International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186

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International Journal ofRock Mechanics & Mining Sciences

1365-16

http://d

n Tel.:

E-m

journal homepage: www.elsevier.com/locate/ijrmms

Study of pressure distribution within the crushed zone in the contact areabetween rock and disc cutters

Jamal Rostami n

Department of Energy & Mineral Engineering, Pennsylvania State University, University Park, PA 16802-5000, USA

a r t i c l e i n f o

Article history:

Received 11 August 2011

Received in revised form

10 June 2012

Accepted 27 July 2012Available online 5 November 2012

Keywords:

Tunnel Boring Machines (TBM)

Mechanical Excavation

Cutting Forces

Disc Cutters

Rock Fragmentation

Rock Indentation

09/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.ijrmms.2012.07.031

þ1 814 863 7606; fax: þ1 814 865 3248.

ail address: rostami@psu.edu

a b s t r a c t

Estimation of the cutting forces acting on a disc cutter while cutting rock has been used for cutterhead

design and performance prediction of tunnel boring machines (TBMs) and various other mechanical

excavators. The cutting forces are the result of the pressure in the contact area between the disc cutter

and rock surface. This paper presents the result of direct measurement of pressure in this contact area

and observed pressure distribution patterns when cutting various rock types. The results show that the

pressure within the contact zone is more concentrated, and the actual pressurized area is smaller, than

the size of the contact zone that has been assumed by previous models. The measurements also show

that there are areas of no pressure within the theoretical contact zone both in front of the highly

concentrated pressure zone and behind it. This indicates that the peak stresses within the cutter ring

are higher than normally expected, and explains the reason for some observed behavior of various disc

cutters in the field.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Cutting of rock with mechanical tools has become moreprevalent in various underground and surface operations due tothe trends for mechanization and increased utilization of mechan-ical excavators in mining and civil projects. In hard rock applica-tions, the most efficient and most popular cutting tool is singledisc cutters. Disc cutters are used frequently on different types ofhard rock machines and have become the standard tools on hardrock TBMs. As such, estimating the forces for cutting a certaintype of rock with disc cutter is an essential part of TBM cutter-head design and performance prediction. The estimated cuttingforces can be utilized for different purposes and providing anaccurate force estimates depend on the level of understanding ofthe cutting process.

The process of crushing of the rock under an indentor or acutting tool is a very complex phenomenon and has remained oneof the fields that is not well understood. This is mainly due tolimited measurements that can be and have been made withinthe pressure bubble or crushed zone, which develops directlyunderneath the cutter and controls the process of rock breakage.Very few direct measurement of pressure distribution in thecrushed zone has been made. The study of pressure profile inthe indentation tests under a spherical indentor by Gertsch [1] is

ll rights reserved.

one of the few studies of this kind. This is while there are studiesof indentation forces where a generalized pressure has beenassumed in the crushed zone [2].

While some headways has been made in recent years insimulation of the rock cutting with numerical models such asdiscrete element or finite element methods, these models dorequire a reality check, which is direct measurement of stressesunder the cutting tool for verification. The application of thenumerical analysis methods based on continuum media forsolving this problem has had very limited success so far,owning to the disintegration of the rock particles whichundermines the basic assumption of most of these methods, thatis the continuity of the media. This limit has been partiallyaddressed by some of the recent improvements in FEM modelsby introducing the capability to simulate failure within therock [3,4] through removal of the elements within area of veryhigh stress.

Meanwhile, discrete element analysis (DEA or DEM) is almosta natural choice of numerical modeling when it comes to rockfragmentation since it allows for separate bodies to be attachedand separated and thus allows for modeling a crack growth. Thesimulations of rock cutting by DEA models have been veryintriguing and seem realistic. Examples of simulation runs byPFC and PFC3D for rock cutting show the potentials of thisapproach [5–7]. However, these models need to be coupled withactual measurements for calibration to allow for simulation ofrock cutting in the same rock type under different tool or cuttinggeometry. As it stands, estimation of cutting forces by simulation

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186 173

of rock fragmentation through numerical modeling based on rockphysical properties, such as measured rock strength, can lead tomajor errors.

Magnitude and composition of the cutting forces on a singledisc is a function of many parameters such as rock type and itsmechanical characteristics, planes of weakness and discontinu-ities, geometry of the disc, spacing between the cuts, depth ofpenetration, and machine specifications. There have been manystudies of rock cutting forces acting on disc cutters in various rocktypes and models for prediction of these forces for various endpurposes. This paper will review the background of rock cuttingprocess by disc cutters and the basic theories behind the existingmodels and force estimation formulae. The efforts made tomeasure the pressure in the rock-disc contact area and the resultsof the measurements will be discussed. Finally an equation forestimation of cutting forces acting on disc cutters will be offered.The highlight of the study, which is the measured pressuredistribution in the contact area between the disc and rock surface,will be presented and its practical implications are reviewed.Detailed information and all background data for this study canbe found in [8].

