Junfeng Gun,Yulong Zhng ,Jingfeng Meng ,Xinhu Yo ,Lielie Li,b ,Shunghu He
a School of Civil Engineering and Communication,North China University of Water Resources and Electric Power,Zhengzhou 450045,China
b State Key Laboratory of Hydraulics and Mountain River Engineering,Sichuan University,Chengdu 610065,China
Keywords:Rock Size effect Boundary effect Structural geometry parameter Fracture toughness Tensile strength
ABSTRACT This paper develops a model that only requires two sets of small-size rock specimens with the ratio of the structural geometry parameter maximum to minimum ae,max:ae,min ≥3:1 to determine the rock fracture and strength parameters without size effect and predict the actual structural performance of rock.Regardless of three-point-bending,four-point-bending,or a combination of the above two specimen types,fracture toughness KIC and tensile strength ft of rock were determined using only two sets of specimens with ae,max:ae,min ≥3:1.The values KIC and ft were consistent with those determined using multiple sets of specimens.The full structural failure curve constructed by two sets of small-size specimens with ae,max:ae,min ≥3:1 can accurately predict large-size specimens fracture failure,and±10%upper and lower limits of the curve can encompass the test results of large-size specimens.The peak load prediction curve was constructed by two sets of specimens with ae,max:ae,min ≥3:1,and±15%upper and lower limits of the peak load prediction curve can cover the small-size specimen tests data.The model and method proposed in this paper require only two sets of small-size specimens,and their selection is unaffected by the specimen type,geometry,and initial crack length.
The true fracture and strength parameters of rock should be constant and not affected by test method,specimen type,geometry,and size,etc.However,many rock material test studies have shown that the rock fracture and strength parameters determined under laboratory conditions have apparent size and shape effects[1–15].The test results of the material parameters vary with test method,specimen type,geometry,and size,etc.[16–26].In early research on the size effect of rock and concrete material parameters,scholars focused on the change in‘‘absolute size”and believed that the size effect of the material parameter test results was mainly caused by a change in the specimen size.For example,the size effect model(SEM)proposed by Bazˇant used geometrically similar specimens with the maximum depth to minimum depth ratio of 4:1 to eliminate the influence of specimen size changes on the determination of material parameters.However,for rock and concrete materials containing rock grains or aggregate,the so-called size effect on material properties is not a simple ‘‘absolute size effect”,but a ‘‘relative size effect” coupled with multiple factors such as specimen size,loading type,grain size,initial crack length,etc.
The relative size of a rock specimen under laboratory conditions,that is,W/gav(the ratio of the specimen depthWto the average grain sizegav) is relatively small (W/gav≤20) [27].Therefore,fracture failure is quasi-brittle fracture rather than linear elastic fracture.Determining the rock’s actual fracture and strength parameters without size effect from small-size specimens controlled by quasi-brittle fracture is problematic.There are two broad categories for models for determining fracture parameters without size effect based on the small-size specimens in the laboratory.The first category considers the influence of the fracture process zone of the small-size specimen.The equivalent crack lengthacis used to replace the initial crack lengtha0,ac=a0+Δafic,which is substituted into the linear elastic fracture mechanics calculation formula to obtain the equivalent fracture toughness.
The second category considers small-size specimens’ structural behavior (such as peak load) as measured by tests.The fracture toughnessKICis determined by the extrapolation method based on the basic analytical solution of the fracture model.The boundary effect model(BEM)proposed by Hu[28–30]belongs to the second category,and fracture toughnessKICand tensile strengthftwithout size effect are determined by the peak loads of the concrete specimens controlled by quasi-brittle fracture based on the extrapolation method.
Recent studies focused on the characteristics of small specimens in the laboratory and proposed the concept of‘‘relative size”(the ratio of specimen depthWto the average grain sizegavof rock)[31–39].At the meso-level,the vital influence of rock grain size on crack growth has been considered.Based on the basic theory of the boundary effect,a model and method that simultaneously determines fracture toughnessKICand tensile strengthftwithout size effect have been developed and verified by concrete [31–36],cement mortar[37],rock [27],and metal[38,39] tests.This developed model has proven to be suitable for the analysis of different geometrical types,such as geometrically similar and nongeometrically similar,and different specimen types such as three-point-bending (3-p-b) and wedge-splitting (WS).
The structural geometry parameteraein the boundary effect model is a coupling quantity of a specimen depthW,initial crack lengtha0,and specimen loading type parameterY(α),reflecting the common influence of the above 3 factors on the fracture performance of the specimen.The structural geometry parameteraecan simultaneously consider the influences of the front boundarya0and the back boundaryW-a0on the fracture failure of a specimen or structure.Therefore,to study the size effect of rock material parameters,it is of more physical significance to replace absolute size (specimen depthW) with the structural geometry parameterae.
In previous studies on the variations of fracture parameters of rock and concrete materials determined by non-geometrically similar specimens,the ratio of initial crack lengtha0and depth of specimenW,α(a0/W),or the ratio of the effective length of specimenSand depth of specimenW,S/W,is usually taken as the variation parameter.Dong et al.conducted fracture tests on 5 sets of concrete specimens with different strengths [40].Each set of specimens were non-geometrically similar specimens withW=100 mm,α=0.2,0.3,or 0.4.The corresponding structural geometry parametersaewere 6.38,6.27,and 5.74 mm,respectively,andae,max:ae,min=1.11.Although α changed from 0.2 to 0.4,aechanged very little.The reasonable fracture toughnessKICand tensile strengthftwere unable to be determined based on the boundary effect model.Yin et al.[41] conducted 6 sets of concrete fracture tests with differentS/W,whereW=150 mm,α=0.4,S/W=2,2.5,3,4,5,and 6,with correspondingaeof 7.46,7.74,8.02,8.61,8.75,and 8.90 mm,respectively,andae,max:ae,min=1.19.AlthoughS/Wchanged from 2 to 6,aeessentially did not change.The reasonable material parametersKICandftwere unable to be determined based on the boundary effect model.Therefore,the conclusion given in the literature [41] that the determined fracture parameters do not vary withS/Wis worthy of discussion because of the coupling effect of its geometry,crack length,loading type,etc.,unchanged.It can be seen from the above that for non-geometrically similar specimens that if the variation range of the structural geometry parameteraeis small due to the improper selection of α orS/Wof the specimens,the true material parameters cannot be effectively obtained even with a large number of test data.Therefore,it will be of great practical significance if the key specimens or the law of the specimen combination for determining the rock material parameters can be found;that is,the minimum amount of experimental data,in theory,can be used to determine the true material parameters of the rock.
