Fictitious Crack Model Research Articles

Experimental and numerical investigations on rate-dependent fracture behavior of concrete-rock interface

To ensure the safety and stability of the concrete gravity dams built in the seismic regions, this study investigated the rate-dependent fracture behavior of the concrete-rock interfaces with different roughness degrees. Firstly, direct tensile tests and three-point bending tests were conducted to measure the mechanical and fracture properties of the concrete-rock interface with three roughness degrees, i.e. the 4 × 4 interface, natural interface, and 7 × 7 interface, and under four strain rates, i.e. 10-5/s, 10-4/s, 10-3/s, and 10-2/s. By employing the fictitious crack model and initial fracture toughness-based crack propagation criterion, numerical simulations were conducted to analyze the crack propagation processes of the concrete-rock interfaces under different strain rates. The results indicated that the tensile strength and initial fracture toughness exhibited an obvious increasing tendency with the increase of the strain rates, and they showed a nearly linear relationship with the logarithms of the strain rates. Prediction models were proposed to calculate the tensile strength and initial fracture toughness of the concrete-rock interface under different strain rates. In addition, the initial fracture toughness-based crack propagation criterion was proved to be applicable to analyze the crack propagation processes of the concrete-rock interfaces under both quasi-static and seismic strain rates. It was found that the fully-formed fracture process zone lengths and the corresponding crack length decreased obviously with the increase of the strain rates, indicating that the boundary effect of the concrete became stronger under the higher strain rates, which approached to the fracture behaviors of the large-size specimens.

Analytic solution of pull-out failure on bond-slip relationship between deformed rebar and ultra-high-performance concrete

Numerous studies have explored the bond performance between deformed rebar and ultra-high-performance concrete (UHPC), yet an analytical solution to the bond-slip relationship remains elusive. This study presents an analytical solution of pull-out failure for the staged bond-slip constitutive relationship between rebar and UHPC, taking into account the meso-scale bond properties of fibers. The critical point coordinates for different stages are determined by the analytical solution. Utilizing the elastic mechanic solution of thick-walled cylinder under uniform internal pressure, the fictitious crack model, and the fiber-matrix discrete model, analytical solutions and simplified formulas for bond stress and slip at the critical points are provided respectively. The simplified formulas for predicting bond characteristics are proposed. The bond-slip test data are composed of 107 groups from 15 literatures, and the predictions of different formulas are compared. The average ratios of the theoretical ultimate bond stress and peak slip to the test results are 0.995 and 1.033, with a standard deviation of 0.179 and 0.231, and the error is generally within 20%. Compared with the bond-slip relationship, it is more reasonable that the ascending and descending sections are exponential. Moreover, the parameters in the bond-slip relationship can be calculated by the theoretical predictions of bond characteristics at each critical point.

Analytical method for predicting mode I crack propagation process of concrete under low-cycle fatigue loading

The fatigue failure of concrete structures is preceded by the formation of microcracks and the propagation of macrocracks. Therefore, the fatigue fracture properties of concrete should be investigated thoroughly to guarantee the safety of structures. In this paper, an analytical method for predicting the mode I crack propagation in concrete under low-cycle fatigue loading is proposed according to the fictitious crack model, where the fatigue tension-softening constitutive relationship is introduced, and the initial fracture toughness-based crack propagation criterion is applied. The fatigue crack propagation process can be determined by solving two governing equations. To validate the effectiveness of the method, the fatigue crack propagation in three-point bending (TPB) beams is calculated, and the results are compared with the simulated and experimental results. It is shown that, although the linearization assumption of the crack opening displacement (COD) along the depth direction leads to slight differences between the analytical and simulated results, there is still a reasonable agreement between the analytical and experimental results. The effectiveness of the analytical method is verified. Therefore, when Young’s modulus, initial fracture toughness, and fatigue tension-softening constitutive relationship are determined, the mode I fatigue crack propagation in concrete can be predicted based on the analytical method. It is expected that the method will contribute to the safety assessment of concrete structures under low-cycle fatigue loading.

