Incrementally non‐linear plasticity applied to rock joint modelling

Rock joint constitutive modelling is discussed through two new rock joint constitutive relations and a discrete numerical model. Regarding the constitutive relations, we emphasise the number of ‘tensorial zones’, that is, domains of constitutive incremental linearity; they involve four zones for the first (called ‘quadrilinear’) and an infinite number for the second one (called ‘incrementally nonlinear’). Using these formulations, a large class of loading paths can be considered. Hardening through shearing and relations between the normal and tangential directions of the joint (e.g., dilatancy) can be described. Their predictive abilities are checked. Plastic features are included even if the relations are defined outside the elasto‐plastic formalism. These relations obey, hence, the physical evidence as the plastic limit criterion and flow rule. The flow rule is nonassociated, and the corresponding link with the nonsymmetry of the constitutive matrix is examined. Comparisons between the two relations and the discrete numerical model, that is, a direct numerical simulation, which is fundamentally different, also are discussed within the context of infilled rock joints. Copyright © 2011 John Wiley & Sons, Ltd.     

Explicit solutions for the instantaneous undrained contraction of hollow cylinders and spheres in porous elastoplastic medium

In this article we present closed‐form solutions for the undrained variations in stress, pore pressure, deformation and displacement inside hollow cylinders and hollow spheres subjected to uniform mechanical pressure instantaneously applied to their external and internal boundary surfaces. The material is assumed to be a saturated porous medium obeying a Mohr–Coulomb model failure criterion, exhibiting dilatant plastic deformation according to a non‐associated flow rule which accounts for isotropically strain hardening or softening. The instantaneous response of a porous medium submitted to an instantaneous loading is undrained, i.e. without any fluid mass exchange. The short‐term equilibrium problem to be solved is now formally identical to a problem of elastoplasticity where the constitutive equations involve the undrained elastic moduli and particular equivalent plastic parameters. The response of the model is presented (i) for extension and compression undrained triaxial tests, and (ii) for unloading problems of hollow cylinders and spheres through the use of appropriately developed closed‐form solutions. Numerical results are presented for a plastic clay stone with strain hardening and an argilite with strain softening. The effects of plastic dilation, of the strain softening law and also of geometry of the cavity on the behaviour of the porous medium have been underlined. Analytical solutions provide valuable benchmarks enabling various numerical methods in undrained conditions with a finite boundary to be verified. Copyright © 2002 John Wiley & Sons, Ltd.     

颗粒物质在等比例应变加载下的分散性失稳模式

由地下水引起的静力液化可能是边坡失稳的隐含机制之一,松砂在不排水剪切条件下可能发生静力液化,密实的颗粒集合体在特定的应变路径下也会出现相似的现象,即试样整体发生急剧的失稳,应力状态尚处于峰值强度线以内。该种失稳模式称为分散性失稳,是为了强调失稳模式中没有出现应变局部化或者剪切带。采用连续-离散耦合分析方法,研究由不规则形状颗粒组成的密实集合体在等比例应变加载路径下的力学特性。根据Hill的材料失稳理论,当试样的应力增量 和应变增量 对应的2阶功 为负时,试样即发生不可逆的整体失稳破坏。以根据不同等比例应变路径得到 曲线为界,在 平面内将试样的应力状态分为剪缩区、剪胀-稳定区和剪胀-非稳定区,连接不同围压下试样发生分散性失稳时的应力状态形成失稳线发现,峰值强度线高于临界状态线,临界状态线高于失稳线。     

Application of the exact constitutive relationship of modified Cam clay to the undrained expansion of a spherical cavity

The modified Cam clay (MCC) model is used to study the response of virgin‐compressed clay subjected to undrained triaxial compression. The MCC constitutive relationship is obtained in a closed form. Both elastic and plastic deviatoric strains are considered in the analysis. The solution allows to obtain total and effective stress paths followed by the clay in undrained spherical expansion. Pore water pressures are determined from the difference between total and effective mean stresses. For illustration purposes, the analysis is also applied to the well‐known reconstituted normally consolidated London clay and the results are compared with the recently published data obtained by a numerical approach. In addition, the Almansi large strains are used in the analysis, as these allow to obtain limit expansion and pore pressures, whereas both small‐strain and logarithmic‐strain approaches do not permit to determine them. Copyright © 2010 John Wiley & Sons, Ltd.     

