A phase-field model for simulating the propagation behavior of mixed-mode cracks during the hydraulic fracturing process in fractured reservoirs

A novel phase-field model for the propagation of mixed-mode hydraulic fractures,characterized by the formation of mixed-mode fractures due to the interactions between fluids and solids,is proposed.In this model,the driving force for the phase field consists of both tensile and shear components,with the fluid contribution primarily manifesting in the tension driving force.The displacement and pressure are solved simultaneously by an implicit method.The numerical solution's iterative format is established by the finite element discretization and Newton-Raphson(NR)iterative methods.The correctness of the model is verified through the uniaxial compression physical experiments on fluid-pressurized rocks,and the limitations of the hydraulic fracture expansion phase-field model,which only considers mode I fractures,are revealed.In addition,the influence of matrix mode II fracture toughness value,natural fracture mode II toughness value,and fracturing fluid injection rate on the hydraulic fracture propagation in porous media with natural fractures is studied.

Simulation of Corrosion-Induced Cracking of Reinforced Concrete Based on Fracture Phase Field Method

Accurate simulation of the cracking process caused by rust expansion of reinforced concrete(RC)structures plays an intuitive role in revealing the corrosion-induced failure mechanism.Considering the quasi-brittle fracture of concrete,the fracture phase field driven by the compressive-shear term is constructed and added to the traditional brittle fracture phase field model.The rationality of the proposed model is verified by a mixed fracture example under a shear displacement load.Then,the extended fracture phase model is applied to simulate the corrosion-induced cracking process of RC.The cracking patterns caused by non-uniform corrosion expansion are discussed for RC specimens with homogeneous macroscopically or heterogeneous with different polygonal aggregate distributions at the mesoscopic scale.Then,the effects of the protective layer on the crack propagation trajectory and cracking resistance are investigated,illustrating that the cracking angle and cracking resistance increase with the increase of the protective layer thickness,consistent with the experimental observation.Finally,the corrosion-induced cracking process of concrete specimens with large and small spacing rebars is simulated,and the interaction of multiple corrosion cracking is easily influenced by the reinforcement spacing,which increases with the decrease of the steel bar interval.These conclusions play an important role in the design of engineering anti-corrosion measures.The fracture phase field model can provide strong support for the life assessment of RC structures.

Numerical manifold method for thermo-mechanical coupling simulation of fractured rock mass

As a calculation method based on the Galerkin variation,the numerical manifold method(NMM)adopts a double covering system,which can easily deal with discontinuous deformation problems and has a high calculation accuracy.Aiming at the thermo-mechanical(TM)coupling problem of fractured rock masses,this study uses the NMM to simulate the processes of crack initiation and propagation in a rock mass under the influence of temperature field,deduces related system equations,and proposes a penalty function method to deal with boundary conditions.Numerical examples are employed to confirm the effectiveness and high accuracy of this method.By the thermal stress analysis of a thick-walled cylinder(TWC),the simulation of cracking in the TWC under heating and cooling conditions,and the simulation of thermal cracking of the SwedishÄspöPillar Stability Experiment(APSE)rock column,the thermal stress,and TM coupling are obtained.The numerical simulation results are in good agreement with the test data and other numerical results,thus verifying the effectiveness of the NMM in dealing with thermal stress and crack propagation problems of fractured rock masses.

Improved Staggered Algorithm for Phase-Field Brittle Fracture with the Local Arc-Length Method

The local arc-length method is employed to control the incremental loading procedure for phase-field brittle fracture modeling.An improved staggered algorithm with energy and damage iterative tolerance convergence criteria is developed based on the residuals of displacement and phase-field.The improved staggered solution scheme is implemented in the commercial software ABAQUS with user-defined element subroutines.The layered system of finite elements is utilized to solve the coupled elastic displacement and phase-field fracture problem.A one-element benchmark test compared with the analytical solution was conducted to validate the feasibility and accuracy of the developed method.Our study shows that the result calculated with the developed method does not depend on the selected size of loading increments.The results of several numerical experiments show that the improved staggered algorithm is efficient for solving the more complex brittle fracture problems.

