Numerical Simulation of Fracking in Shale Rocks: Current State and Future Approaches

Extracting gas from shale rocks is one of the current engineering challenges but offers the prospect of cheap gas. Part of the development of an effective engineering solution for shale gas extraction in the future will be the availability of reliable and efficient methods of modelling the development of a fracture system, and the use of these models to guide operators in locating, drilling and pressurising wells. Numerous research papers have been dedicated to this problem, but the information is still incomplete, since a number of simplifications have been adopted such as the assumption of shale as an isotropic material. Recent works on shale characterisation have proved this assumption to be wrong. The anisotropy of shale depends significantly on the scale at which the problem is tackled (nano, micro or macroscale), suggesting that a multiscale model would be appropriate. Moreover, propagation of hydraulic fractures in such a complex medium can be difficult to model with current numerical discretisation methods. The crack propagation may not be unique, and crack branching can occur during the fracture extension. A number of natural fractures could exist in a shale deposit, so we are dealing with several cracks propagating at once over a considerable range of length scales. For all these reasons, the modelling of the fracking problem deserves considerable attention. The objective of this work is to present an overview of the hydraulic fracture of shale, introducing the most recent investigations concerning the anisotropy of shale rocks, then presenting some of the possible numerical methods that could be used to model the real fracking problem.

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title: An approach for assessing engineering risk from shale gas wells in the United States
abstract: In response to a series of “energy crises” in the 1970s, the United States government began investigating the potential of unconventional, domestic sources of energy to offset imported oil. Hydraulic fracturing applied to vertical tight sand and coal bed methane wells achieved some degree of success during a period of high energy prices in the early 1980s, but shale gas remained largely untapped until the late 1990s with the application of directional drilling, a mature technology adapted from deepwater offshore platforms that allowed horizontal wells to penetrate kilometers of organic-rich shale, and staged hydraulic fracturing, which created high permeability flowpaths from the horizontal wells into a much greater volume of the target formations than previous completion methods. 
 
These new engineering techniques opened up vast unconventional natural gas and oil reserves, but also raised concerns about potential environmental impacts. These include short-term and long-term impacts to air and water quality from rig operations, potential migration of gas, fluids and chemicals through the ground, and effects on small watersheds and landscapes from roads, pads and other surface structures. 
 
Engineering risk assessment commonly uses integrated assessment models (IAMs), which define sources of risk from features, events and processes. The risk from each system element is assessed using high-fidelity models. Output from these is simplified into reduced-order models, so that a large, integrated site performance assessment can be run using the IAM. The technique has been applied to engineered systems in geologic settings for sequestering carbon dioxide, and it is also applicable to shale gas, albeit with some modifications of the various system elements. 
 
Preliminary findings indicate that shale gas well drilling and hydraulic fracturing techniques are generally safe when properly applied. Incident reports recorded by state environmental agencies suggest that human error resulting from the disregard of prescribed practices is the greatest cause of environmental incidents. This can only be addressed through education, regulations and enforcement.

title: Oil and gas wells and their integrity : implications for shale and unconventional resource exploitation
abstract: Data from around the world (Australia, Austria, Bahrain, Brazil, Canada, the Netherlands, Poland, the UK and the USA) show that more than four million onshore hydrocarbon wells have been drilled globally. Here we assess all the reliable datasets (25) on well barrier and integrity failure in the published literature and online. These datasets include production, injection, idle and abandoned wells, both onshore and offshore, exploiting both conventional and unconventional reservoirs. The datasets vary considerably in terms of the number of wells examined, their age and their designs. Therefore the percentage of wells that have had some form of well barrier or integrity failure is highly variable (1.9%–75%). Of the 8030 wells targeting the Marcellus shale inspected in Pennsylvania between 2005 and 2013, 6.3% of these have been reported to the authorities for infringements related to well barrier or integrity failure. In a separate study of 3533 Pennsylvanian wells monitored between 2008 and 2011, there were 85 examples of cement or casing failures, 4 blowouts and 2 examples of gas venting. In the UK, 2152 hydrocarbon wells were drilled onshore between 1902 and 2013 mainly targeting conventional reservoirs. UK regulations, like those of other jurisdictions, include reclamation of the well site after well abandonment. As such, there is no visible evidence of 65.2% of these well sites on the land surface today and monitoring is not carried out. The ownership of up to 53% of wells in the UK is unclear; we estimate that between 50 and 100 are orphaned. Of 143 active UK wells that were producing at the end of 2000, one has evidence of a well integrity failure.

title: Reservoir modeling of shale formations
abstract: Economic production from shale has been intimately tied to hydraulic fracturing since the first signs of success in Barnet Shale in the late 90s. The introduction of horizontal wells and multi-stage hydraulic fracturing was met by a huge move by operators toward developing shale formations that were mainly ignored in the past. Today using pad drilling, multiple horizontal wells share surface facilities and infrastructure, a development that minimizes the industry's environmental footprint. To understand production from shale reservoirs one must understand the network of natural fractures in the shale and the role of hydraulically induced fractures and their interaction. Hydraulic fracturing has been around and been studied by engineers for decades. Analytical, numerical and data-driven models have been built to explain their behavior and contribution to flow. Contribution of natural fracture networks to storage and flow in carbonate (and some sandstone) reservoirs had led to the development of techniques to study and model them. Since they are the predominant source of porosity and permeability in shale, more attention has been focused on their characteristics in the recent years. Studies of methane production from coal seams in the mid 80s provided insights on sorption as a storage mechanism and desorption and diffusion as a transport phenomenon in reservoirs that came to be known as CBM (Coalbed Methane). Today, production from shale is mainly modeled based on lessons learned in the past several decades where all the above techniques are integrated to create the modern shale reservoir models. The coupling of hydraulic fractures and natural fracture networks and their integration and interaction with the shale matrix remains the major challenge in reservoir simulation and modeling of shale formations. This article reviews the methods used by scientists and engineers in recent years to understand the complexities associated with production from shale. This will shed light on the commonly held belief amongst some of the best minds in reservoir engineering (those that have been intimately involved in modeling production from shale) that there is much to be learned about this complex resource and that our best days in understanding and modeling how oil and gas are produced from shale are still ahead of us.

title: Clay mineral diagenesis in sedimentary basins — a key to the prediction of rock properties. Examples from the North Sea Basin
abstract: Dissolution of feldspar and mica and precipitation of kaolinite require a through flow of meteoric water to remove cations such as Na+ and K+ and silica. Compaction driven pore-water flow is in most cases too slow to be significant in terms of transport of solids. The very low solubility of Al suggests that precipitation of new authigenic clay minerals requires unstable Al-bearing precursor minerals. Chlorite may form diagenetically from smectite and from kaolinite when a source of Fe and Mg is present. In the North Sea Basin, the main phase of illite precipitation reducing the quality of Jurassic reservoirs occurs at depths close to 4 km (130-140°C) but the amount of illite depends on the presence of both kaolinite and K-feldspar. Clay mineral reactions in shales and sandstones are very important factors determining mechanical and chemical compaction and are thus critical for realistic basin modelling.

title: Principal aspects of compaction and fluid flow in mudstones
abstract: Compaction and fluid flow in sedimentary basins are usually modelled as mechanical compaction assuming that the main driving force is the effective stress. However, in the deeper part of sedimentary basins (> 2–3 km, 70–100°C), compaction also involves dissolution and precipitation of minerals, and these processes are strongly controlled by temperature and to a lesser extent by variations in effective stress. The total pore-water flux is relatively independent of the permeability and pressure gradients because it is controlled by the rate of compaction. The main effect of permeability variations is the focusing of the compaction-driven flow. Fluids are also released by petroleum generation and dehydration of clay minerals; both these processes are also essentially temperature dependent. If the minimum permeability in one sedimentary layer (seal) falls below a critical value and there is little lateral drainage, the pressure will build up to fracture pressure. Pore pressures and fluid flow cannot be calculated from the matrix permeability of the rock when fracture pressures are reached, because much of the flow occurs along the generated fractures. The permeability produced by fracturing is a dynamic variable adjusting itself to the flux; it is not a rock property like the intergranular permeability. The intergranular permeability of shale samples before fracturing is also difficult to measure reliably in the laboratory as a result of unloading. This makes pressure prediction using one-dimensional pressure modelling uncertain. The prediction of permeability distributions in three dimensions is much more difficult.

title: Definition and practical application of mudstone porosity–effective stress relationships
abstract: The relationship developed in soil mechanics between void ratio and vertical effective stress is a simple but practical way of describing the one-dimensional, mechanical compaction of fine-grained, clastic mudstones. The compression coefficients (e100 and ß) that define this relationship are strongly influenced by grain size, which can be simply described by the sediment's clay content. In this paper, data both from the soil mechanics literature and from geological samples from the North Sea and Gulf of Mexico are used to construct the relationship between clay content and compression coefficients. The two datasets yield different values for the coefficients, but the authors believe that the coefficients derived from the large geological database should be used to describe geological compaction. Regression of the geological data generates the following relationships between clay content and compression coefficients: \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[e_{100}{\,}={\,}0.3024{\,}+{\,}1.6867clay{\,}+{\,}1.9505clay^{2}\] \end{document} \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[{\beta}{\,}={\,}0.0407{\,}+{\,}0.2479clay{\,}+{\,}0.3684clay^{2}\] \end{document} There is excellent agreement between true porosities and porosities calculated using the standard soil mechanics compaction equation with these coefficients. This indicates that the mechanical compaction of natural mudstones to 40 MPa can be adequately described using the soil mechanics approach. Practical application of the work is a two-stage process which first involves the evaluation of clay content and, thus, compression coefficients directly from wireline log data. These data can then be used to (a) help define effective stress–porosity inputs for basin models and (b) estimate mudstone pore pressures. Examples of pore pressure estimates are shown from west Africa, the North Sea and the Gulf of Mexico.

title: Mechanism of burial metamorphism of argillaceous sediment: 1. Mineralogical and chemical evidence
abstract: A detailed mineralogical and chemical investigation has been made of shale cuttings from a well (Case Western Reserve University Gulf Coast 6) in Oligocene-Miocene sediment of the Gulf Coast of the United States. The 10-µm fractions from the 1,250- to 5,500-m stratigraphic interval were analyzed by x-ray diffraction. Major mineralogic changes with depth take place over the interval 2,000 to 3,700 m, after which no significant changes are detectable. The most abundant mineral, illite/smectite, undergoes a conversion from less than 20 percent to about 80 percent illite layers over this interval, after which the proportion of illite layers remains constant. Over the same interval, calcite decreases from about 20 percent of the rock to almost zero, disappearing from progressively larger size fractions with increasing depth; potassium feldspar (but not albite) decreases to zero; and chlorite appears to increase in amount. Variations in the bulk chemical composition of the shale with depth show only minor changes, except for a marked decrease in CaO concomitant with the decrease in calcite. By contrast, the <0.1-µm fraction (virtually pure illite/smectite) shows a large increase in K2O and Al2O3 and a decrease in SiO2 The atomic proportions closely approximate the reaction smectite + Al+3 + K+ = illite + Si+4. The potassium and aluminum appear to be derived from the decomposition of potassium feldspar (and mica?), and the excess silicon probably forms quartz. We interpret all the major mineralogical and chemical changes as the response of the shale to burial metamorphism and conclude that the shale acted as a closed system for all components except H2O, CaO, Na2O, and CO2. Compositional changes in the shale as a function of metamorphic grade closely parallel compositional changes in shale as a function of geologic age.

title: NATURE OF MIXED-LAYER CLAYS AND MECHANISMS OF THEIR FORMATION AND ALTERATION
abstract: ▪ Abstract Mixed-layer clay minerals are intermediate products of reactions involving pure end-member clays. They come from natural environments ranging from surface to low-grade metamorphic and hydrothermal conditions. Most often mixed layering is essentially two component, but more complicated interstratifications have also been documented. Variable tendency to form regular 1:1 interstratifications has been observed and explanations of this phenomenon have been proposed. Mixed-layer clays are either di- or trioctahedral; di/trioctahedral interstratifications are rare. Most mixed-layer clays contain smectite or vermiculte as a swelling component. Exceptions are all trioctahedral: serpentine/chlorite in low-temperature environments, and mica/chlorite and talc/chlorite at high temperatures. Solid state transformation and dissolution/crystallization are the two mechanisms responsible for the formation of different mixed-layer clays. In general, the weathering reactions that produce mixed layering are revers...

title: Diagenetic reorientation of phyllosilicate minerals in Paleogene mudstones of the Podhale Basin, southern Poland
abstract: We used high-resolution X-ray texture goniometry to quantify changes in the mm-scale orientation of phyllosilicate minerals in a suite of Paleogene mudstones from the Podhale Basin in southern Poland. The sample set covers an estimated range of burial depths between 2.4 and 7.0 km, corresponding to a temperature range of 60–160°C. Although mechanical compaction has reduced porosities to ∼10% in the shallowest samples, the phyllosilicate fabric is only modestly aligned. Coarser-grained (>10 µm) detrital chlorite and mica appear to be more strongly aligned with (001) parallel to bedding, suggesting their deposition as single grains rather than as isotropic flocs or aggregates. From 2.4 to 4.6 km, R0 illite-smectite with 40–50% illite layers changes to R1 illite-smectite with 70–80%) illite layers. At the same time kaolinite is lost and diagenetic chlorite is formed. The mineralogical changes are accompanied by a strong increase in the alignment of illite-smectite, chlorite, and detrital illite, parallel to bedding and normal to the presumed principal effective stress. We propose that the development of a more aligned I-S fabric results from the dissolution of smectite and the growth of illite with (001) normal to the maximum effective stress. Water released by illitization may act as a lubricant for the rotation of all platy minerals into nanoporosity transiently formed by the illitization reaction. At greater depths and temperatures, further illitization is inhibited through the exhaustion of K-feldspar. After the cessation of illitization, a further 2.4 km of burial only results in a small increase in phyllosilicate alignment. At such small values for porosity and pore size, increasing stress does not substantially reorient phyllosilicates in the absence of mineralogical change.

title: Fabric anisotropy induced by primary depositional variations in the silt: clay ratio in two fine-grained slope fan complexes: Texas Gulf Coast and northern North Sea
abstract: article i nfo High-resolution X-ray texture goniometry (HRXTG) has been used to quantify the alignment (anisotropy) of phyllosilicate minerals in a series of diagenetically altered mudstone samples from the deep subsurface of the Texas Gulf Coast and the northern North Sea. The sampled formations have undergone a simple burial history with no overprint of tectonic deformation that may have influenced the phyllosilicate orientation. Observations have been made in two constrained temperature windows 148-150 °C (North Sea data) and 169-210 °C (Texas Gulf Coast) wherein less than 20% smectite remains in the mixed layer phase illite- smectite. Quantitative X-ray powder diffraction (XRPD) shows quartz (which is dominantly detrital) and phyllosilicate content having an inverse relationship. Quantified illite-smectite fabric intensity (phyllosi- licate alignments) ranges from low alignment values of 2.42 m.r.d. (multiples of a random distribution) to high alignment values 6.75 m.r.d. and chlorite+kaolinite ranges from 1.80 to 6.46 m.r.d. We propose that the broad range of phyllosilicate fabric intensity observed reflects the range of quartz/phyllosilicate ratios that are the imprint of depositional setting, bedform truncation, and initial grain size. Detrital quartz acts to disrupt fabric intensity by being non-platy, producing areas sheltered from effective stress ('pressure shadows'), and acting as a matrix support. There is no evidence that bioturbation is a significant control on phyllosilicate alignment in these quartz/phyllosilicate mixtures. These findings have important implications for the role of grain-scale alignment on seismic anisotropy in mudstone systems. Fabric intensity in post- diagenetic samples can be predicted from (1) quartz content and (2) total clay content where:

title: Preferred Orientation of Phyllosilicates in Gulf Coast Mudstones and Relation to the Smectite-Illite Transition
abstract: Development of preferred orientations of illite-smectite (I-S) has been studied using X-ray diffraction (XRD) texture goniometry to produce pole figures for clay minerals of a suite of 16 mudstone samples from a core from the Gulf Coast. Samples represent a compaction-loading environment in which the smectite-to-illite (S-I) transition occurs. In five shallow, pre-transition samples, there is no significant preferred orientation for smectite-rich I-S. Development of preferred orientation of I-S, although weak, was first detected at depths slightly less than that of the S-I transition. The degree of preferred orientation, which is always bedding-parallel, increases rather abruptly, but continuously, over a narrow interval corresponding to the onset of the S-I transition, then continues to strengthen only slightly with increasing depth. The degree of post-transition preferred orientation is also dependent on lithology, where the preferred orientation is less well-defined for quartz-rich samples.Previously obtained transmission electron microscope (TEM) data define textures consistent with the change in orientation over many crystallites. The smectite in pre-transition rocks consists largely of anastomosing, “wavy” layers with variable orientation and whose mean orientation is parallel to bedding, but which deviate continuously from that orientation. This results in broad, poorly defined peaks in pole figures. Post-transition illite, by contrast, consists of thin, straight packets, with most individual crystallites being parallel or nearly parallel to bedding. This results in pole figures with sharply defined maxima. By analogy with development of slaty cleavage in response to tectonic stress during metamorphism, the S-I transition is marked by dissolution of smectite and neocrystallization of illite or I-S locally within the continuous “megacrystals” of smectite. The transition is inferred to have some component of mechanical rotation of coherent illite crystals within a pliant matrix of smectite. The data suggest that change in orientation and coalescence of clay packets plays an important role in the formation of the hydraulic seal required for overpressure generation.

title: Clay mineral diagenesis and quartz cementation in mudstones: The effects of smectite to illite reaction on rock properties
abstract: The Late Cretaceous to Early Tertiary sequence of the Voring and More Basins from the Norwegian Sea has been examined with respect to mineralogy based on 319 cutting samples from five wells. A clear relationship between mineralogy and well log data is demonstrated. A significant change with respect to velocity, porosity and density occurs within the depth interval corresponding to 80–90 °C. At shallow depths/temperatures (

title: Quartz cementation in Late Cretaceous mudstones, northern North Sea: Changes in rock properties due to dissolution of smectite and precipitation of micro-quartz crystals
abstract: Late Cretaceous mudstones from two wells located in the northern North Sea and the Norwegian Sea have been examined with respect to quartz cement. Two different types of quartz cement (Type 1 and Type 2) have been identified using SEM/EDS/CL-analysis of drill-bit cuttings at depths 2370–2670 m (80–85 °C). Type 1 appears as relatively large aggregates (30–100 μm) of depth/temperature related crypto- or microcrystalline to macrocrystalline irregular quartz cement formed by local re-crystallization of biogenic silica. The CL-responses of Type 1 quartz cement give a clear indication of an authigenic origin. Type 2 quartz cement represents relatively high amounts of extremely fine-grained micro-sized (1–3 μm) crystals embedded as discrete, short chains or small clusters/nests within the illitized clay matrix. The CL-responses of micro-quartz crystals indicate an authigenic origin. The micro-quartz is most probably sourced from silica released during the smectite to illite dissolution–precipitation reaction. The petrographic evidence indicates that most of the silica released by the smectite to illite reaction has not been exported out of the mudstones. The silica released produce a subtle inter-connected micro-quartz network interlocked with aggregates of micro-quartz and authigenic clay crystals. This micro-quartz cementation process causes a significant and sharp change in the mudstone stiffness at the onset of the chemical compaction regime. This is indicated by an abrupt increase in well log velocity (Vp) and change in seismic facies close to 2500 m (80/85 °C).

title: An evaluation of temperature scales for silica diagenesis in diatomaceous sequences including a new approach based on the Miocene Monterey Formation, California
abstract: Geologic relations indicate that silica phases transformed in the Monterey Formation in two zones that persist over a narrow depth/temperature range and do not stratigraphically overlap. The wide and overlapping range of reported temperatures of these transformations is mainly a result of the many uncertainties inherent in the different methods used to estimate temperature and does not indicate that phases transform throughout these ranges. Our approach to a reliable temperature scale for silica diagenesis combines as empirical zonation of silica phases with temperature calibration from a sequence at maximum temperature and depth of burial.

title: Silica diagenesis; II, General mechanisms
abstract: Amorphous silica phases (opal-A) precipitate in nature due to the formation of dense colloids in supersaturated alkaline aqueous solutions with low relative concentrations of other ions. Opal-A dissolves and yields solutions of still relatively high silica content. In pore waters containing abundant cations, open framework polymers form which flocculate to yield opal-CT. Opal-CT becomes increasingly ordered, primarily due to preferential growth of cristobalite relative to tridymite and crystal size increase. Opal-CT dissolves to yield pore waters of low silica concentration, which allows slow growth of quartz crystals from monomeric solution. The quartz crystals then slowly increase in size and crystallinity. Carbonates appear to enhance opal-CT formation, possibly due to the activity of positively charged hydroxyl complexes. Hence, polymerization in relatively pure systems is involved in opal-A formation and polymerization in impure systems is involved in opal-CT formation, and slow growth from monomeric (low silica concentration) solutions is involved in precipitation of quartz in sedimentary realms.

title: Pressure solution during diagenesis
abstract: Pressure solution is a durable problem in geology. It has engendered continuing arguments on its mechanism, thermodynamic basis, and geo­ logical significance. Recently, it has drawn renewed attention both for its important role in the diagenesis of sedimentary rocks and for its relation to rock deformation. Pressure solution is defined as a process by which grains dissolve at intergranular or intercrystalline contacts. This process is presumably related to the higher solubility under nonhydrostatic stress at the contacts than at free grain surfaces. The process often, although not always, is accompanied by reprecipitation at adjacent free grain surfaces. Nevertheless, it is not justifiable to treat pressure solution as a recrys­ tallization process. Two common types of pressure solution are recognized: intergranular pressure solution and stylolitization. The former, occurring at individual grain contacts, was first described by Sorby (1863) based on experiments and theoretical predictions by Thompson ( l862a,b). The latter produces "a surface or contact that is marked by an irregular and interlocking penetration of the two sidcs, and is supposedly formcd diagenctically by differential vertical movement under pressure, accompanied by solution" (Bates & Jackson 1980). Here we use the term stylolite for a surface that involves an aggregate of grains (the "aggregate stylolite" of Park & Schot 1968), originating exclusively by dissolution under nonhydrostatic stress.

title: From Oil-Prone Source Rock to Gas-Producing Shale Reservoir - Geologic and Petrophysical Characterization of Unconventional Shale Gas Reservoirs
abstract: Many currently producing shale-gas reservoirs are overmature oil-prone source rocks. Through burial and heating these reservoirs evolve from organic-matter-rich mud deposited in marine, lacustrine, or swamp environments. Key characterization parameters are: total organic carbon (TOC), maturity level (vitrinite reflectance), mineralogy, thickness, and organic matter type. Hydrogento-carbon (HI) and oxygen-to-carbon (OI) ratios are used to classify organic matter that ranges from oil-prone algal and herbaceous to gas-prone woody/coaly material. Although organic-matter-rich intervals can be hundreds of meters thick, vertical variability in TOC is high ( 50% of the total porosity, and these pores may be hydrocarbon wet, at least during most of the thermal maturation process. A full understanding of the relation of porosity and gas content will result in development of optimized processes for hydrocarbon recovery in shale-gas reservoirs.

title: Unconventional shale-gas systems: The Mississippian Barnett Shale of north-central Texas as one model for thermogenic shale-gas assessment
abstract: Shale-gas resource plays can be distinguished by gas type and system characteristics. The Newark East gas field, located in the Fort Worth Basin, Texas, is defined by thermogenic gas production from low-porosity and low-permeability Barnett Shale. The Barnett Shale gas system, a self-contained source-reservoir system, has generated large amounts of gas in the key productive areas because of various characteristics and processes, including (1) excellent original organic richness and generation potential; (2) primary and secondary cracking of kerogen and retained oil, respectively; (3) retention of oil for cracking to gas by adsorption; (4) porosity resulting from organic matter decomposition; and (5) brittle mineralogical composition. The calculated total gas in place (GIP) based on estimated ultimate recovery that is based on production profiles and operator estimates is about 204 bcf/section (5.78 × 10 9 m 3 /1.73 × 10 4 m 3 ). We estimate that the Barnett Shale has a total generation potential of about 609 bbl of oil equivalent/ac-ft or the equivalent of 3657 mcf/ac-ft (84.0 m 3 /m 3 ). Assuming a thickness of 350 ft (107 m) and only sufficient hydrogen for partial cracking of retained oil to gas, a total generation potential of 820 bcf/section is estimated. Of this potential, approximately 60% was expelled, and the balance was retained for secondary cracking of oil to gas, if sufficient thermal maturity was reached. Gas storage capacity of the Barnett Shale at typical reservoir pressure, volume, and temperature conditions and 6% porosity shows a maximum storage capacity of 540 mcf/ac-ft or 159 scf/ton.

title: Some pragmatic perspectives in source rock geochemistry
abstract: Conceptual advances in petroleum source rock geochemistry made during the last decade are reviewed and some problem areas in analytical geochemistry are outlined. Areas of discussion include views of source rocks based on assessment of hydrocarbon expulsion, kinetic models of kerogen maturation/vitrinite reflectance evolution and a discussion of the data dilemma facing modern source rock geochemistry. The increased application of petroleum geochemistry to the quantitative solution of geological problems is stressed.

title: Simple kinetic models of petroleum formation. Part III : Modelling an open system
abstract: The movement of petroleum from organic-rich source beds to potential carrier beds is the net result of two processes: expulsion—the release of generated petroleum from kerogen within the source bed; and primary migration — flow of a petroleum phase through the inorganic void networks of the source bed and all other intervening fine-grained rocks. Expulsion is by far the more important process within the source bed. Orgas , a simple zero-dimensional model of the source rock as an open system, explicitly models expulsion, concurrent with oil and gas generation and oi-lgas cracking, as a storage/threshold problem. Oil and gas are retained (adsorbed/absorbed) within kerogen until their concentrations exceed the respective sorptive capacity of the residual organic carbon (0.1 and 0.02 g g CK −1 ). The initial hydrogen index (Hl 0 ) controls the important balance between potential sorbate (petroleum) and sorbant (residual carbon). Using simple geochemical measurements, the initial organic carbon is divided into initial oil, inert kerogen and oil- and gas-generative (reactive) kerogen components. reactive kerogens are assigned to one of five global kinetic organofacies with pre-determined kinetic parameter sets governing separately rates of oil and gas generation. Kinetic parameters governing the rates of oil cracking within the source rock are predicted from Hl 0 . The two most important determinants of source rock behaviour are organofacies and Hl 0 . At a reference constant heating rate of 2°C Ma −1 , the projected oil expulsion threshold for typical organofacies members increases in the order A-DE (100, 110, 120, 135°C); F typically expels no oil. An order of magnitude increase (decrease) in heating rate increases (decreases) these thresholds by ca. 15°C. Typical cumulative gas mass fractions in their ultimate expulsion products are (A–F): 0.25, 0.3, 0.25, 0.65 and 1.0 g g −1 . A value of 200 mg HC g C −1 is an approximate minimum Hl 0 prerequisite for oil expulsion, irrespective of absolute petroleum potential (P 0 ). As a family, coals vary dramatically in their expulsion behaviour as their Hl 0 ranges both above and velow this value.  for expulsive source rocks, movement within the source bed is limited only at low P 0 . A source rocks total genetic potential (P 0 , thickness) becomes important during migration outside the source bed: it potentially restricts the primary migration distance and the excess of petroleum available for secondary migration and entrapment. We evaluate primary migration approximately using a small migration loss factor (no more than about 10% saturation of porosity or 1 mg g −1 ), coupled with an understanding of fluid potential gradients which determine migration direction within the stratigraphic architecture. Ultimately, kerogen degradation and oil to gas cracking in the source rock are the rate-limiting processes in petroleum accumulation.