2. Background

Disc cutters have been in use for number of years, but themajor practical use of the disc cutters go back to its application ona TBM in mid 1950s. Through the years there has been a series ofimprovements in the profile and bearing capacity of the discs. Themost common disc cutters used on TBMs today is the 432 mm(17 in.) discs with constant cross section (CCS) profile. Recently,the use of 483 mm (19 in.) cutters have received more tractiondue to the higher cutter life in the field, especially in very hardrocks and difficult ground conditions that makes cutter change atthe face more difficult and risky [9,10]. A review of the disctechnology and discussion of various disc sizes and profiles can befound in [11].

Many researchers have worked on the issue of rock cutting withdisc cutters over the past few decades. The focus of the work rangedfrom the simple indentation of discs into the rocks to evaluate rockbrittleness, to estimating cutting forces for cutting rock with certainmechanical properties. Examples of the works on indentation processgoes back to early researchers [12,13] who studied the indentation ofa rigid body into the rock using theories of elasticity. Lawn and Swain[14] offered a review of the state of stresses within the rock underpointed load. This work was followed by many others including thatof Pang et al., Lindqvist, and Cook et al. [2,15,16], who studied theprocess of indentation, development of the crushed zone, and theinitiation and propagation of radial cracks around the crushed zone.

The studies on the rock cutting with disc cutters were primarilyconducted by running full scale tests in the laboratory using actualdisc cutters on linear or rotary cutting machines. This includes thework by Ozdemir et al. [17] and Roxborough and Philips [18], whodeveloped theoretically derived predictor equations for estimatingcutting forces, which did not include the effects of spacing. Roxbor-ough and Phillips [18] developed equations for cutting forces of a V-shape disc using the UCS and disc diameter and tip edge angle. Thiswork was followed by Howarth and Roxborough [19] to include theeffects of joints and encounter them into the equations. Bilgin[20] investigated the mechanical cutting characteristics of somemedium and high strength Igneous rocks. Ozdemir [21] devel-oped a set of predictor equations for V-shape cutters which usedthe cutting geometry (S, P) along with rock uniaxial compressive(UCS) and shear strength to calculate cutting forces (normal androlling) for both new and worn cutter. Sanio [22] also introduceda model for estimating the cutting forces on V-shaped disc cutters

using fracture toughness indices of the rock and accounting forexistence of joints in the rock mass. Sato et al. [23,24] followedSanio’s work and using the same approach, but on a rotary cuttingmachine and offered new equations for force estimation.

Rostami [25,26] offered a set of equations for estimation of thecutting forces acting on disc cutter for constant cross section disccutters (CCS). The original formula was based on various pressuredistributions in the contact zone between the rock and cutters fordifferent cutter profiles. This work was followed by additionaltesting and measurements on the disc cutters and introduction ofa new and dimensionally balanced formula for estimation of disccutting forces [8,27]. These formulas are better known as CSMTBM performance prediction model and have subsequently beenstudied and amended to include additional factors such as rockbrittleness represented by rock indentation or better known aspunch penetration tests [28]. Cheema [29] introduced the conceptof rock mass boreability index based on the CSM model and use ofrock mass classification system. Ramezanzadeh et al. [30] offeredan adjustment factor to account for rock mass properties inconjunction with the CSM model. Gong et al. [31,32] looked intothe impact of joint orientation on performance of TBMs and disccutters through numerical modeling and impact of rock masscharacteristics. Balci and Bilgin [33] compare the measured forcesbetween the small and full scale cutting tests and Balci [34] hascompared the results of rock cutting with disc cutters and TBMfield performance.

The review of available literature shows that there is sub-stantial amount of research on estimation of the cutting forcesacting on disc cutters in various rock types, but not much work onthe measurement of contact pressures between the disc cutterand rock and current paper is an attempt to address this issue.

3. Pressure distribution within the crushed zone

The process of indentation of a cutting tool into the rock startswith the development of a pressure bubble or crushed zone underthe tool. This zone transfers the cutting forces to the rock andcomprises a fine powder immediately adjacent to the tool, whichquickly transitions to coarse grains and to the fractured surround-ing areas and ultimately to the body of the rock. Study of thecrushed zone is important in that the dust generated in theprocess of rock cutting is primarily the result of formation of thiscrushed zone. Therefore, the smaller the crushed zone, the betterit is from the dust generation viewpoint. Obviously, creation offiner particles takes more energy, thus smaller volume of crushedzone is also preferred from cutting efficiency point of view.Moreover, the crushed zone and distribution of the pressurewithin this zone determines the magnitude and direction of thecutting forces. Rostami [25,26] offered a review of variouspressure distribution models within the crushed zone, under thedisc cutters. The full scale testing performed as part of theaforementioned studies showed that the distribution of pressurein this zone is a function of the rock properties and moreimportantly, the disc cutter profile as represented by the tipwidth.

In the initial studies, the contact zone or contact area betweenthe disc cutter and rock surface was estimated based on availabletheories and cutting geometry as shown in Fig. 1. This figure is ageneralized pressure distribution within the contact area basedon the cutting geometry, including the disc diameter and depth ofpenetration. The size of nominal contact area could be simplycalculated as

Ac ¼ RTj, where j¼ cos�1 p

R�p

� �ð1Þ

Fig. 2. The anticipated pressure distribution that could meet the boundary conditions.

