A Bounding Surface Plasticity Model for Intact Rock Exhibiting Size-Dependent Behaviour

A new constitutive model for intact rock is presented recognising that rock strength, stiffness and stress–strain behaviour are affected by the size of the rock being subjected to loading. The model is formulated using bounding surface plasticity theory. It is validated against a new and extensive set of unconfined compression and triaxial compression test results for Gosford sandstone. The samples tested had diameters ranging from 19 to 145 mm and length-to-diameter ratios of 2. The model captures the continuous nonlinear stress–strain behaviour from initial loading, through peak strength to large shear strains, including transition from brittle to ductile behaviour. The size dependency was accounted for through a unified size effect law applied to the unconfined compressive strength—a key model input parameter. The unconfined compressive strength increases with sample size before peaking and then decreasing with further increasing sample size. Inside the constitutive model two hardening laws act simultaneously, each driven by plastic shear strains. The elasticity is stress level dependent. Simple linear loading and bounding surfaces are adopted, defined using the Mohr–Coulomb criterion, along with a non-associated flow rule. The model simulates well the stress–strain behaviour of Gosford sandstone at confining pressures ranging from 0 to 30 MPa for the variety of sample sizes considered.     

Influence of Lode angle‐dependent failure criteria on shear localization analysis in sand

Geotechnical experiments show that Lode angle‐dependent constitutive formulations are appropriate to describe the failure of geomaterials. In the present study, we have adopted one such class of failure criteria along with a versatile constitutive relationship to theoretically analyze the effects of Lode angle on localized shear deformation or shear band formation in loose sand for both drained and undrained conditions. We determine the variation in the possible stress states for shear localization due to the introduction of Lode angle by considering the localized deformation as a bifurcation problem. Further, similar bifurcation analysis is performed for the stress states along a specific loading path, namely, plane strain compression at the constitutive level. In addition, the plane strain compression tests have been simulated as a boundary value finite element problem to see how Lode angle affects the post‐localization response. Results show that the inclusion of a Lode angle parameter within the failure criterion has considerable effects on the onset, plastic strain, and propagation of shear localization in loose sand specimens. For drained condition, we notice early inception of shear localization and multiple band formation when the Lode angle‐dependent failure criterion is used. Undrained localization characteristics, however, found to be independent of Lode angle consideration.     

Discrete element method modeling of inherently anisotropic rocks under uniaxial compression loading

A new numerical approach is proposed in this study to model the mechanical behaviors of inherently anisotropic rocks in which the rock matrix is represented as bonded particle model, and the intrinsic anisotropy is imposed by replacing any parallel bonds dipping within a certain angle range with smooth‐joint contacts. A series of numerical models with β = 0°, 15°, 30°, 45°, 60°, 75°, and 90° are constructed and tested (β is defined as the angle between the normal of weak layers and the maximum principal stress direction). The effect of smooth‐joint parameters on the uniaxial compression strength and Young's modulus is investigated systematically. The simulation results reveal that the normal strength of smooth‐joint mainly affects the behaviors at high anisotropy angles (β > 45°), while the shear strength plays an important role at medium anisotropy angles (30°–75°). The normal stiffness controls the mechanical behaviors at low anisotropy angles. The angle range of parallel bonds being replaced plays an important role on defining the degree of anisotropy. Step‐by‐step procedures for the calibration of micro parameters are recommended. The numerical model is calibrated to reproduce the behaviors of different anisotropic rocks. Detailed analyses are conducted to investigate the brittle failure process by looking at stress‐strain behaviors, increment of micro cracks, initiation and propagation of fractures. Most of these responses agree well with previous experimental findings and can provide new insights into the micro mechanisms related to the anisotropic deformation and failure behaviors. The numerical approach is then applied to simulate the stress‐induced borehole breakouts in anisotropic rock formations at reduced scale. The effect of rock anisotropy and stress anisotropy can be captured. Copyright © 2015 John Wiley & Sons, Ltd.     

Modeling of elastic anisotropy due to one-dimensional plastic consolidation of clays

Experiments indicate that in one-dimensionally consolidated natural clays the elastic anisotropy is much stronger than the plastic strain anisotropy. Moreover, the elastic anisotropy appears to be dependent on the pre-consolidation strain. Coupled elasto-plastic constitutive law is shown to be able to simulate these anisotropy effects of natural clay deposits. In this law the elastic potential is not only a function of stress, but additionly of the plastic strain. The plastic strain comprises the geological process of pre-consolidation idealized as an one-dimensional plastic straining as well as a mechanically induced strain due to engineering activity. Calibration of the model and simulation of some stress paths are presented and related to the classical experimental results by Mitchell (1972).     

