Flaw Tip Research Articles

Promoted cutting and energy transition efficiency by an asymmetric cutter

Laboratory and numerical tests were performed to understand the rock breakage mechanism of a CCS (constant cross section) cutter and an asymmetric cutter. The laboratory and numerical tests consistently show that the asymmetric cutter frequently generates larger breakage areas and consumes less indentation energy than the CCS cutter, but there is a defect spacing of 70 mm. Thus, the asymmetric cutter frequently has a higher cutting efficiency than the CCS cutter. In addition, the numerical tests reveal that the CCS cutter first generates a plastic zone and subsequently forms cracks to connect with the defect tips. In this process, a large proportion of the energy is wasted in the particle friction in the plastic zone. However, the asymmetric cutter tends to form much smaller plastic zones, which makes it difficult to generate particle friction. Only a small amount of friction energy is consumed in the later crack propagation process. Thus, the asymmetric cutter has a higher energy transition efficiency than the CCS cutter.

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.

Compression-induced failure characteristics of brittle flawed rocks: Mechanical confinement-dependency

The flaw tips in brittle rocks are often the sources of crack initiation and growth due to the stress concentration, which commonly governs the rock strength. However, a unified framework identifying the compression-induced crack types, ultimate failure patterns and the cracking levels of brittle flawed rocks under different mechanical confinements is not yet available. This study conducts the laboratory compression experiments with the AE monitoring to explore the failure characteristics of flawed limestone and its confinement-dependency. Four new crack types including loop crack, secondary transverse crack, near-field transverse crack and far-field transverse crack are found experimentally, and then a modified crack type classification strategy is proposed. Four failure patterns including the σ1-axisymmetric flaw-disturbed spalling for uniaxial compression, the σ3-transverse-symmetric flaw-disturbed spalling for biaxial compression, the σ1 −axisymmetric flaw-disturbed shearing for conventional triaxial compression, and the mixed σ3-transverse-symmetric flaw-disturbed shearing and σ2-transverse-symmetric flaw-disturbed spalling for true triaxial compression, are documented for the first time. Moreover, an acousto-mechanics-based classification methodology of rock cracking levels is established, as well as an AF (average frequency)-RA (rising angle)-based Kernel density estimation method for interpreting the rock cracking nature and the strength mechanism. This paper gets insights into the mechanical confinement-dependency of the rock failure characteristics incorporating the pre-existing flaws and help interpret the field observations.

Crack propagation process in double-flawed granite under compression using digital image correlation method and numerical simulation

The existence of flaws seriously weakens the rock strength, directly affects the crack expansion morphology, and indirectly affects the slope stability. In this study, uniaxial compression tests were carried out on double-flawed granite to investigate the effects of flaw angle on its compressive strength and crack coalescence based on DIC technology and PFC2D numerical simulation. The result indicates that the UCS values of flawed specimens at angles of 15°, 30°, 45°, and 60° are approximately 25.5–51.7% lower than that of the intact specimen. The failure strength of the double-flawed sample increases with the increase of flaw inclination angle, and the fracture morphology shifts from no expansion to split expansion. When the flaw tip strain of each flaw sample exceeded 0.6%, the stress concentration was generated at the flaw tip, and the sample began to appear macroscopic large cracks. DIC technique can well observe the crack initiation and propagation process, recording inclined shear strain bands at flaw tips, tensile strain bands in the middle of flaws, and tensile shear O-shaped strain ring in rock bridge areas. Numerical simulations using PFC2D software were carried out and showed a good agreement with the physical results. In addition, the effect of structural inclination on specimen failure strength is clearly explained by the theory of compressive damage based on the projected size of flaws. The deformation of rock mass is not a simple material deformation but is composed of material deformation and structural deformation. These experimental and numerical results enhance our understanding of crack initiation and coalescence characteristics, aiding in the analysis of rock structure stability in scenarios such as excavated underground openings, slopes, and tunneling construction, where pre-existing cracks or step-path fractures are pivotal to structural integrity.

