Multi-head neural networks for simulating particle breakage dynamics

The breakage of brittle particulate materials into smaller particles under compressive or impact loads can be modelled as an instantiation of the population balance integro-differential equation.In this paper,the emerging computational science paradigm of physics-informed neural networks is studied for the first time for solving both linear and nonlinear variants of the governing dynamics.Unlike conventional methods,the proposed neural network provides rapid simulations of arbitrarily high resolution in particle size,predicting values on arbitrarily fine grids without the need for model retraining.The network is assigned a simple multi-head architecture tailored to uphold monotonicity of the modelled cumulative distribution function over particle sizes.The method is theoretically analyzed and validated against analytical results before being applied to real-world data of a batch grinding mill.The agreement between laboratory data and numerical simulation encourages the use of physics-informed neural nets for optimal planning and control of industrial comminution processes.

Particle breakage evolution of coral sand using triaxial compression tests

Particle breakage continuously changes the grading of granular materials and has a significant effect on their mechanical behaviors.Revealing the evolution pattern of particle breakage is valuable for development and validation of constitutive models for crushable materials.A series of parallel triaxial compression tests along the same loading paths but stopped at different axial strains were conducted on two coral sands with different particle sizes under drained and undrained conditions.The tested specimens were carefully sieved to investigate the intermediate accumulation of particle breakage during the loading process.The test results showed that under both drained and undrained conditions,particle breakage increases continuously with increasing axial strain but exhibits different accumulating patterns,and higher confining pressures lead to greater particle breakage.Based on the test results,the correlations between particle breakage and the stress state as well as the input energy were examined.The results demonstrated that either the stress state or input energy alone is inadequate for describing the intermediate process of particle breakage evolution.Then,based on experimental observation,a path-dependent model was proposed for particle breakage evolution,which was formulated in an incremental form and reasonably considers the effects of the past breakage history and current stress state on the breakage rate.The path-dependent model successfully reproduced the development of particle breakage during undrained triaxial compression using the parameters calibrated from the drained tests,preliminarily demonstrating its effectiveness for different stress paths.

Study on the particle breakage of ballast based on a GPU accelerated discrete element method

Breakage of particles will have greatly influence on mechanical behavior of granular material(GM)under external loads,such as ballast,rockfill and sand.The discrete element method(DEM)is one of the most popular methods for simulating GM as each particle is represented on its own.To study breakage mechanism of particle breakage,a cohesive contact mode is developed based on the GPU accelerated DEM code-Blaze-DEM.A database of the 3D geometry model of rock blocks is established based on the 3D scanning method.And an agglomerate describing the rock block with a series of non-overlapping spherical particles is used to build the DEM numerical model of a railway ballast sample,which is used to the DEM oedometric test to study the particles’breakage characteristics of the sample under external load.Furthermore,to obtain the meso-mechanical parameters used in DEM,a black-analysis method is used based on the laboratory tests of the rock sample.Based on the DEM numerical tests,the particle breakage process and mechanisms of the railway ballast are studied.All results show that the developed code can better used for large scale simulation of the particle breakage analysis of granular material.

A study on particle breakage behavior during pile penetration process using acoustic emission source location

Particle breakage is a common occurrence in granular systems when the external stress exceeds the individual particle strength.A large number of experimental evidences suggested that particle breakage may significantly influence the soil behavior.In the case of pile foundations,the subsoil below the pile tip experiences considerable high stress and consequently prone to break.Due to the lack of sufficient understanding on particle breakage mechanism,there is currently no consentaneous theoretical background for particle breakage analysis during the pile penetration process.This study aims to clarify the location of particle breakage and its evolving characteristics with the aid of acoustic emission(AE)source location method.The spatial distribution of AE hypocenters is interpreted to be associated with the mechanism of particle breakage.Results showed that the AE sources were not uniformly distributed,but concentrated within certain zones below the pile tip.This AE concentration zone was pushed downward with the advancing pile tip,and its distance from the real time pile tip position decreased after certain depth of pile penetration.The location of particle breakage interpreted from AE source location was verified with posttest excavations and the insights on the particle breakage evolution zone were further discussed.

