Coarse-grained local and objective continuum description of three-dimensional granular flows down an inclined surface

Abstract

Dry, frictional, steady-state granular flows down an inclined, rough surface are studied with discrete particle simulations. From this exemplary flow situation, macroscopic fields, consistent with the conservation laws of continuum theory, are obtained from microscopic data by time-averaging and spatial smoothing (coarse-graining). Two distinct coarse-graining length scale ranges are identified, where the fields are almost independent of the smoothing length w. The smaller, sub-particle length scale, w d, resolves layers in the flow near the base boundary that cause oscillations in the macroscopic fields. The larger, particle length scale, w ≈ d, leads to smooth stress and density fields, but the kinetic stress becomes scale-dependent; however, this scale-dependence can be quantified and removed. The macroscopic fields involve density, velocity, granular temperature, as well as strain-rate, stress, and fabric (structure) tensors. Due to the plane strain flow, each tensor can be expressed in an inherently anisotropic form with only four objective, coordinate frame invariant variables. For example, the stress is decomposed as: (i) the isotropic pressure, (ii) the "anisotropy" of the deviatoric stress, i.e., the ratio of deviatoric stress (norm) and pressure, (iii) the anisotropic stress distribution between the principal directions, and (iv) the orientation of its eigensystem. The strain rate tensor sets the reference system, and each objective stress (and fabric) variable can then be related, via discrete particle simulations, to the inertial number, I. This represents the plane strain special case of a general, local, and objective constitutive model. The resulting model is compared to existing theories and clearly displays small, but significant deviations from more simplified theories in all variables -on both the different length scales. C 2013 AIP Publishing LLC. [http://dx

From discrete particles to continuum fields in mixtures

“…As coarse-graining function, we use a Lucy polynomial [7,3] with cutoff radius c and width (or standard deviation) w = c/2. To satisfy mass balance, the partial velocity is…”

“…The IFD can also be incorporated into the stress, yielding an extended stress definition. The approach has been carefully studied in several publications: In [3], it was shown how to define macroscopic fields that are independent of the coarse-graining width. This approach was successfully applied to flows near boundaries/discontinuities, as well as layered flows [4].…”

General scaling relations for locomotion in granular media

“…Granular materials display a diversity of phenomena that require special constitutive treatment to model, such as history-and preparation-dependent strengthening and dilation [2][3][4], flow anisotropy and normal stress differences [5][6][7], nonlinear rate-sensitive yielding [8][9][10][11], and nonlocality due to the finite size of grains [12][13][14][15][16]. Obtaining pragmatic scaling relations from these models or a combination thereof has some inherent difficulties: (1) A quantitative model that attempts to capture the various granular complexities invariably requires more material parameters.…”

Numerical investigation of granular flow in a shear cell

“…The analysis of the force structures further helps to understand the microscopic origins for the macroscopic phase/regime transition in granular flows. An extension of the I-rheology involving also the tensorial nature of strain, stress, and structure will shed further light onto the interesting behavior of powders and grains [11].…”