Anisotropic Nonlinear Elastic Model for Particulate Materials

This paper presents the development of an elastic model for particulate materials based on micromechanics considerations. A particulate material is considered as an assembly of particles. The stress–strain relationship for an assembly can be determined by integrating the behavior of the interparticle contacts in all orientations and using a static hypothesis which relates the average stress of the granular assembly to a mean field of particle contact forces. Hypothesizing a Hertz–Mindlin law for the particle contacts leads to an elastic nonlinear behavior of the particulate material, we were able to determine the elastic constants of the granular assembly based on the properties of the particle contacts. The numerical predictions, compared to the results obtained during experimental studies on different granular materials, show that the model is capable of taking into account both the influence of the inherent anisotropy and the influence of the stress-induced anisotropy for different stress conditions.     

Elastic Model for Partially Saturated Granular Materials

This paper presents the development of an elastic model for partially saturated granular materials based on micromechanical factor consideration. A granular material is considered as an assembly of particles. The stress-strain relationship for an assembly can be determined by integrating the behavior at all interparticle contacts and by using a static hypothesis, which relates the average stress of the granular assembly to a mean field of particle contact forces. As for the nonsaturated state, capillary forces at grain contacts are added to the contact forces created by an external load. These are then calculated as a function of the degree of saturation, depending on the grain size distribution and on the void ratio of the granular assembly. Hypothesizing a Hertz-Mindlin law for the grain contacts leads to an elastic nonlinear behavior of the particulate material. The prediction of the stress-strain model is compared to experimental results obtained from several different granular materials in dry, partially saturated and fully saturated states. The numerical predictions demonstrate that the model is capable of taking into account the influence of key parameters, such as degree of saturation, void ratio, and mean stress.     

Critical State of Granular Materials Based on the Sliding-Rolling Theory

By representing the assembly by a simplified column model, a constitutive theory was recently developed for a two-dimensional assembly of rods. This theory, referred to as the sliding-rolling theory, is extended in this paper to represent the triaxial stress-strain behavior of granular materials. The sliding-rolling theory provides a dilatancy rule and an expression for the slope of the line of zero dilatancy in the stress space. These rules are then combined with triaxial observations to provide a microstructural interpretation of the critical state of granular materials. According to the theory, the slope of the critical state line in the stress space depends on the interparticle friction angle and the degree of contact normal anisotropy. To verify the basic ideas of the sliding-rolling theory, numerical experiments are conducted using the discrete-element method on three-dimensional assemblies of spheres.     

Micromechanical Analysis for Interparticle and Assembly Instability of Sand

Instability of granular material may lead to catastrophic events such as the gross collapse of earth structures, and thus it is an important topic in geotechnical engineering. In this paper, we adopt the micromechanics approach for constitutive modeling, in which the soil is considered an assembly of particles, and the stress-strain relationship for the assembly is determined by integrating the behavior of the interparticle contacts in all orientations. Although analyses regarding material instability have been extensively studied for a soil element at the constitutive level, it has not been considered at the interparticle contact level. Through an eigenvalue analysis, two modes of instability are identified at the local contact level: the singularity of tangential stiffness matrix and the loss of positiveness of second-order work. The constitutive model is applied to simulate drained and undrained triaxial tests on Toyoura sand of various densities under various confining pressures. The predictions are compared with experimentally measured instability at the assembly level. The modes of stability at the interparticle contact level and their relations to the overall instability of the assembly are also analyzed.     

Effect of Interparticle Friction on the Cyclic Behavior of Granular Materials Using 2D DEM

This study presents the influence of the interparticle friction angle on the cyclic behavior of granular materials using the two-dimensional (2D) discrete-element method (DEM). The numerical sample was modeled with oval-shaped particles, whereas the isotropically compressed dense sample was prepared from the initial sparse sample using periodic boundaries. Biaxial cyclic shear tests were simulated with different interparticle friction angles. It was noted that the width of the stress-strain cyclic loops becomes thin when the interparticle friction angle increases. It was also noted that the induced fabric anisotropy is more pronounced during unloading than loading. Moreover, a strong correlation between macro- and microquantities was observed for strong contacts during cyclic loading.     

