Additive rheology of complex granular flows

Origins of complexity in the rheology of Soft Earth suspensions

When wet soil becomes fully saturated by intense rainfall, or is shaken by an earthquake, it may fluidize catastrophically. Sand-rich slurries are treated as granular suspensions, where the failure is related to an unjamming transition, and friction is controlled by particle concentration and pore pressure. Mud flows are modeled as gels, where yielding and shear-thinning behaviors arise from inter-particle attraction and clustering. Here we show that the full range of complex flow behaviors previously reported for natural debris flows can be reproduced with three ingredients: water, silica sand, and kaolin clay. Going from sand-rich to clay-rich suspensions, we observe continuous transition from brittle (Coulomb-like) to ductile (plastic) yielding. We propose a general constitutive relation for soil suspensions, with a particle rearrangement time that is controlled by yield stress and jamming distance. Our experimental results are supported by models for amorphous solids, suggesting that the paradigm of non-equilibrium phase transitions can help us understand and predict the complex behaviors of Soft Earth suspensions.

Unbiased construction of constitutive relations for soft materials from experiments via rheology-informed neural networks

The ability to concisely describe the dynamical behavior of soft materials through closed-form constitutive relations holds the key to accelerated and informed design of materials and processes. The conventional approach is to construct constitutive relations through simplifying assumptions and approximating the time- and rate-dependent stress response of a complex fluid to an imposed deformation. While traditional frameworks have been foundational to our current understanding of soft materials, they often face a twofold existential limitation: i) Constructed on ideal and generalized assumptions, precise recovery of material-specific details is usually serendipitous, if possible, and ii) inherent biases that are involved by making those assumptions commonly come at the cost of new physical insight. This work introduces an approach by leveraging recent advances in scientific machine learning methodologies to discover the governing constitutive equation from experimental data for complex fluids. Our rheology-informed neural network framework is found capable of learning the hidden rheology of a complex fluid through a limited number of experiments. This is followed by construction of an unbiased material-specific constitutive relation that accurately describes a wide range of bulk dynamical behavior of the material. While extremely efficient in closed-form model discovery for a real-world complex system, the model also provides insight into the underpinning physics of the material.

Collapse dynamics and deposition morphology of low-viscocohesive granular columns on a rough horizontal surface

Using the three-dimensional discrete element method, we numerically investigate the collapse dynamics and deposition morphology of low-viscocohesive granular columns on a rough-horizontal plane by systematically varying a broad range of values of the initial column aspect ratio, cohesive stress, and liquid viscosity. The results show that the kinetic energy, half runout time, and runout distance increase with increasing the initial column aspect ratio but decrease with increasing the cohesive and viscous effects of the binding liquid, while the toe angle and deposit height decrease with increasing the aspect ratio and increase with increasing cohesive stress and liquid viscosity. Remarkably, by defining a dimensionless scaling number that incorporates the Bond number and initial column aspect ratio, this allows us to nicely describe the kinetic energy, half runout time, deposition height, runout distance, and toe angle. These unified descriptions may provide insights into the physical properties of the collapse dynamics and deposition morphology of low-viscocohesive granular columns, leading to good explanations of the complex properties of natural disaster events.

The combined effect of cohesion and finite size on the collapse of wet granular columns

The collapse of low-saturation liquid-containing granular materials is prevalent in nature and industrial processes, and understanding the associated transient dynamics is extremely important for exploring such complex flow processes. In this paper, the collapse of a finite-size wet granular column is systematically studied and the determinants affecting its dynamics are analyzed based on the discrete element model for wet particles and the corresponding small-scale experiments. With the aid of parametric analysis, the dimensionless cohesion parameter containing the system size and grain-scale bond number is proposed, and its relevance in characterizing column stability and collapse dynamics of wet granular materials is further confirmed. For the collapse of wet granular columns with a fixed aspect ratio, the initial height contained in the cohesion parameter is verified to be a manifestation of the finite size effect, which is present in a wet granular collapse and is coupled with the cohesive effect. Such a coupling effect is taken into account in our proposed scaling laws that can be applied to uniformly describe the deposit morphology of wet granular columns after collapse.

Secondary flow in ensembles of nonconvex granular particles under shear

Studies of granular materials, both theoretical and experimental, are often restricted to convex grain shapes. We demonstrate that a nonconvex grain shape can lead to a qualitatively novel macroscopic dynamics. Spatial crosses (hexapods) are continuously sheared in a split-bottom container. Thereby, they develop a secondary flow profile that is completely opposite to that of rod-shaped or lentil-shaped convex grains in the same geometry. The crosses at the surface migrate towards the rotation center and sink there mimicking a "reverse Weissenberg effect." The observed surface flow field suggests the existence of a radial outward flow in the depth of the granular bed, thus, forming a convection cell. This flow field is connected with a dimple formed in the rotation center. The effect is strongly dependent on the particle geometry and the height of the granular bed.

