Steady State Rheology of Homogeneous and Inhomogeneous Cohesive Granular Materials
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Granular Matter (2020) 22:14 https://doi.org/10.1007/s10035-019-0968-5 ORIGINAL PAPER Steady state rheology of homogeneous and inhomogeneous cohesive granular materials Hao Shi1 · Sudeshna Roy1 · Thomas Weinhart1 · Vanessa Magnanimo1 · Stefan Luding1 Received: 23 July 2018 / Published online: 19 December 2019 © The Author(s) 2019 Abstract This paper aims to understand the efect of diferent particle/contact properties like friction, softness and cohesion on the compression/dilation of sheared granular materials. We focus on the local volume fraction in steady state of various non- cohesive, dry cohesive and moderate to strong wet cohesive, frictionless-to-frictional soft granular materials. The results from (1) an inhomogeneous, slowly sheared split-bottom ring shear cell and (2) a homogeneous, stress-controlled simple shear box with periodic boundaries are compared. The steady state volume fractions agree between the two geometries for a wide range of particle properties. While increasing inter-particle friction systematically leads to decreasing volume fractions, the inter-particle cohesion causes two opposing efects. With increasing strength of cohesion, we report an enhancement of the efect of contact friction i.e. even smaller volume fraction. However, for soft granular materials, strong cohesion causes an increase in volume fraction due to signifcant attractive forces causing larger deformations, not visible for stif particles. This behaviour is condensed into a particle friction—Bond number phase diagram, which can be used to predict non-monotonic relative sample dilation/compression. Keywords Granular materials · Cohesion · Friction · Granular rheology · Phase diagram · Homogeneous simple shear · Split bottom shear cell · Contact model · Inhomogeneous shear band · Shear dilatancy · Relative shear dilation/compression 1 Introduction 26, 33], dilatancy [7, 71, 80], shear-band localization [3, 9, 64, 79], history-dependence [67], and anisotropy [28, 37, Granular materials are omnipresent in our daily life and 44, 45] have attracted signifcant scientifc interest over the widely used in various industries such as food, pharmaceuti- past decades. When subjected to external shearing, granular cal, agriculture and mining. Interesting granular phenomena systems exhibit a non-equilibrium jamming transition from like yielding and fowing [2, 29, 41, 56, 70], jamming [6, 22, a static solid-like to a dynamic, liquid-like state [6, 26] and fnally to a steady state [57]. This particular transition drew much attention for dry and wet granular systems in both This article is part of the Topical Collection: Multiscale analysis dilute and dense regimes [8, 14, 38, 39, 43, 46, 49, 58, 62, of particulate micro- and macro-processes. 65, 72, 73]. * Hao Shi Material’s bulk responses like density/shear resistance are [email protected] infuenced strongly by diferent particle properties such as Sudeshna Roy frictional forces, as well as dry or wet cohesion. How the [email protected] microscopic particle properties infuence the granular rheo- Thomas Weinhart logical fow behaviour is still a mystery and thus attracted [email protected] more attention in the last few decades [15–17, 20, 30, 36, Vanessa Magnanimo 42, 51, 52, 63]. Although the infuence of individual parti- [email protected] cle properties is better understood now, there is still very Stefan Luding little known about the combined efect of particle friction [email protected] and cohesion on the rheological behaviour of granular fows [48]. 1 Multi-Scale Mechanics, TFE, ET, MESA+, University of Twente, 7500 AE Enschede, The Netherlands Vol.:(0123456789)1 3 14 Page 2 of 20 H. Shi et al. For dry cohesive granular media, one needs to account for Dilatancy, or shear dilatancy in classical plasticity and the dominant van der Waals force between particles when soil mechanics, has a specifc defnition as the ratio of incre- they are not in contact. This attractive force can be mod- mental volumetric strain to the shear strain. While the “rela- eled by a reversible linear long range interaction [54], if the tive dilatancy” discussed in this work is a dynamic efect that interaction energy and dissipation are matched with the true is distinct from the classical dilatancy since in the steady non-linear force. simple shear fow the volumetric strain is zero. Choosing dif- In contrast, in wet granular media, particles attract each ferent dynamic steady states based on diferent inter-particle other, as liquid bridges cause capillary forces [66, 74, 78]. cohesion, it represents the bulk density change between two The capillary bridge force becomes active at frst contact, diferent steady states. but then is active up to a certain cut-of distance where the The work is structured as follows: in Sect. 2.1, we provide bridge ruptures. This gives asymmetric loading/unload- information on the two simulation geometries; in Sect. 2.2, ing behaviour. Recent studies based on Discrete Element the two cohesive contact models are introduced; in