Particle breakage of sand subjected to friction and collision in drum tests

This paper presents a laboratory experimental study on particle breakage of sand subjected to friction and collision,by a number of drum tests on granular materials(silica sand No.3 and ceramic balls)to investigate the characteristics of particle breakage and its effect on the characteristics of grain size distribution of sand.Particle breakage increased in up convexity with increasing duration of drum tests,but increased linearly with increasing number of balls.Particle breakage showed an increase,followed by a decrease while increasing the amount of sand.There may be existence of a characteristic amount of sand causing a maximum particle breakage.Friction tests caused much less particle breakage than collision tests did.Friction and collision resulted in different mechanisms of particle breakage,mainly by abrasion for friction and by splitting for collision.The fines content increased with increasing relative breakage.Particle breakage in the friction tests(abrasion)resulted in a sharper increase but with a smaller total amount of fines content in comparison with that in the collision tests(splitting).For the collision tests,the fines content showed a decrease followed by an increase as the amount of sand increased,whereas it increased in up convexity with increasing number of balls.The characteristic grain sizes D_(10) and D_(30) decreased in down convexity with increasing relative breakage,which could be described by a natural exponential function.However,the characteristic grain sizes D50 and D60 decreased linearly while increasing the relative breakage.In addition,the coefficients of uniformity and curvature of sand showed an increase followed by a decrease while increasing the relative breakage.

Influence of particle breakage on the isotropic compressibility of sands

Particle breakage is a theme of great focus on affecting the behavior of granular soil.This paper presents an experimental investigation on the influence of particle breakage on the isotropic compressibility of the precrushed sands using a number of drained isotropic consolidation tests.Particle breakage resulted in movement of the compression lines followed by the rebound lines towards a decrease in the void ratio,implying that particle breakage caused a more contractive soil.Particle breakage impaired the bulk deformation modulus by increasing the compression coefficient for the compression behavior of the precrushed sands,but showed a complex effect on the bulk deformation modulus and rebound coefficient for the rebound behavior of the precrushed sands.However,particle breakage caused an increase in the compression indexes of the precrushed sands but showed a complex effect on the rebound indexes of the precrushed sands.In the e-p’plane and the e-logp’plane,the compression lines and rebound lines of the precrushed sands were curved.A generalized model was proposed to straighten the compression and rebound lines of the precrushed sands in the e-(p’p_(a))^(α) plane.Particle breakage resulted in a general rotation and translation of the linear compression and rebound lines of the precrushed sands in the e-(p’p_(a))^(α)plane.The critical state line and isotropic consolidation line on the loosest state of silica sand no.5 were curved in the e-logp’plane but straightened in the e-p’^(α=0.7) plane.In the e-p’^(α=0.7 )plane,a reasonable linear critical state line of silica sand no.5 was proposed by adjusting it to match the isotropic consolidation line on the loosest state.

Modeling for critical state line of granular soil with evolution of grain size distribution due to particle breakage

Determination of the critical state line(CSL)is important to characterize engineering properties of granular soils.Grain size distribution(GSD)has a significant influence on the location of CSL.The influence of particle breakage on the CSL is mainly attributed to the change in GSD due to particle breakage.However,GSD has not been properly considered in modeling the CSL with influence of particle breakage.This study aims to propose a quantitative model to determine the CSL considering the effect of GSD.We hypothesize that the change of critical state void ratio with respect to GSD is caused by the same mechanism that influences of the change of minimum void ratio with respect to GSD.Consequently,the particle packing model for minimum void ratio proposed by Chang et al.(2017)is extended to predict critical state void ratio.The developed model is validated by experimental results of CSLs for several types of granular materials.Then the evolution of GSD due to particle breakage is incorporated into the model.The model is further evaluated using the experimental results on rockfill material,which illustrates the applicability of the model in predicting CSL for granular material with particle breakage.

