Cunbao Li,Donghao Yang,Heping Xie,Li Ren,Jun Wang,*
a Guangdong Provincial Key Laboratory of Deep Earth Sciences and Geothermal Energy Exploitation and Utilization,Institute of Deep Earth Sciences and Green Energy,Shenzhen University,Shenzhen 518060,China
b Shenzhen Key Laboratory of Deep Engineering Sciences and Green Energy,College of Civil and Transportation Engineering,Shenzhen University,Shenzhen 518060,China
c MOE Laboratory of Deep Earth Science and Engineering,Sichuan University,Chengdu 610065,China
Keywords: Size effect Inherent anisotropy Fracture toughness Fracture energy Effective FPZ size Quasi-brittle geomaterials
ABSTRACT Understanding the size effect exhibited by the fracture mechanism of anisotropic geomaterials is important for engineering practice.In this study,the anisotropic features of the nominal strength,apparent fracture toughness,effective fracture energy and fracture process zone (FPZ) size of geomaterials were first analyzed by systematic size effect fracture experiments.The results showed that the nominal strength and the apparent fracture toughness decreased with increasing bedding plane inclination angle.The larger the specimen size was,the smaller the nominal strength and the larger the apparent fracture toughness was.When the bedding inclination angle increased from 0°to 90°,the effective fracture energy and the effective FPZ size both first decreased and then increased within two complex variation stages that were bounded by the 45° bedding angle.Regardless of the inherent anisotropy of geomaterials,the nominal strength and apparent fracture toughness can be predicted by the energy-based size effect law,which demonstrates that geomaterials have obvious quasi-brittle characteristics.Theoretical analysis indicated that the true fracture toughness and energy dissipation can be calculated by linear elastic fracture mechanics only when the brittleness number is higher than 10;otherwise,size effect tests should be adopted to determine the fracture parameters.
Compared with isotropic rocks,anisotropic geomaterials,such as shale[1],exhibit obviously more complicated fracture behavior due to their typical bedding structure [2],which usually complicates crack initiation and propagation in engineering practice [3].Many scholars have applied isotropic or anisotropic linear elastic fracture mechanics(LEFM)to study the fracture toughness,energy and crack propagation paths of anisotropic geomaterials.The results showed that the fracture toughness of anisotropic geomaterials gradually increased when the bedding inclination angle decreased from 90°to 0°[4,5].When the bedding inclination angle was higher than 45°,the crack deviated from the initial direction and mainly propagated in the bedding plane[6].However,the fracture properties of anisotropic geomaterials based on LEFM are obviously size-dependent and shape-dependent due to the presence of the conspicuous fracture process zone(FPZ)near the crack tip[7].The FPZ is produced by the damage and softening of geomaterials under loading.However,these conditions do not conform to the LEFM assumption,i.e.,energy dissipation due to microcrack evolution occurs at one mathematical point [7].This leads to the geomaterial exhibiting a size effect due to the presence of the FPZ.It is worth investigating whether fracture parameters of geomaterials obtained by small-scale laboratory tests can directly reflect the initiation and propagation behavior of cracks in anisotropic geomaterials in large-scale field engineering practice [8].Therefore,an in-depth study of the characteristics and size effects of fractures induced by the geomaterial bedding plane may provide significant guidance for engineering.
At present,the Weibull statistics theory of random strength[9,10],Bažant’s size effect law (Bažant’s SEL) [11] and the theory of crack fractality[12]are widely applied in research on size effects in materials fracture.These theories explain the size effect of materials in terms of probability and statistics,energy release and dissipation,fractal properties of cracks,and the relationships between the strength of materials and specimen size.However,the Weibull statistics theory of random strength has been shown to be unsuitable for recognized quasi-brittle materials such as concrete and rock[13,14].The theory of crack fractality was deduced only from the perspective of geometric analysis,partially verified by experiments without a strict mechanical derivation and energy analysis and shown to be unsuitable for considering the FPZ [15].Bažant’s SEL is widely used for research on the size effects of concrete due to its definite physical meaning and strict mathematical derivation.According to the results of many fracture experiments on concrete,(1) when the specimen size is relatively small,the experimental results are approximately independent of size;(2)when the specimen size is relatively large,the experimental results are close to those predicted by LEFM;and (3) when the specimen size is between the above two cases,the experimental results are very similar to those predicted by Bažant’s SEL [16,17].On the basis of Bažant’s SEL,some different theories of size effects for concrete have been proposed [18,19].Furthermore,Bažant’s SEL has been adopted as a specification by the American Certification Institute (ACI) [20].
