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The aim of the present study is to understand hydraulic fracture propagation behavior in shale formations through numerical simulation. The propagation regime of shale fractures is first analyzed based on shale rock properties and assuming the slick-water fracturing condition holds. Among the formation conditions we discuss in this paper, the transition regime appears as a dominant propagation mechanism. Based on this knowledge, we establish an XFEM-based hydraulic fracture propagation model. The orthotropic nature of shale is taken into account. An iterative approach is successfully used to deal with the solid-fluid interaction problem. The discrete model is verified by several analytical solutions. A Five-stage hydraulic fracturing is simulated to understand the mechanical interaction of fractures with each other. Results show that on-going hydraulic fractures are attracted by the pre-existing hydraulic fractures as a result of the change of direction and magnitude of the local stress. Further, fracture deflections become extensive when the fracture spacing and horizontal stress difference decrease and Young’s modulus ratio increases.

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The hydraulic fracture (HF) morphology and corresponding stimulated reservoir volume (SRV) are significantly dependent on the geomechanical factors of the formation. A better understanding of the hydraulic fracturing mechanism under different reservoir attributes is crucial for fracability evaluation and fracturing treatment optimization. In this work, the geomechanical controls on hydraulic fracturing in a heterogeneous formation and its fracability are investigated using a three-dimensional (3D) fully coupled hydraulic–mechanical–damage (HMD) model. Rock heterogeneity, which causes nonlinear progressive failure behavior, is considered in this model by assuming that the mechanical parameters of elements follow a Weibull distribution. The elastic damage mechanics and Darcy’s law describe the damage process and fluid flow in elements, respectively. The element permeability is dependent on its state, which describes the effect of stress on the seepage field. The HF width is conceptually represented by the aperture of fractures, which depends on the failure mechanism of the damaged element. The coupled equations are solved numerically using the finite element method. The model is verified with experimental results of HF network propagation and multi-fracture interference. Then, a series of numerical simulations were performed to investigate the geomechanical controls of HF geometry and SRV in heterogeneous formations. At last, the optimal conditions for the formation of a complex HF network are further discussed according to the numerical results, based on which an improved fracability index is established. The results show that the numerical model can capture the 3D nature of the HFs and reproduce the HF network propagation and multi-fracture interference process. The complex HFs are more likely to generate in formations with high brittleness, large natural fracture (NF) density, small horizontal stress difference, and small fracture toughness. This study provides a reliable numerical method for hydraulic fracturing simulation and offers some reference for the fracability evaluation and fracturing treatment design in heterogeneous formations.

We present as areas of interest for multifield fracturing thermomechanical fracturing, fluid pressure induced isothermal and nonisothermal fracturing, fracturing due to radiation, drying, hydrogen embrittlement, and fractures induced by chemical effects. We discuss the most appropriate constitutive models for their simulation and choose the cohesive fracture model for quasi-brittle materials. Successively we show governing equations for a thermo-hydro-mechanical problem, which is representative for multifield problems. Possible extensions to more fields are addressed. Then methods for numerical modeling of multifield fracturing are presented and the most representative ones, i.e., interface and embedded discontinuity elements, X-FEM, thick level set and phase field models, and discrete crack approach with adaptive remeshing are discussed in some detail. After incorporating this last method in the governing equations, their numerical solution is shown together with the necessary adaptivity in time and space. This solution is validated. Successively applications to thermomechanical fracture; hydraulic fracturing in case of a pumped well and of 2D and 3D dams; fracturing of drying concrete and of a massive concrete beam and finally mechanical effects of chemical processes in concrete are shown. In the case of the pumped well with constant pumping rate, a comparison between an Extended Finite Element solution and that of the discrete crack approach with adaptive remeshing is made which allows for interesting considerations about the nature of hydraulic fracturing. The examples permit to conclude that with increasing complexity of the multifield problems that of the employed fracture models decreases, i.e., advanced fracture models have to date only been applied to problems with a limited number of fields, mainly displacement, thermal and/or pressure fields. There is hence plenty of room for improvement.

