Three-dimensional DEM investigation of critical state and dilatancy behaviors of granular materials

The critical state is significant to the mechanical behaviors of granular materials and the foundation of the constitutive relations. Using the discrete element method (DEM), the mechanical behaviors of granular materials can be investigated on both the macroscopic and microscopic levels. A series of DEM simulations under true triaxial conditions have been performed to explore the critical state and dilatancy behavior of granular materials, which show the qualitatively similar macroscopic responses as the experimental results. The critical void ratio and stress ratio under different stress paths are presented. A unique critical state line (CSL) is shown to indicate that the intermediate principal stress ratio does not influence the CSL. Within the framework of the unique critical state, the stress–dilatancy relation of DEM simulations is found to fulfill the state-dependent dilatancy equations. As a microscopic parameter to evaluate the static determinacy of the granular system, the redundancy ratio is defined and investigated. The results show that the critical state is very close to the statically determinate state. Other particle-level indexes, including the distribution of the contact forces and the anisotropies, are carefully investigated to analyze the microstructural evolution and the underlying mechanism. The microscopic analysis shows that both the contact orientations and contact forces influence the mechanical behaviors of granular materials.     

Magnetic field models for HD 116458 and HD 126515

We have modeled the magnetic fields of the slowly rotating stars HD 116458 and HD 126515 using the “magnetic charge” technique. HD 116458 has a small angle between its rotation axis and dipole axis (β = 12°), whereas this angle is large for HD 126515 (β = 86°). Both stars can be described with a decentered-dipole model, with the respective displacements being r = 0.07 and r = 0.24 in units of the stellar radius. The decentered-dipole model is able to satisfactorily explain the phase relations for the effective field, B_e(P), and the mean surface field, B_s(P), for both stars, along with the fact that the B_e(P) phase relation for HD 126515 is anharmonic. We discuss the role of systematic measurement errors possibly resulting from instrumental or methodical effects in one or both of the phase relations. The displacement of the dipole probably reflects real asymmetry of the stellar field structure, and is not due to measurement errors. Using both phase relations, B_e(P) and B_s(P), in the modeling considerably reduces the influence of the nonuniform distribution of chemical elements on the stellar surface.     

A monotonic bounding surface critical state model for clays

This study investigates a simple constitutive model based on the critical state framework and bounding surface (BS) plasticity that is suitable for reconstituted clays over a wide range of overconsolidation ratios under monotonic loading. For heavily overconsolidated (OC) clays, rather than using the conventional Hvorslev line, an empirical surface is introduced into the model formulation based on two image points on the BS. The peak strength and the dilatancy of heavily OC clays can thus be predicted satisfactorily. Comparisons with triaxial test data show that the model well captures the peak strength and the dilatancy of heavily OC clays under monotonic loading.     

Fabric evolution within shear bands of granular materials and its relation to critical state theory

In an effort to study the relation of fabrics to the critical states of granular aggregates, the discrete element method (DEM) is used to investigate the evolution of fabrics of virtual granular materials consisting of 2D elongated particles. Specimens with a great variety of initial fabrics in terms of void ratios, preferred particle orientations, and intensities of fabric anisotropy were fabricated and tested with direct shear and biaxial compression tests. During loading of a typical specimen, deformation naturally localizes within shear bands while the remaining of the sample stops deforming. Thus, studying the evolution of fabric requires performing continuous local fabric measurements inside these bands, a suitable task for the proposed DEM methodology. It is found that a common ultimate/critical state is eventually reached by all specimens regardless of their initial states. The ultimate/critical state is characterized by a critical void ratio e which depends on the mean stress p, while the other critical state fabric variables related to particle orientations are largely independent of p. These findings confirm the uniqueness of the critical state line in the e ? p space, and show that the critical state itself is necessarily anisotropic. Additional findings include the following: (1) shear bands are highly heterogeneous and critical states exist only in a statistical sense; (2) critical states can only be reached at very large local shear deformations, which are not always obtained by biaxial compression tests (both physical and numerical); (3) the fabric evolution processes are very complex and highly dependent on the initial fabrics. Copyright © 2010 John Wiley & Sons, Ltd.     

