Expression and Analysis of Joint Roughness Coefficient Using Neutrosophic Number Functions

2017·0.91 MB·

by  Jun Ye

informatio n
Article
Expression and Analysis of Joint Roughness
Coefficient Using Neutrosophic Number Functions
JunYe1,2,*,JiqianChen1,RuiYong1andShiguiDu1
1 KeyLaboratoryofRockMechanicsandGeohazards,ShaoxingUniversity,508HuanchengWestRoad,
Shaoxing312000,China;[email protected](J.C.);[email protected](R.Y.);[email protected](S.D.)
2 DepartmentofElectricalandInformationEngineering,ShaoxingUniversity,508HuanchengWestRoad,
Shaoxing312000,China
* Correspondence:[email protected];Tel.:[phone]
Received:28May2017;Accepted:17June2017;Published:20June2017
Abstract: In nature, the mechanical properties of geological bodies are very complex, and its
variousmechanicalparametersarevague,incomplete,imprecise,andindeterminate. Inthesecases,
wecannotalwayscomputeorprovideexact/crispvaluesforthejointroughnesscoefficient(JRC),
whichisaquitecrucialparameterfordeterminingtheshearstrengthinrockmechanics,butweneedto
approximatethem.Hence,weneedtoinvestigatetheanisotropyandscaleeffectofindeterminateJRC
valuesbyneutrosophicnumber(NN)functions,becausetheNNiscomposedofitsdeterminatepart
andtheindeterminatepartandisverysuitablefortheexpressionofJRCdatawithdeterminateand/or
indeterminateinformation. Inthisstudy,thelowerlimitofJRCdataischosenasthedeterminate
information,andthedifferencebetweenthelowerandupperlimitsischosenastheindeterminate
information. Inthiscase,theNNfunctionsoftheanisotropicellipseandlogarithmicequationofJRC
aredevelopedtoreflecttheanisotropyandscaleeffectofJRCvalues. Additionally,theNNparameter
ψisdefinedtoquantifytheanisotropyofJRCvalues. Then,atwo-variableNNfunctionisintroduced
basedonthefactorsofboththesamplesizeandmeasurementorientation. Further,thechanging
ratesinvarioussamplesizesand/ormeasurementorientationsareinvestigatedbytheirderivative
andpartialderivativeNNfunctions. However,anactualcasestudyshowsthattheproposedNN
functionsareeffectiveandreasonableintheexpressionandanalysisoftheindeterminatevalues
ofJRC.Obviously,NNfunctionsprovideanew,effectivewayforpassingfromtheclassicalcrisp
expressionandanalysestotheneutrosophicones.
Keywords: joint roughness coefficient (JRC); anisotropy; scale effect; neutrosophic number;
neutrosophicnumberfunction
1. Introduction
According to the shear strength formula given by Barton [1] in 1973, the joint roughness
coefficient(JRC)obtainedeffectivelyisquiteacrucialparameterfordeterminingtheshearstrength
inrockmechanics. VariousmethodsfordeterminingtheJRCvalueshavebeenproposedbymany
scholars[2–10]sinceBarton[11]firstlyintroducedtheconceptofJRC.Sincethentherehavealsobeen
manystudiesabouttheanisotropyandscaleeffect,whicharetwoimportantcharacteristicsofJRC
inthesamplesizesandmeasuringorientations. BartonandBandis[12]proposedthatthescaleeffect
of roughness coefficient is one of the main reasons for the scale effect of mechanical behavior of
rockmassthroughmodelteststudies. Duetal.[13]foundthattheundulationamplitudesimplied
the negative scale effect and presented a good logarithmic function relationship with alternation
sizes. Recently,Chenetal.[14]usedthedigitalimageprocessing(DIP)techniquetoretrievethejoint
surfacetopographyandavariogramfunctionforcharacterizingtheanisotropyofthejointsurface
roughness and to estimate the joint aperture. However, existing research methods can only deal
Information2017,8,69;doi:10.3390/info8020069 www.mdpi.com/journal/information
Information2017,8,69 2of13
withthedeterminate/crispfunctionexpressionsofJRCvaluesandanalyzetheiranisotropyandsize
effect,butcannotexpressandhandletheincomplete,uncertain,andimprecisedataofJRC.Dueto
theincompletenessofourobservations,measurements,andestimations,ortheexistingdisturbances
and uncertainties in the real world, we cannot always compute or provide exact/crisp values for
theinhomogeneityoftherockjoints,butweneedtoapproximatetheminindeterminateenvironments.
In this case, Ye et al. [15] established neutrosophic functions (interval functions) of JRC and shear
strength. However,theyonlygavetheinterval/thickfunctionsofJRCandshearstrength,butcannot
expresssuchanindeterminatefunctioncontainingcoefficientsand/orparametersofneutrosophic
numbers(NNs).
Infact,therealwaysexistsdeterminateandindeterminateinformationintherealworld. Hence,
Smarandache [16–18] firstly introduced the concept of NN, which is really suitable for expressing
determinateand/orindeterminateinformation,denotedasz=a+bI.Obviously,theNNzconsists
ofitsdeterminatepartaanditsindeterminatepartbI,whereaandbarerealnumbersandIdenotes
theindeterminacy.Clearly,NNismoresuitablefortheexpressionofdeterminateand/orindeterminate
information in the real world. In recent years, Ye [19] firstly proposed a multiple attribute group
decision-making method under a NN environment and developed the possibility degree ranking
methodforrankingalternatives. Then,Ye[20]presentedabidirectionalprojectionmodelofNNsfor
multipleattributegroupdecision-makingproblemswithNNinformation. Further,Kongetal.[21]and
Ye[22]proposedthecosinesimilaritymeasureofNNsforthemisfirefaultdiagnosisofgasolineengines
andtheexponentialsimilaritymeasureofNNsforthefaultdiagnosisofsteamturbinesunderaNN
environment. Additionally,ChenandYe[23]introducedaprojectionmodelofNNsforthemultiple
attributedecision-makingproblemofclay-brickselectionunderaNNenvironment. Further,Liuand
Liu[24]presentedaNNgeneralizedweightedpoweraveragingoperatorandappliedittomultiple
attributegroupdecision-makingunderaNNenvironment.Ye[25]presentedaNNlinearprogramming
method for NN linear programming problems under an indeterminate environment. All of these
studies have shown the superiority of NNs to express indeterminate information. Unfortunately,
NN functions have not been studied and applied in rock mechanics thus far. Hence, this original
studywillestablishNNoperationsandNNfunctionsofJRCwithdeterminateand/orindeterminate
informationtoexpressandanalyzetheanisotropyandscaleeffectofindeterminateJRCvaluesand
theirchangingratesinmeasuringsizesandorientationsoftestsamplesbytheactualcaseofthenatural
joint surface sample obtained from Changshan County of Zhejiang province in China. In the real
