Fracture Toughness Parameters Research Articles

Quantifying nonlinear fracture parameters in bituminous SCB specimens: A compliance-based approach

Bituminous mixtures containing various defects, such as cracks and voids, exhibit ductile behavior in hot environments and brittle behavior in cold environments. Such materials can only be simulated realistically with the criteria of nonlinear fracture mechanics. In this investigation, the semi-circular bending (SCB) specimens, which have commonly been used to measure linear elastic fracture toughness parameters and fracture energy of asphalt and rock materials up to now, are studied to estimate several nonlinear fracture parameters of bituminous composites such as nonlinear fracture toughness, effective crack extension, critical crack tip opening displacement and brittleness. For this, the compliance functions of SCB specimens based on crack mouth opening displacement and load line displacement are derived by the finite element method at first. Subsequently, eight series of SCB tests in the literature, which were experimented with at low and room temperature environments, are examined by using two different compliance methods in fracture mechanics of quasi-brittle materials. This study's findings reveal that the crack extension in the pre-peak regime for mixes with bituminous at both low and normal temperatures is not statistically negligible and the use of nonlinear fracture toughness is, therefore, a necessity.

Introducing a novel LLD-CMOD equivalence fracture model for quasi-brittle materials exposed to three-point bending

Effective crack models have been preferred in the fracture of concrete thanks to their analytical solution for equivalent effective crack length. Nevertheless, the main hypothesis in the effective crack models, the Load-LLD or Load-CMOD equivalence between a real cohesive material and a corresponding virtual specimen, is controversial. The present study aims to offer a distinct perspective on the effective crack approach. To this end, the Load-LLD or the Load-CMOD equivalence was replaced by the LLD-CMOD equivalence in the proposed fracture model for the notched beams under three-point bending. In the proposed LLD-CMOD equivalence fracture model, three fracture toughness parameters are used, two of them depend on the evolution of the fracture process zone (FPZ) and the other is for the peak load measured. That is, KICin and KICde define the limit of the nonexistent FPZ and the fully developed FPZ, respectively, but KICun corresponds to the peak load only. The test results of the notched beams revealed that KICin and KICun show size independence, but this was not the case for KICde. The proposed fracture model was also compared with the Double-K model through the experimental results from the literature. Accordingly, a good agreement between both models was achieved in terms of the effective crack length and fracture toughness at the peak load.

Effects of adhesive layer thickness on the fracture properties of a concrete-epoxy resin interface

Adhesive layer thickness is a crucial factor that affects the performance of interfaces in cement-based composite structures. However, the influence of adhesive layer thickness on the fracture properties of concrete–epoxy resin remains unclear. In this study, a two-parameter fracture model of a concrete-epoxy resin interface from the perspective of fracture mechanics. Digital image correlation (DIC) technique was used to investigate the evolution of the fracture process zone (FPZ) of nine specimens with adhesive layer thicknesses of 2 mm, 5 mm, and 8 mm, respectively. The interfacial fracture toughness parameters of the concrete-epoxy resin interface were determined based on the four-point bending (FPB) tests. The test results indicated that increasing the adhesive layer thickness changed the failure mode of the interface from approximate-brittle to quasi-brittle. The load-carrying capacity and interfacial stiffness of the concrete-epoxy resin interface decreased by approximately 13.8 % and 61.1 % with an adhesive layer thickness from 2 mm to 8 mm, respectively. The deflection across the specimen increased by approximately 37.2 % with an increase in adhesive layer thickness due to the lower elastic modulus of epoxy resin compare to the concrete. Furthermore, the FPZ for the adhesive layer thickness of 5 mm was the longest; it was approximately 18.7 % longer than that for the test specimen with an adhesive layer thickness of 2 mm. In terms of structural load-bearing capacity and fracture toughness, the concrete-epoxy resin interface exhibited the best working performance when the thickness of the adhesive layer was 5 mm.