ТРІЩИНОСТІЙКІСТЬ БЕТОНІВ З ПОЗИЦІЇ МЕХАНІКИ РУЙНУВАННЯ (ОГЛЯД)

This paper presents the generalized results of the analysisand synthesis of scientific and technical sources of investigation the heavy concrete on the basis of fracture mechanics for the last 25-30 years.New criteria and models for heavy concrete, which were obtained during this period have been described and generalized, in particular, a new deformation model of concrete crack resistance on the basis of fracture mechanics. The analysis of results of experimental and theoretical researches of heavy concrete crack resistance on disk-shaped samples at eccentric stretching by concentrated forces has been presented. A number of methods have been elaborated i.e. a method for experimental determination of the concrete crack resistance characteristics under static loads, a method for determining the length of the fracture zone in the crack, the relationship between the mechanical characteristics of heavy concrete and the parameters of acoustic emission measurement. The method of concrete cracking resistance with the addition of basalt fiber was also developed and the influence of basalt fiber on crack resistance and crack opening was studied in field tests. The generalized results have been presented.It has been stated about the significant development over the last three decades of methods for determining the strength, crack resistance and deformation of concrete in reinforced concrete elements from the standpoint of reinforced concrete mechanics and mechanics of concrete destruction. The conclusions about the expediency of using the Leonov-Panasyuk deformation model for concrete and the prospects for fictitious crack model have been made.On the basis of the generalized data given in the specified publications the technique of technical diagnostics and a technique of repair and restoration works at strengtheningwith use of modern technologies and materials has been developed that gives the chance to increase considerably a resource and reliability of reinforced concrete buildings and constructions

Time-Dependent Fracture Behavior of Rock–Concrete Interface Coupling Viscoelasticity and Cohesive Stress Relaxation

This study investigated the time-dependent crack propagation of the rock–concrete interface under sustained loading. First, sustained loading tests were performed on the composite rock–concrete beams under three-point bending with respect to three interface roughness degrees, i.e., natural, 4×4, and 7×7 interfaces, and three sustained load levels, i.e., 80% of the maximum load, the initial cracking load, and 97% of the maximum load. Then, by employing the Norton-Bailey model and the time-dependent fictitious crack model, the constitutive relationships of bilateral materials and the rock–concrete interface were established and the mechanical responses of the rock–concrete interface under sustained loading were simulated. In addition, a crack propagation criterion proposed in this study was applied in the numerical procedure to simulate the time-dependent crack propagation process under sustained loading. The criterion implied that the interface crack propagated when the difference between the average elastic strain energy densities near the crack tip caused by the external load and cohesive stress exceeded that at the initial fracture status under quasi-static loading. The results indicated that the crack propagation under sustained loading could be interpreted from the view of energy balance between the driving energy from the external load and the resistance energy from the cohesive stress. The time-dependent crack propagation under sustained loading experienced three stages, i.e., decelerated propagation, uniform propagation, and accelerated propagation. A linear relationship between the logarithms of the crack mouth opening rate in the uniform propagation period and the failure time was established to predict the failure time of the rock–concrete interface under sustained loading.

Analysis of bond performance of steel bar in steel-polypropylene hybrid fiber reinforced concrete with partially recycled coarse aggregates

To promote structural sustainability and resource recycling, the bond performance between steel bars and steel-polypropylene hybrid fiber reinforced recycled aggregate concrete (HFRAC) confined with/without stirrups was investigated through the pullout test on 72 specimens. The specimens with varying stirrup spacing, polypropylene fiber volume fraction, and steel fiber volume fraction were designed, and the replacement rate of recycled coarse aggregates was kept at 50% for all specimens. The strain gauges were bonded in the slotted steel bar of specimens for detecting the strain of steel bar during the loading process. According to the measured strain of steel bar, the distribution law of bond stress and relative slip for the specimens with/without stirrups were analyzed. Results showed that the lateral confinement of stirrups dramatically improved the bond stiffness and rendered the distribution of bond stress more uniform than that of specimens without stirrups. The hybridization of steel fiber and polypropylene fiber improved the bond strength of recycled aggregate concrete specimen without/with stirrups, in which the stirrup spacing were taken as 80 mm and 40 mm up to 12%, 8.97% and 5.69%, respectively. Smaller stirrup spacing leaded to a slight reduction in bond strength when 1.5% steel fibers were added into the specimens. Based on the thick-walled cylinder model and fictitious crack model reflecting the softening behavior of HFRAC under tension, the solutions of bond strength of steel bars in HFRAC with/without stirrups were deduced, respectively. Finally, the bond-slip model of steel bars in unconfined/well-confined HFRAC was proposed, which considered the effect of anchorage position.

