Steady State Rheology of Homogeneous and Inhomogeneous Cohesive Granular Materials

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 h.shi‑[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

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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

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(inside which all particles can be assumed to behave similarly), 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 homogeneous 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.

2.1.1 Stress controlled simple shear box (SS)

The frst simple geometry is a cuboid shear volume with Lees-Edwards periodic boundaries in x and y directions [25] and normal periodic boundaries in z direction as shown in Fig. 1 Simulation of a 3D system of polydisperse particles simple Fig. 1. The initial length of each box side is set to L, but shear using Lees–Edwards periodic boundary conditions, with L y controlled to keep the normal stress yy constant the box side length in y-direction, L y , can vary in time. The polydispersed granular sample contains 4096 soft particles. The system is sheared along the x direction with a constant material is confned by gravity between the two concentric 1 shear rate ̇� = 2Vx∕Ly by moving all the particles at each cylinders, the bottom plate, and a free top surface. The grav- −2 timestep to achieve homogeneous shearing. Meanwhile, ity varies from 1 to 50 ms . The bottom plate is split at the normal stress along y direction, yy is kept constant by radius Rs = 85 mm . Due to the split at the bottom, a narrow allowing Ly to dilate/compact, so that it smoothly reaches shear band is formed. It moves inwards and widens towards its steady state [59, 61]. Using this setup, one can keep both the free surface. The flling height ( H ≈ 40 mm) is chosen shear rate and normal stress constant while measuring the such that the shear band does not reach the inner wall. Fig- shear stress responses in both transient and steady state. ure 2 shows the 3D schematic presentation and side view of Although this setup is not achievable in reality, it still repre- the split bottom shear cell geometry with colors blue to red sents typical lab experiments, sand or/and powders sheared indicating low to high kinetic energy of the particles. It is in diferent shear cells. This setup allows the user to explore visible that a wide shear band is formed away from the walls, two variables (shear rate and stress) independently with low which is thus free from boundary efects. computational cost. Therefore, in the current study, we sys- In earlier studies [64, 65], a quarter of this system ◦ ◦ tematically vary the shear rate ̇� = �Vx∕�y and confning ( 0 ≤ ≤ 90 ) was simulated using periodic boundary stress yy to understand the shear fow in both quasi-static conditions. All the data corresponding to diferent gravities and dynamic regimes, as well as the infuence of particle and diferent rotation rates belong to this system. In order to Softness. save computation time, we simulate only a smaller section of ◦ ◦ the system ( 0 ≤ ≤ 30 ) with appropriate periodic bound- 2.1.2 Split‑bottom ring shear cell (SB) ary conditions in the angular coordinate. No noticeable dif- ferences in the macroscopic fow behavior were observed ◦ between simulations performed using a smaller ( 30 ) and a In our present study, we also simulate a shear cell with annu- ◦ lar geometry and a split bottom, as described in detail in [50, larger ( 90 ) section. 64]. The geometry of the system consists of an outer cylinder (radius Ro = 110 mm ) rotating around a fxed inner cylinder 2.2 Force models (radius Ri = 14.7 mm ). We vary the rotation frequency from = 0.06 to 4.71 rad/s (0.01 to 0.75 rotation/s). The granular In our non-cohesive simulations, we use the linear visco- elastic frictional contact model between particle contacts in both geometries mentioned above: Simple Shear Box (SS) and Split Bottom Shear Cell (SB) [31]. 1 ̇� = 1∕2 ̇� The strain-rate defned using the Frobenius norm [23], , The dry cohesive particles are simulated in the geom- is applicable for arbitrary deformation modes [35, 52, 65]. The shear rate ̇� defned here is the same as in [72] as used in most simple shear etry SS using a linear reversible adhesive visco-elastic studies. contact model as shown in Fig. 3a , where we combine

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only difers in the attractive non-contact part compared to the dry cohesive simulations as shown in Eq. (4).

2.2.1 Classical linear contact force

When two particles i, j are in contact, the overlap can be computed as, a a ⋅ =( i + j)−( i − j) (1) with positions i and j , radii ai and aj , for the two primary particles, respectively and the unit vector = ij =( i − j)∕ i − j pointing from j to i. Thus the normal force fn between two particles is simply computed when �> 0 (Fig. 3) as f f k v n = rep ∶= n + n n (2)

with a normal stifness kn , a normal viscosity n and the relative velocity in normal direction vn [31]. On top of this simple elastic contact force law, we sepa- rately add two types of adhesive forces: reversible adhesive Fig. 2 3D Schematic representation of split bottom ring shear cell force and jump-in liquid bridge capillary force. The details (top) and side view, showing shear band formation in the simulation (bottom) [50] of these two adhesive forces are elaborated in the following. 2.2.2 Reversible adhesive force a linear attractive non-contact force with a linear visco- elastic frictional contact force [31] as the sum of Eqs. (2) For the dry granular material, we assume a (linear) van-der- f and (3). This reversible attractive model is chosen as a Waals type long-range adhesive force, a , as shown in Fig. 3a. i j (linearized) representation of the van-der-Waals attractive The adhesive force law between two primary particles and non-contact force. can be written as, And in the geometry SB, we simulate wet cohesive particles using an irreversible adhesive non-contact force with a linear visco-elastic frictional contact force as shown in Fig. 3b, which

Fig. 3 a Reversible adhesive force; b jump-in (irreversible) adhesive capillary force; both combined with a linear visco-elastic contact force as in Eq. (2)

