Dimensionless Stress Intensity Factor 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.

Numerical calibration for fracture parameters of three-point bending semi-circular specimens

This study aims to refine the fracture characterization of three-point bending semi-circular specimens used in rock fracture toughness assessments. The primary objective is to improve the accuracy of such evaluations by developing numerical simulations of specimens with pre-engineered cracks of varying geometries. Numerical simulations were conducted using the finite element method. The interaction integral method was employed to quantify the stress intensity factors (SIFs) and T-stress at crack tips. Initially, the model’s accuracy was validated by replicating stress singularities at crack tips in a benchmark circular disk with a central straight crack. Following validation, dimensionless fracture parameters specific to the three-point bending semi-circular specimens were calibrated. The numerical results demonstrate that the dimensionless stress intensity factor (YI) increases with both the relative crack length (a/R) and the spacing between support points. Notably, for relative crack lengths a/R ≤ 0.5, the dimensionless T-stress assumes negative values, initially decreasing and then increasing as the relative crack length increases. The findings of this study provide valuable insights into the fracture behavior of three-point bending semi-circular specimens with pre-engineered cracks. Based on the observed trends in the dimensionless fracture parameters, it is recommended that relative crack lengths within the range of 0.2–0.6 be used to maintain the accuracy of rock fracture toughness tests. The finite element method used in this study serves as a robust tool for calibrating fracture parameters, thereby laying a strong foundation for the application of these specimens in rock fracture toughness evaluations.

Experimental Study on Fracture Toughness of Shale Based on Three-Point Bending Semi-Circular Disk Samples

A large number of construction practice projects have found that there are many joints and microcracks in rock, concrete, and other structures, which cause the complexity of rock mechanical properties and are the main cause of geological or engineering disasters such as earthquakes, landslides, and rock bursts. To establish a rock fracture toughness evaluation method and understand the distribution range of fracture toughness of Longmaxi Formation shale, this study prepared three-point bending semi-circular disk shale samples of Longmaxi Formation with different crack inclination angles. The dimensionless fracture parameters of the samples, including the dimensionless stress intensity factors of type I, type II, and T-stress, were calibrated using the finite element method. Then, the peak load of the samples was tested using quasi-static loading, and the load–displacement curve characteristics of Longmaxi Formation shale and the variation in fracture toughness with crack inclination angle were analyzed. The study concluded that the specimens exhibited significant brittle failure characteristics and that the stress intensity factor is not the sole parameter controlling crack propagation in rock materials. With an increase in crack inclination angle, the prefabricated crack propagation gradually transitions from being dominated by type I fracture to type II fracture, and the T-stress changes from negative to positive, gradually increasing its influence on the fracture. An excessively large relative crack length increases the error in fracture toughness test results. Therefore, this paper suggests that the relative crack length a/R should be between 0.2 and 0.6. The fracture load distribution range of shale samples with different crack angles is 3.27 kN to 10.92 kN. As the crack inclination angle increases, the maximum load that the semi-circular disk shale samples can bear gradually increases. The pure type I fracture toughness of Longmaxi Formation shale is 1.13–1.38 MPa·m1/2, the pure type II fracture toughness is 0.55–0.62 MPa·m1/2, and the T-stress variation range of shale samples with different inclination angles is −0.49–9.48 MPa.

Using deep learning method to predict dimensionless values of stress intensity factors and T‐stress of edge notch disk bend (ENDB) specimen

AbstractA deep learning (DL) approach is implemented to determine the dimensionless stress intensity factors of mode‐I (YI) and mode‐III (YIII), as well as the normalized T‐stress (T*) of the edge notch disk bend specimen. The deep neural network (DNN) method as the DL approach is used to model the relationship between the geometry parameters of the specimen a/t, S/R, and β as inputs, and YI, YIII, and T* as output variables. To this end, three datasets consisting of 176, 176, and 123 finite element method data points from the previous studies are extracted for YI, YIII, and T*, respectively. The developed DNN models predicted the YI, YIII, and T* with correlations of 0.97003, 0.96918, and 0.97047, respectively. Then partial dependence plot is performed to determine the relationship between YI, YIII, and T* and the geometry parameters, which is useful in predicting and optimizing YI, YIII, and T*.

