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.     

Diffuse failure in geomaterials: Experiments,theory and modelling

This paper presents a synthesis of the works performed by various teams from France, Italy and Canada around the question of second‐order work criterion. Because of the non‐associative character of geomaterials plastic strains, it is now recognized that a whole bifurcation domain exists in the stress space with various possible modes of failure. In a first part these failure modes are observed in lab experimental tests and in discrete element modelling. Then a theoretical study of second‐order work allows to establish a link with the kinetic energy, giving a basis to explain the transition from a prefailure (quasi)static regime to a postfailure dynamic regime. Eventually the main features of geomaterials failure are obtained by applying second‐order work criterion to five different constitutive rate‐independent models—three being phenomenological and two micromechanical. As a whole this paper tries to gather together all the elements for a proper understanding and use of second‐order work criterion in geomechanics. Copyright © 2010 John Wiley & Sons, Ltd.     

Microstructural deformation mechanisms of unsaturated granular soils

A discrete model for unsaturated granular soils has been developed. Three discrete entities have been defined: particles, water menisci and pores. Local interaction forces and water transfer mechanisms have been integrated into a model through the appropriate equilibrium and balance equations. The results of several numerical tests using this model have been described and discussed. Simulations include wetting and drying under load tests, the application of suction cycles and the effect of a deviatoric stress ratio on wetting‐induced collapse. The model reacts just as true granular soil samples behave in laboratory tests. The model provides a new insight into the internal mechanisms leading to large‐scale features of behaviour such as wetting‐induced collapse or the increase in soil strength provided by suction. The paper also stresses that matric suction changes acting on a granular structure are capable of explaining most of the macroscopic features of stress–strain behaviour. Copyright © 2002 John Wiley & Sons, Ltd.     

Material stability analysis of rock joints

For prediction of rockfalls, the failure of rock joints is studied. Considering these failures as constitutive instabilities, a second‐order work criterion is used because it explains all divergence instabilities (flutter instabilities are excluded). The bifurcation domain and the loading directions of instabilities, which fulfill the criterion, are determined for any piecewise linear constitutive relation. The instability of rock joints appears to be ruled by coupling features of the behavior (e.g., dilatancy). Depending on the loading parameters, instabilities can lead to failure, even before the plastic limit criterion. Results for two given constitutive relations illustrate the approach. Some given loading paths are especially considered. Constant volume (undrained) shear and τ‐constant paths are stable or not depending on the link between the deviatoric stress and strain along undrained paths, as found for soils. Some unstable loading paths are illustrated. Along these paths, failure before the plastic limit criterion is possible. The corresponding failure rules are determined. Copyright © 2012 John Wiley & Sons, Ltd.     

Predicting the initiation of static liquefaction of cross‐anisotropic sands under multiaxial stress conditions

Experimental evidence has shown that the liquefaction instability of sands can be affected by its material density, stress state, and inherent anisotropy. In order to predict the initiation of the static liquefaction of inherent cross‐anisotropic sands under multidimensional stress conditions, a rational constitutive model is needed. An elastoplasticity model able to capture the influences of intermediate principal stress ratio (b  = (σ ₂ ? σ ₃)/(σ ₁ ? σ ₃)) and loading direction on stress–strain relationships and volumetric properties was proposed. The yield function was formulated to be controlled by Lode angle, loading direction, and material state; the stress–dilatancy was a material state‐dependent function. After using the existing drained hollow cylinder tests to validate the proposed model, this model was used to simulate the existing undrained hollow cylinder tests. During this simulation, the second‐order work criterion was used to determine the initiation of static liquefaction. The results showed that an increase in both the intermediate principal stress ratio and the loading angle induces a decrease in the second‐order work. Static liquefaction is initiated more easily at a stress state with a large intermediate principal stress ratio and a large loading angle, and the mobilized friction angle at the instability points decreases with the intermediate principal stress ratio and the loading angle. Copyright © 2017 John Wiley & Sons, Ltd.     

