A yield criterion for isotropic and cross‐anisotropic cohesive‐frictional materials

This paper proposes a yield and failure criterion for cohesive and frictional materials. The function is given by the combination of a Lode dependence for the behaviour in the deviatoric plane and a meridian function for the pressure‐dependent behaviour. A variety of shapes can be achieved with the proposed criterion including Lode dependences which are able to reproduce the behaviour of isotropic and cross‐anisotropic materials in the deviatoric plane. The criterion is validated through the comparison with experimental data based on multiaxial experimental tests on clays, sands, rocks and concrete. Finally, the convexity of the criterion is analysed and discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

Failure criteria for cohesive‐frictional materials based on Mohr–Coulomb failure function

This study presents two three‐parameter failure criteria for cohesive‐frictional materials based on the Mohr–Coulomb failure function. One proposed failure criterion can be linked to Mogi's empirical formula and incorporates the well‐known Von‐Mises, Drucker–Prager, and Linear Mogi criteria as special cases. Another one with smooth and convex cross sections contains a general Lode dependence in the deviatoric plane and includes the Matsuoka–Nakai and Lade–Duncan Lode dependences as special cases. The effect of the intermediate principal stress on the strength of the material can be taken into account in both criteria. The proposed criteria are numerically calibrated against polyaxial data sets of many different types of rocks and concrete_. The comparison results show that the performance of the proposed criteria is excellent, and the failure criterion with a general Lode dependence performs better than the other one for concrete. Copyright © 2015 John Wiley & Sons, Ltd.     

A Simplified Failure Criterion for Intact Rocks Based on Rock Type and Uniaxial Compressive Strength

The uniaxial compressive strength (UCS) of intact rock, which can be estimated using relatively straightforward and cost-effective techniques, is one of the most practical rock properties used in rock engineering. Thus, constitutive laws to represent the strength and behavior of (intact) rock frequently use it, along with additional intrinsic rock properties. Although triaxial tests can be employed to obtain best-fit failure criterion parameters that provide best strength predictions, they are more expensive and require time-consuming procedures; as a consequence, they are often not readily available at early stages of a project. Based on the analysis of an extensive triaxial test database for intact rocks, we propose a simplified empirical failure criterion in which rock strength at failure is expressed in terms of confining stress and UCS, with a new parameter which can be directly estimated from the UCS for a specified rock type in the absence of triaxial test data. Performance of the proposed failure criterion is then tested for validation against experimental data for eight rock types. The results show that strengths of intact rock estimated by the proposed failure criterion are in good agreement with experimental test data, with small discrepancies between estimated and measurements strengths. Therefore, the proposed criterion can be useful for preliminary (triaxial) strength estimation of intact rocks when triaxial tests data are not available.     

Instantaneous Friction Angle and Cohesion of 2-D and 3-D Hoek–Brown Rock Failure Criteria in Terms of Stress Invariants

The Mohr–Coulomb (M–C) failure criterion is one of the most widely used failure criteria in rock mechanics, although it has a number of shortcomings such as neglecting the nonlinear strength observed in rock or the effect of the intermediate principal stress σ ₂. Other failure criteria have been proposed to effectively include in the predictions of failure the non-linear response of rock to confinement or the effects of the intermediate principal stress. The M–C criterion is still widely used, and it is arguably the criterion most used in practice. For example, stability evaluations of shallow rock structures such as slopes and foundations are routinely carried out by estimating a friction angle and a cohesion of the rock mass. To include the dependency of cohesion and friction angle on stresses, efforts are being made to estimate equivalent values of the M–C parameters for the range of stresses applicable to a particular design. The paper suggests a new and convenient approach to find the equivalent friction angle and cohesion from any failure criterion that can be expressed in terms of the Nayak and Zienkiewicz’s stress invariants. To demonstrate the capabilities and application of the methodology, the new approach is applied to two failure criteria: the Hoek–Brown (H–B) criterion and the Hoek–Brown and Willam–Warnke (HB–WW) criterion, 2-D and 3-D failure criteria, respectively. Results from the new method, in terms of equivalent friction and cohesion for the H–B criterion, are exactly the same as the results obtained from Balmer’s theory, which confirms the validity of the new method. The predicted equivalent friction and cohesion for the HB–WW criterion show a dependency on σ ₂, which does not occur for a 2-D failure criterion.     