world,wecannotalwayscomputeorprovidedeterminate/crispJRCvaluesfortheJRCcharacteristics,
but we can approximate them to realize the indeterminate expression and analysis of JRC values.
Inthiscase,theNNfunctionsprovideaneffectivemathematicaltoolforpassingfromtheclassical
crispexpressionandanalysistotheneutrosophicones.
The rest of this paper is organized as follows: Section 2 presents some basic concepts and
operations of NNs; Section 3 introduces the JRC data acquisition of natural rock joint surfaces
with an actual case; In Section 4, some elliptic equations with NNs (elliptic NN functions of JRC)
are introduced for the descriptions of the joint profiles in various orientations and their changing
ratesarediscussedbytheirderivativefunctions,andthentheNNparameterψisdefinedtoquantify
thedegreeoftheanisotropy; InSection5,logarithmicequationscontainingNNs(logarithmicNN
functionsofJRC)arepresentedtoanalyzetheJRCvaluesandthechangingratesindifferently-sized
jointsamples;Section6introducesatwo-variableNNfunctionofJRCtodescribeJRCvaluesinvarious
orientationsandsizesanddiscussesitschangingratesbyitspartialderivativefunctions;Section7gives
discussion;Finally,conclusionsandfutureworkaregiveninSection8.
2. SomeBasicConceptsandOperationsofNNs
TheconceptofNNwasfirstlyproposedbySmarandache[16–18],whichconsistsofadeterminate
partandanindeterminatepart. Then,itisdenotedasz=a+bI,whereaandbarerealnumbersandI
Information2017,8,69 3of13
istheindeterminacy,implyingsomeindeterminaterangeI(cid:15)[IL,IU]. Itisobviousthatitcandescribe
determinateand/orindeterminateinformation.
Forexample,assumethataNNisz=8+5I,whereI(cid:15)[0,0.5]. Thus,itsdeterminatepartis8,
itsindeterminatepartis5I,andthenthereisz(cid:15)[8,10.5]forI (cid:15)[0,0.5]. Furthermore,onecanalso
specifysomeindeterminaterangeI(cid:15)[IL,IU]tosatisfytheactualappliedrequirements.
Letz =a +b Iandz =a +b IbetwoNNsforI(cid:15)[IL,IU];theiroperationalrelationsaregiven
1 1 1 2 2 2
asfollows[25]:
z +z = a +a +(b +b )I = [a +a +b IL+b IL,a +a +b IU+b IU] (1)
1 2 1 2 1 2 1 2 1 2 1 2 1 2
z −z = a −a +(b −b )I = [a −a +b IL−b IL,a −a +b IU−b IU] (2)
1 2 1 2 1 2 1 2 1 2 1 2 1 2
z ×z = a a +(a b +a b )I+b b I2
1 2 1 2 1 2 2 1 1 2
 (cid:32) (cid:33) 
(a +b IL)(a +b IL),(a +b IL)(a +b IU),
min 1 1 2 2 1 1 2 2 ,
 (a1+b1IU)(a2+b2IL),(a1+b1IU)(a2+b2IU)  (3)
=  (cid:32) (cid:33) 
 (a +b IL)(a +b IL),(a +b IL)(a +b IU), 
 max 1 1 2 2 1 1 2 2 
(a +b IU)(a +b IL),(a +b IU)(a +b IU)
1 1 2 2 1 1 2 2
z1 = a1+b1I = [a1+b1IL,a1+b1IU]
z2 a2+mb2iIn(cid:16)a1[a+2b+1bI2LI,L,aa12++bb12IILU,]a1+b1IU, a1+b1IU(cid:17), 
=  max(cid:16)aa21++bb21IIUL,aa21++bb21IILL,aa21++bb21IIUU,aa21++bb21IILU(cid:17)  (4)
a2+b2IU a2+b2IL a2+b2IU a2+b2IL
3. JRCDataObtainedfromanActualCase
In real situations, we choose a natural joint surface sample to collect data from Changshan
CountyofZhejiangprovinceinChinaasanactualcase. Thefoliatedslaterockisgrayish,completed,
andhardandhasadensestructure. Itssurfaceundulatesfrom1cmto2cmandislightlyweathered.
Then,asimplemechanicalhandprofilographinventedbyDu[26]wasemployedtorecordthesurface
profilesbyeveryintervalof15◦ in[0◦,360◦]alongthespecifiedlines. Then,weinterceptedsamples
from10cmto100cmatanintervalof10cmineveryorientation. Finally,weusethemodelsproposed
by Zhang et al. [27] to calculate the JRC values of each samples in different sizes and directions.
BythestatisticalanalysisofallJRCvaluesofeverysample, wecanobtaintheaveragevaluesand
standarddeviationsofallsizesineachdirection,whichareusedinthefollowingcaseanalyses.
4. NNFunctionsforDescribingtheAnisotropicCharacteristicsofJRC
4.1. NNFunctionsofJRCinAllOrientations
Here,wetakeasampleinthelengthof20cmasanexample. Allmeasuredvaluesandthemean
values of JRC in every orientation on one polar plot are shown in Figure 1. It can be found that
the JRC values are randomly distributed from 0 to 20 in each orientation. Then, the JRC values
intheorientationsof90◦ and270◦ aresmaller,whiletheJRCvaluesinsomeorientationssuchas0◦
and180◦arerelativelygreater. Performingacalculatedanalysis,theJRCaveragevalueandvarianceof
eachangleorientationareobtainedbythestatisticalanalysis. Forexample,theaveragevalueJRC
mean
andstandarddeviationσofJRCvaluesfor20cmjointsamplesinthe0◦orientationare9.653and1.716,
respectively. Thesmallerstandarddeviationindicatesthatthedatapointstendtothemeanvalue.
Accordingtothestatisticalanalysis,about70%ofthepointsfocusononeareainFigure1,thenmost
ofthebluestarsarecrowdedtogether. Hence,wemaychoosearobustrange(confidencerange)from
JRC − σ(lower limit) to JRC + σ(upper limit) for describing the anisotropic characteristics.
mean mean
InFigure1,itisshownthattheseupperandlowerlimitscanbeapproximatelystructuredasoneellipse
Information2017,8,69 4of13
area. ItisobviousthattheJRCmeanvaluesandmostofthesampledatafallintotheareabetweentwo
redellipses.
Basedontheleastsquaremethod, theupperandlowerlimitsofJRCvaluesarefittedbytwo
ellipseenvelopesforjointsamplesofeachsize. Then,accordingtotheconceptofNNsandthegeneral
parametricequationsoftheellipse,theNNparametricequationsofJRCcanbewrittenas:
(cid:40)
x = (a +b I)cosθ
1 1 ,I ∈ [0,0.5] (5)
y = (a +b I)sinθ
2 2
where a and a are two radiuses of the inside ellipse envelope, and a + b I and a + b I are two
1 2 1 1 2 2
radiusesoftheoutsideellipseenvelope. Throughthestatisticalanalysisofallsizes,theindeterminacy
Ibelongstotheinterval[0,0.5]. Then,b (i=1,2)canbedeterminedbythetwiceofthedifference
i
betweentheradiioftheinsideandoutsideellipseenvelopes.
EachJRCvalueinFigure1isthedistancefromanypointwithinthetwoellipsestothecoordinate
origin. ToexpresstheJRCvaluesinanyorientation,byusingEquations(3)and(5)wecanderive
thefollowingNNfunction:
(cid:113)
JRC(θ) = (cid:112)x2+y2 = [(a +b I)cosθ]2+[(a +b I)sinθ]2
1 1 2 2
(cid:113)
= (a +b I)·(a +b I)cos2θ+(a +b I)·(a +b I)sin2θ
1 1 1 1 2 2 2 2
(cid:113) (6)
= [a2,(a +b /2)2]cos2θ+[a2,(a +b I/2)2]sin2θ
1 1 1 2 2 2
(cid:113) (cid:113)
= [ a2cos2θ+a2sin2θ, (a +b /2)2cos2θ+(a +b /2)2sin2θ]for I ∈ [0,0.5].
1 2 1 1 2 2
InordertoreflectthechangingrateofJRC(θ)inanyorientation;here,wecarriedonthederivative
operationofEquation(6)andobtainthefollowingderivativeEquation(7)soastoanalyzethevariation
situationsofanisotropy:
 
