-a geometry that has resisted modeling efforts for nearly a decade. A key benefit of the model is its simple-to-implement and highly predictive final form, as needed for many real-world applications. G ranular materials are ubiquitous in day-to-day life, as well as central to important industries, such as geotechnical, energy, pharmaceutical, and food processing. In fact, granular matter is second only to water as the most handled industrial material (1), but unlike water, dense granular flows are substantially more complex (2-10). In particular, slowly flowing granular media form clear, experimentally robust features, most notably, shear bands, which can have a variety of possible widths and decay nontrivially into the surrounding quasi-rigid material. However, these behaviors remain poorly understood and have not been rationalized with a universal continuum model, posing a costly problem in industry. Quantitatively describing and predicting dense, well-developed granular flows with a constitutive model that may be applied in arbitrary configurations remains a major open challenge.For many years, mechanicians and materials engineers have approached granular materials modeling from a soil mechanics perspective, grounded in the principles of continuum solid mechanics, invoking various yield criteria and plastic flow relations (11, 12). In contrast, over the past two decades, a resurgence of interest in granular media has arisen among physicists, primarily drawing upon statistical and fluid dynamical approaches (13,14). More recently, drawing upon both schools of thought, granular rheologists have made progress combining a fluid-like, ratedependent flow approach with an appropriate yield criterion. Backed by numerous experiments and a coherent dimensional argument, the key result is the dimensionless relation μ = μ(I), consistent with the seminal work of Bagnold (15), which has become a well-regarded basis for modeling well-developed granular flows in simple shear (9, 16), where μ = τ/P fo...

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Flows of granular media down a rough inclined plane demonstrate a number of nonlocal phenomena. We apply the recently proposed nonlocal granular fluidity model to this geometry and find that the model captures many of these effects. Utilizing the model's dynamical form, we obtain a formula for the critical stopping height of a layer of grains on an inclined surface. Using an existing parameter calibration for glass beads, the theoretical result compares quantitatively to existing experimental data for glass beads. This provides a stringent test of the model, whose previous validations focused on driven steady-flow problems. For layers thicker than the stopping height, the theoretical flow profiles display a thickness-dependent shape whose features are in agreement with previous discrete particle simulations. We also address the issue of the Froude number of the flows, which has been shown experimentally to collapse as a function of the ratio of layer thickness to stopping height. While the collapse is not obvious, two explanations emerge leading to a revisiting of the history of inertial rheology, which the nonlocal model references for its homogeneous flow response.

The destructive growth and collapse of cavitation bubbles are used for therapeutic purposes in focused ultrasound procedures and can contribute to tissue damage in traumatic injuries. Histotripsy is a focused ultrasound procedure that relies on controlled cavitation to homogenize soft tissue. Experimental studies of histotripsy cavitation have shown that the extent of ablation in different tissues depends on tissue mechanical properties and waveform parameters. Variable tissue susceptibility to the large stresses, strains, and strain rates developed by cavitation bubbles has been suggested as a basis for localized liver tumor treatments that spare large vessels and bile ducts. However, field quantities developed within microns of cavitation bubbles are too localized and transient to measure in experiments. Previous numerical studies have attempted to circumvent this challenge but made limited use of realistic tissue property data. In this study, numerical simulations are used to calculate stress, strain, and strain rate fields produced by bubble oscillation under histotripsy forcing in a variety of tissues with literature-sourced viscoelastic and acoustic properties. Strain field calculations are then used to predict a theoretical damage radius using tissue ultimate strain data. Simulation results support the hypothesis that differential tissue

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Experimental observations of the growth and collapse of acoustically and laser-nucleated single bubbles in water and agarose gels of varying stiffness are presented. The maximum radii of generated bubbles decreased as the stiffness of the media increased for both nucleation modalities, but the maximum radii of laser-nucleated bubbles decreased more rapidly than acoustically nucleated bubbles as the gel stiffness increased. For water and low stiffness gels, the collapse times were well predicted by a Rayleigh cavity, but bubbles collapsed faster than predicted in the higher stiffness gels. The growth and collapse phases occurred symmetrically (in time) about the maximum radius in water but not in gels, where the duration of the growth phase decreased more than the collapse phase as gel stiffness increased. Numerical simulations of the bubble dynamics in viscoelastic media showed varying degrees of success in accurately predicting the observations.

Recently, a new nonlocal granular rheology was successfully used to predict steady granular flows, including grain-size-dependent shear features, in a wide variety of flow configurations, including all variations of the split-bottom cell. A related problem in granular flow is that of mechanicallyinduced creep, in which shear deformation in one region of a granular medium fluidizes its entirety, including regions far from the sheared zone, effectively erasing the yield condition everywhere. This enables creep deformation when a force is applied in the nominally quiescent region through an intruder such as a cylindrical or spherical probe. We show that the nonlocal fluidity model is capable of capturing this phenomenology. Specifically, we explore creep of a circular intruder in a two-dimensional annular Couette cell and show that the model captures all salient features observed in experiments, including both the rate-independent nature of creep for sufficiently slow driving rates and the faster-than-linear increase in the creep speed with the force applied to the intruder.Cooperativity is a hallmark of quasi-static, dense granular deformation. At a microscopic level, deformation in granular materials takes place through the rearrangement of clusters of grains. These rearrangement events lead to long-range fluctuations, or agitations, that influence the behavior of nearby clusters, leading to macroscopic manifestations of cooperativity, which are quite varied. Most familiarly, the length-scales associated with velocity fields in dense granular flows depend crucially upon the grain size [1], with grain-size dependent shear band widths being observed in many geometries [2][3][4][5][6][7][8]. Other manifestations of cooperativity include the dependence of volumetric outflow rate on grain size in drainage flows [9,10] and the so-called H stop -effect, in which thin granular layers require greater tilt to flow down an inclined surface [11,12]. A more recently-observed example of cooperativity in dense granular flows is that of mechanicallyinduced creep [13][14][15], understood as follows. When an intruder, such as a sphere, rod, or vane, is placed in a dense granular material, one must apply a force to the intruder that exceeds a critical value in order to move it through the granular media. However, shear deformation far from the intruder enables it to move, or creep, through the granular media for any non-zero value of the applied force -even when this force is less than the critical value. That is to say, flow anywhere in a granular media erases the yield condition everywhere! These cooperative phenomena provide stringent tests for a continuum model of granular flow. Local approaches to continuum modeling of granular materials, such as granular rheology [16][17][18] or soil mechanics [19,20], which relate the stress at a point to the local strain, strain-rate, or locally evolved state variables, are not equipped to address size-dependent manifestations of cooperativity. Recently, we proposed a nonlocal rheology for ...