Rheology of debris flow materials is controlled by the distance from jamming

Debris flows are fast-flowing and dangerous slurries of soil and water that often form when intense rainfall soaks hillsides burned by wildfire. As climate change intensifies this hazard, models capable of predicting failure and flow behaviors are needed. Here we capitalize on recent progress in the physics of dense suspensions, to determine how the physical and chemical composition of natural hillslope soils controls the viscosity and yield stress of debris flows. We show how a simple flow model—previously developed for idealized suspensions—can be extended to highly heterogeneous, natural debris flow materials. This model reconciles previously contradictory observations and could help to improve computer models that assess the hazard potential of debris flows in the field.

Debris flows are dense and fast-moving complex suspensions of soil and water that threaten lives and infrastructure. Assessing the hazard potential of debris flows requires predicting yield and flow behavior. Reported measurements of rheology for debris flow slurries are highly variable and sometimes contradictory due to heterogeneity in particle composition and volume fraction (ϕ) and also inconsistent measurement methods. Here we examine the composition and flow behavior of source materials that formed the postwildfire debris flows in Montecito, CA, in 2018, for a wide range of ϕ that encapsulates debris flow formation by overland flow. We find that shear viscosity and yield stress are controlled by the distance from jamming, Δϕ=ϕm−ϕ, where the jamming fraction ϕm is a material parameter that depends on grain size polydispersity and friction. By rescaling shear and viscous stresses to account for these effects, the data collapse onto a simple nondimensional flow curve indicative of a Bingham plastic (viscoplastic) fluid. Given the highly nonlinear dependence of rheology on Δϕ, our findings suggest that determining the jamming fraction for natural materials will significantly improve flow models for geophysical suspensions such as hyperconcentrated flows and debris flows.

Bulk modulus along jamming transition lines of bidisperse granular packings

We present three-dimensional discrete element method simulations of bidisperse granular packings to investigate their jamming densities ϕ_{J} and dimensionless bulk moduli K as functions of the size ratio δ and the concentration of small particles X_{S}. We determine the partial and total bulk moduli for packings near their jamming densities, including a second transition that occurs for sufficiently small δ and X_{S} when the system is compressed beyond its first jamming transition. While the first transition is sharp, exclusively with large-large contacts, the second is rather smooth, carried by small-large interactions at densities much higher than the monodisperse random packing baseline, ϕ_{J}^{mono}≈0.64. When only nonrattlers are considered, all the effective transition densities are reduced, and the density of the second transition emerges rather close to the reduced baseline, ϕ[over ̃]_{J}^{mono}≈0.61, due to its smooth nature. At size ratios δ≤0.22 a concentration X_{S}^{*} divides the diagram-either with most small particles nonjammed or jammed jointly with large ones. For X_{S}<X_{S}^{*}, the modulus K displays different behaviors at first and second jamming transitions. Along the second transition, K rises relative to the values found at the first transition; however, is still small compared to K at X_{S}^{*}. Explicitly, for our smallest δ=0.15, the discontinuous jump in K as a function of X_{S} is obtained at X_{S}^{*}≈0.21 and coincides with the maximum jamming density where both particle species mix most efficiently. Our results will allow tuning or switching the bulk modulus K or other properties, such as the wave speed, by choosing specific sizes and concentrations based on a better understanding of whether small particles contribute to the jammed structure or not, and how the micromechanical structure behaves at either transition.

Granular effects in sea ice rheology in the marginal ice zone

Sea ice in the marginal ice zone (MIZ) consists of relatively small floes with a wide size span. In response to oceanic and atmospheric forcing, it behaves as an approximately two-dimensional, highly polydisperse granular material. The established viscous-plastic rheologies used in continuum sea ice models are not suitable for the MIZ; the collisional rheology, in which sea ice is treated as a granular gas, captures only one aspect of the granular behaviour, typical for a narrow range of conditions when dynamics is dominated by binary floe collisions. This paper reviews rheology models and concepts from research on granular materials relevant for MIZ dynamics (average stress as a result of 'microscopic' interactions of grains; [Formula: see text] and collisional rheologies). Idealized discrete-element simulations are used to illustrate granular effects and strong influence of the floe size distribution on strain-stress relationships in sheared sea ice, demonstrating the need for an MIZ rheology model capturing the whole range of 'regimes', from quasi-static/dense flow in the inner MIZ to the inertial flow in the outer MIZ. This article is part of the theme issue 'Theory, modelling and observations of marginal ice zone dynamics: multidisciplinary perspectives and outlooks'.