From bifurcation to failure in a granular material: a DEM analysis

The question of diffuse failure in granular media such as soils is treated by numerical modelling with a discrete element model. Two numerical samples are considered, one dense and the other loose. A criterion based on the sign of the second-order work is applied to the numerical model to detect bifurcation points related to diffuse failure. The notions of loss of controllability and loss of sustainability are taken into account to simulate diffuse failure at bifurcation points detected. Therefore, the failures predicted based on the second-order work criterion are numerically verified.     

On incremental non‐linearity in granular media: phenomenological and multi‐scale views (Part I)

On the basis of fundamental constitutive laws such as elasticity, perfect plasticity, and pure viscosity, many elasto‐viscoplastic constitutive relations have been developed since the 1970s through phenomenological approaches. In addition, a few more recent micro‐mechanical models based on multi‐scale approaches are now able to describe the main macroscopic features of the mechanical behaviour of granular media. The purpose of this paper is to compare a phenomenological constitutive relation and a micro‐mechanical model with respect to a basic issue regularly raised about granular assemblies: the incrementally non‐linear character of their behaviour. It is shown that both phenomenological and micro‐mechanical models exhibit an incremental non‐linearity. In addition, the multi‐scale approach reveals that the macroscopic incremental non‐linearity could stem from the change in the regime of local contacts between particles (from plastic regime to elastic regime) in terms of the incremental macroscopic loading direction. Copyright © 2005 John Wiley & Sons, Ltd.     

On the failure of a discrete axial chain using a continualized nonlocal Continuum Damage Mechanics approach

The failure of a discrete elastic‐damage axial system is investigated using both a discrete and an equivalent continuum approach. The Discrete Damage Mechanics approach is based on a microstructured model composed of a series of periodic elastic‐damage springs (axial Discrete Damage Mechanics lattice system). Such a discrete damage system can be associated with the finite difference formulation of a Continuum Damage Mechanics evolution problem. Several analytical and numerical results are presented for the tensile failure of this axial damage chain under its own weight. The nonlocal Continuum Damage Mechanics models examined in this paper are mainly built from a continualization procedure applied to centered or uncentered finite difference schemes. The asymptotic expansion of the first‐order upward difference equations leads to a first‐order nonlocal model, whereas the asymptotic expansion of the centered finite difference equations leads to a second‐order nonlocal Eringen's approach. To complete this study, a phenomenological nonlocal gradient approach is also examined and compared with the first continualization methods. A comparison of the discrete and the continuous problems for the chains shows the effectiveness of the new micromechanics‐based nonlocal Continuum Damage modeling, especially for capturing scale effects. For both continualized approaches, the length scale of the nonlocal models depends only on the cell size, while for the so‐called phenomenological approach, the length scale may depend on the loading parameter. This apparent load‐dependent length scale, already discussed in the literature with numerical arguments, is found to be sensitive to the postulated structure of the nonlocal model calibrated according to a lattice approach. Copyright © 2015 John Wiley & Sons, Ltd.     

土的三维破坏准则在颗粒流模型中的应用

Savage-Hutter(S-H)颗粒流模型用一个只与材料参数相关的土压力系数来描述颗粒流内部的应力状态,不能很好地反映颗粒体在运动时的本构关系。通过引入颗粒体应力与速度梯度之间的关系,得到了一个能够反映颗粒流本构模型的崩塌动力学模型。另外,为解除S-H模型中对横向应力大小的假定,通过引入Von Mises、Drucker-Prager、Mohr-Coulomb和Matsuoka-Nakai等土的三维破坏准则,得到了广义摩擦系数的4种表达形式。该模型的主要优势是通过引入颗粒流的本构关系,能较好地体现颗粒流体在运动中的内在机制,并且材料强度参数简单易知。分析了由Drucker-Prager准则和Mohr-Coulomb准则所得到的材料强度参数,并探讨了广义摩擦系数与应力洛德角等物理量之间的关系。用所建议的模型来模拟颗粒流的运动过程,并将数值计算与试验结果进行对比,发现两者能够较好吻合。     