Compared with other quasi-brittle materials,research related to size effects in rock is mainly focused on the effects of size of the tensile strength and compressive strength during the early stage.Numerous results have shown that the tensile strength and compressive strength of rock decrease with increasing specimen size[21,22].In addition,some recent studies have been performed on the effect of sample size on isotropic rock fracture characteristics[23,24],while there are few publications on the size effect for anisotropic rocks.Only Li et al.[7] used samples with different sizes and three bedding orientations to study the size effect for shale and found that the size-dependent relationships for the nominal strength and apparent fracture toughness were consistent with Bažant’s SEL,which cannot be accurately described by the strength-based criterion and classical LEFM theory.Obviously,current research on the size effect for anisotropic fracture characteristics of geomaterials is not sufficient.
To comprehensively study the size effect of fracture characteristics for anisotropic geomaterials,a set of three-point bending experiments was conducted on notched deep beam(NDB)samples with 4 sizes and 7 bedding inclination angles.Based on the equivalent LEFM and Bažant’s SEL,a detailed method for theoretical analysis was summarized to calculate the nominal strength,apparent fracture toughness,effective fracture energy and effective FPZ size.Then,the effects of anisotropic geomaterial layers on the size effect of some critical fracture mechanical parameters were analyzed.Finally,the size dependence of the apparent fracture toughness and nominal strength was discussed in detail,and the sizes of the inelastic zone calculated by LEFM and the effective FPZ size calculated by Bažant’s SEL were analyzed.
In this study,shale is selected as the typical anisotropic geomaterial [25,26].The shale materials were obtained from an outcrop belonging to the Longmaxi Formation Shale and used in anisotropic fracture size effect experiments.The main mineral components of the Longmaxi shale were quartz(22.5%),dolomite(40.6%),feldspar (8.0%) and illite (20.9%) as well as small amounts of calcite(3.1%),pyrite (3.4%) and chlorite (1.5%).The 5 independent elastic parameters E1,E2,ν12,ν23and G12(defined in rectangular coordinate system o-12 shown in Fig.1a)that characterize the transverse isotropy of the Longmaxi shale were 23,36,0.176,0.185 and 12 GPa,respectively.To reduce the material error,the experimental samples were prepared from the same outcrop.To consider the effect of anisotropy induced by shale bedding on fracture characteristics,mode Ⅰ NDB specimens with 7 bedding orientations(β=0°,15°,30°,45°,60°,75°and 90°)were prepared(Fig.1a,where the dotted line represents the bedding plane:o-xy defines the global coordinate system and o-12 defines the material coordinate system).Considering the effect of shale size on the fracture characteristics,the NDB samples with 7 bedding angles corresponded to 4 sizes of length(L)and height(W)(W=5,10,15 and 20 mm;L=2 W),as shown in Fig.1b.All of the specimens were cut by an ultrafine wire cutting machine to ensure the accuracy of the sample sizes.The radius of the wire was only 200 μm.Thus,the width of the prefabricated crack was less than 300 μm.This study considered only the size effect of two-dimensional geometric similarity;therefore,the thickness B of all samples was 10 mm.The ratios of the precrack length a0and the supporting half span S to the specimen height W of each size specimen were α0=a0/W=0.5 and S/W=0.75.The precracks of all samples were located in the middle of the sample,and the angle between them and the loading direction was 0.To facilitate the subsequent discussion,the samples were numbered as “sample height (W) -bedding inclination (β) -sample number(1,2,3)”before testing.For example,“20-75°-2”means that the height W of the second specimen is 20 mm and its bedding dip angle is 75°.
The AGS-X testing system was used in the anisotropic size effect tests of shale.The upper limit of the instrument’s load was 10 kN,and the load precision was±0.2%.The movable displacement of the indenter was 1000 mm with a displacement monitoring accuracy of 0.033 μm.The loading rate could be adjusted in the range of 0.0001-1000 mm/min.Since the test results are sensitive to the supporting span of the sample support,4 types of three-point bending fixtures with fixed spans were designed and fabricated for the 4 sizes of NDB samples (Fig.2a),and the supporting spans(2S)were 7.5,15,22.5 and 30 mm.During the test,the sample was placed on the support,and then the middle pointer was aligned with the direction of the prefabricated crack(Fig.2b).The indenter was brought into close contact with the sample at a speed of 0.005 mm/min.The displacement and force of the test system were zeroed before formal loading,and then the axial loads were added to the 4 specimens with heights of 5,10,15 and 20 mm at rates of 0.01,0.02,0.03 and 0.04 mm/min in the displacement control mode to ensure that the strain rates for all samples were the same until final failure.During each test,the frequency of data acquisition was 10 Hz,and the applied load and load line displacement were recorded.To reduce experimental error,at least three duplicate tests were conducted for samples with the same size and bedding inclination angle.