The focus of this paper is on constructing the solution for a semi-infinite hydraulic crack for arbitrary toughness, which accounts for the presence of a lag of a priori unknown length between the fluid front and the crack tip. First, we formulate the governing equa-tions for a semi-infinite fluid-driven fracture propagating steadily in an impermeable linear elastic medium. Then, since the pressure in the lag zone is known, we suggest a new inversion of the integral equation from elasticity theory to express the opening in terms of the pressure. We then calculate explicitly the contribution to the opening from the loading in the lag zone, and reformulate the problem over the fluid-filled portion of the crack. The asymptotic forms of the solution near and away from the tip are then dis-cussed. It is shown that the solution is not only consistent with the square root singularity of linear elastic fracture mechanics, but that its asymptotic behavior at infinity is actually given by the singular solution of a semi-infinite hydraulic fracture constructed on the assumption that the fluid flows to the tip of the fracture and that the solid has zero toughness. Further, the asymptotic solution for large dimensionless toughness is derived, including the explicit dependence of the solution on the toughness. The intermediate part of the solution (in the region where the solution evolves from the near tip to the far from the tip asymptote) of the problem in the general case is obtained numerically and relevant results are discussed, including the universal relation between the fluid lag and the toughness.

Modeling hydraulic fracturing in the presence of a natural fracture network is a challenging task, owing to the complex interactions between fluid, rock matrix, and rock interfaces, as well as the interactions between propagating fractures and existing natural interfaces. Understanding these complex interactions through numerical modeling is critical to the design of optimum stimulation strategies. In this paper, we present an explicitly integrated, fully coupled discrete‐finite element approach for the simulation of hydraulic fracturing in arbitrary fracture networks. The individual physical processes involved in hydraulic fracturing are identified and addressed as separate modules: a finite element approach for geomechanics in the rock matrix, a finite volume approach for resolving hydrodynamics, a geomechanical joint model for interfacial resolution, and an adaptive remeshing module. The model is verified against the Khristianovich–Geertsma–DeKlerk closed‐form solution for the propagation of a single hydraulic fracture and validated against laboratory testing results on the interaction between a propagating hydraulic fracture and an existing fracture. Preliminary results of simulating hydraulic fracturing in a natural fracture system consisting of multiple fractures are also presented. Copyright © 2012 John Wiley & Sons, Ltd.

Abstract: This chapter surveys mathematical methods and principal results in the mechanics of fracture. Primary emphasis is placed on the analysis of crack extension as treated through methods of continuum mechanics. Section II begins with relevant concepts and basic equations from the mechanics of solids, including a survey of elasticity and plasticity, and of associated mathematical methods for boundary value problems, such as analytic function theory. Energy comparison methods and the related path-independent energy integral are introduced in this
section; these novel methods of analysis prove to be widely applicable for subsequently treated notch and crack problems. Section III deals with the application of linear elasticity to fracture. Several two-dimensional crack, problems are solved and approximate methods are presented for determination of stress-intensity factors with more complicated geometries. Theories of elastic-brittle fracture are reviewed and the equivalence of Griffith energy balance and cohesive forces approaches
is demonstrated. In addition, dynamic running crack problems, energy
rate computations, and stress concentrations at smooth-ended notches are discussed. Section IV, the longest section, deals with the elastic-plastic and fully plastic analysis of fracture. Here, the small-scale yielding approximation, for which elastic stress-intensity factors govern near tip deformation fields, is presented. Elastic-plastic crack problems in plane strain and plane stress are discussed; while these results are necessarily approximate, further insight is provided by treatment of the simpler antiplane strain case. The incremental and path-dependent nature
of plastic stress-strain relations is shown to lead to a view of fracture as an instability point in a process of continuing crack advance under increasing load. Additional topics in this section include plastic strain concentrations at smooth ended notches, limit analysis of notched bodies, and a brief treatment of separation mechanisms in ductile materials.

