2.1. Flexible fibre model

The flexible fibre model used in this study has been described in our recent work (Guo et al. Reference Guo, Wassgren, Hancock, Ketterhagen and Curtis2013b ). In addition, a comprehensive validation of the mechanical responses of a single fibre, including bending, stretching/compressing and twisting deformation, has been made by comparing the model with analytical solutions (Guo et al. Reference Guo, Wassgren, Hancock, Ketterhagen and Curtis2013b ). For convenience, the model is also summarized here. A fibre is formed by connecting a number of identical spheres in a straight line using virtual elastic bonds, as shown in figure 1. Two ends of a single bond, which are fixed on the centres of two neighbouring spheres, translate and rotate with the spheres. Thus, the relative movement of constituent spheres leads to deformation of the bonds and also the fibre. In return, bond forces/moments are generated and exerted on the spheres to resist the deformation. According to the bonded-particle model by Potyondy & Cundall (Reference Potyondy and Cundall2004), the bond forces and moments can be calculated incrementally as follows:

in which $\text{d}F_{n}^{b},\text{d}F_{t}^{b}$ , $\text{d}M_{n}^{b}$ and $\text{d}M_{t}^{b}$ are the incremental bond normal force, shear force, torsional moment and bending moment respectively, and $\text{d}{\it\delta}_{n}^{b}$ , $\text{d}{\it\delta}_{t}^{b}$ , $\text{d}{\it\theta}_{n}^{b}$ and $d{\it\theta}_{t}^{b}$ are the incremental normal deformation, shear deformation, torsional (normal) angular deformation and bending (tangential) angular deformation respectively of the bond. Thus, $K_{n}^{b}$ , $K_{t}^{b}$ , $K_{tor}^{b}$ and $K_{ben}^{b}$ represent the normal, shear, torsional and bending stiffnesses respectively of the bond. The cylindrical bond of Young’s modulus $E_{b}$ and shear modulus $G_{b}$ has a radius $r_{b}$ , a length $l_{b}$ , a cross sectional area $A={\rm\pi}r_{b}^{2}$ , an area moment of inertia $\text{ }I={\rm\pi}r_{b}^{4}/4$ and a polar area moment of inertia $I_{p}={\rm\pi}r_{b}^{4}/2$ . In the present model, the bond radius $r_{b}$ is assumed to be the same as the constituent sphere radius $r_{s}$ , and the bond Young’s modulus $E_{b}$ and shear modulus $G_{b}$ are correlated through the Poisson’s ratio ${\it\zeta}:E_{b}=2(1+{\it\zeta})G_{b}$ . The virtual bond has no mass. The incremental bond deformation can be obtained from the product of the corresponding relative velocity between two bonded spheres ( $v_{n}^{r}$ , $v_{t}^{r}$ , ${\it\omega}_{n}^{r}$ or ${\it\omega}_{t}^{r}$ ) and the time step $\text{d}t$ .

The movement of individual constituent spheres in each fibre-like particle is governed by Newton’s second law of motion, i.e. 

in which $\boldsymbol{x}_{s}$ and ${\bf\theta}_{s}$ are the translational displacement and angular displacement respectively of the sphere of mass $m_{s}$ and moment of inertia $J_{s}=(2/5)m_{s}r_{s}^{2}$ . The translational motion of the sphere is driven by the contact force $\boldsymbol{F}^{c}$ exerted by the spheres that are not bonded to it or come from a different composite particle, the bond force $\boldsymbol{F}^{b}$ exerted by the bonds connecting to it and the gravitational force $m_{s}\boldsymbol{g}$ . The contact force between the spheres from two different composite particles $\boldsymbol{F}^{c}$ and the bond force $\boldsymbol{F}^{b}$ are calculated incrementally based on (2.1) and (2.2). The moment $\boldsymbol{M}^{c}$ arises from the tangential component of the contact force $\boldsymbol{F}^{c}$ , and $\boldsymbol{M}^{b}$ represents the bond moment induced by the shearing, bending and twisting deformation of the bond. A contact force $\boldsymbol{F}^{c}$ can be decomposed into a normal force and a tangential force. The elastic normal force is computed based on the current normal overlap between two spheres using the Hertz model (Johnson Reference Johnson1985). The tangential interaction consists of static and sliding friction processes. For static friction, the tangential force depends on the current normal force and the tangential loading path according to the Mindlin–Deresiewicz theory (Thornton & Yin Reference Thornton and Yin1991). When the tangential force exceeds the product of the friction coefficient ${\it\mu}$ and the current normal contact force $F_{n}^{c}$ , sliding friction occurs. The tangential force in the sliding process is equal to ${\it\mu}F_{n}^{c}$ according to Coulombic sliding friction theory.

In (2.5), the contact damping force $\boldsymbol{F}_{d}^{c}$ is introduced to account for the energy dissipation when two spheres collide. The normal and tangential contact damping forces can be written as

in which $m_{p}$ is the mass of a bonded particle or fibre. The symbols $K_{n}^{c}$ and $K_{t}^{c}$ represent the contact normal stiffness and the contact tangential stiffness respectively. The expressions of $K_{n}^{c}$ and $K_{t}^{c}$ are given by Thornton & Yin (Reference Thornton and Yin1991) and also summarized in Guo (Reference Guo2010). The variables $v_{n}^{c}$ and $v_{t}^{c}$ represent the normal and tangential components respectively of the relative velocity at the contact point. In (2.6), $\boldsymbol{M}_{d}^{c}$ is the moment arising from the contact damping force $\boldsymbol{F}_{d}^{c}$ .

