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.

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.

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.

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.

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.

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.

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.

Experimental and numerical study of the blast wave decrease using sandwich panel by granular materials core

Among the intrinsic properties of some materials,e.g.,foams,porous materials,and granular materials,are their ability to mitigate shock waves.This paper investigated shock wave mitigation by a sandwich panel with a granular core.Numerical simulations and experimental tests were performed using Autodyn hydro-code software and a shock tube,respectively.The smoothed particle hydrodynamics(SPH)method was used to model granular materials.Sawdust and pumice,whose properties were determined by several compression tests,were used as granular materials in the sandwich panel core.These granular materials possess many mechanisms,including compacting(e.g.,sawdust)and crushing(e.g.,pumice)that mitigate shock/blast wave.The results indicated the ineffectiveness of using a core with low thickness,yet it was demonstrated to be effective with high thickness.Low-thickness pumice yielded better results for wave mitigation.The use of these materials with a core with appropriate core reduces up to 88%of the shock wave.The results of the experiments and numerical simulations were compared,suggesting a good agreement between the two.This indicates the accuracy of simulation and the ability of the SPH method to modeling granular material under shock loading.The effects of grain size and the coefficient of friction between grains have also been investigated using simulation,implying that increasing the grain size and coefficient of friction between grains both reduce overpressure.

Characterization of Elastic Modulus of Granular Materials in a New Designed Uniaxial Oedometric System

A simple uniaxial oedometric system is developed to test the elastic modulus of granular materials. The stress- strain relationship is first measured under conditions of uniaxial compression with additional lateral stress and strain, then the elastic modulus of the material is determined by the linear fitting method. It is found that the modulus is positively correlated to the grain size and negatively correlated to the container size. Arching and dragging are revealed to be the mechanism of such correlations by using the digital image correlation method and the pressure film technology based on the statistical method.

Discrete numerical modeling of granular materials considering crushability

The aim of this study is to numerically investigate the influence of particle breakage on the mechanical behavior of granular materials using a discrete element method(DEM). To enable particle crushing, non-crushable elementary particles are boned together to represents the granular aggregates which can be crushed when the external force exceeds its strength. The flaw of the aggregate was also modeled by randomly distributed void. Single particle crushing tests were carried out to determine the distribution of particle strength. The results of single particle crushing tests illustrate that the simulated single particle fracture strength and pattern agree well with the Weibull's distribution equation.Conventional oedometer tests, drained monotonic and cyclic triaxial tests were also carried out to investigate the crushing of the aggregates and the associated mechanical behaviors. The effect of confining pressure on the crushing of aggregates and the mechanical behavior was also analyzed. It was found that the peak stress and dilation decrease significantly when particle crushing was considered.The deformation behavior of the specimen is essentially controlled by two factors: particle rearrangement-induced dilation and particle crushing-induced contraction. The increase of permanent strain and the reduction of dilation were observed during cyclic loading and they tend to reach a stable state after a certain number of cycles. The crushing of aggregate is most significant in the first two cycles. The results also indicated that for the same axial strain the volumetric strain and the bound breakage in the cyclic loading tests are significantly larger than that in the monotonic loading tests,especially at high cyclic stress ratio.

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.

Experimental study on remodeling strength of granular materials under different loads and lengths of time

Remodeled clay and sand rock specimens were prepared by designing lateral confinement and water drainage experiments based on the stress exerted on granular materials in a waste dump.An in situ test was conducted in an internal waste dump;the physical and mechanical parameters of the remodeled rock mass dumped at different time and depths were measured.Based on statistics,regression analysis was performed with regard to the shearing stress parameters acquired from the two tests.Other factors,such as remodeling pressure(burial depth),remodeling time(amount of time since waste was dumped),and the corresponding functional relationship,were determined.Analysis indicates that the cohesion of the remodeled clay and its remodeling pressure are correlated by a quadratic function but are not correlated with remodeling time length.In situ experimental results indicate that the shear strength of reshaped granular materials in the internal dump is positively correlated with burial depth but poorly correlated with time length.Cohesion Cand burial depth H follow a quadratic function,specifically for a short time since waste has been dumped.As revealed by both in situ and laboratory experiments,the remodeling strength of granular materials varies in a certain pattern.The consistency of such materials verifies the reliability of the remodeling experimental program.

“Relay-mode”promoting permeation of water-based fire extinguishing agent in granular materials porous media stacks

Water-based fire extinguishing agent is the main means to deal with smoldering fires.However,due to the hydrophobic properties of the particle surface,the porous medium channel provide resistance and slow down the extinguishing agent flow during the downward permeation process.To promote the liquid permeation process in such porous media,this work studied liquid imbibition process and analyzed the oscillating and attenuating process of liquid level in capillary channel by theoretical,experimental,and numerical methods.An empirical mathematical equation was proposed to describe the oscillating process,and the effects of the capillary diameter and contact angle parameters on the transportation process were analyzed.Based on this,the“relay-mode”was proposed to promote the liquid transportation forward.Finally,the transient simulation results of liquid permeation in coal stacks showed when the liquid flowed through the channel with changed diameter from large to small ones,the transportation distance was several times longer than that through the unidiameter ones.The trend of liquid“relay-mode”in capillaries can be used to promote the permeation in granular materials porous media stacks.The relevant results also provide new thoughts to develop the water-based fire extinguishing agents and then improve the firefighting efficiency of deep-seated fire in porous media stacks.

