Generalized Maximum Tangential Stress Research Articles

Assessment of fracture criteria for cracking behavior of asphalt concrete using various SCB specimen sizes

AbstractIn this research, five fracture criteria (maximum tangential stress (MTS), generalized maximum tangential stress (GMTS), minimum strain energy density (SED), generalized minimum strain energy density (GSED), and modified maximum tangential stress (MMTS)) were evaluated for asphalt concrete using fracture data from tests conducted under various loading modes and conditions. Tests were performed on semi‐circular bend (SCB) specimens of various sizes. Based on the results, the MTS criterion tends to overestimate fractures, while the GMTS criterion exhibits more precise predictions. The SED and GSED criteria inaccurately predict fractures, especially under certain loading conditions. The MMTS criterion shows superior predictive capability for asphalt concrete fractures. Specimen size influences fracture resistance, with larger specimens exhibiting higher fracture values. Overall, the MMTS criterion closely mirrors the crack growth behavior of asphalt concrete, highlighting its precision in predicting fracture results. The errors between the predictions of the fracture criteria MTS, GMTS, SED, GSED, and MMTS, and the experimental results ranged from 0% to 42.8%, 0% to 22.5%, 0% to 52.7%, 0% to 42.4%, and 0% to 14.5%, respectively.

Study on the dynamic fracture behavior of anisotropic CCNBD shale specimens under different impact angles

Drilling and fracturing are the key technologies for shale gas extraction, while most of the loads generated during rock drilling or fracturing are dynamic. In this study, dynamic impact experiments were conducted on the cracked chevron-notched Brazilian disc shale specimens with different chevron-notched crack (CNC) inclinations (β) and layer inclinations (α) by the split Hopkinson pressure bar. The results show that the dynamic peak load and dissipated energy increases with the increase of β, α, and strain rate, but their changing trends are different. The generation and expansion of cracks during the specimen failure process are mainly affected by β. As β increases, the failure type of the specimen can be divided into pure mode I fracture, mode I-II mixed fracture, pure mode II fracture, mixed tension-shear failure, and Brazilian splitting failure. The increase in strain rate will lead to a decrease in the time for crack initiation and propagation as well as an increase in secondary cracks. In addition, the mode I fracture toughness (KⅠC) and mode II fracture toughness (KⅡC) both grow with the increase of α and strain rate. The KⅡC predicted by the generalized maximum tangential stress criterion (GMST) exhibits more accurate.

Experimental and theoretical investigation of the influence of post-curing on mixed mode fracture properties of 3d-printed polymer samples

This study explores the mechanical properties and fracture characteristics of additively manufactured acrylonitrile butadiene styrene specimens, focusing on the impact of raster angle and post-process heat treatment. To this end, a large number of tensile and semi-circular bending samples with three distinct raster angles of 0/90°, 22/ − 68°, and 45/ − 45° were prepared and exposed to four types of heat treatments with different temperature and pressure conditions. Simultaneously, theoretical models of maximum tangential stress (MTS) and generalized MTS (GMTS) were developed to estimate the onset of specimen fracture under mixed-mode in-plane loading conditions. Recognizing the non-linear behavior within the stress–strain curve of tensile test samples, particularly in the annealed samples, an effort was undertaken to transform the original ductile material into a virtual brittle material through the application of the equivalent material concept (EMC). This approach serves the dual purpose of bypassing intricate and tedious elastoplastic analysis, while concurrently enhancing the precision of the GMTS criterion. The experimental findings have revealed that while the annealing process has a minimal effect on the yield strength, it considerably enhances energy absorption capacity, increases fracture toughness, and reduces the anisotropy. Additionally, the combined EMC-GMTS criterion has demonstrated its capability to predict the failure of the additively manufactured parts with an acceptable level of accuracy.

