Constitutive Model For Granular Materials Research Articles

Remarks on Constitutive Modeling of Granular Materials

In this paper, we provide a brief overview of certain fundamental concepts which can be used to derive constitutive relations for the stress tensor of granular materials. These include concepts such as dilatancy, cohesion, yield criterion, shear banding, etc. The focus will be on the constitutive relations which are used in the so-called ‘frictional flow’ or ‘slow flow’ regime as opposed to the rapid flow regime; in the slow flow regime the material is about to yield or has just yielded and the flow has been initiated. This type of flow occurs in the storage of grains, etc., in silos and bins or hoppers after the valves/gates are opened. The techniques of continuum mechanics are used to discuss constitutive relations where the effects of non-linearities such as yield stress, dilatancy, density gradients, etc., are important.

Data-driven multiscale modelling of granular materials via knowledge transfer and sharing

Machine learning approaches have found immense potential to revolutionise the constitutive modelling of granular materials. However, data scarcity poses a significant challenge to this emerging paradigm. This study aims to tackle this issue by presenting two transfer learning-based strategies that harness well-established constitutive knowledge and similar material data to reduce data demands for data-driven material modelling. The first approach utilises phenomenological constitutive models to generate massive synthetic data which reflect the targeted material behaviour to train a base model. This base model is then repurposed for a new task based on numerical simulation data via transfer learning. The other approach involves using available material data to train a base model, which is then applied to other new materials that are similar but with limited data. The proposed transfer learning methods are tested on both particle-scale simulations of representative volume elements (RVEs) and hierarchical multiscale modelling of boundary value problems (BVPs) of granular materials. The trained data-driven material model is embedded in numerical simulations with the finite element method (FEM) to validate its accuracy, efficiency, and stability. The results demonstrate that transfer learning can effectively achieve high-quality machine learning predictions with limited data. The transfer learning strategy presented in this study is expected to be widely applicable to small data-driven material modelling.

A Hypoplastic Constitutive Model for Granular Materials with Particle Breakage

A Hypoplastic Constitutive Model for Granular Materials with Particle Breakage

Modeling shear-induced solid-liquid transition of granular materials using persistent homology

This paper investigates the transition between the solid and liquid phases of sheared granular materials from the perspective of the contact network. Tools from persistent homology are employed to quantify the dynamics of contact network during the solid-liquid transition from a global perspective, and two important topological invariants, i.e., components and loops, are mainly investigated from discrete numerical simulations. The highly heterogeneous composition of the contact network is revealed, and a rationale partition threshold for distinguishing between strong and weak contact subnetworks can be determined through the emergence and death of these topological invariants. During the shearing process, we recognize mechanical precursors forecasting the occurrence of solid-liquid transition when the assembly is still stable. Furthermore, we provide the panorama of the solid-liquid transition from the evolution of contact network and its homology groups. Finally, this study suggests that the persistent homology method is capable of quantitatively bridging the microscopic dynamics with macroscopic responses through the contact network, which paves an efficient way to further include the evolution of the contact network in the constitutive modeling of granular materials.

A physics-informed deep neural network for surrogate modeling in classical elasto-plasticity

In this work, we present a deep neural network architecture that can efficiently surrogate classical elasto-plastic constitutive relations. The network is enriched with crucial physics aspects of classical elasto-plasticity, including additive decomposition of strains into elastic and plastic parts, and nonlinear incremental elasticity. This leads to a Physics-Informed Neural Network (PINN) surrogate model named here as Elasto-Plastic Neural Network (EPNN). Detailed analyses show that embedding these physics into the architecture of the neural network facilitates a more efficient training of the network with less training data, while also enhancing the extrapolation capability for loading regimes outside the training data. The architecture of EPNN is model and material-independent; it can be adapted to a wide range of elasto-plastic material types, including geomaterials; and experimental data can potentially be directly used in training the network. To demonstrate the robustness of the proposed architecture, we adapt its general framework to the elasto-plastic behavior of sands. We use synthetic data generated from material point simulations based on a relatively advanced dilatancy-based constitutive model for granular materials to train the neural network. The superiority of EPNN over regular neural network architectures is demonstrated through predicting unseen strain-controlled loading paths for sands with different initial densities.