Micromechanical Modeling for Behavior of Cementitious Granular Materials

Crack damage is commonly observed in cementitious granular materials. Previous analytical models based on continuum mechanics have limitations in analyzing localized damages at a microscale level. In this paper, a micromechanics approach is adopted that considers a contract law for the interparticle behavior of two particles connected by a binder. The model is based on the premises that the interparticle binder initially contains microcracks. As a result of external loading, these microcracks propagate and grow. Thus, binders are weakened and fail. Theory of fracture mechanics is employed to model the propagation and growth of the microcracks. The contact law is then incorporated in the analysis for the overall damage behavior of material using a discrete element method. Using this model, the stress-strain behaviors under uniaxial and biaxial conditions were simulated. A reasonable agreement is found between the predictions and experimental results.     

Granular Temperature

Granular materials consist of a large number of small particles arranged in a random way. However, while the geometrical organization of an individual particle assembly is complex, these materials can nevertheless be well characterized by a small number of parameters. Systems of this type are the subject matter of thermodynamics, and so it seems reasonable to suppose that the methods employed in thermodynamics may be able to be applied to characterizing the behavior of particle assemblies. This paper describes an extension of classic thermodynamics to granular assemblies subjected to an energy flux through the granular assembly. By means of a Carnot engine, the granular temperature and entropy of a particle assembly is defined. The principal value of introducing the notion of granular temperature is that it defines a physically realistic internal variable for the granular assembly, which controls the evolution of the system after it is has been perturbed from its equilibrium state. The concept of granular thermodynamic equilibrium is then extended to non‐equilibrium granular thermodynamics. Chemical kinetic theory is employed to describe the relaxation of a granular assembly after it is perturbed by a sudden change in granular temperature. A particle vibration theory has been introduced to explain the behavior of soil subject to a white noise energy flux. While the theory described provides quantitative information about soil behavior, just as importantly, the theory provides a new qualitative insight into soil behavior.     

Microstructural Modeling for Elastic Moduli of Bonded Granules

In this paper, a microgranular mechanics approach is used to derive an expression for the elastic moduli of an assembly of bonded granulates, based on the response of two particles that are connected by an elastic binder. The derived modulus is a function of the particle∕binder modulus, the particle size, the binder thickness, the binder width, the assembly coordination number, the binder content, and the porosity. To demonstrate its applicability, the predicted range of the modulus using the derived model is compared with that measured from experiments. The difference between this method and the traditional homogenization models will be discussed.     

Simple Nonlinear Model for Elastic Response of Cohesionless Granular Materials

A constitutive model based on hyperelasticity is proposed to capture the resilient (elastic) behavior of granular materials. Resilient behavior is a widely accepted idealization of the response of unbound granular layers of pavements, following shakedown. The coupling property of the proposed model accounts for shear dilatancy and pressure-dependent behavior of the granular materials. The model is calibrated using triaxial resilient test data obtained from the literature. A statistical comparison is made between the predictions of the proposed model and a few of the prominent models of resilient response. The proposed coupled hyperelastic model yields a significantly better fit to the experimental data. It also offers a computational efficiency when implemented in a classical nonlinear finite elemental framework.     

Critical Aspects of Alloying of Sintered Steels with Manganese

This study examines the sintering behavior and properties of Fe-0.8Mn-0.5C manganese powder metallurgy steels. The study focuses on the influence of mode of alloying—admixing using either high-purity electrolytic manganese or medium carbon ferromanganese as well as the fully prealloying of water-atomized powder. Three main aspects were studied during the whole sintering process—microstructure development, interparticle necks evolution, and changes in the behavior of manganese carrier particles during both heating and sintering stages. The prealloyed powder shows considerable improvement in carbon homogenization and interparticle neck development in comparison with admixed materials. The first indication of pearlite for the fully prealloyed material was registered at ~1013 K (740 °C) in comparison with ~1098 K (825 °C) in the case of the admixed systems. The negative effect of the oxidized residuals of manganese carrier particles and high microstructure inhomogeneity, which is a characteristic feature of admixed systems, is reflected in the lower values of the mechanical properties. The worst results in this respect were obtained for the system admixed with electrolytic manganese because of more intensive manganese sublimation and resulting oxidation at lower temperatures. According to the results of X-ray photoelectron spectroscopy and high-resolution scanning electron microscopy and energy dispersive X-ray analyses, the observed high brittleness of admixed materials is connected with intergranular decohesion failure associated with manganese oxide formation on the grain boundaries.     

Modeling Stress-Dilatancy for Sand under Compression and Extension Loading Conditions

Experimental results show very different stress-dilatancy behavior for sand under compression and extension loading conditions. In order to describe the behavior in both compression and extension conditions, two separate dilatancy equations are needed; one for each case. Furthermore, the effect of inherent anisotropy has not been considered in those equations. In this paper, a new micromechanics based method is presented, by which the stress dilatancy relation is obtained by considering the slip mechanism of the interparticle contacts in all orientations. The method also accounts for the effect of inherent anisotropy in sand. Experimental results on Hostun sand and LSI-30 sand are used for illustration the proposed method.     