(a2−a2)cosθsinθ ((a +b I)2−(a +b I)2)cosθsinθ
JRC(cid:48)(θ) = (cid:113) 2 1 ,(cid:113) 2 2 1 1  (7)
a2cos2θ+a2sin2θ (a +b I)2cos2θ+(a +b I)2sin2θ
1 2 1 1 2 2
Forexample,whenthelengthL=20cminFigure1,theNNparametricequationswithI(cid:15)[0,0.5]
canbeobtainedbythefollowingsteps:
(1)a =8.3,a =5.7,
1 2
(2)b = (11.5−8.3)×2=6.4,b = (9−5.7)×2=6.6,
1 2
(cid:40)
x = (8.3+6.4I)cosθ
(3) for I ∈ [0,0.5].
y = (5.7+6.6I)sinθ
Thus,wehavethefollowingNNfunctionwithI(cid:15)[0,0.5]:
(cid:113)
JRC(θ) = (cid:112)x2+y2 = [(8.3+6.4I)cosθ]2+[(5.7+6.6I)sinθ]2
(cid:113)
= (8.3+6.4I)·(8.3+6.4I)cos2θ+(5.7+6.6I)·(5.7+6.6I)sin2θ
(cid:113)
= [8.32,(8.3+6.4/2)2]cos2θ+[5.72,(5.7+6.6/2)2]sin2θ
(cid:112) (cid:113)
= [ 8.32cos2θ+5.72sin2θ, (8.3+6.4/2)2cos2θ+(5.7+6.6/2)2sin2θ]
(cid:112) (cid:112)
= [ 68.89cos2θ+32.49sin2θ, 132.25cos2θ+81sin2θ].
InInfofromrmataitoinon 2 021071,7 8, ,8 6, 96 9 5 5o of f1 41 4
JJRRCC(θ(θ))== xx22++yy22== [([8(8.3.3++66.4.4II))ccoossθθ]2]2++[([5(5.7.7++66.6.6II))ssininθθ]2]2
== (8(8.3.3++66.4.4II))⋅⋅(8(8.3.3++66.4.4II))ccooss22θθ++(5(5.7.7++66.6.6II))⋅⋅(5(5.7.7++66.6.6II))ssinin22θθ
== [8[8.3.322,, ( 8(8.3.3++66.4.4//22))22]]ccooss22θθ++[5[5.7.722,, ( 5(5.7.7++66.6.6//22))22]]ssinin22θθ
==[[ 88.3.322ccooss22θθ++55.7.722ssinin22θθ,, (8(8.3.3++66.4.4//22))22ccooss22θθ++(5(5.7.7++66.6.6//22))22ssinin22θθ]]
Informa=t=io[n[20616878,.88.,89699ccooss22θθ++3322.4.499ssinin22θθ,, 113322.2.255ccooss22θθ++8811ssinin22θθ].]. 5of13
TThhuuss, ,i tists N NNN d deerirvivaatitvivee f ufunncctitoionn w witihth I Iϵ ϵ [ 0[0, ,0 0.5.5] ]i sis g giviveenn a ass f ofolllolowwss: :
Thus,itsNNderivativefunctionwithI(cid:15)[0,0.5]isgivenasfollows:
  −−3366.4.400ccoossθθssininθθ −−5511.2.255ccoossθθssininθθ 
JJRRCC'('θ(JθR))C=(cid:48)=(θ) =(6(688(cid:113).8.8(cid:0)969c8c.o−8os93s26c2oθ.4θs02+c+θo3+s32θ23.s42.i4.9n499θsssiniinn222θθθ,,(cid:112), 11313−232.522.152..c225o55scc2cooθoss+sθ22sθ8iθn1+sθ+i8n8121sθsinin22θθ
TThhee c chhaanngginingg r aratete o of fJ RJRCC’(’θ(θ) )i sis s shhoowwnn i nin F Figiguurere 2 2. .I tI ti sis o obbvvioiouuss t hthaat tt hthee c chhaanngginingg c cuurvrveess o of ft hthee
The changing rate of JRC’(θ) is shown in Figure 2. It is obvious that the changing curves of
ininssididee a anndd o ouutstsididee e elllilpipssee e ennvveeloloppeess p preresseennt tp peeriroioddicicitiyty a anndd t hthee c chhaanngginingg r aratetess f afalll li nintoto t hthee r arannggee
theinsideandoutsideellipseenvelopespresentperiodicityandthechangingratesfallintotherange
bbeetwtwebeeeentn wt wteweono n tnweegoganatietvigveae ts isvinienese if nufuennfcuctnitoicontinsos.n .s .
JRJRCC o off LL = = 2 200 c cmm,o,orireienntatatiotionn(°()°)
tetests st asmamplpelses

