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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2015

Cosmic bubble and domain wall instabilities II: fracturing of colliding walls

We study collisions between nearly planar domain walls including the effects of small initial nonplanar fluctuations. These perturbations represent the small fluctuations that must exist in a quantum treatment of the problem. In a previous paper, we demonstrated that at the linear level a subset of these fluctuations experience parametric amplification as a result of their coupling to the planar symmetric background. Here we study the full three-dimensional nonlinear dynamics using lattice simulations, including both the early time regime when the fluctuations are well described by linear perturbation theory as well as the subsequent stage of fully nonlinear evolution. We find that the nonplanar fluctuations have a dramatic effect on the overall evolution of the system. Specifically, once these fluctuations begin to interact nonlinearly the split into a planar symmetric part of the field and the nonplanar fluctuations loses its utility. At this point the colliding domain walls dissolve, with the endpoint of this being the creation of a population of oscillons in the collision region. The original (nearly) planar symmetry has been completely destroyed at this point and an accurate study of the system requires the full three-dimensional simulation.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

The eXtended virtual element method for elliptic problems with weakly singular solutions

This paper introduces a novel eXtended virtual element method, an extension of the conforming virtual element method. The X-VEM is formulated by incorporating appropriate enrichment functions in the local spaces. The method is designed to handle highly generic enrichment functions, including singularities arising from fractured domains. By achieving consistency on the enrichment space, the method is proven to achieve arbitrary approximation orders even in the presence of singular solutions. The paper includes a complete convergence analysis under general assumptions on mesh regularity, and numerical experiments validating the method’s accuracy on various mesh families, demonstrating optimal convergence rates in the L 2 - and H 1 - norms on fractured or L-shaped domains.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Fracture Intensity Impacts on Reaction Front Propagation and Mineral Weathering in Three‐Dimensional Fractured Media

Abstract Studying reaction front propagation in heterogeneous natural settings is challenging, but numerical simulations can provide insight into the varying spatial and temporal scales of reaction front propagation. Here, the impact of increasing fracture intensity on mineral dissolution rates, and the extent of reaction front propagation is investigated using reactive transport simulations in upscaled discrete fracture network domains with varied fracture intensity. Domain‐averaged dissolution rates vary less than 0.5 log units regardless of the fracture intensity, but the spatial distribution of reactions is controlled by the location and number of dead‐end fractures and the number of connected flowpaths through the domain. Higher fracture intensities lead to more weathering in the domain because of more available mineral for water‐rock interactions. We find that reaction fronts propagate through the primary flowpaths in the first 10,000 years of the simulation for a 10‐m length domain, then propagate into secondary flowpaths and dead‐end fractures between 10,000 and 100,000 years, and finally into the matrix over timescales of hundreds of thousands of years. The domain‐averaged reaction rates decrease through time corresponding to a transition from dissolution in advection‐dominated, fast‐flowing pathways, to dissolution in transport‐limited zones of disconnected fractures and matrix. Matrix dissolution, or dissolution under transport‐limited conditions, is the dominant process at late times in these simulations. The results of these simulations recreate the observed paradox found in nature where highly fractured hillslopes tend to be more weathered but have slower weathering rates, while hillslopes with fewer fractures, are less weathered but have higher dissolution rates.