Stress Intensity Factors for Thermoelectric Bonded Materials Weakened by an Inclined Crack

Thermoelectric Bonded Materials (TEBM) weakened by an Inclined Crack Problems (ICP) subjected to remote stress was presented in this study. The problems are addressed by employing the Modified Complex Variable Function (MCVF) method, which incorporates the Continuity Conditions (CC) for the Resultant Electric Force (REF) and Displacement Electric Function (DEF) to formulate the Hypersingular Integral Equations (HSIEs) associated with these problems. By applying the curved length coordinate method, the unknown functions of Crack Opening Displacement (COD), Electric Current Density (ECD), and Energy Flux Load (EFL) are mapped onto the square root singularity function. The resulting equations are then numerically solved using appropriate quadrature formulas, with the traction along the crack utilized as the right-hand term. The obtained COD, ECD and EFL functions is then used to compute the dimensionless Stress Intensity Factors (SIFs) in order to determine the stability behavior of TEBM weakened by an ICP. The numerical results provided demonstrate the dimensionless SIFs at the crack tips. These results exhibit excellent agreement with previous studies conducted on the subject. Additionally, it is observed that the dimensionless SIFs at the crack tips are influenced by factors such as the ratio of Elastic Constants (ECR), the geometry of the cracks, and the coefficients associated with the Electric Current Density (ECD).

Mixed-Mode Crack Propagation in Presence of Vein Fracture in Shale Sample in SCB Test

Mixed-mode fracture toughness refers to the ability of a material to withstand loads and expand under multiple modes of loading. It is usually tested by a semi-circular bend (SCB) to investigate the shale fracture toughness under combined modes of loading, such as tension and shear. When shale samples contain vein fractures, the dimensionless parameters Y1 and Y2 used in the SCB test method can be affected. The finite element method (FEM) was used to conduct a study investigating the impact of the natural fracture angle on the dimensionless parameters Y1 and Y2. The research findings indicate that as the vein fracture angle increases, Y1 and Y2 gradually increase. A construction was made to simulate the propagation path of a mixed-mode crack and compare it under various experimental conditions, utilizing the cohesive zone method (CZM). The results indicate that as the thickness of the vein fracture increases, the propagation path of the fracture becomes more tortuous. When the fracture encounters a vein with lower fracture strength, it will extend the vein fracture propagation. On the other hand, when the fracture encounters a vein with higher fracture strength, it will cross the vein fracture. This research considers the magnitude of the dimensionless stress intensity factor under different vein angles, simulating the crack propagation for varying vein widths and strengths.

Analytical Solution for a 1D Hexagonal Quasicrystal Strip with Two Collinear Mode-III Cracks Perpendicular to the Strip Boundaries

We considered the problem of determining the singular elastic fields in a one-dimensional (1D) hexagonal quasicrystal strip containing two collinear cracks perpendicular to the strip boundaries under antiplane shear loading. The Fourier series method was used to reduce the boundary value problem to triple series equations, then to singular integral equations with Cauchy kernel. The analytical solutions are in a closed form for the stress field, and the stress intensity factors and the energy release rates of the phonon and phason fields near the crack tip are expressed using the first and third complete elliptic integrals. The effects of the geometrical parameters of the crack configuration on the dimensionless stress intensity factors are presented graphically. The studied crack model can be used to solve the problems of a periodic array of two collinear cracks of equal length in a 1D hexagonal quasicrystal strip and an eccentric crack in a 1D hexagonal quasicrystal strip. The propagation of cracks produced during their manufacturing process may result in the premature failure of quasicrystalline materials. Therefore, it is very important to study the crack problem of quasicrystalline materials with defects as mentioned above. It can provide a theoretical basis for the application of quasicrystalline materials containing the above defects.