Towards the understanding of fracture resistance of an ultrahigh-strength martensitic press-hardened steel

Identifying the fracture resistance capabilities and gaining a deeper understanding of the controlling damage mechanisms are important to the development of press-hardened steels (PHS) for automotive applications. In this study, the fracture properties of a novel PHS alloyed with Cr and Si (CrSi-PHS) were assessed by using uniaxial tensile and double-edge notched tensile (DENT) tests. Under uniaxial tension, the fracture resistance is characterized by the fracture strain and work of fracture, and the CrSi-PHS presents a desirable compromise with strength when comparing with other grades of advance high-strength steels (AHSS). The large fracture strain is attributed to the high resistance to damage nucleation even with a high density of grain boundaries. Fracture toughness is characterized by the DENT tests. The fracture toughness parameters of CrSi-PHS are lower than that of the commercial PHS 22MnB5 but comparable to the multiphase AHSSs with lower strength levels. The stress triaxiality near the fatigue pre-crack in DENT tests leads to a change of damage mechanism in CrSi-PHS. The crack propagates discontinuously by joining the micro-cracks and subsequently by the ductile fracture of micro-ligaments. The local plasticity at the crack tip region is suggested to contribute significantly to the energy dissipation. The observations demonstrate a strong dependence of the fracture mode and fracture resistance on the loading condition in the ultrahigh-strength martensitic steels.

Omega and CTOA for dynamic steady-state fracture propagation and their dependency to fracture speed

Dynamic ductile fracture in pressure vessels is an increasingly important topic. Recent catastrophic pressurised pipeline failures highlighted the critical needs of reliable pipeline infrastructures. The ability for line pipe steels to resist fracture is a critical design factor for governing the severity level of fracture failures given that it determines if fracture propagates for several kilometres or is arrested quickly. Developing a reliable fracture toughness test to systematically evaluate new steel products’ quality is therefore important. In the current work, a novel small-scale fracture toughness parameter Omega was implemented and compared to the well-established parameter CTOA, with respect to their performance of characterizing steady-state fracture propagation and their dependencies to Fracture Speed. The current experimental data were collected from a 3-point bend dynamic Drop Weight Tear Test (DWTT) for 9 line pipe steels with different thicknesses and impact conditions. It was found that the Omega gives significantly better description for steady-state fracture propagation; and it is found to be independent of the Fracture Speed. The current study confirmed that in DWTT, CTOA values decrease with increasing Fracture Speed. Finally, a simple analytical model is proposed to describe this dependency (CTOA-Fracture Speed) from a geometrical standpoint.

Determination of fracture toughness of 2.25Cr1Mo0.25V steel based on acoustic emission technique

The correct determination of fracture toughness is dependent on the accurate measurement of the point of crack initiation. In this research work, the AE technique is used to accurately detect the point of crack initiation and determine the initiation fracture toughness value JAE of 2.25Cr1Mo0.25V steel. Moreover, conventional standard procedures such as the J-R curve and stretch zone methods are also used to determine the fracture toughness JQ and JSZW values, respectively. Multiple AE parameters involving the entropy, peak amplitude, count, and energy are extracted from AE signals to identify the damage progression during fracture toughness tests. The results show several critical points associated with the point of plastic yielding, the onset of crack initiation, and the rapid crack growth and final fracture can be accurately identified by AE characteristics. Furthermore, the point of crack initiation recognized by AE is used to determine the JAE value. The fracture toughness JAE values of different specimens are also compared with the corresponding JQ and JSZW values. The JAE values are found to be about 20% lower than the JQ values but are significantly larger than the JSZW(2D) and JSZW(3D) values. It is believed that the fracture toughness parameter JAE determined by the AE technique can provide an important reference for evaluating the fracture toughness of engineering materials.

The effects of high-temperature ion-irradiation on early-stage grain boundaries serrations formation in Ni-based alloys

Nickel-based superalloys display outstanding properties such as excellent creep strength, remarkable fracture toughness parameters, and corrosion resistance. For this reason, Ni-based materials are considered as materials dedicated to the IV generation of nuclear reactors. Although these materials seem promising candidates, their radiation resistance and impact of radiation damage on the deformation mechanism are still not fully understood. In this work, two commercially available nickel-based alloys (Hastelloy X and Haynes 230) were investigated. Structural and mechanical properties have been described by means of SEM/EBSD, TEM, and nanoindentation tests. Radiation damage has been performed by Ar-ion with energy 320 keV with two doses up to 12dpa. Obtained results have revealed a hardening effect for both levels of damage. However, more intensive effects were observed for Hastelloy X. Moreover, a significant change in precipitates' morphology in Hastelloy X has been observed. It has been proposed that structural differences between both alloys determine the type of occurring radiation-induced processes. Excess energy deposited into materials' structure during ion-irradiation can lower the temperature of nucleation of high-temperature phases, which initiates the formation of grain boundary serrations.

