A coupled hydro-mechanical analysis for prediction of hydraulic fracture propagation in saturated porous media using EFG mesh-less method

The details of the Element Free Galerkin (EFG) method are presented with the method being applied to a study on hydraulic fracturing initiation and propagation process in a saturated porous medium using coupled hydro-mechanical numerical modelling. In this EFG method, interpolation (approximation) is based on nodes without using elements and hence an arbitrary discrete fracture path can be modelled.The numerical approach is based upon solving two governing partial differential equations of equilibrium and continuity of pore water simultaneously. Displacement increment and pore water pressure increment are discretized using the same EFG shape functions. An incremental constrained Galerkin weak form is used to create the discrete system of equations and a fully implicit scheme is used for discretization in the time domain. Implementation of essential boundary conditions is based on the penalty method. In order to model discrete fractures, the so-called diffraction method is used.Examples are presented and the results are compared to some closed-form solutions and FEM approximations in order to demonstrate the validity of the developed model and its capabilities. The model is able to take the anisotropy and inhomogeneity of the material into account. The applicability of the model is examined by simulating hydraulic fracture initiation and propagation process from a borehole by injection of fluid. The maximum tensile strength criterion and Mohr-Coulomb shear criterion are used for modelling tensile and shear fracture, respectively. The model successfully simulates the leak-off of fluid from the fracture into the surrounding material. The results indicate the importance of pore fluid pressure in the initiation and propagation pattern of fracture in saturated soils. © 2013 Elsevier Ltd.

Were intercalated komatiites and dacites at the Black Swan nickel sulphide mine, Yilgarn Craton, Western Australia, emplaced as extrusive lavas or intrusive bodies? The significance of breccia textures and contact relationships

Intercalated Archean komatiites and dacites sit above a thick footwall dacite unit in the host rock succession at the Black Swan Nickel Mine, north of Kalgoorlie in the Yilgarn Craton, Western Australia. Both lithofacies occur in units that vary in scale from laterally extensive at the scale of the mine lease to localized, thin, irregular bodies, from > 100 m thick to only centimetres thick. Some dacites are only slightly altered and deformed, and are interpreted to post-date major deformation and alteration (late porphyries). However, the majority of the dacites display evidence of deformation, especially at contacts, and metamorphism, varying from silicification and chlorite alteration at contacts to pervasive low grade regional metamorphic alteration represented by common assemblages of chlorite, sericite and albite. Texturally, the dacites vary from entirely massive and coherent to partially brecciated to totally brecciated. Strangely, some dacites are coherent at the margins and brecciated internally. Breccia textures vary from cryptically defined, to blocky, closely packed, in situ jig-saw fit textures with secondary minerals in fractures between clasts, to more apparent matrix rich textures with round clast forms, giving apparent conglomerate textures. Some clast zones have multi-coloured clasts, giving the impression of varied provenance. Strangely however, all these textural variants have gradational relationships with each other, and no bedding or depositional structures are present. This indicates that all textures have an in situ origin. The komatiites are generally altered and pervasively carbonate veined. Preservation of original textures is patchy and local, but includes coarse adcumulate, mesocumulate, orthocumulate, crescumulate-harrisite and occasionally spinifex textures. Where original contacts between komatiites and dacites are preserved intact (i.e. not sheared or overprinted by alteration), the komatiites have chilled margins, whereas the dacites do not. The margins of the dacites are commonly silicified, and inclusions of dacite occur in komatiite, even at the top contacts of komatiite units, but komatiite clasts do not occur in the dacites. The komatiites therefore were emplaced as sills into the dacites, and the intercalated relationships are interpreted as intrusive. The brecciation and alteration in the dacites are interpreted as being largely due to hydraulic fracturing and alteration induced by contact metamorphic effects and hydrothermal alteration deriving from the intrusion of komatiites into the felsic pile. The absence of autobreccia and hyaloclastite textures in the dacites suggest that they were emplaced as an earlier intrusive (sill?) complex at a high level in the crust.

