Displacement Discontinuity Elements Research Articles

Two-dimensional radial integration displacement discontinuity method with high-order isoparametric elements

To improve the accuracy of Displacement Discontinuity Method and enhance its adaptivity, a general-purpose two-dimensional displacement discontinuity method with both linear and quadratic isoparametric elements has been developed to model engineering discontinuity problems where cracks or cavities are involved. Linear and quadratic isoparametric elements have linear and quadratic distributions of displacement discontinuity, respectively. Both of them belong to the discontinuous element type, in which the geometry shape functions are different from the interpolation shape functions. A new general formulation, based on the boundary integral functions, is given for displacement discontinuity problems with arbitrary boundary shapes. This formulation contains singular and strongly singular integrals which can be evaluated in the sense of Cauchy principal value and Hadamard principal value, respectively. The radial integration technique is applied to perform these singular integrals with sufficiently high accuracy. Five numerical examples, most of which have analytic solutions, are given to illustrate the improved accuracy of the proposed approach. Compared with the constant displacement discontinuity element, the numerical results by using the present isoparametric displacement discontinuity elements show better accuracy.

Pseudo-coupled hydraulic fracturing analysis with displacement discontinuity and finite element methods

Hydraulic fracturing is a widely employed technique in geoengineering. It is the most common stimulation technique to enhance reservoir permeability. Mathematical modeling of hydraulic fracturing must include several coupled processes, leading to complex and computationally expensive solutions, especially in 3D models. This complexity has led researchers to develop computational tools aiming at accuracy and acceptable computational costs. Several coupled and pseudo-coupled hydromechanical schemes have been proposed in association with the Finite Element (FEM) and Boundary Element Methods (BEM). On the other hand, semi-analytical methods apply to specific simple cases and provide approximations to more complex situations. This work proposes using the Displacement Discontinuity Method (DDM) associated with the Finite Element Method to model the hydraulic fracturing process. An explicit two-way pseudo-coupled scheme is developed where the DDM method solves the rock fracturing process. The FEM simulates fluid migration inside the fracture and the porous media. The results of the KGD problem considering impermeable and permeable conditions show good agreement with the proposed methodology. Finally, the accuracy of 2-D analyses while reducing the computational time suggests extending the FEM-DDM method to three-dimensional problems as a promising alternative to more complex fracture analyses.

Fully coupled thermo-hydro-mechanical approach to model fracture response to injection process in enhanced geothermal systems using displacement discontinuity and finite element method

Assessing, predicting, and controlling injection-induced seismicity is a major challenge for developing enhanced geothermal systems (EGS) due to the complexity of coupled thermo-hydro-mechanical (THM) processes. This study aimed to develop a fully coupled numerical model to assess the complex behavior of a low permeable matrix-fracture during non-isothermal single-phase fluid injection process. A thermo-poroelastic displacement discontinuity (DD) method combining with different forms of finite element method are implemented to encapsulate the fractured medium response and transport processes, respectively. The nonlinear characteristics of normal (changing the joint to hydraulic fracture status) and shear (changing the stick to slip fracture status) fracture deformation are taken into account through fracture constitutive relations. Developed numerical approach was applied to simulate cool water injection into fracture/matrix systems, analyzing the role of coupled processes on spatiotemporal variation of matrix-fracture stresses, temperature and pore pressure and assessing induced seismicity (slippage) and permeability alteration. Numerical simulations demonstrate a fully coupled relation between matrix dilation, shrinkage, and non-linear fracture deformation. The redistribution of dynamic and kinematic parameters along with non-linear fracture deformation showed that although the poroelastic effects are dominant in the early stages, thermoelastic effects dominate in the long-term injection stages. • A fully-coupled Thermo-Hydro-Mechanical approach was developed to model injection process in EGS. • Displacement discontinuity and finite element methods were combined to encapsulate fracture deformation and transport processes, respectively. • Fracture constitutive laws were implemented to model non-linear crack deformation. • Poroelastic effects are dominant in the early stages, and thermoelastic effects dominate in the long-term injection stages.

