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.     

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.     

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.     

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.     

Micromechanical Analysis for Interparticle and Assembly Instability of Sand

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

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.     

Undrained Fragility of Clean Sands, Silty Sands, and Sandy Silts

In this paper, intergranular (ec) and interfine (ef) void ratios and confining stress are used as indices to characterize the stress–strain response of gap graded granular mixes. It was found that at the same global void ratio (e) and confining stress, the collapse potential (fragility) of silty sand increases with an increase in fines content (FC) due to a reduction in intergranular contact between the coarse grains. Beyond a certain threshold fines content (FCth), with further addition of fines, the interfine contact friction becomes significant. The fragility decreases and the soil becomes stronger. The value of FCth depends on e and the characteristics of fines and coarse grains. At FCFCth), fine grain friction plays a primary role and dispersed coarse grains provide a beneficial, secondary reinforcement effect. At the same ef, the collapse potential decreases with an increase in sand content. Beyond a certain limiting fines content, the soil behavior is controlled by ef only. An intergranular matrix diagram is presented that delineates zones of different behaviors of granular mixes as a guideline to determine the anticipated behavior of gap-graded granular mixes. New equivalent intergranular contact void ratios, (ec)eq and (ef)eq, are introduced to characterize the behavior of such soils, at FCFCth, respectively.     

Micro-Macro Quantification of the Internal Structure of Granular Materials

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

Microplastic surface deformation of A12219-T851

A model of inhomogeneous plastic deformation in a grain embedded in an elastic matrix is developed and used to examine the highly localized softening of the surface of A12219-T851 during fatigue. The model is used to calculate the stress within plastically deformed grains as a means of obtaining true local stress-strain response. Strains measured within individual grains reveal that the degree of softening increases with grain size. In grains larger than four times the mean grain size, the local yield strength after fatigue is reduced to a third that of the cyclically stable bulk yield strength.     

Laboratory Study of Loose Saturated and Unsaturated Decomposed Granitic Soil

Weathered soils are used extensively as fill materials in slope construction in tropical and subtropical cities such as Hong Kong. The mechanical behavior of loose decomposed fill materials, particularly in the unsaturated state, has not often been investigated and is not yet fully understood. The objective of this study was to understand the mechanical behavior of loose unsaturated decomposed granitic soil and to study the effects of the stress state, the stress path and the soil suction on the stress–strain relationship, shear strength, volume change, and dilatancy via three series of stress path triaxial tests on both saturated and unsaturated specimens. It was found that loose and saturated decomposed granitic soil behaves like clean sands during undrained shearing. Strain-softening behavior is observed in loose saturated specimens. In unsaturated specimens sheared at a constant water content, a hardening stress–strain relationship and volumetric contractions are observed in the considered range of net mean stresses. The suction of the soil contributed little to the apparent cohesion. The angle of friction appeared to be independent of the suction. In unsaturated specimens subjected to continuous wetting (suction reduction) at a constant deviator stress, the volumetric behavior changed from dilative to contractive with increasing net mean stress and the specimen failed at a degree of saturation far below full saturation. It was revealed that the dilatancy of the unsaturated soil depends on the suction, the state, and the stress path.     

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.     

Effect of Particle Grading on the Response of an Idealized Granular Assemblage

The effects of particle-size distribution on a granular assemblage’s mechanical response were studied through a series of numerical triaxial tests using the three-dimensional (3D) discrete-element method. An assemblage was formed by spherical particles of various sizes. A simple linear contact model was adopted with the crucial consideration of varying contact stiffness with particle diameter. Numerical triaxial tests were mimicked by imposing axial compression under constant lateral pressure and constant volume condition, respectively. It was found that an assemblage with a wider particle grading gives more contractive response and behaves toward strain hardening upon shearing. Its critical state locates at a lower position in a void ratio versus mean normal stress plot. Nevertheless, no obvious difference in the critical stress ratio was shown. Model constants in a simple but efficient phenomenologically based granular material model within the framework of critical-state soil mechanics were calibrated from the numerical test results. Results show that some model constants exhibit linear variation with the coefficient of uniformity whereas others are almost independent of particle grading. This investigation provides an opportunity to better understand the implications and meanings of model constants in a phenomenologically based model from the microscale perspective.     

Influence of the localized initial plastic deformation on the effective thermomechanical response of metal-matrix composites

A generalized Eshelby model, allowing interaction among reinforcing particles under a Mori-Tanakalike scheme, is presented. Different inclusion aspect ratios are studied in the elastic and incipient elastoplastic regime for a model SiC-Al composite. The solution of the field equations is obtained via an explicit algorithm that yields the interaction field in terms of the stress and strain variables. The particles and fibers are taken as purely elastic, and the matrix is regarded as elastic-perfectly plastic. Coefficients of thermal expansion (CTE) are calculated both under the assumption of purely elastic response and at the onset of plastic localized deformation. The simulated stress-strain curves show the influence of interaction stresses on macroscopic yield stress for different inclusion aspect ratios, with no consideration of matrix hardening. The model allows a good simulation of the thermomechanical behavior of composite materials and contributes to the understanding of the elastoplastic transition in stress-strain curves. It can also simply explain some of the most distinctive features of the mechanical behavior of composites. The model presents the possibility of controlling many input variables and geometries and simultaneously considering three-dimensional deformation of interacting inclusion-reinforced materials with low computational effort. Comparisons to experimental CTE and residual stresses are provided.     

