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.     

Multiscale insights into classical geomechanics problems

We pay a revisit to some classical geomechanics problems using a novel computational multiscale modelling approach. The multiscale approach employs a hierarchical coupling of the finite element method (FEM) and the discrete element method. It solves a boundary value problem at the continuum scale by FEM and derives the material point response from the discrete element method simulation attached to each Gauss point of the FEM mesh. The multiscale modelling framework not only helps successfully bypass phenomenological constitutive assumptions as required in conventional modelling approaches but also facilitates effective cross‐scale interpretation and understanding of soil behaviour. We examine the classical retaining wall and footing problems by this method and demonstrate that the simulating results can be well validated and verified by their analytical solutions. Furthermore, the study sheds novel multiscale insights into these classical problems and offers a new tool for geotechnical engineers to design and analyse geotechnical applications based directly upon particle‐level information of soils. Copyright © 2015 John Wiley & Sons, Ltd.     

Softening,localization and stabilization: capture of discontinuous solutions in J2 plasticity

This paper exploits the concept of stabilization techniques to improve the behaviour of mixed linear/linear simplicial elements (triangles and tetrahedra) in incompressible or nearly incompressible situations. Elasto-J2-plastic constitutive behaviour has been considered with linear and exponential softening. Two different stabilization methods are used to attain global stability of the corresponding discrete finite element formulation. Implementation and computational aspects are also discussed, showing that a robust application of the proposed formulation is feasible. Numerical examples show that the formulation derived is free of volumetric locking and spurious oscillations of the pressure. The results obtained do not suffer from spurious mesh-size or mesh-bias dependence, comparing very favourably with those obtained with the standard, non-stabilized, approaches. Copyright © 2004 John Wiley & Sons, Ltd.     

Interface and embedded joint methods for modeling dynamic fracture opening by explosive products

Two computational approaches are proposed in the paper to model dynamic fracture opening by explosive products. The first method assumes that the fractures may be modeled using flow elements embedded along the mesh lines. This method models crack opening in a straightforward way by splitting the nodes of the computational grid. It can account for crack branching; however, the crack directions are constrained by existing mesh faces, which may lead to mesh dependence. Also, the stress in flow elements is calculated explicitly separate from the surrounding solid elements that can impose additional limits on the time step stability condition for explicit integration. The second approach uses embedded flow elements to model the cracks. Typical thickness of the cracks is much smaller than the element size. Therefore, gas pressure in the cracks is assumed to be in stress equilibrium with the element stress. To achieve this, the crack thickness and the state of the gas is updated simultaneously with the state of the solid element which contains the crack. Therefore, the time step is controlled by the explicit solver applied for the solid and does not depend on the thickness of the crack. The main disadvantage of the second approach is due to the complexity of modeling multiple intersecting cracks, which go through the same element. We discuss the areas of possible applications of these 2 methods and the ways to improve and enhance them for future practical applications.     

Modelling of coupled fluid‐mechanical problems in fractured geological media using enriched finite elements

Geological environments, such as petroleum reservoirs, normally exhibit physical discontinuities, for example, fractures and faults. Because of the reduced thickness of these discontinuities, finite element formulations with strong discontinuity have been applied to the numerical modelling of geological environments. Until now, two relevant characteristics of petroleum reservoirs have not been addressed by these formulations. The first is the pore pressure jump in the direction normal to a discontinuity in a fluid‐mechanical coupling condition, which is present primarily in sealing faults owing to the contrast of permeability with the porous medium. The absence of this jump can affect the prediction of the deformability of a physical discontinuity. Furthermore, reservoir models frequently use coarse meshes. Thus, the method used to evaluate the pore pressure in the discontinuity may exhibit a strong dependence relative to the mesh refinement. Based on these characteristics, in this study, a formulation of an enriched finite element for application to coupled fluid‐mechanical problems with pre‐existing physical discontinuities saturated by a single fluid is presented. The formulation employs discontinuous interpolation functions and enables the reproduction of jumps of displacement and pore pressure associated with a discontinuity inside the element without the need to discretise it. An approximation to estimate the pore pressure in the discontinuity was developed, one which seeks to minimise the influence of refinement. The element's response is verified by comparison with a one‐dimensional analytical solution and simple examples that are simulated using commercial software. Copyright © 2015 John Wiley & Sons, Ltd.     