Aggregate size effects on fracture behavior of concrete SCB specimens

Fracture in concrete and rock-like materials is the result of applying external loads, initiation, and propagation of the cracks, and finally, creating the failure surface in the material. The most common type of fractures in solid materials are usually formed under combined tensile/shear loading conditions. The stress intensity factors KI and KII, corresponding to opening and sliding modes (I and II), are the main fracture parameters, and their critical values (i.e., fracture toughness) show the material's resistance against the initiation and propagation of cracks. In this paper, the fracture toughness of concrete specimens is determined by considering different aggregate sizes and using the three-point bending test on semi-circular bend (SCB) specimen. The fracture toughness parameters have been determined by different fracture criteria and compared with the values obtained in the laboratory on concrete specimens. The effect of aggregate size on fracture parameters and crack propagation path has been evaluated. To predict thecrack initiation angle and fracture toughness values, themodified maximum tangential stress (MMTS) and extended maximum tangential strain (EMTSN) criteria along with conventional fracture criteriawere used. The results showed, that with the increase in aggregate size, the fracture toughness value under mode I, II, and mixed-mode (I-II) has increased. The laboratory results showed that specimens with aggregate sizes of 0–19 mm have the highest fracture load, and specimens with the aggregate sizes of 0–5 mm have the lowest fracture load

The interplay between constituent material and architectural disorder in bioinspired honeycomb structures

The advent of additive manufacturing has enabled the precise manufacturing of lattice materials at different length scales. While honeycombs are usually constructed from a tessellated array of identical unit cells, most structures in nature tend towards stochastic or disordered tessellation. In this work, we explore the effect of disorder properties of honeycombs, in particular, stiffness, strength, fracture toughness, and crack propagation paths using finite element models. The honeycombs are created using Voronoi tessellations with different levels of disorder measured by a regularity parameter δ, where δ=1 corresponds to a regular hexagonal honeycomb and decreasing values of δ correspond to increasing levels of disorder. We consider three constituent materials: very brittle, quasi-brittle, and very ductile (0.0007–0.25 in failure strain). The results show that pseudo-order realizations with 0.5≤δ≤0.8show the best strength-toughness trade-off. With increased plasticity, the improvement in fracture toughness compared to regular honeycombs can be as high as 50% at the expense of a 20% decrease in strength. This increase in fracture toughness is achieved both at crack initiation with crack blunting intrinsically and extrinsically during crack growth by a combination of crack deflection and crack bridging. Also, a single mathematical variable δ usually referred to as a “regularity parameter” to control cell stochasticity cannot account for the variability in tensile strength and fracture toughness parameters for brittle or quasi-brittle constituent materials and significantly more for a ductile constituent material. Thus, a local parameter for disorder is required to explain the large statistical variance seen in the mechanical properties of disordered honeycomb structures across the same irregularity value.

Mechanical anisotropy and fracture behaviour of bulk ultra-fine grained AA2099 alloy

ABSTRACT In this work, the effect of cryo-deformation and post-deformation annealing process on tensile, anisotropic and fracture behaviour of AA2099 has been investigated. The development of ultrafine-grained (UFG) microstructure is achieved initially by cryorolling (CR) process on the solution-treated (ST) alloy by subjecting it to a true strain value of 2.3. Cryorolled samples are then followed by annealing process at temperatures of 150°C (CR + 150°C) and 200°C (CR + 200°C). A notable improvement in yield strength (~342 MPa), ultimate tensile strength (~472 MPa), hardness (~138 HV) and fracture toughness parameters (K Q (31.8 MPa √ m ), K ee (31 MPa √ m ) and J integral (57.67 kJ/m2)) is recorded, whereas anisotropy increases in the rolling direction and ductility show a significant drop at CR sample condition in comparison with coarse grains at ST condition. When annealing was executed at 150°C and 200°C, notable improvements in tensile, anisotropic and fracture behaviour of UFG AA2099 were observed. There has been a significant enhancement in tensile strength, anisotropy and fracture toughness until temperature reaches 150°C being the highest (~506 MPa), whereas after 150°C, these properties have been shown to decrease notably.