Therefore,this paper proposes that the structural geometry parameteraereplaces the ‘‘absolute sizeW” as the structural parameter to study the size effect of rock material parameters.The objective of this research is to develop theoretical fracture model and specific application method for determining the material parameters (tensile strength and fracture toughness) of rock by using quantitative specimens.Through different specimen sizes,initial crack lengths,and loading types (3-p-b and 4-p-b) of rock specimens fracture tests,the proposed fracture model and specific practical method for determining rock material parametersKICandftwithout size effect from quantitative specimens are studied and proven to be accurate.Finally,the application model and method for determining the true rock material parameters and predicting the true rock structure performance from only two sets of specimens withae,max:ae,min≥3:1 are developed.
2.1.Boundary effect model and structural geometry parameters ae considering the coupling effects of grain size,initial crack length,and specimen size
The recently developed improved boundary effect model [30–39] considers the coupling effects of rock grains,initial crack length,specimen size,etc.,on fracture failure.The fictitious crack growth length Δaficat the peak loadPmaxis related to the characteristic size of the rock grains—the average grain sizegav,and the specific analytical expression of the model as follows:
As shown in Fig.1a,the relative size (W/gav) of the rock specimen under laboratory conditions is small (W/gav<20).The specimen is in a quasi-brittle fracture state,showing nonlinear fracture characteristics.Therefore,the fictitious crack growth length Δaficcannot be ignored,and its influence on the fracture failure of the specimens should be considered.For materials containing rock grains or aggregates,such as rock or concrete,crack growth propagates forward row by row in the model,either moving around or across the grains.If the relative size(W/gav)of a specimen is small,ΔaficatPmaxis limited.As shown in Fig.1a,the irregular distribution of grains of different sizes can be simplified to a regular grain distribution,causing the crack to span one row of grains,Δafic=gav.If the ligament heightW-a0is relatively large,the fracture spans two rows of grains,Δafic=2gav,although it does not span 3 rows of grains.Therefore,we introduce the discrete number β,Δafic=βgav,where β=1.0 to 2.0.If the irregular distribution of the grains is considered,then β=1.5,as shown in Fig.1a.To facilitate the design and application,β=1.0 can meet the design requirements whenW/gav<20.
The analytical solution of the boundary effect model describing material fracture failure is [27–29]:
where σnis the nominal stress considering the influences of the initial crack lengtha0and fictitious crack growth length Δafic;the characteristic crack of a material,which can be determined by fracture toughnessKICand tensile strengthft,and its theoretical expression is:
It can be seen thatis a material parameter and can measure the proportional weight ofKICandftin quasi-brittle fracture failure.aeis structural geometry parameter,and its theoretical expression is:
Fig.1.Improved boundary effect model considering Δafic at Pmax.
where α is the ratio ofa0andW;Y(α) the specimen loading type parameter and is a function of α,which varies with different loading types(3-p-b or 4-p-b)andS/Wof the specimen.According to stress intensity factors handbook [42] (S/W=4 or 2.5 for 3-p-b or 4-p-b,respectively) or the finite element method (S/W=2 for 3-p-b),the specific calculation expression can be obtained by:
In summary,the structural geometry parameteraein the boundary effect model considers the influences of the absolute sizeWand the coupling influence of front boundary (initial crack lengtha0) and back boundary (ligament heightW-a0) of a specimen.Compared with the absolute sizeW,aeis more beneficial in size effect research.
As shown in Fig.1b,Eq.(2) can be changed to regression analysis type:
It can be seen that when σnandaeare determined,the material parameters-fracture toughnessKICand tensile strengthftcan be simultaneously determined using regression analysis.If material parametersKICandfthave been determined,the fracture failure mode (strength control,quasi-brittle fracture,fracture toughness control) and failure state (σnorPmax) of specimens with different structural geometry parametersaecan be predicted using Eq.(9).The relationship between material parameters (KICandft) and structural behavior (Pmax) can then be established also by Eq.(9),as shown in Fig.1b.
Considering the influences of ΔaficatPmax,the specimen front boundarya0,and the back boundaryW-a0,etc.,the calculation diagram of the nominal stress σnfor 3-p-b and 4-p-b type specimens is shown in Fig.2.
Fig.2.Calculation diagram of the nominal stress σn for 3-p-b and 4-p-b.
For the 3-p-b specimen type,the expression of the nominal stress σnis [30–37]:
whereBis the thickness of the specimen.
Fig.3.Simplified two-point method for determining rock material parameters and predicting true structural behavior.
Fig.4.Theoretical relationship between ae and α (=ao/W).
For the 4-p-b specimen type,the expression of the nominal stress σnis [43]:where 2Tis the distance between the two loading points of the 4-pb specimen.
2.2.Two-point (ae,min,ae,max) method for determining rock fracture toughness and tensile strength
When applying the boundary effect model mentioned above,generally,multiple sets of non-geometrically similar specimens with different structural geometry parametersaeare used,and the material parameters without size effect can be accurately determined by regression analysis.The real questions remain in how many sets of the specimen are needed to achieve the best result,and there has not been an in-depth study to determine this.Also,it is unknown how great the difference betweenae(for example,ae,max:ae,min≥3:1)or variation range ofaecan be to the best fit.If representative structural geometry parameters can be determined,the number of specimens required can be reduced,reducing the production cost of specimens and the test workload.This is convenient for popularization and widespread application.
Because of this,this paper proposes a simplified method to determine the material parameters without size effect using two sets of specimens with a significant difference in structural geometry parametersae.The basic theoretical method is shown in Fig.3.The specimens withae,minandae,maxwere selected,and the middleaespecimens were discarded.If there are fixed and unique materialparameters(KICandft),they were determined by regression analysis of the test data based on Eq.(9).Theoretically,the determination effect of the material parameters usingae,minandae,maxis equivalent to that using allaespecimens.