Failure envelopes and plastic potentials for eccentrically loaded surface footings on undrained soil

The failure envelope in V–M space for surface foundations on undrained material under eccentric loading can be determined using an extended version of the scaling (or effective area) concept of Meyerhof. A similar displacement transformation allows production of the plastic potential. The two‐dimensional finite element analyses of fully attached foundations subject to combined vertical (V) and moment (M) loading have been used to calculate appropriate scaling points for deduction of the failure envelope and plastic potential. Failure envelopes and plastic potentials are presented for footings on uniform and non‐uniform undrained material and it is seen that the equivalent ‘critical state’ or ‘parallel point’ lies slightly beneath the peak moment capacity. For accurate prediction of failure envelopes for footings on non‐uniform strength soil, consideration must be made of the apparent reduction of the soil heterogeneity as the area of footing in contact with the soil decreases. Copyright © 2001 John Wiley & Sons, Ltd.     

SPH procedures for modeling multiple intersecting discontinuities in geomaterial

A methodology is developed in SPH framework to analyze the behavior of preexisting multiple intersecting discontinuities or joints in rock material. The procedure does not require any additional unknowns to represent discontinuities and to capture velocity jump across them. Instead, a discontinuity is represented by a set of joint particles placed along the discontinuity plane, in which relative velocity and traction vector is evaluated, obeying the Mohr–Coulomb friction law with zero tension constrain. For failure of continuous rock material, the Drucker–Prager yield criterion with tensile cracking is employed in the elastic‐plastic constitutive model. Free‐sip, no‐slip, and symmetric boundary conditions are also implemented in SPH framework for proper representation of physical system. The paper analyzes behavior of a rock sample having a discontinuity plane under uniaxial loading and compares velocity and stress with a theoretical solution derived considering effective vertical stiffness of the joint planes. The efficacy of the proposed method is successfully demonstrated by solving another two problems of jointed rock mass under uniaxial and gravitational loading conditions.Copyright © 2014 John Wiley & Sons, Ltd.     

Bifurcation and instability modelling by a multimechanism elasto‐plastic model

The bifurcation and instability conditions in geomechanics are closely related to the elasto‐plastic behaviour. In this paper the potential of a multimechanism elasto‐plastic model to predict various modes of failure is examined. First, a brief overview for the essential aspects of the constitutive model and the development of the elasto‐plastic constitutive matrix for this model are presented. Then, numerical simulations of different drained and undrained paths in the axisymmetric and plane‐strain conditions for the Hostun sand are illustrated. These examples confirm the capacity of the model to reproduce instability and strain localization phenomena. The obtained response is in agreement with experimental observations, theoretical developments and numerical analyses existing in the literature. Copyright © 2007 John Wiley & Sons, Ltd.     

Kinematic hardening plasticity formulation of small strain behaviour of soils

An objective of this paper is to demonstrate that the small strain model developed by the authors can be incorporated into the conventional kinematic hardening plasticity framework to predict pre‐failure defor mations. The constitutive model described in this paper is constituted by three elliptical yield surfaces in triaxial stress space. Two inner surfaces are rotated ellipses of the same shape, representing the boundaries of the linear elastic and small strain regions, while the third surface is the modified Cam clay large‐scale yield surface. Within the linear elastic region, the soil behaviour is elastic with cross‐coupling between the shear and volumetric stress–strain components. Within the small strain region, the soil behaviour is elasto‐plastic, described by the kinematic hardening rule with an infinite number of loading surfaces defined by the incremental energy criterion. Within the large‐scale yield surface, the soil behaviour is elasto‐plastic, described by kinematic and isotropic hardening of the small strain region boundary. Since the yield surfaces have different shapes, the uniqueness of the plastic loading condition imposes a restriction on the ratio between their semi‐diameters. The model requires 12 parameters, which can be determined from a single consolidated undrained triaxial compression test. Copyright © 2000 John Wiley & Sons, Ltd.     