In this work, a simple and efficient XFEM approach has been presented to solve 3-D crack problems in linear elastic materials. In XFEM, displacement approximation is enriched by additional functions using the concept of partition of unity. In the proposed approach, a crack front is divided into a number of piecewise curve segments to avoid an iterative solution. A nearest point on the crack front from an arbitrary (Gauss) point is obtained for each crack segment. In crack front elements, the level set functions are approximated
by higher order shape functions which assure the accurate modeling of the crack front. The values of stress intensity factors are obtained from XFEM solution by domain based interaction integral approach. Many benchmark crack problems are solved by the proposed XFEM approach. A convergence study has been
conducted for few test problems. The results obtained by proposed XFEM approach are compared with the
analytical/reference solutions.

In this paper, the extended finite element method (X-FEM) is investigated for the solution of hydraulic fracture problems. The presence of an internal pressure inside the crack is taken into account. Special tip functions encapsulating tip asymptotics typically encountered in hydraulic fractures are introduced. We are especially interested in the two limiting tip behaviour for the impermeable case: the classical LEFM square root asymptote in fracture width for the toughness-dominated regime of propagation and the so-called ⅔ asymptote in fracture width for the viscosity-dominated regime. Different variants of the X-FEM are tested for the case of a plane-strain hydraulic fracture propagation in both the toughness and the viscosity dominated regimes. Fracture opening and fluid pressure are compared at each nodes with analytical solutions available in the literature. The results demonstrate the importance of correcting for the loss of partition of unity in the transition zone between the enriched part and the rest of the mesh. A point-wise matching scheme appears sufficient to obtain accurate results. Proper integration of the singular terms introduced by the enrichment functions is also critical for good performance. Copyright © 2008 John Wiley & Sons, Ltd.

This paper presents a computational technique based on the extended finite element method (XFEM) and the level set method for the growth of biofilms. The discontinuous-derivative enrichment of the standard finite element approximation eliminates the need for the finite element mesh to coincide with the biofilm–fluid interface and also permits the introduction of the discontinuity in the normal derivative of the substrate concentration field at the biofilm–fluid interface. The XFEM is coupled with a comprehensive level set update scheme with velocity extensions, which makes updating the biofilm interface fast and accurate without need for remeshing. The kinetics of biofilms are briefly given and the non-linear strong and weak forms are presented. The non-linear system of equations is solved using a Newton–Raphson scheme. Example problems including 1D and 2D biofilm growth are presented to illustrate the accuracy and utility of the method. The 1D results we obtain are in excellent agreement with previous 1D results obtained using finite difference methods. Our 2D results that simulate finger formation and finger-tip splitting in biofilms illustrate the robustness of the present computational technique. Copyright © 2007 John Wiley & Sons, Ltd.

The flow of a single-phase fluid through a rough-walled rock fracture is discussed within the context of fluid mechanics. The derivation of the cubic law is given as the solution to the Navier-Stokes equations for flow between smooth, parallel plates - the only fracture geometry that is amenable to exact treatment. The various geometric and kinematic conditions that are necessary in order for the Navier-Stokes equations to be replaced by the more tractable lubrication or Hele-Shaw equations are studied and quantified. In general, this requires a sufficiently low flow rate, and some restrictions on the spatial rate of change of the aperture profile. Various analytical and numerical results are reviewed pertaining to the problem of relating the effective hydraulic aperture to the statistics of the aperture distribution. These studies all lead to the conclusion that the effective hydraulic aperture is less than the mean aperture, by a factor that depends on the ratio of the mean value of the aperture to its standard deviation. The tortuosity effect caused by regions where the rock walls are in contact with each other is studied using the Hele-Shaw equations, leading to a simple correction factor that depends on the area fraction occupied by the contact regions. Finally, the predicted hydraulic apertures are compared to measured values for eight data sets from the literature for which aperture and conductivity data were available on the same fracture. It is found that reasonably accurate predictions of hydraulic conductivity can be made based solely on the first two moments of the aperture distribution function, and the proportion of contact area.

