Multi-layer perceptron-based data-driven multiscale modelling of granular materials with a novel Frobenius norm-based internal variable

One objective of developing machine learning(ML)-based material models is to integrate them with well-established numerical methods to solve boundary value problems(BVPs).In the family of ML models,recurrent neural networks(RNNs)have been extensively applied to capture history-dependent constitutive responses of granular materials,but these multiple-step-based neural networks are neither sufficiently efficient nor aligned with the standard finite element method(FEM).Single-step-based neural networks like the multi-layer perceptron(MLP)are an alternative to bypass the above issues but have to introduce some internal variables to encode complex loading histories.In this work,one novel Frobenius norm-based internal variable,together with the Fourier layer and residual architectureenhanced MLP model,is crafted to replicate the history-dependent constitutive features of representative volume element(RVE)for granular materials.The obtained ML models are then seamlessly embedded into the FEM to solve the BVP of a biaxial compression case and a rigid strip footing case.The obtained solutions are comparable to results from the FEM-DEM multiscale modelling but achieve significantly improved efficiency.The results demonstrate the applicability of the proposed internal variable in enabling MLP to capture highly nonlinear constitutive responses of granular materials.

Mobility and dynamic erosion process of granular flow:insights from numerical investigation using material point method

In order to understand the dynamics of granular flow on an erodible base soil,in this paper,a series of material point method-based granular column collapse tests were conducted to investigate numerically the mobility and dynamic erosion process of granular flow subjected to the complex settings,i.e.,the aspect ratio,granular mass,friction and dilatancy resistance,gravity and presence of water.A set of power scaling laws were proposed to describe the final deposit characteristics of granular flow by the relations of the normalized run-out distance and the normalized final height of granular flow against the aspect ratio,being greatly affected by the complex geological settings,e.g.,granular mass,the friction and dilatancy resistance of granular soil,and presence of water in granular flow.An index of the coefficient of friction of granular soil was defined as a ratio of the target coefficient of friction over the initial coefficient of friction to quantify the scaling extent of friction change(i.e.,friction strengthening or weakening).There is a characteristic aspect ratio of granular column corresponding to the maximum mobility of granular flow with the minimum index of the apparent coefficient of friction.The index of the repose coefficient of friction of granular flow decreased gradually with the increase in aspect ratio because higher potential energy of granular column at a larger aspect ratio causes a larger kinetic energy of granular soil to weaken the friction of granular soil as a kind of velocity-related friction weakening.An increase in granular mass reduces gradually the indexes of the apparent and repose coefficients of friction of granular soil to enhance the mobility of granular flow.The mobility of granular flow increases gradually with the decrease in friction angle or increase in dilatancy angle of granular soil.However,the increase of gravity accelerates granular flow but showing the same final deposit profile without any dependence on gravity.The mobility of granular flow increases gradually by lowering the indexes of the apparent and repose coefficients of friction of granular flow while changing the surroundings,in turn,the dry soil,submerged soil and saturated soil,implying a gradually increased excessive mobility of granular flow with the friction weakening of granular soil.Presence of water in granular flow may be a potential catalyzer to yield a long run-out granular flow,as revealed in comparison of water-absent and water-present granular flows.In addition,the dynamic erosion and entrainment of based soil induced by granular flow subjected to the complex geological settings,i.e.,the aspect ratio,granular mass,gravity,friction and dilatancy resistance,and presence of water,were comprehensively investigated as well.

A thermodynamics-based three-scale constitutive model for partially saturated granular materials

A three-scale constitutive model for unsaturated granular materials based on thermodynamic theory is presented.The three-scale yield locus,derived from the explicit yield criterion for solid matrix,is developed from a series of discrete interparticle contact planes.The three-scale yield locus is sensitive to porosity changes;therefore,it is reinterpreted as a corresponding constitutive model without phenomenological parameters.Furthermore,a water retention curve is proposed based on special pore morphology and experimental observations.The features of the partially saturated granular materials are well captured by the model.Under wetting and isotropic compression,volumetric compaction occurs,and the degree of saturation increases.Moreover,the higher the matric suction,the greater the strength,and the smaller the volumetric compaction.Compared with the phenomenological Barcelona basic model,the proposed three-scale constitutive model has fewer parameters;virtually all parameters have clear physical meanings.

