Microplane damage model for jointed rock masses

The paper presents a new microplane constitutive model for the inelastic behavior of jointed rock masses that takes into account the mechanical behavior and geometric characteristics of cracks and joints. The basic idea is that the microplane modeling of rock masses under general triaxial loading, including compression, requires the isotropic rock matrix and the joints to be considered as two distinct phases coupled in parallel. A joint continuity factor is defined as a microplane damage variable to represent the stress‐carrying area fraction of the joint phase. Based on the assumption of parallel coupling between the rock joint and the rock matrix, the overall mechanical behavior of the rock is characterized by microplane constitutive laws for the rock matrix and for the rock joints, along with an evolution law for the microplane joint continuity factor. The inelastic response of the rock matrix and the rock joints is controlled on the microplane level by the stress–strain boundaries. Based on the arguments enunciated in developing the new microplane model M7 for concrete, the previously used volumetric–deviatoric splits of the elastic strains and of the tensile boundary are avoided. The boundaries are tensile normal, compressive normal, and shear. The numerical simulations demonstrate satisfactory fits of published triaxial test data on sandstone and on jointed plaster mortar, including quintessential features such as the strain softening and dilatancy under low confining pressure, as well as the brittle–ductile transition under higher confining pressure, and the decrease of jointed rock strength and Young's modulus with an increasing dip angle of the joint. Copyright © 2014 John Wiley & Sons, Ltd.     

A bounding surface plasticity model for unsaturated soil at small strains

Most existing hydromechanical models for unsaturated soils are not able to fully capture the nonlinearity of stress–strain curves at small strains (less than 1%). They cannot therefore, for example, accurately predict ground movements and the performance of many earth structures under working conditions. To tackle this problem, a state‐dependent bounding surface plasticity model has been newly developed. Particularly, the degradation of shear modulus with strain at small strains ranging from 0.001% to 1% is focused. The proposed model is formulated in terms of mean average skeleton stress, deviator stress, suction, specific volume and degree of saturation. Void ratio‐dependent hydraulic hysteresis is coupled with the stress–strain behaviour. Different from other elastoplastic models for unsaturated soils, plastic strains are allowed inside bounding surfaces. In this paper, details of model formulations and calibration procedures of model parameters are presented. To evaluate the capability of the new model, it is applied to simulate a series of triaxial compression tests on compacted unsaturated silt at various suctions. Effects of suction, drying and wetting as well as net stress on unsaturated soil behaviour are well captured. The model shows good predictions of the degradation of shear modulus with strain over a wide range of strains from 0.001% to 1%. Copyright © 2015 John Wiley & Sons, Ltd.     

Development of a Nonlinear Model for Soft Rock and its Applications

A phenomenological model has been developed for soft rock based on the results of a series of triaxial compression (TC) tests conducted on Kobe sandstone with a very high precision measurement. From the analysis and interpretation of the test results, it has been found that small strain Young’s modulus (E^e) was a function of the major principal stress. E^e for elastic strains of soft rock was assumed to be cross-anisotropic. A damage function has been used to derive the appropriate elastic Young’s modulus when subjected to shear loading. As the basic stress–strain relation, the relationship between the tangent modulus and the shear stress level was used. The differential form of which was subsequently integrated by a 4th order Runge–Kutta solver to obtain the stress–strain relation. The model of soft rock is based on an isotropic strain hardening elasto-plastic framework which takes into account the pressure sensitivity, cross-anisotropy, degradation of Young’s modulus with the degree of mobilized shear stress and the nonlinearity of the shear stress-shear strain relationship. Although the model was developed from the analysis of the TC tests results of Kobe sandstone, it was also applied to the other types of soft rock or stiff geomaterials. Plate loading tests were conducted at a level of 61 m below the ground level at the bottom of a large excavated shaft at four locations. Finally, the model was used to simulate the plate loading test results successfully. This model was successfully calibrated with Akashi sandstone and applied in the simulation for the settlement of Akashi-Kaikyo Bridge piers. The simulations were carried out for both drained and undrained condition by changing the Poisson’s ratio. The layering information beneath the foundations were used in the FEM simulation. The use of very accurate Young’s modulus from the field shear wave velocity test was the key to the successful simulation of the settlement under bridge pier foundations.     