Fig.1.Diagram of the NDB sample details.
Fig.2.Schematic of three-point bending test jigs and setup.
Fig.3a shows the load-displacement curves of the size effect tests of shale for 4 different sizes (taking the bedding inclination angle of 0° as an example).During the loading process,the loaddisplacement curves were similar for samples of different sizes,and the load increased linearly with increasing indenter displacement until the sample was destroyed;the load dropped sharply to 0 when the peak load was reached.It should be noted that this seemingly brittle failure behavior,which was due to the way the applied load was controlled and unstable crack propagation,did not indicate whether the material was brittle.The variation in the peak load with size and bedding inclination angle of shale samples is shown in Fig.3b.Regardless of the test size,the peak load of the shale NDB samples gradually decreased with increasing bedding dip angle,and all specimens showed obvious strength anisotropy.Taking W=20 mm as an example,the peak load reached the maximum value of 598.01 N when the bedding inclination angle was 0°;the minimum value was 357.50 N when the bedding inclination angle was 90°,and the ratio of the maximum to the minimum value was 1.67.Notably,although when the bedding inclination angle was constant the peak load increased with increasing sample size,the peak load difference for different specimen sizes decreased with increasing bedding dip angle.In addition,similar to the work of Li et al.,regardless of sample size,the indenter displacement decreased with increasing bedding inclination angle (the same as the peak load).When the bedding inclination angle was above 45°,cracks mainly propagated along the bedding plane;otherwise,they propagated along the direction through the matrix (Fig.3c,where the direction of the prefabricated crack extension line is denoted by the blue line;the crack propagation mode is illustrated by the red line;the direction of the bedding plane is illustrated by the green line) [27].
The presence of bedding makes shale appear transversely isotropic.Based on the theory of anisotropic LEFM,the apparent stress intensity factor (KI) of shale is a function of the geometric parameters,applied loads,shale elastic parameters,and bedding dip angles of the test specimen [6,27,28].For NDB specimens,the calculation of the mode I apparent stress intensity factor KIis as follows [6,27]:
where P is the applied peak load;a the length of the precrack;W the height of the NDB specimen;B the thickness of the NDB specimen;S the supporting half span;θ the crack inclination angle,which refers to the angle between the precrack of the NDB specimen and the loading direction,and θ of all NDB specimens used in this experiment was 0 to avoid a strong effect of mode II fracture during the test;YIthe dimensionless stress intensity factor of mode I,which was obtained based on the numerical calibration in the study of Li et al.[27];E1,E2,ν12,ν23and G12are 5 independent elastic parameters in the shale material coordinate system (rectangular coordinate system o-12 in Fig.1a).The research results of Bao et al.deduced that the stress intensity factor of mode I KIcould be obtained by the following equation [28]:
where σNrepresents the nominal stress for NDB specimens [28]:
Fig.3.Load displacement curve,peak load and crack growth modes of NDB samples [27].
