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.     

Ultimate State of Samples of Ellipsoids under Different Stress Paths

The ultimate state of granular material was investigated by the discrete element method. This paper discusses a series of numerical triaxial tests on samples of two kinds of ellipsoids. Samples of three different densities were loaded along various stress paths. The ellipsoids were mixed in the proportions 50:50 by weight. The longer particles have an aspect ratio (major length/minor length) of 1.5 and the other particles have an aspect ratio of 1.2. The samples were generated by deposition under gravity to simulate the air-pluviation sample preparation technique. Then they were consolidated isotropically. Fourteen stress-path controlled triaxial tests were conducted numerically for each sample. They are sheared to the ultimate state to determine the friction angle and void ratio at the ultimate state. When samples of different densities were sheared along the same loading path, a unique ultimate state was observed. The initial density does not affect the ultimate state. However, when different loading paths are applied to the same numerical specimen, they do not always arrive at the same critical state in the void-mean stress space.     

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.     

Examination of the Response of Regularly Packed Specimens of Spherical Particles Using Physical Tests and Discrete Element Simulations

Significant insight into the response of granular materials can be gained by coupling accurately controlled physical tests with complementary discrete element simulations. This paper discusses a series of triaxial and plane strain laboratory compression tests on steel spheres with face-centered-cubic and rhombic packings, as well as discrete element simulations of these tests. The tests were performed on specimens of uniform-sized steel balls and on specimens of steel balls with specified distributions of ball diameters. The packing configurations are ideal and differ considerably from real sand specimens, however, studies of such idealized granular materials can yield considerable insight into the response of granular materials and the capability of discrete element simulations to capture the response. The differences in response for the two packing configurations considered illustrate the importance of fabric. The numerical simulations captured the observed laboratory response well if the particle configurations, particle sizes, and boundary conditions were accurately represented. However, the postpeak response is more difficult to capture, and it is shown to be sensitive to the coefficient of friction assumed along the specimen boundaries. The simulations of the tests on the nonuniform-sized specimens demonstrated a clear correlation between strength and coordination number.     

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.     

Implicit Algorithm for Modeling Unsaturated Soil Response in Three-Invariant Stress Space

An implicit integration algorithm has been refined to predict the stress–strain–strength response of unsaturated soil under suction-controlled, multiaxial stress paths that are not achievable in a conventional cylindrical cell. The algorithm supports numerical analyses in a deviatoric plane by using a mixed control constitutive driver, in conjunction with a generalized Cam-Clay model that also incorporates the influence of a third stress invariant, or Lode-angle θ, within a constant-suction scheme. True triaxial data from a previously accomplished series of suction-controlled triaxial compression, triaxial extension, and simple shear tests on 10-cm cubical specimens of silty sand, were used for the tuning and validation of the refined algorithm. The elliptical Willam–Warnke surface was adopted for simulation of unsaturated soil response in three-invariant stress space. Reasonably satisfactory agreement was observed between experimental and predicted deviatoric stress versus principal strain response for different suction states, as well as between experimental and predicted strength loci in a deviatoric plane.     

Constitutive Model of Soil Based on a Dynamical Systems Approach

A soil when sheared ultimately reaches a steady-state condition at which it deforms at a constant shear stress, effective normal stress, and void ratio. Various systems in nature dynamically evolve similarly from some initial condition, to a final steady-state condition. Such systems have been studied using dynamical systems theory. This technical note uses this theory to model monotonic shear of soil as a dynamical system. The principle proposed is simple—the rates of change of the shear stress, effective normal stress, and void ratio are proportional to the applied values of the shear and effective normal stress with the proportionality values decaying with strain until ultimately these proportionality values become zero at the steady-state condition. It provides a well-formed qualitative principle that fits closely the stress-strain-void ratio curves of undrained shear tests on uncemented, resedimented clays at various over consolidated ratios (OCRs), and drained shear tests on sands and silts at various relative densities, for various stress paths including compression, extension from standard triaxial, and true-triaxial tests. For the undrained shear of resedimented clay, these proportionalities and their decay rates vary smoothly with OCR. For drained shear of sand and silt, the model parameters show orderly variation with relative density. Its value lies in that a well-formed qualitative principle derived from the steady-state condition provides an alternate approach to current complex elastoplastic models based on critical state theory.     