JRJRCCmmeaena+nσ+σ 9900 1 515
JRJRCC -σ-σ 112200 6600
mmeaenan
OOutustisdidee
eellilplispese e nevnevleolopepe 1 010
inisnisdidee 115500 3300
eellilplispese e nevnevleolopepe
JRJRCC v avaluleuess 5 5
mmeaenan
118800 00
221100 333300
224400 330000
227700

FFigiguurerFe i g11u. .r PePoo1la.lar r pPpolollaotsrt s poolfo ft dsdirioerfectcditoiironenacatli lo JnRJRaClC J vRvaCaluluveeas,ls u, ueusp,pppueepr rp aeanrndad n lodlowwloeerw re lirlmimliimtist sit osof fo JfRJRJCRC C vvavalaullueuese ss aanannddd ththee
cocorrrerestpshpoeoncndodrinriengsg pe oellnlilpdipisnesge f eiftlitltiitpninsgeg efi entntvivneelgoloepnpevese sol oofp f2 e20s0 co mcfm2 0j ojcoimnint jtso asinamtmpsaplmelesp.s l.e s.
LL = = 2 200 c cmm,t,hthee r aratete o of fc chhaannggee
33
InInssididee e ellilplipssee
22 OOuutstsididee e ellilplipssee
11
θθ
JJ
CJJCJJ00
RR
JJ
-1-1
-2-2
-3-3
00 5500 110000 115500 220000 225500 330000 335500 440000
θθ（（°）°）
FFigiguurere 2 2. .T Thhee c hchaanngginingg c ucurvrvees so of ft hthee l olowweer ra anndd u uppppeer rl ilmimitist sf ofor rJ RJRCC’(’θ(θ).) .
Figure2.ThechangingcurvesofthelowerandupperlimitsforJRC’(θ).

