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英文原文Experimental, numerical and analytical studies on tensile strength of rocksNazife Erarslan , DavidJohnWilliamsGolderGeomechanics Centre, School of Civil Engineering, TheUniversity of Queensland, Brisbane, Qld 4072, AustraliaAbstract: The difculties associated with performing a direct uniaxial tensile test on a rock specimen have led toa number of indirect methods for assessing the tensile strength. This study compares experimentalresults of direct and indirect tensile tests carried out on three rock types: Brisbane tuff, granite andsandstone. Thestandard Brazilian indirect tensile testcaused catastrophic crushing failure of the diskspecimens, due to the stress concentration produced by the line loading applied and exacerbated by thebrittleness of the rock tested, rather than the expected tensile splitting failure initiated by a centralcrack. This nding led to an investigation of the effect of loading conditions on the failure of Braziliandisk specimens using three steel loading arcs of different angle applied to three different rock types,using numerical modeling and analytical results. Numerical modeling studies were also performed toinvestigate theeffect of a pre-existing crack on the stress distribution within Brazilian disk specimens.It was found that there is substantially higher tensile stress concentration at the center of the disk witha pre-existing crack compared with that for a disk without a pre-existing crack. The maximum stressintensity factor (fracture toughness) values at the tip of the central pre-existing cracks were determinedfrom numerical modeling and compared with fracture toughness values obtained experimentally forthe three rock types. It was concluded that a 20loading arc gives the best estimate of the indirecttensile strength.Keywords:Brazilian test;Direct tensile strength of rock;FRANC2D;Indirect tensile strength of rock;Pre-existing crack.1IntroductionThe difculties associated with performing a direct uniaxialtensile test on a rock specimen have led to a number of indirectmethods for assessing the tensile strength. In 1978, the Braziliantest was ofcially proposed by the International Society for RockMechanics (ISRM) as a suggested method for determining thetensile strength of rock materials 1. The Brazilian test, orsplitting tension test, isperformed by applying a concentratedcompressive load across the diameter of a disk specimen. TheBrazilian test is also a suggested method for determining thetensile strength of concrete materials.The Brazilian test has been criticized since itwas initiallyproposed. Fairhurst 2 first discussed the important issue of thevalidity of the Brazilian test. He stated that failure may occuraway from the center of the test disk for small angles of loadingcontact area and also thecalculated tensile strength from aBrazilian test is lower than the true value of thetensile strength.Hondros 3 developed an approach to measure the elasticmodulusand Poissons ratio using a Brazilian disk, and alsoformulated a complete stresssolution for the case of a radialload distributed over a nite circular arc of the disk.Although the Brazilian test has been studied extensively, bothexperimentallyand theoretically, relatively little attention havebeen directed towards researchingthe validity of the test. Severalkey questions remain unresolved: for example, how to guaranteecrack initiation at the center of the specimen (beneath theconcentratedload), how to obtain an accurate representation ofthe tensile strength of the rockfrom the test, and how to obtainclosed-form expressions for the complete stressand strain eldsboth for the Brazilian tests and for the case when the load isapplied as a pressure acting normal to and uniformly across an arcof nite length. Hudson et al. 4 found that the tensile strength ofrock varies considerably whenmeasured by different methodsand that the heterogeneity of the rock tested and the contactcondition between the specimen and the steel platens of thetesting machinewill inuence the tensile strength value obtained.He observed that, In the Brazilian test, it was found that failurealways initiated directly under the loading points if at steelplatens were used to load the specimen, which actually invali-dates the test for the