Peridynamic Study on Fracture Mode and Crack Propagation Path of a Plate with Multiple Cracks Subjected to Uniaxial Tension

How to simulate fracture mode and crack propagation path in a plate with multiple cracks is an attractive but difficult issue in fracture mechanics.Peridynamics is a recently developed nonlocal continuum formulation that can spontaneously predict the crack nucleation,branch and propagation in materials and structures through a meshfree discrete technique.In this paper,the peridynamic motion equation with boundary traction is improved by simplifying the boundary transfer functions.We calculate the critical cracking load and the fracture angles of the plate with multiple cracks under uniaxial tension.The results are consistent with those predicted by classical fracture mechanics.The fracture mode and crack propagation path are also determined.The calculation shows that the brittle fracture process of the plate with multiple cracks can be conveniently and correctly simulated by the peridynamic motion equation with boundary conditions.

Machine learning applications for the prediction of extended length of stay in geriatric hip fracture patients

BACKGROUND Geriatric hip fractures are one of the most common fractures in elderly individuals,and prolonged hospital stays increase the risk of death and complications.Machine learning(ML)has become prevalent in clinical data processing and predictive models.This study aims to develop ML models for predicting extended length of stay(eLOS)among geriatric patients with hip fractures and to identify the associated risk factors.AIM To develop ML models for predicting the eLOS among geriatric patients with hip fractures,identify associated risk factors,and compare the performance of each model.METHODS A retrospective study was conducted at a single orthopaedic trauma centre,enrolling all patients who underwent hip fracture surgery between January 2018 and December 2022.The study collected various patient characteristics,encompassing demographic data,general health status,injury-related data,laboratory examinations,surgery-related data,and length of stay.Features that exhibited significant differences in univariate analysis were integrated into the ML model establishment and subsequently cross-verified.The study compared the performance of the ML models and determined the risk factors for eLOS.RESULTS The study included 763 patients,with 380 experiencing eLOS.Among the models,the decision tree,random forest,and extreme Gradient Boosting models demonstrated the most robust performance.Notably,the artificial neural network model also exhibited impressive results.After cross-validation,the support vector machine and logistic regression models demonstrated superior performance.Predictors for eLOS included delayed surgery,D-dimer level,American Society of Anaesthesiologists(ASA)classification,type of surgery,and sex.CONCLUSION ML proved to be highly accurate in predicting the eLOS for geriatric patients with hip fractures.The identified key risk factors were delayed surgery,D-dimer level,ASA classification,type of surgery,and sex.This valuable information can aid clinicians in allocating resources more efficiently to meet patient demand effectively.

Fatigue Crack Growth Rate of Ti-6Al-4V Considering the Effects of Fracture Toughness and Crack Closure

Fatigue fracture is one of the main failure modes of Ti-6A1-4V alloy,fracture toughness and crack closure have strong effects on the fatigue crack growth（FCG）rate of Ti-6A1-4V alloy.The FCG rate of Ti-6A1-4V is investigated by using experimental and analytical methods.The effects of stress ratio,crack closure and fracture toughness on the FCG rate are studied and discussed.A modified prediction model of the FCG rate is proposed,and the relationship between the fracture toughness and the stress intensity factor（SIF）range is redefined by introducing a correcting coefficient.Notched plate fatigue tests（including the fracture toughness test and the FCG rate test）are conducted to investigate the influence of affecting factors on the FCG rate.Comparisons between the predicted results of the proposed model,the Paris model,the Walker model,the Sadananda model,and the experimental data show that the proposed model gives the best agreement with the test data particularly in the near-threshold region and the Paris region,and the corresponding calculated fatigue life is also accurate in the same regions.By considering the effects of fracture toughness and crack closure,the novel FCG rate prediction model not only improves the estimating accuracy,but also extends the adaptability of the FCG rate prediction model in engineering.