58 GEOSCIENCES↗
DOE OSTI2016

SURROGATE MODEL DEVELOPMENT AND VALIDATION FOR RELIABILITY ANALYSIS OF REACTOR PRESSURE VESSELS

In nuclear light water reactors (LWRs), the reactor coolant, core and shroud are contained within a massive, thick walled steel vessel known as a reactor pressure vessel (RPV). Given the tremendous size of these structures, RPVs typically contain a large population of pre-existing flaws introduced in the manufacturing process. After many years of operation, irradiation-induced embrittlement makes these vessels increasingly susceptible to fracture initiation at the locations of the pre-existing flaws. Because of the uncertainty in the loading conditions, flaw characteristics and material properties, probabilistic methods are widely accepted and used in assessing RPV integrity. The Fracture Analysis of Vessels – Oak Ridge (FAVOR) computer program developed by researchers at Oak Ridge National Laboratory is widely used for this purpose. This program can be used in order to perform deterministic and probabilistic risk-informed analyses of the structural integrity of an RPV subjected to a range of thermal-hydraulic events. FAVOR uses a one-dimensional representation of the global response of the RPV, which is appropriate for the beltline region, which experiences the most embrittlement, and employs an influence coefficient technique to rapidly compute stress intensity factors for axis-aligned surface-breaking flaws. The Grizzly code is currently under development at Idaho National Laboratory (INL) to be used as a general multiphysics simulation tool to study a variety of degradation mechanisms in nuclear power plant components. The first application of Grizzly has been to study fracture in embrittled RPVs. Grizzly can be used to model the thermo-mechanical response of an RPV under transient conditions observed in a pressurized thermal shock (PTS) scenario. The global response of the vessel provides boundary conditions for local 3D models of the material in the vicinity of a flaw. Fracture domain integrals are computed to obtain stress intensity factors, which can in turn be used to assess whether a fracture would initiate at a pre-existing flaw. To use Grizzly for probabilistic analysis, it is necessary to have a way to rapidly evaluate stress intensity factors. To accomplish this goal, a reduced order model (ROM) has been developed to efficiently represent the behavior of a detailed 3D Grizzly model used to calculate fracture parameters. This approach uses the stress intensity factor influence coefficient method that has been used with great success in FAVOR. Instead of interpolating between tabulated solutions, as FAVOR does, the ROM approach uses a response surface methodology to compute fracture solutions based on a sampled set of results used to train the ROM. The main advantages of this approach are that the process of generating the training data can be fully automated, and the procedure can be readily used to consider more general flaw configurations. This paper demonstrates the procedure used to generate a ROM to rapidly compute stress intensity factors for axis-aligned flaws. The results from this procedure are in good agreement with those produced using the traditional influence coefficient interpolation procedure, which gives confidence in this method. This paves the way for applying this procedure for more general flaw configurations.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2015

Initial Probabilistic Evaluation of Reactor Pressure Vessel Fracture with Grizzly and Raven

The Grizzly code is being developed with the goal of creating a general tool that can be applied to study a variety of degradation mechanisms in nuclear power plant components. The first application of Grizzly has been to study fracture in embrittled reactor pressure vessels (RPVs). Grizzly can be used to model the thermal/mechanical response of an RPV under transient conditions that would be observed in a pressurized thermal shock (PTS) scenario. The global response of the vessel provides boundary conditions for local models of the material in the vicinity of a flaw. Fracture domain integrals are computed to obtain stress intensity factors, which can in turn be used to assess whether a fracture would initiate at a pre-existing flaw. These capabilities have been demonstrated previously. A typical RPV is likely to contain a large population of pre-existing flaws introduced during the manufacturing process. This flaw population is characterized stastistically through probability density functions of the flaw distributions. The use of probabilistic techniques is necessary to assess the likelihood of crack initiation during a transient event. This report documents initial work to perform probabilistic analysis of RPV fracture during a PTS event using a combination of the RAVEN risk analysis code and Grizzly. This work is limited in scope, considering only a single flaw with deterministic geometry, but with uncertainty introduced in the parameters that influence fracture toughness. These results are benchmarked against equivalent models run in the FAVOR code. When fully developed, the RAVEN/Grizzly methodology for modeling probabilistic fracture in RPVs will provide a general capability that can be used to consider a wider variety of vessel and flaw conditions that are difficult to consider with current tools. In addition, this will provide access to advanced probabilistic techniques provided by RAVEN, including adaptive sampling and parallelism, which can dramatically decrease run times.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2023

HFTS-1 (Final Report)