where k(α,ρ,ψ) and ξ(α,ρ,ψ) are the dimensionless stress intensity factors,where α represents the dimensionless crack length,α=a/W.ρ and ψ are related to the elastic parameters of the material.According to the research of Bao et al.,the calculations of these parameters are as follows [28]:
where Ex,Ey,vxy,vyx,and Gxy,which are calculated in the Cartesian coordinate system o-xy shown in Fig.1a,are elastic parameters for the geometric direction of the samples.They are related to the material elastic parameters (E1,E2,G12and ν12) of shale and the bedding inclination angle β.The mutual calculation relationship is given as follows [29]:
From the relationship between the stress intensity factor and energy release rate [30],we obtained the following expression:
where G represents the energy release rate under plane stress conditions and g(α,ρ,ψ) represents the dimensionless energy release rate;and E*the effective elastic modulus of the transversely isotropic material.It is different from the elastic modulus of the isotropic material,and its calculation is given as follows:
Meanwhile,the relationship between the dimensionless energy release rate and the dimensionless stress intensity factor is:
Substituting Eq.(8) into Eq.(2),the relationship between the stress intensity factor and the dimensionless energy release rate can be obtained as follows:
Shale has been proven to be a typical quasi-brittle material by many experimental studies.The FPZ at the crack tip cannot be ignored,and it is affected by the size effect.The rock has obvious damage and strain softening characteristics in the FPZ.To better consider the influence of the FPZ,Bažant proposed using equivalent LEFM to describe the fracture behavior for quasi-brittle materials,where the effective crack length a and the effective dimensionless crack length α were defined as follows [7,11,31]:
where a0is the initial fracture length;α0the initial dimensionless fracture length;cfthe length of the effective fracture process area.Substituting Eq.(10)into Eq.(6),the GICcalculation of the effective LEFM fracture energy of the sample when the load is the peak load of the sample is given as follows:
The dimensionless energy release rate g(α) is the key to determining the size effect equation for quasi-brittle materials.By taking different initial dimensionless fracture lengths α0,a series of g(α0) values can be obtained by numerical calculation,and then the expression for g(α)can be obtained by data fitting.A numerical model of the shale NDB sample was constructed by ABAQUS;the constitutive model in ABAQUS was selected as the elastic transverse isotropic model,and then the 5 independent elastic parameters were input according to the experimental data.The stress intensity factor KIwas calculated when the dimensionless fracture length α0gradually increased from 0.2 (in increments of 0.02) to 0.6.The dimensionless energy release rate g(α0)was inversely calculated by Eq.(9).The variation in the dimensionless energy release rate g(α0) with the dimensionless fracture length α at 7 bedding inclination angles for the shale NDB samples is shown in Fig.4a.With increasing α,the dimensionless energy release rate g(α0) shows an increasing trend for a concave curve,and the growth rate increases significantly with increasing α.The bedding dip angle has no effect on the variation trend of g(α0).When the dimensionless crack length α is small,the difference in g(α0)under different bedding inclination angles is small.With the increase in α,the effect of the bedding inclination angle on g(α0) gradually increases.When α is 0.2,0.4,and 0.6,the bedding inclination angle increases from 0°to 90°,and g(α0)increases by 0.02,0.11 and 0.34,respectively (Fig.4a).This result indicates that the value of α has an obvious influence on the anisotropic fracture parameters at the crack tip.
By fitting the variation in g(α) with α under each bedding dip angle shown in Fig.4a,the expression of the dimensionless energy release rate g(α) can be calculated as follows [7,31,32]:
where we can express p(α) (a quartic polynomial function) as follows:
Fig.4.Variation in dimensionless energy release rate with α and fitting of the quartic polynomial p(α) of NDB samples.
Substituting the series of data points (α0,g(α0)) obtained by numerical calculation in Fig.4a into Eq.(14),the corresponding(α0,p(α0))can be obtained.Next,unknown parameters in the equation of p(α) are calculated by fitting.The results are shown in Fig.4b,and the constant terms a1,a2,a3,a4and a5of the function p(α) are listed in Table 1.This function can accurately fit the variation between the dimensionless energy release rate g(α) of the NDB sample and the initial crack length.
The above equation can be rewritten in the following linear form [33]:
In the equation,
The effective dimensionless energy release rate g(α) (Eq.(14))and its derivative (Eq.(16))were determined as shown in Table 1.E*is a function of the known shale material parameters.By testing NDB samples of different scales,the nominal strengths corresponding to a series of different sample sizes were obtained.The slope A and intercept C of the fitted linear equation were obtained by processing the size effect experimental data by Eq.(18).Subsequently,Eq.(19)was used to calculate the shale anisotropic effective LEFM fracture energy GICand effective FPZ size cf.
Table 1 Constant terms of the quartic polynomial function p(α) and its fitting correlation coefficient.
The nominal strength σNuis the basis for calculating the apparent fracture toughness,fracture toughness and size effect law of shale.Fig.5 presents the change in nominal strength σNuof the shale NDB samples with increasing bedding inclination angle and sample size.Fig.5a shows that the variation in the σNuof shale samples of different sizes with bedding inclination angles is basically consistent,and the variation trend can be divided into three distinct stages.Specifically,in stage I,as the bedding inclination angle increases from 0°to 45°,σNudecreases linearly with an aver-age rate of 0.41 MPa/15°.The failures occurring at this stage are mainly caused by initial cracks penetrating the shale matrix.In stage II,as the bedding inclination angle increases from 45° to 60°,the rate of decrease of σNuincreases significantly with increasing bedding inclination angle,which is 1.54 times higher than that of stage I.The penetration of the initial crack transitions from matrix to bedding in this stage.In stage III,as the bedding inclination angle increases from 60° to 90°,the decrease in σNuslows down significantly.At this stage,the σNuof the specimen with a height of 20 mm is reduced by only 11.74%,and failure is mainly caused by the initial crack propagation along the bedding plane.Fig.3a and 5b show that although the peak load increases with increasing shale specimen size,the σNuof shale decreases approximately linearly.Moreover,the bedding inclination angle has little effect on this variation.For example,when the bedding inclination angle is 0°,as the shale specimen height increases from 5 to 20 mm,the average value of σNuof shale decreases from 9.86 to 6.73 MPa,a decrease by 31.74%.Apparently,the nominal strength of shale exhibits evident size effect characteristics.
