analysis of the variation of friction coefficient of sandstone ...2020/06/29 ·...

Research ArticleAnalysis of the Variation of Friction Coefficient of SandstoneJoint in Sliding

Guochao Zhao , Laigui Wang, Na Zhao , Jianlin Yang, and Xilin Li

College of Mechanics and Engineering, Liaoning Technical University, Fuxin 123000, China

Correspondence should be addressed to Na Zhao; [email protected]

Received 29 June 2020; Revised 27 September 2020; Accepted 12 October 2020; Published 29 October 2020

Academic Editor: Weerachart Tangchirapat

Copyright © 2020Guochao Zhao et al.+is is an open access article distributed under the Creative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

+e friction coefficient of rock joints is closely related to the stability of the slope. However, it is difficult to predict the frictioncoefficient due to the influence of surface roughness and mechanical properties of rocks. In this study, we use a method thatcombines theoretical analysis with a sandstone sliding friction test and propose a model to predict the friction coefficient ofSandstone Joint. A sandstone sliding friction test was performed on a self-made reciprocating sliding friction test device. Goodagreement between the estimated values and test values verified the validity of the friction coefficient prediction model. +roughan analysis of the friction coefficient in sandstone sliding, it was established that the larger the wear mass, the larger the frictioncoefficient in sliding, and the larger the wear area, the smaller the friction coefficient. With the cycles increasing of sandstone, thefriction coefficient gradually decreased before finally reaching a stable value. Comparisons between the estimated value and testresults showed that when the wear difference coefficient c� 2.0 and the meshing friction amplification coefficient K� 1.4, theminimum error was 2.89%. +e results obtained are significant in the control of slope sliding.

1. Introduction

Sandstone slopes are common in open pit mining. Sand-stones usually contain joints, which when penetrated causeslope movement due to the slope sliding along the joints.+efriction coefficient of rock joints is closely related to thestability of the slope [1–5], though it is difficult to predict asit is associated with several geometrical and mechanicalproperties, such as surface roughness and asperity strength.When joint is subjected to repeated shearing during theslope sliding process, it becomesmore difficult to predict anyvariations of the friction coefficient [6]. +is creates hiddendangers for mine production and threatens the safety of lifeand property. Understanding any variation of the frictioncoefficient of rock joints plays an important role in thecontrol of slope sliding. Consequently, an analysis of thevariation of the friction coefficient of rock joints is vital if weare to minimize the threat of landslides and other relatednatural disasters.

Friction between two solid surfaces is usually charac-terized by friction coefficient. +e friction angle can also be

defined using the friction coefficient. Over the years, manymodels have been proposed to estimate the composition ofthe friction angle for rock joints. One of the earliest modelsfor friction angle is Patton’s criterion, which demonstratedthat the total friction angle can be separated into compo-nents of the basic angle of friction, peak dilation angle, andcontribution of asperity failures [7]. Maksimovic [8] pro-vided an analytical model to describe the nonlinear failurefor rock discontinuities, which contains the basic angle offriction, roughness angle, and median angle pressure, put-ting forward the expression of the dilatancy angle. Ladanyiand Archambault [9] presented an analytical model thatcontains the ratio of the degraded area of asperities andconsiders the effect of dilatation and failure of asperities onthe shear strength of joints. Saeb [10] and Seidel [11] made acase for factors affecting the failure rate of asperities andmodified the Ladanyi-Archambault model accordingly.Huang et al. [12] put forward a micromechanics model forthe stress-displacement behavior of rock joints. +e modelexplicitly accounted for the influence of the asperity shapeon the deformation and strength of rock joints. +rough the

HindawiAdvances in Civil EngineeringVolume 2020, Article ID 8863960, 12 pageshttps://doi.org/10.1155/2020/8863960

mailto:[email protected]://orcid.org/0000-0002-5209-7348https://orcid.org/0000-0002-7312-6355https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/8863960