On the capillary stress tensor in wet granular materials

This paper presents a micromechanical study of unsaturated granular media in the pendular regime, based on numerical experiments using the discrete element method, compared with a microstructural elastoplastic model. Water effects are taken into account by adding capillary menisci at contacts and their consequences in terms of force and water volume are studied. Simulations of triaxial compression tests are used to investigate both macro and micro‐effects of a partial saturation. The results provided by the two methods appear to be in good agreement, reproducing the major trends of a partially saturated granular assembly, such as the increase in the shear strength and the hardening with suction. Moreover, a capillary stress tensor is exhibited from capillary forces by using homogenization techniques. Both macroscopic and microscopic considerations emphasize an induced anisotropy of the capillary stress tensor in relation with the pore fluid distribution inside the material. Insofar as the tensorial nature of this fluid fabric implies shear effects on the solid phase associated with suction, a comparison has been made with the standard equivalent pore pressure assumption. It is shown that water effects induce microstructural phenomena that cannot be considered at the macro level, particularly when dealing with material history. Thus, the study points out that unsaturated soil stress definitions should include, besides the macroscopic stresses such as the total stress, the microscopic interparticle stresses such as the ones resulting from capillary forces, in order to interpret more precisely the implications of the pore fluid on the mechanical behaviour of granular materials. Copyright © 2009 John Wiley & Sons, Ltd.     

DEM–FEM analysis of soil failure process via the separate edge coupling method

The concurrent multiscale method, which couples the discrete element method (DEM) for predicting the local micro‐scale evolution of the soil particle skeleton with the finite element method (FEM) for estimating the remaining macro‐scale continuum deformation, is a versatile tool for modeling the failure process of soil masses. This paper presents the separate edge coupling method, which is degenerated from the generalized bridging domain method and is good at eliminating spurious reflections that are induced by coupling models of different scales, to capture the granular behavior in the domain of interest and to coarsen the mesh to save computational cost in the remaining domain. Cundall non‐viscous damping was used as numerical damping to dissipate the kinetic energy for simulating static failure problems. The proposed coupled DEM–FEM scheme was adopted to model the wave propagation in a 1D steel bar, a soil slope because of the effect of a shallow foundation and a plane‐strain cone penetration test (CPT). The numerical results show that the separate edge coupling method is effective when it is adopted for a problem with Cundall non‐viscous damping; it qualitatively reproduces the failure process of the soil masses and is consistent with the full micro‐scale discrete element model. Stress discontinuity is found in the coupling domain. Copyright © 2017 John Wiley & Sons, Ltd.     

Investigation of influence of particle characteristics on the non‐coaxiality of anisotropic granular materials using DEM

As a result of deposition process and particle characteristics, granular materials can be inherently anisotropic. Many researchers have strongly suggested that the inherent anisotropy is the main reason for the deformation non‐coaxiality of granular materials. However, their relationships are not unanimous because of the limited understanding of the non‐coaxial micro‐mechanism. In this study, we investigated the influence of inherent anisotropy on the non‐coaxial angle using the discrete element method. Firstly, we developed a new discrete element method approach using rough elliptic particles and proposed a novel method to produce anisotropic specimens. Secondly, the effects of initial specimen density and particle characteristics, such as particle aspect ratio A _m, rolling resistance coefficient β , and bedding plane orientation δ , were examined by a series of biaxial tests and rotational principal axes tests. Findings from the numerical simulations are summarized as follows: (1) the peak internal friction angle ? _p and the non‐coaxial angle i both increase with the initial density, A _m and β , and they both increase initially and then decrease with δ in the range of 0–90°; (2) among the particle characteristics, the influence of A _m is the most significant; and (3) for anisotropic specimens, the non‐coaxial angle can be calculated using the double slip and rotation rate model. Then, an empirical formula was proposed based on the simulation results to depict the relationship between the non‐coaxial angle and the particle characteristics. Finally, the particle‐scale mechanism of non‐coaxiality for granular materials was discussed from the perspective of energy dissipation. Copyright © 2016 John Wiley & Sons, Ltd.     