determination of tensile strength. Wanget al. 5 used specimens with two parallel at ends at the loadingpoints to prevent local crack initiation at the loading points. However, they found that the atness and parallelness of the atends are critical for a successful test.The classical theory 1 assumed that the concentrated load isapplied over an innitesimally small width as a line load, butclearly this would lead to stresses of very high intensity. Theactual loads are not concentrated but are distributed over nitearc of the disk. The tensile strength of a rock disk specimen iscalculated using the equation:（1）where P is the failure load, and d and t are the diameter andthickness of the rockdisk, respectively. When a disk is diame-trically compressed under a line load, thestresses at any point (A)in the disk specimen (Fig. 1) are as follows:（2）where the symbols are dened in Fig. 1.According to 6, if a circular cylinder of radius r is compressedacross its diameter between at surfaces which apply concen-trated loads of W per unit axial length of the cylinder (Fig. 1). Ifthe load is applied to the circumference of thecylinder as apressure p distributed over an arc 2 using shaped platens, so that , then equal biaxial compression exists near the contactswith a value of p 6.The distributed load applied to a disk under diametral com-pression is moredifcult to analyze than that of the concentratedload. Hondros 3 analyzed theBrazilian test for the case of a thindisk loaded by a uniform pressure, appliedradially over a shortstrip of the circumference at each end of the disk. Hondros 3obtained the full-eld stresses using the series expansion techni-que (Eq. (4) and applying these solutions to evaluate Youngsmodulus (E) and Poissons ratio (n) of the specimens for theapplied load, and obtain the strains occurring at the center of thedisk specimens.（3）where p is applied pressure, R is the radius of the disk, r and arethe polar coordinates of a point in disk and is the half centralangle of the applied distributed load (Fig. 2). It can be seen fromEq. (3) that the magnitude of affects the stress distributionwithin disk directly. Recently, Ma and Hung 7 continued andextended Hondros work to successfully obtain the analyticalsolution in explicit form with a simple expression, rather than aseries of equations.In general, tensile failures are most likely to start from theboundary of thespecimens during standard Brazilian indirecttensile tests on brittle materials.However, in the tests describedherein, catastrophic crushing failure developed onstandardBrazilian testing of the brittle Brisbane tuff disk specimens.On the other hand, central cracks were obtained, correspondingto the location of the maximumtensile stress, for loadingBrisbane tuff specimens over an arc length. This is associated with the failure being caused by the horizontal tensile stresses inthe diskspecimen. Thus, the objective of this paper is to critiquethe standard Brazilian testby comparing the results obtainedusing this test with those obtained using a loadedarc, togetherwith a comparison of the experimental results, and the results ofanalytical and numerical modeling.Fig. 1. A disk compressed between the parallel line loading under the Brazilianjaws.Fig. 2. A disk specimen subjected to diametric distributed compression.2Experimental study2.1Indirect tension testsA series of Brazilian disk tests was carried out using specimensprepared from Brisbane tuff, sandstone and granite. Most of thetests were carried out on Brisbanetuff, since it is a host rock ofBrisbanes rst motorway tunnel, CLEM7, fromwhich core sam-ples were obtained. Limited tests were carried out on sandstoneand granite to verify that the results could be more generallyapplied than to asingle rock type. Brisbane tuff behaves in analmost linear elastic manner for asignicant portion of its axialstressstrain curve. The test specimens prepared werestandardBrazilian disks with a diameter of 52 mm and thickness of 26 mm(adiameter: thickness ratio of 0.5). The load was applied by a stiffhydraulic Instron loading frame, with a loading rate suggested by ISRM of 200 N/s 1.Fig. 3. (a) Disk between standard Brazilian jaws, (b) steel loading arcs and (c)disk between loading arcs.Table 1Results of indirect tensile tests on Brisbane tuff disk