Mean stress tensor of discrete particle systems in submerged conditions

The mean stress tensor is essential to investigate the dynamics of granular material. In this paper, we use Hamilton’s principle of least action to derive the averaged stress tensor of discrete granular assemblies subjected to hydraulic force fields, as well as rigorous conditions for a proper definition of the Representative Volume Element (RVE). The main goal behind our efforts is to upscale particle physics into a sound stress tensor for systems involving the complex interaction between grains and water. We identify the contributions from the unbalanced forces, hydraulic forces, gravity, external forces, and particle fluctuation to the mean stress tensor. In doing so, it is convenient to separate the influence of different force fields when the granular system is subjected to complex environments, e.g., subaqueous conditions. The obtained formula is then validated by triaxial test simulations of dry and saturated granular systems using the Discrete Element Method (DEM) and the Lattice-Boltzmann Method (LBM). The results show that the deduced formula can accurately calculate the stress tensor of discrete assemblies with various body-force fields. We used validated DEM-LBM simulations of submerged granular column collapses to explore the physics happening at the grain scale with this mathematical formalism and showcase its potential. We provide a new perspective based on the granular assembly scale to pursue the fluid–solid interaction. Due to the importance of stress analysis in the constitutive modeling of granular materials, this work could help to better obtain the stress–strain relationship of saturated or submerged granular systems.

Deep active learning for constitutive modelling of granular materials: From representative volume elements to implicit finite element modelling

Constitutive relation remains one of the most important, yet fundamental challenges in the study of granular materials. Instead of using closed-form phenomenological models or numerical multiscale modelling, machine learning has emerged as an alternative paradigm to revolutionise the constitutive modelling of granular materials. However, deep neural networks (DNNs) require massive training data and often fail to make credible extrapolations. This study aims to develop a deep active learning strategy to (i) identify unreliable forecasts without knowing the ground truth; and (ii) continuously improve and verify a data-driven constitutive model until the desired generalisation is satisfied. The role of active learning in constitutive modelling is instantiated through three scenarios: (i) off-line strain-stress data pool of granular materials; (ii) interactive constitutive training and strain-stress data labelling; and (iii) finite element modelling (FEM) driven by deep learning-based constitutive models. The results confirm the capability of active learning in advancing data-driven constitutive modelling of granular materials toward developing a faithful surrogate constitutive model with less data. The same active learning strategy can also be applied to other data-centric applications across various science and engineering fields.

Flow and heat transfer simulation in the ADS Dense Granular-flow Target using the Material Point Method

The Dense Granular-flow Target (DGT) proposed by Chinese Academy of Sciences has great potential compared with solid target and liquid metal target. In DGT, the number of metal grains is up to ten million. To study the flow and heat transfer behavior of such a large number of particles, experiments and Discrete Element Method (DEM) simulations are extremely expensive. Therefore, this paper focus on develop a continuum approach suitable for simulating the flow and heat transfer behavior in DGT. This approach is based on the Material Point Method (MPM) and chose the μ(I) rheology as the constitutive model for granular material. It can be concluded from this work that the μ(I) rheology implemented in MPM is able to reproduce quantitatively the granular behavior in hopper flow. The velocity distributions calculated by MPM are in good agreement with those calculated by DEM, although the fluctuation information in particle scale is filtered. The temperature results of MPM are basically consistent with DEM in mesoscopic scale, which is a great improvement compared with the results from uniform velocity estimation. The influences of the effective thermal conductivity, proton current and flow rate on the temperature field are investigated. According to the MPM calculation, the maximum proton current that the target can withstand and the minimum velocity that the granular needs to be met can be estimated quickly. The MPM procedure developed in this paper would be of considerable help in predicting the flow and heat transfer behavior for DGT.

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.

Representation of stress and strain in granular materials using functions of direction

This paper proposes a new way of describing effective stress in granular materials, in which stress is represented by a continuous function of direction in physical space. The proposal provides a rigorous approach to the task of upscaling from particle mechanics to continuum mechanics, but is simplified compared to a full discrete element analysis. It leads to an alternative framework of stress–strain constitutive modelling of granular materials that in particular considers directional dependency. The continuous function also contains more information that the corresponding tensor, and thereby provides space for storing information about history and memory. A work-conjugate set of geometric rates representing strain-rates is calculated, and the fundamental principles of local action, determinism, frame indifference, and rigid transformation indifference are shown to apply. A new principle of freedom from tensor constraint is proposed. Existing thermo-mechanics of granular media is extended to apply for the proposed functions, and a new method is described by which strain-rate equations can be used in large-deformations modelling. The new features are illustrated and explored using simple linear elastic models, producing new results for Poisson’s ratio and elastic modulus. Ways of using the new framework to model elastoplasticity including critical states are also discussed.

A constitutive model for granular materials subjected to a large stress range

Abstract A simple constitutive model is presented to represent the behavior of granular materials subjected to a large stress range by introducing a revised normal compression line with a limit void ratio (eL) in the double logarithmic scale and a state parameter (ξ) to quantify the state of granular materials. In the proposed model, a drop-shaped yield surface is employed and a unified hardening parameter (H) that is a function of the state parameter (ξ) is developed to govern the hardening process. The model is first established in the triaxial compression stress state and then is extended to the general stress state through the transformed stress (TS) method. Adopting a single set of material parameters, the proposed model is verified against the experimental results of various tests on Cambria sand at different confining pressures (0.25–68.9 MPa), including the data of isotropic compression test and drained/undrained triaxial compression/extension tests.