Influence of coupled water and thermal treatments on the fracture characteristics of a typical sandstone

Understanding the fracture behavior of rock after coupled water and thermal environment is important for many geotechnical projects. This study examines the influence of coupled water and thermal treatments on the fracture toughness and characteristics of a typical sandstone under mode I and mode II loading conditions. Notched deep beam (NDB) specimens were utilized and subjected to soaking treatments at various water temperatures (23 °C, 60 °C, and 99 °C). The experimental results indicate a significant reduction in both mode I and mode II fracture toughness values, with reductions ranging from 15.4% to 13.2% for mode I and 26.1% to 8.9% for mode II respectively. As the water temperatures increase, a slightly rising trend is observed in both mode I and mode II fracture toughness within the examined temperature range. Sandstone specimens displayed typical brittle fracture characteristics at lower soaking temperatures. For mode I specimens, an increase in ductility was evident with higher soaking temperatures, while the ductile behavior is less pronounced in the mode II specimens. Based on the Maximum Tangential Stress (MTS) criterion and the Generalized Maximum Tangential Stress (GMTS) criterion, the predicted values of mode II fracture toughness and the fracture process zone (FPZ) were discussed. The results show that both the GMTS and MTS criteria exhibit inaccuracies in predicting the mode II fracture toughness of sandstone treated at different soaking water temperatures. However, the GMTS criterion, which incorporates T-stress, demonstrates smaller errors compared to the MTS criterion. The study shows that the radius r0 of the fracture process zone is not a constant under both mode I and mode II loading conditions. The calculation of the fracture process zone radius r0 in the GMTS criterion requires further theoretical and experimental study.

Mixed mode I/II and I/III fracture toughness investigation for preplaced aggregate concrete mixtures: Experimental study and theoretical prediction

AbstractIn this paper, mixed mode I/II and I/III fracture toughness of preplaced aggregate concrete was studied using an edge‐notched disc bend specimen. The results showed the dependency of the fracture toughness value on the mode mixity. The values of modes I, II, and III fracture toughness were obtained to be equal to , , and , respectively. Several theoretical fracture criteria were examined for predicting the trends of experimental data in the tested preplaced aggregate concrete material. The 3D averaged strain energy density criteria provided the best predictions for the KIIIc/KIc ratio. Mixed mode I/III fracture toughness data were between the predicted envelopes of the maximum tensile stress and 3D maximum tangential strain theories. The generalized maximum tangential stress criterion provided accurate predictions for pure mode II and mixed mode I/II fracture data among the investigated mixed mode fracture criteria. By comparing the preplaced aggregate concrete fracture results with those of other types of concrete, it is observed that preplaced aggregate concrete has acceptable and in‐the‐range fracture resistance.

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.

Wmic-GMTS and Wmic-GMERR criteria for micron-scale crack propagation in red-bed soft rocks under hydraulic action

Micron-scale crack propagation in red-bed soft rocks under hydraulic action is a common cause of engineering disasters due to damage to the hard rock–soft rock–water interface. Previous studies have not provided a theoretical analysis of the length, inclination angle, and propagation angle of micron-scale cracks, nor have they established appropriate criteria to describe the crack propagation process. The propagation mechanism of micron-scale cracks in red-bed soft rocks under hydraulic action is not yet fully understood, which makes it challenging to prevent engineering disasters in these types of rocks. To address this issue, we have used the existing generalized maximum tangential stress (GMTS) and generalized maximum energy release rate (GMERR) criteria as the basis and introduced parameters related to micron-scale crack propagation and water action. The GMTS and GMERR criteria for micron-scale crack propagation in red-bed soft rocks under hydraulic action (abbreviated as the Wmic-GMTS and Wmic-GMERR criteria, respectively) were established to evaluate micron-scale crack propagation in red-bed soft rocks under hydraulic action. The influence of the parameters was also described. The process of micron-scale crack propagation under hydraulic action was monitored using uniaxial compression tests (UCTs) based on digital image correlation (DIC) technology. The study analyzed the length, propagation and inclination angles, and mechanical parameters of micron-scale crack propagation to confirm the reliability of the established criteria. The findings suggest that the Wmic-GMTS and Wmic-GMERR criteria are effective in describing the micron-scale crack propagation in red-bed soft rocks under hydraulic action. This study discusses the mechanism of micron-scale crack propagation and its effect on engineering disasters under hydraulic action. It covers topics such as the internal-external weakening of nano-scale particles, lateral propagation of micron-scale cracks, weakening of the mechanical properties of millimeter-scale soft rocks, and resulting interface damage at the engineering scale. The study provides a theoretical basis for the mechanism of disasters in red-bed soft-rock engineering under hydraulic action.