title: Changes in Type II Kerogen Density as a Function of Maturity: Evidence from the Kimmeridge Clay Formation
abstract: Kerogens were purified from 26 samples of the Kimmeridge Clay Formation over the full range of maturities pertinent to petroleum generation. Most samples comprised >90% amorphous organic matter (AOM). Prior to and during the early phase of petroleum generation, kerogen densities range between 1.18 and 1.25 g cm-3. During peak and late stage petroleum generation, densities increase to ∼1.35 g cm-3 as hydrogen indices decrease from ∼350 to 50 mg HC/g C. The data are qualitatively consistent with the loss of alkyl carbon from kerogen to petroleum and the increased aromatization of remaining carbon. The density increase observed for AOM contrasts with the data for vitrinite, which exhibits a decrease in density at maturity levels relevant to petroleum generation. The contrasting behavior of AOM and vitrinite is thought to reflect the differing structural composition of the two kerogen types, most obviously the greater initial aromaticity of vitrinite.

title: Mechanical properties of organic matter in shales mapped at the nanometer scale
abstract: The mechanical properties of organic matter strongly affect the way shales deform and fracture. However, the way organic matter responds to mechanical stresses is poorly understood, representing a critical obstacle to assessing oil and gas production in shale formations. Little is known about the mechanical properties of organic matter in fine grained rocks primarily because it often occupies tiny nanometer-scale voids between the mineral grains which cannot be accessed using standard mechanical testing techniques. Here, we use a new atomic force microscopy technique (PeakForce QNM™) to map the mechanical properties of organic and inorganic components at the nanometer scale. We find that the method is able to distinguish between different phases such as pyrite, quartz, clays, and organic matter. Moreover, within the organic component Young's modulus values ranged from 0 to 25 GPa; in 3 different samples – all of which come from thermally mature Type II/III source rocks in the dry gas window – a modal value of 15–16 GPa was measured, with additional peaks measured at ≤10 GPa. In addition, the maps suggest that some porous organic macerals possess a soft core surrounded by a harder outer shell 50–100 nm thick. Thus, our results demonstrate that the method represents a powerful new petrographic tool with which to characterize the mechanical properties of organic-rich sedimentary rocks.

title: Mudstone diversity: Origin and implications for source, seal, and reservoir properties in petroleum systems
abstract: Mudstone is the most abundant sedimentary rock and variously acts as sources, seals, and shale gas reservoirs in petroleum systems. Many important physicochemical properties of mudstones are strongly influenced by the mineralogy and size of deposited grains, and by diagenetic changes (precompaction and postcompaction); these are commonly predictable. The diverse composition of mudstones reflects input and hydrodynamic segregation of detrital materials to basins, primary production within basins, and diagenetic processes (both precipitation and dissolution) in the sediment. High-magnification observations both in modern and ancient sediments demonstrate that mudstones are texturally and mineralogically heterogeneous; this variability is not always readily apparent. Although some mud is indeed deposited by suspension settling out of low-energy buoyant plumes, textural analyses reveal that it is commonly dispersed by a combination of waves, gravity-driven processes, and unidirectional currents driven variously by storms and tides. Such dispersal mechanisms mean that muddy successions are typically organized into packages that can be interpreted using sequence stratigraphy. Early bioturbation homogenizes mud, whereas early chemical diagenesis can result in highly cemented zones developing, especially at stratal surfaces. The nature of deeper burial diagenesis, which involves compaction, mineral dissolution, recrystallization, mineral reorientation and lithification, and petroleum generation, is preconditioned by depositional and early diagenetic characteristics of the mud. Although the petrophysical properties of homogeneous mudstones are reasonably well known, the quantitative implications of heterogeneity for petroleum expulsion, retention, petroleum migration, seal capacity, acoustic anisotropy, and identification of shale gas reservoir sweet spots are essentially unexplored. Future work should seek to redress this position.

title: Parasequence types in shelfal mudstone strata—Quantitative observations of lithofacies and stacking patterns, and conceptual link to modern depositional regimes
abstract: Mudstone strata have a vast variety of physical, biogenic, and chemical attributes at the lamina to bed scale (approximately millimeters to decimeters thick). Our observations of more than 7 km of Paleozoic to Cenozoic mudstones revealed ordered patterns in this variety, i.e., recurrent associations of lithofacies, bedding style, sedimentary structures, and stratal architecture at bedset to parasequence scales (approximately decimeters to meters thick). We quantified characteristics of each association and their stacking patterns in vertical succession and linked them to sets of depositional processes. Most shelf mudstone strata appear to have accumulated in one of three facies association successions (FASs) that can be related to depositional regimes through characteristic modes of sediment transport and accumulation, as well as variations in benthic-energy and oxygen levels. We interpret these three FASs as records of mud accumulation on different portions of continental shelves that were dominated by storm waves, river floods, or tidal currents. Each FAS records a distinct parasequence type. This approach can help fully integrate insights from oceanographic studies into more robust interpretations of the rock record and rock-property maps.

title: Lenticular Shale Fabrics Resulting from Intermittent Erosion of Water-Rich Muds—Interpreting the Rock Record in the Light of Recent Flume Experiments
abstract: Lenticular lamination is a fabric that is known from shales of all ages, but its origin and paleoenvironmental significance is poorly understood. We have successfully reproduced this fabric in flume experiments. Beds of water-rich mud were eroded in a flume and yielded sub-millimeter to centimeter-size fragments that can be transported in bedload for distances of ten kilometers or more. Upon redeposition and compaction, these deposits have the same textural qualities as lenticular laminated shales from the rock record. Although accumulation of fecal pellets or abundant burrow tubes in a shale may produce comparable fabrics upon compaction, these can be distinguished from erosion-produced lenticular lamination via petrographic criteria. Lenticular lamination in shales that is due to deposition of water-rich mud fragments most likely records intermittent erosion and transport of surficial muds by currents.

title: Small-scale (<5.0 m) vertical heterogeneity in mudstones: implications for high-resolution stratigraphy in siliciclastic mudstone successions
abstract: 48 mudstone samples, from the Peterborough Member, Oxford Clay Formation, have been analysed to determine the small-scale vertical and lateral lithofacies variability in this succession. The samples were collected every 0.15 m from Saxon and King's Dyke brickpits (Whittlesey, Peterborough, UK) over a short vertical interval (15 m). The sampled interval spans the obductum to the grossouvrei Subzones of the Callovian (Upper Jurassic). Lithofacies present were described using combined field logging, optical and backscattered electron optical techniques. The succession predominantly comprises silt-rich mudstones, with subordinate clay-rich mudstones, shell-beds and shell-pavements. The individual units are variously composed of detrital clay, quartz, pyrite, calcite with minor apatite, kaolinite and glauconite. In some units millimetre-scale upward-fining couplets are present; elsewhere, the original bedding structures have been destroyed by bioturbation. Comparison of successive thin sections reveals that upward-coarsening is present on a metre-scale and that shell-enriched units are located close to the tops of these upward-shoaling units. Comparison of samples collected laterally on a 0.01–1.0 m scale from the same level (3.0 m below the top of Bed 14) indicates that there is lateral lithofacies variability. The small-scale (0.001 m) couplets are interpreted to be beds deposited by waning flow currents. Significant breaks between these depositional events allowed a dysaerobic infauna to colonize the sediment. The metre-scale upward-coarsening intervals which are capped by shell-enriched units are interpreted to be parasequences. The observed small-scale lateral lithofacies variability is interpreted to be a product of small, but significant vertical miscorrelation errors rather that systematic lateral facies changes. These data suggest that producing reliable, bed-scale lateral correlations over distances of more than 1 km in this, and successions like it, will be very difficult.

title: The nano-mechanical morphology of shale
abstract: Shale, the sealing formations in most hydrocarbon reservoirs, is made of highly compacted clay particles of sub-micrometer size, nanometric porosity and different mineralogy. In this paper, we propose and validate a technique to identify the nano-mechanical morphology of such a nanocomposite material. In particular, by means of a massive nanoindentation campaign at two different scales on a large range of shale materials, we show that the highly compacted plate- or sheet-like clay particles have a distinct nano-mechanical morphology with no privileged orientation of the particle-to-particle contact surface, as evidenced by a mechanical percolation threshold of η 0  ⩾ 0.5. Furthermore, the nanoindentation results provide strong evidence that the nano-mechanical elementary building block of shales is transversely isotropic in stiffness, and isotropic and frictionless in strength. These observations lead to a sphere-like mechanical morphology for visibly plate- or sheet-like clay particles. The contact forces between the sphere-like particles activate the intrinsicly anisotropic elastic properties within the clay particles and the cohesive bonds between the clay particles. The mechanical stiffness and strength properties of porous clay scale with the clay packing density toward a unique set of shale-invariant material properties. The determination of mechanical microstructure and invariant material properties are of great importance for the development of predictive microporomechanical models of the stiffness and strength properties of shale. The approach presented here also applies to other chemically and mechanically complex materials exhibiting nanogranular behavior.

title: The effect of the nanogranular nature of shale on their poroelastic behavior
abstract: Natural composite materials are highly heterogeneous porous materials, with porosities that manifest themselves at scales much below the macroscale of engineering applications. A typical example is shale, the transverse isotropic sealing formation of most hydrocarbon bearing reservoirs. By means of a closed loop approach of microporomechanics modeling, calibration and validation of elastic properties at multiple length scales of shale, we show that the nanogranular nature of this highly heterogeneous material translates into a unique poroelastic signature. The self-consistent scaling of the porous clay stiffness with the clay packing density minimizes the anisotropy of the Biot pore pressure coefficients; whereas the intrinsic anisotropy of the elementary particle translates into a pronounced anisotropy of the Skempton coefficients. This new microporoelasticity model depends only on two shale-specific material parameters which neatly summarize clay mineralogy and bulk density, and which makes the model most appealing for quantitative geomechanics, geophysics and exploitation engineering applications.

title: The nanogranular acoustic signature of shale
abstract: A multiscale, micromechanics model has been developed for the prediction of anisotropic acoustic properties of shale. The model is based on the recently identified nanogranular mechanical response of shale through indentation experiments. It recognizes the dominant role of the anisotropic elastic properties of compacted clay in the anisotropic elasticity of shale at different length scales compared to contributions of shape and orientation of particles. Following a thorough validation at multiple length scales using mineral elasticity data, nanoindentation experiment results, and ultrasonic pulse velocity tests, the model predictions compare adequately with measurements on kerogen-free and kerogen-rich shales and shaley sandstones. The acoustic signature of shale thus is found to be controlled by two volumetric parameters that synthesize the porosity and mineralogy information: the clay-packing density and the silt inclusion volume fraction. Through a series of dimensionless isoparametric plots, the micromechanics model predicts trends of increasing elastic anisotropy with increasing clay-packing density (or decreasing porosity), which correspond to the intrinsic mechanical response of unfractured shale, and quantifies the stiffness reduction induced by the presence of kerogen.

title: The effect of particle shape and grain‐scale properties of shale: A micromechanics approach
abstract: Traditional approaches for modeling the anisotropic elasticity response of the highly heterogeneous clay fabric in shale have mainly resorted to geometric factors such as definitions of particles shapes and orientations. However, predictive models based on these approaches have been mostly validated using macroscopic elasticity data. The recent implementation of instrumented indentation aimed at probing nano-scale mechanical behaviors has provided a new context for characterizing and modeling the anisotropy of the porous clay in shale. Nanoindentation experimental data revealed the significant contribution of the intrinsic anisotropy of the solid clay to the measured elastic response. In this investigation, we evaluate both the effects of geometric factors and of the intrinsic anisotropic elasticity of the solid clay phase on the observed anisotropy of shale at multiple length scales through the development of a comprehensive theoretical micromechanics approach. It was found that among various combinations of these sources of anisotropy, the elastic response of the clay fabric represented as a granular ensemble of aligned effective clay particles with spherical morphology and anisotropic elasticity compares satisfactorily to nanoindentation and ultrasonic pulse velocity measurements at nano- and macroscopic length scales, respectively. Other combinations of sources of anisotropy could yield comparable predictions, particularly at macroscopic scales, at the expense of requiring additional experimental data to characterize the morphology and orientations of particles.

title: The nanogranular nature of shale
abstract: Despite their ubiquitous presence as sealing formations in hydrocarbon bearing reservoirs affecting many fields of exploitation, the source of anisotropy of this earth material is still an enigma that has deceived many decoding attempts from experimental and theoretical sides. Sedimentary rocks, such as shales, are made of highly compacted clay particles of sub-micrometer size, nanometric porosity and different mineralogy. In this paper, we present, for the first time, results from a new experimental technique that allows one to rationally assess the elasticity content of the highly heterogeneous clay fabric of shales from nano- and microindentation. Based on the statistical analysis of massive nanoindentation tests, we find (1) that the in-situ elasticity content of the clayfabric at a scale of a few hundred to thousands nanometers is almost an order of magnitude smaller than reported clay stiffness values of clay minerals, and (2) that the elasticity and the anisotropy scale linearly with the clay packing density beyond a percolation threshold of roughly 50%. Furthermore, we show that the elasticity content sensed by nano- and microindentation tests is equal to the one that is sensed by (small strain) velocity measurements. From those observations, we conclude that shales are nanogranular composite materials, whose mechanical properties are governed by particle-to-particle contact and by characteristic packing densities, and that the much stiffer mineral properties play a secondary role.

title: Statistical indentation techniques for hydrated nanocomposites: concrete, bone, and shale
abstract: Concrete, bone and shale have one thing in common: their load-bearing mineral phase is a hydrated nanocomposite. Yet the link between material genesis, microstructure, and mechanical performance for these materials is still an enigma that has deceived many decoding attempts. In this article, we advance statistical indentation analysis techniques that make it possible to assess, in situ, the nanomechanical properties, packing density distributions, and morphology of hydrated nanocomposites. These techniques are applied to identify intrinsic and structural sources of anisotropy of hydrated nanoparticles: calcium–silicate–hydrate (C–S–H), apatite, and clay. It is shown that C–S–H and apatite, the binding phase in, respectively, cement-based materials and bone, are intrinsically isotropic; this is most probably due to a random precipitation and growth process of particles in calcium oversaturated pore solutions, which can also explain the nonnegligible internanoparticle friction. In contrast, the load-bearing clay phase in shale, the sealing formation of most hydrocarbon reservoirs, is found to be intrinsically anisotropic and frictionless. This is indicative of a ‘smooth’ deposition and compaction history, which, in contrast to mineral growth in confined spaces, minimizes nanoparticle interlocking. In all cases, the nanomechanical behavior is governed by packing density distributions of elementary particles delimitating macroscopic diversity.

title: Producing Gas from Its Source
abstract: For help in preparation of this article, thanks to Barbara Anderson, Ridgefield, Connecticut, USA; Walter Arias, Rio de Janeiro, Brazil; Keith Greaves, Salt Lake City, Utah; Valerie Jochen, College Station, Texas; Barbara Marin and Mark Puckett, Houston, Texas; Camron Miller, Oklahoma City, Oklahoma; and Jeron Williamson, Pittsburgh, Pennsylvania. AIT (Array Induction Imager Tool), ClearFRAC, ECLIPSE, ECS (Elemental Capture Spectroscopy sonde), ELANPlus, FiberFRAC, FMI (Fullbore Formation MicroImager), geoVISION, Platform Express and SpectroLith are marks of Schlumberger. Shale, the most abundant of sedimentary rocks, is finally getting its due.

title: Regulating hydraulic fracturing in shale gas plays: The case of Texas
abstract: The ability to economically produce natural gas from unconventional shale gas reservoirs has been made possible recently through the application of horizontal drilling and hydraulic fracturing. This new technique has radically changed the energy future of the United States. The U.S. has shifted from a waning producer of natural gas to a growing producer. The Energy Information Administration forecasts that by 2035 nearly half of U.S. natural gas will come from shale gas. Texas is a major player in these developments. Of the eight states and coastal areas that account for the bulk of U.S. gas, Texas has the largest proved reserves. Texas' Barnett Shale already produces six percent of the continental U.S.' gas and exploration of Texas' other shale gas regions is just beginning. Shale gas production is highly controversial, in part because of environmental concerns. Some U.S. states have put hydraulic fracturing moratoriums in place because of fear of drinking water contamination. The federal government has gotten involved and some states, like Texas, have accused it of overreaching. The contention over shale gas drilling in the U.S. may be a bellwether for other parts of the world that are now moving forward with their own shale gas production.

title: Natural fractures in some US shales and their importance for gas production
abstract: Shale gas reservoirs are commonly produced using hydraulic fracture treatments. Microseismic monitoring of hydraulically induced fracture growth shows that hydraulic fractures sometimes propagate away from the present-day maximum horizontal stress direction. One likely cause is that natural opening-mode fractures, which are present in most mudrocks, act as weak planes that reactivate during hydraulic fracturing. Knowledge of the geometry and intensity of the natural fracture system and the likelihood of reactivation is therefore necessary for effective hydraulic fracture treatment design. Changing effective stress and concomitant diagenetic evolution of the host-rock controls fracture initiation and key fracture attributes such as intensity, spatial distribution, openness and strength. Thus, a linked structural-diagenesis approach is needed to predict the fracture types likely to be present, their key attributes and an assessment of whether they will impact hydraulic fracture treatments significantly. Steep (>75°), narrow ( 100, indicating that the fractures are clustered. These fractures, especially where present in clusters, are likely to divert hydraulic fracture strands. Early, sealed, compacted fractures, fractures associated with deformation around concretions and sealed, bedding-parallel fractures also occur in many mudrocks but are unlikely to impact hydraulic fracture treatments significantly because they are not widely developed. There is no evidence of natural open microfractures in the samples studied.

title: Natural fractures in the Barnett Shale and their importance for hydraulic fracture treatments
abstract: Gas production from the Barnett Shale relies on hydraulic fracture stimulation. Natural opening-mode fractures reactivate during stimulation and enhance efficiency by widening the treatment zone. Knowledge of both the present-day maximum horizontal stress, which controls the direction of hydraulic fracture propagation, and the geometry of the natural fracture system, which we discuss here, is therefore necessary for effective hydraulic fracture treatment design. We characterized natural fractures in four Barnett Shale cores in terms of orientation, size, and sealing properties. We measured a mechanical rock property, the subcritical crack index, which governs fracture pattern development. Natural fractures are common, narrow (0.05 mm; 0.002 in.), sealed with calcite, and present in en echelon arrays. Individual fractures have high length/width aspect ratios (1000:1). They are steep (75), and the dominant trend is west-northwest. Other sets trend north-south. The narrow fractures are sealed and cannot contribute to reservoir storage or enhance permeability, but the population may follow a power-law size distribution where the largest fractures are open. The subcritical crack index for the Barnett Shale is high, indicating fracture clustering, and we suggest that large open fractures exist in clusters spaced several hundred feet apart. These fracture clusters may enhance permeability locally, but they may be problematic for hydraulic fracture treatments. The smaller sealed fractures act as planes of weakness and reactivate during hydraulic fracture treatments. Because the maximum horizontal stress trends northeast-southwest and is nearly normal to the dominant natural fractures, reactivation widens the treatment zone along multiple strands.

title: Computer simulation of hydraulic fractures
abstract: We provide a brief historical background of the development of hydraulic fracturing models for use in the petroleum and other industries. We discuss scaling laws and the propagation regimes that control the growth of hydraulic fractures from the laboratory to the field scale. We introduce the mathematical equations and boundary conditions that govern the hydraulic fracturing process, and discuss numerical implementation issues including: tracking of the fracture footprint, the control of the growth of the hydraulic fracture as a function of time, coupling of the equations, and time-stepping schemes. We demonstrate the complexity of hydraulic fracturing by means of an application example based on real data. Finally, we highlight some key areas of research that need to be addressed in order to improve current models.

title: Plane strain propagation of a hydraulic fracture in a permeable rock
abstract: This paper describes the solution of the plane strain problem of a hydraulic fracture propagating in a permeable, linear elastic medium. The fracture propagation is driven by injection of an incompressible Newtonian fluid at a constant rate. The fracture opening and the fluid pressure are related through an elastic singular integral equation, and the flow of fluid within the fracture is modeled using lubrication theory. The leak-off or infiltration of fracturing fluid into the surrounding medium is treated as a one-dimensional diffusion process. The solution of this problem is restricted to cases where the toughness of the medium and the lag between the fluid front and the fracture tip are both zero. These particular conditions are taken to correspond to limiting cases where the energy rate dissipated in fracturing the medium is negligible compared to viscous dissipation (zero toughness) and the far-field stress perpendicular to the fracture is large (zero lag). The problem is solved numerically, using an explicit time-marching algorithm. A description of the near-tip asymptotic behavior, which is of fundamental importance for the successful convergence of the algorithm, is also included. We obtain the semi-analytic asymptotic solutions corresponding to small and large time, and compare them with the numerical solution, in order to delineate the limits of the propagation regimes.

title: Propagation Regimes of Fluid-Driven Fractures in Impermeable Rocks
abstract: 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 dimen...

title: An approximate solution for toughness-dominated near-surface hydraulic fractures
abstract: Asymptotic solutions for fracture opening, volume and specific surface energy for small and large fracture radii are presented from the literature. By comparison to numerical simulation of static circular fractures subject to constant and uniform internal pressures, it is found that a good approximation for fracture opening, for intermediate fracture radii, is obtained from a power mean (with exponent 1/2) of the small and large fracture radii limiting cases. The power mean equation for fracture opening is used to derive corresponding equations for fracture volume, specific surface energy and mode I and II stress intensities. These are then combined to form an approximate solution to describe the propagation of circular toughness-dominated near-surface hydraulic fractures, suitable for small, large and intermediate fracture radii. The approximate solution is shown to closely approximate results from equivalent numerical simulation.

title: Hydraulic Fracture Propagation with 3-D Leak-off
abstract: Hydraulic fracture models typically couple a fracture elasticity model with a geological reservoir model to forecast the rate of fluid leak-off from the propagating fracture. The most commonly used leak-off model is that originally specified by Carter, which involves the assumption that the fracture is embedded within an infinite homogenous porous medium where flow only occurs perpendicular to the fracture plane. The objectives of this paper are: (1) to show that assuming one-dimensional leak-off can lead to erroneous conclusions, (2) to present a robust numerical methodology for simulating three-dimensional leak-off from propagating hydraulic fractures, and (3) to present and compare a new analytical method based on assuming three-dimensional flow of an incompressible fluid through an incompressible porous formation from a circular planar fracture. Provided the fluid and formation compressibility can be ignored within the reservoir flow model, the three-dimensional leak-off from a circular planar fracture can be written in closed-form as a function, which depends linearly on fracture pressure and radial extent. This simple expression for leak-off can be easily coupled to a range of circular fracture elasticity models. As a comparison example, the Carter model, our new function and a three-dimensional numerical model of the full problem are coupled to the PK-radial fracture model. Comparison with the numerical model shows that our new function overestimates fracture growth during intermediate times but accurately predicts both the early and late-time asymptotic behavior. In contrast, the Carter model fails to replicate both the early and late-time asymptotic behavior. Our new function additionally improves on the Carter model by not requiring the evaluation of convolution integrals and allowing easy evaluation of both the spatial leakage flux distribution across the fracture face and the three-dimensional pressure distribution within the porous formation.

title: An implicit level set method for modeling hydraulically driven fractures
abstract: We describe a novel implicit level set algorithm to locate the free boundary for a propagating hydraulic fracture. A number of characteristics of the governing equations for hydraulic fractures and their coupling present considerable challenges for numerical modeling, namely: the degenerate lubrication equation; the hypersingular elastic integral equation; the indeterminate form of the velocity of the unknown fracture front, which precludes the implementation of established front evolution strategies that require an explicit velocity field; and the computationally prohibitive cost of resolving all the length scales. An implicit algorithm is also necessary for the efficient solution of the stiff evolution equations that involve fully populated matrices associated with the coupled non-local elasticity and degenerate lubrication equations. The implicit level set algorithm that we propose exploits the local tip asymptotic behavior, applicable at the computational length scale, in order to locate the free boundary. Local inversion of this tip asymptotic relation yields the boundary values for the Eikonal equation, whose solution gives the fracture front location as well as the front velocity field. The efficacy of the algorithm is tested by comparing the level set solution to analytic solutions for hydraulic fractures propagating in a number of distinct regimes. The level set algorithm is shown to resolve the free boundary problem with first order accuracy. Further it captures the field variables, such as the fracture width, with the first order accuracy that is consistent with the piecewise constant discretization that is used.

title: Propagation of a hydraulic fracture parallel to a free surface
abstract: 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.

title: A review on phase-field models of brittle fracture and a new fast hybrid formulation
abstract: In this contribution we address the issue of efficient finite element treatment for phase-field modeling of brittle fracture. We start by providing an overview of the existing quasi-static and dynamic phase-field fracture formulations from the physics and the mechanics communities. Within the formulations stemming from Griffith’s theory, we focus on quasi-static models featuring a tension-compression split, which prevent cracking in compression and interpenetration of the crack faces upon closure, and on the staggered algorithmic implementation due to its proved robustness. In this paper, we establish an appropriate stopping criterion for the staggered scheme. Moreover, we propose and test the so-called hybrid formulation, which leads within a staggered implementation to an incrementally linear problem. This enables a significant reduction of computational cost—about one order of magnitude—with respect to the available (non-linear) models. The conceptual and structural similarities of the hybrid formulation to gradient-enhanced continuum damage mechanics are outlined as well. Several benchmark problems are solved, including one with own experimental verification.

title: Cohesive modeling of transverse cracking in laminates under in-plane loading with a single layer of elements per ply
abstract: This study aims to bridge the gap between classical understanding of transverse cracking in cross-ply laminates and recent computational methods for the modeling of progressive laminate failure. Specifically, the study investigates under what conditions a finite element model with cohesive X-FEM cracks can reproduce the in situ effect for the ply strength. It is shown that it is possible to do so with a single element across the thickness of the ply, provided that the interface stiffness is properly selected. The optimal value for this interface stiffness is derived with an analytical shear lag model. It is also shown that, when the appropriate statistical variation of properties has been applied, models with a single element through the thickness of a ply can predict the density of transverse matrix cracks.

title: Dynamic fracture propagation in hydraulic re-fracturing
abstract: Re-fracturing is important for stimulating low-permeability reservoirs, for the effective production duration after the initial fracturing is short and needs to be stimulated again. However, the fracture configuration caused by re-fracturing affects the re-stimulated production greatly. Based on fracture mechanics and hydraulic fracturing theory, a model for dynamic path of fracture propagation during re-fracturing is proposed, and computation shows that stress difference and initiation angle contribute to changes in fracture propagation path. The new fracture changes its direction gradually, until to the direction of the maximum horizontal stress. Horizontal stress difference has great obvious influence on fracture dynamic reorientation, the larger the horizontal stress difference is, the shorter the distance of reoriented fracture away from the maximum horizontal stress will be. The predicted paths are in good accordance with those from the surface micro-seismic monitoring and tri-axial large-scale experiments. The model and conclusions are significant to further understanding of re-fracturing process.

title: Dynamic Fracture Mechanics
abstract: Preface List of symbols 1. Background and overview 2. Basic elastodynamic solutions for a stationary crack 3. Further results for a stationary crack 4. Asymptotic fields near a moving crack tip 5. Energy concepts in dynamic fracture 6. Elastic crack growth at constant speed 7. Elastic crack growth at nonuniform speed 8. Plasticity and rate effects during crack growth Bibliography Index.

title: Dynamic crack tip fields and dynamic crack propagation characteristics of anisotropic material
abstract: Based on mechanics of anisotropic material, the dynamic crack propagation problem of I/II mixed mode crack in an infinite anisotropic body is investigated. Expressions of dynamic stress intensity factors for modes I and II crack are obtained. Components of dynamic stress and dynamic displacements around the crack tip are derived. The strain energy density theory is used to predict the dynamic crack extension angle. The critical strain energy density is determined by the strength parameters of anisotropic materials. The obtained dynamic crack tip fields are unified and applicable to the analysis of the crack tip fields of anisotropic material, orthotropic material and isotropic material under dynamic or static load. The obtained results show Crack propagation characteristics are represented by the mechanical properties of anisotropic material, i.e., crack propagation velocity M and fiber direction α. In particular, the fiber direction α and the crack propagation velocity M give greater influence on the variations of the stress fields and displacement fields. Fracture angle is found to depend not only on the crack propagation but also on the anisotropic character of the material.

