Initial Angle Research Articles

Investigating the influence of geometric configurations and loading modes on mixed mode I/II fracture characteristics of rocks: Part I-Numerical simulation

The influence of geometric configurations and loading modes is a bottleneck that restricts the accurate determination of rock fracture parameters and the precise understanding of fracture mechanisms. Therefore, the phase field method is employed to analyze the influences of specimen geometry and loading modes on the dimensionless stress intensity factors, peak load, fracture toughness, and crack initiation angle during the rock mixed mode I/II fracture process. Three new configurations of specimens are proposed to test rock mixed mode I/II fracture. The research results indicate that as the prefabricated crack inclination angle increases, the peak load of three-point bending type specimen increases, while the peak load of diametric-compression type specimen decreases. Moreover, the influence of geometric configurations on the fracture parameters of three-point bending specimens is greater than that of diametric-compression disk specimens. The research findings of this work can provide basic supporting data for the establishment of mixed mode I/II fracture testing standards.

Crack coalescence and failure behaviors in slate specimens containing a circular cavity and a pre-existing flaw pair under uniaxial compression

The flaws (fractures) widely existing in rock mass pose a threat on the stability of many underground cavities. This study aims at investigating the failure process of a cavity affected by adjacent flaws. A number of slate specimens containing a circular cavity and a pre-existing flaw pair were tested under uniaxial compression. Varied flaw configurations were obtained by changing the flaw inclination angle with respect to horizontal and the spacing between flaw pair and cavity. Three main types of cracks emanated from the pre-existing flaw tips, and played a dominant role in the failure process of cavity, comprising primary wing cracks (tensile cracks) and two types of secondary cracks (quasi-coplanar shear cracks and oblique shear cracks). The initiation and propagation of these cracks were highly dependent on the flaw pair configurations. When the pre-existing flaw pairs were non-parallel to the loading direction, (1) mostly, coalescence occurred both in the tensile and compressive stress concentration regions around cavity; (2) in most cases, the quasi-coplanar and oblique shear cracks were the predominant cracks leading to the coalescence between the compressive stress concentration regions of cavity and the pre-existing flaw pair tips. When the pre-existing flaws were parallel to the loading direction, the combination of inclined shear cracks and tensile cracks or inclined shear cracks only dominated the failure of cavity. The initiation angles of wing cracks, quasi-coplanar shear cracks and oblique shear cracks agreed well with the theoretical predictions by the maximum tangential strain criterion (MTSN), Mohr-Coulomb criterion (M−C) and maximum shear stress criterion (MSS), respectively.

Dynamic behavior of fractured gabbro treated by high temperatures

Rock masses in deep engineering are often subjected to high geothermal effect, and blasting is a key technique for deep rock excavation. It is crucial to study the dynamic characteristics of fractured rock masses under high geothermal conditions and blasting excavation. This paper focuses on fractured gabbro, examining the effects of high temperatures on its surface chromaticity and mass. Dynamic splitting tests were conducted using a Split Hopkinson Pressure Bar (SHPB) apparatus, and the microstructure was analyzed with scanning electron microscopy (SEM). The results indicate that: (1) High temperatures significantly affect the surface chromaticity and mass of gabbro samples. (2) Elevated temperatures lead to the progression from a microstructure with no noticeable microcracks to the development of extensive microcracks and mineral particle fragmentation. (3) The impacts of high temperatures on the splitting strength, peak strain, and dynamic elastic modulus of gabbro samples reach a critical point around 500 °C. At this temperature, the strengthening effect of high temperature is evident, whereas at 800 °C, these properties significantly decrease due to extensive mineral crystal fragmentation and microcrack expansion, weakening the sample's strength, stiffness, and deformation resistance. (4) From 25 °C to 600 °C, the fissure angle significantly influences the splitting strength, peak strain, and dynamic elastic modulus of gabbro. Although the fissure angle still impacts these properties at 800 °C, the overall effect of high temperature is more pronounced. (5) The failure of fractured gabbro samples is primarily characterized by tensile and shear failures, with the crack initiation angle closely related to the fissure angle.