0.45"

9.875"

2.38"

5

1.0

ID=

45

Fig. 3. Picture and drawing of the cross section disc cutter used for experimental

program [8].

Fig. 1. Generalized illustration of the contact area between a disc cutter and

rock [24].

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186174

and Ac is the contact area (mm2 or in2), R is disc cutter radius (mmor inch), T is tip width (mm or inch), j is the angle of thetheoretical contact area (Rad), and p is depth of cutter penetrationor penetration per revolution.

As shown in this figure, various functions can be considered torepresent pressure distribution in the contact zone. Mostresearchers have used a linear increasing pressure profile in thiszone [17], where the contact pressure starts from zero andincreases to its maximum immediately under the disc. The onlycontrol point for verification of different distribution systems isthe location of the resultant force, which is the combination of thenormal and rolling force acting on the disc and the angle of theresultant force with the direction of normal force. By determina-tion of this angle, ‘‘b’’, the ratio of rolling to normal forces can becalculated. This ratio is better known as ‘‘cutting coefficient (CC)’’or ‘‘rolling coefficient (RC)’’, and can be determined for variousdisc cutters and rock types through full scale lab testing.A comparison of measured and calculated values of rollingcoefficient for CCS disc cutters showed that linearly increasingfunction is not the best representation of the pressure distributionin the contact zone under the disc cutter. Instead, the measure-ments indicated that a pressure distribution which is nearlyuniform can be a better representation of the contact stressesbetween the rock and disc.

However, while the assumption of a given contact area and amore uniform pressure distribution could provide a better fit forrolling coefficient measurements, it failed to meet the boundaryconditions at both end of the contact zone. In other words, it isnot logical to have high pressures at the starting point of contactbetween the disc and the rock surface in front of the disc, nor atthe point of departure of the disc from the cut surface. Thus themagnitude of pressure at boundary conditions had to be zero andgradually raising to a peak somewhere in the middle of thecontact zone as shown in Fig. 2. However, there was no otherinformation to allow for determination of the shape of thispressure distribution function and knowing the proportion ofthe normal and rolling forces was insufficient for suchdetermination.

To verify the pressure distribution within the contact zone anddevelop a function to estimate magnitude of pressure at givenpoints in this area, a testing program was devised. The testingincluded installation of a series of strain gages on the disc cutterto sense the load distribution on the contact zone as these gagesapproached, entered and departed from this zone. The assump-tion was that the interpretation of the collected signals from thegages during the cutting process will allow for identification ofthe pressure distribution in this zone. This assumption was

indeed challenged in many levels as the experimental programproceeded and subsequent data analysis presented many unanti-cipated difficulties.

4. Instrumentation of the disc cutter

To measure the pressure distribution within the contact zone,it was deemed necessary to instrument a disc cutter. This was dueto the fact that making such measurements in the rock was notpossible because of creation of cracks that would compromise thecontinuity of the medium in which the measurement was to takeplace. Thus the only logical choice was to measure stress varia-tions within the disc, as close to the tip of the cutter blade aspossible, since the disc would be continuous and behave linearlyas the cutter rolled over the rock surface. For this purpose a disccutter made available by the Robbins Co. was used in the testingprogram. The selected cutter was a so called ‘‘Dura-Cutter’’ or a‘‘long-neck’’ constant cross section disc cutter. The main char-acteristic of the selected cutter is extended length of the bladewith more or less constant thickness as can be seen in Fig. 3.

To measure the stresses within the disc blade, a series of straingages were installed near the tip on both sides of the blade. Straingages used for this purpose were model CEA-06-125WT-350manufactured Vishay Measurement installed on the blade usingM-Bond 610 and cured of two hours at 150 1C. The gages wereinstalled at 13 mm (0.5 in.) from the tip of the blade (to the edgeof the gage) at 9 positions, 151 apart to cover about 1201 (1/3rd) of

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186 175

the total periphery of the disc. After initial measurements andmodeling it was decided that the monitoring of the radial gageswould be sufficient, thus only the radial gages were monitored ina half Wheatstone bridge which combined the measurement of aset of gages on both sides of the blade. Fig. 4 shows the picture ofthe instrumented disc.

Analysis of anticipated stresses within the disc blade caused bya given force at certain location on the blade tip was performed todevelop the force-strain variation or an ‘‘influence’’ function. Thisallowed determining the amount of strains anticipated from agiven load placed at certain angular distance from the center of

Fig. 4. Picture and schematic

Stresses as a Fun

-40,000

-35,000

-30,000

-25,000

-20,000

-15,000

-10,000

-5,000

0

5,000

-30 -25 -20 -15 -10 -5 0

Angle α

Stre

ss (

psi)

Fig. 5. The calculated stresses at the center point of str

the strain gage. The analysis included closed form solution, aswell as numerical modeling by finite element software programALGOR. Fig. 5 shows the comparison between the estimatedstresses within the blade by FEA and theoretical calculations.Fig. 6 shows the influence function generated by the calculationand numerical modeling and they have a perfect match. Thesecalculations were followed by full scale calibration of the instru-mented disc cutter using a hydraulic jack, where the linearity ofeach set of gages for anticipated range of forces was verified. Theresult of calibration closely matched the calculated influencefunction.

of the instrumented disc.

ctin of Angle

5 10 15 20 25 30

(deg)

Stre

ss (

Mpa

)

25

-25

-75

-125

-175

-225

-275

ain gages from theoretical and numerical analysis.