Nonlinear consolidation in randomly heterogeneous highly compressible aquitards

Severe land subsidence due to groundwater extraction may occur in multiaquifer systems where highly compressible aquitards are present. The highly compressible nature of the aquitards leads to nonlinear consolidation where the groundwater flow parameters are stress-dependent. The case is further complicated by the heterogeneity of the hydrogeologic and geotechnical properties of the aquitards. The effect of realistic vertical heterogeneity of hydrogeologic and geotechnical parameters on the consolidation of highly compressible aquitards is investigated by means of one-dimensional Monte Carlo numerical simulations where the lower boundary represents the effect of an instant drop in hydraulic head due to groundwater pumping. Two thousand realizations are generated for each of the following parameters: hydraulic conductivity (K), compression index (C _c), void ratio (e) and m (an empirical parameter relating hydraulic conductivity and void ratio). The correlation structure, the mean and the variance for each parameter were obtained from a literature review about field studies in the lacustrine sediments of Mexico City. The results indicate that among the parameters considered, random K has the largest effect on the ensemble average behavior of the system when compared to a nonlinear consolidation model with deterministic initial parameters. The deterministic solution underestimates the ensemble average of total settlement when initial K is random. In addition, random K leads to the largest variance (and therefore largest uncertainty) of total settlement, groundwater flux and time to reach steady-state conditions.     

Equivalent granular state parameter and undrained behaviour of sand�Cfines mixtures

State parameter defined using void ratio, e, and the steady-state line has been shown to be effective in predicting the undrained behaviour of sand. However, steady-state line for sand with fines is dependent on fines content. To overcome this problem, the concept of equivalent granular void ratio, e*, has been well investigated. However, the conversion from e to e* has been essentially a back-analysis process. A methodology for converting e to e* without the need of a back-analysis process was first presented. The concept of equivalent granular state parameter, ψ*, defined in terms of e*, and equivalent granular steady-state line was then developed. An extensive experimental study was conducted to investigate whether ψ* can capture the effects of fines content, and thus can be used to correlate undrained behaviour of sand–fines mixtures without the need of separately considering the effects of fines content. This study suggested that the effective stress path and deviatoric stress–strain responses in undrained shearing can be correlated with the ψ* value at the start of undrained shearing irrespective of fines content.     

Experimental Characterization of the Shrinkage and Water Retention Behaviour of Tailings from Hard Rock Mines

Mining of hard rock ore deposits produces large amounts of tailings. The safe disposal and management of these tailings require an extensive characterization that should include their drying and desaturation behaviour. Desiccation tests have been performed to characterize the shrinkage response of low plasticity tailings having an initially loose state. The testing procedure developed for this purpose is briefly described here. The main shrinkage tests results are then presented. The experimental data are compared with those obtained from water retention tests performed in a pressure plate with volume change measurements. These two types of results are combined to define unsaturated (drying and shrinkage) relationships in six complementary planes that include the volumetric shrinkage curve and the water retention curve. Specific material characteristics are then determined, including the shrinkage limit w _S, final void ratio e _f, and air entry value ψ _a. Additional tests were also performed to define critical parameters at the initiation of cracking in terms of suction ψ, water content w, and degree of saturation S _r. The original results presented here indicate that the onset of desaturation is closely linked with the end of volumetric straining and with crack initiation. Results also show that the shrinkage limit w _S is a function of the specimen initial water content w ₀. Other related characteristics are also presented and discussed.     

An enhanced constitutive model for crushable granular materials

Studies in the past have tried to reproduce the mechanical behaviour of granular materials by proposing constitutive relations based on a common assumption that model parameters and parameters describing the properties, including gradation of individual grains are inevitably linked. However successful these models have proved to be, they cannot account for the changes in granular assembly behaviour if the grains start to break during mechanical loading. This paper proposes to analyse the relation between grading change and the mechanical behaviour of granular assembly. A way to model the influence of grain breakage is to use a critical state‐based model. The influence of the amount of grain breakage during loading, depending on the individual grain strength and size distribution, can be introduced into constitutive relations by means of a new parameter that controls the evolution of critical state with changes in grain size distribution. Experimental data from a calcareous sand, a quartz sand, and a rockfill material were compared with numerical results and good‐quality simulations were obtained. The main consequences of grain breakage are increased compressibility and a gradual dilatancy disappearance in the granular material. The critical state concept is also enriched by considering its overall relation to the evolution of the granular material. Copyright © 2009 John Wiley & Sons, Ltd.     