Validity for measuring mode II fracture toughness using the short beam in compression method

Mode II fracture toughness is a crucial intrinsic property of rocks, and accurately measuring this parameter is a challenge in rock fracture mechanics. Among the various approaches for measuring the mode II fracture toughness, the short beam in compression (SBC) method is a convenient and effective option. Nevertheless, the validity of measuring the mode II fracture toughness using the SBC method is controversial. This study conducted numerical analyses and experiments on the SBC specimen with different geometries (C/H) to identify the fracture mode and initiation location. In addition, the improved fracture criterion was developed by introducing three components of T-stress to determine the fracture mode at crack tips and the crack initiation angle. The results reveal that the pure shear fracture can occur on notch tips of SBC specimens with C/H=0.1 and 0.2, indicating that the SBC method is valid for measuring the mode II fracture toughness of rocks. The modified fracture criterion can predict the fracture mode and the angle of shear crack initiation observed in the SBC experiment well. Another important finding is that, due to the compressive stress on the notch tip, the measured mode II fracture toughness of rocks using the SBC method should be corrected using the Coulomb friction model. It is also discovered that the length of the fracture process zone and the corrected mode II fracture toughness depend on the value of C/H. A small C/H leads to a significant error between the measured and corrected mode II fracture toughness.

Crack coalescence prediction and load-bearing mechanism of defective specimen based on computer vision recognition model

The coalescence of cracks in rock, triggered by external stress or geological activities, plays a pivotal role in determining the mechanical properties and stability of rock structures. Consequently, the prediction method of crack coalescence become critical tools in ensuring the safety and stability of geotechnical engineering facilities. In this paper, based on the strain field data obtained by digital image correlation (DIC), a set of programs was developed to automatically identify the values and percentage changes of different strain intervals in the strain field of the specimen. Then, a computer vision recognition (CVR) model is established to study the process and prediction of crack coalescence in sandstone specimens with open flaws. This method overcomes the limitations, subjectivity and unpredictability of the traditional method of identifying cracks through artificial vision. The results show that the increase of the flaw width causes the displacement trend to be squeezed and deflected along the width direction, thereby changing the angle of crack initiation and exhibiting different forms of crack coalescence. The increase in the inclination angle of the flaw leads to an increase in the effective compressive area (ECA) of sandstone, and the magnitude of ECA is positively correlated with the uniaxial compressive strength (UCS) of the sandstone. Additionally, the CVR model identifies two types of numerical fluctuation signals before crack coalescence, namely the plastic characteristic signal (PCS) in the early stage of the experiment and the early-warning signal (EWS) near the period of crack coalescence, where EWS can be used as a predictive signal for crack coalescence. The research results provide a new algorithm technical support for the process analysis and prediction of crack coalescence, and provide a basis for early warning of rock mass engineering disasters.

A simple, fast, and robust criterion for predicting fracture load and crack growth path in notched elastoplastic plates under mixed-mode loading

This study assesses the combined in-plane tensile/shear failure of aluminum plates containing notches. With this aim, a numerical method, i.e., the so-called equivalent material concept (EMC) in association with the extended finite element method (XFEM), is introduced to simulate the fracture phenomenon. The predicted failure load, initiation angle, and growth path are compared to experimental results reported for aluminum alloys in the literature to evaluate the functionality of the proposed method. Also, the accuracy of the EMC-XFEM model was compared to the conventional Ductile Damage method. The EMC-XFEM combined model is found to be fast and robust with appropriate accuracy in fracture predictions of ductile samples in comparison with the elastoplastic damage model and experimental data.

Assessing the Effect of Twisting and Twisting Fatigue on ZnO:Al Thin Film Performance on PEN and PET Substrates.

