Size segregation in granular flows is a well-known phenomenon: laboratory experiments consistently show that large particles migrate toward silo walls during filling, while smaller particles concentrate near the center. Paradoxically, field observations in large-scale industrial silos often report the opposite pattern, challenging these findings. We demarcate these patterns through a systematic experimental study spanning a range of dimensionless numbers relevant to bidisperse granular flows in quasi-2D silos under both dry (in air) and immersed (in liquid) conditions, varying container geometry and fluid viscosity. Image analysis reveals that the observed patterns are governed by two key dimensionless parameters: the slenderness of the silo and the Stokes number, which encapsulates the balance between particle inertia and viscous drag. Our results demonstrate the role of fluids on segregation dynamics and provide a unified scaling framework that reconciles laboratory- and field-scale observations in air.

Stresses and pressures are used to represent the hydromechanical state of deformable porous media. Past formulations often adopt the effective stress principle, usually in an empirical and energetically inconsistent way. Using the rigorous hydrodynamic procedure, this study pursues an alternative energy-consistent formulation for the full characterisation of both saturated and unsaturated porous materials. An elastic stress is consistently linked to its energy-conjugated elastic strain and, in the absence of viscous stress, has a structure that was previously interpreted as an effective stress. Here, it is emphasised that this similarity does not imply that the elastic stress is ‘effective’ in the classical sense, namely that it can replace total stress in dry soils to represent the mechanical behaviour of saturated or unsaturated soils. The dependence of the elastic stress on the deformability of the solid is incorporated constitutively using a general elastic strain energy of pressure- and density-dependent media, excluding energy costs from solid density changes due to volumetric elastic straining. By adopting the resulting internal energy that is convex for physically realistic porous materials, the proposed formulation yields a rigorous quantification of the elastic stress, and the pressures of the air, water, and solid required for characterising saturated and unsaturated soils, including the Biot stress correction coefficient for deformable porous media at variable saturation. The formulation also reveals the intrinsic dependence of the stress coefficients on material elasticity and the characteristics of water retention responses.

Small-angle X-ray scattering (SAXS) is a highly effective tool for non-destructive analysis of anisotropy in suspensions and colloids. SAXS can provide information of the size and shape of particles in suspension, and contains information on the ordering and structure of the internal particle arrangement. Such information is particularly relevant for clay materials, whose mechanical response is highly dependent on the microstructure. Here, a new approach was introduced for analysing SAXS images, focusing on the extraction of orientation degrees and preferred angles. This objective characterisation of the internal material fabric was applied to clay slurries sheared in a Couette cell under various shear rates and similar relative liquidity for three different clay minerals: kaolinite, halloysite and Na-montmorillonite. Kaolinite exhibited significantly higher anisotropy compared to halloysite and Na-montmorillonite, and reached its maximum anisotropy under a lower shear rate. Beyond clay science, the methodology developed in this study opens new possibilities in broader materials science and nanotechnology for studying the orientation of other particles and internal fabrics.

Granular flows in ambient fluids exhibit grain-size-dependent segregation, which is difficult to capture efficiently with existing models, especially in large-scale systems involving more than a million grains. We develop a two-way coupled framework that integrates heterarchical granular dynamics (HGD) with a fluid-fraction-weighted incompressible Navier-Stokes solver. This heterarchical granular-fluid dynamics (HGFD) model extends a previous HGD model for quasi-static deformations by introducing inertial, force-balance-driven particle velocities and consistent fluid-solid momentum exchange. The coupling between the inertial HGD and the fluid solver is performed using a staggered explicit sequential scheme and co-located Eulerian fields. The framework is evaluated against experimental data of (i) single-particle settling to verify inertial relaxation, (ii) hindered settling to reproduce concentration-dependent settling and vertical size stratification, and (iii) representative cases covering three reported segregation types to assess regime sensitivity. These results establish HGFD as an efficient and consistent approach for simulating fluid-coupled granular segregation dynamics.

The wide range of behaviour exhibited by granular flows has made them challenging to handle practically. For example, rotary mills used for grinding mining materials operate in the cataracting regime, which combines the whole range of flowing conditions for granular materials, including solid body motion, dense flow, and gaseous collisional regime. A good understanding of the material behaviour in such processes is crucial to optimise industrial operating conditions; however, this is rendered difficult by the large scales of the operations and the complexity of probing the charge inside the mills. In this paper, we present a method allowing for both comprehensive and manageable understanding of the local granular material behaviour, applicable to a wide variety of situations. Inspired by the example case of rotary mills, we perform discrete element numerical simulations of a rotating tumbler with lifters and extract the local fields of interest, such as strains, pressures, and density, through standard coarse-graining methods. Those smooth fields are then appropriately non-dimensionalised, and used as input for a clustering algorithm. The clustering allows pinpointing of distinct zones of behaviour in the tumbler. Crucially, those zones have well-identified physical properties, and are relevant to the whole range of rotation rates. These zones allow us to numerically study how the flow is affected by changes in operating conditions, similar field measurements in industry operating conditions are difficult to acquire due to the opacity of the mill and the harsh environment. We complement the numerical analysis by in-situ measurements in a laboratory scale three-dimensional rotating tumbler. We use recent technological advances in embedded sensors and X-ray dynamic radiography reconstruction to provide new insights into the behaviour of rotary mills, measuring particle trajectories and accelerations directly inside the mill. The analysis and experimental methods presented provide new tools and insights to analyse complex granular flows. They apply to the full range of granular regimes, and can be extended to a broad range of geometries and processes.

