Special Crack Tip Elements Research Articles

An efficient computational framework for height-contained growing and intersecting hydraulic fracturing simulation via SGBEM–FEM

An efficient computational framework to model the growth of intersecting height-contained hydraulic fractures is proposed. The fracture in solids is modeled by the classical theory of linear elastic fracture mechanics, whereas the injection flow in hydraulic fractures is treated as channel flow. The governing equations of fracture mechanics are formulated in terms of weakly-singular weak-form boundary integral equations, with the knowledge of the fracture surface geometry exploited to reduce the integral dimension. The fluid flow within the fracture is treated using finite element method as one dimensional flow in an arbitrarily curved channel. The symmetric Galerkin boundary element method and finite element method are coupled with the traction continuity condition. The model is then formulated in a variational form and augmented with Lagrange multipliers to prescribe displacement constraints on intersections of hydraulic fractures, while fluid constraints are enforced by a consolidation of associated pressure degrees of freedom. Pertinent numerical implementation techniques, including spatial and temporal discretizations, special crack tip element, crack propagation criteria and remeshing strategy, are presented. The proposed framework is verified by comparing to the numerical results from a three dimensional hydraulic fracturing simulator and then verified by comparing the stress intensity factors to the XFEM and Murakami’s results for both star shaped cracks and symmetric branched cracks. The verification shows the utility properly enforces the physical constraints on the intersections of fractures. Three numerical examples are shown to demonstrate the capability of the proposed computational framework modeling the intersecting hydraulic fracturing process in a single or separated height-contained pay zones.

ProCrackPlast: a finite element tool to simulate 3D fatigue crack growth under large plastic deformations

Many structural components and devices in combustion and automotive engineering undergo highly intensive cyclic thermal and mechanical loading during their operation, which leads to low cycle (LCF) or thermomechanical (TMF) fatigue crack growth. This behavior is often characterized by large scale plastic deformations and creep around the crack, so that concepts of linear-elastic fracture mechanics fail. The finite element software ProCrackPlast has been developed at TU Bergakademie Freiberg for the automated simulation of fatigue crack growth in arbitrarily loaded three-dimensional components with large scale plastic deformations, in particular under cyclic thermomechanical loading. ProCrackPlast consists of a bundle of Python routines, which manage finite element pre-processing, crack analysis, and post-processing in combination with the commercial software Abaqus . ProCrackPlast is based on a crack growth procedure which adaptively updates the crack size in finite increments. Crack growth is controlled by the cyclic crack tip opening displacement varDelta CTOD, which is considered as the appropriate fracture-mechanical parameter in case of large scale yielding. The three-dimensional varDelta CTOD concept and its effective numerical calculation by means of special crack-tip elements are introduced at first. Next, the program structure, the underlying numerical algorithms and calculation schemes of ProCrackPlast are outlined in detail, which capture the plastic deformation history along with the moving crack. In all simulations, a viscoplastic cyclic material law is used within a large strain setting. The numerical performance of this software is studied for a single edge notch tension (SENT) specimen under isothermal cyclic loading and compared to common finite element techniques for fatigue crack simulation. The capability of this software is featured in two application examples showing crack growth under mixed-mode LCF and TMF in a typical austenite cast steel, Ni-Resist. In combination with a crack growth law identified in terms of varDelta CTOD for a specific material, the tool ProCrackPlast is able to predict the crack evolution in a 3D component for a given thermomechanical loading scenario.

Numerical calculation of [formula omitted] to simulate fatigue crack growth under large scale viscoplastic deformations

Crack propagation under low cycle fatigue and thermomechanical fatigue is characterized by high plastic and creep strains that extend over large regions around the crack, so that concepts of linear-elastic fracture mechanics cannot be applied. In these cases, the cyclic crack tip opening displacement ΔCTOD is a promising loading parameter to quantify crack growth. In this work, suitable definitions and Finite Element techniques are investigated and compared for an accurate calculation of ΔCTOD under cyclic mechanical and/or thermal loading. A viscoplastic temperature dependent material model of Chaboche-type is used along with large strain settings, specified for the austenitic cast iron Ni-resist. Extensive two-dimensional analyses of Single Edge Notch Tension specimens revealed that collapsed special crack tip elements are superior compared with commonly used regular quadrilateral 8-node elements. At the same level of accuracy of ΔCTOD, they require an about ten times coarser mesh and show less sensitivity w.r.t. element size for both stationary and propagating cracks. In order to simulate fatigue crack growth, an efficient, fully automated FE-technique is developed for an incremental crack propagation by successive remeshing, whereby the deformations and internal state variables are mapped from the old mesh onto the new one. Recommendations are made regarding important numerical control parameters like optimal size of crack tip elements, length of crack growth increment in relation to plastic zone size and ΔCTOD value.