Bythesimilarmethod,wecanobtaintheNNfunctionsofJRC(θ)andJRC’(θ)withI(cid:15)[0,0.5]and
variousscales. Then,alloftheresultsareshowninTables1and2.
Information2017,8,69 6of13
Table1.NNfunctionsofJRC(θ)fortheJRCexpressionsofdifferentsizedjointsamplesandI(cid:15)[0,0.5].
Scale(cm) a1 b1 a2 b2 JRC(θ)
(cid:112) (cid:112)
10 8.6 7.8 5.8 8.4 [ 73.96cos2θ+33.64sin2θ, 156.25cos2θ+100sin2θ]
(cid:112) (cid:112)
20 8.3 6.4 5.7 6.6 [ 68.89cos2θ+32.49sin2θ, 132.25cos2θ+81sin2θ]
(cid:112) (cid:112)
30 8 5.6 5.5 6 [ 64cos2θ+30.25sin2θ, 116.64cos2θ+72.25sin2θ]
(cid:112) (cid:112)
40 7.7 5.6 5.3 5.4 [ 59.29cos2θ+28.09sin2θ, 110.25cos2θ+64sin2θ]
(cid:112) (cid:112)
50 7.4 5.8 5.2 4.6 [ 54.74cos2θ+27.04sin2θ, 106.09cos2θ+56.25sin2θ]
(cid:112) (cid:112)
60 7.3 5.8 5.3 5.0 [ 53.29cos2θ+28.09sin2θ, 104.04cos2θ+60.84sin2θ]
(cid:112) (cid:112)
70 7.2 5.8 5.3 4.6 [ 51.84cos2θ+28.09sin2θ, 102.01cos2θ+57.76sin2θ]
(cid:112) (cid:112)
80 7.0 5.6 5.2 4.4 [ 49cos2θ+27.04sin2θ, 69.04cos2θ+54.76sin2θ]
(cid:112) (cid:112)
90 7.0 5.0 5.1 3.8 [ 49cos2θ+26.01sin2θ, 90.25cos2θ+49sin2θ]
(cid:112) (cid:112)
100 7.0 4.4 5.1 3.4 [ 49cos2θ+26.01sin2θ, 84.64cos2θ+46.24sin2θ]
Table2.NNfunctionsofJRC’(θ)ofdifferentsizedjointsamplesandI(cid:15)[0,0.5].
Scale(cm) JRC’(θ)
(cid:34) (cid:35)
10 (cid:113) −40.32cosθsinθ ,√ −56.25cosθsinθ
(73.96cos2θ+33.64sin2θ 156.25cos2θ+100sin2θ
(cid:34) (cid:35)
(cid:113) −36.40cosθsinθ ,√ −51.25cosθsinθ
20 (68.89cos2θ+32.49sin2θ 132.25cos2θ+81sin2θ
(cid:34) (cid:35)
(cid:113) −33.75cosθsinθ ,√ −44.39cosθsinθ
30 (64cos2θ+30.25sin2θ 116.64cos2θ+72.25sin2θ
(cid:34) (cid:35)
(cid:113) −31.20cosθsinθ ,√ −46.25cosθsinθ
40 (59.29cos2θ+28.09sin2θ 110.25cos2θ+64sin2θ
(cid:34) (cid:35)
(cid:113) −27.70cosθsinθ ,√ −49.84cosθsinθ
50 (54.74cos2θ+27.04sin2θ 106.09cos2θ+56.25sin2θ
(cid:34) (cid:35)
(cid:113) −25.2cosθsinθ ,√ −43.2cosθsinθ
60 (53.29cos2θ+28.09sin2θ 104.04cos2θ+60.84sin2θ
(cid:34) (cid:35)
(cid:113) −23.75cosθsinθ ,√ −44.25cosθsinθ
70 (51.84cos2θ+28.09sin2θ 102.01cos2θ+57.76sin2θ
(cid:34) (cid:35)
(cid:113) −21.96cosθsinθ ,√ −14.28cosθsinθ
80 (49cos2θ+27.04sin2θ 69.04cos2θ+54.76sin2θ
(cid:34) (cid:35)
(cid:113) −22.99cosθsinθ ,√−41.25cosθsinθ
90 (49cos2θ+26.01sin2θ 90.25cos2θ+49sin2θ
(cid:34) (cid:35)
(cid:113) −22.99cosθsinθ ,√ −38.4cosθsinθ
100 (49cos2θ+26.01sin2θ 84.64cos2θ+46.24sin2θ
4.2. AnisotropicParameter
In 2009, Tatone [28] defined a parameter to quantify the anisotropy that is the ratio of
themaximumdirectionalroughnesstotheminimumdirectionalroughnessonthepolarplots.Basedon
the anisotropic definition, we present the NN anisotropic parameter to characterize the degree of
anisotropyforeachsize. Here,themaximumdirectionalroughnessisthesemi-majoraxisofellipse;
theminimumdirectionalroughnessisthesemi-minoraxisoftheellipse. Hence,theNNanisotropic
parameterisdefinedasthefollowingform:
a +b I
ψ = 1 1 ,I ∈ [0,0.5]
a +b I
2 2
Information 2017, 8, 69 7 of 14
In 2009, Tatone [28] defined a parameter to quantify the anisotropy that is the ratio of the
maximum directional roughness to the minimum directional roughness on the polar plots. Based on
the anisotropic definition, we present the NN anisotropic parameter to characterize the degree of
anisotropy for each size. Here, the maximum directional roughness is the semi-major axis of ellipse;
the minimum directional roughness is the semi-minor axis of the ellipse. Hence, the NN anisotropic
parameter is defined as the following form:
+
a b I
ψ= 1 1 ，I∈[0, 0.5]
+
a b I
2 2
Then, according to the Equation (4), the anisotropic parameter ψ is calculated by the following
formula:
 a +b ×0 a +b ×0 a +b ×0.5 a +b ×0.5 
Information2017,8,69 min( 1 1 , 1 1 , 1 1 , 1 1 ), 7of13
a +bI  a +b ×0.5 a +b ×0 a +b ×0.5 a +b ×0 
ψ= 1 1 = 2 2 2 2 2 2 2 2 
Then,accordaing+tbotIheEquationa(4)+,tbhe×an0isotroapi+cpbar×a0metearψ+isbca×lc0u.l5atedaby+thbe×fo0ll.o5win gformula:
2 2 max( 1 1 , 1 1 , 1 1 , 1 1 )
 a +b ×0.5 a +b ×0 a +b ×0.5 a +b ×0  (8)
 2min2( a1+b1×02, a1+2b1×0, a1+2b1×0.25, a1+b1×0.5)2, 2 
=2ψa2a+=1b(cid:104)aa12,++ bb212IIa21=a+b1m,a(cid:105)x(Iaa∈2a2+1++[bb02b2×1,×× 000.0.55.5,a]aa2.21+++bbb221×××000,aaa221+++bbb221×××000...555, aaa122+++bbb122×××000.5)  (8)
2 = 2 2a1 ,22a1+b1 ,I ∈ [0,0.5].
2a2+b2 2a2
For example, when L = 20 cm, we have the following result:
Forexample8,.w3h+e6n.4LI=20cm,2w×e8h.3aveth2e×fo8l.l3ow+i6n.g4result:
ψ= = , =[0.92,2.02], I∈[0, 0.5].
ψ =5.78.+36+.66I.4I =2(cid:20)×5.27×+86..36 ,22××85.3.7+6.4(cid:21) = [0.92,2.02],I ∈ [0,0.5].
5.7+6.6I 2×5.7+6.6 2×5.7
Based on the NN anisotropic parameter ψ, the interval ranges of the anisotropic parameters of
BasedontheNNanisotropicparameterψ,theintervalrangesoftheanisotropicparametersofall
all sizes are shown in Figure 3. Both the upper and lower bounds are fitted by the logarithmic curves,
sizesareshowninFigure3. Boththeupperandlowerboundsarefittedbythelogarithmiccurves,
where the upper curve decreases and the lower curve increases as L increases.
wheretheuppercurvedecreasesandthelowercurveincreasesasLincreases.
2.2
Ψup
2 Ψdodn
1.8
Ψ = 2.478-0.136ln(L)，R = 0.9471
up
1.6
Ψ
1.4
1.2
Ψ = 0.7427+0.05457ln(L)，R = 0.8392
dodn
1
0.8
0 10 20 30 40 50 60 70 80 90 100 110
L（cm）

FFiigguurree 33.. TThhee uuppppeerr aanndd lloowweerr bboouunnddss ooff tthhee ppaarraammeetteerr ψψ ffoorr ddiiffffeerreenntt ssiizzeedd jjooiinntt ssaammpplleess..
FFrroomm FFiigguurree3 3,,w weec acnano botbatianinth tehNe NNNan aisnoistrootproicppica rpaamraemteertψer( Lψ)(cLo)r rceosrpreosnpdoinngditnogL tfoo rL If(cid:15)or[ 0I ,ϵ0 .[50],
a0s.5f]o allso fwolsl:ows:
ψ(ψL()L)=0=.7402.7742+72+.427.427I2−I−(0(.00.504545577++00.1.166228866II))llnnLL (9)
= [0.7427−0.05457lnL,2.478−0.136lnL], I = [0,0.5] (9)
=[0.7427−0.05457lnL, 2.478−0.136lnL], I =[0, 0.5]
5. NNFunctionsforReflectingtheScaleEffectofJointSurfaceRoughness