Quantitative detection of axial defects in pipelines based on modified multi-mode total focusing method (MTFM)

The axial defects in pipelines are hard to detect effectively with the conventional multi-mode total focusing method (MTFM). The inner and outer curved surfaces of pipelines induce beam divergence, reducing focusing performance and introducing imaging distortion and errors. In this paper, the MTFM employed for inspecting flat plates is modified in consideration of the influences of pipe curvature on the ray paths of direct, half-skip and full-skip mode waves. The corresponding travel times are recalculated and used for performing delay-and-sum (DAS) beamforming, achieving the profile reconstruction and quantitative detection of axial defects. The simulation and experiments were implemented on the carbon steel pipeline with 100 mm outer radius and 20 mm wall thickness. The phased array (PA) probe with 32 elements and 5 MHz central frequency and the 55° curved wedge were adopted to detect the inner-wall axial defects in pipeline. The profiles of the planar defects with a length of 5 mm and orientation angles of −45°∼45° were well reconstructed by modified MTFM, and the measurement errors of flaw lengths, orientation angles and tip depths were no more than 16.6 %, 5.1° and 4.0 % after correction, respectively. Further simulated and experimental results indicate that the internal axial defects can also be detected and quantified by modified MTFM. Finally, the influences of pipe curvature on modified MTFM are discussed by simulation.

Numerical characterization on fracture evolution behavior of pre-flawed sandstone under monotonic and multilevel cyclic loading

This study employed an improved stress corrosion model within a three-dimensional particle flow program to investigate the mechanical and cracking behavior of fractured sandstone samples with varying rock bridge angles under both monotonic loading and multilevel cyclic loading. Firstly, using the experimental results from intact sandstone under uniaxial compression, we calibrated the microscopic mechanical parameters of the parallel bond model. Then, the model was validated using the experimental results of fractured sandstone under monotonic and cyclic loading. Finally, the initiation, propagation, and coalescence behavior of cracks in fractured sandstone samples under the above two loading paths, as well as the evolution process of the stress field, displacement field, and force chain distribution, were discussed in detail. The results show that the stress–strain curves, strength, deformation parameters, and macroscopic failure modes obtained from numerical simulations are consistent with those from laboratory mechanical tests. The mechanical behavior of fractured sandstone is influenced by both the rock bridge angle and the loading path. As the rock bridge angle decreases, the mechanical parameters of the sample show an increasing trend, and the influence of rock bridge angle on the failure mode is reduced. Cracks in the samples all initiate at the tips of pre-existing flaws, and the rock bridge angle affects the crack propagation and coalescence process: for sandstones with smaller rock bridge angles, cracks gradually extend to the sample’s ends; for sandstones with larger rock bridge angles, direct coalescence of cracks at the rock bridge is observed, then extending to the sample’s ends. Compared with monotonic loading, the damage degree of sandstone samples under multilevel cyclic loading is more severe, and even obvious block spalling is observed. The experimental and numerical simulation results are expected to improve the understanding of the mechanical behavior and fracture behavior of fractured rocks under monotonic and multilevel cyclic loading.

A true triaxial experiment investigation of the mechanical and deformation failure behaviors of flawed granite after exposure to high-temperature treatment