Boundary Element Analysis for Mode III Crack Problems of Thin-Walled Structures from Micro- to Nano-Scales

This paper develops a new numerical framework for mode III crack problems of thin-walled structures by integrating multiple advanced techniques in the boundary element literature. The details of special crack-tip elements for displacement and stress are derived. An exponential transformation technique is introduced to accurately calculate the nearly singular integral, which is the key task of the boundary element simulation of thin-walled structures. Three numerical experiments with different types of cracks are provided to verify the performance of the present numerical framework. Numerical results demonstrate that the present scheme is valid for mode III crack problems of thin-walled structures with the thickness-to-length ratio in the microscale, even nanoscale, regime.

An interaction integral method coupled with special crack tip elements for evaluation of stress intensity factors

Accurate computation of stress intensity factors (SIFs) for two-dimensional cracks by an interaction integral method combined with special crack tip elements is presented in this paper. SIFs are of great importance to fracture mechanics analysis. Due to special variation of displacement around the crack tip and the singularity and near singularity of integrals in interaction integral method, the results obtained by which are affected by the radius of the closed loop. Thus, a general method for singular integrals and nearly singular integrals is applied in the paper. With the interaction integral method coupled with the proposed special crack tip element, more stable and accurate values of the SIFs can be obtained. Numerical examples have demonstrated the effectiveness of the proposed method.

Fracture mechanics analysis of two-dimensional cracked thin structures (from micro- to nano-scales) by an efficient boundary element analysis

In this paper, the boundary element method (BEM) based on the elasticity theory is developed for fracture analysis of cracked thin structures with the relative thickness-to-length ratio in the micro- or nano-scales. A special crack-tip element technique is employed for the direct and accurate calculation of stress intensity factors (SIFs). The nearly singular integrals, which are crucial in applying the BEM for thin-structural problems, are calculated accurately by using a nonlinear coordinate transformation method. The present BEM procedure requires no remeshing procedure regardless of the thickness of thin structure. Promising SIFs results with only a small number of boundary elements can be achieved with the relative thickness of the thin film is as small as 10−9, which is sufficient for modeling most of the thin bodies as used in, for example, smart materials and micro/nano-electro-mechanical systems.

Fracture analysis of ultra-thin coating/substrate structures with interface cracks

This paper presents a fracture analysis of ultra-thin coating/substrate structures with interface cracks. For this purpose, an advanced boundary element method (BEM) is adopted. To correctly describe the oscillatory displacement and stress fields near the interfacial crack-tip, a new set of novel special crack-tip elements are implemented. The complex interfacial stress intensity factors (SIFs) can be directly and accurately computed at the collocation points extremely close to the crack-tip by using the proposed computational strategy based on the novel special crack-tip elements. Furthermore, the troublesome nearly-singular boundary integrals, which are crucial in the application of the BEM for both crack-like and ultra-thin structural problems, are calculated accurately by using a nonlinear coordinate transformation method. It is shown that the advanced BEM based on the novel special crack-tip elements and appropriate technique for evaluating nearly-singular boundary integrals provides an accurate and robust numerical tool for interfacial crack analysis of ultra-thin coating structures. Accurate and reliable BEM results with only a small number of boundary elements can be obtained for a relative coating-to-substrate thickness as small as 10-9, which is important for modeling ultra-thin coating structures as used in smart materials and micro/nano-electro-mechanical systems (MEMS/NEMS). Several representative numerical examples will be presented and discussed to reveal the effects of the key geometrical and material parameters, the crack configuration and the loading conditions on the interfacial fracture parameters such as energy release rate and complex SIFs

Novel special crack-tip elements for interface crack analysis by an efficient boundary element method