Viscous to Inertial Transition in Dense Granular Suspension

Granular suspensions present a transition from a Newtonian rheology in the Stokes limit to a Bagnoldian rheology when inertia is increased. A custom rheometer that can be run in a pressure- or a volume-imposed mode is used to examine this transition in the dense regime close to jamming. By varying systematically the interstitial fluid, shear rate, and packing fraction in volume-imposed measurements, we show that the transition takes place at a Stokes number of 10 independent of the packing fraction. Using pressure-imposed rheometry, we investigate whether the inertial and viscous regimes can be unified as a function of a single dimensionless number based on stress additivity.

Unified penetration depth of low-velocity intruders into granular packings

Penetration of intruders into granular packings is well described by separately considering the dry or wet case of granular environments in previous experiments and simulations; however, the unified description of such penetration depth in these two granular media remains elusive due to lacking clear explanations about its origins. Based on three-dimensional discrete element method simulations, we introduce a power-law fitting form of the final penetration depth of a spherical intruder with low velocity vertically penetrating into dry and wet granular packings, excellently expressed on a master curve as a power-law function of a dimensionless impact number that is defined as the square root of the ratio between the inertial stress of the intruder and the linear combination of the mean gravitational stress and the cohesive stress exerted on each grain in the packings, as a remarkable extension of the inertial number in dry granular flows. This scaling robustly provides physical insights inherent in the unified description of the material properties of granular packings and the impactor penetration conditions on the final penetration depth in the impact tests, providing evidence of impact properties in different disciplines and applications in science and engineering.

Reorientation of Suspended Ceramic Particles in Robocasted Green Filaments during Drying

This work considers the fabrication of ceramic parts with the help of an additive manufacturing process, robocasting, in which a paste with suspended particles is robotically extruded. Within the final part, the material properties depend on the orientation of the particles. A prediction of the particle orientation is challenging as the part usually undergoes multiple processing steps with varying contributions to the orientation. As the main contribution to the final particle orientation arises from the extrusion process, many corresponding prediction models have been suggested. Robocasting involves, however, further processing steps that are less studied as they have a smaller influence on the orientation. One of the processing steps is drying by natural convection, which follows directly after the extrusion process. A quantification of the reorientation that occurs during drying is mostly unknown and usually neglected in the models. Therefore, we studied the amount of reorientation of suspended particles in robocasted green filaments during drying in detail. For our study, we applied the discrete element method, as it meets various requirements: The exact particle geometry can be resolved precisely; particle–particle interactions can be described; the paste composition is reproduced exactly; the initial particle orientation can be set in accordance with the prediction from the analytical models for the extrusion part; macroscopic force laws exist to represent capillary forces due to the remaining fluid phase that remains during drying. From our study, we concluded that the magnitude of particle reorientation during drying is small compared to the orientation occurring during the extrusion process itself. Consequently, reorientation during drying might further be neglected within analytical orientation prediction models.

Particle-based coarse-grained approach for simulating dry powder inhaler

Drug delivery via dry powder inhaler (DPI) is a complex process affected by multiple factors involving gas and particles. The performance of a carrier-based formulation depends on the release of active pharmaceutical ingredient (API) particles, typically characterized by fine particle fraction (FPF) and dispersion fraction (DF). Computational Fluid Dynamics coupled with Discrete Element Method (CFD-DEM) can capture relevant gas and particle interactions but is computationally expensive, especially when tracking all carrier and API particles. This study assessed the efficacy of two coarse-grained CFD-DEM approaches, the Discrete Parcel Method and the representative particle approach, through highly-resolved CFD-DEM simulations. The representative particle approach simulates all carrier particles and a subset of API particles, whereas the Discrete Parcel Method tracks parcels representing a specified number of carrier or API particles. Both approaches are viable for a small carrier-API size ratio which requires modest degrees of coarse-graining, but the Discrete Parcel Method showed limitations for a large carrier-API size ratio. The representative particle approach can approximate CFD-DEM results with reasonable accuracies when simulations include at least 10 representative API particles per carrier. Using the representative particle approach, we probed powder characteristics that could affect FPF and DF in a model problem and correlated these fractions with the maximum carrier-API cohesive force per unit mass of API particles.

Scaling behavior of the tensile strength of viscocohesive granular aggregates

We numerically analyze the tensile strength of a single wet agglomerate modeled as a viscocohesive aggregate impacting a flat surface by using the discrete-element simulations. The viscocohesive agglomerate composed of primary spherical particles with the inclusion of the interstitial liquid in the form of the capillary bridges characterized by the cohesive and viscous forces between particles is extracted from a cuboidal sample of granular materials by applying a spherical probe. The tensile strength is measured from the impact test of a wet agglomerate by systematically varying different values of the surface tension of the interstitial liquid, the liquid viscosity, and the impact speed. We show that the tensile stress increases immediately when the collision occurs between the agglomerate and the flat surface. The peak of the tensile stress obtained after the collision, then decreases smoothly with increasing the particle movement. The maximum tensile stress is defined to be the tensile strength of such agglomerate. It is remarkable that the normalized tensile strength of such agglomerate can be well described as a function of a dimensionless impact number that incorporates the capillary number and Stokes number (calculated from the surface tension and the viscosity of the liquid and the impact rate of the agglomerate), thus providing the confirmation for the unified representation of the liquid properties and the impact rate of wet granular media.