Work–energy analysis of granular assemblies validates and calibrates a constitutive model

This study presents a rate-independent constitutive model for granular materials which is based on the analysis of discrete element modelling (DEM) results. We examine the elastic and plastic strain rate at any state of shear, and explore how the plastic strain rate and the stress rate are related to some state variables (e.g. stress, void ratio, etc.). The elastic and plastic strain rate are evaluated by assuming that the rate of conservative energy defined in term of elastic strain rate equals the energy rate from work–energy analysis of granular assemblies. The verification of the theoretical work–energy analysis and the acquire of data regarding the mechanical rates at contacts between grains are done through DEM. It is shown that for a representative element volume of granular materials, the external work rate equals the sum of an elastic energy rate and a dissipation rate at contacts. The work rate is either the product of stress and strain rate or the sum of discrete work rate of external forces. Equations regarding the dilatancy, plastic deviatoric strain rate and stress rate are proposed. In the model, a yield surface is not explicitly defined and the deviatoric stress rate is related to not only the elastic deviatoric strain rate, but also a rate of dilatancy. Simulations show that the predictions match both virtual and Toyoura sand satisfactorily in various shear tests within a board range of densities. Additionally, the calibration of parameters is extremely simple. The treatment of cyclic loading condition is also presented.

On feasibility of rate-independent stress paths under proportional deformations within hypoplastic constitutive model for granular materials

We study stress paths that are obtained under proportional deformations within the rate-independent hypoplasticity theory of Kolymbas type describing granular materials like soil and broken rock. For a particular simplified hypoplastic constitutive model by Bauer, a closed-form solution of the corresponding system of non-linear ordinary differential equations is available. Since only negative principal stresses are relevant for the granular body, the feasibility of the solution consistent with physics is investigated in dependence of the direction of a proportional strain path and constitutive parameters of the model.

Constitutive modelling of granular materials using a contact normal-based fabric tensor

This paper presents a fabric tensor-based bounding surface model accounting for anisotropic behaviour (e.g. the dependency of peak strength on loading direction and non-coaxial deformation) of granular materials. This model is developed based on a well-calibrated isotropic bounding surface model. The yield surface is modified by incorporating the back stress which is proportional to a contact normal-based fabric tensor for characterising fabric anisotropy. The evolution law of the fabric tensor, which is dependent on both rates of the stress ratio and the plastic strain, rules that the material fabric tends to align with the loading direction and evolves towards a unique critical state fabric tensor under monotonic shearing. The incorporation of the evolution law leads to a rotational hardening of the yield surface. The anisotropic critical state is assumed to be independent of the initial values of void ratio and fabric tensor. The critical state fabric tensor has the same intermediate stress ratio (i.e. b value) and principal directions as the critical state stress tensor. A non-associated flow rule in the deviatoric plane is adopted, which is able to predict the non-coaxial flow naturally. The stress–strain relation and fabric evolution of model predictions show a satisfactory agreement with DEM simulation results under monotonic shearing with different loading directions. The model is also validated by comparing with laboratory test results of Leighton Buzzard sand and Toyoura sand under various loading paths. The comparison results demonstrate encouraging applicability of the model for predicting the anisotropic behaviour of granular materials.

A constitutive model for granular materials with evolving contact structure and contact forces\u2014part II: constitutive equations

This and the companion paper present a constitutive model for granular materials with evolving contact structure and contact forces, where the contact structure and contact forces are characterised by some statistics of grain-scale entities such as contact normals and contact forces. And these statistics are actually the “fabric” or “force” terms in the “stress–force–fabric” (SFF) equation. The stress–strain response is obtained by inserting the predicted “fabric” or “force” terms from evolution equations into the SFF equation. In the model, the critical state is characterised by two fitting equations and three critical state parameters. A semi-mechanistic analysis is conducted about the change of the contact number and the obtained results are combined with observed phenomena in DEM virtual experiments to give the constitutive equations for the “fabric” terms. The change of fabric anisotropy is related to the strain rate, current fabric anisotropy and also contact forces. The change of coordination number is induced by two terms related to volumetric and shear deformations, and also an additional term related to the change of fabric anisotropy. The constitutive equations regarding the “force” terms are also proposed. All the “fabric” or “force” terms are modelled to tend toward their critial state value, which agrees with Li and Dafalias’s (J Eng Mech 138(3):263–275, 2012. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000324) basic philosophy in their evolution equation for the fabric tensor. These equations along with the SFF equation form a constitutive model.