This paper presents a new set of novel special crack-tip elements for analysis of interface cracks in linear elastic composite bimaterials by using the boundary element method (BEM). Such interface crack problems present some simulation difficulties based on the conventional numerical methods because the asymptotic crack-tip fields in this case exhibit an oscillatory behavior which is quite different from that in a cracked homogeneous material. The novel special crack-tip elements developed in this study contain the correct local asymptotic behavior of the displacement and stress fields at the tip of an interface crack, and can thus appropriately describe the oscillatory near-tip displacement and stress fields for cracked composite bimaterials. The novel special crack-tip elements are implemented into a BEM, which can significantly improve the computational accuracy of the complex stress intensity factor (SIF) for an interface crack even with a relatively coarse mesh. The proposed computational strategy for interface crack analysis based on the novel special crack-tip elements also enables a direct and accurate evaluation of the complex SIF at the collocation point extremely close to the crack-tip. Several numerical examples are presented and discussed to demonstrate that the present method based on the novel special crack-tip elements is pretty simple, highly accurate, and relatively robust with respect to the size of the used crack-tip elements.

A new crack-tip singular element for cracks in three-dimensional elastic bodies

Symplectic analytical singular element (SASE) is a special crack-tip element in the framework of finite element method (FEM) for modelling cracks. A group of SASEs, which have been developed by the authors, for various crack problems almost in two-dimensional (2D), but not touched yet more complex ones, i.e., in three-dimensional (3D) domain. The underlying reason lies in the fact that analytical symplectic eigen solution for 3D elastic crack problem is not yet available, a 3D SASE thus cannot be constructed through a simple generalization from the 2D SASEs. This study aims to fill out this 3D gap, we thus use a trail displacement field to construct a 3D SASE. The derived strain and stress fields still possess singularity in the vicinity of crack-tip. Finite element formulation is derived based on the minimum total potential energy principle. It is found that some of key features of a 2D SASE still exist in the developed 3D one, e.g., the SIFs can be calculated accurately without any post-processing. Numerical examples are considered to show the accuracy and performance of the proposed element.

Applicability of the classical fracture mechanics criteria to predict the crack propagation path in rock under compression

A higher order displacement discontinuity method (HDDM) with quadratic elements is used in this study to evaluate the validity of classical fracture criteria in order to predict crack propagation path under mixed mode loading I–II in the case of a crack in an infinite body. Crack propagation path is predicted here by defining three stress states, uniaxial tensile stress (UTS), uniaxial compressive stress (UCS) and biaxial compressive stress (BCS). Comparing with experimental results, it was found that by increasing the crack inclination angle, the length of crack propagation path increases and the slope of this path decreases. For BCS state, G-criterion almost loses the validity and can be replaced with σ-criterion. S-criterion is strongly dependent on Poisson’s ratio and it is recommended to study fracture under brittle and ductile behaviour. Due to the complexity of crack tip element equations, the effect of this element on the results has been studied in detail and it was found that by considering the mid-point of the last element as a reference point to evaluate displacement discontinuities (Ds and Dn), there is no need to use special crack tip element. This procedure facilitates the modelling and shortens the solution.

Rolling disc model for rock cutting based on fracture mechanics

A rock cutting by a TBM's rolling disc model based on Linear Elastic Fracture Mechanics is considered in this study. From dimensional considerations the factors that primarily influence the normal force exerted by disc cutter mounted on the cutterhead of a rock TBM are: (i) the fracture toughness of the rock itself that controls the propagation of a critical initial crack leading to chipping, (ii) the disc's tip shape that also influences the ratio of normal force to the lateral force and the ratio of the normal force to the rolling force, and (iii) the disc's penetration depth that also depicts together with the disc radius the contact area, and (iv) the spacing between adjacent cuts. The hardness of the rock is then calculated by dividing the normal force with the contact area. These parameters are considered here as the main input parameters for the simulation of the quasi-static propagation of a lateral crack that gives a rock chip starting from the disc towards the adjacent kerf created by the neighboring cutter under the action of a concentrated load. In this first order approach we assume that the size of the starter crack over the penetration depth ratio remains constant. For the accurate computation of the mode-I and mode-II stress intensity factors at each crack growth increment, we have employed the displacement discontinuity numerical method equipped with a special crack tip element based on strain gradient or grade-2 (g2) elasticity theory. Mixed-mode crack growth obeys either the energy balance criterion or maximum tangential tensile stress criterion. The numerical results are then compared with published linear cutting rig test results on four rock types. The accuracy of the maximum tensile tangential stress criterion is better compared to the energy balance criterion. However, even with the simplest self-similar crack growth fracture criterion based on energy balance, we have found acceptable agreement of numerical with experimental results by assuming only one free parameter, namely the mode-I fracture toughness of the rock. It is thus recommended that the rock characterization for disc cutting processes should be done not only by the uniaxial compression and indirect tension tests, but also with a proper fracture mechanics test according to the ISRM Suggested Methods for the determination of the rock fracture toughness.

