A continuum model for granular materials: Considering dilatancy and the Mohr-Coulomb criterion

Summary In this paper we will explore the consequences of the Mohr-Coulomb criterion on the constitutive equation proposed by Rajagopal and Massoudi [1]. This contunuum model which is based on the earlier works of Cowin [2] has also the ability to predict the dilatancy effect which is related to the normal stress effects. At the same time, if a proper representation is given to some of the material parameters, this model would also comply with the Mohr-Coulomb criterion. We also present, as a special case, an exact solution for the case of simple shear flows.     

Understanding granular media: from fundamentals and simulations to industrial application

Academic and industrial research on the behaviour of granular material is increasingly supported by numerical simulations. Such simulation environments are not only a vital tool in design and analysis, but also a key link between physicists and engineers. This Topical Collection includes a range of papers that use simulation to study granular matter from the micro- and macroscopic behaviour of particles to the behaviour of large industrial or geomechanical systems. These contributions enhance our understanding of basic physical effects and processes such as heat transfer, agglomeration and screening but also give essential parameters for the simulation of industrial systems, or, at least, guide the strategies required to provide realistic simulation results. The collection focuses on the presentation of (a) new coupled simulation methods to consider the interaction of granular bodies with structural or fluid systems (b) new findings for the consideration of particle shape and particle size distributions within simulations, (c) acoustic wake agglomeration, and (d) gravitational flow in geomechanics.     

A continuum theory of diatomic solids: viewed as directed media

Summary A generalized continuum model for diatomic solids is presented in this study. Considering the relative displacement of a particle in a cell as a single director, the field and the constitutive equations of diatomic solids are obtained through the use of Toupin's [3] variational principle. Propagation of a longitudinal wave in such a medium is also reported, and the result found here is compared with those of lattice dynamics.     

Constitutive modelling for granular media using an anisotropic distortional yield model

Summary It has been attempted here to model the mechanical response of granular media through a combination of analytical, numerical and experimental techniques. Based on the experimental evidence obtained from a series of triaxial laboratory experiments on specimens made of glass beads, the model attempts to simulate the movement of yield surfaces and the stress-strain relationships. It is found that the yield surface distorts in the direction of loading in a manner analogous to that of metals. An anisotropic distortional yield model is formulated in order to describe the experimental behavior exhibited by these granular media. From the experimental yield surfaces the parameters of the model have been evaluated and a hardening rule, based on the Phillips rule, has been determined. Associativity on the -plane has been observed experimentally. Using these concepts the constitutive formulation has been presented and the stress-strain curves have been generated. From the comparisons of these curves we observe a good correlation between the model and the experimental observations.     

Future continuum models for granular materials in penetration analyses

This paper presents an investigation on future continuum models for granular materials in penetration analyses. A two-dimensional discrete element method has been used to numerically simulate penetration tests on a granular ground. The stress paths of soil elements in the ground have been studied, and then used to highlight the main features of granular materials based on most-advanced knowledge in soil mechanics. The study shows that the penetration makes the soil near the penetrometer undergo a significant changes of stresses in both magnitude and direction. The soil of large deformation rate may arrive at a stress state slightly over the strength envelope obtained from conventional tests. As a result, shear dilatancy, rate-dependency, non-coaxiality and particle crushing are the four main features that future continuum models should capture for granular materials in penetration analyses.     

Modifying continuous-time random walks to model finite-size particle diffusion in granular porous media

The continuous-time random walk (CTRW) model is useful for alleviating the computational burden of simulating diffusion in actual media. In principle, isotropic CTRW only requires knowledge of the step-size, \(P_l\), and waiting-time, \(P_t\), distributions of the random walk in the medium and it then generates presumably equivalent walks in free space, which are much faster. Here we test the usefulness of CTRW to modelling diffusion of finite-size particles in porous medium generated by loose granular packs. This is done by first simulating the diffusion process in a model porous medium of mean coordination number, which corresponds to marginal rigidity (the loosest possible structure), computing the resulting distributions \(P_l\) and \(P_t\) as functions of the particle size, and then using these as input for a free space CTRW. The CTRW walks are then compared to the ones simulated in the actual media. In particular, we study the normal-to-anomalous transition of the diffusion as a function of increasing particle size. We find that, given the same \(P_l\) and \(P_t\) for the simulation and the CTRW, the latter predicts incorrectly the size at which the transition occurs. We show that the discrepancy is related to the dependence of the effective connectivity of the porous media on the diffusing particle size, which is not captured simply by these distributions. We propose a correcting modification to the CTRW model—adding anisotropy—and show that it yields good agreement with the simulated diffusion process. We also present a method to obtain \(P_l\) and \(P_t\) directly from the porous sample, without having to simulate an actual diffusion process. This extends the use of CTRW, with all its advantages, to modelling diffusion processes of finite-size particles in such confined geometries.     

