P-and SV-wave dispersion and attenuation in saturated microcracked porous rock with aligned penny-shaped fractures

P-and SV-wave dispersion and attenuation have been extensively investigated in saturated poroelastic media with aligned fractures.However,there are few existing models that incorporate the multiple wave attenuation mechanisms from the microscopic scale to the macroscopic scale.Hence,in this work,we developed a unified model to incorporate the wave attenuation mechanisms at different scales,which includes the microscopic squirt flow between the microcracks and pores,the mesoscopic wave-induced fluid flow between fractures and background(FB-WIFF),and the macroscopic Biot's global flow and elastic scattering(ES)from the fractures.Using Tang's modified Biot's theory and the mixed-boundary conditions,we derived the exact frequency-dependent solutions of the scattering problem for a single penny-shaped fracture with oblique incident P-and SV-waves.We then developed theoretical models for a set of aligned fractures and randomly oriented fractures using the Foldy approximation.The results indicated that microcrack squirt flow considerably influences the dispersion and attenuation of P-and SV-wave velocities.The coupling effects of microcrack squirt flow with the FB-WIFF and ES of fractures cause much higher velocity dispersion and attenuation for P waves than for SV waves.Randomly oriented fractures substantially reduce the attenuation caused by the FB-WIFF and ES,particularly for the ES attenuation of SV waves.Through a comparison with existing models in the limiting cases and previous experimental measurements,we validated our model.

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.

Identifying the real fracture hidden in rock microcrack zone by acoustic emission energy

Identifying the real fracture of rock hidden in acoustic emission(AE)source clusters(AE-depicted microcrack zone)remains challenging and crucial.Here we revealed the AE energy(representing dissipated energy)distribution rule in the rock microcrack zone and proposed an AE-energy-based method for identifying the real fracture.(1)A set of fracture experiments were performed on granite using wedgeloading,and the fracture process was detected and recorded by AE.The microcrack zone associated with the energy dissipation was characterized by AE sources and energy distribution,utilizing our selfdeveloped AE analysis program(RockAE).(2)The accumulated AE energy,an index representing energy dissipation,across the AE-depicted microcrack zone followed the normal distribution model(the mean and variance relate to the real fracture path and the microcrack zone width).This result implies that the nucleation and coalescence of massive cracks(i.e.,real fracture generation process)are supposed to follow a normal distribution.(3)Then,we obtained the real fracture extension path by joining the peak positions of the AE energy normal distribution curve at different cross-sections of the microcrack zone.Consequently,we distinguished between the microcrack zone and the concealed real fracture within it.The deviation was validated as slight as 1–3 mm.

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.

Calculating the number of radial cracks around a wellbore fractured by liquid CO2 phase transition blasting technology

Integrating liquid CO_(2)phase transition blasting(LCPTB)technology with hydraulic fracturing(HF)methods can help reduce wellbore damage,create multiple radial fractures,and establish a complex fracture network.This approach significantly increases the recovery efficiency of low-permeability oil and gas fields.Accurately calculating the number of fractures caused by LCPTB is necessary to predict production enhancement effects and optimize subsequent HF designs.However,few studies are reported on large-scale physical model experiments in terms of a method for calculating the fracture number.This study analyzed the initiation and propagation of cracks under LCPTB,derived a calculation formula for crack propagation radius under stress waves,and then proposed a new,fast,and accurate method for calculating the fracture number using the principle of mass conservation.Through ten rock-breaking tests using LCPTB,the study confirmed the effectiveness of the proposed calculation approach and elucidated the variation rule of explosion pressure,rock-breaking scenario,and the impact of varying parameters on fracture number.The results show that the new calculation method is suitable for fracturing technologies with high pressure rates.Recommendations include enlarging the diameter of the fracturing tube and increasing the liquid CO2 mass in the tube to enhance fracture effectiveness.Moreover,the method can be applied to other fracturing technologies,such as explosive fracturing(EF)within HF formations,indicating its broader applicability and potential impact on optimizing unconventional resource extraction technologies.

A fracture model for assessing tensile mode crack growth resistance of rocks

Evaluating the fracture resistance of rocks is essential for predicting and preventing catastrophic failure of cracked structures in rock engineering.This investigation developed a brittle fracture model to predict tensile mode(mode I)failure loads of cracked rocks.The basic principle of the model is to estimate the reference crack corresponding to the fracture process zone(FPZ)based on the maximum normal strain(MNSN)ahead of the crack tip,and then use the effective crack to calculate the fracture toughness.We emphasize that the non-singular stress/strain terms should be considered in the description of the MNSN.In this way,the FPZ,non-singular terms and the biaxial stress state at the crack tip are simul-taneously considered.The principle of the model is explicit and easy to apply.To verify the proposed model,laboratory experiments were performed on a rock material using six groups of specimens.The model predicted the specimen geometry dependence of the measured fracture toughness well.More-over,the potential of the model in analyzing the size effect of apparent fracture toughness was discussed and validated through experimental data reported in the literature.The model was demonstrated su-perior to some commonly used fracture models and is an excellent tool for the safety assessment of cracked rock structures.

