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.     

Evaluation of Two Homogenization Techniques for Modeling the Elastic Behavior of Granular Materials

This paper discusses the capabilities of two homogenization techniques to accurately represent the elastic behavior of granular materials considered as assemblies of randomly distributed particles. The stress-strain relationship for the assembly is determined by integrating the behavior of the interparticle contacts in all orientations, using two different homogenization methods, namely the kinematic method and the static method. The numerical predictions obtained by these two homogenization techniques are compared to results obtained during experimental studies on different granular materials. Relations between elastic constants of the assembly, interparticle properties, and fabric parameters are discussed, as well as the capabilities of the models to take into account inherent and 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.     

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.     

Model for Granular Materials with Surface Energy Forces

In light of environmental differences (such as gravitational fields, surface temperatures, atmospheric pressures, etc.), the mechanical behavior of the subsurface soil on the Moon is expected to be different from that on the Earth. Before any construction on the Moon can be envisaged, a proper understanding of soil properties and its mechanical behavior in these different environmental conditions is essential. This paper investigates the possible effect of surface-energy forces on the shear strength of lunar soil. All materials, with or without a net surface charge, exhibit surface-energy forces, which act at a very short range. Although, these forces are negligible for usual sand or silty sand on Earth, they may be important for surface activated particles under extremely low lunar atmospheric pressure. This paper describes a constitutive modeling method for granular material considering particle level interactions. Comparisons of numerical simulations and experimental results on Hostun sand show that the model can accurately reproduce the overall mechanical behavior of soils under terrestrial conditions. The model is then extended to include surface-energy forces between particles in order to describe the possible behavior of lunar soil under extremely low atmospheric pressure conditions. Under these conditions, the model shows that soil has an increase of shear strength due to the effect of surface-energy forces. The magnitude of increased shear strength is in reasonable agreement with the observations of lunar soil made on the Moon’s surface.     

Modeling Liquefaction by a Multimechanism Model

An anisotropic constitutive model was recently presented for describing the stress–strain behavior of granular materials with considerations for the initial and induced anisotropy. The model was developed within the framework of a microstructural theory known as the sliding–rolling theory. The resulting model falls within the definition of multimechanism models. The model was shown to satisfactorily represent the drained and undrained behaviors under monotonic loading. The framework used in the model allows extension to describe the behavior under cyclic loading, which is the subject of the present paper. Specifically, the model is further developed for representing the undrained behavior of granular materials under one- and two-way cyclic loading, some of which cause liquefaction resulting in large strain accumulations and the others lead to limited pore pressure and strain accumulations. The validity of the model is verified using triaxial data on Nevada sand.     

Modeling Cross Anisotropy in Granular Materials

A constitutive model has been developed to capture the behavior of cross-anisotropic frictional materials. The elastoplastic, single hardening model for isotropic materials serves as the basic framework. Based on the experimental results of cross-anisotropic sands in isotropic compression tests, the principal stress coordinate system is rotated such that the model operates isotropically within the rotated framework. Experimental plastic work contours on the octahedral plane are plotted for a series of true triaxial tests on dense Santa Monica Beach sand to study the effects of cross anisotropy on the evolution of yield surfaces. The amount of rotation of the yield and plastic potential surfaces decreases to zero (isotropic state) with loading. The model is constructed for cases where the principal stress and material symmetry axes are collinear and no significant rotation of principal stresses occur. The model incorporates fourteen parameters that can be determined from simple experiments, such as isotropic compression, drained triaxial compression, and triaxial extension tests. A series of true triaxial and isotropic compression tests on dense Santa Monica Beach sand are used as a basis for verification of the capabilities of the proposed model.     

Stochastic Modeling of Load-Induced Settlement in Loose Granular Materials

The response of loose cohesionless granular material to surface applied loads is investigated from the viewpoint of probabilistic mechanics of particulate media. A model is proposed that is based on the combined propagation of intergranular forces and an excess volume of voids. In this regard, it provides a bridge between earlier theories developed independently for the diffusion of stresses and for the propagation of settlements. In its general formulation, the theory can model three-dimensional, transient effects. However, the model is believed to be limited to normally consolidated or noncompacted, fully drained or dry, granular materials that do not exhibit dilatancy effects. The derived numerical modeling of steady state deflection patterns under a rigid footing is found to be in good agreement with x-ray images of laboratory model tests using noncompacted silt. The proposed theory recognizes the discrete and inherently random nature of natural granular materials such as cohesionless soils and builds upon these fundamental characteristics to predict responses of such materials to boundary applied load. This is achieved by modeling intergranular force and excess pore volume propagation as Markovian diffusion-advection processes. This approach, which departs from traditional continuum mechanics models, seems to have potential for addressing some of the challenging aspects of granular material mechanics in lunar or Martian environments.     

