英文文献

modelling of circular sawing system using discretemethodN.E.Yas-ıtlıa,n,F.Bayram a,B.Unver b,Y.O¨zc-elik ba Department of Mining Engineering,Aksaray University,68100Aksaray,Turkeyb Department of Mining Engineering,Hacettepe University,06532Ankara,Turkeya r t i c l e i n f oArticle history:Received12April2011Received in revised form9May2012Accepted19June2012Available online21July2012Keywords:Natural stone processingCircular sawingNumerical modellingPFC3Da b s t r a c tA new method of modelling circular sawblade machines is suggested in this paper.Experimentalstudies and numerical modelling studies were performed on two real marbles,Afyon Violet Marble(AVM)and Afyon White Marble(AWM),and on a limestone,Sivrihisar Beige Limestone(SBL).Experimental studies were carried out using a specially designed,fully automatic circular sawbladestone cutting machine and numerical modelling studies were performed with a commercially availablesoftware called PFC3D.A discrete element model for the sawing process was developed and variousnumerical models were performed for different peripheral speeds and advance rates while at the sametime the actual cutting operations were being carried out in the laboratory.At the end of the study,thenumerical modelling results are found to be quite compatible with the actual results.Consequently,it isshown that numerical modelling can effectively be used for the simulation of the natural stone sawingprocess,and this study paves the way for explaining the sawing mechanism dynamically.&2012Elsevier Ltd.All rights reserved.seen as the most impor-However,due tomanpower,equipment required,andthe high cost of experimental studies,an alternative method hasemerged to reduce or to replace experimental studies in appropriateconditions.The numerical modelling technique has undergone greatdevelopment in recent years,with rapid advances in computertechnology and computers and an increasing executing rate.Thistechnique is now used in many areas of science as well as in mining,particularly as an alternative to experimental studies in rockmechanics.The numerical modelling method has replaced experi-mental studies in many areas because it can be conducted quicklyand cheaply,and is quite compatible with the actual results[1].numerical modelling method is used to determine effec-in a cheaper and more reliable way in rockmechanics;as the parameters it can determine include forces,stresses,deformations and failure zones and it can be used intunnel design,slope stability design,and surface and under-ground mining methods design.studies have been performed related to rock cutting,and excavation.Although many of these studies havebeen based on measurements gathered in situ and in laboratory,nowadays there is an increasing number of studies using numer-ical modelling techniques,particularly in studies relating to rockmechanics looking at the cutting performance of mechanicalcutting edges and at the failure zone and chipping formationcutting operations[2–15].studies have been carried out on circular sawingto determine,for example,an estimation of cuttingforces,an estimation of unit wear on segments,optimal cuttingconditions for particular rocks and the effects of cutting direction[16–27].These studies tend to be theoretical,statistical andexperimental,but some numerical modelling of rock cuttingstudies have been performed by researchers related to indenta-tion and chipping.The aim of this study is to determine thereactional cutting forces on the circular sawing system usingnumerical modelling with PFC3D(Particle Flow Code3Dimen-sion).It also aims to determine suitable operating conditions in arapid and economic manner.2.MethodologyThis study was carried out in two main parts.In thefirst part,twonatural stones known commercially as Afyon Violet Marble(AVM)and Afyon White Marble(AWM)and a limestone known commer-cially as Sivrihisar Beige Limestone(SBL)were brought to thelaboratory and cut with a computer-assisted fully-automatic circularsawing machine.Peripheral speed,sawing rate,depth of cut andreactional cutting forces in3directions were measured during sawingContents lists available at SciVerse ScienceDirectjournal homepage:/locate/ijrmmsInternational Journal ofRock Mechanics&Mining Sciences1365-1609/$-see front matter&2012Elsevier Ltd.All rights reserved./10.1016/j.ijrmms.2012.06.006n Corresponding author.Tel.:þ903822882379;fax:þ903822802298.E-mail address:eyasitli@.tr(N.E.Yas-ıtlı).International Journal of Rock Mechanics&Mining Sciences55(2012)86–96operations by means of load cells,sensors and an analyser mounted on the circular sawing machine.In the second part,numerical modelling of the circular sawing mechanism was performed.The parameters taken from at thefirst part were used as input parameters to the numerical model.The model was created in PFC3D,then the assignment of required data was done and numerical modelling of sawing was performed.Finally,the laboratory results were