title: Mixed mode crack propagation in homogeneous anisotropic solids
abstract: The maximum circumferential tensile stress theory, originally developed for isotropic solids, is extended to an anisotropic one. A two parameter fracture toughness characterization is assumed, and a simplifying relationship among them postulated. The criteria is reformulated in terms of the maximization of the ratio of the maximum circumferential tensile stress (a function of the two stress intensity factors K I and K II ) over the material critical tensile strength (which has a polar variation in terms of two fracture toughnesses). It is found that material anisotropy can strongly deviate the angle of crack extension from the one predicted by current isotropic models.

title: Strain Energy Release Rate for a Crack Under Combined Mode I and Mode II
abstract: The energy release rate for a crack subjected simultaneously to mode I and mode II conditions is computed. The energy was computed by path-independent integrals, using the elastic solution of a deflected crack, having a main branch and a propagation branch. The elasticity solution was obtained from the functional integral equations by the process of iterations. This process leads to a point-wise exact solution in the limit as the propagation branch goes to zero. Using the Griffith-Irwin criterion, incipient paths of propagation of such a crack were obtained from the maximum value of the energy release rate. An experiment, which gives a pure mode II condition at the tip of the crack, was devised. The results were in excellent agreement with the theory. The energy release rate, in parametric form, can be used for any crack subjected to mode I and mode II loading conditions.

title: The Partition of Unity Method
abstract: A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved. This new method can therefore be more efficient than the usual finite element methods. An additional feature of the partition-of-unity method is that finite element spaces of any desired regularity can be constructed very easily. This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers. The basic estimates for a posteriori error estimation for this new method are also proved.

title: THREE-DIMENSIONAL CRACK GROWTH SIMULATION USING BEM
abstract: In this paper an application of dual boundary element method to the analysis of three-dimensional mixed-mode crack growth is presented. The crack growth processes are simulated numerically with an incremental crack-extension analysis based on the minimum strain energy density criterion and a fatigue crack growth law. For each crack extension, the size of the increment (crack growth) and direction of crack growth is evaluated along the crack front. The dual boundary element analysis is applied to perform a single-region analysis and the stress intensity factors are evaluated using a displacement-based method. The crack extension is modelled with the introduction of new boundary elements, removing the requirement of interior remeshing, an intrinsic feature of the dual boundary element method. Results of analysis are presented for several geometries, including planar and non-planar crack growth.

title: Time-harmonic BEM for 2-D piezoelectricity applied to eigenvalue problems
abstract: We will derive the fundamental generalized displacement solution, using the Radon transform, and present the direct formulation of the time-harmonic boundary element method (BEM) for the two-dimensional general piezoelectric solids. The fundamental solution consists of the static singular and the dynamics regular parts; the former, evaluated analytically, is the fundamental solution for the static problem and the latter is given by a line integral along the unit circle. The static BEM is a component of the time-harmonic BEM, which is formulated following the physical interpretation of Somigliana’s identity in terms of the fundamental generalized line force and dislocation solutions obtained through the Stroh–Lekhnitskii (SL) formalism. The time-harmonic BEM is obtained by adding the boundary integrals for the dynamic regular part which, from the original double integral representation over the boundary element and the unit circle, are reduced to simple line integrals along the unit circle. The BEM will be applied to the determination of the eigen frequencies of piezoelectric resonators. The eigenvalue problem deals with full non-symmetric complex-valued matrices whose components depend non-linearly on the frequency. A comparative study will be made of non-linear eigenvalue solvers: QZ algorithm and the implicitly restarted Arnoldi method (IRAM). The FEM results whose accuracy is well established serve as the basis of the comparison. It is found that the IRAM is faster and has more control over the solution procedure than the QZ algorithm. The use of the time-harmonic fundamental solution provides a clean boundary only formulation of the BEM and, when applied to the eigenvalue problems with IRAM, provides eigen frequencies accurate enough to be used for industrial applications. It supersedes the dual reciprocity BEM and challenges to replace the FEM designed for the eigenvalue problems for piezoelectricity.

title: An experimental investigation into dynamic fracture: III. On steady-state crack propagation and crack branching
abstract: This is the third in a series of four papers in which problems of dynamic crack propagation are examined experimentally in large, thin sheets of Homalite-100 such that crack growth in an unbounded plate is simulated. In the first paper crack initiation resulting from stress wave loading to the crack tip as well as crack arrest were reported. It was found that for increasing rates of loading in the microsecond range the stress intensity required for initiation rises markedly. Crack arrest occurs abruptly without any deceleration phase at a stress intensity lower than that which causes initiation under quasi-static loading.In the second paper we analyze the occurrence of micro cracks at the front of the running main crack which control the rate of crack growth. The micro cracks are recorded by real time photography. By the same means it is shown that these micro cracks grow and turn away smoothly from the direction of the main crack in the process of branching.In the present paper we report results on crack propagation and branching. It is found that crack propagation occurs at a constant velocity although the stress intensity factor changes markedly. Furthermore, the velocity is determined by the stress wave induced intensity factor at initiation. The terminal velocity in Homalite-100 was found to be about half the Rayleigh wave speed (0.45 Cr). These observations are analyzed in terms of a microcrack model alluded to in the second paper of this series. A mechanism for crack branching is proposed which considers branching to be a natural evolution from a “cloud” of microcracks that accompany and lead the main crack. These results are believed to apply to quasi-brittle materials other than Homalite-100 and the reasons for this belief are discussed briefly in the first paper of this series.In the final paper of the series the effect of stress waves impinging on the tip of a rapidly moving crack is examined. Waves affect the velocity and the direction of propagation as well as the process of crack branching.

title: An experimental investigation into dynamic fracture: II. Microstructural aspects
abstract: This is the second of a series of four papers in which problems of dynamic crack propagation are examined experimentally in large, thin sheets of Homalite 100 simulating crack growth in an unbounded plate. In the first paper crack initiation resulting from stress wave loading to the crack tip and crack arrest were investigated. It was found that for increasing rates of loading in the micro second range, the stress intensity required for initiation rises markedly. Crack arrest occurs abruptly without any deceleration phase at a stress intensity below that value which causes initiation under quasi-static loading.In this paper we investigate the microscopic processes that control the process of dynamic fracture. Through post-mortem examination of the fracture surfaces it is established that microcracks are nucleated and grown ahead of a propagating crack. A measure of the size of the fracture process zone in which these microcracks are activated is obtained. Real time high speed photomicrography is used to capture the evolution of crack growth and crack branching. Under conditions of low stress intensity the crack front exhibits “thumbnail” curvature similar to the crack front profiles associated with quasi-static fracture processes. At increasing stress intensity the crack front straightens out and is identifiable as a front of multiple microfractures.The third contribution establishes the hitherto unreported occurrence that cracks can propagate rapidly with constant velocity even though the stress intensity factor varies considerably during this propagation. This velocity is determined by the initial stress wave loading on the crack tip and is changed, within limits, only by stress pulses of sufficient magnitude and brevity of rise time.The final paper in the series deals with the effect of stress waves on the behavior of running cracks, in particular with the influence of stress waves on the branching phenomenon. Also, crack curving under transient stress waves is examined. These results are believed to apply to brittle materials other than Homalite 100 and the reasons for this belief are discussed briefly in the first paper of this series.RésuméLe présent mémoire est le deuxième d'une série de quatre, consacrés à l'examen des problèmes de propagation dynamique d'une fissure par voie expérimentale dans des feuillards minces d'Homalite 100 destinés à simuler la croissance d'une fissure dans une tôle non limitée. Dans le premier mémoire, l'amorçage de la fissure qui résulte d'une sollicitation par contrainte ondulée à l'extrémité de cette fissure, ainsi que les problèmes d'arrét de fissure ont été étudiés. On a trouvé que, lorsque la vitesse de mise en charge s'accélère dans la gamme de la microseconde, l'intensité de contrainte requise pour l'amor¢age s'accroit de manière significative. L'arrêt de fissure se produit brutalement sans la moindre phase de décélération à une intensité de contrainte inférieure à la valeur qui provoque l'amor¢age sous des sollicitations quasi-statique.Dans le présent mémoire, on examine le processus microscopique qui régit la rupture dynamique. Grâce à l'examen post-mortem des surfaces de rupture, on établit que des micro-fissures sont créées et croissent en avant d'une fissure en cours de propagation. On obtient une mesure de la taille de la zone concernée par le processus de rupture dans laquelle ces micro-fissures se trouvent activées. L'utilisation est faite de la photo-micrographie à haute vitesse et à temps réel en vue de capter l'évolution de croissance de la fissure ainsi que son arborescence. Sous des conditions de faible intensité de contrainte, le front de fissuration fait état d'une courbure en ongle similaire au profil de front de fissure associé avec un processus de rupture quasi-statique. Lorsque s'accroit l'intensité de contrainte, le front de fissure tend à devenir droit et est identifiable à un front de micro-ruptures multiples.La troisième contribution de cette série rapporte le phénomène jusqu'ici non publié selon lequel des fissures peuvent se propager rapidement avec une vitesse constante même si le facteur d'intensité de contrainte évolve considérablement au course de cette propagation. Cette vitesse est déterminée par la charge ondulante initiale à l'extrémité de la fissure et ne se modifie, du moins dans certaines limites, que par des pulsations de contrainte d'une amplitude suffisante et d'une soudaineté d'apparition.Le dernier des documents de cette série est relatif à l'effet des ondes de contrainte sur le comportement de fissure en cours de propagation en particulier en tenant compte de l'influence des ondes de contraintes sur le phénomène d'arborescence. On examine également la courbure que peut prendre une fissure sous l'effet d'ondes de contrainte transitoire. On pense que ces résultats s'appliquent a des materiaux fragiles autres que l'Homalite 100 et l'on discute brièvement dans le premier document de cette série les raisons de cet avis.

title: Instability in dynamic fracture.
abstract: Cracks in brittle materials have terminal velocities far below theoretical predictions. To address this problem we have investigated the propagation of cracks in a brittle plastic (polymethylmethacrylate). Velocity measurements with resolution an order of magnitude better than past experiments reveal the existence of a critical velocity at which the velocity begins to oscillate, the mean acceleration drops sharply, and a pattern is formed on the fracture surface. Thus the dynamics of cracks may be governed by a dynamical instability.

title: Theory of dynamic crack branching in brittle materials
abstract: The problem of dynamic symmetric branching of a tensile crack propagating in a brittle material is studied within Linear Elastic Fracture Mechanics theory. The Griffith energy criterion and the principle of local symmetry provide necessary conditions for the onset of dynamic branching instability and for the subsequent paths of the branches. The theory predicts a critical velocity for branching and a well defined shape described by a branching angle and a curvature of the side branches. The model rests on a scenario of crack branching based on reasonable assumptions and on exact dynamic results for the anti-plane branching problem. Our results reproduce within a simplified 2D continuum mechanics approach the main experimental features of the branching instability of fast cracks in brittle materials.

title: THE MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS IN BRITTLE FRACTURE
abstract: Publisher Summary In recent years, the interest in the problem of brittle fracture and, in particular, in the theory of cracks has grown appreciably in connection with various technical applications. Numerous investigations have been carried out, enlarging in essential points the classical concepts of cracks and methods of analysis. The qualitative features of the problems of cracks, associated with their peculiar nonlinearity as revealed in these investigations, makes the theory of cracks stand out distinctly from the whole range of problems in terms of the theory of elasticity. The chapter presents a unified view of the way basic problems in the theory of equilibrium cracks are formulated and discusses the results obtained thereby. The object of the theory of equilibrium cracks is the study of the equilibrium of solids in the presence of cracks. However, there exists a fundamental distinction between these two problems, The form of a cavity undergoes only slight changes even under a considerable variation in the load acting on a body, while the cracks whose surface also constitutes a part of the body boundary can expand even with small increase of the load to which the body is subjected.

title: Yielding of steel sheets containing slits
abstract: Y ielding at the end of a slit in a sheet is investigated, and a relation is obtained between extent of plastic yielding and external load applied. To verify this relation, panels containing internal and edge slits were loaded in tension and lengths of plastic zones were measured.

title: The cohesive zone model: advantages, limitations and challenges
abstract: This paper reviews the cohesive process zone model, a general model which can deal with the nonlinear zone ahead of the crack tip––due to plasticity or microcracking––present  Furthermore, the cohesive zone model is able to adequately predict the behaviour of uncracked structures, including those with blunt notches, and not only the response of bodies with cracks––a usual drawback of most fracture models. The cohesive zone model, originally applied to concrete and cementitious composites, can be used with success for other materials. More powerful computer programs and better knowledge of material properties may widen its potential field of application. In this paper, the cohesive zone model is shown to provide good predictions for concrete and for different notched samples of a glassy polymer (PMMA) and some steels. The paper is structured in two main sections: First, the cohesive model is reviewed and emphasis is on determination of the softening function, an essential ingredient of the cohesive model, by inverse analysis procedures. The second section is devoted to some examples of the predictive capability of the cohesive zone model when applied to different materials; concrete, PMMA and steel.

title: Extended Finite Element Method: for Fracture Analysis of Structures
abstract: Preface . Nomenclature . Chapter 1 Introduction. 1.1 ANALYSIS OF STRUCTURES. 1.2 ANALYSIS OF DISCONTINUITIES. 1.3 FRACTURE MECHANICS.  Discrete cracked element. 1.4.5 Singular elements. 1.4.6 Enriched elements.  Elliptical hole. 2.3.4 Westergaard analysis of a sharp crack. 2.4 STRESS INTENSITY FACTOR, K . 2.4.1 Definition of the stress intensity factor. 2.4.2 Examples of stress intensity factors for LEFM. 2.4.3 Griffith theories of strength and energy. 2.4.4 Brittle material. 2.4.5 Quasi-brittle material. 2.4.6 Crack stability. 2.4.7 Fixed grip versus fixed load. 2.4.8 Mixed mode crack propagation. 2.5 SOLUTION PROCEDURES FOR K AND G . 2.5.1 Displacement extrapolation/correlation method. 2.5.2 Mode I energy release rate. 2.5.3 Mode I stiffness derivative/virtual crack model. 2.5.4 Two virtual crack extensions for mixed mode cases. 2.5.5 Single virtual crack extension based on displacement decomposition. 2.5.6 Quarter point singular elements. 2.6 ELASTOPLASTIC FRACTURE MECHANICS (EPFM). 2.6.1 Plastic zone. 2.6.2 Crack tip opening displacements (CTOD). 2.6.3 J integral. 2.6.4 Plastic crack tip fields. 2.6.5 Generalisation of J . 2.7 NUMERICAL METHODS BASED ON THE J INTEGRAL. 2.7.1 Nodal solution. 2.7.2 General finite element solution. 2.7.3 Equivalent domain integral (EDI) method. 2.7.4 Interaction integral method. Chapter 3 Extended Finite Element Method for Isotropic Problems.  A REVIEW OF XFEM DEVELOPMENT. 3.3 BASICS OF FEM. 3.3.1 Isoparametric finite elements, a short review. 3.3.2 Finite element solutions for fracture mechanics.  Partition of unity finite element method. 3.5.4 Generalised finite element method. 3.5.5 Extended finite element method. 3.5.6 Hp-clouds enrichment. 3.5.7 Generalisation of the PU enrichment. 3.5.8 Transition from standard to enriched approximation. 3.6 ISOTROPIC XFEM. 3.6.1 Basic XFEM approximation. 3.6.2 Signed distance function. 3.6.3 Modelling strong discontinuous fields. 3.6.4 Modelling weak discontinuous fields. 3.6.5 Plastic enrichment. 3.6.6 Selection of nodes for discontinuity enrichment. 3.6.7 Modelling the crack.  3.7.1 Governing equation. 3.7.2 XFEM discretization. 3.7.3 Element partitioning and numerical integration. 3.7.4 Crack intersection. 3.8 TRACKING MOVING BOUNDARIES. 3.8.1 Level set method. 3.8.2 Fast marching method. 3.8.3 Ordered upwind method.  A tensile plate with a central crack. 3.9.2 Double edge cracks. 3.9.3 Double internal collinear cracks. 3.9.4 A central crack in an infinite plate. 3.9.5 An edge crack in a finite plate. Chapter 4 XFEM for Orthotropic Problems.  Anisotropic stress functions. 4.2.3 Orthotropic mixed mode problems. 4.2.4 Energy release rate and stress intensity factor for anisotropic. materials. 4.2.5 Anisotropic singular elements. 4.3 ANALYTICAL SOLUTIONS FOR NEAR CRACK TIP. 4.3.1 Near crack tip displacement field (class I). 4.3.2 Near crack tip displacement field (class II). 4.3.3 Unified near crack tip displacement field (both classes).  Governing equation. 4.4.2 XFEM discretization. 4.4.3 SIF calculations. 4.5 NUMERICAL SIMULATIONS. 4.5.1 Plate with a crack parallel to material axis of orthotropy. 4.5.2 Edge crack with several orientations of the axes of orthotropy. 4.5.3 Single edge notched tensile specimen with crack inclination. 4.5.4 Central slanted crack. 4.5.5 An inclined centre crack in a disk subjected to point loads. 4.5.6 A crack between orthotropic and isotropic materials subjected to. tensile tractions. Chapter 5 XFEM for Cohesive Cracks.  Cohesive crack models. 5.2.2 Numerical models for cohesive cracks. 5.2.3 Crack propagation criteria. 5.2.4 Snap-back behaviour. 5.2.5 Griffith criterion for cohesive crack. 5.2.6 Cohesive crack model. 5.3 XFEM FOR COHESIVE CRACKS. 5.3.1 Enrichment functions. 5.3.2 Governing equations. 5.3.3 XFEM discretization. 5.4 NUMERICAL SIMULATIONS. 5.4.1 Mixed mode bending beam. 5.4.2 Four point bending beam. 5.4.3 Double cantilever beam. Chapter 6 New Frontiers.  Stability of interface cracks. 6.2.3 XFEM approximation for interface cracks. 6.3 CONTACT. 6.3.1 Numerical models for a contact problem. 6.3.2 XFEM modelling of a contact problem. 6.4 DYNAMIC FRACTURE. 6.4.1 Dynamic crack propagation by XFEM. 6.4.2 Dynamic LEFM. 6.4.3 Dynamic orthotropic LEFM. 6.4.4 Basic formulation of dynamic XFEM. 6.4.5 XFEM discretization. 6.4.6 Time integration. 6.4.7 Time finite element method.  Time extended finite element method. 6.5 MULTISCALE XFEM. 6.5.1 Basic formulation. 6.5.2 The zoom technique. 6.5.3 Homogenisation based techniques. 6.5.4 XFEM discretization. 6.6 MULTIPHASE XFEM. 6.6.1 Basic formulation. 6.6.2 XFEM approximation. 6.6.3 Two-phase fluid flow. 6.6.4 XFEM approximation. Chapter 7 XFEM Flow.  Defining the model. 7.3.2 Creating the finite element mesh. 7.3.3 Linear elastic analysis. 7.3.4 Large deformation. 7.3.5 Nonlinear (elastoplastic) analysis. 7.3.6 Material constitutive matrix. 7.4 XFEM. 7.4.1 Front tracking. 7.4.2 Enrichment detection. 7.4.3 Enrichment functions. 7.4.4 Ramp (transition) functions. 7.4.5 Evaluation of the B matrix. 7.5 NUMERICAL INTEGRATION. 7.5.1 Sub-quads. 7.5.2 Sub-triangles. 7.6 SOLVER. 7.6.1 XFEM degrees of freedom.  Time integration. 7.6.3 Simultaneous equations solver. 7.6.4 Crack length control. 7.7 POST-PROCESSING. 7.7.1 Stress intensity factor. 7.7.2 Crack growth. 7.7.3 Other applications.

title: Asymptotic analysis of a cohesive crack: 1. Theoretical background
abstract: A method to analyze the evolution of a cohesive crack, particularly appropriate for asymptotic analysis, is presented. Detailed descriptions of the zeroth order and first order asymptotic approaches are given and from these results the far field equivalent elastic crack theorem is derived. An analytically soluble example, the Griffith crack, and a simple model, the Dugdale model, are used to exemplify the results.

title: Effect of in-plane deformation on the cohesive failure of heterogeneous adhesives
abstract: The effect of in-plane deformations on the failure response of heterogeneous adhesives with a second phase of spherical elastic particles is investigated numerically using a 3D cohesive framework. The methodology includes a new interface-enriched generalized finite element scheme for the solution of structural problems with weak discontinuities, allowing for the efficient and accurate prediction of the stress and displacement fields in the adhesive based on finite element meshes that do not conform to the heterogeneities. A rate-dependent isotropic failure model is adopted to capture the failure in the matrix, while the stiff inclusions are assumed to be linearly elastic. Cohesive failure envelopes resulting from the micro-to-macro analysis are extracted for a wide variety of failure mode conditions. A study of 1611the impact of in-plane tensile and shear strains on the macroscopic failure response under tensile (mode I) loading is also presented.

title: Simulation of fracture in heterogeneous elastic materials with cohesive zone models
abstract: In brittle composite materials, failure mechanisms like debonding of the matrix-fiber interface or fiber breakage can result in crack deflection and hence in the improvement of the damage tolerance. More generally it is known that high values of fracture energy dissipation lead to toughening of the material. Our aim is to investigate the influence of material parameters and geometrical aspects of fibers on the fracture energy as well as the crack growth for given load scenarios. Concerning simulations of crack growth the cohesive element method in combination with the Discontinuous Galerkin method provides a framework to model the fracture considering strength, stiffness and failure energy in an integrated manner. Cohesive parameters are directly determined by DFT supercell calculations. We perform studies with prescribed crack paths as well as free crack path simulations. In both cases computational results reveal that fracture energy depends on both the material parameters but also the geometry of the fibers. In particular it is shown that the dissipated energy can be increased by appropriate choices of cohesive parameters of the interface and geometrical aspects of the fiber. In conclusion, our results can help to guide the manufacturing process of materials with a high fracture toughness.

title: A boundary cohesive grain element formulation for modelling intergranular microfracture in polycrystalline brittle materials
abstract: In this paper, a cohesive grain boundary integral formulation is proposed, for simulating intergranular microfracture evolution in polycrystalline brittle materials. Artificially generated polycrystalline microstructures are discretized using the proposed anisotropic boundary element method, considering the random location, morphology and material orientation of each grain. Each grain is assumed as a single crystal with general elastic orthotropic mechanical behaviour. Crack initiation and propagation along the grain boundaries interfaces are modelled using a linear cohesive law, considering mixed mode failure conditions. Furthermore, a non-linear frictional contact analysis is performed over cracked grain interfaces to encounter cases where crack surfaces come into contact, slide or separate. The effect of randomly located pre-existing flaws on the overall behaviour and microcracking evolution of a polycrystalline material is also investigated for different Weibull moduli. The stochastic effects of each grain morphology-orientation, internal friction and randomly distributed pre-existing flaws, under different loading conditions, are studied probabilistically by simulating various randomly generated microstructures.

title: A nonlinear cohesive model for mixed-mode delamination of composite laminates
abstract: An anisotropic nonlinear cohesive model based on continuum damage mechanics is proposed to predict the mixed-mode delamination initiation and growth of composite laminates numerically. A damaged exponential traction–displacement jump relationship is proposed for the cohesive model and an exponential damage evolution law on the evolvement of displacement jump is proposed to describe the irreversible delamination crack propagation. The implementation of cohesive model uses zero-thickness cohesive element and middle-plane isoparametric interpolation technique. The mesh size effect is solved by regulating the delamination fracture toughness by the cohesive length, and the numerical convergence problem for the snap instability is addressed by viscous regularization. Four representative delamination cases of composite laminates are used to validate the proposed cohesive model: (a) double cantilever beam (mode-I), (b) end notch flexure (mode-II), (c) mixed-mode bending fracture, and (d) fixed ratio mixed mode. The effects of the cohesive strengths and mesh sizes on the global load–displacement response for four cases are studied. In addition, the numerical results using proposed cohesive model are also compared with the results obtained by the virtual crack closure technique and other existing cohesive models. The proposed cohesive model is demonstrated to predict the mixed-mode delamination response of composite laminates accurately and effectively.

title: A Stochastic XFEM Model to Study Delamination in PPS/Glass UD Composites: Effect of Uncertain Fracture Properties
abstract: A nonlinear extended finite element (XFEM) modeling framework under a stochastic cohesive zone is presented for realistic prediction of delamination in polyphenylene sulfide (PPS)/glass composites in mode I of fracture. The cohesive zone model adopts damage evolution of the material based on a bilinear traction-separation law, the critical energy release rate and the J-integral method to formulate the delamination interface under stochastic fracture properties. To demonstrate the application of the approach, numerical predictions are compared to experimental data using Double Cantilever Beam (DCB) tests. In particular, it is shown how the XFEM model can be used to capture test non-repeatability due to uncertain fracture properties, which is often the case during the characterization of composites using standard fracture tests.

title: Cohesive-zone-model formulation and implementation using the symmetric Galerkin boundary element method for homogeneous solids
abstract: A new symmetric boundary integral formulation for cohesive cracks growing in the interior of homogeneous linear elastic isotropic media with a known crack path is developed and implemented in a numerical code. A crack path can be known due to some symmetry implications or the presence of a weak or bonded surface between two solids. The use of a two-dimensional exponential cohesive law and of a special technique for its inclusion in the symmetric Galerkin boundary element method allows us to develop a simple and efficient formulation and implementation of a cohesive zone model. This formulation is dependent on only one variable in the cohesive zone (relative displacement). The corresponding constitutive cohesive equations present a softening branch which induces to the problem a potential instability. The development and implementation of a suitable solution algorithm capable of following the growth of the cohesive zone and subsequent crack growth becomes an important issue. An arc-length control combined with a Newton–Raphson algorithm for iterative solution of nonlinear equations is developed. The boundary element method is very attractive for modeling cohesive crack problems as all nonlinearities are located along the boundaries (including the crack boundaries) of linear elastic domains. A Galerkin approximation scheme, applied to a suitable symmetric integral formulation, ensures an easy treatment of cracks in homogeneous media and excellent convergence behavior of the numerical solution. Numerical results for the wedge split and mixed-mode flexure tests are presented.

title: Instability during cohesive zone growth
abstract: Tensile microcracking of quasi-brittle materials is studied by means of micromechanics, based on (i) an elasto-damaging cohesive zone model accounting for cohesive softening and (ii) a dilute distribution of non-interacting microcracks of uniform orientation and size. Considering virgin microcracks (initially without cohesive zones), macroscopic tensile load increase results in growth of cohesive zones ahead of stationary (non-propagating) cracks and, subsequently, in crack propagation which, notably, will be encountered before the cohesive zones are fully developed i.e. onset of instable cohesive zone growth will be encountered at a load level (i) at which tractions are still transmitted across the inner edges of the cohesive zones and (ii) at which the separation at the inner edges of the cohesive zones is smaller than its critical value. Focusing on onset of instable cohesive zone growth, the chosen approach allows for accessing quantities characterizing the stability limit (e.g., the intensity of the macroscopic loading and the opening at the inner edges of the cohesive zones), without raising the need for non-linear Finite Element analyses. It is shown that the tensile macrostrength of materials containing virgin microcracks is larger than the one related to cracks with already initially fully developed cohesive zones, and related strength differences are quantified for a wide class of cohesive softening behavior. The proposed model is validated by comparing model predictions with an exact solution (available for the special case of constant cohesive tractions) and with results from reliable Finite Element analyses. The paper will be of interest for engineers involved in testing and/or in modeling of quasi-brittle media including cementitious materials and rock.

title: Simulation of interlaminar damage in mixed-mode bending tests using large deformation self-adaptive cohesive zones
abstract: Cohesive zone models are common tools for the simulation of delamination since two decades. However, application of these models in a quasi-static finite element framework requires a sufficiently fine discretization to resolve a quasi-brittle process zone. As an alternative to mesh refinement, an adaptive approximation of the growing delamination was obtained in earlier work by enriching the elements in the fracture process zone. Discretization-induced numerical instabilities are thereby avoided and a standard Newton–Raphson iterative scheme remains applicable. This enriched 2D cohesive zone model is here extended to a large deformation self-adaptive finite element framework, which makes it suitable for general engineering applications where geometrical and material non-linearities are expected. Based on recent experimental results from miniaturized mixed-mode bending tests, an irreversible mixed-mode traction–separation law is considered for the simulation of delamination over a wide range of mode mixities. The developed model is used for the simulation of mixed-mode bending tests on bi-material interfaces and the response is compared to the experimental results. Numerical simulations using coarse discretizations in a quasi-static finite element framework show the effectiveness of the self-adaptive cohesive zone model.