A continuum theory for two phase media

Summary A continuum theory of a two phase solid-fluid media is formulated. The basic balance laws for the solid phase as well as for the fluid phase are presented. Based on thermodynamical consideration a set of constitutive equations are derived and the basic equations of motions of the distributed solid and fluid continua are obtained and discussed. It is shown that the theory contains as its special cases, Mohr-Coulomb criterion of limiting equilibrium of granular materials, Saffman theory of dusty gas, as well as Darcy's law of flow through porous media. It is then concluded that the present theory covers the full spectrum of two phase solid-fluid media from low porosity granular media with Darcy's law of fluid motion to low and high concentration two phase flows such as dusty gas and blood flow.     

FE-investigations of a deterministic and statistical size effect in granular bodies within a micro-polar hypoplasticity

The paper deals with numerical investigations of a deterministic and statistical size effect in granular bodies during shearing of an infinite layer under plane strain conditions and free dilatancy. For a simulation of the mechanical behavior of a cohesionless granular material during a monotonous deformation path, a micro-polar hypoplastic constitutive was used which takes into account particle rotations, curvatures, non-symmetric stresses, couple stresses and the mean grain diameter as a characteristic length. The proposed model captures the essential mechanical features of granular bodies in a wide range of densities and pressures with a single set of constants. To describe a deterministic size effect, the calculations were carried out with an uniform distribution of the initial void ratio for four different heights of the granular layer: 5, 50, 500 and 2,000 mm. To investigate a statistical size effect, the distribution of the initial void ratio in infinite granular layers was assumed to be spatially correlated. As only primary stochastic calculations were performed, single examples of different random fields of the initial void ratio were generated. For this purpose a conditional rejection method was used.     

Development of micromechanical models for granular media

Micromechanical analysis has the potential to resolve many of the deficiencies of constitutive equations of granular continua by incorporating information obtained from particle-scale measurements. The outstanding problem in applying micromechanics to granular media is the projection scheme to relate continuum variables to particle-scale variables. Within the confines of a projection scheme that assumes affine motion, contact laws based on binary interactions do not fully capture important instabilities. Specifically, these contact laws do not consider mesoscale mechanics related to particle group behaviour such as force chains commonly seen in granular media. The implications of this are discussed in this paper by comparison of two micromechanical constitutive models to particle data observed in computer simulations using the discrete element method (DEM). The first model, in which relative deformations between isolated particle pairs are projected from continuum strain, fails to deliver the observed behaviour. The second model accounts for the contact mechanics at the mesoscale (i.e. particle group behaviour) and, accordingly, involves a nonaffine projection scheme. In contrast with the first, the second model is shown to display strain softening behaviour related to dilatancy and produce realistic shear bands in finite element simulations of a biaxial test. Importantly, the evolution of microscale variables is correctly replicated. This paper is dedicated to Professor Ching S. Chang on the occasion of his 60th birthday.     

A continuum finite element for single-set jointed media

In response to the need for an advanced computational model for wave propagation in jointed-rock media a new finite element for jointed media with a single set of regularly spaced joints is developed. The element is a numerical implementation of the higher-order homogenization model recently proposed by Murakami and Hegemier. Due to the dispersive effects induced by regularly spaced joints, wave phenomena in jointed media are altered significantly. Therefore, in order to improve the interpretation of seismograms for accurate source identification, it is necessary to develop a higher-order continuum element. The accuracy and efficiency of the new element is investigated by applying it to wave-guide and wave-normal problems of a jointed half-space and by comparing the wave response with that of DYNA2D. The analyses by DYNA2D discretize explicitly the details of the joint microstructure, and are adopted as numerically exact measures for the assessment of the proposed finite element; good correlations were obtained. The validation study also confirmed the importance of wave dispersion for non-linear as well as linear joint responses. Finally, as a more practical application of the proposed element, the problem of a jointed full-space with a cylindrical cavity pressurized by step and pulse loadings was solved. Velocities at several observation points were compared with the numerically exact results of DYNA2D. Similar analyses carried out for elastic isotropic media predicted totally different velocity responses and confirmed the need for the proposed element.     

Micro origins for macro behavior in granular media

We report the latest advances in understanding, characterization and modeling of key micro mechanisms and origins underpinning the interesting and complex macroscopic behavior of granular matter. Included in this Topical Collection are novel theories, innovative experimental tools and new numerical approaches, focusing primarily on three subtopics governing important multiscale properties of granular media: (a) the jamming transition from fluid- to solid-like behavior, critical state flow and wave propagation, (b) the signature of fabric and its evolution for granular media under general loading conditions, and (c) mechanisms like rotation, breakage, failure and aggregation. The significance of these contributions and exploratory future directions pertaining to cross-scale understanding of granular matter are discussed.     