Bond rolling resistance and its effect on yielding of bonded granulates by DEM analyses

A discrete element modelling of bonded granulates and investigation on the bond effect on their behaviour are very important to geomechanics. This paper presents a two‐dimensional (2‐D) discrete element theory for bonded granulates with bond rolling resistance and provides a numerical investigation into the effect of bond rolling resistance on the yielding of bonded granulates. The model consists of mechanical contact models and equations governing the motion of bonded particles. The key point of the theory is that the assumption in the original bond contact model previously proposed by the authors (55th CSCE‐ASCE Conference, Hamilton, Ont., Canada, 2002; 313–320; J. Eng. Mech. (ASCE) 2005; 131 (11):1209–1213) that bonded particles are in contact at discrete points, is here replaced by a more reliable assumption that bonded particles are in contact over a width. By making the idealization that the bond contact width is continuously distributed with the normal/tangential basic elements (BE) (each BE is composed of spring, dashpot, bond, slider or divider), we establish a bond rolling contact model together with bond normal/tangential contact models, and also relate the governing equations to local equilibrium. Only one physical parameter β needs to be introduced in the theory in comparison to the original bond discrete element model. The model has been implemented into a 2‐D distinct element method code, NS2D. Using the NS2D, a total of 86 1‐D, constant stress ratio, and biaxial compressions tests have been carried out on the bonded granular samples of different densities, bonding strengths and rolling resistances. The numerical results show that: (i) the new theory predicts a larger internal friction angle, a larger yielding stress, more brittle behaviour and larger final broken contact ratio than the original bond model; (ii) the yielding stress increases nonlinearly with the increasing value of β, and (iii) the first‐yield curve (initiation of bond breakage), which define a zone of none bond breakage and which shape and size are affected by the material density, is amplified by the bond rolling resistance in analogous to that predicted by the original bond model. Copyright © 2006 John Wiley & Sons, Ltd.     

Micromechanical analysis of the failure process of brittle rock

The failure process of brittle rock submitted to a compression state of stress with different confining pressures is investigated in this paper based on discrete element method (DEM) simulations. In the DEM model, the rock sample is represented by bonding rigid particles at their contact points. The numerical model is first calibrated by comparing the macroscopic response with the macroscopic response of Beishan granite obtained from laboratory tests. After the validation of numerical model in terms of macroscopic responses, the failure process of the DEM model under unconfined and confined compression is studied in micro‐scale in detail. The contact force network and its relation to the development of micro‐cracks and evolution of major fractures are studied. Confining pressure will prohibit the development of tensile cracks and hence alter the failure patterns. An in‐depth analysis of micro‐scale response is carried out, including the orientation distribution and probability density of stress acting on parallel bonds, the effect of particle size heterogeneity on bond breakage and the evolution of fabric tensor and coordination number of parallel bond. The proposed micromechanical analysis will allow us to extract innovative features emerged from the stresses and crack evolution in brittle rock failure process. Copyright © 2014 John Wiley & Sons, Ltd.     

Solid–fluid transition modelling in geomaterials and application to a mudflow interacting with an obstacle

Given the contrasting behaviour observed for geomaterials, for example, during landslides of the flow type, this contribution proposes an original constitutive model, which associates both an elasto‐plastic relation and a Bingham viscous law linked by a mechanical transition criterion. This last is defined as the second‐order work sign for each material point, which is a general criterion for divergence instabilities. Finite element method with Lagrangian integration points is chosen as a framework for implementing the new model because of its well‐known ability to deal with both solid and fluid behaviours in large deformation processes. A first boundary model considering a sample of initially stable soil, a slope and an obstacle is performed. The results show the power of the constitutive model because the consistent evolution of initiation, propagation and arrest of the mudflow is described. A parametric study is led on various plastic and viscous parameters to determine their influence on the flow development and arrest. Finally, forces against the obstacle are compared with good agreement with those of other authors for the same geometry and a pure viscous behaviour. Copyright © 2014 John Wiley & Sons, Ltd.     