specimensSpecimenRecorded maximum load (kN)Standard Brazilian jaws15Loading arc20Loading arc30Loading arcReplicate 125.0012.5017.0621.10Replicate 216.7716.3919.8224.60Replicate 315.4315.6520.2321.13Replicate 421.0014.7019.4122.17Average19.6014.8119.2022.30Standard deviation4.341.691.641.64Four series of indirect tension tests were conducted, with:(1) standard Brazilian jaws, (2) 15steel loading arcs, (3) 20steelloading arcs and (4) 30steel loading arcs. Up to four repetitionswere carried out. The steel loading arcswere machined fromstandard mild steel, as recommended by ISRM 1 (Fig. 3).The tensile strength of the rock specimens tested using thestandard Brazilianjaws was calculated using the formula given byISRM 1. Since the loadingboundaries of the steel loading arcsare different from that of the standard Brazilianjaws, the formulagiven by ISRM 1 cannot be used to calculate the indirect tensilestrength of specimens under loading arcs. The tensile strength ofthe samples testedunder angled loading arcs was calculated fromEq. (3) to nd the stresses at verynear the center using Hondrosequation 3.The details of the tests and test results are given in Table 1. Themaximumrecorded ultimate load was obtained using a loadingarc with. However, the highest standard deviation of theultimate loads was obtained from the standard Brazilian testresults.请键入文字或网站地址，或者上传文档。In this study, the main focus was to achieve tensile failure dueto a centralcrack along the vertical diameter of the disk, which isassumed to be the region ofmaximum indirect tensile stress5,8,18. However, most experimental studiescarried out usingstandard Brazilian jaws cause cracks to initiate just under theloading points 810. In the present tests on Brisbane tuff,loading with Brazilianjaws caused catastrophic crushing failureof the disk specimens (Fig. 4a). Testingusing 15loading arcscaused a single crack that diverged from the vertical loading axis,together with secondary cracks. Testing using 20loading arcscaused a singlevertical, central crack (Fig. 4c). Testing using 30loading arcs caused an arrested, vertical, central crack (Fig. 4d).To further investigate the role of diametral load in contributingto the determination of indirect tensile strength, sandstone andgranite disk specimens were used beside Brisbane tuff specimens.The granite sample, having a uniaxial compressive strength (UCS)of 210 MPa,was obtained from a quarry at Keperra in Brisbane,while the sandstone sample,having a UCS of 37 MPa, wasobtained from Helidon, near Toowoomba, west ofBrisbane. Onthe other hand, the UCS strength of Brisbane tuff is between thoseworocks, UCS= 90 MPa.2.2Direct tension testsA series of direct tension tests was carried out on Brisbane tuff.The diameterof direct tension test specimens was same as thatfor the indirect tension test specimens. The direct tension testspecimens were cylindrical core samples with adiameter of52 mm and a length of 135 mm (a length:diameter ratio of 2.59).Inorder to apply direct tension, two cylindrical steel caps werecemented to the endsof the specimens using a high strength gap-lling epoxy paste Megapoxy PM (Fig.7). The dimensions of metalcaps and specimen preparation were in accordance with ISRMstandards 1. Specimen preparation is key in conducting a directtension test. A linkage system was used to transfer tensile loadfrom the Instronloading frame to the specimen (Fig. 7).The test results are given in Table 3. Fivereplicate specimenswere tested. However, torsion effects were observed in two testspecimens due to misalignment of the load transfer system, andonly the threesuccessful replicate tests are reported.Fig. 4. Failed Brisbane tuff specimens under: (a) standard Brazilian jaws, (b) 15 loading arc, (c) 20loading arc and (d) 30loading arc.Fig. 5. Failed sandstone specimens under: (a) standard Brazilian jaws, (b) 15loading arc, (c) 20loading arc and (d) 30loading arc.32D FEM modeling of Brazilian disk specimensA series of two-dimensional nite element analyses wereconducted to betterunderstand the stress distribution within adisk specimen under different indirecttension loading modes.The analyses were carried out using FRANC2D (FRactureAnalysisCode). In the numerical modeling, the specimens were assumedto becontinuous, isotropic and homogeneous elastic bodies.Based on