Anisotropy of elasticity and fabric of granular soils

Anisotropy of elasticity is a very important feature of granular soils. In this paper, numerical experiments using discrete element method were performed to emulate drained triaxial tests and simple shear tests at different stress levels. From these numerical experiments the macroscopic elasticity parameters were determined. The results show that at isotropic stress states the stiffness of the numerical specimen increases, while the Poisson’s ratio decreases with increasing confining pressure. The small strain shear modulus of the numerical specimen agrees well with the laboratory experimental results on a specimen with similar conditions. At anisotropic stress states, there is a threshold stress ratio (\({ SR}_{\mathrm{th}}\)), which characterizes the degrees of stiffness change and fabric change during the shearing. When the stress ratio (SR) is less than \({ SR}_{\mathrm{th}}\), the microscopic contact number does not change and its distribution remains nearly isotropic, while the distribution of contact forces change and become anisotropic to resist the applied anisotropic stress. Therefore the stiffness anisotropy of the specimen mainly results from the anisotropy of contact forces. When SR is larger than \({ SR}_{\mathrm{th}}\), however, the contact number decreases significantly in the minor principal stress direction resulting in the fabric anisotropy, along with the adjustments of contact forces. The stiffness anisotropy of the specimen results from both the fabric anisotropy and the contact force anisotropy. It also indicates that the stress normalized stiffness may be used as an index of the degree of fabric anisotropy. Moreover, the Poisson’s ratio of the specimen increases continuously with increasing stress ratio and its anisotropy can be approximately related to the stiffness anisotropy.     

Macroscopic model with anisotropy based on micro–macro information

Physical experiments can characterize the elastic response of granular materials in terms of macroscopic state variables, namely volume (packing) fraction and stress, while the microstructure is not accessible and thus neglected. Here, by means of numerical simulations, we analyze dense, frictionless granular assemblies with the final goal to relate the elastic moduli to the fabric state, i.e., to microstructural averaged contact network features as contact number density and anisotropy. The particle samples are first isotropically compressed and then quasi-statically sheared under constant volume (undrained conditions). From various static, relaxed configurations at different shear strains, infinitesimal strain steps are applied to “measure” the effective elastic response; we quantify the strain needed so that no contact and structure rearrangements, i.e. plasticity, happen. Because of the anisotropy induced by shear, volumetric and deviatoric stresses and strains are cross-coupled via a single anisotropy modulus, which is proportional to the product of deviatoric fabric and bulk modulus (i.e., the isotropic fabric). Interestingly, the shear modulus of the material depends also on the actual deviatoric stress state, along with the contact configuration anisotropy. Finally, a constitutive model based on incremental evolution equations for stress and fabric is introduced. By using the previously measured dependence of the stiffness tensor (elastic moduli) on the microstructure, the theory is able to predict with good agreement the evolution of pressure, shear stress and deviatoric fabric (anisotropy) for an independent undrained cyclic shear test, including the response to reversal of strain.     

Scan line void fabric anisotropy tensors of granular media

Soil fabric anisotropy tensors are related to the statistical distribution of orientation of different microstructural vector-like entities, most common being the contact normal vectors between particles, which are extremely difficult to determine for real granular materials. On the other hand, void fabric based tensors can be determined by image based quantification methods of voids (graphical approaches), which are well defined and easy to apply to both physical and numerical experiments. A promising void fabric characterization approach is based on the scan line method. Existing scan line based definitions of void fabric anisotropy tensors are shown analytically to inherit a shortcoming, since numerous small void segments in a sample have an inordinate contribution towards unwarranted isotropy. Discrete Element Method (DEM) of analysis subsequently confirms this analytical proof. The fact that such scan line void fabric tensor definitions yield acceptable results when used in conjunction with physical image-based measurements, is shown to be attributed to the natural “cut off” of smaller void segments that occurs during such measurements. This is the motivation to propose using the existing definition of void fabric tensors, with exclusion of void segments less than a “cut off” value associated with an internal length of the granular assembly. In addition, an entirely new void fabric tensor was introduced using the squared length, instead of the length of a void segment, as the weighting factor for the definition of the scan line void fabric tensor. It was found by means of DEM analysis that both alternative definitions are void of the aforementioned shortcoming and compatible with existing image quantification methods of void fabric anisotropy.     

