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Velocity fields

Kinematic fields (particle translations and rotations) at the particle scale in sheared granular materials have complex patterns that have been much less investigated than contact force distributions in exception to shear localization, which has for a long time been associated with failure at peak stress ratio. DEM simulations show that even at early stages of shear deformation, well-organized microbands of intense shearing occur despite overall homogeneous boundary conditions (Kuhn 1999; Lesniewska and Wood 2009). These microbands evolve rapidly in space and time, and the shear bands at larger strains seem to arise as a result of their coalescence.

Shear localization in biaxial compression simulated by the contact dynamics method.

Grain rotations and rolling contacts play a crucial role in the local kinematics of granular materials. Shear zones are generally marked by intense particle rotations (Oda et al. 1982; Kuhn and Bagi 2004). Since rolling contacts dissipate much less energy than sliding contacts, grain motions in quasi-static shear occur mostly by rolling. Sliding contacts are actually a consequence of the frus- tration of particle rotations in the sense that all contacts within a loop of contiguous particles cannot be simultaneously in rolling state. The basic structure of a loop of three grains illustrates well this property (Tordesillas 2007). For this reason, it has been argued that such mesoscopic structures evolve, and their statistics are correlated with plastic hardening and softening of granular materials. In general, long-range correlations, such as those reflected in the structure of force chains, indicate that single-contact models cannot fully capture the local behavior. A correlation length of the order of 10 grain diameters is observed in forces (Staron et al. 2005). As an internal length scale, it can be related to the thickness of shear bands.

Average velocity field around a particle in simple shear.

Another important feature of particle velocity fields is that, as a result of steric exclusions, the particle velocities have a non- affine fluctuating component of zero mean with respect to the background shear flow (Radjai and Roux 2002; Peters and Walizer 2013; Combe et al. 2015). These fluctuating velocities have a scaling behavior which is very similar to those of fluid turbulence and were therefore dubbed granulence by Radjai and Roux (2002). In particular, the velocity probability density functions undergo a transition from stretched exponential to Gaussian as the time resolution is increased, and the spatial power spectrum of the velocity field obeys a power law, reflecting long- range correlations and the self-affine nature of the fluctuations. These observations contradict somehow the conventional approach, which disregards kinematic fluctuations in macroscopic modeling of plastic flow in granular media. The long-range correlations of velocity fluctuations may be at the origin of the observed dependence of shear stress on the higher-order gradients of shear strain, implying that granular materials are not simple materials in the sense of Noll (Kuhn 2005; Noll 1958).

Fluctuating velocities in a sheared layer of glass beads.

Nonaffine velocity field in a sheared granular material.

Nongaussian broadening of probability distribution of particle velocities at small integration time scales.

A key aspect of granular kinematics is its discontinuous evolution. The contacts between particles have a short lifetime and new contacts are constantly formed as the particles move. The overall picture is that more contacts are gained along the directions of contraction and lost along the directions of extension (Rothenburg and Bathurst 1989). A detailed balance equation can be written for the evolution of contacts along different directions. Only in the critical state, the rates of gain and loss are equal (Radjai et al. 2004, 2012). However, this is only an average picture and the effects of the chaotic fluctuating velocity field are not yet well understood (Pouragha and Wan 2016). They may well behave as a noise or control the fabric evolution under complex loading conditions. When the direction of shear is reversed, for example, there is a clear asymmetry between the gain and loss processes, which leads to a decrease of the coordination number whereas the void ratio decreases at the same time (Radjai and Roux 2004). Such effects control the nonlinear behavior of granular materials under complex loading paths.

Further reading:

Turbulentlike fluctuations in quasistatic flow of granular media

Rheology, force transmission, and shear instabilities in frictional granular media from biaxial numerical tests using the contact dynamics method

Model for granular texture with steric exclusion

Intermittent flow of a collection of rigid frictional disks in a vertical pipe