Simulation of drilling‐induced compaction bands using discrete element method

Rock failure is observed around boreholes often with certain types of failure zones, which are called breakouts. Laboratory‐scale drilling tests in some high‐porosity quartz‐rich sandstone have shown breakouts in the form of narrow localized compacted zones in the minimum horizontal stress direction. They are called fracture‐like breakouts. Such compaction bands may affect hydrocarbon extraction by forming barriers that inhibit fluid flow and may also be a source of sand production. This paper presents the results of numerical simulations of borehole breakouts using 3D discrete element method to investigate the mechanism of the fracture‐like breakouts and to identify the role of far‐field stresses on the breakout dimensions. The numerical tool was first verified against analytical solutions. It was then utilized to investigate the failure mechanism and breakout geometry for drilled cubic rock samples of Castlegate sandstone subjected to different pre‐existing far‐field stresses. Results show that failure occurs in the zones of the highest concentration of tangential stress around the borehole. It is concluded that fracture‐like breakout develops as a result of a nondilatant failure mechanism consisting of localized grain debonding and repacking and grain crushing that lead to the formation of a compaction band in the minimum horizontal stress direction. In addition, it is found that the length of fracture‐like breakouts depends on both the mean stress and stress anisotropy. However, the width of the breakout is not significantly changed by the far‐field stresses. Copyright © 2013 John Wiley & Sons, Ltd.     

基于微观力学机制的各向异性结构性砂土的本构模型研究

在岩土破损力学基础上,基于微观破损机制,提出了考虑各向异性的结构性砂土本构理论。采用Lade-Duncan强度准则考虑中主应力对抗剪强度的影响;采用考虑颗粒排列组构的各向异性状态变量A反映各向异性对土体强度和变形的影响;通过相似扩大重塑土的屈服面反映结构性对土性的影响;通过引入非相关联流动法则考虑各向异性和结构性对土体塑性变形的影响。同时,将基于微观力学机制的损伤演化规律引入结构性土的硬化规律;该硬化规律同时考虑了塑性体积应变和剪切应变对各向异性结构性土强度的影响。然后将该模型用于模拟室内三轴压缩试验,初步验证了该模型的合理性和适用性。     

基于Cosserat连续体的颗粒破碎影响研究

颗粒破碎对颗粒材料宏观力学行为有重要影响。 结合Hardin的破碎经验公式,将表征破碎程度的破碎参量与Cosserat连续体的内部长度参数相关联,形成一个基于Cosserat连续体且能考虑颗粒破碎的弹塑性模型。数值算例主要考察了颗粒破碎对颗粒材料承载能力、塑性应变及局部化行为的影响,数值结果表明,颗粒破碎主要发生在剪切带内,颗粒破碎使得剪切带明显变窄且剪切带内外等效塑性应变梯度明显增大。     

位移控制加载、内压及侧压作用下平面应变含孔洞土样应变场观测

利用研制的平面应变模型加载及观测系统,研究了位移控制加载条件下含孔洞土样的破坏过程。孔洞内压及侧压由气囊施加。利用数字图像相关方法,获得应变场。通过布置测线的方式,研究了剪切带内外最大剪切应变及主应变轴偏转角的分布及演变规律。得到了下列结果。随着纵向应变的增加,含孔洞土样的最大剪切应变的分布经历由均匀变形向局部化变形的转化过程。对于剪切带外的测点,随着远离孔洞表面,最大剪切应变有增加的趋势,这与剪切带内损伤导致带外弹性应变降低（卸荷）有关。在孔洞表面附近,最大剪切应变变化复杂,这与应变弹性成分的下降与塑性成分的增加的博弈有关。对于剪切带尖端的测点,其和前后测点相比,水平线应变发生突增（剧烈膨胀）,垂直线应变的值发生突降（挤压程度剧烈降低）,剪切应变的值发生突增（剪切变形剧烈增加）,由此导致主应变轴偏转角的值发生突增,达到30°。当纵向应变相同时,侧压或内压的增加对含孔洞土样的纵向应力有增强作用。     

