Itai Einav

The role of self-organization during confined comminution of granular materials.

During confined comminution of granular materials a power-law grain size distribution (gsd) frequently evolves. We consider this power law as a hint for fractal topology if self-similar patterns appear across the scales. We demonstrate that this ultimate topology is mostly affected by the rules that define the self-organization of the fragment subunits, which agrees well with observations from simplistic models of cellular automata. There is, however, a major difference that highlights the novelty of the current work: here the conclusion is based on a comprehensive study using two-dimensional ‘crushable’ discrete-element simulations that do not neglect physical conservation laws. Motivated by the paradigm of self-organized criticality, we further demonstrate that in uniaxial compression the emerging ultimate fractal topology, as given by the fractal dimension, is generally insensitive to alteration of global index properties of initial porosity and initial gsd. Finally, we show that the fractal dimension in the confined crushing systems is approached irrespective of alteration of the criteria that define when particles crush.

Fracture propagation in brittle granular matter

It is nearly a century since Alan Arnold Griffith developed his energy criterion for the fracture propagation of cracks in ‘near-continuous’ solids. Needless to say that his celebrated work has revolutionized the world of material science. In a very succinct way, Griffith connected between three important aspects of the fracture process: (i) the material, (ii) the stress level, and (iii) the geometry of the crack. Nothing similar was developed for brittle granular matter, although in these materials fracture propagates in the sense of comminution. Recently, I have developed an energy theory, called breakage mechanics, based on the concept of breakage. However, the analogy between the mechanics of breakage and fracture is missing. Here I establish this relation using energy principles and derive a critical comminution pressure for brittle granular materials. This critical pressure is surprisingly complementary to Griffiths critical tensile stress for near-continuous materials. This step enables for the first time to apply the principles of fracture mechanics to all disciplines dealing with confined particles comminution such as geophysics, geology, geotechnical engineering, mineral processing, agriculture and food industry, pharmaceutics and powder technology.

Soil mechanics: breaking ground

In soil mechanics, students models are classified as simple models that teach us unexplained elements of behaviour; an example is the Cam clay constitutive models of critical state soil mechanics (CSSM). ‘Engineers models’ are models that elaborate the theory to fit more behavioural trends; this is usually done by adding fitting parameters to the students models. Can currently unexplained behavioural trends of soil be explained without adding fitting parameters to CSSM models, by developing alternative students models based on modern theories? Here I apply an alternative theory to CSSM, called ‘breakage mechanics’, and develop a simple students model for sand. Its unique and distinctive feature is the use of an energy balance equation that connects grain size reduction to consumption of energy, which enables us to predict how grain size distribution (gsd) evolves—an unprecedented capability in constitutive modelling. With only four parameters, the model is physically clarifying what CSSM cannot for sand: the dependency of yielding and critical state on the initial gsd and void ratio.

Effects of surface structure deformation on static friction at fractal interfaces

The evolution of fractal surface structures with flattening of asperities was investigated using isotropically roughened aluminium surfaces loaded in compression. It was found that asperity amplitude, mean roughness and fractal dimension decrease through increased compressive stress and number of loading events. Of the samples tested, surfaces subjected to an increased number of loading events exhibited the most significant surface deformation and were observed to exhibit higher levels of static friction at an interface with a single-crystal flat quartz substrate. This suggests that the frequency of grain reorganisation events in geomaterials plays an important role in the development of intergranular friction. Fractal surfaces were numerically modelled using Weierstrass– Mandelbrot-based functions. From the study of frictional interactions with rigid flat opposing surfaces it was apparent that the effect of surface fractal dimension is more significant with increasing dominance of adhesive mechanisms.

Standing jumps in shallow granular flows down smooth inclines

The shapes of standing jumps formed in shallow granular flows down an inclined smooth-based chute are analysed in detail, by varying both the slope and mass discharge. Laboratory tests and analytic jump solutions highlight two important transitions. First, for dense flows at high mass discharge, we observe a transition between steep jumps and more diffuse jumps. The traditional shallow-water equation offers a valid prediction for the thickness of the steep water-like jumps. Diffuse frictional jumps require a more general equation accounting for the forces acting inside the jump volume. Second, moving from dense to dilute flows produces another transition between incompressible and compressible jumps. The observed jump height decrease may be reproduced for a more dilute incoming flow by including experimentally measured density variation in the jump equation. Finally, we briefly discuss the likely relevance to avalanche protection dam design that currently utilises traditional shock equations for incompres...