Microstructure Evolution and Fracture Toughness Behaviour of Cryorolled LM6 Al Alloy

In the present work, the effect of cryo-rolling on the processed LM6 alloy samples has been studied. The solution treated (ST) sample of LM6 alloy has been processed through cryo-rolling with reduction of its thickness with values such as 30%, 40%, and 75%. One of the key material properties i.e., fracture toughness has been studied and equivalent energy fracture toughness ( ) is being evaluated according to the ASTM E 992 standard. The microstructure evolution after processing through cryorolling (CR) has been carried out with the help of optical microscopy and Scanning Electron Microscopy (SEM). Then, the calculated values of fracture toughness parameter i.e., equivalent energy fracture toughness ( is being correlated with the microstructure evolution after processing of LM6 alloy. It was found out that there is an improvement in equivalent energy fracture toughness ( ) as the reduction values increases. The 75% CR sample showed great increment of 67% as compared to ST alloy sample. The microstructure evolution also signifies the mix-mode fracture visualized through Scanning Electron Microscopy (SEM) and as the reduction values increase, the ductile fracture zone dominance increases on brittle fracture zone indicating there is improvement in fracture toughness of the ultra-fined grain LM6 alloy due to the grain refinement, dislocation strengthening and grain boundary strengthening.

Analysis of Inclined Crack in Aluminium Alloy Plate Subjected to Three-Point Bending Load Repaired with Carbon Fiber Reinforced Polymer

If the attention is not paid to the crack in a structure, then it could suddenly propagate at a rapid rate and rip apart the structures. A small crack needs urgent attention and repair since replacing the parts with a small crack is not economically feasible at all the time. Repairs were used to be carried out through rivets, welding and nut-bolts, but recently composite materials are showing promising results in this field. Since composite material are anisotropic in nature their application needs careful study about the loading pattern on the repaired structure. In this study, Carbon Fiber Reinforced Polymer (CFRP) was used as a composite material to repair Aluminium alloy specimens. These specimen were subjected to a three-point bending load to investigate the effectiveness of CFRP. By using innovative ply drop technique and design of experiment a configuration was selected to sustain three-point bending load. To suppress the CFRP’s peeling off tendency, attention was given to the interfacial shear stress rather than to the fracture toughness parameter.

Fracture toughness determination of porcine muscle tissue based on AQLV model derived viscous dissipated energy

The ability of soft collagenous tissue (SCT) to withstand propagation of a defect in the presence of a macroscopic crack is termed the ’fracture toughness parameter’. In soft tissues not undergoing significant plastic deformation, it is purported that a considerable amount of additional energy is dissipated during failure processes, due to viscoelasticity. Hence the total work, measured experimentally during failure, is the sum of fracture and viscoelastic energies. Previous authors have aimed to apply constitutive modeling to describe viscoelastic hysteresis for fracture toughness determination with a tendency of models to either over or underestimate the viscous energy. In this study, the fracture toughness of porcine muscle tissue is determined using two strategies. Firstly, it was determined experimentally by calculation of the difference in dissipated energy of notched and unnotched tissue specimens undergoing cyclic ’triangular wave’ excitation with increasing strain levels in uniaxial tension. The second strategy involved the extension and use of the adaptive quasi-linear viscoelastic model (AQLV) to model cyclic loading (model parameters were obtained from a previous study) and sequentially the dissipated energy was calculated. The mean value of the dissipated energy based on the AQLV approach was then subtracted from the total dissipated energy of notched porcine muscle tissue samples to determine the fracture toughness. The mean experimental viscous dissipated energy ratio was 0.24 ± 0.04 in the experimental approach, compared to 0.28 ± 0.03 for the AQLV model. Fracture toughness determined experimentally yielded 0.84 ± 0.80 kJ/m2, and 0.71 ± 0.76 kJ/m2 for the AQLV model, without a significant difference (p = 0.87). Hence, the AQLV model enables a reasonable estimation of viscous dissipated energy in porcine muscle tissue with the advantage to perform tests only on notched specimens, instead of testing additional unnotched samples. Moreover, the AQLV model will help to better understand the constitutive viscoelastic behaviour of SCTs and might also serve as a basis for future fracture toughness determination with constitutive model simulations.