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frictional force in the tangential direction. The tangential 0 if � ≤ −�a; f = −ka� − f max if − � ≤ �<0; force is calculated in a similar fashion like the normal force a c a a (3) f = k ⎧ max � ≥ through a “spring-dashpot” model ( t t t ) and coupled to −fa if 0 ≤ C ⎪ the normal force through Coulomb’s law, ft fs ∶= sfrep , ⎨ max a a where for the sliding case one has dynamic friction with ⎪ a ∶= f ∕k k ≤ C rep with ⎩the range of interaction a c , where c is the ft ft ∶= df . In this study, we use p = s = d . For a f max adhesive strength of the material and a is the (constant) non-adhesive contact, frep = fn > 0 , and the tangential force �>0 adhesive force magnitude, active for the overlap in is active and calculated based on the history in tangential addition to the contact force. direction. For an adhesive contact, Coulomb’s law has to = 0 −f max f rep When , the force is a . The adhesive force a be modifed in so far that f is replaced by fn − fa . In the − is active when particle overlap is greater than a , when it current model, the reference for a contact is no longer the ka starts increase/decrease linearly with slope c independent of zero force level, but it is the adhesive, attractive force level the two particles approaching and separating. In the current [31, 32]. study, this contact model is applied in the case of homogeneous stress controlled simple shear box simulations. 2.3 Time scales and dimensionless numbers 2.2.3 Jump‑in liquid bridge capillary force Dimensional analysis is often used to defne the characteristic time scales for diferent physical phenomena that The capillary attractive force between two particles in a wet involved in the system. Even in a homogeneously deform- granular bulk is modeled with a discontinuous, irreversible ing granular system, the deformation of individual grains is attractive law as shown in Fig. 3b. The jump-in capillary not homogeneous. Due to geometrical and local parametric force can be simply written as: constraints at grain scale, grains are not able to displace 0 if �<0, approaching; as afne continuum mechanics dictates they should. The −f max f = a if − S ≤ �<0, separation; fow or displacement of granular materials on the grain a (1+1.05S̄+2.5S̄ 2) c (4) scale depends on the timescales for the local phenomena ⎧ −f max if � ≥ 0 ⎪ a and interactions. Each time scale can be obtained by scal- ⎨ ing the associated parameter with a combination of particle ⎪S̄ = S (r∕V ) where⎩ b is the normalized separation distance, diameter dp and material/particle density p . While some S =− r being√ the separation distance, the reduced radius of the time scales are globally invariant, others are varying V and b the volume of liquid bridges. The maximum capil- locally. The dynamics of the granular fow can be character- S = 0 f max = 2 r cos lary force at contact ( ) is given by a s , ized based on diferent time scales depending on local and with the surface tension of the liquid, s , and the contact global variables. First, we defne the time scale related to angle, [54]. contact duration of particles which depends on the contact There is no attractive force before the particles come frst stifness kn as given by [65]: into contact; the adhesive force becomes active and sud- −f max = 0 3 denly drops to a negative value, a , when (the pdp liquid bridge is formed). Note that this behavior is defned tk = (5) kn here only during frst approach of the particles. We assume the model to be irreversible: the forces will not follow the In the special case of a linear contact model, this is invari- same path, i.e. the attractive force is active until the liquid ant and thus represents a global time scale. Two other time =−S bridge ruptures at c . This attractive force is following scales are globally invariant, the cohesive time scale tc , i.e. Willet’s capillary bridge model [78], as explained in [54]. the time required for a single particle to traverse a length Similar to the reversible model, we combine the attrac- scale of dp∕2 under the action of an attractive force, and the tive capillary adhesive model with the linear visco-elastic gravitational time scale tg , i.e. the elapsed time for a single contact model defned in Sect. 2.2.1 and use this combined particle to fall through half its diameter dp under the infu- model to simulate wet granular materials under shear in the ence of the gravitational force. The time scale tc could vary split bottom shear cell. locally depending on the local capillary force fc . However, the capillary force is weakly afected by the liquid bridge 2.2.4 Friction volume while it strongly depends on the surface tension of the liquid s . This leads to the cohesion time scale as a global Introducing the additional cohesive force in the normal parameter given by: direction between the two particles also influences the

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4 infuence of the timescale related to the granular temperature pdp tc = (6) on such dense granular fow and thus the tT is not included f max a in the dimensionless study. 2 max The frst dimensionless number is the Inertial number, with maximum adhesive force fa between two particles. which is the ratio between tp and t ̇� , The global time scales for granular fow are complemented by locally varying time scales. Granular materials subjected I = ̇� dp∕ p∕�p (10) to strain undergo constant rearrangement and thus the contact network re-arranges (by extension and compression and The second Softness characterizes how “soft” the system by rotation) with a shear rate time scale related to the local is and is the square of the ratio between contact collision shear rate feld: timescale tk and pressure timescale tp, 1 t = p∗ = pd ∕k ̇� ̇� (7) p n (11) the shear rate is high in the shear band and low outside, The last one studied in this paper which is important to so that also this time scale varies between low and high, characterize the cohesiveness in the whole system is Bond respectively. number. It is also the square of the ratio between pressure t t The time for rearrangement of the particles under a cer- timescale p and cohesion timescale c, tain pressure constraint is driven by the total local pressure Bo = f max∕pd2 a p (12) p. This microscopic local time scale based on pressure is: Interestingly, all three dimensionless ratios are based on the p t = d common time scale tp . Thus, the time scale related to con- p p p (8) fning pressure is important in every aspect of the granular fow [52, 65]. As the shear cell has an unconfned top surface, where the Even though not used, for completeness, we also pro- ∗ 2 pressure vanishes, this time scale varies locally from very pose the dimensionless granular temperature, T =(tp∕tT ) , low (at the base) to very high (at the surface). which is much smaller than 1 in all data analyzed further Finally, one has to also look at the granular temperature 2 on. time scale due to the velocity fuctuations (T g ∝ (vi − v̄) ) among all the particles which is of signifcance of determin- ∑ 2.4 Simulation parameters ing how “hot” the system is [13, 34, 72, 75]: 2.4.1 Fixed particle parameters dp tT = T (9) g The fxed input parameters for the two diferent simulation √ set-ups are summarized in Table 1. In the liquid bridge cap- Dimensionless numbers in fuid and granular mechanics are illary contact model, there are two extra input parameters a set of dimensionless quantities that have a dominant role compared to the dry adhesive model: contact angle and liq- in describing the fow behavior. These dimensionless num- uid bridge volume. These two parameters are kept constant bers are often defned as the ratio of diferent time scales or for all the simulations. Apart from that, we use restitution forces, thus signifying the relative dominance of one phe- coefcient, e = 0.8 to compute the normal viscosity in the nomenon over another. In general, we expect four time scales dry adhesive model but directly input the normal viscosity (t p , tc , t ̇� and tk ) to infuence the rheology of our system. Note in the liquid bridge capillary contact model, which gives the that among the four time scales discussed here, there are six same damping in the two contact models. possible dimensionless ratios of diferent time scales. We propose three of them that are sufcient to defne the rheol- 2.4.2 Control parameters ogy that describes our results. We do not expect a strong Apart from the fxed particle parameters, there are also several control parameters which we vary to explore their infu- 2 Alternative Inertial numbers could be defned using the bulk den- ences. For instance, varying shear rate, pressure, gravity and � 1 2 sity, so that I = I � = ̇� dp∕ p∕�p� , or the strain rate ( ̇� = ∕ ̇� ), cohesion, we explore our rheology towards diferent iner- ̂ 2 � ̂� 2 tia, softness and cohesion regimes. The details of parameter so that I = I∕ =√̇� dp∕ p∕�p , or both so that I = I �∕ , which ranges are summarized in Table 2. will lead to small diferences in ftted characteristic √numbers. For sake of brevity, we use the defnition in Eq. (10).

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Table 1 Summary and Parameter Symbol Simple shear (dry) Split-bottom (wet) numerical values of fxed input particle parameters used in the Average diameter dp 2.2 mm 2.2 mm DPM simulations Polydispersity w = rmax∕rmin 2 and 3 2 3 3 Particle density p 2000 kg/m 2000 kg/m 2 2 Normal stifness kn 100 kg/s 100 kg/s

Tangential stifness kt∕kn 2/7 2/7 Rolling/torsion stifness kr∕kn 2/7 2/7 Inter-particle friction p 0 to 1 0.01 Rolling friction r 0 0 Torsion friction t 0 0 Restitution coefcient e 0.8 0.8 Normal viscosity n 0.002 kg/s 0.002 kg/s Friction viscosity fr∕ n 2/7 2/7 Rolling viscosity ro∕ n 2/7 2/7 Torsion viscosity to∕ n 2/7 2/7 Adhesion stifness kadh∕kn 1 [-] ◦ Contact angle [-] 20 Liquid bridge volume Vb [-] 75 nl Rupture distance Sc [-] 4.95 nm