On the degradation of granular materials due to internal erosion

A new state-based elasto-plastic constitutive relationship along with the discrete element model is established to estimate the degradation of granular materials due to internal erosion.Four essential effects of internal erosion such as the force network damage and relaxation are proposed and then incorporated into the constitutive relationship to formulate internal erosion impacts on the mechanical behavior of granular materials.Most manifestations in the degradation of granular materials,such as reduction of peak strength and dilatancy are predicted by the modified constitutive relationship in good agreement with the discrete element method（DEM）simulation.In particular,the sudden reduction of stress for conspicuous mass erosion in a high stress state is captured by force network damage and the relaxation mechanism.It is concluded that the new modified constitutive relationship is a potential theory to describe the degradation of granular materials due to internal erosion and would be very useful,for instance,in the prediction and assessment of piping disaster risk during the flood season.

Editorial: Computational mechanics of granular materials

Most of granular materials are highly heteroge- neous, composed of voids and particles with different sizes and shapes. Geological matter, soil and clay in nature, geo-structure, concrete, etc. are practical ex- amples among them. From the microscopic view, a lo- cal region in the medium is occupied by particles with small but finite sizes and granular material is naturally modeled as an assembly of discrete particles in contacts On the other hand, the local region is identified with a material point in the overall structure and this discon- tinuous medium can then be represented by an effective continuum on the macroscopic level

Dynamics of a Rotating Sphere on Free Surface of Vibrated Granular Materials

We investigate the rotational dynamics of a low-density sphere on the free surface of a vertically vibrated granular material（VGM）. The dynamical behavior of the sphere is influenced by the external energy input from an electromagnetic shaker which is proportional to ε,where ε is equal to the ratio between the square of the dimensionless acceleration Γ and the square of the vibration frequency f of the container. Empirical results reveal that as the VGM transits from local-to-global convection,an increase in ε generally corresponds to an increase in the magnitudes of the rotational ω（RS） and translational v（CM） velocities of the sphere, an increase in the observed tilting angle θ（bed） of the VGM bed, and a decrease in the time t（wall） it takes the sphere to roll down the tilted VGM bed and hit the container wall. During unstable convection, an increase in ε results in a sharp decrease in the sphere＇s peak and mean ω（RS）,and a slight increase in t（wall）.For the range of ε values covered in this study, the sphere may execute persistent rotation, wobbling or jamming, depending on the vibration parameters and the resulting convective flow in the system.

New double yield surface model for coarse granular materials incorporating nonlinear unified failure criterion

A new double-yield-sarface （DYS） model was developed to characterize the strength and deformation behaviors of coarse granular materials （CGMs）. Two kinds of deformation mechanisms, including the shear and compressive plastic deformation, were taken into account in this model, These two deformation mechanisms were described by the shear and compressive yield functions, respectively. The Lode angle dependent formulations of proposed model were deduced by incorporating a 3D nonlinear unified failure criterion. Some comparisons were presented between the numerical predictions of proposed model and test data of true triaxial tests on the modeled rockfills. The model predictions are in good agreement with the test data and capture the strain hardening and plastic volumetric dilation of CGMs. These findings verify the reasonability of current DYS model, and indicate that this model is well suited to reproduce the stress-strain-volume change behavior of CGMs in general.

The Material Deformation and Internal Structure Development of Granular Materials under Different Cyclic Loadings

Common structures in engineering such as slopes,roadbeds,ballasts,etc.,are closely related to granular materials.They are usually subjected to long-term cyclic loads.This study mainly focused on the mechanical behaviors of randomly arranged granular materials before they reach a stable state under different cyclic loads.The variation of the maximum axial strain and the influence of CSR(cyclic stress ratio)were analyzed.The energy consumed in each cycle under constant confining stress loading condition is significantly greater than that of the fixed wall loading condition.The internal deformation evolution of granular materials is studied in detail.The deformation mode of granular material under cyclic loading at different positions inside the material is different according to the strain variation.In addition,the strain,force chain structure and contact force magnitude are combined to explore their effects on local deformation of granular materials under cyclic loading.From the perspective of the deformation form,the material sample can be divided into several regions,and the ability to adjust particle positions determines the deformation mode of different regions.The changes of local strain with the cyclic loading also reflect the contribution of particle displacements to the evolution ofmicrostructure.This research will provide insights into the understanding of granular materials behaviors under cyclic loading.

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.