title: Fracture in quasi-brittle materials: a review of continuum damage-based approaches
abstract: Starting from the early ideas on smeared-crack modelling of quasi-brittle materials (concrete, rock) as pioneered in the late 1960s various refinements are discussed such as the shear retention factor, the tension-stiffening concept and the tension-softening concept. Next, smeared-crack models are put in the unified context of damage mechanics together with the multiple-smeared crack approach and the (closely) related microplane damage models. The concept of fracture energy is introduced for tensile and compressive loadings and is also elaborated for reinforced concrete. A concise summary is given of recent finite element concepts for cohesive-zone models (fracture energy models).

title: Cohesive models for quasistatic cracking in inelastic solids
abstract: In has been shown that slow stable cracking process can be sustained by strain-softening materials such as cementitious composites, e.g., rock, concrete, mortars, ceramics and others. A mathematical model is proposed based on the theory of quasi-static crack extension governed by Wnuk's criterion of final stretch (equivalent to the CTOA condition). Requirement of self-similarity of the crack opening profile, maintained during the quasistatic growth phase, leads to a differential equation defining the material resistance to sustained crack extension, as a function of crack growth increment, material properties such as tearing modulus, strain-softening parameter, and the size of the process zone.The equations derived on the basis of the step-like distribution of the restraining stress which operates within a structured end zone associated with a moving crack, are compared against the Wnuk-Rice-Sorensen equation to ran R-curve in an ideal elasto-plastic material, and with the more recent results of Wnuk and Hunsacharoonroj  pertaining to strain-hardening materials which obey the Ramberg-Osgood power law.Present results suggest that the nature of constitutive equations substantially influence composite fracture resistance as measured by the R-curve. Ability to absorb energy and the ensuing resistance to crack growth can be significantly enhanced when the mechanisms of energy dissipation within the matrix are properly understood. This work establishes a “bridge” between the continuum mechanics and micromechanics of deformation and meture processes by providing a relation between material fracture toughness and its constitutive equations supplemented by certain microstructural characteristics.

title: Determination of quasibrittle fracture law for cohesive crack models
abstract: A basic and necessary ingredient of cohesive crack models, so widely used for numerical quasibrittle fracture simulations, is the softening function which characterizes the property of concrete-like materials. In this paper, a simple procedure to determine the softening parameters for such functions is proposed. In addition to knowledge of the tensile strength—typically obtainable within about 20% from a cylinder splitting strength (Brazilian) test—the only other requirement is knowledge of the load-displacement curve, as would be obtained from a standard three-point bending test. The method is particularly suitable for materials, such as fiber reinforced concrete, which exhibit a long tail in their load-displacement response. Although suitable for identifying both nonlinear and piece-wise linear relations, we illustrate applications of the scheme using a bilinear law.

title: Modelling large crack propagation: from gradient damage to cohesive zone models
abstract: Continuum Damage Mechanics and cohesive zone models are both prone to model large crack propagation inside quasi-brittle materials. Comparing the advantages of the formulations, the latter could be advantageously applied in cases where the crack path is known a priori while the former implicitly encompasses crack path prediction but requires more complex computations involving nonlocal interactions. In order to assess the acceptability of such a hierarchy for industrial studies, it is necessary that the predictions coincide quantitatively. Starting from a gradient damage model in a one-dimensional context, a cohesive law is derived as the asymptotic response of the damage model for vanishing nonlocal length scale. The cohesive law is hence independent of the nonlocal length scale, which is consistent with the fact that it ignores details characteristic of the “best-estimate” damage approach. Besides, the existence of such a limit ensures that the damage model is not much sensitive to a small nonlocal length scale, which then appears rather as a numerical regularisation parameter. A numerical comparison between the damage model and its asymptotic cohesive law is then carried out for large bi-dimensional crack propagation. The computed responses remain close to each other although some small discrepancies arise probably related to the damage spread resulting from the stress distribution in the vicinity of the crack tip: the hierarchy strategy is thus validated.

title: Cohesive zone size of microcracks in brittle materials
abstract: Consideration of cohesive microcracks in continuum micromechanics is a challenging task since a lot of applications (such as, e.g., estimation of the stiffness of a microcracked solid) require a priori knowledge of the size of the cohesive zone. The latter, however, can be determined analytically only for the special case of Barenblatt–Dugdale cracks, i.e. for cracks with spatially constant cohesive tractions. Herein, we deal with the general case of spatially non-constant cohesive tractions: Generalizing the Barenblatt–Dugdale approach, we consider that each crack is surrounded by a plane annular cohesive zone characterized by a constitutive softening law (introduced as a power law) relating the vector of cohesive tractions to the displacement discontinuity. The size of this cohesive zone is then estimated using the theorem of minimum potential energy, based on a class of kinematically admissible displacement fields.

title: A simple J-integral governed bilinear constitutive relation for simulating fracture propagation in quasi-brittle material
abstract: The present paper develops a bilinear constitutive relation for material at fracture tip based on the cohesive zone model and the J-integral theory. In this bilinear constitutive relation, the stress linearly decreases rather than suddenly drops down to zero at the post-peak stage. The slope of stress degradation is related to the critical J-integral of material and the element size. By this bilinear constitutive relation, the tensile and mixed fracture propagation can be well simulated for the strain energy release rate can be preserved with element size decreasing. The simulation results are almost free of the element size sensitivity. The present model and its implementation are very simple. For the elements at fracture tip, the derived bilinear constitutive relation is adopted while for the rest elements, the ideal elastic-brittle model is adopted. For the fracture propagation is represented by the deterioration of stiffness through constitutive relation, no separate fracture criterion is needed, which makes the evaluation of the fracture propagation by separate fracture criterion and the mesh modification after each load step unnecessary. The simulation examples of tensile and mixed fracture suggest that the present method is valid and highly efficient.

title: Experimental and numerical study of the dependency of interface fracture in concrete–rock specimens on mode mixity
abstract: The interface between the concrete and the rock is usually considered the weakest link in concrete structures built on rock foundations. The fracture behaviour at the concrete–rock interface is influenced by many factors e.g. the material properties of the individual constituents, the fracture process zone at the interface and the mode mixity ratio. This paper investigates the dependency of the fracture behaviour of concrete–rock interfaces on the mode mixity ratio using experimental and numerical methods. The experimental program involves four-point-shearing of concrete–rock composite beams. It is designed to test a wide range of mode mixity ratio. Using linear elastic fracture mechanics theory, the fracture toughness and the fracture energy are first quantified in terms of the mode mixity ratio. The scaled boundary finite element method, which is known for its accuracy in modelling fracture, is used to compute the fracture toughness and fracture energy. Next, the crack propagation process of the concrete–rock composite beam is modelled using nonlinear fracture mechanics theory. The scaled boundary finite element method is coupled with interface elements to model the fracture process zone, which is a characteristic of fracture in quasi-brittle materials such as concrete and rock. A revised scaled boundary finite element method formulation using generalized coordinates is used to model the cohesive tractions. The cohesive crack at the interface is assumed to propagate when either the Mode 1 or the Mode 2 stress intensity factors change sign. A simple remeshing algorithm is used to propagate the crack at the interface. The numerical simulations are validated by the experimental measurements. The simulated crack propagation processes are described in terms of the mode mixity ratio.

title: Linear Elastic and Cohesive Fracture Analysis to Model Hydraulic Fracture in Brittle and Ductile Rocks
abstract: Hydraulic fracturing technology is being widely used within the oil and gas industry for both waste injection and unconventional gas production wells. It is essential to predict the behavior of hydraulic fractures accurately based on understanding the fundamental mechanism(s). The prevailing approach for hydraulic fracture modeling continues to rely on computational methods based on Linear Elastic Fracture Mechanics (LEFM). Generally, these methods give reasonable predictions for hard rock hydraulic fracture processes, but still have inherent limitations, especially when fluid injection is performed in soft rock/sand or other non-conventional formations. These methods typically give very conservative predictions on fracture geometry and inaccurate estimation of required fracture pressure. One of the reasons the LEFM-based methods fail to give accurate predictions for these materials is that the fracture process zone ahead of the crack tip and softening effect should not be neglected in ductile rock fracture analysis. A 3D pore pressure cohesive zone model has been developed and applied to predict hydraulic fracturing under fluid injection. The cohesive zone method is a numerical tool developed to model crack initiation and growth in quasi-brittle materials considering the material softening effect. The pore pressure cohesive zone model has been applied to investigate the hydraulic fracture with different rock properties. The hydraulic fracture predictions of a three-layer water injection case have been compared using the pore pressure cohesive zone model with revised parameters, LEFM-based pseudo 3D model, a Perkins-Kern–Nordgren (PKN) model, and an analytical solution. Based on the size of the fracture process zone and its effect on crack extension in ductile rock, the fundamental mechanical difference of LEFM and cohesive fracture mechanics-based methods is discussed. An effective fracture toughness method has been proposed to consider the fracture process zone effect on the ductile rock fracture.

title: The Effective Fracture Toughness in Hydraulic Fracturing
abstract: This paper examines the effective fracture toughness approach which is used in hydraulic fracturing in order to explain the high net-pressures that are often observed in field operations. The effective fracture toughness is calculated using a fully deterministic elasto-plastic hydraulic fracturing model. Rock is modelled by Mohr–Coulomb flow theory of plasticity for cohesive-frictional dilatant material. Fluid flow is modelled by lubrication theory. A cohesive crack model which takes into account the softening behaviour of rocks is employed as the propagation criterion. The fully coupled model is solved numerically by the finite element method and the effective fracture toughness is calculated using the path independent J-integral. The results show that plastic yielding near the tip of a propagating fracture provides an effective shielding, resulting in an increase in the rock effective fracture toughness by more than an order of magnitude. It is demonstrated that an elastic model based on the concept of effective fracture toughness matches the results of plasticity quite well. The effective fracture toughness increases with formation yielding, which is influenced by the deviator of the in-situ stresses, the rock strength, the elastic modulus and the pumping parameters. Tables of effective fracture toughness for a representative set of physical parameters are presented.

title: Multiple scale analysis of heterogeneous elastic structures using homogenization theory and voronoi cell finite element method
abstract: This paper deals with the development of a multiple scale finite element method by combining the asymptotic homogenization theory with Voronoi cell (VCFEM) for microstructural modeling. The Voronoi cell finite element model originates from Dirichlet tessellation of a representative material element or a base cell in the microstructure. Homogenized material coefficients for a global displacement finite element model are generated by VCFEM analysis using periodic boundary conditions on the base cell. Following the macroscopic analysis, the local VCFEM analysis is implemented to depict the true evolution of microstructural stresses and strains. Various numerical examples are executed for validating the effectiveness of VCFEM macro-micro modeling of elastic materials. The effect of size, shape, orientation and distribution of heterogeneities on the local and global response are examined.

title: Multiscale modeling of cohesive geomaterials with a polycrystalline approach
abstract: The main objective of this paper is to investigate the macroscopic elastic-plastic behaviors of a class of cohesive geomaterials with the aid of classical polycrystalline schemes. Specific local constitutive equations are proposed to describe the typical features of geomaterials. The local yield criterion for crystallographic sliding systems takes into account the pressure sensitivity and a non-associated plastic potential is introduced to properly describe the plastic dilatancy. Consequently, the concentration law is also modified in order to establish the relationship between the macroscopic stress tensor and the local stress tensor in each mineral grain. Computational aspects associated with the numerical implementation of polycrystalline model are revisited and discussed. The proposed model is applied to a typical polycrystalline rock, granite. After the identification of material parameters, its validity is verified through comparisons between model's predictions and experimental data on both conventional and true triaxial compression tests.

title: The determination of the elastic field of an ellipsoidal inclusion, and related problems
abstract: It is supposed that a region within an isotropic elastic solid undergoes a spontaneous change of form which, if the surrounding material were absent, would be some prescribed homogeneous deformation. Because of the presence of the surrounding material stresses will be present both inside and outside the region. The resulting elastic field may be found very simply with the help of a sequence of imaginary cutting, straining and welding operations. In particular, if the region is an ellipsoid the strain inside it is uniform and may be expressed in terms of tabulated elliptic integrals. In this case a further problem may be solved. An ellipsoidal region in an infinite medium has elastic constants different from those of the rest of the material; how does the presence of this inhomogeneity disturb an applied stress-field uniform at large distances? It is shown that to answer several questions of physical or engineering interest it is necessary to know only the relatively simple elastic field inside the ellipsoid.

title: The elastic field outside an ellipsoidal inclusion
abstract: The results of an earlier paper are extended. The elastic field outside an inclusion or inhomogeneity is treated in greater detail. For a general inclusion the harmonic potential of a certain surface distribution may be used in place of the biharmonic potential used previously. The elastic field outside an ellipsoidal inclusion or inhomogeneity may be expressed entirely in terms of the harmonic potential of a solid ellipsoid. The solution gives incidentally the velocity field about an ellipsoid which is deforming homogeneously in a viscous fluid. An expression given previously for the strain energy of an ellipsoidal region which has undergone a shear transformation is generalized to the case where the region has elastic constants different from those of its surroundings. The Appendix outlines a general method of calculating biharmonic potentials.

title: THEORETICAL PREDICTION OF PLASTIC STRAINS OF POLYCRYSTALS
abstract: Plastic strain is considered a consequence of slipping on some of the 12 slip systems of facecentered-cubic single crystals. The polycrystal is assumed to be macroscopically homogeneous and initially isotropic -- that is, the crystal orientations, sizes, shapes and locations are random and uncorrelated. The rotations of the grains during deformation are neglected, and the average plastic strain in grains of a given orientation is calculated on the basis of the assumption that the grains under consideration are spherical and elastically isotropic.

title: A multiscale cohesive zone model and simulations of fractures
abstract: In this work, a novel multiscale cohesive zone model is proposed, in which the bulk material is modeled as a local quasi-continuum medium that obeys the Cauchy–Born rule while the cohesive force and displacement relations inside the cohesive zone are governed by a coarse grained depletion potential. The interface depletion potential is constructed based on the Derjaguin approximation of nonlocal colloidal interactions. By doing so, the interface constitutive descriptions are made genetically consistent with the bulk constitutive relations that are enriched from underneath atomistic structure. The method provides an effective means to describe properties of material inhomogeneities such as grain boundaries, bi-material interfaces, slip lines, and inclusions, etc. We have developed and implemented the proposed multiscale cohesive zone model in a cohesive finite element weak Galerkin formulation, and we have applied it to simulate dynamic fracture problems in solids. The numerical simulation results have shown that the method has successfully captured the phenomenon of spall fracture during simulations of impacts and penetrations.

title: The Quasicontinuum Method: Overview, applications and current directions
abstract: The Quasicontinuum (QC) Method, originally conceived and developed by Tadmor, Ortiz and Phillips  in 1996, has since seen a great deal of development and application by a number of researchers. The idea of the method is a relatively simple one. With the goal of modeling an atomistic system without explicitly treating every atom in the problem, the QC provides a framework whereby degrees of freedom are judiciously eliminated and force/energy calculations are expedited. This is combined with adaptive model refinement to ensure that full atomistic detail is retained in regions of the problem where it is required while continuum assumptions reduce the computational demand elsewhere. This article provides a review of the method, from its original motivations and formulation to recent improvements and developments. A summary of the important mechanics of materials results that have been obtained using the QC approach is presented. Finally, several related modeling techniques from the literature are briefly discussed. As an accompaniment to this paper, a website designed to serve as a clearinghouse for information on the QC method has been established at www.qcmethod.com. The site includes information on QC research, links to researchers, downloadable QC code and documentation.

title: Flow and transport in unsaturated fractured rock: effects of multiscale heterogeneity of hydrogeologic properties.
abstract: The heterogeneity of hydrogeologic properties at different scales may have different effects on flow and transport processes in a subsurface system. A model for the unsaturated zone of Yucca Mountain, Nevada, is developed to represent complex heterogeneity at two different scales: (1) layer scale corresponding to geologic layering and (2) local scale. The layer-scale hydrogeologic properties are obtained using inverse modeling, based on the available measurements collected from the Yucca Mountain site. Calibration results show a significant lateral and vertical variability in matrix and fracture properties. Hydrogeologic property distributions in a two-dimensional, vertical cross-section of the site are generated by combining the average layer-scale matrix and fracture properties with local-scale perturbations generated using a stochastic simulation method. The unsaturated water flow and conservative (nonsorbing) tracer transport through the cross-section are simulated for different sets of matrix and fracture property fields. Comparison of simulation results indicates that the local-scale heterogeneity of matrix and fracture properties has a considerable effect on unsaturated flow processes, leading to fast flow paths in fractures and the matrix. These paths shorten the travel time of a conservative tracer from the source (repository) horizon in the unsaturated zone to the water table for small fractions of total released tracer mass. As a result, the local-scale heterogeneity also has a noticeable effect on global tracer transport processes, characterized by an average breakthrough curve at the water table, especially at the early arrival time of tracer mass. However, the effect is not significant at the later time after 20% tracer mass reaches the water table. The simulation results also verify that matrix diffusion plays an important role in overall solute transport processes in the unsaturated zone at Yucca Mountain.

title: Multiscale, Multiphysics Network Modeling of Shale Matrix Gas Flows
abstract: We present a pore network model to determine the permeability of shale gas matrix. Contrary to the conventional reservoirs, where permeability is only a function of topology and morphology of the pores, the permeability in shale depends on pressure as well. In addition to traditional viscous flow of Hagen–Poiseuille or Darcy type, we included slip flow and Knudsen diffusion in our network model to simulate gas flow in shale systems that contain pores on both micrometer and nanometer scales. This is the first network model in 3D that combines pores with nanometer and micrometer sizes with different flow physics mechanisms on both scales. Our results showed that estimated apparent permeability is significantly higher when the additional physical phenomena are considered, especially at lower pressures and in networks where nanopores dominate. We performed sensitivity analyses on three different network models with equal porosity; constant cross-section model (CCM), enlarged cross-section model (ECM) and shrunk length model (SLM). For the porous systems with variable pore sizes, the apparent permeability is highly dependent on the fraction of nanopores and the pores’ connectivity. The overall permeability in each model decreased as the fraction of nanopores increased.

title: A multiscale modeling of damage and time-dependent behavior of cohesive rocks
abstract: The present paper deals with a micromechanical approach to modeling the time-dependent mechanical behavior of a class of cohesive geomaterials. The considered material is Callovo-Oxfordian argillite, which is mainly composed of three constituents: an elastoviscoplastic clay matrix, elastic quartz minerals, and elastic damaged calcite grains. The macroscopic constitutive law is obtained by adapting the incremental method proposed by Hill (; 13:89-101). Its unified formulation allows a description of not only the time-dependent behavior of the argillite but also its elastoplastic damage response. The developed model is first validated by comparison with finite element solutions and then it is applied to the prediction of argillites' macroscopic responses in connection with their mineralogical compositions. The validity of the model is checked through comparisons between the model's predictions and experimental data.

title: A two-scale poromechanical model for cohesive rocks
abstract: This paper presents a computational homogenization approach in the framework of poromechanics. A fully coupled hydromechanics problem is formulated at the macroscopic scale. The constitutive equations are replaced by results of numerical computations on a Representative Elementary Volume in order to obtain the overall stress of the mixture as well as its transmissivity properties. At the microscale, the material is assumed to be composed of an assembly of hyperelastic grains connected by cohesive interfaces. These interfaces are also channels of a network where the fluid can percolate. The fluid acts on the boundaries of the grains and influences the behavior of the cohesive interfaces. Conversely, the opening of the interfaces induces changes in the transmissivity properties of the corresponding channels. This yields a fully coupled hydromechanical problem at the microscopic scale. The finite element method is considered for the numerical solutions at both scales, the present approach extending the purely mechanical FE2 scheme to the coupled hydromechanical framework. The local macroscopic behavior resulting from the homogenization scheme is illustrated on different numerical tests. The results clearly show the coupling between damage and fluid permeability in the overall response, as a consequence of the small-scale interaction between the action of the percolating fluid, the deformation of the solid skeleton, and the failure of the cohesive interfaces.

title: A multi-scale rate dependent crack model for quasi-brittle heterogeneous materials
abstract: A multi-scale numerical approach for modeling cracking in heterogeneous quasi-brittle materials under dynamic loading is presented. In the model, a discontinuous crack model is used at macro-scale to simulate fracture and a gradient-enhanced damage model has been used at meso-scale to simulate diffuse damage. The traction-separation law for the cohesive zone model at macro-scale is obtained from the meso-scale through the discontinuous computational homogenization method. An implicit time integration is used to solve the dynamic problem at the macro-scale while the meso-scale model is solved as a quasi-static problem. The effect of crack opening rate on the macro cohesive law is taken into account by relating the material properties of the meso-scale model to the macro crack opening rate. The objectivity of the model response with respect to the representative volume element size is demonstrated for wave propagation problems. The model is verified by comparison with a direct numerical simulation.

title: Boundary element methods for potential problems
abstract: The boundary element formulation of potential problems is presented using weighted residual techniques. The paper emphasizes the simplicity of the boundary methods and the way in which they can be applied in engineering. The advantage of using this method in preference to finite elements is discussed in the applications.

title: Rapid solution of integral equations of classical potential theory
abstract: An algorithm is described for rapid solution of classical boundary value problems (Dirichlet an Neumann) for the Laplace equation based on iteratively solving integral equations of potential theory. CPU time requirements for previously published algorithms of this type are proportional to n2, where n is the number of nodes in the discretization of the boundary of the region. The CPU time requirements for the algorithm of the present paper are proportional to n, making it considerably more practical for large scale problems.

title: Traction boundary elements for cracks in anisotropic solids
abstract: A general mixed boundary element approach based on displacement and traction integral equations for anisotropic media is presented. Integration of the singular and hypersingular kernels along general quadratic line elements is carried out by analytical transformation of the integrals into regular ones, which are numerically evaluated, plus simple singular integrals with known analytical solution. This is achieved by the simple election of an integration variable, which is consistent with that of the anisotropic fundamental solution. The generality of the method allows for the use of curved elements and discontinuous quarter-point elements to represent Fracture Mechanics problems. Stress Intensity Factors are accurately computed from the crack opening displacement at the nodes of the quarter-point element. Several examples, including curved crack geometries and different material properties are presented.

title: Crack propagation by element-free Galerkin methods
abstract: Element free Galerkin methods (EFG) are gridless methods for solving partial differential equations which employ moving least square interpolants for the trial and test functions. EFG methods require only nodes and a description of the external and internal boundaries and interfaces of the model; no element connectivity is needed. The implementation of EFG to arbitrary crack growth in static problems such as fatigue is described. Numerical examples show that accurate stress intensity factors can be obtained by this method without any enrichment of the displacement field by a near crack tip singularity and that crack growth can be easily modeled since it requires almost no remeshing.

title: Fracture of magnetoelectroelastic composite materials using boundary element method (BEM)
abstract: The behavior of cracked linear magnetoelectroelastic solids is analysed by means of the dual Boundary Element Method (BEM) approach. Media possessing fully coupled piezoelectric, piezomagnetic and magnetoelectric effects are considered. An explicit 2-D Green’s function in terms of the extended Stroh formalism for magnetoelectroelastic full-plane under static loading is implemented. Hypersingular integrals arising in the traction boundary integral equations are computed through a regularization technique. Evaluation of fracture parameters directly from computed nodal values is discussed. The stress intensity factors (SIF), the electric displacement intensity factor (EDIF), the magnetic induction intensity factor (MIIF) as well as the mechanical strain energy release rate (MSERR) are evaluated for different crack configurations in both finite and infinite solids subjected to in-plane combined magnetic–electric–mechanical loading conditions. The accuracy of the boundary element solution is confirmed by comparison with selected analytical solutions in the literature. The new results that can be of interest in the design and maintenance of novel magnetoelectroelastic devices are also discussed.

title: Recent Advances and Emerging Applications of the Boundary Element Method
abstract: Sponsored by the U.S. National Science Foundation, a workshop on the boundary element method (BEM) was held on the campus of the University of Akron during September 1–3, 2010 (NSF, 2010, “Workshop on the Emerging Applications and Future Directions of the Boundary Element Method,” University of Akron, Ohio, September 1–3). This paper was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts with a brief introduction to the BEM. Then, new developments in Green's functions, symmetric Galerkin formulations, boundary meshfree methods, and variationally based BEM formulations are reviewed. Next, fast solution methods for efficiently solving the BEM systems of equations, namely, the fast multipole method, the pre-corrected fast Fourier transformation method, and the adaptive cross approximation method are presented. Emerging applications of the BEM in solving microelectromechanical systems, composites, functionally graded materials, fracture mechanics, acoustic, elastic and electromagnetic waves, time-domain problems, and coupled methods are reviewed. Finally, future directions of the BEM as envisioned by the authors for the next five to ten years are discussed. This paper is intended for students, researchers, and engineers who are new in BEM research and wish to have an overview of the field. Technical details of the BEM and related approaches discussed in the review can be found in the Reference section with more than 400 papers cited in this review.

title: Analysis of anisotropic Kirchhoff plates using a novel hypersingular BEM
abstract: In this article a hypersingular boundary element method (BEM) for bending of thin anisotropic plates is presented. A new complex variable fundamental solution is implemented in the algorithm. For spatial discretization a collocation method with discontinuous quadratic elements is adopted. The domain integrals arising from the transversely applied load are transformed analytically into boundary integrals by means of the radial integration technique. The considered numerical examples prove that the novel BEM formulation presented in this study is much more efficient than previous formulations developed for the analysis of this kind of problems.

title: Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems
abstract: In this paper the dual reciprocity boundary element method in the Laplace domain for anisotropic dynamic fracture mechanic problems is presented. Crack problems are analyzed using the subregion technique. The dynamic stress intensity factors are computed using traction singular quarter-point elements positioned at the tip of the crack. Numerical inversion from the Laplace domain to the time domain is achieved by the Durbin method. Numerical examples of dynamic stress intensity factor evaluation are considered for symmetric and non-symmetric problems. The influence of the number of Laplace parameters and internal points in the solution is investigated.

title: Two-dimensional time-harmonic BEM for cracked anisotropic solids
abstract: A mixed time-harmonic boundary element procedure for the analysis of two-dimensional dynamic problems in cracked solids of general anisotropy is presented. To the author's knowledge, no previous BE approach for time-harmonic two-dimensional crack problems in anisotropic solids exists. In the present work, the fundamental solution is split into the static singular part plus dynamic regular terms. Hypersingular integrals associated to the singular part in the traction boundary integral equation are transformed, by means of a simple change of variable, into regular ones plus very simple singular integrals with known analytical solution. Subsequently, only regular (frequency dependent) terms have to be added to the regularized static fundamental solution in order to solve the dynamic problem. The generality of this procedure permits the use of general straight or curved quadratic boundary elements. In particular, discontinuous quarter-point elements are used to represent the crack-tip behavior. Stress intensity factors are accurately computed from the nodal crack opening displacements at discontinuous quarter-point elements. The efficiency and robustness of the present time-harmonic BEM are verified numerically by several test examples. Results are also obtained for more complex configurations, not previously studied in the literature. They include curved crack geometry.

title: A comparative study of three BEM for transient dynamic crack analysis of 2-D anisotropic solids
abstract: Three different boundary element methods (BEM) for transient dynamic crack analysis in two-dimensional (2-D), homogeneous, anisotropic and linear elastic solids are presented. Hypersingular traction boundary integral equations (BIEs) in frequency- domain, Laplace-domain and time-domain with the corresponding elastodynamic fundamental solutions are applied for this purpose. In the frequency-domain and the Laplace-domain BEM, numerical solutions are first obtained in the transformed domain for discrete frequency or Laplace-transform parameters. Time-dependent results are subsequently obtained by means of the inverse Fourier-transform and the inverse Laplace-transform algorithm of Stehfest. In the time-domain BEM, the quadrature formula of Lubich is adopted to approximate the arising convolution integrals in the time-domain BIEs. Hypersingular integrals involved in the traction BIEs are computed through a regularization process that converts the hypersingular integrals to regular integrals, which can be computed numerically, and singular integrals which can be integrated analytically. Numerical results for the dynamic stress intensity factors are presented and discussed for a finite crack in an infinite domain subjected to an impact crack-face loading.

