A generalised Modified Cam clay model for clay and sand incorporating kinematic hardening and bounding surface plasticity

This paper proposes a simple non-associated Modified Cam clay model suitable for clay and sand. The yield surface is taken to be that of Modified Cam clay, which is a simple ellipse. The modified model reduces the amount of shear strain predicted, and for clay requires no new parameters because the flow rule uses a well established empirical result. For sand, the critical state frictional dissipation constant is required in addition to the stress ratio at the peak of the yield surface. This permits realistic modelling of the undrained behaviour of sand in states looser and denser than critical. The model resembles more sophisticated models with yield surfaces of more complex shapes, but is much simpler. More realistic behaviour could be obtained by assuming a yield surface with the same form as the potential if required. The model is suitable for incorporating kinematic hardening for the modelling of cyclic loading of clay. In addition, bounding surface plasticity can be included to distinguish between compacted and overconsolidated sand. The contribution in this paper is therefore to provide a generalised simple model based on Modified Cam clay.The authors are grateful to Mr C.D. Khong for discussions on the bounding surface formulation of the CASM model.     

考虑相变状态的较密实饱和砂土弹塑性模型

在三轴试验中,较密实砂土是很难最后到达临界状态的。该文在Yu等提出的CASM模型基础上,提出了适合模拟较密实饱和砂土的弹塑性模型。该模型有两方面的改进:一方面,模型将剪胀应力比M_d引入剪胀方程;另一方面,模型中硬化参数采用与应力路径无关的参数来替代CASM模型中的塑性体应变增量。经试验结果验证,该模型较适合模拟饱和中密或密实砂土的剪胀特性,同时也能较好地体现中密或密砂剪切过程中的硬化及软化现象。模型共10个参数,用常规三轴试验就可获取。     

A simple non-associated flow model for sand

 This paper proposes a simple non-associated plasticity model for sand. The yield surface is taken to be a member of a recently derived family of yield loci, requiring the specification of a single parameter in addition to the stress ratio at the peak value of deviatoric stress on the yield surface in deviatoric:mean effective stress space. This simple equation, can easily be fitted to given sand data. The flow rule also has a simple equation, such that the critical state is not at the top of the yield locus in stress space. The equation of the flow rule requires the specification of the critical state dissipation constant, plus one additional parameter. This permits realistic modelling of the undrained behaviour of sand in states looser and denser than critical. The parameter controlling the flow rule can, for convenience, be taken to be equal to the parameter governing the shape of the yield surface. However, since the two parameters are not required to be equal, the flow rule can easily be adjusted to model more accurately the rate of change of direction of the plastic strain increment vector with changing stress ratio around the yield surface. The model resembles more complex models based on the mathematical theory of envelopes, but the equations of the yield loci and flow rules are much simpler. The contribution in this paper is therefore to provide a model similar to those derived based on micro mechanical considerations, but which is more useful to geotechnical engineers, in that the number of parameters is kept to a minimum, the constitutive equations are simple, and the flow rule can easily be controlled. The model is easy to apply in geotechnical analysis, and would be easy to implement in a finite element program. Received: 11 January 2002     

Critical state behaviour of granular materials from isotropic and rebounded paths: DEM simulations

This paper presents the results of numerical simulations using the three-dimensional discrete element method (DEM) on the critical state behaviour of isotropically compressed and rebounded assemblies of granular materials. Drained and undrained (constant volume) numerical simulations were carried out. From these numerical simulations of drained and undrained tests, it has been shown that the steady state is same as the critical state. Critical state for both isotropically compressed and rebounded assemblies form unique curved line that can be approximated by a bilinear line as proposed by Been et al. [Géotechnique 41(3): 365–381, 1991]. Further more, evolution of the internal variables such as average coordination number and induced anisotropy coefficients during shear deformation has been studied.     

Probing into the strain induced anisotropy of Hostun RF loose sand

Several recent linear drained preloading histories with fixed direction were especially designed to study the effects of strain induced anisotropy of loose Hostun RF sand in the compression side of the classical triaxial plane. Nearly identical void ratio and a same initial isotropic stress state prior to the final undrained shearing in compression are the requirements of the experimental program to take into account only the deviatoric strain histories. The effects of previous deviatoric strain histories on the undrained response of loose Hostun RF sand are identified: mainly the progressive transformation of a contractive and unstable behaviour of very loose sand into a dilative and stable behaviour of dense-like sand by previous linear drained history, while remaining in the same state of loose density. Experimental data evidence the directional dependency of the initial gradient of the effective stress paths, independent of the length of the approaching linear stress paths; the large common non linear effective stress response up to the deviatoric stress peak; the progressive appearance of the dilatancy domain and the unexpected evolution of the undrained behaviour of loose and presheared sand. The paper provides new insights into the mechanisms of strain induced anisotropy of loose sand created by simple linear triaxial stress paths from an isotropic stress state.     

