arXiv subjects

Raphael Blumenfeld

Publications and source records attributed to Raphael Blumenfeld.

At least 37 records · Page 2Linked to original sources

arXiv2019

Structural Evolution of Granular Systems: Theory

A general theory is developed for the evolution of the cell order (CO) distribution in planar granular systems. Dynamic equations are constructed and solved in closed form for several examples: systems under compression; dilation of very dense systems; and the general approach to steady state. We find that all the steady states are stable and that they satisfy detailed balance-like condition when the CO$\,\leq 6$. Illustrative numerical solutions of the evolution are shown. Our theoretical results are validated against an extensive simulation of a sheared system. The formalism can be readily extended to other structural characteristics, paving the way to a general theory of structural organisation of granular systems.

cond-mat.soft↗

arXiv2018

Stress - Strain Rate Relation in Plug-Free Flow of Dense Granular Matter

We derive the macroscopic stress tensor for plug-free dense granular flow, using a first-principles coarse-graining of the inter-granular forces. The derivation is based on the assumption, which defines the da Vinci Fluid model, that the inter-granular interactions are dominated by normal contact forces and solid friction. An explicit form for the stress -- strain rate relation is obtained, providing closure for the flow equations of dense granular fluids and improves on existing models, which are also based on solid friction as the dominant drag mechanism.

cond-mat.stat-mech↗

arXiv2017

Theory-based design of sintered granular composites triples three-phase boundary in fuel cells

Solid-oxide fuel cells produce electric current from energy released by a spontaneous electrochemical reaction. The efficiency of these devices depends crucially on the microstructure of their electrodes and, in particular, on the three-phase boundary (TPB) length, along which the energy-producing reaction occurs. We present a systematic maximisation of the TPB length as a function of four readily-controllable microstructural parameters, for any given mean hydraulic radius, which is a conventional measure of the permeability to gas flow. We identify the maximising parameters and show that the TPB length can be increased by a factor of over 300% compared to current common practices. We support this result by calculating the TPB of several numerically simulated structures. We also compare four models for a single intergranular contact in the sintered electrode and show that the model commonly used in the literature is oversimplified and unphysical. We then propose two alternatives.

cond-mat.mtrl-sci↗

arXiv2017

Affine and topological structural entropies in granular statistical mechanics: explicit calculations and equation of state

We identify two orthogonal sources of structural entropy in rattler-free granular systems - affine, involving structural changes that only deform the contact network, and topological, corresponding to different topologies of the contact network. We show that a recently developed connectivity-based granular statistical mechanics separates the two naturally by identifying the structural degrees of freedom with spanning trees on the graph of the contact network. We extend the connectivity-based formalism to include constraints on, and correlations between, degrees of freedom as interactions between branches of the spanning tree. We then use the statistical mechanics formalism to calculate the partition function generally and the different entropies in the high-angoricity limit. We also calculate the degeneracy of the affine entropy and a number of expectation values. From the latter, we derive an equipartition principle and an equation of state relating the macroscopic volume and boundary stress to the analogue of the temperature, the contactivity.

cond-mat.stat-mech↗

arXiv2017

Fundamental structural characteristics of planar granular assemblies: self-organisation and scaling away friction and initial state

We identify the fundamental factors determining the microstructural (MS) statistics of granular systems, using numerical experiments on 2D assemblies of polydisperse frictional discs and studying the emergent properties of quadrons (Qs), the basic structural elements of granular solids, whose statistics are universal-like[1]. The dependence of the structures and of the packing fraction on friction and initial state are analysed and we report the following. 1. Derivation of an analytical formula for the mean Q volume in terms of: the mean coordination number, the packing fraction and the rattlers fraction. 2. Derivation of a unique, initial-state-independent, relation between the mean coordination number and the rattler-free packing fraction. We support the relation with data for a range of different systems. 3. Collapse of the Q volume distributions of all systems onto one curve, having an exponential tail. 4. Decomposition of the Q volumes distribution into conditional distributions, each of these also collapsing onto a single curve. 5. The mean Q volume decreases with increasing inter-granular friction coefficients, an effect prominent in high order cells. We argue that this phenomenon is due to an increased probability of stable irregularly-shaped cells and test this by a herewith developed free cell analytical model. We conclude that, in principle, the MS characteristics are governed mainly by the packing procedure, while effects of friction and initial states can be scaled away. Yet, mechanical constraints limit slightly occurrence of small Q volumes in large cells, which does depend on friction. We quantify this deviation from exact collapse for these cells. 6. We argue that our results support the view that ensemble granular statistical mechanics does not satisfy the uniform measure assumption of conventional statistical mechanics.

