Mingdong Wei,Feng Dai,Yi Liu,Ruochen Jiang

State Key Laboratory of Hydraulics and Mountain River Engineering,College of Water Resources and Hydropower,Sichuan University,Chengdu,610065,China

Keywords:Brittle fracture model Fracture toughness Maximum normal strain (MNSN)Fracture process zone (FPZ)Size effect

ABSTRACT Evaluating the fracture resistance of rocks is essential for predicting and preventing catastrophic failure of cracked structures in rock engineering.This investigation developed a brittle fracture model to predict tensile mode (mode I) failure loads of cracked rocks.The basic principle of the model is to estimate the reference crack corresponding to the fracture process zone (FPZ) based on the maximum normal strain(MNSN) ahead of the crack tip,and then use the effective crack to calculate the fracture toughness.We emphasize that the non-singular stress/strain terms should be considered in the description of the MNSN.In this way,the FPZ,non-singular terms and the biaxial stress state at the crack tip are simultaneously considered.The principle of the model is explicit and easy to apply.To verify the proposed model,laboratory experiments were performed on a rock material using six groups of specimens.The model predicted the specimen geometry dependence of the measured fracture toughness well.Moreover,the potential of the model in analyzing the size effect of apparent fracture toughness was discussed and validated through experimental data reported in the literature.The model was demonstrated superior to some commonly used fracture models and is an excellent tool for the safety assessment of cracked rock structures.

1.Introduction

The initiation and propagation of cracks in rocks may cause premature failure of structures in rock engineering.A reliable design of rock engineering structures requires an understanding of the crack growth resistance and the critical failure conditions of cracked rocks (You et al.,2022).For this reason,linear elastic fracture mechanics (LEFM),a theory specifically developed to analyze the effects of cracks,has been widely introduced into rock engineering(Yin et al.,1990;Akbardoost et al.,2014;Zhang,2002,2020;Bahmani and Nemati,2021).The application of this methodology is usually based on the determination of the stress intensity factor(SIF),which can describe the elastic stress and strain fields around the crack tip.Within this framework,structure failure is expected to be triggered by the SIF reaching a critical value under external loads.The critical value is called fracture toughness.Since rock materials generally exhibit weaker strength behaviors under tension than shear and compression(Xu and Dai,2018),and mixedmode cracks in the rock engineering field often gradually evolve into tensile (mode I) cracks during the propagation process (Aliha et al.,2021a,b;Liu and Dai,2021),the mode I fracture toughness(KIc)is an essential material parameter in LEFM and the related rock engineering applications.

Unfortunately,a large number of studies have shown that in many cases,LEFM cannot produce satisfactory fracture prediction for rocks,and theKIcvalue measured by laboratory tests is usually not a constant material property (Dwivedi et al.,2000;Cui et al.,2010).For example,theKIcmeasurement of rocks usually depends largely on the size and geometry of the specimen,as shown in Table 1.The inapplicability of LEFM in rock fracture characterization is widely attributed to the fracture process zone (FPZ),a damaged area containing crack-bridging and micro-cracking,localized at the crack tip,and related to the heterogeneity of the microstructure (Baˇzant and Kazemi,1990;Wei et al.,2018a).As illustrated in Fig.1a,the part other than the macrocrack of the structure is considered linearly elastic and free of any damage in LEFM.However,real-life fracturing of a quasi-brittle solid,such as rocks,is a process where microcracks continuously form at the crack tip and then aggregate into macrocracks (Fig.1b).When thesize of the FPZ is minimal compared with the structure size,the fracture behavior approaches that described by LEFM.However,if the FPZ occupies all or most of the ligament ahead of the crack tip,the failure is more suitable to be analyzed using a strength or yield criterion (Baˇzant and Kazemi,1990).To deal with engineering practice where the FPZ size is between these two situations,some nonlinear fracture mechanics models or improved LEFM methods have been proposed,including the fictitious crack model(Hillerborg et al.,1976),the crack band model (Baˇzant and Oh,1983),the two-parameter model (Jenq and Shah,1985),the effective crack model(Nallathambi and Karihaloo,1986),the size effect model (Baˇzant and Kazemi,1990),the double-K model (Xu and Reinhardt,1999),and the boundary effect model (Hu and Duan,2008).

Table 1 Some representative data reported in the literature showing the test method dependence of rock fracture toughness.

Fig.1.The difference in the crack propagation process between (a) the assumption in the LEFM theory and (b) real-life fracturing of rock-like materials.

The FPZ can be considered a factor outside the LEFM framework and that causes LEFM to poorly represent quasi-brittle fracture problems.On the other hand,when applying LEFM,engineers usually only focus on the SIF and ignore the higher-order stress terms in the well-known Williams series expansion for describing the crack tip stress.This operation may lead to considerable errors once the higher-order stress terms also significantly affect the stress,strain and strain energy density near the crack tip (for example,in small-scale fracture problems).Indeed,many recent publications have demonstrated the importance of non-singular stress terms for mode I and mixed-mode fractures (Berto et al.,2016;Wei et al.,2017,2018b;Mirsayar et al.,2018a,b;Aliha and Mousavi,2020).For example,based on the generalized strain energy density criterion,Ayatollahi et al.(2016a) revealed that theT-stress affects the apparent fracture toughness (Ka) and fracture trajectory of quasi-brittle solid samples under pure mode I loading,where theT-stress represents the first non-singular stress term in the Williams series expansion(Aliha et al.,2021b).Mirsayar (2015) applied the extended maximum tangential strain criterion to predict fracture initiation in cracked solids and deduced that theT-stress influencesKaas it contributes to the maximum tangential strain at the crack tip.Aliha et al.(2012)incorporated the higher-order termsA3into the maximum tangential stress criterion and predicted well the difference in apparent fracture toughness between a center cracked Brazilian disc limestone specimen and an edge cracked semicircular bend (SCB) limestone specimen.