An Experimental Study on the Determination of Shale KIC by Semi-Disk Three-Point Bending

In order to accurately test the KIC of the vertical stratification direction of shale, a semi-circular bending specimen with a linear chevron notch ligament (LCNSCB) was designed. The minimum dimensionless stress intensity factor (Y*min) of the LCNSCB specimen was calculated by the finite element method and the slice synthesis method, respectively. Two sets of prefabricated samples of the LCNSCB specimen under arrester and divider mode were used to conduct three-point bending loading experiments. The dispersion of the measured KIC value of the specimens was analyzed by standard deviation and coefficient of variation, and the reason that the KIC dispersion of specimens in divider mode was larger than in arrester mode was discussed. Compared with the experimental data of the existing literature, the data of this experiment shows that the LCNSCB specimen can avoid the disadvantage of lower measured KIC values due to a larger fracture processing zone featured in the CSTSCB and CCNBD specimens, combined with the merits of a shorter fracture processing zone of the SR or CR specimens, and the render measured the KIC value to be closer to the material’s true fracture toughness value. The narrow ligament of the LCNSCB specimen has a favorable crack propagation guiding effect, can generate consistent KIC values, and could be used to accurately test the fracture toughness of rock material in vertical bedding direction.

Derivation of hyper-singular integral equations for thermoelectric bonded materials featuring a crack parallel to interface

In this paper, the derivation of hyper-singular integral equations (HSIEs) for thermoelectric bonded materials (TEBM) featuring a crack parallel to interface subject to in-plane shear stress τ∞xy was intensively studied. Generally, stress intensity factors (SIFs) were calculated using HSIEs with the help of modified complex stress variable function (MCSVF), and continuity conditions of the resultant electric force and displacement electric function. The unknown crack opening displacement (COD) function, electric current density, and energy flux load are mapped into the square root singularity function using the curved length coordinate method as the right-hand term. This unknown function is then used to compute the dimensionless SIFs in order to determine the stability behavior of TEBM featuring a crack parallel to interface subject to in-plane shear stress τ∞xy. Numerical results of the dimensionless SIFs at all the crack tips are presented. Our results are totally in good agreement with those of the previous works. It is observed that the dimensionless SIFs at the crack tips depend on the elastic constants ratio, the crack geometries, the electric conductivity, and the thermal expansion coefficients.

Anti-plane problem of a nano-sharp crack in one-dimensional hexagonal piezoelectric quasicrystals with the electrically semi-permeable condition

Based on the Gurtin–Murdoch surface/interface theory and the complex potential theory, the problem of an electrically semi-permeable nano-sharp crack in one-dimensional (1D) hexagonal piezoelectric quasicrystals is analyzed. By constructing conformal mapping of the sharp crack, the analytic solutions of the electroelastic field are determined. Meanwhile, the field intensity factors and the energy release rate (ERR) under the electric boundary conditions of semi-permeable are obtained. Numerical examples are given to analyze the effects of the crack size, dielectric constant, electric loading, and mechanic loadings on the dimensionless field intensity factors and the dimensionless ERR. The results show that the impact of the surface effect becomes weaker along with the growth of the crack length, but increases with the increase in the crack angle. The change of the dielectric coefficient has little influence on the dimensionless stress intensity factors (SIFs), but has a strong influence on the dimensionless electric displacement intensity factor (EDIF). The obtained results are helpful to promote the development of nano-quasicrystal composite mechanics and provide an important theoretical support for the design and application of nano-QC materials.

Static and dynamic fracture behavior of rock-concrete bi-material disc with different interface crack inclinations

The fracture behavior of rock-concrete bi-material structures with interface crack is of great significance to the stability of construction projects in rock engineering. Dynamic testing of cracked straight-through Brazilian disc (CSTBD) samples was performed using a split-Hopkinson pressure bar (SHPB) system. The fracture development of rock-concrete bi-material was monitored by the digital image correlation (DIC) technique. A static fracture test using the MTS322 hydraulic servo-controlled testing system was also carried out for comparison. The results show that three typical fracture patterns can be identified for CSTBD samples under both static and dynamic loading. At a smaller interface inclination angle, the disk samples fracture along the interface, while at a larger interface inclination angle, the samples suffer tensile fracture along the loading direction. The increase of strain rate leads to an increase in the number of secondary cracks. As the inclination of interface increases from 0° to 75°, the dissipated energy increases gradually. When the inclination exceeds 75°, the dissipated energy turns to weaken. The difference in peak dissipated energy between different inclination angles weakens as the strain rate increases. The dimensionless stress intensity factors (SIFs) at the two crack tips are different, and increasing the crack length-diameter ratio will enlarge the difference. The interface inclination angle has a significant effect on both the static and dynamic fracture toughness of CSTBD samples. As the inclination of the interface increases, the difference between mixed-mode static and dynamic fracture toughness also increases. The dependence of dynamic fracture toughness on inclination is significantly higher than that of static fracture toughness, and the ratio between dynamic and static fracture toughnesses increases with increasing loading rate.