Equivalent continuum strain calculations based on 3D particle kinematic measurements of sand

Constitutive modeling of granular materials has been a subject of extensive research for many years. While the calculation of the Cauchy stress tensor using the discrete element method has been well established in the literature, the formulation and interpretation of the strain tensor are not as well documented. According to Bagi, 1 researchers mostly adopt well‐known continuum or discrete microstructural approaches to calculate strains within granular materials. However, neither of the 2 approaches can fully capture the behavior of granular materials. They are considered complementary to each other where each has its own strengths and limitations in solving granular‐mechanics problems. Zhang and Regueiro 2 proposed an equivalent continuum approach to calculating finite strain measures at the local level in granular materials subjected to large deformations. They used three‐dimensional discrete element method results to compare the proposed strains measures. This paper presents an experimental application of the Zhang and Regueiro 2 approach using three‐dimensional synchrotron microcomputed tomography images of a sheared Ottawa sand specimen. Invariant Eulerian finite strain measures were calculated for representative element volumes within the specimen. The spatial maps of Eulerian octahedral shear and volumetric strain were used to identify zones of intense shearing within the specimen and compared well with maps of incremental particle translation and rotation for the same specimen. The local Eulerian volumetric strain was compared to the global volumetric strains, which also can be considered as an averaging of all local Eulerian volumetric strains.     

Discrete element model for crack propagation in brittle materials

We propose a discrete element model for brittle rupture. The material consists of a bidimensional set of closed‐packed particles in contact. We explore the isotropic elastic behavior of this regular structure to derive a rupture criterion compatible to continuum mechanics. We introduce a classical criterion of mixed mode crack propagation based on the value of the stress intensity factors, obtained by the analysis of two adjacent contacts near a crack tip. Hence, the toughness becomes a direct parameter of the model, without any calibration procedure. We verify the consistency of the formulation as well as its convergence by comparison with theoretical solutions of tensile cracks, a pre‐cracked beam, and an inclined crack under biaxial stress. Copyright © 2015 John Wiley & Sons, Ltd.     

扩展有限元法在岩石断裂力学中的应用

岩石裂纹的扩展是一个经典的不连续问题,常规有限元方法难以实现裂纹扩展过程的仿真模拟。扩展有限元法(XFEM)实现了计算网格与不连续面相互独立,因此模拟移动的不连续面时无需对网格进行重新剖分。本文介绍了XFEM基本原理和岩石断裂力学常用判据,尝试对岩石类材料单缝Ⅰ型三点弯曲、单缝剪切和双缝平板实验进行模拟。分析结果表明:扩展有限元模拟岩石类材料断裂问题不受网格划分限制,裂纹以实际应力场分布随机扩展;直观地给出岩样的微裂纹产生、演化,直至完全破坏的全过程,并与实验结果吻合。该方法能够应用到岩石断裂力学方面的研究,模拟岩石类材料的宏细观破坏过程,为解决复杂问题提供了方便的途径。     

Numerical modeling of dynamic rock fracture with a combined 3D continuum viscodamage‐embedded discontinuity model

This paper deals with numerical modeling of dynamic failure phenomena in rate‐sensitive quasi‐brittle materials, such as rocks, with initial microcrack populations. To this end, a continuum viscodamage‐embedded discontinuity model is developed and tested in full 3D setting. The model describes the pre‐peak nonlinear and rate‐sensitive hardening response of the material behavior, representing the fracture‐process zone creation, by a rate‐dependent continuum damage model. The post‐peak response, involving the macrocrack creation accompanied by exponential softening, is formulated by using an embedded displacement discontinuity model. The finite element implementation of this model relies upon the linear tetrahedral element, which seems appropriate for explicit dynamic analyses involving stress wave propagation. The problems of crack locking and spreading typical of embedded discontinuity models are addressed in this paper. A combination of two remedies, the inclusion of viscosity in the spirit of Wang's viscoplastic consistency approach and introduction of isotropic damaging into the embedded discontinuity model, is shown to be effective in the present explicit dynamics setting. The model performance is illustrated by several numerical simulations. In particular, the dynamic Brazilian disc test and the Kalthoff–Winkler experiment show that the present model provides realistic predictions with the correct failure modes and rate‐dependent tensile strengths of rock at different loading rates. The ability of initial embedded discontinuity populations to model the initial microcrack populations in rocks is also successfully tested. Copyright © 2016 John Wiley & Sons, Ltd.     