Analogously to contact damping, bond damping is proposed to take into account the energy dissipation due to elastic wave propagation within a particle. Thus, the bond damping forces and moments are assumed to follow

The bond damping coefficient ${\it\beta}_{b}$ varies in the range 0–1 and determines the energy dissipation rate. The bond damping force $\boldsymbol{F}_{d}^{b}$ and bond damping moment $\boldsymbol{M}_{d}^{b}$ (based on (2.9)–(2.12)) are added to (2.5) and (2.6).

Figure 2.

Numerical model of shear flow with flexible fibres (solid volume $\text{fraction}=0.1$ ). The different coloured fibres are for identification purposes only.

2.2. Computational set-up of shear flow

As shown in figure 2, a specified number of fibres were generated in a shear cell, with the fibres randomly positioned and oriented. A procedure for the generation of a densely packed bed of elongated particles in a confined space is provided in Guo et al. (Reference Guo, Wassgren, Ketterhagen, Hancock, James and Curtis2012). Shear was initiated by applying a linear $x$ -velocity ( $V_{x}$ ) profile in the $y$ direction with zero velocity at the centre,

in which the applied shear strain rate ${\it\gamma}$ can be expressed as ${\it\gamma}=2U/L$ , where $L$ represents the distance between the top and bottom boundaries and $U$ is the magnitude of the particle $x$ -velocity at the top and bottom boundaries. In (2.13), $y$ is the $y$ coordinate of the mass centre of a fibre and $y_{c}$ is the $y$ coordinate of the centre of the shear cell. At the beginning of a simulation, each fibre has an initial $x$ -velocity based on its mass centre $y$ coordinate using (2.13). The initial translational velocities in the $y$ and $z$ directions and initial angular velocities are all set to zero for each fibre. Periodic boundary conditions were applied in the $x$ and $z$ directions, and a Lees–Edwards boundary condition (Lees & Edwards Reference Lees and Edwards1972) was adopted in the $y$ direction. Shear flows were performed in the absence of gravity and an interstitial fluid. The volume of shear cell was fixed, and consequently the average solid volume fraction was fixed during the shear flow for a given number of fibres. In the present simulations, the measurement of quantities, such as stress, kinetic energy, potential energy (PE) and deforming parameter, is made when the shear flow reaches steady state. At steady state, these quantities remain approximately constant with limited fluctuations (Guo et al. Reference Guo, Wassgren, Ketterhagen, Hancock, James and Curtis2012).

A shear cell with dimensions $[x,y,z]$ of $[15.12d_{v},15.12d_{v},7.56d_{v}]$ , in which $d_{v}$ is the equivalent sphere volume diameter of a fibre, was employed in this study. According to a shear cell size sensitivity test in our recent work (Guo et al. Reference Guo, Wassgren, Ketterhagen, Hancock, James and Curtis2012), reliable solid-phase stress results can be obtained at a relatively small computational cost using a shear cell of the same size for flows of rigid rod-like particles. The material and operating parameters used in the simulations are listed in table 1. Using a base case set of simulation conditions, the effects of bond stiffness, bond damping, interfibre friction, fibre aspect ratio and solid volume fraction on fibre flow behaviour have been explored.

Table 1.

Material properties and operating parameters used in the simulations. AR, fibre aspect ratio.

3.1. Mechanical energies

As external forces are applied to bulk materials, particles gain kinetic energy as they are propelled to translate and rotate. If the particles are flexible, significant PE can also be generated as they undergo large deformation. Meanwhile, these mechanical energies can be dissipated during particle collision and vibration processes. The kinetic energy of a fibre, $E_{kin}$ , can be decomposed into global kinetic energy (GKE), $E_{kin}^{glo}$ , and local kinetic energy (LKE), $E_{kin}^{loc}$ ,

in which the GKE, $E_{kin}^{glo}$ , is given by the total mass, $m_{p}$ , and the translational velocity of the mass centre of the fibre ( $O$ ), $v_{O}$ , as

The LKE, $E_{kin}^{loc}$ , which describes the kinetic energy relative to the mass centre of the fibre, is given by

in which $N_{s}$ is the number of constituent spheres in the fibre, $v_{si}^{r}$ is the relative velocity of the constituent sphere $i$ of mass $m_{si}$ to the centre of mass of the fibre, and $J_{si}$ and ${\it\omega}_{si}$ are the moment of inertia and angular velocity (in the global frame of reference) respectively of the constituent sphere $i$ . Based on (3.2) and (3.3), the GKE is caused by the translation of the fibre and the LKE is generated by the rotation and vibration of the fibre.

Elastic PE, $E_{pot}$ , is generated due to the elastic deformation of bonds and can be expressed as

in which $N_{b}$ is the number of bonds in a fibre. In addition, ${\it\delta}_{ni}^{b}$ , ${\it\delta}_{ti}^{b}$ , ${\it\theta}_{ni}^{b}$ and ${\it\theta}_{ti}^{b}$ represent the current normal deformation, shear deformation, torsional angular deformation and bending angular deformation respectively of the bond $i$ .