Reservatório de gás não convencional: história de caso na Bacia do Recôncavo - Brasil

O principal objetivo deste trabalho foi à realização de um estudo geológico sobre os folhelhos fraturados da Formação Candeias que ocorrem em diferentes partes da Bacia do Recôncavo, sendo enfatizadas suas principais relações com a acumulação de petróleo. Este trabalho contempla, ainda, uma revisão bibliográfica acerca da evolução da campanha exploratória nesta bacia, desde seus primórdios, na década de trinta, além de uma análise sob as perspectivas na área de exploração e produção. A Bacia do Recôncavo, situada na região Nordeste do Brasil, está inserida numa grande bacia tafrogênica (Rifte abortado Recôncavo-Tucano-Jatobá), originada nos momentos iniciais de ruptura do paleocontinente Gondwana ocidental, no Eocretáceo, sob os campos de tensões que produziram o Oceano Atlântico. Em termos estratigráficos, a bacia guarda um importante registro geológico, compreendido por depósitos continentais (lacustres, fluvio-deltáicos, neríticos, eólicos e leques aluviais), sendo subdividido em seis seqüências sedimentares, referentes aos estágios pré-rifte e rifte. As principais acumulações de petróleo na bacia podem ser agrupadas em três grandes sistemas: pré-rifte, rifte-Candeias e rifte-Ilhas, representando plays exploratórios distintos. As perspectivas futuras para a Bacia do Recôncavo devem considerar a importância da identificação de novos plays exploratórios, através da caracterização de novos intervalos geradores e reservatórios, como é o caso dos folhelhos fraturados e do shale gas. Não menos importante é o desenvolvimento de novas técnicas de produção, aumentando o fator de recuperação de campos produtores.

Subsurface to Surface Correlation of the Tensleep Sandstone in the West Flank of the Pryor Mountains in Carbon County, Montana

The Pennsylvanian Tensleep Sandstone is an eolian and nearshore marine/sabka quartz arenite unit with prominent outcrops along the western Pryor/Bighorn Mountain front east of Red Lodge, MT. Regionally, the formation represents one of the largest ergs in the global geologic record. High permeability makes it an important oil and gas reservoir and aquifer in south central Montana and throughout much of Wyoming. The Tensleep Sandstone’s high percentage of quartz content and grain roundness, due to its eolian origin, makes it a prospective source for natural proppant sand. Three continuous 4-inch cores were obtained during a cooperative project between Montana Tech and industry partners. Using stratigraphic sections, cores, thin sections, and x-ray fluorescence (XRF) analysis, the usefulness and economic feasibility of the Tensleep Sandstone as a minable hydraulic fracture proppant was explored. Usefulness depends on cementation, grain shape, grain size, and depth from surface of the prospective zone. Grain shape and size were determined by thin sections, sieving, and stereomicroscope analysis. Analysis of 20 disaggregated sand samples has shown that as much as 30 percent of the grain sizes fall between 30-50 mesh (medium- to finegrained sand size) and about 45 percent of the grain sizes fall between 70–140 mesh (very fine-grained sand to coarse silt), grain sizes appropriate for some hydraulic fracture operations. Core descriptions and XRF data display the distribution of lithology and cementation. Core descriptions and XRF data display the distribution of lithology and cementation. Elemental (XRF) analyses help to delineate more pure quartz sands from those with grain fractions reflecting fine-grained clastic and evaporitic inputs. The core and nearby stratigraphic sections are used to quantify the amount of overburden and the 3 amount of resource in the area. Initial results show favorable crush strength and useable grain size and shape.

Aplicación de la técnica de inyección del terreno por fracturación hidráulica a las cimentaciones de puentes en circunstancias de especial dificultad