title: Analysis of cracked magnetoelectroelastic composites under time-harmonic loading
abstract: This paper presents a numerical model for the analysis of cracked magnetoelectroelastic materials subjected to in-plane mechanical, electric and magnetic dynamic time-harmonic loading. A traction boundary integral equation formulation is applied to solve the problem in combination with recently obtained time-harmonic Green’s functions (Rojas-Diaz et al., 2008). The hypersingular boundary integral equations appearing in the formulation are first regularized via a simple change of variables that permits to isolate the singularities. Relevant fracture parameters, namely stress intensity factors, electric displacement intensity factor and magnetic induction intensity factor are directly evaluated as functions of the computed nodal opening displacements and the electric and magnetic potentials jumps across the crack faces. The method is checked by comparing numerical results against existing solutions for piezoelectric solids. Finally, numerical results for scattering of plane waves in a magnetoelectroelastic material by different crack configurations are presented for the first time. The obtained results are analyzed to evaluate the dependence of the fracture parameters on the coupled magnetoelectromechanical load, the crack geometry and the characteristics of the incident wave motion.

title: Hypersingular BEM for dynamic fracture in 2-D piezoelectric solids
abstract: Fracture behavior of piezoelectric solids under time-harmonic loading is numerically analyzed in this paper. A 2-D boundary element method (BEM) based on both displacement and traction boundary integral equations is presented. The time-harmonic Green’s functions for the infinite plane are split into singular plus regular terms, the singular ones coinciding with the static Green’s functions. In this manner the singular and hypersingular integrals arising in the formulation may be treated by the same simple regularization procedure proposed by the authors for static piezoelectricity. Quarter-point elements are used for the direct evaluation of stress and electric displacement intensity factors from nodal values. Several numerical examples for the scattering of waves by different crack configurations including branched and curved cracks and crack interaction problems are given to demonstrate the performance of the proposed method.

title: A hypersingular time‐domain BEM for 2D dynamic crack analysis in anisotropic solids
abstract: A hypersingular time-domain boundary element method (BEM) for transient elastodynamic crack analysis in two-dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack-faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time-stepping scheme is obtained to compute the unknown boundary data including the crack-opening-displacements (CODs). Special crack-tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time-domain BEM.

title: A 2D hypersingular time-domain traction BEM for transient elastodynamic crack analysis
abstract: A two-dimensional (2D) hypersingular time-domain traction BEM for transient elastodynamic crack analysis is presented in this paper. Time-domain traction boundary integral equations (BIEs) are applied for this purpose, to both crack-faces and external boundaries of the cracked solid. A numerical solution procedure is developed for solving the hypersingular time-domain traction BIEs. A time-stepping scheme in conjunction with a collocation method is applied. The time-stepping scheme uses a linear temporal shape-function which makes an analytical time-integration of the convolution integrals possible. As spatial shape-functions, two different shape-functions are adopted: a square-root crack-tip shape-function for elements adjacent to crack-tips and a constant shape-function for elements away from crack-tips. In this manner, the local behavior of the crack-opening-displacements (CODs) at crack-tips is correctly approximated by the square-root crack-tip shape-function. This allows us to accurately compute the dynamic stress intensity factors from the CODs. Suitable methods for evaluating the hypersingular and the strongly singular integrals are presented. Numerical examples for unbounded and bounded solids with cracks are given to show the efficiency and the accuracy of the time-domain BEM for transient elastodynamic crack analysis.

title: CRACK TIP FINITE ELEMENTS ARE UNNECESSARY
abstract: It is shown that a singularity occurs in isoparametric finite elements if the mid-side nodes are moved sufficiently from their normal position. By choosing the mid-side node positions on standard isoparametric elements so that the singularity occurs exactly at the corner of an element it is possible to obtain quite accurate solutions to the problem of determining the stress intensity at the tip of a crack. The solutions compare favourably with those obtained using some types of special crack tip elements, but are not as accurate as those given by a crack tip element based on the hybrid principle. However, the hybrid elements are more difficult to use.

title: Hypersingular quarter‐point boundary elements for crack problems
abstract: The present paper deals with the study and effective implementation for Stress Intensity Factor computation of a mixed boundary element approach based on the standard displacement integral equation and the hypersingular traction integral equation. Expressions for the evaluation of the hypersingular integrals along general curved quadratic line elements are presented. The integration is carried out by transformation of the hypersingular integrals into regular integrals, which are evaluated by standard quadratures, and simple singular integrals, which are integrated analytically. The generality of the method allows for the modelling of curved cracks and the use of straight line quarter-point elements. The Stress Intensity Factors can be computed very accurately from the Crack Opening Displacement at collocation points extremely close to the crack tip. Several examples with different crack geometries are analyzed. The computed results show that the proposed approach for Stress Intensity Factors evaluation is simple, produces very accurate solutions and has little dependence on the size of the elements near the crack tip.

title: A path-independent integral and the approximate analysis of strain
abstract: An integral is exhibited which has the same value for all paths surrounding a class of notches in two-dimensional deformation fields of linear or non-linear elastic materials. The integral may be evaluated almost by inspection for a few notch configurations. Also, for materials of the elastic- plastic type (treated through a deformation rather than incremental formulation) , with a linear response to small stresses followed by non-linear yielding, the integral may be evaluated in terms of Irwin's stress intensity factor when yielding occurs on a scale small in comparison to notch size. On the other hand, the integral may be expressed in terms of the concentrated deformation field in the vicinity of the notch tip. This implies that some information on strain concentrations is obtainable without recourse to detailed non-linear analyses. Such an approach is exploited here. Applications are made to: Approximate estimates of strain concentrations at smooth ended notch tips in elastic and elastic-plastic materials, A general solution for crack tip separation in the Barenblatt-Dugdale crack model, leading to a proof of the identity of the Griffith theory and Barenblatt cohesive theory for elastic brittle fracture and to the inclusion of strain hardening behavior in the Dugdale model for plane stress yielding, and An approximate perfectly plastic plane strain analysis, based on the slip line theory, of contained plastic deformation at a crack tip and of crack blunting.

title: Fracture Mechanics Analysis of Cracked Discs of Anisotropic Rock Using the Boundary Element Method
abstract: This paper is the second of a series of two papers dealing with the determination of the deformability, tensile strength and fracturing of anisotropic rocks by diametral compression (Brazilian test) of discs of rock. It is shown how a new formulation of the Boundary Element Method (BEM), proposed recently by the authors, can be used to determine the stress intensity factors (SIFs) and the fracture toughness of anisotropic rocks from the results of diametral compression tests on initially cracked discs. Crack initiation angles and propagation paths can also be predicted using a numerical procedure based on the BEM and maximum tensile stress criterion. Numerical examples of calculation of mixed mode SIFs are presented for both isotropic and anisotropic media. The calculated SIFs for the special isotropic case are found to be in good agreement with those reported by previous authors. Diametral loading tests were conducted on Cracked Straight Through Brazilian Disc (CSTBD) specimens of a shale in order to evaluate its fracture toughness, the angle of crack initiation and the path of crack propagation. It was found that the numerical simulations of crack initiation and propagation in the CSTBD specimens of the shale were in good agreement with the experimental observations.

title: Determination of Fracture Toughness of Anisotropic Rocks by Boundary Element Method
abstract: SummaryThis paper presents a systematic procedure for determining fracture toughness of an anisotropic marble using the diametral compression test (Brazilian test) with a central crack on the discs. Additionally, a novel formulation to increase the accuracy in Stress Intensity Factor (SIF) calculations using Boundary Element Method (BEM) is applied to determine the stress intensity factors and the fracture toughness of anisotropic rocks under mixed-mode loading. The numerical results show that the SIFs for both isotropic and anisotropic problems are in good agreement with those reported by previous authors. The marble with clear white-black foliation from Hualien (in eastern Taiwan), was selected for the Brazilian tests. Diametral loading was conducted on the Cracked Straight Through Brazilian Disc (CSTBD) specimens to evaluate their fracture toughness. In addition, a new fracture criterion was developed to predict pure mode I, pure mode II or mixed mode (I–II) fracture toughness of the anisotropic marble. The new fracture criterion is based on the examination of mode I, mode II and mixed mode (I–II) fracture toughness for different crack angles and anisotropic orientation.

title: Modeling crack propagation path of anisotropic rocks using boundary element method
abstract: In a cracked material, the stress intensity factors (SIFs) at the crack tips, which govern the crack propagation and are associated with the strength of the material, are strongly affected by the crack inclination angle and the orientation with respect to the principal direction of anisotropy. In this paper, a formulation of the boundary element method (BEM), based on the relative displacements of the crack tip, is used to determine the mixed-mode SIFs of isotropic and anisotropic rocks. Numerical examples of the application of the formulation for different crack inclination angles, crack lengths, and degree of material anisotropy are presented. Furthermore, the BEM formulation combined with the maximum circumferential stress criterion is adopted to predict the crack initiation angles and simulate the crack propagation paths. The propagation path in cracked straight through Brazilian disc specimen is numerically predicted and the results of numerical and experimental data compared with the actual laboratory observations. Good agreement is found between the two approaches. The proposed BEM formulation is therefore suitable to simulate the process of crack propagation.

title: Dual boundary element incremental analysis of crack propagation
abstract: This paper describes an application of the dual boundary element method to the analysis of mixed-mode crack growth in linear elastic fracture mechanics. Crack-growth processes are simulated with an incremental crack-extension analysis based on the maximum principal stress criterion which is expressed in terms of the stress intensity factors. For each increment of the crack extension, the dual boundary element method is applied to perform a single-region stress analysis of the cracked structure and the J-integral technique is used to compute the stress intensity factors. When the crack extension is modelled with new boundary elements, remeshing is not required because of the single-region analysis, an intrinsic feature of the dual boundary element method. Results of this incremental crack-extension analysis are presented for several geometries. The analysis of fatigue crack-growth is introduced as a post-processing procedure on the crack-extension results and an example is presented.

title: Cohesive crack growth modelling based on an alternative nonlinear BEM formulation
abstract: Fracture mechanics and crack propagation problems have been widely studied by the scientific community in recent years, because crack growth phenomenon can explain the failure of structures. In order to model accurately the structural behaviour of complex engineering structures numerical techniques are required. In this regard, the boundary element method (BEM) has been widely used to solve complex engineering problems, especially those where its mesh dimension reduction includes advantages on the modelling. This paper addresses the analysis of crack propagation in quasi-brittle materials using an alternative BEM formulation. In this type of problem, the damaged zone ahead the crack tip is modelled based on the fictitious crack model. Therefore, the residual resistance of the damaged zone is represented by cohesive stresses, which tends to close the crack faces. The alternative BEM formulation proposed aims at modelling the cohesive stresses using the domain term of the direct integral representation. This term is degenerated, in order to be non null only at the fictitious crack path. As a result of the domain term manipulation, appears a dipole of stresses, which will govern the cohesive stresses. It leads a nonlinear formulation, where the cohesive stresses are determined according the crack opening displacement. Linear, bi-linear and exponential cohesive laws are used to model the residual resistance of the cohesive zone. The results achieved by the proposed formulation are compared with experimental and numerical ones available in literature, in order to validate and prove its robustness and accuracy.

title: A single-domain dual-boundary-element formulation incorporating a cohesive zone model for elastostatic cracks
abstract: A cracked elastostatic structure is artificially divided into subdomains of simpler topology such that the well-developed classic dual integral equations can be applied appropriately to each domain. Applying the continuity and equilibrium conditions along artificial boundaries and properties of the integral kernels a single-domain dual-boundary-integral equation formulation is derived for a cracked elastic structure. A cohesive zone model is used to model the crack tip processes and is coupled with the single-domain dual-boundary-integral equation formulation; the resulting nonlinear equations are solved using the iterative method of successive-over-relaxation. The constitutive law used for a crack includes three parts: a law relating cohesive force to crack displacement difference when a crack is opening, a characterization of tangential interaction between crack surfaces when the crack surfaces are in contact, and a maximum principal stress criterion of crack advance. Incorporation of local unloading effect of the cohesive zone material has enabled a simulation of fracture with initial damage, partial development of the failure process zone at structural instability and multiple crack interaction. Some of the features of the method are demonstrated by considering three examples. The first problem is a single-edge-cracked specimen that exhibits a snap-back instability. The second example is the development of wing cracks from an angled crack under compression. The last example demonstrates the capability to consider mixed-mode crack growth and interaction of cracks. Thus, the problem of crack growth has been reduced to the determination of the cohesive model for the fracture process.

title: Crack growth analysis in concrete using boundary element method
abstract: In this paper, the dual boundary element method is extended to include the non-linear behaviour of cracking in concrete. The fictitious crack model is used to simulate the pure mode I and mixed mode crack propagation in concrete. The fictitious crack in the fracture zone in front of the real crack tip is replaced by the closing forces acting on both crack surfaces. When the force at the fictitious crack tip exceeds the maximum tensile strength of the concrete, the crack will propagate perpendicular to the maximum principal stress. The numerical simulation is controlled through the crack length as a monotonic increasing function during the fracture process. Three-point bending and four-point shear specimens are used to check the numerical results for mode I and mixed mode, respectively. Under the crack length controlled scheme, the snapback instability is observed in the mixed mode problem.

title: Boundary element analysis of the pullout behaviour of an anchor bolt embedded in concrete
abstract: A boundary element formulation for the analysis of pullout behaviour of an anchor bolt embedded in concrete is presented. The pullout analysis involves modelling two different bodies (i.e. anchor head and concrete) which are in contact over a certain region. The fracture of concrete is represented by the fictitious crack model (FCM) in which the fracture zone is replaced by applying closing forces on both crack surface. The FCM in conjunction with the boundary element method (BEM) allows the simulation of crack growth in concrete in a straight forward manner without remeshing. The crack path need not be known in advance as it is calculated during the iteration process. The numerical results obtained are compared with the round-robin analysis of pullout test proposed by RILEM TC 90-FMA.

title: Crack Growth Analysis in Reinforced Concrete Using BEM
abstract: A boundary-element formulation for modeling the nonlinear behavior of reinforced concrete is presented. The influence of reinforcements on the concrete is considered as a distribution of forces over the region of attachment. The solution for the attachment forces is obtained from the condition that the deformations of the concrete and the reinforcement under the action of the external loading are compatible. The yielding of reinforcement is considered when the total force at any section of the reinforcement is greater than the yielding force and is assumed to be broken when the strain reaches the maximum strain. The fracture of concrete is simulated using the fictitious crack model in which the fracture zone is replaced by closing forces acting on both crack surfaces. In using the boundary-element method to simulate cracks, the crack path is not known in advance because it can be calculated during the iteration process, and then the need of remeshing becomes obsolete. The numerical results obtained are compared to the finite-element method analysis and experimental results.

title: Fracture mechanics analysis of cracking in plain and reinforced concrete using the boundary element method
abstract: Boundary element formulations for modelling the nonlinear behaviour of concrete are reviewed. The analysis based on the dual boundary element method (BEM) to represent the cracks in concrete is presented. The fictitious crack is adopted to represent the fracture process zone in concrete. The influence of reinforcements on the concrete is considered as a distribution of forces over the region of attachment. The yielding of reinforcement is considered when the total force at any section of the reinforcement is greater than the yielding force and is assumed to be broken when the strain reaches the maximum strain. In using the BEM to simulate cracks, the crack path need not be known in advance since it can be calculated during the iteration process and as such remeshing becomes obsolete. The numerical results obtained are compared to the FEM analysis.

title: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
abstract: We present an iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from t...

title: A fast multipole boundary integral equation method for crack problems in 3D
abstract: This paper discusses a three-dimensional fast multipole boundary integral equation method for crack problems for Laplace's equation. The proposed implementation uses collocation and piecewise constant shape functions to discretise the hypersingular boundary integral equation for crack problems. The resulting numerical equation is solved with GMRES (generalised minimum residual method) in connection with FMM (fast multipole method). It is found that the obtained code is faster than a conventional one when the number of unknowns is greater than about 1300.

title: Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3D
abstract: Fast multipole method (FMM) has been developed as a technique to reduce the computational cost and memory requirements in solving large-scale problems. This paper discusses an application of FMM to three-dimensional boundary integral equation method for elastostatic crack problems. The boundary integral equation for many crack problems is discretized with FMM and Galerkin's method. The resulting algebraic equation is solved with generalized minimum residual method (GMRES). The numerical results show that FMM is more efficient than conventional methods when the number of unknowns is more than about 1200 and, therefore, can be useful in large-scale analyses of fracture mechanics.

title: GPU-accelerated indirect boundary element method for voxel model analyses with fast multipole method
abstract: An indirect boundary element method (BEM) that uses the fast multipole method (FMM) was accelerated using graphics processing units (GPUs) to reduce the time required to calculate a three-dimensional electrostatic field. The BEM is designed to handle cubic voxel models and is specialized to consider square voxel walls as boundary surface elements. The FMM handles the interactions among the surface charge elements and directly outputs surface integrals of the fields over each individual element. The CPU code was originally developed for field analysis in human voxel models derived from anatomical images. FMM processes are programmed using the NVIDIA Compute Unified Device Architecture (CUDA) with double-precision floating-point arithmetic on the basis of a shared pseudocode template. The electric field induced by DC-current application between two electrodes is calculated for two models with 499,629 (model 1) and 1,458,813 (model 2) surface elements. The calculation times were measured with a four-GPU configuration (two NVIDIA GTX295 cards) with four CPU cores (an Intel Core i7-975 processor). The times required by a linear system solver are 31 s and 186 s for models 1 and 2, respectively. The speed-up ratios of the FMM range from 5.9 to 8.2 for model 1 and from 5.0 to 5.6 for model 2. The calculation speed for element-interaction in this BEM analysis was comparable to that of particle-interaction using FMM on a GPU.

title: An adaptive dual-information FMBEM for 3D elasticity and its GPU implementation
abstract: Combined the boundary element method (BEM) with the fast multipole method (FMM), the fast multipole BEM (FMBEM) is proposed to solve large scale problems. A key issue the FMBEM has to address is the element integrals, which usually consumes much time when the FMM for N -body problems is directly used. In order to accelerate element integrals, we present an adaptive FMBEM with a particular dual-information tree structure which contains both node and element information, and use it for 3D elasticity in this paper. In our adaptive FMBEM, the Multipole Expansions (ME), Moment-to-Local (M2L) translation, Local Expansions (LE), and the Near Field Direct Computation (NFDC) are level independent so that they are suitable for parallel computing. The examples show that the time of ME and NFDC in our FMBEM is almost 1/3 and 1/2 compared with that in a node-based FMBEM which deals with FMBEM in a particle interaction mode. We develop two GPU parallel strategies to accelerate the processes of ME, M2L and NFDC and implement them on a NVIDIA GTX 285 GPU, and the speedups to an Intel Core2 Q9550 CPU using 4 cores can reach 10.7 for ME, 16.2 for M2L, and 3.6 for NFDC.

title: Fast Multipole Boundary Element Method: Theory and Applications in Engineering
abstract: 1.  Conventional BEM for potential problems 3. Fast multipole BEM for potential problems 4. Elastostatic problems 5. Stokes flow problems 6. Acoustic wave problems.

title: Improved implementation and robustness study of the X-FEM for stress analysis around cracks
abstract: Numerical crack propagation schemes were augmented in an elegant manner by the X-FEM method. The use of special tip enrichment functions, as well as a discontinuous function along the sides of the crack allows one to do a complete crack analysis virtually without modifying the underlying mesh, which is of industrial interest, especially when a numerical model for crack propagation is desired. This paper improves the implementation of the X-FEM method for stress analysis around cracks in three ways. First, the enrichment strategy is revisited. The conventional approach uses a 'topological' enrichment (only the elements touching the front are enriched). We suggest a 'geometrical' enrichment in which a given domain size is enriched. The improvements obtained with this enrichment are discussed. Second, the conditioning of the X-FEM both for topological and geometrical enrichments is studied. A preconditioner is introduced so that 'off the shelf' iterative solver packages can be used and perform as well on X-FEM matrices as on standard FEM matrices. The preconditioner uses a local (nodal) Cholesky based decomposition. Third, the numerical integration scheme to build the X-FEM stiffness matrix is dramatically improved for tip enrichment functions by the use of an ad hoc integration scheme. A 2D benchmark problem is designed to show the improvements and the robustness.

title: Solving a large dense linear system by adaptive cross approximation
abstract: An efficient algorithm for the direct solution of a linear system associated with the discretization of boundary integral equations (in two dimensions) is described without having to compute the complete matrix of the linear system. This algorithm is based on the unitary-weight representation, for which a new construction based on adaptive cross approximation is proposed. This low rank approximation uses only a small part of the entries to construct the adaptive cross representation, and therefore the linear system can be solved efficiently.

title: Numerical analysis of multi-crack large-scale plane problems with adaptive cross approximation and hierarchical matrices
abstract: The problem of interaction of large number of cracks in a plate is considered by the method of singular integral equations (SIE). The corresponding system of SIE is solved by using Gauss–Chebyshev quadratures, which results in a large system of linear algebraic equations. The solution of the latter employs the adaptive cross approximation (ACA) technique that has not previously been applied for studying multi-crack large-scale plane problems. Therefore, several benchmarks problems with large number of cracks modelling periodical arrangements have been tested to investigate performance of the method; these include arrays of collinear cracks, parallel cracks, and double network of parallel cracks. Comparisons with analytical and numerical periodical solutions available for the mentioned cases reveal high accuracy and fast performance of the method. It is also applied for studying effective characteristics of bodies with up to 20,000 cracks and for accurate modelling of interaction of a macrocrack with thousands of microcracks.

title: A fast hierarchical dual boundary element method for three-dimensional elastodynamic crack problems
abstract: In this work a fast solver for large-scale three-dimensional elastodynamic crack problems is presented, implemented, and tested. The dual boundary element method in the Laplace transform domain is used for the accurate dynamic analysis of cracked bodies. The fast solution procedure is based on the use of hierarchical matrices for the representation of the collocation matrix for each computed value of the Laplace parameter. An ACA (adaptive cross approximation) algorithm is used for the population of the low rank blocks and its performance at varying Laplace parameters is investigated. A preconditioned GMRES is used for the solution of the resulting algebraic system of equations. The preconditioners are built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. An original strategy, based on the computation of some local preconditioners only, is presented and tested to further speed up the overall analysis. The reported numerical results demonstrate the effectiveness of the technique for both uncracked and cracked solids and show significant reductions in terms of both memory storage and computational time.

title: Adaptive cross-approximation applied to the solution of system of equations and post-processing for 3D elastostatic problems using the boundary element method
abstract: We present the hierarchical matrix (H-matrix) technique combined with the adaptive cross-approximation (ACA) applied to a three-dimensional (3D) elastostatic problem using the boundary element method (BEM). This is used in structural geology and geomechanics for the evaluation of the deformation and perturbed stress field associated with surfaces of displacement discontinuity. Such optimization significantly reduces (i) the time and memory needed for the resolution of the system of equations, but more importantly (ii) the time needed for the post-processing at observation points where the deformation and the perturbed stress field are evaluated. Specifically, it is shown that the H-matrix structure used with the ACA, clearly captures the kernel smoothness during the post-processing stage according to the field point positions, and optimizes the computation accordingly. Combined with the parallelization on multi-core processors, this technique allows intensive computations to be done on personal desktop and laptop computers. Numerical simulations are presented, showing the advantages of such optimizations compared to the standard method.

title: Investigation of 3D crack propagation problems via fast BEM formulations
abstract: AbstactThe simulation of 3D fatigue crack growth is investigated and in combination with suitable experiments it is aimed to identify a reliable crack growth criterion. To perform the crack growth simulation as effectively as possible the boundary element method (BEM) in terms of the 3D Dual BEM is utilized. The relevant boundary integral equations are evaluated in the framework of a collocation procedure. To compress the system matrix and to speed-up the solution procedure the adaptive cross approximation (ACA) method is applied. It is an algebraic technique acting purely on the system matrix itself. The efficiency of this procedure with respect to crack problems will be shown on both a standard fracture specimen and an industrial application.

title: Developing new enrichment functions for crack simulation in orthotropic media by the extended finite element method
abstract: New enrichment functions are proposed for crack modelling in orthotropic media using the extended finite element method (XFEM). In this method, Heaviside and near-tip functions are utilized in the framework of the partition of unity method for modelling discontinuities in the classical finite element method. In this procedure, by using meshless based ideas, elements containing a crack are not required to conform to crack edges. Therefore, mesh generation is directly performed ignoring the existence of any crack while the method remains capable of extending the crack without any remeshing requirement. Furthermore, the type of elements around the crack-tip remains the same as other parts of the finite element model and the number of nodes and consequently degrees of freedom are reduced considerably in comparison to the classical finite element method. Mixed-mode stress intensity factors (SIFs) are evaluated to determine the fracture properties of domain and to compare the proposed approach with other available methods. In this paper, the interaction integral (M-integral) is adopted, which is considered as one of the most accurate numerical methods for calculating stress intensity factors.

title: ENRICHED ELEMENT-FREE GALERKIN METHODS FOR CRACK TIP FIELDS
abstract: The Element-Free Galerkin (EFG) method is a meshless method for solving partial di⁄erential equations which uses only a set of nodal points and a CAD-like description of the body to formulate the discrete model. It has been used extensively for fracture problems and has yielded good results when adequate refinement is used near the crack tip, but stresses tend to be oscillatatory near the crack tip unless substantial refinement is used. An enriched EFG formulation for fracture problems is proposed. Two methods are used: (1) adding the asymptotic fields to the trial function and (2) augmenting the basis by the asymptotic fields. A local mapping of the enriched fields for curved cracks is also described. Results show that both methods greatly reduce stress oscillations and allow the calculation of accurate stress intensity factors with far fewer degrees of freedom.

title: Fracture analysis for orthotropic cracked plates
abstract: The present investigation is intended to study the elastostatic fracture response of an orthotropic cracked plate subjected at infinity to a biaxial uniform load. The effects of orthotropy and load biaxiality on the near crack tip elastic fields is pointed out. Considering the non-singular terms, the angle of incipient crack propagation and the fracture loci are obtained. For this purposes the strain energy density theory and the maximum circumferential stress theory are considered and extended to orthotropic materials. A numerical analysis is performed for a wide range of orthotropic material properties and applied loads.

title: On the construction of blending elements for local partition of unity enriched finite elements
abstract: For computational efficiency, partition of unity enrichments are preferably localized to the sub-domains where they are needed. It is shown that an appropriate construction of the elements in the blending area, the regionwhere the enriched elements blend to unenriched elements, is often crucial for good performance of local partition of unity enrichments. An enhanced strain formulation is developed which leads to good performance; the optimal rate of convergence is achieved. For polynomial enrichments, it is shown that a proper choice of the finite element shape functions and partition of unity shape functions also improves the accuracy and convergence. The methods are illustrated by several examples. The examples deal primarily with the signed distance function enrichment for treating discontinuous derivatives inside an element, but other enrichments are also considered. Results show that both methods provide optimal rates of convergence.

title: A corrected XFEM approximation without problems in blending elements
abstract: The extended finite element method (XFEM) enables local enrichments of approximation spaces. Standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured. In elements that blend the enriched areas with the rest of the domain problems arise in general. These blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate. A modification of the XFEM approximation is proposed in this work. The enrichment functions are modified such that they are zero in the standard elements, unchanged in the elements with all their nodes being enriched, and varying continuously in the blending elements. All nodes in the blending elements are enriched. The modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements. The corrected XFEM is applied to problems in linear elasticity and optimal convergence rates are achieved.