Understanding the mechanical and deformation failure behaviors of heated flawed granite under true triaxial stresses is of great significance for evaluating the wellbore stability in developing hot dry rock resources. In this study, true triaxial experiments are conducted on flawed granite specimens of a varying flaw overlapping length lfo with and without high temperature (HT) treatment, with the emphases on the crack development under true triaxial stresses and its correlations with the strength, plastic anisotropy and failure forecast. Investigations demonstrate that HT, σ2 (intermediate principal stress) and lfo, significantly affect the mechanical properties of flawed granite. Due to strong 3D confinement, the loop crack, the secondary transverse crack and the far-field crack are frequently induced in flawed granite, unlike experiments of uniaxial, biaxial and conventional triaxial compression. When increasing lfo, the rock bridge undergoes the stress amplification which tendentiously drives the internal crack coalescence and typically causes a reduced strength. The PSIRs (Plastic Strain Increments Ratios) analysis, first extended to the true-triaxial experimental data, reveals that the 3D stress-induced extensional/shear fracturing from the flaw tips in σ2-direction is the mechanism of the plastic anisotropy generated. Besides, an attempt to anticipate the failure time in the framework of the time‐reversed Omori’s law suggests a poor predictability by using the acoustic emission but a higher-quality forecast (being more precise and less biased) by using the dissipated energy.

Experimental and numerical investigation of the fracture behavior in granite specimens with two co-planar side flaws under biaxial loading

It is well known that the structural plane plays a decisive role in the mechanical properties of rock mass. To better understand the cracking mechanism and stress characteristics of co-planar intermittent double-flawed granite specimens with various dip angles (Abbreviated as flawed specimens), two co-planar flaws were prefabricated on both sides of the granite, and biaxial compression tests were conducted. The failure characteristics and local microstrain were recorded by the camera and strain acquisition instrument. The evolution law of mechanical parameters of flawed specimens was analyzed. Simultaneously, a two-dimensional particle flow code (PFC2D) was used to establish a biaxial compression model of flawed specimens. The crack initiation and propagation of the specimens were explained by the crack hot spot, the evolution law of crack, and the evolution law of the number of microcracks. It is found that (a) as flaw dip angle (α) increases, peak strength and elastic modulus initially decrease and then increase, with reductions ranging from 17.6% to 46.1% and 15.6% to 48.2%, respectively. (b) Increasing lateral stress (σ2) triggers a shift in failure mode from a tensile-shear mixed failure to a shear failure. The cohesion (c) and friction angle (φ) of flawed specimens were notably smaller compared to those of intact specimens, with reductions correlating with the α. (c) Flawed specimens exhibit pronounced contact force concentration and stress concentration at the flaw tip. The analysis of the microstrain confirms stress concentration at the flaw tip, leading to failure. However, various σ2 induces the differences in the magnitude and extent of tip stress.

Fractures of low-k materials in a RF package with integrated passive device based on TGV

The radio frequency (RF) chips and passive devices integrated on the through-glass-via (TGV) substrate meets the demands of miniaturization, high performance and low losses in the application. The RF chip and integrated passive devices (IPDs) are interconnected electrically by a redistribution layer (RDL) on the TGV substrate with the isolation low-k materials. The low-k materials, however, are susceptible to fracture during the thermal process in packaging due to their weak mechanical properties. In this study, the fractures of the low-k material were studied by experiments and the finite element analysis (FEA) for a RF package with integrated passive device based on TGV. The mechanical properties of the low-k material used in the FEA were tested by fabricating freestanding low-k films using microfabrication techniques. Then the fracture behaviors of the low-k material in the package and its impact factors under thermal loadings were examined. The impact factors, such as the initial defect location, direction and length, were investigated by evaluating the stress intensity factors (SIFs) at the defect tips. The results revealed that the most hazardous location in the low-k material is the region below the micro-joint of RF chip. The vertical defects along thickness in low-k film are more likely to propagate than horizontal ones. The SIF value increases linearly with the defect length both in heating and cooling conditions.

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.