Correlative Factors Research of Tibial Shaft Fracture with Ipsilateral Posterior Malleolar Crack Fractures

Background:Missed diagnosis rate of spiral tibia shaft fracture with posterior malleolus crack fracture(PMCF)is high in the clinical.However,the mechanism and related factors of fracture are still unclear.Moreover,PMCF has been observed in other types of tibial shaft fractures.Objective:To explore the correlative factors of tibial shaft fracture with ipsilateral PMCF,decrease the rate of clinical missed diagnosis,strengthen the effective fixation of PMCF,and reduce the incidence of traumatic arthritis.Methods:From September 2014 to May 2019,we collected 137 tibiofibular fracture.Only 68 cases involved in ankle joint CT examination and were retrospectively analyzed.The patients were divided into posterior malleolus group(30 cases)and non-posterior malleolus group(38 cases)according to whether come up PMCF or not.The posterior malleolus group contained 24 males and 6 females,27-77(47.57±11.79)years old,the non-posterior malleolus group contained 23 males and 15 females,18-85(48.71±13.84)years old.The gender,age,location,fibula fracture and tibial shaft fracture classification were observed for univariate and multivariate analysis.Results:The probability of PMCF was higher with right tibial shaft fracture(OR=3.6995%CI:1.13-12.08 P<0.05);the probability of PMCF following distal fibular fracture was higher than that without fibular fracture(OR=11.3695%CI:1.72-75.05 P<0.05);the probability of PMCF with type A tibial shaft fracture was higher than type C(OR=4.8295%CI:1.19-19.58 P<0.05).Conclusion:Right type A tibial shaft fracture accompanied by distal fibular fracture are very important factors related to PMCF,which needs highly attention to avoid clinical missed diagnosis.

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.

Effect of heat treatment on stress corrosion cracking, fracture toughness and strength of 7085 aluminum alloy

The influences of heat treatment on stress corrosion cracking （SCC）, fracture toughness and strength of 7085 aluminum alloy were investigated by slow strain rate testing, Kahn tear testing combined with scanning electron microscopy （SEM） and transmission electron microscopy （TEM）. The results show that the fracture toughness of T74 overaging is increased by 22.9% at the expense of 13.6% strength, and retrogression and reaging （RRA） enhances fracture toughness 14.2% without reducing the strength compared with T6 temper. The fracture toughness of dual-retrogression and reaging （DRRA） is equivalent to that of T74 with an increased strength of 14.6%. The SCC resistance increases in the order： T6〈RRA〈DRRA≈T74. The differences of fracture toughness and SCC were explained on the basis of the role of matrix precipitates and grain boundary orecioitates.

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.

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.

Calculation Model of Equivalent Strength for Induced Crack Based on Double-K Fracture Theory and Its Optimizing Setting in RCC Arch Dam

By means of fracture testing on roller-compacted concrete （RCC） three-point bending beams with two different specimen sizes, the P-CMOD complete curve for RCC was gained. Furthermore, by applying double-K fracture theory, KiniⅠC,KunⅠC, as well as the critical effective crack length and the critical crack tip opening displacement, were evaluated. Based on the double-K fracture parameters above, the calculation model of equivalent strength for induced crack was established, thus the calculation method on its initiation, stable propagation and unstable fracture was ascertained. Moreover, the finite element simulation analysis of stress field in ShaPai arch dam and the on-site observational splaying points of induced crack at different altitudes validated the reliability of the model. Finally, crack inducer′s optimal setting in RCC arch dam was studied. It improves the design level of induced crack in RCC arch dam and satisfies the necessity of engineering practice.

Characterizing the influence of stress-induced microcracks on the laboratory strength and fracture development in brittle rocks using a finite-discrete element method-micro discrete fracture network FDEM-μDFN approach