A single numerically efficient equation for approximating the Mohr–Coulomb and the Matsuoka–Nakai failure criteria with rounded edges and apex

This paper presents a novel formulation for defining soil failure. It plots in the principal stress space as a surface with the shape ranging between an approximation of the Matsuoka–Nakai and of the Mohr–Coulomb criteria depending on the value of a single parameter. The new function can be used as a replacement of the original equations of these well‐established criteria for implementing in a program for numerical analyses, and it is particularly effective for approximating the Matsuoka–Nakai criterion. Both the Mohr–Coulomb and the Matsuoka–Nakai failure criteria present numerical difficulties during implementation and also at run‐time. In the case of the Matsuoka–Nakai, the new formulation plots in the first octant only, whereas the original criterion plots in all octants, which causes severe convergence problems particularly for those Gauss points with low stress state, such as those on the side of a shallow footing. When the shape parameter is set to reproduce the Mohr–Coulomb failure criterion, on the other hand, the new formulation plots as a pyramid with rounded edges. Moreover, as the new function is at least of class C², the second derivatives are continuous, thus ensuring quadratic convergence of the Newton's method used within the integration scheme of the constitutive law. The proposed formulation can also provide both sharp and rounded apex of the surface at the origin of the stress space by setting accordingly one additional parameter. Copyright © 2013 John Wiley & Sons, Ltd.     

An Empirical Failure Criterion for Intact Rocks

The parameter m _i is an important rock property parameter required for use of the Hoek–Brown failure criterion. The conventional method for determining m _i is to fit a series of triaxial compression test data. In the absence of laboratory test data, guideline charts have been provided by Hoek to estimate the m _i value. In the conventional Hoek–Brown failure criterion, the m _i value is a constant for a given rock. It is observed that using a constant m _i may not fit the triaxial compression test data well for some rocks. In this paper, a negative exponent empirical model is proposed to express m _i as a function of confinement, and this exercise leads us to a new empirical failure criterion for intact rocks. Triaxial compression test data of various rocks are used to fit parameters of this model. It is seen that the new empirical failure criterion fits the test data better than the conventional Hoek–Brown failure criterion for intact rocks. The conventional Hoek–Brown criterion fits the test data well in the high-confinement region but fails to match data well in the low-confinement and tension regions. In particular, it overestimates the uniaxial compressive strength (UCS) and the uniaxial tensile strength of rocks. On the other hand, curves fitted by the proposed empirical failure criterion match test data very well, and the estimated UCS and tensile strength agree well with test data.     

A relation between safety factors with respect to strength and height of slopes

Conventional slope stability analysis is usually based on the linear Mohr–Coulomb failure criterion utilizing the notion of safety factors with respect to shear strength, and one of the available slice methods. Failure criteria of most soils are not linear, and it is possible to show that this non-linearity has a very significant effect on calculated safety factors. The present work is based on a non-linear failure criterion, which appears to fit the experimental information better than Mohr–Coulomb. All slice methods utilize various kinematical and static assumptions, which cannot be rationally justified. The present work is based on rigorous variational approach to slope stability analysis, which does not employ any kinematical and static assumptions. Safety factors with respect to shear strength are useful abstractions, but physical significance of results based on them is clear only at failure when they are equal to 1 (at any other value of the safety factor with respect to strength results of the analysis correspond to fictitious material with a modified shear strength function). In the present note, we use the variational analysis in order to establish a simple analytical relation between safety factors with respect to strength and height. These two safety factors provide alternative measures for the stability of a given slope; but the safety factor with respect to height appears to have clearer physical interpretation.     

Active earth pressure analysis based on normal stress distribution function along failure surface in soil obeying nonlinear failure criterion

In this paper, the normal stress distribution function along the failure surface in soil obeying general nonlinear failure criterion is first derived, and then a method for calculating the static and seismic active earth pressure for soils obeying nonlinear failure criteria is proposed. The method proposed yields not only the active earth pressure and its action point, but also the failure surface and the stress distribution along it. Results of the calculated examples by the proposed method are more reasonable than those reported in the literature. The influence of the parameters of the nonlinear failure criterion on the active earth pressures is also investigated.     