Finite Element Simulations on Failure Behaviors of Granular Materials with Microstructures Using a Micromechanics-Based Cosserat Elastoplastic Model

This paper presents a micromechanics-based Cosserat continuum model for microstructured granular materials.By utilizing this model,the macroscopic constitutive parameters of granular materials with different microstructures are expressed as sums of microstructural information.The microstructures under consideration can be classified into three categories:a medium-dense microstructure,a dense microstructure consisting of one-sized particles,and a dense microstructure consisting of two-sized particles.Subsequently,the Cosserat elastoplastic model,along with its finite element formulation,is derived using the extended Drucker-Prager yield criteria.To investigate failure behaviors,numerical simulations of granular materials with different microstructures are conducted using the ABAQUS User Element(UEL)interface.It demonstrates the capacity of the proposed model to simulate the phenomena of strain-softening and strain localization.The study investigates the influence of microscopic parameters,including contact stiffness parameters and characteristic length,on the failure behaviors of granularmaterials withmicrostructures.Additionally,the study examines themesh independence of the presented model and establishes its relationship with the characteristic length.A comparison is made between finite element simulations and discrete element simulations for a medium-dense microstructure,revealing a good agreement in results during the elastic stage.Somemacroscopic parameters describing plasticity are shown to be partially related to microscopic factors such as confining pressure and size of the representative volume element.

Evaluation of the coefficient of lateral stress at rest of granular materials under repetitive loading conditions

Although the internal stress state of soils can be affected by repetitive loading,there are few studies evaluating the lateral stress(or K_(0))of soils under repetitive loading.This study investigates the changes in K_(0) and directional shear wave velocity(V_(s))in samples of two granular materials with different particle shapes during repetitive loading.A modified oedometer cell equipped with bender elements and a diaphragm transducer was developed to measure the variations in the lateral stress and the shear wave velocity,under repetitive loading on the loading and unloading paths.The study produced the following results:(1)Repetitive loading on the loading path resulted in an increase in the K_(0) of test samples as a function of cyclic loading number(i),and(2)Repetitive loading on the unloading path resulted in a decrease in K_(0) according to i.The shear wave velocity ratio(i.e.V_(s)(HH)/V_(s)(VH),where the first and second letters in parentheses corresponds to the directions of wave propagation and particle motion,respectively,and V and H corresponds to the vertical and horizontal directions,respectively)according to i supports the experimental observations of this study.However,when the tested material was in lightly over-consolidated state,there was an increase in K_(0) during repetitive loading,indicating that it was the initial K_(0),rather than the loading path,which is responsible for the change in K_(0).The power model can capture the variation in the K_(0) of samples according to i.Notably,the K_(0)=1 line acts as the boundary between the increase and decrease in K_(0) under repetitive loading.

Modelling of the variation of granular base materials resilient modulus with material characteristics and humidity conditions

This study aims to quantify the susceptibility of granular materials used in pavements to changes in moisture content and propose a correlation model to incorporate this susceptibility into seasonal analyses.The fines content and the percentage of fractured coarse aggregates were identified as direct indicators of the resilient modulus susceptibility to changes in water content.The results showed that the percentage of fractured coarse aggregates particles(FR)has a more significant impact on the resilient modulus(Er)of crushed granular materials used in pavement construction than the combined indicator of the fines content and sample volumetrics(nf).Crushed granular materials with a higher percentage of fractured coarse aggregates are relatively insensitive to changes in the degree of saturation,but become more sensitive as the fine fraction porosity decreases.An adjusted model was proposed based on the existing formulation,but considers a complex parameter to describe and adjust the sensitivity of base granular materials to variations in moisture content with respect to fabrication charac-teristics,fines content and volumetric properties.The model shows that the variation of Er values is below10%for fully crushed granular materials.However,it reaches approximately±12%for materials with 75%of crushed coarse aggregates andþ40%and-25%for materials with FR=50%.This model could help select good ag-gregates characteristics and adjust grain-size distribution for environments where significant moisture content variations can occur in the pavement system,such as in the Province of Quebec(Canada).As it is based on pa-rameters that can be easily determined or estimated,it also represents a valuable tool for detailed design and analysis that can consider material characteristics.