Quantification and Simulation of Particle Kinematics and Local Strains in Granular Materials Using X-Ray Tomography Imaging and Discrete-Element Method

Microfeatures of granular materials have significant effects on their macrobehaviors. Unfortunately, three-dimensional (3D) quantitative measurements of microfeatures are rare in literature because of the limitations of conventional techniques in obtaining microquantities such as microdisplacements and local strains. This paper presents a new method for quantifying the particle kinematics and local strains for a soft confined compression test using X-ray computed tomography and compares the experimental measurements with the simulated results using the discrete-element method (DEM). The experimental method can identify and recognize 3D individual particles automatically, which is essential for quantifying particle kinematics and local strains. 3D DEM simulations of the soft confined compression test were performed by using spherical particles and irregular particles. The simulated global deformations and particle translations that were based on irregular particles showed better agreement with the experimental measurements than those that were based on spherical particles. The simulated movements of spherical particles were more erratic, and the material composed of spherical particles showed larger vertical contraction and radial dilation.     

Elastoplastic Model for Clay with Microstructural Consideration

Clay material can be considered as a collection of clusters, which interact with each other mainly through mechanical forces. From this point of view, clay is modeled by analogy to granular material in this paper. An elastoplastic stress-strain relationship for clay is derived by using the granular mechanics approach developed in previous studies for sand. However, unlike sand, clay deformation is generated not only by the mobilizing but also by compressing clusters. Thus, in addition to the Mohr-Coulomb’s plastic shear sliding and a dilatancy type flow rule, a plastic normal deformation has been modeled for two clusters in compression. The overall stress-strain relationship can then be obtained from the mobilization and compressing of clusters through a static hypothesis of the macro-micro relations. The predictions are compared with the experimental results for clay under both drained and undrained triaxial loading conditions. Three different types of clay, including remolded and natural clay, have been selected to evaluate the model’s performance. The comparisons verify that this model is capable of accurately reproducing the overall behavior of clay, which accounts for the influence of key parameters such as void ratio and mean stress. A section of this paper is devoted to show the model’s capability of considering the influence of inherent anisotropy on the stress-strain response under undrained triaxial loading conditions.     

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.     

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.     

Microscopic Evaluation of Strain Distribution in Granular Materials during Shear

The evolution of local strains during shear of particles of a granular material is presented in this paper. A cylindrical specimen composed of 6.5-mm spherical plastic particles was loaded under an axisymmetric triaxial loading condition. Computed tomography (CT) was used to acquire three-dimensional images of the specimen at three shearing stages. The high-resolution CT images were used to identify the 3D coordinates of 400 particles. Nine strain components (normal, shear, and rotation), rotation angles, and local dilatancy angles for particle groups were calculated, and their frequency distribution histograms are presented and discussed. It was found that there is no preferred shear direction, and the standard deviation values for shear strain components (εxy, εxz, and εyz) were almost equal for the specific test shearing stage. Shear strains as high as 25.6% were recorded for some particle groups. Furthermore, granular particle groups rotated in the 3D space with almost equal amounts of rotation strains when loaded under axisymmetric triaxial condition. Rotation strain values are very close to the corresponding shear strains. Compared to particle sliding, rotation plays a major role in the shearing resistance of granular materials. The cumulative vertical rotation angles can be as high as 38° and the horizontal rotation angles have values as high as 60°. The statistical distributions of the local dilatancy angle (ψ1) of particle groups were calculated and found to be increasing as shearing continues. The “global” dilatancy angle value is very close to the mean local ψ1 during the first stage of shearing (i.e, when global εz = ?7.3%)     

Observations of Stresses and Strains in a Granular Material

The use of glass ballotini as a granular material provides the opportunity to simultaneously study internal stress fields and internal fields of deformation as a sample is submitted to boundary perturbations. Digital image correlation makes use of the visible fabric of the material to deduce a field of displacements from one digital photographic image to the next. If the glass granules are immersed in a fluid having the same refractive index, then observation with polarized light exploits the photoelastic properties of the glass to reveal information about the stresses. Again, comparison of digital photographs enables changes in stress conditions from one image to the next to be discovered. Tests performed in a simple loading device which forces rotation of principal axes in parts of the granular mass are presented to demonstrate the unique potential of this dual experimental configuration.     