A unified bounding surface plasticity model for unsaturated soils

A unified constitutive model for unsaturated soils is presented in a critical state framework using the concepts of effective stress and bounding surface plasticity theory. Consideration is given to the effects of unsaturation and particle crushing in the definition of the critical state. A simple isotropic elastic rule is adopted. A loading surface and a bounding surface of the same shape are defined using simple and versatile functions. The bounding surface and elastic rules lead to the existence of a limiting isotropic compression line, towards which the stress trajectories of all isotropic compression load paths approach. A non‐associated flow rule of the same general form is assumed for all soil types. Isotropic hardening/softening occurs due to changes in plastic volumetric strains as well as suction for some unsaturated soils, enabling the phenomenon of volumetric collapse upon wetting to be accounted for. The model is used to simulate the stress–strain behaviour observed in unsaturated speswhite kaolin subjected to three triaxial test load paths. The fit between simulation and experiment is improved compared to that of other constitutive models developed using conventional Cam‐Clay‐based plasticity theory and calibrated using the same set of data. Also, the model is used to simulate to a high degree of accuracy the stress–strain behaviour observed in unsaturated Kurnell sand subjected to two triaxial test load paths and the oedometric compression load path. For oedometric compression theoretical simulations indicate that the suction was not sufficiently large to cause samples to separate from the confining ring. Copyright © 2005 John Wiley & Sons, Ltd.     

Mechanical Behaviour of Reservoir Rock Under Brine Saturation

Acoustic emissions (AE) and stress–strain curve analysis are well accepted ways of analysing crack propagation and monitoring the various failure stages (such as crack closure, crack initiation level during rock failure under compression) of rocks and rock-like materials. This paper presents details and results of experimental investigations conducted for characterizing the brittle failure processes induced in a rock due to monocyclic uniaxial compression on loading of two types of sandstone core samples saturated in NaCl brines of varying concentration (0, 2, 5, 10 and 15 % NaCl by weight). The two types of sandstone samples were saturated under vacuum for more than 45 days with the respective pore fluid to allow them to interact with the rocks. It was observed that the uniaxial compressive strength and stress–strain behaviour of the rock specimens changed with increasing NaCl concentration in the saturating fluid. The acoustic emission patterns also varied considerably for increasing ionic strength of the saturating brines. These observations can be attributed to the deposition of NaCl crystals in the rock’s pore spaces as well some minor geo-chemical interactions between the rock minerals and the brine. The AE pattern variations could also be partly related to the higher conductivity of the ionic strength of the high-NaCl concentration brine as it is able to transfer more acoustic energy from the cracks to the AE sensors.     

An elastoplastic constitutive model for frozen saline coarse sandy soil undergoing particle breakage

The mechanical property of frozen saline sandy soil is complicated due to its complex components and sensitivity to salt content and confining pressure. Thus, a series of triaxial compression tests were carried out on sandy samples with different Na₂SO₄ contents under different confining pressures to explore the effects of particle breakage, pressure melting, shear dilation and strain softening or hardening. The test results indicate that the stress–strain curves exhibit strain softening/hardening phenomena when the confining pressures are below or above 6 MPa, respectively. A shear dilation phenomenon was observed in the loading process. With increasing confining pressure, the strength firstly increases and then decreases. By taking into consideration the changes between the grain size distributions before and after triaxial compression tests, a failure strength line incorporating the influences of both particle breakage and pressure melting is proposed. In order to describe the deformation characteristics of frozen saline sandy soil, an elastoplastic incremental constitutive model is established based on the test results. The proposed model considers the plastic compressive, plastic shear and breakage mechanisms by adopting the non-associated flow rule. The breakage mechanism can be reflected by an index related to the initial, current and ultimate grain size distributions. The hardening parameters corresponding to compressive and shear mechanisms consider the influence of particle breakage. Then the effect of particle breakage on both the stress–strain and volumetric strain curves is analyzed. The calculated results fit well with the test results, indicating that the developed constitutive model can well describe the mechanical and deformation features of frozen saline sandy soil under various stress levels and stress paths. In addition, the strain softening/hardening, contraction, high dilation and particle breakage can be well captured.