3.2. Deforming parameter

A fibre can experience bending, twisting and axial stretching deformations, among which the bending deformation may be the most significant for elongated fibres. In order to quantify the extent of the bending deformation of a fibre $i$ , a deforming parameter $S_{i}$ is defined as

in which ${\it\theta}_{ij}$ is the angle between two neighbouring bonds (separated by a length of one sphere diameter) and $K$ is the number of angles ${\it\theta}_{ij}$ in the fibre $i$ . As illustrated in figure 3, the maximum angle ${\it\theta}_{max}$ between such two bonds is equal to $2{\rm\pi}/3$ . For a system of $N$ fibres, the mean value of the deforming parameters of all the fibres, $S_{mean}$ , can be used to characterize the extent of fibre deformation for the system,

The mean deforming parameter $S_{mean}$ varies in the range [0, 1], and a larger value of $S_{mean}$ indicates larger average bending deformation of fibres.

Figure 3.

Illustration for the definition of the fibre deforming parameter.

3.3. Solid-phase stress

The solid-phase stress tensor of a bulk system can be expressed as the sum of a kinetic or streaming component and a collisional component. Following Campbell & Gong (Reference Campbell and Gong1986), the kinetic component, which arises from momentum transfer by fibres as they move through the bulk material, is given by

in which $\langle \,\rangle$ represents an average over time and volume, and $\boldsymbol{C}=\boldsymbol{c}_{i}-\langle \boldsymbol{c}\rangle$ is the fluctuating velocity of fibre $i$ . The average velocity $\langle \boldsymbol{c}\rangle$ is the local steady-state shear velocity (i.e. the average velocity in the shear layer where fibre $i$ resides). The collisional component, which arises from momentum transfer from one point to another in the material by interparticle collisions, is written as

where $\boldsymbol{F}_{\boldsymbol{i}}^{\boldsymbol{j}}$ is the contact force exerted on fibre $i$ by fibre $j$ , and $\boldsymbol{l}_{\boldsymbol{i}\boldsymbol{c}}$ is the vector pointing from the mass centre of fibre $i$ to the contact point. Similarly, $\boldsymbol{F}_{\boldsymbol{j}}^{\boldsymbol{i}}$ is the contact force exerted on fibre $j$ by fibre $i$ , and $\boldsymbol{l}_{\boldsymbol{j}\boldsymbol{c}}$ is the vector pointing from the mass centre of fibre $j$ to the contact point. For contact between fibres $i$ and $j$ , according to Newton’s third law, $\boldsymbol{F}_{\boldsymbol{j}}^{\boldsymbol{i}}=-\boldsymbol{F}_{\boldsymbol{i}}^{\boldsymbol{j}}$ , so that (3.8) can be rewritten as

in which $\boldsymbol{l}_{\boldsymbol{i}\boldsymbol{j}}$ represents the vector from the mass centre of fibre $i$ to the mass centre of fibre $j$ . In this study, the measurements of fibre system stresses are made by calculating (3.7) and (3.9) and summing them.

Figure 4.

Snapshots from a collinear collision between two flexible fibres arranged with their two major axes perpendicular to each other. An initial velocity is given to the fibre on the left-hand side: (a) before collision, (b) during collision, (c) after collision.

5.1. Frictionless fibres

Fibre alignment and deformation

Fibre deformation has a significant impact on the microstructure of a fibre system. The studies in § 4 show that fibre deformation strongly depends on the bond stiffness and bond damping for a collinear collision, as shown in figure 4. The effect of solid volume fraction on the extent of fibre deformation in shear flows is investigated in this section, and the effect of interfibre friction is discussed in § 5.2.

Snapshots of shear flows of frictionless flexible fibres at low, intermediate and high solid volume fractions ( $v=0.05$ , 0.3 and 0.466) are shown in figure 7. In a dilute flow with a solid volume fraction of $v=0.05$ , significant fibre bending deformation and vibration are observed after collisions. As the solid volume fraction increases, the deformation of fibres becomes very limited, and the elongated fibres tend to align. For the roughly straight fibres, one may define the line connecting two spheres at each end of a bonded-sphere fibre as the major axis of the fibre. Then, the orientation of an approximately straight fibre can be described using two angles: ${\it\alpha}$ , the angle between the major axis of the fibre and the horizontal plane ( $x{-}z$ plane), and ${\it\phi}$ , the angle between the projection of the major axis on the $x{-}z$ plane and the $x$ -axis, as illustrated in figure 8(a). The probability density functions of the fibre orientation in terms of ${\it\alpha}$ and ${\it\phi}$ are shown in figures 8(b) and 8(c) respectively. In figure 8(b), it is evident that both flexible and rigid fibres tend to align with a small positive angle (approximately $6.5^{\circ }$ ) from the shear plane (i.e.  $x{-}z$ plane). This alignment phenomenon is similar to previous observations (Reddy, Kumaran & Talbot Reference Reddy, Kumaran and Talbot2009; Campbell Reference Campbell2011; Börzsönyi et al. Reference Börzsönyi, Szabó, Törös, Wegner, Török, Somfai, Bien and Stannarius2012a ,Reference Börzsönyi, Szabó, Wegner, Harth, Török, Somfai, Bien and Stannarius b ; Guo et al. Reference Guo, Wassgren, Hancock, Ketterhagen and Curtis2013a ; Hua et al. Reference Hua, Curtis, Hancock, Ketterhagen and Wassgren2013). As shown in figure 8(c), a narrow symmetric probability density function of ${\it\phi}$ is obtained with the peak value at approximately ${\it\phi}=0^{\circ }$ , indicating that the projections of the major axes of the fibres on the $x{-}z$ plane tend to align within a small positive or negative angle from the flow direction (i.e. $x$ direction).