Fabric,force and strength anisotropies in granular materials: a micromechanical insight

In micromechanics, the stress–force–fabric (SFF) relationship is referred to as an analytical expression linking the stress state of a granular material with microparameters on contact forces and material fabric. This paper employs the SFF relationship and discrete element modelling to investigate the micromechanics of fabric, force and strength anisotropies in two-dimensional granular materials. The development of the SFF relationship is briefly summarized while more attention is placed on the strength anisotropy and deformation non-coaxiality. Due to the presence of initial anisotropy, a granular material demonstrates a different behaviour when the loading direction relative to the direction of the material fabric varies. Specimens may go through various paths to reach the same critical state at which the fabric and force anisotropies are coaxial with the loading direction. The critical state of anisotropic granular material has been found to be independent of the initial fabric. The fabric anisotropy and the force anisotropy approach their critical magnitudes at the critical state. The particle-scale data obtained from discrete element simulations of anisotropic materials show that in monotonic loading, the principal force direction quickly becomes coaxial with the loading direction (i.e. the strain increment direction as applied). However, material fabric directions differ from the loading direction and they only tend to be coaxial at a very large shear strain. The degree of force anisotropy is in general larger than that of fabric anisotropy. In comparison with the limited variation in the degree of force anisotropy with varying loading directions, the fabric anisotropy adapts in a much slower pace and demonstrates wider disparity in the evolution in the magnitude of fabric anisotropy. The difference in the fabric anisotropy evolution has a more significant contribution to strength anisotropy than that of force anisotropy. There are two key parameters that control the degree of deformation non-coaxiality in granular materials subjected to monotonic shearing: the ratio between the degrees of fabric anisotropy and that of force anisotropy and the angle between the principal fabric direction and the applied loading direction.     

On the influence of inter-particle friction and dilatancy in granular materials: a numerical analysis

Mechanical behavior of granular soils is a classic research realm but still yet not completely understood as it can be influenced by a large number of factors, including confining pressure, soil density, loading conditions, and anisotropy of soil etc. Traditionally granular materials are macroscopically regarded as continua and their particulate and discrete nature has not been thoroughly considered although many researches indicate the macro mechanical behavior closely depends on the micro-scale characteristics of particles. This paper presents a DEM (discrete element method)-based micromechanical investigation of inter-particle friction effects on the behavior of granular materials. In this study, biaxial DEM simulations are carried out under both ‘drained’ and ‘undrained’ (constant volume) conditions. The numerical experiments employ samples having similar initial isotropic fabric and density, and the same confining pressure, but with different inter-particle friction coefficient. Test results show that the inter-particle friction has a substantial effect on the stress-strain curve, peak strength and dilatancy characteristics of the granular assembly. Clearly, it is noted that apart from the inter-particle friction, the shear resistance is also contributed to the dilation and the particle packing and arrangements. The corresponding microstructure evolutions and variations in contact properties in the particulate level are also elaborated, to interpret the origin of the different macro-scale response due to variations in the inter-particle friction.     

On micromechanical characteristics of the critical state of two-dimensional granular materials

In micromechanics of quasi-static deformation of granular materials, relationships are investigated between the macro-scale, continuum-mechanical characteristics, and the micro-scale characteristics at the particle and interparticle contact level. An important micromechanical quantity is the fabric tensor that reflects the distribution of contact orientations. It also contains information on the coordination number, i.e. the average number of contacts per particle. Here, the focus is on characteristics of the critical state in the two-dimensional case. Critical state soil mechanics is reviewed from the micromechanical viewpoint. Two-dimensional discrete element method (DEM) simulations have been performed with discs from a fairly narrow particle-size distribution. Various values for the interparticle friction coefficient and for the confining pressure have been considered to investigate the effect of these quantities on critical state characteristics (shear strength, packing fraction, coordination number and fabric anisotropy). Results from these DEM simulations show that a limiting fabric state exists at the critical state, which is geometrical in origin. The contact network tessellates the assembly into loops that are formed by contacts. For each loop, a symmetrical loop tensor is defined, based on its contact normals. This loop tensor reflects the shape of the loop. An orientation is associated with each loop, based on its loop tensor. At the critical state, the frequencies with which loops with different number of sides occur depend on the coordination number. At the critical state, these loops have, on average, the following universal characteristics, i.e. independent of the coordination number: (1) loops with the same number of sides and orientation have identical anisotropy of the loop tensor, (2) the anisotropy of the loop tensor depends linearly on the number of sides of the loop, (3) the distribution of loop orientations is identical, (4) Lewis’s law for the loop areas, which is a linear relation between the number of sides of loops and their area, is satisfied (not exclusively at the critical state) and (5) the areas of the loops do not depend on their orientation.     