Spectral BEM for the Analysis of Wave Propagation and Fracture Mechanics

This paper presents a spectral boundary element formulation for analysis of structures subjected to dynamic loading. Two types of spectral elements based on Lobatto polynomials and Legendre polynomials are used. Two-dimensional analyses of elastic wave propagation in solids with and without cracks are carried out in the Laplace frequency domain with both conventional BEM and the spectral BEM. By imposing the requirement of the same level of accuracy, it was found that the use of spectral elements, compared with conventional quadratic elements, reduced the total number of nodes required for modeling high-frequency wave propagation. Benchmark examples included a simple one-dimensional bar for which analytical solution is available and a more complex crack problem where stress intensity factors were evaluated. Special crack tip elements are developed for the first time for the spectral elements to accurately model the crack tip fields. Although more integration points were used for the integrals associated with spectral elements than the conventional quadratic elements, shorter computation times were achieved through the application of the spectral BEM. This indicates that the spectral BEM is a more efficient method for the numerical modeling of structural health monitoring (SHM) processes, in which high-frequency waves are commonly used to detect damage, such as cracks, in structures.

On the accuracy of higher order displacement discontinuity method (HODDM) in the solution of linear elastic fracture mechanics problems

The higher order displacement discontinuity method (HODDM) utilizing special crack tip elements has been used in the solution of linear elastic fracture mechanics (LEFM) problems. The paper has selected several example problems from the fracture mechanics literature (with available analytical solutions) including center slant crack in an infinite and finite body, single and double edge cracks, cracks emanating from a circular hole. The numerical values of Mode I and Mode II SIFs for these problems using HODDM are in excellent agreement with analytical results (reaching up to 0.001% deviation from their analytical results). The HODDM is also compared with the XFEM and a modified XFEM results. The results show that the HODDM needs a considerably lower computational effort (with less than 400 nodes) than the XFEM and the modified XFEM (which needs more than 10000 nodes) to reach a much higher accuracy. The proposed HODDM offers higher accuracy and lower computation effort for a wide range of problems in LEFM.

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.

Simulating the propagation of hydraulic fractures from a circular wellbore using the Displacement Discontinuity Method

Hydraulic fracturing as applied to well stimulation is analyzed. Propagation of one or more hydraulic fractures from a wellbore that is oriented with either the maximum or intermediate principal in situ stress is studied. A Higher Order Displacement Discontinuity Method (HODDM) with special crack tip elements is used for achieving higher accuracy in simulation of the process. The effect of fluid pressure inside pre-existing fractures is considered. A uniformly distributed internal pressure, which implies injection of a fluid with a very low viscosity is assumed. On the onset of initiation, the crack may reorient itself as it extends from the wellbore, until it becomes aligned with the preferred direction for crack propagation relative to the far-field stresses. Propagation of multiple cracks has been considered to study their interaction. In such cases, some cracks are unable to extend at all, or they arrest after limited propagation, but others can propagate in length relative to earlier developed fractures and even may intersect each other. Also two common perforation phases are studied and best orientations for each one is determined.

Experimental and Numerical Simulation of the Microcrack Coalescence Mechanism in Rock-Like Materials

Rocks and rock-like materials frequently fail under compression due to the initiation, propagation and coalescence of the pre-existing microcracks. The mechanism of microcrack coalescence process in rock-like materials is experimentally and numerically investigated. The experimental study involves some uniaxial compression tests on rock-like specimens specially prepared from portland pozzolana cement, mica sheets and water. The microcrack coalescence is studied by scanning electron microscopy on some of the prepared thin specimens. It is assumed that the mica sheets play the role of microcracks within the specimens. Some analytical and numerical studies are also carried out to simulate the experimentally observed microcrack coalescence phenomena within the specimens. A higher-order indirect boundary element method known as higher-order displacement discontinuity method implementing special crack tip elements to treat the singularities of stress and displacement fields near the crack ends is used to estimate the Mode I and Mode II stress intensity factors at the microcrack tips. The maximum tangential stress fracture criterion is implemented in a sophisticated computer code using the linear elastic fracture mechanics theory and the propagation and coalescence of random microcracks within the rock-like specimens are numerically simulated based on an iterative algorithm. The proposed analyses are validated by comparing the corresponding experimental and numerical results of microcrack coalescence phenomena of rock-like materials.