Micromechanical Analysis of the Shear Behavior of a Granular Material

A distinct element analysis of the behavior of a granular material was performed by simulating direct shear tests of a dense and a loose 2D sample of 1,050 cylinders. Macroscopic results exhibit typical features of the shear response of granular materials: a perfect plasticity state that does not depend on the initial density, a peak stress and a dilatant behavior in the case of the dense sample, and a contractant behavior of the loose sample. A micromechanical analysis of the shear behavior was carried out based on the simulation results. Using the particle displacements and rotations, a shear band is located within the sample. Special attention is focused on the evolution of particle∕particle contact orientation as well as on the direction of particle∕particle contact forces. The shear process induces a clear change of contact and contact force orientations. A strong correlation between the induced anisotropy of the microstructure and the macroscopic loading is evident in the simulation results.     

State Pressure Index for Modeling Sand Behavior

The effort to model sand behavior within the framework of critical-state soil mechanics would benefit from a state variable that relates the current void ratio and mean pressure of the soil to its void ratio and mean pressure at the critical state. In this paper we propose a state pressure index, Ip, which is defined as the ratio of the current mean pressure to the mean pressure at the critical state that corresponds to the current void ratio. Using this state pressure index, a bounding surface hypoplasticity model for sand is modified so that the phase transformation and failure stress ratios both depend on Ip and merge into the critical-state stress ratio at failure. The Ip dependency introduced enables use of a single set of model constants in modeling sand behavior for various initial confining pressures and densities under both undrained and drained conditions. Dilatancy, strain softening, and strain hardening are simulated for both loose and dense sands. Simulations from the modified model are compared with results of laboratory tests of drained and undrained triaxial compression.     

Coupled Continuum-Discrete Model for Saturated Granular Soils

A coupled hydromechanical model was used to analyze the mesoscale pore fluid flow and microscale solid phase deformation of saturated granular soils. The fluid motion was idealized using averaged Navier–Stokes equations, and the discrete element method was employed to model the assemblage of solid particles. The fluid–particle interactions were quantified using established semiempirical relationships. Simulations were conducted to investigate the three-dimensional response of sandy deposits when subjected to critical and overcritical upward pore fluid flow. These simulations revealed complex response patterns after the onset of quicksand conditions and provided valuable insight into the associated mechanisms. The employed model provides an effective tool to assess the microscale mechanisms and characteristics of the partially drained response of saturated granular media.     

Prediction of Permanent Deformations in Pavements Using a High-Cycle Accumulation Model

The present paper discusses the application of a high-cycle accumulation (HCA) model originally developed for sand for the prediction of permanent deformations in an unbound granular material (UGM) used for base and subbase layers in pavements. Cyclic triaxial tests on precompacted samples of an UGM have been performed in order to validate and calibrate the model. The stress amplitude, the initial density, and the average stress were varied. The test results are compared to those of air-pluviated samples of sand (subgrade material). Some significant differences in the behavior of both materials under cyclic loading are outlined. It is demonstrated that the functions describing the intensity of accumulation can be maintained for an UGM with different material constants, but that the flow rule must be generalized in order to describe the anisotropy. Recalculations of the laboratory tests show a good prediction of the modified HCA model.     

Influence of Stress Ratio and Stress Path on Behavior of Loose Decomposed Granite

This paper presents results from four series of triaxial compression tests of loosely compacted decomposed granite (DG) or silty sand on both isotropically and anisotropically consolidated specimens. These tests included undrained tests, drained tests with constant deviator stress, and a decreasing mean effective stress path. The silty sand possessed high compressibility during isotropic compression. The observed high compressibility is probably attributed to the loose soil structure created by using the moist tamping method and the presence of crushable feldspar in the soil. Static liquefaction behavior and the so-called “reversed” sand behavior were observed in all undrained tests. This “reversed” sand behavior can be readily explained by the high compressibility of DG leading to the nonparallel and converging nature of the initial state line and the critical state line. Preshearing resulted in a more brittle response in the postpeak behavior. The higher the initial stress ratio (ηc), the smaller the ductility. Structural collapse of DG was observed. This collapse is characterized by a sudden large increase in both the axial and contractive volumetric strains. The mobilized angles of friction at collapse range from 31.8° to 38.7°, which are smaller than the critical state angle (?col′), but higher than the mobilized friction angle of the instability line (28.1°) determined by the isotropically consolidated undrained tests. A trilinear approximate relationship can be found between ?col′ and ηc and a liquefaction potential index is introduced to provide a simple preliminary design parameter for static liquefaction and instability prone slopes.     