考虑塑性应变率梯度的单轴压缩岩样轴向响应

基于剪切应变率梯度格式,采用解析方式研究了岩石材料在单轴压缩条件下的应变软化的结构响应。根据非局部连续介质模型,提出了一维二阶剪切应变率梯度格式。非局部剪切应变率与局部剪切应变率及其二阶梯度有关。将经典塑性理论中的局部剪切应变率替换为非局部剪切应变率,可以直接得到局部剪切应变率的封闭解析解,而不必通过将局部剪切应变对时间求导获得。通过对局部剪切应变率积分,得到了沿剪切带方向的相对剪切速度。试件峰值强度后的端部速度由弹性及塑性两部分构成。前一部分由虎克定律描述;后一部分与相对剪切速度有关。对弹性及塑性两部分速度求和,得到了单轴压缩岩样剪切破坏问题轴向响应的解析式。研究表明：试样高度越大、内部长度越小、剪切软化模量越大及泊松比越小,则岩样的轴向响应倾向于脆性。根据岩样与矿柱的相似性,岩样响应倾向于脆性,意味着矿柱将失去稳定性,发生矿柱岩爆。目前的基于剪切应变率梯度格式的主要优点是简洁。     

On double shearing in frictional materials

This paper evaluates the mechanical behaviour of yielding frictional geomaterials. The general Double Shearing model describes this behaviour. Non‐coaxiality of stress and plastic strain increments for plane strain conditions forms an important part of this model. The model is based on a micro‐mechanical and macro‐mechanical formulation. The stress–dilatancy theory in the model combines the mechanical behaviour on both scales. It is shown that the general Double Shearing formulation comprises other Double Shearing models. These models differ in the relation between the mobilized friction and dilatancy and in non‐coaxiality. In order to describe reversible and irreversible deformations the general Double Shearing model is extended with elasticity. The failure of soil masses is controlled by shear mechanisms. These shear mechanisms are determined by the conditions along the shear band. The shear stress ratio of a shear band depends on the orientation of the stress in the shear band. There is a difference between the peak strength and the residual strength in the shear band. While peak stress depends on strength properties only, the residual strength depends upon the yield conditions and the plastic deformation mechanisms and is generally considerably lower than the maximum strength. It is shown that non‐coaxial models give non‐unique solutions for the shear stress ratio on the shear band. The Double Shearing model is applied to various failure problems of soils such as the direct simple shear test, the biaxial test, infinite slopes, interfaces and for the calculation of the undrained shear strength. Copyright © 2006 John Wiley & Sons, Ltd.     

AUS: Anisotropic undrained shear strength model for clays

A total stress model applicable to clays under undrained conditions is presented. The model involves three strength parameters: the undrained shear strengths in triaxial compression, triaxial extension, and simple shear. The amount of physical anisotropy implied by the model is a function of the relative magnitude of these three strengths assuming a Mises-type plastic potential. Elastoplastic deformation characteristics below failure are accounted for by a hardening law requiring two additional parameters that can be related to the axial strains halfway to failure in triaxial compression and extension. Finally, elasticity is accounted for by Hooke law. The result is a relatively simple model whose parameters can all be inferred directly from a combination of in situ and standard undrained laboratory tests. The model is applied to a problem involving the horizontal loading of a monopile foundation for which full scale tests have been previously conducted. The model shows good agreement with the measured data.     

Mesoscale numerical study on the evolution of borehole breakout in heterogeneous rocks

Inherent heterogeneity of a rock strongly affects its mechanical behavior. We numerically study the mechanisms governing the initiation, propagation, and ultimate pattern of borehole breakouts in heterogeneous rocks. A two-dimensional finite element model incorporating material heterogeneity is established to systematically examine the effects of several key factors on borehole failure, including borehole diameter, far-field stress, and rock heterogeneity. The inherent heterogeneity of a rock is explicitly characterized by prescribing the rock mechanical properties of mesoscale elements statistically obeying the Weibull distribution. Elastic damage mechanics is used to represent the constitutive law of the mesoscale element. We find that borehole diameter reduction remarkably changes the crack failure from tensile to shear and elevates the critical hydrostatic pressure. Far-field stress anisotropy strongly affects the shape of the borehole breakout. Rock heterogeneity dictates the location of the preferred crack under the hydrostatic stress, which leads to local stress concentration, and determines the types of breakouts around the borehole. Our findings facilitate in-depth understanding of the classic borehole stability problems in heterogeneous rocks.     

Gradient plasticity modelling of strain localization in granular materials

The flow stress in the yield surface of plastic constitutive equation is modified with a higher order gradient term of the effective plastic strain to model the effect of inhomogeneous deformation in granular materials. The gradient constitutive model has been incorporated into the finite element code ABAQUS and used to simulate biaxial shear tests on dry sand. It is shown that the shape of the post-peak segment of the load displacement curve predicted by the numerical analysis is dependent on the mesh size when gradient term is not used. Use of an appropriate gradient coefficient is shown to correct this and predict a unique shape of the load displacement curve regardless of the mesh size. The gradient coefficient required turns out to be approximately inversely proportional to the mesh elemental area. Use of the strain gradient term is found to diffuse the concentration of plastic strains within shear band resulting in its consistent width. The coefficient of the higher gradient term appears as a function of the grain size, the mean confining stress, and the plastic softening modulus. Copyright © 2004 John Wiley & Sons, Ltd.     