Figures of equilibrium inside a gravitating ring and the limiting oblateness of elliptical galaxies

A new class of figures of equilibrium for a rotating gravitating fluid located inside a gravitating ring or torus is studied. These figures form a family of sequences of generalized oblate spheroids, in which there is for any value of the tidal parameter α in the interval 0 ≤ \(0 \leqslant \frac{\alpha }{{\pi G\rho }} \leqslant 0.1867\) ≤ 0.1867 a sequence of spheroids with oblatenesses emin (α) ≤ e ≤ e_max (α). A series of classicalMaclaurin spheroids from a sphere to a flat disk is obtained for α = 0. At intermediate values 0 < α ≤ α_max, there are two limiting non-rotating spheroids in each sequence. When α = α_max, the sequence degenerates into a single non-rotating spheroid with e_cr ≈ 0.9600, corresponding to the maximum oblateness of E7 elliptical galaxies. The second part of the paper considers the influence of rings of dark matter on the dynamics of elliptical galaxies. It is proposed that the equilibrium of an oblate isolated non-rotating galaxy is unstable, and it cannot be supported purely by anisotropy of the stellar velocity dispersion. A ring of dark matter can stabilize a weakly rotating galaxy, supplementing standard dynamical models for such stellar systems. In order for a galaxy to acquire appreciable oblateness, the mass of the ring must be an order of magnitude higher than the mass of the galaxy itself, consistent with the ratios of the masses of dark and baryonic matter in the Universe. The influence of massive external rings could shed light on the existence of galaxies with the critical oblateness E7.     

Minimum velocity dispersion in stable stellar disks. Numerical simulations

N-body dynamical simulations are used to analyze the conditions for the gravitational stability of a three-dimensional stellar disk in the gravitational field of two rigid spherical components—a bulge and halo whose central concentrations and relative masses vary over wide ranges. The number of point masses N in the simulations varies from 40 to 500 000 and the evolution of the simulated systems is followed over 10–20 rotation periods of the outer edge of the disk. The initially unstable disks are heated and, as a rule, reach a quasi-stationary equilibrium with a steady-state radial-velocity dispersion c_r over five to eight turns. The radial behavior of the Toomre stability parameter Q_T(r) for the final state of the disk is estimated. Simple models are used to analyze the dependence of the gravitational stability of the disk on the relative masses of the spherical components, disk thickness, degree of differential rotation, and initial state of the disk. Formal application of existing, analytical, local criteria for marginal stability of the disk can lead to errors in c_r of more than a factor of 1.5. It is suggested that the approximate constancy of Q_T?1.2–1.5 for r?(1–2)×L (where L is the radial scale of disk surface density), valid for a wide range of models, can be used to estimate upper limits for the mass and density of a disk based on the observed distributions of the rotational velocity of the gaseous component and of the stellar velocity dispersion.     

Deformation mechanism of strain localization in 2D numerical interface tests

Heterogeneity arises in soil subjected to interface shearing, with the strain gradually localizing into a band area. How the strain localization accumulates and develops to form the structure is crucial in explaining some significant constitutive behaviors of the soil–structural interface during shearing, for example, stress hardening, softening, and shear-dilatancy. Using DEM simulation, interface shear tests with a periodic boundary condition are performed to investigate the strain localization process in densely and loosely packed granular soils. Based on the velocity field given by grains’ translational and rotational velocities, several kinematic quantities are analyzed during the loading history to demonstrate the evolution of strain localization. Results suggest that tiny concentrations in the shear deformation have already been observed in the very early stage of the shear test. The degree of the strain localization, quantified by a proposed new indicator, α, steadily ascends during the stress-hardening regime, dramatically jumps prior to the stress peak, and stabilizes at the stress steady state. Loose specimen does not develop a steady pattern at the large strain, as the deformation pattern transforms between localized and diffused failure modes. During the stress steady state of both specimens, remarkable correlations are observed between α and the shear stress, as well as between α and the volumetric strain rate.     