In these brittle fracture criteria,the control parameters for triggering fracture are usually the stress,strain or strain energy density at the crack tip.These crack-tip parameters are described according to the elastic field at the crack tip and essentially determined from LEFM-based formulae (Razavi et al.,2018).Although these criteria consider higher-order terms to some extent,they might not fully consider the impact of the FPZ.More specifically,these fracture criteria might be inapplicable to the case where the FPZ is large enough to alter the crack-tip stress,strain and strain energy density distributions significantly or even beyond the K-dominant region (or the equivalent considering non-singular items).As for the nonlinear fracture mechanics models or improved LEFM methods (Hillerborg et al.,1976;Jenq and Shah,1985;Nallathambi and Karihaloo,1986),most of them do not explicitly reflect the consideration of the non-singular terms on their theoretical basis or on their application to engineering practices.Moreover,most of them require relatively complex experiments to calibrate model parameters.

In this study,to predict the fracture loads of cracked rocks,we establish a brittle fracture model that can explicitly consider the FPZ and non-singular terms,which can be deemed the influencing factors outside and inside the LEFM framework,respectively.The organization of this article is as follows.Section 2 introduces the principle and detailed derivation process of our model,in which the FPZ and the corresponding equivalent crack length are estimated from the maximum normal strain (MNSN)ahead of the crack tip,and the critical SIF is determined based on the concept of an effective crack.We emphasize in the model that the MNSN should consider both singular and non-singular terms.In this way,the effects of FPZ,non-singular terms and biaxial stress state at the crack tip on the fracture behavior are considered.To validate the model,we perform mode I fracture experiments using rock specimens of several geometries in Section 3.The apparent fracture toughness(Ka)measured from these different specimens is utilized to assess the proposed model in Section 4.For comparison,the maximum circumferential strain fracture criterion is also applied to predict theKaresults.The proposed model is further discussed in Section 5.Section 6 summarizes the whole study.

2.Derivation of the fracture model

In this section,we first briefly introduce the maximum normal stress criterion for the estimation of the FPZ length ahead of the crack tip (referred to as “FPZ estimation” later) in rock-like materials and the idea of plastic zone length correction for metal materials considering crack-tip stress redistribution.Then,we propose a method for the FPZ estimation based on the MNSN.Subsequently,we establish a fracture model by combining the MNSN-based FPZ estimation and the effective crack concept.

2.1.The maximum normal stress criterion for the FPZ estimation

The maximum normal stress criterion was proposed by Schmidt(1980) and applied by many studies (Tutluoglu and Keles,2011;Ayatollahi et al.,2016a;Pakdaman et al.,2019)to determine the FPZ length of rock-like materials.

For a two-dimensional cracked body,stresses at the crack tip can be expressed as

where (r,θ) are the polar coordinates (Fig.2a);KIandKIIare the mode I and mode II SIFs;σθθand σrrare the tangential and radial stresses,respectively;Tis theT-stress,the first non-singular stress term in the Williams series expansion;andOθθ(r1/2),Orθ(r1/2) andOrr(r1/2) are the higher-order terms,theoretically involving an infinite number of stress terms in the Williams expansion.

For mode I fracture,the normal and the radial stresses ahead of the crack tip can be written as

In the maximum normal stress criterion,the FPZ is assumed to be caused by the excessive tensile stress at the crack tip (Fig.2b).The FPZ length along the crack propagation direction(lFPZ1,a firstorder estimation without considering crack-tip stress redistribution) is determined by

where σtis the tensile strength.

When neglecting the higher-order terms,lFPZ1can be expressed as

This equation is somewhat similar to the plastic zone length prediction (Eq.(8)) for metals based on the von Mises yield criterion and the plane stress condition.

where σyis the yield strength of metals.

Fig.2.(a) Stress components at a crack tip in polar coordinates,and (b) The maximum normal stress criterion to estimate the FPZ of a rock-like material.

2.2.The FPZ correction considering crack-tip stress redistribution

The above plastic zone estimation for metals (Eq.(8)) is conservative since it is obtained from an elastic stress field.As illustrated in Fig.3a,once yielding appears,stress redistribution must occur to satisfy the balance in the normal direction of the crack.It is generally assumed that the area below segmentCEFequals that below segmentADB(i.e.SbelowCEF=SbelowADB)andSbelowEF=SbelowDB(Anderson,2004).According to these relationships,SbelowCE=SbelowADand Eq.(9) can be obtained.

Eq.(10) indicates that the revised plastic zone length considering the stress adjustment due to plastic yield is twice the plastic zone length estimated without considering the stress adjustment.

