Ken Kamrin

A predictive, size-dependent continuum model for dense granular flows

Dense granular materials display a complicated set of flow properties, which differentiate them from ordinary fluids. Despite their ubiquity, no model has been developed that captures or predicts the complexities of granular flow, posing an obstacle in industrial and geophysical applications. Here we propose a 3D constitutive model for well-developed, dense granular flows aimed at filling this need. The key ingredient of the theory is a grain-size-dependent nonlocal rheology—inspired by efforts for emulsions—in which flow at a point is affected by the local stress as well as the flow in neighboring material. The microscopic physical basis for this approach borrows from recent principles in soft glassy rheology. The size-dependence is captured using a single material parameter, and the resulting model is able to quantitatively describe dense granular flows in an array of different geometries. Of particular importance, it passes the stringent test of capturing all aspects of the highly nontrivial flows observed in split-bottom cells—a geometry that has resisted modeling efforts for nearly a decade. A key benefit of the model is its simple-to-implement and highly predictive final form, as needed for many real-world applications.

Effective slip boundary conditions for arbitrary periodic surfaces: the surface mobility tensor

In a variety of applications, most notably microfluidics design, slip-based boundary conditions have been sought to characterize fluid flow over patterned surfaces. We focus on laminar shear flows over surfaces with periodic height fluctuations and/or fluctuating Navier scalar slip properties. We derive a general formula for the ‘effective slip’, which describes equivalent fluid motion at the mean surface as depicted by the linear velocity profile that arises far from it. We show that the slip and the applied stress are related linearly through a tensorial mobility matrix, and the method of domain perturbation is then used to derive an approximate formula for the mobility law directly in terms of surface properties. The specific accuracy of the approximation is detailed, and the mobility relation is then utilized to address several questions, such as the determination of optimal surface shapes and the effect of random surface fluctuations on fluid slip.

Continuum modelling and simulation of granular flows through their many phases

We propose and numerically implement a constitutive framework for granular media that allows the material to traverse through its many common phases during the flow process. When dense, the material is treated as a pressure sensitive elasto-viscoplastic solid obeying a yield criterion and a plastic flow rule given by the

Two-temperature continuum thermomechanics of deforming amorphous solids

\mu(I)

Mechanistic Origin of the Ultrastrong Adhesion between Graphene and a-SiO2: Beyond van der Waals.

inertial rheology of granular materials. When the free volume exceeds a critical level, the material is deemed to separate and is treated as disconnected, stress-free media. A Material Point Method (MPM) procedure is written for the simulation of this model and many demonstrations are provided in different geometries. By using the MPM framework, extremely large strains and nonlinear deformations, which are common in granular flows, are representable. The method is verified numerically and its physical predictions are validated against known results.

Intrusion rheology in grains and other flowable materials

Abstract There is an ever-growing need for predictive models for the elasto-viscoplastic deformation of solids. Our goal in this paper is to incorporate recently developed out-of-equilibrium statistical concepts into a thermodynamically consistent, finite-deformation, continuum framework for deforming amorphous solids. The basic premise is that the configurational degrees of freedom of the material – the part of the internal energy/entropy that corresponds to mechanically stable microscopic configurations – are characterized by a configurational temperature that might differ from that of the vibrational degrees of freedom, which equilibrate rapidly with an external heat bath. This results in an approximate internal energy decomposition into weakly interacting configurational and vibrational subsystems, which exchange energy following a Fourier-like law, leading to a thermomechanical framework permitting two well-defined temperatures. In this framework, internal variables, that carry information about the state of the material equilibrate with the configurational subsystem, are explicitly associated with energy and entropy of their own, and couple to a viscoplastic flow rule. The coefficients that determine the rate of flow of entropy and heat between different internal systems are proposed to explicitly depend on the rate of irreversible deformation. As an application of this framework, we discuss two constitutive models for the response of glassy materials, a simple phenomenological model and a model related to the concept of Shear-Transformation-Zones as the basis for internal variables. The models account for several salient features of glassy deformation phenomenology. Directions for future investigation are briefly discussed.