This paper studies the effects of chemical, elastic and interfacial energies on the equilibrium morphology of misfit particles due to phase separation in binary alloys under chemo-mechanical equilibrium conditions. A continuum framework that governs the chemo-mechanical equilibrium of the system is first developed using a variational approach by treating the phase interface as a sharp interface endowed with interfacial excess energy. An extended finite element method (XFEM) in conjunction with the level set method is then developed to simulate the behaviors of the coupled chemo-mechanical system. The coupled chemo-mechanics model together with the numerical techniques developed here provides an efficient simulation tool to predict the equilibrium morphologies of precipitates in phase separate alloys.

Successfully creating multiple hydraulic fractures in horizontal wells is critical for unconventional gas production economically. However, it will not be possible to fully optimize multiple fracture stimulations until accurate methods are available to directly predict fracture geometry.
In this paper, a novel Fracture Propagation Model (FPM) is developed to simulate multiple hydraulic fracture propagation simultaneously in a stage of the horizontal wellbore. This model couples fracture deformation with fluid flow in the fractures and the horizontal wellbore. The displacement discontinuity method is used to describe the geometry of multiple fractures, considering mechanical interaction between the fractures. Fluid flow and pressure drop in the fractures are determined by the Hagen Poiseuille law. Friction pressure drop in the wellbore and perforation zones is taken into account by applying Kirchoff's first and second laws. The fluid flow rates and pressure compatibility are maintained between the wellbore and the multiple fractures by using Newton's numerical method. The model accurately predicts multiple fracture geometry and computed non-planar fracture trajectories. Shear stress distributions indicate areas likely to have induced micro-seismicity.
The simulation results of FPM can serve as guidance for shale gas simulation and multiple fracture treatments. Also, the results from the model highlight conditions under which restricted width occurs that could lead to proppant screenout. This wok can help operators to design multiple fractures to enhance recovery in unconventional gas reservoirs.

This paper presents a general overview of hydraulic fracturing models developed and applied to simulation of complex fractures in naturally fractured shale reservoirs. It discusses the technical challenges involved in modeling complex hydraulic fracture networks, the interaction between a hydraulic fracture and a natural fracture, and various models and modeling approaches developed to simulate hydraulic fracture–natural fracture interaction, as well as the induced large scale complex fractures during fracturing treatments.

Physical accuracy, numerical stability, and computational speed are critical factors in the simulation of collisions. The impulse-based method models collisions in a system of rigid bodies in a relatively reliable and fast manner. In the present paper, evidence is presented for the energy-conserving and momentum-conserving properties of the method. Two different impulse-based approaches are validated using numerical tests. A necessary condition is proposed for the impulse-based method to be energy conservative. Results indicate that the impulse-based method for collision simulation, which satisfies the proposed condition, is energy conservative. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, a fully coupled numerical model is developed for the modeling of the hydraulic fracture propagation in porous media using the extended finite element method in conjunction with the cohesive crack model. The governing equations, which account for the coupling between various physical phenomena, are derived within the framework of the generalized Biot theory. The fluid flow within the fracture is modeled using the Darcy law, in which the fracture permeability is assumed according to the well-known cubic law. By taking the advantage of the cohesive crack model, the nonlinear fracture processes developing along the fracture process zone are simulated. The spatial discretization using the extended finite element method and the time domain discretization applying the generalized Newmark scheme yield the final system of fully coupled nonlinear equations, which involves the hydro-mechanical coupling between the fracture and the porous medium surrounding the fracture. The fluid leak-off and the length of fracture extension are obtained through the solution of the resulting system of equations, not only leading to the correct estimation of the fracture tip velocity as well as the fluid velocity within the fracture, but also allowing for the eventual formation of the fluid lag. It is illustrated that incorporating the coupled physical processes, i.e. the solid skeleton deformation, the fluid flow in the fracture and in the pore spaces of the surrounding porous medium and the hydraulic fracture propagation, is crucial to provide a correct solution for the problem of the fluid-driven fracture in porous media, where the poroelastic effects are significant.