Prediction of Mode I Fracture Toughness of Shale Specimens by Different Fracture Theories Considering Size Effect

In this study, mode I fracture tests on cracked straight-through Brazilian disc (CSTBD) and notched semi-circular bend (NSCB) shale specimens with different sizes were conducted to investigate the difference between maximum tangential stress fracture criterion and the size effect law (SEL) model in predicting apparent fracture toughness (Ka) of shale. In addition, the effects of specimen size and geometry on the Ka and the selection of fracture criterion on the prediction of the inherent fracture toughness (KIc) were also studied. The results show that the Ka increases with the increase of specimen size, and the difference between KIc of shale specimens with different sizes predicted by the fracture process zone length determined by the further improved maximum tangential stress (FIMTS) criterion is the smallest. For the prediction of Ka of NSCB specimen, the results predicted by the FIMTS criterion are the closest to the tested fracture toughness. However, the effect of SEL model applied to the prediction of Ka of NSCB specimens is poor. The effective establishment of SEL model requires high accuracy for test data, especially for the configuration with large variation of the dimensionless stress intensity factor (Y*) with normalized crack length (α).

Initiation and Fracture Characteristics of Different Width Cracks of Concretes under Compressional Loading

A stress concentration at a crack tip may cause fracture initiation even under low-stress conditions. The maximum axial stress theory meets the challenges of explaining the fracture propagation of a non-closed fracture of cracked concretes under compressional loading. Uniaxial loading tests of single-crack concrete specimens were carried out and a numerical simulation of fracture propagation under uniaxial compression was performed. The radial shear stress criterion for a mode-II fracture is proposed to examine the stress intensity factor (SIF) of the pre-crack specimens under compressional loading. When the maximum radial shear stress at the crack tip is larger than the maximum axial tensile stress, and the maximum dimensionless SIFs can satisfy frθmax/fθmax > 1 and frθmax/fθmax > KIIC/KIC (fθmax=KIe/σyπa and frθmax=KIIe/σyπa are maximum dimensionless mode-I and mode-II SIFs, respectively), the crack will extend along the direction of the maximum radial shear stress. The influence of the single-crack angle and width on the mechanical properties of the specimens was examined. The experimental and numerical results indicate that the existence of cracks can considerably weaken the strength of the specimen. The distribution and width of the cracks had a significant effect on the specimen strength. The strength of the concrete specimen initially decreased and then increased with increasing fracture angle. The failure mechanism and rupture angle of pre-crack brittle material while considering crack width will be discovered.

Effect of Mechanical Loadings on Two Unequal Slanted Cracks Length in Bi-Materials Plate

Although a lot of crack problems in bi-materials plate were previously treated, few solutions are available under mechanical loadings, arbitrary crack lengths and material combinations. In this paper the dimensionless stress intensity factors (SIFs) of two slanted cracks in the upper plate of bi-materials are considered under mechanical loadings with varying the crack length and material combinations systematically. In order to calculate the dimensionless SIFs accurately, the hypersingular integral equations (HSIEs) was formulated by using the modified complex potentials (MCP) function. The details numerical results of the dimensionless SIFs are given in tabular form and graphical presentations. Comparisons with the existing exact solutions show that the numerical results in this paper have high accuracy. Our results are described with clarifying the effect of the mechanical loadings, bi-elastic constant ratio and element size of cracks on the dimensionless SIFs.