Table 2 Summary of variable SS SB control parameters used in two s−1 Pa max ms−2 Nm−1 geometries: simple shear box ̇� ( ) p ( ) fa (N) (rad/s) g ( ) s ( ) (SS) and split bottom shear cell (SB) Input range 0.0001–1 125–4000 0–0.01 0.063–4.712 1–50 0–0.3 ∗ ∗ Dimensionless I p Bo I p Bo numbers Input range 0.001–1 0.001–0.1 0–5 0.001–0.1 0.002–0.05 0–4

To study the efect of inertia and contact stifness on the corresponding to a liquid saturation of 8% of the volume of ◦ dry-non-cohesive materials rheology in the geometry SS, the pores and the contact angle is fxed at 20 . The cho- −1 −1 we vary the shear rate, ̇� , between 0.0001 and 1 s , as sen values of surface tensions are between 0 and 0.3 Nm . well as the pressure, p, between 125 and 4000 Pa. Thus, we The rotation rate, = 0.063 rad/s (0.01 rps) and the grav- −2 obtain Inertial numbers, I, between 0.001 and 1; Softness, ity, g = 9.81 ms are kept constant for all the wet cohesive ∗ p , between 0.001 and 0.1. simulations. For the dry-cohesive simulations in the same set-up, we Note that for all the simulations performed using geom- −1 keep the shear rate, ̇� = 0.005 s , pressure, p = 500 Pa, and etry SB, the inter-particle friction is kept constant at max vary only the maximum adhesion force, fa , between 0 p = 0.01 , while for the simulations performed in geometry and 0.01 N. This leads to the range of Bond number, Bo, SS, we vary p from 0 to 1. Therefore, for the comparison between 0 and 5. between the two geometries, we choose only the data with For the dry non-cohesive simulations using the geometry p = 0.01 , and in each simulation performed, the Coulomb SB, we vary the rotation rate, , between 0.063 to 4.712 friction static and dynamic are always kept the same and rad/s (0.01 and 0.75 rotation/s), of the outer wall to vary the referred to by the inter-particle friction, p = s = d. system inertia and therefore match the case of the simple shear box. And to change the pressure in this free surface system, we achieve low or high pressure by varying the grav- 3 Rheology −2 ity, g, between 1 and 50 ms . In the case of wet granular material in SB, we vary the Our frst goal is to check the consistency of bulk densities max intensity of the maximum capillary force fa = 2 rs cos , from two diferent geometries: SS and SB when using the by varying the surface tension of the liquid s , while keep- same non-cohesive contact model. And the second goal is to ing the volume of liquid bridges Vb constant at 75 nl, investigate how diferent cohesive contact models infuence

1 3 14 Page 8 of 20 H. Shi et al. the bulk density at steady state. Therefore, in Sect. 3.1, we check whether the shear behaviour of the non-cohesive granular media is the same in the two diferent geometries, by comparing the steady state volume fractions. And then in Sect. 3.2, we compare the rheological behaviours of cohesive granular materials using the reversible adhesive contact model (stress controlled simple shear box) and irreversible liquid bridge capillary contact model (split bottom shear cell). In addition, we also check the validity of existing frictional rheology and consummate it towards cohesive materials. Finally, in order to achieve our second goal, we explain the interplay between particle friction and cohesion in Sect. 3.3.

3.1 Non‑cohesive granular materials Fig. 4 Volume fraction, , plotted against Inertial number, I, using stress controlled simple shear box (SS) and split bottom shear cell 0.01 (SB) with p = . For SS, two sets of parameters are chosen: (1) ∗ For dry granular materials, which have rather large particle fx Softness at p = [0.0025, 0.005, 0.01], vary shear rate ̇� between Bo ≈ 0 0.0001 and 1 (black squares); (2) fx shear rate at ̇� = [0.0005, 0.005, size and negligible attractive forces ( ). The rheology ∗ 0.05], vary Softness p between 0.001 and 0.1 (blue circles). For SB, (i.e. the equations of state for volume fraction and macro- the range of values of gravity g and rotation frequency are given in scopic friction) depend mainly on the Inertial number, I, Table 2 and the cloud of points represent center of shear band data ∗ and the Softness, p , which are the competition between the from 12 SB simulations. The lines are the predictions Eq. (13), ftted 2 0.65 2.25 ∗ 0.27 t ∕t (t ∕t ) using SS data, with 0 = , I = and p = . Solid and time scales p ̇� and k p , respectively. The dependence ∗ ∗ dashed lines represent constant ̇� and p , respectively of the macroscopic friction coefcient = ∕p on p and I has been studied earlier [52, 65] as summarized in Appendix 1. In order to complete the rheology for soft, compressible moderate inertial regime ( I < 0.25 ≈ I�∕10 ) and low to ∗ ∗ particles, a relation for the volume fraction, , as function of moderate overlaps between particles ( p < 0.03 ≈ p� ∕10 ) pressure and shear rate is still missing for cohesive frictional with only few data outside this regime [35, 59]. materials. In previous work [35], the following dependency Note that in Eq. (13), we do not include inter-particle fric- was reported: tion p as functional variable although the volume fraction (I, p∗)= g (I)g (p∗)≈ (1 − I∕I )(1 + p∗∕p ∗) depends on p . We consider inter-particle friction as a mate- 0 I p 0 (13) rial micro-parameter like polydispersity or restitution coef- fcient, which is diferent from the dimensionless state vari- with the critical or steady state volume fraction under shear, ables in our functions, e.g. shear rate and pressure. However, i.e., the limit for vanishing pressure and Inertial number, we did explore this dependency in detail in "Appendix 2". 0 =0.01 = 0.64 . Luding et al. [35] found that the typical p In Fig. 4, we plot the volume fraction in steady state as shear rate for which shear dilation would turn to extreme ̂� a function of Inertial number for both the stress controlled fuidization is I = 0.85 (for more details on number, see Eq. � simple shear box (SS) as well as the split bottom shear cell (10) and Table 3), and the typical pressure level for which = 0.01 ∗ (SB) with the same friction coefcient p . For the p = 0.33 ∗ Softness leads to too huge densities is . Note that case of SS, when we we keep Softness p constant and vary both correction functions are frst order Taylor expansions only shear rate (black squares and dashed lines), the volume with respect to full, higher order functions, i.e. they are valid fractions decrease with Inertial number, as the increase of only for sufciently small arguments. In slow, quasi-static shear rate leads to higher collisional/dynamic pressure. Cor- fows in the split bottom shear cell simulations, weak dila- respondingly in the case of SB, when we vary the rotation tion is observed, i.e., no strong dependence of on the local velocity (red triangles), the volume fraction follows the same shear rate [35]. On the other hand, large Inertial numbers trend and the data collapse well with the data from SS but fully fuidize the system so that the rheology should be that with slightly more scatter due to the local, small volumes of a granular fuid, for which kinetic theory applies. Small used for averaging. The red triangles are not following any ∗ pressure, i.e. small overlaps have little efect, while too large single of the dashed lines since both I and p are small but pressure would lead to enormous overlaps, for which the neither is constant. When we fx shear rate and vary only contact model and the particle simulation with pair forces pressure (blue circles and solid lines), the larger pressures become questionable. Therefore, we focus on the linear lead to considerable compaction. expansion as shown in Eq. (13), valid in the quasi-static to