Despite the long history of hydraulic fracturing, the optimal number of fracturing stages during multi-stage fracture stimulation in horizontal wells can be a complex subject. In addition to the increased expense of multistage fracturing in horizontal wells, increasing the number of fracturing stages does not always correlate with a commensurate increase in production. The problem is the application of a uniform fracture stimulation design to all stages with no consideration for geological variations along the wellbore. The result is an inefficient use and costly waste of energy and water. Optimization of the fracturing process requires an understanding of the cause-and-effect relationship between fracturing parameters and local geological properties at a given location along the wellbore. Realizing that the generalized rock mechanics theories and hypotheses are not truly applicable to fractured and laminated shales, quantifiable impacts of a shale’s geomechanical and depositional features are a prerequisite for design and implementation of optimized hydraulic fractures. The overarching goal of this project is to understand and define the relationships of shale geology and fracture dynamics using detailed field data that includes coring of the fracture domain. Finally, resource recovery from shale formations is estimated to be less than fifteen percent. Research proposed in this project will establish the foundations for investigating enhanced recovery techniques for increased resource recovery in existing fracture treated wells. Natural gas as an EOR fluid is in the initial stages of being used broadly in the oil and gas industry and as such, many aspects of the process need to be researched and addressed prior to widespread acceptance. In addition, re-fracturing techniques, particularly using cemented liners also need to be further investigated. This project explores several of these EOR techniques in greater detail, not just at the theoretical level, but through actual field test sites. This report delves into the research program, identified tasks, several studies, observations, results, and conclusions under the Hydraulic Fracturing Test Site (HFTS) project. This test project is subdivided into three distinct phases and each phase is discussed in detail under separate sections. The original test site for Phase 1 as well as the Phase 2 site are in Midland Basin in Texas and operated by Laredo Petroleum. In the first two phases, the program targeted the Wolfcamp formation in Midland Basin. Phase 3 of the study targeted the EagleFord play in South Texas with Devon Energy as the operator. Significant conclusions from this study include: 1) lateral and vertical fracture growth mapping as well as proppant transport behavior, 2) Use of field gas and its applicability for huff n puff EOR injection in the Midland Basin, 3) Development and validation of proppant log, both at core scale as well as in-fill well implementation by analyzing drilling mud return samples, 4) Applicability of liner refracturing including best practices for enhanced unit level productivity. This report will detail all these studies as well as technical results and conclusions from said studies. These include general subsurface characterization work, completion designs as implemented, studies looking at fracture geometry, high resolution microseismic study, advanced crosswell strain mapping studies including drainage characterization, core characterization results, proppant analysis (core and observation wells), geochemistry, etc. to name a few.

02 PETROLEUM↗
DOE OSTI2022

Reactive Transport Modeling of Mineral Precipitation and Carbon Trapping in Discrete Fracture Networks

In this study we use numerical experiments to analyze reactive flow and transport behavior in discrete intersecting fracture networks, focusing on (a) how reaction-induced changes in physical and chemical properties affect flow connectivity and (b) how fracture networks developed in the Earth's critical zone contribute to carbon sequestration via mineral weathering reactions. In the first part of the study, we used two-dimensional reactive flow and transport simulations to analyze the impacts of mixing in a natural discrete fracture network. We concluded that reaction-induced changes can substantially alter the flow connectivity, especially at fracture intersections. The second set of simulations considered the problem of natural weathering of fractured mafic and ultramafic rocks in the partially saturated Earth's critical zone as a function of infiltration rates, fracture permeability, and partially saturated flow parameters. As a model system, we considered an incongruent reaction network with dissolution of forsterite and precipitation of magnesite. The behavior is complex in terms of the rate-controlling processes because of the multicomponent nature of the system as shown by the grid Peclet number: the CO 2 behavior is gas diffusion-controlled in the partially saturated zone, while the rate of water flow via the Damkӧhler number controls Mg 2+ transport through the fracture network. The amounts of carbon that can be trapped are modest, but the naturally fractured domain considered here provides a useful “base case” against which various engineered solutions can be compared.

58 GEOSCIENCES↗
DOE OSTI2024

Fracture mechanics approach to TRISO fuel particle failure analysis

Weibull stress-based methods for failure probability assessment have been developed and analyzed to assess the integrity of tristructural isotropic (TRISO) fuel particles during fuel life cycles and accident operating conditions. While simple, these methods entail a number of drawbacks when stress concentrates near crack tips, including finite element mesh size dependency when the Weibull stress is averaged over the finite element domain. Fracture mechanics approaches involving the use of interaction integrals eliminate this lack of mesh convergence and produce consistent fracture predictions. In this work, we use an interaction integral approach to computing stress intensity factors for a crack in the inner pyrolytic carbon layer perpendicular to the silicon carbide layer, which is simplified representation of a failure mode in TRISO particles. Further, the interface between these two TRISO layers has been shown to become porous, which we simulate by considering a transition of mechanical properties over such porous length, i.e. the layers are modeled as a functionally graded material. Aspects such as porosity and thermal and irradiation eigenstrains are considered in computing the stress intensity factor from a fracture mechanics approach and compared with the known Weibull stress failure approach. The methodology introduced in this paper enables a more general fracture probability assessment in TRISO particles and eliminates the need to identify best suited parameters when using local or averaged stress-based failure criterion. Finally, the numerical sensitivity studies show how parameters such as the porous transition zone length, the material stiffness, and creep affect the probability of TRISO fuel particle failure.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2025

