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 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.     

Axisymmetric compression of a Mohr–Coulomb medium around a circular hole

An analytical solution is presented for the stress and strain fields in a Mohr–Coulomb material in plane strain around a circular hole when it is compressed by an axisymmetric far-field pressure. It is shown that several solutions arise involving one to three plastic zones depending on the values of Poisson's ratio and the friction angle. The solution chosen for presentation was obtained and used to validate the functioning of the Mohr–Coulomb yield condition that was added to the NONSAP finite element code. Stress and strain field comparisons are made.     

A new method to calculate the equivalent Mohr–Coulomb friction angle for cohesive and frictional materials

In this note, a new method to calculate the equivalent Mohr–Coulomb friction angle ?′_mc for cohesive and frictional materials is presented. This method makes a transformation from the failure surface for cohesive materials to the failure surface for cohesionless materials and obtains ?′_mc as well as the principal stress ratio σ′₁/σ′₃ for cohesionless materials in the transformed space first, then obtains ?′_mc for cohesive materials by linking σ′₁/σ′₃ in the transformed space and in the original space. In the application example, an analytical solution of the invariant stress ratio L is derived from the failure function in the transformed space. The influence of the intermediate effective principal stress σ′₂ is also demonstrated using the already calculated ?′_mc. Copyright © 2010 John Wiley & Sons, Ltd.     

Mohr–Coulomb MiniCLoE model Uniqueness and localization studies,links with normality rule

This paper is devoted to a parametric study of a plane Mohr–Coulomb CLoE model. As CLoE models are designed with a consistency condition, it is possible to define a normality condition and to study its consequences. The positiveness of the second order work which implies the uniqueness of the solution of a small strain boundary value problem is studied firstly. Then the localization criterion is also studied. It is proved that normality has consequences similar to those for classical elasto plastic models. However if induced anisotropy is introduced in the hypoplastic CLoE model, some conclusions are no longer true. Finally plane strain experimental data are used to identify the parameters of the model. Copyright © 2002 John Wiley & Sons, Ltd.     

Observations on the extended Matsuoka–Nakai failure criterion

The equivalent Mohr–Coulomb (M‐C) friction angle ? (J. Geotech. Eng. 1990; 116 (6):986–999) of the extended Matsuoka–Nakai (E‐M‐N) criterion has been examined under all possible stress paths. It is shown that ? depends only on the ratio of cohesion to confining stress c^′/σ and the frictional angle ?, where ? is the friction angle measured in triaxial compression (or extension) to which the E‐M‐N surface is fitted. It is also shown that ? is independent of c^′, when σ=0 and of σ when c^′=0, with the former representing an upper bound and the latter a lower bound of ? for any particular stress path. The closest point projection method has also been implemented successfully with the E‐M‐N criterion, and plane strain and axisymmetric element tests performed to verify some theoretical predictions relating to failure and post‐yielding behavior. Finally, a bearing capacity problem was analyzed using both E‐M‐N and M‐C, highlighting the conservative nature of M‐C for different friction angles. Copyright © 2009 John Wiley & Sons, Ltd.     

The critical state behaviour of barodesy compared with the Matsuoka–Nakai failure criterion

Barodesy is a new approach to constitutive modelling of soil. It is based on Goldscheider's principles and maps stretching directions onto corresponding stress directions with the help of a simple exponential function. This mapping also determines a critical state surface in principal stress space. The article investigates this surface and relates it to the well‐known Matsuoka–Nakai failure criterion. It turns out that the difference between these two surfaces is negligible for practical applications. Copyright © 2011 John Wiley & Sons, Ltd.     

A fully convex reformulation of the original Matsuoka–Nakai failure criterion and its implicit numerically efficient integration algorithm

This paper presents a reformulation of the original Matsuoka–Nakai criterion for overcoming the limitations which make its use in a stress point algorithm problematic. In fact, its graphical representation in the principal stress space is not convex as it comprises more branches, plotting also in negative octants, and it does not increase monotonically as the distance of the stress point from the failure surface rises. The proposed mathematical reformulation plots as a single, convex surface, which entirely lies in the positive octant of the stress space and evaluates to a quantity which monotonically increases as the stress point moves away from the failure surface. It is an exact reproduction, and not an approximated one, of the only significant branch of the original criterion. It is also suitable for shaping in the deviatoric plane the yield and plastic potential surfaces of complex constitutive models. A very efficient numerical algorithm for the implicit integration of the proposed formulation is presented, which enables the evaluation of the stress at the end of each increment by solving a single scalar equation, both for associated and non‐associated plasticity. The algorithm can be easily adapted for other smooth surfaces with linear meridian section. Finally, a close expression of the consistent Jacobian matrix is given for achieving quadratic convergence in the external structural newton loop. It is shown that all this results in extremely fast solutions of boundary value problems. Copyright © 2013 John Wiley & Sons, Ltd.     

最大往返剪切作用面上应变破坏标准的应力条件

通过一系列的动三轴试验，探讨以应变为破坏标准的最大往返剪切作用面上的破坏应力条件。试验结果表明：破坏应力条件仍符合张克绪提出的公式，参数k,c是破坏振次与破坏应变标准的二元线性函数。     

A simple method to describe three-dimensional anisotropic failure of soils

In order to describe the anisotropic failure of soils caused by the internal fabric, isotropic failure criterion should be generalized to be anisotropic. This paper achieves the generalization by introducing a simple method, called anisotropic transformed stress method, which apparently differs from the common way. Physical interpretation of this method are analyzed further. Using this method, many existing isotropic criteria can become anisotropic in the same way, and will be expressed by a unified formula finally. To verify this method, anisotropic Unified Strength Criterion is used to predict the peak strength of anisotropic soils in different loading conditions.     