The prediction of equivalent granular steady state line of loose sand with fines

Void ratio has been used as a state variable for predicting the liquefaction behaviour of soils under the critical state, sometimes also referred to as the steady state, framework. Recent publications show that void ratio may not be a good parameter for characterising sand with fines because the steady state line (or curve) in the e-log(p′) space moves downward with increase in fines content until it reaches a threshold value referred to as the threshold fines content (TFC). Recently, an alternative state variable, referred to as the equivalent granular void ratio, has been proposed to resolve this problem. To calculate this alternative state variable, an additional parameter ‘b’ is needed. This parameter ‘b’ represents the fraction of fines that actively participate in the force structure of the solid skeleton. However, predicting the ‘b’ value is problematic. This paper examines the factors affecting the ‘b’ value based on published work on binary packing. This leads to a simple semi-empirical equation for predicting the ‘b’ value based on fines size and fines content. The proposed equations were evaluated with published data sets. Then, the concept of an equivalent granular steady state line is proposed. This concept was used to predict the location of SSLs for sand with different fines content from either the SSL of clean sand or the SSL of sand with a given fines content. The predictions agree well with experimental results.     

控制应力路径散体材料真三轴试验强度特征的离散元模拟与细观机制分析

通过一系列真三轴离散元数值试验,模拟了不同应力路径下的等b试验中散体材料的强度特征。根据模拟结果详细地分析了三维应力条件下中主应力和应力路径对散体材料峰值强度的影响,研究了峰值摩擦角、峰值应力比的变化规律,并根据真应力的概念和组构张量的演化结果分析了散体材料的强度成因。研究表明,在不同类型的数值试验中峰值偏应力随b参数的变化规律不同,但采用初始围压归一化后的应力-应变曲线规律一致。峰值强度线的斜率只与b值有关而与应力路径无关,且随着b值的增加,峰值应力比qf /pf逐渐减小,数值模拟结果与室内试验结果吻合较好;随着应变的发展,数值试样的组构也随之发生变化,产生了明显的应力诱发各向异性;散体的强度为颗粒摩擦及材料各向异性共同作用的结果;理论上,组构比-应力比坐标系中破坏点位置仅取决于颗粒摩擦角 ,而数值模拟结果与理论值的差异源于颗粒间咬合和滚动摩擦的影响,其影响与颗粒表面摩擦系数有关,也受空间应力状态的影响。     

Why the OCR may reduce the small strain shear stiffness of granular materials?

The small strain shear stiffness G₀ of the soil is of interest and importance in both theory and practice. It is expected that for granular materials G₀ would slightly increases with over-consolidation ratio (OCR). However, laboratory tests indicate that G₀ may decrease with increasing OCR, especially for loose specimens, which is counterintuitive. To explore the underlying mechanism, discrete element method (DEM) is used to investigate the effect of OCR on G₀. The DEM simulations successfully capture the laboratory observations. The analyses at the particulate level reveal that the decrease in small strain stiffness is mainly due to the decreases in coordination number and the uniformity of contact force distribution during unloading process.     

Evolution of the shape parameter in the Weibull distribution for brittle rocks under uniaxial compression

Applying the statistical damage theory based on the Weibull distribution to describe rock deformation and failure processes is an important development in rock mechanics. The shape parameter of the Weibull distribution, m, determines the basic shape of the distribution curve; additionally, it also represents a physical characteristic which can be applied when constructing rock constitutive models. To study the evolution of m during rock failure when applying the Weibull distribution to rock mechanics, uniaxial cyclic loading tests of shale specimens were conducted and previous rock mechanics experiments under different temperatures and loading rates were reviewed. The results indicate that m varied with the accumulation of damage but was almost constant between the volume expansion point and the peak strength point of each specimen. Combined with previous studies about the accelerated failure behavior of rocks, we conclude that between the volume expansion point and the peak strength point, the mechanical behavior of the rock fracture process did not change significantly. Based on the characteristics of m at different damage stages during the rock failure process, ranges of m values at different damage stages are proposed. The conclusions reached in this study may be used as an important reference for theoretical research on rock mechanics.     