Heterogeneity is an inherent component of rock and may be present in different forms including mineralheterogeneity, geometrical heterogeneity, weak grain boundaries and micro-defects. Microcracks areusually observed in crystalline rocks in two forms： natural and stress-induced; the amount of stressinducedmicrocracking increases with depth and in-situ stress. Laboratory results indicate that thephysical properties of rocks such as strength, deformability, P-wave velocity and permeability areinfluenced by increase in microcrack intensity. In this study, the finite-discrete element method （FDEM）is used to model microcrack heterogeneity by introducing into a model sample sets of microcracks usingthe proposed micro discrete fracture network （mDFN） approach. The characteristics of the microcracksrequired to create mDFN models are obtained through image analyses of thin sections of Lac du Bonnetgranite adopted from published literature. A suite of two-dimensional laboratory tests including uniaxial,triaxial compression and Brazilian tests is simulated and the results are compared with laboratory data.The FDEM-mDFN models indicate that micro-heterogeneity has a profound influence on both the mechanicalbehavior and resultant fracture pattern. An increase in the microcrack intensity leads to areduction in the strength of the sample and changes the character of the rock strength envelope. Spallingand axial splitting dominate the failure mode at low confinement while shear failure is the dominantfailure mode at high confinement. Numerical results from simulated compression tests show thatmicrocracking reduces the cohesive component of strength alone, and the frictional strength componentremains unaffected. Results from simulated Brazilian tests show that the tensile strength is influenced bythe presence of microcracks, with a reduction in tensile strength as microcrack intensity increases. Theimportance of microcrack heterogeneity in reproducing a bi-linear or S-shape failure envelope and itseffects on the mechanisms leading to spalling damage near an underground opening are also discussed.

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.

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.

COMPUTER SIMULATED ANALYSES ON DEFORMATION AND FRACTURE OF NON-CRACKED AND CRACKED SPECIMENS

A unified damage and fracture model,the combinatory work density model,which is suitable for ei- ther non-cracked body or cracked body has been suggested.In the present paper,the deformation and fracture of the two kinds of tensile spceimen and TPB specimen made of 40Cr steel have been simulated by using the new mod- el together with the large èlastic-plastic deformation finite element method.The results give a good picture of the whole deformation and fracture processes of the specimens in experiments;especially,the results on the TPB specimen can be used to obtain the relationship between load and displacement at the loading point P-Δ,and between crack ex- tension and displacement at the loading point Δα-Δ,the resistance curve J_R-Δa and the fracture toughness J_(IC).All the results are in remarkable agreement with those obtained by experiments.Therefore the model suggested here can be used to simulate crack initiation and propagation in non-cracked body and fracture initiation and crack stable propa- gation in cracked body.

ON PROBLEMS OF FRACTURE OF MATERIALS IN COMPRESSION ALONG TWO INTERNAL PARALLEL CRACKS

The fracture of materials under the action of compressive forces, directed along cracks which are parallel in plane can,lot be described within the framework of the linear fracture mechanics. The criteria of fracture of the Griffith-Irvin or COC type, used in classical linear fracture mechanics, are not applicable in this problem, since these forces have no influence on stress intensity coefficients and on values of cracks opening([1, 2]). The problems of such a class may be described only by using new approaches. One of possible approaches is presented by the first author, which involves using linearized relations, derived from exact non-linear equations of deformable solid body mechanics([3, 4, 5]). It should be remarked here that this approach has been widely used in problems of deformable bodies stability. As a criterion of the initiation of fracture the criterion of local instability near defects of the crack type is used. In these cases the process of loss of stability initiates the fracture process.

CRACK INITIATION POINT DETERMINATION AND DYNAMIC FRACTURE TOUGHNESS FOR CHARPY PRE-CRACKED SPECIMEN

The method to detect the crack initiation point of Charpy pre-cracked specimen under dy- namie loading was studied using strain gauge.The load-time curve and nominal strain-time curve at the crack tip for impact testing specinens may be shnultaneously measured by twin-channel oscilloscope with high speed sampling and diskette storing.Based on the dynam- ic finite element simulation of impact response of Charpy specimen,the measuring method of dynamic fracture toughness was analysed and some problems in previous dynamic fracture toughness measurement were discussed.

Towards fully automatic modelling of the fracture process in quasi-brittle and ductile materials:a unified crack growth criterion

Fully automatic finite element(FE) modelling of the fracture process in quasi-brittle materials such as concrete and rocks and ductile materials such as metals and alloys,is of great significance in assessing structural integrity and presents tre-mendous challenges to the engineering community. One challenge lies in the adoption of an objective and effective crack propagation criterion. This paper proposes a crack propagation criterion based on the principle of energy conservation and the cohesive zone model(CZM) . The virtual crack extension technique is used to calculate the differential terms in the criterion. A fully-automatic discrete crack modelling methodology,integrating the developed criterion,the CZM to model the crack,a simple remeshing procedure to accommodate crack propagation,the J2 flow theory implemented within the incremental plasticity framework to model the ductile materials,and a local arc-length solver to the nonlinear equation system,is developed and im-plemented in an in-house program. Three examples,i.e.,a plain concrete beam with a single shear crack,a reinforced concrete(RC) beam with multiple cracks and a compact-tension steel specimen,are simulated. Good agreement between numerical predictions and experimental data is found,which demonstrates the applicability of the criterion to both quasi-brittle and ductile materials.