compared with the results obtained from modelling.3.Experimental studies and circular sawing operationsThe experimental study was conducted in three stages.In thefirst stage,the real marble samples AVM and AWM,and the limestone SBL were prepared to dimensions of200Â300Â500mm3.In the second stage,the rock mechanics tests were performed in order to determine the physical and mechanical properties of the natural stones.The results of these tests are given in Table1.In the third stage,the sawing operations were carried out at different peripheral speeds and advance rates.puter-assisted fully-automatic circular sawing machineA computer-assisted fully automatic circular sawing machine was designed and the machine parameters that are effective during the sawing process,such as the effective forces in the X, Y and Z directions,were identified,and the controlling sawing conditions(sawing rate,peripheral speed of the saw,sawing depth,etc)were provided[28].Due to a special platform located on the top of the wagon,the three-dimensional reactional forces that occur during the sawing process could be measured by load cells.In order to identify the horizontal reactional forces(X and Y)on this platform,two load cells are used for each direction,which,together with the three load cells used to identify the vertical reactional force(Z),gives a total of seven load cells.The four load cells used for the identification of the horizontal reactional forces are Flintec types, UB6code,with a capacity of100kg(tension type),and the three load cells used for the identification of the vertical reactional forces are also Flintec types,SB14code,with a capacity of200kg (beam type).The load cells are able to measure to70.01kg sensitivity.The fully-automatic circular sawing machine can be controlled either manually or by computer.An automation programme was developed to convey the necessary parameters(advance rate, peripheral speed,sawing depth,sawing width,number of saw-ings,etc)to the machine via computer during the sawing,and to monitor and record the three-dimensional reactional sawing force data that occur during sawing.3.2.Sawing operationsThe sawing operations were performed under24sawing conditions at four different peripheral speeds(40–70m/s)and six different advance rates(400–900mm/min)for each natural stone with the saw at a constant depth of60mm.For each sawing condition,the sawing operations were repeated5times.During the sawing operations,some data,such as height of sample,depth of cut,peripheral speed and advance rate,were entered into the software that controlled the machine;then the sawing was carried out in the form of a channel and was repeatedfive times.After the sawing operation,four data per second obtained from the load cells were recorded on a computer.The horizontal and vertical reactional forces that occurred during this sawing operation can be seen in Fig.1.At the end of the sawing operations,it was found that the reactional forces did not show a regular trend in the horizontal directions(X and Y);therefore,the analysis of these forces was not taken into account.In this study,only the vertical forces that occurred in the Z direction were examined in detail.With the start of the sawing operation in Region I,the sawblade is in contact with the block and the reactional cutting forces begin.Maximum contact between the block and the sawblade occurs in Region II,where the vertical reactional cutting forces reach their maximum values.The vertical reactional cut-ting forces remain at approximately the same level in Region II. With the sawblade leaving the block in Region III,the reactional cutting forces reduce suddenly.Due to the minimal contact between the block and the sawblade,the reactional cutting forces remain at approximately at the same level.When the whole sawblade leaves the block,the sawing operation stops;conse-quently,recording of the data also stops.The blocks used in the laboratory are500mm in length.How-ever,this would be much longer in plant.Therefore,in order to apply the results of this study to a plant scale,the reactional cutting forces obtained from sawing operation were evaluated for the whole line(the total of Regions I,II and III)and for just the middle region (Region II)separately,and the analysis was carried out in this scope.3.2.1.Reactional cutting forces for different peripheral speedsThe averages of the vertical reactional forces obtained for the whole cutting line and for the middle region at40m/s peripheral speed are presented graphically in Fig.2for AVM.The average vertical reactional forces along the whole line and middle region for the lowest advance rate of400mm/min are63.43N and 117.68N,respectively.For the highest advance rate of900mm/ min,the average vertical reactional forces for whole line and for middle region are176.4N and374.16N,respectively.The average vertical reactional forces between400mm/min and900mm/min change in these values.As seen in Fig.2,the