Figure 7.

Snapshots from shear flows of frictionless flexible fibres at low, intermediate and high solid volume fractions $v$ ( ${\it\mu}=0$ , ${\it\beta}_{c}=1.63\times 10^{-2}$ , ${\it\beta}_{b}=0$ , $E_{b}=2.6\times 10^{4}~\text{Pa}$ , $\mathit{AR}=6$ ). The different coloured fibres are for identification purposes only: (a) $v=0.05$ , (b) $v=0.3$ , (c) $v=0.466$ .

Figure 8.

(a) Description of elongated particle orientation using ${\it\alpha}$ , the angle between the major axis of the particle and the horizontal plane ( $x{-}z$ plane), and ${\it\phi}$ , the angle between the projection of the major axis on the $x{-}z$ plane and the $x$ -axis. (b,c) Probability density function of the fibre orientation in terms of the angle ${\it\alpha}$ and the angle ${\it\phi}$ respectively. The flexible fibres have the properties ${\it\mu}=0$ , ${\it\beta}_{c}=1.63\times 10^{-2}$ , ${\it\beta}_{b}=0$ , $E_{b}=2.6\times 10^{4}~\text{Pa}$ , $\mathit{AR}=6$ .

The variation of the deforming parameter ( $S_{mean}$ ) with the solid volume fraction ( $v$ ) for the frictionless fibres with and without bond damping ( ${\it\beta}_{b}$ ) is plotted in figure 9. Without bond damping ( ${\it\beta}_{b}=0$ ), the deforming parameter generally decreases with increasing solid volume fraction. This trend may be attributed to the alignment of fibres at high solid volume fractions, as shown in figure 7. At low solid volume fractions, fibre bending deformation is promoted as two fibres contact with a significant angle between long axes. However, at high solid volume fractions, the alignment of fibres leads to the dominance of end–end contact, for which one end of a fibre touches the end of another fibre, and parallel contact, for which the long axes of two contacting fibres are nearly parallel to each other. The bending deformation of fibres is prohibited by end–end contact and parallel contact. As a result, the deformation of frictionless fibres is reduced due to the fibre alignment as the solid volume fraction increases.

Figure 9.

Deforming parameter as a function of solid volume fraction with and without bond damping ( ${\it\mu}=0$ , ${\it\beta}_{c}=1.63\times 10^{-2}$ , $E_{b}=2.6\times 10^{4}~\text{Pa}$ , $\mathit{AR}=6$ ).

As discussed in § 4, in the presence of bond damping, the vibration of fibres decays and the fibre shape can quickly recover after collisions. As a result, smaller deforming parameters are obtained with bond damping ( ${\it\beta}_{b}=4.2\times 10^{-2}$ ) compared with those without bond damping ( ${\it\beta}_{b}=0$ ), as shown in figure 9. However, an increase in the deforming parameter at high solid volume fractions is observed with bond damping. The collision frequency increases and the time between two consecutive collisions decreases as the solid volume fraction increases. Thus, the time for a vibrating fibre to recover its shape due to bond damping can be longer than the time between two consecutive collisions at high solid volume fractions. As a result, the effect of bond damping on the fibre deformation and vibration is reduced at high solid volume fractions, due to the fact that the time duration between two consecutive collisions is too short for a fibre to recover its shape via bond damping.

Fibre mechanical energies

As discussed previously, kinetic energy reflects translational motion, rotational motion and vibration of fibres, and PE is associated with fibre deformation (see § 3.1). In addition, the conversion between kinetic and potential energies and the energy dissipation have a significant impact on the effective coefficient of restitution for a collision between two fibres (see § 4). Hence, the fibre mechanical energies may provide an important perspective for understanding particle motion, deformation and interaction. As a result, the effects of fibre flexibility, bond damping and solid volume fraction on different forms of mechanical energy, including local and global kinetic energies and PE, are examined in this section.

The average elastic PE, LKE, GKE and TME per fibre, normalized by $m_{p}d_{v}^{2}{\it\gamma}^{2}$ , are plotted as a function of solid volume fraction for shear flows of rigid and flexible fibres in figure 10(a–d). The elastic energy, LKE and GKE are defined in § 3.1, and the TME is the sum of these three energies. Due to the fact that the elastic energy due to particle deformation does not exist for the rigid fibres, the normalized average elastic energies per fibre ( $E_{pot}/(m_{p}d_{v}^{2}{\it\gamma}^{2})$ ) are plotted only for the flexible fibres in figure 10(a). Bond damping leads to a reduction in fibre vibration and consequently dissipation of elastic energy, as shown in figure 5(d). As a result, smaller elastic energy is obtained for flexible fibres with bond damping compared with those without bond damping. The average elastic energy per fibre without bond damping generally decreases with increasing solid volume fraction, while the average elastic energy per fibre with bond damping initially decreases and then increases with increasing solid volume fraction. These trends in elastic energy are similar to those of the deforming parameter (figure 9). This behaviour is not surprising considering that the elastic energy is determined by the bond deformation, as described in (3.4).