Analytical solution of the bond strength between ribbed steel bar and engineered cementitious composites considering the fiber bridging effect

AbstractThis paper investigated the analytical solution of the bond strength between ribbed steel bar and engineered cementitious composites (ECC). In this method, ECC cover was regarded as a thick‐walled cylinder subjected to the uniform internal pressure which consisted of an elastic uncracked outer ring and a partially cracked inner ring. For the uncracked outer ring, it can be solved by the elastic mechanics theory. To solve the partially cracked inner ring, two key problems that had to be investigated were the crack width and the tensile stress–crack width relationship of ECC. According to the steady‐state cracking mode, the hoop deformation of ECC was expressed by the equal distribution and then the crack width can be determined. Besides, the fictitious crack model in fracture mechanic was introduced to describe the crack propagation process. Then a trilinear model was applied to reflect the tensile stress–crack width relationship of ECC materials, which considered the combined action of aggregate bridging and fiber bridging on the crack surface. Finally, the comparison between the calculation results and experimental results was carried out to prove the validity of the analytic method of bond strength proposed in this paper. Results indicated that the analytical solution and the experimental data fitted in well with each other. Then parameter analyses were conducted based on the analytical method proposed in this paper.

A three-dimensional deformable spheropolyhedral-based discrete element method for simulation of the whole fracture process

A deformable spheropolyhedral-based discrete element method (DS-DEM) is proposed for predicting the evolution of fracture by coupling the finite element method (FEM) with the spheropolyhedral-based discrete element method (DEM). Under the framework of the proposed method, each rock mass is discretized into finite elements, and zero thickness joint elements are inserted along the boundary of finite element elements. The fictitious crack model and the Mohr-Coulomb failure criterion are employed to determine the failure state and failure mode of joint elements. The contact interaction between discrete particles and fracture surfaces is captured by the spheropolyhedral-based DEM. The method is verified by particle stacking test, three-point bending beam test, Brazilian disc tensile strength test, crack propagation in a single notched beam test, four-point bending beam test, and sphere impact test. The results of numerical simulations indicate that the proposed method is adaptable in simulating the whole fracture process of quasi-brittle materials including the initiation and propagation of cracks, as well as the collision and deformation of fragments.

Simplified equations for determining double-k fracture parameters of concrete for compact tension test

Polynomial equations in non-dimensional form for various fracture parameters of double-K fracture model for compact tension specimen have been derived and presented in this paper. These equations can be used for computing different double-K fracture parameters of concrete for known material properties and specimen size having relative size of initial crack length of 0.3 without involving much complexity in numerical computations. Values of peak load and corresponding crack opening displacement as necessary to compute the double-K fracture parameters of concrete have been derived from the established fictitious crack model in the present study. A simplified equation in non-dimensional form between peak load and critical crack opening displacement as obtained from a fictitious crack model has also been presented.

An improved ring test to assess cracking resistance of concrete under restrained shrinkage

Self-restraint caused by differential shrinkage along the radius of a concrete ring has great driving effect on the restrained cracking in the traditional ring test. In order to estimate cracking potential of concrete under restrained shrinkage, i.e. steel ring restraint effect, an improved circular/elliptical ring test was proposed by pre-setting a 6 mm long real crack at the inner circumference of the concrete specimen. Meanwhile, a numerical study based on fracture theory was conducted to investigate the cracking mechanism of the improved ring test. By introducing the fictitious crack model, the crack initiation age and propagation process in the ring specimen were simulated using the initial fracture toughness-based cracking criterion. The driving forces of the shrinkage cracking, provided by self-restraint and steel ring restraint, were analyzed and the influence of the pre-set crack length on the cracking age was discussed. The results indicated that, different from the traditional ring test, the shrinkage crack firstly initiated from the pre-set crack and then unstably propagated toward the outer circumference of the concrete in the improved method. In the crack propagation process, the self-restraint provided cracking insistence rather than driving force on the shrinkage cracking due to the bending effect along the radial direction caused by non-uniform moisture distribution in concrete. Therefore, the driving force of the shrinkage cracking was totally provided by the steel ring restraint and would overcome the cracking resistances from concrete material and non-uniform shrinkage. Furthermore, the advantage of the elliptical geometry in increasing the restraining level was still available for the improved ring test with a pre-set crack, which could make the shrinkage cracking occur in advance and enhance the steel ring restraint.