Asweknow,theroughnesscoefficientofarockmassstructuredecreaseswiththeincreaseofsize,
whichiscalledthescaleeffect. Inthissection,wediscussthisissue.
Whentheorientationis15◦,theJRCvaluesontheincreaseofthejointsamplesizesareshown
in Figure 4, which apparently exhibit a negative scale effect. Then, the fitting curves present
thelogarithmicfunctionswithmonotonously-decreasingtrends,andthenboththeupperandlower
boundsarematchedwell.
Thus,theNNfunctionJRC(L)anditsderivativefunctionJRC’(L)forI(cid:15)[0,0.5]aregivenas:
JRC(L) = (a +b I)+(a +b I)lnL = (a +a lnL)+(b +b lnL)I
1 1 2 2 1 2 1 2 (10)
= [a +a lnL,a +a lnL+(b +b lnL)/2],I ∈ [0,0.5],
1 2 1 2 1 2
a +b I
JRC(cid:48)(L) = 2 2 ,I ∈ [0,0.5] (11)
L
Information 2017, 8, 69 8 of 14
5. NN Functions for Reflecting the Scale Effect of Joint Surface Roughness
As we know, the roughness coefficient of a rock mass structure decreases with the increase of
size, which is called the scale effect. In this section, we discuss this issue.
When the orientation is 15°, the JRC values on the increase of the joint sample sizes are shown
in Figure 4, which apparently exhibit a negative scale effect. Then, the fitting curves present the
logarithmic functions with monotonously-decreasing trends, and then both the upper and lower
bounds are matched well.
Thus, the NN function JRC(L) and its derivative function JRC’(L) for I ϵ [0, 0.5] are given as:
JRC(L)=(a +bI)+(a +b I)lnL=(a +a lnL)+(b +b lnL)I
1 1 2 2 1 2 1 2
(10)
=[a +a lnL, a +a lnL+(b +b lnL)/2], I∈[0, 0.5],
1 2 1 2 1 2
a +b I
JRC'(L)= 2 2 ,I∈[0, 0.5] (11)
L
Inwforhmeartieo na21 0a1n7,d8 ,a629 are the fitting parameters of the lower bounds and b1 and b2 are the twice of 8thofe1 3
difference of the lower and upper bounds. In the NN function JRC(L), a1 + a2lnL is the determinate
part and (b1 + b2lnL)I is the indeterminate part.
whereFao1r aexnadmap2lea, rien tFhiegufirtet i4n,g a1p aanradm a2e taerres 9o.3f7t5h eanldo w−0e.r39b9o9u, nredsspeacntdivebl1y.a Tnhdenb,2 ba1 raentdh eb2 tcwanic ebeo f
thoebdtaifinfeerde nacse foolflothwesl:o werandupperbounds. IntheNNfunctionJRC(L),a +a lnListhedeterminate
1 2
partand(b +b lnL)Iistheindeterminatepart.
1 2 b =2×(16.74−9.375)=14.73
For example, in Figure 4, a1 and a are 9.375 and −0.3999, respectively. Then, b and b can
 1 2 1 2
b =2×(−1.653+0.3999)=−2.5062
beobtainedasfollows: 
(cid:40) 2
b =2×(16.74−9.375) =14.73
1
According to Equations (10) and (11), we have the following results:
b =2×(−1.653+0.3999) = −2.5062
2
JRC(L)=9.375−0.3999lnL+(14.73−2.5062lnL)I
AccordingtoEquations(10)and(11),wehavethefollowingresults:

=[9.375−0.3999lnL, 16.74−1.653lnL], θ=15

JRC(L) =9.375−0.3999lnL+(14.73−2.5062lnL)I
= [9.−3705.3−9909.3−9929.5ln06L2,1I6.74−11..665533lnL]0,.39θ9=915◦
JRC'(L)= = − , −
 
JRC(cid:48)(L) = −0.3999L−2.5062I =(cid:20)−1L.653,−0.39L99(cid:21)
L L L
In Figure 5, the shaded part is the changing rate range of JRC’(L). It is monotonously increasing
on tIhneF inigcurereas5e, othf eL.s Chaledaerdlyp, tahret isscathlee ecfhfeacnt gpihnegnroamteernaonng ies omfoJrReC o’b(Lv)i.oIutsi sinm somnaoltloern osiuzselsy. increasing
ontheBiyn ctrheea sseimofilLa.rC mleeatrhloy,dt,h feitsticnagle reefsfueclttsp ohfe noothmeer noornieinstamtioorneso cbavni oaulssoin bsem oablltearinseidze, st.he fitting
paraBmyethteerssi,m NilNar fmunecthtioodn,sf iftotirn JgRrCe(sLu)l,t saonfdo dtheerrivoartiievnet aftuionnctsiocanns afolsro JbReCo’(bLt)a ianreed t,atbhuelfaittteidn ginp aTraabmleest e3r s,
NaNndfu 4n. ctionsforJRC(L),andderivativefunctionsforJRC’(L)aretabulatedinTables3and4.
θ = 15°
14
JRC = 16.74-1.653*lnL,R2 = 0.9669 JRCmean+σ
12 up JRCmean-σ
JRC
mean
10 fitting curve (up)
fitting curve (dodn)
8
JRC JRCdodn = 9.375-0.3999*lnL,R2 = 0.8966
6
4
2
0
0 20 40 60 80 100 120
L(cm)
Information 2017, 8, FxFi FgigOuuRrer Pe4E 4.E.J RJRR RCCE vVvaIaElluWuee ss ooff ddiiffffeerreenntt ssiizzeedd jjooiinntt ssaammpplleess iinn tthhee oorriieennttaatitoionn oof f00°.◦ . 9 of 14