Limit-State Curve of Base-Course Material and Its Relevance for Resilient Modulus Testing

The limit state characteristics of base-course granular materials were obtained using a typical triaxial testing equipment devoted to the measurement of resilient modulus. Accurate monitoring of axial strain during isotropic and anisotropic compression was used to determine the stress conditions where significant irrecoverable strains occur for samples prepared by static compression, Proctor rammer, and vibratory compaction. The limit state curve is highly anisotropic, centered about the q/p = 1 line. It is sensitive to sample preparation technique and fines content. The Strategic Highway Research Program (SHRP) procedure corresponds to stress paths during conditioning and repeated loading that remain within the limit state curve of the control base course material containing 3.5% fines. The resilient modulus values reflect henceforth the behaviour of the same material with its original particle contact distribution. The Laboratoires des Ponts et Chaussées (LCPC) procedure is characterized by stress paths that cross the original limit state curve of the Proctor compacted samples. Particle contact distribution changes thus continuously as the limit state curve expands in response to the various stress paths used in this procedure. The resilient modulus values correspond to samples with different fabrics. A simple procedure based on isotropic loading has been proposed for the determination of a simplified limit state curve of base course materials with the intent of specifying the testing conditions for obtaining adequate resilient modulus values.     

Contraction, Dilation, and Failure of Sand in Triaxial, Torsional, and Rotational Shear Tests

The response of a saturated fine sand (Nevada sand No. 120) with relative density Dr ≈ 70% in drained and undrained conventional triaxial compression and extension tests and undrained cyclic shear tests in a hollow cylinder apparatus with rotation of the stress directions was studied. It was observed that the peak mobilized friction angle for this dilatant material was different in undrained and drained tests; the difference is attributed to the fact that the rate of dilation is smaller in an undrained test than it is in a drained test. Consistent with the findings of others, the material is more resistant to undrained cyclic loading for triaxial compression than for triaxial extension. In rotational shear tests in which the second invariant of the deviatoric stress tensor is held constant, the shear stress path (after being normalized by the mean normal effective stress) approached an envelope that is comparable but not identical in shape to a Mohr-Coulomb failure surface. As the stress path approached the envelope, the shear end deviatoric strains continued to increase in an unsymmetrical smooth spiral path. During the rotational shear tests, the direction of the deviatoric strain-rate vector (deviatoric strain increment divided by the magnitude of change in Lode angle) was observed to be about midway between the deviatoric stress increment vector and the normal to a Mohr-Coulomb failure surface in the deviatoric plane. The stress ratio at the transition from contractive to dilative behavior (i.e., “phase transformation”) was also observed to depend on the direction of the stress path; therefore this stress ratio is not a fundamental property. Results from torsional hollow cylinder tests with rotation of stress directions are presented in new graphical formats to help understand and interpret the fundamental soil behavior.     

Performance of a Three-Dimensional Hvorslev–Modified Cam Clay Model for Overconsolidated Clay

It is well established that critical state soil mechanics provides a useful theoretical framework for constitutive modeling of soil. Most of the critical state models, including the popular modified Cam clay (MCC) model, predict soil behavior in the subcritical region fairly well. However, the predictions for heavily overconsolidated soils, in the supercritical region, are not so satisfactory. Furthermore, the critical state models were developed from triaxial test data and extension of these models into three-dimensional (3D) stress space has not been investigated thoroughly. In the present work, experiments were carried out to obtain stress–strain behavior of overconsolidated soil in triaxial compression, extension, and plane strain conditions. A novel biaxial device has been developed to conduct the plane strain tests. The experimental results were used to formulate Hvorslev–MCC model which has MCC features in the subcritical region and Hvorslev surface in the supercritical region. The model was generalized to 3D stress space using the Mohr–Coulomb failure criterion. A comparison of the model predictions with test results has indicated that the Hvorslev–MCC model performs fairly well up to the peak supercritical point, during which deformations are fairly uniform and the specimens remain reasonably intact. Limitations of this simple model in predicting postpeak localization are also discussed. The model’s predictions for volumetric response in different shear modes seem to agree reasonably well with test results.     

Behavior of Ellipsoids of Two Sizes

The influence of particle shape on granular material response is examined by using the discrete element method. Triaxial drained and undrained tests were performed on specimens of ellipsoids of two sizes. The triaxial test boundary conditions were simulated with a recently developed boundary mechanism. Different loading paths including axial compression, axial extension, lateral compression, and true extension were employed. The specimens were composed of 1,170 ellipsoids having two types of particles. The specimen is made up of 50% by weight of Type I particles that have an aspect ratio of 1.2. The aspect ratio of the Type II particles varies between 1.5 and 2. The specimens were consolidated isotropically before shearing. Comparing with the behavior of specimens of mono-size particles, a higher friction angle and a more complex particle shape effect were observed. The friction angles from the drained tests (axial extension, true extension, and lateral compression tests) were similar and the values are higher than that of the axial compression test. All simulated results are in good agreement with laboratory observation of sands.     