above study, it was found that a number of factors influencethe friction coefficient, among which the dilatancy angle iscritical, and consequently it has been extensively studied.Schneider [13] studied the effect of roughness on the di-latancy angle of rough joints and presented a negative ex-ponential model to describe the evolution law of the jointdilatancy angle. In other works, Plesha et al. [14,15] deduceda constitutive law for the behavior of geologic discontinuitieswith dilatancy and contact surface degradation, based oncyclic shear test results of artificial joints. Leong and Ran-dolph [16] investigated the degradation of surface roughnessusing the wear theory and proved that it could be applied todescribe the complex behavior of two sliding bodies. Leeet al. [17] found that the degradation of asperities undercyclic shear loading also followed the exponential degra-dation laws for the asperity angle and proposed an elasto-plastic constitutivemodel, which considered the degradationof second-order asperities. Homand et al. [18] analyzed thevariation of the joint surface during cyclic shearing anddefined the degradation degree by the change of surface areabefore and after shearing and studied the evolution of initialjoint roughness during the course of shearing. Liu et al. [19]proposed a generalized damage model for a residual form oftooth-asperity and established the mathematical relation-ships between the snipped rate of tooth-asperity, dilatancyrate, angle of base friction, and average dilatancy angle,separately. Hong et al. [20] conducted a series of direct sheartests on artificial rock joint surfaces and obtained the spatialdistribution and statistical parameters of degradationroughness by analyzing the damage area of a specimen aftershearing. Liu et al. [21] discussed the fatigue damagemechanism of rock joints with first- and second-ordertriangular asperities under a prepeak cyclic load.

Previous studies have shown that surface roughness isthe main factor affecting the dilatancy angle, and so manyresearchers have focused on the relationship between thejoint surface roughness and dilatancy angle. Barton [22,23]proposed an empirical model in which he creatively in-troduced the joint roughness coefficient used to describe theroughness of joint surfaces with different morphologies.Grasselli et al. [24,25] conducted a large number of directshear tests on rock joints and obtained the statisticalfunction relationship between the effective shear angle of themicroelement of the joint surface and its correspondingcontact area. In doing so, they established the relationshipbetween three-dimensional morphology parameters and thefriction coefficient.+e study of the relationship between thejoint surface morphology parameters and dilatancy anglehas drawn increasing attention [26–28].

Although the above studies have conducted experimentsand theoretical analyses on the composition of the frictionangle, the change of dilatancy angle, and the relationshipbetween the surface roughness and the dilatancy angle, therehas been less research conducted on the variation of thefriction coefficient of rock joints during the sliding process.Yet, the variation of the friction coefficient of rock joint is ofgreat significance to the control of slope sliding and theproduction safety of mines. In this paper, a prediction modelfor the friction coefficient of Sandstone Joint is proposed,

combining theoretical analysis with sliding friction tests ofsandstone, starting from the single sliding process ofsandstone and extending through to multiple sliding pro-cesses. +e sliding friction test on a self-made reciprocatingsliding friction test device was performed. +e variation ofthe friction coefficient of Sandstone Joint was investigated,so as to verify the rationality and correctness of the frictioncoefficient prediction model, which uses the surfaceroughness, wear area, and wear mass as variables, and thento determine the model parameters.

2. Prediction Friction Coefficient Model ofSandstone Joint in Sliding

Joints develop widely on natural and open slopes, Figure 1(a)illustrates an example of a joint at the top of a slope inAishihik River bank, Canada (D. S. 1 is the number of thecrack group) [29], and Figure 1(b) is an example to illustratea joint of a slope south of Xilinhaote open-pit mine. As canbe seen from Figure 1, the joint surface is usually rough.During slope sliding, the highest asperities are sheared offand ground flat, and as sliding develops the second-highestasperities are sheared off and ground flat, with this cyclebeing repeated again and again. +e asperities on the surfaceof Sandstone Joint wear each other during the slidingprocess, which makes the prediction of the friction coeffi-cient of Sandstone Joint very complicated. +erefore, theSandstone Joint was used as the research object. +e in-teraction between the rigid asperities and asperities on thesurface of Sandstone Joint was used to simulate the slopesliding process.

2.1. Prediction Friction CoefficientModel of Sandstone Joint inthe First Sliding. It is assumed that the Sandstone Jointsurface was covered with a series of equally spaced conicalasperities of different heights. For the first sliding process,the interaction between the asperities of height Hn with therigid asperities was the meshing friction. +e upper part ofthe asperities was sheared off, with the height removed by thefirst sliding being hm. +e wear process of the asperities isshown in Figure 2.

To simplify the derivation process, the following basicassumptions were proposed:

(1) +e adjacent asperities that come into contact areindependent of each other during each slidingprocess

(2) +e height and cone angle of the asperities whichproduce meshing friction are the same during eachsliding process

Accordingly, Patton [7] conducted a large number oftests on regular joints and proposed the shear stress cal-culation model.+e shear force in sliding can be rewritten as

τ � σn tan ϕb + i( , (1)

where τ is the shear stress, σn is the normal stress, ϕb is thebasic friction angle, and i is the dilatancy angle.