Mode I/II cracking behavior of additively manufactured interpenetrating phase composites (IPC), an experimental and theoretical study

An interpenetrating phase composite (IPC) material combines the advantages of being made from materials for an intended behavior. 3D-printed parts have received attention for rapid prototyping of complex structures; however, their porous inherent makes them susceptible to cracking. Polymer adhesives can be used to overcome these issues to increase their strength and durability. In this paper, IPCs were developed using 3D-printed scaffolds and epoxy resin matrix, and their fracture behavior was examined. The samples were printed with 25, 50, 75, and 100 infill percentages and tested in the full range of mode I-II fracture. From the results, the promising behavior of IPCs can be seen. Where the 3d printed samples with 100, 75, 50, and 25 infills show 2.4, 1.8, 0.8, and 0.3 MPa√m mode-I fracture toughness, IPCs made from samples with 75, 50, and 25 infills, show 2.0, 2.2 and 2.7 MPa√m, respectively. The same trends are also seen in the full range of mode I-II fractures. The generalized maximum tangential stress (GMTS) criterion accurately predicted KIIc/KIc ratios among the investigated mixed-mode fracture criteria. The evaluations indicated that the discrepancy between the measurements is mainly due to the sign and magnitude of the T-stress parameter.

Initiation and growth of fatigue cracks in sheets with U-shaped notches in the first and mixed modes of fracture

Initiation and growth of fatigue cracks in sheets with U-shaped notches in the first and mixed modes of fracture

Experimental Study on Mixed-Mode (I–II) Fracture Toughness of Freshwater Ice

In recent years, the issue of aircraft icing has gained widespread recognition. The breaking and detachment of dynamic ice can pose a threat to flight safety. However, the shedding and fracture mechanisms of dynamic ice are unclear and cannot meet the engineering needs of ice-shedding hazard assessment. Therefore, studying the fracture toughness of ice bodies has extremely important practical significance. To address this issue, this article uses a centrally cracked Brazilian disk (CCBD) specimen to measure the pure mode I toughness and pure mode II fracture toughness of freshwater ice at different loading rates. The mixed-mode (I–II) fracture characteristics of ice are discussed, and the experimental results are compared and analyzed with the theoretical values of the generalized maximum tangential stress (GMTS) criterion considering the influence of T-stress. The results indicated that as the loading rate increases, the pure mode I toughness and pure mode II fracture toughness of freshwater ice decrease, and the fracture toughness of freshwater ice is more sensitive to the loading rate. In terms of fracture criteria, the theoretical value of the ratio of pure mode II fracture toughness to pure mode I fracture toughness based on the GMTS criterion is in good agreement with the experimental value, while the theoretical value based on the maximum tangential stress (MTS) criterion deviates significantly from the experimental value, indicating that the GMTS criterion considering the influence of T-stress can better predict the experimental results.

Fracture analysis of Brazilian circular hole disk under mixed mode loading

The paper presents an investigation into a Brazilian disk specimen featuring a concentric circular hole, accompanied by two internal edge radial cracks. By introducing rotations of the cracks relative to the vertical axis, a combination of fracture modes I and II is induced. Employing the finite element method, we compute stress intensity factors related to fracture modes I and II, alongside T-stresses, while varying geometric parameters such as inner and outer diameters, crack length-to-ring width ratio, and crack inclination angle. Experimental analyses of ebonite's fracture toughness in the mixed mode are conducted, subjecting samples to static loading until complete failure. Each test provides the angle at which crack initiation occurs and the corresponding critical load. Three distinct fracture criteria - the generalized maximum tangential stress criterion (GMTS), extended maximum tangential strain criterion (EMTSN), and generalized strain energy density (GSED) criterion - are employed to predict fracture direction and critical load. The outcomes exhibit good correspondence between experimental critical loads and theoretical predictions.