For rock materials,the crack tip produces microcracks instead of yielding,as schematically shown in Fig.3b.Theoretically,the stress in the FPZ must be lower than the tensile strength.Therefore,the stress redistribution must be greater in rocks compared with the situations of elastoplastic materials,and the revised estimation of the FPZ length (lFPZ2) considering stress redistribution must be larger than twice the first-order FPZ length (lFPZ1):

Obviously,a larger first-order estimation of the FPZ length usually corresponds to a larger actual FPZ:

2.3.The MNSN criterion for FPZ estimation

The maximum normal stress criterion (Schmidt,1980) only considers the influence of the normal stress on mode I fracture.This may cause some errors in the FPZ estimation,because the crack tip area is actually in a biaxial stress state and many studies have demonstrated the influence of the biaxial stress state on material failure.The radial stress ahead of the crack tip affects the tensile strain and strain energy density there.In addition,theT-stress has also been identified to play a key role in the fracture behaviors of rock-like materials,even for mode I (Schmidt,1980;Aliha et al.,2012;Ayatollahi et al.,2016a;Mirsayar et al.,2018a,b;Bahmani et al.,2020).When only focusing on the normal stress,the effect of theT-stress on mode I fracture cannot be considered(see Eq.(4)).Therefore,to accommodate the contributions of theT-stress and the radial stress,we utilize the MNSN ahead of the crack tip to estimate the FPZ.By imitating Eq.(6),the first-order estimation of the FPZ based on the MNSN can be expressed as

whereEandvare the Young’s modulus and Poisson’s ratio,respectively;and εθθcis the critical value of the MNSN.Since the present study aims to propose a brittle fracture model to predict the mode I fracture resistance of rocks in the brittle regime,the nominal normal strain at material failure can be determined from the uniaxial tensile test or the Brazilian-type indirect tensile test using Eqs.(14)and(15),as long as the specimen fails in the brittle regime.

For a cracked rock sample of a given material,geometry and size,its apparent fracture toughness and tensile strength always have the following relationship:

Fig.3.(a)The first and the revised estimations of the plastic zone length for metals,and(b)The FPZ adjustment considering crack-tip stress redistribution for rock-like materials.

wheremdepends on the specimen size and geometry as well as the material,=1 MPa,and=1 MPa m0?5.

Since the strain at specimen failure andKaare proportional to the failure load,the following equation holds

whereg1(r) is a function of the distance to the crack tip (MPa-1m-0.5).

Substituting Eqs.(15) and (17) into Eq.(13),one can obtain the first-order FPZ estimation as a function of σt/Kaandm:

2.4.The strain-based effective crack (SEC) model

Irwin(1948)considered the effect of the plastic zone by defining the effective crack,which is equal to the actual crack plus a correction amount resulted from the plastic zone.Later,an ASTM metal fracture test standard (ASTM E561,1987astm:1987) also recommends taking the effective crack as the sum of the initial crack,visible crack extension,and a plastic zone adjustment.The concept of the effective crack has also been widely introduced into fracture problems of rock-like materials (Labuz et al.,1991).In the double-K model,the effective crack length is employed to determine the unstable fracture toughness,i.e.the SIF corresponding to unstable crack growth.Baˇzant and Kazemi (1990) also stated that the FPZ approximates an effective (or elastically equivalent) crack.In the study by Hu and Duan(2008),the FPZ also corresponds to a reference crack (aFPZ),which is proportional to the FPZ size.Based on similar ideas,the following relationship holds in our model:

According to Eqs.(12) and (19),we assume

where ρ represents the ratio of the reference crack (aFPZ) to the first-order FPZ estimation (lFPZ1,without considering the microcracking-induced stress redistribution) obtained using the MNSN criterion.ρ is a key parameter to calibrate when applying our model,as demonstrated in the following sections.Note thatlFPZ1does not consider the stress redistribution,and it may be much smaller than the actual FPZ and the effective crack length.

Therefore,the critical effective crack length (aec) is written as

whereacis the critical crack length determined without considering the FPZ.

Then,the critical effective SIF (Ke,also called the effective fracture toughness) can be obtained by

where σcis the nominal failure stress,andaecis the critical effective crack length normalized by a geometric parameter of the specimen(e.g.specimen radius in this study).

Herein,we introduce a parameter to present the ratio between the apparent fracture toughness and the effective fracture toughness:

The k value can reflect the influence of the FPZ on a fracture toughness measurement.Theoretically,it should be between 0 and 1.The closer the k value is to 1,the smaller the influence of the FPZ on the fracture problem,and the more appropriate to regard the measuredKavalue as a representative material parameter.

Because the radial stress,T-stress and other non-singular terms are considered when determininglFPZandaec,and the calculation ofKeaccommodates the effect of the FPZ,this modified LEFM model is considered promising to address fracture problems of rock-like materials.A potential application of this model is to useKeof the material and k values of different cracked rock structures as bridges to correlate the failure loads of the cracked rock structures.For this purpose,ρ orKeshould be first calibrated for the studied rock material through some benchmark experiments(e.g.CB and SCB tests in this study),and then the prediction of the mode I fracture load for any given cracked body of the rock material can be realized.

Fig.4.Schematics of the fracture specimens used in the experiments.VNSRB represents the V-notched short rod bend.

