Shear-Jammed, Fragile, and Steady States in Homogeneously Strained Granular Materials

We study the jamming phase diagram of sheared granular material using a novel Couette shear setup with a multiring bottom. The setup 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 setup 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 ${\varphi }_{\mathrm{SJ}}\approx 0.75$, the smallest packing fraction supporting a shear-jammed state. We observe fragile states created under large shear strain over a range of $\varphi <{\varphi }_{\mathrm{SJ}}$. We also find a transition in the character of the quasistatic steady flow centered around ${\varphi }_{\mathrm{SJ}}$ on the yield curve as a function of packing fraction. Near ${\varphi }_{\mathrm{SJ}}$, the average contact number, fabric anisotropy, and nonrattler fraction all show a change of slope. Above ${\varphi }_F\approx 0.7$ the steady flow shows measurable deviations from the basal linear shear profile, and above ${\varphi }_b\approx 0.78$ the flow is localized in a shear band.

Figure 1

(a) The jamming phase diagram in the shear stress $\tau$ and packing fraction $\varphi$ plane adapted from [24]. Only the $\varphi <{\varphi }_J^0$ part of the diagram is shown. The yield stress curve and the jamming curve are highlighted in blue and yellow, respectively. (b) Schematic of the multiring Couette setup. Twenty-one horizontal concentric rings rotate step wisely to quasistatically shear bidisperse photoelastic discs. For each shear step, the ring at position $r$ rotates by an arc length $d(r)$. The nominal shear strain is defined as $\gamma =d(r)/r$. (c) Particle displacements in radial ($u_r$) and azimuthal ($u_{\theta }$) directions in a shear step for a dilute system ($\varphi =0.57$). The dashed line is the linear basal profile $d(r)$. After each shear step the system is imaged in uv light (e) and in polarized green light (d).

Figure 2

(a),(b) The pressure $P$ and the nonrattler contact number $Z_{\mathrm{nr}}$ vs $\gamma$ for different $\varphi$ during forward shear. The dashed black curve in (b) plots the exponential fit by Eq. (1) for $Z_{\mathrm{nr}}(\gamma )$ with $\varphi =0.78$. The blue dashed line shows the ${\gamma }_{\mathrm{st}}$ value for this run. (c) Strain needed to reach the steady regime ${\gamma }_{\mathrm{st}}(\varphi )$. The black line is a linear fit ${\gamma }_{\mathrm{st}}\propto (\varphi -{\varphi }_0)$ for $\varphi >0.72$. (d) $P$ vs $\gamma$ for a typical reverse shear test ($\varphi =0.781$) with forward shear strain ${\gamma }_{\max }$. (e) The minimum pressure, $P_{\min }$, during the reverse shear vs ${\gamma }_{\max }$ for $\varphi =0.781$. ${\gamma }_{\mathrm{SJ}}$ is the minimum ${\gamma }_{\max }$ for which $P_{\min }>P_{\text{noise}}=0.3\mathrm{N}/\mathrm{m}$ [31].

Figure 3

(a) Strain needed to create fragile state, ${\gamma }_F$ (blue), and SJ state, ${\gamma }_{\mathrm{SJ}}$ (black). The minimum packing fraction for the SJ states is ${\varphi }_{\mathrm{SJ}}\approx 0.745$. The blue and black solid curves are fitted with Eq. (2). In (a), (c), and (d), solid gray circles are raw data and open circles are averaged data. (b) The jamming phase diagram in the $(P,\varphi )$ plane built from our data. The dynamic unjammed (uJ), fragile ($F$), and shear jammed (SJ) states are separated by the yield stress curve $P_{\mathrm{st}}$ and the jamming curve $P(\gamma ={\gamma }_{\mathrm{SJ}})$. The dark gray region below the noise level $0.3\mathrm{N}/\mathrm{m}$ indicates static unjammed states. ${\varphi }_F$ is the minimum packing fraction for fragile states. (c),(d) The steady state nonrattler contact number $Z_{\mathrm{nr},\mathrm{st}}$, nonrattler fraction $f_{\mathrm{nr},\mathrm{st}}$ and fabric anisotropy ${\rho }_{\mathrm{st}}$ obtained from Eq. (1). Note the change in slope near ${\varphi }_{\mathrm{SJ}}$ in all three cases. The red dashed line in (c) shows a linear fit using data above ${\varphi }_{\mathrm{SJ}}$. (e) Surface plot of all static states measured during forward shear experiments in the space of $P$, $\varphi$, and inverted strain $1/\gamma$ space. Smooth curves join states accessed in a single run. States are labeled using the same color code as in (b).

Figure 4

(a),(b) Polarized images showing force networks of typical steady states with packing fraction 0.72 and 0.78. (c) Physical shear strain per step averaged over steady states, ${ϵ}_{r\theta ,\mathrm{st}}(r)$ for different $\varphi$, labeled by the color bar. The black dashed curve shows the basal profile with linear nominal strain $\gamma$. $r_{\mathrm{bulk}}=7d$ is where ${ϵ}_{r\theta ,\mathrm{st}}$ vanishes for $\varphi >{\varphi }_b$. (d) ${ϵ}_{\mathrm{bulk}}$, defined as the averaged ${ϵ}_{r\theta ,\mathrm{st}}$ for $r>r_{\mathrm{bulk}}$, drops at ${\varphi }_{\mathrm{SJ}}$ and vanishes at ${\varphi }_b$. The same figure plots the width of shear zone $w$, defined as the range of $r$ that ${\epsilon }_{r\theta ,\mathrm{st}}$ is nonzero. $w(\varphi )$ drops to $r_{\mathrm{bulk}}$ at $\varphi \approx {\varphi }_b$.