pH-SWITCHABLE DYNAMIC BINARY COMPLEXES AS VISCOSITY MODIFYING AGENTS

The study of supramolecular structures has been gaining a lot of momentum in recent years due to their potential application in several interesting areas such as medicine, catalysis, energy, oil and gas, etc. Hydraulic fracturing is one area where the ongoing demand for novel viscosity modifiers has prompted researchers to explore various facets of chemistry. In this study, a dynamic binary complex (DBC) solution is presented, with the potential to be useful in the hydraulic fracturing domain. It has a supramolecular structure that is formed by the self-assembly of oleic acid and diethylenetriamine into an elongated network under alkaline conditions. With less than 2 wt% of constituents in aqueous solution, a viscous gel that exhibits high viscosities even under shear was formed. It was sensitive to pH and salinity, and its zero-shear viscosity could be tuned by a factor of ~280 by changing the pH. Furthermore, its viscous properties were pronounced in the presence of salt. Sand settling tests revealed its potential to hold up sand particles for extended periods of time, longer than 24 hours. In conclusion, this DBC solution system has potential to be utilized as a smart pH-switchable hydraulic fracturing fluid that can be prepared using seawater.

02 PETROLEUM↗
DOE OSTI2021

Supramolecular dynamic binary complexes with pH and salt-responsive properties for use in unconventional reservoirs

Hydraulic fracturing of unconventional reservoirs has seen a boom in the last century, as a means to fulfill the growing energy demand in the world. The fracturing fluid used in the process plays a substantial role in determining the results. Hence, several research and development efforts have been geared towards developing more sustainable, efficient, and improved fracturing fluids. Herein, we present a dynamic binary complex (DBC) solution, with potential to be useful in the hydraulic fracturing domain. It has a supramolecular structure formed by the self-assembly of low molecular weight viscosifiers (LMWVs) oleic acid and diethylenetriamine into an elongated entangled network under alkaline conditions. With less than 2 wt% constituents dispersed in aqueous solution, a viscous gel that exhibits high viscosities even under shear was formed. Key features include responsiveness to pH and salinity, and a zero-shear viscosity that could be tuned by a factor of ~280 by changing the pH. Furthermore, its viscous properties were more pronounced in the presence of salt. Sand settling tests revealed its potential to hold up sand particles for extended periods of time. In conclusion, this DBC solution system has potential to be utilized as a smart salt-responsive, pH-switchable hydraulic fracturing fluid.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

A study of changes in deep fractured rock permeability due to coupled hydro-mechanical effects

This paper presents a numerical study of the hydro-mechanical behaviour of a fractured rock domain at 1000. m depth below the land surface as a function of different levels of fluid pore pressure. A 2D fractured rock domain is adopted based on data obtained from outcrop mapping, displaying multiple fracture sets, fracture intersections, dead-end and curved fractures. A continuum based numerical model is used to evaluate the effects of compressive boundary stresses, cracking by tension failure in the intact rock and fractures and shear displacement along fractures on its equivalent permeability. Two in situ stress boundary conditions are considered: an isotropic case SR1 with the two horizontal boundary compressive stresses having the same magnitude, and an anisotropic case SR2 with the ratio between these compressive stress components set to be 2. In the SR2 case, changes in the local stress and stress ratio distributions due to different fluid pore pressure levels are anisotropic and more significant than in the SR1 case, because of tension failures in the intact rock forming bridges between fractures. These failure regions opened new flow connections between fractures and thereby caused important anisotropic changes in the flow paths, and significant decrease in local gradients of fluid pore pressure. The equivalent permeability increases sharply when the fluid pore pressure is approximately 90% of the magnitude of the minimum stress at the boundaries of the fractured rock domain. Results show that the equivalent permeability of the fractured rock domain is most sensitive to the fractures normal stiffness, the permeability of the tension failure regions and the power-law exponent for permeability change.