The change in specimen depthWor initial crack lengtha0can cause a change inae.Therefore,when the simplified two-point method is applied,the selection of α (a0/W) is more critical.
Three typical cases with different loading types andS/Wwere selected:3-p-b,S/W=4;3-p-b,S/W=2.5;4-p-b,S/W=4.The theoretical relationship betweenaeand α is shown in Fig.4.aefirst increases and then decreases with an increase in α.Also,α corresponding to the maximum structural geometry parametersae,maxin the 3 cases above are 0.22,0.23,0.25,respectively.α corresponding to the minimum structural geometry parametersae,minis 0 and 1.
In this study,three-point-bending (3-p-b) and four-pointbending (4-p-b) fracture tests on non-geometrically similar small-size rock specimens with a depth ofW=40 mm,and threepoint-bending (3-p-b) fracture tests on large-size rock specimens withW=425 mm were carried out.These were then combined with the geometrically similar specimens fracture tests performed by other scholars to verify the rationality and applicability of the proposed model and method.
3.1.Experimental study
3.1.1.Material
The rock material used in this study is commercial granite plates(sesame ash granite)acquired in Shanghai,China.Its related typical technical properties are listed in Table 1.The statistical analysis results of the grain size of the granite are shown in Fig.5.The grain size distribution range of the granite is 1.5–5 mm,mainly concentrated in the 1.5–3 mm range and accounting for 89.29% of the statistical sample.The average granite sizegavis 2.39 mm.
Table 1Technical properties of the sesame ash granite.
Fig.5.Sesame Ash Granite grain size statistics.
Fig.6.Design of specimen type and size.
3.1.2.Specimen type and size
Three series of specimens were designed for this experiment.The specimen types and sizes are shown in Fig.6,whereLis the length of the specimen;Sthe effective length;Wthe depth;Bthe thickness;a0the initial crack length;and α the ratio of the initial crack lengtha0and depth of specimenW.
The first series are non-geometrically similar three-pointbending (3-p-b) specimens.Specimens size:W=40 mm,S=160 mm,B=30 mm.The initial crack lengtha0was set as follows:1,2,4,8,10,12,16,20,24,26,or 28 mm.α(a0/W)was set as follows:0.025,0.05,0.1,0.2,0.3,0.4,0.5,0.6,or 0.7.Four specimens were manufactured for each α (a0/W).The second series consisted of non-geometrically similar four-point-bending (4-p-b) specimens.Specimens sizes wereW=40 mm,S=160 mm,T=30 mm,andB=30 mm.The initial crack lengtha0was set as follows: 1,2,4,8,10,12,16,20,24,26,or 28 mm.α(a0/W)was set as follows:0.025,0.05,0.1,0.2,0.3,0.4,0.5,0.6,or 0.7.Four specimens were manufactured for eachα(a0/W).The third series consisted of large-size threepoint-bending (3-p-b) test specimens.Specimen sizes wereW=425 mm,S=850 mm,B=30 mm.α was set to 0.2 and 4 specimens were manufactured for testing.The notches of all specimens were prepared using a rock cut machine with a 2 mm-thick diamond saw,and the notches tip radius is about 1 mm.
The actual measured specimen sizes are listed in Tables 2–4.The specimen depthWratio of the third series to the first and sec-ond series was 11:1,and the volume ratio reached 56:1.The first and second series of specimens can be regarded as the laboratory level,while the third series of specimens is the actual structural level.
Table 2Details of 3-p-b small-size granite specimens.
Table 3Details of 4-p-b small-size granite specimens.
Table 4Details of 3-p-b large-size granite specimens.
3.1.3.Test method
The three series of specimens were tested on an SHT4605 electro-hydraulic servo universal testing machine with a measuring range of up to 600 kN.A constant load rate of 10 and 100 N/s was used for small-size specimens (W=40 mm) and large-size specimens (W=425 mm),respectively.The peak loadPmaxof each specimen was recorded during the test.The actual loading situation is shown in Fig.7,and the small-size specimens before and after the tests are shown in Fig.8.
Fig.7.Actual loading diagram of specimens.
Fig.8.Small-size granite specimens before and after testing (W=40 mm).
3.2.Determination of KIC and ft of rock using non-geometrically similar specimens of this test
3.2.1.Determination of KIC and ft of rock using 3-p-b specimens
As shown in Table 2 and Fig.6,there are 11 kinds of α(a0/W)for non-geometrically similar 3-p-b small-size specimens(W=40 mm)with a varied range of 0.025–0.7.The variation range of the structural geometry parameteraeof the specimens is 0.84–2.56 mm.The structural geometry parameteraeincreases with an increase in α when α=0.025–0.25 from 0.84 to 2.56 mm.Conversely,aedecreases with an increase in α when α=0.3–0.7 from 2.47 to 1.15 mm.Therefore,for the structural geometry parameteraeof 3-p-b small-size specimens,the minimum valueae,minis 0.84 mm and the maximum valueae,maxis 2.56 mm.For the 3-pb specimens with α values of 0.025–0.25,although the number of specimens and the number of α sets are relatively small,the variation range ofaeis relatively large and changes about 3 times,covering the variation range ofaecorresponding to α=0.025–0.7.Conversely,for the 3-p-b specimens with α=0.3–0.7,although the number of specimens and the number of α sets are relatively large,the variation range ofaeis relatively tiny and changes about 2 times.For rock specimens with onlyae,minandae,max,aealso has the most extensive variation range.
Four different 3-p-b specimen combinations were used to study the influences of the number of specimens and the variation range ofaeon the determination of rock material parameters (fracture toughnessKICand tensile strengthft).Combination 1:ae=0.84–2.56 mm,α=0.025–0.7(11 sets of α),a total of 44 specimens;Combination 2:ae=0.84–2.56 mm,α=0.025–0.25(5 sets of α),a total of 20 specimens;Combination 3:ae=1.15–2.47 mm,α=0.3–0.7(6 sets of α),a total of 24 specimens;Combination 4:ae,min=0.84 mm (2 specimens) andae,max=2.56 mm (2 specimens),a total of 4 specimens.
Combination 4 is the method proposed in this study,consisting of theae,minandae,max.Combinations 1–3 belong to the previous method,including multiple sets of specimens.Again,the goal is to minimize the number of needed specimens to determine reasonable material parameters while reducing the test workload.
