Fracture Nucleation Research Articles

Emergence of tension-compression asymmetry from a complete phase-field approach to brittle fracture

The classical variational approach to brittle fracture propagation does not distinguish between strain energy accumulation in tension versus compression and consequently results in physically unrealistic cracking under compression. A variety of energy splits have been proposed as a possible remedy. However, a unique energy split that can describe this asymmetry for general loading conditions has not been found. The main objective of this paper is to show that a complete phase-field theory of brittle fracture nucleation and propagation, one that accounts for the material strength at large, can naturally capture the tension-compression asymmetry without an energy split. One such theory has been recently proposed by Kumar et al. (2018). Over the past few years, several studies have shown that this theory is capable of accurately describing fracture nucleation and propagation for materials soft and hard under arbitrary monotonic loading conditions. However, a systematic study of the tension-compression asymmetry that emerges from this theory has not yet been reported. This paper does precisely that. In particular, this paper reports a comprehensive study of crack propagation in two problems, one involving a symmetric tension-compression state and the other involving larger compressive stresses at the crack tip. The results are compared with popular energy splits used in literature. The results show that, remarkably, for the second problem, only the complete theory is able to produce experimentally consistent results.

Investigation of microscale brittle fracture opening in diamond with olivine inclusion using XFEM and cohesive zone modeling

Inclusions trapped in diamonds are a fundamental source of information to probe the Earth’s interior, provided that the pressure conditions at which the diamond grew are correctly determined.This study explores the traditional assumptions in geothermobarometry for olivine-in-diamond host-inclusion systems by employing extended finite element methods (XFEM) and cohesive zone models (CZM) to quantify the contributions of brittle fractures to the relaxation of the residual stress of inclusions. Our analysis was performed assuming that the host-inclusion system does not contain fluids and that the unfractured minerals are elastically isotropic. Our models show that the damage initiation is solely dependent on the shape of the inclusion and on the fracture strength of the diamond host, while the fracture nucleation is influenced by both the size of the inclusion and the toughness of the diamond. Our findings indicate that, in dry systems, the amount of relaxation of residual stress of the inclusion due to the opening of brittle fractures is much lower than that due to the elastic interaction between the host and the inclusion. Moreover, the pressure release due to fractures is not substantially affected by the shape of the inclusion. We also show that the total relaxation of the residual pressure due to the combined effect of the elastic interaction and of the brittle deformation is lower than what is observed in natural samples, even when assuming fracture strength and toughness lower than those reported from experiments on single crystals of diamond. Such discrepancies suggest that in natural olivine-diamond systems additional mechanisms such as viscous or plastic deformation and/or the presence of preexisting defects and fluids in the host might play a relevant role in the relaxation of the residual stress. These findings underscore the need for advanced numerical tools that consider the complex interplay of the geometry of the host-inclusion system, the fracture properties, and the presence of fluids and defects in order to build more accurate models to constrain the geological history of diamonds.

Nucleation of Fracture: The First-Octant Evidence Against Classical Variational Phase-Field Models

Abstract As a companion work to [1], this article presents a series of simple formulae and explicit results that illustrate and highlight why classical variational phase-field models cannot possibly predict fracture nucleation in elastic brittle materials. The focus is on “tension-dominated” problems where all principal stresses are nonnegative, that is, problems taking place entirely within the first octant in the space of principal stresses.

Dynamic in-situ CT imaging to study meso-structure evolution and fracture mechanism of shale during thermal-mechanical coupling condition