the results of UCStesting of Brisbane tuff, a Youngsmodulus of 22 GPa and a Poissons ratio of 0.24were adopted.Load was applied as a traction pressure, derived fromexperimen-tally obtained failure load e.g. 83 MPa for =15loading arc, overtheprojected width of the loaded section of the disk. In allsimulations, the base of the disk was xed in both x and ydirections, and thewidth of the loaded section under the steelarcs was held constant. For the standardBrazilian simulation, theline load was applied over a 1 mm width by assumingadistributed load over 2= 2.Since compressive and indirectly produced tensile stressesalong the loadedvertical diameter are of the most interest, thehorizontal stress, , distributions for the different loading modelsare shown in Figs. 811. In all simulations, localmaximumcompressive stresses developed under the loaded section anddisappeared away from the loaded section. In general, there is atensile zone in thecenter of the disks. The tensile zone was mostconcentrated for the steel loadingdisks with 2=30 (Fig. 11).The standard Brazilian simulations produced the highestcompressive stress(Fig. 8a)， and a high tensile stress zone closerto the loaded section (Fig. 8a and b).This may help to explain whytensile cracks tend to start beneath the line load instandardBrazilian tests. For the standard Brazilian simulation the highestmaximumprincipal stress was tensile and appeared from justunder the compressive zone andextending over the upper half ofthe disk (Fig. 8b).The classical theory assumes that the standard Brazilian jawsapply a line load，but clearly this would lead to applied stresses ofvery great intensity. In reality, applied stresses are distributed.Fig. 12 shows the tensile stress distributions alongthe horizontaldiameter (AX) under the standard Brazilian jaws and the loadingarcs.The maximum tensile stress appears at the center of the diskin all cases, and thestandard Brazilian jaws produced the highestmaximum. The lowest maximum tensile stress was obtained forthe steel loading arcs with 2=15. It is seen thatthe tensilestress increases and the tensile zone is conned to a narrowerregion in the verticaldirection when 2increases (Fig. 12).Fig. 6. Failed granite specimens under: (a) standard Brazilian jaws, (b) 15 loading arc, (c) 20loading arc and (d) 30loading arc.Table 2 Results of indirect tensile tests on sandstone and granite disk specimens.Loading modeAverage recorded maximum load (kN)SandstoneGraniteStandardBrazilian jaws8.535.315Loading arc8.833.520Loading arc10.539.530Loading arc11.946.73.1Stress distributions in a disk loaded by a loading arcAn analytical solution is available for the calculation of boththe horizontaland vertical stress distributions in a disk loadedacross its vertical diameter by aloading arc 3. Under plane stress(disk) conditions, the theoretical horizontaltensile stress alongthe vertical diameter shown in Fig. 2 is given by the followingequation 22:In Fig. 13, the theoretical values of the stresses obtained usingEq. (3) are compared with the numerical modeling resultsobtained using FRANC2D, in order to validate the numericalresults. The theoretical horizontal stress distributions areallsimilar and are reasonably consistent with the numerical results,although thereare clear discrepancies towards the boundaries.The reason for these discrepanciesmay be differences in the assumedboundary conditions for the two types ofanalyses, and/or mesh andgeometry specications in the numerical simulations. Asshown inFig. 13, tensile stresses reach a maximum at the center of the disk (r=0)and persist over more than half of the diameter of thespecimen. Because thecompressive stress regions occur under theloaded boundaries, the numerical valuesat the upper boundaryare larger than those at the lower, xed boundary. In general,theanalytically calculated tensile stress distributions are more uniformthan thenumerically calculated distributions. The numerical simula-tions produceincreasingly uniform tensile stresses of lower magni-tude with decreasing loading arc angle (Fig. 13). Details of thesimulations are given in Table 4.Table 3 Results of direct tensile tests on Brisbane tuff.SpecimenRecorded maximum load (kN)Direct tensile strength (MPa)Replicate 113.26.22Replicate 212.25.74Replicate 310.234.98Average11.885.65Table 