Elastoplastic analysis of jointed rocks using a coupled finite element and boundary element method

This paper presents a coupled, elastoplastic, finite element and boundary element method for the two-dimensional, non-linear analysis of anisotropic jointed rock. The non-linear and anisotropic behaviour of a jointed rock mass is simulated by representing the mass as an equivalent anisotropic, elastoplastic continuum, so that the influence of the jointing system is ‘smeared’ across the continuum, i.e. the individual joints are not modelled as discrete entities. Numerical examples have been solved to verify the capability, accuracy and efficiency of the present technique. The proposed technique has also been applied to the analysis of tunnel excavation problems in plane strain. The effects of anisotropy and non-linearity of the jointed rock mass during excavation have been investigated in some detail.     

A hybrid discrete-continuum approach to model hydro-mechanical behaviour of jointed rocks

A hybrid discrete-continuum approach has been presented in this paper to simulate water flow in the near and far fields of deformable fractured rocks. In the near field, the discrete model is used; while in the far field, the equivalent continuum model is employed. The discrete element method (with the static relaxation algorithm) is used in the near field and the boundary integral equation method in the far field. Along the interface of these two domains, both mechanical and hydraulic compatibility conditions are satisfied. Fully coupled hydro-mechanical analysis can be conducted in the combined near and far fields. Application to a dam foundation problem has demonstrated the capability of the developed approach.     

A coupling scheme for boundary and finite elements using a joint element

A coupling scheme for boundary and finite elements using a joint element is proposed. The scheme is based on the finite element discretization (i.e. the equivalent FE procedure). Footing problems founded on multiple layers are solved. Furthermore, the scheme is extended to a non-linear analysis.     

A superposition scheme to obtain fundamental boundary element solutions in multi‐layered elastic media

A superposition scheme is proposed to obtain a fundamental solution for boundary elements in multi‐layered elastic media. A three‐layered elastic region is obtained by superposing two sets of bonded half‐planes and subtracting one infinite plane. Therefore, the solution for an element in a layered media can be expressed in terms of bonded half‐plane solutions and an infinite‐plane solution. The major advantages of this superposition scheme are: (1) it is unnecessary to introduce elements at the interface, (2) it can be extended to higher‐order element, and (3) it may be applicable to three dimensions easily. The accuracy and performance of the developed model is illustrated by two examples. For the problem of a pressurized two‐dimensional crack within a three‐layered system, the comparison with other numerical results shows the model is quite accurate and efficient. The model is also used for a study of a practical two‐dimensional mining problem in South Africa, i.e. stoping through a dyke with material properties different from the host rock. Copyright © 2000 John Wiley & Sons, Ltd.     

Testing and evaluation of strength and deformation behaviour of jointed rocks

The objective of the paper is to derive the strength and modulus properties of rockmass as a function of intact rock strength and joint factor. The joint factor reflects the combined effect of joint frequency, joint inclination and joint strength. A study for the strength and deformation characteristics of jointed rock is done by conducting standard laboratory tests on cylindrical specimens of plaster of Paris after introducing artificial joints. The specimens having one to four joints at different inclinations which vary from 0° to 90° were tested at different confining conditions. The test results were examined to understand the effect of joint frequency and joint inclination on the strength and deformation behaviour of rock mass. Empirical correlations were developed for prediction of the uniaxial compressive strength and elastic modulus of jointed rocks. Results are compared with the earlier work on jointed specimens covering a wide variety of rocks. So, knowing the intact rock properties and the joint factor, the jointed rock properties can be estimated. These relations can be used for developing an equivalent continuum model for rock mass for handling boundary value problems. A failure criterion as proposed by Ramamurthy (1993 Ramamurthy, T. 1993. “Strength and modulus response of anisotropic rocks”. In Comprehensive rock engineering, Edited by: Hudson, J.A. Vol. 1, 313–329. Oxford: Pergamon Press.  [Google Scholar]) has been validated from these experimental results.     