Temperature has a significant impact on the occurrence characteristics and flow properties of oil and gas in shale reservoirs at in-situ conditions. Under thermal–mechanical coupling conditions, the macroscopic deformation instability characteristics and the evolution of the internal microstructure of shale are of great importance for the exploitation of shale oil and gas. However, most studies on the macroscopic and microscopic characteristics of shale only analyze the deformation and instability mechanisms of shale from observations of the post-failure microscopic structures. This paper employs a self-developed THM-CT coupling test system to study the macroscopic deformation characteristics of shale under triaxial compression at real-time temperatures, and simultaneously uses in-situ CT imaging technology to characterize the internal microstructure of shale under different stress levels. Real-time CT porosity obtained via in-situ CT imaging technology represents the mesoscopic structural parameters, together with macroscopic deformation characteristics, delineate the four stages of the entire deformation process in shale: initial compression, linear elastic deformation, yielding deformation, and post-failure. The deformation after peak-stress (σf) in shale is 1–3 times that before peak-stress (σf). The dissolution and softening of clay minerals within the internal micro-lamination of shale, along with the weakening of the internal structure, result in a decline in mechanical parameters such as stress strength (σf) and elastic modulus (E) with increasing temperature. The reconstruction of three-dimensional μCT digital volumes further reconstructs the internal fracture spatial structure under different stress levels during the deformation process, precisely characterizing the dynamic evolution of internal fracture nucleation, expansion, and coalescence during the whole test process. The failure characteristics of shales display notable heterogeneity and anisotropy. Internally, fractures experience a sequence of initiation, formation of vertical and oblique main fractures along the axial direction, and crossing connections between fractures. During test process, multiple occurrences of horizontal secondary micro-fractures extend along weakly cemented lamination, playing a connecting role between the main fractures. Importantly, a real-time quantitative relationship between the macroscopic and microscopic mechanics of rock has been established by linking the mesoscopic damage factor with the applied stress. It was found that the damage factor increases according to a power-law relationship with stress in the yield deformation phase. The variation in the mesoscopic damage factor serves as an intuitive index reflecting the systematic failure of rock, providing an early warning for the impending failure of shale.

Modeling abscission of cacti branches

During evolution, various functional principles have evolved that allow plants to create predetermined breaking points for the spatially defined abscission of organs. In the plant family of cacti, some species, such as Cylindropuntia bigelovii, have fragile branch–branch junctions that serve vegetative reproduction, while in other species, such as Opuntia ficus-indica, they are very stable. The fracture behavior of these junctions has been thoroughly characterized anatomically and mechanically, the data being the prerequisite for the performance of cactus-inspired phase field simulations. We have found that models composed of homogeneous materials or material systems with low elastic modulus contrast (analogous to Cylindropuntia bigelovii) exhibit a fracture mode where cracks initiation occurs at the epidermis of the junction notch. In comparison, heterogeneous material systems with high elastic modulus contrast (similar to Opuntia ficus-indica) show fracture nucleation along the inner vascular bundles, with an increase in the maximum fracture energy by a factor of 2.2. In the high contrast heterogeneous models, the V-notch and stiffening of the dermal tissue (“periderm”) have a negligible effect on their fracture behavior. In addition, the fracture morphologies of these models resemble the rough junction fracture sites found experimentally. The knowledge gained about the geometric influences and the importance of the contrasts in the mechanical properties of the individual materials in the overall system can be transferred as functional principles to bioinspired engineering composites in order to program their fracture behavior.

Thermo-mechanical modelling of spalling around the deposition boreholes in an underground nuclear waste repository during its thermal phase

This paper presents a three-dimensional numerical analysis of multiple fracture growth leading to the development of excavation disturbed zones and spalling around deposition boreholes in a geological disposal facility. The development of fracture patterns is simulated with the Imperial College Geomechanics Toolkit, a finite-element based simulator that can model the simultaneous nucleation, growth, and coalescence of multiple fractures in quasi-brittle rock. In these simulations, fractures develop due to the stress concentrations around the borehole wall, caused by the local in situ stresses, and due to the thermal stresses caused by the radioactive decay of the waste. Fracture patterns, and the extent of the spalled zone, are computed after the borehole drilling, heating, and cooling stages, at the Forsmark repository site in Sweden. The effect of temperature on the nucleation and growth of spalling fractures, as well as on the reactivation of pre-existing fractures, is assessed qualitatively, by comparing fracture patterns, and quantitatively, in terms of the maximum spalling depth, width, and increase in the total fractured surface area. Overall, the simulations presented herein indicate that thermal spalling will increase the depths (away from the borehole) and angular widths of the spalled zone, but is not likely to lead to major increases in fracture aperture, and concomitant increases in hydraulic transmissivity and permeability of the spalled zone, above that which has already been caused by mechanical spalling.

