Shear jammed, fragile, and steady states in homogeneously strained granular materials

TL;DR

This study introduces a novel experimental setup to explore the jamming phase diagram of sheared granular materials, revealing detailed insights into shear-jammed, fragile, and steady states, and identifying critical packing fractions.

Contribution

The paper presents a new Couette shear setup enabling precise measurements of granular states and phase boundaries, including the first experimental observation of fragile states under shear.

Findings

Identification of the shear-jamming transition at φ_SJ ≈ 0.74.

Observation of fragile states at φ < φ_SJ under large shear strain.

Detection of flow deviations and shear band formation at higher packing fractions.

Abstract

We study the jamming phase diagram of sheared granular material using a novel Couette shear set-up with multi-ring bottom. The set-up uses small basal friction forces to apply a volume-conserving linear shear with no shear band to a granular system composed of frictional photoelastic discs. The set-up can generate arbitrarily large shear strain due to its circular geometry, and the shear direction can be reversed, allowing us to measure a feature that distinguishes shear-jammed from fragile states. We report systematic measurements of the stress, strain and contact network structure at phase boundaries that have been difficult to access by traditional experimental techniques, including the yield stress curve and the jamming curve close to $\phi_{SJ}\approx 0.74$, the smallest packing fraction supporting a shear-jammed state. We observe fragile states created under large shear strain over a range of $\phi < \phi_{SJ}$. We also find a transition in the character of the quasi-static steady flow centered around $\phi_{SJ}$ on the yield curve as a function of packing fraction. Near $\phi_{SJ}$, the average contact number, fabric anisotropy, and non-rattler fraction all show a change of slope. Above $\phi_{F}\approx 0.7$ the steady flow shows measurable deviations from the basal linear shear profile, and above $\phi_c\approx 0.78$ the flow is localized in a shear band.

Equations

$$Q=Q_{st}+c\ast e^{-\gamma /{\gamma }_c}$$

$${\gamma }_{SJ}(\varphi )={\gamma }_b{\left[\ln \left(\frac{{\varphi }_J^0-{\varphi }_{SJ}}{\varphi -{\varphi }_{SJ}}\right)\right]}^{\alpha }$$

$$\hat{R}=\frac{1}{S}\sum _{i,j}{\vec{r}}_{ij}\otimes {\vec{r}}_{ij}=\frac{1}{S}\sum _{i,j}\left(\begin{array}{cc}{r}_{ij,x}{r}_{ij,x}& r_{ij,x}{r}_{ij,y}\\ r_{ij,y}{r}_{ij,x}& r_{ij,y}{r}_{ij,y}\end{array}\right)$$

Full text

Granular matter, Shear jamming, Strain amplitude, Couette multi-ring bottom geometry, Photoelasticity

When a granular material prepared in a stress free state is sheared, it can make a transition into a mechanically stable state through a process known as shear jamming Bi et al. (2011). Shear jamming occurs in many different systems, including glasses Urbani and Zamponi (2017), suspensions Han et al. (2016); Majumdar et al. (2017); Han et al. (2018); James et al. (2018); Chen et al. (2019); Seto et al. (2019) and dry granular matter with Howell et al. (1999); Bi et al. (2011); Ren et al. (2013); Zheng et al. (2014); Wang et al. (2018); Otsuki and Hayakawa or without Chen et al. (2018); Kumar and Luding (2016); Bertrand et al. (2016); Baity-Jesi et al. (2017) friction. In 2011, Bi $et~{}al.$ Bi et al. (2011) provided a jamming phase diagram (Fig. 1(a)) that extended the Liu-Nagel framework Liu and Nagel (1998) by including a region of shear-jammed (SJ) states for frictional granular materials at finite shear stress with packing fractions $\phi$ between a critical value $\phi_{SJ}$ and $\phi_{J}^{0}$, the isotropic jamming packing fraction for frictionless particles. Starting from a stress free state, applying shear strain $\gamma$ can lead to two different types of jammed states: fragile (F) states that are only stable for compatible loads, and SJ states that are stable to reverse shear Cates et al. (1998); Bi et al. (2011). A minimum shear strain $\gamma_{SJ}(\phi)$ is needed to create a SJ state for fixed $\phi$. In the past decade, many efforts have focused on explaining the origin of rigidity in sheared granular matter with $\phi$ close to the high packing fraction portion of the jamming curve (the yellow curve in Fig. 1(a)) Bi et al. (2011); Kumar and Luding (2016); Wang et al. (2018); Vinutha and Sastry (2016); Sarkar et al. (2013); Sarkar and Chakraborty (2015); Sarkar et al. (2016). However, less attention has been paid to other parts of the phase diagram, in particular to the yield stress curve, which is important for the rheology of dense granular flow, or to the jamming curve close to the critical packing fraction $\phi_{SJ}$, where the relation between the shear strain $\gamma$ and jamming has not been experimentally determined.