Numerical simulation study on particle breakage behavior of granular materials in confined compression tests

In order to study the fragmentation law,the confined compression experiment of granular assemblies has been conducted to explore the particle breakage characteristic by DEM approach in this work.It is shown that contact and contact force during the loading process gradually transform from anisotropy to isotropy.Meanwhile,two particle failure modes caused by different contact force states are analyzed,which are single-through-crack failure and multi-short-crack failure.Considering the vertical distribution of the number of cracks and the four characteristic stress distributions(the stress related to the maximum contact force,the major principal stress,the deviatoric stress and the mean stress),it is pointed out that the stress based on the maximum contact force and the major principal stress can reflect the distribution of cracks accurately.In addition,the size effect of particle crushing indicates that small size particles are prone to break.The lateral pressure coefficient of four size particles during the loading process is analyzed to explain the reason for the size effect of particle breakage.

DEM Modeling of Particle Breakage in Silica Sands under One-Dimensional Compression

A Discrete Element Method （DEM） model is developed to study the particle break- age effect on the one-dimensional compression behavior of silica sands. The ＇maximum tensile stress＇ breakage criterion considering multiple contacts is adopted to simulate the crushing of circular particles in the DEM. The model is compared with published experimental results. Com- parison between the compression curves obtained from the numerical and experimental results shows that the proposed method is very effective in studying the compression behavior of silica sands considering particle breakage. The evolution of compression curves at different stress levels is extensively studied using contact force distribution, variation of contact number and particle size distribution curve with loading. It is found that particle breakage has great impact on com- pression behavior of sand, particularly after the yield stress is reached and particle breakage starts. The crushing probability of particles is found to be macroscopically affected by stress level and particle size distribution curve, and microscopically related to the evolutions of contact force and coordination number. Once the soil becomes well-graded and the average coordination number is greater than 4 in two-dimension, the crushing probability of parent particles can reduce by up to 5/6. It is found that the average contact force does not always increase with loading, but increases to a peak value then decreases once the soil becomes more well-graded. It is found through the loading rate sensitivity analysis that the compression behavior of sand samples in the DEM is also affected by the loading rate. Higher yield stresses are obtained at higher loading rates.

Effect of particle breakage on the behavior of simulated angular particle assemblies

Many attempts have been made to find various relationships for different parameters and some kinds of constitutive models for studying the behavior of particulate media. All these models are based on concepts of continuous media. Using a numerical method such as discrete element method, one can figure out what is happening through a discontinuous media where soil particles play the main role in introducing the shear strength and deformation characteristics. The behavior of the media with breakable particles is studied in this paper and compared with that of the assembly with non-breakable particles. In this paper, the hyperbolic elastic model is investigated for the assembly of polygon shaped particles in two different test series. In addition, evolution of different macro parameters of the assembly such as volume strain, angle of friction, angle of dilatancy and elastic modulus is studied during the simulation tests both for non-breakable and breakable soil particles. At the end, a parametric study is performed on the effect of strength of particle breakage on the assembly behavior.

A constitutive model incorporating particle breakage for gravelly soil-structure interface under cyclic loading

Under cyclic loading,particle breakage occurs at gravelly soil-structure interface,resulting in the decrease of interface strength and the increase of normal displacement.Based on the theory of critical state soil mechanics,the modified Cam-Clay model(MCC)was extended to the plane strain condition of the interface,the state parameter was introduced and the influence of particle breakage on the critical state line was considered,and the cyclic constitutive model for gravelly soil-structure interface considering particle breakage was established by using the non-associated flow rule.Then,the established cyclic constitutive model was used to simulate large-scale cycle direct shear tests of Zipingpu rockfill-steel interface and Zipingpu rockfill-concrete interface under constant normal load(CNL)and constant normal stiffness(CNS),respectively.The simulation results show that under the CNL cyclic loading path,there is little difference between the cyclic shear stress considering particle breakage and that without particle breakage,but the normal displacement considering particle breakage is larger than that without particle breakage,and the difference increases with the increasing number of cycles and normal stress;Under the CNS cyclic loading path,with the increase of the number of cycles,the cyclic shear stress and cyclic normal stress considering particle breakage is significantly smaller than that without particle breakage,and the shear contraction of normal displacement becomes more obvious.In general,the simulation results are closer to the experimental results when particle breakage is considered.