La implantación de una gran obra de paso facilita en multitud de ocasiones la vida de aquellos que aprovechen las ventajas que ofrece esta estructura. Sin embargo, el camino que lleva a su construcción supone importantes y tortuosos retos cuando el terreno de cimentación presenta condiciones geotécnicas no compatibles con las grandes cargas que el viaducto necesita transmitirle para su buen funcionamiento. El trabajo que desarrolla esta Tesis Doctoral proporciona una herramienta eficaz y económica, por lo reducido de su extensión y medios, que permite allanar el camino que acomete la construcción de una estructura de tal envergadura. Mediante el análisis de la problemática del terreno y de las distintas soluciones de cimentación empleadas en la actualidad, se conduce al lector hacia una técnica de cimentación innovadora que combina la técnica del micropilotaje, para la canalización de las cargas estructurales, junto a la técnica de la inyección de fracturación hidráulica, que mejorará el terreno de implantación de los micropilotes, conformando así un medio capaz de recibir y transmitir grandes cargas en cualquier tipo de terreno. La técnica ya empleada del micropilotaje, por su trabajo esencialmente axil, requiere, aunque sin problema, la constitución de sistemas de fuerzas que equilibren el sistema de cargas provenientes del viaducto. Pero su capacidad resistente viene condicionada por el terreno circundante. Cuando la roca rodea el micropilote, su empleo se realiza sin problema y sin necesidad de mejorar el terreno. Pero sin terreno consistente, el empleo de inyecciones de fracturación hidráulica a través de los propios micropilotes, no sólo mejorará la capacidad resistente de los micropilotes, aumentando la inercia necesaria cuando las cargas son de origen sísmico o estructural ferroviario, sino que resolverá y eliminará los problemas de estabilidad que presentan las laderas que frecuentemente deben recibir las cargas de la obra de paso. Tras recoger el análisis ya realizado en el Trabajo de Investigación, donde se justificaba la alta capacidad resistente de un micropilote con terreno circundante mejorado por la inyección, en la actual Tesis se emplean modelos matemáticos sobre un caso real de viaducto sometido a un gran sismo e implantado en una enorme quebrada (500 m) de Colombia, sujeta a lluvias torrenciales y en la zona de mayor sismicidad del país. Con ello se comprueba la estabilización que se alcanza en el terreno de cimentación con el empleo de esta técnica de transmisión de carga y mejora del terreno. De esta forma se completa un ciclo que justifica las bondades de esta combinación de técnicas de cimentación, pero se abren las puertas a nuevos entornos de aplicación, como edificios antiguos de cualquier tipo que requieran recalces, y no sólo en la implantación de grandes obras de paso. ABSTRACT The establishment of a large bridge represents, in many cases, a better life for those who can take profit of the advantages provided by that structure. Nevertheless, the process of building this structure has to overcome important and difficult circumstances whenever the geotechnical conditions of the bridge site are not adequate to carry the large loads transmitted by the bridge structure. This study develops a method both effective and economical, due to the extension and means necessary for its application, which allows to solve properly the foundation of a structure of that importance. Considering the geotechnical problems inherent to the bridge site, along with the different foundation solutions that are presently used and their limitations, the study leads the reader to an innovative technique which combines the micropile system, for transmission of the structural loads of the bridge, with the technique of hydraulic fracture grouting for improvement of the ground around the micropiles, allowing to both stabilizing and transmitting large loads in any kind of ground. It is well known that the micropiles work axially, and this condition requires an adequate distribution of those units, in order to properly absorb the load system introduced by the viaduct. The resistance of the bridge foundation is, in any case, provided by the ground. When rock is encountered, the micropiles have been successfully used without improving the ground. However, as it is shown in this study, by using the micropiles as sleeve-pipes for hydraulic fracture grouting, not only the micropile resistance can be improved in any ground, but it is possible to develop grouted “solids” in the ground, whose inertia allows to absorb actions of structural and seismic origin. Additionally, as it is shown and analyzed in the Thesis, the ground improvement can give an adequate safety factor to the slopes frequently encountered in bridge sites. In order to properly justify those advantages of combining micropiles with ground improvement through fracture grouting around the micropiles, mathematical models have been developed and applied to a real case of a cable-stayed bridge installed on a very large ravine (500 m) in Colombia, located in the highest seismic zone of the country, and subject to torrential rains. The results of this numerical analysis show the high safety condition provided by the ground improvement to the viaduct site. In conclusion, the Thesis shows the important improvement that can be provided by the combination of micropiles and soil improvement, through fracture grouting, to the problem of founding bridges. However, it can be understood that this technique could be applied successfully to underpinning buildings, specially old buildings of any type, apart from its use in bridge foundations.

Efficient Computation of Electromagnetic Waves in Hydrocarbon Exploration Using the Improved Numerical Mode Matching (NMM) Method

In this study, we developed and improved the numerical mode matching (NMM) method which has previously been shown to be a fast and robust semi-analytical solver to investigate the propagation of electromagnetic (EM) waves in an isotropic layered medium. The applicable models, such as cylindrical waveguide, optical fiber, and borehole with earth geological formation, are generally modeled as an axisymmetric structure which is an orthogonal-plano-cylindrically layered (OPCL) medium consisting of materials stratified planarly and layered concentrically in the orthogonal directions.