Experimental study on Mode-I fracture toughness using Chevron Straight-notched Semi-circular Bend (CSNSCB) method

Rock fracture toughness is an important parameter for the engineering geological design. In this present study, chevron straight-notched semi-circular bend (CSNSCB) specimens with wide-flat liner chevron ligament (Type-A) and long-narrow liner chevron ligament (Type-B) were designed and prepared using wire cutting method. Mode-I fracture toughness KIC values and dimensionless stress intensity factor Y* were calculated. Moreover, these two types rock failure properties and KIC values variation features were both analyzed. The obtained results indicate that the width of precast artificial crack is small which can ensure the flatness and symmetry of cutting plane and then the tested Stress Intensity Factor (SIF) which has smaller deviation is more accurate and reliable. The Y* values for each type of configuration were calculated using Finite Element Method (FEM) and Slice Synthesis Method (SSM) and they firstly had a decreased trend and then an increased trend with the increasing of normalized crack length α, which represent the transition from stable crack propagation (α > αm) to unstable crack propagation (α < αm). Compared to Type-A specimens, Type-B specimens had the calculated minimum dimensionless stress intensity factor Y*min with smaller deviation and average KIC values with smaller variation coefficients. Moreover, Type-A specimen has the average fractal dimension D value of 1.2209 which is 4.79% larger than that of 1.1651 for Type-B specimen, indicating its more complex crack generation and rougher fracture surfaces. That is to say, Type-B specimen has the advantages of small crack FPZ and crack propagation without deviation as well as smooth fracture surface. Since it can offset the harmful influence of rock anisotropy and good crack guided effect, its KIC value is much satisfactory and closer to the true fracture toughness of rock-like materials.

Experimental analysis of the pseudo-compact tension (pCT) testing configuration using two alternative sample geometries

The pseudo-compact tension (pCT) method recently proposed by Muñoz-Ibáñez et al. (2020) is a satisfactory approach to measure mode I fracture toughness (KIC) in rocks and other materials using disc-shaped samples loaded under pure tensile conditions. In contrast to other methods, such as the semi-circular bend (SCB) suggested by the ISRM (2014), the pCT test provides with good control after peak load, making it possible to further characterize the processes involved in fracture propagation. In this work we assess the influence of the testing configuration at the onset of unstable crack propagation. In order to extend the pCT concept to complementary geometries with potential interest we studied an alternative to the SCB specimen, which we call pseudo-SCB (pSCB). To compute KIC in this configuration we have derived the corresponding dimensionless stress intensity factor function (Y’) based on the finite element method. The results show that the pSCB test provides with consistent values of KIC and it also allows to control the propagation of the crack beyond peak load, which reinforces the idea that the loading conditions may be a more determinant factor than the sample geometry in controlling post-peak behaviour. In addition, an expression of Y’ is presented for cubic samples tested using the pCT approach. This configuration may be useful for testing other materials amenable of moulding such as mortar, concrete, ceramics, etc.

V-notch crack toughness of PMMA by strain gages and its comparison with caustics results under impact loading

Strain gages and dynamic caustics were used in this experiment to measure the V-notch crack initiation toughness and propagation toughness of polymethyl methacrylate (PMMA) specimens. Furthermore, a method of pasting strain gages on V-notch PMMA specimens for three-point bending test was proposed. The specimens were divided into five groups according to the opening angle (ω), and each group corresponded to two pasting methods: acute-angle pasting method and obtuse-angle pasting method. The results of notch stress intensity factors (NSIFs) obtained by strain gage measurement were consistent with the results calculated using the dynamic caustics method. The concept of the dimensionless notch stress intensity factor (DNSIF) was introduced for a better analysis of the effect of opening angles on cracking. The experimental results revealed that the initiation of V-notch crack in the specimens were inhibited more when ω was larger than 90° than for small opening angles. Under this condition, the specimens displayed higher cracking initiation toughness. Furthermore, the time required for crack initiation was longer, and the cracking was instantaneous. An analysis of the vicinity of the V-notch crack tip showed that with the increase in ω, the stress component decreased, and the rate of decrease became higher. This also explains the phenomenon wherein the specimens transformed from being easy to crack to not being so as ω increased. The experimental results showed that the crack propagation velocity and propagation toughness were also affected by the opening angle of the V-notch crack.