Intergranular fracture, grain-boundary structure, and dislocation-density interactions in FCC bicrystals

A dislocation-density based crystalline plasticity (DCP) and nonlinear finite element (FE) analysis were used to predict, and fundamentally understand how and why fracture nucleation and propagation are related to the interrelated microstructural mechanisms of dislocation-density pileups, GB structure, orientation, and total and partial dislocation density interactions within and adjacent for a random low angle grain boundary (LAGB) and a random high angle GB (HAGB). The GB orientations and structures were obtained from micropillar experiments, such that LAGBs and the HAGBs can be accurately represented and used for the modeling predictions. The normal stress, density of pileups, and dislocation-density accumulation along and within the GB were higher for the low angle GB bicrystal. These interrelated phenomena delineate how fracture for high angle GBs nucleate and propagate at lower nominal strains than the lower angle GB bicrystal case. These predictions underscore how fundamental mechanisms can be identified and used to understand how failure nucleates and propagates for different GB structures and orientations.

Numerical Modelling of the Influence of Tidal Stresses on Fracture Patterns on the Surface of Europa

AbstractJupiter’s satellite Europa is covered by an ice shell exhibiting many surface features, including linear structures called lineae, which in this work are treated as fractures. A three-dimensional finite-element simulator is used to numerically model fracture nucleation, growth, and interaction, assuming the ice is an isotropic, linear elastic medium. Tidal stresses are exerted upon the ice through the Jovian orbital relationship. These stresses are calculated using a closed-form model derived from first principles. The fractures grow in response to stress concentration around their tips, and a damage criterion models the weakening of the ice matrix. Three-dimensional non-planar multiple fracture growth is modelled as a function of geometric multi-modal stress intensity factors computed at the fracture tips. Fracturing is evaluated over multi-scale periods, from days to millions of years, thus capturing multiple tidal effects. Fracture behaviour is modelled across the Europan surface in one domain. The patterns are dense clusters of lineae about the stress maxima with diffuse fracturing in outlying regions. Fractures are also modelled in the vicinity of subsurface meltwater lenses, where fractures form parallel to the surface in contrast to the usual perpendicular orientation. The resultant fracture patterns are qualitatively compared against images from NASA’s Galileo mission. This work contributes to the understanding of Europan lineae by illustrating how they behave in a fracture mechanics framework, and suggests interesting results regarding lineae interaction with meltwater lenses. This work is also a proof of concept for this modelling approach, and will serve as the framework for future work.

How are microstructural defect interactions linked to simultaneous intergranular and transgranular fracture modes in polycrystalline systems?

The major objective of this investigation is to fundamentally understand and predict how intergranular (IG) and transgranular (TG) fracture modes nucleate and propagate in f.c.c. polycrystalline systems due to defects, such as total and partial dislocation densities and grain boundary (GB) structures and misorientations. A dislocation density crystalline plasticity (DCP) formulation based on the evolution and interaction of total and partial dislocation densities was integrated with a recently developed fracture approach to investigate the fracture nucleation and propagation of simultaneous multiple fracture events, including both IG and TG fracture events. The aggregate grains and GB orientations and morphologies are based on EBSD measurements that are representative of polycrystalline copper aggregates with a broad range of random high angle and low angle GBs. The predictions indicate that dislocation density pileups induce IG fracture due to interrelated stress, slip, and total and partial dislocation density accumulations and interactions for both high angle GBs (HAGBs) and low angle GBs (LAGBs). TG fracture nucleation and propagation occurred due to normal stress accumulations, which exceeds the fracture stress, along cleavage planes. Furthermore, it is shown how IG cracks transition from the GB plane to the cleavage planes as cracks nucleate and propagate. Accumulated plastic zones due to different slip system activities can impede and blunt crack propagation and fronts. These plastic zones form due to high Lomer and Shockley partial dislocation densities. These predictions, which are consistent with experimental observations, provide a fundamental understanding of how simultaneous failure modes initiate and propagate for physically representative microstructures.