A critical evaluation of cortical bone fracture toughness testing methods

Cortical bone fracture mechanics which quantifies the tissue's resistance to fracture is widely regarded as important to finding key determinants of bone fragility and fracture. Currently, the most widely used fracture mechanics approach is the J-integral resistance (J-R) curve as defined in ASTM E1820 standard. This standard employs an unloading compliance (UC) method to estimate crack extension, necessary for fracture toughness and resistance curve (R-curve) quantification. Further, this UC method requires a series of unload-reload cycles to be conducted during the fracture test. However, cortical bone violates some assumptions on which the UC method is based, which are: no energy loss during the unload-reload cycles and any change in unloading compliance is only due to crack extension. Consequently, the aim of this study was to examine the impact of the UC method on the accuracy of fracture toughness measurement for bovine cortical bone. Ten pairs of single edged notched bend specimens were prepared from the posterior diaphysis of bovine tibiae and underwent three-point bending fracture tests. The paired specimens were divided into two groups: a cyclic loaded group and a monotonic loaded group. Further, crack extension was determined by the UC method for the cyclic group and by an optical method for both the cyclic and monotonic groups. From these, three different approaches were used to generate J-R curves from which three fracture toughness parameters were computed and compared between the three approaches. This comparison allowed the impact of crack extension estimation by the UC method as well as the unload-reload cycles on the accuracy of the fracture toughness measures to be assessed. Results show that the UC method underestimates crack extension by an average error of 73%. In addition, the combined effects from crack extension estimation using the UC method and the unload-reload cycles lead to a significant overestimation of the specimen's fracture toughness measures. This highlights the need for more studies to establish a standardized approach to cortical bone fracture testing.

Fractal characteristic of corroded steel surface and application to the fracture analyses

To analyze the effects of morphological characteristics on fracture behaviors of corroded steel, the morphology analysis, tensile test and corresponding finite element (FE) simulation were systematically performed to investigate the morphological evolution and the degradation of mechanical property of corroded steel. The results indicated that the size parameters (C and W) followed the Chi-square distribution, and the fractal character was obviously observed on corroded steel, especially for critical corrosion pit. The fractal dimension of critical corrosion pit (D*) was a key factor influencing the mechanical property and fracture behavior for corroded steel due to the advantage in systematacially reflecting the morphological characteristics, the agreement with the various mechanical parameters was obvious more than C or W of critical corrosion pit. Comparing the toughness parameters (α and η) among different corroded specimens, the effect of the oxide passivation layer on fracture toughness properties of corroded steel was revealed, and modified. Finally, the characterization of D* on fracture toughness parameters (α, η, effective plastic strain εpcritical and stress triaxiality T) of corroded steel was discussed

Fracture behaviour and toughening mechanisms of dry and wet collagen

The growing interest to the use of collagen films for biomedical applications motivates the analysis of their fracture behaviour in different environments. Studies revealed the decreased mechanical strength and stiffness as well as increased plasticity in water compared to collagen specimens tested in air. However, the fracture behaviour of pure collagen films in both air and water has not been reported so far. In this paper, the entire process of mode-I loading of single-edge notched tension (SENT) specimens was recorded and analysed. In case of in-air (dry) specimens, cracks propagated rapidly in a brittle fashion while large plastic deformations were observed in aqua prior to failure due to crack opening and a blunting mechanism in wet specimens. The fracture-toughness parameters for pure collagen in air and in aqua were estimated using linear-elastic (KI and GI) and elasto-plastic (JI) fracture-mechanics approaches, respectively, following the force-displacement response and deformational behaviour. GIC and JI were 1365 ± 112 J/m2 and 2500 ± 440 J/m2, respectively. Scanning electron microscopy was used to observe the structural changes linked to collagen fibrils in the crack-tip area and the fracture surface. For in-air specimens, the former mostly exhibited extrinsic toughening (usually at micro scale) acting behind the crack-tip, while in-aqua intrinsic toughening acting ahead of a crack tip was found. Fractography of in-air specimens showed no occurrence of voids while multiple micro-voids were found for in-aqua specimens. Statement of significanceThe fracture toughness and crack propagation of both mineralised (bone, dentine) and non-mineralised (skin) tissues has been extensively investigated over the past decades. Though these tissues are rich in collagen, the fracture properties of pure collagen have not been quantified yet at macroscale. Considering the applications of collagen films in tissue regeneration, it is essential to perform investigations of their fracture behaviour in both dry and wet conditions. Determining the effect of environment on the fracture behaviour of collagen and understanding its toughening mechanism are essential for prevention of failures during application. Moreover, this would give an insight for fabrication of tougher collagen-based biomaterials for biomedical uses.