In this report, several important improvements have been made to extend applications of this efficient solver to the anisotropic OCPL medium. The formulas for anisotropic media with three different diagonal elements in the cylindrical coordinate system are deduced to expand its application to more general materials. The perfectly matched layer (PML) is incorporated along the radial direction as an absorbing boundary condition (ABC) to make the NMM method more accurate and efficient for wave diffusion problems in unbounded media and applicable to scattering problems with lossless media. We manipulate the weak form of Maxwell's equations and impose the correct boundary conditions at the cylindrical axis to solve the singularity problem which is ignored by all previous researchers. The spectral element method (SEM) is introduced to more efficiently compute the eigenmodes of higher accuracy with less unknowns, achieving a faster mode matching procedure between different horizontal layers. We also prove the relationship of the field between opposite mode indices for different types of excitations, which can reduce the computational time by half. The formulas for computing EM fields excited by an electric or magnetic dipole located at any position with an arbitrary orientation are deduced. And the excitation are generalized to line and surface current sources which can extend the application of NMM to the simulations of controlled source electromagnetic techniques. Numerical simulations have demonstrated the efficiency and accuracy of this method.

Finally, the improved numerical mode matching (NMM) method is introduced to efficiently compute the electromagnetic response of the induction tool from orthogonal transverse hydraulic fractures in open or cased boreholes in hydrocarbon exploration. The hydraulic fracture is modeled as a slim circular disk which is symmetric with respect to the borehole axis and filled with electrically conductive or magnetic proppant. The NMM solver is first validated by comparing the normalized secondary field with experimental measurements and a commercial software. Then we analyze quantitatively the induction response sensitivity of the fracture with different parameters, such as length, conductivity and permeability of the filled proppant, to evaluate the effectiveness of the induction logging tool for fracture detection and mapping. Casings with different thicknesses, conductivities and permeabilities are modeled together with the fractures in boreholes to investigate their effects for fracture detection. It reveals that the normalized secondary field will not be weakened at low frequencies, ensuring the induction tool is still applicable for fracture detection, though the attenuation of electromagnetic field through the casing is significant. A hybrid approach combining the NMM method and BCGS-FFT solver based integral equation has been proposed to efficiently simulate the open or cased borehole with tilted fractures which is a non-axisymmetric model.

A multi-scaling approach to predict hydraulic damage of poromaterials

The purpose of this paper is to introduce the concept of hydraulic damage and its numerical integration. Unlike the common phenomenological continuum damage mechanics approaches, the procedure introduced in this paper relies on mature concepts of homogenization, linear fracture mechanics, and thermodynamics. The model is applied to the problem of fault reactivation within resource reservoirs. The results show that propagation of weaknesses is highly driven by the contrasts of properties in porous media. In particular, it is affected by the fracture toughness of host rocks. Hydraulic damage is diffused when it takes place within extended geological units and localized at interfaces and faults.

Effects of Fracture Seepage on the Stability of Landslide during Reservoir Water Level Fluctuation

The hydraulic conductivity function of fractures is a key scientific question to describe and reveal the process and the role of water seepage reasonably. In this paper, the generation technology of random fracture network and the latest numerical computation method for equivalent permeability tensor of fracture network are applied to analyze the landslide located at Wangjiayuanzi in Wanzhou District of Chongqing by simulating the changes of the seepage field caused by the running of the Three Gorges Reservoir. The influences of the fracture seepage on the seepage field and stability of the landslide were discussed with emphasis. The results show that the fractures existing in the soil increase the permeability coefficient of the landslide body and reduce the delay time of the underground water level in the landslide which fluctuates relative to the water level of reservoir，that causes the safe coefficient of the slope changes more gently than that of the same slope without fractures. It means, if only water level fluctuating condition is concerned, the fractures existing in the soil plays a positive role to the stability of slopes.

Fracture strength of flared bovine roots restored with different intraradicular posts

Objective: The aim of this study was to evaluate the fracture strength and failure mode of flared bovine roots restored with different intraradicular posts. Material and Methods: Fifty bovine incisors with similar dimensions were selected and their roots were flared until 1.0 mm of dentin wall remained. Next, the roots were allocated into five groups (n=10): GI-cast metal post-and-core; GII-fiber posts plus accessory fiber posts; GIII-direct anatomic post; GIV-indirect anatomic post and GV-control (specimens without intraradicular post). A polyether impression material was used to simulate the periodontal ligament. After periodontal ligament simulation, the specimens were subjected to a compressive load at a crosshead speed of 0.5 mm/min in a servo-hydraulic testing machine (MTS 810) applied at 135 to the long axis of the tooth until failure. The data (N) were subjected to ANOVA and Tukey's post-hoc test (alpha=0.05). Results: GI and GIV presented higher fracture strength (p<0.05) than GII. GIII presented intermediate values without statistically significant differences (p>0.05) from GI, GII and GIV. Control specimens (GV) produced the lowest fracture strength mean values (p<0.05). Despite obtaining the highest mean value, GI presented 100% of unfavorable failures. GII presented 20% of unfavorable failures. GIII, GIV and GV presented only favorable failures. Conclusions: Although further in vitro and in vivo studies are necessary, the results of this study showed that the use of direct and indirect anatomic posts in flared roots could be an alternative to cast metal post-and-core.