BEM analysis for curved cracks

The boundary integral equations (BIE) and displacement discontinuity methods (DDM) are formulated for solution of smooth curved crack problems within 2D elasticity and Reissner's plate bending theory. The equivalence between the direct boundary element method and the indirect boundary element method is shown here. The Chebyshev polynomials of the second kind are employed to solve the integral equations numerically. This enables determination of the stress intensity factors at the crack tips directly5 by the coefficients of Chebyshev polynomials. In the DDM, the coefficients of stress influence for the constant displacement discontinuity element are derived in closed form for the Reissner's plate bending problem. The degree of convergence of BIE is demonstrated with increasing numbers of the collocation points. Comparison is made with the analytical solutions from the stress intensity factor handbook for an embedded circular crack in a flat plate under tensile force. High accuracy of the presented numerical solutions makes possible to use them as benchmarks to evaluate the degree of accuracy for any numerical algorithms.

Displacement discontinuity method for cohesive crack propagation

Displacement discontinuity method (DDM) can conveniently solve fracture mechanics problems because the discontinuous displacement field across the fracture surfaces can be explicitly captured by the displacement discontinuity elements. However, the potential of DDM on solving crack propagation problems in quasi-brittle materials has not been covered in the past. In this paper, the DDM is extended to take into account the fracture process zone in quasi-brittle materials such as concrete by incorporating an iterative algorithm to account for the cohesive stress vs crack opening relation. Tensile strength criterion and fracture toughness criterion for crack propagation are used respectively for two corresponding types of crack tip element-parabolic displacement discontinuity element (PDDE) and root-r displacement discontinuity element (RRE). Numerical simulations of the wedge splitting test and three-point bending of notched beam are performed. The results are found to compare well with experimental and finite element results simulations.

A boundary element/finite difference analysis of subsidence phenomenon due to underground structures

Analysis of the stresses, displacements, and horizontal strains of the ground subsidence due to underground excavation in rocks can be accomplished by means of a hybridized higher order indirect boundary element/finite difference (BE/FD) formulation. A semi-infinite displacement discontinuity field is discretized (numerically) using the cubic displacement discontinuity elements (i.e. each higher order element is divided into four sub-elements bearing a cubic variation in the displacement discontinuities). Then the classical finite difference formulation (i.e. the backward, central, and forward finite difference formulations) is hybridized using the boundary element formulation, enabling us to obtain the nodal tangential stresses and horizontal strains along the elements. Several example problems are solved numerically, and the results obtained are then compared with their corresponding results available in the literature. These comparisons show the effectiveness and validness of the proposed method. A classical practical problem is also used to verify the applicability of the hybridized method.

A pseudo-3D model for hydraulic fracture growth in a layered rock

This paper presents a new pseudo-3D (P3D) model for a hydraulic fracture growing in a layered rock with contrasts in both material properties and in situ stresses. In the model, the vertically planar fracture is divided along the lateral direction into cells. Within each cell, the cross-sectional deformation is plane strain, and the fluid pressure is allowed to vary vertically. The cells are discretized by displacement discontinuity elements that are formulated to include the elastic layered effect. The fluid flow in the cell is in two directions. Along the central part, which is of uniform pressure, the fluid flow is lateral, corresponding to the main component of fluid transport. Near the vertical fracture edge of a cell, the flow can be vertical and is generated by the vertical pressure gradient. This part of the cell is called the filling part. When the pressure in the filling part reaches the level equal to that in the central part, the flow direction switches from vertical to lateral. The filling and central parts both contribute to fracture height growth. The proposed P3D problem is solved in a coupled manner that accounts for the two-directional flow and cross-sectional deformation through a two-loop iterative method. In the outer loop, the fluid storage of the central part is updated by satisfying mass conservation in the lateral direction, and in the inner loop, the cross-sectional elastic deformation and the influxes to the filling parts are found by satisfying energy minimization subject to an equality constraint on the central-part volume of a cell. The results of pressure ad fracture width at a given elapsed time are thus obtained. After that, fracture growth in both lateral and vertical directions is controlled by the fracture toughness criterion based on linear elasticity. In describing the P3D model, the governing equations are provided and their dimensionless forms are derived. The numerical algorithm used for solving the P3D problem is also described. Numerical examples are presented, including a constant-height fracture, a radial one and asymmetric fractures in three-layered rocks. Comparisons of our results are made with other published results and good agreements between them are found.