CO2注入岩体的热-气-应力耦合二维弹塑性有限元分析

目前国内关于CO2地质埋存课题的研究已经起步,其中与岩石力学有关的工作急需加强。为此,考虑到向含气地层中注入CO2时岩石、气体和温度的相互作用,将CO2视为理想气体,使用Drucker-Prager屈服准则和‘无拉应力’判据,建立了一个热-气-应力耦合模型并研制了相应的二维有限元程序。假定了一个由下部注入层和上覆冠石层组成的CO2埋存地质系统,以此为数值模拟的对象,分析了CO2在不同的注入速率和注入时间条件下岩体的中的位移、应力、受拉与塑性破坏区的变化和分布情况,结果显示,为保证CO2注入岩体的稳定,应优择最佳的注入速率和注入时间。     

A micromechanical finite element model for linear and damage‐coupled viscoelastic behaviour of asphalt mixture

This study presents a finite element (FE) micromechanical modelling approach for the simulation of linear and damage‐coupled viscoelastic behaviour of asphalt mixture. Asphalt mixture is a composite material of graded aggregates bound with mastic (asphalt and fine aggregates). The microstructural model of asphalt mixture incorporates an equivalent lattice network structure whereby intergranular load transfer is simulated through an effective asphalt mastic zone. The finite element model integrates the ABAQUS user material subroutine with continuum elements for the effective asphalt mastic and rigid body elements for each aggregate. A unified approach is proposed using Schapery non‐linear viscoelastic model for the rate‐independent and rate‐dependent damage behaviour. A finite element incremental algorithm with a recursive relationship for three‐dimensional (3D) linear and damage‐coupled viscoelastic behaviour is developed. This algorithm is used in a 3D user‐defined material model for the asphalt mastic to predict global linear and damage‐coupled viscoelastic behaviour of asphalt mixture. For linear viscoelastic study, the creep stiffnesses of mastic and asphalt mixture at different temperatures are measured in laboratory. A regression‐fitting method is employed to calibrate generalized Maxwell models with Prony series and generate master stiffness curves for mastic and asphalt mixture. A computational model is developed with image analysis of sectioned surface of a test specimen. The viscoelastic prediction of mixture creep stiffness with the calibrated mastic material parameters is compared with mixture master stiffness curve over a reduced time period. In regard to damage‐coupled viscoelastic behaviour, cyclic loading responses of linear and rate‐independent damage‐coupled viscoelastic materials are compared. Effects of particular microstructure parameters on the rate‐independent damage‐coupled viscoelastic behaviour are also investigated with finite element simulations of asphalt numerical samples. Further study describes loading rate effects on the asphalt viscoelastic properties and rate‐dependent damage behaviour. Copyright © 2006 John Wiley & Sons, Ltd.     

非饱和地质体中水-应力耦合二维有限元方法及对锚杆支护的分析

从建立应力平衡方程、水连续性方程入手,采用Galerkin方法将各控制方程分别在空间域和时间域进行离散,开发出了一个用于分析饱和-非饱和岩土介质中水-应力耦合弹塑性问题的二维有限元程序。然后就一个假定的处于渗流场中的饱和-非饱和土体使用预应力锚杆的支护效果问题进行了模拟计算。结果表明,在有地下水赋存的饱和-非饱和条件下,预应力锚杆的作用主要体现在减少了土体中的塑性区,而对位移的约束依部位而异,其对渗流场的影响不大。     

A fracture mapping and extended finite element scheme for coupled deformation and fluid flow in fractured porous media