     

Capturing the complete stress–strain behaviour of jointed rock using a numerical approach

This paper presents the results of a series of numerical experiments using the synthetic rock mass (SRM) approach to quantify the behaviour of jointed rock masses. Field data from a massive sulphide rock mass, at the Brunswick mine, were used to develop a discrete fracture network (DFN). The constructed DFN model was subsequently subjected to random sampling whereby 40 cubic samples, of height to width ratio of two, and of varying widths (0.05 to 10 m) were isolated. The discrete fracture samples were linked to 3D bonded particle models to generate representative SRM models for each sample size. This approach simulated the jointed rock mass as an assembly of fractures embedded into the rock matrix. The SRM samples were submitted to uniaxial loading, and the complete stress–strain behaviour of each specimen was recorded. This approach provided a way to determine the complex constitutive behaviour of large‐scale rock mass samples. This is often difficult or not possible to achieve in the laboratory. The numerical experiments suggested that higher post‐peak modulus values were obtained for smaller samples and lower values for larger sample sizes. Furthermore, the observed deviation of the recorded post‐peak modulus values decreased with sample size. The ratio of residual strength of rock mass samples per uniaxial compressive strength intact increases moderately with sample size. Consequently, for the investigated massive sulphide rock mass, the pre‐peak and post‐peak representative elemental volume size was found to be the same (7 × 7 × 14 m). Copyright © 2015 John Wiley & Sons, Ltd.     

Viscoelastic damage model for asphalt concrete

The strain rate-dependent mechanical behavior of asphalt concrete was characterized using unconfined compression tests carried out at different loading rates. It was shown that at high strain rates, the elastic deformation and peak axial stress are highly sensitive to strain rate. Both increase as the strain rate increases. At very low strain rates, elastic response and unconfined compressive strength are relatively independent of the loading rate. Based on the experimental observations, a simple viscoelastic damage model is proposed for the strain rate-dependent unconfined compression behavior of asphalt concrete. In the model, strain rate response is modeled by a two-component viscoelastic model consisting of a constant elastic modulus and a viscous modulus that is related by a power-law function to the axial strain rate. Failure and strain softening are modeled via a damage formulation where damage evolution in the asphalt concrete is given by a simple form of the Weibull distribution function. The model was shown to be capable of describing the strain rate-dependent deformation, compressive strength, strain-softening and creep behavior of asphalt concrete. The model is relatively simple and requires only five material parameters.     

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.     

赤泥改性黄土的基本工程性质研究

为探究赤泥改性黄土的基本工程特性,进行无侧限抗压试验、直剪试验及渗透试验,得出赤泥改性黄土的最佳配比,建立了抗剪强度、无侧限抗压强度与电阻率的经验公式。通过动三轴试验,分析了循环动荷载下改性黄土的累积塑性变形、动弹性模量及浸出毒性。结果表明,随赤泥含量增加,改性黄土无侧限抗压强度和黏聚力都先增大后减小,当赤泥含量为15%时其28 d无侧限抗压强度达到3.5 MPa,较不掺入赤泥时提高约34.7%;改性黄土的渗透系数随赤泥含量增加先快速降低,至赤泥含量5%后趋于稳定;改性黄土的抗剪强度、无侧限抗压强度与电阻率呈良好的线性关系;改性黄土在循环动荷载下的临界动应力达600 kPa,最大动弹性模量比未改性黄土高出6倍,且其浸出液不具有浸出毒性。     