Soil mechanics: breaking ground

In soil mechanics, student's models are classified as simple models that teach us unexplained elements of behaviour; an example is the Cam clay constitutive models of critical state soil mechanics (CSSM). 'Engineer's models' are models that elaborate the theory to fit more behavioural trends; this is usually done by adding fitting parameters to the student's models. Can currently unexplained behavioural trends of soil be explained without adding fitting parameters to CSSM models, by developing alternative student's models based on modern theories?Here I apply an alternative theory to CSSM, called 'breakage mechanics', and develop a simple student's model for sand. Its unique and distinctive feature is the use of an energy balance equation that connects grain size reduction to consumption of energy, which enables us to predict how grain size distribution (gsd) evolves-an unprecedented capability in constitutive modelling. With only four parameters, the model is physically clarifying what CSSM cannot for sand: the dependency of yielding and critical state on the initial gsd and void ratio.     

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 bounding surface plasticity model for cyclic loading of granular soils

A constitutive model for describing the stress–strain behaviour of granular soils subjected to cyclic loading is presented. The model is formulated using bounding surface theory within a critical state framework. A single set of material parameters is introduced for the complete characterization of the constitutive model. The shape of the bounding surface is based on experimental observations of undrained stress paths for loose samples. A mapping rule which passes through stress reversal points is introduced to depict the stress–strain behaviour during unloading and reloading. The effect of particle crushing is considered through a modified critical state line. Essential features of the model are validated using several experimental data from the literature. Both drained and undrained loading conditions are considered. The characteristic features of behaviour in granular soils subjected to cyclic loading are captured. Copyright © 2005 John Wiley & Sons, Ltd.     

DEM simulation of the undrained shear behavior of sand containing dissociated gas hydrate

A numerical simulation technique is proposed using discrete element method (DEM) to study the undrained shear behavior of sand containing dissociated gas hydrate. Dissociation of gas hydrate in sand samples under undrained conditions is simulated first, followed by undrained bi-axial compression of these samples, which consist of three phases, i.e., sand particles, water, and methane gas. The simulations demonstrate that hydrate dissociation under undrained conditions generates significant excess pore pressure and volumetric dilation, the magnitudes of which are found to be comparable with those predicted by theoretical analysis. During the undrained bi-axial compression, samples with different initial degrees of hydrate saturation exhibit markedly different shear behavior. Complete strain softening behavior, i.e., static liquefaction, is observed for the samples with relatively high initial degrees of hydrate saturation, while other samples reach quasi-steady state followed by strain hardening at large strains. The observed static liquefaction is believed to be a combined result of the looser sand structure induced by hydrate dissociation, and the essentially stiff modulus of fluid under high pore pressure, despite the existence of gas in sand pores. The influence of initial sample porosity and particle shape is also investigated.     

A laboratory investigation on an undisturbed silty sand from a slope prone to landsliding

A laboratory investigation is presented for undisturbed samples of a silty sand under saturated conditions. The soil was sampled from test pits south of Rüdlingen in North–East Switzerland, where a landslide triggering experiment was carried out on a steep forest slope. The aim of the work was to characterise the behaviour of the soil in triaxial tests, in the light of the possible failure mechanisms of the slope. Conventional drained and undrained triaxial tests were conducted to detect critical state conditions as well as peak shear strength as a function of confining pressure. Soil specimens were also exposed to stress paths simulating in situ water pressure increase to study the stress–strain response and to enhance the ability to predict failure conditions more accurately in the future. Possible unstable response along the stress paths analysed was investigated by means of second order work and strain acceleration. The results show that temporary unstable conditions may be encountered for this soil at stress ratios below ultimate failure and even below critical state line, depending on void ratio, drainage conditions and time dependent compressibility. A modified state parameter is explored as a potentially useful tool to discriminate conditions leading to eventual collapse.     