This paper presents a fracture mapping (FM) approach combined with the extended finite element method (XFEM) to simulate coupled deformation and fluid flow in fractured porous media. Specifically, the method accurately represents the impact of discrete fractures on flow and deformation, although the individual fractures are not part of the finite element mesh. A key feature of FM‐XFEM is its ability to model discontinuities in the domain independently of the computational mesh. The proposed FM approach is a continuum‐based approach that is used to model the flow interaction between the porous matrix and existing fractures via a transfer function. Fracture geometry is defined using the level set method. Therefore, in contrast to the discrete fracture flow model, the fracture representation is not meshed along with the computational domain. Consequently, the method is able to determine the influence of fractures on fluid flow within a fractured domain without the complexity of meshing the fractures within the domain. The XFEM component of the scheme addresses the discontinuous displacement field within elements that are intersected by existing fractures. In XFEM, enrichment functions are added to the standard finite element approximation to adequately resolve discontinuous fields within the simulation domain. Numerical tests illustrate the ability of the method to adequately describe the displacement and fluid pressure fields within a fractured domain at significantly less computational expense than explicitly resolving the fracture within the finite element mesh. Copyright © 2013 John Wiley & Sons, Ltd.     

A 2D coupled hydro-thermal model for the combined finite-discrete element method

Based on the combined finite-discrete element method (FDEM), a two-dimensional coupled hydro-thermal model is proposed. This model can simulate fluid flow and heat transfer in rock masses with arbitrary complex fracture networks. The model consists of three parts: a heat conduction model of the rock matrix, a heat-transfer model of the fluid in the fracture (including the heat conduction and convection of fluid), and a heat exchange model between the fluid and rock at the fracture surface. Three examples with analytical solutions are given to verify the correctness of the coupled model. Finally, the coupled model is applied to hydro-thermal coupling simulations of a rock mass with a fracture network. The temperature field evolution, the effect of thermal conductivity of the rock matrix thermal conductivity and the fracture aperture on the outlet temperature are studied. The coupled model presented in this paper will enable the application of FDEM to study rock rupture driven by the effect of hydro-thermo-mechanical coupling in geomaterials such as in geothermal systems, petroleum engineering, environmental engineering and nuclear waste geological storage.

     

Evolution of elastic properties in finite poroplasticity and finite element analysis

The formulation of the poroelastoplastic constitutive equations at large strains of a fully saturated material is performed focusing on the usually ignored influence of large strain plasticity on the poroelastic properties. A micromechanics approach allows to take into account the evolution of the microstructure geometry which in turn induces a coupling between elasticity and plasticity. Such a coupling results in an additional term in the macroscopic Cauchy stress rate equation derived from inclusion‐based estimates that leads to a modified Jaumann derivative. The pressure rate equation is also analysed. The finite element discretization of finite poroplasticity is then presented taking into account the elasticity–plasticity coupling. Application to the consolidation situation shows that the coupling may lead to non‐negligible effects. Copyright © 2002 John Wiley & Sons, Ltd.     

A non‐linear coupled finite element–boundary element model for the prediction of vibrations due to vibratory and impact pile driving

This paper presents a non‐linear coupled finite element–boundary element approach for the prediction of free field vibrations due to vibratory and impact pile driving. Both the non‐linear constitutive behavior of the soil in the vicinity of the pile and the dynamic interaction between the pile and the soil are accounted for. A subdomain approach is used, defining a generalized structure consisting of the pile and a bounded region of soil around the pile, and an unbounded exterior linear soil domain. The soil around the pile may exhibit non‐linear constitutive behavior and is modelled with a time‐domain finite element method. The dynamic stiffness matrix of the exterior unbounded soil domain is calculated using a boundary element formulation in the frequency domain based on a limited number of modes defined on the interface between the generalized structure and the unbounded soil. The soil–structure interaction forces are evaluated as a convolution of the displacement history and the soil flexibility matrices, which are obtained by an inverse Fourier transformation from the frequency to the time domain. This results in a hybrid frequency–time domain formulation of the non‐linear dynamic soil–structure interaction problem, which is solved in the time domain using Newmark's time integration method; the interaction force time history is evaluated using the θ‐scheme in order to obtain stable solutions. The proposed hybrid formulation is validated for linear problems of vibratory and impact pile driving, showing very good agreement with the results obtained with a frequency‐domain solution. Linear predictions, however, overestimate the free field peak particle velocities as observed in reported field experiments during vibratory and impact pile driving at comparable levels of the transferred energy. This is mainly due to energy dissipation related to plastic deformations in the soil around the pile. Ground vibrations due to vibratory and impact pile driving are, therefore, also computed with a non‐linear model where the soil is modelled as an isotropic elastic, perfectly plastic solid, which yields according to the Drucker–Prager failure criterion. This results in lower predicted free field vibrations with respect to linear predictions, which are also in much better agreement with experimental results recorded during vibratory and impact pile driving. Copyright © 2008 John Wiley & Sons, Ltd.     