Simple model for wet granular beds subjected to tapping

We present a simple model to describe the response of a granular bed to a tapping-like excitation when grains can form capillary bridges due to the presence of interstitial liquid. We implement a pseudodynamic simulation of adhesive hard disks. The packing fraction and coordination number after the steady state of the tapping process has been reached are compared for different tapping intensities and liquid contents. We find some contrasting behavior with dry systems and qualitative agreement with experimental data.     

From liquid to solid bonding in cohesive granular media

We study the transition of a granular packing from liquid to solid bonding in the course of drying. The particles are initially wetted by a liquid brine and the cohesion of the packing is ensured by capillary forces, but the crystallization of the solute transforms the liquid bonds into partially cemented bonds. This transition is evidenced experimentally by measuring the compressive strength of the samples at regular intervals of times. Our experimental data reveal three regimes: (1) Up to a critical degree of saturation, no solid bonds are formed and the cohesion remains practically constant; (2) The onset of cementation occurs at the surface and a front spreads towards the center of the sample with a nonlinear increase of the cohesion; (3) All bonds are partially cemented when the cementation front reaches the center of the sample, but the cohesion increases rapidly due to the strengthening of cemented bonds. We introduce a model based on a parametric cohesion law at the bonds and a bond crystallization parameter. This model predicts correctly the phase transition and the relation between microscopic and macroscopic cohesion.     

Numerical analyses of braced excavation in granular grounds: continuum and discrete element approaches

Numerical simulations have been extensively used in braced excavation design. However, previous analyses indicate that the universally adopted constitutive models such as Mohr–Coulomb (M–C) model and Drucker–Prager (D–P) model need to be further clarified due to the unsatisfactory prediction of the ground deformation. This study focuses on the features that future continuum models should capture for braced excavation in granular ground. For this purpose, a simplified braced excavation in granular ground was simulated using the distinct element method (DEM). The same excavation case was also simulated by the Finite Difference Method (FDM) using M–C and D–P model to check their applicability. The excavation was 7.5 m in depth and was braced at the level of $-1.5\,\text{ m}$ . The results indicate that the DEM simulation can reproduce the main responses of granular ground during excavation; the excavation initiates failure at the excavation depth of 5.0 m and evolves into total failure at the depth of 7.5 m; two types of stress paths in front of and behind the wall are observed, respectively; obvious principal stress rotations of soils are recognized. Compared with DEM results, M–C and D–P model can generally predict excavation responses qualitatively but under-estimate the ground deformation and internal forces of the wall. This is due to the incapability of the two continuum models to capture the mechanical behavior of granular material under complicated stress conditions in braced excavation. Based on these observations and comparisons, three features are emphasized for future continuum models: stress path dependency, non-coaxiality, and shear dilatancy.     

A continuum phase field model for fracture

A phase field model based on a regularized version of the variational formulation of brittle fracture is introduced. The influences of the regularization parameter that controls the interface width between broken and undamaged material and of the mobility constant of the evolution equation are studied in finite element simulations. A generalized Eshelby tensor is derived and analyzed for mode I loading in order to evaluate the energy release rate of the diffuse phase field cracks. The numerical implementation is performed with finite elements and an implicit time integration scheme. The configurational forces are computed in a postprocessing step after the coupled problem of mechanical balance equations and the evolution equation is solved. Some of the numerical results are compared to analytical results from classical Griffith theory.     

Mechanism of avalanche precursors in inclining granular layers using a continuum model obtained by discrete element method

How do large shear stress and stress ratio peaks, which are avalanche precursors, occur in granular layers in an inclining box and why do the precursors have periodicities? The answers are our main objectives. The large and small shear stress peaks which are constituted by the large and small sticks and slips are found in our simulation results. Small sticks and slips, which occur after the large slip, change the granular structure until the stress ratio becomes equal to or larger than the frictional coefficient of granular materials. Here the next large slip with the large stick and slip, which shows the precursor, occurs. The precursors which are constituted by the series of the large stick–slip events have naturally periodicities because the stick–slip events are oscillation phenomena. Our 3-dimensional simulation results using the smoothed particle hydrodynamics (SPH) method based on our constitutive equations obtained by the 3-dimensional discrete element method (DEM) present the precursors in the bulk and on the surface and show the difference between them. The precursors are also found in the longitudinal and the depth directions of the granular layer. The simulated periodicities agree with the published experimental results.