The symmetry of mobility laws for viscous flow along arbitrarily patterned surfaces

The origin of the ultrastrong adhesion between graphene and a-SiO2 has remained a mystery. This adhesion is believed to be predominantly van der Waals (vdW) in nature. By rigorously analyzing recently reported blistering and nanoindentation experiments, we show that the ultrastrong adhesion between graphene and a-SiO2 cannot be attributed to vdW forces alone. Our analyses show that the fracture toughness of the graphene/a-SiO2 interface, when the interfacial adhesion is modeled with vdW forces alone, is anomalously weak compared to the measured values. The anomaly is related to an ultrasmall fracture process zone (FPZ): owing to the lack of a third dimension in graphene, the FPZ for the graphene/a-SiO2 interface is extremely small, and the combination of predominantly tensile vdW forces, distributed over such a small area, is bound to result in a correspondingly small interfacial fracture toughness. Through multiscale modeling, combining the results of finite element analysis and molecular dynamics simulations, we show that the adhesion between graphene and a-SiO2 involves two different kinds of interactions: one, a weak, long-range interaction arising from vdW adhesion and, second, discrete, short-range interactions originating from graphene clinging to the undercoordinated Si (≡Si·) and the nonbridging O (≡Si-O·) defects on a-SiO2. A strong resistance to relative opening and sliding provided by the latter mechanism is identified as the operative mechanism responsible for the ultrastrong adhesion between graphene and a-SiO2.

Eulerian Method for Multiphase Interactions of Soft Solid Bodies in Fluids

The interaction of intruding objects with deformable materials is a common phenomenon, arising in impact and penetration problems, animal and vehicle locomotion, and various geo-space applications. The dynamics of arbitrary intruders can be simplified using Resistive Force Theory (RFT), an empirical framework originally used for fluids but works surprisingly well, better in fact, in granular materials. That such a simple model describes behavior in dry grains, a complex nonlinear material, has invigorated a search to determine the underlying mechanism of RFT. We have discovered that a straightforward friction-based continuum model generates RFT, establishing a link between RFT and local material behavior. Our theory reproduces experimental RFT data without any parameter fitting and generates RFT’s key simplifying assumption: a geometry-independent local force formula. Analysis of the system explains why RFT works better in grains than in viscous fluids, and leads to an analytical criterion to predict RFT’s in other materials.

Some exact properties of the effective slip over surfaces with hydrophobic patternings

Generalizations of the no-slip boundary condition to allow for slip at a patterned fluid-solid boundary introduce a surface mobility tensor, which relates the shear traction vector tangent to the mean surface to an apparent surface velocity vector. For steady, low-Reynolds-number fluid motions over planar surfaces perturbed by arbitrary periodic height and Navier slip fluctuations, we prove that the resulting mobility tensor is always symmetric, which had previously been conjectured. We describe generalizations of the results to three other families of geometries, which typically have unsteady flow.

Symmetry relations in viscoplastic drag laws

We introduce an Eulerian approach for problems involving one or more soft solids immersed in a fluid, which permits mechanical interactions between all phases. The reference map variable is exploited to simulate finite-deformation constitutive relations in the solid(s) on the same fixed grid as the fluid phase, which greatly simplifies the coupling between phases. Our coupling procedure, a key contribution in the current work, is shown to be computationally faster and more stable than an earlier approach, and admits the ability to simulate both fluid--solid and solid--solid interaction between submerged bodies. The interface treatment is demonstrated with multiple examples involving a weakly compressible Navier--Stokes fluid interacting with a neo-Hookean solid, and we verify the methods convergence. The solid contact method, which exploits distance-measures already existing on the grid, is demonstrated with two examples. A new, general routine for cross-interface extrapolation is introduced and used as part of the new interfacial treatment.