A two-parameter expression for failure surfaces

The Mohr–Coulomb, Tresca and Von Mises criteria are classical failure criteria, widely accepted and used for various materials. To take into account the influence of intermediate principal stress on the strength of soil, Bishop, Lade–Duncan and Matsuoka proposed criteria in terms of three principal stresses or three stress invariants. This note describes an expression with two parameters for modeling the shape of the failure surface in the octahedral plane. By studying the roles of these two parameters and comparing the new criterion with the aforementioned criteria, the advantage and flexibility of the proposed function is explored. Application of the function is demonstrated by fitting the new surface to experimental data for various soils.     

Upper bound finite element analysis of slope stability using a nonlinear failure criterion

In this study, upper bound finite element (FE) limit analysis is applied to stability problems of slopes using a nonlinear criterion. After formulating the upper bound analysis as the dual form of a second-order cone programming (SOCP) problem, the stress field and corresponding shear strength parameters can be determined iteratively. Thus, the nonlinear failure criterion is represented by the shear strength parameters associated with stress so that the analysis of slope stability using a nonlinear failure criterion can be transformed into the traditional upper bound method with a linear Mohr–Coulomb failure criterion. Comparison with published solutions illustrates the accuracy and feasibility of the proposed method for a simple homogeneous slope stability problem. The proposed approach is also applied to a seismic stability problem for a rockfill dam to study the influence of different failure criterions on the upper bound solutions. The results show that the seismic stability coefficients obtained using two different nonlinear failure criteria are similar but that the convergence differs significantly.     

基于弹性应变能的岩石强度准则

为建立更符合岩石屈服与破坏机制的强度准则,基于能量转化是岩石屈服与破坏的本质属性,采用试验与理论分析相结合的方法,对岩石屈服与破坏准则进行了研究。以岩石强度与整体破坏准则为基础,通过引入弹性应变能释放分散系数,建立基于弹性应变能强度准则;分别采用M-C准则、Murrell准则、三剪能量准则、统一能量准则、三维H-B强度准则及基于弹性应变能岩石强度准则对盐岩和花岗岩的破坏强度进行了计算。结果表明,基于弹性应变能岩石强度准则的计算结果与试验值比较吻合（尤其是真三轴试验条件下）,并且分析了产生上述结果的内在机制。所建立的强度准则仅需测定常规岩石力学参数（单轴抗压强度与泊松比）,物理力学意义明确,对于定量描述岩石的屈服与破坏特性具有重要的意义。     

Improved finite element–based limit equilibrium method for slope stability analysis by considering nonlinear strength criteria and its application in assessing the anchoring effect

The shear strength envelope of most geotechnical materials is nonlinear. In this study, the finite element–based limit equilibrium method (FELE) is improved to assess the stability of a prescribed slip surface that obeys a nonlinear failure criterion. Two nonlinear failure criteria, namely, the Barton criterion and the generalized Hoek-Brown (GHB) criterion, are implemented. A power curve model that can perfectly fit the envelope of the GHB criterion is proposed to explicitly express the shear strength from the normal stress. The algorithm of the improved FELE, which includes a nonlinear criterion, is explained in detail, and the fast convergence of the method is checked by evaluating examples. The numerical examples show that, for a smooth slip surface, FELE can calculate a greater factor of safety (FOS) than that calculated using the Morgenstern-Price (MP) method when the strength criterion of slip surface is nonlinear. In addition, FELE can determine the bell-shaped distribution of the increase in normal stress caused by anchoring measures, which cannot be considered in the MP method.     

A discrete numerical method for brittle rocks using mathematical programming

A computational formulation of discrete simulations of damage and failure in brittle rocks using mathematical programming methods is proposed. The variational formulations are developed in two and three dimensions. These formulations naturally lead to second-order cone programs and can conveniently be solved using off-the-shelf mathematical programming solvers. Pure static formulations are derived so that no artificial damping parameters are required. The rock is represented by rigid blocks, with interfaces between blocks modelled by zero-thickness springs based on the rigid-body–spring network method. A modified Mohr–Coulomb failure criterion is proposed to model the failure of the interfaces. When the interface’ strength limits are reached, a microscopic crack forms and its strength is irreversibly lost. The microscopic elastic properties of the springs are related to the observed elastic behaviour of rocks with the developed empirical equations. The program is first validated with three simple tests. Then, numerical uniaxial and biaxial compression tests and the Brazilian tests are conducted. Furthermore, the proposed approach is employed to study the rock crack propagation and coalescence using cracked Brazilian disc test. The results are in good agreements with reported experimental data, which shows its potential in modelling mechanical behaviour of brittle rocks.     

