Experimental study on yielding and failure of an anisotropic clay

The effect of initial and stress-induced anisotropies on yielding and failure of an anisotropic clay has been studied experimentally by loading soil samples along different stress paths under triaxial stress. Five orientations of bedding planes of clay fabric were tested. In all the loading paths, it was observed that the strain increment vectors were not coaxial with the stress vectors. The degree of disassociation depended on initial fabric anisotropy and stress-induced anisotropy. The test results indicated that for the inherently anisotropic material, there is no particular requirement (a) for the yield and plastic potential surfaces to coincide, and (b) for an associated flow rule to hold.     

Coupled effects of capillary suction and fabric on the strength of moist granular materials

This paper discusses the coupled effects of capillary suction and fabric on the behavior of partially saturated granular materials at pendular state when discrete liquid bridges form around particle contacts. Experimental results show that the soil–water characteristic curves of granular materials are affected by the internal structure formed during reconstitution of the specimen. The effect of capillary suction on the shear strength of moist sand varies with the direction of shearing relative to the bedding plane which is generally perpendicular to the major principal direction of the fabric tensor. When treating capillary attraction as interparticle forces at particle contacts, a micromechanics analysis shows that the coupling between capillary-attracting forces and fabric results in an additional stress tensor, which describes the anisotropic effect of capillary suction on the behavior of moist sand.     

Micromechanical behavior of granular materials with inherent anisotropy under cyclic loading using 2D DEM

This paper presents the micromechanical behavior of granular materials due to different initial inherent anisotropic conditions during cyclic loading using the discrete element method (DEM). Oval particles were used to model the samples. Three samples, with three different inherent anisotropic conditions based on the particle’s bedding direction, were prepared and subjected to biaxial cyclic loading. The differences in the inherent anisotropic conditions of the samples affect the stress–strain-dilative behavior of granular materials. The width of the stress–strain cyclic loops decreases as the preferred bedding angle changes from vertical to horizontal. Contact fabric evolution is found to be dependent on the inherent anisotropic fabric of the sample during loading and unloading. The fabric anisotropy is dominant for horizontal particle bedding at the end of loading and for vertical particle bedding at the end of unloading. A change in fabric anisotropy is observed only for the first few loading–unloading cycles for the given conditions depicted in the present study.     

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.     

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.     

3D experimental quantification of fabric and fabric evolution of sheared granular materials using synchrotron micro-computed tomography

Many experimental studies have demonstrated that mechanical response of granular materials is highly influenced by micro-structural fabric and its evolution. In the current literature, quantification of fabric and its evolution has been developed based on micro-structural observations using Discrete Element Method or 2D experiments with simple particle shapes. The emergence of X-ray computed tomography technique has made quantification of such experimental micro-structural properties possible using 3D high-resolution images. In this paper, synchrotron micro-computed tomography was used to acquire 3D images during in-situ conventional triaxial compression experiments on granular materials with different morphologies. 3D images were processed to quantify fabric and its evolution based on experimental measurements of contact normal vectors between particles. Overall, the directional distribution of contact normals exhibited the highest degree of isotropy at initial state (i.e., zero global axial strain). As compression progressed, contact normals evolved in the direction of loading until reaching a constant fabric when experiments approached the critical state condition. Further assessment of the influence of confining pressure, initial density state, and particle-level morphology on fabric and its evolution was formed. Results show that initial density state and applied confining pressure significantly influence the fabric-induced internal anisotropy of tested specimens at initial states. Relatively, a higher applied confining pressure and a looser initial density state resulted in a higher degree of fabric-induced internal anisotropy. Influence of particle-level morphology was also found to be significant particularly on fabric evolution.     

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.     