饱和砂土基于相变状态的不排水本构模型

饱和砂土基于相变状态的改进本构模型能较好地描述砂土不排水情况下的应力-应变关系。通过改进模型中相变状态参量的计算方法,将e-lgp°表达式加以改进,通过详细的阐述,提出e_pt-lgp的关系式,并通过三轴压缩试验的数据对提出的关系式加以验证。应用新的关系式计算不排水条件下试样的相变孔隙比e_pt,进而计算出模型中的相变状态参量。改进后的不排水本构模型的计算结果与不排水三轴压缩的试验结果能较好地吻合,即验证了本构模型的有效性,也验证了e_pt-lgp关系式的合理性,使原有的边界面本构模型更好地描述不排水的情况下、不同密度和围压状态下饱和砂土的力学行为。     

砂土统一本构模型研究及其三维数值实现

基于广义塑性理论与临界状态概念,研究提出了一个统一三维砂土本构模型,通过一组参数实现了砂土由压缩至剪切过程中状态参量的统一表述。基于ABAQUS提供的用户自定义材料子程序UMAT接口,利用Fortran语言编程实现了该三维弹塑性本构关系模型在软件中的二次开发。分别利用Toyoura砂、Fuji River砂以及Tokachi砂的剪切试验数据与数值模拟结果进行对比,结果表明:提出的有限元计算模型可以有效反映加载过程中不同围压和砂土初始密度对应力-应变曲线的影响,能够准确描述密砂的剪胀特性与应变软化特性以及松砂的剪缩特性与应变硬化特性,从而更加真实地反映三维应力状态下土的变形和强度特性。研究成果进一步扩展了ABAQUS在岩土工程中的应用范围,能够为岩土工程领域的数值分析计算提供更加快捷的解决方案。     

地震液化的突变模型研究

能量和标准贯入击数是用于液化评价的两个综合性指标。以经过持时修正的能量函数和经过粘粒及上覆压力修正的标贯击数为控制变量,以振动孔隙水压力增长为状态变量, 根据液化的物理意义、室内试验所得到的孔压发展曲线的特点和国内外实测液化资料,建立了基于修正Zeeman 突变方程的孔压发展模型。与目前广泛使用的传统尖点突变模型相比,修正模型增加了一项和三个系数,这样做既保留了原方程中状态变量与其中一个控制变量成线性关系这一优良性质,又能够适应于准确模拟孔压与液化发展。在此基础上,利用液化资料,基于统计学中控制“弃真”概率的方法,运用Matlab 逐步逼近确定了所有的六个待定未知量, 确定了分叉曲线和新旧坐标之间的转换关系。检验结果表明了所建议模型的合理性。     

DEM of triaxial tests on crushable sand

This paper presents simulations of high-pressure triaxial shear tests on a crushable sand. The discrete element method is used, featuring a large number of particles and avoiding the use of agglomerates. The triaxial model features a flexible membrane, therefore allowing realistic deformation, and a simple breakage mechanism is implemented using the octahedral shear stress induced in the particles. The simulations show that particle crushing is essential to replicate the realistic behaviour of sand (in particular the volumetric contraction) in high-pressure shear tests. The general effects of crushing during shear are explored, including its effects on critical states, and the influence of particle strength and confining pressure on the degree of crushing are discussed.     

A coupled FEM/DEM approach for hierarchical multiscale modelling of granular media

A hierarchical multiscale framework is proposed to model the mechanical behaviour of granular media. The framework employs a rigorous hierarchical coupling between the FEM and the discrete element method (DEM). To solve a BVP, the FEM is used to discretise the macroscopic geometric domain into an FEM mesh. A DEM assembly with memory of its loading history is embedded at each Gauss integration point of the mesh to serve as the representative volume element (RVE). The DEM assembly receives the global deformation at its Gauss point from the FEM as input boundary conditions and is solved to derive the required constitutive relation at the specific material point to advance the FEM computation. The DEM computation employs simple physically based contact laws in conjunction with Coulomb's friction for interparticle contacts to capture the loading‐history dependence and highly nonlinear dissipative response of a granular material. The hierarchical scheme helps to avoid the phenomenological assumptions on constitutive relation in conventional continuum modelling and retains the computational efficiency of FEM in solving large‐scale BVPs. The hierarchical structure also makes it ideal for distributed parallel computing to fully unleash its predictive power. Importantly, the framework offers rich information on the particle level with direct link to the macroscopic material response, which helps to shed lights on cross‐scale understanding of granular media. The developed framework is first benchmarked by a simulation of single‐element drained test and is then applied to the predictions of strain localisation for sand subject to monotonic biaxial compression, as well as the liquefaction and cyclic mobility of sand in cyclic simple shear tests. It is demonstrated that the proposed method may reproduce interesting experimental observations that are otherwise difficult to be captured by conventional FEM or pure DEM simulations, such as the inception of shear band under smooth symmetric boundary conditions, non‐coaxial granular response, large dilation and rotation at the edges of shear band and critical state reached within the shear band. Copyright © 2014 John Wiley & Sons, Ltd.     