We presented a finite-element-based algorithm to simulate plane-strain, straight hydraulic fractures in an impermeable elastic medium. The algorithm accounts for the nonlinear coupling between the fluid pressure and the crack opening and separately tracks the evolution of the crack tip and the fluid front. It therefore allows the existence of a fluid lag. The fluid front is advanced explicitly in time, but an implicit strategy is needed for the crack tip to guarantee the satisfaction of Griffith's criterion at each time step. We enforced the coupling between the fluid and the rock by simultaneously solving for the pressure field in the fluid and the crack opening at each time step. We provided verification of our algorithm by performing sample simulations and comparing them with two known similarity solutions. Copyright © 2012 John Wiley & Sons, Ltd.

The extended finite element method enhances the approximation properties of the finite element space by using additional enrichment functions. But the resulting stiffness matrices can become ill-conditioned. In that case iterative solvers need a large number of iterations to obtain an acceptable solution. In this paper a procedure is described to obtain stiffness matrices whose condition number is close to the one of the finite element matrices without any enrichments. A domain decomposition is employed and the algorithm is very well suited for parallel computations. The method was tested in numerical experiments to show its effectiveness. The experiments have been conducted for structures containing cracks and material interfaces. We show that the corresponding enrichments can result in arbitrarily ill-conditioned matrices. The method proposed here, however, provides well-conditioned matrices and can be applied to any sort of enrichment. The complexity of this approach and its relation to the domain decomposition is discussed. Computation times have been measured for a structure containing multiple cracks. For this structure the computation times could be decreased by a factor of 2. Copyright © 2010 John Wiley & Sons, Ltd.

We describe coupled algorithms that use the Extended Finite Element Method (XFEM) to solve the elastic crack component of the elasto-hydrodynamic equations that govern the propagation of hydraulic fractures in an elastic medium. With appropriate enrichment, the XFEM resolves the Neumann to Dirichlet (ND) map for crack problems with O(h2)O(h2) accuracy and the Dirichlet to Neumann (DN) map with O(h)O(h) accuracy. For hydraulic fracture problems with a lag separating the fluid front from the fracture front, we demonstrate that the finite pressure field makes it possible to use a scheme based on the O(h2)O(h2) XFEM solution to the ND map. To treat problems in which there is a coalescence of the fluid and fracture fronts, resulting in singular tip pressures, we developed a novel mixed algorithm that combines the tip width asymptotic solution with the O(h2)O(h2) XFEM solution of the ND map away from the tips. Enrichment basis functions required for these singular pressure fields correspond to width power law indices λ>12, which are different from the index λ=12 of linear elastic fracture mechanics. The solutions obtained from the new coupled XFEM schemes agree extremely well with those of published reference solutions.

Hydraulic fracturing in shale gas reservoirs has often resulted in complex fracture network growth, as evidenced by microseismic monitoring. The nature and degree of fracture complexity must be clearly understood to optimize stimulation design and completion strategy. Unfortunately, the existing single planar fracture models used in the industry today are not able to simulate complex fracture networks.
A new hydraulic fracture model is developed to simulate complex fracture network propagation in a formation with preexisting natural fractures. The model solves a system of equations governing fracture deformation, height growth, fluid flow, and proppant transport in a complex fracture network with multiple propagating fracture tips. The interaction between a hydraulic fracture and pre-existing natural fractures is taken into account by using an analytical crossing model and is validated against experimental data. The model is able to predict whether a hydraulic fracture front crosses or is arrested by a natural fracture it encounters, which leads to complexity. It also considers the mechanical interaction among the adjacent fractures (i.e., the "stress shadow" effect). An efficient numerical scheme is used in the model so it can simulate the complex problem in a relatively short computation time to allow for day-to-day engineering design use.
Simulation results from the new complex fracture model show that stress anisotropy, natural fractures, and interfacial friction play critical roles in creating fracture network complexity. Decreasing stress anisotropy or interfacial friction can change the induced fracture geometry from a bi-wing fracture to a complex fracture network for the same initial natural fractures. The results presented illustrate the importance of rock fabrics and stresses on fracture complexity in unconventional reservoirs. They have major implications on matching microseismic observations and improving fracture stimulation design.