Gravel hardness effect on compaction characteristics of gravelly soil

The compaction characteristics of gravelly soil are affected by gravel hardness.To investigate the evolution and influencing mechanism of different gravel hardness on the compaction characteristics of gravelly soil,heavy compaction tests and crushing tests were conducted on gravelly soils with gravels originated from hard,soft and extremely soft rocks.According to orthogonal experiments and variance analysis,it was found that hardness has a significant impact on the maximum dry density of gravelly soil,followed by gravel content,and lastly,moisture content.For gravel compositions with an average saturated uniaxial compressive strength less than 60 MPa,the order of compacted maximum dry density is soft gravels>hard gravels>extremely soft gravels.Each type of gravelly soil has a threshold for gravel content,with 60%for hard and soft gravels and 50%for extremely soft gravels.Beyond these thresholds,the compacted dry density decreases significantly.There is a certain interaction between hardness,gravel content,and moisture content.Higher hardness increases the influence of gravel content,whereas lower hardness increases the influence of moisture content.Gravelly soils with the coarse aggregate(CA)between 0.7 and 0.8 typically achieve higher dry densities after compaction.In addition,the prediction equations for the particle breakage rate and CA ratio in the Bailey method were proposed to estimate the compaction performance of gravelly soil preliminarily.The results further revealed the compaction mechanism of different gravelly soils and can provide reference for subgrade filling construction.

Determination of critical state line(CSL)for silty-sandy iron ore tailings subjected to low-high confining pressures

The disposal of filtered tailings in high dry stacks can induce particle breakage,changing the material's behaviour during the structure's lifetime.The grading changes influence material properties at the critical state,and this is not mature for intermediate artificial soils(tailings)in a broad range of confining pressures.In this paper,it aims to describe the behaviour of iron ore tailings in a spectrum of confining pressures broader than the reported in previous studies.A series of consolidated drained(CD)triaxial tests was carried out with confining pressures ranging from 0.075 MPa to 120 MPa.These results show that the amount of breakage plays an essential role in the response of iron ore tailings.The existence of curved critical state line(CSL)in both specific volume(ν)-logarithm of mean effective stress(p′)and deviatoric stress(q)-mean effective stress(p′)planes,and different responses in the deviatoric stress-axial strain-volumetric strain curves were verified.An inverse S-shaped equation was proposed to represent the silty-sandy tailings'behaviour up to high pressures onν-lnp′plane.The proposed equation provides a basis for enhancing constitutive models and considers the evolution of the grading up to severe loading conditions.The adjustment considered three regions with different responses associated with particle breakage at different pressure levels.

Dilation and breakage dissipation of granular soils subjected to monotonic loading

Dilation and breakage energy dissipation of four different granular soils are investigated by using an energy balance equation. Due to particle breakage, the dilation curve does not necessarily pass through the origin of coordinates. Breakage energy dissipation is found to increase significantly at the initial loading stage and then gradually become stabilised. The incremental dissipation ratio between breakage energy and plastic work exhibits almost independence of the confining pressure. Accordingly, a plastic flow rule considering the effect of particle breakage is suggested. The critical state friction angle is found to be a combination of the basic friction between particles and the friction contributed by particle breakage.

Packing,compressibility,and crushability of rockfill materials with polydisperse particle size distributions and implications for dam engineering

In rockfill dam engineering,particle breakage of rockfill materials is one of the major factors resulting in dam settlement.In this study,one-dimensional compression tests on a series of coarse granular materials with artificially-graded particle size distributions(PSDs)were carried out.The tests focused on understanding the role of initial PSDs in the dense packing density,compressibility and crushability of coarse granular materials.The effects of fractal dimension(D)and size polydispersity(θ)of PSDs were quantitatively analyzed.Two different loading stages were identified from the logarithms of the stress-strain relationships,with the turning point marked as the yield stress.A similar effect of initial PSDs was observed on the packing density and low-pressure modulus of coarse granular materials.The packing density and low-pressure modulus increased monotonically withθ,and their peak values were attained at a D value of approximately 2.2.However,there was no unique correspondence between the dense packing density and low-pressure modulus.The particle breakage was influenced differently by the initial PSDs,and it decreased with the values of D andθ.The emergence of the unique ultimate state was also identified from both the compression curves and PSDs of the samples after the tests.The potential implications of the test results in the design of both low and high rockfill dams were also demonstrated.