Three dimensional fabric evolution of sheared sand

Granular particles undergo translation and rolling when they are sheared. This paper presents a three-dimensional (3D) experimental assessment of fabric evolution of sheared sand at the particle level. F-75 Ottawa sand specimen was tested under an axisymmetric triaxial loading condition. It measured 9.5?mm in diameter and 20?mm in height. The quantitative evaluation was conducted by analyzing 3D high-resolution x-ray synchrotron micro-tomography images of the specimen at eight axial strain levels. The analyses included visualization of particle translation and rotation, and quantification of fabric orientation as shearing continued. Representative individual particles were successfully tracked and visualized to assess the mode of interaction between them. This paper discusses fabric evolution and compares the evolution of particles within and outside the shear band as shearing continues. Changes in particle orientation distributions are presented using fabric histograms and fabric tensor.     

Micro-scale modeling of anisotropy effects on undrained behavior of granular soils

This paper presents a micro-scale modeling of fabric anisotropy effects on the mechanical behavior of granular assembly under undrained conditions using discrete element method. The initial fabrics of the numerical samples engendered from the deposition under gravity are measured, quantified and compared, where the gravitational field can be applied in different directions to generate varying anisotropy orientations. The samples are sheared under undrained biaxial compression, and identical testing conditions are applied, with samples having nearly the same anisotropy intensities, but with different anisotropy directions. The macroscopic behaviors are discussed for the samples, such as the dilatancy characteristics and responses at the critical state. And the associated microstructure changes are further examined, in terms of the variables in the particulate scale, with the focus on the fabric evolution up to a large deformation reaching the critical state. The numerical analysis results compare reasonably well with available experimental data. It is also observed that at critical state, in addition to the requirements by classical critical state theory, a unique fabric structure has also been developed, and might be independent of its initial fabric. This observation is coincided with the recent theoretical achievement of anisotropic critical state theory. Finally, a general framework is introduced for quantifying and modeling the anisotropy effects.     

A flow rule incorporating the fabric and non-coaxiality in granular materials

Deficiencies of constitutive models in prediction of dilatancy are often attributed to simplifications associated with flow rules such as assumptions of isotropy and coaxiality. It is thus proposed here to develop a comprehensive flow rule for granular materials by including the effect of fabric and without the assumption of coaxiality. A second-order tensor is introduced as a fabric for the distribution of contact normals and contact forces. By using the energy principle in micro-mechanical scale and a suitable dissipation mechanism in granular materials, a stress-dilatancy relation is obtained. Fabric plays a “bridge-like” role in the dilatancy and non-coaxiality. Non-coaxialities between stress-strain-fabric are attributed to the non-coaxiality between stress-fabric and strain-fabric. In this formulation the constants for modeling fabric depend on non-coaxiality of the system rather than the history that determines such a state. Ability of this stress-fabric-dilatancy for modeling the non-coaxiality shows that this relation can predict the behavior of granular materials in the presence of the rotation of principal stress axes.     