 = 15°
0
-0.02
-0.04
-0.06
)L-0.08
’(C
R -0.1
J
-0.12
-0.14
-0.16
-0.18
0 10 20 30 40 50 60 70 80 90 100 110
L(cm)
FFigiguurere5 5..T Thheec chhaannggiinngg rraatteess ooff uuppppeerr aanndd lloowweerr lilmimitists oof fJRJRCC’(’L()L i)ni nvavrairoiuosu ssizsiezse. s.
Table 3. NN functions of JRC(L) for I ϵ [0, 0.5] corresponding to the JRC values of the joint samples in
different orientations.
Orientation (°) a1 b1 a2 b2 JRC(L)
0 9.487 11.246 −0.5415 −1.253 [9.4870.5415lnL,15.111.168lnL]
15 9.375 14.73 −0.3999 −1.653 [9.3750.3999lnL,16.741.653lnL]
30 9.22 15.34 −0.4101 −2.5498 [9.220.4101lnL,16.891.685lnL]
45 8.83 12.78 −0.4538 −1.8364 [8.830.4538lnL,15.221.372lnL]
60 7.432 14.736 −0.3859 −2.1142 [7.4320.3859lnL,14.81.443lnL]
75 6.454 13.952 −0.2913 −2.3814 [6.4540.2913lnL,13.431.482lnL]
90 4.447 15.086 0.1112 −2.5844 [4.4470.1112lnL,11.991.181lnL]
105 6.18 13.62 −0.2751 −2.1678 [6.180.2751lnL,12.991.359lnL]
120 7.748 15.544 −0.359 −2.712 [7.7480.359lnL,15.521.715lnL]
135 9.009 10.482 −0.6697 −1.4586 [9.0090.6697lnL,14.251.399lnL]
150 9.454 15.032 −0.5928 −2.4184 [9.4540.5928lnL,16.971.802lnL]
165 10.49 12.58 −0.823 −1.746 [10.490.823lnL,16.781.696lnL]
180 8.705 12.55 −0.4852 −1.6456 [8.7050.4852lnL,14.981.308lnL]
195 8.905 14.79 −0.5113 −2.3994 [8.9050.5113lnL,16.31.711lnL]
210 9.395 11.33 −0.7162 −1.5076 [9.3950.7162lnL,15.061.47lnL]
225 8.614 9.932 −0.6189 −1.1822 [8.6140.6189lnL,13.581.21lnL]
240 7.112 10.756 −0.4301 −1.6358 [7.1120.4301lnL,12.491.248lnL]
255 5.94 11.88 −0.2837 −1.8705 [5.940.2837lnL,11.881.219lnL]
270 4.659 12.042 −0.0052 −1.7540 [4.6590.0052lnL,10.680.882lnL]
285 6.779 6.502 −0.5106 −0.5216 [6.7790.5106lnL,10.030.771lnL]
300 7.978 10.324 −0.5043 −1.5754 [7.9780.5043lnL,13.141.292lnL]
315 9.2 14.78 −0.6081 −2.2038 [9.20.6081lnL,16.591.71lnL]
330 8.792 13.676 −0.5216 −2.1808 [8.7920.5216lnL,15.631.612lnL]
345 9.491 12.583 −0.6661 −2.0618 [9.4910.6661lnL,15.761.697lnL]
Table 4. NN functions of JRC’(L) for I ϵ [0, 0.5] in different orientations.
Orientation (°) JRC’(L)
0 [1.168/L,0.5415/L]
15 [1.653/L,0.3999/L]
30 [0.4101/L,1.685/L]
Information2017,8,69 9of13
Table3. NNfunctionsofJRC(L)forI(cid:15)[0,0.5]correspondingtotheJRCvaluesofthejointsamples
indifferentorientations.
Orientation(◦) a1 b1 a2 b2 JRC(L)
0 9.487 11.246 −0.5415 −1.253 [9.487−0.5415lnL,15.11−1.168lnL]
15 9.375 14.73 −0.3999 −1.653 [9.375−0.3999lnL,16.74−1.653lnL]
30 9.22 15.34 −0.4101 −2.5498 [9.22−0.4101lnL,16.89−1.685lnL]
45 8.83 12.78 −0.4538 −1.8364 [8.83−0.4538lnL,15.22−1.372lnL]
60 7.432 14.736 −0.3859 −2.1142 [7.432−0.3859lnL,14.8−1.443lnL]
75 6.454 13.952 −0.2913 −2.3814 [6.454−0.2913lnL,13.43−1.482lnL]
90 4.447 15.086 0.1112 −2.5844 [4.447+0.1112lnL,11.99−1.181lnL]
105 6.18 13.62 −0.2751 −2.1678 [6.18−0.2751lnL,12.99−1.359lnL]
120 7.748 15.544 −0.359 −2.712 [7.748−0.359lnL,15.52−1.715lnL]
135 9.009 10.482 −0.6697 −1.4586 [9.009−0.6697lnL,14.25−1.399lnL]
150 9.454 15.032 −0.5928 −2.4184 [9.454−0.5928lnL,16.97−1.802lnL]
165 10.49 12.58 −0.823 −1.746 [10.49−0.823lnL,16.78−1.696lnL]
180 8.705 12.55 −0.4852 −1.6456 [8.705−0.4852lnL,14.98−1.308lnL]
195 8.905 14.79 −0.5113 −2.3994 [8.905−0.5113lnL,16.3−1.711lnL]
210 9.395 11.33 −0.7162 −1.5076 [9.395−0.7162lnL,15.06−1.47lnL]
225 8.614 9.932 −0.6189 −1.1822 [8.614−0.6189lnL,13.58−1.21lnL]
240 7.112 10.756 −0.4301 −1.6358 [7.112−0.4301lnL,12.49−1.248lnL]
255 5.94 11.88 −0.2837 −1.8705 [5.94−0.2837lnL,11.88−1.219lnL]
270 4.659 12.042 −0.0052 −1.7540 [4.659−0.0052lnL,10.68−0.882lnL]
285 6.779 6.502 −0.5106 −0.5216 [6.779−0.5106lnL,10.03−0.771lnL]
300 7.978 10.324 −0.5043 −1.5754 [7.978−0.5043lnL,13.14−1.292lnL]
315 9.2 14.78 −0.6081 −2.2038 [9.2−0.6081lnL,16.59−1.71lnL]
330 8.792 13.676 −0.5216 −2.1808 [8.792−0.5216lnL,15.63−1.612lnL]
345 9.491 12.583 −0.6661 −2.0618 [9.491−0.6661lnL,15.76−1.697lnL]
Table4.NNfunctionsofJRC’(L)forI(cid:15)[0,0.5]indifferentorientations.
Orientation(◦) JRC’(L)
0 [−1.168/L,−0.5415/L]
15 [−1.653/L,−0.3999/L]
30 [−0.4101/L,−1.685/L]
45 [−1.372/L,−0.4538/L]
60 [−1.443/L,−0.3659/L]
75 [−1.482/L,−0.2913/L]
90 [−1.181/L,0.1112/L]
105 [−1.359/L,−0.2751/L]
120 [−1.715/L,−0.359/L]
135 [−1.399/L,−0.6697/L]
150 [−1.802/L,−0.5982/L]
165 [−1.696/L,−0.823/L]
180 [−1.308/L,−0.4852/L]
195 [−1.711/L,−0.5113/L]
210 [−1.47/L,−0.7162/L]
225 [−1.21/L,−0.6189/L]
240 [−1.248/L,−0.430/L]
255 [−1.219/L,−0.2337/L]
270 [−0.8822/L,−0.005181/L]
285 [−0.7714/L,−0.5106/L]
300 [−1.292/L,−0.5043/L]
315 [−1.71/L,−0.6081/L]
330 [−1.612/L,−0.5216/L]
345 [−1.697/L,−0.666/L]
Information2017,8,69 10of13
6. NNFunctionofJRCwiththeTwoVariablesθandL
Information 2017, A8,s 6a9n alyzedabove, wehaveobtainedtheanisotropyandthescaleeffectofJRC,respectively, 11 of 14
whicharetheimportantcharacteristicsoftherockjointsamples. Inthissection,weshalldescribe
By thaeN pNarfutinaclt idonerwivitahtithvee towfo Evqauriaabtileosnθ (a1n2d),L w,uen dcearnN oNbteanivnir tohnem cehntasn. ging rate functions of the scale
effect and theT aonfiinsdottrhoepreyla: tionshipbetweentheJRCvaluesandthetwovariablesθandL,throughnumerical
analyses,thetrigonometricfunctionpolynomialofθandLcanfitintotwosmoothsurfacesoftheupper
andlowerboundsinthethree-dim∂eJnRsiCona(lθco,Lor)dinatesyst1em,asshowninFigure6. Itisobviousthat
up =−1.42 ,
allofthepointsofmeanvaluesfallbetw∂eeLntheupperandLlowersu rfaces. Thatistosay,theuncertain
(13)
rangealmostcontainsallJRCvalue∂sJ.RThCus,th(θe,fiLtt)ingequations1basedontheupperandlowersurfaces
ofJRCvaluesarepresentedasfollows: down =−0.4536 ,
∂L L
∂JRC J(RθC,(Lθ),L)down=−0.4536lnL+1.382sinθ+3.866sin2θ+1.039cos3θ −32.03sin4θ −
up 22.65s=in25.θ0+88c5o.s8θ6si−n16θ8.+70534.4c5ossinθ7sθi−nθ10−96.4.s1in883θc−os326θ.4ssiinn9θθ++2608.738.1si2ns1i0nθ3−θcosθ
∂θ
7.551cos11θ −11.04sin12θ+6.802cos13θ+4.334sin15θ+8.869,
−233.45sin4θcosθ−1531.8sin5θcosθ+820.4sin6θcosθ+3553.6sin7θcosθ
JRC(θ,L) =−1.42lnL+2.08sinθ −9.352sin2θ+2.061cos3θ+71.78sin4θ −46.69sin5θ
−734.76siunp8θcosθ−3831sin9θcosθ+205.48sin10θcosθ+1546.8sin11θcosθ
−255.3sin6θ+117.2sin7θ+444.2sin8θ −81.64sin9θ −383.1sin10θ −18.68cos11θ+