Application of a unified linear yield criterion in plane strain to study the strength theory effect in geotechnical engineering

The plane strain condition is a common, but polyaxial stress state for geotechnical structure designs, in which the selection of an appropriate yield or failure criterion is crucial to reasonably account for the intermediate principal stress. Under plane strain condition, a unified linear yield criterion for seven commonly used geotechnical yield criteria is presented in conjunction with the inductive method. These seven yield criteria considered in this study are the Mohr–Coulomb, Tresca, Drucker–Prager, Mogi–Coulomb, Extended Matsuoka–Nakai, Extended Lade–Duncan criteria, and the Unified Strength Theory. The generalized analytical solutions for earth pressure of retaining walls, critical load of strip foundations as well as stress and displacement of circular tunnels are derived on the basis of the proposed unified yield criterion, and their respective theoretical significance is analyzed. Thereafter, the critical load of strip foundations obtained herein is compared with two numerical results from the literature. Furthermore, the effect of strength theory on result differences of the three typical geotechnical problems by simply selecting constants, which conform to different yield criteria, is explored through a parametric study. It is found that the proposed unified yield criterion is convenient for investigating analytical solutions of the aforementioned geotechnical structures. The strength theory effect due to adopting different yield criteria is considerably significant, which cannot be ignored. Additionally, recommendations are provided on how to make use of these seven yield criteria for an optimum design.     

A new rock mass failure criterion for biaxial loading conditions

To simulate brittle rocks, a mixture of glastone, sand and water was used as a model material. Thin galvanized sheets of thickness 0.254 mm were used to create joints in blocks made out of the model material. To investigate the failure modes and strength, both the intact material blocks as well as jointed model material blocks of size 35.6 × 17.8 × 2.5 cm having different joint geometry configurations were subjected to uniaxial and biaxial compressive loadings. A new intact rock failure criterion is proposed at the 3-D level. This criterion is validated for biaxial loading through laboratory experimental results obtained on intact model material blocks. Results obtained from both the intact and jointed model material blocks are used to develop a strongly non-linear new rock mass failure criterion for biaxial loading. In this failure criterion, the fracture tensor component is used to incorporate the directional effect of fracture geometry system on jointed block strength. The failure criterion shows the important role, the intermediate principal stress plays on rock mass strength.     

Application of the Christensen Failure Criterion to Intact Rock

The Christensen criterion, originally introduced in materials science, has a simple mathematical form and uniaxial tensile and compressive strength as the only parameters, making it an attractive candidate for rock engineering purposes. In this study, the applicability of the criterion to rock materials is examined. Explicit equations for application of the criterion under biaxial, triaxial compression, triaxial extension, and polyaxial states of stresses are derived. A comprehensive strength data set including the results of tests on synthetic rock, chert dyke, Carrara marble and Westerly granite is utilized to examine the accuracy of the Christensen criterion to the failure of rock material. The two surprising findings about the Christensen criterion are the zero values of tensile strength and the very low slopes of the failure envelope obtained from fitting analyses for chert dyke and Westerly granite. It is shown that the two problems are interrelated and the values of tensile strength tend to zero to produce higher slopes. It is then mathematically proven that the maximum initial slope of the Christensen failure envelope is limited to 4 in triaxial compression and 2.5 in triaxial extension which is considerably lower than the slope of experimental data. The accuracy of the Christensen criterion was found to be significantly lower than the well-established Hoek–Brown criterion. The circular π-plane representations and brittle-to-ductile transition limits from the Christensen criterion are also inconsistent with the observed behavior of rocks.     