The two most commonly used methods to model the behaviour of granular flows are discrete element and continuum mechanics simulations. These approaches concentrate on the deterministic description of particle or bulk material motion. Unlike these approaches, this paper introduces an alternative model that describes the stochastic dynamics of the void spaces under the action of gravity. The model includes several key phenomena which are observed in deforming granular media, such as segregation, mixing, and an angle of repose. These mechanisms are modelled heterarchically using both spatial and microstructural internal coordinates. Key aspects of the model include its ability to describe both stable and flowing states of granular media based on a solid fraction cut-off, and the influence of particle size on flow, segregation, and mixing. The model is validated with simulations of column collapse and silo discharge.

The development of rate- and state-dependent friction laws offered important insights into the key physical mechanisms of the frictional behavior of fault gouges and their seismic cycle. However, past approaches were specifically tailored to address the problem of fault shearing, leaving questions about their ability to comprehensively represent the gouge material under general loading conditions. This work establishes an alternative approach for developing a physical friction law for fault gouges that is grounded on the rigour of the hydrodynamic procedure with two-scale temperatures through Terracotta, a thoroughly robust constitutive model for clay in triaxial loading conditions. By specifying the model for direct shearing, the approach yields an alternative friction law that readily captures the frictional dynamics of fault gouges, including explicit dependencies on gouge layer thickness, normal stress, and solid fraction. Validated against available laboratory experiments, the friction law retains the original predictive capabilities of Terracotta in triaxial conditions and explains the rate-and-state, dilatational behavior of fault gouges in direct shear conditions. Finally, when the Terracotta friction law is connected to a spring-dashpot representation of the host rock, the combined model predicts an elastic buildup precursor to the onset of and subsequent seismicity, with results closely reflecting experimental evidence and field observations. While this study focuses on clay-rich gouges, the approach and findings are expected to offer much wider implications to a variety of materials.

Semi-autogenous grinding (SAG) mills are commonly used in the mineral processing industry to effectively grind large particles into finer ones suitable for further processing. Despite their effectiveness in comminution, these mills are highly inefficient in terms of energy consumption. Understanding this inefficiency using population-balance models is challenging, as those are not informed by granular interactions and physics involving the interplay of crushing, segregation, and mixing. Conversely, using discrete-element simulation models to study SAG mills is also limited in scope since at any given time these mills can contain eight orders of magnitude more particles than those models can handle. Here, we address these challenges by developing a new comminution approach based on heterarchy. Expanding upon the previously developed heterarchical model for comminution in rotary mills, we incorporate the dynamics of grinding media consisting of steel balls, offering a comprehensive understanding of comminution within a SAG mill. Employing this new comminution model, we further analyse the effect of the different mill operating parameters - mill speed, fill fraction, and ball-to-charge ratio - on the comminution behaviour. This model brings us one step closer to understanding energy consumption in SAG mills.

High pressure grinding rolls (HPGR) are commonly used for material grinding in industries such as mineral processing, cement production, and chemical manufacturing. Comminution modelling for these devices is typically performed using either continuum-based population balance models or particle-based numerical tools such as the discrete element method (DEM). However, population balance models overlook the complex interplay of particle crushing, segregation, and mixing that occurs during comminution, limiting their effectiveness in accurately predicting device performance under varying operating conditions. Similarly, DEM models are limited by their high computational cost, which restricts their ability to track only a limited number of particles and simulate their sequential crushing. Here, we overcome the limitations of these traditional modelling approaches by exploring a novel heterarchical model for HPGR comminution. The heterarchical model captures the physics of particle crushing, segregation, and mixing while efficiently handling an arbitrarily large number of particles. Therefore, this model allows for the prediction and analysis of HPGR performance by tracking the evolving particle size distribution at any point in space and time.

