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Register a new userSnells Law for Shear Zone Refraction in Granular Materials
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We present experiments on slow shear flow in a split-bottom linear shear cell, filled with layered granular materials. Shearing through two different materials separated by a flat material boundary is shown to give narrow shear zones, which refract at the material boundary in accordance with Snells law in optics. The shear zone is the one that minimizes the dissipation rate upon shearing, i.e.a manifestation of the principle of least dissipation. We have prepared the materials as to form a granular lens. Shearing through the lens is shown to give a very broad shear zone, which corresponds to fulfilling Snells law for a continuous range of paths through the cell.
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Snells law, which encompasses both refraction and total internal reflection (TIR), provides the foundation for ray optics and all lens-based instruments, from microscopes to telescopes. Refraction results when light crosses the interface between media of different refractive index, the dimensionless number that captures how much a medium retards the propagation of light. In this work, we show that the motion of self-propelled particles moving across a drag discontinuity is governed by an analogous Snells law, allowing for swimmer ray optics. We derive a variant of Snells law for neutral swimmers moving across media of different viscosities. Just as the ratio of refractive indexes sets the path of a light ray, the ratio of viscosities is shown to determine the trajectories of swimmers. We find that the magnitude of refraction depends on the swimmers shape, specifically the aspect ratio, as analogous to the wavelength of light. This enables the demixing of a polymorphic, many-shaped, beam of swimmers into distinct monomorphic, single-shaped, beams through a viscosity prism. In turn, beams of monomorphic swimmers can be focused by spherical and gradient viscosity lenses. Completing the analogy, we show that the shape-dependence of the TIR critical angle can be used to create swimmer traps. Such analogies to ray optics suggest a universe of new devices for sorting, concentrating, and analyzing microscopic swimmers is possible.
In order to understand the nature of friction in closely-packed granular materials, a discrete element simulation on granular layers subjected to isobaric plain shear is performed. It is found that the friction coefficient increases as the power of the shear rate, the exponent of which does not depend on the material constants. Using a nondimensional parameter that is known as the inertial number, the power-law can be cast in a generalized form so that the friction coefficients at different confining pressures collapse on the same curve. We show that the volume fraction also obeys a power-law.
We report on experiments that probe the stability of a two-dimensional jammed granular system formed by imposing a quasistatic simple shear strain $gamma_{rm I}$ on an initially stress free packing. We subject the shear jammed system to quasistatic cyclic shear with strain amplitude $deltagamma$. We observe two distinct outcomes after thousands of shear cycles. For small $gamma_{rm I}$ or large $deltagamma$, the system reaches a stress-free, yielding state exhibiting diffusive strobed particle displacements with a diffusion coefficient proportional to $deltagamma$. For large $gamma_{rm I}$ and small $deltagamma$, the system evolves to a stable state in which both particle positions and contact forces are unchanged after each cycle and the response to small strain reversals is highly elastic. Compared to the original shear jammed state, a stable state reached after many cycles has a smaller stress anisotropy, a much higher shear stiffness, and less tendency to dilate when sheared. Remarkably, we find that stable states show a power-law relation between shear modulus and pressure with an exponent $betaapprox 0.5$, independent of $deltagamma$. Based on our measurements, we construct a phase diagram in the $(gamma_{rm I},deltagamma)$ plane showing where our shear-jammed granular materials either stabilize or yield in the long-time limit.
We report a new lift force model for intruders in dense, granular shear flows. Our derivation is based on the thermal buoyancy model of Trujillo & Hermann[L. Trujillo and H. J. Herrmann, Physica A 330, 519 (2003).], but takes into account both granular temperature and pressure differences in the derivation of the net buoyancy force acting on the intruder. In a second step the model is extended to take into account also density differences between the intruder and the bed particles. The model predicts very well the rising and sinking of intruders, the lift force acting on intruders as determined by discrete element model (DEM) simulations and the neutral-buoyancy limit of intruders in shear flows. Phenomenologically, we observe a cooling upon the introduction of an intruder into the system. This cooling effect increases with intruder size and explains the sinking of large intruders. On the other hand, the introduction of small to mid-sized intruders, i.e. up to 4 times the bed particle size, leads to a reduction in the granular pressure compared to the hydrostatic pressure, which in turn causes the rising of small to mid-sized intruders.
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 $phi_capprox 0.78$ the flow is localized in a shear band.
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