XFEM formulation with sub-interpolation,and equivalence to zero-thickness interface elements

This paper describes a particular formulation of the extended finite element method (XFEM) specifically conceived for application to existing discontinuities of fixed location, for instance, in geological media. The formulation is based on two nonstandard assumptions: (1) the use of sub-interpolation functions for each subdomain and (2) the use of fictitious displacement variables on the nodes across the discontinuity (instead of the more traditional jump variables). Thanks to the first of those assumptions, the proposed XFEM formulation may be shown to be equivalent to the standard finite element method with zero-thickness interface elements for the discontinuities (FEM+z). The said equivalence is theoretically proven for the case of quadrangular elements cut in two quadrangles by the discontinuity, and only approximate for other types of intersections of quadrangular or triangular elements, in which the XFEM formulation corresponds to a kinematically restricted version of the corresponding interface plus continuum scheme. The proposed XFEM formulation with sub-interpolation, also helps improving spurious oscillations of the results obtained with natural interpolation functions when the discontinuity runs skew to the mesh. A possible explanation for these oscillations is provided, which also explains the improvement observed with sub-interpolation. The paper also discusses the oscillations observed in the numerical results when some nodes are too close to the discontinuity and proposes the remedy of moving those nodes onto the discontinuity itself. All the aspects discussed are illustrated with some examples of application, the results of which are compared with closed-form analytical solutions or to existing XFEM results from the literature.     

3‐D mesoscale simulation of crack‐permeability coupling in the Brazilian splitting test

In this paper, mesoscale hydromechanical simulations are performed to study (1) fracture features and (2) crack‐gas permeability coupling evolution in the context of the tensile splitting test. The mesostructure is based on a 2‐phase 3‐D representation of heterogeneous materials, such as concrete, where stiff aggregates are embedded into a mortar matrix. To take into account these heterogeneities without any mesh adaptation, a weak discontinuity is introduced into the strain field. In addition, a strong discontinuity is also added to take into account microcracking. This mechanical model is cast into the framework of the enhanced finite element method. Concerning the coupling with gas permeability, a double‐porosity method is used to simulate the flow through the cracks and the porosity. The apparent gas permeability is afterwards evaluated by a homogenization method. On the basis of finite element simulations, influence of aggregate size on ultimate crack opening, macroscopic ultimate tensile stress, total dissipated energy, and gas permeability evolution is numerically investigated. Furthermore, gas permeability evolution is also compared with experimental results from the literature. In addition, in the spirit of a sequential multiscale approach, macroscale gas permeability equations are identified from the hydromechanical results coming from the mesoscale computations. These equations lead to a relation between macroscale gas permeability evolution and crack opening. Besides, we show how the aggregate size influences the percolation threshold and that after this threshold, a cubic relation between macroscale gas permeability and crack opening is obtained.     

Coupled HM analysis using zero‐thickness interface elements with double nodes. Part I: Theoretical model

In recent years, the authors have proposed a new double‐node zero‐thickness interface element for diffusion analysis via the finite element method (FEM) (Int. J. Numer. Anal. Meth. Geomech. 2004; 28 (9): 947–962). In the present paper, that formulation is combined with an existing mechanical formulation in order to obtain a fully coupled hydro‐mechanical (or HM) model applicable to fractured/fracturing geomaterials. Each element (continuum or interface) is formulated in terms of the displacements (u) and the fluid pressure (p) at the nodes. After assembly, a particular expression of the traditional ‘u–p’ system of coupled equations is obtained, which is highly non‐linear due to the strong dependence between the permeability and the aperture of discontinuities. The formulation is valid for both pre‐existing and developing discontinuities by using the appropriate constitutive model that relates effective stresses to relative displacements in the interface. The system of coupled equations is solved following two different numerical approaches: staggered and fully coupled. In the latter, the Newton–Raphson method is used, and it is shown that the Jacobian matrix becomes non‐symmetric due to the dependence of the discontinuity permeability on the aperture. In the part II companion paper (Int. J. Numer. Anal. Meth. Geomech. 2008; DOI: 10.1002/nag.730 ), the formulation proposed is verified and illustrated with some application examples. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive finite element analysis of mode I fracture in cement‐based materials

