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.     

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.     

On failure criteria for anisotropic cohesive‐frictional materials

Anisotropic failure criteria are formulated using two different approaches. The first one employs a spatial distribution of strength parameters and defines the failure condition in terms of traction components acting on the critical plane. The second one incorporates a microstructure tensor and the relevant mixed invariants. Both formulations are illustrated by some numerical examples. In particular, the variation of strength with orientation of the sample is examined for a series of uniaxial compression tests. Copyright © 2001 John Wiley & Sons, Ltd.     

Three‐dimensional shakedown solutions for anisotropic cohesive‐frictional materials under moving surface loads

Previous work on three‐dimensional shakedown analysis of cohesive‐frictional materials under moving surface loads has been entirely for isotropic materials. As a result, the effects of anisotropy, both elastic and plastic, of soil and pavement materials are ignored. This paper will, for the first time, develop three‐dimensional shakedown solutions to allow for the variation of elastic and plastic material properties with direction. Melan's lower‐bound shakedown theorem is used to derive shakedown solutions. In particular, a generalised, anisotropic Mohr–Coulomb yield criterion and cross‐anisotropic elastic stress fields are utilised to develop anisotropic shakedown solutions. It is found that shakedown solutions for anisotropic materials are dominated by Young's modulus ratio for the cases of subsurface failure and by shear modulus ratio for the cases of surface failure. Plastic anisotropy is mainly controlled by material cohesion ratio, the rise of which increases the shakedown limit until a maximum value is reached. The anisotropic shakedown limit varies with frictional coefficient, and the peak value may not occur for the case of normal loading only. Copyright © 2013 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.     

Scratch hardness–strength solutions for cohesive‐frictional materials

In this paper we develop analytical solutions for scratch hardness–strength relations for cohesive‐frictional materials of the Mohr–Coulomb and Drucker–Prager type. Based on the lower bound yield design approach, closed‐form solutions are derived for frictionless scratch devices, and validated against computational upper bound and elastoplastic finite element solutions. The influence of friction at the blade–material interface is also investigated, for which a simple computational optimization is proposed. Illustrated for scratch tests on cement paste, we show that the proposed solutions provide a convenient way to determine estimates of cohesion and friction parameters from scratch data, and may serve as a benchmark to identify the relevance of strength models for scratch test analysis. Copyright © 2011 John Wiley & Sons, Ltd.     

Strain‐softening analysis of a spherical cavity

This paper studies the excavation of a spherical cavity subjected to hydrostatic initial stresses in the infinite homogeneous and isotropic rock mass with strain‐softening Mohr–Coulomb (M‐C) and Hoek–Brown (H‐B) behaviors. Numerical solutions of the spherical cavity are obtained and the application to determining stress–strain curve of strain‐softening M‐C and H‐B rock mass is studied. A closed‐form solution for the elastic–brittle–plastic medium is introduced first, and then a numerical procedure that simplifies the strain‐softening process into a series of brittle–plastic ones is presented. The approach is validated against the facts that the strain‐softening process evolves into a brittle–plastic one when the softening slope is very steep, whereas it evolves into an elasto‐plastic one when the softening slope approaches zero. Numerical solutions for the prediction of displacements and stresses around the spherical cavity in the strain‐softening M‐C and H‐B rock mass are presented. On the basis of the analysis of the spherical cavity in strain‐softening rock mass, the stress–strain relationship at an infinitesimal cube around the cavity is obtained and discussed with different evolution laws for the strength parameters considered. Copyright © 2011 John Wiley & Sons, Ltd.     

Approximate model for blunt objects indenting cohesive‐frictional materials

An approximate two‐dimensional model for indentation of blunt objects into various types of rigid‐perfectly plastic cohesive‐frictional material is derived. Particular emphasis is placed on considering indentation as a process involving evolution of the boundary of material displaced by the indenter. Force–penetration relationships are obtained by an incremental approach utilizing key kinematic and static information from indentation of a flat punch. Albeit approximate, the proposed model applies to arbitrary indenter geometry and weightless or ponderable cohesive‐frictional materials exhibiting associated or non‐associated plastic flow. Two specific indenter geometries, the cylinder and blunt wedge, are explored in detail. Favorable agreement is found between the analytic results and those obtained using the finite element method (FEM). For both the wedge and cylinder, it is further shown that accurate analytic expressions relating indentation force explicitly to penetration can be derived. In the case of the wedge and weightless material, predictions of indentation force obtained from the derived expressions are very close to those computed from implicit equations available in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Damage‐viscoplastic consistency model for rock fracture in heterogeneous rocks under dynamic loading