Viscosity of cohesive granular flows

Cohesive granular materials such as wet sand, snow, and powders can flow like a viscous liquid. However, the elementary mechanisms of momentum transport in such athermal particulate fluids are elusive. As a result, existing models for cohesive granular viscosity remain phenomenological and debated. Here we use discrete element simulations of plane shear flows to measure the viscosity of cohesive granular materials, while tuning the intensity of inter-particle adhesion. We establish that two adhesion-related, dimensionless numbers control their viscosity. These numbers compare the force and energy required to break a bond to the characteristic stress and kinetic energy in the flow. This progresses the commonly accepted view that only one dimensionless number could control the effect of adhesion. The resulting scaling law captures strong, non-Newtonian variations in viscosity, unifying several existing viscosity models. We then directly link these variations in viscosity to adhesion-induced modifications in the flow micro-structure and contact network. This analysis reveals the existence of two modes of momentum transport, involving either grain micro-acceleration or balanced contact forces, and shows that adhesion only affects the latter. This advances our understanding of rheological models for granular materials and other soft materials such as emulsions and suspensions, which may also involve inter-particle adhesive forces.

Time scales and rheology of visco-cohesive granular flows

In the presence of viscous and cohesive interactions between particles, a granular flow is governed by several characteristic time and stress scales that determine its rheological properties (shear stress, packing fraction, effective viscosities). In this paper, we revisit and extend the scaling arguments used previously for dry cohesionless granular flows and suspensions. We show that the rheology can be in principle described by a single dimensionless control parameter that includes all characteristic times. We also briefly present simulation results for 2D sheared suspensions and 3D wet granular flows where the effective friction coefficient and packing fraction are consistently described as functions of this unique control parameter.

Shear-induced diffusion: the role of granular clusters

This paper is concerned with the physical mechanisms controlling shear-induced diffusion in dense granular flows. The starting point is that of the granular random walk occurring in diluted granular flows, which underpins Bagnold’s scaling relating the coefficient of self-diffusion to the grain size and shear rate. By means of DEM simulations of plane shear flows, we measure some deviations from this scaling in dense granular flows with and without contact adhesion. We propose to relate these deviations to the development of correlated motion of grains in these flows, which impacts the magnitude of grain velocity fluctuations and their time persistence.

Flows of cohesive granular media

Cohesive granular media have broad applications in industries. However, our understanding of their flow behavior is still limited compared to dry granular media, although rich knowledge about their static and plastic properties has been gained. In this paper, we provide some insights into the flow behavior of cohesive granular media from our recent numerical studies using an inclined plane and a plane shear cell. We evidence that the cohesive nature of flows is significantly affected by material properties of the particles like stiffness and inelasticity in addition to the inter-particle adhesion and introduce the concept of “effective” adhesion, which incorporates the effects of these three variables. We propose constitutive relations involving dimensionless inertial number and “effective” cohesion number, based on the “effective” adhesion to describe the rheology. We also show that increasing adhesion increases the hysteresis in granular media, evidencing the existence of a prominent shear weakening branch in the friction coefficient versus inertial number rheological curve. Moreover, we reveal that this increasing hysteresis gives rise to the increasing occurrence of shear banding instability, pointing to the increasing possibility of jamming in cohesive granular media. Finally, we present a promising experimental approach to investigate the flow behavior of cohesive granular materials, based on a simple method of preparing a long time stable medium with a controlled adhesion between particles.

The role of inter-particle friction on rheology and texture of wet granular flows

In order to get insight into the rheology and texture of rough unsaturated granular flows, we study the effects of the inter-particle friction coefficient on the macroscopic attributes and the texture variables of steady-state shearing flow of wet granular materials by relying on three-dimensional (3D) particle dynamics simulations. The macroscopic attributes are characterized by the macroscopic friction coefficient, macroscopic cohesion, and the packing fraction. The microstructural variables are characterized by the fabric and force anisotropies, the coordination number, and the stress transmission ratio. We show that the macroscopic observables behave as a function of the inertial number as a dry case for different values of the inter-particle friction coefficient. In particular, the macroscopic friction coefficient increases and the packing fraction decreases rapidly for small values of the friction coefficient, then they almost reach plateaus for higher values of the friction coefficient. Interestingly, all the macroscopic observables nicely behave as a function of the small values of the friction coefficient. Similarly, we also observe these characteristics for the fabric and force anisotropies and the coordination number as well as the stress transmission ratio which reflects the intermediate relationship between the microstructure and the mechanical behavior of such flows.