Investigation of mechanical properties in thin spray-on liner coated sandstone: Comparative study of unfilled and filled flaws

Thin spray-on liners (TSLs) are increasingly recognized as a potent skin support technology in underground engineering applications, particularly in mining. To better understand the coating and filling effects of TSL on the mechanical properties of the flawed specimens, a series of uniaxial compression tests were conducted on TSL-coated sandstone with unfilled and filled flaws. The experimental results showed that compared with the flawed specimen, the strength and peak strain of the TSL-coated and filled specimens increased, and the degree of improvement was more evident for the filled specimen coated with TSL. The specimens coated with TSL exhibited higher total, elastic, and dissipation energies compared to the uncoated specimens. Additionally, a greater proportion of the total input energy was transformed into dissipation energy after coating. For the filled specimens, more dissipation energy was generated during the failure process owing to the friction of the flaw surface provided by the filling effect. Moreover, the crack initiation and propagation behaviors of the flawed and filled sandstone specimens coated with TSL were analyzed by numerical simulation. For the specimens with an inclination angle of 45°, the TSL provided confinement to inhibit the initiation and propagation of tensile cracks, resulting in delayed crack initiation. The TSL filling plays a role in sharing part of the stress on the flaw surface, which weakens the stress concentration at the flaw tip. The effect of the TSL on the stress at the flaw tip was analyzed using a theoretical solution. The variation in SIF KII of the specimens with different flaw inclination angles was studied using a theoretical solution. Compared with the flawed specimens, the stress concentration in the filled flawed specimen was lower and the confinement effect could enhance the decrease in the flaw tip stress.

Numerical investigation on the influence of secondary flaw lengths on the mechanical characteristics and cracking behaviour of red sandstone containing orthogonal cross flaws

AbstractFlaw length has a significant effect on the cracking behaviour of fractured rock. PFC2D was used to simulate the uniaxial compression of red sandstone samples with secondary flaw lengths L2 of 0 mm, 5 mm, 10 mm, 15 and 20 mm under different primary flaw angles α(α = 0°, 15°, 30°, 45°, 60°, 75°, and 90°). Based on the simulation results, the effects of the secondary flaw length on the mechanical parameters, crack initiation location and crack evolution process of the fractured rock were analysed. Then, a method for judging the initiation location of the first cracks was proposed. Finally, the characteristics of the particle displacement field at preexisting flaw tips during first and secondary crack initiation were analysed. The results of this study are helpful for more deeply understanding the cracking behaviour of fractured rock.

Mechanical and fracturing characteristics of defected rock-like materials under biaxial compression

Various types of discontinuities have great influences on rock stability in underground excavation, as engineering disasters can be triggered by crack growth from flaws. Motivated by this concern, quasi-static biaxial compression tests are conducted on rock-like cement mortar materials. These tests examine the effect of shape, orientation, and interaction of pre-existing defects, representing natural or preconditioning fractures. Mechanical and fracturing characteristics are analyzed using stress-strain response, acoustic emission (AE), digital image correlation (DIC), and synchrotron X-ray computed tomography (CT) techniques. It could be found that the confinement increases the sample peak stress and restricts the development of cracks in intact samples. In hole samples, cracks initiate from the boundary of the hole, and the application of the confining pressure restricts the tensile cracks originating at the roof and floor of the hole to some extent. The hole-flaw interaction changes stress distribution and cracking behaviours. Cracks initiating from intersected flaw tips with a high inclination angle coalesce with the hole, while parallel independent flaws block the propagation of the cracks initiating from the hole boundary. Besides, it is proved that the integrated analysis framework, coupled with acoustic emission information interpretation, is feasible to accurately investigate fracturing characteristics in samples with complex pre-existing defects. The findings would benefit underground civil and resource-related projects.

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.