cond-mat.soft↗

arXiv2017

Bending back stress chains and unique behaviour of granular matter in cylindrical geometries

We analyse the general solutions for the stress field in planar annuli of isostatic media, a model often used for marginally rigid granular materials in Couette cells. We demonstrate that these solutions are much richer than in rectangular symmetries. Even for uniform media, stress chains are found to curve, broaden away from the stress source, attenuate and leak stress into a cone of influence. Most spectacularly, stress chains may bend back and transmit forces oppositely to the original direction. None of these phenomena arises in solutions for uniform media in Cartesian coordinates. We further analyse non-uniform media, which exhibit chain branching and stress leakage from the chains. These results are directly relevant to the many experiments on granular materials, carried out in Couette cells. They also shed light on, and are supported by, hitherto unexplained experimental observations of curved and back-bending chains, which we point out. In particular, we use our results to provide a new interpretation for the pattern of slip lines observed experimentally.

cond-mat.soft↗

arXiv2016

Vertical dynamics of a horizontally-oscillating active object in a 2D granular medium

We use a DEM simulation and analytical considerations to study the dynamics of a self-energised object, modelled as a disc, oscillating horizontally within a two-dimesional bed of denser and smaller particles. We find that, for given material parameters, the immersed object (IO) may rise, sink or not change depth, depending on the oscillation amplitude and frequency, as well as on the initial depth. With time, the IO settles at a specific depth that depends on the oscillation parameters. We construct a phase diagram of this behaviour in the oscillation frequency and velocity amplitude variable space. We explain the observed rich behaviour by two competing effects: climbing on particles, which fill voids opening under the disc, and sinking due to bed fluidisation. We present a cavity model that allows us to derive analytically general results, which agree very well with the observations and explain quantitatively the phase diagram. Our specific analytical results are the following. (i) Derivation of a critical frequency, $f_c$, above which the IO cannot float up against gravity. We show that this frequency depends only on the gravitational acceleration and the IO size. (ii) Derivation of a minimal amplitude, $A_{min}$, below which the IO cannot rise even if the frequency is below $f_c$. We show that this amplitude also depends only on the gravitational acceleration and the IO size. (iii) Derivation of a critical value, $g_c$, of the IO's acceleration amplitude, below which the IO cannot sink. We show that the value of $g_c$ depends on the characteristics of both the IO and the granular bed,as well as on the initial IO's depth.

cond-mat.soft↗

arXiv2016

Modifying continuous-time random walks to model finite-size particle diffusion in granular porous media

The continuous-time random walk (CTRW) model is useful for alleviating the computational burden of simulating diffusion in actual media. In principle, isotropic CTRW only requires knowledge of the step-size, $P_l$, and waiting-time, $P_t$, distributions of the random walk in the medium and it then generates presumably equivalent walks in free space, which are much faster. Here we test the usefulness of CTRW to modelling diffusion of finite-size particles in porous medium generated by loose granular packs. This is done by first simulating the diffusion process in a model porous medium of mean coordination number, which corresponds to marginal rigidity (the loosest possible structure), computing the resulting distributions $P_l$ and $P_t$ as functions of the particle size, and then using these as input for a free space CTRW. The CTRW walks are then compared to the ones simulated in the actual media. In particular, we study the normal-to-anomalous transition of the diffusion as a function of increasing particle size. We find that, given the same $P_l$ and $P_t$ for the simulation and the CTRW, the latter predicts incorrectly the size at which the transition occurs. We show that the discrepancy is related to the dependence of the effective connectivity of the porous media on the diffusing particle size, which is not captured simply by these distributions. We propose a correcting modification to the CTRW model -- adding anisotropy -- and show that it yields good agreement with the simulated diffusion process. We also present a method to obtain $P_l$ and $P_t$ directly from the porous sample, without having to simulate an actual diffusion process. This extends the use of CTRW, with all its advantages, to modelling diffusion processes of finite-size particles in such confined geometries.