Fig.5.Variation trend of the nominal strength σNu of the shale NDB specimen with the bedding inclination angle and specimen size.
The fracture toughness,KIC,which characterizes the inherent ability of materials to resist crack initiation and propagation,is a key mechanical parameter.Most analyses of shale fracture test data are based on LEFM,and the apparent fracture toughness,KICA,is the basis for exploring the real fracture toughness,KIC,of materials.Fig.6 shows the variation in the shale KICAwith various bedding inclination angles and sample sizes.The changes in KICAfor all of the shale NDB specimens exhibit typical anisotropic characteristics caused by bedding planes regardless of the size of the specimen.Taking the specimen with height W=5 mm as an example,the shale KICAreaches the maximum value (1.08 MPa.m1/2) when the bedding inclination angle is 0°.With increasing bedding inclination angle,the average value of KICAdecreases nonlinearly.When the bedding inclination angle is 90°,KICAreaches the minimum value (0.75 MPa.m1/2),and the ratio of the maximum value to the minimum value is 1.44.The ratio increases successively to 1.52,1.58 and 1.60 when the height W of the specimen is 10,15 and 20 mm,respectively.Notably,although KICAdemonstrates the same trend with the increase in bedding inclination angle at all specimen sizes,the rate of decrease differs.Especially when the angle is between 45°and 60°,the rate of decrease of the specimen KICAwith a height of 5 mm is significantly smaller than that of the other three sizes.Figs.5 and 6 show that under the same bedding dip angle,with the increase in specimen size,the shale KICAshows an overall tendency to increase,which is contrary to the law of nominal strength change with sample size.However,different bedding dip angles have varying degrees of influence on the amplification of KICAproduced by increasing specimen size.As the height of the specimen increases from 5 to 20 mm,the shale specimen KICAwith bedding inclination angles of 0°,15°,30°,45°,60°,75°and 90°increases by 36.63%,32.13%,33.11%,36.49%,22.40%,18.34% and 22.00%,respectively.This also indicates that the KICAof the shale matrix is more sensitive to size.Notably,for shale samples with heights of 15 and 20 mm,there is little difference in the values of the shale KICAwith different bedding angles,indicating that with increasing specimen size,the influence of size on KICAgradually decreases,which can provide a certain reference for the selection of specimen size for the shale size effect test.Overall,the shale KICApresents obvious characteristics of anisotropy and size effects.Typical LEFM theory could not meet the needs of analyzing the anisotropic fracture behavior of shale in this study.
Fig.6.Variation in the KICA of shale NDB samples under different bedding inclination angles and sample sizes.
Fig.7.Linear regression analysis of the size effect law with different bedding inclination angles and sample sizes.
Table 2 Fracture parameters calculated based on Bažant’s SEL.
Fig.8.Variation in the effective LEFM fracture energy GIC of shale NDB samples with different bedding inclination angles.
According to the data in Table 2,Fig.8 shows the variation in the shale GICwith different bedding inclination angles calculated based on Bažant’s SEL.The GICof the shale NDB specimen presents complex fluctuation characteristics with increasing bedding inclination angle.The GICreaches the maximum value (99.42 N/m)when the bedding angle is 0°,while it reaches the minimum value(39.38 N/m) when the bedding angle is 90°,which indicates that more energy is required for crack initiation and propagation along the direction through the matrix.Between the maximum and minimum values,the GICof shale decreases nonlinearly with the bedding inclination angle,which is quite different from the variation in the energy release rate calculated based on LEFM [27].When the bedding inclination angle gradually increases from 0° to 45°,the GICfirst decreases and then increases,reaching the first minimum value of 77.82 N/m when the bedding angle is 15°and then reaching the maximum value (83.10 N/m) when the bedding angle is 45°;when the bedding inclination angle increases from 45° to 60°,the GICrapidly decreases to 47.91 N/m,a sharp reduction of 42.34%.The reason for this decrease may be that when the bedding inclination angle is 60°,there are already crack initiation and propagation along weak bedding planes,and the energy required for propagation is significantly reduced.When the bedding inclination angle reaches 75°,the GICdecreases by 17.68% compared with the sample with a bedding inclination angle of 60° and drops to 39.44 N/m,which is only 0.15% larger than the GICwith a bedding inclination angle of 90°.Clearly,the anisotropy of the GICis more complex than those of the KICAand nominal strength.This may be attributed to the parameters controlling the fracture energy,such as the effective FPZ and the effective elastic modulus,which have a complex functional relationship with the bedding inclination angle.The variation in these parameters leads to a complicated variation trend of the fracture energy.