Fracture behaviors of HVFA-SCC mixed with seawater and sea-sand under three-point bending

This paper aims to study the fracture behaviors of high-volume fly ash-self-compacting concrete (HVFA-SCC) mixed with seawater and sea-sand (SWSS) or freshwater and river sand (FWRS). Three-point bending test were performed on 24 notched beams fabricated with varying in replacement ratio of fly ash (0%, 30%, 50%, and 70%) and the type of water and sand (SWSS and FWRS). The initial and unstable fracture toughness of these test specimens are determined by the double- K fracture model. The effect of fly ash replacement ratio and type of water and sand on the fracture parameters is analyzed and discussed. In addition, the cohesive fracture toughness of all the test specimens is calculated by using Gauss–Chebyshev integral method and the weight function method based on the bilinear tensile softening curve given in CEP-FIP Model Code. A comparison of fracture toughness parameters of determined from the experimental approach and analytical approaches is presented in these SCC specimens. Results show that SCC mixed with SWSS replacing FWRS can improve the unstable fracture toughness and fracture energy, and decrease its brittleness behavior. The cohesive fracture toughness of SWSS-SCC specimens is underestimated by these analytical methods based on the tensile softening curve given in CEP-FIP Model Code.

Investigation and optimization of Cracked Aluminium Alloy plate Restored for Fatigue Loading Application

Crack propagation could be catastrophic, and it needs the urgent attention of the user. A cracked structure needs to be repaired or replaced at the earliest. Costs associated with the complete replacement of the part paves the way for the former option, especially in the aircraft industry. Composite materials have good directional properties in the strength and toughness compared to the conventional materials like metals, alloys and plastics. In this study carbon fibre reinforced polymer (CFRP) has been used to repair the inclined crack in aluminium alloy plate having 1.6 mm thickness. Repaired aluminium alloy specimens were subjected to fatigue load to investigate the effectiveness of CFRP. By using the ply drop technique, the strength of the damaged structure has been restored. Various options have been evaluated by conducting the design of experiment. A peeling off tendency of the CFRP patch was observed during the study. To suppress this peeling off tendency, interfacial shear stress between CFRP and the aluminium was studied more intensely rather than the fracture toughness parameters. To select the optimum configuration MCDM optimisation techniques were used and numerical solutions were validated by lab experiments.

Investigation and optimisation of cracked aluminium alloy plate restored for fatigue loading application

Crack propagation could be catastrophic, and it needs the urgent attention of the user. A cracked structure needs to be repaired or replaced at the earliest. Costs associated with the complete replacement of the part paves the way for the former option, especially in the aircraft industry. Composite materials have good directional properties in the strength and toughness compared to the conventional materials like metals, alloys and plastics. In this study carbon fibre reinforced polymer (CFRP) has been used to repair the inclined crack in aluminium alloy plate having 1.6 mm thickness. Repaired aluminium alloy specimens were subjected to fatigue load to investigate the effectiveness of CFRP. By using the ply drop technique, the strength of the damaged structure has been restored. Various options have been evaluated by conducting the design of experiment. A peeling off tendency of the CFRP patch was observed during the study. To suppress this peeling off tendency, interfacial shear stress between CFRP and the aluminium was studied more intensely rather than the fracture toughness parameters. To select the optimum configuration MCDM optimisation techniques were used and numerical solutions were validated by lab experiments.

Comments on “Essential work of fracture analysis for surface modified carbon fiber/polypropylene composites with different interfacial adhesion” [Polymer Composites 41 (9) (2020): 3541–3551

AbstractIn this paper, some comments on the study “Essential work of fracture analysis for surface modified carbon fiber/polypropylene composites with different interfacial adhesion” Polymer Composites 41 (9) (2020): 3541–3551, have been discussed. The comments are mostly concentrated on the technique utilized for the assessment of the parameters of fracture toughness. The essential work of fracture (EWF) approach was employed for all the samples to characterize the fracture manner, while it appears that some of the samples did not meet the prerequisites of the EWF method. These prerequisites are as follows: the ligament length must be completely yielded before the onset of crack propagation, self‐similarity between load–displacement curves is a requirement, and fracture must occur under plane stress conditions. Therefore, the information, data, and results obtained by the authors are questionable.