Experimental measurements of the phase boundaries in the jamming phase diagram are challenging because it is hard to create SJ states without the formation of a shear band and the associated heterogeneities in the packing fraction $\phi$ and strain field Veje et al. (1999); Zhang et al. (2010); Moosavi et al. (2013); Coussot et al. (2002); Fenistein and van Hecke (2003); Ren et al. (2013). In 2013, Ren $et~{}al.$ Ren et al. (2013) developed a multi-slat, simple shear setup that avoids shear banding, which revealed a distinction between F and SJ states Sarkar et al. (2013, 2016). However, their multi-slat setup had a strain limit ($\sim 60\%$) Ren et al. (2013), and thus could not access the yield stress curve or the SJ states near $\phi_{SJ}$, where $\gamma_{SJ}$ keeps growing as $\phi\rightarrow\phi_{SJ}^{+}$ Bi et al. (2011); Bertrand et al. (2016); Kumar and Luding (2016); Zheng et al. (2014).

In this letter we solve this challenge using a multi-ring Couette shear set-up, which applies a linear shear strain field using basal friction forces to drive the system until it becomes shear-jammed. This form of driving may be thought of as a physical implementation of the algorithm used in certain athermal, quasistatic (AQS) simulations Maloney and Lemaître (2006); Vinutha and Sastry (2016). With our apparatus, we can also keep shearing the jammed system using boundary racks to measure the yield stress curve. By shearing a layer of photoelastic disks, we for the first time experimentally map out the phase boundaries in the jamming phase diagram close to $\phi_{SJ}$, including the yield stress curve and the jamming curve. We find that fragile states exist below $\phi_{SJ}$ that were not included in the traditional phase diagram Bi et al. (2011). Moreover, we find two transitions on the yield stress curve: ($i$) above $\phi_{F}\approx 0.7$, the steady states no longer deform linearly under shear, and ($ii$) above $\phi_{c}\approx 0.78$ their deformation field becomes localized. We relate those transitions to the contact network structures.

Experiments – The experiments are carried out with a novel multi-ring Couette shear set-up shown in Fig. 1(b), which quasi-statically and linearly shears a 2D granular medium composed of bidisperse photoelastic discs with friction coefficient $0.9$ and diameters $1.59$ cm and $1.27$ cm (denoted as $d$) 111See Supplemental Material at [URL will be inserted by publisher] for (1) more details of the set-up, (2) a stress strain curve of a single particle, (3) the pressure calibration, (4) the fabric tensor calculation, (5) raw data for reverse shear tests and (6) the choice of noise level.. The ratio of the numbers of big and small particles is $1/3$. Particles have reflective paint on their bases to enable reflective photoelasticimetry Puckett and Daniels (2013); Zhao et al. (2017); Daniels et al. (2017); Zadeh et al. . The total number of particles is varied from $1447$ to $2101$, which corresponds to $0.56<\phi<0.82$. The Couette set-up consists of $21$ independently controlled concentric rings. The $1.2$ cm wide rings rotate collectively, providing weak frictional forces to the particles sitting on them. Although essential to perform the linear shear, the magnitude of basal friction is $\sim 8$ times smaller than the typical contact forces measured in the SJ states on the jamming curve (Fig. 1(a)). Particles are constrained radially by outer and inner toothed boundaries of radius $r_{out}=35.5~{}cm$ and $r_{in}=8.7~{}cm$. The outer boundary rotates with the rings and the inner boundary is fixed.