2 Advances in Civil Engineering

If the left and right sides of equation (1) are divided byσn, the following expression for the friction coefficient of aSandstone Joint is obtained:

μ1 � tan ϕb + i1( , (2)

in which i1 denotes the initial dilatancy angle of SandstoneJoint during the first sliding process.

Equation (2) implies that the total friction angle is thesum of the two components listed above.

In order to investigate the contribution of the compo-nents of the friction angle, Maksimovic [8] conducted a largenumber of direct shear experiments on rock joints and foundthat, under low normal stress, the base friction angleremained unchanged and that the dilatation angle had agreater impact on the friction angle, implying that the di-latation angle was the main factor affecting the frictioncoefficient. +e dilatation of Sandstone Joint is accompaniedby the wear of asperities during the sliding process. Leongand Randolph [16] proposed a model to describe the fric-tional resistance of two sliding bodies and demonstrated that

the wear theory can be applied to describe the complexbehavior of two sliding bodies. Consequently, wear mass andwear area were used to calculate the dilatancy angle.

In the first sliding, the removal height of the asperities onthe surface of Sandstone Joint is expressed as follows:

h1 �3m1ρA1

. (3)

In the first sliding, the average base radius of the as-perities on the surface of Sandstone Joint can be written as

r1 �

��A1

N1π

. (4)

In the first sliding, the initial dilatancy angle of Sand-stone Joint is determined as

i11 � arctanh1

2r1� arctan

3m1N1(1/2)π(1/2)

2ρA1(3/2)

⎛⎝ ⎞⎠, (5)

D.S. 1

D.S. 3

D.S. 2

(a)

Discontinous joints

Discontinous joints

Discontinous joints

(b)

Figure 1: Photographs of typical examples of rock slope joints. (a) A joint at the top of a slope in Aishihik River bank. (b) A joint of a southslope of Xilinhaote open-pit mine.

Sliding direction

Rigid asperities

Sandstone asperities

hmhm+1hm+2

Hn–1 Hn–1

Meshing friction regionResidual friction region�e meshing and residual friction region

Hn Hn

Figure 2: Schematic diagram of the asperities wear process.

Advances in Civil Engineering 3

where ρ is the density of sandstone, A1 is the wear area in thefirst sliding,m1 is the wear mass in the first sliding, and N1 isthe number of asperities inWear Zone 1. ρ is a constant, andwear area A1 and wear mass m1 can be measured during thesliding process, so it is a feasible method to predict thefriction coefficient of Sandstone Joint using wear area andwear mass. +e number of asperities N1 is related to A1. +evalue ofN1 can be corrected by the results of a sliding frictiontest of the Sandstone Joint.

2.2. Prediction Friction CoefficientModel of Sandstone Joint inthe Second Sliding. In the second sliding, for asperities ofheight Hn on the surface of the Sandstone Joint, the heightremoved was hm+1 due to the residual friction interactionwith the rigid asperities resulting in the angle of the as-perities being passivated and worn flat. For asperities ofheight Hn−1, the height removed was hm+1 due to themeshing friction interaction with the rigid asperities. +einteraction between the asperities on the surface of theSandstone Joint and rigid asperities was meshing frictionand residual friction during the sliding process. In order tocalculate the friction coefficient of the wear zone, Ladanyiand Archambault [9] divided the wear zone into a slidingzone and shear zone and expressed the shear strength usingthe shear area ratio. Consequently, the failure rate of as-perities on the surface of the Sandstone Joint was introducedto calculate the friction coefficient. +e change of the wearzone on the surface of the Sandstone Joint during the secondsliding process is shown in Figure 3.

As can be seen from Figure 3, the friction coefficient ofthe Sandstone Joint in the second sliding can be divided intotwo parts: one is the meshing friction coefficient corre-sponding to Wear Zone 2 (solved for as per Section 2.1), andthe other is the residual friction coefficient corresponding toWear Zone 1. +e asperities on the surface of Wear Zone 1had residual friction interaction with the rigid asperities, thewear area did not change, and the asperities were worn flat.+e surface roughness was also reduced to Saij (Sa is thearithmetic mean deviation of the surface height, with thefirst subscript indicating the number of new wear zone andthe second representing the sliding number). At the sametime, the initial dilatancy angle i11 becomes the residualdilatancy angle i12 (the first subscript indicates the numberof the new wear zone and the second represents the slidingnumber). Considering that the surface roughness of WearZone 1 decreased and part of the asperities did not havecontact with the rigid asperities during the sliding process,the contribution of the meshing friction coefficient of WearZone 2 to the total friction coefficient was reduced.+erefore, we introduced the meshing friction amplificationcoefficient K to modify the friction coefficient of theSandstone Joint during the second sliding as follows:

μ2 � Ka2 tan ϕb + i22( + a1 tan ϕb + i12( , (6)

and the failure rate of asperities on the surface ofSandstone Joint is

a1 �A1

A1 + A2,

a2 �A2

A1 + A2,

(7)

where A1 is the new wear area during the first sliding and A2is the new wear area during the second sliding.

In equation (6), i12 is the residual dilatancy angle, in-dicating that the initial dilatancy angle i11 on Wear Zone 1attenuates during the second sliding. In order to analyze thevariation of the dilatancy angle with sliding, Schneider [13]and Plesha [14] conducted a large number of direct shearexperiments and found the attenuation of the dilatancy angleto be a negative exponential. +e surface roughness of theSandstone Joint on the wear area is accompanied by theattenuation of the dilatancy angle during the sliding process;thus, the attenuation of the dilatancy angle is expressed bythe variation of surface roughness. +e relationship betweeni12 and i11 can be expressed as

i12 � i11e− Samax− Sa12( )/ Samax− Samin( )( ), (8)

where Sa12 is the surface roughness of Wear Zone 1 after thesecond sliding, Samax is the maximum surface roughness ofthe new wear zone, and Samin is the optimal roughness. Inthe second sliding, the friction coefficient of the SandstoneJoint can be expressed as

μ2 � Ka2 tan ϕb + i22(

+ a1 tan ϕb + i12e− Samax− Sa12( )/ Samax− Samin( )( ) .

(9)

+e total wear mass in the second sliding is

m2 � md2 + mr2, (10)

An

A2

An–1

A3

A1

. . .

. . .

. . .

. . .

Figure 3: Schematic diagram of change of wear zone on thesurface.


where m2 is the total wear mass in the second sliding, md2 isthe meshing wear mass in the second sliding, and mr2 is theresidual wear mass in the second sliding. Considering thedifference in the contribution of meshing friction and re-sidual friction to total wear, the wear mass on Wear Zone 2with the total wear mass has the following relationship:

md2 � cm2a2. (11)

+e wear difference coefficient c is related to lithologyand surface roughness. +e initial dilatation angle of theSandstone Joint in Wear Zone 2 is

i22 � arctan3N2

(1/2)π(1/2)ca2m22ρA2

(3/2) , (12)

where N2 is the number of asperities in Wear Zone 2.Analysis of the second sliding process found that the

friction coefficient was the combination of the meshingfriction coefficient and residual friction coefficient. +e at-tenuation of the initial dilatancy angle was characterized bythe variation of the surface roughness, and the residualfriction coefficient was calculated by introducing the failurerate of asperities on the surface. At the same time, the weardifference coefficient c was introduced to calculate themeshing friction coefficient, and the meshing friction am-plification coefficient K was introduced to correct the fric-tion coefficient of the Sandstone Joint.

2.3. Prediction Friction CoefficientModel of Sandstone Joint intheNth Sliding. For the Nth sliding process, the change of thewear zone on the surface is shown in Figure 3, and theinteraction is the same as that described in Section 2.2. +efriction coefficient can be divided into two parts: one is themeshing friction part corresponding to the new zone n, ofwhich the asperities on the surface are sheared off, and theother is the residual friction corresponding toWear Zones 1,2, . . ., n-1. Similarly, for the residual friction in the first n-1zones, the wear area remains unchanged, and the asperitiesare worn flat; the surface roughness decreases, and the re-sidual dilatancy angle is ijn (j� 1, 2, . . ., n-1).

In the Nth sliding, the friction coefficient of the Sand-stone Joint can be developed as

μn � Kan tan ϕb + inn(

+ n−1

j�1aj tan ϕb + ijne

Samax− Sajn( / Samax− Samin( )( .

(13)

+e Jth failure rate of asperities on the surface of theSandstone Joint after n cycles is

aj �Aj

nj�1 Aj

. (14)

+e Jth dilatancy angle of the Sandstone Joint after ncycles is

ijn � arctan3cNj

(1/2)π(1/2)mjaj2ρAj

(3/2) , j � 1, 2, 3, . . . ,n.