Strain energy dependent numerical simulation of rock fracture initiation and propagation using soundless cracking demolition agents (SCDA): Effects of SCDA-filled rock joints

Soundless Cracking Demolition Agents (SCDA) are gaining traction as an innovative technique for controlled rock fracturing in applications such as underground energy, mineral extraction, and excavation. This paper presents a novel numerical approach to simulate SCDA-charged rock fracture initiation using the strain energy density of SCDA. This approach eliminates the need for prior knowledge of the peak expansive pressure developed in a rock, a common requirement in conventional SCDA modelling. An empirically derived model is presented which captures the rate of energy release in SCDA as a function of applied strain to the surrounding material. The model was implemented in the 3-dimensional Distinct Element Code by applying an exponential strain-rate decay function. The proposed fracture stimulation method accurately mimics the expansion behaviour and finite strain energy release from SCDA expansion. The method was extended to simulate SCDA expansion in rock joints. The SCDA expansion method was assessed by creating a numerical model in which a central injection well was intersected by a horizontal joint and SCDA expansion was simulated in the injection well and the joint. The simulations were verified against laboratory fracture experiments. The model was extended to evaluate the fracturing performance of SCDA under varying joint roughness and confining pressures. SCDA expansion in rock joints introduces an additional compressive stress field that inhibits fracture initiation and propagation near the joint. Increasing fracture roughness suppressed fracture growth, but led to more fracture nucleation points around the injection well and shear damage at the joint interface. The simulation method employed shows sensitivity to confining pressures and simulates the fracturing potential of SCDA in different rocks subjected to confining pressures without requiring further calibration of the SCDA expansion parameters. The proposed numerical simulation method improves the accuracy in investigating the fracturing potential of expansive agents over existing simulation methods for SCDA.

A multiscale preconditioner for crack evolution in porous microstructures: Accelerating phase‐field methods

AbstractPhase‐field methods are attractive for simulating the mechanical failure of geometrically complex porous microstructures described by 2D/3D x‐ray CT images in subsurface (e.g., CO storage) and manufacturing (e.g., Li‐ion battery) applications. They capture the nucleation, growth, and branching of fractures without prior knowledge of the propagation path or having to remesh the domain. Their drawback lies in the high computational cost for the typical domain sizes encountered in practice. We present a multiscale preconditioner that significantly accelerates the convergence of Krylov solvers in computing solutions of linear(ized) systems arising from the sequential discretization of the momentum and crack‐evolution equations in phase‐field methods. The preconditioner is an algebraic reformulation of a recent pore‐level multiscale method (PLMM) by the authors and consists of a global preconditioner and a local smoother . Together, and attenuate low‐ and high‐frequency errors simultaneously. The proposed , used in the momentum equation only, is a simplification of a recent variant proposed by the authors that is much cheaper and easier to deploy in existing solvers. The smoother , used in both the momentum and crack‐evolution equations, is built such that it is compatible with and more robust and efficient than black‐box smoothers like ILU(). We test and systematically for static‐ and evolving‐crack problems on complex 2D/3D porous microstructures, and show that they outperform existing algebraic multigrid solvers. We also probe different strategies for updating as cracks evolve and show the associated cost can be minimized if is updated adaptively and infrequently. Both and are scalable on parallel machines and can be implemented non‐intrusively in existing codes.

On the possibility of detecting potential fracture nuclei in the 09G2S steel by magnetic measurements

The paper reports the results of measuring the magnetic characteristics (coercive force, residual induction, and maximum magnetic permeability) of a specimen made of the 09G2S low-carbon steel in different sectors of its gauge length, depending on the value of tensile load. Measurements are made in both major and minor magnetization reversal cycles. The principal possibility of using the coercive force in the minor cycles of the magnetization reversal of plastically deformed steel products in weak fields to detect potential fracture nucleation at early stages is experimentally shown.

Control of Mechanical and Fracture Properties in Two-Phase Materials Reinforced by Continuous, Irregular Networks.

Composites with high strength and high fracture resistance are desirable for structural and protective applications. Most composites, however, suffer from poor damage tolerance and are prone to unpredictable fractures. Understanding the behavior of materials with an irregular reinforcement phase offers fundamental guidelines for tailoring their performance. Here, the fracture nucleation and propagation in two phase composites, as a function of the topology of their irregular microstructures is studied. A stochastic algorithm is used to design the polymeric reinforcing network, achieving independent control of topology and geometry of the microstructure. By tuning the local connectivity of isodense tiles and their assembly into larger structures, the mechanical and fracture properties of the architected composites are tailoredat the local and global scale. Finally, combining different reinforcing networks into a spatially determined meso-scale assembly, it is demonstrated how the spatial propagation of fracture in architected composite materials can be designed and controlled a priori.