For each experiment, a stress-free random configuration is prepared. The quasi-static linear shear is then applied in a stepwise manner. For each step, the ring at radial position $r$ rotates through an arc length $d(r)=\gamma r$. The function $d(r)$ sets the ‘basal profile’ and $\gamma$ is called the ‘shear strain’ by analogy with traditional simple shear Ren et al. (2013). We note that $\gamma$ is not the physical shear strain, i.e., the off-diagonal element of the strain tensor, $\varepsilon_{r\theta}=\partial_{r}d(r)-d(r)/(r+r_{in})=\gamma r_{in}/(r+r_{in})$ Landau and Lifshitz (1986). During a rotation step, in which $\delta\gamma=0.6\%$, the shear rate is $\dot{\gamma}\sim 10^{-3}s^{-1}$. After each step, the rings stop for $10$ s to let the system reach a static state. As plotted in Fig. 1(c), for a dilute system, the azimuthal particle displacements $u_{\theta}$ per step follow $d(r)$, and the radial displacements $u_{r}$ fluctuate around zero. No shear band is observed. We apply large forward strains to measure the yield stress curve, and the strain direction is then reversed to distinguish fragile and shear-jammed states.

The system is sequentially lit from the top by circular polarized green light, and from the side by ultra-violet (UV) light Note (1). Between two consecutive shear steps, after reaching a static state, the system is imaged (Canon EOS 70D, $5472\times 3648$ px2) through a circular polarizer with UV and polarized lights. UV images (Fig. 1(e)) give particle positions. The polarized images (Fig. 1(d)) give stress and contact information. We measure the pressure $P$, defined as the trace of the force moment tensor Bi et al. (2011); Ren et al. (2013), using the averaged squared intensity gradient Howell et al. (1999); Ren et al. (2013); Daniels et al. (2017); Zadeh et al. ; Zhao et al. (2019, ) of the polarized image Note (1). A sheared system must develop a non-zero $P$ to resist finite shear stress $\tau$. We also measure the non-rattler contact number $Z_{nr}$, defined as the mean contact number among stressed grains Majmudar et al. (2007); Bi et al. (2011); Daniels et al. (2017) (see Daniels et al. (2017) for a detailed description of the detection algorithm), the non-rattler fraction $f_{nr}$, defined as the number fraction of stressed grains, and the fabric anisotropy $\rho$, defined as the ratio between the difference and the sum of the eigenvalues of the fabric tensor Note (1).

Results – Figures 2(a) and (b) show pressure $P$ and non-rattler contact number $Z_{nr}$ versus shear strain $\gamma$, for typical runs with different $\phi$. For a given $\phi$, after a transient growth regime, both $Z_{nr}$ and $P$ fluctuate around constant values that define the yield stress curve. We refer the associated stress as the “steady states” stress. We find that $Z_{nr}$ can be fitted to:

[TABLE]

where $Q$ can be $Z_{nr}$, $f_{nr}$ or $1-\rho$, and $Q_{st}$, $c$ and $\gamma_{c}$ are fit parameters. An example fit for $Z_{nr}(\gamma)$ with $\phi=0.76$ is plotted in Fig. 1(b). We find that the steady regime has been reached at $\gamma_{st}\equiv 3\gamma_{c}$ for all state variables, where $\gamma_{c}$ is obtained from the fits for $Z_{nr}$. Figure 2(c) shows $\gamma_{st}(\phi)$, where a linear fit $\gamma_{st}\propto(\phi-\phi_{0})$ for $\phi>0.72$ gives $\phi_{0}=0.84\pm 0.02$, close to the frictionless isotropic jamming density O’Hern et al. (2003). The slope is $-1545\pm 427$ (%).

We identify a system as shear-jammed (SJ) if under reverse shear the pressure never drops below the noise threshold $P_{noise}=0.3$ N/m Note (1), which indicates that the system resists the reversed stress rather than simply allowing a reversion to a stress-free (unjammed) state. Figure 2(d) shows the evolution of $P$ during a shear cycle for a system with $\phi=0.781$. Figure 2(e) plots the dependence of the minimum pressure $P_{min}$ during reverse shear on the maximum forward shear strain $\gamma_{max}$, from which we extract the minimum strain, $\gamma_{SJ}$, required to create a SJ state. We find no SJ state for $\phi=0.74$ even when $\gamma_{max}\gg\gamma_{st}$ Note (1). For $\phi=0.75$ $\gamma_{SJ}\approx\gamma_{st}$. The minimum packing fraction that supports shear jamming must lie between these two values: $\phi_{SJ}=0.745\pm 0.005$. Figure 3(a) plots the relation between $\gamma_{SJ}$ and $\phi$, which can be fitted using a form suggested by Kumar and Luding (2016),

[TABLE]

where $\phi_{SJ}=0.745$ is preset and the fit parameters are $\alpha=0.68\pm 0.11$, $\gamma_{b}=64\pm 6(\%)$ and $\phi_{J}^{0}=0.820\pm 0.005$.