Stress-induced anisotropy in sand under cyclic loading

The anisotropy of a granular material’s structure will influence its response to applied loads and deformations. Anisotropy can be either inherent (e.g. due to depositional process) or induced as a consequence of the applied stresses or strains. Discrete element simulations allow the interactions between individual particles to be explicitly simulated and the fabric can be quantified using a fabric tensor. The eigenvalues of this fabric tensor then give a measure of the anisotropy of the fabric. The coordination number is a particle scale scalar measure of the packing density of the material. The current study examines the evolution of the fabric of a granular material subject to cyclic loading, using two-dimensional discrete element method (DEM) simulations. Isotropic consolidation modifies and reduces the inherent anisotropy, but anisotropic consolidation can accentuate anisotropy. The ratio of the normal to shear spring stiffness at the particle contacts in the DEM model affects the evolution of anisotropy. Higher ratios reduce the degree of anisotropy induced by anisotropic consolidation. The anisotropy induced by cyclic loading depends on the amplitude of the loading cycles and the initial anisotropy.     

Stress-chain based micromechanics of sand with grain shape effect

The mechanical behaviors of granular media are controlled by grain properties and microstructure. The primary property of granular media is denoted by its grain shape, grain size distribution, stiffness, and interparticle friction. The grain shape itself is of particular importance. Microstructures are formed in the connection paths of contact points between grains. In this paper, the deformation of granular materials with different grain shapes was simulated using two-dimensional DEM under different stress-levels and densities. After analyzing the results, the authors investigated fabric changes. The evolution rule of stress-induced anisotropy and its limitation as well as the existence of a critical state of fabric are revealed.     

Effect of fabric anisotropy on shear localization in sand during plane strain compression

Summary The paper focuses on the effect of fabric anisotropy on shear localization in cohesionless granular materials. For the numerical simulation, a hypoplastic constitutive model was used. In order to take into account a characteristic length of the micro-structure, the constitutive model was extended to include the second gradient of the Euclidian norm of the deformation rate. The hypoplastic model captures the salient features of granular bodies in a wide range of density and pressure with a single set of parameters. Transversal isotropy is described by the dyadic product of the normal vector of the space orientation of the plane of symmetry. FE-simulations of plane strain compression under constant lateral pressure were carried out with a medium dense specimen for both uniform and stochastic distribution of the initial void ratio. The effect of the direction of the bedding plane and the initial void ratio distribution on the load-deformation behavior was investigated. Moreover, the location, thickness and inclination of the shear zone were also analyzed.     

Fabric rates of elliptical particle assembly in monotonic and cyclic simple shear tests: a numerical study

This paper aims to investigate the evolutions of microscopic structures of elliptical particle assemblies in both monotonic and cyclic constant volume simple shear tests using the discrete element method. Microscopic structures, such as particle orientations, contact normals and contact forces, were obtained from the simulations. Elliptical particles with the same aspect ratio (1.4 and 1.7 respectively for the two specimens) were generated with random particle directions, compacted in layers, and then precompressed to a low pressure one-dimensionally to produce an inherently anisotropic specimen. The specimens were sheared in two perpendicular directions (shear mode I and II) in a strain-rate controlled way so that the effects of inherent anisotropy can be examined. The anisotropy of particle orientation increases and the principal direction of particle orientation rotates with the shearing of the specimen in the monotonic tests. The shear mode can affect the way fabric anisotropy rate of particle orientation responds to shear strain as a result of the initial anisotropy. The particle aspect ratio exhibits quantitative influence on some fabric rates, including particle orientation, contact normal and sliding contact normal. The fabric rates of contact normal, sliding contact normal, contact force, strong and weak contact forces fluctuate dramatically around zero after the shear strain exceeds 4 % in the monotonic tests and throughout the cyclic tests. Fabric rates of contact normals and forces are much larger than that of particle orientation. The particle orientation based fabric tensor is harder to evolve than the contact normal or contact force based because the reorientation of particles is more difficult than that of contacts.     

Determining particulate–solid interphase strength using shear-induced anisotropy

The resistance to shear deformation developed by a granular material layer in contact with a topographically rough natural or manufactured solid material surface is critical to the stability of a variety of composite systems. By using discrete-element method numerical simulations, we show that evolution of fabric and contact force anisotropy at the boundary between the surface and the granular media controls shear behavior. Full mobilization of granular material strength occurs when the contact force anisotropy developed at the interface is equal to the maximum contact force anisotropy of the granular media.     