Application of modified fracture criteria incorporating T-stress for various cracked specimens under mixed mode I-II loading

Accurate assessment of fracture behavior and life prediction of cracked materials based on the mixed mode fracture criteria is of utmost importance in fracture mechanics. In this study, the modified fracture criteria incorporating T-stress for mixed mode I-II cracks were comprehensively reviewed. Additionally, a comparative analysis was conducted between the experimental results obtained from five different cracked configurations and the theoretical predictions according to these modified criteria. Moreover, a brief discussion was held regarding the effect of T-stress on both crack initiation angle and fracture toughness, accompanied by providing some suggestions. The results revealed that the predictive accuracy of modified criteria exhibited noticeable variations across different cracked configurations due to the significant disparities in both the sign and magnitude of T-stress around the crack tip. However, it was observed that the predicted curves based on the modified fracture criteria for the semi-circular bend and edge-crack triangular specimens had remarkable similarities owing to their similar biaxial stress ratio B. When the mode II loading dominated, the generalized maximum tangential stress, generalized minimum strain energy density and generalized averaged strain energy density criteria were suitable for the semi-circular bend, edge-crack triangular, central crack rectangular and short bend beam specimens with positive T-stresses, whereas the maximum tangential strain, generalized maximum tangential stress and generalized maximum tangential strain energy density criteria could provide better theoretical predictions for central cracked Brazilian disk specimens having a negative T-stress.

On the decay of fracture toughness of sandstone subjected to freeze–thaw cycles

AbstractIn cold regions, progressive cracking, caused by freeze–thaw cycles, plays an important role in deteriorating engineering rock mass. Employing central cracked Brazilian disks (CCBDs) of sandstone, this paper aims at investigating the variations of falling rate among different modes of fracture toughness exposed to freeze–thaw cycles and the applicability of fracture criterion in predictions. Results revealed that different modes of fracture toughness exhibited comparable descending trends after varied freeze–thaw cycles. There was an agreement between the predictions via the generalized maximum tangential stress criterion (GMTS) and the test results. In specified conditions, the GMTS criterion incorporating fracture process zone of 0 freeze–thaw cycle can be applied to estimations for different freeze–thaw cycles. With increasing freeze–thaw cycles the fractal dimensions of mode I and mixed‐mode I–II fracture surfaces displayed a growing trend, indicating that the fracture surfaces became rougher and more complicated.

Investigation of Constraint Effect and Fracture Mode for Mixed Mode Inclination Surface Crack in Infinite Plate Under Compression

Abstract The stress field, constraint effect, and fracture mode transition at crack tip of mixed mode I-II-III inclination surface crack under compression have been investigated. The effects of geometrical configurations (relative crack depth and aspect ratio), friction coefficient, and biaxial scale factor on stress intensity factor (KII and KIII) and in-plane constraint parameter T-stress are quantitatively studied, the stress field at different crack inclination angles under tension and compression are compared, the failure mode at special locations along crack front of inclination surface crack is analyzed according to the generalized maximum tangential stress criterion (GMTS). The relative crack depth has slight effect on stress intensity factor and T-stress, and aspect ratio has a significant effect on stress intensity factor and T-stress. The friction coefficient decreases the magnitude of stress intensity factor and increases the magnitude of T-stress, the greater the crack inclination angle is, the more pronounced the effect is when crack inclination angle greater than 30 deg. The stress distribution around crack tip under tension and compression is completely different. At free surface, the crack will failure in-plane shear mode II sliding crack, and at the deepest part of crack, the crack will start as out-plane shear mode III tearing crack under compression.