Constitutive modelling of idealised granular materials using machine learning method

Predicting the constitutive response of granular soils is a fundamental goal in geomechanics.This paper presents a machine learning(ML)framework for the prediction of the stress-strain behaviour and shearinduced contact fabric evolution of an idealised granular material subject to triaxial shearing.The MLbased framework is comprised of a set of mini-triaxial tests which provide a benchmark for the setup and validation of the discrete element method(DEM)model of the granular materials,a parametric DEM simulation programme of virtual triaxial tests which provides datasets of micro-and macro-mechanical information,as well as a multi-layer perceptron(MLP)neural network which is trained and tested using the DEM-based datasets.The ML model only requires the initial void ratio of the granular sample as the input for predicting its constitutive response.The excellent agreement between the ML model prediction and experimental test and DEM simulation results indicates that the MLebased modelling approach is capable of capturing accurately the effects of initial void ratio on the constitutive behaviour of idealised granular materials,bypassing the need to incorporate the complex micromechanics underlying the macroscopic mechanical behaviour of granular materials.Lastly,a detailed comparison between the used MLP model and long short-term memory(LSTM)model was made from the perspective of technical algorithm,prediction accuracy,and computational efficiency.

Fluidization and Transport of Vibrated Granular Matter:A Review of Landmark and Recent Contributions

We present a short retrospective review of the existing literature about the dynamics of(dry)granular matter under the effect of vibrations.The main objective is the development of an integrated resource where vital information about past findings and recent discoveries is provided in a single treatment.Special attention is paid to those works where successful synthetic routes to as-yet unknown phenomena were identified.Such landmark results are analyzed,while smoothly blending them with a history of the field and introducing possible categorizations of the prevalent dynamics.Although no classification is perfect,and it is hard to distillate general properties out of specific observations or realizations,two possible ways to interpret the existing results are defined according to the type of forcing or the emerging(ensuing)regime of motion.In particular,first results concerning the case where vibrations and gravity are concurrent(vertical shaking)are examined,then the companion situation with vibrations perpendicular to gravity(horizontal shaking)is described.Universality classes are introduced as follows:(1)Regimes where sand self-organizes leading to highly regular geometrical“pulsating”patterns(thin layer case);(2)Regimes where the material undergoes“fluidization”and develops an internal multicellular convective state(tick layers case);(3)Regimes where the free interface separating the sand from the overlying gas changes inclination or develops a kind a patterned configuration consisting of stable valleys and mountains or travelling waves;(4)Regimes where segregation is produced,i.e.,particles of a given size tend to be separated from the other grains(deep containers).Where possible,an analogy or parallelism is drawn with respect to the companion field of fluid-dynamics for which the assumption of“continuum”can be applied.

The Effects of the Particle Size Ratio on the Behaviors of Binary Granular Materials

The particle size ratio(PSR)is an important parameter for binary granular materials,which may aect the microstructure and macro behaviors of granular materials.However,the eect of particle ratio on granular assemblies with dierent arrangements is still unclear.To explore and further clarify the eect of PSR in dierent packing structures,three types of numerical samples with regular,layered,and random packing are designed.Numerical results show that PSR has signi􀀀cant eects on binary granular samples with regular packing.The larger the PSR,the stronger the strength,the larger the modulus,and the smaller the angle between the shear band and the load direction.And a theoretical solution of the peak stress ratio vs.PSR is obtained for regular packing,and the results by DEM are in good agreement with the theoretical solution.Under layered packing,PSR has little eect on peak stress ratio due to similar microstructure obtained with the changing of PSR.The modulus slightly increased with the increase of PSR.Under random packing with small grain content of 50%,PSR has little eect in the range of 0.5–0.9,but in a larger range,larger PSR leads to greater modulus.

Effect of granular shape on radial segregation in a two-dimensional drum

Granular segregation is widely observed in nature and industry.Most research has focused on segregation caused by differences in the size and density of spherical grains.However,due to the fact that grains typically have different shapes,the focus is shifting towards shape segregation.In this study,experiments are conducted by mixing cubic and spherical grains.The results indicate that spherical grains gather at the center and cubic grains are distributed around them,and the degree of segregation is low.Through experiments,a structured analysis of local regions is conducted to explain the inability to form stable segregation patterns with obviously different geometric shapes.Further,through simulations,the reasons for the central and peripheral distributions are explained by comparing velocities and the number of collisions of the grains in the flow layer.

Particle breakage of granular materials during sample preparation

Particle breakage is commonly observed in granular materials when subjected to external loads. It was found that particle breakage would occur during both sample preparation and loading stages. However,main attention was usually paid to the particle breakage behaviour of samples during loading stage. This study attempts to explore the breakage behaviour of granular materials during sample preparation.Triaxial samples of rockfill aggregates are prepared by layered compaction method to achieve different relative densities. Extents of particle breakage based on the gradings before and after test are presented and analysed. It is found that particle breakage during sample preparation cannot be ignored. Gradings after test are observed to shift away from the initial grading. Aggregates with larger size that appear to break are more than the smaller-sized ones. Irrespective of the initial gradings, an increase in the extent of particle breakage with the increasing relative density is observed during sample preparation.

