A new numerical procedure for elasto-plastic analysis of a circular opening excavated in a strain-softening rock mass

A simple numerical procedure for calculating the distribution of stresses and radial displacements around a circular tunnel excavated in a strain-softening Mohr–Coulomb or generalized Hoek–Brown rock mass is proposed. The problem is considered as axisymmetric, i.e. the initial stress state is assumed to be hydrostatic and the rock mass is said to be isotropic. By invoking the finite difference approximation of the equilibrium and compatibility equations, the increments of stresses and strains for each ring, starting from the outmost one for which boundary conditions are known a priori, are calculated in a successive manner. In the proposed approach, the potential plastic zone is divided into a finite number of concentric rings whose thicknesses are determined internally to satisfy the equilibrium equation. For the strain-softening behavior, it is assumed that all the strength parameters are a linear function of deviatoric plastic strain. Several illustrative examples are given to demonstrate the performance of the proposed method. For the brittle–plastic case, the results show a very good agreement with the closed-form solution. For strain-softening cases, the predictions by the proposed method are also in good agreement with the known rigorous numerical solutions. It is shown that the approximate solution converges to the exact solution when the increment of stress for each ring becomes smaller. The influence of the strength parameter ‘a’, appearing in the generalized Hoek–Brown criterion, on the elasto-plastic solutions is examined through the establishment of ground reaction curves and the discussion for the locations of the plastic radii.     

M-C与D-P屈服准则计算参数的能量等效方法及误差分析

 基于应变比能的概念,在偏平面内建立了分别进入Mohr-Coulomb(M-C)和Drucker-Prager(D-P)屈服面时所需形状改变比能的差值公式,获得M-C材料参数向D-P计算参数等效的统一性方案。可为各种参数匹配方法提供统一的计算公式、误差估计和能量解释。在此基础上讨论各种参数匹配方法的差异仅在于能量误差控制条件不同,且偏平面内能量误差零点可用Lode角或应力路径表示,此外,还讨论能量误差零点时对应Lode角的多值性及系列D-P准则等价性的条件。     

饱和黏土中柱孔三维弹塑性扩张机制研究

 目前饱和黏土中柱孔弹塑性扩张问题解答均假定土体为理想弹塑性材料,因而柱孔扩张过程中土体剪切变形和剪切破坏的一致性无法体现,且既有研究成果无法综合考虑超固结比及土体的三维强度特性,使得理论计算值与实际存在较大偏差。为得出饱和黏土中柱孔不排水扩张问题的精确解答,采用基于SMP准则改进的修正剑桥模型(SMP-MCC),充分考虑土体三维强度特性和超固结比,在转换应力空间使用相关联流动法则,进而将柱孔扩张问题归结为一阶偏微分方程组的求解问题。在此基础上,根据弹塑性边界处的应力和位移条件,采用应力转换方法在转换应力空间建立偏微分方程组的定解条件,并通过数值方法得到柱孔扩张问题的精确解答,通过与修正剑桥模型解答相对比,研究柱孔扩张过程中孔壁周围土体应力的变化规律。结果表明,由于提出的三维弹塑性精确解答充分考虑了土体的三维强度特性,因而可以更加真实地反应柱孔扩张过程中土体应力场的变化规律。     

致密砂岩抗剪参数与H-B参数间关系研究

以致密砂岩三轴测试为基础,建立Hoek-Brown(H-B)参数反算模型,应用4种基于不同假设H-B准则方法对岩石抗剪参数进行估值并修正,与Mohr-Coulomb(M-C)准则结果对比。结果表明,修正后两种准则估值结果极为接近。将该方法应用于Shirahama砂岩,评价结果与试验结果较为一致,表明基于反算法建立的H-B参数与岩体抗剪参数间定量关系可信。岩体声学参数与抗剪参数间关系的进一步探索对预测岩体破坏及采矿安全具参考价值。     