Analysis of Fracture Characteristics of Ore Rock Based on GMTS Criterion

In this paper, the Tongkaiyu ore-rock is taken as the research object, aiming to analyze the fracture characteristics of the mixed type I, II and I–II faults in ore-rock by using the appropriate fracture criterion, and analyze the fracture development mode. Considering the non-singular term (T stress) in the formula of stress field sequence at crack tip, the generalized maximum tangential stress (GMTS) criterion is established, which is consistent with the boundary conditions of the central straight Brazilian disk (CSTBD) specimen. Through CSTBD splitting test and three-dimensional particle flow code (PFC3D) numerical simulation, the Angle between the central direct joint and the loading direction was changed, and the fracture toughness and fracture development law of type I, type II and type I–II mixed fracture were obtained. The theoretical and simulation results show that, compared with the maximum tangential stress (MTS) criterion, the GMTS criterion has an advantage in judging the fracture characteristics of mixed types I, II and I–II in ore rock.

Effects of Support Friction on Mixed-Mode I/II Fracture Behavior of Compacted Clay Using Notched Deep Beam Specimens under Symmetric Fixed Support

This paper investigates the effects of support friction on mixed-mode I/II fracture behavior of compacted clay using notched deep beam (NDB) specimens under symmetric fixed support. Numerical models of 330 NDB specimens were established considering the crack inclination angle, crack length, support span, and support friction coefficient, and the normalized fracture parameters (YI, YII, and T*) of NDB specimens were calibrated. The numerical results showed that the values of YI, YII, and T* decreased at different degrees after considering the support friction. Notably, the support friction coefficient could significantly change the loading pattern at the crack tip. To verify this phenomenon, 12 compacted clay NDB specimens were prepared, and a mixed-mode I/II fracture test was performed under fixed support conditions; the phenomenon of asymmetric crack propagation was studied. The test data were processed using the numerical calibration results of YI, YII, and T* with and without consideration of friction. Afterward, the test data were compared and analyzed by combining the generalized maximum tangential stress (GMTS) and the maximum tangential stress (MTS) criteria. The analysis indicated that the real fracture characteristics of compacted clay NDB specimens could not be reflected when conducting mixed-mode I/II fracture tests under symmetric fixed support conditions if the test results were analyzed by YI, YII, and T* without considering support friction, as in previous studies.

Mixed mode I/II fracture behavior of CSTBD sandstone specimen under different loading angles

Cracks are often exposed to mixed mode I/II loading because of the random direction of crack relative to the external load, causing rock failure in rock engineering. However, many mixed mode I/II fracture properties of rock with different loading angles (β) are still poorly understood. To investigate the influence of β on mixed mode I/II fracture behavior, cracked straight through Brazilian disc tests were performed on sandstone under various β at the loading rate of 0.2 kN/s. The results show that the effect of β on the peak load and crack propagation velocity is slight. The mode I stress intensity factor (SIF) decreases from positive to negative with increasing β, while mode II SIF first increases and then decreases as β exceeds approximately 30°. The mixed mode I/II fracture toughness increases linearly with increasing β. When β exceeds 60°, the crack initiation location shifts from the center of the semi-circular notch tip to the surface of preset crack. For non-tip cracking, the crack propagates toward the loading point in a direction approximately perpendicular to the preset crack. The threshold β for it decreases with increasing internal friction coefficient and relative crack length. Moreover, the mixed-mode fracture surface becomes smoother as the mode I component contribution drops. There are remarkable differences in the prediction of fracture initiation angles and SIFs of various rock types applying the generalized maximum tangential stress criterion. The findings of this study could help understand the mixed mode I/II fracture observed in rock engineering.

Multi-fidelity model using GRNN and ANFIS algorithms-based fracture criterion for predicting mixed-mode I-II of sugarcane leaves/epoxy composite

Artificial intelligence plays a huge role in solving engineering problems. In the fracture mechanics field, artificial intelligence is used to predict fracture behavior or parameters, based on testing data, which is a destructive type of testing, so it is difficult to get a large amount of data for the learning modeling process. Therefore, this study presents artificial intelligence modeling to predict the fracture toughness by reducing the use of the actual testing fracture toughness data, while replacing the data with a sufficient quantity of data by fracture toughness prediction from the fracture criterion, for which artificial intelligence was modeled by the concept called the “multi-fidelity model”. The concept of the modeling begins with calculating and predicting the difference between the actual data set and the criterion data set, then combining the data sets, which is the data from the criterion performed mathematically with the data from the above difference prediction to the actual data set to create a model based on the increased data volume of combining the two data sets. In this study, the mixed-mode I-II fracture toughness of sugarcane leaves/epoxy composite with different mixing conditions was performed on an inclined crack specimen via the three-point bending test configuration used as the actual data set, while the generalized maximum tangential stress criterion (GMTS), which is preliminary fracture toughness, was used as criterion data. The multi-fidelity model is proposed via different artificial intelligence algorithms, namely the general regression neural network (GRNN) and the adaptive neuro-fuzzy inference system (ANFIS). Once the modeling process was complete, the performance metrics showed a clear increase in the efficiency of multi-fidelity models compared to the original artificial intelligence models.