Structural Reliability of pressure tube of a PHWR with volumetric flaws

The volumetric flaws can get generated in the pressure tubes (PT) of PHWR (made of Zr-2.5Nb material) during the operation of the reactor. The PT material has hydrogen present during the fabrication. Additionally, the zirconium material has a tendency to pick up hydrogen from the surrounding environment and can cause initiation of a crack through delayed hydride cracking (DHC) mechanism. This crack can grow due to the DHC and has the potential to cause leakage or break of the PT. The crack formation occurs only when a set of conditions are met. The model for the crack initiation ahead of the tip of a volumetric flaws is prescribed by the CSA Standard N285.8-15. The formation of the crack is dependent of the geometry of PT, geometry of the flaw (length, depth and root radius), applied loading (due to internal pressure, residual stresses), hydrogen concentration in the material and the material tensile and fracture properties. Most of these parameters are well defined. However, some of them like flaw geometry and material properties are not known with certainty. The structural reliability based methods quantify the uncertainty using probability distributions and present results in form of probability estimates. In this work, the probability of initiation of DHC ahead of a volumetric flaw and its subsequent growth is reported. Monte Carlo based simulation method along with variance reduction techniques is used to predict the probability. This study is useful to establish the Leak-Before-Break behaviour of the PT.

A FINITE ELEMENT ANALYSES OF THE EFFECT OF HYDROSTATIC PRESSURE IN A THIN ADHESIVE LAYER ON ITS FRACTURE UNDER MIXED-MODE LOADING

The problem of deformation of a sample, which is a composition of orthotropic bodies bound by an adhesive layer, realizing a mixed loading mode I + II for the layer material in the vicinity of a crack-like defect, is considered. A crack-like defect in a composite was considered both in the form of a mathematical section and a physical section, the thickness of which was considered as a linear parameter. A finite element solution with an elastic description of materials in a state of plane deformation is used. To find critical states, the values of specific elastic energy in the tip of a crack-like defect were analyzed. So, for the mathematical section, the values of energy were associated with stress intensity factors and for the physical section with the energy product in the form of the product of a linear parameter and the average specific free energy on the face of a dead-end finite element of the adhesive layer. Finding the characteristics averaged over the layer thickness was determined within the framework of the Neuber – Novozhilov approach. For a small finite value of the linear parameter the convergence of the energy product to the critical value of the specific elastic energy found for the crack model in the form of a mathematical cut is shown for equivalent loading of the sample with a physical cut and the linear parameter tends to zero. For an adhesive layer of finite thickness an energy fracture criterion is considered which takes into account the loosening of the material due to hydrostatic pressure. The loosening of the material was taken into account by the parameter which was determined from the solution of the problem of normal tension of the adhesive layer by the cantilevers of the two-cantilever beam taking into account the critical values of the specific elastic energy. A comparison is made of the calculated critical external load for the formulation of the problem using a layer of finite thickness and the classical representation of a crack-like defect in the form of a mathematical section and a layer of zero thickness under the appropriate failure criteria.

Quantitative investigation of the cracking mechanism of 3D sand-printed rock containing a fold flaw

Natural discontinuities originating inside rocks are often distributed in folds rather than simple straights. To represent the details of fold-shaped flaws, rock-like specimens containing a fold flaw are created by 3D sand printing, and cases involving four undulation angles (α) are considered. Uniaxial compression tests, digital image correlation (DIC), and the extended finite element method (XFEM) on rock-like models are conducted to investigate the influence of undulation angles on the cracking mechanism of the specimens. The 3D sand-printed specimen has obvious advantages in simulating natural rocks. Two types of crack initiation modes are observed: tip cracking and nontip cracking. DIC-based measurements and XFEM simulations show that the maximum value of maximum principal stress and strain occur near when α = 0° and 10°, while they appear in the flaw inflexion when α = 20° and 30°, which explains the difference in crack initiation. Two types of cracks are identified based on the displacement vector analysis: tensile and horsetail cracks. The crack initiation position of these two cracks is precisely determined by extracting strain data around the flaw tip. The undulation angle cannot alter the final coalescence, which is induced by the connection of horsetail cracks with the flaw.