The energy release in the effective FPZ is an important reason for the size effect.According to Table 2,Fig.9 shows the variation in the effective FPZ size cfof shale NDB samples with different bedding inclination angles.Bordered by the bedding inclination angle of 45°,the cfpresents two distinct stages.The shale cfwith bedding inclination angles less than 45° is significantly larger than that of shale with bedding inclination angles of 60°,75° and 90°.This is because when the bedding inclination angle is less than 45°,fracture initiation and propagation mainly occur in the shale matrix.However,when the bedding inclination angle is above 60°,cracks mainly propagate in the weak bedding plane.Specifically,the cffirst decreases and then increases as the bedding inclination angle changes from 0° to 45°.The minimum value of this stage(0.41 mm) is obtained when the bedding inclination angle is 15°,and then the maximum value(1.17 mm)is obtained when the bedding inclination angle is 45°.When the bedding inclination angle changes from 60° to 90°,the variation characteristics of the cfare generally consistent with those when the bedding inclination angle increases from 0° to 45°.The minimum value (0.41 mm) in this stage appears when the bedding dip angle is 75°,which is also the minimum value among all different bedding angles.Figs.8 and 9 show that the variations in the cfand GICwith the bedding inclination angle are similar,but the bedding dip angles corresponding to their peak values are different.The reason for this could be that the GICis the root cause of the cf.However,due to the influence of the bedding effect,the anisotropic elastic parameter E* of shale is quite different at different bedding inclination angles,and the fracture process zone width may differ,resulting in various bedding dip angles corresponding to the maximum and minimum values of the cfand GIC.
Fig.9.Variation in the effective fracture process zone length cf of shale NDB samples with different bedding inclination angles.
As described in Sections 4.1 and 4.2,significant size effects were exhibited in the shale nominal strength and apparent fracture toughness.Data analysis was performed on the nominal strength of shale obtained from the test based on Bažant’s SEL (Eq.(13))to better characterize this behavior,as shown in Fig.10.It can be intuitively identified that the nominal strength of shale NDB specimens cannot be predicted by LEFM at different bedding dip angles and specimen sizes (brittle behavior,inclined asymptote with a slope of-1/2 in Fig.10),nor can it be predicted by strength theory(ductility behavior,horizontal asymptote in Fig.10);the nominal strength of shale NDB samples is in the transition region between strength theory and LEFM theory predictions and is in good agreement with Bažant’s SEL curve,indicating that the fracture behavior of shale is quasi-brittle.It should be emphasized that although the anisotropic properties of the shale itself have a significant effect on the nominal strength(Fig.5),they have no effect on the applicability of Bažant’s SEL because the effective elastic modulus and the dimensionless energy release rate fully consider the anisotropic effect of shale.Hence,Bažant’s SEL can better predict the size effect on fracture behavior for both quasi-brittle isotropic and anisotropic materials.
The variation in the shale brittleness number under different bedding inclination angles and sizes is shown in Fig.11 to better discuss the influence of shale anisotropy on the brittleness number.Similar to the variation in the effective LEFM fracture energy and effective FPZ size with the bedding inclination angle,the variation in the brittleness number of the shale NDB samples with the bedding inclination angle can be divided into two stages with a bedding inclination angle of 45° as the boundary.When the bedding inclination angle is above 45°,the brittleness number of the sample is significantly greater than that at a bedding inclination angle less than or equal to 45°.The brittleness number reaches the maximum value when the bedding inclination angle is 75°and the minimum value when the bedding angle is 45°.Interestingly,the anisotropic properties of shale have different effects on samples of different sizes.With increasing size,the difference in shale brittleness numbers under different bedding inclination angles increases.When the sample height is 5 mm,the difference between the maximum and minimum brittleness numbers is 1.20,while when the sample height is 20 mm,the difference increases by 4.02 times.Furthermore,Figs.11 and 9 show that the variation in the brittleness number with the bedding inclination angle is opposite to the trend for the effective FPZ size.The brittleness number increases as the effective FPZ size decreases,which is closer to the LEFM asymptote in Fig.10.Hence,the brittleness number has a good correlation with the effective fracture process interval,which also confirms that the presence of the FPZ (strain softening zone) is the reason for the size effect [11,15].