1 3 Page 9 of 20 14 Steady state rheology of homogeneous and inhomogeneous cohesive granular materials

pressure increase in the system or the particles becoming softer [61, 72]; in either case the particles will overlap more thus leading to an increase in the volume fraction. In the same Figs. 4 and 5, we also plot the prediction of Eq. (13) ftted using data from SS and with the ftting parameters summarized in Table 3. The coefcients for p = 0, 0.01 and 0.5 are from the ftting whereas the others are from estimation of the curve ftting, which are further tested and confrmed in this paper. Since the two systems have similar rheological behaviour, one would expect the rheological model developed from SB to capture also the behaviour of SS and vice versa. The prediction of ftted function looks promising and it captures well both the dependencies on inertia and Softness as shown in dashed and solid ∗ Fig. 5 Volume fraction, , plotted against Softness, p , for the same lines, respectively. All the ftting deviations are within 5%, data as in Fig. 4 even for data with relatively high inertia ( I > 0.25 ). How- ever, when the volume fraction gets too low (below 0.5), Table 3 The ftting coefcients of the non-cohesive rheological the predictions of our rheological model deviate from the model for in Eq. (13) for various p simulation data because the system goes from the dense state ∗ p 0 I p towards a loose granular fuid, which our model does not take into account. Alternative rheological models for this 0.00 0.656 1.960 0.271 transition region are reviewed in [72], while the appropriate 0.01 0.653 2.246 0.272 model for the liquid/gas state is standard kinetic theory as 0.01 in Ref. [34, 35, 52] 0.65 2.11* 0.33 studied in [1, 5, 12, 18]. This indicates the limits of using 0.05 0.643 3.171 0.274 the rheological model Eq. (13); it predicts well moderate 0.10 0.634 3.959 0.276 to dense granular fows with fnite contact stresses, but 0.20 0.619 4.806 0.280 not dynamic, less dense granular gases. Note that here in 0.30 0.610 5.166 0.283 Table 3, we have also included the I = 2.11 from [34, 35, 0.40 0.603 5.319 0.285 3 52], which is very close to our I = 2.246 obtained from 0.50 0.595 5.383 0.287 the SS setup. 0.70 0.591 5.422 0.291 0.90 0.585 5.429 0.293 3.2 Cohesive granular materials 1.00 0.583 5.430 0.293

Note that in this study, we use shear rate ̇� and particle density p in For cohesive granular materials, attractive forces enhance the Inertial number, whereas strainrate ̇� and bulk density b = p the local compressive stresses acting on the particles, thus 0.01 are used in other literature [34, 35, 52] for p = . All the coef- ∗ leading to an increase in the volume fraction. On the other fcients are ftted using data with �>0.5 , I < 0.25 and p < 0.03 . Note the * symbol indicating the diferent defnition as discussed in hand, rough and frictional particles will display a stronger the main text and Eq. (10) dilatancy, i.e, a reduced volume fraction under shear at steady state, compared to their initial (over-consolidated) volume fraction. We systematically vary the inter-particle ∗ In order to focus on the dependency of on Softness p , friction ( p from 0 to 1) and then for each inter-particle fric- in Fig. 5, we plot the same data as in Fig. 4 against Softness. tion value we vary cohesion (Bo from 0 to 5) to study the For the case of SS, when the shear rate ̇� is kept constant volume fraction variations in steady state. Then we intro- ∗ while Softness p is varying (blue circles), we observe an duce a generalized rheological model involving cohesion, increase of volume fraction with Softness. For the corre- as based on evidence from the simulations. sponding case in SB, when we keep the rotation velocity The Bond number quantifes the competition between constant and vary gravity (brown crosses), the volume frac- cohesion and pressure. Low Bond numbers refer to weak t ≪ t tions follow the same trend as in the case of SS and increase cohesion, or p c , while high Bond numbers indicate the with Softness in a linear relation as specifed in [65]. Dif- ferent gravity data from the split bottom shear cell collapse 3 Note the actual value for I in the references was 0.85. This dif- also well with the simple shear box data but with slightly Î ∗ ference is due to their defnition of , see Eq. (10) and its footnote, more scatter. The increase in Softness p can be due to the which requires to translate their value by a factor of 2∕ ≈ 2.5 to ours. √ 1 3 14 Page 10 of 20 H. Shi et al.

Fig. 6 The SS volume fraction as a function of the Bond number Fig. 7 The scaled volume fraction, i.e. the cohesion term, Bo for diferent inter-particle friction coefcients p . The shear rate gc = ∕( 0gI gp) as a function of the Bond number Bo for diferent −1 ̇� = 0.005 s and pressure p = 500 Pa are kept constant for all the inter-particle friction coefcients as shown in the legend. The lines ∗ simulations shown here, so that I = 0.02 and p = 0.01 . The lines are the predictions of the ftted Eq. (15) and the data are the same as 0, 0.1 are the predictions plotted only for three cases of p = and 0.5, in Fig. 6 with solid lines indicating Eq. (16) while dashed lines including the p dependency as in Eq. (1B) including Eq. (1B) over-predict the volume fractions at the steady state and deviate more towards higher Bond numbers. opposite. The variation of Bond number in the current study The details of how the rheological model is modifed/gener- covers a wide range of cohesion intensities by comparing the alized will be discussed in the following. max 2 maximum pressure contributed by cohesion, fa ∕dp , to the In cohesive fows, attractive forces enhance the compres- actual pressure. sive pressure acting on the particles. This can be quantifed as follows: the net pressure can be split into two compo- 4 3.2.1 Efects of cohesion for diferent particle friction nents, p = prep − pcoh, denoting the respective contributions of repulsive and cohesive contact forces. The ratio between In Fig. 6, we plot the steady state volume fraction against the total cohesive contribution and the total pressure is the Bond number for samples with diferent inter-particle given by the local Bond number, Bo = pcoh∕p , and thus friction using the SS geometry. As expected, for very low prep =(1 + Bo)p , which represents the dimensionless over- inter-particle friction ( p ⩽ 0.01 ), we observe a continu- lap. As the geometrical compression (deformation at each ous increase in the volume fraction with Bo. However, for contact) is related to the repulsive stress, it is the repulsive higher inter-particle friction ( p ⩾ 0.05 ), increasing dila- pressure prep that has to be considered in the Softness fac- tancy is observed with increasing Bo up to values around tor gp to correct this cohesion induced bulk density change. 2, an efect which we will discuss later. When the Bond However, this (1 + Bo) correction is not considered inside number becomes larger, the volume fraction increases with the Inertial number I, because the dominating timescale increasing Bo. When we fx cohesion (Bo ≈ const.) and vary here is the pressure time scale tp that correlates to the total the particle friction, the volume fraction always decreases pressure p in the system not the repulsive part prep . Thus, with increasing particle friction; this indicates that particle using Eq. (13), the modifed Softness correction for cohesive friction leads to shear dilation, not to compression. Thus, systems is given as: we assume that the change in volume fraction is due to both g = g (p∗, Bo)=g (p∗ )=1 +(1 + Bo)p∗∕p ∗ compression of soft particles in presence of cohesion and the p p p rep (14) dilatancy due to structural changes in presence of friction. For non-cohesive systems, Bo = 0 , gp is consistent with the We also added here the predictions of our proposed rheo- non-cohesive rheological function. This modifed pressure is logical models as in Eq. (16) and Eq. (1B) with solid and similar in spirit to the modifed Inertial number as presented dashed lines, respectively. The predictions of Eq. (16) are in in [4], which takes into account the cohesive contribution very good agreements with the actual data, but the lines of to stress. A similar modifcation of pressure in the Inertial number would require an additional parameter but would only reduce I, and thus is not considered here. 4 For sign convention, we defne repulsive force as positive and attractive forces as negative, see Fig. 3.