The above derivation process indicates that the established theoretical fracture model can consider the FPZ(reflected by using effective crack length to calculateKIc),the non-singular stress terms including theT-stress,the biaxial stress state of the crack tip (reflected in the use of the crack-tip MNSN),and the stress redistribution(by introducing the parameter ρ that needs to be calibrated to experimental data).This is different from related previous studies,which often tend to emphasize the importance of only one of those factors.For example,studies based on the maximum normal stress criterion have usually overlooked the contributions of the radial stress and the stress redistribution to the fracture behavior of rocks (Tutluoglu and Keles,2011;Wei et al.,2016).In the following sections,we use fracture specimens of several geometries to present the detailed process of applying this SEC fracture model to predict the mode I fracture resistance of cracked rocks.

3.Fracture experiments

To check the applicability of the SEC fracture model,we performed rock fracture experiments using four types of fracture experiments.Because some of these fracture experiments involve varying boundary conditions or specimen sizes,a total of six sets of apparent fracture toughness measurement data were obtained.

3.1.Brief introduction of the four types of fracture experiments

A CB method was proposed by Ouchterlony (1988) and subsequently recommended by the International Society for Rock Mechanics (ISRM) forKIctesting of rocks (Ouchterlony,1982).The configuration of the CB specimen is shown in Fig.4,whereDis the diameter of the specimen,a0is the initial notch length,and θ is the angle between the V-shape notches.For the standard CB configuration recommended by the ISRM,θ=90o,a0/R=0.3 andS/R=3.33,whereSis half of the support span.In addition toS/R=3.33,the case ofS/R=1.5 was also considered in our experiments.

For such chevron-notched specimens loaded in mode I,the crack(lengtha)would initiate from the tip of the ligament and then extend along the ligament.Once reaching a critical length(ac),the crack growth becomes unstable,and the loading force drops.The fracture toughness can be determined according to the peak force(Pmax)and the geometry of the specimen witha=ac,which can be expressed as

whereY*is usually called the critical normalized SIF,which is a coefficient depending on the specimen geometry and boundary conditions(e.g.support span).For chevron-notched specimens,Y*is the minimum of the normalized SIFs corresponding to different crack lengths and can be determined using finite element modeling.

An SCB specimen geometry,proposed by Chong and Kuruppu(1984),was also recommended by the ISRM forKIcmeasurements(Kuruppu et al.,2014).As shown in Fig.4,the SCB specimen contains a straight-through edge notch.In our SCB experiments,S/Ris always equal to 0.8,and α=a/R=0.5.

Using the failure load (Pmax) recorded in the test,KIccan be determined via the following equations (Kuruppu et al.,2014;Ayatollahi et al.,2016b):

whereBis the specimen thickness.

The experiments also involved a CNSCB specimen configuration(Fig.4),which has been developed by Kuruppu (1997) and has received increasing attention recently(Chang et al.,2002;Mahdavi et al.,2015;Ayatollahi et al.,2016b;Mahdavi et al.,2020;Wei et al.,2016;Wei et al.,2021).The geometric parameters for our CNSCB specimens are as follows:α0(=a0/R)=0.7184,α1(=a1/R)=0.2729,αB(=B/R)=0.8,S/R=0.8,αS(=RS/R)=0.8,whereRSdenotes the radius of the saw used for notching.The principle of CNSCB forKIctesting is similar to that of CB.For this specimen configuration,KIccan be calculated using the following formula:

The VNSRB specimens (Wei et al.,2018c) were tested in the experiments as well.Unlike CNSCB and CB,VNSRB can test the resistance of crack growth along the axial direction of rock cores,as illustrated in Fig.4.For our VNSRB specimens,α0(=a0/R)=0.3,α1(=a1/R)=1.3 and αB(=H/R)=1.6.KIcof the VNSRB specimen is determined using

whereHis the height of the short rod specimen.

As indicated above,the value ofY*is essential for calculatingKIc.Moreover,the normalized SIFs corresponding to effective crack lengths are required in the following sections.Therefore,we calibrated these important parameter values for the test specimens through finite element modeling based on linear elastic analysis.

Fig.5.Representative numerical models used to calibrate the SIFs of the tested chevron notched specimens: (a) CB and (b) VNSRB.

Fig.5 shows representative numerical models built in ABAQUS to simulate the CB and VNSRB tests.Due to symmetry,only a quarter of CB and VNSRB were modeled (as shown in Fig.5,each finite element model has two surface sets with the boundary conditions considering symmetry).To facilitate the modeling,we adopted the sub-modeling technique.First,a full model was established to determine the full-field displacement of the specimen,and then the local area containing the crack front was cut into a sub-model for further analysis.In the sub-modeling,the displacement computed using the full model was applied to the boundary of the sub-model as the external load.The models were meshed with 20-node quadratic brick,reduced integration elements,except that special elements with nodes at quarter-point positions were adopted to simulate the crack-tip stress singularity.Moreover,concentric ring meshes centered around the crack fronts were employed to help determine the SIF through the contour integral method.