Figure 10.

Average mechanical energies per fibre as a function of solid volume fraction for shear flows of rigid and flexible fibres ( ${\it\mu}=0$ , ${\it\beta}_{c}=1.63\times 10^{-2}$ , $\mathit{AR}=6$ ): (a) elastic potential energy (PE), (b) local kinetic energy (LKE), (c) global kinetic energy (GKE), (d) total mechanical energy (TME).

The LKE represents the kinetic energy caused by the rotation and vibration of the fibre (see § 3.1). The flexible fibres can rotate and vibrate, while the rigid fibres only rotate and do not vibrate. As a result, the flexible fibres without bond damping, which vibrate vigorously, can have larger average LKEs than the rigid fibres, as shown in figure 10(b). However, when bond damping is introduced to the flexible fibres, the LKE, like elastic energy, can also be damped rapidly during vibration (see figure 5 d). Thus, the flexible fibres with bond damping have smaller average LKEs than both the flexible fibres without bond damping and the rigid fibres.

The GKE is determined by the translational velocity of a fibre (3.2). For dilute systems, transient and binary collisions are dominant. After a binary collision, the GKE is proportional to $(1+e^{2})$ according to (4.7). Since the effective coefficient of restitution $e$ decreases as follows: rigid particle ${>}$ flexible particle without bond damping ${>}$ flexible particle with bond damping (figure 6), the same rank ordering is obtained for the GKE in dilute systems ( $v<0.2$ ; figure 10 c). For dense granular systems, enduring and multiple interparticle contacts become dominant, and the motion of a fibre, which is constrained by neighbouring fibres, follows the local mean flow. Thus, the translational velocities of fibres are determined mainly by the specified linear velocity profile and shear rate. As a result, the GKEs remain nearly the same for the rigid fibres and flexible fibres with and without bond damping at solid volume fractions greater than 0.2.

The average TME per fibre, which is the sum of the average elastic energy, LKE and GKE per fibre, is plotted in figure 10(d). In dilute systems ( $v<0.2$ ), the average TME decreases sharply with increasing solid volume fraction. Flexible fibres without bond damping have greater average TMEs than rigid fibres due to the large elastic energy and large LKE. Flexible fibres with bond damping can have smaller average TMEs than rigid fibres since the elastic energy, LKE and GKE could all be reduced by the bond damping. In dense systems ( $v>0.2$ ), the average TME changes slightly with increasing solid volume fraction, and the average TMEs of flexible fibres without bond damping are slightly larger than those of rigid fibres and flexible fibres with bond damping. At large solid volume fractions ( $v>0.2$ ), the nearly constant TME is attributed to the dominance of GKEs, which are at least one order of magnitude larger than the elastic energies and LKEs.

Bulk stresses

The stresses of bulk materials are required to determine constitutive relations for granular flows. Hence, the effects of fibre flexibility and bond damping on the stresses are also explored in the present work. Figure 11 shows the shear stress, normalized by ${\it\rho}_{p}d_{v}^{2}{\it\gamma}^{2}$ , as a function of solid volume fraction for shear flows of rigid fibres and flexible fibres with and without bond damping. Similar trends are observed for both the rigid and the flexible fibres. As the solid volume fraction increases, the normalized shear stress initially decreases due to the reduction in mean free path (leading to a reduction in velocity fluctuations) and then increases due to the increase in collision frequency. Compared with rigid fibres, smaller stresses are obtained for flexible fibres without bond damping, and even smaller stresses are obtained with bond damping. One of the major reasons for the stress difference between the rigid fibres and flexible fibres is due to the difference in the effective coefficient of restitution $e$ . As shown in figure 6, the effective coefficients of restitution $e$ of the flexible fibres without bond damping are smaller than that of the rigid fibres due to the conversion of part of the kinetic energy to PE. In addition, even smaller $e$ values are obtained for the flexible fibres with bond damping due to energy dissipation in the vibration of fibres. According to granular kinetic theory for spherical particles (Lun et al. Reference Lun, Savage, Jeffrey and Chepurniy1984), the granular temperature $T$ is proportional to $1/(1-e^{2})$ , the shear stress ${\it\sigma}_{xy}$ is proportional to $\sqrt{T}$ and the normal stress ${\it\sigma}_{yy}$ is proportional to $T$ . Therefore, smaller effective coefficients of restitution $e$ lead to lower granular temperatures, and consequently smaller stresses. In addition, compared with the rigid particles, the deformation of flexible fibres allows the fibres to be rearranged in a more restrictive space and therefore smaller contact forces are obtained.

Figure 11.

Normalized shear stress as a function of solid volume fraction for shear flows of rigid and flexible fibres ( ${\it\mu}=0$ , ${\it\beta}_{c}=1.63\times 10^{-2}$ , $\mathit{AR}=6$ ).