58 GEOSCIENCES↗
DOE OSTI2025

Finite domain solution of a hydraulic fracture in a permeable rock

In this work, we present a domain-based algorithm to simulate the propagation of a plane-strain hydraulic fracture in a zero-toughness permeable elastic medium. The algorithm utilizes a domain-based method to solve the elasticity equation and integrates a multi-scale tip asymptote, which is particular to hydraulic fractures, into this framework. This integration is key to accurately model the energy dissipation and the fluid leak-off in the fracture tip region. The algorithm combines a 2D finite volume method (FVM) for solving the elasticity equation with a 1D FVM for solving the nonlinear lubrication equation. Incorporating the far-field asymptotics and using a moving-mesh scheme reduces the computational burden while improving the accuracy of the scheme. The paper concludes with an analysis of the numerical results. This study demonstrates the potential of this domain-based approach for modeling hydraulic fractures in poroelastic media.

Domain-based method↗
DOE OSTI2017

A numerical manifold method model for analyzing fully coupled hydro-mechanical processes in porous rock masses with discrete fractures

A numerical manifold method (NMM) model was developed for fully coupled analysis of hydro-mechanical (HM) processes in porous rock masses with discrete fractures. Using an NMM two-cover-mesh system of mathematical and physical covers, fractures are conveniently discretized by dividing the mathematical cover along fracture traces to physical cover, resulting in a discontinuous model on a non-conforming mesh. In this model, discrete fracture deformation (e.g. open and slip) and fracture fluid flow within a permeable and deformable porous rock matrix are rigorously considered. For porous rock, direct pore-volume coupling was modeled based on an energy-work scheme. For mechanical analysis of fractures, a fracture constitutive model for mechanically open states was introduced. For fluid flow in fractures, both along-fracture and normal-to-fracture fluid flow are modeled without introducing additional degrees of freedom. When the mechanical aperture of a fracture is changing, its hydraulic aperture and hydraulic conductivity is updated. At the same time, under the effect of coupled deformation and fluid flow, the contact state may dynamically change, and the corresponding contact constraint is updated each time step. Therefore, indirect coupling is realized under stringent considerations of coupled HM effects and fracture constitutive behavior transfer dynamically. To verify the new model, examples involving deformable porous media containing a single and two sets of fractures were designed, showing good accuracy. Finally, the model was applied to analyze coupled HM behavior of fractured porous rock domains with complex fracture networks under effects of loading and injection.

58 GEOSCIENCES↗
DOE OSTI2017

The effects of nearby fractures on hydraulically induced fracture propagation and permeability changes

Fracture propagation caused by hydraulic fracturing operations can be significantly influenced by adjacent fractures. In this paper, we present a detailed coupled hydro-mechanical analysis to study the effects of nearby natural fractures on hydraulically induced fracture propagation and changes in fracture permeability. Two rock domains were considered in comparison: FD1, with one fracture, and FD2, with two adjacent parallel or non-parallel fractures. It was assumed that water injection occurred in a borehole that intersected the single fracture in FD1 and one of the two fractures in FD2. Simulations were made for a time period of 3 h with an injection period of 2 h followed by 1 h of shut-in. An elastic-brittle model based on the degradation of material properties was implemented in a 2D finite-difference scheme and used for elements of the intact rock subjected to tension and shear failure. The intact rock was considered to have a low but non-negligible permeability. A verification study against analytical solutions showed that the fracture propagation and stress concentrations due to differential boundary stresses could be accurately represented by our model. Next, a base case was considered, in which the stress ratio (SR) between the magnitudes of the maximum and minimum boundary stresses, the permeability k R of the intact rock and the initial permeability k TF of the tension failure regions were fixed. In FD2, the distance d F between the two natural fractures defined by the closest distance was also fixed. The results showed that in both rock domains, the fracture started to propagate when the pore pressure was approximately 85% of the magnitude of the minimum boundary stress. The propagation of a single fracture was significantly greater than the propagation of a double fracture system because, in the latter case, the pore pressure decreased when the two fractures connected. As a result, changes in permeability in FD2 were smaller than in FD1. At the end of injection, the maximum ratios between the final and initial permeability of the natural fractures were found to be approximately 3 and 2 for rock domains FD1 and FD2, respectively. For non-parallel fractures, the controlling factor for fracture propagation was the separation between the tips of the pressurised fracture and the neighbouring non-pressurised fracture. A sensitivity study was conducted to study the influence of the key parameters d F , SR, k R and k TF on the simulation results. Lastly, fracture propagation showed more sensitivity to dF and SR than to the other parameters.