Stress‐dependent elastic wave velocity of microfractured sandstone

A model for the stress‐dependent elastic wave velocity response of fractured rock mass is proposed based on experimental evidence of stress‐dependent fracture normal and shear stiffness. Previously proposed models and previous experimental studies on stress‐dependent fracture stiffness have been reviewed to provide a basis for the new model. Most of the existing stress‐dependent elastic wave velocity models are empirical, with model parameters that do not have clear physical meanings. To propose the new model, the rock mass is assumed to have randomly oriented microscopic fractures. In addition, the characteristic length of microfractures is assumed to be sufficiently short compared to the rock mass dimensions. The macroscopic stress‐dependent elastic wave velocity response is assumed to be attributed to the stress dependency of fracture stiffness. The stress‐dependent fracture normal stiffness is defined as a generalized power law function of effective normal stress, which is a modification of the Goodman's model. On the other hand, the stress dependency of fracture shear stiffness is modeled as a linear function of normal stress based on experimental data. Ultrasonic wave velocity responses of a dry core sample of Berea sandstone were tested at effective stresses ranging from 2 to 55 MPa. Visual observation of thin sections obtained from the Berea sandstone confirms that the assumptions made for microstructure of rock mass model are appropriate. It is shown that the model can describe the stress‐dependent ultrasonic wave velocity responses of dry Berea sandstone with a set of reasonable material parameter values. Published 2013. This article is a U.S. Government work and is in the public domain in the USA.     

Delayed plastic model for time‐dependent behaviour of materials

A delayed plastic model, based on the theory of plasticity, is proposed to represent the time‐dependent behaviour of materials. It is assumed in this model that the stress can lie outside the yield surface and the conjugate stress called static stress is defined on the yield surface. The stress–strain relation is calculated based on the plastic theory embedding the static stress. Thus, the stress–strain relation of the model practically corresponds to that of the inviscid elastoplastic model under fairly low rate deformation. The delayed plastic model is coupled with the Cam‐clay model for normally consolidated clays. The performance of the model is then examined by comparing the model predictions with reported time‐dependent behaviour of clays under undrained triaxial conditions. It is shown that the model is capable of predicting the effect of strain rate during undrained shear and the undrained creep behaviour including creep rupture. Copyright © 2001 John Wiley & Sons, Ltd.     

高应力卸荷条件下砂岩扩容特征及其剪胀角函数

为研究高应力水平条件下岩石卸荷扩容特性及其剪胀角变化规律,开展了若干不同初始围压水平的三轴峰前卸围压试验,同时进行了相应围压水平的常规三轴压缩试验。基于试验成果分析了卸荷应力路径对砂岩扩容的影响效应,总结了卸荷应力路径下剪胀角变化规律,进而提出了基于卸荷应力路径的剪胀角函数,并给出了其数值实现方法,最后通过对三轴峰前卸围压试验的数值模拟,验证了所提出的剪涨角函数的可行性与合理性。研究表明：（1）加载路径时,不同围压水平下峰前扩容体积应变值相差不大,而卸荷应力路径时,随着围压水平的增加,峰前扩容体积应变从最小值3.15×10?3增加到最大值9.65×10?3,卸荷路径下的峰前扩容体积应变约为加载路径的1.1～4.0倍;（2）卸荷应力路径下偏应力峰值对应的体积应变基本为0或接近于0,而加载应力路径下偏应力峰值对应的体积应变均为负值;（3）卸荷应力路径条件下峰前扩容体积应变在总扩容体积应变中所占比例显著大于加载路径;（4）两种应力路径下,剪胀角随塑性剪切应变增加均经历了先增加后减小的过程,并且低围压工况下偏应力峰值对应的剪胀角和最大剪胀角均大于高围压工况;（5）与加载路径相比,卸荷路径下的剪胀角更快达到剪胀角最大值,并且其偏应力峰值对应的剪胀角和最大剪胀角更大;（6）基于砂岩三轴卸荷试验成果,采用线性拟合方法得出了以围压和峰后塑性剪切应变增量为自变量的剪胀角函数,基于该函数与应变软化本构模型的砂岩三轴峰前卸围压试验的数值模拟结果与试验结果吻合较好,表明该函数能较好地描述三轴峰前卸围压条件下砂岩的扩容特性。以上结论可为深部地下工程变形预测、稳定性分析与支护设计提供理论基础。     

Analysis of borehole failure modes and pore pressure effects

The theoretical background and finite element simulations of Red Wildmoor sandstone hollow-cylinder experiments performed under various axial and radial loading conditions are presented together with borehole failure predictions. A non-linear, stress dependent elastic, Mohr–Coulomb elastoplastic constitutive model generalized for three-dimensional Cosserat continua is employed. The failure predictions are based on the bifurcation theory and incorporate the effect of microstructure. Two bifurcation instability criteria are formulated, one for lateral and one for axial failure modes. The effect of fluid flow on borehole failure is investigated with reservoir depletion and drawdown simulations and the results are corroborated through experimental data.     