Manifestation of particle morphology on the mechanical behaviour of granular ensembles

This paper presents the effect of particle morphology (grain shape) on the mechanical response of granular materials. Two model systems with extreme differences in morphology were selected (spherical glass ballotini and angular sand) for this experimental programme. A series of hollow cylinder torsion tests were conducted in this programme under monotonic drained conditions on specimens reconstituted to the same relative density. Tests were conducted under different intermediate principal stress ratio (b) on both the model materials. The glass ballotini shows increased dilation at the outset of the test, however, at large strains, the particle rearrangement in the sand and the increased interlocking leads to higher strength at the critical state. The effect of individual particle morphology is manifested in both the increased friction angle and a larger sized failure locus in stress space with increase in angularity. The stresses developed in these two model materials are also accompanied by intriguing volume change behaviour. The glass ballotini despite a lower strength presents a predominantly dilative response immaterial of the ‘b’ value, while the angular sand shows increased strength at large strains, while showing a contractive response. These results allow incorporation of particle morphology effects at the ensemble level in plasticity based constitutive models.     

Logical inference for inverse problems

Estimating a deterministic single value for model parameters when reconstructing the system response has a limited meaning if one considers that the model used to predict its behaviour is just an idealization of reality, and furthermore, the existence of measurements errors. To provide a reliable answer, probabilistic instead of deterministic values should be provided, which carry information about the degree of uncertainty or plausibility of those model parameters providing one or more observations of the system response. This is widely-known as the Bayesian inverse problem, which has been covered in the literature from different perspectives, depending on the interpretation or the meaning assigned to the probability. In this paper, we revise two main approaches: the one that uses probability as logic, and an alternative one that interprets it as information content. The contribution of this paper is to provide an unifying formulation from which both approaches stem as interpretations, and which is more general in the sense of requiring fewer axioms, at the time the formulation and computation is simplified by dropping some constants. An extension to the problem of model class selection is derived, which is particularly simple under the proposed framework. A numerical example is finally given to illustrate the utility and effectiveness of the method.     

A comparative study of prototype and model nailed timber joints

A series of prototype and model nailed timber joints were tested to determine their load-deformation characteristics. The load-slip curves were fitted to three different empirically derived mathematical expressions containing arbitrary constants. The results were correlated with the grading modulus (which is the stiffness, of the boards on the flat measured under centre point loading over a0.915 m span). The results showed that the load-slip behaviour of the nailed joints could be characterised by relatively simple empirical curves based on an exponential function with good agreement being obtained for loads up to half the maximum. Only one arbirary constant in the exponential expression showed reasonable correlation with the grading modulus, the other(s) appear(s) to define the shape of the curve but not its maximum. Good model-prototype similitude can be obtained for either maximum load or for load-slip behaviour, but not both.     

连续排水边界条件下考虑自重的地基一维固结分析

基于连续排水边界条件,建立考虑自重的地基一维固结方程,利用有限正弦Fourier变换方法,给出了解析解,然后通过数值方法获得不排水对称面,分析了界面参数和自重系数对不排水对称面的影响。针对实际工程中水平排水砂垫层铺设位置的优化问题,通过建立考虑土体自重下的固结数学模型,运用半数值半解析方法获得了土体在不同时间因数下设置排水砂垫层的最优位置。分析了水平排水砂垫层设置位置对地基固结度的影响,并进一步给出不同土体界面参数下砂垫层最优铺设位置与时间的关系图。结果表明:考虑土体自重的不排水对称面随时间变化,自重系数越大,不排水对称面越靠近土体底部;最优砂垫层铺设位置是随时间不断变化的。最后,通过一个算例表明考虑优化砂垫层对于固结时间效率的提高具有明显效果。     

On the quasi-static granular convective flow and sand densification around pile foundations under cyclic lateral loading

The saturated sand surrounding an offshore pile foundation under quasi-static cyclic lateral load can show the physical phenomena of macromechanical densification and convective granular flow. Based on the results from physical model tests at different geometrical scales, this paper provides a certain quantification of such phenomena and discusses their causes and consequences. The progressive sand densification leads to subsidence of the soil surface and a significant stiffening of the pile behaviour. Conversely, the ratcheting convective motion of two closed cells of soil beneath the pile-head is responsible for an endless grain migration at the soil surface, the inverse grading of the convected material and a direct shear of the sand at the distinct boundary of the revolving soil domain. In this respect, and from a macromechanical perspective considering the soil as a continuum, it appears that the convecting material tends to follow gradient lines of shear stress during its ratcheting motion. Concluding the paper, the practical relevance of these phenomena and their extrapolation to other conditions are briefly discussed.