Fracture of two three-dimensional parallel internal cracks in brittle solid under ultrasonic fracturing

Similar to hydraulic fracturing(HF), the coalescence and fracture of cracks are induced within a rock under the action of an ultrasonic field, known as ultrasonic fracturing(UF). Investigating UF is important in both hard rock drilling and oil and gas recovery. A three-dimensional internal laser-engraved crack(3D-ILC) method was introduced to prefabricate two parallel internal cracks within the samples without any damage to the surface. The samples were subjected to UF. The mechanism of UF was elucidated by analyzing the characteristics of fracture surfaces. The crack propagation path under different ultrasonic parameters was obtained by numerical simulation based on the Paris fatigue model and compared to the experimental results of UF. The results show that the 3D-ILC method is a powerful tool for UF research.Under the action of an ultrasonic field, the fracture surface shows the characteristics of beach marks and contains powder locally, indicating that the UF mechanism includes high-cycle fatigue fracture, shear and friction, and temperature load. The two internal cracks become close under UF. The numerical result obtained by the Paris fatigue model also shows the attraction of the two cracks, consistent with the test results. The 3D-ILC method provides a new tool for the experimental study of UF. Compared to the conventional numerical methods based on the analysis of stress-strain and plastic zone, numerical simulation can be a good alternative method to obtain the crack path under UF.

HIGH TEMPERATURE CRACK PROPAGATION AND FRACTURE OF SUPERALLOYS

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CRACK GROWTH BEHAVIOR AND FRACTURE FEATURE OF GH2132 UNDER CREEP-FATIGUE CONDITIONS

The superalloy GH2132 is equivalent to A286. The tests were carried out at 550°C under various cyclic frequencies (hold time) and load levels, and the fracture surfaces were examined by using a scanning electron microscope. It was shown that the fracture mode of creep-fatigue and the effect of cyclic loading on crack growth change with the growth of crack and the increase of net-section stress, and both are reversed when the net-section stress is up to the yield stress of material. When σn0.2, cracking is predominantly cyclic-dependent transgranular and cyclic loading accelerates creep crack growth, whereas when σn>σ0.2, the case is reversed.

Use of double edge-cracked Brazilian disk geometry for compression-shear fracture investigation of rock

A new specimen geometry-the double edge-cracked Brazilian disk and a relevant fracture analysis byweight function method are proposed for the investigation of rock fracture caused by compression-shear loading. Notonly can the mixed mode fracture with any ratio of KⅠ/KⅡ be achieved, but also the pure mode Ⅱ crack extensioncan be obtained. The combined mode fracture analysis for this geometry shows that diametral compression in the far-field can induce a compression-shear stress state in the singular stress field ahead of crack tips. Experimental investi-gations conducted on marble specimens show that the pure mode Ⅱ crack extension can be obtained when the dimen-sionless crack length a＞0.7 and the inclined crack angle 5°≤ψ≤40°. Normalized mode Ⅰ and mode Ⅱ stress inten-sity factors decrease from -0.45 and 2.47 at ψ=5° to -1.65 and 1.52 at ψ=40°, respectively. The strains at threepoints of specimen are also measured in order to investigate the influence of stress singularity on initial crack exten-sion. The results show that the principal orientations of strain at three points are very stable in the loading process.The derived formulae are quite explicit, and the specimen geometry is easy to fabricate and convenient to achieve thepure mode Ⅱ crack extension. Therefore, it can hopefully be used to obtain mode Ⅱ fracture toughness of rock.

Influence of gravel content and cement on conglomerate fracture

Tight reservoirs are typically developed by horizontal wells and multi-stage hydraulic fracturing.The conglomerate reservoir is one type of tight reservoirs,which is different from homogeneous rock,such as tight sandstone.This is because that the existence of gravels makes conglomerate have strong hetero-geneity.Thus,it is difficult to grasp the fracture mechanism and the law of fracture propagation of conglomerate,which limits the efficient development of the conglomerate reservoir.In this paper,the fracture characteristics and factors influencing the fracturing of Mahu conglomerate were studied by uniaxial compression,acoustic emission monitoring and X-ray computed tomography(CT)scanning experiments.The results show that the fracture characteristics of conglomerates are influenced by the gravel content and cement.The conglomerate in the study area is mainly divided into carbonate cemented conglomerate and clay cemented conglomerate.The fracture complexity of carbonate cemented conglomerate first increases and then decreases with increasing gravel content.However,for clay cemented conglomerates,the fracture complexity increases over the gravel content.The crack development stress is a significant parameter in the crack assessment of conglomerates.This study is useful to understand the influence of meso-fabric characteristics of conglomerates on their fracturing and crack evolution and guides the design of hydraulic fracturing.