Effects of confining pressure and loading path on deformation and strength of cohesive granular materials: a three-dimensional DEM analysis

This paper is devoted to numerical analysis of strength and deformation of cohesive granular materials. The emphasis is put on the study of effects of confining pressure and loading path. To this end, the three-dimensional discrete element method is used. A nonlinear failure criterion for inter-granular interface bonding is proposed, and it is able to account for both tensile and shear failure for a large range of normal stress. This criterion is implemented in the particles flow code. The proposed failure model is calibrated from triaxial compression tests performed on representative sandstone. Numerical results are in good agreement with experimental data. In particular, the effect of confining pressure on compressive strength and failure pattern is well described by the proposed model. Furthermore, numerical predictions are studied, respectively, for compression and extension tests with a constant mean stress. It is shown that the failure strength and deformation process are clearly affected by loading path. Finally, a series of numerical simulations are performed on cubic samples with three independent principal stresses. It is found that the strength and failure mode are strongly influenced by the intermediate principal stress.

     

模拟岩石中裂纹扩展连接的近场动力学方法

脆性岩石材料在压应力作用下常出现两类裂纹：翼型张拉裂纹和次生剪切裂纹。近场动力学是一种新型的无网格数值计算方法。在近场动力学理论中,采用积分形式的控制方程代替微分形式的控制方程使得该数值算法在断裂问题上具有独特的优势。将Mohr-Coulomb准则和最大主应力准则引入非普通“态”基近场动力学理论中,分别用于模拟材料常见的压剪和张拉破坏。这种扩展的非普通“态”基近场动力学可以有效地模拟脆性岩石材料在多种受力状态下的裂纹起裂、扩展和连接问题。通过5个不同的数值算例说明该数值算法在处理脆性岩石材料断裂问题的有效性和准确性。首先,通过模拟含圆孔的弹性板拉伸数值试验说明该数值算法的有效性和准确性。其次,数值模拟了简单三点弯曲试验以及不使用其他外部准则条件下动荷载作用下裂纹的分叉试验,所得结果与其他试验结果或数值结果相吻合,从而验证了该理论的有效性。然后,模拟了包含斜裂纹的巴西圆盘试验,裂纹扩展路径和计算所得的断裂韧度同样吻合于试验结果。最后,模拟了单轴压缩状态下,预制裂纹试样的裂纹扩展和连接问题。将该数值算法与试验结果对比表明,所提出的数值方法可以模拟和预测岩石类材料的张拉和压剪裂纹的起裂、扩展和连接行为。     

一种适用于各向异性土体的破坏准则

自然界的土体通常具有各向异性的特点,而传统的破坏准则大多只适用于各向同性的土体。结合应力张量和反映材料各向异性状态的组构张量,定义了修正偏应力及其不变量,提出了适用于各向异性土体材料的破坏准则。给出了共轴条件下正交各向异性和横向各向异性材料在一般应力空间的破坏曲线以及不同应力区中主应力系数b与摩擦角的关系曲线,并分析了它们的特性以及与各向同性材料相应曲线的区别。通过与真三轴试验数据的比较,表明该准则能很好地描述各向异性土体材料的强度特点。     

Strength criterion for cross-anisotropic sand under general stress conditions

By incorporating the fabric effect and Lode’s angle dependence into the Mohr–Coulomb failure criterion, a strength criterion for cross-anisotropic sand under general stress conditions was proposed. The obtained criterion has only three material parameters which can be specified by conventional triaxial tests. The formula to calculate the friction angle under any loading direction and intermediate principal stress ratio condition was deduced, and the influence of the degree of the cross-anisotropy was quantified. The friction angles of sand in triaxial, true triaxial, and hollow cylinder torsional shear tests were obtained, and a parametric analysis was used to detect the varying characteristics. The friction angle becomes smaller when the major principal stress changes from perpendicular to parallel to the bedding plane. The loading direction and intermediate principal stress ratio are unrelated in true triaxial tests, and their influences on the friction angle can be well captured by the proposed criterion. In hollow cylinder torsional shear tests with the same internal and external pressures, the loading direction and intermediate principal stress ratio are related. This property results in a lower friction angle in the hollow cylinder torsional shear test than that in the true triaxial test under the same intermediate principal stress ratio condition. By comparing the calculated friction angle with the experimental results under various loading conditions (e.g., triaxial, true triaxial, and hollow cylinder torsional shear test), the proposed criterion was verified to be able to characterize the shear strength of cross-anisotropic sand under general stress conditions.