The apparent friction coefficient, defined as the ratio of shear to normal stress magnitudes ( $|{\it\sigma}_{xy}|/{\it\sigma}_{yy}$ ), is plotted as a function of solid volume fraction in figure 12. Granular kinetic theory predicts that the apparent friction coefficient, $|{\it\sigma}_{xy}|/{\it\sigma}_{yy}$ , is proportional to $1/\sqrt{T}$ . Thus, as the effective coefficient of restitution $e$ decreases, the granular temperature $T$ decreases, leading to an increase in the apparent friction coefficient. As a result, larger apparent friction coefficients are observed for flexible fibres, particularly with bond damping, compared with rigid fibres. In general, the apparent friction coefficient decreases with increasing solid volume fraction, as the fibres tend to align and move in horizontal layers (see figure 7). However, a much steeper decrease is observed for the flexible fibres with bond damping, and the apparent friction coefficients of rigid and flexible fibres tend to converge at large solid volume fractions. These observations indicate that the fibre flexibility and bond damping have a smaller impact on the apparent friction coefficient in dense flows than in dilute flows.

Figure 12.

The apparent friction coefficient as a function of solid volume fraction for rigid and flexible fibres without friction ( ${\it\mu}=0$ , ${\it\beta}_{c}=1.63\times 10^{-2}$ , $\mathit{AR}=6$ ).

According to (1.1), the normalized normal stress ${\it\sigma}_{yy}$ (i.e. pressure $P$ ) can be correlated with the inertial number $I$ as follows:

Therefore, the solid volume fraction and bulk friction coefficient can also be expressed as a function of the inertial number.

Figure 13.

Snapshots from shear flows of flexible fibres with a friction coefficient of ${\it\mu}=0.5$ at various solid volume fractions ( ${\it\beta}_{c}=1.63\times 10^{-2}$ , $E_{b}=2.6\times 10^{4}~\text{Pa}$ , ${\it\beta}_{b}=0$ , $\mathit{AR}=6$ ). The different coloured fibres are for identification purposes only: (a) $v=0.05$ , (b) $v=0.3$ , (c) $v=0.466$ .

Figure 14.

Effect of friction on fibre deformation ( ${\it\beta}_{c}=1.63\times 10^{-2}$ , $E_{b}=2.6\times 10^{4}~\text{Pa}$ , ${\it\beta}_{b}=0$ , $\mathit{AR}=6$ ).

5.2. Effect of interfibre friction

Previous studies of rigid rod-like particles (Guo et al. Reference Guo, Wassgren, Hancock, Ketterhagen and Curtis2013a ) indicate that interparticle friction has a significant effect on the particle flow pattern and solid-phase stresses. Subsequently, it is interesting to examine the effect of friction on flexible fibre flows. Snapshots of shear flows of flexible fibres with an interfibre friction of ${\it\mu}=0.5$ at various solid volume fractions are shown in figure 13. At an intermediate solid volume fraction ( $v=0.3$ ), the fibres with friction tend to align with a small angle from the flow direction, as observed for frictionless fibres (figure 7 b). However, large fibre deformation is observed at the high solid volume fraction of $v=0.466$ (see figure 13 c) due to the fact that the frictional forces can cause strong contacts and large fibre bending. A comparison of deforming parameters with and without friction is made in figure 14. The deforming parameter decreases monotonically with increasing solid volume fraction for the frictionless flexible fibres, while a U-shaped trend is observed for the flexible fibres with friction. At small solid volume fractions ( $v<0.3$ ), smaller deforming parameters are obtained with friction than without friction, due to the dissipation of kinetic energy by sliding friction. However, at large solid volume fractions ( $v>0.3$ ), larger deforming parameters are obtained with friction compared with the without friction case, which is consistent with the observations of fibre deformation in figures 7(c) and 13(c).

Figure 15.

The average mechanical energies per fibre as a function of solid volume fraction for shear flows of rigid and flexible fibres with and without friction ( ${\it\beta}_{c}=1.63\times 10^{-2}$ , $E_{b}=2.6\times 10^{4}~\text{Pa}$ , ${\it\beta}_{b}=0$ , $\mathit{AR}=6$ ): (a) elastic potential energy (PE), (b) local kinetic energy (LKE), (c) global kinetic energy (GKE), (d) total mechanical energy (TME).

The average elastic PE, LKE, GKE and TME per fibre, normalized by $m_{p}d_{v}^{2}{\it\gamma}^{2}$ , are plotted as a function of solid volume fraction for shear flows of rigid and flexible fibres with and without friction in figure 15. Since the rigid fibres have no elastic energies, only the elastic energies of flexible fibres are plotted in figure 15(a). Similarly to the deforming parameter (figure 14), the elastic energy also shows a U-shaped trend with increasing solid volume fraction for the flexible fibres with friction. The similarity reflects the dependence of the elastic energy on the fibre deformation. U-shaped trends are also observed for the LKEs of rigid and flexible fibres with friction, as shown in figure 15(b). For the flexible fibres with friction, the increased number of contacts at large solid volume fractions leads to larger fibre deformation, larger PE and stronger fibre vibration. For rigid fibres with friction, the increased number of contacts at large solid volume fractions can increase the rotation of fibres, causing the increase in the LKE. The average GKEs of rigid and flexible fibres with and without friction are plotted in figure 15(c). It can be seen that the interfibre friction leads to smaller GKEs at small solid volume fractions ( $v<0.1$ ), due to the dissipation of kinetic energy through sliding friction. In dense flows of rigid fibres with friction, the fibres have a good chance of interlocking and forming agglomerates that possess large contact forces under the combined effects of high fibre stiffness and large friction. The shear-induced breakup of fibre agglomerates can result in high fibre velocities. As a result, large GKEs are obtained for the rigid fibres with friction at large solid volume fractions ( $v>0.3$ ). The average TMEs are plotted in figure 15(d). At small solid volume fractions ( $v<0.2$ ), smaller TMEs are obtained for the fibres with friction compared with the without friction case due to the fact that the friction can cause a reduction in the elastic energy, LKE and GKE in dilute flows. At large solid volume fractions, large TMEs are obtained for the fibres with friction. The large TME of rigid fibres with friction is mainly due to the large GKE caused by the strong contacts of fibres (figure 15 c), while the large TME of flexible fibres with friction is primarily attributed to the large elastic energy induced by the large fibre deformation (figure 15 a).