cond-mat.stat-mech↗

arXiv2016

Failure of the Volume Function in Granular Statistical Mechanics and an Alternative Formulation

We first show that the currently accepted statistical mechanics for granular matter is flawed. The reason is that it is based on the volume function, which depends only on a minute fraction of all the structural degrees of freedom and is unaffected by most of the configurational microstates. Consequently, the commonly used partition function underestimates the entropy severely. We then propose a new formulation, replacing the volume function with a ${\it connectivity}$ function that depends on all the structural degrees of freedom and accounts correctly for the entire entropy. We discuss the advantages of the new formalism and derive explicit results for two- and three-dimensional systems. We test the formalism by calculating the entropy of an experimental two-dimensional system, as a function of system size, and showing that it is an extensive variable.

cond-mat.stat-mech↗

arXiv2016

Statistical mechanics of dense granular fluids - contacts as quasi-particles

A new first-principles statistical mechanics formulation is proposed to describe slow and dilated granular fluids, where prolonged intergranular contacts vitiate collision theory. The contacts, where all the important physics takes place, are regarded as quasi-particles that can appear and disappear. A contact potential, $ν$, is defined as a measure of the fluctuations and the mean coordination number per particle and its fluctuations are calculated as a function of it. This formulation extends the Edwards statistical mechanics to slow dynamic systems and converges to it when the motion stops. The theory is applied to a model system of a simply sheared granular material in the limit of small confining stress. The dependence of the contact potential on the shear rate is derived, making it possible to calibrate $ν$ experimentally and predict the coordination number distribution as a function of the shear rate. Setting next $ν=0$ as the jamming point, where the critical mean coordination number is $z_c$, a finite value is found for the shear rate there, $\dotγ_c$. A simple mean field theory then yields the scaling of the mean coordination number and its fluctuations near $\dotγ_c$. Existing results in the literature appear to support the predictions.

cond-mat.soft↗

arXiv2015

Structural evolution of granular systems: Theory

A first-principles theory is developed for the general evolution of a key structural characteristic of planar granular systems - the cell order distribution. The dynamic equations are constructed and solved in closed form for a number of examples: dense systems undergoing progressive compaction; initial dilation of very dense systems; and the approach to steady state of general systems. It is shown that the convergence to steady state is exponential, except when contacts are only broken and no new contacts are made, in which case the approach is algebraic in time. Where no closed form solutions are possible, illustrative numerical solutions of the evolution are shown. These show that the dynamics are sensitive to the cell event rates, which are process dependent. The formalism can be extended to other structural characteristics, paving the way to a general theory of structural organisation of granular systems, parameterised by the contact event rates.

cond-mat.mtrl-sci↗

arXiv2014

Flow equations for dense granular fluids: New insight from a first-principles derivation

We present a first-principles theory for plug-free dense granular flow. This is done by coarse-graining directly the microscopic dynamics and deriving an explicit relation between the macroscopic stress and strain rate tensors. The newly derived relation not only differs significantly from that of the existing empirical models for such flows but it also provides a novel understanding of the effect of rigid-like rotational regions in the flow.

cond-mat.soft↗

arXiv2013

On universal structural characteristics of granular packs

Dependence of structural self-organization of granular materials on preparation and grain parameters is key to predictive modelling. We study 60 different mechanically equilibrated polydisperse disc packs, generated numerically by two protocols. We show that, for same-variance disc size distributions (DSDs): 1. the mean coordination number of rattler-free packs vs. the packing fraction is a function independent of initial conditions, friction and the DSD: 2. all quadron volume and cell order distributions collapse to universal forms, also independent of the above. We conclude that, contrary to common wisdom, equilibrated granular structures are determined mainly by the packing protocol and higher moments of the DSD.