1 3 Page 11 of 20 14 Steady state rheology of homogeneous and inhomogeneous cohesive granular materials

Table 4 List of rheological correction functions on as in Eq. (16) Thus, cohesion contributes to the initial decrease and the for application in a continuum model subsequent increase in the volume fraction correction of gc Dimensionless numbers Corrections for sheared material. In order to quantify this dependence, the correction function gc of the Bond number, Bo, is given Inertial number (I) gI = 1 − I∕I ∗ ∗ ∗ by a fourth degree polynomial function: Softness ( p ) gp = 1 +(1 + Bo)p ∕p 2 3 4 2 3 4 Bond number (Bo) gc = 1 + p1Bo + p2Bo + p3Bo + p4Bo gc(Bo)=1 + p1Bo + p2Bo + p3Bo + p4Bo (15)

where, p1, p2, p3 and p4 depend on p . The actual ftted values for gc with diferent inter-particle friction, p , are sum- Table 5 The ftting coefcients of 4th degree polynomial in Eq. (15) marized in Table 5. With increasing p , the polynomial con- p p1 p2 p3 p4 stants p1, p3 decrease, but p2, p4 increase. The lines in Fig. 7 represent Eq. (15), where a promising collapse between this 0.00 – 0.0175 – 0.0060 0.0029 – 0.0003 prediction and the simulation data is observed. For low par- 0.01 – 0.0237 – 0.0056 0.0034 – 0.0004 ticle friction, e.g. p = 0.01 , the volume fraction decreases 0.05 – 0.0394 0.0030 0.0015 0.0002 gradually with Bo, while for higher particle friction, e.g. 0.10 – 0.0564 0.0134 – 0.0011 0.00003 p = 0.5 , the volume fraction decreases more sharply with 0.20 – 0.0845 0.0310 – 0.0058 0.0005 Bo. We note that also frictionless particles ( p = 0 ) under 0.30 – 0.1091 0.0480 – 0.0107 0.0010 shear, the correction gc decreases with increasing Bo, which 0.40 – 0.1195 0.0542 – 0.0125 0.0011 suggests that cohesion causes the structural changes that 0.50 – 0.1359 0.0668 – 0.0162 0.0015 relate to shear dilation. In other words, on top of the efect in 0.70 – 0.1516 0.0779 – 0.0195 0.0018 gp , cohesion could make frictionless particles stick together 0.90 – 0.1494 0.0733 – 0.0176 0.0016 and form clusters, which results in an increase of the bulk 1.00 – 0.1511 0.0746 – 0.0181 0.0017 shear resistance (data not shown here). The sample has to dilate to reduce its shear resistance to compensate this efect. Thus the role of contact friction in enhancing dilation is not In Fig. 7, we plot the same data as in Fig. 6 to obtain the a solitary efect, also cohesion is afecting the system behav- functional form of the cohesive correction gc : the normal- g = ∕[ g (I)g (p∗, Bo)] iour: one should not neglect the role of either. Frictional par- ized volume fraction, c 0 I p as function ticles need at least four mechanical contacts to form a stable Bo of the Bond number, , in order to isolate the efect of structure while frictionless particles need six contacts. The cohesion on the dilatancy of soft particles. The normalized higher the friction, the higher the probability that a stable local volume fraction helps isolate the efect of cohesion packing can be formed at lower volume fractions with less Bo g only. From low to moderate , the correction c decreases average contacts/coordination number. Bo with increase of . An increase of the repulsive force Comparing to the previous works [52, 65], a more com- between two contacting particles also leads to an increase plete rheology including the cohesion/Bond number infu- of the sliding limit tangential forces, resulting in an enhance- ence on the volume fraction is introduced as: ment of the role of friction. For Bo ⩾ 3 , gc increases from ∗ ∗ its minimum at Bo ≈ 3 , since the attractive force is so high (I, p , Bo)= 0gI(I)gp((1 + Bo)p )gc(Bo) (16) that sample dilation is compensated by compression. Note p∗ → 0 that compression is prevailing for soft particles but is neg- In case of rigid particles, , Eq. (16) reduces to ∗ = (I, 0, Bo)= g g ligible when p ≈ 0 , in the limits of low confning stress or stiﬀ 0 I c . The details of each contribution infnite stifness, when the local volume fraction is expected are summarized in Table 4. to monotonically decrease with Bo. Some previous studies ∗ have confrmed the negligible efect of confning stress p 3.2.2 Generalized rheology when using stif particles [4, 21]. Here we assume that the frictional contributions are the Now we go further and include data from SB with a difer- same for both cohesive and non-cohesive materials. There- ent contact model to compare with the data from SS. As fore, we use the coefcients of the non-cohesive material: discussed earlier in Sect. 2.2, we use two diferent contact ∗ 0, I and p from Table 3. The local volume fraction is models for the attractive forces in the two systems, SS and scaled by gI as in Eq. (13) and gp as modifed in Eq. (14). In SB. Thus, the question arises whether the two models are such a way, we hope to remove the efects of particle inertia infuencing both systems in the same way. Since we dis- as well as particle Softness. cuss the granular rheology for soft particles, our second focus would be the effects of cohesion (Bond number) on the volume fraction, which are the increase due to the

1 3 14 Page 12 of 20 H. Shi et al.