reactional forces increase linearly as the advance rate increases.From the results,it can be seen that the vertical reactional cutting forces have values approximately twice as high in the cases where the advance rate is at its lowest than where it is at its highest at constant peripheral speeds.The reason why the reactional cutting force increases as the advance rate increases at a constant peripheral speed is that the amount of material that will be cut in a unit of time,or,in other words,the sawblade’s amount of contact with the block in a specific unit ofTable1Physical and mechanical properties of the natural stones.Sample Natural Stone Lithologic UVW WA P UCS E UTS H S No Trade Name Name(gr/cm3)(%)(%)(MPa)(GPa)(MPa)–1Afyon Violet Real Marble 2.670.060.1573.9417.6 5.4942 2Afyon White Real Marble 2.650.050.1551.4515.5 6.2249 3Sivrihisar Beige Limestone 2.700.120.2270.0022.27.0562UVW:unit volume weight,WA:water absorption,P:porosity,UCS:uniaxial compressive strength,E:elasticity modulus,UTS:uniaxial tensile strength,H S:shore scleroscope hardness.N.E.Yas-ıtlıet al./International Journal of Rock Mechanics&Mining Sciences55(2012)86–9687time,increases,and,therefore,the cutting force increases.Conversely,at a constant advance rate,since the peripheral speed increases as the sawblade’s contact with the block in any unit of time decreases,the cutting force decreases.As a result of the sawing operation for AVM,linear relationships are obtained with R 2¼0.945–0.987for the whole cutting line and R 2¼0.903–0.979Fig.1.Reactional forces in three directions during sawing.I.Region:Sawing starts and sawblade enters the block with the wagon advancing,II.Region:Whole sawblade isin the block and contact between the block and the saw is at its maximum and III Region:The sawblade starts to leave the block and the whole sawblade is out of the block as the wagon advances.050100150200250300050100150200250300350400Peripheral Speed : 40 m/sPeripheral Speed : 50 m/sPeripheral Speed : 60 m/sPeripheral Speed : 70 m/sAdvance Speed (mm/min)V e r t i c a l R e a c t i o n a l C u t t i n g F o r c e (N )V e r t i c a l R e a c t i o n a l C u t t i n g F o r c e (N )V e r t i c a l R e a c t i o n a l C u t t i n g F o r c e (N )V e r t i c a l R e a c t i o n a l C u t t i n g F o r c e (N )2004006008001000Advance Speed (mm/min)2004006008001000Advance Speed (mm/min)2004006008001000Advance Speed (mm/min)2004006008001000Fig.2.Average of the vertical reactional forces occurring in the whole cutting line and in the central region for AVM.N.E.Yas -ıtlıet al./International Journal of Rock Mechanics &Mining Sciences 55(2012)86–968850100150200250050100150200250300350050100150200250300Peripheral Speed : 40 m/sPeripheral Speed : 50 m/sPeripheral Speed : 60 m/sPeripheral Speed : 70 m/sAdvance Speed (mm/min)V e r t i c a l R e a c t i o n a l C u t t i n g F o r c e (N )V e r t i c a l R e a c t i o n a l C u t t i n gF o r c e (N )V e r t i c a l R e a c t i o n a l C u t t i n g F o r c e (N )V e r t i c a l R e a c t i o n a l C u t t i n g F o r c e (N )Advance Speed (mm/min)Advance Speed (mm/min)Advance Speed (mm/min)Fig.3.Average of vertical reactional forces occurring in whole cutting line and in the central region for AWM.050100150200250300350400450050100150200250300350050100150200250300350050100150200250300Peripheral Speed : 40 m/sPeripheral Speed : 50 m/sPeripheral Speed : 60 m/sPeripheral Speed : 70 m/s Advance Speed (mm/min)V e r t i c a l R e a c t i o n a l C u t t i n g F o r c e (N )V e r t i c a l R e a c t i o n a l C u t t i n g F o r c e (N )V e r t i c a l R e a c t i o n a l C u t t i n g F o r c e (N )V e r t i c a l R e a c t i o n a l C u t t i n g F o r c e (N )Advance Speed (mm/min)0Advance Speed (mm/min)2004006008001000Advance Speed (mm/min)Fig.4.Average of vertical reactional forces occurring in whole cutting line and in the central region for SBL.N.E.Yas -ıtlıet al./International Journal of Rock Mechanics &Mining Sciences 55(2012)86–9689for the middle region.The same process was carried out for AWM and SBL(Figs.3and4)and it was found that for AWM R2¼0.939–0.990for the whole cutting line and R2¼0.964–0.991 for the middle region,while for SBL R2¼0.989–0.997for the whole cutting line and R2¼0.903–0.979for the middle region. 