title: eXtended Finite Element Method for fracture characterization of adhesive joints in pure mode I
abstract: Adhesive-bonding for the unions in structures is gaining momentum over welding, riveting and fastening. It is vital for the design of bonded structures the availability of reliable damage models to predict their fracture behaviour.  Zone Models (CZM’s) have been extensively used in the past, taking advantage of traction-separation laws between stresses and relative displacements to simulate crack growth along predefined paths. The eXtended Finite Element Method (XFEM) is a recent improvement of the Finite Element Method (FEM) that relies on damage laws based on the bulk strength of materials for damage initiation and strain for the assessment of failure, rather than the tipping tractions and tensile/shear relative displacements used for the CZM’s. Compared to the FEM, XFEM excels in allowing discontinuities to grow within bulk solids along an arbitrary path. This work aims to assess the viability of the XFEM to predict the fracture behaviour of a thin layer of adhesive between stiff and compliant adherends. To build the XFEM damage laws, the fracture toughness in pure mode I (GIc) and tensile cohesive strength (
				 σ n 0 ) of the two adhesives were initially determined by Double-Cantilever Beam (DCB) and bulk tensile tests, respectively. Particular emphasis was given to GIc, as this is the most influent parameter for the numerical predictions. The numerical simulations of the DCB tests with the proposed laws matched accurately the experimental load–displacement (P–δ) curves, which validated the analysis procedure. The accuracy of the data reduction methods for calculation of GIc was checked by comparison of the input values in the simulations with the results of GIc calculations, with good results.

title: X-FEM a good candidate for energy conservation in simulation of brittle dynamic crack propagation
abstract: This paper is devoted to the simulation of dynamic brittle crack propagation in an isotropic medium. It focuses on cases where the crack deviates from a straight-line trajectory and goes through stop-and-restart stages. Our argument is that standard methods such as element deletion or remeshing, although easy to use and implement, are not robust tools for this type of simulation essentially because they do not enable one to assess local energy conservation. Standard cohesive zone models behave much better when the crack’s path is known in advance, but are difficult to use when the crack’s path is unknown. The simplest method which consists in placing the cohesive segments along the sides of the finite elements leads to crack trajectories which are mesh-sensitive. The adaptive cohesive element formulation, which adds new cohesive elements when the crack propagates, is shown to have the proper energy conservation properties during remeshing. We show that the X-FEM is a good candidate for the simulation of complex dynamic crack propagation. A two-dimensional version of the proposed X-FEM approach is validated against dynamic experiments on a brittle isotropic plate.

title: Extended finite element method for cohesive crack growth
abstract: The extended finite element method allows one to model displacement discontinuities which do not conform to interelement surfaces. This method is applied to modeling growth of arbitrary cohesive cracks. The growth of the cohesive zone is governed by requiring the stress intensity factors at the tip of the cohesive zone to vanish. This energetic approach avoids the evaluation of stresses at the mathematical tip of the crack. The effectiveness of the proposed approach is demonstrated by simulations of cohesive crack growth in concrete.

title: New anisotropic crack-tip enrichment functions for the extended finite element method
abstract: In this paper, the extended finite element method (X-FEM) is implemented to analyze fracture mechanics problems in elastic materials that exhibit general anisotropy. In the X-FEM, crack modeling is addressed by adding discontinuous enrichment functions to the standard FE polynomial approximation within the framework of partition of unity. In particular, the crack interior is represented by the Heaviside function, whereas the crack-tip is modeled by the so-called crack-tip enrichment functions. These functions have previously been obtained in the literature for isotropic, orthotropic, piezoelectric and magnetoelectroelastic materials. In the present work, the crack-tip functions are determined by means of the Stroh’s formalism for fully anisotropic materials, thus providing a new set of enrichment functions in a concise and compact form. The proposed formulation is validated by comparing the obtained results with other analytical and numerical solutions. Convergence rates for both topological and geometrical enrichments are presented. Performance of the newly derived enrichment functions is studied, and comparisons are made to the well-known classical crack-tip functions for isotropic materials.

title: Cohesive and non-cohesive fracture by higher-order enrichment of XFEM
abstract: A comprehensive study is performed on the use of higher-order terms of the crack tip asymptotic fields as enriching functions for the eXtended FEM (XFEM) for both cohesive and traction-free cracks. For traction-free cracks, the Williams asymptotic field is used to obtain highly accurate stress intensity factors (SIFs), directly from the enriched degrees of freedom without any post-processing. The low accuracy of the results of the original research on this subject by Liu et al.  is remedied here by appropriate modifications of the enrichment scheme. The modifications are simple and can be easily included into an XFEM computer code. For cohesive cracks, the relevant asymptotic field is used, and two widely used criteria including the SIFs criterion and the stress criterion are examined for the crack growth simulation. Both linear and nonlinear cohesive laws are used. For the stress criterion, averaging is avoided due to the highly accurate crack tip approximation because of the higher-order enrichment. Then, a modified stress criterion is proposed, which is shown to be applicable to a wider class of problems. Several numerical examples, including straight and curved cracks, stationary and growing cracks, single and multiple cracks, and traction-free and cohesive cracks, are studied to investigate in detail the robustness and efficiency of the proposed enrichment scheme.

title: Two-Dimensional Mode I Crack Propagation in Saturated Ionized Porous Media Using Partition of Unity Finite Elements
abstract: Shales, clays, hydrogels, and tissues swell and shrink under changing osmotic conditions, which may lead to failure. The relationship between the presence of cracks and fluid flow has had little attention. The relationship between failure and osmotic conditions has had even less attention. The aim of this research is to study the effect of osmotic conditions on propagating discontinuities under different types of loads for saturated ionized porous media using the finite element method (FEM). Discontinuous functions are introduced in the shape functions of the FEM by partition of unity method, independently of the underlying mesh. Damage ahead of the crack-tip is introduced by a cohesive zone model. Tensile loading of a crack in an osmoelastic medium results in opening of the crack and high pressure gradients between the crack and the formation. The fluid flow in the crack is approximated by Couette flow. Results show that failure behavior depends highly on the load, permeability, (osmotic) prestress and the stiffness of the material. In some cases it is seen that when the crack propagation initiates, fluid is attracted to the crack-tip from the crack rather than from the surrounding medium causing the crack to close. The results show reasonable mesh-independent crack propagation for materials with a high stiffness. Stepwise crack propagation through the medium is seen due to consolidation, i.e., crack propagation alternates with pauses in which the fluid redistributes. This physical phenomenon challenges the numerical scheme. Furthermore, propagation is shown to depend on the osmotic prestressing of the medium. This mechanism may explain the tears observed in intervertebral disks as degeneration progresses.

title: A continuous-discontinuous approach to simulate physical degradation processes in porous media
abstract: A macroscopic framework for the simulation of physical degradation processes in quasi-brittle porous materials is proposed. The framework employs the partition of unity (PU) concept and introduces a cohesive zone model, capturing the entire failure process starting from the growth and coalescence of micro-defects until the formation of macro-cracks. The framework incorporates the interaction between the failure process and the heat and mass transfer in the porous medium. As an example, physical degradation of an outside render is studied. The analysis illustrates that both material and interface failure can be investigated with this formulation. Depending on the boundary conditions, either one dominant crack or a network of small cracks is formed.

title: Recent efforts toward modeling interactions of matrix cracks and delaminations: an integrated XFEM-CE approach
abstract: This paper presents a novel integrated approach using the extended finite element method (XFEM) and cohesive elements (CEs) for modeling three-dimensional (3D) delaminations, matrix cracks, and their interactions in progressive failure of composite laminates. In the proposed XFEM-CE approach, the matrix cracks are explicitly modeled by the XFEM through nodal enrichment and element partition, and the inter-ply delaminations are modeled by cohesive elements through traction–separation law. An XFEM-based cohesive element enrichment scheme is developed in order to properly model the interactions between the delaminations and matrix cracks. It is critical to enrich the cohesive elements at the local juncture where cracks meet, in order to obtain the correct crack path interactions. Examples are presented for failure analysis of double-cantilever-beam and end-notch-flexure laminates with different layups. The results show strong explicit matrix crack–delamination interactions in these laminates. This work presents another significant development of a computational platform for realistic modeling of progressive damage in composites.

title: New crack‐tip elements for XFEM and applications to cohesive cracks
abstract: An extended finite element method scheme for a static cohesive crack is developed with a new formulation for elements containing crack tips. This method can treat arbitrary cracks independent of the mesh and crack growth without remeshing. All cracked elements are enriched by the sign function so that no blending of the local partition of unity is required. This method is able to treat the entire crack with only one type of enrichment function, including the elements containing the crack tip. This scheme is applied to linear 3-node triangular elements and quadratic 6-node triangular elements. To ensure smooth crack closing of the cohesive crack, the stress projection normal to the crack tip is imposed to be equal to the material strength. The equilibrium equation and the traction condition are solved by the Newton–Raphson method to obtain the nodal displacements and the external load simultaneously. The results obtained by the new extended finite element method are compared to reference solutions and show excellent agreement.

title: An XFEM/Spectral element method for dynamic crack propagation
abstract: A high-order extended finite element method based on the spectral element method for the simulation of dynamic fracture is developed. The partition of unity for the discontinuous displacement is constructed by employing p order spectral element. This method shows great advantages in the simulations of moving crack and mixed mode crack. The numerical oscillations are effectively suppressed and the accuracy of computed stress intensity factors and crack path are improved markedly. Furthermore the simulation results show that p-refinement is more effective in improving the stress contour near the crack tip than h-refinement. The well known form of the explicit central difference method is used and the critical time step for this method is investigated. We find that by using lumped mass matrix the critical time step Δtc for this high-order extended finite element is almost independent of the crack position.

title: Time dependent crack tip enrichment for dynamic crack propagation
abstract: We study several enrichment strategies for dynamic crack propagation in the context of the extended finite element method and the effect of different directional criteria on the crack path. A new enrichment method with a time dependent enrichment function is proposed. In contrast to previous approaches, it entails only one crack tip enrichment function. Results for stress intensity factors and crack paths for different enrichments and direction criteria are given.

title: Dynamic analysis of fixed cracks in composites by the extended finite element method
abstract: This paper is dedicated to simulation of dynamic analysis of fixed cracks in orthotropic media using an extended finite element method. This work is in fact an extension to dynamic problems of the recently developed orthotropic extended finite element method for fracture analysis of composites. In this method, the Heaviside and near-tip enrichment functions are used in the framework of the partition of unity for modeling crack discontinuity and crack-tip singularities within the classical finite element method. In this procedure, elements that include a crack are not required to conform to crack edges. Therefore, mesh generation can be performed without any need to comply to crack edges and the method is capable of modeling the crack propagation without any remeshing. To determine the fracture properties, mixed-mode dynamic stress intensity factors (DSIFs) are evaluated by means of domain separation integral (J′-integral) method. Results of the proposed method are compared with other available analytical and computational results.

title: Fracture analysis of composites by time independent moving-crack orthotropic XFEM
abstract: Time-independent orthotropic enrichment functions are introduced for dynamic propagation analysis of moving cracks in composites by the extended finite element method (XFEM). The proposed enrichment functions are derived from the analytical solutions for a moving/propagating crack in orthotropic media, and can be considered as a new extension to the available XFEM techniques for dynamic analysis of stationary and moving cracks in orthotropic materials. They are included within the framework of partition of unity and XFEM to enhance the accuracy of basic FEM solution near a moving crack tip in orthotropic media. The method allows for analysis of the whole crack propagation pattern on an unaltered finite element mesh, which is independently defined from the existence of any predefined crack or its propagation path. A combination of dynamic crack initiation toughness and crack orientation along the maximum circumferential stress is used to design a relatively simple and efficient formulation. Dynamic stress intensity factors (DSIFs) are evaluated by means of the domain separation integral method and the dynamic energy release rate. The time dependent XFEM equations are constructed by discretizing the standard weak formulation of the governing elastodynamics equation. They are solved by the unconditionally stable Newmark time integration scheme. A number of benchmark and test problems are simulated and the results are compared with the available reference results to illustrate the accuracy and efficiency of the proposed scheme.

title: An XFEM method for modeling geometrically elaborate crack propagation in brittle materials
abstract: We present a method for simulating quasistatic crack propagation in 2-D which combines the extended finite element method (XFEM) with a general algorithm for cutting triangulated domains, and introduce a simple yet general and flexible quadrature rule based on the same geometric algorithm. The combination of these methods gives several advantages. First, the cutting algorithm provides a flexible and systematic way of determining material connectivity, which is required by the XFEM enrichment functions. Also, our integration scheme is straightforward to implement and accurate, without requiring a triangulation that incorporates the new crack edges or the addition of new degrees of freedom to the system. The use of this cutting algorithm and integration rule allows for geometrically complicated domains and complex crack patterns.

title: Energy-based modeling of cohesive and cohesionless cracks via X-FEM
abstract: Numerical analyses of structures made of brittle materials such as glass or quasi-brittle materials such as concrete or ceramics require robust models for the opening and propagation of cracks which adequately represent the discontinuous character of the fracture process and consider adequately cohesive forces acting within the fracture process zone. The extended finite element method (X-FEM) is capable to model cracks as propagating discontinuities within the structure independent of the finite element discretization. Nevertheless, the analysis of crack propagation using discrete crack models crucially depends on the crack growth criterion. In this paper, a global energy-based method is proposed for the determination of the crack propagation length as well as for the crack propagation direction. The method is formulated within an X-FEM-based analysis model leading to a variational formulation in terms of displacements, crack lengths and crack angles. Both cohesive and cohesionless cracks are considered. The constitutive model for cohesive cracks considers the transfer of residual stresses parallel as well as orthogonal to the crack faces. For the representation of the displacement field in the vicinity of the tip of cohesive cracks, enriched approximations which do not lead to a singularity of the stresses at the crack tip are proposed in the paper. After linearization, the proposed energy-based finite element crack propagation model results in a coupled system of equations analogous to a multifield problem. The model is verified by means of comparisons with crack growth analyses based on linear elastic fracture mechanics using the maximum circumferential stress criterion for the determination of the crack propagation direction. Furthermore, two examples, characterized by Mode I and Mixed-Mode fracture, respectively, are used to study the performance of the developed energy-based crack propagation model.

title: A continuous–discontinuous approach to simulate fracture processes in quasi-brittle materials
abstract: A macroscopic framework for the simulation of failure processes in quasi-brittle materials is proposed. The framework employs the partition of unity (PU) concept and introduces a new cohesive zone model, capturing the transition between the initial continuum state and the final discrete state. The model is generic in a sense that it allows extension of most continuum models to a discontinuous framework in an efficient and robust way, thereby adding the effect of macro-crack formation by the growth and coalescence of micro-defects. Both material failure and interface failure can be studied with this formulation.

title: A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics
abstract: We introduce a novel enriched Boundary Element Method (BEM) and Dual Boundary Element Method (DBEM) approach for accurate evaluation of Stress Intensity Factors (SIFs) in crack problems. The formulation makes use of the Partition of Unity Method (PUM) such that functions obtained from a priori knowledge of the solution space can be incorporated in the element formulation. An enrichment strategy is described, in which boundary integral equations formed at additional collocation points are used to provide auxiliary equations in order to accommodate the extra introduced unknowns. In addition, an efficient numerical quadrature method is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals. Finally, results are shown for mixed mode crack problems; these illustrate that the introduction of PUM enrichment provides for an improvement in accuracy of approximately one order of magnitude in comparison to the conventional unenriched DBEM.

title: Element‐free Galerkin methods
abstract: An element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems. In this method, moving least-squares interpolants are used to construct the trial and test functions for the variational principle (weak form); the dependent variable and its gradient are continuous in the entire domain. In contrast to an earlier formulation by Nayroles and coworkers, certain key differences are introduced in the implementation to increase its accuracy. The numerical examples in this paper show that with these modifications, the method does not exhibit any volumetric locking, the rate of convergence can exceed that of finite elements significantly and a high resolution of localized steep gradients can be achieved. The moving least-squares interpolants and the choices of the weight function are also discussed in this paper.

title: A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics
abstract: A local symmetric weak form (LSWF) for linear potential problems is developed, and a truly meshless method, based on the LSWF and the moving least squares approximation, is presented for solving potential problems with high accuracy. The essential boundary conditions in the present formulation are imposed by a penalty method. The present method does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the “energy”. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. No post-smoothing technique is required for computing the derivatives of the unknown variable, since the original solution, using the moving least squares approximation, is already smooth enough. Several numerical examples are presented in the paper. In the example problems dealing with Laplace & Poisson's equations, high rates of convergence with mesh refinement for the Sobolev norms ||·||0 and ||·||1 have been found, and the values of the unknown variable and its derivatives are quite accurate. In essence, the present meshless method based on the LSWF is found to be a simple, efficient, and attractive method with a great potential in engineering applications.

title: Imposing essential boundary conditions in mesh-free methods
abstract: Imposing essential boundary conditions is a key issue in mesh-free methods. The mesh-free interpolation does not verify the Kronecker delta property and, therefore, the imposition of prescribed values is not as straightforward as for the finite element method. The aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in Galerkin based mesh-free methods. Special attention is paid to the mesh-free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the Galerkin weak form. Particular examples are used to analyze and compare their performance in different situations.

title: Meshless methods: An overview and recent developments
abstract: Meshless approximations based on moving least-squares, kernels, and partitions of unity are examined. It is shown that the three methods are in most cases identical except for the important fact that partitions of unity enable p-adaptivity to be achieved. Methods for constructing discontinuous approximations and approximations with discontinuous derivatives are also described. Next, several issues in implementation are reviewed: discretization (collocation and Galerkin), quadrature in Galerkin and fast ways of constructing consistent moving least-square approximations. The paper concludes with some sample calculations.

title: Classification and Overview of Meshfree Methods
abstract: This paper gives an overview over Meshfree  Starting point is an extended and modified classification of Meshfree Methods due to three aspects: The construction of a partition of unity, the choice of an approximation either with or without using an extrinsic basis and the choice of test functions, resulting into a collocation, Bubnov-Galerkin or Petrov-Galerkin Meshfree  Most of the relevant Meshfree Methods are described taking into account their different origins and viewpoints as well as their advantages and disadvantages. Typical problems arising in meshfree methods like integration, treatment of essential boundary conditions, coupling with mesh-based methods etc.~are discussed. Some valuing comments about the most important aspects can be found at the end of each section. This text was revised in 2004

title: Meshless methods: A review and computer implementation aspects
abstract: The aim of this manuscript is to give a practical overview of meshless methods (for solid mechanics) based on global weak forms through a simple and well-structured MATLAB code, to illustrate our discourse. The source code is available for download on our website and should help students and researchers get started with some of the basic meshless methods; it includes intrinsic and extrinsic enrichment, point collocation methods, several boundary condition enforcement schemes and corresponding test cases. Several one and two-dimensional examples in elastostatics are given including weak and strong discontinuities and testing different ways of enforcing essential boundary conditions.

title: Fracture and crack growth by element free Galerkin methods
abstract: Element free Galerkin (EFG) methods are methods for solving partial differential equations that require only nodal data and a description of the geometry; no element connectivity data are needed. This makes the method very attractive for the modeling of the propagation of cracks, as the number of data changes required is small and easily developed. The method is based on the use of moving least-squares interpolants with a Galerkin method, and it provides highly accurate solutions for elliptic problems. The implementation of the EFG method for problems of fracture and static crack growth is described. Numerical examples show that accurate stress intensity factors can be obtained Without any enrichment of the displacement field by a near-crack-tip singularity and that crack growth can be easily modeled since it requires hardly any remeshing.

title: Continuous meshless approximations for nonconvex bodies by diffraction and transparency
abstract: Continuous meshless approximations are developed for domains with non-convex boundaries, with emphasis on cracks. Two techniques are developed in the context of the element-free Galerkin method: a transparency method wherein smooth approximations are generated by making boundaries partially transparent, and a diffraction method, where the domain of influence wraps around a concave boundary. They are compared to the original method based on the visibility criterion in which the approximations are discontinuous in the vicinity of nonconvex boundaries. The performance of the methods is compared using two elastostatic examples: an infinite plate with a hole and a crack problem. The continuous approximations show only moderate imporvement in accuracy over the discontinous approximations, but yield significant improvements for enhanced bases, such as crack-tip singular functions.

title: Element-free galerkin methods for static and dynamic fracture
abstract: Element-free Galerkin (EFG) methods are presented and applied to static and dynamic fracture problems. EFG methods, which are based on moving least-square (MLS) interpolants, require only nodal data; no element connectivity is needed. The description of the geometry and numerical model of the problem consists only of a set of nodes and a description of exterior boundaries and interior boundaries from any cracks. This makes the method particularly attractive for growing crack problems, since only minimal remeshing is needed to follow crack growth. In moving least-square interpolants, the dependent variable at any point is obtained by minimizing a function in terms of the nodal values of the dependent variable in the domain of influence of the point. Numerical examples involving fatigue crack growth and dynamic crack propagation are presented to illustrate the performance and potential of this method.

title: Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment
abstract: This paper presents a three-dimensional, extrinsically enriched meshfree method for initiation, branching, growth and coalescence of an arbitrary number of cracks in non-linear solids including large deformations, for statics and dynamics. The novelty of the methodology is that only an extrinsic discontinuous enrichment and no near-tip enrichment is required. Instead, a Lagrange multiplier field is added along the crack front to close the crack. This decreases the computational cost and removes difficulties involved with a branch enrichment. The results are compared to experimental data, and other simulations from the literature to show the robustness and accuracy of the method.

title: Meshless and Generalized Finite Element Methods: A Survey of Some Major Results
abstract: In this lecture, we discuss Meshless and Generalized Finite Element Methods. We survey major results in this area with a unified approach.

title: The partition of unity finite element method: Basic theory and applications
abstract: The paper presents the basic ideas and the mathematical foundation of the partition of unity finite element method (PUFEM). We will show how the PUFEM can be used to employ the structure of the differential equation under consideration to construct effective and robust methods. Although the method and its theory are valid in n dimensions, a detailed and illustrative analysis will be given for a one-dimensional model problem. We identify some classes of non-standard problems which can profit highly from the advantages of the PUFEM and conclude this paper with some open questions concerning implementational aspects of the PUFEM.

title: A clumped particle model for rock
abstract: Discrete element (particle) modeling is now often used to simulate the behavior of rock. While the method is attractive because it does not require the formulation of complex constitutive models, it requires extensive calibration with measured macro-scale results to determine the particle–contact parameters that will predict the macro-scale response. This calibration is usually carried out using the laboratory uniaxial compression test. The behavior of intact rock in compression is complex, resulting in a nonlinear failure envelope with high frictional resistance and a ratio of tensile to uniaxial compression strength of approximately 0.05–0.1. An extensive series of calibrations were carried out using a particle code to determine if the micro-parameters determined from uniaxial compression tests could be used to predict the intact rock response regardless of stress path, i.e., tension to triaxial compression. It is determined that the particle code cannot predict the failure envelope measured on laboratory samples, unless significant modifications are made. This study shows that adjusting the particle parameters had little effect on the macro-scale properties of compression and tension tests, but using a clumped-particle geometry improves the predictive capabilities of the particle code significantly. For both Lac du Bonnet granite and a weak synthetic rock, the particle code calibrated to uniaxial tests using the clumped-particle geometry predicts both the stress–strain behavior and the complete nonlinear failure envelope.

title: Simulations of instability in dynamic fracture by the cracking particles method
abstract: Crack instabilities and the phenomenon of crack speed saturation in a brittle material (PMMA) are studied with a meshfree cracking particle method. We reproduce the experimental observation that the computed terminal crack speeds attained in PMMA specimens are substantially lower than the Rayleigh wave speed; the computed crack speeds agree quite well with the reported experimental results. We also replicate repetitive microcrack branching along with the increased rate of energy dissipation after attainment of a critical crack speed, even in the absence of microstructural defects. We show that the presence of microdefects changes the response only a little. The computations reproduce many of the salient features of experimental observations.

title: Extended meshfree methods without branch enrichment for cohesive cracks
abstract: An extended meshless method for both static and dynamic cohesive cracks is proposed. This new method does not need any crack tip enrichment to guarantee that the crack closes at the tip. All cracked domains of influence are enriched by only the sign function. The domain of influence which includes a crack tip is modified so that the crack tip is always positioned at its edge. The modification is only applied for the discontinuous displacement field and the continuous field is kept unchanged. In addition to the new method, the use of Lagrange multiplier is explored to achieve the same goal. The crack is extended beyond the actual crack tip so that the domains of influence containing the crack tip are completely cut. It is enforced that the crack opening displacement vanishes along the extension of the crack. These methods are successfully applied to several well-known static and dynamic problems.

title: A new partition of unity finite element free from the linear dependence problem and possessing the delta property
abstract: Partition of unity based finite element methods (PUFEMs) have appealing capabilities for p-adaptivity and local refinement with minimal or even no remeshing of the problem domain. However, PUFEMs suffer from a number of problems that practically limit their application, namely the linear dependence (LD) problem, which leads to a singular global stiffness matrix, and the difficulty with which essential boundary conditions can be imposed due to the lack of the Kronecker delta property. In this paper we develop a new PU-based triangular element using a dual local approximation scheme by treating boundary and interior nodes separately. The present method is free from the LD problem and essential boundary conditions can be applied directly as in the FEM. The formulation uses triangular elements, however the essential idea is readily extendable to other types of meshed or meshless formulation based on a PU approximation. The computational cost of the present method is comparable to other PUFEM elements described in the literature. The proposed method can be simply understood as a PUFEM with composite shape functions possessing the delta property and appropriate compatibility.

title: A meshless method with enriched weight functions for three-dimensional crack propagation
abstract: The propagation of cracks in three dimensions is analysed by a meshless method. The cracks are modelled by a set of triangles that are added when the propagation occurs. Since the method is meshless, no remeshing of the domain is necessary during the propagation. To avoid using a large number of degrees of freedom, the stress singularity along the front of the cracks is taken into account by an enrichment of the shape functions of the meshless method by means of appropriate weight functions. This enrichment technique is an extension of the technique that proved to be successful in two dimensions in a previous paper. Several algorithms for efficiently implementing the meshless method in three dimensions are detailed. The accuracy of the enrichment is first assessed on simple examples and some results of non-planar propagation of multiple cracks are then presented.

title: A study of the representation of cracks with level sets
abstract: The level set method has been used for a few years to represent cracks in fracture mechanics simulations instead of an explicit description of the cracks faces geometry. This paper studies in detail the shape of the level set functions around a crack in two dimensions that is propagating with a sharp kink, obtained both with level set update methods found in the literature and with several innovative update methods developed by the author. A criterion based on the computation of a J integral of a virtual displacement field obtained with the values of the level set functions is proposed in order to assess the quality of these update methods. With the help of this criterion, two optimal approaches are identified, which predict an accurate evolution of the crack with smooth and consistent level set functions. These methods are then applied in three dimensions to the case of an initially penny-shaped crack that propagates out of its plane.

title: NON-PLANAR 3D CRACK GROWTH BY THE EXTENDED FINITE ELEMENT AND LEVEL SETS- PART II: LEVEL SET UPDATE
abstract: We present a level set method for treating the growth of non-planar three-dimensional cracks.The crack is defined by two almost-orthogonal level sets (signed distance functions). One of them describes the crack as a two-dimensional surface in a three-dimensional space, and the second is used to describe the one-dimensional crack front, which is the intersection of the two level sets. A Hamilton–Jacobi equation is used to update the level sets. A velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface. The technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity. This displacement field is constructed directly in terms of the level sets, so the discretization by finite elements requires no explicit representation of the crack surface. Numerical experiments show the robustness of the method, both in accuracy and in treating cracks with significant changes in topology.

title: Fast Marching Methods
abstract: Fast Marching Methods are numerical schemes for computing solutions to the nonlinear Eikonal equation and related static Hamilton--Jacobi equations. Based on entropy-satisfying upwind schemes and fast sorting techniques, they yield consistent, accurate, and highly efficient algorithms. They are optimal in the sense that the computational complexity of the algorithms is O(N log N), where N is the total number of points in the domain. The schemes are of use in a variety of applications, including problems in shape offsetting, computing distances from complex curves and surfaces, shape-from-shading, photolithographic development, computing first arrivals in seismic travel times, construction of shortest geodesics on surfaces, optimal path planning around obstacles, and visibility and reflection calculations. In this paper, we review the development of these techniques, including the theoretical and numerical underpinnings; provide details of the computational schemes, including higher order versions; and demonstrate the techniques in a collection of different areas.