Microseismic Monitoring of the Fracture Nucleation Mechanism and Early Warning for Cavern Rock Masses

The rock mass is susceptible to instability and damage during cavern construction. The blast-induced cracking process of the rock mass contains a wealth of information about the precursors of instability, and the identification of fracture nucleation signals is a prerequisite for effective hazard warning. A laboratory mechanical test and microseismic (MS) monitoring were carried out in the Baihetan Cavern to investigate the fracture nucleation process in the rock mass. MS monitoring shows that pre-existing microcracks were closed or new cracks were generated under the action of high stress, which caused the migration of microcracks. As the crack density increases, the fracture interaction gradually increases. The study of the rock fracture nucleation mechanism helps to reveal the MS sequences during the rock fracture process, and the fore-main shock was found in the MS sequence during access tunnel excavation. This study can effectively provide guidance for the early warning of rock mass failure and the stability analysis of underground caverns under blasting excavation disturbance.

A phase‐field model for hydraulic fracture nucleation and propagation in porous media

AbstractMany geo‐engineering applications, for example, enhanced geothermal systems, rely on hydraulic fracturing to enhance the permeability of natural formations and allow for sufficient fluid circulation. Over the past few decades, the phase‐field method has grown in popularity as a valid approach to modeling hydraulic fracturing because of the ease of handling complex fracture propagation geometries. However, existing phase‐field methods cannot appropriately capture nucleation of hydraulic fractures because their formulations are solely energy‐based and do not explicitly take into account the strength of the material. Thus, in this work, we propose a novel phase‐field formulation for hydraulic fracturing with the main goal of modeling fracture nucleation in porous media, for example, rocks. Built on the variational formulation of previous phase‐field methods, the proposed model incorporates the material strength envelope for hydraulic fracture nucleation through two important steps: (i) an external driving force term, included in the damage evolution equation, that accounts for the material strength; (ii) a properly designed damage function that defines the fluid pressure contribution on the crack driving force. The comparison of numerical results for two‐dimensional test cases with existing analytical solutions demonstrates that the proposed phase‐field model can accurately model both nucleation and propagation of hydraulic fractures. Additionally, we present the simulation of hydraulic fracturing in a three‐dimensional domain with various stress conditions to demonstrate the applicability of the method to realistic scenarios.

An enriched mixed finite element model for the simulation of microseismic and acoustic emissions in fractured porous media with multi‐phase flow and thermal coupling

AbstractA novel mixed enriched finite element model is developed for coupled non‐linear thermo‐hydro‐mechanical simulation of fractured porous media with three‐phase flow and thermal coupling. Simulation of induced acoustic emission (AE) and microseismic emission (ME) due to tensile fracturing and shear slip instability of pre‐existing fracture interfaces is carried out and the numerical results of the emitted signals are analysed. The mathematical model is based on the generalized Biot's theory for coupled interaction of solid and fluid phases. A computationally robust non‐linear solver is developed to handle the severe non‐linearities arising from fluid saturations, relative permeabilities of fluids, constitutive models of interfaces and convective thermal coupling. To model pre‐existing natural fractures and faults, discrete fracture propagation and nucleation of cracks (micro‐cracking) independently of the original mesh topology, a local Partition‐of‐Unity (PU) finite element method, namely, the Phantom Node Method (PNM) is implemented. The cohesive fracture modelling scheme is implemented to account for the non‐linear behaviour of fracturing and localization, and to rectify the non‐physical stress singularity condition at the fracture tip. Effects of different system parameters on fracturing, shear‐slip instability and the associated induced AEs and MEs are investigated through various numerical results.

How can machine learning be used for accurate representations and predictions of fracture nucleation in zirconium alloys with hydride populations?