In this work, fragile (F) states refer to states with non-zero pressure ($P>P_{noise}$) and have $P_{min}<P_{noise}$ at some point in the reverse shear process. As shown in Fig. 3(a), we find $\gamma_{F}$, the minimum strain required to create a fragile state, also follows Eq. 2. In this fit, we take $\phi_{J}^{0}=0.82$ from the previous fit, and we determine $\phi_{F}$, the minimum packing fraction for fragile states, from the fit, obtaining $\phi_{F}=0.706\pm 0.003$ along with $\gamma_{b}=19\pm 2$ (%) and $\alpha=0.86\pm 0.12$. We also note, however, that the divergence predicted by Eq. 2 near $\phi_{SJ}$ and $\phi_{F}$ is not clearly seen in our data. Below $\phi_{F}$, the steady state pressure falls to a plateau value near the noise level.

Figure 3(b) shows the experimentally constructed jamming phase diagram in the $(P,\phi)$ space. The yield stress curve is the $P_{st}(\phi)$ curve, showing the average steady state pressure for each $\phi$. $P_{st}$ increases monotonically from $\phi_{F}$ and appears to have an inflection point at $\phi_{SJ}$. However, for the steady states above $\phi\approx 0.78$, the pressure of some particles becomes so large that their photoelastic fringes can not be resolved, likely leading to artificially low pressure measurements. $P_{st}(\phi)$ also separates SJ states and the dynamic unjammed states, which have non-zero shear rates. The jamming curve is also plotted as the $P(\phi,\gamma_{SJ})$ curve, which consists of the pressure value for each $\phi$ at the jamming strain $\gamma_{SJ}$. The gray region below $P_{noise}$ refers to the static unjammed states without measurable stress. Figure 3(e) extends (b) by including the inverted strain axis and plots all the static states measured during the forward shear process in the $(P,\phi,1/\gamma)$ space, highlighting their dependence on the driving strain $\gamma$. A state is labeled SJ when the shear strain exceeds $\gamma_{SJ}$ determined using Eq. 2. All static SJ (green), F (red) and unjammed (gray) states lie approximately on a smooth surface in the 3D space.

To quantify the contact network structure on the yield stress curve, we measure $Z_{nr,st},f_{nr,st}$ and $\rho_{st}$, which are obtained from fits to the form of Eq. (1). Figures 3(c) and (d) show a change in slope in all three state variables at a packing fraction slightly above $\phi_{SJ}$. The red dashed line in Fig. 3(c) is the linear fit using data with $\phi>\phi_{SJ}$, which highlights the change in behavior at $\phi_{SJ}$. Figures 4(a) and (b) show two polarized images taken from the steady regime with packing fractions $0.72$ and $0.78$, showing typical force network in F and SJ states.

When the system is shear-jammed, the basal friction becomes unimportant, and the particle displacement field deviates from the basal profile. Based on the azimuthal displacement field per shear step averaged over the steady states, $u_{\theta,st}(r)$, we calculate the off-diagonal element of the strain tensor $\varepsilon_{r\theta,st}(r)=\partial_{r}u_{\theta,st}(r)-u_{\theta,st}(r)/(r+r_{in})$ Landau and Lifshitz (1986), which gives the mean physical shear strain field for steady states (Fig. 4(c)). We also measure the width of the shear zone $w$, which is the $r$ value beyond which $\varepsilon_{r\theta,st}$ becomes smaller than the noise level 0.02%. Figure 4(d) shows $w(\phi)$ (in red), which jumps discontinuously near $\phi_{c}\approx 0.78$, below which $w=r_{out}-r_{in}\approx 20d$. Above $\phi_{c}$, $w\approx 7d$ , denoted $r_{bulk}$ in Fig. 4(c). The local packing fraction in this shear band is also smaller than the global value. The part of the system with $r>r_{bulk}$ just rotates as a solid with the moving outer boundary in the steady states for $\phi>\phi_{c}$. We also calculate $\varepsilon_{bulk}$, which is the averaged $\varepsilon_{r\theta,st}$ for $r>r_{bulk}$. Figure 4(d) shows $\varepsilon_{bulk}$ starts to drop at $\phi_{F}$ and becomes zero near $\phi_{c}$.