(15)Based on the assumption that the asperities shear off and

are worn flat and that new asperities shear off in a cyclicalfashion on the surface of the Sandstone Joint in sliding, thewear area Ai and wear massmi which can be easily measuredduring the sliding process were used to calculate the frictioncoefficient of the Sandstone Joint.

A prediction model was proposed for the friction co-efficient of the Sandstone Joint with surface roughness Saij,wear area Ai, and wear mass mi, and the wear differencecoefficient c and the meshing friction amplification coeffi-cient K were introduced to modify the prediction model ofthe Sandstone Joint friction coefficient. In order to verify thecorrectness of the model and determine the values of thenumber of asperities Ni, the basic friction angle φb, the weardifference coefficient c, and the meshing friction amplifi-cation coefficient K, a reciprocating sliding friction test wasperformed on the surface of the Sandstone Joint.

3. Reciprocating Sliding FrictionTest of Sandstone

In order to verify the correctness of the model, a recipro-cating sliding friction test of the sandstone was carried outusing the self-made reciprocating sliding friction test device,and the surface roughness, wear mass, and wear area weremeasured. For calculating the experimental value of thefriction coefficient of Sandstone Joint, the shear stress insandstone sliding was recorded.

3.1. Joint Specimen Preparation. Artificial rock joints havebeen widely used to investigate their friction characteristicsin order to better reflect the change of surface roughnessfrom initial sliding to stable sliding [17,20,25,26]. To obtainartificial rock joints, the sandstone block was split (withdimensions of 20× 20× 40mm), and then the splitting testwas carried out on the sandstone block with the testingmachine, and the joint specimen’s dimensions are20× 20× 20mm. +e density of the specimen was 2.36 g/cm3, and the tensile strength was 5.68MPa. Before beingsubjected to the reciprocating sliding friction test, the surfaceof the specimen was colored with black ink, such that thesurface damage zone could be easily identified by comparingthe specimen surface before and after the test.

3.2. Test Procedure. +e reciprocating sliding friction testwas carried out between the diamond lapping and theSandstone Joint surface on the self-made reciprocatingsliding friction test device, as shown in Figure 4. +e particlesize of the diamond lapping was 120mesh, with a crosssection of 150× 80mm. +e diamond lapping was adheredto the upper friction box, which was fixed to the verticalbeam (without sliding). +e cross section of the sandstonespecimen was 20× 20mm, and it was installed in the lower


friction box. +e lower friction box was connected to theguide rail. During the shearing process, the specimen re-ciprocated uniform motion between the initial position andmaximum displacement (50mm), and the sliding speed ofthe guide rail was 10mm/s. +e total weight of the upperfriction box and diamond lapping was 4573 g, cross sectionarea of sandstone specimen was 4 cm2, and normal pressurethat the specimen was subjected to during the test was0.11MPa.

Pressure sensors were installed on both sides of thespecimen, and data of the shear stress were collected by thepressure sensors and output to a computer via a dynamicstrain gauge (sampling frequency: 20Hz) during the slidingprocess.

+e test value of the friction coefficient of the SandstoneJoint is determined by the following:

μi �τiσn

, (16)

where σn is the normal pressure and τi is the maximum shearstress in the Nth sliding.

+e specimen was disassembled every time it slipped,and images of the specimen surface were taken with a high-resolution camera. +en, the wear zone was traced withdrawing software, and the wear area Ai was calculated viaphoto image analysis. +e electronic analytical balanceFA1004 (measuring accuracy 0.1mg) was used to weigh themass of the specimen. +e measuring datum was repeatedthree cycles, and an average was applied, and the wear massis denoted as mi. A laser confocal scanning microscopeOLS4000 was used to test the surface morphology of the newwear zones and the existing wear zones, respectively. Each ofthe worn zones had three locations selected on which toperform the surface roughness test (an average of a set ofnumbers) and the surface roughness was denoted as Saij (thefirst subscript indicates the number of the new wear zone,and the second represents the sliding number, i≤ j). In orderto reduce the deviation of results, the sandstone specimenwas slipped ten cycles in total during the test.