4 Comparison ofexperimental,numerical and analytical results for Brisbanetuff.Loading modeExperimental ultimate load (kN)Projectedwidthofloaded section of disk (mm)Theoretical tensile stress at center of disk (MPa)Numerical tensile stress at center of disk (MPa)StandardBrazilianjaws19.601.009.2012.3615Loading arc14.816.806.616.8820Loading arc19.209.078.218.7930Loading arc22.3013.608.858.98Recentresearches 16,17 show that the formula for calculating the indirect tensilestrength from the failure loads obtainedby Brazilian tensile testing might not beapplicable for rockscontaining minerals with different strengths, awsand cracks.Inprevious sections, both numerical and analytical solutions showedthat the stressdistribution in a disk specimen depends strongly onthe loading conditions.However, in both numerical and analyticalsolutions rock specimen is assumed tobe homogenous, isotropicand linear elastic. Since pre-existing or newly-initiatedaws andcracks affect the stress distribution in the specimens, such specimens cannot be assumed to comply with this assumption. Further-more, closed-formanalytical solution assumes that failure occurs inpure tension. Some of the available studies show that failure mayoccur in both tension and shear 15,16. Thedecrease in tensilestrength of an actual brittle material is due to the presence ofpre-existing cracks in the material 11. Inglis 20 and Grifth 21 werethe rstresearchers to realize the signicance of pre-existing cracksacting as precursors to failure.Fig. 7. (a) Prepared specimens before direct tensile testing, (b) direct tensileloading test apparatus and (c) failed specimens after direct tensile testing.In second series of numerical simulations, vertically-aligned2 mm centralpre-existing non-cohesive cracks were placed alongthe vertical diameter of the diskin FRANC2D simulations to assesstheir effect on the calculated tensile strength(Fig. 14).The calculated tensile stresses across the horizontal diameter ofthe disk,at the tip of the pre-existing central cracks, are shown inFig. 15. It can be seen thatthere are much higher tensile stressesdue to the presence of a central pre-existingcrack, compared withnone (Fig. 12). The highest tensile stress at the central pre-existingcrack was obtained for the standard Brazilian jaws.The crack tip stress and displacements can be characterizedby the stressintensity factor, K, which is dependent on themagnitude of remote stress eld, ,and radius of crack, a, i.e. Since crack propagation was not considered inthesimulations to avoid using any propagation criteria, the stressintensity factor atthe tip of the crack is not calculated byFRANC2D. Instead, it was calculated usingthe above equationfor the mode I stress intensity factor, for a central crack undertensile stress without internal pressure acting on the crack sur-face 10. It is knownthat K is for an innite plate containing acrack. However, Gross et al.s 23boundary collocation techniquewas used for a specimen of nite size. The correction factor wasfound by a polynomial of forth order and mode I (tension)stressintensity factor.The calculated stress intensity factor values at the tip of thepre-existing crackunder the standard Brazilian jaws, and the 2=30,2=20and 2=15loadingarc simulations are 1.9MPa, 1.34 MPa, 1.21 MPa and 0.67 MPa,respec-tively. These values are the maximum stress intensity factors(fracturetoughness) values before any crack propagation.The mode I fracture toughness of Brisbane tuff was foundexperimentally tobe 1.18 MPa, using one of the InternationalSociety of Rock Mechanics suggested methods, the Crack ChevronNotched Brazilian Disk (CCNBD) 12. CCNBD specimens have thesame diameter and thickness as standard Brazilian disk speci-mens. Thus, consideration of fracture toughness in the investiga-tion of the indirect tensile strength of rocks is quite helpful andmeaningful. The numericalstress intensity value closest to theexperimental fracture toughness of 1.18MPa,was obtained forthe 2=20 loading arc （1.21MPa）. In order to support thatresult, one of the well known empirical relation between the modeI fracturetoughness, KIC, and the tensile strength of rocks, can beused 13. The mode I fracture toughness, KIC, can be calculatedusing:（7）According to Eq. (7), the expected indirect tensile strength is8.13 MPa,calculated using the experimental mode I fracturetoughness value of 1.18 MPa.Whittaker et al. 11 found that fracture toughness is dependent on otherphysico-mechanical parameters of rocks, such astheir tensile strength, compressivestrength, and Poissons ratio.The empirical relationship between mode I fracturetoughnessand the tensile strength of brittle rocks is 11 （8）According to Eq. (8), the expected indirect tensile strength is 8.5 MPa calculated using the experimental mode I fracture toughness value of 1.18 MPa. The numerical indirect tensile strengthvalue closest to those calculated using Eqs.