Microscopic studies using advanced experimental techniques have provided better insight into the fracture mechanisms in cement‐based materials. A clear understanding of fracture mechanisms is critical for the development of rigorous computational models for analysing fracture. Fracture analysis is usually carried out by finite element method. Accuracy of FE analysis depends upon the choice of mesh and for the predictions to be reliable, discretization errors are to be minimized. In cohesive crack approach, the non‐linearity is limited to the boundary conditions along the geometric discontinuity while the bulk of the material retains its elastic nature. The paper presents a mesh‐adaptive strategy based on ZZ error estimator to model discrete crack propagation in cement‐based materials. Examples of simulations have demonstrated the potential of the mesh‐adaptive technique in modelling the evolution of the localized strain profiles as well as failure of concrete test specimen. Copyright © 2001 John Wiley & Sons, Ltd.     

Weak discontinuity in porous media: an enriched EFG method for fully coupled layered porous media

In this paper, a series of multimaterial benchmark problems in saturated and partially saturated two‐phase and three‐phase deforming porous media are addressed. To solve the process of fluid flow in partially saturated porous media, a fully coupled three‐phase formulation is developed on the basis of available experimental relations for updating saturation and permeabilities during the analysis. The well‐known element free Galerkin mesh‐free method is adopted. The partition of unity property of MLS shape functions allows for the field variables to be extrinsically enriched by appropriate functions that introduce existing discontinuities in the solution field. Enrichment of the main unknowns including solid displacement, water phase pressure, and gas phase pressure are accounted for, and a suitable enrichment strategy for different discontinuity types are discussed. In the case of weak discontinuity, the enrichment technique previously used by Krongauz and Belytschko [Int. J. Numer. Meth. Engng., 1998; 41:1215–1233] is selected. As these functions possess discontinuity in their first derivatives, they can be used for modeling material interfaces, generating only minor oscillations in derivative fields (strain and pressure gradients for multiphase porous media), as opposed to unenriched and constrained mesh‐free methods. Different problems of multimaterial poro‐elasticity including fully saturated, partially saturated one, and two‐phase flows under the assumption of fully coupled extended formulation of Biot are examined. As a further development, problems involved with both material interface and impermeable discontinuities, where no fluid exchange is permitted across the discontinuity, are considered and numerically discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Three‐dimensional simulations of tensile cracks in geomaterials by coupling meshless and finite element method

Failure in geotechnical engineering is often related to tension‐induced cracking in geomaterials. In this paper, a coupled meshless method and FEM is developed to analyze the problem of three‐dimensional cracking. The radial point interpolation method (RPIM) is used to model cracks in the smeared crack framework with an isotropic damage model. The identification of the meshless region is based on the stress state computed by FEM, and the adaptive coupling of RPIM and FEM is achieved by a direct algorithm. Mesh‐bias dependency, which poses difficulties in FEM‐based cracking simulations, is circumvented by a crack tracking algorithm. The performance of our scheme is demonstrated by two numerical examples, that is, the four‐point bending test on concrete beam and the surface cracks caused by tunnel excavation. Copyright © 2014 John Wiley & Sons, Ltd.     

数值流形法在处理强奇异性问题时的网格无关性

数值流形方法（NMM）的最大优势在于可以统一地处理岩土力学中的连续和非连续变形问题。它在求解断裂力学问题时无需强制裂纹与数学网格保持一致,非常适合应用于岩土工程中由连续到非连续的破坏过程模拟。在裂纹扩展过程中,裂纹与数学网格的相对位置将会是任意的,如裂纹尖端可能落在网格内部、网格节点上或网格边上等。因此,对同一条裂纹,通过旋转和移动数学网格构造了它们之间的这种相对位置关系以及一些可能对计算结果产生影响的极端情况,并以应力强度因子作为衡量标准,研究了NMM在处理线弹性断裂力学问题时的网格依赖性。研究表明,NMM即使在处理强奇异性问题时依然有着很好的网格无关性,进一步证实了它在模拟裂纹扩展问题时的鲁棒性。     

An approach to calculating large strain accumulation for discrete element simulations of granular media