Kinematic hardening of soft clay in simple shear

In a separate paper, the authors have proposed a normalized, non-degrading form of the shear stress–shear strain relationship for undrained, cyclic simple shear of soft clay. This relationship is described in the present paper, and it is seen to include a single fatigue parameter—the mean effective stress. Application of the relationship therefore requires knowledge of the history of the mean effective stress during any loading history. The present paper proposes an effective stress path model which may be used for prediction of this history. The model is developed within the framework of bounding surface kinematic and isotropic hardening plasticity. It incorporates an isotropic hardening bounding surface, and a kinematic hardening yield surface, in which the elastic region vanishes, and so the yield surface reduces to the stress point. The normalized shear stress–shear strain relationship, developed on the basis of Iwan's model, is used to establish the shape of the cap of the bounding surface. A new translation rule is also incorporated in the model, allowing improved prediction of stress path development within the bounding surface during regular or irregular cyclic loading. Use of the proposed model to simulate the behaviour of soft clay in laboratory undrained cyclic simple shear tests shows excellent qualitative agreement, with most of the major features of the actual behaviour being predicted.     

循环载荷下含水砂岩裂纹演化与损伤特征试验研究

基于矿井地下水库岩体频繁受到矿震、采动应力等循环荷载扰动这一工程背景,在实验室条件下开展了不同含水率砂岩单轴及循环加卸载试验,采用数字散斑技术,揭示了不同含水率砂岩裂纹扩展及破坏规律,基于电镜扫描微观分析,获得了循环载荷下不同含水砂岩微观劣化机制。试验结果表明：单轴及循环加卸载条件下,随着含水率增大,砂岩峰值强度均逐渐降低。干燥砂岩峰值轴向应变经历初始变形、等速变形、加速变形和失稳破坏4个阶段,含水砂岩经历初始变形、等速变形和失稳破坏3个阶段;随着含水率的增大,对应阶段内峰值轴向应变逐渐减小。通过变形率分析法验证发现,水对砂岩的变形记忆特性并无影响。单轴循环加卸载条件下,砂岩的破坏模式由干燥时的张拉−劈裂破坏逐渐向拉−剪混合破坏过渡,至饱和状态时呈现单一的剪切破坏。电镜扫描结果表明,随含水率增大,破裂结构面逐渐由光滑结构、浑圆状结构、片状结构向完全破碎结构过渡。随着含水率增大,绝对损伤参数不断增大,从侧面反映水岩耦合损伤的正相关性;累计损伤参数累积速率更快,同一循环周次下,累计损伤参数也更大。     

Mechanical behaviors of conjugate-flawed rocks subjected to coupled static–dynamic compression

Conjugate flaws widely exist in rock masses and play a significant role in their deformation and strength properties. Understanding the mechanical behaviors of rock masses containing conjugate flaws is conducive to rock engineering stability assessment and the related supporting design. This study experimentally investigates the mechanical properties of conjugate-flawed sandstone specimens under coupled static–dynamic compression, thereby providing insight into how conjugate fractures interact to produce tracing tensional joints. Results indicate that the coupled compressive strength and the dynamic elastic modulus of conjugate-flawed rock specimens show remarkable loading rate dependence. For a fixed strain rate, the specimen with a static pre-stress equal to 60% of its uniaxial compressive strength has the highest coupled strength. Besides, both higher static pre-stress and strain rate can induce smaller mean fragment size and greater fractal dimension of the specimen, corresponding to a more uniform distribution of the broken fragments with smaller sizes. When the static pre-stress is lower than 80%UCS, the flawed specimen under a higher strain rate is characterized by higher absorbed energy. However, when the pre-stress equals 80%UCS, the value of the energy absorbed by the specimen in the dynamic loading process is negative due to the release of the preexisting considerable elastic strain energy input from the static pre-loading. As for the failure modes, cracks always penetrate the preexisting ipsilateral flaw tips to form anti-wing cracks. Under dynamic loading, the conjugate-flawed specimen generally shows tensile failure at a low strain rate, while the shear failure dominates at a high strain rate. In addition, based on progressive failure processes of the conjugate-flawed rock specimens, the evolution of tracing tensional joints in the field is discussed.