Considering that the relative size of 3-p-b specimens in this test is small,i.e.,W/gav=16.74 (W=40 mm,gav=2.39 mm),then the discrete number β corresponding to the fictitious crack growth length Δaficat peak loadPmaxwas taken as 1–2.To study the influence of Δafic=0(without considering fictitious crack growth length)on the determination of material parameters,for Combination 1,β was taken as 0,1,1.5,and 2.For Combinations 2 to 4,β was taken as 1,1.5,and 2.The rock material parameters determined by regression analysis are shown in Fig.9 and Table 5.
As shown in Fig.9 and Table 5,for the specimen Combination 1,the determined rock material parameters areKIC=2.11 MPa∙m1/2andft=19.61 MPa when β=0,andKIC=2.11–2.58 MPa∙m1/2andft=11.55–14.59 MPa when β=1–2.The determined rock tensile strengthftis relatively large when β=0 (no fictitious crack growth length Δaficat peak loadPmax).
For specimen Combination 2,KIC=1.91–2.39 MPa∙m1/2andft=12.60–13.74 MPa when β=1–2.For specimen Combination 3,aechanges about 2 times,KIC=1.91–2.58 MPa∙m1/2andft=12.6–15.62 MPa when β=1 and 1.5.However,KICandftcannot be obtained through regression analysis when β=2.For specimen Combination 4,aechanges about 3 times.When β=1–2,the rock material parametersKIC=1.69–2.11 MPa∙m1/2andft=11.95–13.11 MPa were determined using only four specimens.The values ofKICandftare consistent with those determined using Combination 1(44 specimens)or Combination 2(20 specimens).Therefore,for 3-p-b rock specimens,only two sets (ae,max:ae,min=3:1) of specimens are needed,and the rock material parametersKICandftcan be determined based on regression analysis.It can be seen from Table 5 that the results of the rock material parameters are essentially consistent when β=1–2.Therefore,for the convenience of application,β can be taken as 1.
3.2.2.Determination of KIC and ft of rock using 4-p-b specimens
As shown in Table 3 and Fig.6,there are 11 kinds of α(a0/W)for non-geometrically similar 4-p-b small-size specimens(W=40 mm),with a varied range of 0.025–0.7.The variation range of the structural geometry parameteraeof the specimens is 0.86–2.92 mm.The structural geometry parameteraeincreases from 0.86 to 2.91 mm with an increase in α when α=0.025–0.2.When α=0.25–0.7,aegenerally tends to decrease with the increase of α,from 2.92 to 1.20 mm.Due to the error of specimen production,the specimen with the maximum structural geometry parameter corresponds to α of 0.3.Therefore,for the 4-p-b specimens,ae,min=0.86 mm,andae,max=2.92 mm.For 4-p-b specimens with α=0.025–0.2,although the number of specimens and the number of α sets are relatively small,the variation range ofaeis relatively large and changes about 3.4 times,basically covering the variation range ofaecorresponding to α=0.025–0.7.Conversely,for the 4-p-b specimens with α=0.25–0.7,although the number of specimens and the number of α sets are relatively large,the variation range ofaeis relatively small and changes about 2.4 times.Similarly,rock specimens with onlyae,minandae,max,aealso have the most extensive variation range.
Table 5Rock material parameters (KIC and ft) determined using different 3-p-b specimen combinations.
Furthermore,for the 4-p-b specimens,to study the influence of the number of specimens and the variation range ofaeon the determination of rock material parameters (fracture toughnessKICand tensile strengthft),4 different specimen combinations were used to determine the rock material parameters,Combinations 5 to 8.Combination 5:ae=0.86–2.92 mm,α=0.025–0.7 (11 sets of α),a total of 44 specimens;Combination 6:ae=0.86–2.91 mm,α=0.025–0.2 (4 sets of α),a total of 16 specimens;Combination 7:ae=1.20–2.92 mm,α=0.25–0.7 (7 sets of α),a total of 28 specimens;Combination 8:ae,min=0.86 mm (2 specimens) andae,max=2.92 mm (2 specimens),a total of 4 specimens.
Fig.9.Determination of rock material parameters (KIC and ft) using different 3-p-b specimen combinations.
For Combination 1,β was taken as 0,1,1.5,and 2.For Combinations 2 to 4,β was taken as 1,1.5,and 2.The rock material parameters determined by regression analysis are shown in Fig.10 and Table 6.
As shown in Fig.10 and Table 6,for specimen Combination 5,the determined rock material parameters areKIC=2.11 MPa∙m1/2andft=17.96 MPa when β=0,andKIC=2.11–2.39 MPa∙m1/2andft=10.85–13.48 MPa when β=1–2.Overall,the tensile strengthftis relatively large when β=0(no fictitious crack growth length Δaficat peak loadPmax),and the values ofKICandftare almost consistent with the results determined by all 3-p-b specimens (Combination 1) when β=1–2.
For specimen Combination 6,KIC=1.83–2.24 MPa∙m1/2andft=11.70–12.80 MPa when β=1–2.For specimen Combination 7,aechanges about 2.4 times.KIC=1.91–2.39 MPa∙m1/2andft=11.87–14.59 MPa when β=1 and 1.5;KIC=4.47 MPa∙m1/2andft=10.10 MPa when β=2;the fracture toughnessKIC(β=2) is relatively large for Combination 7.For specimen Combination 8,aechanges about 3.4 times.When β=1–2,the rock material parametersKIC=1.75–2.24 MPa∙m1/2andft=11.87–13.02 MPa was determined using only 4 specimens.These values ofKICandftare consistent with those determined using Combination 5 (44 specimens) or Combination 6 (16 specimens).Therefore,for 4-p-b rock specimens,only two sets(ae,max:ae,min≥3:1)of specimens need to be used,and the rock material parametersKICandftcan be determined using regression analysis.It can be seen from Table 6 that the results of the rock material parameters are essentially consistent when β=1–2.Therefore,for the convenience of application,β can be taken as 1.