20 15 10 5 0 5 10 15 205

0

5

10

15

Position Angle (deg)

13.5359

-0.28562

Out

put V

olta

ge (

mV

)er 200.5,

0.001

2020

Output Signal of the Circuit Estimated by FEM Analysis

-0.014

-0.012

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

3020100-10-02-30

Angle (deg)

Vol

tage

(V

)

Calibration Simulation

Fig. 6. Plot of the influence function (anticipated strain for a given load versus location of the load). top: by closed form solutions; bottom: by FEA analysis.

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186176

5. Analysis to determine pressure distribution from gage signals

The theory behind the analysis of measured signals to backcalculate the original pressure or load distribution that was imposedon the disc blade in the contact zone was basically the principal ofsuperposition. This simply means that if a series of loads were to beapplied on the tip of the blade at known locations, the strainmeasured at a certain location on the blade (e.g., the center point ofstrain gages) are the cumulative strains caused by each individualload (Fig. 7). Thus, as the gages get close and pass through the contactarea, a signal will be recorded which can be later used to determinethe variation of the pressure components imposed on the cutter bladeas a series of loads spread within the contact point (Fig. 8).

In this analysis, there are certain control parameters, includingthe cutting forces expressed in terms of normal and rolling forces.The relationship between the cutting forces and the pressure orload distribution in the contact area can be expressed as follows:

FN ,FR½ � ¼ L1 L2 :: LN� �

cosy1

cosy2

:

cosyN

siny1

siny2

:

sinyN

266664

377775¼

XN

1Licosyi,

XN

1Li sinyi

h i

ð2Þ

where FN is the normal force, FR is the rolling force, Li is the load atpoint i¼PiTRd yi, Pi is the pressure at point i, T is the tip width ofthe cutter, R is the cutter radius, y is the angle, and N is thenumber of loads/discrete areas.

The other control parameter is the measured signal whichcan be related to the load distribution through the followingequation:

Sj ¼X

iEij ¼

XiLif ðXijÞ ð3Þ

where f(Xij) is the influence function reflecting the contribution ofthe load Li on the gage j given the spatial location of the load andthe gage. This is equation can be expressed in vector or matrixform as follows:

S!¼ L!�_C ð4Þ

where S is the vector of measurements (force, output signal, etc.),L is the load vector/string, one column matrix containing Li’s, andC is the coefficient matrix, containing entries that are calculatedfrom the influence function. Therefore, the ‘‘ij’’ entry of the ‘‘C’’ isf(Xij). This system of equations can be solved since the matrix ‘‘C’’has rank N. Assuming that Vector ‘‘L’’ has the size of N, for asimple case, the size of vector ‘‘S’’ is the same and the coefficientmatrix C is then N by N, f(Xii)¼ f(0)¼1.

-1000

0

1000

2000

3000

4000

5000

6000

7000

Distance/Position Angle (Deg)

Out

put S

igna

l.

-500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

-25 -20 -15 -10 -5 0 5 10 15 20 25

-25 -20 -15 -10 -5 0 5 10 15 20 25

Distance/Position Angle (Deg)

Out

put S

igna

l.

Fig. 7. The principal of superposition as applied to anticipated signal measured from the strain gage for (a) uniform, (b) for a variable load distribution.

Fig. 8. The schematic drawing of the strain gage moving through the contact zone.

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186 177

Given the form of the influence function, the coefficient matrixwould look as follows:

C ¼

1 0:999 0:998 0

0:999 1 0:999 : :

0:998 0:999 : : 0:998

: : : 0:999

0 : 0:998 0:999 1

26666664

37777775

ð5Þ

The implication of this setting was that the load distributioncould be broken to a finite number of locations, in this case ninecomponents, to see the snap shot of load distributions at givenpoint in time, if the impact of load distribution on all 9 gagelocations were to be analyzed at the same time. The alternativeapproach was to use the data from the same gage at certain timeintervals as it approached the contact zone and passed through it.The implication of the second setting was that the load distribu-tion must have stayed constant to allow the subsequent analysis.In this case the data interrogation rate could be set to pickup certain number of samples within the given time spanto match the number of load distribution components for thecoefficient matrix to be N by N and thus allow for solvingthe system of equations. The later approach was selected for theanalysis.

Obviously, the load distribution is constantly changingas new cracks develop and crushed zone undergoes loadingand unloading cycles. The objective of the analysis usingthe later setting was to be able to determine an averageload distribution system that could allow for understandingthe general load distribution that could yield the averagenormal and rolling forces in that period of time. However,the actual conditions did not lend itself to application ofeither of these methods and a different analysis had to bedeveloped for subsequent data analysis as will be explainedlater.