     

A 1D Hypoplastic constitutive model for expansive soils

This paper presents a simple hypoplastic constitutive model that describes the essential features of the material behaviour of partially saturated clayey soils observed in oedometric compression tests. The model is formulated in terms of net stress and degree of saturation. The total strain rate is decomposed into a portion related to the changes in saturation and a portion for the evolution of net stress. However, no distinction is made between plastic and elastic strains. With this strain rate decomposition, the maximum swelling strain/stress are obtained by simulating wetting processes under constant stress/strain conditions. In addition to the void ratio, the model includes two scalar variables to track the loading history (preloading). The calibration of the model constants using common laboratory tests is discussed. Confined and unconfined swelling tests under oedometric conditions with subsequent loading and unloading phases carried out on three different materials were satisfactorily simulated by the model. Its promising results call for an extension to a 3D formulation.

     

岩石裂纹损伤及围压对剪切破坏影响研究

岩石内部细观裂纹的存在,对压缩作用下岩石剪切断裂的宏观现象有着重要的影响。然而,能够通过解析解阐释细观裂纹几何特性、围压等影响因素对压缩作用下剪切断裂面角度变化趋势的研究很少。基于Ashby模型中提出的裂纹尖端应力强度因子,提出了一种改进的考虑裂纹角度影响的应力强度因子表达式。利用该改进的应力强度因子表达式,推出了一个可以预测岩石峰值强度的裂纹扩展、应变与应力之间的本构关系。结合本构关系的峰值强度与摩尔-库仑失效准则,得到了岩石损伤与内摩擦角、黏聚力、剪切强度及失效断裂面角度之间的理论关系;讨论了围压、裂纹尺寸、角度及摩擦系数对岩石宏观剪切断裂面角度的影响,通过试验结果验证了模型合理性。结果表明：随着损伤增大,内摩擦角、黏聚力及剪切强度不断减小;随着围压增大、摩擦系数增大和初始裂纹尺寸减小,剪切断裂面角度不断增大;随着裂纹角度增大,剪切断裂纹面角度先减小后增大。     

基于考虑动力本构的连续−非连续方法的单轴压缩岩样开裂过程模拟

动载作用下岩石的破坏规律研究对于众多地质灾害的机制分析和预防具有重要的理论及实际意义。鉴于数值模拟研究的优势,应大力发展适于岩石动力断裂过程模拟的数值方法。在自主开发的拉格朗日元与离散元耦合连续−非连续方法的基础上,采用朱−王−唐本构模型取代了广义胡克定律,发展了考虑动力本构的连续−非连续方法,其正确性通过模拟不同加载速度时砂岩试样的单轴压缩试验进行了验证。通过统计裂缝区段数目随着岩样的纵向应变的演化规律,并监测岩样左、右对称线上多个测点的最小主应力的演化规律,开展了不同加载速度时单轴压缩花岗岩试样的变形−开裂过程研究,阐明了岩样的开裂机制。研究发现,剪裂缝以雁列式展布,整体上形成剪切带。随着时步数目的增加,各测点的最小主应力均呈波动下降−震荡上升的变化趋势。震荡上升阶段对应岩样的应变软化阶段。测点分离后最小主应力的震荡幅度较大,这是由于节点分离和单元接触激发了较大的应力波。剪切带尖端的最小主应力集中会使测点发生剪切分离。当岩样的三角块向下楔入时,下方测点的应力状态类似于紧凑拉伸试验进而发生拉伸分离。     