(7) and (8) wasobtained for the 2=20loading arc (8.79 MPa, Table 4).Mode I fracture toughness values for granite and sandstone CCNBD specimens were found to be 2.5MPaand 0.8MPa,respectively. From a seriesof numerical simulations for 2=20loading arcs, using the physico-mechanicalparameters of sand-stone and granite, the maximum stress intensity values at the tipof the pre-existing cracks were found to be 2.6MPaand0.6MPa, respectively.Thus,2=20loading arc simulationsagain gave fracture toughness valuesconsistent with the experi-mental values.Fig.8 Standard Brazilian jaws imulations: (a) horizontal stress distribution(MPa), (b)Maximum principal stress distribution and (c) tensile stress bars (+: tensile, -: compression). Fig.9. 15Loading arcs imulations; (a) horizontal stress distribution(MPa) and (b) tensile stress bars (+: tensile, -: compression).Fig. 10. 20Loading arcs imulations: (a) horizontal stress distribution(MPa) and (b) tensile stress bars(+: tensile, -: compression). Fig. 11. 30Loading arcs imulations: (a) horizontal stress distribution(MPa) and (b) tensile stress bars(+: tensile, -: compression). 4 ConclusionsThe conclusions arising from the experimental, numerical andanalytical studies are given in the following points.From the experimental studies on Brisbane tuff specimens, itwas concludedthat both tensile fracturing and indirect tensilestrength of rocks are strongly dependent on the type of diametricloads. The maximum recorded failure load, corresponding to thenest and most centrally-located crack, was obtained with a 2=30loading arc.The highest standard deviation of the failure loads was obtained using the standardBrazilian jaws, whichresulted in catastrophic specimen failure.A general increase in failure load was obtained with increasingangle ofloading arc for all three rock types tested.The direct tensile strength of Brisbane tuff was found to belower than the indirect tensile strengths, due to the difculty ofeliminating torsional effects in the direct tensile test.Numerical simulations performed using FRANC2D enhancedunderstandingof the tensile stress distributions in the disksunder different loading conditions.Numerical simulations indicated that the maximum tensilestress appears at the center of the disk under the standardBrazilian jaws, while the lowest maximumtensile stress appearsat the center of the disk under a 2= 15loading arc.It was found that while the theoretical horizontal stressdistributions along thevertical diameter are reasonably consis-tent with the numerical results, the analytically calculated tensilestresses are less uniform.Numerical simulations incorporating pre-existing cracksshowed much higherand more concentrated tensile stresses atthe tips of these cracks.Experimentally obtained fracture toughness values of thethree rock typeswere compared with the fracture toughnessvalues at the tips of the pre-existingcracks obtained fromnumerical simulations. It was concluded that the best agreementwas obtained from the 2=20loading arc simulations. In order to support that result, well known empirical relationshipsbetween fracture toughness and tensile strength of rocks wereapplied.References 1 ISRM. International society for rock mechanics. Suggested methods fordetermining tensile strength of rock materials. Int J Rock Mech Min SciGeomechAbstr 1978;15:99103.2 Fairhurst C. On the validity of the Brazilian test for brittle materials. IntJRock Mech Min Sci 1964;1:53546.3 Hondros G. The evaluation of Poissons ratio and the modulus of materialsofa low tensile resistance by Brazilian (indirect tensile) test with particularreference to concrete. Aust J Appl Sci 1959;10:24368.4 Hudson JA, Brown ET, Rummel F. The