Three-Dimensional Elastic Analysis of Rock Excavations by Using the g2 Constant Displacement Discontinuity Method

A fast computational code is presented that is dedicated for the elastic analysis of three-dimensional excavations and cracks in rocks. The problem is solved on the boundaries that are discretized with a new triangular leaf constant displacement discontinuity element with one collocation point. The creation of the new triangular element was inspired from Mindlin's special version of grade-2 or strain-gradient elasticity theory (second gradient of displacement, g2). This element is characterized by a much better measure of the average stress at the center of gravity of the triangular element compared to that of the classical elasticity element close to regions with stress or strain gradients (e.g. notches, cracks etc). In a verification stage, the accuracy of the computational algorithm for the pressurized penny-shaped and mixed-mode elliptical crack problems that have analytical solutions is demonstrated. More specifically, it is shown that the average error of the crack tip Stress Intensity Factor predicted by the gradient modified method for nine discretizations of varying density is around 3.5% with a máximum error of 5%, while the constant displacement discontinuity element displays errors varying around 14%. Moreover, the new method preserves the simplicity and hence the high speed of the constant displacement discontinuity with only one collocation point per element, but it is far more efficient compared to it, especially close to the crack tips and corners of excavations where the displacement and stress gradients are highest.

Analyses of Inclined Cracks Neighboring Two Iso-Path Cracks in Rock-Like Specimens Under Compression

Crack propagation process in pre-cracked rock like specimens has been studied experimentally and numerically considering three cracks in the middle part of each specimen. The rock-like specimens are specially prepared from Portland pozzolana cement, fine sands and water. These pre-cracked cylindrical specimens (each containing a single inclined crack in the neighborhood of two iso-path cracks) are experimentally tested under compressive loading. The same problems are numerically simulated by a modified displacement discontinuity method using higher order displacement discontinuity elements and higher order special crack tip elements for crack tip treatment to increase the accuracy of the Mode I and Mode II stress intensity factors obtained based on linear elastic fracture mechanics theory. The crack propagation and coalescence paths of the inclined crack are estimated by implementing a suitable iteration algorithm of incremental crack length extension in a direction predicted by using the maximum tangential stress criterion. The numerical and analytical crack extension analyses are compared which are in good agreement and show the validity, applicability and accuracy of the present work.

Crack analysis of pre-cracked brittle specimens under biaxial compression

Crack propagation process in rock like specimens has been studied experimentally and numerically considering both; a single crack and double cracks in the center of specimens. The rock-like specimens are specially prepared from Portland Pozzolana Cement (PPC), fine sands and water. These precracked cylindrical specimens are experimentally tested under compressive loading. Effect of confining pressure on the crack propagation path of specimens with single crack is explained first. In these experiments, it has been observed that the crack propagation path is diverted from its rout due to lateral confinement compared to that of the uniaxially loaded pre-cracked experiments. The same experiments are numerically simulated by an indirect boundary element method using higher order displacement discontinuity elements. Effect of lateral stress on the crack propagation of specimens containing single and double cracks have been studied and compared with the experimental results showing a good agreement between them. Effect of confinement on the cracks coalescence in the specimens containing two cracks is also investigated. It has been observed that wing cracks are produced at the first stage of compression and start their propagation toward the direction of uniaxial compressive loading in the absence of confinement. In the presence of confining pressure the propagating wings are diverted related to the uniaxially loaded cracks and showing a mixed mode crack propagation phenomenon due to both induced tension and shear. It is also observed that the crack coalescence path is affected a little bit (due to confinement) compared to that of wing cracks.

Propagation mechanism of neighboring cracks in rock-like cylindrical specimens under uniaxial compression

The mechanism of cracks propagation and coalescence of neighboring cracks existing in precracked rock like cylindrical specimens has been studied experimentally and numerically by considering multiple cracks in the middle part of each specimen. The pre-cracked cylindrical specimens of rock like materials are experimentally tested under uniaxial compression in a rock mechanics laboratory. Then these experiments are numerically simulated by a modified higher order displacement discontinuity method (HODDM) using cubic displacement discontinuity elements and three special crack tip elements for crack tip behavior to increase the accuracy of the Mode I and Mode II stress intensity factors (SIFs) obtained based on linear elastic fracture mechanics (LEFM) theory. The crack propagation and coalescence paths of the internal inclined crack are estimated by implementing a suitable iteration algorithm of incremental crack length extension in a direction predicted by using the maximum tangential stress criterion. The numerical and experimental results are compared showing the validity, applicability and accuracy of the present work. Finally, a numerical simulation has been accomplished to study the effect of interaction among neighboring cracks on the SIFs.