Zirconium alloys are critical material components of systems subjected to harsh environments such as high temperatures, irradiation, and corrosion. When exposed to water in high temperature environments, these alloys can thermo-mechanically degrade by forming hydrides that have a crystalline structure that is different from that of zirconium. Cracks can nucleate near these hydrides; hence, these hydrides are a direct link to fracture failure and overall large inelastic strain deformation modes. To fundamentally understand and predict these microstructural failure modes, we interrogated a finite-element database that was deterministically tailored and generated for large strain-dislocation-density crystalline plasticity and fracture modes. A database of 210 simulations was created to randomly sample from a group of microstructural fingerprints that encompass hydride volume fraction, hydride orientation, grain orientation, hydride length, and hydride spacing for a hydride that is physically representative of an aggregate of a hydride population. Machine learning approaches were then used to understand, identify, and characterize the dominant microstructural mechanisms and characteristics. We first used fat-tailed Cauchy distributions to determine the extreme events. A multilayer perceptron was used to learn the mechanistic characteristics of the material response to predefined strain levels and accurately determine the critical fracture stress response and the accumulated shear slips in critical regions. The predictions indicate that hydride volume fraction, a population-level parameter, had a significant effect on localized parameters, such as fracture stress distribution regions, and on the accumulated immobile dislocation densities both within the face centered cubic hydrides and the hexagonal cubic packed h.c.p. matrix.

Proposal of a hydrogen embrittlement index for a martensitic advanced high-strength steel

This paper proposes a definition of a Hydrogen Embrittlement Index for assessing structural components made by a martensitic advanced high-strength steel. A material model, including a damage model and a fracture criterion, was developed to reproduce the tensile behaviour until fracture nucleation under different hydrogen concentrations. The analysis of the tests with different specimen geometries demonstrated that the maximum principal strain at fracture is correlated to hydrogen concentration, and it can be used to define the embrittlement index for quantifying the hydrogen susceptibility of material in a wide range of notch severity.

Effect of a singular planar heterogeneity on tensile failure

Many rocks contain planar heterogeneities, in the form of open fractures, veins and/or stylolites, but scarce data exist on how strength and fracture pattern formation is affected by the presence of a singular planar heterogeneity in an otherwise uniform matrix. The mechanics of stylolite-bearing and/or fractured limestone is of interest to several engineering applications, from quarries to subsurface gas or geothermal reservoirs. We have performed Brazilian Disc tests on pre-fractured Indiana limestone samples and Treuchtlinger Marmor discs which contain cohesive stylolites, investigating Brazilian test Strength and the resulting fracture pattern. All experiments were filmed, and where possible analyzed with particle image velocimetry. When viewed in 2D, the planar discontinuity was set at different rotation angles compared to the principal loading direction, where perpendicular to the loading direction is defined as 0⁰. The results show that all samples are weaker than their intact counterparts. For the pre-fractured Indiana limestone, there is 10–75% angle-dependent weakening. However, in the samples with a stylolite, strength is weakened by 35–75%, independent of direction. Several new cracks appeared when fracturing a stylolite-sample, where the orientation is heavily influenced by the stylolite orientation. The fracture pattern and associated stress drops are more complex for high angles. In these samples always more than one fracture formed, whereas in pre-fractured samples usually only one new fracture formed. This suggests a potential for more permeability increase when hydrofracturing a stylolite-rich interval. Comparison with Finite Element Models indicates that this difference in fracture pattern is caused by the strength contrast between the anastomosing stylolite zone and the matrix material, leading to stress concentrations effects. This causes (micro-) fracture nucleation to occur locally, promotes fracture coalescence and fracture growth at lower overall sample-load conditions compared to intact samples.

Effect of fluid pressure on adhesive wear of spherical contact

Lubrication with fluid between two rough surfaces has been extensively studied, while the present work provides an additional principle for its effect on adhesive wear reduction associated with the fluid pressure. In this work, a finite element model for elastoplastic spherical contact with fluid pressure applied on surface is established, considering the realistic ductile failure criteria. It is found that as the fluid pressure increases, the location of fracture nucleation in the front contact region moves towards the contact surface. Two wear modes, namely the detaching and scratching modes, are identified by defining a critical dimensionless fluid pressure. Accordingly, the wear coefficient decreases as the fluid pressure increases, serving as the additional principle for wear reduction in lubrication systems.