This study examines the electromechanical characteristics of aluminium-doped zinc oxide (AZO) films. The films were produced using the RF magnetron sputtering process with a consistent thickness of 150 nm on various polymer substrates. The study focuses on assessing the electro-mechanical failure processes of coated segments using flexible substrates, namely polyethylene naphthalate (PEN) and polyethylene terephthalate (PET), with a specific emphasis on typical cracking and delamination occurrences. This examination involves conducting twisting deformation together with using standardised electrical resistance measurements and optical microscope monitoring instruments. It was found that the crack initiation angle is mostly dependent on the mechanical mismatch between the coating and substrate. Higher critical twisting angle values are observed for the AZO/PEN film during twisting testing. Relative to the perpendicular plane of the untwisted sample, it was found that cracks initiated at a twist angle equal to 42° ± 2.1° and 38° ± 1.7° for AZO/PEN and AZO/PET, respectively, and propagated along the sample length. SEM images indicate that the twisting motion results in deformation in the thin film material, leading to the presence of both types of stress in the film structure. These discoveries emphasise the significance of studying the mechanical properties of thin films under different stress conditions, as it can impact their performance and reliability in real-world applications. The electromechanical stability of AZO was found to be similar on both substrates during fatigue testing. Studying the electromechanical properties of various material combinations is important for selecting polymer substrates and predicting the durability of flexible electronic devices made from polyester.

Strain-rate effect on mechanical properties of rock-like specimens with narrow crack under uniaxial compression

Uniaxial compression tests were carried out on rock-like specimens containing prefabricated narrow crack under various strain rates. Experimental observations and numerical results show that narrow crack will close or partially close or expand under low compressive stress. Due to the unidirectional loading of loading device, the deformation degree of crack tip near the loading end was different with that of crack tip near the bearing end. With the increase of strain rate, the cracking modes can be divided into 3 categories, near vertical splitting failure affected by horizontal crack (α = 0°), anti-N-wing crack (15°≤α ≤ 60°) and wing crack (60°<α ≤ 90°). Both of the initiation angle θ1 of crack tip near the loading end and θ2 of crack tip near the bearing end decreased with the increase of α and strain rate. Specially, when α = 90°, both θ1 and θ2 rose with the increase of strain rate. Moreover, the value of θ2 was greater than that of θ1 under the same α value because of the different deformation characteristics at two tips. Besides, the mechanical properties of fissured rock including peak compressive stress, residual compressive stress, strain at peak stress, elastic modulus and post-peak softening modulus were significantly influenced by the strain rate and inclination angles.

Experimental and numerical study on the damage evolution and acoustic emission multi-parameter responses of single flaw sandstone under uniaxial compression

Cracks are generally the weakest position in engineering rock mass. It is of great practical significance to study the influence of cracks on the fracture process of rock mass and the precursor criterion of failure for disaster prevention and mitigation projects. In this paper, particle flow program (PFC2D) is used to analyze the mechanical strength and damage evolution of fractured rock. The multi-parameter stage characteristics of acoustic emission (AE) and the difference of fracture precursor information of fractured rock mass under uniaxial compression (UC) are explored. The results are as follows: (1) The strength and modulus of elasticity of fractured rock samples are smaller than that of intact rock mass, and increase in quadratic function with the increase of fracture angle, and the strength of B-90 sample is closest to intact rock samples; (2) The PFC analysis shows that the cracks in the defective rock samples all originate from the tip of the prefabricated crack, and the crack initiation angle is basically in the range of 60°∼90°. The loading process mainly produces tensile damage, and the number of tensile cracks are far more than that of shear cracks. Compressive stress and tensile stress increase with the increase of load, but the force chain of failure surface is the smallest after complete failure of rock; (3) The acoustic emission ringing number, RA/AF value, b value and entropy value of sandstone have obvious stage characteristics, the main frequency signal is mainly low frequency signal, and large-scale cracks are mainly produced in the failure process. The sudden increase point of AE ringing times and RA/AF value accumulation curve can be used as critical failure point. The fluctuation of B value and entropy value reflects the instability of fracture development, and “cliff-like” descending point and ascending point can be regarded as the precursor of critical failure of rock; (4) The failure precursor of acoustic emission parameters of sandstone appears in the peak load period of 85 %–98 %. The order of response time of different parameters is: cumulative RA/AF value, cumulative event number, entropy value, B value. The response time of B-45 and B-15 is earlier, while that of B-60 and B-30 is later. The research results are expected to deepen the understanding of the relevant standards of damage evolution and failure precursors of defective rocks in engineering applications.