4.Numerical modelling studiesNumerical modelling studies were carried out using a com-mercially available software called PFC3D,which was developed by Itasca Consulting,and which works on the principle of distinct element modelling.4.1.Distinct element method and numerical modelling softwareof PFC3DThe distinct element method(DEM)is a numerical method which models solid objects using round,spherical and polygon-shaped particles.The basic approach and analysis of DEM was first developed by Cundall[29].It has been used in various disciplines for modelling micro-and macro-sized materials[30]. The most commonly known and used DEM programs are UDEC, 3DEC,PFC2D,PFC3D and DDA[31–33].DEM allowsfinite displacements and rotations of discrete bodies,including complete detachment,and recognises new contacts automatically during the calculation processes.In DEM, the interactions between the particles are treated as a dynamic process with states of equilibrium developing whenever the internal forces balance.The calculations performed in the DEM alternate between the application of Newton’s second law to the particles and a force-displacement law at the contacts.PFC3D is a three-dimensional numerical modelling pro-gramme that analyses the interactions of objects with each other using spherical particles.It models the mechanical behaviours of the objects according to dynamic behaviour.While the behaviour of the particles is determined using Newton’s Law of Movement as the basis,the interactions among particles are defined using contact models.The contact model in PFC3D is a soft contact method,which allows very little overlapping in a fairly small area.Particles must be in contact with each other in modelling studies performed in thefield of rock mechanics.Contact models known as the Bonded Particle Model(BPM)are used among the particles.BPM provides both a scientific tool to investigate the micromechanisms that combine to produce complex macroscopic behaviours and an engineering tool to predict these macroscopic behaviours[34].The emergence,development and interaction of microcracks affect the mechanical behaviours of rocks.Therefore, BPM is used to model the mechanical behaviours of the spherical particles of both regular and irregular sizes that are bonded to each other from a contact point.The term particle,as used here, has a different meaning to its more common definition in thefield of mechanics;here it means a body whose dimensions are negligible and that,therefore,occupies only a single point in space.Solid particles affect each other only by thin contact and they havefinite normal and shear stiffnesses.The normal stiff-ness,K n,is a secant modulus in that it relates to total displace-ment and force.The shear stiffness,k s,on the other hand,is a tangent modulus in that it relates to incremental displacement and force(Eq.(1)).An upper-case K will be used to denote a secant modulus,and a lower-case k will be used to denote a tangent modulus.The mechanical behaviour of this system is defined by the movement of each particle,force and moment affected at each contact.Newton’s Law of Motion provides the basic relationships between the force arising as a result of a particle’s movement and moments[34]:F i¼F niþF sið1ÞF ni¼K n U n n ið2ÞD F si¼Àk s D U s ið3ÞThere are two types of elements defined in PFC3D:particle and wall elements.The wall in the programme forms a solid object that keeps the particles together at the beginning of the model.In this study,particles are located inside a cube formed by the wall elements.Moreover,a circular wall representing the sawblade is also used in the modelling study.In PFC3D,particles interact both with each other and with the wall.4.2.Parallel bonded modelNumerical modelling studies have been carried out using contact models,including in PFC3D.Although there are a lot of models in PFC3D,the Parallel Bonded Model(PBM)was chosen in order to define and model connected materials with each other.The PBM describes the constitutive behaviour of afinite-sized piece of cementatious material deposited between two particles. These bonds establish an elastic interaction between particles. Thus,the existence of a parallel bond does not preclude the possibility of slip.Parallel bonds can transmit both forces and moments between particles.Thus,parallel bonds may contribute to the resultant force and moment acting on two bonded particles.Relative motion at the point of contact(occurring after the parallel bond has been created)causes a force and a moment to develop within the bond material as a result of the parallel-bond stiffnesses.This force and moment act on the two bonded particles and can be related to maximum normal and shear stresses acting within the bond material at the bond periphery.A parallel bond is defined by the followingfive parameters: normal and shear stiffness n and s(stress/displacement); normal and shear strength s c and t c;and bond disc radius R. These parameters are specified by the pb_kn,pb_ks,pb_n strength, pb_s strength and pb_radius keywords.If the maximum tensile stress exceeds the normal strength(s max Z s c)or the maximum shear stress exceeds the shear strength(t max Z t c),then the parallel bond breaks.4.3.Calibration of micromechanical parameters of rock specimensThe discrete element model can be regarded as micromecha-nical material model,with the contact model parameters being micromechanical parameters.Assuming adequate micromecha-nical parameters the required macroscopic rock properties can be obtained.The most important macroscopic rock properties include;the Elasticity modulus(E),Poisson’s ratio(n),uniaxial