This paper presents a damage‐viscoplastic consistency model for numerical simulation of brittle fracture in heterogeneous rocks. The model is based on a combination of the recent viscoplastic consistency model by Wang and the isotropic damage concept with separate damage variables in tension and compression. This approach does not suffer from ill‐posedness, caused by strain softening, of the underlying boundary/initial value problem since viscoplasticity provides the regularization by introducing a length scale effect under dynamic loading conditions. The model uses the Mohr–Coulomb yield criterion with the Rankine criterion as a tensile cut‐off. The damage law in compression is calibrated via the degradation index concept of Fang and Harrison. Thereby, the model is able to capture the brittle‐to‐ductile transition occurring in confined compression at a certain level of confinement. The heterogeneity of rock is accounted for by the statistical approach based on the Weibull distribution. Numerical simulations of confined compression test in plane strain conditions demonstrate a good agreement with the experiments at both the material point and structural levels as the fracture modes are realistically predicted. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Friction effect on stresses in ensiled granular media

We develop, in this paper, an analytical approach of continuous medium type which improves the theory of Janssen and other existing approaches of the same type. This approach allows to calculate the stresses in any point of the ensiled granular medium and to represent as well qualitatively as quantitatively the stress saturation effect. The influence of the friction angles is studied.     

Multi‐slicing strategy for the three‐dimensional discontinuity layout optimization (3D DLO)

Discontinuity layout optimization (DLO) is a recently presented topology optimization method for determining the critical layout of discontinuities and the associated upper bound limit load for plane two‐dimensional and three‐dimensional (3D) problems. The modelling process (pre‐processing) for DLO includes defining the discontinuities inside a specified domain and building the target function and the global constraint matrix for the optimization solver, which has great influence on the the efficiency of the computation processes and the reliability of the final results. This paper focuses on efficient and reliable pre‐processing of the discontinuities within the 3D DLO and presents a multi‐slicing strategy, which naturally avoids the overlapping and crossing of different discontinuities. Furthermore, the formulation of the 3D discontinuity considering a shape of an arbitrary convex polygon is introduced, permitting the efficient assembly of the global constraint matrix. The proposed method eliminates unnecessary discontinuities in 3D DLO, making it possible to apply 3D DLO for solving large‐scale engineering problems such as those involving landslides. Numerical examples including a footing test, a 3D landslide and a punch indentation are considered, illustrating the effectiveness of the presented method. © 2016 The Authors. International Journal for Numerical and Analytical Methods in Geomechanics published by John Wiley & Sons Ltd.     

Elastic‐plastic solutions for expanding cavities embedded in two different cohesive‐frictional materials

An analytical solution of cavity expansion in two different concentric regions of soil is developed and investigated in this paper. The cavity is embedded within a soil with finite radial dimension and surrounded by a second soil, which extends to infinity. Large‐strain quasi‐static expansion of both spherical and cylindrical cavities in elastic‐plastic soils is considered. A non‐associated Mohr–Coulomb yield criterion is used for both soils. Closed‐form solutions are derived, which provide the stress and strain fields during the expansion of the cavity from an initial to a final radius. The analytical solution is validated against finite element simulations, and the effect of varying geometric and material parameters is studied. The influence of the two different soils during cavity expansion is discussed by using pressure–expansion curves and by studying the development of plastic regions within the soils. The analytical method may be applied to various geotechnical problems, which involve aspects of soil layering, such as cone penetration test interpretation, ground‐freezing around shafts, tunnelling, and mining. © 2014 The Authors. International Journal for Numerical and Analytical Methods in Geomechanics published by John Wiley & Sons Ltd.     

A discrete thermodynamic approach for anisotropic plastic–damage modeling of cohesive‐frictional geomaterials

A discrete plastic–damage model is developed for cohesive‐frictional geomaterials subjected to compression‐dominated stresses. Macroscopic plastic strains of material are physically generated by frictional sliding along weakness planes. The evolution of damage is related to the evolution of weakness planes physically in connection with the propagation of microcracks. A discrete approach is used to account for anisotropic plastic flow and damage evolution, by introducing two stress invariants and one plastic hardening variable for each family of sliding weakness planes. Plastic flow in each family is coupled with damage evolution. The proposed model is applied to typical geomaterials and comparisons between numerical predictions and experimental data are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite volume–based modeling of flow‐induced shear failure along fracture manifolds

In this paper, a numerical model to predict flow‐induced shear failure along pre‐existing fractures is presented. The framework is based on a discrete fracture representation embedded in a continuum describing the damaged matrix. A finite volume method is used to compute both flow and mechanical equilibrium, whereas specifically tailored basis functions are used to account for the physics at discontinuities. The failure criterion is based on a maximum shear strength limit, which changes with varying compressive stress on the fracture manifold. The displacements along fracture manifolds are obtained such that force balance is achieved under conditions, where shear stress of the failing fracture segment is constrained to the maximum shear strength at the segment. Simultaneously, the fluid pressure is computed independently of the shear slip. A relaxation model approach is used to obtain the maximum shear limit on the fracture manifold, which leads to grid convergence.     