A 3D fourth order fabric tensor approach of anisotropy in granular media

We propose a micromechanical approach for granular media, with a particular account of the texture-induced anisotropy and of the strain localization rule. The approach is mainly based on the consideration of a fourth order fabric tensor able to capture general anisotropy which can be induced by complex distribution of contacts. Incorporation of this fourth order fabric tensor in a suitable homogenization scheme allows to determine the corresponding macroscopic elastic properties of the granular material. For this purpose, in addition to the classical Voigt upper bound, a new kinematics-based localization rule is proposed. It generalizes the one formulated by Cambou et al. [B. Cambou, Ph. Dubujet, F. Emeriault, F. Sidoroff, Eur. J. Mech. A/Solids 14 (1995) 225–276] in the case of an isotropic contact distribution. The results of the complete model compare well to numerical simulations results when available [C.S. Chang, C.L. Liao, Appl. Mech. Rev. 47 (1 Part 2) (1994) 197–207] (case of isotropic distribution of contacts). Finally, the interest of the fourth order fabric tensor based approach combined with the proposed localization rule is shown for different distributions of contacts by comparing its predictions to those given by a second order fabric tensor approach.     

Investigation of the fabric evolution and the stress-transmission behaviour of sands based on X-ray μCT images

This paper presents the use of X-ray micro-tomography (X-ray $\mu$CT) and image processing and analysis techniques to investigate the stress transmission and buckling of inter-particle contacts within a granular material. A triaxial testing of a miniature Leighton Buzzard sand (LBS) sample was carried out with full-field in-situ X-ray $\mu$CT scanning. High-spatial-resolution CT images of the sample were acquired at several loading stages of the test. Image processing and analysis techniques were used to quantify the inter-particle contact evolution (contact gain, contact loss and contact movement), fabric, contact duration and buckling of stress-transmission contacts based on the CT images. The results indicated that contact gain and loss, and contact movement played two competing roles in determining the overall fabric evolution of the sample. Contacts with a longer duration were more likely to orient in the major principal stress direction and form a stress-transmission contact network. A gradual decrease in the buckling rate of the stress-transmission contacts was observed outside of the shear band, and a relatively stable buckling rate was observed within the shear band during the shear. The results suggested that jamming occurred outside of the shear band and that unjamming occurred within the shear band.     

Influence of fabric on liquefaction and densification potential of cohesionless sand

For simple shearing under constant pressure, the effects of fabric on liquefaction and densification potentials of saturated cohensionless granular materials are examined theoretically and experimentally. The fabric is identified with the distribution of the dilatancy angles (the angle between the sliding and the macroscopic shearing directions), and the influence of prestraining on this distribution and hence on the macroscopic sample behavior is studied. It is shown that prestraining with zero residual stress can reduce resistance to liquefaction by one or even two orders of magnitude, although the sample density and other conditions are kept the same. The micromechanical features responsible for this and related behaviors are examined in some detail. Finally, some tentative results on the effect of the inherent anisotropy that is produced during sample preparation are reported, showing that a method which yields samples more resistive in triaxial cyclic tests may provide samples less resistive in cyclic shearing.     

Discrete element method investigation on thermally-induced shakedown of granular materials

Temperature change, as a common kind of internal perturbation performed on granular materials, has a significant effect on the bulk properties of granular materials. However, few studies on thermally-induced shakedown under a long-term thermal cycling were reported. In this work, the discrete element method was used to give insight into the thermally-induced shakedown on the fabric and stress states within non-cohesive, frictional granular assemblies. Assemblies were submitted to thermal cycling at a stationary boundary condition after experiencing a one-dimensional compression. Evolution of coordination number, entropy and anisotropy was investigated as well as boundary forces and contact forces. At the same time, effects of the heating rate, the initial vertical load and the magnitude of temperature change were examined. It demonstrates that thermal cycling induces a significant force relaxation within granular materials, while the corresponding granular fabric has a small change. In addition, the entropy and anisotropy decreases with thermal cycling. Moreover, the initial vertical load can constrain the development of thermally-induced fabric change, thereby limiting force relaxation to some degree. Both high heating rate and larger magnitudes of temperature change contribute to more significant force relaxation. However, they cause smaller fabric changes even though they provide larger perturbations.