−220.48sin12θcosθ12+81.93s7in.3124θs+in1164.θ96ccooss1θ3θ, +9.156sin15θ+15.74.

∂JTRhCen,th(θeN,LN)functionJRC(θ,L)forI(cid:15)[0,0.5]isgivenasfollows:
down =1.382cosθ+7.732cosθsinθ−3.117cos2θsinθ−128.12sin3θcosθ
∂θ
JRC(θ,L)=(−0.4536−1.9328I)lnL+(1.382+1.396I)sinθ+(3.866−26.436I)sin2θ+
−113.25sin4θcosθ+515.16sin5θcosθ+374.15sin6θcosθ−875.2sin7θcosθ
(1.039+2.044I)cos3θ+(−32.03+207.62I)sin4θ+(−22.65−48.08I)sin5θ+
−327.6s(i8n58.8θ6c−os6θ82+.326I0)s3i.n86sθin+9(5θ3c.4o5s+θ1+278.35I.)0s6in17sθin+10(−θ1c0o9s.4θ+−1113072..24I8)ssinin8θ11+θcosθ
(12)
(−36.4−90.48I)sin9θ+(60.38−886.96I)sin10θ+(−7.551−22.258I)cos11θ+
−88.426sin12θcosθ+65.01sin14θcosθ.
(−11.04+279.88I)sin12θ+(6.802+20.316I)cos13θ+(4.334+9.644I)sin15θ+
(8.869+13.742I),forI=[0,0.5].
Then, their changing rates are shown in Figures 7 and 8.

Figure6.CurvedsurfaceareaofJRCvaluescorrespondingtobothLandθ.
Figure 6. Curved surface area of JRC values corresponding to both L and θ.
BythepartialderivativeofEquation(12),wecanobtainthechangingratefunctionsofthescale
effectandtheanisotropy:
0 ∂JRC∂uLp(θ,L) = −1.42L1, (13)
∂JRCdown(θ,L) = −0.45361,
∂L L
-0.02
-0.04
L-0.06
∂
)/
L
，θ-0.08
(
C
R
J -0.1
∂
-0.12
-0.14
-0.16
0 10 20 30 40 50 60 70 80 90 100 110
L（cm）

Figure 7. The changing rates of the scale effect.