A numerical study of flexural buckling of foliated rock slopes

The occurrence of foliated rock masses is common in mining environment. Methods employing continuum approximation in describing the deformation of such rock masses possess a clear advantage over methods where each rock layer and each inter‐layer interface (joint) is explicitly modelled. In devising such a continuum model it is imperative that moment (couple) stresses and internal rotations associated with the bending of the rock layers be properly incorporated in the model formulation. Such an approach will lead to a Cosserat‐type theory. In the present model, the behaviour of the intact rock layer is assumed to be linearly elastic and the joints are assumed to be elastic–perfectly plastic. Condition of slip at the interfaces are determined by a Mohr–Coulomb criterion with tension cut off at zero normal stress. The theory is valid for large deformations. The model is incorporated into the finite element program AFENA and validated against an analytical solution of elementary buckling problems of a layered medium under gravity loading. A design chart suitable for assessing the stability of slopes in foliated rock masses against flexural buckling failure has been developed. The design chart is easy to use and provides a quick estimate of critical loading factors for slopes in foliated rock masses. It is shown that the model based on Euler's buckling theory as proposed by Cavers (Rock Mechanics and Rock Engineering 1981; 14 :87–104) substantially overestimates the critical heights for a vertical slope and underestimates the same for sub‐vertical slopes. Copyright © 2001 John Wiley & Sons, Ltd.     

NGI‐ADP: Anisotropic shear strength model for clay

Many geotechnical problems involve undrained behavior of clay and the capacity in undrained loading. Most constitutive models used today are effective stress based and only indirectly obtain values for the undrained shear strength. To match the design profiles of undrained shear strengths, in active (A), direct simple shear (D) and passive (P) modes of loading are complicated. This paper presents the elastoplastic constitutive model NGI‐ADP which is based on the undrained shear strength approach with direct input of shear strengths. Consequently, exact match with design undrained shear strengths profiles is obtained and the well‐known anisotropy of undrained shear strength and stiffness is accounted for in the constitutive model. A non‐linear stress path‐dependent hardening relationship is used, defined from direct input of failure strains in the three directions of shearing represented by triaxial compression, direct simple shear and triaxial extension. With its clear input parameters the model has significant advantages for design analysis of undrained problems. The constitutive model is implemented, into finite element codes, with an implicit integration scheme. Its performance is demonstrated by a finite element analysis of a bearing capacity problem. Copyright © 2011 John Wiley & Sons, Ltd.     

Modeling progressive failures in rock slopes with non‐persistent joints using the numerical manifold method

Rock slope failure is a complex process that usually involves both opening/sliding along pre‐existing discontinuities as well as fracturing of intact rock bridges. Discontinuity persistence is an important factor governing rock slope instabilities. However, traditional slope failure analysis assumes persistent discontinuities, and rock slope fails along a predefined persistent continuous potential failure surface because of the limitations of the analysis tools. This paper proposes the numerical manifold method (NMM) incorporated with a Mohr–Coulomb criterion‐based fracturing algorithm to simulate the progressive failure of rock slopes with non‐persistent joints. Detailed fracturing algorithm is first presented. Then, the NMM enabling fracturing is calibrated through simulating an edge‐cracked plate and the Brazilian test. Lastly, the developed code is applied to investigate the failure process of rock slopes involving non‐persistent joints. Numerical results indicate that the proposed method can capture the opening/sliding along existing discontinuities, the fracturing in intact rock bridges and the final kinematic release. Progressive slope failure is well exhibited. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability analysis of tunnels driven in jointed rocks by means of a homogenized limit analysis approach

Conceived as a potential alternative to the classical design methods employed for analyzing the stability of underground structures driven in jointed rocks, the homogenization approach stems from the heuristic idea that, from a macroscopic point of view, a rock mass cut by a network of joints may be perceived as a homogenized continuum. The strength properties of the latter can be theoretically obtained from the failure conditions of its individual constituents: rock matrix and joint interfaces. At the material level, the limit analysis reasoning is used in the context of homogenization to formulate the homogenized strength criterion of a jointed rock mass in the particular situation of a single set of parallel joints. As it could be expected, the obtained closed‐form expressions show the strength anisotropy induced by joint preferential orientation. The support functions (π functions) associated with the homogenized strength criterion are also determined in both plane strain and three‐dimensional cases. This criterion is then applied to the investigation of stability analysis of a tunnel excavated in a jointed rock mass. Upper bounds estimated of the stability factor are derived from the implementation of the kinematic approach directly on the homogenized underground structure. Finally, the approach is applied to analyze and discuss the collapse of the Pinheiros subway station (São Paulo, Brazil). Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