DEM simulation of the shear behaviour of breakable granular materials with various angularities

Particle shape is an important factor that affects particle breakage and the mechanical behaviour of granular materials. This report explored the effect of angularity on the mechanical behaviour of breakable granular materials under triaxial tests. Various angular particles are generated using the quasi-spherical polyhedron method. The angularity α is defined as the mean exterior angle of touching faces in a particle model. A breakable particle is constructed as an aggregate composed of coplanar and glued Voronoi polyhedra. After being prepared under the densest conditions, all assemblies were subjected to triaxial compression until a critical state was reached. The macroscopic characteristics, including the shear strength and dilatancy response, were investigated. Then, particle breakage characteristics, including the extent of particle breakage, breakage pattern and correlation between the particle breakage and energy input, were evaluated. Furthermore, the microscopic characteristics, including the contact force and fabric anisotropy, were examined to probe the microscopic origins of the shear strength. As α increases, the peak shear strength increases first and then remains constant, while the critical shear strength generally increases. Assemblies with larger angularity tend to cause more serious particle breakage. The relative breakage is linearly correlated with α under shear loading. Compared with unbreakable particles, the peak shear strength and the critical volumetric strain decline, and the degree of decline linearly increases with increasing α.     

Investigating the micromechanical evolutions within inherently anisotropic granular materials using discrete element method

This paper investigates the mechanical behavior of inherently-anisotropic granular materials from macroscopic and microscopic points of view. The study is achieved by simulating biaxial compression tests performed on granular assemblies by using numerical discrete element method. In the same category of numerical studies found in the literature, the simulations were performed by considering elliptical/oval particles. In the present study, however, the shape of particles is considered as convex polygons, which mostly resembles real sand grains. Particle assemblies with four different bedding angles were tested. Similar to what observed in experiment, inherent anisotropy has a significant effect on macroscopic mechanical behavior of granular materials. The shear strength and dilative behavior of assemblies were found to decrease as the bedding angle increases. Evolution of the microstructure of all samples and the influence of bedding angle on the fabric and force anisotropy during loading process were investigated. It is seen that the microscopic evolutions in the fabric can justify well the macroscopic behavior of granular assemblies. It is found that the long axis of particles tend to be inclined perpendicular to the loading axis, which results in generating more stable column-like microstructures in order to transfer the applied load. Moreover, the number of contacts as well as the magnitude of forces among particles varies in different directions during the loading process and the initial anisotropy condition totally evolves due to the induced anisotropy within samples.     

Macro deformation and micro structure of 3D granular assemblies subjected to rotation of principal stress axes

This paper presents a numerical investigation on the behavior of three dimensional granular materials during continuous rotation of principal stress axes using the discrete element method. A dense specimen has been prepared as a representative element using the deposition method and subjected to stress rotation at different deviatoric stress levels. Significant plastic deformation has been observed despite that the principal stresses are kept constant. This contradicts the classical plasticity theory, but is in agreement with previous laboratory observations on sand and glass beads. Typical deformation characteristics, including volume contraction, deformation non-coaxiality, have been successfully reproduced. After a larger number of rotational cycles, the sample approaches the ultimate state with constant void ratio and follows a periodic strain path. The internal structure anisotropy has been quantified in terms of the contact-based fabric tensor. Rotation of principal stress axes densifies the packing, and leads to the increase in coordination numbers. A cyclic rotation in material anisotropy has been observed. The larger the stress ratio, the structure becomes more anisotropic. A larger fabric trajectory suggests more significant structure re-organization when rotating and explains the occurrence of more significant strain rate. The trajectory of the contact-normal based fabric is not centered in the origin, due to the anisotropy in particle orientation generated during sample generation which is persistent throughout the shearing process. The sample sheared at a lower intermediate principal stress ratio \((b=0.0)\) has been observed to approach a smaller strain trajectory as compared to the case \(b=0.5\), consistent with a smaller fabric trajectory and less significant structural re-organisation. It also experiences less volume contraction with the out-of plane strain component being dilative.     

Numerical analysis of critical state behaviors of granular soils under different loading conditions

Discrete element method is used to simulate granular assembly behaviors with different initial conditions under three different loading conditions—plain strain, conventional triaxial compression, and direct shear. Different deformation modes of specimens with different conditions are presented. Some important parameters of the critical state theory are investigated. Uniqueness of the critical state line is checked which shows that there is no a unique critical state line for specimens with different initial void ratios under different loading conditions. Frictional angles and dilation angles of specimens with different conditions at critical state are compared. Void ratios and coordination numbers of specimens at critical state are studied. Anisotropies of the particle orientation and normal contact force at initial state, critical state, as well as the evolutions during shearing are analyzed. The anisotropy is shown to have significant effects on the soil behaviors and is related to the non-uniqueness of the critical state line. The developed numerical models can be used to study the micromechanics and microstructure of the specimen subjected to different loading conditions in the future.     