Numerical Modeling of Nonhomogeneous Behavior of Structured Soils during Triaxial Tests

The nonhomogeneous behavior of structured soils during triaxial tests has been studied using a finite element model based on the Structured Cam Clay constitutive model with Biot-type consolidation. The effect of inhomogeneities caused by the end restraint is studied by simulating drained triaxial tests for samples with a height to diameter ratio of 2. It was discovered that with the increase in degree of soil structure with respect to the same soil at the reconstituted state, the inhomogeineities caused by the end restraint will increase. By loading the sample at different strain rates and assuming different hydraulic boundary conditions, inhomogeneities caused by partial drainage were investigated. It was found that if drainage is allowed from all faces of the specimen, fully drained tests can be carried out at strain rates about ten times higher than those required when the drainage is allowed only in the vertical direction at the top and bottom of the specimen, confirming the findings of previous studies. Both end restraint and partial drainage can cause bulging of the triaxial specimen around mid-height. Inhomogeneities due to partial drainage influence the stress–strain behavior during destructuring, a characteristic feature of a structured soil. With an increase in the strain rate, the change in voids ratio during destructuration reduces, but, in contrast, the mean effective stress at which destructuration commences was found to increase. It is shown that the stress–strain behavior of the soil calculated for a triaxial specimen with inhomogeneities, based on global measurements of the triaxial response, does not represent the true constitutive behavior of the soil inside the test specimen. For most soils analyzed, the deviatoric stress based on the global measurements is about 25% less than that for the soil inside the test specimen, when the applied axial strain is about 30%. Therefore it can be concluded that the conventional global measurements of the sample response may not accurately reflect the true stress–strain behavior of a structured soil. This finding has major implications for the interpretation of laboratory triaxial tests on structured soils.     

Engineering Properties of Grouted Sands

A comparison of the behavior of uncemented and grouted sands is presented. Four sands (Fontainebleau sand and three types of alluvial deposits of the Seine River) were tested. Specimens of grouted sands were prepared in the laboratory by injection of very fine cement or mineral grouts. An initial series of unconfined uniaxial compression tests and tensile tests was performed to highlight the effect of some key factors (mainly the cement-to-water ratio of the grout and the relative density of the granular skeleton) on the strength of the grouted sands. Subsequent triaxial tests showed that when a soil is impregnated by either a very fine cement grout or a mineral grout, both stiffness (secant stiffness or small-strain stiffness) and strength of the soil improve. Similar trends were observed for the behavior of both uncemented and grouted sands. The behavior of grouted sands can be roughly reproduced by applying a linear elastic, perfectly plastic model with a nonassociated Mohr–Coulomb yield criterion whose parameters can be easily determined. Finally, preliminary recommendations are proposed relative to improvements ratios of the parameters of this simple constitutive model that is still commonly used in geotechnical engineering.     

Stress distribution in particulate-reinforced metal-matrix composites subjected to external load

Finite element calculations were carried out to study the stress distribution in particulate SiC-reinforced Al metal-matrix composites (MMCs) subjected to external load. The results showed that, in addition to the effect of the aspect ratio of the reinforcement, particle distribution and grouping behavior-clustering, resulting from the casting process in the present materials, also contribute considerably to the nonuniform stress distribution in the material. The triaxial stress state around a particle or inside a particle cluster may change the von Mises stress and result in plastic deformation features, such as strain localization. Furthermore, the relative orientation of the particle distribution to the external load was found to be important to the stress distribution. As the applied stress increases, the state of triaxial stress inside a cluster appears to promote the early particle cracking, interface debonding, and void formation in the ductile matrix. Experimental observations of slip band distribution and other features in compression tests are consistent with the calculated results.     

Liquefaction Resistance of Sandy Soils under Partially Drained Condition

In order to simulate the effect of drainage on soils adjacent to gravel drains that are installed as countermeasure against liquefaction, several series of cyclic triaxial tests were performed on saturated sands under partially drained conditions. The condition of partial drainage under cyclic loading was simulated in the laboratory using triaxial testing equipment installed with a drainage control valve to precisely regulate the volume of water being drained from test specimens. Effects of both drainage conditions and loading frequencies on cyclic response were incorporated through the coefficient of drainage effect, α*. Experimental results showed that for sand exhibiting strain softening, the partially drained response was controlled by the critical effective stress ratio while for sand showing strain hardening behavior, the controlling factor was the phase transformation stress ratio. Moreover, test results indicated that the minimum liquefaction resistance under partially drained conditions can be used as a parameter to describe the liquefaction resistance of sands improved by the gravel drain method. From these results, a simplified procedure for designing gravel drains based on the factor of safety (FL) concept was proposed.