基于能量原理盐岩的强度与破坏准则

讨论了盐岩变形过程中可释放弹性应变能与耗散能内在关系,并给出了基于弹性应变能的破坏准则。为了验证该准则的可行性,一方面利用Mohr准则、D-P准则、基于能量原理强度准则对围压分别为1、5、10、15 MPa一组三轴压缩试验进行了研究;另一方面,应用该准则对来源于其他文献的三轴试验数据就行了研究。研究结果表明,由于该准则仅需确定常规岩石力学参数,物理力学意义明确,试验值与理论值的相对误差较小,因此,该准则可作为盐岩破坏的判据。     

SMP准则在计算煤柱极限强度中的应用

基于黏性材料的SMP（spatially mobilized plane）破坏准则,分析了煤柱在三维应力状态下的应力问题,并推导了煤柱极限强度的计算公式。通过改变煤体的内摩擦角? 和凝聚力c,与Mohr-Coulomb准则下煤柱极限强度以及简化的A. H. Wilson公式得到的煤柱极限强度进行对比,结果表明,Mohr-Coulomb准则和A.H.Wilson公式由于忽略了煤柱中主应力的影响,其得到的极限强度计算公式低估了煤柱实际极限强度,说明基于黏性材料的SMP准则下的煤柱极限强度计算符合实际,对于分析煤柱的稳定性具有一定的现实指导意义。     

Lower bound limit analysis of an anisotropic undrained strength criterion using second‐order cone programming

In this paper, the formulation of the lower bound limit analysis of an anisotropic undrained strength criterion using second‐order cone programming is described. The finite element concept was used to discretize the soil mass into 3‐noded triangular elements. The stress field was modeled using a linear interpolation within the elements while stress discontinuities were permitted to occur at the shared edges of adjacent elements. An elliptical yield criterion was adopted to model the anisotropic undrained strength of the clay. A statically admissible stress field was defined by enforcing the equilibrium equations within all triangular elements and along all shared edges of adjacent elements, stress boundary conditions, and no stress violation of the anisotropic strength envelope cast in the form of a conic quadratic constraint. The lower bound solution of the proposed formulation was solved by second‐order cone programming. The proposed formulation of the anisotropic undrained strength criterion was validated through comparison of the model's predictions with the known exact solutions of strip footings, and was applied to solve undrained stability of a shallow unlined square tunnel. Computational performance between the proposed approach of second‐order cone programming and linear programming was examined and discussed.     

Prediction of fracture trajectory in anisotropic rocks using modified maximum tangential stress criterion

A new criterion to predict crack propagation trajectory in anisotropic rocks with incorporating the concept of T-stress in formulating stress field near the crack tip was developed. The developed criterion along with enrichment functions and interaction integral in the extended finite element method (XFEM) framework made a sophisticated tool in modeling fracturing process in anisotropic media. Numerical results indicated that stress intensity factors considerably depend on orientation of anisotropy axes and ratio of the elastic modulus. The proposed formulation for anisotropic media provides a more accurate prediction of crack propagation trajectory compared with conventional methods, especially in mixed mode conditions.     

Stress‐dependent hardening and failure surfaces of dry sand

During several triaxial compression experiments on plastic hardening, softening, and failure properties of dense sand specimens, it was found on various stress paths that the size of the failure surface was not constant. Instead, it changed depending on the current state of hydrostatic pressure. This finding is in contrast to the standard opinion consisting of the fact that the failure surface remains constant, once it has been reached during an experiment or in situ. In general, the behaviour of cohesionless granular‐material‐like sand is somehow characterised in between fluid and solid, where the solid behaviour results from the angle of internal friction and the confining pressure. Although the friction angle is an intrinsic material property, the confining pressure varies with the boundary conditions, thus defining different solid properties like plastic hardening, softening, and also failure. Based on our findings, it was the goal of the present contribution to introduce an improved setting for the plastic strain hardening and softening behaviour including the newly found yield properties at the limit state. For the identification of the material parameters, a complete triaxial experimental analysis of the tested sand is given. The overall elasto‐plasticity concept is validated by numerical computations of several laboratory foundation‐ and slope‐failure experiments. The performance of the proposed approach is compared with the standard concept of a constant failure surface, where the corresponding yield surfaces are understood as contours of equivalent plastic work or plastic strain. Copyright © 2012 John Wiley & Sons, Ltd.     

岩石变形破坏的熵突变过程与破坏判据

在岩石的变形破坏过程中,当进入不稳定的破裂发展阶段之后,系统不断调整结构抵抗外力的扰动,裂纹向局域集中的有序方向发展,应变能不断耗散,并以应变能的耗散为媒介,使系统与外扰动所追加的负熵流产生能量流通,也使系统熵值和系统的维数逐次降低,这一阶段岩石系统远离平衡态,岩石的破坏是系统熵的突变过程。基于这一认识,并在验证应变能分布与结构块度分布模式的一致性的基础上,推导出包含了结构因素和能量分布的熵表达式;对熵表达式进行平衡分析获得局部突变的分岔集,得到了岩石局部破坏的熵折迭突变破坏准则;同时,探讨了熵表达式所表征的结构有序度的尖点突变性,解出岩石系统的分岔集,这个分岔集就是岩石系统熵突变的整体破坏准则。