Fracture strength of bovine incisors after intra-radicular treatment with MTA in an experimental immature tooth model

Aim To evaluate, using an experimental immature tooth model, the fracture resistance of bovine incisors submitted to different reinforcement treatments with mineral trioxide aggregate (MTA).Methodology An immature tooth model was created by sectioning the coronal and apical portions of 40 bovine incisors 8 mm above and 12 mm below the cementoenamel junction. The root canals were irrigated with 1.0% sodium hypochlorite. They were enlarged both coronally and apically using number 703 carbide burs (ISO: 500-104-168-007-021) and their internal diameter was standardized to 2.1 mm. The specimens were assigned to four groups (n = 10): GI-control (without filling); GII-apical MTA plug + filling with gutta-percha and endodontic sealer; GIII-filling with MTA; GIV-apical MTA plug + filling with MTA + metallic post (Reforpost I). A polyether impression material was used to simulate the periodontal ligament. The specimens were submitted to a compressive load at a crosshead speed of 0.5 mm min(-1) in a servo-hydraulic universal testing machine (MTS 810) applied at 45 degrees to the long axis of the tooth until failure. Data were submitted to statistical analysis by the Kruskal-Wallis test at 5% significance level.Results GIV presented the highest fracture resistance (32.7N) and differed significantly from the other groups (P < 0.05). No statistically difference was found between GII (16.6N) and GIII (23.4N) (P > 0.05). GIII had a significantly higher fracture resistance than GI (P < 0.05).Conclusions the use of MTA + metallic post as an intra-radicular reinforcement treatment increased the resistance to fracture of weakened bovine teeth in an experimental immature tooth model.

Fracture resistance of simulated immature teeth after different intra-radicular treatments

The aim of this study was to evaluate the fracture resistance of simulated immature teeth after different intra-radicular treatments. Crowns and roots of bovine incisors were cut transversally and removed to simulate immature teeth. Root canal preparation and flaring were performed using a bur in crown-apex and apex-crown direction. The samples were distributed into 5 groups (n=10): Positive control (PoC) - no root canal flaring or filling; Negative control (NeC) - teeth were sectioned and their root canals were flared; Direct anatomical glass fiber post (RaP) - #2 Reforpost main glass fiber post relined with composite resin; Double tapered conical glass fiber posts (ExP) - #3 Exacto glass fiber post; and #2 Reforpost main glass fiber + Reforpin accessory glass fiber posts (RrP). In RaP, ExP and RrP, 4.0-mm apical plugs were done with MTA Angelus. The specimens were embedded in polystyrene resin inside cylinders and the periodontal ligament was simulated with a polyether-based impression material. The specimens were submitted to compressive fracture strength test (0.5 mm/min at 135° relative to the long axis of the tooth) in a servo-hydraulic mechanical testing machine MTS 810. Data were subjected to one-way ANOVA and Dunnett's C or Tukey's tests (α=0.05). The control groups (PoC and NeC) showed lower fracture strength than the experimental groups. NeC presented the lowest resistance and ExP presented the highest resistance among the experimental groups. The flaring procedures produced a detrimental effect on the fracture resistance of the bovine teeth. Glass fiber intra-radicular posts increased significantly the fracture resistance of simulated immature teeth.

The Influence of Fractures on Vulnerability to Pollution of a Carbonate Aquifer: Monte Conero (Italy)