Crack propagation mechanism in bedded rock with parallel flaws: Insights from moment tensor inversion

Studying defects’ effect on bedded rock mechanical properties and failure mechanisms is essential for slope and underground excavation engineering. Acoustic emission helps analyze microcrack generation in rocks, shedding light on quasi-brittle material failure. However, previous studies have shown that the inhomogeneous geometry of defect-containing specimens complicates the localization of acoustic emission events, and some events may go undetected by sensors. This study uses discrete element modeling to examine bedded rock failure with parallel defects. We differentiate damage between acoustic emission events and macro fractures by assessing forces on failure sources. Our findings are as follows: (1) Bedding influences crack propagation direction but has limited impact on final failure. Cracks merging and penetrating in the bridge area are critical for catastrophic failure of samples. (2) Shear failure accumulates at the bridge area and defect tip, resulting in fragmented macroscopic failure. Tensile failure occurs during nucleation region development, leading to a more regular macroscopic failure. (3) High-magnitude events primarily arise from microcrack merging and connections, while crack propagation struggles to generate high-magnitude events. Most acoustic emission events involve 1∼4 microcracks, regardless of bedding angles or defects. The acoustic emission simulation method based on moment tensor inversion presents a novel approach for quantitatively determining the crack propagation law and failure mechanism of rocks with defects.

Damage effect and progressive failure mechanism of sandstone considering different flaw angles under dynamic loading

In many engineering applications, it is essential to have information about rocks that inherently contain pre-existing flaws under dynamic loading conditions. Dynamic impact tests are conducted on samples with varying flaw angles using the split Hopkinson pressure bar (SHPB) test system and the Digital Image Correlation system (DIC). The characteristics of the samples after dynamic loading, including dynamic strength, energy dissipation, and fractal fracture, are compared and analyzed. As the flaw angle increases, the peak stress and strain exhibit a typical V-shaped pattern, reaching the minimum value at 30°, and the initial initiation position shifts from the flaw tips to the middle of the flaw. Failure modes can be divided into three modes depending on the flaw angle. The progressive failure process, taking into account the heterogeneity of the rock, is demonstrated by developing an elastic damage constitutive model that uses dynamic compression and tensile tests to parameterize it. As the flaw angle increases, the initial damage zone also moves from the flaw tips to the middle of the flaw. Failures around the hole with redistributed stress are observed, and the failure mechanisms can be explained with the aid of strain energy density (SED). Using fracture mechanics, the analytical solution of stress around the flaw is provided, and the variation of crack initiation angle, stress distribution, and energy dissipation under different flaw angles is theoretically explained, which is in good agreement with the experimental and simulated results.

Numerical modeling of crack propagation from open and closed flaws in rock

The traditional fracture criterion performs well when calculating crack propagation under tensile conditions, but a zigzag crack propagation path was obtained under compression, which is inconsistent with experimental results. This study indicates that the positive and negative oscillations of the mode-II stress intensity factor (KII) under compressive loading during crack propagation is the reason for the zigzag trajectory of crack propagation. Such oscillation of KII can be effectively eliminated when the non-singular stress (T-stress) at the flaw tips is considered, and thus, the path of crack propagation can also be smoothed. However, due to the complicated calculations of the stress intensity factors and T-stress at each step of crack propagation, it is not suitable for simulation in fast, complex environments and multi-flaws. Also, the lack of a shear fracture criterion in traditional fracture methods leads to difficulty in studying shear crack propagation in rock. Therefore, a strength-based localized maximum stress (SLMS) criterion was proposed in this study to model both tensile and shear crack propagations in rock more efficiently, conveniently, and accurately. Then, the crack propagation processes in both plate and Brazilian disc specimens with a single flaw under uniaxial and biaxial compression were modeled to investigate the effects of the flaw inclination angle, friction coefficient, and loading level on the crack propagation path. The influence of the contact friction between the flaw surfaces on the contact status and the crack propagation path during the crack propagation process was also analyzed. Also, the crack propagation in a plate with a single flaw under biaxial tension was modeled, which shows bifurcation crack propagation when the lateral tension is several times larger than the axial tension. All the modeled results indicated that the proposed SLMS criterion can get better results when modeling crack propagations for open or closed flaws under tension or compression conditions.