張雙樓煤礦2.4Mta新井設計含5張CAD圖.zip,張雙樓,煤礦,2.4,Mta,設計,CAD 英文原文 Experimental, numerical and analytical studies on tensile strength of rocks Nazife Erarslan , DavidJohnWilliams GolderGeomechanics Centre, School of Civil Engineering, TheUniversity of Queensland, Brisbane, Qld 4072, Australia Abstract: The dif?culties 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 20°loading 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. 1Introduction The dif?culties 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 of?cially 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 Poisson’s 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 in?uence 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 in?nitesimally 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 de?ned 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 moredif?cult 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 [3]obtained the full-?eld stresses using the series expansion techni-que (Eq. (4)) and applying these solutions to evaluate Young’smodulus (E) and Poisson’s 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 study 2.1Indirect tension tests A 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 ofBrisbane’s ?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 asigni?cant portion of its axialstress–strain 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 1 Results of indirect tensile tests on Brisbane tuff disk specimens Specimen Recorded maximum load (kN) Standard Brazilian jaws 15°Loading arc 20°Loading arc 30°Loading arc Replicate 1 25.00 12.50 17.06 21.10 Replicate 2 16.77 16.39 19.82 24.60 Replicate 3 15.43 15.65 20.23 21.13 Replicate 4 21.00 14.70 19.41 22.17 Average 19.60 14.81 19.20 22.30 Standard deviation 4.34 1.69 1.64 1.64 Four series of indirect tension tests were conducted, with:(1) standard Brazilian jaws, (2) 15°steel loading arcs, (3) 20°steelloading arcs and (4) 30°steel 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 Hondros’equation [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. 請鍵入文字或網(wǎng)站地址，或者上傳文檔。 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 stress[5,8,18]. However, most experimental studiescarried out usingstandard Brazilian jaws cause cracks to initiate just under theloading points [8–10]. In the present tests on Brisbane tuff,loading with Brazilianjaws caused catastrophic crushing failureof the disk specimens (Fig. 4a). Testingusing 15°loading arcscaused a single crack that diverged from the vertical loading axis,together with secondary cracks. Testing using 20°loading arcscaused a singlevertical, central crack (Fig. 4c). Testing using 30°loading 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 tests A 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) 20°loading arc and (d) 30°loading arc. Fig. 5. Failed sandstone specimens under: (a) standard Brazilian jaws, (b) 15°loading arc, (c) 20°loading arc and (d) 30°loading arc. 32D FEM modeling of Brazilian disk specimens A 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 Young’smodulus of 22 GPa and a Poisson’s ratio of 0.24were adopted.Load was applied as a traction pressure, derived fromexperimen-tally obtained failure load e.g. 83 MPa for β=15°loading 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, the‘‘line load’’ was applied over a 1 mm width by assumingadistributed load over 2α= 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. 8–11. 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 2α=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 2α=15°. It is seen thatthe tensilestress increases and the tensile zone is con?ned to a narrowerregion in the verticaldirection when 2αincreases (Fig. 12). Fig. 6. Failed granite specimens under: (a) standard Brazilian jaws, (b) 15° loading arc, (c) 20°loading arc and (d) 30°loading arc. Table 2 Results of indirect tensile tests on sandstone and granite disk specimens. Loading mode Average recorded maximum load (kN) Sandstone Granite StandardBrazilian jaws 8.5 35.3 15°Loading arc 8.8 33.5 20°Loading arc 10.5 39.5 30°Loading arc 11.9 46.7 3.1Stress distributions in a disk loaded by a loading arc An 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 speci?cations 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. Specimen Recorded maximum load (kN) Direct tensile strength (MPa) Replicate 1 13.2 6.22 Replicate 2 12.2 5.74 Replicate 3 10.23 4.98 Average 11.88 5.65 Table 4 Comparison ofexperimental,numerical and analytical results for Brisbanetuff. Loading mode Experimental ultimate load (kN) Projectedwidthofloaded section of disk (mm) Theoretical tensile stress at center of disk (MPa) Numerical tensile stress at center of disk (MPa) StandardBrazilianjaws 19.60 1.00 9.20 12.36 15°Loading arc 14.81 6.80 6.61 6.88 20°Loading arc 19.20 9.07 8.21 8.79 30°Loading arc 22.30 13.60 8.85 8.98 Recentresearches [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-initiated?aws 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 Grif?th [21] werethe ?rstresearchers to realize the signi?cance 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 tensil