due to increasing shear rate are removed. For the case of SS, the scaled volume fraction increases with Softness, and the same trend is observed for the case of SB, regardless of non-cohesive or cohesive data. The scaled local volume fractions ∕[ 0gI(I)gc(Bo)] ∗ increase with the repulsive pressure (1 + Bo)p , and all the data are found to collapse on a single increasing trend that is gp . Therefore we plot the predictions of our rheological functions as red and black solid lines in Fig. 8a. We see the two lines difer slightly from each other and we associate this to the small diference in the particle size distributions used in the two geometries, the simple shear box has a sample with polydispersity w = 3 , while the split bottom shear cell (a) has w = 2 . Nevertheless, this diference is minimal and the global trend is well captured using the proposed rheological model. Note here, for the cohesive SB data, the Inertial ∗ number I ranges from 0.0004 to 0.0013 and the Softness p ranges from 0.002 to 0.012, which difers from the cohesive ∗ SS data with constant I and p . If one expects the correction ∗ function gc to be strongly afected by diferent I and p , then these infuences would show up in the data of SB where a wider range of Inertial numbers and Softness is covered. However, the infuence/diference is not observed, therefore, the correction function gc if at all, is weakly afected by the ∗ dimensionless numbers I and p , apart from the dominant dependence on the Bond number Bo. When we look closer at the cohesive SS data as shown in Fig. 8b, we observe that the data for �p > 0.01 deviate from (b) the prediction for p = 0.01 (black line), which is captured ∗ using the proper p (p) from Table 3 ( p = 1 is shown as Fig. 8 a The scaled volume fraction, i.e. the repulsive stress term, dashed line). As we control all the samples having the same ∗ 1 ∗ gp = ∕( 0gI gc) as a function of the cohesive softness p ( + Bo) p for the data shown here, the cause of the increase in vol- and b zoom in to the simple shear data with diferent inter-particle ume fraction can only be from the increase of cohesion (Bo), friction coefcients and Bond numbers. Diferent symbols represent diferent particle friction, where small black diamonds represent local which is the increase of the attractive forces among the soft data from the split bottom shear cell (SB) for diferent Bond num- particles leading to compression. The repulsive component ∗ bers using a wet cohesive capillary bridge force model (Fig. 3b) and of pressure is solely contributed by p for non-cohesive = 0.01 p , while all other are from the homogeneous stress con- materials. For cohesive material, the net repulsive contribu- trolled simple shear box (SS). The red circles are the simple shear data using the normal visco-elastic contact model with no cohesive tion is due to the cohesive (tensile) as well as the pressure forces involved but varying shear rate and confning stress. The rest of (compressive) forces, and this efect is found to be independ- the SS data are performed with diferent inter-particle friction coef- ent of inter-particle friction. For Bond number up to 2, all fcients and Bond numbers using the aforementioned linear reversible the data are collapsing on a single trend with less than 2% adhesive contact model (Fig. 3a). The black solid and dashed lines are the predictions of our rheological model Eq. (14) ftted using SS deviations relative to their absolute values, while the high 0.01 data with p = and 1, respectively, while the red line is based on Bond number data are deviating more from this trend, but SB data with the same inter-particle friction as reported in [35] still within 5% deviations up to Bond number 5. Here, the model predictions from this work are given as lines for two values of friction: black solid line for p = 0.01 and black compression and decrease due to the structural change dashed line for p = 1 . Both lines predict the corresponding enhanced by friction. First, we illustrate our data in a slightly dataset very well. The red line is slightly of due to a difer- diferent way in Fig. 8a, plotting the scaled volume fraction ent polydispersity in [35]. ∕[ 0gI(I)gc(Bo)] as a function of the repulsive pressure ∗ (1 + Bo)p . Unlike the previous section, here we use the function gc from Eq. (15) for the scaling. In such a way, dilation due to an increasing Bond number and inertial efects

1 3 Page 13 of 20 14 Steady state rheology of homogeneous and inhomogeneous cohesive granular materials

(a)

(b)

Fig. 10 The diferences of the steady state volume fractions plotted against a Bond number, Bo, for datasets with fxed inter-particle (b) friction, p , and b inter-particle friction, p , for datasets with fxed Bond number, Bo, as shown in the legend. Lines are the predictions of our proposed model in Eq. (16) Fig. 9 a The change of steady state volume fraction as shown in the color bar, plotted against Bond number Bo and inter-particle friction p ; b continuous phase diagram created from data in a with red and blue indicating relative compression and dilation, respectively. ∗ Bond numbers, defning the cohesive efect on the steady The Inertial number I = 0.02 and Softness p = 0.01 are kept constant for all the simulations shown here state volume fraction: ∗ ∗ = ( p, I, p , Bo)− ( p, I, p ,0) (17) 3.3 The combined efect of inter‑particle friction where positive indicates compression and negative indi- and cohesion cates dilation of the sample relative to the cohesionless case. In Fig. 9a, we plot the dependence of on both inter- As mentioned earlier, cohesion among particles can lead to particle friction and Bond number. Using all points with either compression or dilation of steady granular fow rela- colors indicating relative compression and dilation, we fur- tive to its non-cohesive limit. Which mechanism dominates ther interpolate , and obtain the phase diagram as shown depends on the interplay between inter-particle friction and in Fig. 9b. For weak friction and low to moderate cohe- cohesion. In order to investigate how cohesion works with sion, Bo < 0.1 , volume fraction changes are inconspicuous friction, we choose the SS set-up and systematically vary ( ≈ 0 ). For very high cohesion, compression is observed both inter-particle friction and cohesion to check their com- ( �� > 0 ), while relative dilation is observed ( �� < 0 ) at bined infuences. For each inter-particle friction, we use the intermediate Bond numbers and strong enough inter-particle ( , I, p∗, Bo = 0) steady state volume fraction p as reference friction. However, when the role of friction is almost negligi- and subtract it from the steady state volume fraction of fnite ble ( �p < 0.05 ), we observe a purely compressive efect with

1 3 14 Page 14 of 20 H. Shi et al. increasing Bo. This further confrms that sufcient inter- particle friction leads to relative dilation of cohesive fow, where cohesion plays an important role in enhancing the dilation efect, but turning to compaction if large enough. Although the phase diagram ofers us a nice overview of the relative compression and dilation behaviour, we further study the data more quantitatively by fxing one parameter and looking at the infuence of the second parameter. There- fore, in Fig. 10a and b, we plot the volume fraction for constant inter-particle friction and Bond number, respectively. When we fx the inter-particle friction and increase Bo, we observe negligible volume fraction change ( ≈ 0 ) up to Bo = 0.1 . If we increase the Bond number further, for low inter-particle friction ( p = 0.01 ), we see a monotonically increasing trend, while for moderate to high inter-particle (a) friction ( �p > 0.05 ), we get frst a decrease then an increase, where the point of infection depends on the inter-particle friction: higher inter-particle friction moves this point towards higher Bond number. This could be explained by the infuence of friction: high inter-particle friction with moderate cohesion leads to strong relative dilation, but one needs even stronger cohesion to compensate the dilation efect and turn the bulk back into compaction. The predictions of the rheological model are plotted as lines. Instead of using the coefcients from p dependency ftting in "Appendix 2", here the model parameters (Tables 3 and 5) from the actual data ftting are used. For all the inter-particle frictions presented here, the predictions of the model are quite good up to Bond numbers around 1, then the model over-predicts the diferences of the volume fractions, and the over-prediction increases with the inter-particle friction. Nevertheless, our (b) model still predicts the volume fraction at steady state quite well, the highest deviation between the model prediction Fig. 11 The same data as in Fig. 9a plotted with a diferent color- and the actual data point is around 4% of the actual volume a bar and b the phase diagram with a sharp red/blue-relative compres- fraction value measured from the simulation. Because we sion/dilation transition. The vertical lines are the boundaries between 0.027 focus on a much smaller scale of the relative here (-0.06 to zones 1–2 and 2–3, at p = and 0.4, respectively 0.08), the diferences between the line and the actual data points only appear large. When we fx the Bond number and increase the inter- . Note that here we focus on an even smaller variation of particle friction as shown in Fig. 10b, we observe the turn- volume fraction (− 0.04 < �� < 0.01 ) and thus the devia- ing point more clearly. If the cohesion between particles tions are acceptable. is very small ( Bo = 0.1 ), the volume fraction is not chang- Although we have shown our data in the above men- ing much with inter-particle friction (relative to the Bo = 0 tioned two possible ways, it is still interesting to look fur- case). However, for stronger cohesion between particles ther into this inconspicuous volume fraction change (green) ( Bo = 0.1, 0.5 and 1), we see some relative compression region in Fig. 9b and fnd out where the transition between (positive values) for low inter-particle friction, which turns relative compression and dilation happens. In Fig. 11, into strong relative dilation (negative values) with increas- we narrow down this transition by applying a two-color ing inter-particle friction. Similarly, the predictions of the map. Using the data shown in Fig. 11a, a sharp transition rheological model are also plotted here. As we pointed out line between relative compression and dilation is recon- earlier, for large Bond number, the prediction quality gets structed and highlighted in Fig. 11b. In the parameter range lower. Our model tends to over-predict the steady state vol- ( 0 ⩽ p ⩽ 1 and 0 ⩽ Bo ⩽ 5 ), three regimes can be identi- ume fraction at higher Bond numbers, which results in larger fed: i) pure relative compression for �p ≲ 0.027 ; ii) relative compression followed by relative dilation for increasing Bo

