Kamrin Group
Continuum modeling from solids to fluids

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Nonlinear Granular Elasto-Plasticity

DEM result


continuum silo

Top:  The pricipal stress vector field in a steadily flowing silo as obtained in a full discrete particle simulation (c/o Chris Rycroft).  Bottom: The principal stress field predicted by the elasto-plastic continuum model.


The work of many French physicists dealing with granular rheology hit a high point in the summer of 2006 with the results of P. Jop et al. which showed that a Bingham fluid flow model was in fact able to describe a 3D granular flow.  On another front, Y. Jiang and M. Liu detailed a functioning granular elasticity model in the winter of that year which determines stresses in a static granular assembly.  Our work combines both of these models into one universal granular constitutive law, capable of predicting both flowing regions and stagnant zones simultaneously in any arbitrary flow geometry.   The model is implemented as a user material subroutine in the Finite Element Method software package ABAQUS/Explicit.  The combined effect of elastic stresses and plastic yielding resolves features such as shear bands which many previously believed to be outside the realm of a continuum model. This work adheres to finite deformation elasto-plasticity theory, a framework which has developed through the ages to become a highly rigorous and complex mathematical science.  For more information, see K. Kamrin, Int. J. Plasticity (2010).


Finite-Difference Method for Solid Deformation



Both results above computed using a Cartesian finite-difference grid. Top: The scalar displacement field for a hyperelatsic washer after large circular shear of the inner wall.  Bottom: The scalar displacement of a hyperelastic disk (in red) upon being stretched into a triangular shape.


Solid constitutive laws are naturally descibed in Lagrangian frame, where local deformation measurements are most easily accessible.  Even so, there is a desire from some communities for methods that can capture the essential physics of general solid deformation but in an Eulerian framework, which can be discretized and simulated as a finite-difference scheme.  This work presents such an alternative, by way of a surrogate quantity called the reference map.  The reference map is a vector field that appropriately reflects the deformation history, and evolves under an explicit update law on the grid.  Finite-difference of the reference map can be used to approximate the deformation gradient, which is key to solid response.  This field is in direct analogy to the roll the velocity field plays in simulating fluid stress on a fixed computational grid.  For more on this method, see K. Kamrin and J.-C. Nave, arXiv:0901.3799.

Fixed Grid Fluid/Solid Interaction Algorithm

The finite-difference algorithm previously described for solid deformation bears a number of similarities to an explicit Navier-Stokes algorithm.  Exploiting the likeness of these methods, one can write a simple code on a single fixed grid, where both fluid and solid phases coexist and interact.  The interface between phases is described by a level-set function.  Under a smeared treatment, the properties of the material change smoothly but rapidly from fluid to solid near the zero level-set.  In the fluid phase, the material stress is derived from the finite-difference velocity gradient, and in the solid phase the reference map is used to compute the stress.  Incompressibility of both phases can be upheld using an Eulerian splitting method (i.e. the projection method).  Current work is underway to sharpen the treatment of the interface by employing an analogue of the Ghost Fluid Method for fluid/fluid interaction. Under this approach, a subgrid routine would be used to apply the appropriate jump conditions at the interface.






Two snapshots of flimsy elastic disk falling and bouncing in a water tank.  Blue vectors are the velocity field (images c/o collaborator Yang Zhang).


Effective Slip Boundary Conditions



Stokes fluid is sheared over an arbitrary (periodic) textured surface. The "effective slip" velocity us is computed from the surface corrugations and can be used as a macroscopic boundary condition.


Recent work in the fabrication of microfluidic devices has motivated general interest in quantifying the effects of surface texture on bulk low-Reynolds-number fluid flow. In non-trivial ways, wall patterning can be used to impede or expedite adjacent fluid motion, causing non-negligible far-away effects. As such, improved models for textured walls could have impact beyond microfluidics, helping optimize the design of apparati for fluid-like materials in macroscopic environments. One basic goal is to extract a simple boundary condition for the smooth, mean surface, that mimics the effects of the actual condition along the true, corrugated surface. For Stokes fluid, tensorial boundary conditions have been suggested relating the stress traction vector along the mean surface to an “effective slip” velocity. We have carried out an analytical/perturbative study of the Stokes equations along surfaces of small height and/or Navier slip-length fluctuations. This has resulted in a second-order formula, complete with error bound, that converts any such surface to an effective tensorial mobility law on the mean surface.  The law can be used to substitute the rough boundary with an approriate slip on the mean surface. We have followed up with additional analytical results pertaining to the symmetry of mobility tensors for arbitrary patterned surfaces.  For more information, see K. Kamrin, M. Z. Bazant, and H. A. Stone, JFM (2010) and K. Kamrin and H. A. Stone Phys. Fluids (2011).




Stochastic Flow Rule

This work describes a multi-scale approach for granular flow, where the flow is viewed as a sequence of many localized collective grain displacements.  Clustered grain displacements are dictated by the motion of meso-sized "spots", which perform a random walk through the material packing.  The Spot Model was first proposed by Bazant in 2000 to model granular flow through a silo apparatus.  However, the original model lacks of any mechanics, thereby constricting general applicability to other flow environments.  The Stochastic Flow Rule (SFR) attempts to generalize the Spot Model by deriving spot parameters directly from the material stresses.   This extends the spot concept beyond silo flow to any geometry with a computable stress field.  To approximate the stress profile in a slow flowing granular assembly, we utilize the Slip-Line Theory of solid mechanics.  The SFR then describes quantitatively how to convert the slip-line field and stresses into the necessary parameters to fully characterize a spot's random trajectory through the material and generate a steady flow profile.  For more information, see K. Kamrin and M. Z. Bazant, Phys. Rev. E, (2007).




Top:  A spot displacement causes collective grain motion.  Bottom: SFR is used to guide spot trajectories to produce the net flow.