Through the modeling,the normalized SIFs at different crack lengths,the critical normalized SIF (Y*) and the corresponding critical normalized crack length(αc=ac/R)can be determined.For example,Fig.6 gives the results for the CB specimen withS/R=1.5 and the VNSRB specimen.Values of (ac,Y*) are (0.545,9.227),(0.545,3.743),(0.592,2.642) and (0.485,6.513) for the CB (S/R=3.33),CB(S/R=1.5),VNSRB and CNSCB specimens,respectively.Based on least-squares polynomial fits to the modeling results,the normalized SIFs of these specimens at different crack lengths are expressed as

Based on Eqs.(24),(25),(27)and(28),the influences of FPZs on theKIcmeasurements using these specimen geometries can be expressed as

3.2.Laboratory tests and results

The rock material tested in this study is fine-grained sandstone quarried from Sichuan Province,China.The indirect tensile strength,Young’s modulus and Poisson’s ratio of the sandstone were measured to be 5.24 MPa,8.13 GPa and 0.17,respectively,by following the ISRM suggested methods (ISRM,1978;Bieniawski and Bernede,1979).The sandstone cores used for specimen preparation were drilled from a block with relatively uniform texture and no visible cracks.The rock cores are in two sizes: 50 mm or 100 mm in diameter.As shown in Table 2,the 50 mm diameter cores were used for all the four specimen geometries,and 100 mm diameter cores for larger-size SCB specimens.The radii of all the specimens satisfied the ISRM suggested size requirement of “l(fā)ager than 10 times the grain size” (Kuruppu et al.,2014).Similar specimen sizes are common in rock fracture experiments(Chang et al.,2002;Akbardoost et al.,2014).

The sandstone cores were further processed on a numerically controlled machine tool.First,we cut the core blocks into the discs or short rods with design dimensions,i.e.200 mm in length for the CB specimens,40 mm in height for the VNSRB specimens,20 mm in thickness for the CNSCB and small-size SCB specimens,and 40 mm in thickness for the large-size SCB specimens.Then,the specimens were notched using circular saw blades with thicknesses of～0.6 mm;the notch width of all the specimens was～0.8 mm.The normalized initial notch length (α0=a0/R) of the SCB specimens was 0.5.For the chevron-notched specimens,the initial and the final notch lengths were described in Section 3.1.A total of 36 specimens were prepared for this experimental campaign.The number of each type of specimens is detailed in Table 2.

The experiments were performed using a servo-controlled material testing system combined with a three-point bending fixture to aid loading (Fig.7).The normalized support span (S/R)was set to 3.33 or 1.5 in the CB experiments and 0.8 in the SCB,CNSCB and VNSRB experiments,as summarized in Table 2.Vaseline was smeared between the support/loading rollers and the specimen to reduce friction.The loading of all experiments was controlled by the load point displacement until the specimen fractured.The loading rate was set at 0.01 mm/s for all the specimens such that failure always occurred in several seconds or within 20 s of initial load application,satisfying the requirement of no more than 20 s suggested by the ISRM(Fowell,1995).This loading rate is appropriate to avoid causing significant creep damage or imposing dynamic effects on the rock failure process.The failure loads (i.e.maximum loading forces) were recorded by a load cell with a capacity of 50 kN and an accuracy of ±0.1%.

Table 2 A brief summary of the fracture experiments conducted in this study.

Fig.6.The SIF calibration results for representative specimens: (a) CB under S/R=1.5,and (b) VNSRB.

Fig.7.Assembly of the specimens in the three-point bending fixture.

All the specimens showed mode I failure.Several broken specimens are displayed in Fig.8.In general,the fracture surfaces of the specimens are relatively flat,and no visible secondary cracks or particle shedding were observed.The apparent fracture toughness(Ka)results are summarized in Table 3.To facilitate the comparison of theKaresults from the different specimens,all the data are drawn in Fig.9.TheKavalues obtained by the CB,CNSCB and VNSRB experiments are very close,about 0.66-0.68 MPa m0.5,while the SCB specimens produce relatively lowKavalues.Moreover,all the data points from the CB experiments are higher than those from the small-size SCB experiments,strongly reflecting the geometry dependence of the apparent fracture toughness.By the way,the phenomenon observed here that CB yields higherKathan SCB is consistent with that reported in the literature (Chang et al.,2002;Funatsu et al.,2015).On the other hand,the small-size SCB specimens have more conservativeKavalues than the large-size SCB specimens,demonstrating the size effect of the apparent fracture toughness.

Table 3 Failure loads and apparent fracture toughness data obtained in the experiments.

Fig.8.Representative fractured specimens collected after the experiments.

4.Assessment of the proposed model based on the experimental results

In this section,the proposed fracture model is assessed based on the experimental results.The main steps are summarized in Fig.10.First,the MNSNs on the ligaments of the specimens are required to characterize the FPZs in the specimens.Thus,we determined the MNSN distributions ahead of the critical cracks of the specimens from the finite element modeling presented in Section 3.1.Fig.11 displays the MNSN results in a generalized form,assuming that the apparent fracture toughness(Ka)of each specimen is unknown.As can be expected,the MNSN decreases with the distance to the crack tip.Through curve-fitting,the crack-tip MNSN distributions can be expressed as

ThelFPZ1values of the specimens can also be determined as the functions of their respectiveKavalues according to Eqs.(13)-(15).