Study on the initiation criterion for double symmetric radial perforations emanating from circular hole under high water pressure considering T stress

Hydraulic fracturing is common technique to break the rock mass down under high water pressure. In this paper, the tensile failure of double symmetric radial perforations emanating from circular hole under far-field compression and high water pressure is studied. And the existing theoretical model does not consider the unequal water pressure applied on the surfaces of perforation and circular hole, it also neglects the effect of T stress, so the maximum tangential stress (MTS) criterion could be revised with considering T stress. Firstly, considering the unequal water pressure on the perforation surfaces and circular hole, the stress intensity factors (SIFs) are derived with conformal mapping method and compared with the semi-analytical solution. Secondly, the terms of T stress are derived by analogy to the open-crack under high water pressure. On this basis, the revised MTS criterion is obtained, and its validity is verified with model tests. Finally, the effects of the lateral pressure coefficient k, minimum horizontal principal stress σh(σh/σH remains constant), radius of circular hole a, hydraulic coefficient λ and radius of process zone rc on the initiation angle −θc, critical water pressure Pc and SIFs are analyzed respectively. It is found that −θc always increases firstly and then decreases with increasing the inclination angle α, whereas Pc firstly decreases and then increases as a/(c-a) varies from 0 to 1 and increases as α increases in the range of a/(c-a) = 1∼+∞. Then −θc increases with increasing σh and k, whereas Pc can intersect each other with increasing k and increases with increasing σh. As c-a remains constant, −θc increases firstly and then decreases with increasing the ratio a/c, whereas Pc keeps decreasing. Also −θc and Pc increase with decreasing λ. As rc increases, −θc decreases and Pc increases. As for SIFs, when α increases, KI decreases and KII increases firstly, then decreases. As α remains constant, KI increases firstly, then decreases with increasing k and KII keeps increasing. In the range of a/(c-a) = 0 ∼ 1, KI decreases with increasing a, and KII keeps increasing. Within the range of a/(c-a) = 1∼+∞, KI increases firstly, then decreases with increasing a, the increment of KII is not obvious. And KI decreases with decreasing λ.

Crack-tip plastic zone and initiation mechanism of rock based on Eshelby stress

The crack initiation and propagation mechanism of rock is significantly related to characteristics of the crack-tip plastic zone due to energy dissipation. In this paper, conservation integral and configurational stress/force is derived by the Noether’s method. The crack-tip configurational stress field is then deduced and analyzed. Additionally, the Drucker–Prager yield criterion of configurational stress is derived and it is used to predict the crack-tip plastic zone of rock. The fracture criteria of rock are also developed based on the critical area and minimum polar radius of the crack-tip plastic zone. Subsequently, the effects of bedding dip angles and porosity on rock mixed mode fracture are investigated, and the fracture resistance of rock is assessed using the resistivity and wave velocity. It is shown that the configurational force as the crack driving force is induced by the invariant transformation of an action integral with respect to translation of a defect. It is also found that the crack-tip plastic zone assessed by Drucker–Prager yield criterion of Cauchy stress is discontinuous at the crack surface while that predicted by yield criterion of configurational stress is continuous. The present fracture criteria take into account the internal friction angle, and the predicted initiation angles and fracture loads are in good agreement with those determined by the maximum tensile stress (MTS) fracture criterion and experiments. The fracture loading envelope decreases with the porosity, bedding dip angles and resistivities of rock, but increases with wave velocity.