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186178

6. Full scale testing of the instrumented disc cutter

After installation of the instrumented disc cutter on the linearcutting machine (LCM) and calibration of the system for eachindividual gage set, as well as machines triaxial load cell, the fullscale trials commenced. Fig. 4 shows the configuration of theinstrumented disc with a pair of cables on each side coiled aroundthe hub of the cutter to allow for the rotation of the disc duringthe test. Since the length of the cuts were less than 1 m, the discrotated less than one full rotation and thus the cable could bewound around the hub and unwound as the disc returned to itsoriginal position.

Fig. 9. The picture and plot of force variation and the output signal

To verify the response of the system, a steel plate was cast ontop of the sample box and instrumented disc was rolled over thesteel plate to see the response of the system and the gages to thepoint contact between the cutter blade and the steel plate. Fig. 9shows the picture of the sample as well as the plot of the forcesand output signals from the strain gages. As anticipated, thesignals were similar to the shape of the point contact influencefunction and corresponding to the magnitude of the forces for agiven point in time.

With the verification of the system response as a whole, thefull scale rock cutting tests using the instrumented disc cutterwas commenced. The testing included three different rock types

of the gages as a function of time for the test on the steel plate.

Table 1Test matrix for all the rock types used in the experiments.

Rock Type Penetration

2.5 mm (0.100) 5 mm (0.200) 7.5 mm (0.300) Notes

Test series name (Robbins cutter instrumentation & rock & test set) (No of Passes), Number of data lines pre path

Indiana limestone RCIIL01 (3), 11 RCIIL02 (3), 19 RCIIL03 (3), 10 Medium strength rock with UCS of �50 MPa

Colorado red granite RCIRG01 (3), 9 RCIRG02 (3), 12 RCIRG03 (3), 12 Hard rock with UCS in the 140 MPa range

Umettela basalt. RCIBS01 (4), 12 RCIBS02 (4), 15 RCBSG03 (3), 12 Hard volcanic rock with UCS in 180 MPa range

Spacing was fixed at 75 mm (3 in.).

Fig. 10. Picture of the instrumented disc after testing, with some damaged gages.

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186 179

and cut spacing and penetration as summarized in Table 1. Threedifferent rock types were selected for testing from a set of well-known and tested rock samples available, including IndianaLimestone, Colorado Red Granite, and Umittela Basalt [35,36].Testing was performed at spacing of 75 mm (3 in.) and penetra-tions of 2.5, 5, and 7.5 mm (0.1, 0.2, 0.3 in.). Higher penetrationwas deemed inappropriate since it could harm the gages. Mean-while, during the testing on the granite and basalt samples somerock fragment did penetrate the gage shields and rip the gagesand permanently damage them, thus rendering them broken anduseless. Fig. 10 shows the picture of the instrumented disc and adamaged gage. Consequently, some of the gages were taken out ofcalculations and subsequent analysis as the testing progressed.

Figs. 11–13 shows examples of force and gage responses forcutting limestone, granite, and basalt, respectively. As can be seenin these figures, while the gage responses in limestone samplewere very well behaved, the gage responses in granite and basaltsamples were very variable and with rather irregular shape.

7. Data analysis

Once the recording of the forces and gage responses wereavailable, a program had to be devised to look at the various loaddistributions that could generate the recorded signals. The idea-lized theoretical approach through the use of commercial solversto solve the systems of equations indicated in Eq. (4) to determinethe load vector from the coefficient matrix and signal vector wassoon proved to be inapplicable. This was done both on individualgage response as well as cumulative and averaged gage responsesignal and the answers were the same. While one can list variousreasons (non linearity of the system, variation of speed of cut thuslocation pointers, rapid change in load configuration, etc.) for the

unstable solutions rendered by the solvers, the end result wasthat the load distribution could not be back calculated from thesignals.

As a result, other approaches were examined to seek a solutionfor this problem. This included singular value decomposition,wavelets, and application of various solvers, where none couldoffer a sensible solution and reasonable load distribution. Thus adifferent approach was used to analyze the data and that wasapproximation method.

In this scheme, the signals from various gages were super-imposed using the position pointer in the data columns, andaveraged over the span of the time series used to evaluate thesignals. This was practically performed by snapping the data setsfor each gage in the vicinity of the contact zone and superimposing the signals relative to a reference point for a giventime span, and averaging the values of measured signals for everygiven time steps. The data sampling rate was 2000 Hz and atnominal cutting speed of about 50 mm/s, each time step corre-sponded to travel distance of 0.025 mm (0.01 in.) or 0.0671 ofrotation on the disc. A set of 500 time steps deemed sufficient tocapture the load distribution within the anticipated contactarea for various depths of penetration. Thus for each lineof cut, the signal responses were reduced to an average signalwhich presented the measurement of all the gages. To allow thesesignals from various cuts to be compared in their shape, thesignal was normalized relative to its peak value. Thus thisprocedure generated a hypothetical gage response signal thatvaried between 0 and 1. Fig. 14 shows examples of cumulativeand normalized signals for various rock types. As one canobserve, the shape of average signal seem very logical for thelimestone sample, whereas it takes an unusual shape for basaltsample.

