A hierarchy of granular continuum models: Why flowing grains are both simple and complex

EPJ Web of Conferences 140 , 01007 (2017 ) DOI: 10.1051/epjconf/201714001007 Powders & Grains 2017 A hierarchy of granular continuum models: Why flowing grains are both simple and complex Ken Kamrin1 , 1 Massachusetts Institute of Technology, Department of Mechanical Engineering, Cambridge, MA 02139 Abstract. Granular materials have a strange propensity to behave as either a complex media or a simple media depending on the precise question being asked. This review paper offers a summary of granular flow rheologies for well-developed or steady-state motion, and seeks to explain this dichotomy through the vast range of complexity intrinsic to these models. A key observation is that to achieve accuracy in predicting flow fields in general geometries, one requires a model that accounts for a number of subtleties, most notably a nonlocal effect to account for cooperativity in the flow as induced by the finite size of grains. On the other hand, forces and tractions that develop on macro-scale, submerged boundaries appear to be minimally affected by grain size and, barring very rapid motions, are well represented by simple rate-independent frictional plasticity models. A major simplification observed in experiments of granular intrusion, which we refer to as the ‘resistive force hypothesis’ of granular Resistive Force Theory, can be shown to arise directly from rate-independent plasticity. Because such plasticity models have so few parameters, and the major rheological parameter is a dimensionless internal friction coefficient, some of these simplifications can be seen as consequences of scaling. 1 Introduction Granular materials have a well-deserved reputation as a complex rheological media [1]. Much debate still exists on what is the constitutive relation for granular flow. Dry granular systems display history- and preparation-dependent strengthening and dilation [2–4], flow anisotropy and normal stress differences [5–7], nonlinear rate-sensitive yielding [8–11], and nonlocality due to the finite size of grains [12–16]. All these phenomena depend sensitively on grain characteristics such as shape and size distribution, frictional properties, and stiffness [10, 17–20]. Continuum models have been proposed of varying complexity to represent these phenomena, with some taking a simpler form while representing fewer effects, and others more complicated forms with the tradeoff of capturing more of the subtleties. Rather than choosing a particular model, the purpose of this review is to highlight the benefits of each model depending on the problem at hand. We will focus on well-developed flow phenomena. Interestingly, the complexity of granular materials is somewhat problem specific. For motivation, let us compare granular phenomena to a relatively standard material model, say, a linear viscous fluid, whose shear stress τ and strain-rate γ̇ are related by τ = ηγ̇ for viscosity η. In inhomogeneous flow geometries, where the strain-rate field is not uniform, steady granular flow profiles display a characteristic size-effect whereby the size of shear features is influenced directly by the grain size rather than just the  e-mail: local homogenized fields, such as stress [10, 21, 22]. In this sense, granular phenomenology is more complex than a viscous fluid, in which the flow rule is fully local; i.e. stress at a position gives flow at that position. Indeed, to obtain predictive stress and flow fields in a granular material, one needs a nonlocal framework to account for flow cooperativity induced by the finite size of grains. However, if the question is reduced, the modeling complexity needed to obtain an adequate solution reduces enormously. For example, consider the problem of determining the resistive force on an arbitrary rigid solid object being dragged through a bed of grains. This problem has direct relevance in granular-solid interactions, such as locomotion in granular media. Here, the quantity being desired is a global net force, rather than spatial fields of flow and/or stress. By reducing the question in this way, it turns out that very simple models provide an accurate solution. In particular, reduced-order resistive force models from viscous flow theory can be translated to the granular problem, and often work better than in the original viscous fluid application [23–25]. It would seem that in these kinds of problems, interestingly, granular media may behave simpler than a viscous fluid. Herein, we discuss the behaviors of granular media in terms of a nested family of continuum models. The basic features of granular flows can be predicted with simple frictional plastic models containing only a single (dimensionless) internal friction coefficient [26–28]. However, obtaining details at the next order of precision often incurs a rather large leap in modeling complexity, although certain specific observables are affected less than others by © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 140 , 01007 (2017 ) DOI: 10.1051/epjconf/201714001007 Powders & Grains 2017 We now summarize three granular flow models, each adding more complexity to the one before it. which model is used. We discuss the Nonlocal Granular Fluidity (NGF) model as an example of a more complex model [12–14, 29], which augments the frictional plastic form with both rate-dependence and particle size dependence. We begin by describing the continuum models at hand, and then follow up with demonstrations focusing on weighing the need for simpler models, which are more amenable to analytical tools, with the need for more detailed forms that have greater field accuracy but at a larger computational cost. Frictional plasticity: In the well-developed limit, transient effects cease and we can express the Drucker-Prager failure criterion as μ ≡ τ/p = μ s if γ̇ > 0, and τ ≤ μ s p otherwise. Here, μ s is an internal friction coefficient, deemed constant in this model. The above scalar relation combined with codirectionality and momentum balance provides a closed system of equations. Note this is a non-associated plasticity model. It should also be clarified that because we consider cohesionless grains, we must supply additional physics to ensure the pressure stays non-negative, i.e. no tension. We can achieve this by requiring 2 Nested Family of Flow Models The simplest continuum models for granular flow date back to the ideas of Coulomb in the 1700’s. They are based on the concept of a frictional yield criterion, where the shear stress needed to achieve plastic flow grows linearly with the applied pressure. These notions have been extended to three-dimensions by Mohr and further simplified by Drucker and Prager [30] where they are common tools in geotechnical engineering [26, 31]. To write a steady flow model based on these principles, we first assume a standard momentum balance ρv̇ = ∇ · (...truncated)