Fracture features of brittle coal under uniaxial and cyclic compression loads

Under the efects of complex geological and stress environments,burst hazards continue to be a major challenge for underground space utilization and deep resources exploration as its occurrence can lead to personnel causalities,equipment damage and structural collapse.Considering the stress path experienced by in-situ coal body,cyclic loading appears in quite various forms for instance shearer cutting,overlying strata breakage,hydro-fracturing and blasting,during tunnel,mining and underground space utilizing process.The stability of the underground coal body subject to periodic loading/unloading stress is extremely important for maintain the function of designed engineering structure for waste storage,safe mining,roadway development,gas recovery,carbon sequestration and so on.The mechanical properties of hard rock subject to cyclic fatigue loads has been intensively investigated by many researchers as the rock burst induced by supercritical loads has long been a safety risk and engineering problems for civil and tunneling engineering under deep overburden.More recently,the mechanical properties of coal samples under cyclic fatigue loads is investigated from the aspect of hysteresis,energy dissipation and irreversible damage as the burst hazards of brittle coal is rising in many countries.However,the crack propagation and fracture pattern of brittle coal need more research to understand the micro mechanism of burst incubation subject to cyclic fatigue loads as brittle coal can store more elastic strain energy and rapidly release the energy when its ultimate strength once reached.This research studied the internal crack status corresponding to diferent cyclic fatigue loading stage of brittle coal samples.The AE monitoring was applied during the uniaxial and cyclic loading process of brittle coal samples to record the crack intensity of samples at diferent loading stages.The damage evolution curve corresponding to loading status was then determined.The fracture pattern of coal samples determined by micro-CT scan was observed and discussed.It has been found by this paper that brittle coal of uniaxial compression tests demonstrated sudden failure caused by major splitting fracture while that of cyclic fatigue tests experienced progressive failure with mixture fracture network.

Experimental Study of Mode-I and Mode-II Interlaminar Fracture Characteristics of Poplar LVL

Fracture is a common failure form of poplar laminated veneer lumber(LVL).In the present work,we performed an experimental study on the mode-I along-grain interlaminar fracture,mode-I cross-grain interlaminar fracture,and mode-II interlaminar fracture of poplar LVL.We investigated stress mechanisms,failure modes,and fracture toughness values of the different fracture types.The experimental results revealed that the crack in the mode-I along-grain interlaminar fracture specimen propagated along the prefabricated crack direction,and the crack tip broke.The mode-I cross-grain interlaminar fracture specimen had cracks in the vertical direction near the prefabricated crack.In the mode-II interlaminar fracture specimen,cracks appeared along the initial prefabricated crack direction.The load–displacement curves of these three specimens were linear in the early stage of loading.With the increase in the load,a nonlinear segment appeared before crack propagation and a descending segment appeared after crack propagation.The nonlinear segments of the mode-I along-grain interlaminar fracture and mode-II interlaminar fracture were very short,and cracks expanded quickly after their initiation,resulting in brit-tle fracture.The nonlinear segment of the mode-I cross-grain interlaminar fracture was long,resulting in plastic failure.The average toughness values of the mode-I along-grain interlaminar fracture,mode-I cross-grain inter-laminar fracture,and mode-II interlaminar fracture were 15.43,270.15,and 39.72 MPa·mm^(1/2),respectively.

Experimental study of multi-timescale crack blunting in hydraulic fracture

Hydraulic fracture is important in unconventional oil and gas exploration.During the propagation of the hydraulic fracture,the crack tip is blunted due to the development of the process zone in the near-tip area.In this study,the blunting of the hydraulic fracture in polymethyl methacrylate specimens due to multi-timescale stress concentration is investigated.The ratio of the initiation toughness to the arrest toughness of the blunted hydraulic fracture is measured using both the dynamic and the static methods.Results show that a hydraulic fracture can be blunted with the time span of stress concentration from 1 ms to 600 s.It is also shown that the blunting of hydraulic fracture is a highly localized process.The morphology of the blunted crack depends on the stress distribution in the vicinity of the crack tip.