4. Results and Discussion

4.1. Analysis Variation of Wear Zone on the Surface ofSandstone Joint. During the process of sliding, the asperitieson the surface of the Sandstone Joint were often worn. Forthe study of the failure characteristics of asperities, Hongperformed a large number of direct shear tests on rock jointsand analyzed the degradation mechanism by using thefailure characteristics of asperities after shearing [20]. So, weanalyzed the sliding process through changes of the wearzone in this study. +e surfaces of the Sandstone Joint afterthe 1st, 5th, and 10th cycles are shown in Figure 5. +e blackareas on the surface indicated the undamaged areas, ofwhich the surface was rough and there were many asperities.+e white areas underwent shearing off or were worn andbecame relatively flat.

In Figure 6, the variation of the damage zones on thesurface of the Sandstone Joint by sliding cycle is shown. +erange of the red line in the figure shows the newly damagedzone, indicating that meshing friction occurred in the zone.+e range of the blue line shows the damage augmentationzone, which indicates that meshing friction and residualfriction occurred in the zone. +e green line shows thedamage invariant zone, indicating that residual frictionoccurred in the zone. As can be seen from Figure 6(a), thewear zones are within the red line, indicating that meshingfriction occurred in the zone and that the asperities weresheared off. Figure 6(b) shows that the wear zones in thefigure are within the range of the red line, green line, andblue line, respectively. +e red line shows that meshingfriction occurred in the zone, and the asperities were shearedoff, while the green line shows that the interaction in thezone was residual friction, the asperities were worn flat, andthe surface roughness decreased. +e blue line shows thatmeshing friction and residual friction occurred together inthe zone. As a result of the residual friction, the overallheight of the new surface dropped, and new asperitiesaround it engaged with the diamond lapping. +e interac-tion between the asperities on the surface of the SandstoneJoint and the diamond lapping in Figures 6(c)–6(j) is thesame as 6(b). +erefore, sandstone sliding is a cyclicalprocess of asperities shearing off, being worn flat, with newasperities shearing off. Consequently, the process of sand-stone sliding in the prediction model was validated.

4.2. Analysis of the Variation of Sandstone Joint FrictionCoefficient and Determination of Model Parameters

4.2.1. Test Results and Determination of Surface RoughnessParameters. In Figure 7, the surface morphology of posi-tions A and B of Figure 5(a), using the laser confocalscanning microscope OLS4000, is shown. +e surfaceroughness test results are shown in Table 1. +e rate of thenew wear area on the surface of the Sandstone Joint insliding was calculated using Figure 6 and expressed by thefollowing:

Pressure sensorUpper friction box

Controller

Guide rail

Vertical beam Lower friction box

Specimen base

Figure 4: Photograph of reciprocating sliding friction test device.


Ari �Ai

Ac, (17)

where Ai is the wear area added to the surface of SandstoneJoint during the first sliding process and Ac is the cross-

sectional area. +e test results of the wear mass and the newwear area rate are shown in Table 1.

As can be seen from Table 1, the surface roughness of thewear area gradually decreased with increasing cycles. Be-cause the surface of the Sandstone Joint did not match the

Wear zone

Sliding direction

(a)

Wear zone

Sliding direction

(b)

Wear zone

Sliding direction

(c)

Figure 5: Surface diagrams of the Sandstone Joint after the (a) 1st cycle, (b) 5th cycle, and (c) 10th cycle.

(a) (b) (c) (d) (e)

(f ) (g) (h) (i) (j)

Figure 6: Variation of wear zones on the surface of the Sandstone Joint with cycles.

694.8

555.9

416.9

277.9

139.0

00 640 1280 1920 2560

695347.5256019201280640

Sliding direction

Z (μ

m)

X (μ

m)

(a)

0640

12801920

2560

569284.5

25601920

1280

640

Z (μm

)

X (μ

m)

Sliding direction

568.2

454.5

340.9

227.3

113.7

0

(b)

Figure 7: Surface morphology after the 1st sliding. (a) Surface topography of position A after the 1st sliding. (b) Surface topography ofposition B after the 1st sliding.


surface of the diamond lapping, the interaction between theasperities was meshing friction when sliding occurred, andthe asperities on the surface of the Sandstone Joint weresheared off, resulting in decreasing surface roughness. Whenthe worn zone continued to slide for five cycles, the asperitieson the surface of the Sandstone Joint were worn flat, and thesurface roughness finally reached a stable value. On the basisof the test results, the maximum surface roughness of thenew wear zone was 69.88 μm, and the maximum surfaceroughness was 70 μm. If the new wear zone continued toslide for five cycles, the surface roughness would be 3031 μm,and the optimal roughness would be 30 μm.