Study on the mechanism of energy evolution and bearing degradation in pre-cracked sandstone under non uniform cyclic loading

The engineering rock mass is affected by the original cracks, which increases the uncertainty of instability and failure mechanisms. Under the action of excavation unloading or other disturbance loads, its bearing properties are more complex and the deformation characteristics is more special, which has always been a difficult issue in the analysis of rock mechanics. In order to explore the propagation and failure mechanism of spatial micro cracks in fractured rock masses, acoustic emission (AE) experiment of sandstone was carried out under pre-static loading and low-frequency stepwise cyclic loading–unloading. The research results indicate that the peak strain and crack initiation stress of rocks with different pre-cracked positions have the highest differences of 12.62% and 18.65%, respectively. The difference between peak stress and crack initiation strain is within 4%. The pre-cracked position mainly affects the peak strain and crack initiation stress of pre-cracked rocks, and its effect on peak stress and crack initiation strain is limited. The crack initiation angle of prefabricated cracks is mainly distributed within the range of 45° − 90°, and it decreases with the increase of quantities of prefabricated cracks. The crack initiation angle of rocks with prefabricated three cracks is 30% lower than that of single crack. The research results have deepened the analysis of energy evolution and deformation damage of fractured rock masses.

Quantitative Analysis and 3D Visualization of Crack Behavior in 3D-Printed Rock-Like Specimens with Single Flaw Using In-Situ Micro-CT Imaging

3D printing technology allows for precise control of preparing complex geometries and internal defects in printed rock analogs, while in-situ Micro-CT imaging enables real-time observation of crack behavior. The combination of these technologies offers a new research approach for studying rock crack behavior. In this study, 3D-printed rock-like specimens containing a pre-existing flaw were prepared using a gypsum powder-based 3D printer. An advanced in-situ Micro-CT system equipped with a loading device was used to quantitatively and visually investigate the crack behavior in 3D-printed specimens under uniaxial compression testing. 2D CT images obtained from in-situ compression testing at different deformations could be used to reconstruct a 3D model and visually identify the crack patterns of the extracted cracks in 3D-printed specimens. The initiation angle of cracks, volume of the pre-existing flaw, volume of newly formed cracks, and damage value with respect to strains were analyzed to quantitatively investigate crack behavior. The results indicated that within the 3D-printed specimens, tensile cracks were first initiated near the internal flaw, followed by the occurrence of shear cracks or tensile-shear mixed cracks at the flaw tips. Additionally, there was a negative linear correlation between the initiation angle of newly formed cracks and the initial flaw angle. For flaw angles in the range of 0° ≤ α ≤ 45°, a higher number of newly formed cracks were observed in the 3D-printed specimens, and the rates of increase in crack volume and damage values with strain were faster. However, for flaw angles in the range of 45° < α ≤ 90°, the results showed the opposite trend. Furthermore, through comparison with the crack behavior of natural rocks containing a single flaw, it was found that the failure modes and crack behavior of the 3D-printed specimens exhibit certain similarities with natural rocks.

Cross-Crack Interaction and Initiation Mechanism Under Hydro-Mechanical Coupling

In this paper, the equation for computing the maximum Stress Intensity Factor (SIF) of cross-crack under hydraulic-mechanical coupling was derived by the dislocation method and the integral equation, and the influence of cross-crack geometric parameters (branch-crack length, pinch angle, inclination angle) on the interacting SIF was analyzed, and its influence law was obtained. Using the maximum shear and tensile SIF ratio criterion as a basis, the fracture criterion of hydraulic-mechanical coupling cross-crack was established, and the initiation process of hydraulic-mechanical coupled red sandstone cross-crack was predicted (including the cross-crack initiation angle, initiation load, and initiation mechanism). Results showed that for cross-crack with the inclination angle fixed ([Formula: see text]), [Formula: see text] of crack tip C decreases slowly. For cross-crack with change of [Formula: see text]. [Formula: see text] of crack tip A increases, and crack tip C decreases with [Formula: see text]. For the cross-crack hydraulic-mechanical coupling test, as the cross-crack angle [Formula: see text] grows, the fracture initiation stress first reduces before increasing. Initiation mechanisms are Mode I fracture and initiation angles are usually negative. The hydraulic-mechanical coupling cross-crack initiation criterion is validated by the experimental findings, which align with the expected outcomes.