cond-mat.soft↗

arXiv2012

Inter-dependence of the volume and stress ensembles and equipartition in statistical mechanics of granular systems

We discuss the statistical mechanics of granular matter and derive several significant results. First, we show that, contrary to common belief, the volume and stress ensembles are inter-dependent, necessitating the use of both. We use the combined ensemble to calculate explicitly expectation values of structural and stress-related quantities for two-dimensional systems. We thence demonstrate that structural properties may depend on the angoricity tensor and that stress-based quantities may depend on the compactivity. This calls into question previous statistical mechanical analyses of static granular systems and related derivations of expectation values. Second, we establish the existence of an intriguing equipartition principle - the total volume is shared equally amongst both structural and stress-related degrees of freedom. Third, we derive an expression for the compactivity that makes it possible to quantify it from macroscopic measurements.

cond-mat.soft↗

arXiv2012

Theory of strains in auxetic materials

Auxetic materials, or negative-Poisson's-ratio materials, are important technologically and fascinating theoretically. When loaded by external stresses, their internal strains are governed by correlated motion of internal structural degrees of freedom. The modelling of such materials is mainly based on ordered structures, despite existence of auxetic behaviour in disordered structures and the advantage in manufacturing disordered structures for most applications. We describe here a first-principles expression for strains in disordered such materials, based on insight from a family of 'iso-auxetic' structures. These are structures, consisting of internal structural elements, which we name 'auxetons', whose inter-element forces can be computed from statics alone. Iso-auxetic structures make it possible not only to identify the mechanisms that give rise to auxeticity, but also to write down the explicit dependence of the strain rate on the local structure, which is valid to all auxetic materials. It is argued that stresses give rise to strains via two mechanisms: auxeton rotations and auxeton expansion / contraction. The former depends on the stress via a local fabric tensor, which we define explicitly for 2D systems. The latter depends on the stress via an expansion tensor. Whether a material exhibits auxetic behaviour or not depends on the interplay between these two fields. This description has two major advantages: it applies to any auxeton-based system, however disordered, and it goes beyond conventional elasticity theory, providing an explicit expression for general auxetic strains and outlining the relevant equations.

cond-mat.mtrl-sci↗

arXiv2010

Da Vinci Fluids, catch-up dynamics and dense granular flow

We introduce and study a da Vinci Fluid, a fluid whose dissipation is dominated by solid friction. We analyse the flow rheology of a discrete model and then coarse-grain it to the continuum. We find that the model gives rise to behaviour that is characteristic of dense granular fluids. In particular, it leads to plug flow. We analyse the nucleation mechanism of plugs and their development. We find that plug boundaries generically expand and we calculate the growth rate of plug regions. In systems whose internal effective friction coefficient is relatively uniform we find that the linear size of plug regions grows as (time)$^{1/3}$. The suitability of the model to granular materials is discussed.

cond-mat.soft↗

arXiv2010

Plug flow formation and growth in da Vinci Fluids

A new, da Vinci, fluid is described as a model for flow of dense granular matter. We postulate local properties of the fluid, which are generically different from ordinary fluids in that energy is dissipated by solid friction. We present the equation of flow of such a fluid and show that it gives rise to formation and growth of plug flow regions, which is characteristic of flow of granular matter. Simple explicit examples are presented to illustrate the evolution of plug flow regions.

cond-mat.soft↗

arXiv2005

Geometric partition functions of cellular systems: Explicit calculation of the entropy in two and three dimensions

A method is proposed for the characterisation of the entropy of cellular structures, based on the compactivity concept for granular packings. Hamiltonian-like volume functions are constructed both in two and in three dimensions, enabling the identification of a phase space and making it possible to take account of geometrical correlations systematically. Case studies are presented for which explicit calculations of the mean vertex density and porosity fluctuations are given as functions of compactivity. The formalism applies equally well to two- and three-dimensional granular assemblies.

cond-mat.soft↗