A kinematic hardening constitutive model for sands: the multiaxial formulation

This paper explores the possibility of using well-accepted concepts—Mohr-Coulomb-like strength criterion, critical state, existence of a small strain elastic region, hyperbolic relationship for representing global plastic stress–strain behaviour, dependence of strength on state parameter and flow rules derived from the Cam-Clay Model—to represent the general multiaxial stress–strain behaviour of granular materials over the full range of void ratios and stress level (neglecting grain crushing). The result is a simple model based on bounding surface and kinematic hardening plasticity, which is based on a single set of constitutive parameters, namely two for the elastic behaviour plus eight for the plastic behaviour, which all have a clear and easily understandable physical meaning. In order to assist the convenience of the numerical implementation, the model is defined in a ‘normalized’ stress space in which the stress–strain behaviour does not undergo any strain softening and so certain potential numerical difficulties are avoided. In the first part the multiaxial formulation of the model is described in detail, using appropriate mixed invariants, which rationally combine stress history and stress. The model simulations are compared with some experimental results for tests on granular soils along stress paths lying outside the triaxial plane over a wide range of densities and mean stresses, using constitutive parameters calibrated using triaxial tests. Furthermore, the study is extended to the analysis of the effects induced by the different shapes of the yield and bounding surfaces, revealing the different role played by the size and the curvature of the bounding surface on the simulated behaviour of completely stress- and partly strain-driven tests. Copyright © 1999 John Wiley & Sons, Ltd.     

Modeling cyclic shearing of sands in the semifluidized state

Liquefaction is associated with the loss of mean effective stress and increase of the pore water pressure in saturated granular materials due to their contractive tendency under cyclic shear loading. The loss of mean effective stress is linked to loss of grain contacts, bringing the granular material to a “semifluidized state” and leading to development and accumulation of large cyclic shear strains. Constitutive modeling of the cyclic stress-strain response in earthquake-induced liquefaction and post-liquefaction is complex and yet very important for stress-deformation and performance-based analysis of sand deposits. A new state internal variable named strain liquefaction factor is introduced that evolves at low mean effective stresses, and its constitutive role is to reduce the plastic shear stiffness and dilatancy while maintaining the same plastic volumetric strain rate in the semifluidized state. This new constitutive ingredient is added to an existing critical state compatible, bounding surface plasticity reference model, that is well established for constitutive modeling of cyclic response of sands in the pre-liquefaction state. The roles of the key components of the proposed formulation are examined in a series of sensitivity analyses. Their combined effects in improving the performance of the reference model are examined by simulating undrained cyclic simple shear tests on Ottawa sand, with focus on reproducing the increasing shear strain amplitude as well as its saturation in the post-liquefaction response.     

The impact of reflection effects on the parameters of the old pre-cataclysmic variables MS Peg and LM Com