The influence of particle-size distribution on critical state behavior of spherical and non-spherical particle assemblies

This paper presents an investigation into the effects of particle-size distribution on the critical state behavior of granular materials using discrete element method (DEM) simulations on both spherical and non-spherical particle assemblies. A series of triaxial test DEM simulations examine the influence of particle-size distribution (PSD) and particle shape, which were independently assessed in the analyses presented. Samples were composed of particles with varying shapes characterized by overall regularity (OR) and different PSDs. The samples were subjected to the axial compression through different loading schemes: constant volume, constant mean effective stress, and constant lateral stress. All samples were sheared to large strains to ensure that a critical state was reached. Both the macroscopic and microscopic behaviors in these tests are discussed here within the framework of the anisotropic critical state theory. It is shown that both OR and PSD may affect the response of the granular assemblies in terms of the stress–strain relations, dilatancy, and critical state behaviors. For a given PSD, both the shear strength and fabric norm decrease with an increase in OR. The critical state angle of shearing resistance is highly dependent on particle shape. In terms of PSD, uniformly distributed assemblies mobilize higher shear strength and experience more dilative responses than specimens with a greater variation of particle sizes. The position of the critical state line in the e–p′ space is also affected by PSD. However, the effects of PSD on critical strength and evolution of fabric are negligible. These findings highlight the importance of particle shape and PSD that should be included in the development of constitutive models for granular materials.     

The effects of texture in titanium alloys for engineering components under fatigue

The mechanical response of textured Ti 6/4 plate material is assessed through an evaluation of monotonic properties under tension and torsion loading and fatigue testing of plain section and notched specimen geometries. Significant variations in modulus, yield strength, ultimate tensile strength and ductility are demonstrated for testpieces taken from the plate materials parallel to either the transverse or longitudinal rolling direction. Cyclic performance is also shown to be sensitive to orientation with different cyclic stress–strain curves defined in the two orientations. The relationship between the principal stress axis and the dominant basal plane texture is shown to control fatigue crack initiation lives and the ultimate mode of fracture. Whilst loading parallel to the transverse direction offers the strongest monotonic and cyclic stress–strain response, fatigue tests performed on specimens orientated parallel to the longitudinal rolling direction provide the optimum cyclic life. These effects are discussed with reference to the inherent, anisotropic mechanical response of α+β titanium alloys, which results from the hexagonal crystallographic form of the α phase and the availability of preferential slip systems. It is argued that the anisotropic response could be utilised to an engineering advantage by matching critical stressing directions to the specific properties offered by the texture.     

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.     

Dependence of elastic constants of an anisotropic porous material upon porosity and fabric

The elastic properties of an anisotropic porous material can be represented as functions of the material's solid volume fraction (or porosity) and the principal diameters of the material's fabric ellipsoid. The fabric ellipsoid is a measure of the anisotropy of the microstructure of a material. The definitions and measurement techniques for fabric ellipsoids in granular materials, foams, cancellous bone, and rocks are discussed. The principal results presented in this work are algebraic expressions for the dependence of the orthotropic elastic constants upon both solid volume fraction and the fabric ellipsoid.     

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.     

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.     

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.     

Probability density distribution of the orientation of strength-controlling flaws from multiaxial loading using the unit-sphere stochastic strength model for anisotropy

Models that predict the failure probability of monolithic glass and ceramic components under multiaxial loading have been developed by authors such as Batdorf, Evans, and Matsuo. These “unit-sphere” failure models assume that the strength-controlling flaws are randomly oriented, noninteracting planar microcracks of specified geometry but of variable size. The purpose of this paper is to describe a formulation of the probability density distribution of the orientation of critical strength-controlling flaws that results from an applied load. This distribution is a function of the multiaxial stress state, the shear sensitivity of the flaws, the Weibull modulus, and the strength anisotropy. Examples are provided showing the predicted response on the unit sphere for various stress states for isotropic and transversely isotropic (anisotropic) materials—including the most probable orientation of critical flaws for offset uniaxial loads with strength anisotropy. The author anticipates that this information could be used to determine anisotropic stiffness degradation or anisotropic damage evolution for individual brittle (or quasi-brittle) composite material constituents within finite element or micromechanics-based software.