单轴压缩非贯通节理岩石尖端塑性区及弹塑性断裂模型研究

基于Drucker-Prager（下简称D-P）准则,建立压缩载荷作用下的非贯通节理岩石的弹塑性断裂模型。针对节理岩石小范围屈服翼裂纹尖端塑性区,推导了D-P屈服准则的纯I、纯II及I、II复合型3种翼裂纹无量纲塑性区径长函数,并与Mises准则的塑性区进行对比;结果表明,D-P准则的I型和复合型塑性区较Mises屈服准则的塑性区大,且其II型及I、II复合型塑性区在翼裂纹上下表面不连续。进一步,引入断裂软化因子以表征节理岩石裂隙断裂扩展后的断裂软化规律,考虑非贯通节理岩石复合型断裂软化,是由于节理尖端翼裂纹应变能密度超过最小应变能密度导致其成核扩展引起的,提出用应变能密度的指数函数形式表征断裂软化变量的演化;塑性屈服函数采用Borja等的应力张量3个不变量的硬化/软化函数,反映塑性内变量及应力状态对硬化函数的影响;建立节理岩石的弹塑性断裂本构关系及其数值算法,并用回映隐式积分算法编制了弹塑性断裂模型的程序。以单轴压缩下非贯通节理岩石为例,分析岩石断裂韧度、节理摩擦系数和节理倾角等参数的影响,结果表明,所提出的弹塑性断裂模型与数值和试验结果比较吻合。     

非共面复合加载条件下桶形基础稳定性研究

采用荷载-位移联合搜索方法,对于饱和软黏土地基上裙板式桶形基础在水平荷载H、力矩M与扭矩T的非共面复合加载条件下的稳定性进行了比较系统的三维有限元分析,主要探讨了在不同荷载空间内桶形基础的破坏包络轨迹。在分析中,地基土服从基于Tresca屈服准则的理想弹塑性本构关系。计算结果表明,与传统共面复合加载情况相比,非共面复合加载条件下桶形基础的破坏包络面形状有较大差异。非共面复合加载情况下,包络面形状不依赖于基础埋深比,可用简单的圆或椭圆方程进行描述。根据有限元计算结果,将已有的三自由度破坏包络面方程扩展到六自由度复合加载空间内,进而初步建议了一个六自由度破坏包络面方程,并对于实际工程中常用的桶形基础埋深比,给出了方程中偏心度参数取值的确定方法。该方程可用来合理评价复杂荷载条件下软黏土地基上桶形基础承载力。     

Bifurcation modeling in geomaterials: From the second‐order work criterion to spectral analyses

The present paper investigates bifurcation analysis based on the second‐order work criterion, in the framework of rate‐independent constitutive models and rate‐independent boundary‐value problems. The approach applies mainly to nonassociated materials such as soils, rocks, and concretes. The bifurcation analysis usually performed at the material point level is extended to quasi‐static boundary‐value problems, by considering the stiffness matrix arising from finite element discretization. Lyapunov's definition of stability (Annales de la faculté des sciences de Toulouse 1907; 9 :203–274), as well as definitions of bifurcation criteria (Rice's localization criterion (Theoretical and Applied Mechanics. Fourteenth IUTAM Congress, Amsterdam, 1976; 207–220) and the plasticity limit criterion are revived in order to clarify the application field of the second‐order work criterion and to contrast these criteria. The first part of this paper analyses the second‐order work criterion at the material point level. The bifurcation domain is presented in the 3D stress space as well as 3D cones of unstable loading directions for an incrementally nonlinear constitutive model. The relevance of this criterion, when the nonlinear constitutive model is expressed in the classical form (dσ = Mdε) or in the dual form (dε = Ndσ), is discussed. In the second part, the analysis is extended to the boundary‐value problems in quasi‐static conditions. Nonlinear finite element computations are performed and the global tangent stiffness matrix is analyzed. For several examples, the eigenvector associated with the first vanishing eigenvalue of the symmetrical part of the stiffness matrix gives an accurate estimation of the failure mode in the homogeneous and nonhomogeneous boundary‐value problem. Copyright © 2008 John Wiley & Sons, Ltd.     