4.2.2. Analysis of the Variation of Friction Coefficient. It canbe seen from Figure 8 that the correlation of the wear massand the friction coefficient is consistent. +e larger the wearmass, the larger the friction coefficient, while the change ofthe wear area and the friction coefficient is the reverse. As thewear area increased, the friction coefficient gradually de-creased. When the wear area increased to a certain value, thefriction coefficient decreased slowly before finally reaching aconstant value.+ewear area on the surface of the SandstoneJoint was small in the initial stages of sliding, while themeshing friction occurred between the asperities. +efriction coefficient was only the meshing friction coefficient.+e meshing friction coefficient transformed into a com-bination of meshing friction coefficient and residual frictioncoefficient with an increase in cycles. +e area of the residualfriction zone gradually increased, the proportion of theresidual friction coefficient increased, the roughness de-creased continuously, and the residual friction coefficientdecreased. Due to the fact that the height and cone angle ofthe asperities on the new wear zone surface of the SandstoneJoint were inconsistent, the meshing friction force generatedby the top of the shearing asperity was different, whicheventually led to a local jump in the friction coefficient. Atthis time, the surface of the Sandstone Joint and the diamondlapping gradually changed from the initial mismatch tomutual matching, and the contribution of the meshingfriction coefficient and the residual friction coefficient to thefriction coefficient was alternately dominant. As the numberof cycles continued to increase, the wear area also increased.When the wear area reached a certain value, the residual

friction gradually dominated, and the friction coefficientreached a relatively stable state.

4.2.3. Correction of Sandstone Joint Friction CoefficientModel Parameters. Because the sandstone sliding process inthe model agrees well with test results, it is expected that themodel can give good predictions of the friction coefficient ofSandstone Joint. +e undetermined parameters in the modelwere corrected by using the test values of ten friction co-efficients. +e number of asperities N1 and the basic frictionangle φb were obtained from the results of the first slidingfriction test. +e wear difference coefficient c and themeshing friction amplification coefficient K were correctedusing the results of the sliding friction test (over 2–10 cycles).+e friction coefficient depends on Ni, and if Ni is deter-mined, φb can also be determined. Meanwhile, c and K varywith Ni, so the value of Ni is first determined.

+e values of c and K are strongly influenced by Ni. +ecurve of the initial dilatancy angle versus the number ofasperities in the first sliding is shown in Figure 9. In thisfigure, when the value of N1 is 1–20, the range of i11 isbetween 4.5 and 19.3°; when N1 is 21–40, the range of i11 is

Table 1: Wear mass (m), total wear area rate (Ar), and surface roughness of wear zone (Saij) with cycles.

Number of cycles m (mg) Ar (%)Saij (μm)

Sa1j Sa2j Sa3j Sa4j Sa5j Sa6j Sa7j Sa8j Sa9j Sa10j1 20.2 10.89 69.882 15.3 13.56 41.37 57.453 17.3 16.37 37.62 39.88 62.764 16.9 19.09 33.96 35.64 38.84 46.735 16.4 21.51 31.84 32.39 35.75 37.52 63.896 15.2 23.40 30.64 31.18 33.62 34.47 45.12 55.747 15.8 25.08 30.18 30.43 31.26 31.93 38.53 40.53 59.948 15.2 26.54 29.87 30.65 30.88 30.45 32.49 36.28 41.08 65.389 14.1 28.17 30.24 30.26 30.19 30.21 31.34 32.37 35.33 38.66 56.6310 15.3 29.54 30.17 30.15 30.44 30.39 30.82 31.03 31.29 33.59 37.42 60.42

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Test valueWear mass

Wear area ratio (%)

Fric

tion

coeffi

cien

t

14

15

16

17

18

19

20

21

Wea

r mas

s (m

g)

10 15 20 25 30

Figure 8: Variation of friction coefficient test value and wear masswith wear area ratio.


between 19.8 and 26.4°; when N1 is 41–60, the range of i11 isbetween 26.7 and 31.3°; when N1 is larger than 60, i11 isgreater than 31.5°.