For research on granular materials, establishing a method to calculate continuum strain from particle displacements is necessary for understanding the material behaviour at macro-level and developing continuum constitutive models. Existing methods are generally based on constructing a mesh or background grid to calculate strain from particle motions. These methods offer rigorous ways to measure strain for granular materials; however, they suffer from several problems such as mesh distortion and lacking grid-to-particle strain mapping procedure, which hinders their capability of calculating strain accumulation during large deformation processes of granular media. To address this issue, this study proposes a new strain calculation method for discrete element simulations of granular materials. This method describes a particle assembly as an equivalent continuum system of material points, each of which corresponds to a particle centre and represents a continuous region with its initial volume/area presumably equal to the volume/area of Voronoi cells generated in accordance with the particle assembly configuration. Smooth Particle Hydrodynamics (SPH) interpolation functions are then employed to calculate strain for these material points. This SPH-based method does not require any mesh or background grid for computation, leading to advantages in calculating strain accumulation under large deformation. Simulations of granular materials in both uniform and heterogeneous gradations were carried out, and strain results obtained by the proposed method indicate good agreements with analytical and numerical solutions. This demonstrates its potential for strain calculations in discrete element simulations of granular materials involving large deformations and/or large displacements.     

Modelling strain localization in granular materials using micropolar theory: mathematical formulations

It has been known that classical continuum mechanics laws fail to describe strain localization in granular materials due to the mathematical ill‐posedness and mesh dependency. Therefore, a non‐local theory with internal length scales is needed to overcome such problems. The micropolar and high‐order gradient theories can be considered as good examples to characterize the strain localization in granular materials. The fact that internal length scales are needed requires micromechanical models or laws; however, the classical constitutive models can be enhanced through the stress invariants to incorporate the Micropolar effects. In this paper, Lade's single hardening model is enhanced to account for the couple stress and Cosserat rotation and the internal length scales are incorporated accordingly. The enhanced Lade's model and its material properties are discussed in detail; then the finite element formulations in the Updated Lagrangian Frame (UL) are used. The finite element formulations were implemented into a user element subroutine for ABAQUS (UEL) and the solution method is discussed in the companion paper. The model was found to predict the strain localization in granular materials with low dependency on the finite element mesh size. The shear band was found to reflect on a certain angle when it hit a rigid boundary. Applications for the model on plane strain specimens tested in the laboratory are discussed in the companion paper. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite element analysis of two cylindrical expansion problems involving nearly incompressible material behaviour

The displacement formulation of the finite element method is well suited to the analysis of elasto-plasticity problems involving compressible material behaviour, but it is well known that numerical difficulties occur when the material is incompressible or nearly incompressible. The effect of these additional constraints depends on both element formulation and mesh topology. A two-dimensional plane strain finite element formulation suitable for the solution of problems involving large strains and displacements (but small rotations) based on the isoparametric approach is described. The kinematics of deformation are defined in terms of the Eulerian strain rates that are invariably used in small strain analysis; the formulation therefore retains some of the character of small strain theory but includes additional geometrically non-linear terms. The results of a series of plane strain finite element analyses of two cylindrical expansion problems are presented. These results confirm the previously observed trend that as Poisson's ratio approaches 0·5 then the quality of the calculated stress deteriorates. The study also indicates that the solution quality depends increasingly on mesh topology as perfect incompressibility is reached.     

基于局部单元劈裂的FEM/DEM自适应分析方法

为了模拟岩体中裂纹的萌生、扩展,Munjiza提出了有限元法/离散元法(FEM/DEM)耦合分析方法。因为裂纹是沿单元边界进行扩展的,亦即裂纹扩展具有网格依赖性,为获得较好的裂纹扩展形态,需要划分密集的初始网格。为解决上述难题,基于FEM/DEM耦合分析方法,提出了基于局部单元动态劈裂的FEM/DEM自适应分析方法,以克服裂纹扩展形态对网格的依赖性。该方法在最初建模时无需划分很密的初始网格,随着荷载的施加,对裂纹尖端附近的局部单元进行动态劈裂,为裂纹的后续扩展提供了更多可能的扩展方向,使得裂纹扩展不必沿着初始网格的单元边界扩展,即可以沿着单元内部进行扩展,裂纹扩展形态更为平滑,与实际情况更为接近。同时相对原FEM/DEM耦合分析方法一开始就划分很密的网格而言,新方法可以划分较为稀疏的初始网格,计算成本降低。最后,通过巴西劈裂算例与原FEM/DEM耦合分析方法对比,分析表明,新方法在一定程度上克服了裂纹扩展形态对初始网格的依赖性。