As collections of grains flow, free-surface deformations often develop. These typically suggest the presence of secondary flows, smaller in magnitude than the primary motion but driving complex three-dimensional internal structures. While one can infer such behaviour from boundaries or simulations, we have not previously been able to directly observe secondary flows experimentally. In this paper we present an experimental confirmation of secondary kinematics within granular media using dynamic x-ray radiography, without needing to stop motion for tomography. Specifically, we create a bulldozing mechanism of conveyor-driven grains. This generates a non-uniform, indented free-surface, hinting that secondary mechanisms are at play alongside the primary regime. Discrete element method simulations are shown to be consistent with this secondary-flow explanation. We then probe further experimentally using two perpendicular x-ray source/detector pairs to measure the velocity inside the bulk. This indeed unveils a complex three-dimensional flow pattern that deviates from the primary vertical planes and must include vortices and convection rolls. This advancement is pertinent for industrial and natural scenarios where grains impact obstacles, and has broader relevance for studying the rheology associated with secondary flows in other amorphous materials such as emulsions, pastes and colloids.

The degradation of porous media can trigger devastating events, from sinkhole formation to ice-shelve breakouts. In a recent study, regularly recurring creep-quake collapse cycles were identified in cereals, whose progression was related to partial saturation. This Letter reveals such patterns in other materials, including sand and ice, but crucially, we demonstrate that they do not necessitate partial saturation. Conversely, we experimentally and theoretically show how the mechanism underpinning the creep-quake collapse cycles can develop in any porous medium, as long as it is simultaneously subjected to compression and a nonuniform degradation field, let that be a hydraulic, thermal, or chemical field. Our results highlight the ubiquity of creep-quake collapse cycles and the importance of the underlying degradation processes that activate this mechanism in any porous medium. Given the vast interest in porous media, this understanding is anticipated to have significant implications in the agricultural, pharmaceutical, and environmental sciences.

The rheology of granular materials is commonly described using the inertial number (I) as a measure of their fluidity. As the ratio between the time of macroscopic deformation and the time of microscopic rearrangement, I depends on the shear strain rate within the medium, which can vary in space. With dry granular media being opaque, experimental measurements of the field of inertial numbers were mainly limited to twodimensional (2D) observations along transparent walls in three-dimensional (3D) systems. This work addresses this gap using dynamic x-ray rheography of continuously flowing granular materials through an open channel conveyor belt setup. The granular medium is driven towards a perpendicular wall and forms a steady heap, for which the velocity field along the belt direction is measured in 3D. When used to estimate the shear strain rate, it is shown that, under the heap, a marked 3D pattern develops in the spatial distribution of I. The experimental findings are cross-validated with discrete element method simulations.

Heterarchical Granular Dynamics (HGD) represents a new approach to modelling the complex behaviour of granular materials across multiple scales. By modelling granular materials stochastically, HGD has enabled the simulation of granular phenomena ranging from simple systems to highly dynamic processes such as comminution in rotary mills. This summary reviews the foundational assumptions and key mechanisms underpinning HGD. Unlike the discrete element method, HGD efficiently scales to industrially relevant numbers of particles, and unlike traditional continuum methods, it retains microstructural information. HGD continues to evolve as a versatile framework for understanding granular dynamics in both natural and industrial contexts.

Laboratory description of clay normally distinguishes the scale of atoms from the scale of clay particles and aggregates. Contemporary constitutive models for clay tend to ignore this scale separation, and rather focus on phenomenology. By considering scale separation, this paper introduces a robust physics-based phenomenological constitutive model for clay that qualitatively captures their broad spectrum of rate-dependent mechanical features. The model is derived using the thoroughly rigorous hydrodynamic procedure. While some imagine that by considering rigour and physics, their models would get complicated, the resulting set of equations reveal a surprising degree of simplicity. The derivation strongly benefits from the principle of two-stage irreversibility, which describes energy flow within the material from the continuum scale down to the atomistic micro-scale, through the meso-scale of clay aggregates. While thermal and meso-related temperatures capture atomistic and clay aggregate fluctuating motions, a sink term from the latter to the former underpins the direction of the energy flow. The model's standout feature is in pinpointing new transport coefficients that drive both volumetric and shear plastic flows in a thermodynamically coupled manner. A novel scheme is then proposed to calibrate these coefficients from conventional steady-state observations. Thanks to the formulation the model shows a remarkable level of predictiveness, despite being relatively simple mathematically. In particular, the model readily explains the broad spectrum of rate-dependent phenomena during transient loading, along with creep and relaxation processes. Given the generality of hydrodynamics, it is anticipated that the new model could be expanded to capture fluid-solid transitions between liquid-like soft mud and plastic-like stiff clay responses, contingent on water content variations.