Expressions for predicting the elasticity modulus of materials reinforced by second-phase grains

In the present study, expressions for predicting the elasticity modulus of the materials reinforced by the second-phase grains, which may be referred to as the granular composite for the sake of simplicity, are developed based on the solution of the overmatching problem in elastic mechanics. Taking the voids or defects in materials as the second phase with zero elasticity modulus, one can easily obtain the expressions for predicting the elasticity modulus and the threshold of the elastic percolation failure of materials containing voids. The values of the elasticity modulus of the granular composite and the materials containing random voids and the values of the threshold for the elastic percolation failure of the materials with voids predicted by using the above-mentioned expressions are in good agreement with results given in available literature.     

Analytical Elastic Solution Based on Fourier Series for a Laterally Confined Granular Column

This paper presents an analytical solution methodology for the complete stress and displacement fields of a laterally confined granular column loaded from the top end. The granular column is idealized as a homogeneous isotropic elastic medium with Coulomb’s friction at the lateral boundary. The solution methodology consists of an analytical procedure that incorporates a potential approach with trigonometric series and Bessel functions, finite Fourier transforms and the superposition method, and an iterative algorithm to satisfy the Coulomb’s friction condition at the lateral boundary. Stress and displacement fields are computed for a specific example and found completely consistent with corresponding finite element results. Key characteristics, computational errors, the convergence behavior, and restrictions of the present approach are discussed. The methodology developed herein can be beneficially applied in the validation process of numerical simulation techniques in granular mechanics such as finite or discrete element methods.     

Sample Size Effects on Constitutive Relations of Granular Materials—A Numerical Simulation Study with Two-Dimensional Flow of Disks

A simple shear flow of granular materials can have a range of behavior from a rate-independent plastic material to a rate-dependent viscous material. From physical experiments and computer simulations, it is known that this constitutive relation is a consequence of the shear rate, the solid concentration, and the micromechanical properties of the particles. Simple shear tests of granular materials show that in the rate-independent case, a shearing granular assembly forms crystallized regions and shearing occurs locally in a narrow band. In the rate-dependent case, these crystallized zones “melt” and the whole granular assembly participates in the shear motion. This study utilizes computer simulation to address yet another effect on the constitutive relation: the sample size. A 2D uniform disk assembly is simulated using periodic boundary conditions. Through investigating details of the kinematics the source of the transition is examined from a rate-independent to a rate-dependent fluid as the sample size increases. Results of this study have implications on the design of equipment handling granular materials. That is, rheological properties of a granular flow can change due to a change of the relative size of the equipment and the grains.     

Micro-Macro Quantification of the Internal Structure of Granular Materials

We have attempted a multiscale quantification of the internal structure of granular materials. The internal structure of granular materials, i.e., the geometrical information on granular particles and their spatial arrangement, was described mathematically on the particle scale using Voronoi–Delaunay tessellations. These tessellations were further modified into two cell systems: a solid cell system and a void cell system, with the internal supporting structure properly reflected. By doing so, the two cell systems were geometrically and physically significant. Taking solid/void cells as the microscopic basic elements, the behavior of granular materials was expressed as the volumetric average of the microcell behavior. Macroscopically, the internal structure could be characterized by the statistical measures from the geometry of the microcells. Our approach was used to investigate the anisotropic behavior of granular materials. A study on the void cells explains how the spatial arrangement affects the strength and dilatancy of granular materials. A new anisotropic fabric tensor was defined based on the void cell anisotropy. The correlation between the anisotropic fabric tensor and the macro behavior of granular materials was verified with numerical simulations. The results showed that the new material anisotropic tensor is a more effective definition than the existing ones based on particle orientations and contact normals.     