3.2.3.Determination of KIC and ft of rock using 3-p-b and 4-p-b specimens simultaneously
For the combination of different specimen types(3-p-b and 4-pb),to study the influence of the number of specimens and the variation range ofaeon the determination of rock material parameters(fracture toughnessKICand tensile strengthft),6 different specimen combinations were used to determine the rock material parameters,Combinations 9 to 14.Combination 9:ae=0.84–2.92 mm (3-p-b,α=0.025–0.7;4-p-b,α=0.025–0.7),a total of 88 specimens;Combination 10:ae=0.84–2.91 mm (3-p-b,α=0.025–0.25;4-p-b,α=0.025–0.2),a total of 36 specimens;Combination 11:ae=1.15–2.92 mm(3-p-b,α=0.3–0.7;4-p-b,α=0.25–0.7),a total of 52 specimens;Combination 12:ae=0.84–2.91 mm (3-p-b,α=0.025,0.25;4-p-b,α=0.025,0.2),a total of 16 specimens;Combination 13:ae,min(3-p-b,2 specimens) andae,max(4-p-b,2 specimens),a total of 4 specimens;Combination 14:ae,min(4-pb,2 specimens) andae,max(3-p-b,2 specimens),a total of 4 specimens.
Combinations 13 and 14 are the methods proposed in this study,using only 4 samples and the maximum and minimumae.For Combination 9,β was taken as 0,1,1.5,and 2.For Combinations 10 to 14,β was taken as 1,1.5,and 2.The rock material parameters determined by regression analysis are shown in Fig.11 and Table 7.
As shown in Fig.11 and Table 7,the determined rock material parameters for the specimen Combination 9 areKIC=2.00 MPa∙m1/2andft=18.90 MPa when β=0,andKIC=2.00–2.24 MPa∙m1/2andft=11.40–14.29 MPa when β=1–2.The tensile strengthftis relatively large when β=0 (no fictitious crack growth length Δaficat peak loadPmax),and the values ofKICandftare almost consistent with the results determined by all 3-p-b specimens (Combination 1) or all 4-p-b specimens (Combination 5) when β=1–2.
For specimen Combination 10,KIC=1.83–2.24 MPa∙m1/2andft=12.22–13.36 MPa when β=1–2.For specimen Combination 11,aechanges about 2.4 times,KIC=1.75–2.11 MPa∙m1/2andft=12.60–15.62 MPa when β=1 and 1.5,KIC=3.16 MPa∙m1/2andft=10.66 MPa when β=2.Overall,the fracture toughnessKIC(β=2)is relatively large.For specimen Combination 12,aechanges about 3.4 times.When β=1–2,the rock material parametersKIC=1.91–2.39 MPa∙m1/2andft=11.87–13.02 MPa.
For the method proposed in this study,Combination 13:KIC=1.75–2.24 MPa∙m1/2andft=11.95–13.02 MPa;Combination 14:KIC=1.75–2.11 MPa∙m1/2andft=11.95–13.02 MPa.Overall,the values ofKICandftfor both combinations are essentially consistent with those determined using Combination 5 (44 specimens) or Combination 6 (16 specimens).Therefore,for different specimen types (3-p-b and 4-p-b),only two sets of specimens (4 specimens total) withae,max:ae,min≥3:1 are required,and then rock material parameterKICandftcan be determined by regression analysis.
Through the comprehensive comparison of Tables 5–7,based on the methods proposed in this study(ae,minandae,max),whether it is a 3-p-b (Combination 4) or 4-p-b (Combination 8) type specimen or a combination of 3-p-b and 4-p-b(Combination 13 and 14)type specimen,the determined material parametersKICandftare consistent when the discrete number β=1–2.Therefore,the method proposed in this study is not affected by specimen types (3-p-b and 4-p-b).
Table 6Rock material parameters (KIC and ft) determined using different 4-p-b specimen combinations.
3.3.Determination of KIC and ft of rock using geometrically similar specimens in other literature
3.3.1.Determination of KIC and ft of rock using geometrically similar specimens in literature [8]
Tarokh et al.[8]conducted 3-p-b fracture tests on geometrically similar rock specimens,consisting of Rockville Granite with an average grain sizegavof 10 mm.The rock specimens used in the test have 4 depths,W=50.8,101.6,203.2,and 406.4 mm.The ratio of the effective length of specimenSand depth of specimenW(S/W)was 2.5.The specimen thicknessBwas 30 mm.The ratio of the initial crack lengtha0and depth of specimenW,α=a0/W=0.2,and relative sizeW/gav=5–41.
Fig.10.Determination of rock material parameters (KIC and ft) using different 4-p-b specimen combinations.
The structural geometry parameteraeof the rock specimens with 4 depths (W=50.8,101.6,203.2 and 406.4 mm) were 2.74,5.47,10.95,21.89 mm,respectively,andae,min=2.74 mm,ae,max=21.89 mm.To study the influence of the number of specimens on the determination of rock material parameters (fracture toughnessKICand tensile strengthft),two different geometrically similar specimen combinations were used to determine the rock material parameters.Combination 1:ae=2.74–21.89 mm(W=50.8,101.6,203.2 and 406.4 mm),a total of 8 specimens;Combination 2:ae,min(2.74 mm,W=50.8 mm,2 specimens) andae,max(21.89 mm,W=406.4 mm,2 specimens),a total of 4 specimens.
Combination 2 is the method in this paper.For specimen Combination 1 and Combination 2,β was taken as 1,1.5,and 2.The rock material parameters determined by regression analysis are shown in Fig.12 and Table 8.
As shown in Fig.12 and Table 8,the rock material parameters determined by specimen Combination 1 (8 specimens) areKIC=2.00–2.24 MPa∙m1/2andft=8.03–11.32 MPa (β=1–2).The rock material parameters determined by specimen Combination 2 (4 specimens,i.e.,the method proposed here) areKIC=2.11–2.58 MPa∙m1/2andft=7.20–10.31 MPa (β=1–2).The rock material parametersKICandftdetermined by the 2 specimen combinations are consistent with the results ofKIC=1.89 MPa∙m1/2andft=9.81-MPa as determined by the size effect of the model given in the literature [8].