A comparison of two damage models for inverse identification of mode Ⅰ fracture parameters: Case study of a refractory ceramic

Fracture behavior of refractories influences their durability in high-temperature applications to a great extent. The fictitious crack model has been used for simulation of the fracture process of refractories and concrete materials. The present study investigates the effect of the lower post-failure stress limit of the softening law in the fictitious crack model by comparing an in-house developed subroutine for damaged elasticity model with the concrete damaged plasticity model implemented in Abaqus. The numerical wedge splitting tests show that in the case of brittle materials, the lower post-failure stress limit defined in the concrete damaged plasticity model resulted in energy consumption for crack propagation exceeding the defined fracture energy (114% higher in the case of a brittleness number of 4.4). Therefore, the developed damaged elasticity model allows for a more accurate simulation of fracture since the lower post-failure stress limit was decreased to 0.0001% of the tensile strength. Moreover, an inverse evaluation of the fracture parameters of an alumina spinel refractory material supported the developed model.

The Overview of Fracture Mechanics Models for Concrete

Abstract Fracture mechanics of concrete is a complex matter still thoroughly researched from different angles. It is not an easy task to describe fracture process in concrete, as there are many factors affecting crack development and propagation. Practical applications of fracture mechanics could allow engineers to design concrete structures more effectively and safely. At the minimum, it could help estimate the “safe” period of time left before the unstable, dangerous crack propagation. This utilitarian goal was the reason for many researchers to invent numerous theoretical models in order to describe the crack occurrence better. However, dealing with various analytical problems was not a simple matter and thus existing models of fracture mechanics for concrete have different limitations. Over the years first fracture theories for concrete were reviewed repeatedly. All of these investigations lead to modifications of older models in order to overcome found drawbacks, which proved not to be an easy task. Recently, new approaches to fracture analyses seemed to produce promising results, like universal size effect law (USEL) or modified two parameter fracture model (MTPM) with alternative ways for evaluating fracture parameters. In the paper some of them will be discussed together with other fracture models, starting from some of the very first ones introduced for concrete, like fictitious crack model (FCM) and crack band model (CBM).

Fracture Models and Effect of Fibers on Fracture Properties of Cementitious Composites-A Review.

Cementitious composites have good ductility and pseudo-crack control. However, in practical applications of these composites, the external load and environmental erosion eventually form a large crack in the matrix, resulting in matrix fracture. The fracture of cementitious composite materials causes not only structural insufficiency, but also economic losses associated with the maintenance and reinforcement of cementitious composite components. Therefore, it is necessary to study the fracture properties of cementitious composites for preventing the fracture of the matrix. In this paper, a multi-crack cracking model, fictitious crack model, crack band model, pseudo-strain hardening model, and double-K fracture model for cementitious composites are presented, and their advantages and disadvantages are analyzed. The multi-crack cracking model can determine the optimal mixing amount of fibers in the matrix. The fictitious crack model and crack band model are stress softening models describing the cohesion in the fracture process area. The pseudo-strain hardening model is mainly applied to ductile materials. The double-K fracture model mainly describes the fracture process of concrete. Additionally, the effects of polyvinyl alcohol (PVA) fibers and steel fibers (SFs) on the fracture properties of the matrix are analyzed. The fracture properties of cementitious composite can be greatly improved by adding 1.5–2% PVA fiber or 4% steel fiber (SF). The fracture property of cementitious composite can also be improved by adding 1.5% steel fiber and 1% PVA fiber. However, there are many problems to be solved for the application of cementitious composites in actual engineering. Therefore, further research is needed to solve the fracture problems frequently encountered in engineering.

Stiffness Assessment of Cracked Reinforced Concrete Beams Based on a Fictitious Crack Model

A fictitious crack model is introduced into cracked reinforced concrete beams to assess the changing beam stiffness under loads. Firstly, nonlinear concrete stress distributions near cracks are built based on the model. Then the stress of the steel bar at the cracked section is considered as cohesive stress. The concrete and steel stresses are substituted into the equilibrium equations of forces to solve the concrete stress. Based on the solution, the section inertias are estimated by iterating the calculation of the cracking open displacement, and finally the beam stiffness is assessed. Experimental data from seven concrete beams after cracking are adopted to validate the effectiveness of the proposed method, and the results show that the fictitious cracks ahead of actual cracks increase their depth with the load, which will raise the neutral axis and change the inertias of cracked sections and their neighboring sections. These changes are taken into account in the stiffness assessment, so the results predicted by the proposed method are shown to coincide well with the nonlinear deflections measured in the experiments.

