Patterning of a cohesionless granular layer under pure shear

The response of a thin layer of granular material to an external pure shear imposed at its base is investigated. The experiments show that, even for noncohesive materials, the resulting deformation of the material is inhomogeneous. Indeed, a novel smooth pattern, consisting of a periodic modulation of the shear deformation of the free surface, is revealed by an image-correlation technique. These observations are in contrast with the previous observation of the fracture pattern in cohesive granular materials subjected to stretching. For cohesive materials, the instability is due to the weakening of the material which results from the rupture of capillary bridges that bond the grains to one another. For noncohesive materials, the rupture of the capillary bridges cannot be invoked anymore. We show that the instability results from the decrease of friction on shearing. PACS: 89.75.Kd: Pattern formation in complex systems; 83.60.Uv: Rheology: fracture; 45.70.Qj: Pattern formation in granular matter

Figure 1

The experimental setup. The four arms of a latex membrane are driven by computer-controlled actuators. At center, arrows show the displacements of the membrane. Pure shear strain is achieved to better than $1\%$ over a surface area of 50 ${\mathrm{cm}}^2$. Inset: Typical modulation of shear strain at the free surface (brass beads, $d=75$–106 $\mu \mathrm{m}$, $h=3\mathrm{mm}$, $\theta =0.14$ and $R_H=39$%).

Figure 2

(a) Pattern structure for various $h$ and humidity. $R_H$ (${\sigma }_s$): 39% (17 Pa), 74% (37 Pa), 89% (144 Pa), scale bar $=1\mathrm{cm}$. (b) Pattern wavelength, $\lambda$, for increasing $h$ at various cohesion (relative humidity, $R_H$). The lines are from Eq. (4) with parameters reported in Table 1 (glass beads, $d=0$–45 $\mu \mathrm{m}$, $\mathrm{\Delta }\theta =1.5\times {10}^{-3}$).

Figure 3

(a) Pattern structure for various $h$ and humidity $R_H$ (${\sigma }_s$): 39% (0.5 Pa), 74% (0.8 Pa), 89% (1.9 Pa), scale bar $=1\mathrm{cm}$. (b) Wavelength $\lambda$ for increasing $h$ at various cohesion. The lines are from Eq. (4) with parameters reported in Table 1 (glass beads, diameter 150–200 $\mu \mathrm{m}$, $\mathrm{\Delta }\theta =1.5\times {10}^{-3}$).

Figure 4

Surface shear, ${\epsilon }_{xy}$, and vorticity, ${\omega }_z$, fields for two distinct values of thickness for $\theta =0.083$ (glass beads, diameter 150–200 $\mu \mathrm{m}$, $\mathrm{\Delta }\theta =1.5\times {10}^{-3}$, $R_H=39\%$).

Figure 5

Amplitude $<\mathrm{\Delta }d>$ of the modulation vs. shear-strain $\theta$. (a) $R_H=39\%$ and (b) $R_H=74\%$ (Glass beads, diameter 0–45 $\mu \mathrm{m}$, $\mathrm{\Delta }\theta =1.5\times {10}^{-3}$).

Figure 6

Vorticity fields ${\omega }_z$ for brass particles. (a) $d=75$–106 $\mu \mathrm{m}$ ($h=5\mathrm{mm}$, $\theta =0.14$, and $R_H=39$%). (b) Wavelength $\lambda$ vs. thickness $h$, ($\square$) $d=75$–106 $\mu \mathrm{m}$ and ($\circ$) $d=212$–300 $\mu \mathrm{m}$. The lines are from Eq. (4) with parameters reported in Table 2 (brass beads, $\mathrm{\Delta }\theta =3.6\times {10}^{-3}$, $\theta =0.14$, and $R_H=39$%).