Yet the average signal could not be processed through the saidequation system and generate an ‘‘average’’ load distribution forthe rock type. To address this issue a reverse calculation approachwas used. In this approach, a series of known load distributionswere defined and the resulting gage signal response was calcu-lated. Then these calculated responses were compared withaverage signals from measurements to see which pre-determined load distribution could generate an output signalwith minimal differences with the measured signal. The pre-determined load distribution systems were then given a casenumber and the top ten closest distribution cases to the measuredsignal was recorded for each cut line. Then a statistical programwas used to seek the highest number of occurrence or selection ofthe load distribution cases for the entire test based on frequencyand rank of the case. This method was considered to allow forapproximation of a reasonable load distribution that could mimicthe average load distribution in the contact zone. The magnitudeof the resultant forces for each case could be compared with thatof measured average cutting forces to derive the correct

150

125

175

100

75

50

25

0

-25

-50

Forc

e (k

N)

Forc

e (k

N)

0.5

1

0

-0.5

-1

-1.5

-2

-2.5

-3.5

-4.5

-5

-4

-3

Fig. 11. Plot of an example force variation chart and the output signal of the gages as a function of time for the testing in Indiana limestone.

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186180

magnification of the pre-determined distribution that couldmatch the actual force measurements.

The pre-determined loading cases included a series of differentload distribution scenarios including a uniform distribution (rec-tangles), linear distribution with peaks varying from one end tothe other, exponential distribution, power function, etc. Since thespan or distance between the beginning and end of the loadingpoints was unknown, the algorithm started from a set with onlyfive loading points (1.25 mm¼0.05 in.¼0.3351 wide) and at eachstep, added 10 points (2.5 mm–0.1 in.¼0.671) to the span. Thiswas continued to cover a range of loading spans from 5 to 245points representing the loading span from 0.031 to 171 angles.Within the loading area (span) the magnitude of the loads varieddepending on the selected type of distribution mentioned earlier.The total length of the vector representing the load distributionwas selected to be 500 points or 125 mm (5 in. 341), which is overtwice the angular distance between the gages. The actual loadingpoints were placed at the center of the vector, with zeros beforeand after the loading span.

Fig. 15 shows the examples predetermined load distributionscenarios used in the analysis to find which load distribution canclosely mimic the shape of the average normalized measuredgage signals. All together, the total number of predeterminedload vectors considered was 385 cases. For each load vector,the expected output signal was calculated using the equationsmentioned earlier. These signals were normalized to offset the

variation in the load values and spans. Therefore, the entries ofpredetermined output signals varied between 0 and 1 as well.

The actual calculations and analysis were performed in aMathCad code controlled by an Excel spreadsheet throughdynamic data exchange (DDE) program. The program openedthe data files and used proper calibration factors to determine thecutting forces and gage output signals. The result of the analysisincluding the average/min/max cutting forces, standard deviation,specific energy, rolling coefficient, and top ten best matches inload distribution was reported back to Excel sheet and recoded foreach line. This process was repeated for all the lines in the test foreach rock type and penetration value. Then the statistical analysiswas performed to identify the loading scenario that best repre-sented and matched the measured strain gage signals.

8. Discussion of the test results

The result of the testing and subsequent analysis was veryinteresting and eye opening. While the cutting forces werefollowing the anticipated trend (Fig. 16), the load distributionmeasurements were a different story. First issue that wasobserved was the small footprint of loading area compared tothe theoretical/nominal contact area calculated from cuttinggeometry. While the shape of the approximated load distributionconfiguration was obviously not the exact shape of the true

250

200

150

100

50

0

-50

-100

Forc

e (k

N)

Forc

e (k

N)

2

0

-2

-4

-6

-8

-10

Fig. 12. Plot of an example force variation chart and output signal of the gages as a function of time for the testing in red Granite. (For interpretation of the references to

color in this figure legend, the reader is referred to the web version of this article.)

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186 181

load distribution in the contact zone (due to computationissues discussed before), the span of the output signal and thepossible or closest replica of load distribution simply indicatedthat the width or extent of the actually loaded zone withinthe contact area is relatively small. Fig. 17 shows the approxi-mated load distribution within the contact area for the threerock types tested. As can be seen, in each rock as the depthof penetration increases the area/span of the loading zoneincreases.

The alignment of the resultant forces from the assumedpredetermined load distribution with that measured cuttingforces by the LCM load cell (based on the rolling coefficient orratio of rolling ‘‘FR‘‘ to normal ‘‘FN’’ forces) shows that this area ofconcentrated loading occurs in the middle of the theoreticalcontact area. This means that there is an area of no load withinthe theoretical contact area in front of the high pressure zone, andan area of no loading directly underneath the disc, as can be seenin Fig. 18. An evidence for validity of this conclusion is theobservation of disc cutting process using high speed camera.The close examination of the direction of movement of dust in thecutting process shows that in some occasions, there is a portion ofthe dust that gushes out underneath the cutter, in the directionopposite to the direction disc movement. This indicates that thecrushed zone directly beneath the disc undergoes a cycle ofpressure relief by releasing some of the crushed material behind

the disc. This process is less visible in softer rocks as the crushedmaterial seem to squeeze out to the lower pressure zone behindthe track of the disc.