Alternative Definition of Particle Rolling in a Granular Assembly

The paper presents a definition of rolling between a pair of two-dimensional or three-dimensional particles with a compliant contact. The definition of rolling movement is based upon the shapes of the objects’ surfaces as described with differential geometry. A pseudoinverse of the surface curvatures is used for producing a rolling vector that is tangent to the two objects at their contact. Matrix expressions are presented for the efficient computation of the vector. The rolling vector is objective and is independent of the reference points that are used to track the particle motions. The definition of rolling is applied in a discrete element method simulation of the triaxial compression of three large, dense cubic assemblies: one packing of spherical particles and two packings of nonspherical particles. At small strains, particle rolling was slightly less with the nonspherical particles, but the packing with the greatest coordination number had much less rolling.     

Modeling Permanent Deformation of Unbound Granular Materials under Repeated Loads

Finite-element analysis on a pavement structure under traffic loads has been a viable option for researchers and designers in highway pavement design and analysis. Most of the constitutive drivers used were nonlinear elastic models defined by empirical resilient modulus equations. Few isotropic/kinematic hardening elastoplastic models were used but applying thousands of repeated load cycles became computationally expensive. In this paper, a cyclic plasticity model based on fuzzy plasticity theory is presented to model the long-term behavior of unbound granular materials under repeated loads. The discussion focuses on the model parameters that control long-term behavior such as elastic shakedown. The performance of the constitutive model is presented by comparing modeled and measured permanent strain at various numbers of load cycles. Calculated resilient modulus from the complete stress-strain curve is also discussed.     

Wave Velocities in Granular Materials under Microgravity

Velocities of primary (P) and shear (S) waves in granular materials are highly dependent on confining stress. These wave velocities are related to mechanical properties of the materials such as stiffness, density, and stress history. Measurements of the wave velocities using piezoelectric sensors provide scientists and engineers a technique for nonintrusive characterization of those mechanical properties. For aerospace engineering, measuring the wave velocities under microgravity, which simulates low loading and stress conditions, has a number of potential applications. It can help the understanding of the soil mechanics and the development of appropriate materials handling technologies in extraterrestrial environments, which will be crucial to meeting NASA’s future space exploration goals. This paper presents the technique and results of experiments conducted at NASA Glenn Research Center using the 2.2?s drop tower. Velocities of P and S waves in three sizes of glass beads and one size of alumina beads were measured under initially dense or loose compaction states. It was found that under microgravity, the wave signals were significantly weaker and the velocities were much slower. The material that makes up the beads has a strong influence on the wave velocities as well. The initial compaction state also has some influence on the wave velocities.     

Modeling Nonlinear Anisotropic Elastic Properties of Unbound Granular Bases Using Microstructure Distribution Tensors

The resilient properties of unbound aggregate bases are important parameters in the design of asphalt pavements. Previous studies have shown that these resilient properties exhibit nonlinear and transverse anisotropic characteristics. The paper in hand presents a micromechanics-based approach to model the nonlinear and anisotropic properties of unbound aggregate bases. The anisotropic behavior is captured using two microstructure parameters representing the preferred orientation of aggregate particles, and the ratio of the normal contact stiffness to shear contact stiffness among particles. The nonlinear response is modeled using a relationship that relates the shear modulus to particle packing, material properties, particle size, and confining pressure. The micromechanics model is used to represent the resilient properties for a total of 18 different combinations of material conditions with different aggregate types, moisture contents, and gradation characteristics. Anisotropic and nonlinear resilient properties were measured at ten different stress states for each of the material conditions. The results presented in this paper show that the micromechanics model is capable of successfully representing the experimental measurements.