Since CB and SCB have corresponding ISRM suggested methods,and the fracture toughness results of the CB (S/R=3.33) and SCB(R=25 mm)experiments show a considerable difference,they are selected to calibrate ρ.As illustrated in Fig.10,the nominal MNSN ahead of the crack tip of the CB (S/R=3.33) specimen can be determined with the measuredKavalue.By assuming an arbitrary value of ρ,Keof the rock can be determined from the CB(S/R=3.33)specimen when using the effective crack length to calculate the fracture toughness.Moreover,k of the SCB (R=25 mm) specimen can be determined as a function of itsKabased on the assumed ρ value and Eq.(34)(because the effective crack length is related toKaand the normalized SIFs at any crack lengths are already known as Eq.(26) for the SCB specimen).Now onlyKais unknown in the equation of k=Ka/Kefor the SCB specimen,and thusKaof the SCB specimen can be solved.In this way,by using the measuredKavalue of the CB (S/R=3.33) specimens,Kaof the SCB (R=25 mm)specimens can be predicted based on the proposed SEC fracture model.The predictions under different values of ρ are presented in Fig.12a,where the percentage error is defined as the ratio of the difference between theoretical prediction and experimental measurement to the measured value.Thus,ρ can be determined to be 7 for the sandstone material by the trial-and-error method.When a smaller ρ value is assumed,the effect of the FPZ cannot be considered thoroughly,and the SEC-based prediction overestimatesKaof the small-size SCB specimen.If too large a value of ρ is presumed,the FPZ effect is overestimated,and the predictedKafor the SCB test is lower than the experimental measurement.By following the above procedures,Karesults in the other tests can also be predicted with the calibrated model.

Fig.12b depicts theKapredictions for the CB(S/R=1.5),CNSCB,VNSRB and SCB(R=50 mm)experiments using the proposed SEC fracture model.For comparison,the maximum tangential strain fracture criterion (referred to as the “MTSNF” fracture criterion in this study) is also used to predict theKaresults,and the corresponding predictions are presented in Fig.12b as well.The MTSNF criterion assumes that crack initiation is triggered when the MNSN atrcfrom the crack tip reaches a critical strain value εθθc(Mirsayar,2015;Ghouli et al.,2018;Shahryari et al.,2021).Herein,we consider two approaches (I and II) to implement the MTSNFcriterion-based prediction.In Approach I,we userc=(Ka/σt)2/(2π) and our CB (S/R=3.33) experiments to calibratercand εθθc,which are determined to be 2.59 mm and 4.47×10-4,respectively.This formula is commonly used in the literature to estimaterc.For Approach II,rcand εθθc are calibrated using Eqs.13-15.Their values are determined to be 0.544 mm and 1.01×10-3,respectively,based on Approach II and our CB (S/R=3.33) experiments.

Fig.9.Summary of the apparent fracture toughness obtained in the experiments.

Fig.12b indicates that the SEC model performs well,and the percentage errors of the predictions are within ±5%.Compared with the SEC model,the MTSNF criterion causes higher prediction errors,especially for SCB and CNSCB.Moreover,when using the MTSNF criterion for fracture prediction,Approach II seems to be more suitable than Approach I to determinercand εθθc.This is presumably because the determination ofrcin Approach II considers the non-singular strain terms.

5.Discussion

5.1.Characteristics of the proposed model

In the literature,some models have been developed to delineate the effect of the FPZ on the fracture behavior of rock-like materials.These models have,respectively,achieved satisfactory results for some specific fracture problems.For example,the size effect model(Baˇzant and Kazemi,1990)and the boundary effect model(Hu and Duan 2008)have shown great vitality in studying the size effect on the fracture resistance of various rock-like materials.The double-K model (Xu and Reinhardt 1999;Xu et al.,2016) presents unique advantages when dealing with the fracture problems of some important engineering structures like high concrete dams and liquid retaining structures,for which crack initiation is also crucial in addition to unstable crack propagation.In most of the models,the calibration of model parameters is not easy.For the cohesive crack model (Hillerborg et al.,1976),the constitutive behavior of solids is characterized by the traction-displacement curve,which usually requires tedious and elaborate experiments to describe accurately.As for those commonly used models involving the concept of the effective crack,the effective crack length is usually determined directly based on laboratory measurement methods,such as the compliance method(Ouchterlony,1982)and the digital image correlation method (Sutton et al.,1983;Yan et al.,2021).A potential inconvenience of applying these models is that,although researchers can measure the critical effective crack lengths in laboratory benchmark experiments,such as by measuring the compliance,and then determine the effective fracture toughness,it is usually impossible to pre-determine the critical effective crack of an engineering structure in the field or conduct on-site compliance testing on the structure.

In the proposed SEC model,the reference crack corresponding to the FPZ is estimated based on the MNSN ahead of the crack tip,and the fracture toughness is calculated using the effective crack.The model circumvents complex measurements to describe the material constitutive behavior,e.g.the traction-displacement curve.Moreover,when applying this model to fracture prediction of in situ cracked engineering structures with noticeable FPZs,it is unnecessary to perform on-site measurements,such as compliance testing,to characterize the effective cracks in the structures.This is because the determination oflFPZ1and the effective crack in the SEC model only requires the elastic crack-tip strain field,which can be easily obtained by finite element analysis.In addition,since the description of the MNSN involves the non-singular stress/strain terms and the biaxial stress state at the crack tip,the SEC model explicitly includes consideration of these factors.Admittedly,the inability to reflect the constitutive behavior of materials can also be regarded as a weakness of the proposed brittle fracture model.