Figure 16.

The normalized shear stress as a function of solid volume fraction in shear flows of rigid and flexible fibres with and without friction ( ${\it\beta}_{c}=1.63\times 10^{-2}$ , $E_{b}=2.6\times 10^{4}~\text{Pa}$ , ${\it\beta}_{b}=0$ , $\mathit{AR}=6$ ).

The variation of normalized shear stress with solid volume fraction for shear flows of rigid and flexible fibres with and without friction is plotted in figure 16. The insets show the networks of force chains for the frictionless and frictional flexible fibres at a solid volume fraction of $v=0.466$ . A force chain is plotted by connecting the centres of two contacting constituent spheres from different fibres using a line, and the thickness of the line is scaled by the magnitude of the contact force. The effect of friction on the stress (figure 16) is similar to the effect of friction on the mechanical energies (figure 15). Compared with the fibres without friction, smaller stresses are obtained at small solid volume fractions and much larger stresses are observed at large solid volume fractions for the fibres with friction. At small solid volume fractions, the kinetic stress, which is determined by the fluctuating velocities, dominates the total stress. The reduction in GKE due to sliding friction (figure 15 c) leads to a decrease in the fluctuating velocities and consequently the kinetic stresses. At large solid volume fractions, the enhanced interfibre contacts, as indicated by the dense network of force chains, lead to larger stresses for the frictional fibres. In addition, a less pronounced stress increase at high solid volume fractions is observed for the frictional flexible fibres than the frictional rigid fibres. The apparent friction coefficients of the rigid and flexible fibres with and without friction are plotted in figure 17. Similarly to the rigid fibres, interfibre friction leads to a significant increase in apparent friction coefficient for the flexible fibres. In addition, the apparent friction coefficient decreases as the solid volume fraction increases.

Figure 17.

The apparent friction coefficient as a function of solid volume fraction for rigid and flexible fibres with and without friction ( ${\it\beta}_{c}=1.63\times 10^{-2}$ , $E_{b}=2.6\times 10^{4}~\text{Pa}$ , ${\it\beta}_{b}=0$ , $\mathit{AR}=6$ ).

5.3. Jamming in dense flows

In frictional fibre flows at high solid volume fractions, long lasting, dense and system spanning force chains are formed to strongly resist the flow, causing a sharp stress increase (see figure 16). This fluid-like to solid-like phase transition phenomenon is defined as jamming. The dependence of the jamming onset on the interfibre friction, fibre surface roughness, bond Young’s modulus (or stiffness) and fibre aspect ratio is explored in this section.

Figure 18 shows the normalized shear stress, coordination number and apparent friction coefficient as a function of interfibre friction coefficient ${\it\mu}$ for dense shear flows with $v=0.466$ . The coordination number is defined as the average number of contacting neighbours per fibre, i.e.  $(\sum _{i=1}^{N}N_{i}^{n})/N$ , in which $N_{i}^{n}$ is the number of neighbouring fibres in contact with the fibre $i$ and $N$ is the total number of fibres in the system. Two types of fibre with the same aspect ratio of $\mathit{AR}=6$ have been considered: (i) a rough fibre, which is formed by gluing six identical spheres in a line with the distance between two neighbouring spheres equal to the diameter of the sphere (no sphere–sphere overlap), and (ii) a smoothed fibre, which is formed by gluing eleven identical spheres in a line with the distance between two neighbouring sphere centres equal to the radius of the sphere (allowing sphere–sphere overlaps). For rough fibres, rapid increases in the shear stress, coordination number and apparent friction coefficient with increasing ${\it\mu}$ are obtained when ${\it\mu}$ is less than 0.4, while these three quantities tend to saturate when ${\it\mu}$ is greater than 0.4. The saturation of the apparent friction coefficient (or the internal friction angle) with increasing ${\it\mu}$ was also observed in spherical particle systems (Thornton Reference Thornton2000; Suiker & Fleck Reference Suiker and Fleck2004; Ketterhagen et al. Reference Ketterhagen, Curtis, Wassgren and Hancock2009). Suiker & Fleck (Reference Suiker and Fleck2004) and Ketterhagen et al. (Reference Ketterhagen, Curtis, Wassgren and Hancock2009) argued that the saturation is caused by the rotation of particles, because the apparent friction coefficient increases sharply with increasing ${\it\mu}$ when particle rotation is inhibited. In the present work, it is observed that in dense flows, the fibres with friction are interlocked to form agglomerates, which can rotate en masse. The rotation of fibre agglomerates may play a similar role to the rotation of spheres in the previous work (Suiker & Fleck Reference Suiker and Fleck2004; Ketterhagen et al. Reference Ketterhagen, Curtis, Wassgren and Hancock2009) in facilitating shear deformation of granular assemblies.