1 3 Page 15 of 20 14 Steady state rheology of homogeneous and inhomogeneous cohesive granular materials and then relative compression again for 0.027 ≲�p ≲ 0.40 ; bottom shear cell. Independent of the type of cohesive iii) pure relative dilation for �p ≳ 0.40 (this value depends model, if the two systems are in same steady state, e.g., on the maximal Bo ≈ 5 chosen). In regime i), the friction same Inertial number, Softness, friction and Bond num- efect is almost negligible such that cohesion dominates ber, the steady state volume fractions from the two geom- the fow behaviour leading only to relative compression. etries agree well in the range of parameters studied. Our In regime ii), both friction and cohesion afect the fow extended rheological model is thus applicable in diferent behaviour: When cohesion is low, it has an almost negli- systems. The fact that diferent contact models can be uni- gible relative compression efect. Increasing cohesion, the fed by the Bond number indicates that the macroscopic relative dilation due to friction is enhanced by cohesion. But rheological behaviour of the steady state volume fraction when cohesion is strong ( Bo > 1 ), the relative compression depends on cohesion intensity but not on the microscopic efect dominates again. In regime iii), the fow shows almost origin of cohesion between particles, whereas macroscopic only relative dilation, mostly contributed by friction, up to friction dose, see Appendix 1. Bo ⩽ 5 . If cohesion is extremely strong, ( Bo > 5 ), we expect Interestingly, when investigating the efects of cohesive that relative compression contributed by cohesion domi- models, we discovered that cohesion can either contribute nates the system again. However, this is beyond the scope to a decrease or an increase of the steady state volume of this study since our simulations are not homogeneous fraction of sheared materials, relative to the non-cohesive with such high cohesion. reference case, enhancing or counter-acting the dilation efect of the inter-particle friction. Using the extended rheological model, we can successfully distinguish the two micro-mechanical mechanisms of cohesion: compression 4 Conclusion and outlook from the increase in the normal contact forces and relative dilation from enhancing both frictional forces as well as We have extended an existing rheological model that pre- structural stability. dicts the relation between volume fraction, Inertial number A phase diagram reveals how the combinations of these and Softness [52], including friction and cohesion depend- two particle parameters lead to sample compression or encies. We have calibrated this extended model without/ dilation in steady state shear, relative to a non-cohesive with cohesion using two diferent simulation geometries: case. In addition, a sharp interface between compression a homogeneous stress controlled simple shear box and an and dilation on our phase diagram allows to categorize inhomogeneous split bottom shear cell. Furthermore, two the explored parameter space into three regimes: i) pure geometries are featured with two diferent cohesive contact relative compression for �p ≲ 0.027 ; ii) non-monotonic models (dry, reversible vs. wet, irreversible). We systemati- behaviour with Bo: relative compression followed by rela- cally varied the inter-particle friction and cohesion in the tive dilation for higher Bo and then relative compression stress controlled simple shear box with shear rate and normal again, for 0.027 ≲�p ≲ 0.40 ; and iii) pure relative dilation stress fxed. We report an interesting interplay between the for �p ≳ 0.40. inter-particle friction and cohesion, as represented in phase The present paper is an extension of former works [52, diagrams. This allows the prediction of compression-dila- 65] on rheological modeling, but with a deeper insight on tion behaviour relative to non-cohesive reference systems the infuence of friction and cohesion. It could be enriched in steady states. by exploring more closely how the micro-structure [64] Besides extending the rheological model towards cohe- is infuenced by the combination of inter-particle friction sive-frictional granular media, we had two main goals: (1) and cohesion or vice-versa. Furthermore, extending the to understand the relation between microscopic properties rheological model towards the intermediate to low volume such as inter-particle friction or/and cohesion and macro- fraction regime, where most dense rheological models fail scopic, bulk properties such as volume fraction; (2) to check but kinetic theory works well, is still a great challenge the validity of our rheological model in both systems i.e., [72]. This will involve the granular temperature results to confrm if the homogeneous representative elementary in at least one more relevant dimensionless number in the volumes (REVs) represent the center of a shear band in an rheology. Moreover, comparing the stress controlled sys- inhomogeneous system (and maybe even the data away from tem used here to a volume controlled system as in [72] is the center). For completeness, the rheological model for the still ongoing research and will be addressed in the future. macroscopic friction is given in "Appendix 1". Furthermore, we introduce a reversible van der Waals Acknowledgements Helpful discussions with A. Singh, D. Vescovi and M.P. van Schrojenstein-Lantman are gratefully acknowledged. This type cohesive force in the simple shear box and an irre- work was fnancially supported by the STW Project 12272 ’Hydro- versible liquid bridge type cohesive force in the split dynamic theory of wet particle systems: Modeling, simulation and validation based on microscopic and macroscopic description’ and the

1 3 14 Page 16 of 20 H. Shi et al.

T-MAPPP project of the European-Union-funded Marie Curie Initial Training Network FP7 (ITN607453); see http://www.t-mappp.eu/ for more information.

Compliance with ethical standards

Conflict of Interest The authors declare that they have no confict of interest.

Open Access This article is distributed under the terms of the Crea- tive Commons Attribution 4.0 International License (http://creativeco mmons.org/licen ses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. (a) Appendix 1: Macroscopic friction coefcient

For a full constitutive law of steady state rheology, one also needs to take into account the shear resistance as quantifed by the macroscopic friction coefcient, = ∕yy . This has been already developed in a previous work, in which the classical − I rheology on hard spheres was generalized for soft, cohesive granular fows [52], involving both dry [64] and wet [52, 65] granular materials. The trends caused by diferent mechanisms are combined and shown to col- lectively contribute to the rheology [52] as multiplicative functions given by: ∗ ∗ ∶= ( p, I, p , Bo)= (I)fp(p )fBo(Bo) (1A) (b) where (I)= 0fI(I) (see Table 6 or [52]) and all functions depend on one dimensionless number, while their coef- Fig. 12 The macroscopic friction coefcient, , plotted against a p∗ cients depend on the particle friction coefcient, see Table 7. inertial number, I, b softness, , using the stress controlled simple shear box and the split bottom shear cell from the same simulations The rheological function Eq. (1A) is slightly simpler than as shown in Fig. 4. The lines are the predictions of Eq. (1A) ftted 0.1645 0.3499 0.0818 the general form in previous work [52], since we have taken using SS data, with 0 = , inf = , I = and p∗ 0.4950 out the contributions of fg and fq due to absence of free = boundaries and too small Inertial numbers in the simple shear box. Remarkably, all the functions were (to leading macroscopic friction is ruled by a power law. The details order) linear in ratios of time scales and in particular, the of each contribution are given in Tables 6 and 7 and then macroscopic friction increases linearly to frst order with explained next. Note that here we limit our ftting range as the Bond number [52]. However, based on the dry, cohesive ∗ following: �>0.5 , 0.005 < I < 0.5 and p < 0.1 . The only data (from SS) shown in the main text for , we propose diferent range used here compared with ftting is the lim- here a new modifcation for the Bond number contribution I 1 its of the inertial number , due to the granular temperature as fBo = 1 + 1Bo , where we claim that the increase of and its consequent creep efect, as described in detail in [52]. Table 6 List of rheological correction functions on for application in a continuum model, see Eq. (1A) Non‑cohesive slightly frictional material Dimensionless numbers Corrections