5.2.Analysis of the specimen geometry and size dependence of Ka based on the proposed model

The dependence ofKaon the geometry and size of cracked rock structures has attracted considerable research attention.In this section,we further investigate this issue based on the proposed model and the above specimen geometries to enhance relevant insights.

As indicated in Section 4 and Eq.(18),the first-order FPZ estimation for a specimen is correlated withm(related to the ratio of σttoKa)of the specimen,and the effective crack length is a function ofmand ρ (see Eq.(20)).Thus,Keof the test material can also be expressed as functions ofmand ρ because the normalized SIFs at any crack lengths for the specimen geometries are known(Eqs.29-32).Therefore,Ka/Ke(i.e.k)of the specimens with these geometries depends on their respectivemand ρ values of the rock.Fig.13 depicts k values of the specimens under various ρ andmvalues.Note that ρ is considered a material property in the proposed model,and different materials may have inconsistent values of ρ.Thus,Fig.13 aims to present the correspondence between k andmfor the specimen geometries under different materials (with inconsistent ρ values),not limited to the sandstone tested in this paper.

Becausemis related to the specimen geometry and cannot represent a material property,another parameter(n)is introduced below to facilitate comparing theKIcmeasurements using these different specimen geometries on the same material.

wherenis related to the ratio of the effective fracture toughness to the tensile strength.BecauseKeis deemed a material property since it takes the FPZ into account,nis more suitable thanmto be treated as a material property.Note that although bothnand ρ are constants for a given rock material,ρ cannot be determined just fromn,σtandKe.Only when the apparent fracture toughness of a fracture specimen with a definite geometry and size is also known,ρ can be solved.Therefore,different rocks may have different combinations ofnand ρ.

Based on Fig.13 andn=mk,the quantitative relationship between k andncan be obtained;the results are depicted in Fig.14.The figure indicates that the dependence ofKaon specimen geometry is related to the properties of the rocks.In other words,the difference between theKavalues measured by different specimen geometries is not constant for rocks with inconsistentnor ρ values.For a constant ρ value,k positively correlates withnfor all the specimen geometries.This is because a largernvalue corresponds to a higher tensile strength relative to the fracture toughness;with a relatively high tensile strength,the rock specimen has a small FPZ,and the measured apparent fracture toughness is close to the effective fracture toughness.For a constantnvalue,k negatively correlates with ρ.This is because a larger ρ value means that the FPZ behaves as a longer reference crack in the fracture process of the specimen and affects theKIcmeasurement more significantly.

Fig.10.A flowchart of the main steps to assess the proposed model based on the experimental results.

Interestingly,for given values ofnand ρ,the k values for the CB,CNSCB and VNSRB specimen geometries are always close to each other and are all higher than those of the SCB specimen geometry.This indicates that for a given rock material,CB,CNSCB and VNSRB produce higher apparent fracture toughness than SCB.The reason is that theKIcmeasurements using these chevron-notched threepoint bend specimens show less sensitivity to the normalized reference crack length (aFPZ/R) than the straight-notched threepoint bend specimens(Wei et al.,2016).In other words,for a given value ofaFPZ/R,neglecting the FPZ andaFPZwill make theKIcmeasurement using the SCB specimen underestimate the rock fracture toughness more significantly than using the CB,CNSCB and VNSRB specimens.This can also be understood from an energetic standpoint.For the SCB and CNSCB specimens with the same thickness (B) andlFPZ,if the fracture parameters (e.g.fracture toughness and fracture energy)are calculated without considering the energy consumed in the FPZ,more energy is neglected for the SCB specimen than the CNSCB specimen.This is because the width of the crack front of SCB(equal toB)is larger than that of the critical crack front of CNSCB(less thanB);thus,a larger volume of the FPZ is actually neglected in the straight-notched SCB specimen than the chevron-notched CNSCB specimen.

To further illustrate how the model specifically achieves the consideration of the size effect,we compare the FPZ-related reference cracks and the k values for the CB (S/R=3.33),SCB,CNSCB and VNSRB specimen geometries,supposing that their radii vary from 25 mm to 100 mm.The results are depicted in Figs.15 and 16.These comparisons are based on ρ=7.Note that the MNSN distributions of the 25 mm radius specimens are available in Fig.10.Those of the specimens with other sizes can be determined by similar finite element analysis or,more conveniently,through simple conversions on account of the theory of elasticity.Then,the results exhibited in Figs.15 and 16 can be obtained.

Fig.11.The MNSN distributions on the ligaments of the fracture specimen geometries at their critical cracks.The apparent fracture toughness is assumed unknown.FEM represents the finite element method.

Fig.12.(a)Calibration of the model parameter ρ for the sandstone using Ka of CB(S/R=3.33)to predict Ka of SCB(R=25 mm),and(b)Percentage errors of the predictions yielded from the SEC model and the MTSNF criterion on the fracture resistance of the tested specimens.