Concluding discussion– We set up a multi-ring Couette device that uses small basal friction to drive a 2D granular medium in a way that maintains a linear shear strain profile until the system becomes jammed, allowing us to probe the jamming transition close to $\phi_{SJ}$. The set-up subsequently shears the jammed system using the boundary racks, allowing a study of the yield stress curve for a wide range of packing fractions. Finally, reversing the direction of the drive allows us to distinguish shear-jammed (SJ) from fragile (F) states.

We systematically measured the phase boundaries in the jamming phase diagram, including close to $\phi_{SJ}$, leading to the following key observations: ($i$) In our system $\phi_{SJ}\approx 0.75$, whose value may depend on the friction coefficient $\mu$, polydispersity, and particle shape, though we expect the qualitative features of the jamming phase diagram to be the same. ($ii$) The SJ strain $\gamma_{SJ}$ is well fit by a stretched logarithmic function of $\phi$. The measured exponent $\alpha=0.68\pm 0.11$ is in quantitative agreement with the exponent $\alpha=1/1.37\approx 0.73$ measured from simulation of sheared 3d frictionless soft spheres Kumar and Luding (2016). The same form, but with $\alpha=1$, has also been observed in experiments on shear-thickening suspensions Han et al. (2018). ($iii$) We observe fragile states below $\phi_{SJ}$, which are not included in the traditional phase diagram Bi et al. (2011). In our system, small basal friction forces and particle deformability may be crucial for stabilizing the fragile force network. ($iv$) On the yield stress curve, for increasing packing fraction, we find that $P_{st}$ has an inflection point at $\phi_{SJ}$ and that $Z_{nr,st}$, $\rho_{st}$ and $f_{nr,st}$ all show a change of slope near $\phi_{SJ}$, suggesting a physical transition in the nature of the steady states.

We also find that the quasi-static steady flow field changes from the non-localized basal profile for systems with $\phi<\phi_{F}\approx 0.7$ to a localized shear band for $\phi>\phi_{c}\approx 0.78$, where $\phi_{F}<\phi_{SJ}<\phi_{c}$. The coexistence of a solid and fluid phase in slowly sheared dense granular matter has been reported in many systems Coussot et al. (2002); Debregeas et al. (2001); Varnik et al. (2003); Losert et al. (2000); Schall and van Hecke (2010); Moosavi et al. (2013); Veje et al. (1999). In this work we characterize the contact network associated with the different quasistatic steady flow regimes. When $\phi=\phi_{c}$, the steady states have $\rho_{st}\approx 0.05$ and $f_{nr,st}\approx 1$, showing a nearly isotropic, fully percolated contact network. Notably, $\phi_{c}\approx\phi_{J}^{\mu}$ with $\mu\approx 0.9$, where $\phi_{J}^{\mu}$ is the isotropic jamming packing fraction with friction coefficient $\mu$ Silbert (2010). We also note that $Z_{nr,st}(\phi_{SJ})\approx 3.4$, similar to the mean contact number observed when a strong force network percolates in both principal directions in biaxial experiments Bi et al. (2011), and $Z_{nr,st}(\phi_{c})\approx 3.9$, close to the isostatic value for ideal frictionless disks van Hecke (2009); O’Hern et al. (2003).

The results suggest several directions for further study. First, our shear device can generate other basal profiles Zhao et al. (2017) to study how shear jamming affects the granular rheology for shear fields found in real world applications. Second, the set-up can create a controlled shear band, providing a new technique to study the generation and evolution of shear bands in dense granular flow.