New shakedown criterion and permanent deformation properties of unbound granular materials

Unbound granular material specifications for road pavements in Australia are primarily based on physical material specification rather than mechanical characterisation. This simplified approach does not reflect the actual material performance under repeated dynamic traffic loads. There is a little information available on the influence of the local crushed rock properties and compacted layer properties on permanent deformation (PD). This study aims to characterise the local unbound granular materials in Victoria according to their PD behaviour under repeated loads and to develop a suitable shakedown criterion that could describe the PD of the tested materials to simplify the flexible pavement design. Repeated-load triaxial tests were conducted over several samples with a range of moisture contents, gradations, densities, and stress conditions. The laboratory test results showed that PD behaviour was influenced by several factors. In addition, the tested subbase-specified unbound granular materials reflect high PD resistance that is almost equivalent to basequality unbound granular materials. This may indicate that current requirements for the subbase-quality unbound granular materials are over-prescribe. Moreover, as the existing shakedown criterion was not applicable for the multi-stage repeated-load triaxial test and the local tested materials, a new shakedown criterion and new boundaries are proposed based on the PD behaviour. In the proposed criterion, the shakedown ranges are identified based on the curve angle of the PD vs. logarithm of the number of loading cycles, and this new criterion was validated using several materials from existing literature. The local tested base and subbase materials can be assigned as Range A when PD\1%, Range B when 1%\PD\3%, and Range C when PD[3%. The proposed criterion could provide a useful and quick approach to assess the PD of the unbound granular materials with both single and multistages of stresses.

Effect of vertical stress rest period on deformation behaviour of unbound granular materials:Experimental and numerical investigations

Repeated load triaxial test is used to assess the deformation behaviour of unbound granular materials(UGMs) in flexible road pavements. Repeated load pulse characteristics(i.e. shape, loading period and rest period) are the stress configurations used in the experimental set-up to simulate the passing axle loads. Some researchers and standard testing protocols suggest a rest period of varying durations after a loading phase. A thorough review of existing literature and practises has revealed that there is no agreement about the effect of the rest period of vertical stress pulse on the deformation behaviour of the UGMs. Therefore,the main objective of this study is to investigate the effect of repeated stress rest period on the deformation behaviour of UGMs experimentally. Experiments are conducted, both with and without rest period, using basalt and granite crushed rocks from Victoria, Australia. Furthermore, in order to gain insight into the effect of the rest period, finite element modelling is also developed. Both the experimental and modelling results show that the rest period has a noticeable effect on both resilient and permanent deformation behaviours of UGMs. It is, therefore, recommended to take extra precautions while adopting a particular standard testing protocol and to supplement the results by additional tests with different loading configurations.

Deep Learning Predicts Stress–Strain Relations of Granular Materials Based on Triaxial Testing Data

This study presents an AI-based constitutive modelling framework wherein the prediction model directly learns from triaxial testing data by combining discrete element modelling(DEM)and deep learning.A constitutive learning strategy is proposed based on the generally accepted frame-indifference assumption in constructing material constitutive models.The low-dimensional principal stress-strain sequence pairs,measured from discrete element modelling of triaxial testing,are used to train recurrent neural networks,and then the predicted principal stress sequence is augmented to other high-dimensional or general stress tensor via coordinate transformation.Through detailed hyperparameter investigations,it is found that long short-term memory(LSTM)and gated recurrent unit(GRU)networks have similar prediction performance in constitutive modelling problems,and both satisfactorily predict the stress responses of granular materials subjected to a given unseen strain path.Furthermore,the unique merits and ongoing challenges of data-driven constitutive models for granular materials are discussed.

The measure of friction angles for different types of granular material

The aim of this research is to deepen the knowledge of the role of friction on the dynamics of granular media; in particular the friction angle is taken into consideration as the physical parameter that drives stability, motion and deposition of a set of grains of any nature and size. The idea behind this work is a question: is the friction angle really that fundamental and obvious physical parameter which rules stability and motion of granular media as it seems from most works which deal with particle dynamics? The experimental study tries to answer this question with a series of laboratory tests, in which different natural and artificial granular materials have been investigated in dry condition by means of a tilting flume. The characteristic friction angles, both in deposition(repose) and stability limit(critical) conditions, were measured and checked against size, shape, density and roughness of the considered granular material. The flume tests have been preferred to "classical" geotechnical apparatuses(e.g. shear box) since the flume experimental conditions appear closer to the natural ones of many situations of slope stability interest(e.g. a scree slope). The results reveal that characteristic friction angles depend on size and shape of grains while mixtures of granules of different size show some sorting mechanism with less clear behaviour.