Micromechanics of breakage in sharp-edge particles using combined DEM and FEM

By combining DEM （Discrete Element Method） and FEM （Finite Element Method）, a model is established to simulate the breakage of twodimensional sharp-edge particles, in which the simulated particles are assumed to have no cracks. Particles can, however, crush during different stages of the numerical analysis, if stress-based breakage criteria are fulfilled inside the particles. With this model, it is possible to study the influence of particle breakage on macro- and micro-mechanical behavior of simulated angular materials. Two series of tests, with and without breakable particles, are simulated under different confining pressures based on conditions of biaxial tests. The results, presented in terms of micromechanical behavior for different confining pressures, are compared with macroparameters. The influence of particle breakage on microstructure of sharp-edge materials is discussed and the related confining pressure effects are investigated. Breakage of particles in rockfill materials are shown to reduce the anisotropy coefficients of the samples and therefore their strength and dilation behaviors.

Grain crushing in geoscience materials-Key issues on crushing response,measurement and modeling:Review and preface

Grain crushing is commonly encountered in deep foundation engineering,high rockfill dam engineering,railway engineering,mining engineering,coastal engineering,petroleum engineering,and other geoscience application.Grain crushing is affected by fundamental soil characteristics,such as their mineral strength,grain size and distribution,grain shape,density and specimen size,and also by external factors including stress magnitude and path,loading rate and duration,degree of saturation,temperature and geochemical environment.Crushable material becomes a series of different materials with the change in its grading during grain crushing,resulting in a decrease in strength and dilatancy and an increase in compressibility.Effects of grain crushing on strength,dilatancy,deformation and failure mechanisms have been extensively investigated through laboratory testing,discrete element method(DEM)modelling,Weibull statistics,and constitutive modelling within the framework of the extended crushing-dependent critical state theory or the energy-based theory.Eleven papers summarized in this review article for this special issue addressed the above issues in grain crushing through the advanced testing and modelling.

Numerical Investigation of the Cushion and Size Effects During Single-Particle Crushing via DEM

This paper uses the discrete element method to model the size and cushion effects during single-particle crushing tests.We propose simplified numerical modeling to examine the effects of particle size and coordination number on particle breakage behavior.We validate the proposed modeling by comparing the numerical results with the experimental data reported in the literature,in terms of the variability of particle tensile strength and axial force-displacement responses.Based on the numerical results,it is clear that a larger particle size entails a higher tensile strength with a larger discreteness.In addition,the characteristic tensile strength increases linearly with an increasing coordination number.Moreover,smaller particles are more susceptible to the cushion effect than larger particles.The numerical results also indicate that an increasing coordination number induces a more ductile mode of failure.Based on these results,we propose an empirical equation for calculating tensile strength,incorporating both the cushion effect and the size effect.

Effect of stress path on the mechanical properties of calcareous sand

Calcareous sand is widely observed in the foundation of off-shore infrastructure.Although a lot of research has been carried out on the mechanical properties of calcareous sand,study into the influence of the stress–strain path on the mechanical behaviour of calcareous sand is very limited.In this study,a series of triaxial tests were performed on calcareous sand under three different stress paths.The particle morphology of calcareous sand before and after the tests,the stress–strain relationship under different stress paths,and the char-acteristics of shear strength and deformation were investigated.The results show that the consolidation pressure and stress path have significant effects on the volume strain,strength,and particle breakage of calcareous sand.In addition,the underlying mechanisms of the different behaviours of calcareous sand observed in this study were discussed.