We have determined the main parameters of the old precataclysmic variable stars MS Peg and LM Com. The radial velocities of the components, reflection effects in the spectra, and light curves of the systems are studied based on model stellar atmospheres subject to external irradiation. Forty-seven moderate-resolution spectra for MS Peg and 57 for LM Com obtained with the 6-m telescope of the Special Astrophysical Observatory are used to derive the refined orbital periods of 0.1736660 days and 0.2586873 days, respectively; the orbital eccentricities do not exceed e=0.04. The mass (M _w =0.49e_⊙) and radius (e _w =0.015R_⊙) of the MS Peg primary calculated using the gravitational redshift correspond to those for a cooling carbon white dwarf with a thin hydrogen envelope. The parameters of the red dwarf (M _r =0.19M_⊙, T_eff=3560 K, R _r =0.18R_⊙) are close to those derived from evolutionary tracks for main-sequence M stars with solar chemical composition. The radius (R _r =0.22R_⊙) and temperature (T_eff=3650 K) of the LM Com secondary exceed theoretical estimates for main-sequence stars with masses of M _r =0.17M_⊙. The luminosity excess of the red dwarf in LM Com can be explained by a prolonged (T>5×10⁶ yrs) relaxation of the M star to its normal state after the binary leaves the common-envelope stage. For both systems, theoretical U, B, V, and R light curves and spectra calculated using the adopted sets of parameters are generally consistent with the observations. This confirms the radiative origin of the hot spots, the unimportance of horizontal radiative transport, and the absence of large-scale velocity fields with high values (V_trans>50 km/s) at the surfaces of the secondaries. Most of the emission lines in the spectra of these objects are formed under conditions close to thermalization, enabling modeling of their pro files in an LTE approximation. A strong λ3905 Å emission line has been identified as the 3s²3p4s 1P⁰-3s²3p² 1S SiI λ3905.52 Å line formed in the atmosphere of the hot spot. The observed intensity can be explained by non-LTE “superionization” of SiI atoms by soft UV radiation from the white dwarf. We suggest a technique for identifying binaries whose cool components are subject to UV irradiation based on observations of λ3905 Å emission in their spectra.     

Fabric evolution of granular materials along imposed stress paths

The stress–strain behavior of a granular material is dominated by its internal structure, which is related to the spatial connectivity of particles, and the force chain network. In this study, a series of discrete element simulations were carried out to investigate the evolution of internal structure and force chain networks in initially isotropic granular materials along various imposed stress paths. The fabric tensor of the strong sub-network, which is the bearing network toward loading, can be related to the applied stresses uniquely. The principal directions of fabric tensor of the strong sub-network coincide with those of stress tensor during the loading process in the Lode coordinate system. The fabric of the whole contact network in the pre- and post-peak deformation stages can be related to the applied stresses as \(q_{\phi } = B\left( {q/p} \right)^{z}\) (B and z are constants depending on loading condition, such as the stress paths and mean stress level) and \(\phi_{1} :\phi_{2} :\phi_{3} \approx \left( {\sigma_{1} } \right)^{0.4} :\left( {\sigma_{2} } \right)^{0.4} :\left( {\sigma_{3} } \right)^{0.4}\), respectively. At the critical stress state, the deviator of fabric tensor of the strong sub-network is much larger than that of the whole contact network. When plotted on the π-plane, the fabric state of the strong sub-network can be expressed as a Lade’s surface, while the fabric state of the whole network corresponds to an inverted Lade’s surface.     

A many-year photometric and spectroscopic variability study of planetary nebulae (1968–2008). The young planetary nebula IC 4997

We present the results of our many-year (1968–2008) photoelectric photometry and absolute spectrophotometry of the young variable planetary nebula IC 4997, performed in uniform systems. Integrated line energy fluxes in the range 3727–5007 Å are tabulated, along with the integrated (star + nebula) continuum flux at 4500 Å, and time variations investigated. We study the time behavior of fluxes in the hydrogen, HeI, [OIII], [OII], [NeIII] emission lines. Variations of the line intensity ratio R = F([OIII]4363 Å)/F(Hγ) between 1938 and 2008 are presented. We estimate the nebula’s electron density N _e and electron temperature T _e for 1972–1992: the mean nebula N _e increased by the factor of five, from ～4 × 10⁵ to ～2 × 10⁶ cm^?3, while the mean T _e increased from 12 000 to 14 000 K. The color temperature of the exciting star, measured from the 4000–4900 Å continuum, increased from 37 000–40 000 K to 47 000 K during the time interval covered by our observations, as is confirmed by the growth of ionization of HeI, NeIII, and OIII (from the 4363 Å line). V-band variations of the object’s integrated (nebula + star) light with an amplitude about 0.5 ^m can essentially be fully explained as being due to changing contributions from variable fluxes in the [OIII] 5007 and 4959 Å lines. The V magnitude in 2009 (after 40 years) happened to be the same as at the start of our observations in 1968. A period of the order of 50 years can also be noted in log R(t). This may provide evidence for binarity of the central star and be related to its orbital period.