Table 2 is the summary of measured footprint and location ofloading zone as well as non-loading zones for testing in differentrock types and at various depths of penetration. The table showsthat the span or extent of the loading zone in general increaseswith the depth of penetration. This trend is more pronounced inlimestone and is logical. The width of the loading zone in harderrock, and especially in basalt was much smaller than limestone. Inthe case of basalt, the extent of the loading zone showed littlesensitivity to the depth of penetration, which is somewhatsurprising. Needless to say that the amount of available data atthis stage is very limited thus one should be careful in general-izing the conclusions.

The implication of this finding is rather important. First itinvalidates the initial assumptions about the size and extent ofthe contact zone. Second, it shows that the stresses within thering are more concentrated than initially assumed. In fact some ofthe observations in the field of cutter rings mushrooming orgetting loose on the hub could be a good evidence of thisphenomenon. In these cases, the cutter ring was designed for anaverage stress in the contact area where the actual stresses werein fact higher due to loading area being smaller than thetheoretical contact zone.

350

300

250

200

150

100

50

0

-50

-100

5

0

-5

-10

-15

-20

Forc

e (k

N)

Forc

e (k

N)

Fig. 13. Plot of an example force variation chart and the output signal of the gages as a function of time for the testing in umittela basalt.

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186182

These observations open the door to the new interpretations ofthe cutting process. The no loading zone in front of the disc couldsimply be justified by creation of the chips ahead of the disc as thearea of high pressure is created and the wave, however slowly,moves forward. The actual cutting/crushing occurs in a zonewhere the disc engages the rock and starts the penetrationphenomenon by creation of the pressure bubble. The pressureincreases to a peak, and so is presumably the depth of the crushedzone, and gradually decreases as the pressure of the crushed zoneis relieved by creation of the chips on the sides or the movementof the fine particles towards the back, where there is no pressure.Hence, the flow of the fines towards the back of the disc in thearea directly underneath the disc forms a none- loading area inthis zone.

9. Estimation of the cutting forces

The findings of the experimental program and testing did notlend itself to develop a new model for estimation of the cuttingforces since there are too many unknowns at this stage to allowfor development of a general model. This includes the shape,magnitude, and location of the loading zone as well as the impactof rock properties on the pressure within the crushed zone. Thus,to address the need for development of a simple formula forestimation of the cutting forces acting on disc cutter in differentrock types, a different approach was taken. For this purpose, a

database of rock cutting forces which was initially developed by[25] was extended to include addition full scale cutting test dataand was subsequently used to derive new formulas for estimationof normal and rolling forces through statistical analysis. Moredetailed information and the actual database of cutting forces canbe found in [8]. The database included disc sizes from 125 to483 mm in diameter, 7.5 to 25 mm in tip width and rocks fromvery soft 50 MPa to nearly 300 MPa in compressive strength. Thepenetrations varied from 2 to 30 mm and spacing of 12 to150 mm was also included in the database. The initial formulaswere slightly modified to obtain a dimensionally correct equationfor force estimation that can be used with any unit system asfollows:

FN ¼ TRjPr ð6Þ

Pr ¼ C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSsc

2st

jffiffiffiffiffiffiRTp

3

sð7Þ

where R is the disc cutter radius (mm or inch), T is the disc cuttertip width (mm or inch), j is the angle of theoretical contact area,Pr is the nominal average pressure (psi or MPa), S is the cutspacing (mm or inch), sc is the uniaxial compressive strength ofthe rock (MPa or psi), st is the Brazilian Tensile Strength ofthe rock (MPa or psi), C is a constant, and for the general case(not rock type specific), C¼2.12.

Fig. 14. Plot of an example normalized combined gage signals for the testing in (a) limestone, (b) granite and (c) basalt.

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186 183

The rolling forces can be estimated by using the rolling coeffi-cient, which is the ratio of rolling to normal forces, as follows:

RC ¼FR

FNð8Þ

In order to estimate rolling coefficient, there are severalreliable formulations. For example:

RC ¼ tanj2

� �ð9Þ

0

20

40

60

80

100

120

140

160

180

200

0 1 2 3 4 5 6 7 8 9

Nor

mal

For

ce (k

N)

Penetration (mm)

Umettela Basalt

Colo. Red GraniteIndiana Limestone

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9

Rol

ling

Forc

e (k

N)

Penetration (mm)

Umettela BasaltColo. Red GraniteIndiana Limestone

-10

-5

0

5

10

15

20

25

30

35

40

45

0 1 2 3 4 5 6 7 8 9

Side

For

ce (k

N)

Penetration (mm)

Umettela Basalt

Colo. Red GraniteIndiana Limestone

0

5

10

15

20

25

0 5 10 15 20 25 30 35

Spec

ific

Ener

gy (k

W-h

r/m

3 )

S/P Ratio

Umettela BasaltColo. Red GraniteIndiana Limestone

Fig. 16. Summary of the test results for normal, rolling, and side forces as well as specific energy for various rock types.