Fig.10.Relationship between the nominal strength of the NDB shale samples and Bažant’s SEL.
Fig.11.Brittleness number of shale NDB samples under different bedding inclination angles and sample sizes based on Bažant’s SEL.
As mentioned in Section 4.2,the apparent fracture toughness KICAcalculated by LEFM theory is dependent on the specific sample size and geometry.However,the true fracture toughness KICis a material property that is independent of the test method and sample size[7].Understanding the connections between them is critical to the analysis of shale size effects.The fracture toughness KICof shale can be calculated according to Bažant’s SEL formula,KIC=(E*GIC)1/2.According to the data in Table 2,the fracture toughness values KICof shale NDB samples with bedding angles of 0°,15°,30°,45°,60°,75° and 90° are 1.763,1.585,1.568,1.549,1.145,1.023 and 1.017 MPa.m1/2,respectively.The apparent fracture toughness KICAand fracture toughness KICshow the same trends:the present values decrease as the bedding inclination angle increases.However,the value of KICAis significantly smaller than that of KIC.According to Bažant’s SEL (Eq.(12)),the relationship between the apparent fracture toughness KICA,fracture toughness KICand brittleness number δ can be expressed by the following equation [7]:
where KICA/KICis the normalized apparent fracture toughness.
The variations in the KICA/KICwith the brittleness number δ on a linear scale are shown in Fig.12.The KICA/KICof Longmaxi shale is in good agreement with the SEL prediction curve,and the value of the KICA/KICincreases with increasing δ.When the brittleness number δ approaches infinity,the KICA/KICvalue increases continuously and finally converges to 1,indicating that when the dimension of the tested specimen δ is large enough,the KICAcan be directly regarded as the KICof the material.Generally,when the brittleness number satisfies 0.1<δ<10,the shale has quasi-brittle fracture behavior [7],and the size of the FPZ of shale cannot be ignored.However,when δ≥10,the fracture parameters calculated based on LEFM can be considered the real fracture parameters of the material.The brittleness numbers of all shale samples are in the transition region of 0.64-7.41 in this test,and thus,the KICAof the NDB sample of Longmaxi shale measured in this study can hardly reflect the true fracture parameters of the shale.Figs.11 and 12 show that in the shale fracture parameter test conducted in this study,the KICAmeasured for the largest samples at these angles was closer to the KICbecause of the larger brittleness number at bedding inclination angles of 60°,75° and 90°.In contrast,the brittleness number was small when the bedding inclination angle was 45°,and the KICAwas quite different from the KIC.Therefore,it is necessary to consider the effect of the bedding dip angle on shale fracture parameters in practical engineering applications.
Fig.12.Relationship between the normalized apparent fracture toughness KICA/KIC of shale samples and the brittleness number δ.