compressive strength(UCS)and uniaxial tensile strength(UTS). These properties were used in calibrating the micromechanical model used in this work.The discrete element model can also be calibrated using other macroscopic rock properties such as the shear strength,the angle of internal friction or fracture toughness [35,15,36].For this purpose,as seen in Fig.5,prismatic and cylindrical specimens were created according to the procedure mentioned in the PFC3D manual[30].Since it is very difficult to determine microscopic properties at particle level by experimen-tal measurements,it is essential to establish a correlation between the properties of the bulk materials and the properties at the particle level to validate the particle properties used in the modelling.During the calibration process,the micro-properties were varied systematically using the trial and error method.The axial stress versus the axial strain,and the axial stress versusN.E.Yas-ıtlıet al./International Journal of Rock Mechanics&Mining Sciences55(2012)86–96 90lateral strain curves were plotted after each trial and checked toascertain whether the curves exhibit the macro-properties.This process was repeated until the response of the model achieved a good agreement with the response of the rock sample.On completion of the calibration,the laboratory and simulation results were matched.The calibration results after modelling of E ,UCS ,UTS and Poisson’s ratio test are given in Table 2.In addition,in order to realistically represent circular sawing,the appropriate particle radii should be determined.During the calibration studies the particle radius was also changed usingthe trial and error method and for the sawing modelling the particle radius was found to be 1.5mm [37].The BPM parameters at particle level used for the numerical tests,listed in Table 3,could be introduced in the subsequent modelling of circular sawing.4.4.Constitution of the modelsThe numerical modelling of sawing operation in PFC3D consists of seven parts:determination of the restrictions,deter-mination of the material properties,formation of the model geometry and particles,definition of the relations between particles,determination of the boundary and initial conditions,initial running of the programme and monitoring of the responses of the model,re-evaluation of the model and undertaking any necessary modifications,and obtaining the results.The determination of the restrictions and the material proper-ties refers to the assumptions made during the numerical studies implemented with PFC3D.One restriction was to define appro-priate sawblade properties.The circular sawblade placed on the model was defined as being of very strong material behaving as a rigid cutting tool.The material properties for each natural stone sample to be entered into the programme were determined during the calibration of the micromechanical parameters.By forming the model’s geometry and particles,the shape of the model and the desired size of particles can be explained.The procedure for the definition of the relationships between particles means entering the bonding properties of the particles into the model in order to observe the interactions between the particles that have bonded or after bond breakage.These are the fundamental actions to constitute the model.After this process,the system will be modelled but further procedures still need to be undertaken.First of all,the installation of the sawblade and definition of the cutting condition such as different peripheral speed and advance rate of the sawblade was expressed as boundary and initial conditions should be entered into the model at boundaries.After these processes,the model will be run and the results willbeFig.5.General procedure for calibration of micromechanical properties of rock specimens.Table 2Main mechanical properties of rock specimens in the macro-test and DEM.Mechanical propertiesAVMAWMSBLExperiment resultsDEM model results Experiment results DEM model results Experiment results DEM model results UCS s c (MPa)73.9472.9251.4547.9270.0072.88Elasticity modulus E (GPa)17.6016.6815.5017.4022.2023.30Poisson’s ratio n 0.240.240.230.230.240.24UTS s t (MPa)5.499.996.225.537.056.74Table 3DEM parameters for rock specimens.AVM AWM SBL ParticlesParallel bondParticlesParallel bondParticlesParallel bondr ¼2.67g/cm 3l ¼1r ¼2.65g/cm 3l _¼1r ¼2.70g/cm 3l _¼1k n /k s ¼2.5n =s ¼2.5k n /k s ¼2.5n =s ¼2.5k n /k s ¼2.5n =s ¼2.5E c ¼18.9GPac ¼18.9GPaE c ¼17.0GPac ¼17.0GPa E c ¼23.0GPaE c _¼23.0GPa R max /R min ¼1R min ¼1.5mm s n _¼62712MPa R max /R min ¼1R min ¼1.5mm s n ¼3977MPa R max /R min ¼1R min ¼1.5mms n _¼60712MPa m ¼0.5s s _¼62712MPam ¼0.5s s ¼3977MPam ¼0.5s s _¼60712MPar density of the particles,k n /k s ratio of normal to shear stiffness of the particles,E c Elasticity modulus of the particles,R max /R min ratio of particles’maximum radius to minimum radius,m friction factor,l radius multiplier of the parallel-bond,n =s