title: Solidification microstructures and solid-state parallels: Recent developments, future directions
abstract: Rapid advances in atomistic and phase-field modeling techniques as well as new experiments have led to major progress in solidification science during the first years of this century. Here we review the most important findings in this technologically important area that impact Our quantitative understanding of: (i) key anisotropic properties of the solid-liquid interface that govern solidification pattern evolution, including the solid-liquid interface free energy and the kinetic coefficient; (ii) dendritic solidification at small and large growth rates. with particular emphasis on orientation selection; (iii) regular and irregular eutectic and peritectic microstructures; (iv) effects of convection on microstructure formation; (v) solidification at a high Volume fraction of solid and the related formation of pores and hot cracks: and (vi) solid-state transformations as far as they relate to solidification models anti techniques. In light of this progress, critical issues that point to directions for future research in both solidification and solid-state transformations are identified.

title: Three-dimensional phase-field simulation of micropore formation during solidification: Morphological analysis and pinching effect
abstract: A three-dimensional (3-D) multiphase-field model has been developed in order to study the formation of a micropore constrained to grow in a solid network (i.e. pinching effect). The model accounts for the pressure difference due to capillarity between liquid and gas, the equilibrium condition at triple (solid–liquid–pore) lines, and the partitioning and diffusion of dissolved gases such as hydrogen. From the predicted 3-D morphology of the pore, entities such as the interfacial shape distribution are plotted and analyzed. It is shown that the mean curvature of the pore–liquid surface, and thus also the pressure inside the pore, is uniform. The results are then compared with analytical pinching models. While predicting a similar trend, analytical models tend to underestimate the pore curvature at high solid fractions. Despite the complex morphology of pores reconstructed using high-resolution X-ray tomography, the present phase-field results suggest that a simple pinching model based on a spherical tip growing in between remaining liquid channels is a fairly good approximation.

title: Phase field method to optimize dielectric devices for electromagnetic wave propagation
abstract: We discuss a phase field method for shape optimization in the context of electromagnetic wave propagation. The proposed method has the same functional capabilities as the level set method for shape optimization. The first advantage of the method is the simplicity of computation, since extra operations such as re-initialization of functions are not required. The second is compatibility with the topology optimization method due to the similar domain representation and the sensitivity analysis. Structural shapes are represented by the phase field function defined in the design domain, and this function is optimized by solving a time-dependent reaction diffusion equation. The artificial double-well potential function used in the equation is derived from sensitivity analysis. We study four types of 2D or 2.5D (axisymmetric) optimization problems. Two are the classical problems of photonic crystal design based on the Bloch theory and photonic crystal wave guide design, and two are the recent topics of designing dielectric left-handed metamaterials and dielectric ring resonators.

title: Phase-field simulation of anisotropic crack propagation in ferroelectric single crystals: effect of microstructure on the fracture process
abstract: Crack propagation during the indentation test of a ferroelectric single crystal is simulated using a phase-field model. This model is based on variational formulations of brittle crack propagation and domain evolution in ferroelectric materials. Due to the high compressive stresses near the indenter contact faces, a modified regularized formulation of the variational brittle fracture is coupled with the material model to prevent crack formation and interpenetration in the compressed regions. The simulation results show that the radial cracks perpendicular to the poling direction of the material propagate faster than the parallel ones, which is in agreement with experimental observations. This anisotropy in the crack propagation is due to interactions between the material microstructure and the radial cracks, as captured by the phase-field simulation.

title: A real-space non-local phase-field model of ferroelectric domain patterns in complex geometries
abstract: Ferroelectric perovskites are used in various transducer, memory and optical applications due to their attractive properties. In these applications, the ferroelectric materials often have complex geometries and electrode arrangements, and are subjected to domain switching. Therefore, it is important to understand the domain pattern that forms in these geometries. However, models of domain evolution often make assumptions like periodicity and complete shielding which make them unrealistic for these applications. In this paper, we develop a phase-field approach that can study domain patterns and domain evolution in complex geometries with no a priori assumption on geometry, electrode arrangement, and dielectric properties. The key idea is a boundary element method to resolve the electrostatic fields. We illustrate the method by examining the closure domains that form at a free surface and domain switching under cyclic electric field in a device with interdigitated electrodes.

title: Revisiting brittle fracture as an energy minimization problem
abstract: A variational model of quasistatic crack evolution is proposed. Although close in spirit to Griffith’s theory of brittle fracture, the proposed model however frees itself of the usual constraints of that theory : a preexisting crack and a well-defined crack path. In contrast, crack initiation as well as crack path can be quantified, as demonstrated on explicitly computable examples. Furthermore the model lends itself to numerical implementation in more complex settings.

title: Numerical experiments in revisited brittle fracture
abstract: The numerical implementation of the model of brittle fracture developed in Francfort and Marigo (1998.  (8), 1319–1342) is presented. Various computational methods based on variational approximations of the original functional are proposed. They are tested on several antiplanar and planar examples that are beyond the reach of the classical computational tools of fracture mechanics.

title: Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments
abstract: This paper presents a modified regularized formulation of the Ambrosio–Tortorelli type to introduce the crack non-interpenetration condition in the variational approach to fracture mechanics proposed by Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem.  We focus on the linear elastic case where the contact condition appears as a local unilateral constraint on the displacement jump at the crack surfaces. The regularized model is obtained by splitting the strain energy in a spherical and a deviatoric parts and accounting for the sign of the local volume change. The numerical implementation is based on a standard finite element discretization and on the adaptation of an alternate minimization algorithm used in previous works. The new regularization avoids crack interpenetration and predicts asymmetric results in traction and in compression. Even though we do not exhibit any gamma-convergence proof toward the desired limit behavior, we illustrate through several numerical case studies the pertinence of the new model in comparison to other approaches.

title: A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits
abstract: The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. Following our recent work [C. Miehe, F. Welschinger, M. Hofacker, Thermodynamically-consistent phase field models of fracture: Variational principles and multi-field fe implementations,  Engineering DOI:10.1002/nme.2861] on phase-field-type fracture, we propose in this paper a new variational framework for rate-independent diffusive fracture that bases on the introduction of a local history field. It contains a maximum reference energy obtained in the deformation history, which may be considered as a measure for the maximum tensile strain obtained in history. It is shown that this local variable drives the evolution of the crack phase field. The introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology. In particular, it allows for the construction of a new algorithmic treatment of diffusive fracture. Here, we propose an extremely robust operator split scheme that successively updates in a typical time step the history field, the crack phase field and finally the displacement field. A regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added. The proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples.

title: Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations
abstract: The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase-field. In this paper, we outline a thermodynamically consistent framework for phase-field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that Γ-converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase-field. Here, we propose alternative rate-independent and viscous over-force models that ensure the local growth of the phase-field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase-field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase-field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase-field formulations of fracture by means of representative numerical examples.

title: Fractographic aspects of crack branching instability using a phase-field model.
abstract: A phase-field model of a crack front propagating in a three-dimensional brittle material is used to study the fractographic patterns induced by the branching instability. The numerical results of this model give rise to crack surfaces that are similar to those obtained in various experimental situations. Depending on applied loading configurations and initial conditions, we show that the branching instability is either restricted to a portion of the crack front or revealed through quasi-two-dimensional branches. For the former, the crack front leaves on the main broken surface either aligned or disordered parabolic marks. For the latter, fractography reveals the so called échelon cracks showing that branching instability can also induce crack front fragmentation.

title: Simulation of damage evolution in composites : A phase-field model
abstract: In this study, a phase-field model is introduced for the simulation of damage evolution in composite systems. The damage evolution in discontinuously reinforced and laminated composites, including elastic–plastic cases, was studied parametrically in order to establish its viability. The algorithm offers significant advantages to describe the microstructure and topological changes associated with the damage evolution in comparison to conventional simulation algorithms, due to the absence of formal meshing. The known damage mechanisms for specific composite systems naturally evolve from the algorithm with a simple evolution equation.

title: A continuum phase field model for fracture
abstract: A phase field model based on a regularized version of the variational formulation of brittle fracture is introduced. The influences of the regularization parameter that controls the interface width between broken and undamaged material and of the mobility constant of the evolution equation are studied in finite element simulations. A generalized Eshelby tensor is derived and analyzed for mode I loading in order to evaluate the energy release rate of the diffuse phase field cracks. The numerical implementation is performed with finite elements and an implicit time integration scheme. The configurational forces are computed in a postprocessing step after the coupled problem of mechanical balance equations and the evolution equation is solved. Some of the numerical results are compared to analytical results from classical Griffith theory.

title: Phase field approximation of dynamic brittle fracture
abstract: Numerical methods that are able to predict the failure of technical structures due to fracture are important in many engineering applications. One of these approaches, the so-called phase field method, represents cracks by means of an additional continuous field variable. This strategy avoids some of the main drawbacks of a sharp interface description of cracks. For example, it is not necessary to track or model crack faces explicitly, which allows a simple algorithmic treatment. The phase field model for brittle fracture presented in Kuhn and Müller ( 2010) assumes quasi-static loading conditions. However dynamic effects have a great impact on the crack growth in many practical applications. Therefore this investigation presents an extension of the quasi-static phase field model for fracture from Kuhn and Müller ( 2010) to the dynamic case. First of all Hamilton’s principle is applied to derive a coupled set of Euler-Lagrange equations that govern the mechanical behaviour of the body as well as the crack growth. Subsequently the model is implemented in a finite element scheme which allows to solve several test problems numerically. The numerical examples illustrate the capabilities of the developed approach to dynamic fracture in brittle materials.

title: On numerical aspects of phase field fracture modelling
abstract: Recently, a continuum model for crack propagation in brittle visco-elastic materials has been presented (M. Fleck et al.,  (2011)), in which fracture is described as an elastically induced nonequilibrium interfacial pattern formation process. The underlying mesoscopic description of a propagating crack results in a complicated moving boundary problem, that allows the self-consistent determination of the crack velocity as well as the entire time-dependent crack surface. Here we focus on the numerical aspects tied to the approximative solution of the arising moving boundary problem and discuss several strategies to develop efficient numerical schemes applying to it: i) a finite difference scheme, ii) a scheme based on the finite element method and iii) a scheme based on the multipole expansion method. The pros and cons of each method are elucidated in detail. Further, we present a comprehensive view on how the different approaches can be employed to validate each other in comparative studies, which is then illustrated for the example of crack propagation under mode I loading.

title: A phase-field description of dynamic brittle fracture
abstract: In contrast to discrete descriptions of fracture, phase-field descriptions do not require numerical tracking of discontinuities in the displacement field. This greatly reduces implementation complexity. In this work, we extend a phase-field model for quasi-static brittle fracture to the dynamic case. We introduce a phase-field approximation to the Lagrangian for discrete fracture problems and derive the coupled system of equations that govern the motion of the body and evolution of the phase-field. We study the behavior of the model in one dimension and show how it influences material properties. For the temporal discretization of the equations of motion, we present both a monolithic and staggered time integration scheme. We study the behavior of the dynamic model by performing a number of two and three dimensional numerical experiments. We also introduce a local adaptive refinement strategy and study its performance in the context of locally refined T-splines. We show that the combination of the phase-field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions.

title: A geometrically nonlinear phase field theory of brittle fracture
abstract: Phase field theory is developed for solids undergoing potentially large deformation and fracture. The elastic potential depends on a finite measure of elastic strain. Surface energy associated with fracture can be anisotropic, enabling description of preferred cleavage planes in single crystals, or isotropic, applicable to amorphous solids such as glass. Incremental solution of the Euler–Lagrange equations corresponds to local minimization of an energy functional for the solid, enabling prediction of equilibrium crack morphologies. Predictions are in close agreement with analytical solutions for pure mode I or pure mode II loading, including the driving force for a crack to extend from a pre-existing plane onto a misoriented cleavage plane. In an isotropic matrix, the tendency for a crack to penetrate or deflect around an inclusion is shown to depend moderately on the ratio of elastic stiffness in matrix and inclusion and strongly on their ratio of surface energy. Cracks are attracted to (shielded by) inclusions softer (stiffer) than the surrounding matrix. The theory and results apparently report the first fully three-dimensional implementation of phase field theory of fracture accounting for simultaneous geometric nonlinearity, nonlinear elasticity, and surface energy anisotropy.

title: On three-dimensional modelling of crack growth using partition of unity methods
abstract: This paper reviews different crack tracking techniques in three-dimensions applicable in the context of partition of unity methods, especially meshfree methods. Issues such as describing and tracking the crack surface are addressed. A crack tracking procedure is proposed in detail and implemented in the context of the extended element-free Galerkin method (XEFG). Several three-dimensional cracking examples are compared to other results from the literature or the experimental data and show good agreement.

title: Phase field modeling of crack propagation
abstract: Fracture is a fundamental mechanism of materials failure. Propagating cracks can exhibit a rich dynamical behavior controlled by a subtle interplay between microscopic failure processes in the crack tip region and macroscopic elasticity. We review recent approaches to understand crack dynamics using the phase field method. This method, developed originally for phase transformations, has the well-known advantage of avoiding explicit front tracking by making material interfaces spatially diffuse. In a fracture context, this method is able to capture both the short-scale physics of failure and macroscopic linear elasticity within a self-consistent set of equations that can be simulated on experimentally relevant length and time scales. We discuss the relevance of different models, which stem from continuum field descriptions of brittle materials and crystals, to address questions concerning crack path selection and branching instabilities, as well as models that are based on mesoscale concepts for crack tip scale selection. Open questions which may be addressed using phase field models of fracture are summarized.

title: Phase-field models for brittle and cohesive fracture
abstract: In this paper we first recapitulate some basic notions of brittle and cohesive fracture models, as well as the phase-field approximation to fracture. Next, a critical assessment is made of the sensitivity of the phase-field approach to brittle fracture, in particular the degradation function, and the use of monolithic versus partitioned solution schemes. The last part of the paper makes extensions to a recently developed phase-field model for cohesive fracture, in particular for propagating cracks. Using some simple examples the current state of the cohesive phase-field model is shown.

title: Three-dimensional brittle fracture : configurational-force-driven crack propagation
abstract: This paper presents a computational framework for quasi-static brittle fracture in three-dimensional solids. The paper sets out the theoretical basis for determining the initiation and direction of propagating cracks based on the concept of configurational mechanics, consistent with Griffith's theory. Resolution of the propagating crack by the FEM is achieved by restricting cracks to element faces and adapting the mesh to align it with the predicted crack direction. A local mesh improvement procedure is developed to maximise mesh quality in order to improve both accuracy and solution robustness and to remove the influence of the initial mesh on the direction of propagating cracks. An arc-length control technique is derived to enable the dissipative load path to be traced. A hierarchical hp-refinement strategy is implemented in order to improve both the approximation of displacements and crack geometry. The performance of this modelling approach is demonstrated on two numerical examples that qualitatively illustrate its ability to predict complex crack paths. All problems are three-dimensional, including a torsion problem that results in the accurate prediction of a doubly-curved crack.

title: A computational framework of three-dimensional configurational-force-driven brittle crack propagation
abstract: We consider a variational formulation of quasi-static brittle fracture and develop a new finite-element-based computational framework for propagation of cracks in three-dimensional bodies. We outline a consistent thermodynamical framework for crack propagation in elastic solids and show that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius–Planck inequality in the sense of Coleman’s method. Consequently, the crack propagation direction associated with the classical Griffith criterion is identified by the material configurational force which maximizes the local dissipation at the crack front. The variational formulation is realized numerically by a standard spatial discretization with finite elements which yields a discrete formulation of the global dissipation in terms configurational nodal forces. Therefore, the constitutive setting of crack propagation in the space-discretized finite element context is naturally related to discrete nodes of a typical finite element mesh. In a consistent way with the node-based setting, the discretization of the evolving crack discontinuity is performed by the doubling of critical nodes and interface facets of the mesh. The crucial step for the success of this procedure is its embedding into an r -adaptive crack-facet reorientation procedure based on configurational-force-based indicators in conjunction with crack front constraints. We propose a staggered solution procedure that results in a sequence of positive definite discrete subproblems with successively decreasing overall stiffness, providing a robust algorithmic setting in the postcritical range. The predictive capabilitiy of the proposed formulation is demonstrated by means of representative numerical simulations.

title: A robust algorithm for configurational-force-driven brittle crack propagation with R-adaptive mesh alignment
abstract: The paper considers a variational formulation of brittle fracture in elastic solids and proposes a numerical implementation by a finite element method. On the theoretical side, we outline a consistent thermodynamic framework for crack propagation in an elastic solid. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius–Planck inequality in the sense of Coleman's method. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip and minimizes the incremental energy release. On the numerical side, we exploit this variational structure in terms of crack-driving configurational forces. First, a standard finite element discretization in space yields a discrete formulation of the global dissipation in terms configurational nodal forces. As a consequence, the constitutive setting of crack propagation in the space-discretized finite element context is naturally related to discrete nodes of a typical finite element mesh. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity is performed by the doubling of critical nodes and interface segments of the mesh. Critical for the success of this procedure is its embedding into an r-adaptive crack-segment reorientation procedure with configurational-force-based directional indicator. Here, successive crack releases appear in discrete steps associated with the given space discretization. These are performed by a staggered loading–release algorithm of energy minimization at frozen crack state followed by the successive crack releases at frozen deformation. This constitutes a sequence of positive-definite discrete subproblems with successively decreasing overall stiffness, providing an extremely robust algorithmic setting in the postcritical range. We demonstrate the performance of the formulation by means of representative numerical simulations.

title: A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization
abstract: A variational formulation of quasi-static brittle fracture in elastic solids at small strains is proposed and an associated finite element implementation is presented. On the theoretical side, a consistent thermodynamic framework for brittle crack propagation is outlined. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius–Planck inequality. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip. On the numerical side, we first consider a standard finite element discretization in the two-dimensional space which yields a discrete formulation of the global dissipation in terms of configurational nodal forces. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity for two-dimensional problems is performed by the doubling of critical nodes and interface segments of the mesh. A crucial step for the success of this procedure is its embedding into a r-adaptive crack-segment re-orientation algorithm governed by configurational-force-based directional indicators. Here, successive crack propagation is performed by a staggered loading-release algorithm of energy minimization at frozen crack state followed by nodal releases at frozen deformation. We compare results obtained by the proposed formulation with other crack propagation criteria. The computational method proposed is extremely robust and shows an excellent performance for representative numerical simulations.

title: Mechanics in Material Space: with Applications to Defect and Fracture Mechanics
abstract: 1 Mathematical Preliminaries.- 1.1 General Remarks.- 1.2 What is a Conservation Law?.- 1.3 Trivial Conservation Laws.- 1.4 System with a Lagrangian Noether's Method.- 1.5 System without a Lagrangian Neutral-Action Method.- 1.6 Discussion.- 2 Linear Theory of Elasticity.- 2.1 General Remarks.- 2.2 Elements of Linear Elasticity.- 2.3 Conservation Laws of Linear Elastostatics.- 2.4 Alternative Derivations of Conservation Laws.- 3 Properties of the Eshelby Tensor.- 3.1 General Remarks 81.- 3.2 Physical Interpretation of the Components of the Eshelby Tensor.- 3.3 Invariants, Principal Values, Principal Directions and Extremal Values of the Eshelby Tensor.- 4 Linear Elasticity with Defects.- 4.1 General Remarks.- 4.2 Path-Independent Integrals and Energy-Release Rates.- 4.3 Example: Hole-Dislocation Interaction.- 4.4 Path-Independent Integrals of Fracture Mechanics.- 5 Inhomogeneous Elastostatics.- 5.1 General Remarks.- 5.2 Symmetry Transformations.- 5.3 The Homogeneous Case.- 5.4 The Inhomogeneous Case.- 5.5 Relation to Stress-Intensity Factors.- 5.6  6 Elastodynamics.- 6.1 General Remarks.- 6.2 Time t as an Additional Independent Variable.- 6.3 Convolution in Time.- 6.4 Domain-Independent Integrals.- 6.5 Energy-Release Rates.- 6.6 Wave Motion.- 7 Dissipative Systems.- 7.1 General Remarks.- 7.2 Diffusion Equation.- 7.3 Non-Linear Wave Equation.- 7.4 Viscoelasticity.- 8 Coupled Fields.- 8.1 General Remarks.- 8.2 Piezoelectricity.- 8.3 Thermoelasticity.- 8.4 Mechanics of a Porous Medium.- 9 Bars, Shafts and Beams.- 9.1 General Remarks.- 9.2 Elements of Strength-of-Materials.- 9.3 Balance and Conservation Laws for Bars and Shafts.- 9.4 Balance and Conservation Laws for Beams.- 9.5 Energy-Release Rates and Stress-Intensity Factors.- 9.6 Examples.- 10 Plates and Shells.- 10.1 General Remarks.- 10.2 Plate Theories.- 10.3 Conservation Laws for Elastostatics of Mindlin Plates.- 10.4 Reduction to the Classical Theory.- 10.5 Conservation Laws for Shells.- Appendix A.- Conservation Laws for Inhomogeneous Bars under Arbitrary Axial Loading.- Appendix B.- B.1 Elastodynamics of Inhomogeneous Bernoulli-Euler Beams.- B.2 Reduction to

title: Pseudomomentum and material forces in nonlinear elasticity: variational formulations and application to brittle fracture
abstract: SummaryThis work examines critically the various formulations of the balance of linear momentum innonlinear inhomogeneous elasticity. The corresponding variational formulations are presented. From the point of view of the theory of elastic inhomogeneities, the most interesting formulations are those which, being either completely material or mixed-Eulerian, exhibit explicitly the inhomogeneities in the form ofmaterial forces. They correspond to the balance ofpseudomomentum, a material covector which is seldom used but which we show to play a fundamental role in the Hamiltonian canonical formulation of nonlinear elasticity. The flux associated with pseudomomentum is none other than theEshelby material tensor. Applying this formulation to the case of an elastic body containing a crack of finite extent, the notion of suction force acting at the tip of the crack follows while afracture criterion à la Griffith can be deduced from a variational inequality. Possible extensions to higher-grade elastic materials and inelastic materials are indicated as well as the role played by pseudomomentum in the quantization of elastic vibrations.

title: Variational principles of nonlinear fracture mechanics
abstract: SummaryA variational principle of total energy is formulated for finite strain statics of a hyperelastic body whose initial configuration contains a gap. From this principle statical equations and boundary conditions for the gapped body are derived. The equilibrium condition at a gap tip is associated with the well-knownJ-integrals. By including the reaction of inertia and the flux of kinetic energy the principle of total energy is transformed to the variational inequality of evolution for dynamics of a hyperelastic body with a propagating crack. A closed system of dynamical equations, boundary conditions and additional conditions on the unknown contact crack surfaces and crack tip is obtained. As example the antiplane shear of an infinite gapped body is considered.

title: Studies in elastic fracture mechanics based on the material force method
abstract: The object of this work is to discuss a further improvement of the material force method for non-linear hyperelastostatic fracture mechanics. We investigate the accuracy of the material force method within a 'modified boundary layer'-formulation using a Ramberg-Osgood material type for the sake of comparison. The proposed improvement leads to a reliable and very accurate method to compute the vectorial J-integral in fracture mechanics.

title: Adaptive strategies and error control for computing material forces in fracture mechanics
abstract: The concept of material forces pertains to a variation of the inverse motion map while the placement field is kept fixed. From the weak formulation of the self-equilibrating Eshelby (material) stress tensor it turns out that the classical J-integral formulations in fracture mechanics are just special cases due to the choice of particular weight functions. In this contribution, we discuss a posteriori error control of the material forces as part of an adaptive strategy to reduce the discretization error to an acceptable level. The data of the dual problem involves the quite non-conventional tangent stiffness of the (material) Eshelby stress tensor with respect to a variation of the (physical) strain field. The suggested strategy is applied to the common fracture mechanics problem of a single-edged crack, whereby different strategies for computing the J-integral are compared. We also consider the case in which the crack edges are not parallel, i.e. a notch.

title: On Configurational forces in the context of the finite element method
abstract: The theory of configurational forces is briefly recast, together with the underlying balance laws. It is shown, that in the case of a homogeneous body without body forces the additional balance laws are identically satisfied if the standard force balance holds. In approximate solutions, for example obtained by finite elements, the equilibrium is not satisfied exactly, thus configurational forces occur on discretization nodes. An implementation of the configurational force balance into the finite element scheme is presented. The use of configurational forces is discussed with three main aspects. It is demonstrated how configurational forces can be used to check and to improve the finite element solution. Examples from fracture mechanics and problems with material inhomogeneities are discussed.

title: Application of material forces to hyperelastostatic fracture mechanics. II. Computational setting
abstract: The concern of this work is a novel algorithmic treatment of hyperelastostatic fracture mechanics problems consistent to the notion of material forces within the geometrically nonlinear setting of continuum mechanics. To this end, we consider the continuum mechanics of material forces, as outlined in Part I of this work (P. Steinmann, ), which act, contrary to the common physical forces, on the material manifold or rather in the material space. In the sequel it is proposed to discretize the corresponding quasi-static balance of pseudo momentum by a standard Galerkin finite element procedure. As a result we obtain global discrete node point quantities, the material node point forces, which prove to be of the same qualitative and quantitative importance for the assessment of fracture mechanics problems as the classical J -integral.

title: On material forces and finite element discretizations
abstract: The idea of using material forces also termed configurational forces in a computational setting is presented. The theory of material forces is briefly recast in the terms of a non-linear elastic solid. It is shown, how in a computational setting with finite elements (FE) the discrete configurational forces are calculated once the classical field equations are solved. This post-process calculation is performed in a way, which is consistent with the approximation of the classical field equations. Possible physical meanings of this configurational forces are discussed. A purely computational aspect of material forces is pointed out, where material forces act as an indicator to obtain softer discretizations.

title: Theory and numerics for finite deformation fracture modelling using strong discontinuities
abstract: A general finite element approach for the modelling of fracture is presented for the geometrically non-linear case. The kinematical representation is based on a strong discontinuity formulation in line with the concept of partition of unity for finite elements. Thus, the deformation map is defined in terms of one continuous and one discontinuous portion, considered as mutually independent, giving rise to a weak formulation of the equilibrium consisting of two coupled equations. In addition, two different fracture criteria are considered. Firstly, a principle stress criterion in terms of the material Mandel stress in conjunction with a material cohesive zone law, relating the cohesive Mandel traction to a material displacement ‘jump’ associated with the direct discontinuity. Secondly, a criterion of Griffith type is formulated in terms of the material-crack-driving force (MCDF) with the crack propagation direction determined by the direction of the force, corresponding to the direction of maximum energy release. Apart from the material modelling, the numerical treatment and aspects of computational implementation of the proposed approach is also thoroughly discussed and the paper is concluded with a few numerical examples illustrating the capabilities of the proposed approach and the connection between the two fracture criteria.

title: A framework for fracture modelling based on the material forces concept with XFEM kinematics
abstract: A theoretical and computational framework which covers both linear and non-linear fracture behaviour is presented. As a basis for the formulation, we use the material forces concept due to the close relation between on one hand the Eshelby energy–momentum tensor and on the other hand material defects like cracks and material inhomogeneities. By separating the discontinuous displacement from the continuous counterpart in line with the eXtended finite element method (XFEM), we are able to formulate the weak equilibrium in two coupled problems representing the total deformation. However, in contrast to standard XFEM, where the direct motion discontinuity is used to model the crack, we rather formulate an inverse motion discontinuity to model crack development. The resulting formulation thus couples the continuous direct motion to the inverse discontinuous motion, which may be used to simulate linear as well as non-linear fracture in one and the same formulation. In fact, the linear fracture formulation can be retrieved from the non-linear cohesive zone formulation simply by confining the cohesive zone to the crack tip. These features are clarified in the two numerical examples which conclude the paper.

title: On the numerical modelling of quasi-static crack growth in linear elastic fracture mechanics
abstract: In this paper we present a strategy for the simulation of a propagating crack under mixed mode linear elastic conditions using a discontinuous finite element method. A key issue is to accurately compute the incremental change in the kink angle of the propagating crack during subsequent steps. We have chosen to work with a domain formulation of the material force vector as a criteria for the propagation direction. We describe the theoretical background together with the numerical implementation in detail and show some results for different loading conditions.