Simulating the Crack Propagation Mechanism of Pre-Cracked Concrete Specimens Under Shear Loading Conditions

The mechanism of crack propagation in concrete specimens containing cracks under shear loading conditions is studied. The shear box test of pre-cracked (double edge cracks) concrete specimens is carried out under laboratory conditions. The higher order displacement discontinuity formulation and the special crack tip elements for the treatment of crack ends is used to numerically simulate the crack propagation mechanism of brittle solids under direct shear loading. A special modeling technique is also proposed to take into account the effect of crack overlapping on the fracturing process of the bridge area in between the two parallel cracks. In this study, the wing cracks are produced at the first stage of loading and continued their propagation paths toward the shear loading direction. The crack propagation path of the double edge cracked specimens in the bridge area is mostly affected by ligament angles and crack length whereas the shear strength is closely related to the failure pattern. The coalescence mechanism of cracks indicated that the pre-cracked concrete failure occurs in mixed mode in case of non-overlapping cracks configuration and in tensile mode for the overlapping cracks. Finally, comparing the numerical and experimental results validated the crack propagation modeling and verified the accuracy and efficiency of the proposed numerical method.

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.

A boundary element analysis of crack-propagation mechanism of micro-cracks in rock-like specimens under a uniform normal tension

In this work, the mechanism for fracture of brittle substances such as rocks under a uniform normal tension is considered. The oriented straight micro-cracks are mostly created in all the polycrystalline materials resulting from the stress concentrations. The present work focuses on the interactions of the pre-existing micro-cracks, which can grow and propagate within a rock-like specimen. The micro-crack initiation and propagation in rock-like specimens is investigated using the Fortran Code TDDCRACK 2D , which is a 2D displacement discontinuity method (DDM) for crack analysis, a boundary element computer code based on the linear elastic fracture mechanics (LEFM) theory. In the present work, a higher order DDM is used to implement special crack tip elements for estimation of the stress intensity factors (SIFs) and crack initiation angles for the wing-crack problems initiated at different angles from the original micro-crack tips in an infinite specimen under a uniform tension.

Analysis of cracks in 3D piezoelectric media with various electrical boundary conditions

A weakly singular, symmetric Galerkin boundary element method capable of solving problems of isolated cracks in three-dimensional, linear anisotropic piezoelectric, infinite media with various types of crack-face boundary conditions including impermeable, permeable, semi-permeable, and the energetically consistent boundary condition introduced by Landis (Int J Solids Struct 41:6291–6315, 2004) is established. The key governing boundary integral equation used in the formulation possesses several crucial features including its desirable symmetric weak-form, weakly singular nature, and ability to treat general material anisotropy, arbitrary crack configurations and any type of boundary condition on the crack surface. The positive consequence of utilizing the singularity-reduced integral equations in the modeling, is that all involved singular integrals can be interpreted in the sense of Riemann and their validity requires only continuous crack-face data allowing $$\hbox {C}^{0}$$ -interpolation functions to be employed everywhere in the numerical discretization. Special crack-tip elements with appropriate square-root functions are adopted in a local region along the crack front to accurately approximate the relative crack-face displacement and electric potential. With use of these crack-tip elements, the stress and electric intensity factors can be extracted directly in terms of crack-front nodal data. A system of nonlinear algebraic equations resulting from semi-permeable and energetically consistent boundary conditions is solved by standard Newton–Raphson iterative scheme. Various numerical examples of both planar and non-planar cracks under different types of electrical boundary conditions are considered and the proposed technique is found promising and computationally robust. In addition, it was determined that using crack-tip elements along the crack front significantly enhances the computational performance and that the stress and electric intensity factors can be obtained accurately using relatively coarse meshes.