The carbonate outcrops of the anticline of Monte Conero (Italy) were studied in order to characterize the geometry of the fractures and to establish their influence on the petrophysical properties (hydraulic conductivity) and on the vulnerability to pollution. The outcrops form an analog for a fractured aquifer and belong to the Maiolica Fm. and the Scaglia Rossa Fm. The geometrical properties of fractures such as orientation, length, spacing and aperture were collected and statistically analyzed. Five types of mechanical fractures were observed: veins, joints, stylolites, breccias and faults. The types of fractures are arranged in different sets and geometric assemblages which form fracture networks. In addition, the fractures were analyzed at the microscale using thin sections. The fracture age-relationships resulted similar to those observed at the outcrop scale, indicating that at least three geological episodes have occurred in Monte Conero. A conceptual model for fault development was based on the observations of veins and stylolites. The fracture sets were modelled by the code FracSim3D to generate fracture network models. The permeability of a breccia zone was estimated at microscale by and point counting and binary image methods, whereas at the outcrop scale with Oda’s method. Microstructure analysis revealed that only faults and breccias are potential pathways for fluid flow since all veins observed are filled with calcite. According this, three scenarios were designed to asses the vulnerability to pollution of the analogue aquifer: the first scenario considers the Monte Conero without fractures, second scenario with all observed systematic fractures and the third scenario with open veins, joints and faults/breccias. The fractures influence the carbonate aquifer by increasing its porosity and hydraulic conductivity. The vulnerability to pollution depends also on the presence of karst zones, detric zones and the material of the vadose zone.

On the loading-rate dependence of the Al 7017-T73 fracture-initiation toughness

While static fracture toughness is a widely studied and standardised parameter, its dynamic counterpart has not been exhaustively examined. Therefore, in this research a series of quasi-static and different loading-rate dynamic tests were carried out to determine the evolution of fracture toughness with the velocity of the application of the load on aluminium 7017-T73 alloy. Three-point bending tests of pre-fatigued standard specimens (ASTM E399) at four loading-rates were carried out. The experiments were conducted by employing the subsequent apparatus ordered from lowest to highest load application velocity: a servo-hydraulic universal testing machine, a free-drop tower, a modified Split Hopkinson Pressure Bar and an explosive load testing device. In order to perform the dynamic fracture toughness tests, it was necessary to design and develop some experimental devices. The fracture-initiation toughness of the aluminium 7017-T73 alloy did not exhibit a significant variation for the studied cases. As a conclusion, the research showed that fracture-initiation toughness remained constant regardless of the velocity at which the load was applied.

Application of Numerical Methods to Study Arrangement and Fracture of Lithium-Ion Microstructure

The focus of this work is to develop and employ numerical methods that provide characterization of granular microstructures, dynamic fragmentation of brittle materials, and dynamic fracture of three-dimensional bodies.

We first propose the fabric tensor formalism to describe the structure and evolution of lithium-ion electrode microstructure during the calendaring process. Fabric tensors are directional measures of particulate assemblies based on inter-particle connectivity, relating to the structural and transport properties of the electrode. Applying this technique to X-ray computed tomography of cathode microstructure, we show that fabric tensors capture the evolution of the inter-particle contact distribution and are therefore good measures for the internal state of and electronic transport within the electrode.

We then shift focus to the development and analysis of fracture models within finite element simulations. A difficult problem to characterize in the realm of fracture modeling is that of fragmentation, wherein brittle materials subjected to a uniform tensile loading break apart into a large number of smaller pieces. We explore the effect of numerical precision in the results of dynamic fragmentation simulations using the cohesive element approach on a one-dimensional domain. By introducing random and non-random field variations, we discern that round-off error plays a significant role in establishing a mesh-convergent solution for uniform fragmentation problems. Further, by using differing magnitudes of randomized material properties and mesh discretizations, we find that employing randomness can improve convergence behavior and provide a computational savings.

The Thick Level-Set model is implemented to describe brittle media undergoing dynamic fragmentation as an alternative to the cohesive element approach. This non-local damage model features a level-set function that defines the extent and severity of degradation and uses a length scale to limit the damage gradient. In terms of energy dissipated by fracture and mean fragment size, we find that the proposed model reproduces the rate-dependent observations of analytical approaches, cohesive element simulations, and experimental studies.

Lastly, the Thick Level-Set model is implemented in three dimensions to describe the dynamic failure of brittle media, such as the active material particles in the battery cathode during manufacturing. The proposed model matches expected behavior from physical experiments, analytical approaches, and numerical models, and mesh convergence is established. We find that the use of an asymmetrical damage model to represent tensile damage is important to producing the expected results for brittle fracture problems.

The impact of this work is that designers of lithium-ion battery components can employ the numerical methods presented herein to analyze the evolving electrode microstructure during manufacturing, operational, and extraordinary loadings. This allows for enhanced designs and manufacturing methods that advance the state of battery technology. Further, these numerical tools have applicability in a broad range of fields, from geotechnical analysis to ice-sheet modeling to armor design to hydraulic fracturing.