THERMAL FRACTURE OF FUNCTIONALLY GRADED PLATE WITH PARALLEL SURFACE CRACKS

This work examines the fracture behavior of a functionally graded material （FGM） plate containing parallel surface cracks with alternating lengths subjected to a thermal shock. The thermal stress intensity factors （TSIFs） at the tips of long and short cracks are calculated using a singular integral equation technique. The critical thermal shock △Tc that causes crack initiation is calculated using a stress intensity factor criterion. Numerical examples of TSIFs and △Tc for an Al2O3/Si3N4 FGM plate are presented to illustrate the effects of thermal property gradation, crack spacing and crack length ratio on the TSIFs and △Tc. It is found that for a given crack length ratio, the TSIFs at the tips of both long and short cracks can be reduced significantly and △Tc can be enhanced by introducing appropriate material gradation. The TSIFs also decrease dramatically with a decrease in crack spacing. The TSIF at the tips of short cracks may be higher than that for the long cracks under certain crack geometry conditions. Hence, the short cracks instead of long cracks may first start to grow under the thermal shock loading.

PROBABILISTIC MODELING OF BRITTLE FRACTURE WITH PRIOR DUCTILE CRACK EXTENSION

A computation framework for brittle fracture which incorporates weakest link statistics and a microme- chanics model reflecting reflecting local damage of the material is described.The Weibull stress W emerges as a probabilistic fracture parameter to define the condition leading material failure. Unstable crack propa- gation occurs at a critical value of W which may be attained paior to or following some amount of duc- tile crack extension. A realistic model of ductile crack growth using the computation cell methodology is used to define the evolution of near tip stress fields during crack extension. An application of proposed framework to predict the measured geometry and ductile tearing effects on the statistical distributio of fracture toughness for the pipe line steel welded joint is described.

Fracture behaviors of columnar jointed rock mass using interface mechanics theorem

For a special geological structure of columnar jointed rock mass(CJRM),its mechanical properties are strongly affected by the columnar joints.To describe the fracture behaviors of CJRM using the basic theories of interface mechanics for composite materials,the interface stresses of the vertical and horizontal joints,which are the two primary joints in the CJRM under triaxial compression,are studied,and their mathematical expressions are derived based on the superposition principle.Based on the obtained interface stresses of the vertical and horizontal joints in the CJRM,the crack initiation of the joint interface in the CJRM is studied using the maximum circumferential stress theory in fracture mechanics.Moreover,based on this investigation,the fracture behaviors of CJRM are analyzed.According to the results of similar material physical model tests for the CJRM,the theoretical study is verified.Finally,the influence of the mechanical parameters of the CJRM on the joint interface stress is discussed comprehensively.

New role of α phase in the fracture behavior and fracture toughness of a β-type bio-titanium alloy

The role of α precipitates formed during aging in the fracture toughness and fracture behavior of β-type bio-titanium alloy Ti–29Nb–13Ta–4.6Zr(TNTZ) was studied. Results showed that the fracture toughness of the TNTZ alloy aged at 723 K decreases to the minimum of 72.07–73.19 kJ·m^(-2)when the aging time is extended to 4–8 h and then gradually increases and reaches 144.89 kJ·m^(-2)after 72 h. The decrease in fracture toughness within the aging time of 4–8 h is caused by the large stress concentration at the tip of acicular α precipitates with a high aspect ratio and the preferential crack propagation along the inhomogeneous acicular α precipitates distributed in “V-shape” and “nearly perpendicular shape”. When the aging time is extended to 8–72 h, the precrack tip is uniformly blunted, and the crack is effectively deflected by α precipitates with multi long axis directions, more high homogeneity, low aspect ratio, and large number density. Analysis of the effect of αprecipitates on the fracture behavior suggested that the number of long axis directions of α precipitates is the key controlling factor for the fracture behavior and fracture toughness of the TNTZ alloy aged for different times.

SIF-based fracture criterion for interface cracks

The complex stress intensity factor K governing the stress field of an interface crack tip may be split into two parts, i.e.,■ and s^（-iε）, so that K = ■ s^（-iε）, s is a characteristic length and ε is the oscillatory index. ■ has the same dimension as the classical stress intensity factor and characterizes the interface crack tip field. That means a criterion for interface cracks may be formulated directly with■, as Irwin（ASME J. Appl. Mech. 24：361–364, 1957） did in 1957 for the classical fracture mechanics. Then, for an interface crack,it is demonstrated that the quasi Mode I and Mode II tip fields can be defined and distinguished from the coupled mode tip fields. Built upon SIF-based fracture criteria for quasi Mode I and Mode II, the stress intensity factor（SIF）-based fracture criterion for mixed mode interface cracks is proposed and validated against existing experimental results.