Hygrothermal degradation of MWCNT/epoxy brittle materials under I/II combined mode loading conditions: An experimental, micro structural and theoretical study

The effects of carboxylated multi-walled carbon nanotubes (MWCNTs) are studied experimentally and theoretically on mixed-mode (I/II) fracture behavior of epoxy resin at the presence of hygrothermal aging conditions. In this paper, the mixed-mode fracture of MWCNT/epoxy is experimentally measured using Short Beam Bending (SBB) samples with different crack inclination angles (α) ranging from pure mode I (i.e., α = 0°) to pure mode II (i.e., α = 39°). To examine the effects of hygrothermal aging conditions and water barrier properties of MWCNTs, four groups of samples including a group of neat epoxy specimens, and three groups of MWCNT/epoxy specimens (with 0.1, 0.3, and 0.5 wt% of MWCNTs) were manufactured and immersed into hot deionized water for 15 days at 60 °C temperature and similarly four groups were placed at room temperature conditions (26 °C and 25–35% relative humidity) for 15 days. According to the experimental results, toughening the epoxy resin by incorporating nanotubes with 0.1, 0.3, and 0.5 wt% of MWCNTs improved the fracture toughness at both environmental conditions, but the highest fracture toughness was obtained for composition containing 0.3 wt% nanoparticles. The variations of fracture loads were dependent on the ambient environment, MWCNTs content, and mode mixity. The relationship between the obtained experimental results, microstructure and morphology of fracture in the tested samples was also investigated. In addition, the load-carrying capacity of the SBB samples made of the neat and MWCNTs modified epoxy material was also predicted well using a two-term based fracture model called Generalized Maximum Tangential Stress (GMTS) criterion.

Numerical & experimental assessment of mixed-modes (I/II) fracture of PMMA/hydroxyapatite nanocomposite

Nowadays, polymer-based nanocomposites widely used in biocompatible restorative materials and medical equipment and polymer-based nanocomposites are the most important of them. One of the most useful biocompatible polymers in this field is polymethyl methacrylate (PMMA). In this research, the experimental and numerical analysis of the fracture behaviors of PMMA nanocomposites that are reinforced with hydroxyapatite (HA) nanoparticles are investigated under pure mode I, pure mode II and mixed-mode (I/II) loading. For this purpose, notched semi-circular bend (SCB) specimens containing various Wt. % of HA nanoparticles and different notch angles were fabricated. Then, the mechanical properties of the specimens and the fracture toughness of the notched specimens were investigated. To evaluate the accuracy of nanomaterial distribution, scanning electron microscopy (SEM) images are analyzed and it was observed that at Wt. % of more than 1.5%, specimens with more cumulative masses of hydroxyapatite nanoparticles have more strength reduction. Also, to investigate the fracture behaviors of the produced specimens, the failure criteria of maximum tangential stress (MTS) and generalized maximum tangential stress (GMTS) were evaluated and it was observed that the GMTS criterion is more in line with the experimental results. In addition, the results showed that by increasing the Wt. % of HA nanoparticles from zero to 1% mass, the KIC value increased compared to the pure PMMA, whereas, by increasing this value to 1.5% mass, the fracture toughness decreased. Finally, the fracture initiation and propagation are evaluated using FEM and XFEM numerical methods and the values of stress intensity factors and crack growth path are predicted and compared with experimental values. The obtained numerical results are in good agreement with the experimental results.