Elastic–brittle–plastic analysis of circular openings in Hoek–Brown media

A closed-form solution is presented in this paper for the prediction of displacements around circular openings in a brittle rock mass subject to a hydrostatic stress field. The rock mass is assumed to be governed by Hoek–Brown yield criterion and a non-associated flow rule is used. For the elastic–brittle–plastic analysis of circular openings in an infinite Hoek–Brown medium, the existing analytical solutions were found to be incorrect. The present closed-form solution is based on a theoretically consistent method and the solution does not require the use of any numerical method.The present closed-form solution was validated by using the finite element method. In the finite element analysis, the infinite boundary was simulated “exactly” by using the newly developed elastic support method. Several cases were analyzed and the present closed-form solutions for stresses and displacements were found to be in an excellent agreement with those obtained by using the finite element method.     

基于FLAC3D的空间滑动面准则计算格式与实现

 通过求解一元三次方程得出塑性因子 ,推导空间滑动面(SMP)准则在FLAC3D中的计算格式。基于FALC3D提供的二次开发平台,通过编制C++代码形成dll文件,将SMP准则导入FLAC3D中,实现在FALC3D软件中SMP准则的二次开发,并给出编程基本过程。通过进行真三轴数值试验、厚壁圆筒受内压的解析解与数值解的对比、边坡稳定性的计算验证模型开发的可行性。分别采用莫尔–库仑准则和SMP准则分析一边坡稳定性,得出SMP准则与莫尔–库仑准则计算边坡稳定有相似的破坏过程,由于SMP准则考虑了中间主应力对材料屈服面的影响,前者的安全系数高于后者;在相同折减系数下,边坡的塑性区和位移量均小于莫尔–库仑准则计算的结果。     

Analytical solutions for the stresses and deformations of deep tunnels in an elastic-brittle-plastic rock mass considering the damaged zone

The excavation impact (e.g. due to blasting, TBM drilling, etc.) induces an excavation damaged or disturbed zone around a tunnel. In this regard, in drill and blast method, the damage to the rock mass is more significant. In this zone, the stiffness and strength parameters of the surrounding rock mass are different. The real effect of a damage zone developed by an excavation impact around a tunnel, and its influence on the overall response of the tunnel is of interest to be quantified. In this paper, a fully analytical solution is proposed, for stresses and displacements around a tunnel, excavated in an elastic–brittle–plastic rock material compatible with linear Mohr–Coulomb criterion or a nonlinear Hoek–Brown failure criterion considering the effect of the damaged zone induced by the excavation impact. The initial stress state is assumed to be hydrostatic, and the damaged zone is assumed to have a cylindrical shape with varied parameters; thus, the problem is considered axial-symmetric. The proposed solution is used to explain the behavior of tunnels under different damage conditions. Illustrative examples are given to demonstrate the performance of the proposed method, and also to examine the effect of the damaged zone induced by the excavation impact. The results obtained by the proposed solution indicate that, the effects of the alteration of rock mass properties in the damaged zone may be considerable.     

基于应变空间弹塑性本构模型的FLAC~(3D)二次开发研究

基于应变空间理论,选择不同的塑性势函数(M-C关联准则和D-P非关联准则),对Mohr-Coulomb本构模型给出了它的数值计算格式,同时阐述了编制的关键技术和具体实施方法。采用FLAC3D二次开发平台UDM,在UserMohr模型的基础上使用Visual Studio 2005编译得到本构模型的动态链接库文件(DLL),实现了应变空间中弹塑性本构模型的二次开发。并利用地下盐穴静力溶腔过程模拟的验证算例进行对比分析。结果表明,计算结果与FLAC3D自带Mohr-Coulomb本构模型及解析解在弹塑性条件下进行比较,差值均很小,从而验证了程序的正确性。该程序可以为地下盐穴储库的溶腔模拟和长期运行流变分析提供一种可选择的数值模拟方法。     

An analytical solution to wellbore stability using Mogi-Coulomb failure criterion