Understanding heavy mineral enrichment using a three‐dimensional numerical model

Layered deposits of relatively light and heavy minerals can be found in many aquatic environments. Quantification of the physical processes which lead to the fine‐scale layering of these deposits is often limited with flumes or in situ field experiments. Therefore, the following research questions were addressed: (i) how can selective grain entrainment be numerically simulated and quantified; (ii) how does a mixed bed turn into a fully layered bed; and (iii) is there any relation between heavy mineral content and bed stability? Herein, a three‐dimensional numerical model was used as an alternative measure to study the fine‐scale process of density segregation during transport. The three‐dimensional model simulates particle transport in water by combining a turbulence‐resolving large eddy simulation with a discrete element model prescribing the motion of individual grains. The granular bed of 0·004 m in height consisted of 200 000 spherical particles (D50 = 500 μ m). Five suites of experiments were designed in which the concentration ratio of heavy (5000 kg m⁻³) to light particles (i.e. 2560 kg m⁻³) was increased from 6%, 15%, 35%, 60% to 80%. All beds were tested for 10 sec at a predefined flow speed of 0·3 m sec⁻¹. Analysis of the particle behaviour in the interior of the beds showed that the lighter particles segregated from the heavy particles with increasing time. The latter accumulated at the bottom of the domain, forming a layer, whereas the lighter particles were transported over the layer forming sweeps. Particles below the heavy particle layer indicated that the layer was able to armour the particles below. Consequentially, enrichment of heavy minerals in a layer is controlled by the segregation of a heavy mineral fraction from the light counterpart, which enhances current understanding of heavy mineral placer formation.     

Pore‐scale modeling of fluid‐particles interaction and emerging poromechanical effects

A micro‐hydromechanical model for granular materials is presented. It combines the discrete element method for the modeling of the solid phase and a pore‐scale finite volume formulation for the flow of an incompressible pore fluid. The coupling equations are derived and contrasted against the equations of conventional poroelasticity. An analogy is found between the discrete element method pore‐scale finite volume coupling and Biot's theory in the limit case of incompressible phases. The simulation of an oedometer test validates the coupling scheme and demonstrates the ability of the model to capture strong poromechanical effects. A detailed analysis of microscale strain and stress confirms the analogy with poroelasticity. An immersed deposition problem is finally simulated and shows the potential of the method to handle phase transitions. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on the generation of periodic granular microstructures based on grain size distributions

This short communication discusses an algorithm suited for the generation of periodic microstructures of granular media. Its particular features are a user‐defined grain size distribution, a representative volume element which is intrinsically periodic ab initio and a user‐defined termination criterion, controlled by an increase of volume fraction. For low densities our particle packings resemble fluids or gases, while we aim to reach for rather dense particle packings, similar to granular solids. The generated microstructures can thus be readily incorporated into large multiscale simulations, e.g. on the integration point level of a finite element analysis of a particular sand or concrete. The individual grain size distribution of the granular medium is incorporated through the introduction of different growth rates governing the final particle size distribution. We briefly sketch the generation of the representative volume element within a serial event‐driven scheme and demonstrate how periodic boundary conditions are ensured throughout the representative volume element generation process. The potential of the suggested algorithm will be illustrated through the generation of two different periodic multi‐disperse microstructures. They are based on different given grain size distributions, one for a quartz sand with a low non‐uniformity index and one for concrete aggregates classified as A32 by the German standard norm DIN 1045 to have a rather large variation in grain size. Copyright © 2007 John Wiley & Sons, Ltd.     

Bifurcation and second‐order work in geomaterials

In this paper, the ability of a material rate‐independent system to evolve toward another mechanical state from an equilibrium configuration, with no change in the control parameters, is investigated. From a mechanical point of view, this means that the system can spontaneously develop kinetic energy with no external disturbance from an equilibrium state, which corresponds to a particular case of bifurcation. The existence of both conjugate incremental strain and stress such that the second‐order work vanishes is established as a necessary and sufficient condition for the appearance of this bifurcation phenomenon. It is proved that this fundamental result is independent of the constitutive relation of the rate‐independent material considered. Then the case of homogeneous loading paths is investigated, and, as an illustration, the subsequent results are applied to interpret the well‐known liquefaction observed under isochoric triaxial loading conditions with loose granular materials. Copyright © 2006 John Wiley & Sons, Ltd.