基于RA/AF的高温后砂岩破裂特征识别研究

开展岩石高温损伤破裂信息识别研究对地热开发和煤炭地下气化等具有重要的指导意义。通过不同高温处理后砂岩在破坏过程中声发射参数RA(上升时间/振幅)和AF(平均频率)值变化,研究高温后砂岩内部不同类型裂纹的发展演化规律。结果表明:砂岩在整个加载过程中以拉张裂纹为主,当加载应力超过峰值应力的80%后,剪切裂纹所占比例迅速增大,可将此作为砂岩发生破坏的前兆;当加热温度超过600℃时,剪切裂纹所占比例上升,超过800℃,剪切裂纹所占比例迅速下降,600℃和800℃可作为砂岩损伤突变的阈值温度;高温后位错塞积现象增多,砂岩塑性特征增强,拉张裂纹所占比例增大。研究成果将对高温作用后岩石破裂失稳前兆信息的识别提供重要的理论基础。      

Poro‐elasto‐plastic response of an unconsolidated formation confined with stiff seal rocks under radial injection

In this article, we evaluate geomechanics of fluid injection from a fully penetrating vertical well into an unconsolidated formation confined with stiff seal rocks. The coupled behavior of an isotropic, homogeneous sand layer is studied under injection pressures that are high enough to induce plasticity yet not fracturing. Propagation of the significant influence zone surrounding the injection borehole, quantified by the extent of the plastic domain in the elasto‐plastic model, is examined for the first time. First, a new fully coupled axisymmetric numerical model is developed. A comprehensive assessment is performed on pore pressures, stresses/strains, and failure planes during the entire transient period of an injection cycle. Results anticipate existence of five distinctive zones in terms of plasticity state: liquefaction at wellbore; two inner plastic domains surrounding the wellbore, where failure occurs along two planes and major principal stress is in vertical direction; remaining of the plastic domain, where formation fails along one plane and major principal stress is in radial direction; and a non‐plastic region. Failure mechanism at the wellbore is found to be shear followed by liquefaction. Next, a novel methodology is proposed based on which new weakly coupled poro‐elasto‐plastic analytical solutions are derived for all three stress/strain components. Unlike previous studies, extension of the plastic zone is obtained as a function of injection pressure, incorporating plasticity effects on the subsequent elastic domain. Solutions, proven to be a good approximation of numerical simulations, offer a huge advantage as the run time of coupled numerical simulations is considerably long. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical modeling of plastic deformation and failure around a wellbore in compaction and dilation modes

In many wellbore stability analyses, the ability to forecast both the occurrence and extent of plastic deformation and failure hinges upon a fundamental understanding of deformation mode and failure mechanism in the reservoir rock. This study focuses on analyzing plastic zones, localized deformations, and failures around a borehole drilled overbalanced or underbalanced through a highly porous rock formation. Based on several laboratory experiments, porous rocks are prone to deform under both shear-induced dilation and shear-enhanced compaction mechanisms depending on the stress state. The shapes of the deformation and failure patterns around the borehole are shown, depending on the initial stress state and the local stress paths. The inquiry of the local stress paths in the near-wellbore zone facilitates the understanding of the reasons for different types of failure mechanisms, including the mixed-mode and the plastic deformation structures. The modification of the 2D plane strain condition by imitating third stress in the numerical scheme helps us bring the stress paths closer to the real state of loading conditions. Our modeling reveals that the transition from isotropic to anisotropic stress state is accompanied by an increase in the deviatoric part of effective shear tensor that leads to the development of inelastic deformation, degradation, and subsequent rock failure. Particular interest is devoted to the modeling of strain localization especially in compaction mode around a wellbore and computing the amount of stress concentration at the tips of dog-eared breakouts. Stress concentration can result in a change in irreversible deformation mode from dilatancy to compaction, elucidating the formation of the shear-enhanced compaction phenomenon at the failure tips in the direction of the minimum horizontal stress.