Theoretical and experimental study on mixed-mode I + II fracture characteristics of thermally treated granite under liquid nitrogen cooling

Liquid nitrogen fracturing technology is widely used in the exploitation of hot dry rock (HDR) geothermal resources. It is necessary to understand the fracture characteristics of hot dry rocks under liquid nitrogen cooling. In this study, the mixed-mode I + II fracture characteristics of the thermally treated granite under liquid nitrogen cooling were investigated. The results indicate that as the exposed temperature increases, the number of internal microcracks increases, while the fracture energy and fracture toughness of the granites decrease. From 25 ℃ to 600 ℃, mode I fracture toughness decreases from 1.286MP·m1/2 to 0.612MP·m1/2; mode Ⅱ fracture toughness decreases from 0.830MP·m1/2 to 0.413MP·m1/2; mixed-mode Ⅰ+Ⅱ fracture toughness decreases from 1.082 MP·m1/2 to 0.530MP·m1/2 at Me = 0.705 and from 0.914 MP·m1/2 to 0.445 MP·m1/2 at Me = 0.427, respectively. Additionally, the intergranular fracture behavior increases as temperature increases, resulting in an increase in irregularity of fracture surfaces. The generalized maximum tangential stress (GMTS) criterion is used for predicting the fracture characteristics (fracture initiation angle and fracture toughness) of thermal damaged granites. The accuracy of the GMTS criterion in predicting crack resistance when mode II loading dominates has been improved by considering the variation of critical damage radius.

A FEA-Based Fast AC Steady-State Algorithm for a Voltage-Driven PMSM by a Novel Voltage-Flux-Driven Model

Finite element analysis (FEA) has been used more frequently than ever before in electrical machine design covering different disciplines such as electromagnetics, rotor dynamics, thermodynamics etc. The accuracy and speed of FEA are in high demands, especially when the time-stepping FEA is employed in the electromagnetic design of the machine. In this paper, to accelerate numerical transient in FEA of a voltage-driven permanent magnet synchronous machine (PMSM), a fast AC steady-state algorithm has been proposed, in which the voltage supply is converted into the stator flux linkage and the machine can be regarded as driven by the flux linkage directly. The electromagnetic transient is eliminated by injecting additional voltage signals to compensate the oscillating stator flux linkage in transient and the time-stepping error caused by the backward Euler in FEA. And the additional voltage signals will finally disappear after the steady state is achieved within about a half electrical time period. In addition, the maximum torque per ampere (MTPA) control logic can also be realized by adjusting the initial phase angle of the voltage supply based on the vector diagram. The simulation speed of the proposed method in this paper was compared to other traditional models and the accuracy was verified by experimental results.

Effect of high-temperature and strain rate on the mechanical and cracking behaviors of flawed sandstone under dynamic impact loading

A deep understanding of how flawed rock cracks under high temperatures and dynamic impacts is crucial for safe deep rock engineering design and operation. Dynamic impact tests were conducted on sandstones with various flaw inclinations using a Split Hopkinson Press bar (SHPB) system at room temperature and after high-temperature treatment at 600 °C. The results show that the dynamic peak stress and energy consumption of rock were reduced by thermal treatment. The study identified the circular tensile- and compressive-tangential stress zones around the flaw boundary using elastic theory. The strain accumulation zones gradually shift from the tensile- to the compressive-tangential stress zones as the flaw inclination increases, resulting in a larger crack initiation angle and opening displacement, while also constraining the propagation path. Additionally, the high-temperature treatment strengthened the deformation in the circular tensile-tangential stress zone. In contrast to the severe failure of specimens at room temperature, the clear failure pattern of the specimens after high-temperature treatment merely consists of wing cracks. Finally, this study presents a detailed theoretical explanation of the cracking behavior under various conditions, which is consistent with the experimental results. This research enhances the understanding of dynamic crack propagation in deep high-temperature rocks and provides scientific references for engineering safety design and protection.