Sect. 2.3 Method (DEM) confrmed that the specifc choice of capil- the important dimensionless number and their related time lary bridge models (CBMs) has no marked infuence on the scales are elaborated; and in Sect. 2.4, the input parameters hydrodynamics of granular fow [11, 24] for small volumes are given. Section 3 is devoted to the discussion of major of interstitial liquids. In the present paper, our focus is thus fndings of rheological modelling with a focus on the com- to investigate the efect of two qualitatively diferent cohe- bined infuences of several particle parameters, while con- sion mechanisms, dry vs. wet, on the bulk density of cohe- clusions and outlook are presented in Sect. 4. sive granular materials. We quantify the cohesion associated with diferent contact models by the dimensionless Bond number, Bo, which is defned as the ratio of time scales 2 Simulation methods related to confning stress and adhesive force. In this way, we generalize the existing rheological models with cohesion The Discrete Particle Method (DPM) or Discrete Element dependencies. Method (DEM) is a family of numerical methods for simu- For dry, non-cohesive but frictional granular materials, lating the motion of large numbers of particles. In DPM, the the bulk density under steady shear decreases [17]. Cohe- material is modeled as a fnite number of discrete particles sive grains are more sensitive to stress intensity as well as with given micro-mechanical properties. The interactions direction, and exhibit much larger variations in their bulk between particles are treated as dynamic processes with densities [55]. Dry granular media with median particle size states of equilibrium developing when the internal forces below 30 μ m fow less easily and show a bulk cohesion due balance. As previously stated, granular materials are consid- to strong van der Waals interactions between particle pairs ered as a collection of discrete particles interacting through [60]. Unsaturated granular media with interstitial liquid in contact forces. Since the realistic modeling of the shape and the form of liquid bridges between particle pairs can also deformations of the particles is extremely complicated, the display bulk cohesion which can be strongly infuenced by grains are assumed to be non-deformable spheres which are the fow or redistribution of the liquid [47, 53]. Fournier allowed to overlap [32]. The general DPM approach involves et al. [10] observed that wet systems are signifcantly less three stages: (1) detecting the contacts between elements; (2) dense than dry granular materials even for rather large par- calculating the interaction forces among grains; (3) comput- ticles. The packing density is only weakly dependent on ing the acceleration of each particle by numerically inte- the amount of wetting liquid, because the forces exerted by grating Newton’s equations of motion after combining all the liquid bridges are very weakly dependent on the bridge interaction forces. This three-stage process is repeated until volume [78]. In general, one expects that the steady state the entire simulation is complete. Based on this fundamental local volume fraction decreases with increasing either par- algorithm, a large variety of codes exist and often difer only ticle friction or cohesion [16, 69]. This decrease of volume in terms of the contact model and some techniques used in fraction is related to both structural changes and increasing the interaction force calculations or the contact detection. bulk friction of the materials [64]. However, Roy et al. [51] After the discrete simulations are fnished, there are two reported an opposing increase of local steady state volume popular ways to extract relevant continuum quantities for fraction proportional with cohesive strength. Thus whether fow description such as stress, bulk density from discrete the particle cohesion enhances or suppresses the frictional particle data. The traditional one is ensemble averaging of efects that causes dilatancy or compression remains still a “microscopic” simulations of homogeneous small samples, a debate. Therefore, the second focus of this paper is to under- set of independent RVEs [19, 27, 40]. A recently developed stand the interplay between particle friction and cohesion alternative is to simulate an inhomogeneous geometry where on the steady state volume fraction of the granular systems dynamic, fowing zones and static, high-density zones coex- under shear. ist. By using adequate local averaging over a fxed volume 1 3 Page 3 of 20 14 Steady state rheology of homogeneous and inhomogeneous cohesive granular materials (inside which all particles can be assumed to behave simi- larly), continuum descriptions in a certain parameter range can be obtained from a single set-up [30, 35, 64, 65]. 2.1 Geometries In this study, we use MercuryDPM [68, 76, 77], an open- source implementation of the Discrete Particle Method (DPM) to simulate granular fow in two geometries: a homo- geneous stress controlled simple shear box (RVEs) and an inhomogeneous split bottom shear cell, where the local stress is given by the weight of the particles above.