3.3.2.Determination of KIC and ft of rock using geometrically similar specimens in literature [2]
Bazˇant et al.[2]conducted 3-p-b fracture tests on geometrically similar rock specimens of Indiana limestone with a grain size of 0.1–1.5 mm.The rock specimens used in the test had four different depths,W=12.7,25.4,50.8,and 101.6 mm.The ratio of the effective length of specimenSand depth of specimenW (S/W) was 4.The specimen thicknessBwas 12.7 mm.The ratio of the initial crack lengtha0and depth of the specimenW(α=a0/W)was 0.4.
The structural geometry parameteraeof the rock specimens with the 4 different depths (W=12.7,25.4,50.8 and 101.6 mm)were 0.75,1.44,2.93,and 5.86 mm,respectively,whereae,min=0.75 mm andae,max=5.86 mm.To study the influence of the number of specimens on the determination of the rock material parameters(fracture toughnessKICand tensile strengthft),two different geometrically similar specimen combinations were used to determine the rock material parameters.Combination 1:ae=0.75–5.86 mm (W=12.7,25.4,50.8 and 101.6 mm),a total of 12 specimens;Combination 2:ae,min(0.75 mm,W=12.7 mm,3 specimens) andae,max(5.86 mm,W=101.8 mm,3 specimens),a total of 6 specimens.
Combination 2 is the method proposed in this paper.For specimen Combination 1 and Combination 2,β was taken as 1,1.5,and 2.The rock material parameters determined by regression analysis are shown in Fig.13 and Table 9.
As shown in Fig.13 and Table 9,the rock material parameters determined by the specimen Combination 1 (12 specimens) areKIC=1.04–1.24 MPa∙m1/2andft=6.36–7.86 MPa (β=1–2).Based on the method proposed in this paper,the rock material parameters determined by specimen Combination 2 (6 specimens total) areKIC=1.03–1.26 MPa∙m1/2andft=6.26–8.11 MPa (β=1–2).The rock material parametersKICandftdetermined by the two specimen combinations are consistent with the results ofKIC=0.969–1.13 M Pa∙m1/2andft=6.5 MPa as determined based on the size effect model given in the literature [2].
Table 7Rock material parameters (KIC and ft) determined using different specimen combinations with 3-p-b and 4-p-b.
Table 8Rock material parameters (KIC and ft) determined using different geometrically similar specimen combinations from the literature [8].
Table 9Rock material parameters (KIC and ft) determined by using different geometrically similar specimen combinations from the literature [2].
3.4.Summary of the determination of rock material parameters based on the two-point method
From the above analysis,for non-geometrically similar rock specimens with relative sizeW/gav<20,whether 3-p-b specimens,4-p-b specimens,or a combination of the two specimen types,there only needs to be two sets of small-size specimens (4 specimens in total) with structural geometry parameter ratio ofae,max:ae,min≥3:1.Based on the improved boundary effect model considering the influence of rock average grain sizegav,the fracture toughnessKICand tensile strengthftof rock can be determined simultaneously through regression analysis.Similarly,for geometrically similar specimens,only two sets of specimens withae,max:ae,min≥3:1 are needed to obtain reasonable rock material parametersKICandftbased on the improved boundary effect model.The method proposed in this study is not affected by specimen type (3-p-b and 4-p-b) or geometrical type (non-geometric similarity and geometrical similarity).For the convenience of application,the discrete number β can be taken as 1.
If the material parameters (fracture toughnessKICand tensile strengthft) are determined,the full structural failure curve can be established based on Eq.(9).Substituting the determinedKICandftinto Eq.(9),the relationship between the nominal stress σnand the structural geometry parameteraecan be obtained.Takingaeas the abscissa,σnas the ordinate,and the coordinate in logarithmic form,the full structural failure curve can be established according to the relationship between σnandae.
To describe the discreteness of the test data,the determined material parametersKICandftwere set with upper and lower limits(for example,±20% upper and lower limits);here,the upper limit was 1.2KIC,1.2ft,and the lower limit was 0.8KIC,0.8ft.In addition,by substituting the determined material parametersKICandftinto Eq.(3),the characteristic crackcan be obtained.Furthermore,according to the ratios(tensile strength control),(quasi-brittle fracture)and(fracture toughness control),the fracture region of a specimen or structure can be identified [34,37,38].
Fig.11.Determination of rock material parameters (KIC and ft) using different specimen combinations with 3-p-b and 4-p-b.
Fig.11 (continued)
4.1.Full structural failure curve of rock constructed by 3-p-b specimens
Based on the method proposed in this study,the fracture toughnessKICand tensile strengthftof rock were determined by two sets of small-size 3-p-b specimens (Combination 4 above) with structural geometry parameter ratioae,max:ae,min≥3:1.The full structural failure curve of rock constructed using the rock material parametersKIC=2.11 MPa∙m1/2andft=13.13 MPa(β=1)determined by the specimen Combination 4 is shown in Fig.14.
As shown in Fig.14,the solid red symbols represent the two sets of test data points(ae,minandae,max)of specimen Combination 4.The full structural failure curve of rock was constructed from these two sets of data points.The open symbols represent test data points that were not used to construct the full structural failure curve of rock.The yellow open symbols are the small-size threepoint-bending (3-p-b) rock specimens data points (40 test data points).The blue open symbols are the large-size 3-p-b rock specimens(4 test data points).The small-size 3-p-b rock specimens are highly heterogeneous based upon their small relative size(W/gav=16.74),and the test data points show a certain degree of dispersion,but they are essentially within the ±20% upper and lower limits of the curve.For large-size 3-p-b rock specimens(W=425 mm),the full structural failure curve constructed by the two sets of the small-size 3-p-b specimens accurately predict its fracture failure.The test data points are all within the ±10% upper and lower limits of the curve.In addition,the small-size 3-p-b specimens and the large-size 3-p-b specimens in this test are in quasi-brittle fracture region (0.1 <<10).
4.2.Full structural failure curve of rock constructed using 4-p-b specimens
Based on the method proposed in this study,the fracture toughnessKICand tensile strengthftof rock were determined using the two sets of small-size 4-p-b specimens (Combination 8) with a structural geometry parameter ratio ofae,max:ae,min≥3:1.The fullstructural failure curve of a rock constructed using the rock material parametersKIC=2.24 MPa∙m1/2andft=13.02 MPa(β=1)is shown in Fig.15.
Fig.12.Determination of rock material parameters (KIC and ft) using different geometrically similar specimen combinations from the literature [8].