单轴压缩条件下含三叉裂隙类岩石试样力学特性的细观研究

三叉裂隙是自然界普遍存在的一种岩体缺陷形式,其对岩体的力学特性有重要影响。对含预制三叉裂隙的水泥砂浆试样进行室内单轴压缩试验,配合使用摄像机拍摄裂纹的起裂、扩展、贯通过程,通过数字图像技术处理获取试样的应变场云图,并结合PFC2D程序研究不同?、? 条件下试样的强度特征、裂纹模式和裂纹演化扩展规律。研究表明：三叉裂隙对试样单轴抗压强度有明显的削弱作用。当? 恒定为120°时,试样在? = 30°时达到最大抗压强度;当? 恒定为90°时,随?增大,试样抗压强度呈先减小后增大的趋势,且当? = 45°时达到最大抗压强度。试样产生的裂纹可分为3类,分别是张拉型裂纹(Ⅰ型裂纹)、剪切型裂纹(Ⅱ型裂纹)、混合型裂纹(Ⅲ型裂纹)。这3类裂纹通常从裂隙尖端开始产生,并且Ⅰ型裂纹沿加载方向扩展,通常未扩展至试样边界;Ⅱ型和Ⅲ型裂纹通常与加载方向呈一定角度扩展至试样边界。通过对裂纹的几何形态和组成宏观裂纹的微裂纹成分的分析,得知导致含三叉裂隙试样在单轴压缩条件下失效的是张拉破坏。数字图像技术得到的应变云图表明,当载荷达到一定阶段,裂隙尖端出现应力集中,微破裂开始发育并聚集成微破裂区,微破裂区扩大产生宏观裂纹。通过对主应变和剪应变云图分析,发现导致试样失效的是张拉破坏,剪应变在裂纹扩展过程中的影响较小。     

Fibre reinforced sands: from experiments to modelling and beyond

Based on hypotheses derived directly from experimental observations of the triaxial behaviour, a constitutive model for fibre reinforced sands is built in this paper. Both the sand matrix and the fibres obey their own constitutive law, whereas their contributions are superimposed using a volumetric homogenization procedure. The Severn‐Trent sand model, which combines well‐known concepts such as critical state theory, Mohr‐Coulomb like strength criterion, bounding surface plasticity and kinematic hardening, is adopted for the sand matrix. Although the fibres are treated as discrete forces with defined orientation, an equivalent continuum stress for the fibre phase is derived to allow the superposition of effects of sand and fibres. The fibres are considered as purely tensile elements following a linear elastic constitutive rule. The strain in the fibres is expressed as a fraction of the strain in the reinforced sample so that imperfect bonding is assumed at the sand‐fibre interface. Only those fibres oriented within the tensile strain domain of the sample can mobilize tensile stress—the orientation of fibres is one of the key ingredients to capture the anisotropic behaviour of fibre reinforced soil that is observed for triaxial compression and extension loading. A further mechanism of partition of the volume of voids between the fibres and the sand matrix is introduced and shown to be fundamental for the simulation of the volumetric behaviour of fibre‐reinforced soils. Copyright © 2012 John Wiley & Sons, Ltd.     

Microplane constitutive model for porous isotropic rocks

The paper deals with constitutive modelling of contiguous rock located between rock joints. A fully explicit kinematically constrained microplane‐type constitutive model for hardening and softening non‐linear triaxial behaviour of isotropic porous rock is developed. The microplane framework, in which the constitutive relation is expressed in terms of stress and strain vectors rather than tensors, makes it possible to model various microstructural physical mechanisms associated with oriented internal surfaces, such as cracking, slip, friction and splitting of a particular orientation. Formulation of the constitutive relation is facilitated by the fact that it is decoupled from the tensorial invariance restrictions, which are satisfied automatically. In its basic features, the present model is similar to the recently developed microplane model M4 for concrete, but there are significant improvements and modifications. They include a realistic simulation of (1) the effects of pore collapse on the volume changes during triaxial loading and on the reduction of frictional strength, (2) recovery of frictional strength during shearing, and (3) the shear‐enhanced compaction in triaxial tests, manifested by a deviation from the hydrostatic stress–strain curve. The model is calibrated by optimal fitting of extensive triaxial test data for Salem limestone, and good fits are demonstrated. Although these data do not cover the entire range of behaviour, credence in broad capabilities of the model is lend by its similarity to model M4 for concrete—an artificial rock. The model is intended for large explicit finite‐element programs. Copyright © 2002 John Wiley & Sons, Ltd.