A particle-water based model for water retention hysteresis

A particle–water discrete element based approach to describe water movement in partially saturated granular media is presented and tested. Water potential is governed by both capillary bridges, dominant at low saturations, and the pressure of entrapped air, dominant at high saturations. The approach captures the hysteresis of water retention during wetting and drainage by introducing the local evolution of liquid–solid contact angles at the level of pores and grains. Extensive comparisons against experimental data are presented. While these are made without the involvement of any fitting parameters, the method demonstrates relative high success by achieving a correlation coefficient of at least 82%, and mostly above 90%. For the tested materials with relatively mono-disperse grain size, the hysteresis of water retention during cycles of wetting and drainage has been shown to arise from the dynamics of solid–liquid contact angles as a function of local liquid volume changes.

How rotational vortices enhance transfers

Inspired by recent observations of granular flow, we examine how rotational vortices contribute to heat or mass transfer enhancement in a fluid. We use a tracer method to simulate both diffusion and advection in systems of differing intrinsic diffusivities D0, vortex sizes R, vortex rotation frequencies f, and vortex lifetimes l. The results reveal that these systems exhibit an effective diffusive behavior, characterized by an effective diffusivity Deff. A striking finding is the existence of two regimes, dichotomised by the Peclet number Pe = R2f/D0. When the Peclet number is less than one, there is no transfer enhancement, Deff = D0. For higher values, vortices produce some transfer enhancement with a corresponding power law Deff/D0 ≈ Pen. The power n ranges from a lower bound of 0.5 for stationary vortices of lifetime infinity, to an upper bound of 1 for vortices of lifetimes shorter than half a rotation. This difference is attributed to two different internal mechanisms involving the coupling of diffu...

A scaling law for heat conductivity in sheared granular materials

We investigate the heat transfer through the contact network of a sheared granular material using the standard Discrete Element Method. Elastic and frictional grains are subjected to steady and uniform plane shear. The effective conductivity tensor is expressed through the sum of contact conductances, which enables instantaneous measurements without simulating the actual heat transfer. We show that the conductivity i) does not depend on the inertial number I which controls the shear state, ii) increases with grain deformations (higher confining stress or lower Youngs modulus) and iii) decreases for higher friction coefficients. We extract a robust semi-empirical scaling which quantitatively relates the conductivity to the contact density, Youngs modulus and the part of the stresses that carries only the normal forces.

Continuous hyperplastic models for overconsolidated clays

This paper presents four constitutive models for prediction of the behaviour of overconsolidated clays under triaxial loading conditions. Their formulations are based on a new continuous hyperplastic framework. Continuous hyperplasticity allows the derivation of plasticity models which simulate smooth stress-strain behaviour, have continuous memory, and are guaranteed to obey the laws of thermodynamics. All four models are capable of simulating the triaxial stress-strain behaviour of overconsolidated clays in a wide strain range. The ability of the models to predict the stress path dependency of the clay behaviour is evaluated against experimental data obtained on undisturbed Laval samples of Bothkennar clay.

Internal relaxation time in immersed particulate materials.

We study the dynamics of the static-to-flow transition in a model material made of elastic particles immersed in a viscous fluid. The interaction between particle surfaces includes their viscous lubrication, a sharp repulsion when they get closer than a tuned steric length, and their elastic deflection induced by those two forces. We use soft dynamics to simulate the dynamics of this material when it experiences a step increase in the shear stress and a constant normal stress. We observe a long creep phase before a substantial flow eventually establishes. We measure the change in volume (dilatancy) and find that during the creep phase, it does not change significantly. We find that the typical creep time relies on an internal relaxation process, namely, the separation of two particles driven by the applied stress and resisted by the viscous friction. The present mechanism should be relevant for granular pastes, living cells, emulsions, and wet foams.