Fig.13.Determination of rock material parameters (KIC and ft) by using different geometrically similar specimen combinations from the literature [2].
As shown in Fig.15,the solid red symbols represent the two sets of test data points(ae,minandae,max)of specimen Combination 8,and the full structural failure curve of rock is constructed from these two sets of data points.The open symbols represent test data points that were not used to construct the full structural failure curve of rock.The yellow open symbols are the small-size 4-p-b rock specimens data points (40 test data points).The blue open symbols are the test data points of large-size three-point bending rock specimens(4 test data points).The small-size 4-p-b rock specimens are highly heterogeneous because of their small relative size(W/gav=16.74).The test data points also show a certain degree of dispersion,but they are essentially within the ±20% upper and lower limits of the curve.For the large-size 3-p-b rock specimens(W=425 mm),the full structural failure curve of rock constructed by two sets of small-size 4-p-b specimens accurately predicts its fracture failure.Furthermore,the test data points are all within the ±10% upper and lower limits of the curve.Finally,the smallsize 4-p-b specimens and the large-size 3-p-b specimens in this test are in quasi-brittle fracture region (0.1
Fig.14.Full structural failure curve of rock constructed using specimen Combination 4.
Fig.15.Full structural failure curve of rock constructed by specimen Combination 8.
4.3.Full structural failure curve of rock constructed by 3-p-b and 4-p-b specimens
Based on the method proposed in this study,the fracture toughnessKICand tensile strengthftof the rock were determined by two sets of specimens (Combinations 13 and 14) with different specimen types (3-p-b and 4-p-b).The rock material parameters determined by Combinations 13 and 14 were used to construct the full structural failure curves of rock,as shown in Fig.16.The full structural failure curve of rock constructed with the rock material parametersKIC=2.24 MPa∙m1/2andft=13.02 MPa (β=1)determined by specimen Combination 13 is shown in Fig.16a.The full structural failure curve of rock constructed with rock material parametersKIC=2.11 MPa∙m1/2andft=13.02 MPa (β=1) determined by specimen Combination 14 is shown in Fig.16b.
As shown in Fig.16,the solid red symbols represent the two sets of test data points(ae,minandae,max)of specimen Combination 13 or 14,and the full structural failure curve of rock is constructed from these two sets of data points.The open symbols represent test data points that were not used to construct the full structural failure curve of rock.The yellow open symbols are the small-size rock specimens (3-p-b and 4-p-b,total 84 test data points).The blue open symbols are the large-size 3-p-b rock specimens (4 test data points).The small-size rock specimens are also highly heterogeneous,and the test data points show a certain degree of dispersion,but they are basically within the ±20% upper and lower limits of the curve.For the large-size 3-p-b rock specimens (W=425 mm),the full structural failure curve of rock constructed by the two sets of small-size specimens with different specimen types accurately predict its fracture failure,and the test data points are all within the ±10% upper and lower limits of the curve.In addition,the small-size specimens and the large-size 3-p-b specimens in this test are in quasi-brittle fracture region (0.1
4.4.Full structural failure curve of rock constructed by geometrically similar specimens in literature [8]
Based on the method proposed in this study,the fracture toughnessKICand tensile strengthftof rock were determined using only two sets of specimens (Combination 2) in literature [8].The full structural failure curve of rock constructed using the rock material parametersKIC=2.11 MPa∙m1/2andft=10.31 MPa (β=1) is determined by the geometrically similar specimen Combination 2 in literature [8] is shown in Fig.17.
Fig.16.Full structural failure curve of rock constructed using different specimen types (3-p-b and 4-p-b).
As shown in Fig.17,the solid symbols represent the two sets of test data points (ae,minandae,max),and the full structural failure curve of rock was constructed from these two sets of data points.The open symbols represent test data points that were not used to construct the full structural failure curve of rock (4 test data points).It can be seen from Fig.17 that the full structural failure curve of rock constructed by the 4 specimens can accurately predict the fracture failure of the specimens,and the ±20% upper and lower limits cover all the test data points (8 test data points).The geometrically similar specimens used are all in the quasibrittle fracture region (0.1<<10).
4.5.Full structural failure curve of rock constructed by geometrically similar specimens in literature [2]
Based on the method proposed in this study,the fracture toughnessKICand tensile strengthftof rock were determined using two sets of specimens (Combination 2) from the literature [2].The full structural failure curve of a rock constructed using the rock material parametersKIC=1.12 MPa∙m1/2andft=7.05 MPa (β=1) determined by the geometrically similar specimen Combination 2 from the literature [2] is shown in Fig.18.
Fig.17.Full structural failure curve of rock constructed using the geometrically similar specimen Combination 2 from the literature [8].
As shown in Fig.18,the solid symbols represent the two sets of test data points (ae,minandae,max),and the full structural failure curve of rock was constructed from these two sets of data points.The open symbols represent test data points that were not used to construct the full structural failure curve of rock (6 test data points).It can be seen from Fig.18 that the full structural failure curve of rock constructed by the 6 specimens accurately predicts the fracture failure of specimens,and the ±10% upper and lower limits cover all the test data points (12 test data points).The geometrically similar specimens used are all in the quasi-brittle fracture region (0.1<<10).
For non-geometrically similar specimens,the specimen depthWis a constant,and it can be seen from Eq.(4) that the structural geometry parameteraeis a function of α (a0/W).If the material parameters (fracture toughnessKICand tensile strengthft) are determined,the determination relation of peak loadPmaxand α(a0/W) of the 3-p-b specimens can be obtained by combining Eqs.(4),(9) and (10).Similarly,by combining Eqs.(4),(9),and (11),we can obtain the determining relation ofPmaxand α of the 4-pb specimens.Based on the determined relationship,the relationship curve betweenPmaxand α can predict the peak loadPmaxof specimens or structures.Based on theKICandft(β=1) determined by the proposed method (ae,minandae,max),the peak loadsPmaxof small non-geometrically similar rock specimens (W=40 mm) with different loading types (3-p-b and 4-p-b) were predicted.
Fig.18.Full structural failure curve of rock constructed using geometrically similar specimen Combination 2 from the literature [2].