This paper treats the propagation of hydraulic fractures of limited vertical extent and elliptic cross-section with the effect of fluid loss included. Numerical and asymptotic approximate solutions in dimensionless form give the fracture length and width at any value of time or any set of physical parameters. The insight provided by The dimensionless parameters. The insight provided by The dimensionless results and approximate solutions should be useful in the design of fracture treatments.
Introduction
The theory and practice of hydraulic fracturing has been reviewed by Howard and Fast. Therefore, we confine our discussion of previous investigations to those pertinent to the present study of the propagation of vertical fractures. propagation of vertical fractures. An important theoretical result is Carter's formula for the area of a fracture of constant width formed by injection at constant rate with fluid lost to the formation. For a vertical fracture of constant height, Carter's formula gives fracture length as a function of time. In general, Carter's assumption of constant width is not realistic. However, at large values of time the effect of this assumption becomes insignificant since the effect of fluid loss dominates. The width of a vertical fracture was first investigated by Khristianovic and Zheltov under the assumption that the width does not vary in the vertical direction. Thus, a state of plane strain prevails in horizontal planes and the width can be prevails in horizontal planes and the width can be determined as the solution of a plane elasticity problem. An approximate solution is found in Ref. 3 problem. An approximate solution is found in Ref. 3 upon neglect of fluid loss, fracture volume change, and pressure variation along the fracture. The fracture length is determined by the condition of finite stress at the fracture tip. Baron et al. and Geertsma and de Klerk have included the effect of fluid loss in the approach of Ref. 3. Geertsma and de Klerk give simple approximate formulas for fracture length and width. A different approach to the determination of fracture width was taken by Perkins and Kern. They considered a vertically limited fracture under the assumption of plane strain in vertical planes perpendicular to the fracture plane. The perpendicular to the fracture plane. The cross-section of the fracture is found to be elliptical, and the maximum width decreases along the fracture according to a simple formula that contains the fracture length. In the derivation of this formula, fluid loss and fracture volume change are neglected in the continuity equation and the fracture length is not determined. In a subsequent application, a "reasonable" fracture length was assumed. Carter's formula for length and the width formula of Perkins and Kern are both cited by Howard and Fast, and combined use of the two formulas is believed to be common practice. The present theoretical investigation is concerned with vertically limited fractures of the type studied by Perkins and Kern. However, we include the effects of fluid loss and fracture volume change in the continuity equation. Consequently, fracture length is determined as part of the solution. General results for the variation of fracture width and length with time are obtained in dimensionless form by a numerical method. In addition, asymptotic solutions are derived for large and small values of time. The small-time solution is also the exact solution for the case of no fluid loss to the formation. For large values of time our asymptotic formula for fracture length is identical with Carter's formula at large time. Our large-time formula for fracture width differs from the formula of Perkins and Kern by a numerical factor that varies along the fracture. In comparison with our formula, this formulas overestimates the width by 12 percent at the well and 24 percent at the midlength of the fracture. At early times Perkins and Kern's formulas for width in terms of length is again a fair approximation to our result. However, our formula for length differs from Carter's formula, which is not applicable since the neglected width variation is important at early times. The results for the width of a vertically limited fracture as obtained here and in Ref. 6 differ from the results for vertically constant fractures.
SPEJ
P. 306

This paper reviews recent results of a research program aimed at developing a theoretical framework to understand and predict the different modes of propagation of a fluid-driven fracture. The research effort involves constructing detailed solutions of the crack tip region, developing global models of hydraulic fractures for plane strain and radial geometry, and identifying the parameters controlling the fracture growth. The paper focuses on the propagation of hydraulic fractures in impermeable rocks. The controlling parameters are identified from scaling laws that recognize the existence of two dissipative processes: fracturing of the rock (toughness) and dissipation in the fracturing fluid (viscosity). It is shown that the two limit solutions (corresponding to zero toughness and zero viscosity) are characterized by a power law dependence on time and that the transition between these two asymptotic solutions depends on a single number, which can be chosen to be either a dimensionless toughness or a dimensionless viscosity. The viscosity- and toughness-dominated regime of crack propagation are then identified by comparing the general solutions with the asymptotic solutions. This analysis yields the ranges of the dimensionless parameter for which the solution can be approximated for all practical purposes either by the zero toughness or by the zero viscosity solution.