Slope stability analysis for undrained loading conditions

The current state of art for limit equilibrium analysis of slope stability problems lacks a satisfactory procedure for stability evaluation under general, rapid (undrained) loading conditions. Some procedures are available for the analysis of rapid drawdown, but these suffer from several shortcomings and, furthermore, are not applicable to other types of rapid loading. An approach is presented which overcomes these limitations. The approach integrates four components-establishment of soil behaviour on the basis of laboratory testing, estimation of steady-state conditions in the slope using a boundary value analysis, estimation of distribution of undrained strength in the slope using undrained stress paths, and identification of the critical slip surface followed by calculation of its factor of safety. The approach is illustrated through its application to the stability analysis of an earth dam under rapid drawdown and earthquake conditions.     

Modelling large plastic deformations of cohesive soils using sequential limit analysis

This paper introduces sequential limit analysis (SLA) as a method for modelling large plastic deformations of purely cohesive materials such as undrained clay. The method involves solving a series of consecutive small‐deformation plastic collapse problems using finite element limit analysis, thus ensuring high levels of accuracy, efficiency, and robustness. The techniques needed to develop an SLA implementation for two‐dimensional (plane strain) problems are described in detail, including model geometry updating routines, treatment of rigid bodies, interfaces and boundaries, and periodic remeshing and interpolation of field variables. A simple total stress‐based constitutive model is used to account for strain softening and strain rate effects. Extensive verifications and validations are performed using analytical solutions and physical model test results, comparing both collapse loads and failure mechanisms, to demonstrate the effectiveness of the SLA approach. Additional solution quality checks on the bracketing discrepancy between lower‐bound and upper‐bound limit analysis solutions, and on the incompressibility of the rigid‐plastic material, are also presented.     

Failure Loci of Some Igneous and Metamorphic Rocks

Summary Experimental evidence from true triaxial tests on dense rocks are analysed with emphasis on the failure modes of these materials under multiaxial loading, ambient temperature and external pressure. The strong dependence of the modes of fracture on the secondary components of applied stresses, and especially on the intermediate principal stress, indicated that the failure surface of these brittle materials may be appropriately described by a failure tensor polynomial criterion. As such, the elliptic paraboloid failure criterion was found to conveniently describe their mode of failure, by considering also the severe influence of anisotropy of the material.  For this purpose, a method developed recently (Theocaris and Panagiotopoulos, 1995a, 1995b) was applied, defining anisotropic hardening plasticity through an appropriate sequence of anisotropic elasticity problems. Assuming a particular path of loading or unloading, we measured the instantaneous tension and compression yield stresses along the transient principal-stress directions. These parameters completely define the instantaneous state of anisotropy of the body for the corresponding loading step, by applying the theory of the elliptic paraboloid failure locus (EPFS) (Theocaris, 1989a). A parameter identification problem was formulated on the constitutive expressions for this most general failure criterion. Then, by applying convenient constraints derived from the EPFS theory, which serve as filters throughout the whole procedure, the characteristic values of terms defining the variable components of the failure tensor polynomial were calculated, as the material was continuously loaded from the elastic into the plastic region and up to the ultimate failure load. Accurate simple tests in uniaxial tension and compression provided sufficient data for the definition of the yield loci of the material, at the considered loading step. These tests may be complemented with biaxial and triaxial modes of loading of the specimens. The results improve the accuracy and sensitivity of the method. All such data were used as input values, for establishing the mode of plastic deformation of the body during particular loading paths.  Moreover, the method employed allows the complete definition of the components of the failure, H, and the strength differential effect, h, tensors at each loading step. These quantities define completely the failure tensor polynomial for each material. Therefore, it presents the important advantage over other experimental methods by clearly indicating the parts contributed to the failure mode (either by plasticity, or by the strength differential effect) and their evolution during plastic deformation.  As convenient prototype materials for testing the method, specimens of metamorphic rocks such as Westerly granite (G), or quartzite (Q) were selected. Interesting results concerning the mechanical and especially the failure modes of such materials were obtained. Furthermore, the mechanical tests indicated clearly some basic properties of these materials as concerns the mode of their structure.