Crack mechanisms in concrete – from micro to macro scale

The paper discusses a fictitious crack model of concrete in tension proposed by Hillerborg. This model presents a concept that illustrates the mechanism of crack initiation and its propagation in concrete on meso-level. It has proven to be a very useful tool for practical use, for both numerical and experimental research. The model was derived from findings on crack mechanisms on more advanced micro- and macro-scale, as presented in this paper. One of the paramount issues regarding crack analysis is the influence of aggregate size on mechanical and fracture parameters of concrete, and also on micro-crack development and associated macro-crack formation. Although significant progress in recognizing crack mechanisms in concrete has been achieved, there are still some aspects that should be studied in depth, for example the role of aggregate particles on crack development. This problem is analysed in the paper as well.

Fracture of warm S2 columnar freshwater ice: size and rate effects

Large scale laboratory experiments on size and rate effects on the fracture of warm columnar freshwater ice have been conducted with floating edge-cracked rectangular plates loaded at the crack mouth. The largest test plate size had dimensions of 19.5m x 36m. The overall crack-parallel dimension covered a size range of 1:39, possibly the largest for ice tested under laboratory conditions. The loading rates applied led to test durations from fewer than 2 seconds to more than 1000 seconds, leading to an elastic response at the highest rates to a viscoelastic response at the lower rates. Methods for both the linear elastic fracture mechanics (LEFM) and a non-linear viscoelastic fictitious crack model (VFCM) were derived to analyze the data and calculate values for the apparent fracture toughness, crack opening displacement, stress-separation curve, fracture energy, and size of the process zone near a crack tip. Issues of notch sensitivity and minimum size requirements for polycrystalline homogeneity were addressed. Both size and rate effects were observed, as well as how these two factors are interrelated in the fracture of columnar freshwater ice. There was a size effect at low rates but no size effect at high rates. There was a rate effect for the larger test sizes but a weaker or no rate effect for the smallest test size.

Experimental Study on Crack Propagation of Concrete Under Various Loading Rates with Digital Image Correlation Method

The quantificational exploration of the propagation law of fracture process zone (FPZ) is of great importance to the research on concrete fracture. This paper performed fracture experiments on pre-cracked concrete beams under various loading rates. Digital image correlation (DIC) method was applied to obtain the whole field displacement of concrete in the fracture test. The crack opening displacement (COD) and the evolution of FPZ were determined based on the whole field displacement. The results show that the length of FPZ first increases and then decreases with the development of the effective crack length and the maximum length of FPZ is about 60 mm. It can be found that the length of FPZ corresponding to the peak load decreases with the increase of loading rates. Based on the fictitious crack model, a bilinear softening model was established. According to the proposed model, the mechanical behavior and the propagation law of FPZ were analyzed. The bilinear softening model can reflect the microcrack development and the aggregate interlocking in the FPZ.

SIF-based fracture criterion of rock-concrete interface and its application to the prediction of cracking paths in gravity dam

Experimental tests were conducted on the composite rock-concrete specimens with granite, sandstone, and C30/C50 concrete to study the interfacial fracture process under three-point bending and four-point shear conditions. A unified interfacial crack initiation criterion expressed by the stress intensity factors (SIFs) was fitted by the SIFs corresponding to the interfacial crack initiation based on the experimental data. By combining with the fictitious crack model, the crack initiation criterion can be transformed into the crack propagation criterion by considering the contributions of the cohesive forces in the fracture process zone. By assessing the relationships of the interfacial propagation criterion and the maximum circumferential stress criteria of the rock and concrete, the potential interfacial crack propagation paths can be predicted. Furthermore, numerical analyses were carried out by introducing these criteria to simulate the complete fracture process of the rock-concrete interface, where the predicted P-CMOD curves and crack trajectories showed good agreements with the experimental data. Finally, by taking a classic gravity dam in practical engineering as an example, the effects of the water levels, initial crack length and crack propagation length on the fracture behaviour of concrete, rock and their interface were investigated. The whole fracture processes for various fracture parameters of the rocks, concretes and their interfaces were simulated numerically. The results indicated that the employed method was effective for the safety assessment of the gravity dam and the application of these propagation criteria is convenient because only the initial fracture toughnesses of the rocks, concretes and their interfaces are required.