Data-driven models based on deep learning algorithms intend to overcome the limitations of traditional constitutive modelling by directly learning from data. However, the need for extensive data that collate the full state of the material is hindered by traditional experimental observations, which typically provide only small data - sparse and partial material state observations. To address this issue, we develop a novel deep learning algorithm referred to as Neural Integration for Constitutive Equations to discover constitutive models at the material point level from scarce and incomplete observations. It builds upon the solution of the initial value problem describing the time evolution of the material state, unlike the majority of data-driven approaches for constitutive modelling that require large data of increments of state variables. Numerical benchmarks demonstrate that the method can learn accurate, consistent, and robust constitutive models from incomplete, sparse, and noisy data collecting simple conventional experimental protocols.

Elastic wave speeds are fundamental in geomechanics and have historically been described by an analytic formula that assumes linearly elastic solid medium. Empirical relations stemming from this assumption were used to determine nonlinearly elastic stiffness relations that depend on pressure, density, and other state variables. Evidently, this approach introduces a mathematical and physical disconnect between the derivation of the analytical wave speed (and thus stiffness) and the empirically generated stiffness constants. In our study, we derive wave speeds for energy-conserving (hyperelastic) and non-energy-conserving (hypoelastic) constitutive models that have a general dependence on pressure and density. Under isotropic compression states, the analytical solutions for both models converge to previously documented empirical relations. Conversely, in the presence of shear, hyperelasticity predicts changes in the longitudinal and transverse wave speed ratio. This prediction arises from terms that ensure energy conservation in the hyperelastic model, without needing fabric to predict such an evolution, as was sometimes assumed in previous investigations. Such insights from hyperelasticity could explain the previously unaccounted-for evolution of longitudinal wave speeds in oedometric compression. Finally, the procedure used herein is general and could be extended to account for other relevant state variables of soils, such as grain-size, grain-shape, or saturation.

Rotary mills aim to effectively reduce the size of particles through a process called comminution. Modelling comminution in rotary mills is a challenging task due to substantial material deformation and the intricate interplay of particle kinematics of segregation, mixing, crushing, and abrasion. Existing particle-based simulations tend to provide predictions that cannot cope with the large number of particles within rotary mills, their wide range of sizes, and the physics dictating the crushing of individual particles. Similarly, there is currently no deterministic modelling means to determine the evolving population of particle sizes at any point in time and space within the mill. The aim of this two-part contribution is to address these gaps by advancing a framework for a novel stochastic comminution model for rotary mills, which has a well-defined deterministic continuum limit and can cope with arbitrarily large numbers of particles. This work describes the basic physics and structure of the new model within a heterarchical framework for ball and autogenous grinding mills. The primary focus of this Part I paper is to develop a computational model for the integration of motion of material along streamlines inside a mill. Coupled to this process is the kinetic physics dictating particle crushing. In a subsequent work, Part II, segregation and mixing will be added to this model such that realistic behaviour from the mill can be observed.

Semi-autogeneous grinding mills are used widely to reduce the size of mined ores and extract minerals of interest. However, these mills are extremely energy consuming. To reduce the energy wasted by mostly moving and shearing instead of grinding the material, a good understanding of the behaviour of the charge in the mill is crucial. In this paper, we first establish different smooth fields of internal variables such as density, strain rates, and pressure, for a rotary mill simulated using discrete element modelling. As these fields are interrelated, we then use a clustering method to identify well-defined zones of distinct behaviour within the mill. This approach allows us to analyse and summarise the behaviour of complex flows as a set of localised zones with well identified physical properties. These zones resemble the ones which were previously sketched intuitively based on experimental observations for specific parameters, including the rotation rate and filling fraction of the charge. On the other hand, the dependence of those zones on the operating parameters of the mill provides a direct interpretation of the effects of adjusting the available operating conditions in the field. The method can be readily generalised to other geometries and particle technologies, and can therefore provide a convenient approach for both comprehensive and manageable understanding of material behaviour in a wide range of applications.

Elastic wave speeds are crucial in geomechanics for measuring elastic stiffness. Traditionally, these speeds are calculated using an analytic formula based on a linearly elastic solid medium. However, empirical evidence suggests that stiffness is state-dependent, creating a mismatch between the theoretical assumptions and the observed stiffness constants. Recent advancements have addressed this gap by deriving wave speeds for hyperelastic (energy conserving) and hypoelastic (non-energy-conserving) constitutive models that are dependent on pressure and density. These new derivations align with conventional empirical findings for isotropic states. However, the hyperelastic model predicts variations in the ratio of longitudinal to transverse wave speeds, as observed in experiments and discrete element simulations. This prediction emerges from energy-conservation terms in the model, eliminating the need for fabric assumptions previously considered in research. Our study expands on this by exploring various density-dependent relationships and extending wave speed calculations to saturated granular media. This research enhances our understanding of granular media stiffness in dry and saturated scenarios, offering new insights for interpreting wave speed experiments.