Analytical methods to estimate the hydraulic conductivity of jointed rocks

Hydrogeology Journal - The hydraulic conductivity of jointed rocks is one of the main input parameters to predict water inflow to engineering structures that are located in the jointed rocks....     

A complete formulation of an indirect boundary element method for poroelastic rocks

Rocks are naturally filled with cracks and pores that are saturated with one or more fluid phases. Many problems in rock mechanics, petroleum engineering, geophysics, etc. deal with cracks and discontinuities in rock formations. These problems should consider effects of a porous medium. Displacement discontinuity method (DDM) as an indirect boundary element method is particularly ideal for problems involving fractures and discontinuities. However, the DDM in its original form is limited to elastic problems. The paper uses a fundamental solution of a point displacement discontinuity in poroelastic medium to obtain the solution for a poroelastic DDM. Then it introduces a numerical formulation and implementation for the poroelastic DDM in a code named CEP-DDM (Constant Element Poroelastic DDM). The accuracy and validity of the proposed solution and the newly developed code are verified by two analytical solutions, another numerical solution, and some field measurements. These results showed good agreement between CEP-DDM and other methods’ results. The verifications prove the accuracy and applicability of the proposed numerical model in a wide range of real-world problems.     

A new approach to modeling perpendicular fabrics in porphyritic plutonic rocks using the finite element method

A growing body of field evidence indicates that hypersolidus fabrics preserved in syntectonic plutons are likely to have formed in highly crystallized ‘rigid sponge’ magma. This paper demonstrates that such magma could be idealized as a rheological solid and that the development of non-coaxial fabrics in plutonic rocks can very conveniently be modeled in the framework of solid mechanics. Using the finite element method (FEM), we modeled two strain regimes of small magnitudes (plane-strain horizontal simple shear with the shear strain γ of up to 0.30 and plane-strain pure shear of up to 15% shortening) superposed onto vertically oriented and variously spaced elastic phenocrysts set in the viscoelastic matrix. In the simple shear regime, the phenocrysts slightly rotate toward the shear plane, while the principal strain directions in the matrix are instantaneously oriented at an angle of about 45° or less to the phenocryst fabric. Simple shear thus can only lead to the formation of oblique phenocryst and matrix fabrics. By contrast, the vertical phenocryst fabric is maintained in the pure shear regime, and a new horizontal fabric can develop almost instantaneously in the matrix even for small amounts of superposed shortening (5% shortening after 10 ky in our model). We conclude that such a mechanism can easily produce perpendicular hypersolidus fabrics in plutonic rocks and that only a very short time span (first thousands of years) is required to develop magmatic fabric in a pluton for ‘normal’ rates (10⁻¹⁵ to 10⁻¹³ s⁻¹) of tectonic deformation.     

Guidelines for using the finite element method to predict one-dimensional consolidation behaviour

Several finite element schemes, based both on the diffusion and coupled approaches, have been implemented in computer programs and a comparative study carried out to investigate the numerical performance of each scheme. Factors such as stability, convergency, accuracy, computational time and the effects of wide variations in soil parameters (eg laminated soils) have been examined. The study indicates that the numerical performance of each scheme is controlled by a non-dimensional parameter and guidelines have been suggested which allow accurate and economic solutions to be obtained.     