Quantification of Doublet Vector Distribution of Granular Materials

Soil fabric quantities such as doublet vector and branch vector distributions are important in doublet mechanics and micromechanics modeling of soil behavior. The quantification of these quantities is therefore imperative in applying and verifying these modeling techniques. This paper presents the development of a new method to quantify the doublet distribution of granular assemblies and some experimental results. The method treated doublet vectors as random vectors and built up the integrated distribution function from the distribution functions of the projections of the random vectors in the three orthogonal subspaces. A mechanism to evaluate the doublet vector distribution from sectional images was developed for the case of granular assemblies composed of spherical or subspherical particles. A procedure was also developed to assess the integrated distribution function through evaluating the projection distribution functions in two orthogonal subspaces for the case of axial-symmetric distribution of doublet vectors of spherical particles. The comparison of the doublet distribution of a dense Ottawa sand specimen, quantified by the proposed procedure, with Oda's experimental observations on contact normal distributions indicated a qualitative consistency. Doublet distribution of spherical assemblies is the same as the branch vector distribution and the contact normal distribution when the nearest neighbors are in contact, and therefore can be applied in both doublet mechanics and micromechanics modeling. The developed procedure is not scale-dependent and can be used to quantify these distributions of particulate materials in different length scales.     

Two-Dimensional Physical and Numerical Modeling of a Pile-Supported Earth Platform over Soft Soil

This paper focuses on the mechanisms occurring in a granular earth platform over soft ground improved by rigid piles. Two-dimensional physical model experiments were performed using the Schneebeli’s analogical soil to investigate the load transfer mechanisms by arching and the settlement reduction and homogenization. Experimental outputs are compared to results obtained on a numerical model using a plane strain continuum approach. The impact of the constitutive model complexity to simulate the platform material behavior was first assessed, but no large difference was recorded. As far as the proposed model, which takes the main features of the observed behavior satisfactorily into account, the numerical procedure could be validated and the parametric studies extended numerically. Both approaches of this study underlined the main geometrical and geotechnical parameters which should inevitably be taken into account in a simplified design method, namely the capping ratio, the platform height, and the platform material shear strength.     

Effect of Contact Force Models on Granular Flow Dynamics

The contact force model consisting of a linear spring dashpot with a frictional glider has been widely adapted to simulate granular flows. Real contact mechanics between two solid bodies is very complicated. Extensive theoretical and experimental studies exist for binary contacts. Very little work has been reported that addresses the effect of contact mechanics on the bulk behavior of granular materials. We first briefly summarize the difference of binary contacts between a linear spring–dashpot model and the Hertzian nonlinear spring with two nonlinear dashpot models. We then compare the constitutive behaviors of a granular material using a linear and a nonlinear model. The stress- and strain-rate relation in simple shear flow and the resulting coordination number are calculated using the discrete element method. It is found that although at the grain level binary contact between two particles depends on whether a linear or a nonlinear model is used, the bulk behavior of granular materials is qualitatively similar with either model.     

Sliding and Rolling Constitutive Theory for Granular Materials

By analyzing a microelement based on four spheres, equations governing the equilibrium of the microelement are developed. By examining these equations more closely, two primary mechanisms of failure of the microelements, one based on particle sliding and the other based on particle rolling, are identified. For each primary mechanism, two separate mechanisms, one based on collapse of the microelement in the vertical direction and the other based on collapse in the horizontal direction, are recognized. With the aid of these concepts, constitutive equations are developed for a two-dimensional assembly of granular particles. The assembly is considered to consist of four-sphere microelements. Taking the plastic strain to be a consequence of the collapse of some of the microelements, equations are developed for plastic strain. The formulation yields loading criteria and flow directions. With suitable hardening rules, it is shown that the microstructural model is capable of simulating most of the salient features of the stress-strain behavior of granular materials. In particular, the stress-dilatancy relation, taking into consideration phenomena of phase transformation, and critical state failure are simulated satisfactorily.     

Damage Theory Based on Composite Mechanics

A new theory of composite damage mechanics is developed. A material with damage is considered as a composite comprised of two different phases (called matrix and inclusion). Both phases are linearly elastic isotropic materials. The matrix is considered as the intact material, and the inclusion is the damaged material. Three different composite models, Voigt (parallel), Reuss (serial), and generalized self-consistent (spherical), are introduced for three types of damage distributions. These composite models are usually used for initial tangential modulus of a composite material, here we use them for secant modulus of a distressed material. Since the parallel and the serial models represent the upper and lower bounds for stiffness of materials, the composite damage theory obtains the upper and lower bounds for postpeak stress and the level of damage for the material beyond the elastic limit. The spherical model is in between the two bounds. Depending on the “elastic limit” of the inclusion, the theory can be used to describe elastic perfectly plastic behavior, strain hardening, and strain softening. Two different degradations, the linear and exponential degradations of the stress–strain response curve are introduced. The two degradation models are used in two different failure surfaces, i.e., Tresca and Mohr–Coulomb failure surfaces, to predict the postpeak behavior of distressed material.