I ∞∕ 0−1 Inertial number ( ) fI = 1 + First we compare the data for non-cohesive slightly fric- 1+I ∕I tional material using the above mentioned two shear cell p∗ f = 1 − [(1 + Bo)p∗∕p ∗]0.5 Softness ( ) p setups and show the results in Fig. 12. We see an inverse Bo 1 Bond number ( ) fBo = 1 + 1Bo trend of macroscopic friction compared to the trend of vol- Bond number (Bo) in [52] fc = 1 + aBo ume fraction in Fig. 4. The macroscopic friction increases

1 3 Page 17 of 20 14 Steady state rheology of homogeneous and inhomogeneous cohesive granular materials

with Inertial number but decreases with Softness. The lines are the prediction of Eq. (1A) with Bond number equal to zero. The prediction is accurate when the Iner- tial number I is less than 0.5, but deviating for larger I. This can be explained by the considerable dilation of our constant stress shear box setup. When the system goes to very high Inertial numbers, the stress contribution from the kinetic part increases substantially and the granular bulk dilates in order to keep the stress constant; a much reduced bulk density is the consequence. Furthermore, we also compare the results between the two diferent setups: SS and SB, the center of shear band data of SB agrees very (a) well with SS data, which confrms that the shear stress (macroscopic friction) responses are also the same when using the same material (same contact model) independent of the boundary conditions.

Cohesive materials

In Fig. 13, we plot the macroscopic friction, , against Bond number, Bo, with the predictions of our proposed rheological model. For the sake of clearness, we only plot data of three inter-particle friction coefcients instead of all the data we have. With increase of cohesion, the macroscopic friction stays almost constant for Bo < 1 , but increases when the cohesion gets stronger. High attractive forces between particles lead to a more stable packing, thus higher resistance to shear. We have also included the predictions from both (b) our modifed rheological model and the model from previous work [52], shown as solid and dashed lines, respectively. The latter can only capture the behaviour at low Bond number ( Bo < 1 ), while our modifed model agrees very well with the SS data up to Bo < 5 . A possible reason that the previous linear model agrees well with the split bottom shear cell data is that they were ftting a data cloud, see Fig. 13c, which could introduce higher deviations. The data from simple shear box are cleaner and allow us to look at the infuence of cohesion more closely, Fig. 13c nevertheless shows a signifcant qualitative diference between wet and dry cohesive models. All the model parameters are summarized in Table 7, the diference in the coefcients (related to cohesion) from the two setups are due to diferent contact models. (c) If we look at all the friction data (not shown), we observe that the macroscopic friction increases first with inter- Fig. 13 a The macroscopic friction coefcient as a function of Bo = 0, 0.01 particle friction coefcient, but then decreases slightly for Bond number for inter-particle friction coefcients p � > 0.3 and 0.5 from SS data. The Inertial number I = 0.02 and Softness p . This non-monotonic trend might be caused by ∗ p = 0.01 are kept constant for all the simulations shown here. Solid the friction induced, cohesion enhanced micro-structural lines are the predictions of Eq. (1A) with fBo , while dashed lines are anisotropy. The shear stress also shows similar trends as f predictions using the linear form c from [52]. b The same data as the macroscopic friction, whereas the pressure is changing in a but relative to 0 and in logarithmic scale. c The comparison 0.01 between SS and SB data for p = only continuously. The granular temperature time scale, tT is comparable to the shear time scale t ̇� but slightly smaller, while they

1 3 14 Page 18 of 20 H. Shi et al.

Table 7 The ftting coefcients ∗ p 0 inf I p 1 1 a (here) a (in [52]) of the rheological model Eq. (1A) for macroscopic friction. 0.00 (SS) 0.1432 0.3281 0.0645 0.4222 0.3136 1.6475 0.2820 [-] The numbers from [52] have 0.01 (SS) 0.1645 0.3499 0.0818 0.4950 0.4490 1.5218 0.4280 [-] additional correction terms for small I, but ignored for our 0.01 (SB) 0.15 0.42 0.14 0.9 [-] [-] [-] 1.47 ftting here. inf and I are not 0.50 (SS) 0.3530 0.7148 0.7184 0.3905 0.2551 1.5125 0.2496 [-] independent from each other 0.5 0.005 0.5 ∗ 0.1 and thus can vary strongly All the coefcients are ftted using SS data with �> , < I < and p < dependent on the available range of data

Table 8 The summary of ftting coefcients in Eq. (1B) Q Q 0 Coefcients ∞ 0 p RMS Residuals

0 0.5854 0.6551 0.2756 0.00058 I 5.4310 1.9604 0.1166 0.00002 ∗ p 0.2969 0.2714 0.5054 0.00002 − − p1 0.1588 0.0174 0.2952 0.00321 − p2 0.0816 0.0088 0.3159 0.00355 − p3 0.0204 0.0042 0.3355 0.00130 − p4 0.0019 0.0004 0.3489 0.00014

Appendix 2: The infuence of inter‑particle (a) friction on the coefcients in the rheological model of volume fraction

For completeness, one has to also look at the dependency of the inter-particle friction p on the ftting coefcients obtained in our proposed rheological model as presented in Tables 3 and 5. The function used to describe this dependency is:

0 (− p∕ p ) Q = Q∞ +(Q0 − Q∞)e (1B)

where Q represents the coefcients obtained from the rheological model, Q0 and Q∞ represent the values of the coef- 0 fcient at zero and infnite p , respectively. While p refers to the typical inter-particle friction at which the change of (b) coefcient starts saturating with the increase of p. For all the model coefcients, our proposed dependency laws ft well most of the data points with very small values Fig. 14 The efect of inter-particle friction, p , on the ftting coef- a of Root-Mean-Square (RMS) residuals, see the summarized fcient of the dry cohesive model, 0 , in Eq. (13); b the ftting coef- fcient from the cohesive model, p1 , in Eq. (15). The lines are the pre- details in Table 8. For the sake of brevity, we only plot here dictions of Eq. (1B) and details are explained in the main text and two coefcients out of seven due to the similar behaviour summarized in Table 8 among all the coefcients. The results are shown in Fig. 14 with the efects of p on 0 and p1 in (a) and (b), respectively. 0 In the case of 0 , it decreases with characteristic around are both much larger than the pressure time scale tp , which p indicates that neither granular temperature nor shear are the 0.28, then the decrease becomes weaker and saturates 0 I dominating mechanisms for the friction trend, for the range towards large p . The characteristic p for the changes of p∗ p of parameters studied here. is smaller but larger for . For the cohesive coefcient 1 , the decrease becomes weaker at p around 0.29, and this saturation turning points are very similar for all the other

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