Deep wellbores/boreholes are generally drilled into rocks for oil and gas exploration, monitoring of tectonic stresses purposes. Wellbore and tunnel in depth are generally in true triaxial stress state, even if the ground is under axisymmetric loading condition. Stability of such wellbores is very critical and collapse of wellbore must be avoided. Mogi-Coulomb failure criterion is a better representation of rock strength under true triaxial condition. In this paper, an analytical solution is proposed using Mogi-Coulomb failure criterion. The solution is obtained for rock mass exhibiting elastic-perfectly plastic or elastic-brittle-plastic behaviour considering in-plane isotropic stresses. The proposed solution is then compared with exact analytical solution for incompressible material and experimental results of thick-wall cylinder. It is shown that the results obtained by the proposed analytical solution are in good agreement with the experimental results and exact analytical solution. A reduction of about 13%–20% in plastic zone from the proposed closed-form solution is observed, as compared to the results from the finite element method (FEM) based Mohr-Coulomb criterion. Next, the influences of various parameters such as Poisson's ratio, internal pressure (mud weight), dilation angle, and out-of-plane stress are studied in terms of stress and deformation responses of wellbore. The results of the parametric study reveal that variation in the out-of-plane stress has an inverse relation with the radius of plastic zone. Poisson's ratio does not have an appreciable influence on the tangential stress, radial stress and radial deformation. Dilation angle has a direct relation with the deformation. Internal pressure is found to have an inverse relation with the radial deformation and the radius of plastic zone.     

拉压模量不同的剪胀土体中的球孔扩张问题

土体具有不同的拉压模量和剪胀特性。为使球形孔扩张问题的解答更符合实际情况,根据不同模量弹性理论和非关联Mohr-Coulomb流动法则研究了球形孔扩张时弹性区和塑性区的应力和位移,得到了弹性区和塑性区的径向应力和环向应力的解析解以及最大塑性区半径与扩张压力的计算表达式。探讨了不同拉压模量之比、拉压泊松比和剪胀角对球形孔扩张的扩张压力及塑性区半径的影响。结果表明:拉压模量比、拉压泊松比和剪胀角对球孔扩张时塑性区的开展和扩张后土体的应力有重要影响,计算中忽略这些参数将带来较大的误差。     

地下圆形洞室塑性回填效应的解析

本文对洞壁和衬砌间采用塑性材料回填的地下圆形隧道,由莫尔-库仑(Mohr-Coulomb)屈服准则,求得岩体、塑性回填材料与衬砌的内力及位移场的解析解。由某工程的计算值与实测值的符合程度表明,采用某种低模量的回填材料回填的方法,可以有效地减少衬砌的应力与位移。说明在一定条件下,这是一种有前途的工程措施。     

Shear strength criteria for rock,rock joints,rockfill and rock masses: Problems and some solutions

Although many intact rock types can be very strong, a critical confining pressure can eventually be reached in triaxial testing, such that the Mohr shear strength envelope becomes horizontal. This critical state has recently been better defined, and correct curvature or correct deviation from linear Mohr–Coulomb (M-C) has finally been found. Standard shear testing procedures for rock joints, using multiple testing of the same sample, in case of insufficient samples, can be shown to exaggerate apparent cohesion. Even rough joints do not have any cohesion, but instead have very high friction angles at low stress, due to strong dilation. Rock masses, implying problems of large-scale interaction with engineering structures, may have both cohesive and frictional strength components. However, it is not correct to add these, following linear M-C or nonlinear Hoek–Brown (H-B) standard routines. Cohesion is broken at small strain, while friction is mobilized at larger strain and remains to the end of the shear deformation. The criterion ‘c then σ_n tan φ’ should replace ‘c plus σ_ntan φ’ for improved fit to reality. Transformation of principal stresses to a shear plane seems to ignore mobilized dilation, and caused great experimental difficulties until understood. There seems to be plenty of room for continued research, so that errors of judgement of the last 50 years can be corrected.     