This paper presents an enriched finite element method to simulate the growth of cracks in linear elastic, aerospace composite materials. The model and its discretisation are also validated through a complete experimental test series. Stress intensity factors are calculated by means of an interaction integral. To enable this, we propose application of (1) a modified approach to the standard interaction integral for heterogeneous orthotropic materials where material interfaces are present; (2) a modified maximum hoop stress criterion is proposed for obtaining the crack propagation direction at each step, and we show that the “standard” maximum hoop stress criterion which had been frequently used to date in literature, is unable to reproduce experimental results. The influence of crack description, material orientation along with the presence of holes and multi-material structures are investigated. It is found, for aerospace composite materials with E1/E2 ratios of approximately 10, that the material orientation is the driving factor in crack propagation. This is found even for specimens with a material orientation of 90°, which were previously found to cause difficulty in both damage mechanics and discrete crack models e.g. by the extended finite element method (XFEM). The results also show the crack will predominantly propagate along the fibre direction, regardless of the specimen geometry, loading conditions or presence of voids.

Strain singularities appear in many linear elasticity problems. A very fine mesh has to be used in the vicinity of the singularity in order to obtain acceptable numerical solutions with the finite element method (FEM). Special enrichment functions describing this singular behavior can be used in the extended finite element method (X-FEM) to circumvent this problem. These functions have to be known in advance, but their analytical form is unknown in many cases. Li et al. described a method to calculate singular strain fields at the tip of a notch numerically. A slight modification of this approach makes it possible to calculate singular fields also in the interior of the structural domain. We will show in numerical experiments that convergence rates can be significantly enhanced by using these approximations in the X-FEM. The convergence rates have been compared with the ones obtained by the FEM. This was done for a series of problems including a polycrystalline structure. Copyright © 2010 John Wiley & Sons, Ltd.

Determining the lifetime of solder joints subjected to thermomechanical loads is crucial to guarantee the quality of electronic devices. The fatigue process is heavily dependent on the microstructure of the joints. We present a new methodology to determine the lifetime of the joints based on microstructural phenomena. Random microstructures are generated to capture the statistical variety of possible microstructures and crack growth calculations are performed. The extended finite element method is used to solve the structural problem numerically which allows a complete automation of the process. Numerical examples are given and compared to experimental data.

Extensions of a new technique for the finite element modelling of cracks with multiple branches, multiple holes and cracks emanating from holes are presented. This extended finite element method (X-FEM) allows the representation of crack discontinuities and voids independently of the mesh. A standard displacement-based approximation is enriched by incorporating discontinuous fields through a partition of unity method. A methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed. Computation of the stress intensity factors (SIF) in different examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique. Copyright © 2000 John Wiley & Sons, Ltd.

A method for modelling the growth of multiple cracks in linear elastic media is presented. Both homogeneous and inhomogeneous materials are considered. The method uses the extended finite element method for arbitrary discontinuities and does not require remeshing as the cracks grow; the method also treats the junction of cracks. The crack geometries are arbitrary with respect to the mesh and are described by vector level sets. The overall response of the structure is obtained until complete failure. A stability analysis of competitive cracks tips is performed. The method is applied to bodies in plane strain or plane stress and to unit cells with 2–10 growing cracks (although the method does not limit the number of cracks). It is shown to be efficient and accurate for crack coalescence and percolation problems. Copyright © 2004 John Wiley & Sons, Ltd.

This paper provides experimental confirmation of the opening asymptotes that have been predicted to develop at the tip of fluid-driven cracks propagating in impermeable brittle elastic media. During propagation of such cracks, energy is dissipated not only by breaking of material bonds ahead of the tip but also by flow of viscous fluid. Theoretical analysis based on linear elastic fracture mechanics and lubrication theory predicts a complex multiscale asymptotic behavior of the opening in the tip region, which simplifies either as or as power law of the distance from the tip depending on whether the dominant mechanism of energy dissipation is bond breaking or viscous flow. The laboratory experiments entail the propagation of penny-shaped cracks by injection of glycerin or glucose based solutions in polymethyl methacrylate (PMMA) and glass specimens subjected to confining stresses. The full-field opening is measured from analysis of the loss of intensity as light passes through the dye-laden fluid that fills the crack. The experimental near-tip opening gives excellent agreement with theory and therefore confirms the predicted multi-scale tip asymptotics.