Figure 18.

(a) Shear stress and coordination number and (b) apparent friction coefficient as a function of interfibre friction coefficient, ${\it\mu}$ ( $v=0.466$ , $\mathit{AR}=6$ , ${\it\beta}_{c}=1.63\times 10^{-2}$ , $E_{b}=2.6\times 10^{4}~\text{Pa}$ , ${\it\beta}_{b}=0$ ).

Large fibre deformation (figure 13 c), large stress and a dense network of force chains (figure 16) are observed for dense flows of rough fibres with ${\it\mu}=0.5$ at $v=0.466$ . However, for dense flows of smoothed fibres with the same friction coefficient and concentration ( ${\it\mu}=0.5$ and $v=0.466$ ), much smaller shear stress, coordination number and apparent friction coefficient are obtained compared with rough fibres, as shown in figure 18. The smoothed surface leads to a reduced chance of fibre interlocking and consequently reduced fibre contacts. As shown in figure 19, the smoothed fibres tend to align in the flow direction with a small number of contact force chains scattered throughout the shear cell. Hence, smoothing the fibre surface, as well as reducing the friction coefficient, decreases the tangential contacts between fibres.

Figure 19.

A snapshot and corresponding force chains of dense shear flows with smoothed fibres ( $v=0.466$ , $\mathit{AR}=6$ , ${\it\mu}=0.5$ , ${\it\beta}_{c}=1.63\times 10^{-2}$ , $E_{b}=2.6\times 10^{4}~\text{Pa}$ , ${\it\beta}_{b}=0$ ). The different coloured fibres are for identification purposes only.

The bond Young’s modulus $E_{b}$ determines the stiffness of fibres. The dependence of shear stress, coordination number and apparent friction coefficient on the bond Young’s modulus for dense flows of frictional rough fibres ( ${\it\mu}=0.5$ and $v=0.466$ ) is shown in figure 20. In general, the shear stress and coordination number both increase as $E_{b}$ increases. In dense flows in a shear cell with a fixed volume, larger fibre stiffness leads to increased contact forces and contact number. It is found that softer frictional fibres tend to roll into round agglomerates under shear, while stiffer fibres are more resistant to bending, remain straight and tend to align with a small angle from the flow direction. The alignment of elongated particles can lead to a reduction in the apparent friction coefficient, as also reported in Guo et al. (Reference Guo, Wassgren, Hancock, Ketterhagen and Curtis2013a ). This reduction occurs because the alignment of the major axes of elongated particles affects the contact force transmission: the normal force transmission (perpendicular to the particle major axis and flow direction) is more favoured than the shear force transmission (parallel to the particle major axis and flow direction). As a result, the extent of increase in the normal stress ${\it\sigma}_{yy}$ is greater than the extent of increase in the shear stress ${\it\sigma}_{xy}$ as the bond Young’s modulus $E_{b}$ increases. Thus, the apparent friction coefficient, i.e. the ratio of ${\it\sigma}_{xy}$ to ${\it\sigma}_{yy}$ , decreases as $E_{b}$ increases (figure 20 b). Further, the shear stress, coordination number and apparent friction coefficient of the flexible fibres all approach the values for the rigid fibres as the bond stiffness increases.

Figure 20.

(a) Shear stress and coordination number and (b) apparent friction coefficient as a function of bond Young’s modulus ( $v=0.466$ , $\mathit{AR}=6$ , ${\it\mu}=0.5$ , ${\it\beta}_{c}=1.63\times 10^{-2}$ , ${\it\beta}_{b}=0$ ).

The effect of the fibre aspect ratio AR on the shear stress, coordination number and apparent friction coefficient in dense flows of frictional rough fibres ( ${\it\mu}=0.5$ and $v=0.4$ ) is shown in figure 21. In dense frictional flows, the elongation of fibres increases the possibility for the fibres to touch their neighbours, therefore increasing contact forces and contact number. As a result, both shear stress and coordination number increase with increasing AR (figure 21 a). It is observed that sharp increases in the shear stress and coordination number are obtained with intermediate fibre aspect ratios of $4<\mathit{AR}<6$ , while slow increases are obtained with smaller values of $2<\mathit{AR}<4$ and larger values of $6<\mathit{AR}<8$ . The rapid increase in stress at intermediate ARs may indicate that a critical AR exists for a given solid volume fraction. Jamming occurs with the formation of a dense contact force network at the critical AR. However, the stress remains at a low level before jamming occurs (with smaller ARs), and the stress is large but increases more slowly with AR after the contact force network and jamming are established (with larger ARs). As the fibre AR increases, the alignment of stiff fibres in the flow direction is enhanced. As discussed above, normal force transmission becomes more favoured than shear force transmission due to the alignment, and a larger extent of increase is obtained for the normal stress ${\it\sigma}_{yy}$ than the shear stress ${\it\sigma}_{xy}$ with increasing AR. Thus, the apparent friction coefficient decreases as the AR increases, as shown in figure 21(b).

Figure 21.

(a) Shear stress and coordination number and (b) apparent friction coefficient as a function of fibre AR ( $v=0.4$ , ${\it\beta}_{c}=1.63\times 10^{-2}$ , $E_{b}=1.0\times 10^{8}~\text{Pa}$ , ${\it\beta}_{b}=0$ , ${\it\mu}=0.5$ ).