Hoek-Brown强度准则研究进展与应用综述

1980年E. Hoek和E. T. Brown提出了Hoek-Brown(H-B)强度准则,已充分得到岩石力学与工程研究者的认同,并进行研究和应用。首先系统地阐述H-B强度准则研究进展：E. Hoek和E. T. Brown对H-B强度准则的研究成果、三维H-B强度准则、H-B强度准则岩石和岩体参数研究、考虑层状节理的H-B强度准则及其参数的各向异性研究。再对过去30 a国内外基于H-B强度准则工程应用的成果进行总结。最后对笔者所开展的H-B强度准则最新研究工作进行介绍：提出一种真正意义上的广义三维H-B强度准则,并采用3种Lode角函数进行屈服面修正,使其可以直接应用于后续本构模型建立和数值软件嵌入;采用三维颗粒流模型进行微观数值建模,对H-B强度准则岩石和岩体参数开展微观研究并建立多尺度联系,为参数的确定提供更加可靠的依据。     

Analytical solution for a circular opening in an elastic–brittle–plastic rock

This paper deals with the analytical solutions for the prediction of displacements around a circular opening in an elastic–brittle–plastic rock mass compatible with a linear Mohr–Coulomb or a nonlinear Hoek–Brown yield criterion. Three different cases of definitions for elastic strains in the plastic region, used in the existing solutions, are mentioned. The closed-form analytical solutions for the displacement in the plastic region are derived on a theoretically consistent way for all the cases by employing a non-associated flow rule. The results of the dimensionless displacements are compared using the data of the soft and hard rocks to investigate the effect of different definitions for elastic strains with the dilation angle.     

Numerical evaluation of strength and deformability of fractured rocks

Knowledge of the strength and deformability of fractured rocks is important for design, construction and stability evaluation of slopes, foundations and underground excavations in civil and mining engineering. However, laboratory tests of intact rock samples cannot provide information about the strength and deformation behaviors of fractured rock masses that include many fractures of varying sizes, orientations and locations. On the other hand, large-scale in situ tests of fractured rock masses are economically costly and often not practical in reality at present. Therefore, numerical modeling becomes necessary. Numerical predicting using discrete element methods(DEM) is a suitable approach for such modeling because of their advantages of explicit representations of both fractures system geometry and their constitutive behaviors of fractures, besides that of intact rock matrix. In this study, to generically determine the compressive strength of fractured rock masses, a series of numerical experiments were performed on two-dimensional discrete fracture network models based on the realistic geometrical and mechanical data of fracture systems from feld mapping. We used the UDEC code and a numerical servo-controlled program for controlling the progressive compressive loading process to avoid sudden violent failure of the models. The two loading conditions applied are similar to the standard laboratory testing for intact rock samples in order to check possible differences caused by such loading conditions. Numerical results show that the strength of fractured rocks increases with the increasing confning pressure, and that deformation behavior of fractured rocks follows elasto-plastic model with a trend of strain hardening. The stresses and strains obtained from these numerical experiments were used to ft the well-known Mohr-Coulomb(MC) and Hoek-Brown(H-B) failure criteria, represented by equivalent material properties defning these two criteria. The results show that both criteria can provide fair estimates of the co     

岩石强度准则的数学形式和参数确定的研究

基于6种岩石的试验数据研究6种常规三轴强度准则。利用偏差平方和最小即最小二乘法确定强度准则中的待定系数,误差较大的个别数据使拟合曲线整体偏离多数正常数据;将强度准则变化后线性回归的结果可能完全失真;建议以拟合偏差的绝对值之和最小为目标直接搜索强度准则中的待定参数。剪应力–正应力的Mohr强度准则所陈述的力学含义并非真实,且不易确定其中的待定参数,建议使用显式的主应力强度准则。含有单一参数的抛物线准则可以描述多种岩石的强度特征,优于含有2个待定参数的Hoek-Brown和Coulomb准则;含有3个参数的指数强度准则具有较小的平均拟合偏差,且可以凸显异常数据点,优于二次多项式准则与广义Hoek-Brown准则。     

Residual stresses in press-braked stainless steel sections,I: Coiling and uncoiling of sheets