Fig.15 shows that the reference crack is related to thenvalue of the rock material for all the specimen geometries.A largernvalue means a higher tensile strength relative to the fracture toughness,and thus,helps to suppress microcracking at the crack tip before macrocrack propagation.More importantly,the FPZ-related reference cracks of different geometries are inconsistent for a given rock material (i.e.nis fixed).Moreover,as the specimen size increases,the ratio of the reference crack to the specimen radius gradually decreases until it can be ignored.This observation is in line with the insights into the FPZ and corresponding reference crack in the literature (Baˇzant and Kazemi,1990;Hu and Duan,2008;Akbardoost et al.,2014;Ayatollahi and Akbardoost,2014).Fig.16 plots the change of k with the specimen size (R) and thenvalue of the rock.The k value for all the specimen geometries increases with the increases ofRandn,and eventually tends to 1.For a given specimen geometry,the sensitivity of k toRis inconsistent at differentnvalues;the rock with a highernvalue shows less dependence of the apparent fracture toughness on the specimen size.In addition,the geometry of the specimen also affects the apparent fracture toughness and its size dependence.For example,k is more heavily dependent onRfor SCB than CB.This is because theKIcmeasurement using SCB is more sensitive toaFPZ/Rthan using the other three chevron-notched specimen geometries,as explained above.Figs.15 and 16 can also provide the following information: when applying the SEC model to rock fracture prediction,the ρ value should be calibrated (i) through the specimen geometries whose apparent fracture toughness is significantly different in sensitivity to the FPZ,such as CB and SCB investigated herein,or(ii)by conducting fracture experiments at multiple scales using the specimen geometry whose apparent fracture toughness is significantly sensitive to specimen size,such as SCB.

To further examine the applicability of the SEC model,especially in assessing the size effect on the apparent fracture toughness,we have employed the model to analyze experimental data reported by Berto et al.(2016).In their mode I fracture experiments,Kaof Guiting limestone(σt=2 MPa)was tested using SCB(S/R=0.43 and α=0.3)specimens atR=25,50,75 and 150 mm.Based on the SEC model,k values of the different-size SCB limestone specimens and the ratio between theirKaresults can be estimated.Then,using theKavalue measured from the 25 mm radius SCB specimen as a baseline,Karesults of the SCB specimens at the other three radii can be predicted.The experimental data and the prediction obtained with ρ=7.5 are compared in Table 4.ForR=50,75 and 150 mm,the prediction errors are only 9.9%,-2.5% and -2.9%,respectively,manifesting the good performance of the model.

Table 4 The apparent fracture toughness of limestone reported by Berto et al.(2016)and the prediction output from the proposed SEC model.

6.Conclusions

A brittle fracture model is proposed to analyze the mode I fracture resistance of rocks.The FPZ and the corresponding reference crack in the model are estimated by the MNSN criterion,which assumes that the FPZ is caused by excessive tensile strain at the crack tip.The fracture toughness is determined using the effective crack.In addition,it is emphasized that the MNSN should consider both singular and non-singular terms.The principle of the model is explicit and easy to apply.To check the applicability of the model,laboratory fracture experiments were conducted on sandstone using six groups of specimens with inconsistent geometries or sizes.The following conclusions can be drawn:

(1) The proposed fracture model can consider the contributions of the FPZ,the non-singular terms,the biaxial stress state and the microcracking-induced stress/strain redistribution at the crack tip to the fracture resistance of rocks.

Fig.13.Variation of k versus m for the specimen geometries with (a) ρ=1,(b) ρ=2,(c) ρ=3,and (d) ρ=4.

Fig.14.Variation of k versus n for the specimen geometries with (a) ρ=1,(b) ρ=2,(c) ρ=3,and (d) ρ=4.

Fig.15.Comparison of the FPZ-related reference crack between the fracture specimen geometries((a)CB with S/R=3.33,(b)SCB,(c)CNSCB,and(d)VNSRB)for different specimen sizes and n values of rocks.

Fig.16.Comparison of the k value between the fracture specimen geometries((a)CB with S/R=3.33,(b)SCB,(c)CNSCB,and(d)VNSRB)for different specimen sizes and n values of rocks.

(2) The characterization of the reference crack in the model does not require complex laboratory or field measurements;only the linear elastic strain field at the crack tip and the ratio(ρ)of the reference crack to the MNSN-based first-order FPZ estimation are needed,where ρ can be calibrated by mode I fracture experiments with several common specimen geometries,such as CB and SCB tests conducted according to the relevant ISRM suggested methods.

(3) The apparent fracture toughness predicted by the model is in good agreement with the experimental measurement.The proposed model has a better predictive effect than the maximum tangential strain fracture criterion.In addition,the potential of the model in predicting the size effect of apparent fracture toughness is discussed and verified by experimental data reported in the literature.

(4) The analysis based on the proposed model indicates that the apparent fracture toughness shows less dependence on the geometry and size of the specimen for the rock material with a higher tensile strength relative to the effective fracture toughness or with a lower value of ρ.

Declaration of competing interest

The authors declare that they have no known competing financial interestsor personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors thank the financial support from the Key Program of National Natural Science Foundation of China(Grant No.52039007)and the Youth Science and Technology Innovation Research Team Fund of Sichuan Province (Grant No.2020JDTD0001).