Enriched finite element procedures for analyzing decoupled bolts installed in rock mass

Numerical procedures are developed to analyze interaction between fully grouted bolts and rock mass using ‘enriched finite element method (EFEM)’. A solid element intersected by a rock bolt along any arbitrary direction is termed as ‘enriched’ element. The nodes of an enriched element have additional degrees of freedom for determining displacements, stresses developed in the bolt rod. The stiffness of the enriched element is formulated based on properties of rock mass, bolt rod and grout, orientation of the bolt and borehole diameter. Decoupling at grout–bolt interface and elasto‐plastic behavior of rock mass have also been incorporated into the EFEM procedures. The results of this method are compared with analytical pull‐out test results presented by Li and Stillborg (Int. J. Rock Mech. Min. Sci. 1999; 36 :1013–1029). In addition, a numerical example of a bolted tunnel is provided to demonstrate the efficacy of the proposed method for practical applications. Copyright © 2010 John Wiley & Sons, Ltd.     

Evaluation of a constitutive model for clays and sands: Part I – sand behaviour

This paper evaluates the performance of a generalized effective stress soil model for predicting the rate independent behaviour of freshly deposited sands, while a companion paper describes model capabilities for clays and silts. Most material parameters can be obtained from standard laboratory data, including hydrostatic or one‐dimensional compression, drained and undrained triaxial shear testing. A compilation of data on compression behaviour allows for estimation of compression parameters when this type of data is not available. Extensive comparisons of model predictions with measured data from undrained triaxial shear tests shows that the model gives excellent predictions of the transition from dilative to contractive shear response as the confining pressure and/or the initial formation void ratio increases. A parametric study of drained response shows that the model describes realistically the variation of peak friction angle and dilation rate as a function of confining pressure and density when compared with an empirical correlation valid for many sands. The proposed formulation predicts a unique critical state locus for both drained and undrained triaxial testing which is non‐linear over the entire range of stresses and is in excellent agreement with recent experimental data. Overall, the model provides excellent predictions of the stress–strain–strength relationships over a wide range of confining pressures and formation densities. Copyright © 2002 John Wiley & Sons, Ltd.     

Localized failure of fibre‐reinforced elastic–plastic materials subjected to plane strain loading

We consider discontinuous bifurcations as the indicator of a localized failure for a class of composites that are characterized by elastic fibres reinforcing an elastic–plastic matrix. A macroscopic tangent stiffness tensor for the fibre‐reinforced composite is developed by consistently homogenizing the contribution of fibres in a spherical representative volume element. Analytical solutions are derived for the critical hardening modulus and corresponding bifurcation directions for the case of plane strain loading. Properties of the solutions are further illustrated on the example of the non‐associated Drucker–Prager model at onset of yielding. Results show that presence of fibres decreases the critical hardening modulus, thus inhibiting the onset of strain localization. The rate of decrease in the critical hardening modulus is the highest for pure shear, followed by uniaxial tension, uniaxial compression, biaxial tension and biaxial compression. The main fibre parameters that control the onset of strain localization are their volumetric content and their stiffness modulus whereby very stiff fibres can produce the most significant decrease in the critical hardening modulus, especially for the state of biaxial tension. The critical hardening modulus for the non‐associated Drucker–Prager model exhibits a full range of localization modes including compaction bands, dilation bands, and transition in the form of shear bands regardless of the presence of fibres. Presence of fibres affects bifurcation directions, except in the case when Poisson's ratio of the matrix is equal to 0.25. The results demonstrate stabilizing effects of fibres by which they provide the control against the onset of strain localization. Copyright © 2006 John Wiley & Sons, Ltd.     

A finite strain closed‐form solution for the elastoplastic ground response curve in tunnelling

The ground response to tunnel excavation is usually described in terms of the characteristic line of the ground (also called ‘ground response curve’, GRC), which relates the support pressure to the displacement of the tunnel wall. Under heavily squeezing conditions, very large convergences may take place, sometimes exceeding 10–20% of the excavated tunnel radius, whereas most of the existing formulations for the GRC are based on the infinitesimal deformation theory. This paper presents an exact closed‐form analytical solution for the ground response around cylindrical and spherical openings unloaded from isotropic and uniform stress states, incorporating finite deformations and linearly elastic‐perfectly plastic rock behaviour obeying the Mohr–Coulomb failure criterion with a non‐associated flow rule. Additionally, the influence of out‐of‐plane stress in the case of cylindrical cavities under plane‐strain conditions is examined. The solution is presented in the form of dimensionless design charts covering the practically relevant parameter range. Finally, an application example is included with reference to a section of the Gotthard Base tunnel crossing heavily squeezing ground. The expressions derived can be used for preliminary convergence assessments and as valuable benchmarks for finite strain numerical analyses. Copyright © 2014 John Wiley & Sons, Ltd.