58 GEOSCIENCES↗
DOE OSTI2021

Seismic Diffusivity and the Influence of Heterogeneity on Injection‐Induced Seismicity

Abstract The spatiotemporal patterns of injection‐induced seismicity (IIS) are commonly interpreted with the concept of a triggering front, which propagates in a diffusion‐like manner with an associated diffusivity parameter. Here, we refer to this diffusivity as the “seismic diffusivity.” Several previous studies implicitly assume that seismic diffusivity is equivalent to the effective hydraulic diffusivity of the subsurface, which describes the behavior of the mean pressure field in heterogeneous porous media. Seismicity‐based approaches for hydraulic characterization or simulations of IIS using domains of homogeneous equivalent porous media are implicitly based on this assumed equivalence. However, seismicity is expected to propagate with the threshold triggering pressure, and thus not be controlled by the evolution of the mean pressure field. We present numerical simulations of fluid injection to compare the seismic and effective hydraulic diffusivities in heterogeneous formations (including fractured rock). The numerical model combines uncoupled, linear pressure diffusion with the Mohr‐Coulomb failure criterion to simulate IIS. We demonstrate that connected pathways of relatively high hydraulic diffusivity in heterogeneous media (particularly in fractured rock domains) allow the threshold triggering pressure to propagate more rapidly than predicted by the effective hydraulic diffusivity. As a result, the seismic diffusivity is greater than the effective hydraulic diffusivity in heterogeneous porous media, possibly by an order of magnitude or more. Additionally, we present a case study of IIS near Soultz‐sous‐Forêts where seismic diffusivity is found to be at least one order of magnitude larger than the effective hydraulic diffusivity.

Haagenson, Ryan↗
DOE OSTI2024

Finite domain solution of a KGD hydraulic fracture in the viscosity-dominated regime

This paper describes a numerical algorithm for solving the classic problem of a plane strain (KGD) fracture propagating in an impermeable elastic medium with zero toughness. The method, which takes advantage of the self-similar nature of the solution, combines a domain-based scheme to solve the elasticity equations and a finite volume method to solve the nonlinear lubrication equation. This work represents a first step towards developing a model able to account for pore pressure diffusion in the medium and corresponding poroelastic effects, noting that these processes are more efficiently solved using a domain-based rather than a boundary integral method. To enhance the efficiency and accuracy of the numerical scheme, the far-field crack asymptotics is embedded in the discretized elastic relationship between the fluid pressure and the crack opening, while the coupled fluid-solid tip asymptote is enforced in a weak form when solving the nonlinear lubrication equation. The proposed technique yields results that closely match the analytical solution, even with a coarse mesh. This approach offers potential for addressing more complex hydraulic fracturing problems in the future.

Domain-based method↗
NASA NTRS1991

An equivalent domain integral method for three-dimensional mixed-mode fracture problems

A general formulation of the equivalent domain integral (EDI) method for mixed mode fracture problems in cracked solids is presented. The method is discussed in the context of a 3-D finite element analysis. The J integral consists of two parts: the volume integral of the crack front potential over a torus enclosing the crack front and the crack surface integral due to the crack front potential plus the crack face loading. In mixed mode crack problems the total J integral is split into J sub I, J sub II, and J sub III representing the severity of the crack front in three modes of deformations. The direct and decomposition methods are used to separate the modes. These two methods were applied to several mixed mode fracture problems, were analyzed, and results were found to agree well with those available in the literature. The method lends itself to be used as a post-processing subroutine in a general purpose finite element program.

Shivakumar, K. N.↗