On the linear elastic responses of the 2D bonded discrete element model

The bonded discrete element model (DEM) is a numerical tool that is becoming widely used when studying fracturing, fragmentation, and failure of solids in various disciplines. However, its abilities to solve elastic problems are usually overlooked. In this work, the main features of the 2D bonded DEM which influence Poisson's ratio and Young's modulus, and accuracy when solving elastic boundary value problems, are investigated. Outputs of numerical simulations using the 2D bonded DEM, the finite element method, a hyper elasticity analysis, and the distinct lattice spring model (DLSM) are compared in the investigation. It is shown that a shear interaction (local) factor and a geometric (global) factor are two essential elements for the 2D bonded DEM to reproduce a full range of Poisson's ratios. It is also found that the 2D bonded DEM might be unable to reproduce the correct displacements for elastic boundary value problems when the represented Poisson's ratio is close to 0.5 or the long-range interaction is considered. In addition, an analytical relationship between the shear stiffness ratio and the Poisson's ratio, derived from a hyper elasticity analysis and applicable to discontinuum-based models, provides good agreement with outputs from the 2D bonded DEM and DLSM. Finally, it is shown that the selection of elastic parameters used the 2D bonded DEM has a significant effect on fracturing and fragment patterns of solids.     

On the use of discrete element modelling to simulate the viscoelastic deformation behaviour of an idealized asphalt mixture

The use of discrete-element modelling (DEM) to simulate the behaviour of a highly idealized bituminous mixture under uniaxial and triaxial compressive creep tests is investigated in this paper. The idealized mixture comprises single-sized spherical particles (sand) mixed with bitumen and was chosen so that the packing characteristics are known and the behaviour of the mixture is dominated by the bitumen. The bitumen is represented as shear and normal (tensile and compressive) contact stiffnesses. Numerical sample preparation procedures for specimens containing spherical particles or clumps have been developed to ensure that the final specimen is isotropic and has the correct volumetric proportions. An elastic contact was used for the compressive normal contact stiffness and a viscoelastic contact was used for shear and tensile normal contact stiffness. Simulation results show that the idealized mixture is found to dilate when the ratio of compressive to tensile contact stiffness increases as a function of loading time. Uniaxial and triaxial viscoelastic simulations have been performed to investigate the effect of stress ratio on dilation and the numerical results have been verified with experimental data. The effects of introducing a proportion of frictional contacts and a more complex particle shape (clump) on dilation have been examined.     

A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics

In this paper, we present a computational framework for the simulation of coupled flow and reservoir geomechanics. The physical model is restricted to Biot’s theory of single-phase flow and linear poroelasticity, but is sufficiently general to be extended to multiphase flow problems and inelastic behavior. The distinctive technical aspects of our approach are: (1) the space discretization of the equations. The unknown variables are the pressure, the fluid velocity, and the rock displacements. We recognize that these variables are of very different nature, and need to be discretized differently. We propose a mixed finite element space discretization, which is stable, convergent, locally mass conservative, and employs a single computational grid. To ensure stability and robustness, we perform an implicit time integration of the fluid flow equations. (2) The strategies for the solution of the coupled system. We compare different solution strategies, including the fully coupled approach, the usual (conditionally stable) iteratively coupled approach, and a less common unconditionally stable sequential scheme. We show that the latter scheme corresponds to a modified block Jacobi method, which also enjoys improved convergence properties. This computational model has been implemented in an object-oriented reservoir simulator, whose modular design allows for further extensions and enhancements. We show several representative numerical simulations that illustrate the effectiveness of the approach.     

A novel nonlinear solution for the polygon scaled boundary finite element method and its application to geotechnical structures

The polygon scaled boundary finite element method is semi-analytical and known for its high precision. However, the material nonlinearity cannot be maintained because this method uses an analytical solution in the radial direction. In this paper, a novel nonlinear algorithm is developed by introducing internal Gaussian points over a subdomain. The response of nonlinearity for a concrete-faced rockfill dam is modeled. The results correspond well with the results from finite element modelling, which demonstrates the method can be used to describe the nonlinear characteristics of geomaterials. Furthermore, this method offers promising flexibility for analyzing complex geometries without decreasing the precision.