ratio of normal to shear stiffness of the parallel bond,s n tensile strength of the parallel-bond,s s shear strength of the parallel-bond.N.E.Yas -ıtlıet al./International Journal of Rock Mechanics &Mining Sciences 55(2012)86–9691investigated.If there is an error,the source of the problem will befound and the necessary changes can be made.4.5.Model geometry and particle assignmentAs a result of the calibration of micromechanical parameters,the appropriate particle size was found to be a 1.5mm radius,and so this was used during sawing modelling.The length of the marble block cut in numerical modelling was taken as 300mm.The depth of cut was taken as 60mm and the block height chosen was 100mm.The model width was selected as 6mm to comply with the segment width (5.6mm)and the channel width (6mm)formed during cutting.The model is surrounded by an element defined as a wall in the PFC3D programme.In order to determine the reactional cutting force at the bottom of the model,29measurement spheres were located (Fig.6).Various parameters within a PFC3D model can be measured over a given spherical volume.The user specifies the location and size of the measurement sphere.These spheres can be defined to measure porosity,stress,strain rate,coordination number and force.A circular wall of 500mm diameter was located in the model to represent the circular sawblade (Fig.6).During the calculation of the data taken from the modelling studies,normalisation between the experimental and numerical data was carried out for the whole line due to the difference between the length of the blocks used in experimental study (500mm)and numerical study (300mm).A total of 8276parti-cles were used in the model.4.6.Circular sawing modelling studiesAfter building the sawing model in PFD3D,different sawing conditions were modelled in parallel with laboratory studies.For this purpose,three separate models were constituted,one for each of the natural stones.Each model was run for peripheral speeds of 40–70m/s,and for advance rates of 400–900mm/min.24sawing operations were performed for each natural stone and,in total,72sawing operations were performed for all the natural stones.The numerical modelling performed in this study gave similar results to the actual sawing operation carried out at the plant.The only difference between them was the movement of the block.In the case of the numerical modelling,the block was kept still and the sawblade was moved.This difference that does not lead to any difference in the character of sawing operation,may be attributedto fact that to move lots of particles makes the modelling process slower.Therefore,resolving this by making this compromise was more practical.Moving of 8276particles would also cause the model to progress very slowly and hence the solution would certainly take a longer time to find.After the cutting model was created,the circular sawblade was placed on the model as shown in Fig.7and the simulation process was started after entering the parameters of sawing conditions into the programme.The reactional cutting forces obtained during the sawing process are given in Fig.8.In vertical direction these forces start to increase as the sawblade enters the block.After a while they increase to their maximum value and then stay the same for the whole of the middle region.As the sawblade leaves the block,the forces fall suddenly and continue to decrease until the sawblade completely leaves the block.The reactional cutting force data are divided into the same two regions as the laboratory study:the whole cutting line and the middle region.Calculation of the data was done by taking the average values for the middle region of the block and for the whole cutting line.The laboratory sawing operations showed that the reactional forces in the horizontal directions (X and Y )did not have a regular trend,therefore these data were not taken into account in the analysis.Consequently,only the vertical forces that occurred in the Z direction were examined in detail in this modelling study.4.6.1.Modelling of sawing operations for various peripheral speed conditionsThe averages of the vertical reactional forces obtained in the whole cutting line and in the middle region for 40m/s peripheral speed for AVM are presented graphically in Fig.9.When the advance rate is 400mm/min,the average vertical reactional forces for whole line and for middle region are 63.59N and 123.94N,respectively.For the 900mm/min advance rate,the average vertical reactional forces for whole line and for the middle region are 159.85N and 339.45N,respectively.100 m m300 mmMeasurement Spheres500 mmRotation Direction Advance DirectionFig.6.Block model was created in the PFC3D programme.Ball breaking from block Resultant reactional forces affecting at cutting lineFig.7.Locating the circular wall used as a circular saw and running the model.N.E.Yas -ıtlıet al./International Journal of Rock Mechanics &Mining Sciences 55(2012)86–9692。