controlled failure of rock discs andringsloaded in diametral compression. Int J Rock Mech Min Sci 1972;9(2):2414.5 Wang QZ, Jia M, Kou SQ, Zhang ZX, Lindquist PA. The attened Brazilian discspecimen used for testing elastic modulus, tensile strength andfracturetoughness of brittle rocks: analytical and numerical results. Int J RockMechMin Sci 2004;41:24553.6 Jaeger JC, Cook NGW. Fundamentals of rock mechanics. London:Chapmanand Hall; 1976.7 Ma CC, Hung KM. Exact full-eld analysis of strain and displacement forcircular disks subjected to partially distributed compressions. Int J Rock MechMinSci 2008;50:27592.8 Yu Y, Zang JX, Zang J. A modied Brazilian disc tension. Int J RockMech MinSci 2009;46:4215.9 Hudson JA, Brown ET, Rummel F. The controlled failure of rock discs andrings, loaded in diametrical compression. Int J Rock Mech Min Sci GeomechAbstr 1972;9:2418.10 Markides CF, Pazis DN, Kourkoulis SK. Closed full-eld solutions forstressesand displacements in the Brazilian disk under distributed radial load. IntJRock Mech Min Sci 2010;47:23747.11 Whittaker BN, Singh RN, Sun G. Rock fracture mechanicsprinciples,designand applications. Amsterdam: North-Holland; 1992.12 ISRM co-ordinator: Fowell RJ. Suggested method for determiningmode Ifracture toughness using cracked chevron notched Brazilian disk (CCNBD)specimens. Int J Rock Mech Min Sci Geomech Abstr 1995;32(1):5764.13 Zang ZX. An empirical relation between mode I fracture toughness andtensile strength of rocks. Int J Rock Mech Min Sci 2002;39:4016.14 Okubo S, Fukui K. Complete stressstrain curves for various rock typesinuniaxial tension. Int J Rock Mech Min Sci Geomech Abstr 2000;33:54956.15 Van de Steen B, Vervoort A, Napier JAL. Observed and simulatedfracturepattern in diametrically loaded discs of rock material. Int J Fract2005;131:3552.16 Lanaro F, Sato T, Stephansson S. Microcrack modelling of Braziliantensiletests with the boundary element method. Int J Rock Mech Min Sci2009;46:45061.17 Cai M, Kaiser PK. Numerical simulations of the Brazilian test and tensilestrength of anisotropic rocks and rocks with pre-existing cracks. Int J RockMechMin Sci 2004:41. paper 2B03.18 Yanagidani T, Sano O, Terada M, Ito I. The observation of cracks propagatingin diametrically-compressed rock disks. Int J Rock Mech Min SciGeomechAbstr 1978;15:22535.19 Mellor M, Hawkes I. Measurement of tensile strength by diametralcompres-sion of discs and annuli. J Eng Geol 1971;5:173225.20 Inglis CE. Stresses in a plate due to the presence of cracks and sharpcorners.Trans Inst Naval Archit 1913;55:219.21 Grifth AA. The phenomena of rupture and ow in solids. Philos Trans RSocLondon 1920;A221:163.22 Sarris E, Agioutantis Z, Kaklis K, Kourkoulis S. Numerical simulation ofthecracked Brazilian disk under diametral compression. In: Proceedings ofseventh international workshop on bifurcation, instabilities, degradation ingeomechanics. Hania; 1316 June 2005.23 Gross B, Srawley JE, Brown WF. Stress intensity factors for asingle-edge-notch specimen by boundary collocation of a stress function. NASA TND-2395; 1964.中文译文岩石的抗拉强度实验，数值模拟和分析研究Nazife Erarslan , DavidJohnWilliams澳大利亚昆士兰大学，土木工程学院，戈尔德力学中心，布里斯班，QLD4072。摘要：由于夹持试样的困难，确定岩石的抗拉强度一般不采用直接拉伸试验，而多采用各种间接的方法。本文研究比较了三种不同类型的岩石在进行直接和间接拉伸测试的实验结果：分别为布里斯班凝灰岩，花岗岩和砂岩。标准巴西圆盘试件间接拉伸试验，应用和测试，表明其破坏不是因为线载荷应力集中，是因为加载点处的应力集中，岩石的脆性加剧造成了灾难性的圆盘标本的加载破坏。这一发现导致圆盘加载不同的中心角角度，适用于三种不同的岩石类型，采用数值模拟和分析巴西标本加载破坏条件的影响进行调查。数值模拟研究，还进行预先存在的裂纹巴西圆盘试样内的应力分布的影响进行调查。预先存在的裂纹平盘与没有预先存在的中心直裂纹平盘相比，在预先存在的裂纹平盘中它被发现在磁盘的中心有较高的拉伸应力。在中央预先存在的裂缝尖端的最大动态应力强度因子（断裂韧性）值确定数值模拟，并与三大岩石类型实验获得的断裂韧性值相比。得出的结论是，当圆盘试样加载中心角为20时，其间接抗拉强度达到最大值。关键词：巴西圆盘试验、岩石直接拉伸强度、二维有限元建模软件、岩石间接拉伸强度、预先存在的裂纹1简介由于夹持试样的困难，确定岩石的抗拉强度一般不采用直接拉伸试验，而多采用各种间接的方法。1978 年，国际岩石力学学会（ISRM）推荐将巴西圆盘试验作为测试岩石的抗拉强度的间接方法。巴西圆盘测试（或剥离拉伸测试）是采用集中压缩载荷整个圆盘标本直径的方法来测试岩石的抗拉强度。巴西圆盘测试，也被作为建议的方法，来确定混凝土材料的拉伸强度。因为巴西圆盘测试式最早被提出的，现在它已被修正。fairhurst2首先讨论了巴西圆盘测试的有效性的重要问题。他说，“从测试圆盘加载接触面积小角度的中心，可能会发生破坏。”，而且巴西圆盘试验测试的抗拉强度值低于其抗拉强度的真实值。Multibits3提出了一种通过巴西圆盘试验来衡量弹性模量和泊松比的方法，还制定了完整的在应力分布在有限的圆盘圆弧径向负荷的情况下的解决方案。虽然巴西圆盘测试已被广泛研究在实验和理论上，但只有相对较少的关注是针对其研究测试的有效性。几个关键性的问题仍然没有得到解决：例如，如何保证在试样中心（集中载荷）下，如何获取准确地表述测试的岩石拉伸强度的中心裂纹萌生，以及如何获得完整的应力的封闭形式表达和巴西测试应变场的情况下，当负载作为行事的正常压力均匀地有限长弧应用。Hudson等 4发现，当岩石的拉伸强度差别很大时，用不同的方法测试和计量岩石试样和试验机的钢压板之间的接触条件的异质性将影响到其抗拉强度值。他指出，在巴西圆盘测试中，如果扁钢压板被用来装入标本，加载破坏总是直接由加载点引起的，值实际上是无效的测定拉伸强度的测试。王等人5使用一种试件，在其加载点的两端有两个平行平面，来防止在加载点的地方产生裂纹。