Simulation of crack coalescence mechanism underneath single and double disc cutters by higher order displacement discontinuity method

The present research is focused on the numerical crack coalescence analysis of the micro-cracks and cracks produced during the cutting action of TBM disc cutters. The linear elastic fracture mechanics (LEFM) concepts and the maximum tangential stress criterion are used to investigate the micro crack propagation and its direction underneath the excavating discs. A higher order displacement discontinuity method with quadratic displacement discontinuity elements is used to estimate the stress intensity factors near the crack tips. Rock cutting mechanisms under single and double type discs are simulated by the proposed numerical method. The main purposes of the present modeling are to simulate the chip formation process of indented rocks by single and double discs. The effects of specific disc parameters (except speed) on the thrust force F r, the rolling force F r, and the specific energy E S are investigated. It has been shown that the specific energy (energy required to cut through a unit volume of rock) of the double disc is less than that of the single disc. Crack propagation in rocks under disc cutters is numerically modeled and the optimum ratio of disc spacing S to penetration depth P d (i.e. S/P d ratio) of about 10 is obtained, which is in good agreement with the theoretical and experimental results cited in the literature.

Simulating the effect of disc erosion in TBM disc cutters by a semi-infinite DDM

Underground excavation in rocks can be accomplished in several steps. The cutting action of mechanical tools such as disc cutters used in tunnel boring machines (TBM) is one of the most basic steps. Indentation of TBM disc cutters into the rock can produce rock chips in different scales. This phenomenon involves the production of micro cracks, their propagating, and coalescing process to form macro cracks. The present research focuses on the concepts of linear elastic fracture mechanics (LEFM) and maximum tangential stress criterion to investigate the micro crack propagation and its direction of growth in rocks underneath the TBM disc cutters. A modified higher order semi-infinite displacement discontinuity method (HSDDM) with third order displacement discontinuity elements (cubic elements) is used to estimate the stress intensity factor in the fractured rocks underneath a single disc cutter. To reduce the errors caused by stress and displacement singularities near the crack tip, three special crack tip elements have been implemented in the HSDDM2D computer code. TBM disc cutters will be eroded after a period of working. Therefore, the numerical simulation of eroded and not eroded discs has been proposed in this study. To simulate the eroded disc, four small elements have been modeled to generate the curvature of the cutter tip, which in turn reduces the cutting efficiency and increases required specific energy for a typical disc cutter.

A stress and displacement discontinuity element method for elastic transversely anisotropic rock

SUMMARYThe paper presents closed‐form solutions for stress and displacement influence functions for stress discontinuity (SD) and displacement discontinuity (DD) elements, for a two‐dimensional plane‐strain elastic, transversely anisotropic medium. The solutions for SD elements are based on Kelvin's problem and for DD elements on the concept of dipoles. Stress and displacement influence functions are derived for the following elements: constant SD, linear SD, constant DD, linear DD, square root DD, parabolic DD, constant DD surface, and linear DD surface elements. The formulations are incorporated into FROCK, a hybridized boundary element method code, and are validated by providing comparisons between the results from FROCK and the finite element code ABAQUS. A limited parametric analysis shows the effects of slight anisotropy on the stress field around the tip of a crack and of the orientation of the crack with respect to the axes of elastic symmetry. Copyright © 2014 John Wiley & Sons, Ltd.

Displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space

This paper describes a displacement discontinuity method for modeling axisymmetric cracks in an elastic half-space or full space. The formulation is based on hypersingular integral equations that relate displacement jumps and tractions along the crack. The integral kernels, which represent stress influence functions for ring dislocation dipoles, are derived from available axisymmetric dislocation solutions. The crack is discretized into constant-strength displacement discontinuity elements, where each element represents a slice of a cone. The influence integrals are evaluated using a combination of numerical integration and a recursive procedure that allows for explicit integration of hyper- and Cauchy singularities. The accuracy of the solution at the crack tip is ensured by adding corrective stresses across the tip element. The method is validated by a comparison with analytical and numerical reference solutions.

Modeling of concrete cracking—A hybrid technique of using displacement discontinuity element method and direct boundary element method

This paper presents a new approach by making use of a hybrid method of using the displacement discontinuity element method and direct boundary element method to model concrete cracking by incorporating fictitious crack model. Fracture mechanics approach is followed using the Hillerborg's fictitious crack model. A boundary element based substructure method and a hybrid technique of using displacement discontinuity element method and direct boundary element method are compared in this paper. In order to represent the process zone ahead of the crack, closing forces are assumed to act in such a way that they obey a linear normal stress-crack opening displacement law. Plain concrete beams with and without initial crack under three-point loading were analyzed by both the methods. The numerical results obtained were shown to agree well with the results from existing finite element method. The model is capable of reproducing the whole range of load–deflection response including strain-softening and snap-back behavior as illustrated in the numerical examples.

The G2 constant displacement discontinuity method – Part I: Solution of plane crack problems

A new constant displacement discontinuity element was presented in a previous paper applied initially for the numerical solution of either isolated straight cracks or for co-linear cracks of the three fundamental deformation modes I, II and III due to the special form of the solution. It was based on the strain-gradient elasticity theory in its simplest possible Grade-2 variant. The assumption of the G2 expression for the stresses has resulted to a better average stress value at the mid-point of the straight displacement discontinuity compared to the classical elasticity solution. This new element gave considerably better predictions of the stress intensity factors compared to the constant displacement discontinuity element and the linear displacement discontinuity element. Moreover, it preserved the simplicity and hence the high speed of computations. In this Part I, the solution for this element is extended for the analysis of cracks of arbitrary shape in an infinite plane isotropic elastic body and it is validated against three known analytical solutions.

The G2 constant displacement discontinuity method – Part II: Solution of half-plane crack problems

In the previous Part I, the G2 constant displacement discontinuity element was presented that is dedicated for the fast (only one collocation point per element), stable and accurate numerical solution of modes I, II and III cracks of arbitrary shape in an infinite plane isotropic elastic body. Herein, another G2 constant displacement discontinuity element is constructed for the case of cracks in the half-plane. It is successfully validated against existing semi-analytical and numerical solutions of crack problems in the half-plane.

A G2 constant displacement discontinuity element for analysis of crack problems

A new constant displacement discontinuity (CDD) element is presented for the numerical solution of Mode I, II and III crack problems, based on the strain-gradient elasticity theory in its simplest possible Grade-2 (second gradient of strain or G2 theory) variant. The accuracy of the proposed new element is demonstrated herein in a first attempt only for isolated straight cracks or for co-linear straight cracks for which closed form solutions exist. It is shown that the results based on this new element are in good agreement with the exact solutions. Moreover, the new method preserves the simplicity and hence the high speed of the CDD method originally proposed by Crouch with only one collocation point per element for plane crack problems, but it is far more efficient compared to it, especially close to the crack tips where the displacement and stress gradients are highest.

Multiphase poroelastic modeling in semi-space for deformable reservoirs

Field observations suggest that while pressure depletion is a local process that occurs dominantly in reservoirs being exploited, it triggers a redistribution of effective stress in a more extensive domain that involves not just the adjacent strata, but the entire overburden and surrounding rocks out to considerable distances. In an infinite or semi-infinite domain, a petroleum reservoir can be simulated as a displacement discontinuity, permitting use of an efficient displacement discontinuity boundary element method to calculate surrounding stress and displacement changes. A 3D poroelastic finite element method is used to account for the local reservoir model. By coupling the displacement discontinuity and finite element models, a 3D poroelastic reservoir in an infinite or semi-infinite domain is simulated. The numerical model has been verified and numerical examples are given. Results show that the relative deformation properties of surrounding strata have significant effects on pressure drawdown and reservoir deformation during production, and thus should be taken into account.