5.1.Peak load prediction based on 3-p-b specimens
Based on the method proposed in this paper,the peak load prediction curves of small-size rock specimens(W=40 mm)were constructed from the rock material parametersKIC=2.11 MPa∙m1/2andft=13.13 MPa (β=1) determined by two sets of small-size 3-p-b specimens (Combination 4),as shown in Fig.19.
Fig.19.Peak load prediction curves of small-size rock specimens constructed by specimen Combination 4.
Fig.20.Peak load prediction curves of small-size rock specimens constructed by specimen Combination 8.
As shown in Fig.19,the solid red symbols represent the two sets of test data points (ae,minandae,max,four test data points) of specimen Combination 4,and the peak load prediction curve was constructed from these two sets of data points.The open symbols represent test data points that were not used to construct the peak load prediction curve.The peak load prediction curves of the 3-p-b and 4-p-b specimens are solid and dash lines,respectively.The small-size rock specimens are heterogeneous,as noted by their relatively small size (W/gav=16.74).The test results ofPmaxshow a certain degree of dispersion,but ±15% upper and lower limits of the prediction curve essentially cover all test data points (88 test data points).
5.2.Peak load prediction based on 4-p-b specimens
Based on the method proposed in this paper,the peak load prediction curves of small-size rock specimens(W=40 mm)were constructed from the rock material parametersKIC=2.24 MPa∙m1/2andft=13.02 MPa (β=1) determined using two sets of small-size 4-p-b specimens (Combination 8),as shown in Fig.20.
As shown in Fig.20,the solid red symbols represent the two sets of test data points(ae,minandae,max,4 test data points)of specimen Combination 8.The peak load prediction curve was constructed from these two sets of data points.The open symbols represent test data points that were not used to construct the peak load prediction curve.The peak load prediction curves of the 3-p-b and 4-p-b specimens are solid and dash lines,respectively.For 3-pb and 4-p-b small-size specimens,±15% upper and lower limits of the prediction curve essentially encompass all test data points(88 test data points).
5.3.Peak load prediction based on 3-p-b and 4-p-b specimens
Based on the method proposed in this study,the rock material parameters determined by Combinations 13 and 14 were used to construct the peak load prediction curves of small-size rock specimens (W=40 mm),as shown in Fig.21.The peak load prediction curves constructed with rock material parametersKIC=2.24-MPa∙m1/2,ft=13.02 MPa (β=1) determined by specimen Combination 13 are shown in Fig.21a.The peak load prediction curves constructed with rock material parametersKIC=2.11 MPa∙m1/2,ft=13.02 MPa (β=1) determined by specimen Combination 14 are shown in Fig.21b.
Fig.21.Peak load prediction curves of small-size rock specimens constructed using different specimen types (3-p-b and 4-p-b).
As shown in Fig.21a,the solid red symbols represent the two sets of test data points(ae,minandae,max,4 test data points)of specimen Combination 13.The peak load prediction curve was constructed from these two sets of data points.The open symbols represent test data points that were not used to construct the peak load prediction curve.The peak load prediction curves of the 3-p-b and 4-p-b specimens are solid and dash lines,respectively.For 3-pb and 4-p-b small-size specimens,±15% upper and lower limits of the prediction curves cover all the test data points (88 test data points) essentially.
As shown in Fig.21b,the solid red symbols represent the two sets of test data points(ae,minandae,max,4 test data points)of specimen Combination 14.The peak load prediction curve was constructed from these two sets of data points.For 3-p-b and 4-p-b small-size specimens,±15% upper and lower limits of the prediction curve cover essentially all test data points (88 test data points).
In this paper,the structural geometry parameteraeis an essential parameter for studying the size effect on rock material parameters (fracture toughnessKICand tensile strengthft),and a simplified two-point method was developed to determineKICandftof rock.Through a series of fracture tests and analysis of published test data,the rationality and applicability of the proposed model and method were verified.The conclusions are as follows:
(1) This study focused on the influence of relative size,using the structural geometry parameteraeto consider the coupling effect of the specimen depthW,the initial cracka0,and the front and back boundaries of the specimen.These are different from the vital parameter of the existing size effect model—the absolute sizeW.The influence of the different variation ranges of the structural geometry parameteraeon the determination of the rock material parameters was systematically analyzed by comparing the variation range of the determined material parameters.It is recommended to use rock non-geometrically similar specimens withae,max:ae,min≥3:1 as the design specimens for determining the true material parameters of rock.It is different from theWmax:Wmin≥ 4:1 geometrically similar specimens required by the existing size effect model.
(2) The method proposed (ae,maxandae,min) in this paper is not affected by the specimen type and geometry.For smallsize rock specimens in this test,whether 3-p-b specimens,4-p-b specimens or a combination of the two specimen types,reasonable rock material parametersKICandftcan be determined using only two sets specimens withae,max:ae,min≥3:1,and the results of the above three cases are consistent.Whether non-geometrically similar specimens or geometrically similar specimens,reasonable rock material parametersKICandftwere determined using only two sets specimens withae,ma:ae,min≥3:1,which is consistent with the results determined using all or multiple sets of specimens.
(3) Based on the method proposed (ae,maxandae,min) in this paper,whether for non-geometrically similar specimens or geometrically similar specimens,the full structural failure curve of rock can be constructed from two sets of specimens.For the non-geometrically similar specimens in this test,the full structural failure curve of a rock was constructed from two sets of small-size rock specimens (W=40 mm) withae,max:ae,min≥3:1.The ±20% upper and lower limits of the curve encompass the test data points of small-size specimens (W=40 mm),and the ±10% upper and lower limits can cover the test data points of large-size specimens(W=425 mm).
(4) Based on the method proposed in this paper (ae,maxandae,min),whether 3-p-b specimens,4-p-b specimens or a combination of the two specimen types,the peak load prediction curves constructed by two sets of specimens accurately predict the peak loads of small-size rock specimens(W=40 mm).The±15%upper and lower limits of the prediction curve can encompass the vast majority of the data points.
Acknowledgements
This research was supported by National Natural Science Foundation of China(No.52179132);Program for Science&Technology Innovation Talents in Universities of Henan province (No.20HASTIT013);and Sichuan University,State Key Lab Hydraul&Mt River Engn (No.SKHL2007).
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