Stress and fabric in granular material

It has been well recognized that, due to anisotropic packing structure of granular material, the true stress in a specimen is different from the applied stress. However, very few research efforts have been focused on quantifying the relationship between the true stress and applied stress. In this paper, we derive an explicit relationship among applied stress tensor, material-fabric tensor, and force-fabric tensor; and we propose a relationship between the true stress tensor and the applied stress tensor. The validity of this derived relationship is examined by using the discrete element simulation results for granular material under biaxial and triaxial loading con- ditions.

Mathematical Model for the Drying Process of Granular Materials in a Fluidized Bed

The experiment of granular materials, barley, drying in a fluidized bed was carried out to investigate the influence of the factors, inlet air temperature, air moisture, bed height and original moisture content of the dried materials on drying process. Based on the experimental data, a corresponding mathematical model is presented. As a conclusion, a higher inlet air temperature and a reasonable bed height should be used so as to increase the dring rate and to improve the product quality.

Investigations of the effects of particle morphology on granular material behaviors using a multi-sphere approach

This article studies the influences of particle morphology on the behaviors of granular materials at both macroscopic and microscopic levels based on the discrete element method(DEM).A set of numerical tests under drained triaxial compression was performed by controlling two morphological descriptors,i.e.ratio of the smallest to the largest pebble diameter,x,and the maximum pebbleepebble intersection angle,b.These descriptors are vital in generating particle geometry and surface textures.It was found that the stress responses of all assemblies exhibited similar behavior and showed post-peak strainsoftening.The normalized stress ratio and volumetric strains flatten off and tended to reach a steady value after an axial strain of 40%.While the friction angles at peak state varied with different morphological descriptors,the friction angles at critical state showed no significant variation.Moreover,evolution of the average coordination numbers showed a dramatic exponential decay until an axial strain of about 15%after which it stabilized and was unaffected by further increase of axial strain.In addition,stress ratio q/p and strong fabric parameter fs d=fs m were found to follow an approximately linear relationship for each assembly.These findings emphasized the significance of the influences of particle morphology on the macroscopic and microscopic responses of granular materials.

Investigation on relations between grain crushing amount and void ratio change of granular materials in one-dimensional compression and creep tests

Grain crushing plays an important role in one-dimensional (1D) compression and creep behaviors of granular materials under high stress. It is clear that the macro-properties of granular materials are closely related to the micro-fracture properties of grains in 1D compression and creep tests. In this paper, a series of 1D compression and creep tests were performed on Ottawa sand to investigate the deformation and grain crushing properties of granular materials, and it shows that the void ratio is correlated to the grain crushing amount (the quantity of crushed grains) for granular materials subjected to grain crushing. The test results, combining with the existing test data related to grain crushing of granular materials, were used to verify the relation. Moreover, the implications of these relations on the yield of granular material, and the equivalent effect of stress and time in changing soil fabric are presented.

Applicability of discrete element method with spherical and clumped particles for constitutive study of granular materials

Discrete element method(DEM)has been intensively used to study the constitutive behaviour of granular materials.However,to what extent a real granular material can be reproduced by virtual DEM simulations remains unclear.This study attempts to answer this question by comparing DEM simulations with typical features of experimental granular materials.Three groups of models with spherical and clumped particles are investigated from four perspectives:(i)deviatoric stress and volumetric behaviour;(ii)critical state behaviour;(iii)stress-dilatancy relationship;and(iv)the evolution of principal stress ratio against axial strain.The results demonstrate that DEM with spherical or clumped particles is capable of qualitatively describing macroscopic deviatoric stress responses,volumetric behaviour,and critical state behaviour observed in experiments for granular materials.On the other hand,some qualitative deviations between experiments and the investigated DEM simulations are also observed,in terms of the stress-dilatancy behaviour and principal stress ratio against axial strain,which are proven to be critical for constitutive modelling.The results demonstrate that DEM with spherical or clumped particles may not necessarily fully capture experimental features of granular materials even from a qualitative perspective.It is thus encouraged to thoroughly validate DEM with experiments when developing constitutive models based on DEM observations.