title: On energetic changes due to configurational motion of standard continua
abstract: The appropriate thermodynamic setting for the combined configurational and deformational motion of standard continua is discussed in this paper. A key ingredient is an absolute (fixed) reference configuration, relative to which configurational and deformational changes (rates) with respect to the material (undeformed) and spatial (deformed) configurations, respectively, can be described in a unified fashion. In particular, we are interested in formulating the local as well as global measures of the energy dissipation due to configurational changes of a given physical system. It is believed that the presentation in this paper provides the following advantages: a unified kinematic and thermodynamic setting of the configurational and deformational motions, in particular the generic balance law accounting for configurational flux, increases the general clarity; the separation of the total dissipation in terms of the intrinsic change in elastic energy and in terms of the material dissipation that is induced by configurational changes becomes transparent. All results are obtained without any restrictions on dynamics, thermomechanical couplings, etc.

title: Fracture mechanical behaviour of visco-elastic materials: application to the so-called dwell-effect
abstract: The material force approach is an efficient, elegant, and accepted means to compute the J-integral as a fracture mechanical parameter for elastic and inelastic materials. With the formulation of a multiplicative split of the deformation gradient at hand, rate-dependent (visco-elastic) materials described for example by the physically based Bergstrom-Boyce model can be investigated. For these investigations, the so-called material volume forces have to be computed in order to separate the driving forces acting on the visco-elastic zone around the crack tip from the driving forces acting on the crack tip itself, representing the crack driving force. To illustrate the effectiveness of this approach, the so-called dwell-effect of elastomeric materials is investigated.

title: Material forces for inelastic models at large strains: application to fracture mechanics
abstract: Elastomeric materials show a wide range of different elastic and inelastic properties. Additionally, this class of materials is subjected to large deformations. Considering all these effects, fracture mechanical investigations are very challenging tasks and cannot be performed with standard approaches. Effects of inhomogeneities and discontinuities such as cracks can be investigated with the so-called material force approach in an efficient and elegant way. For comprehensive investigations of inelastic materials, the complete balance of the material motion problem has to be formulated. In this case, the material volume forces depend on the internal history variables which are required for the inelastic constitutive model. This paper derives a general formulation for rate-dependent and rate-independent inelastic materials based on a multiplicative split of the deformation gradient to cover viscoelastic and elastoplastic materials at finite deformations.

title: Evaluation of energy contributions in elasto-plastic fracture: A review of the configurational force approach
abstract: This paper discusses the role of the material or rather configurational force approach in elastic–plastic materials with a pre-cracked configuration and gives an overview of some recent numerical investigations of the crack tip field. On the theoretical side, a consistent thermodynamic framework for the combined configurational and deformational motion in elasto-plastic continua at small and finite strains is discussed. Furthermore, the study researches the separation of the total dissipation in terms of the change in elastic energy and in terms of the material dissipation by a configurational change obtained from the global energy momentum balance. On the numerical side, an equivalent general expression of the vectorial material forces is derived from the weak form of the energy momentum balance. For the sake of simplicity, all results are obtained neglecting dynamic and thermo-mechanical phenomena. The computations are applied to a stationary crack in a circular pre-cracked domain and a compact tension specimen under plasticity condition. The results show that the material force approach remains path independent only if all components of the momentum balance equation are properly included into the corresponding variational formulation. In addition, the cohesive fracture theory is combined with the material force approach in order to increase the clarity of the interpretation of the approach in engineering applications. Correspondingly, the results obtained from the compact tension specimen with three different free energy functions are compared to the conventional J-integral method and to experimental results available from a previous study. The contributions of this study are threefold. First, the path dependency of the material force approach in elasto-plastic continua is found to be considerably depending on the so-called material body forces. Secondly, interpretation of the induced material dissipation forces in the definition of the crack driving forces is not explicitly clear but they play an important role in case of path independency. Finally, with further analyses on compact tension examples, it is shown that the introduced energy functions in the material momentum balance yield a difference for the evaluation of the material force approach and the traditional J-integral.

title: On a solution strategy for energy-based mesh optimization in finite hyperelastostatics
abstract: The objective of this work is the development of a numerical solution strategy for energy-based mesh optimization in finite hyperelastostatics. In finite element computations that rely on the principle of minimum potential energy, the variational principle itself provides the basis for r-adaptive methods. The numerical solution can be improved by further minimizing the discrete potential energy with respect to the material node point positions. In this paper, we regard the mesh optimization as a nonlinear minimization problem with equality and inequality constraints. The equality constraints correspond to the spatial equilibrium condition, whereas the inequality constraints are given by the natural restriction that material elements with a negative volume (Jacobian) are inadmissible. Based on this interpretation, we develop a stable numerical solution strategy in which two approaches of nonlinear programming are combined. Applying a barrier method, the minimization problem is transformed into a sequence of problems without inequality constraints. Each problem of the sequence is solved by means of a Newton scheme that operates on the constrained surface given by the spatial equilibrium condition.

title: NUMERICAL MODELLING OF DISCONTINUA
abstract: Discrete element methods are numerical procedures for simulating the complete behaviour of systems of discrete, interacting bodies. Three important aspects of discrete element programs are examined: (1) the representation of contacts; (2) the representation of solid material; and (3) the scheme used to detect and revise the set of contacts. A proposal is made to define what constitutes a discrete element program, and four classes of such programs are described: the distinct element method, modal methods, discontinuous deformation analysis and the momentum‐exchange method. Several applications and examples are presented, and a list is given of suggestions for future developments.

title: A bonded-particle model for rock
abstract: A numerical model for rock is proposed in which the rock is represented by a dense packing of non-uniform-sized circular or spherical particles that are bonded together at their contact points and whose mechanical behavior is simulated by the distinct-element method using the two- and three-dimensional discontinuum programs PFC2D and PFC3D. The microproperties consist of stiffness and strength parameters for the particles and the bonds. Damage is represented explicitly as broken bonds, which form and coalesce into macroscopic fractures when load is applied. The model reproduces many features of rock behavior, including elasticity, fracturing, acoustic emission, damage accumulation producing material anisotropy, hysteresis, dilation, post-peak softening and strength increase with confinement. These behaviors are emergent properties of the model that arise from a relatively simple set of microproperties. A material-genesis procedure and microproperties to represent Lac du Bonnet granite are presented. The behavior of this model is described for two- and three-dimensional biaxial, triaxial and Brazilian tests and for two-dimensional tunnel simulations in which breakout notches form in the region of maximum compressive stress. The sensitivity of the results to microproperties, including particle size, is investigated. Particle size is not a free parameter that only controls resolution; instead, it affects the fracture toughness and thereby influences damage processes (such as notch formation) in which damage localizes at macrofracture tips experiencing extensile loading.

title: Simulation of blasting induced rock motion using spherical element models
abstract: Control of the rock motion associated with blasting can have significant economic benefits. For example, surface coal mining can be made more efficient if the overburden material can be cast further with explosives, leaving less work for mechanical equipment. The final muck pile shape in every type of surface and underground blasting is controlled by the blasting induced motion of the rock. A theoretically sound method of predicting rock motion will be beneficial to understanding the blasting process.

title: An alternating digital tree (ADT) algorithm for 3D geometric searching and intersection problems
abstract: A searching algorithm is presented for determining which members of a set of n points in an N dimensional space lie inside a prescribed space subregion. The algorithm is then extended to handle finite size objects as well as points. In this form it is capable of solving problems such as that of finding the objects from a given set which intersect with a prescribed object. The suitability of the algorithm is demonstrated for the problem of three dimensional unstructured mesh generation using the advancing front method.

title: A fast contact detection algorithm insensitive to object sizes
abstract: Presents a new contact detection algorithm based on double‐ended spatial sorting (DESS) that is insensitive to variations in object size. It was developed to address the problems that arise when objects with non‐spherical geometry and non‐uniform sizes are simulated using discrete element techniques. The algorithm is applicable to general spatial reasoning problems. While techniques based on spatial hashing (sometimes called bining methods) perform well for objects of similar size, they degrade significantly when the objects vary in size. The DESS algorithm overcomes this problem by using a spatial sorting technique applied to both ends of the object’s projection along each orthogonal axis. Discrete element test simulations comparing DESS and spatial hashing (NBS) are detailed. The results demonstrate that when object sizes vary significantly (size ratios greater than 8:1), DESS outperforms NBS up to around 100,000 objects. It is noted, however, that the superior scaling properties of NBS will always outperform DESS for some large numbers of objects.

title: Influence of the Main Contact Parameters in Finite Element Analysis of Elastic Bodies in Contact
abstract: One of the key issues in solving contact problems is the correct choice of the contact parameters. The contact stiffness, the penetration limit and the contact algorithm are some of the parameters that have to be adjusted. There are no methodologies available in the literature for choosing the contact parameters, relying only on the user experience for this important task. In this work we investigate how the contact parameters behave in a commercial finite element analysis software. We will show that while the contact stiffness has great influence on the finite element analysis, other parameters will not affect it significantly. Some contact examples are shown to illustrate the performance of the contact parameters during the solution of a contact problem.

title: A new discrete fracture modelling approach for rock masses
abstract: A method for modelling discrete fracture in rock masses under tensile and compressive stress fields is presented, based on a Mohr–Coulomb failure surface in compression and three independent anisotropic rotating crack models in tension. Extension fracturing is modelled by coupling the softening of the anisotropic rotating crack failure criterion to the compressive plastic strain evolution. An explicitly time-integrated coupled discrete element/finite element approach is employed with an explicit Lagrangian contact algorithm to enforce non-penetration of the surfaces created when the tensile strength is depleted. The geomechanical model is applied to naturally fractured rock masses, which are characterised by field mapping and borehole data integrated in a stochastic 3D discrete fracture network model. This approach maximises the utility of the field data and provides a direct approach to the determination of rock mass strength and deformability. It also provides a realistic insight into complex failure me...

title: The stability of a laminated Voussoir beam: Back analysis of a historic roof collapse using DDA
abstract: The stability of a horizontally bedded and vertically jointed roof, referred to here as a laminated Voussoir beam, is studied using careful documentation of a historic roof collapse, which occurred in an ancient underground water storage reservoir dated back to ca. 1000 B.C. The roof of the opening failed immediately after the excavation leaving a dome shaped loosened zone, with a span of 7 m and height of 2.5 m, consisting of horizontal beds and vertical joints with mean spacing of 50 and 25 cm, respectively. The ancient engineers erected a massive pillar at the center of the dome in order to passively support the failed roof and the opening remained in service for several hundred years following the failure. Analysis of the roof using iterative Voussoir beam procedure [Beer, G. and Meek, J. L., Design curves for roofs and hanging walls in bedded rock based on Voussoir beam and plate solutions.  1982, 91 , A18–22.] shows that the roof was more sensitive to failure by shear along the abutments rather than by crushing at hinge zones and that the required friction angle ( φ req. ) for stability would have been 36°. The available friction angle is estimated between 38.6° and 46.4° and, therefore, the result of the iterative solution is considered unconservatively wrong. Results of DDA [Shi, G. -H. and Goodman, R. E., Two-dimensional discontinuous deformation analysis.   1989, 13 , 359–380; Shi, G. -H., Block System Modeling by Discontinuous Deformation Analysis . Computational Mechanics Publications, Southampton, 1993, ] however indicate that φ req. must have been greater than 60°, a shear strength which was not available at the time of construction, thus the immediate failure. Using DDA we further demonstrate that φ req. is related to joint spacing (or block length) in a parabolic function: with increasing joint spacing φ req. decreases to a minimum value and than increases with further increase in joint spacing. This result is attributed to the interaction of two competing forces: one is the stabilizing axial thrust which increases with increasing moment arm in individual blocks (a function of joint spacing or block length), the other is the destabilizing vertical load which increases with increasing weight of overlying blocks.

title: Stability of shallow karstic caverns in blocky rock masses
abstract: The limiting relationship is explored between underground opening span and required cover height for stability in blocky rock masses characterized by a network of horizontal bedding planes and vertical joints. Understanding this relationship is crucial for the design of mining excavations in karstic terrain as typically encountered in carbonate rock masses. We perform numerical analysis of multiple roof spans vs. cover height geometries using the DDA method to obtain the boundary curve between stable and unstable opening geometries. Our results indicate that for cavern spans of up to 18 m a low cover height vs. opening span ratio of h/B =0.33 is sufficient for stability. For spans greater than 18 m the demand for cover height rapidly increases and it appears to stabilize at h/B =1.0 for B =26 m and above. To validate our numerical analysis results, a unique case study is analyzed wherein a 40 m span karstic cavern, the Ayalon cave, has been preserved below an active open pit mine in central Israel with cover height of only 30 m, thus rendering the cave marginally stable according to our model prediction. Indeed there is ample evidence of partial collapse of the roof in the cave. The predictive capability of our model is further confirmed using two additional case studies in blocky rock masses, each of which possesses very different mechanical parameters such as intact rock strength, density, and deformability, suggesting that our model predictions remain valid as long the rock mass maintains a “blocky” structural configuration.

title: Review of validation of the discontinuous deformation analysis (DDA) method
abstract: Over the last decade, researchers in the discontinuous deformation analysis (DDA) community have dedicated a great deal of effort to document the accuracy of the method by performing validation studies. This paper contains a summary of more than 100 published and unpublished validation studies which comprise the body of DDA validation information to which the authors have access. The studies are grouped into three general categories: (a) validation with respect to analytical solutions, (b) validation with respect to results of other numerical techniques, and (c) validation with respect to laboratory and field data. Three general techniques for validation are described: qualitative assessment visually examining runtime behaviour of simulations, semi-quantitative assessment comparing numerical results of simulations, and quantitative where numerical simulation results are evaluated in detail with respect to similar analytical, laboratory or field results. We find that for many of the problems addressed by the papers in this review, DDA performs more than adequately for engineering analysis.

title: Tunnel roof deflection in blocky rock masses as a function of joint spacing and friction - A parametric study using discontinuous deformation analysis (DDA)
abstract: The stability of underground openings excavated in a blocky rock mass was studied using the discontinuous deformation analysis (DDA) method. The focus of the research was a kinematical analysis of the rock deformation as a function of joint spacing and friction. Two different opening geometries were studied: (1) span B  =  h t ; (2) B  = 1.5 h t ; where the opening height was h t  = 10 m for both configurations. Fifty individual simulations were performed for different values of joint spacing and friction angle. It was found that the extent of loosening above the excavation was predominantly controlled by the spacing of the joints, and only secondarily by the shear strength. The height of the loosening zone h r was found to be dependent upon the ratio between joint spacing and excavation span S j / B : (1) h r B for S j / B  ⩽ 2/10; (2) stable arching within the rock mass for S j / B  ⩾ 3/10. The results of this study provide explicit correlation between geometrical features of the rock mass, routinely collected during site investigation and excavation, and the expected extent of the loosening zone at the roof, which determines the required support.

title: Combined three-dimensional lattice Boltzmann method and discrete element method for modelling fluid-particle interactions with experimental assessment
abstract: A general algorithmic framework is established in this paper for numerical simulations of three-dimensional fluid–particle interaction problems with a large number of moving particles in turbulent flows using a combined lattice Boltzmann method (LBM) and discrete element method (DEM). In this approach, the fluid field is solved by the extended three-dimensional LBM with the incorporation of the Smagorinsky turbulence model, while particle interactions are modelled by the DEM. The hydrodynamic interactions between fluid and particles are realized through the extension of an existing two-dimensional fluid–particle hydrodynamic interaction scheme. The main computational aspects comprise the lattice Boltzmann formulation for the solution of fluid flows, the incorporation of a large eddy simulation-based turbulence model within the framework of the three-dimensional LBM for turbulent flows, the moving boundary condition for hydrodynamic interactions between fluid and moving particles, and the discrete element modelling of particle-particle interactions. To assess the solution accuracy of the proposed approach, a much simplified laboratory model of vacuum dredging systems for mineral recovery is employed. The numerical results are compared with the experimental data available. It shows that the overall correspondence between numerical results and experimental measurements is good and thus indicates, to a certain extent, the solution accuracy of the proposed methodology.

title: YADE‐OPEN DEM: an open‐source software using a discrete element method to simulate granular material
abstract: YADE‐OPEN DEM is an open‐source software based on the discrete element method, (DEM) which uses object oriented programming techniques. The purpose of this paper is to describe the software architecture. The DEM chosen uses position, orientation, velocity and angular velocity as independent variables of simulated particles which are subject to explicit leapfrog time‐integration scheme (Lagrangian method). The three‐dimensional dynamics equations based on the classical Newtonian approach for the second law of motion are used. The track of forces and moments acting on each particle is kept at every time step. Contact forces depend on the particle geometry overlap and material properties. The normal, tangential and moment components of interaction force are included. An effort is undertaken to extract the underlying object oriented abstractions in the DEM. These abstractions are implemented in C++, conform to object oriented design principles and use design pa...

title: Y-Geo: New Combined Finite-Discrete Element Numerical Code for Geomechanical Applications
abstract: AbstractThe purpose of this paper is to present Y-Geo, a new numerical code for geomechanical applications based on the combined finite-discrete element method (FDEM). FDEM is an innovative numerical technique that combines the advantages of continuum-based modeling approaches and discrete element methods to overcome the inability of these methods to capture progressive damage and failure processes in rock. In particular, FDEM offers the ability to explicitly model the transition from continuum to discontinuous behavior by fracture and fragmentation processes. Several algorithmic developments have been implemented in Y-Geo to specifically address a broad range of rock mechanics problems. These features include (1) a quasi-static friction law, (2) the Mohr-Coulomb failure criterion, (3) a rock joint shear strength criterion, (4) a dissipative impact model, (5) an in situ stress initialization routine, (6) a material mapping function (for an exact representation of heterogeneous models), and (7) a tool to i...

title: Numerical Models in Discontinuous Media: Review of Advances for Rock Mechanics Applications
abstract: The paper presents a description of the methods used to model rock as discontinuous media. The objective of the work is to bring to the geomechanics community recent advances in numerical modeling in the field of rock mechanics. The following methods are included: (1) the distinct element method; (2) the discontinuous deformation analysis method; and (3) the bonded particle method. A brief description of the fundamental algorithms that apply to each method is included, as well as a simple case to illustrate their use.

title: Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces
abstract: Some materials may naturally form discontinuities such as cracks as a result of deformation. As an aid to the modeling of such materials, a new framework for the basic equations of continuum mechanics, called the "peridynamic" formulation, is proposed. The propagation of linear stress waves in the new theory is discussed, and wave dispersion relations are derived. Material stability and its connection with wave propagation is investigated. It is demonstrated by an example that the reformulated approach permits the solution of fracture problems using the same equations either on or off the crack surface or crack tip. This is an advantage for modeling problems in which the location of a crack is not known in advance.

title: A meshfree method based on the peridynamic model of solid mechanics
abstract: An alternative theory of solid mechanics, known as the peridynamic theory, formulates problems in terms of integral equations rather than partial differential equations. This theory assumes that particles in a continuum interact with each other across a finite distance, as in molecular dynamics. Damage is incorporated in the theory at the level of these two-particle interactions, so localization and fracture occur as a natural outgrowth of the equation of motion and constitutive models. A numerical method for solving dynamic problems within the peridynamic theory is described. Accuracy and numerical stability are discussed. Examples illustrate the properties of the method for modeling brittle dynamic crack growth.

title: Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot
abstract: This study presents an analysis approach based on a merger of the finite element method and the peridynamic theory. Its validity is established through qualitative and quantitative comparisons against the test results for a stiffened composite curved panel with a central slot under combined internal pressure and axial tension. The predicted initial and final failure loads, as well as the final failure modes, are in close agreement with the experimental observations. This approach demonstrates the capability of the PD approach to assess the durability of complex composite structures.

title: Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites
abstract: We propose a computational method for a homogenized peridynamics description of fiber-reinforced composites and we use it to simulate dynamic brittle fracture and damage in these materials. With this model we analyze the dynamic effects induced by different types of dynamic loading on the fracture and damage behavior of unidirectional fiber-reinforced composites. In contrast to the results expected from quasi-static loading, the simulations show that dynamic conditions can lead to co-existence of and transitions between fracture modes; matrix shattering can happen before a splitting crack propagates. We observe matrix–fiber splitting fracture, matrix cracking, and crack migration in the matrix, including crack branching in the matrix similar to what is observed in recent dynamic experiments. The new model works for arbitrary fiber orientation relative to a uniform discretization grid and also works with random discretizations. The peridynamic composite model captures significant differences in the crack propagation behavior when dynamic loadings of different intensities are applied. An interesting result is branching of a splitting crack into two matrix cracks in transversely loaded samples. These cracks branch as in an isotropic material but here they migrate over the “fiber bonds”, without breaking them. This behavior has been observed in recent experiments. The strong influence that elastic waves have on the matrix damage and crack propagation paths is discussed. No special criteria for splitting mode fracture (Mode II), crack curving, or crack arrest are needed, and yet we obtain all these modes of material failure as a direct result of the peridynamic simulations.

title: A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media
abstract: A new material model for the dynamic fracture analysis of anisotropic materials has been proposed within the framework of the bond-based peridynamic theory. This model enables predicting complex fracture phenomena such as spontaneous crack nucleation and crack branching, curving and arrest, a capability inherited from the bond-based peridynamic theory. An important feature of the model is that the bond properties, i.e. the stiffness constant and critical stretch, are continuous functions of bond orientation in the principal material axes. This facilitates fracture analysis of anisotropic materials with random orientations, such as polycrystalline microstructures. Elastic and fracture behaviour of the model has been verified through simulating uniaxial tension of a composite plate and fracture of a cortical bone compact tension specimen, and making quantitative comparisons to analytical and experimental data. To further demonstrate the capabilities of the proposed model, dynamic fracture of a polycrystalline microstructure (alumina ceramic) has been simulated. The influence of the grain boundary and grain interior fracture energies on the interacting and competing fracture modes of polycrystalline materials, i.e. intergranular and transgranular fracture, has been studied.

title: Peridynamic States and Constitutive Modeling
abstract: A generalization of the original peridynamic framework for solid mechanics is proposed. This generalization permits the response of a material at a point to depend collectively on the deformation of all bonds connected to the point. This extends the types of material response that can be reproduced by peridynamic theory to include an explicit dependence on such collectively determined quantities as volume change or shear angle. To accomplish this generalization, a mathematical object called a deformation state is defined, a function that maps any bond onto its image under the deformation. A similar object called a force state is defined, which contains the forces within bonds of all lengths and orientation. The relation between the deformation state and force state is the constitutive model for the material. In addition to providing a more general capability for reproducing material response, the new framework provides a means to incorporate a constitutive model from the conventional theory of solid mechanics directly into a peridynamic model. It also allows the condition of plastic incompressibility to be enforced in a peridynamic material model for permanent deformation analogous to conventional plasticity theory.

title: A non-ordinary state-based peridynamic method to model solid material deformation and fracture
abstract: In this paper, we develop a new non-ordinary state-based peridynamic method to solve transient dynamic solid mechanics problems. This new peridynamic method has advantages over the previously developed bond-based and ordinary state-based peridynamic methods in that its bonds are not restricted to central forces, nor is it restricted to a Poisson’s ratio of 1/4 as with the bond-based method. First, we obtain non-local nodal deformation gradients that are used to define nodal strain tensors. The deformation gradient tensors are used with the nodal strain tensors to obtain rate of deformation tensors in the deformed configuration. The polar decomposition of the deformation gradient tensors are then used to obtain the nodal rotation tensors which are used to rotate the rate of deformation tensors and previous Cauchy stress tensors into an unrotated configuration. These are then used with conventional Cauchy stress constitutive models in the unrotated state where the unrotated Cauchy stress rate is objective. We then obtain the unrotated Cauchy nodal stress tensors and rotate them back into the deformed configuration where they are used to define the forces in the nodal connecting bonds. As a first example we quasi-statically stretch a bar, hold it, and then rotate it ninety degrees to illustrate the methods finite rotation capabilities. Next, we verify our new method by comparing small strain results from a bar fixed at one end and subjected to an initial velocity gradient with results obtained from the corresponding one-dimensional small strain analytical solution. As a last example, we show the fracture capabilities of the method using both a notched and un-notched bar.

title: Non-ordinary state-based peridynamic analysis of stationary crack problems
abstract: An implicit implementation of the non-ordinary state-based peridynamics formulation for quasi-static linearly elastic solids is presented. Emphasis is placed on assessing the accuracy of the numerical scheme in the vicinity of the crack front and other sources of stress concentration. We also present a comparative study of methods used to control the zero-energy modes inherent in the nonlocal definition of the strain tensor and reduce the spurious oscillations present particularly in regions of high strain gradients. The accuracy of the peridynamics scheme, including the impact of the lattice spacing and configuration, is assessed by performing an analysis of the near-tip stress and displacement fields (K-fields) for 2D problems. The manuscript also summarizes a verification study based on the classical 3D penny-shaped crack problem and a validation study of a 3D notched fracture specimen.

title: Higher‐order extended finite elements with harmonic enrichment functions for complex crack problems
abstract: In this paper, we analyze complex crack problems in elastic media using harmonic enrichment functions in a higher-order extended finite element implementation. The numerically computed enrichment function of a crack is the solution of the Laplace equation with discontinuous Dirichlet boundary condition along the crack, and its interaction with branches or other cracks is realized by imposing vanishing Neumann boundary conditions along those cracks. The classical finite element displacement approximation is enriched by adding the enrichment function of a crack through the framework of partition of unity. A nested subgrid mesh is used in the Laplace solve with a rasterized approximation of a crack, which simplifies the numerical integration—no partitioning of finite elements is required. Harmonic enrichment functions readily permit the extension to handle multiple interacting and branched cracks without any special treatment around the junction points. Several numerical examples are presented that arm the accuracy and eectiveness of the method when applied to complex crack

title: Modeling of dynamic crack branching by enhanced extended finite element method
abstract: The conventional extended finite element method (XFEM) is enhanced in this paper to simulate dynamic crack branching, which is a top challenge issue in fracture mechanics and finite element method. XFEM uses the enriched shape functions with special characteristics to represent the discontinuity in computation field. In order to describe branched cracks, it is necessary to set up the additional enrichment. Here we have developed two kinds of branched elements, namely the “element crossed by two separated cracks” and “element embedded by a junction”. Another series of enriched degrees of freedom are introduced to seize the additional discontinuity in the elements. A shifted enrichment scheme is used to avoid the treatment of blending element. Correspondingly a new mass lumping method is developed for the branched elements based on the kinetic conservation. The derivation of the mass matrix of a four-node quadrilateral element which contains two strong discontinuities is specially presented. Then by choosing crack speed as the branching criterion, the branching process of a single mode I crack is simulated. The results including the branching angle and propagation routes are compared with that obtained by the conventionally used element deletion method.

title: Studies of dynamic crack propagation and crack branching with peridynamics
abstract: In this paper we discuss the peridynamic analysis of dynamic crack branching in brittle materials and show results of convergence studies under uniform grid refinement (m-convergence) and under decreasing the peridynamic horizon (δ-convergence). Comparisons with experimentally obtained values are made for the crack-tip propagation speed with three different peridynamic horizons. We also analyze the influence of the particular shape of the micro-modulus function and of different materials (Duran 50 glass and soda-lime glass) on the crack propagation behavior. We show that the peridynamic solution for this problem captures all the main features, observed experimentally, of dynamic crack propagation and branching, as well as it obtains crack propagation speeds that compare well, qualitatively and quantitatively, with experimental results published in the literature. The branching patterns also correlate remarkably well with tests published in the literature that show several branching levels at higher stress levels reached when the initial notch starts propagating. We notice the strong influence reflecting stress waves from the boundaries have on the shape and structure of the crack paths in dynamic fracture. All these computational solutions are obtained by using the minimum amount of input information: density, elastic stiffness, and constant fracture energy. No special criteria for crack propagation, crack curving, or crack branching are used: dynamic crack propagation is obtained here as part of the solution. We conclude that peridynamics is a reliable formulation for modeling dynamic crack propagation.

title: Predicting crack initiation and propagation using XFEM, CZM and peridynamics: A comparative study
abstract: This study presents a comparison of extended finite elements (XFEM), cohesive zone model (CZM) and the peridynamic theory (PD). By comparisons against two experimental benchmark studies, the capability of these techniques to predict dynamic fracture is demonstrated through both qualitative and quantitative observations.