然而，他们发现，对于一个成功的测试，其两端平行平面的平整度和平行性是至关重要的因素。经典理论1假设，集中应力载荷作用在线路负荷极小的宽度，但显然，这将导致非常高强度的压力。实际载荷并不集中，但通过有限的圆盘弧分布。只要圆盘试样从中心处开始破坏，就可以不考虑应力集中现象对拉伸强度测试的影响岩盘试样的拉伸强度计算公式： （1）式中：P破坏载荷；d圆盘试样的直径；t圆盘试样的厚度。直径为d 的圆盘受到一对力P 的作用，P 的作用线过圆心O，圆盘中任一点A 的正应力，和剪应力的表达式分别为：（2）上述公式中以拉应力为负，压应力为正。直径BC 两端因受集中力，存在应力集中现象。根据圣维南原理，在远离B 点、C 点处，应力集中的影响可以不计。当，为0 时，由式(1)得到圆盘受压直径BC 上的水平正应力分布情况，可知它是均匀的拉应力，其值为2P /(d)。直径BC 上的最小压应力在圆盘中心，为6P /(d)，是BC 直线上拉应力的3 倍。（3）式中：P施加压力；R圆盘的半径；r，在圆盘上点的极坐标；平台中心角。从式（3）中可以看出的大小直接影响圆盘内的应力分布。最近，马和洪7继承和发展了Multibits所做的工作，他们成功地获得一个简单的表达，而不是一系列的方程式中明确的形式解析解。在一般情况下，拉伸破坏最有可能从巴西圆盘脆性材料间接拉伸试验标准试样的边界开始的。然而，在本文所述的测试中，灾难性的破碎失败是有脆性材料布里斯班凝灰巴西圆盘标本检测标准制定的。另一方面，中央裂缝，相应的最大拉应力的位置，加载布里斯班弧长凝灰岩标本。这是与在圆盘标本被横向拉伸应力造成破坏。因此，本文的目的是用使用加载弧获得的连同这个测试的实验结果进行比较，分析和数值模拟，得到的结果进行比较，批判巴西的标准测试。图1 加载模型为集中线点荷载图2加载模型为弧形加载2实验研究2.1间接张力试验用布里斯班凝灰岩，砂岩和花岗岩进行了一系列巴西圆盘试验。大多数测试是用布里斯班凝灰岩进行的，因为它是布里斯班的第一个高速公路隧道，CLEM7，从中获得岩心样品的围岩。对砂岩和花岗岩进行有限的测试验证，结果可能会更普遍应用，而不是一个单一的岩石类型。布里斯班凝灰岩表现在其轴向应力应变曲线的弹性方式为很大一部分几乎是线性的。试样准备了一个直径为52毫米和一个26毫米的厚度标准巴西磁盘（直径：厚度比为0.5）。负载采用坚硬的液压英斯特朗加载框架，由国际岩石力学学会建议的装载率为200 N / S1。图3 （a）标注加载圆盘（b）钢加载弧（c）钢加载弧之间的圆盘表1 布里斯班凝灰岩圆盘试件间接拉伸试验的结果试件最大负载载荷 (kN)标准试件中心角15中心角20中心角30试件 125.0012.5017.0621.10试件216.7716.3919.8224.60试件315.4315.6520.2321.13试件421.0014.7019.4122.17平均值19.6014.8119.2022.30标准偏差4.341.691.641.64四大系列的间接张力试验进行：（1）标准的巴西试件，（2）中心角为15的试件，（3）中心角为20的试件，（4）中心角为30的试件。每种状态下试件的数量不少于四个重复进行试验。钢装载弧标准是低碳钢加工，有国际岩石力学学会1标准确定的，如图3所示。岩石标本的拉伸强度测试，使用标准的巴西圆盘试件是由国际岩石力学学会1所提供的公式计算。由于圆盘中心角不同于标准的巴西圆盘试件，由国际岩石力学学会所提供的的公式不能用于计算间接拉伸载荷作用下的强度。分别计算不同加载中心角下样品的抗拉强度，通过Hondros准则3得知中心点处的压拉力影响比较显著。表1给出了测试和测试结果的具体数值。由表1得知当圆盘中心角时测试得到了最大极限载荷。然而，从巴西标准试件测试结果获得了极限载荷的最高标准偏差值。在这项研究中，主要重点是实现拉伸破坏，由于中央裂纹沿着垂直圆盘直径的方向，这被认为是地区最大的间接拉伸应力5,8,18。然而，大多数的使用标准的巴西圆盘试实验研究表明，试样在集中载荷作用处会首先屈服碎裂8-10。在布里斯班凝灰岩加载荷加载，目前测试造成圆盘标本灾难性的破坏（图4a）。在测试中使用15载荷弧造成偏离垂直加载轴，二次裂纹单裂纹。在测试中使用20载荷弧造成一个垂直中心裂纹（图4c）。在测试中使用30载荷弧造成一个被捕捉的垂直中心裂纹（图4d）。为了进一步探讨在径向载荷的作用，间接拉伸强度的测定，除了布里斯班凝灰岩标本之外，还用了砂岩和花岗岩圆盘盘标本。花岗岩样品中，从布里斯班的Keperra的采石场获得的样本，其单轴抗压强度（UCS）为210兆帕；而砂岩样品中，从布里斯班以西图沃柏附近的Helidon采石场获得的样本，其单轴抗压强度（UCS）为37兆帕。另一方面，布里斯班凝灰岩的单轴抗压强度值介于两者之间，其值为90兆帕。2.2直接拉伸实验使用布里斯班凝灰岩进行了一系列的直接拉伸试验。试验中直接拉伸试样的直径与间接拉伸试样的直径是相同的。直接拉伸试样呈圆柱形岩芯样品，直径52毫米，厚度为135毫米（厚径比为2.59）。为了适用于直接张力，两个圆柱形钢帽巩固试样的两端，并使用高强度的间隙充填环氧树脂粘贴Megapoxy PM（图7）。金属盖和样品制备的尺寸是按照国际岩石力学学会标准1确定的。样品制备是进行直接拉伸试验的关键。一种连锁系统被用来传输拉伸载荷，载荷来源于从斯特朗加载帧的标本（图7）。表3给出了测试结果。五各复制标本进行了测试。然而，在两个试样的观察中出现了扭转效应，由于错误的荷载传递系统，只有三个试样成功的复制了测试报告。图4布里斯班岩巴西圆盘试件加载破坏情况：(a)标准巴西试件(b) 15圆盘中心角(c) 20圆盘中心角(d) 30圆盘中心角图5布里斯班岩巴西圆盘试件加载破坏情况：(a)标准巴西试件(b) 15圆盘中心角(c) 20圆盘中心角(d) 30圆盘中心角图6巴西圆盘试件加载破坏情况：(a)标准巴西试件(b) 15圆盘中心角(c) 20圆盘中心角(d) 30圆盘中心角3二维有限元建模的巴西圆盘试件标本进行了一系列的二维有限元分析，以便更好地了解不同的间接张力负荷模式下的应力分布在磁盘标本。使用FRANC2D（断裂分析代码）进行了分析。在数值模型中，标本被认为是连续的，各向同性和均质的弹性体。基于布里斯班凝灰岩的单轴抗压强度测试，一个弹性模量22 GPa和泊松比0.24的测试结果被采用。载荷被应用为牵引压力来自试验获得破坏载荷，例如= 15的加载弧，其单轴抗压强度值为83兆帕，超过预计的圆盘加载的宽度。在所有的模拟中，圆盘试件被固定在X和Y方向，并且钢加载弧的宽度保持不变。以标准巴西试件模拟，线载荷在下列情况被应用：超过1毫米的宽度由假设分布式负载超过2= 2。沿加载垂直直径方向，由于压缩和间接产生的拉应力，水平应力，其不同负荷模型的分布在图8-11所示。在一般情况下，有拉伸应力体现在圆盘的中心区。当钢加载弧的中心角2= 30是其应力最为集中，如图11所示。标准巴西模拟产生了最大的压应力（如图8a所示），高拉应力区主要集中于接近加载点的位置（如图8a和图8b所示）。这可能有助于解释为什么拉伸裂纹往往开始在巴西标准测试线载荷之下。标准巴西模拟最高的最大主应力是拉伸强度，其体现在压应力区，扩展于圆盘的上半部分（如图8b所示）。古典理论假设认为标准巴西圆盘试件加载线载荷，但显然，这会导致很大的施加压力。在现实中，应用应力分布如图12所示。图12显示了标准巴西试件和在加载弧情况下沿水平直径（AX）的拉伸应力分布情况。在所有情况下，圆盘的最大拉应力出现在中心区域，标准巴西圆盘获得了拉应力的最大值，而当钢加载加载中心角为2=15时获得了拉应力的最小值。这很明显的可以看出，随着中心角2的增加，拉伸应力也随之增大，拉伸区被限制在垂直方向的一片下载区域内，如图12所示。表2 砂岩和花岗岩磁盘标本间接拉伸试验的结果加载模式记录的平均最大负荷（kN）砂岩花岗岩标准试件8.535.315加载角8.833.520加载角10.539.530加载角11.946.73.1在磁盘上的应力分布加载加载弧一种解析解可用于计算一个在其垂直直径的方向加载加载弧的圆盘试件的水平和垂直应力分布3。平面应力状态下（圆盘试件）的条件下，沿垂直直径的理论水平拉应力如图2所示。其计算公式如下式所示22：表3 布里斯班凝灰岩直接拉伸试验的结果试件标本记录的最大负荷 (kN)直接拉伸强度 (MPa)试件 113.26.22试件 212.25.74试件310.234.98平均值11.885.65如图13所示，将使用方程式（3）求得的应力的理论值与使用二维有限元建软件模数值模拟的结果进行比较，来以验证数值结果。虽然有明显的边界走向的差异，但是其理论水平应力分布都是相似的，并与数值模拟结果是一致的。通过分析在数值模拟中网格和几何规格的两种规格，发现这些差异的原因可能是在假定的边界条件上的差异。正如图13所示，在圆盘试件的中心处（r=0），拉应力达到最大值，并坚持超过一半以上的试样直径。由于压应力区域发生在加载边界处，在上边界的数值大于较低，固定边界的数值。在一般情况下，分析计算的拉伸应力分布要比数值计算分布均匀。在数值模拟中，随着载荷加载角的加载幅度越来越小，其数值模拟产生的拉伸应力分布越来越均匀（如图13所示）。数值模拟结果的详情如表4所示。表4 布里斯班凝灰岩的实验，数值模拟和分析结果的比较加载模式试验的极限荷载值 (kN)预计宽度的圆盘加载截面 (mm)圆盘中心的理论拉应力值 (MPa)圆盘中心的实际拉应力值(MPa)标准巴西试件19.601.009.2012.3615加载角14.816.806.616.8820加载角19.209.078.218.7930加载角22.3013.608.858.98最近的研究16,17表明，由间接抗拉强度的计算公式求得的巴西圆盘试件拉力试验破坏荷载值，可能无法适用于含有不同优势，缺陷和裂缝的矿物岩石。在前面的章节中，数值模拟和分析解决方案表明圆盘标本中的应力分布，在很大程度上取决于加载条件的不同。然而，在数值模拟和分析解决方案岩石标本被认为是均匀地，各向同性地线性弹性体。由于预先有的或新发起的缺陷和裂缝影响标本中的应力分布，这些标本不能被认为符合这个假设。此外，封闭形式的解析解，假设认为在纯张力作用情况下发生