Numerical Solution for Crack Phenomenon in Dissimilar Materials under Various Mechanical Loadings

A new mathematical model is developed for the analytical study of two cracks in the upper plane of dissimilar materials under various mechanical loadings, i.e., shear, normal, tearing and mixed stresses with different geometry conditions. This problem is developed into a new mathematical model of hypersingular integral equations (HSIEs) by using the modified complex potentials (MCPs) function and the continuity conditions of the resultant force and displacement with the crack opening displacement (COD) function as the unknown. The newly obtained mathematical model of HSIEs are solved numerically by utilizing the appropriate quadrature formulas. Numerical computations and graphical demonstrations are carried out to observe the profound effect of the elastic constants ratio, mode of stresses and geometry conditions on the dimensionless stress intensity factors (SIFs) at the crack tips.

Experimental Investigation on the Tensile and Fracture Properties of Burst‐Prone Coal under Quasistatic Condition

To study the tensile and fracture properties of the specimen under the quasistatic loading, the Brazilian disc splitting method and the notched semicircular bend (NSCB) method were used to test the tensile properties of coal specimens, and the fracture properties of NSCB specimens with different notch depths were tested and analyzed. The applicability of plane strain fracture toughness KIC and J‐integral fracture toughness in evaluating the fracture properties of coal specimens was discussed. The influence of notch depth on the fracture toughness measurement of the NSCB specimen was studied. Combined with the surface strain monitoring of specimens during loading and the industrial CT scanning image of damaged specimens, the deformation characteristics of coal specimen under loads and the distribution law of crack after failure were analyzed. The results show that the NSCB test is suitable for measuring the tensile strength of a coal specimen; when the dimensionless notch depth is β = 0.28, the dispersion of plane strain fracture toughness KIC of the NSCB specimen is the smallest. Besides, the plane strain fracture toughness of coal is obviously affected by the notch depth and dimensionless stress intensity factor. The J‐integral fracture toughness can be used to effectively evaluate the fracture performance of specimens.

Experimental study on dynamic fracture behaviors of Beishan NSCB and CCNSCB granite specimens under different loading rates

Beishan area in Gansu province of China has been listed as a major candidate for underground constructions of high-level radioactive waste repository. Therefore, to explore the dynamic responses of Beishan granite is important to the related excavation engineering. This paper investigates the rate dependent effect on the dynamic fracture behaviors of the Beishan granite. The notched semi-circular bend (NSCB) and cracked chevron notched semi-circular bend (CCNSCB) specimens are prepared with speckles uniformly distributed on surface for digital image correlation (DIC) image processing. An ultra-high-speed camera is employed for the modified split Hopkinson pressure bar (SHPB) system to capture the complete cracking process of the rock specimen. The dimensionless stress intensity factors (SIFs) of NSCB and CCNSCB specimens are calculated numerically by the J-integral method in ABAQUS simulation software. To systematically reveal the dynamic fracture mechanisms of Beishan granite specimens, the key dynamic fracture parameters including the dynamic fracture toughness, the evolution of dynamic stress intensity factor (DSIF) during fracturing and the dynamic fracture energy are obtained for further investigations and comparisons. The results show that the dynamic fracture toughness of NSCB and CCNSCB specimens exhibits a distinct rate dependence-the fracture toughness increases with the loading rate rises. Furthermore, the toughness value of the CCNSCB specimen are mostly higher than that of the NSCB specimen due to a more severe stress concentration level near the notch tip. The evolutions of DSIFs of Beishan NSCB and CCNSCB specimens are quite similar to each other, and the corresponding variation curves are all subjected to three typical developing phases until specimens are totally fractured. The dynamic fracture energy of NSCB and CCNSCB specimens is relatively consistent and shows a positive relationship with the loading rate. However, the fracture energy absorption rate (defined as the ratio between the fracture energy and the energy generated in the incident bar) gradually decreases with the loading rate rising, indicating a lower loading rate may be more suitable to improve rock fragmentation efficiency.