A constitutive model for granular materials with evolving contact structure and contact forces\u2014Part I: framework

This and the companion paper present a constitutive model for granular materials with evolving contact structure and contact forces, where the contact structure and contact forces are characterised by some statistics of grain-scale entities such as contact normals and contact forces. And these statistics are actually the “fabric” or “force” terms in the “stress–force–fabric” (SFF) equation. The stress–strain response is obtained by inserting the predicted “fabric” or “force” terms from evolution equations into the SFF equation. Discrete element modelling is used to verify the slightly modified SFF equation and also to obtain the data of how the contact structure and contact forces evolve in various loading paths. It is demonstrated that a normalised contact force is a better measure of the contact forces in polydisperse granular assemblies and strong contacts should be contacts with larger normalised contact forces. The modified SFF equation is shown to predict the stress accurately. The constitutive equations regarding the response of the contact structure and contact forces are presented and they along with the SFF equation form a constitutive model, which is found capable of capturing the observed phenomena correctly and predicting the mechanical response in various loading conditions. The model is shown to be an extension to the hypoplastic models with more state variables.

Genetic variants implicated in the pathogenesis of delusional disorder: A systematic review of evidence-to-date

In this paper, the effects of the intermediate stress ratio, i.e., b-value (b = (σ2 − σ3)/(σ1 − σ3)), on the contact normal-based fabric evolution of granular material, are incorporated into an extant hybrid fabric evolution law. The new evolution law is validated by Discrete Element Method (DEM) simulation results under monotonic shearing with different b-values. Predictions of the proposed generalized fabric evolution law agree well with the DEM simulation results. This evolution law can be widely used for constitutive modelling of granular materials, considering the effects of b-value in a general geomechanical three-dimensional stress space.

The influence of particle-size distribution on critical state behavior of spherical and non-spherical particle assemblies

This paper presents an investigation into the effects of particle-size distribution on the critical state behavior of granular materials using discrete element method (DEM) simulations on both spherical and non-spherical particle assemblies. A series of triaxial test DEM simulations examine the influence of particle-size distribution (PSD) and particle shape, which were independently assessed in the analyses presented. Samples were composed of particles with varying shapes characterized by overall regularity (OR) and different PSDs. The samples were subjected to the axial compression through different loading schemes: constant volume, constant mean effective stress, and constant lateral stress. All samples were sheared to large strains to ensure that a critical state was reached. Both the macroscopic and microscopic behaviors in these tests are discussed here within the framework of the anisotropic critical state theory. It is shown that both OR and PSD may affect the response of the granular assemblies in terms of the stress–strain relations, dilatancy, and critical state behaviors. For a given PSD, both the shear strength and fabric norm decrease with an increase in OR. The critical state angle of shearing resistance is highly dependent on particle shape. In terms of PSD, uniformly distributed assemblies mobilize higher shear strength and experience more dilative responses than specimens with a greater variation of particle sizes. The position of the critical state line in the e–p′ space is also affected by PSD. However, the effects of PSD on critical strength and evolution of fabric are negligible. These findings highlight the importance of particle shape and PSD that should be included in the development of constitutive models for granular materials.

On a fabric evolution law incorporating the effects of b-value

In this paper, the effects of the intermediate stress ratio, i.e., b-value (b = (σ2 − σ3)/(σ1 − σ3)), on the contact normal-based fabric evolution of granular material, are incorporated into an extant hybrid fabric evolution law. The new evolution law is validated by Discrete Element Method (DEM) simulation results under monotonic shearing with different b-values. Predictions of the proposed generalized fabric evolution law agree well with the DEM simulation results. This evolution law can be widely used for constitutive modelling of granular materials, considering the effects of b-value in a general geomechanical three-dimensional stress space.

μ-GM: A purely micromechanical constitutive model for granular materials

A strictly micromechanical framework is proposed for the constitutive modelling of granular materials as a variant to existing micromechanically-enriched continuum models. The theory hinges on the fact that the constitutive behaviour of granular materials can be formally written via an incrementally non-linear expression of stress and strain derived from an exhaustive set of key microvariables that statistically describe contact normals and force networks at the lower scale. The model also incorporates key physics uncovered by micromechanical findings of granular material behaviour such as dissipative and non-dissipative deformational mechanisms, including the existence of a unique surface, named Stable Evolution State as a bounding surface to quasi-static stable evolutionary states that generalizes the well-known critical state in soil mechanics. The proposed model, herein coined as μ-GM, acts as a surrogate for a full-blown discrete element method model in that it includes only three intrinsic material parameters complemented with three calibration ones to reveal most salient microstructural information and their evolution during loading history. As such, it can be readily incorporated into standard finite element simulations of boundary-value problems with computations to the resolution of the microscale. The present model currently formulated in 2D conditions can be readily extended to 3D to explore more complex problems with rotation of principal stresses.