Appendix A Supplemental Material

Details of the experimental set-up

Figure 5(a) shows a detailed view of the experimental set-up. Figure 5(b) shows a top view of the set-up with particles. Figure 5(c) shows a detailed view of the rack gear fixed around the inner wall of the Couette cell. A similar rack is also fixed inside the outer wall. The racks allow for strong friction forces at the inner and outer boundaries, which in enable boundary driven shear to occur at pressures high enough that basal friction alone is not effective.

Details of the imaging system

Figure 6(a) presents the imaging system used to measure stresses on individual particles Zhao et al. (2017). This imaging system is similar to the one described in Ref. Puckett and Daniels (2013). The particles are lit from the top by green flat lights laying just next to the camera. A first polarizer is set between these lights and the particles. The polarized light passes through the particles. This light, after going through each particle, is reflected by the reflective paint at the bottom of the particles.The reflected light passes again through the particles, passes through a second polarizer with the same polarization direction as the first, and is imaged by a camera. For each shear step the system is lit from the edges and the center of the rings by ultra-violet (UV) light and then from the top by polarized green light and imaged by a camera with a crossed polarizer. Figure 6(c) and (e) show example UV and polarized images. Figure 6(b) and (d) show side views of the shear set-up under UV and green polarized light.

Contact force law and the pressure measurement calibration

Figure 7(a) plots the relation between normal contact force magnitude and particle deformation. $\delta/r$ is the ratio of the change of diameter under force $F$ divided by the original particle radius. In this work, we can measure only $F<0.5N$, leading to less than 3% relative deformation $\delta/r$. Values of $F$ are measured with respect to a measured noise threshold. For forces in this regime, $F$ is proportional to the square of the deformation: $F\propto(\delta/r)^{\alpha}$, with $\alpha\approx 2$ for large and small particles. Details of the fits can be found in the caption of Fig. 7.

The pressure $P$, as defined by the trace of the force moment tensor, is equivalent to the averaged pressure on each particle. For a single particle, $P=\sum_{i=1}^{z}F_{i}/2\pi r$, where $z$ is the number of contacts, $F_{i}$ is the normal component of the $ith$ contact force and $r$ is the particle radius. We measure $P$ using a previously developed “$G^{2}$” technique Howell et al. (1999); geng2001_prl; Daniels et al. (2017) . $G^{2}$ is the sum of the squared gradient of the intensity over the pixels inside the particle. We calibrate the relation between $P$ and $G^{2}$ using the diametric test shown in Fig. 7(a). As shown in Fig. 7(b-c), for both small and large disks, $P=kG^{2}$ when $P<15N/m$, which corresponds to $F<0.5N$. For larger forces, the photoelastic fringes cannot be resolved well with the current imaging system. The values of $k$ can be found in the caption of Fig. 7. As mentioned in the main text, this limit is reached at the yield stress curve for $\phi>0.77$, but does not affect other results.

**Calculation of the fabric anisotropy

**

In this work, the fabric tensor is calculated in order to better mimic a simple shear system. Let $S$ denote the area of the container, and let $\vec{r}_{ij}=r_{ij,x}\hat{e}_{t}+r_{ij,y}\hat{e}_{r}$ be the branch vector pointing from the center of $i$th particle to the contact between $i$th and $j$th particles. $\hat{e}_{t}$ and $\hat{e}_{r}$ are the unit tangential and radial vectors in a system centered on the $i$th particle. The fabric tensor is defined as

[TABLE]

where the summation is over all contacting pairs $(i,j)$. The fabric anisotropy is $\rho=(R_{1}-R_{2})/(R_{1}+R_{2})$, where $R_{1}$ and $R_{2}$ are eigenvalues of $\hat{R}$.

**Raw data of the reverse shear tests

**

Figure 8(a) shows a typical run at the largest packing fraction where no shear-jammed state is observed. Even though the system has reached the steady flow regime and supports a nonzero pressure, it remains fragile.

Note that for small strains, where $P<P_{noise}$, $P$ increases slowly due to the growth of weak force chain segments that do not connect the inner and outer boundaries of the system. Basal friction and imperfections in the ring alignments are two sources of this effect. The value of $P_{noise}$ is chosen empirically by examining photoelastic images to make sure that at least one force chain connects the inner to the outer boundary for $P>P_{noise}$. A small change in the choice of $P_{noise}$ does not alter the qualitative features of the measured phase boundaries.