The inelastic zone or FPZ at the crack tip is the main area of energy dissipation when cracks initiate and propagate.The FPZ introduced in Section 3 can be calculated by size effect theory in quasi-brittle fracture mechanics.In LEFM,there is no concept of a fracture process zone,and there is an inelastic zone near the crack tip that satisfies the strength criterion.Based on the characteristics of the stress field near the crack tip of anisotropic materials calculated by LEFM [30] and the Drucker-Prager (DP) yield criterion theory[34],Li et al.calculated the radius r of the inelastic zone as follows [27]:
where c is the cohesion force;φ the internal friction angle;ϑ the angle between the extension line of the crack tip and the r direction;u,v both the real part of the Re function (a complex variable function);μk(k=1,2) the parameters calculated by the roots Tkof the material characteristic equation a11T4+(2a12+a66)T2+a22=0 of anisotropic shale;and aijthe flexibility coefficient for anisotropic materials.The equations for μk(k=1,2) and aijare given as follows[27,35]:
Based on the values of the cohesion force c and internal friction angle φ at various bedding angles obtained by Li et al.[27],the inelastic zone lengths r of shale NDB specimens on the initial crack tip extension line(i.e.,ϑ=0°)under different bedding dips and sample sizes were calculated.The variation in the inelastic zone length r of shale samples under different bedding inclination angles and sample sizes was compared with the effective FPZ size cfobtained based on Bažant’s SEL,which is shown in Fig.13.Obviously,the length r of the inelastic zone when the sample height W is 5 mm is much smaller than those of the other three samples,and the average value of r increases with increasing sample size.When the heights of the specimens are 15 and 20 mm,the difference in the values of r is small,mainly because the difference in the apparent fracture toughness between the two specimen sizes is small(Fig.6).Interestingly,although the r values of shale samples with different sizes and bedding dip angles are quite different from the value of cf,they show the same variation with the bedding angle,and both have two obvious stages:first decreasing and then increasing.When the bedding angle is 0°,15°and 30°,the length of r is less than those of the cf.However,when the bedding angle is 60°,75°and 90°,the length of r of the same specimen size is larger than the length of the cf.When the bedding angle is 45°,the average values of r of the 4 sizes of shale samples are almost equal to the cf,with a difference of only 8.28%.According to the above analysis,for the shale fracture test sample size in this study,when the initial crack mainly extends through the matrix(the bedding inclination angle is less than 45°),the calculation based on LEFM underestimates the size of the energy dissipation area at the crack tip.In contrast,when the initial crack mainly spreads along the bedding plane (the bedding inclination angle is above 45°),the calculation based on LEFM overestimates the size of the energy dissipation area at the crack tip.
Fig.13.Comparison between the size of the inelastic zone near the crack tip obtained by LEFM and the effective FPZ size obtained by Bažant’s SEL for NDB shale samples.
To study the effects of the inherent bedding structure and specimen size on the fracture behavior of anisotropic geomaterials,a series of anisotropic fracture tests were systematically carried out on NDB specimens with 7 bedding inclination angles (β=0°,15°,30°,45°,60°,75° and 90°) and 4 sizes (W=5,10,15 and 20 mm).Using Equivalent LEFM and Bažant’s SEL theory,the variations in the nominal strength,apparent fracture toughness,effective LEFM fracture energy and effective FPZ size for different specimen sizes and bedding dip angles were analyzed in detail.The size effects on the nominal strength and the apparent fracture toughness were discussed,and the differences between the sizes of the inelastic zone near the crack tip calculated by LEFM and the effective FPZ sizes were compared and analyzed.The main conclusions are described as follows:
(1) The nominal strength and the apparent fracture toughness of shale exhibited a nonlinear decreasing trend with increasing bedding dip angle under different specimen sizes,and both reached minimum values and maximum values when the bedding inclination angles were 90° and 0°,respectively.However,with increasing specimen size,the nominal strength showed a gradually decreasing trend,while the apparent fracture toughness gradually increased.
(2) When the bedding inclination angle increased from 0° to 90°,the effective LEFM fracture energy and the effective FPZ size calculated by Bažant’s SEL both decreased first and then increased within two complex variation stages that were bounded by the bedding angle of 45°.The reason for this could be that there is a nonlinear coupling relationship between the stress field near the crack tip and the shale material parameters due to the bedding dip angle,which affects the energy dissipation near the crack tip.
(3) Regardless of the specimen size and bedding dip angle,the nominal strength and the apparent fracture toughness could be accurately fitted with Bažant’s SEL,which indicated that shale had quasi-brittle fracture characteristics,and its fracture behavior could not be described by the strength-based criterion or classic LEFM in this study.
(4) With varying bedding dip angles,the difference between the apparent fracture toughness and true fracture toughness of shale is significantly different.Only when the brittleness index is greater than 10 is the apparent fracture toughness close to the true fracture toughness.In actual engineering practice,it is necessary to consider the effect of the bedding inclination angle and specimen size on the fracture toughness of shale.
(5) The inelastic zone sizes near the crack tip calculated by LEFM of shale samples with different sizes and bedding dip angles are quite different from the value of effective FPZ sizes calculated by SEL.However,they show the same variation under the changes in bedding inclination angle.The applicability and validity of the energy dissipation zone calculated based on LEFM is affected by the bedding plane of shale and specimen size.
Acknowledgments
This work was financially supported by the National Natural Science Foundation of China (Nos.U22A20166,51904190,12172230,11872258 and U19A2098),the Department of Science and Technology of Guangdong Province (No.2019ZT08G315) and MOE Laboratory of Deep Earth Science and Engineering (No.DESE202102).
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