Numerical simulation of wedges slamming non-Newtonian fluids based on SPH method

Current research on non-Newtonian fluids using Smoothed Particle Hydrodynamics (SPH) has primarily focused on dynamic behavior, with limited exploration into interactions between rigid bodies and non-Newtonian fluids. Leveraging the SPH method's effectiveness in addressing free surface flow challenges, a two-dimensional numerical model of a rigid body impacting non-Newtonian fluids based on the Herschel-Bulkley-Papanastasiou (HBP) constitutive model is developed within the SPH framework. The study integrates experimental and numerical findings from non-Newtonian dam break scenarios, comparing particle distribution and the evolution of the waterfront over time to ensure accurate representation of non-Newtonian fluid behavior. While ensuring the convergence of SPH results, the research simulates the impact of a wedge on the surface of non-Newtonian fluids. This comparison of the slamming load exerted by non-Newtonian fluids on the wedge serves to validate the accuracy of the numerical method. Furthermore, the study explores the effects of varying initial velocities and roll angles of wedges slamming into specific non-Newtonian fluids. It analyzes fluid jet phenomena and changes in slamming loads.

Experimental study of the effect of crack distribution on the failure mechanism of sandstone specimens based on inclination angles and number of parallel flaws

Discontinuous joints are prevalent in engineered rock masses and play a significant role in the stability of the rock mass. This study aims to analyze the impact of the inclination angle and number of prefabricated flaws on the crack evolution and failure pattern of sandstone specimens. Uniaxial compression tests, along with acoustic emission technology and digital image technology, were employed to monitor and analyze the effects. The findings indicate that: (1) With the increase in the flaw inclination angle, the damage mode of the specimen transitions from tensile to compressive-shear failure. The localized high-strain region on the surface of the specimen predicts the propagation path for the formation of macroscopic cracks. (2) When the number of prefabricated flaws is small, the flaws mainly expand through tensile wing cracks. As the number of flaws increases, the inner flaw tip does not produce cracks. Instead, the failure of the entire specimen occurs along the direction of the outer flaw's tensile wing crack, with the inner flaw running through it. (3) The winged tensile crack is the first crack to appear in all rock samples, regardless of the flaw initiation angles. Finally, the stress intensity factor at the crack tip under uniaxial compression conditions, without considering the closure effect, was expressed based on fracture mechanics theory. The crack initiation angle was then calculated. The results of the theoretical calculation of the initiation angle were found to be consistent with the test results. These research findings can serve as theoretical references and provide insights into the failure mechanisms of cracked rocks and the development of disaster control methods in rock engineering.

A research report on the phase-domain trajectories of fault recording and their mathematical models

In currently known theories and algorithms of fault recording analysis and application, there is little literature where fault feature parameters are mined deeply from electrical physical quantities themselves. In this report the calculation method of the phase angles of the reference point and sample points is obtained, which ensures the correspondence of the time-domain waveform of digital fault recording with its phase-domain trajectory. The relationship between the initial phase angle of a sinusoid and the position of its trajectory, the characteristics of the trajectories of three-phase short-circuit currents containing dc components and the mathematical models of these trajectories are analyzed. Taking a rectangular wave current and a triangle wave current as example, the characteristics of the trajectories of non-sinusoidal and periodical waveforms containing harmonics are analyzed as well as their mathematical models. The research results show that the relationship between the initial phase angle of a sinusoid and the position of its circular trajectory is definite; the dc components have no impact on the positions of the trajectories of the short-circuit currents, but on their sizes and shapes; the harmonics have an impact on the shapes of the trajectories, and their positions are decided by the initial phase angles of the ac fundamental components in the waveforms. Subsequent study of the algorithm of fault recording analysis will be spread based on the contents in this report.