A technique is presented to model arbitrary discontinuities in the finite element framework by locally enriching a displacement-based approximation through a partition of unity method. This technique allows discontinuities to be represented independently of element boundaries. The method is applied to fracture mechanics, in which crack discontinuities are represented using both a jump function and the asymptotic near-tip fields. As specific examples, we consider cracks and crack growth in two-dimensional elasticity and Mindlin–Reissner plates. A domain form of the J-integral is also derived to extract the moment intensity factors. The accuracy and utility of the method is also discussed.

The recently developed techniques for modelling cracking within the finite element (FE) framework which use meshes independent of the crack configuration and thus avoid remeshing are reviewed. They combine the traditional FE method with the partition of unity method for modelling individual cracks, intersecting or branching cracks, as well as cracks emanating from holes or other internal interfaces. Numerical integration for the enriched elements, linear dependence and the corresponding solution techniques for the discretized system of equations, as well as the accuracy of the crack tip fields are addressed. Future improvements of the techniques as well as their applications are discussed.

We present in this paper recent achievements realised on the application of strain smoothing in finite elements and propose suitable extensions to problems with discontinuities and singularities. The numerical results indicate that for 2D and 3D continuum, locking can be avoided. New plate and shell formulations that avoid both shear and membrane locking are also briefly reviewed. The principle is then extended to partition of unity enrichment to simplify numerical integration of discontinuous approximations in the extended finite element method. Examples are presented to test the new elements for problems involving cracks in linear elastic continua and cracked plates. In the latter case, the proposed formulation suppresses locking and yields elements which behave very well, even in the thin plate limit. Two important features of the set of elements presented are their insensitivity to mesh distortion and a lower computational cost than standard finite elements for the same accuracy. These elements are easily implemented in existing codes since they only require the modification of the discretized gradient operator, B.

This paper analyses the plane strain problem of a fracture, driven by injection of an incompressible viscous Newtonian fluid, which propagates parallel to the free surface of an elastic half-plane. The problem is governed by a hyper-singular integral equation, which relates crack opening to net pressure according to elasticity, and by the lubrication equations which describe the laminar fluid flow inside the fracture. The challenge in solving this problem results from the changing nature of the elasticity operator with growth of the fracture, and from the existence of a lag zone of a priori unknown length between the crack tip and the fluid front. Scaling of the governing equations indicates that the evolution problem depends in general on two numbers, one which can be interpreted as a dimensionless toughness and the other as a dimensionless confining stress. The numerical method adopted to solve this non-linear evolution problem combines the displacement discontinuity method and a finite difference scheme on a fixed grid, together with a technique to track both crack and fluid fronts. It is shown that the solution evolves in time between two asymptotic similarity solutions. The small time asymptotic solution corresponding to the solution of a hydraulic fracture in an infinite medium under zero confining stress, and the large time to a solution where the aperture of the fracture is similar to the transverse deflection of a beam clamped at both ends and subjected to a uniformly distributed load. It is shown that the size of the lag decreases (to eventually vanish) with increasing toughness and compressive confining stress.

51210006 & 51234006) Authors also thank Dr

- Nos

Nos.51210006 & 51234006). Authors also thank Dr. Elizaveta Gordeliy, Dr. Anthony R. Lamb,
424
and Dr. Andrew Bunger for their valuable discussion.

Global energy minimization for multiple fracture growth

- D Sutula
- S Bordas

Sutula, D. and S. Bordas, Global energy minimization for multiple fracture growth, 2013.

Global Energy Minimization for Multi-crack Growth in Linear Elastic Fracture using the Extended 488 Finite Element Method

- D Sutula
- S Bordas

Sutula, D. and S. Bordas, Global Energy Minimization for Multi-crack Growth in Linear Elastic Fracture using the Extended
488
Finite Element Method. 2014.