The manufacturing process of cold-formed thin-walled steel members induces cold work which can be characterized by the co-existent residual stresses and equivalent plastic strains and has a significant effect on their structural behaviour and strength. The present paper and the companion paper are concerned with the prediction of residual stresses and co-existent equivalent plastic strains in stainless steel sections formed by the press-braking method. This manufacturing process consists of the following two distinct stages: (i) coiling and uncoiling of the sheets, and (ii) press-braking operations. This paper presents an analytical solution for the residual stresses and co-existent equivalent plastic strains that arise from the first stage. In the analytical solution, the coiling–uncoiling stage is modelled as an inelastic plane strain pure bending problem; the stainless steel sheets are assumed to obey Hill’s anisotropic yield criterion with isotropic hardening to account for the effects of material anisotropy and nonlinear stress–strain behaviour. The accuracy of the solution is demonstrated by comparing its predictions with those obtained from a finite element analysis. The present analytical solution and the corresponding analytical solution for press-braking operations presented in the companion paper form an integrated analytical model for predicting residual stresses and equivalent plastic strains in press-braked stainless steel sections.     

Residual stresses in press-braked stainless steel sections,II: Press-braking operations

The manufacturing process of cold-formed thin-walled steel members induces cold work which can be characterized by the co-existent residual stresses and equivalent plastic strains and has a significant effect on their structural behaviour and strength. The present paper and the companion paper are concerned with the prediction of residual stresses and co-existent equivalent plastic strains in stainless steel sections formed by the press-braking method. This manufacturing process consists of the following two distinct stages: (i) coiling and uncoiling of the sheets, and (ii) press-braking operations. This paper first presents an analytical solution for the residual stresses and the co-existent equivalent plastic strains that arise from the second stage while a corresponding analytical solution for the first stage is presented in the companion paper. In both solutions, plane strain pure bending is assumed and the effect of material anisotropy is taken into account. On the basis of these two analytical solutions, an analytical model is presented to predict residual stresses and equivalent plastic strains in press-braked stainless sections. The predictions of the analytical model are shown to be in close agreement with results from a finite element-based method, demonstrating the validity and accuracy of the analytical model. The analytical model provides a much simpler method for the accurate prediction of residual stresses and equivalent plastic strains in different parts of a press-braked stainless steel section than a finite element-based method.     

不同胶结半径的粒间胶结拉伸和压缩试验研究

为在三维离散元模型中建立考虑胶结半径的胶结破坏准则,在已有10 mm胶结半径的三维铝球胶结试验的基础上,继续对胶结半径为8,12,14 mm的胶结铝球进行拉伸和压缩试验。结合10 mm胶结半径的试验数据,总结出不同胶结半径的胶结铝球拉伸与压缩的强度规律。试验结果表明:高铝水泥试样的拉伸强度与压缩强度随着胶结半径的增大而非线性增大;位移曲线在拉伸荷载下表现为脆性破坏,在压缩荷载下表现脆塑性和塑性破坏,并且压缩荷载下发生脆塑性破坏的概率随着胶结半径的减小而增大。     

扩孔理论在非饱和黄土劈裂注浆中的应用

针对扩孔理论在非饱和黄土劈裂注浆中的应用,将孔周围土体分为弹性区和塑性区,在弹性区中假设服从小变形理论,在塑性区中假设服从大变形理论,结合黏性土一般强度准则(SMP准则、Lade-Duncan准则、M-C准则),推导了不同理论下黄土地区在不排水条件下的劈裂注浆压力预估公式。利用现场注浆试验获得了压力P及注浆量Q的时程曲线,在P-Q-t三维散点图基础上,将理论计算结果与现场注浆数据对比分析,结果表明：基于黏性土一般强度准则推导的黄土劈裂注浆压力预估公式中,由于Land-Duncan准则及SMP准则考虑了中主应力的因素,较传统Mohr-Cunlomb模型的计算结果更加精确。对黄土注浆压力机理分析可知,当压力不大时,浆液对土体以挤密作用为主,当超过劈裂压力时,土体出现劈裂。