0 100 200 300 400 5000

0.5

1

1.5

Lu,1 j

Lu,5 j

Lu,10 j

Lu,15 j

Lu,20 j

Lu,24 j

j0 100 200 300 400 500

0

0.5

1

1.5

Lt,1 j

Lt,5 j

Lt,10 j

Lt,15 j

Lt,20 j

Lt,24 j

j

0 50 100 150 200 250 300 350 400 450 5000

50

100

150

200

250

Lh,24 j

Lh,49 j

Lh,74 j

Lh,99 j

Lh,124 j

Lh,149 j

Lh,174 j

Lh,199 j

Lh,224 j

Lh,249 j

Lh,274 j

j0 100 200 300 400 500

0

0.2

0.4

0.6

0.8

1

Lp,1 j

Lp,3 j

Lp,5 j

Lp,7 j

Lp,9 j

Lp,11 j

Lp,13 j

Lp,15 j

Lp,17 j

Lp,19 j

j

Fig. 15. Plot of various examples of pre-determined load distributions, the result of which was compared to the measured strain gage signals.

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186184

160 180 200 220 240 260 280 300 320 3400

5000

1104

1.5104

2104

2.5104

3104

.CIL01

.CIL02

.CIL03

l160 180 200 220 240 260 280 300 320 340

0

1104

2104

3104

4104

5104

6104

7104

8104

.CRG01

.CRG02

.CRG03

l

160 180 200 220 240 260 280 300 320 3400

2104

4104

6104

8104

1105

1.2105

1.4105

.CBS01

.CBS02

.CBS03

l

Fig. 17. The approximated load distributions within the contact zone for various rock types and depth of penetration. (unit of the load is lb/in¼0.17 N/mm). (a) Limestone,

(b) Colorado Red Granite and (c) Umittela Basalt.

Fig. 18. The postulated load distribution in the contact zone.

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186 185

where p is the penetration, D is the cutter diameter, and j is theangle of the arc of contact.

As noted before, this formula has been in use for over a decadewith reasonable degree of success and more detailed information

on the use of the cutting forces for TBM cutterhead design orperformance predictions can be found in Rostami [37].

10. Conclusions

Direct measurement of the pressure or load distributionwithin the contact zone between disc cutter and rock has provedthat the actual loading zone is smaller than the anticipated ortheoretical contact area. This indicates that higher stresses couldbe experienced in this zone, beyond what was previouslyexpected and used in many of the models. Much more workneeds to be done to fully understand the behavior of rock in thecrushed zone and surely additional measurements are needed ondifferent cutter profiles and rock types to allow for developmentof a coherent theory for the development of the crushed zone, itsshape and load distributions, and its relationship with variousrock properties. However, it seems like the simpler formulas canstill be used to estimate the cutting forces for practical uses. Withso many parameters to define the shape, extent, location, andmagnitude of the components of the load distribution, it isunlikely that the true load distribution configuration within thecontact area can be used for estimation of cutting forces in nearfuture. Meanwhile, study of the load distribution in the contactarea can be used for refining the numerical modeling of the

Table 2Summary of the results for load distribution analysis.

Test Penetration Theoretical contact area Actual loading area Front non loadinga Rear non loadingb

(mm) (in) Points Deg. Rad Pt Deg. % Pt Deg. % Pt Deg. %

IL01 2.5 0.1 131 8.8 0.15 102 6.8 78 18 1.2 14 11 0.74 8

IL02 5 0.2 184 12.5 0.22 131 8.8 71 44 2.9 24 9 0.6 5

IL03 7.5 0.3 226 15.3 0.27 182 12.2 81 37 2.5 16 7 0.47 3

RG1 2.5 0.1 131 8.8 0.15 54 3.6 41 62 4.2 47 15 1 11

RG02 5 0.2 184 12.5 0.22 72 4.8 45 84 5 40 28 1.9 15

RG03 7.5 0.3 226 15.3 0.27 84 5.5 37 111 7.6 49 31 2.1 14

Bs01 2.5 0.1 131 8.8 0.15 52 3.5 40 27 1.8 21 52 3.5 39

BS02 5 0.2 184 12.5 0.22 74 5 40 70 4.7 38 39 2.6 22

BS03 7.5 0.3 226 15.3 0.27 74 5 33 88 5.9 39 65 4.3 28

a The front non loading contact is between the start of the contact area to the actual loading area.b The rear non loading contact is between the loading area and the vertical line.

J. Rostami / International Journal of Rock Mechanics & Mining Sciences 57 (2013) 172–186186

cutting process and for development of more efficient disc cutterprofiles.

Acknowledgement

The author acknowledges the efforts and guidance Dr. LeventOzdemir who has served as academic advisor for completion of thisstudy as part of a PhD thesis at the Colorado School of Mines. Also,the assistance and support of Earth Mechanics Institute (EMI) staff,including Brian Asbury are much appreciated and acknowledged.This study was partially supported by the Robbins Company of KentWashington through their financial support and providing the disccutter used in the testing program, for which I am grateful.

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