Equivalent Stress Equation for Unsaturated Soils. II: Solid-Porous Model

Based on the study of the equilibrium of the particles of a soil showing a bimodal structure and subject to certain suction, it was possible to establish an analytical expression for the value of Bishop’s parameter χ (see the companion paper). This parameter can be written as a function of the saturated fraction and the degree of saturation of the unsaturated fraction of the soil. However, the determination of these last two parameters cannot be made from current experimental procedures. Therefore, a solid-porous model simulating the structure of the soil is proposed herein and used to determine these parameters. The data required for the solid-porous model are obtained from the grain and pore size distributions, void ratios, and secondary soil–water retention curves of the soil. The plots of the deviator stress versus equivalent stress shows a unique failure line for a series of tests performed at different confining net stresses and suctions, confirming that the proposed equivalent stress equation is adequate to predict the shear strength of unsaturated soils. It also results in different strengths for wetting and drying, as the experimental evidence suggests.     

Micromechanical Modeling for Behavior of Cementitious Granular Materials

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

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.     

Microstructural evolution and superplastic deformation behavior of fine grain 5083Al

The microstructural evolution during superplastic deformation of a fine grain Al-4.7 pct Mg alloy (5083Al) has been studied quantitatively. Starting from an average grain size of 7 μm, grain growth was monitored in this alloy both under static annealing and with concurrent superplastic deformation at a high test temperature of 550°C. Grain size was averaged from measurements taken in longitudinal, transverse, and thickness directions and was found to grow faster during concurrent superplastic deformation than for static annealing. A grain growth law based on an additive nature between time-based and strain-based growth behavior was used to quantify the dynamics of concurrent grain growth. The extent of void formation during deformation was quantified as the area fraction of voids on L-S planes. This void fraction, referred to as the cavity area percent, was recorded at several levels of strain for specimens deformed at two different strain rates. A constitutive equation incorporating this grain growth data into the stress-strain rate data, determined during the early part of deformation, was generated and utilized to model the superplastic tensile behavior. This model was used in an effort to predict the stress-strain curves in uniaxial tension under constant and variable strain rate conditions. Particular attention was paid to the effects of a rapid prestrain rate on the overall superplastic response and hardening characteristics of this alloy.     

Suction Stress Characteristic Curve for Unsaturated Soil

The concept of the suction stress characteristic curve (SSCC) for unsaturated soil is presented. Particle-scale equilibrium analyses are employed to distinguish three types of interparticle forces: (1) active forces transmitted through the soil grains; (2) active forces at or near interparticle contacts; and (3) passive, or counterbalancing, forces at or near interparticle contacts. It is proposed that the second type of force, which includes physicochemical forces, cementation forces, surface tension forces, and the force arising from negative pore-water pressure, may be conceptually combined into a macroscopic stress called suction stress. Suction stress characteristically depends on degree of saturation, water content, or matric suction through the SSCC, thus paralleling well-established concepts of the soil–water characteristic curve and hydraulic conductivity function for unsaturated soils. The existence and behavior of the SSCC are experimentally validated by considering unsaturated shear strength data for a variety of soil types in the literature. Its characteristic nature and a methodology for its determination are demonstrated. The experimental evidence shows that both Mohr–Coulomb failure and critical state failure can be well represented by the SSCC concept. The SSCC provides a potentially simple and practical way to describe the state of stress in unsaturated soil.     

The effects of triaxial stress on void growth and yield equations of power-hardening porous materials

In engineering applications, especially for ductile fracture of materials, nucleation, growth and coalescence of voids have often been observed. Currently there is an increase in interest for the effects of voids on the behaviour of engineering materials. In this paper, by the method of combining micro- and macro-parameters, the effects of triaxial stress on the rates of void growth and yield equations are presented for porous materials with power-hardening. The relations between triaxial stress and the rates of void growth for different n-values and yield equations with different n-values and void volume fractions are discussed. Following results have been obtained: For a porous material with power-hardening, the yield equation can be approximately expressed by an elliptical equation in equivalent stress and triaxial stress. Both the long half-axis and the short half-axis of the elliptical equation are functions of the void volume fraction for a given hardening exponent. The triaxial stress has a strong effect on the growth rates of voids. For linear hardening materials, the relation between the growth rate of voids and the triaxial stress is linear. For elastic/perfectly plastic materials with a small void volume fraction, the growth rate of voids can be described in relation to the triaxial stress with an exponential function. The results from this paper are compared with theoretical results from other researchers for elastic/perfectly plastic materials. A good agreement is shown.