Mode I Non-Linear Fracture Model: Cases on Concrete and Fiber Reinforced Concrete

Model fraktur ragam I non-linier telah banyak digunakan untuk memperoleh faktor intensitastegangan KSIc dan perpindahan bukaan ujung retak CTODc sebagai kriteria fraktur untuk beton danbeton serat. Beberapa model fraktur ragam I non-linier terdahulu antara lain Model Retak Fiktif olehHillerborg, (1976), Model Pita Retak oleh Bazant (1983, 1986), Model Dua-Parameter oleh Jenq danShah (1986), Model Penjalaran Retak Mode I oleh Zhang dan Li (2005), dan Model Kerusakan Non-Lokal oleh Ferrara dan Prisco (2005). Tulisan ini mengimplementasikan model fraktur ragam I nonlinierpada 2 kasus. Kasus pertama diimplementasikan pad beton sedangkan kasus keduadiimplementasikan pada beton serat. Kedua kasus tersebut akan memperoleh nilai faktor intensitastegangan KSIc dan perpindahan bukaan ujung retak CTODc. Kasus 1 adalah kasus benda uji balokbeton bertakik model fraktur ragam I non-linier dan kasus 2 adalah beton serat tak hingga modelfraktur ragam I non-linier. Kasus 1 menghasilkan nilai faktor intensitas tegangan KSIc sebesar 15.078MPa mm-1/2 dan perpindahan bukaan ujung retak CTODc sebesar 0.023 mm. Kasus 1 menghasilkannilai faktor intensitas tegangan KSIc sebesar 3.917.10-4 MPa mm-1/2 dan perpindahan bukaan ujung retak CTODc sebesar –1.994.10-4 mm. Secara umum, keberadaan serat sangat mempengaruhi solusianalitis. Tulisan ini memperoleh kesimpulan sebagai berikut: (1) Model fraktur ragam I non-linierdapat digunakan untuk memperoleh faktor intensitas tegangan KSIc dan perpindahan bukaan ujungretak CTODc sebagai kriteria fraktur untuk beton dan beton serat, (2) Perilaku fraktur beton seratadalah spesifik dibandingkan beton karena adanya fenomena penjembatanan serat, (3) Dalamperhitungan hasil faktor intensitas tegangan KSIc dan perpindahan bukaan ujung retak CTODc akanberlebihan bila traksi serat diabaikan dan kurang bila Zona Proses Fraktur diabaikan, (4) Akan sangatbaik bila mengkombinasikan Kasus 1 dan Kasus 2 bersama-sama untuk memperoleh nilai faktorintensitas tegangan KSIc dan perpindahan bukaan ujung retak CTODc dengan memperhatikankeberadaan serat dalam komposit matriks berserat.

An analytical method for predicting mode-I crack propagation process and resistance curve of rock and concrete materials

In this paper, taking into account the fictitious crack model and the closure effect of the crack opening displacement (COD) induced by the cohesive force, an analytical method with the crack mouth opening displacement (CMOD) as the control parameter is proposed to predict the mode-I crack propagation process of rock and concrete materials by using stress intensity factor (SIF)-based criteria. With this method, the load-crack mouth opening displacement (P-CMOD) curve, crack extension resistance curve (KR-curve), and fracture process zone (FPZ) length during crack propagation are analytically obtained. The validity of the proposed analytical method is then verified with the experimental results of three-point bending tests on rock and concrete beams. Therefore, given the tensile strength ft, Young’s modulus E, initial fracture toughness KICini, and fracture energy GF, the complete mode-I crack propagation process of rock and concrete beams under three-point bending can be predicted accurately using the proposed analytical method without iteration in calculating the cohesive stress. Finally, the influence of the closure effect of the COD induced by the cohesive force on the P-CMOD curve, KR-curve, and FPZ length is discussed. The results show that although ignoring the closure effect of the COD induced by the cohesive force has little effect on the shape of the P-CMOD curve, the value of crack extension resistance and the maximum FPZ length are significantly underestimated, and the calculation error increases with the crack propagation and then decreases gradually when the FPZ length reaches the maximum.