According to equation (2), the following can beobtained:

ϕb + i11 � arctan μ1. (18)

Based on the test results, the test value of the frictioncoefficient was 0.731 during the first sliding, and, accordingto equation (18), the friction angle of Sandstone Joint was36.2°. According to the experimental results [8,10–12], forthe same specimen, the basic friction angle of rock joints isunchanged, which is always larger than 20°; then i11≤ 16.2°,so the value range of N1 is between 1 and 20. According tothe results of the first sliding friction test, the rate of newwear area Ar1 was 10.89%. Since the value of N1 is related tothe wear area A1, it is assumed that the value of N1 is theinteger percentage of Ar1; that is, N1 � 11, and i11, obtainedfrom equation (5), is 14.7°.+is agrees well with the results ofthe dilatancy angle, obtained by Huang, who performed thedirect shear test on joints with an inclination angle of 15°under 0.1MPa of normal stress [12]. +e values of thenumber of asperities in the 2–10 cycles of sliding were theinteger percentage of Ari. Using equation (18), φb � 21.5°.

Ignoring the wear difference and the contribution of themeshing friction coefficient to the total friction coefficient,that is, c� 1 and K� 1, the estimated values obtained bysubstituting the 2nd–10th test results of the wear mass, weararea, and surface roughness were entered into the frictioncoefficient prediction model and then compared with thetest values. +e curve of the estimated values and the testvalues with cycles is shown in Figure 10.

Figure 10 is a comparison diagram of the friction co-efficient between the test value and the estimated value withcycles. It can be seen from the figure that the change of thetwo curves had the same trend, and both of them weregradually reduced with the cycles increasing, so the proposed

model can give a good prediction of friction coefficient. Inthe fourth cycle, the friction coefficient test value had a jumpphenomenon, and the friction coefficient test value increasedby 5.80%; in the seventh cycle, the friction coefficient testvalue had a jump phenomenon, and the friction coefficienttest value increased by 1.12%. +e amplitude of the frictioncoefficient jump in the seventh cycle was lower than that inthe fourth cycle. It shows that the error caused by height andcone angle of the asperity on the surface of the SandstoneJoint has a gradually reduced influence on the overall changetrend of friction coefficient with the cycles increasing. +etest value of the friction coefficient will gradually stabilizefrom local fluctuations in the initial stage. +erefore, theproposed model can give a good prediction of the frictioncoefficient. A further investigation regarding the accuracy ofpredicting the friction coefficient (the average estimationerror) was used as a precision index as follows:

δ �19

10

i�2

μitest − μiestimated

μitest

× 100%, (19)

where δ is the average estimation error, μitest is the test valueof the friction coefficient in the ith sliding, and μiestimated is theprediction value estimated by the friction coefficient pre-diction model in the ith sliding. When c� 1 and K� 1, theerror of the model is 12.44% and the accuracy of the model ispoor. In order to reduce the error of the model, the values ofc and K are further analyzed.

Since c is the internal term of equation (6), the value of chas a great influence on K, so the value of c is analyzed first.According to equation (11),

ca2 < 1. (20)

So, the range of c is between 0 and 5. When c takes ondifferent values, according to equation (15), the change curveof the initial dilatancy angle inn with cycles can be derived, asshown in Figure 11. From this figure, inn decreased with

0

5

10

15

20

25

30

35

40

Initi

al d

ilata

ncy

angl

e (de

gree

)

Number of asperities 0 20 40 60 80 100

Figure 9: Changes of the initial dilatancy angle versus the numberof asperities in the first sliding.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fric

tion

coeffi

cien

t

Number of cycles

TestEstimated

1 2 3 4 5 6 7 8 9 10 11

Figure 10: Comparison between the test value and the estimatedvalue with cycles.


increasing cycles, and the larger the value of c, the larger inn,with K being constant, and the larger the value of c, thegreater the coefficient of friction. +e value of inn wascorrected by μitest, and it was found that when c> 2.0, theestimated value of the meshing friction coefficient was 0.843,which was quite different from the actual value. Hence, thevalue of cwas between 1 and 2. We then tried to calculate theerror with a larger interval value of K between 1 and 5 in thecase of c� 1.5 and found that, with K> 3, the error betweenthe estimated value and the test value was larger. In order toreduce the error, the error of the model was further cal-culated, the error results of which are shown in Table 2.

From Table 2, it can be seen that when the value of c isbetween 1.0 and 1.1 and the value of K is larger, the error issmaller. When the value of c is between 1.2 and 1.7 and thevalue of K is 1.4–2.0, the error is smaller. When the valueof c is between 1.8 and 2.0 and the value of K is smaller, theerror is smaller. As such, the value of K affects the spatialposition of the friction coefficient. When the value of K isapproximately 1.4, the estimated value of the frictioncoefficient is close to the spatial position of the test value.

0

10

20

30

40

50

Initi

al d

ilata

ncy

angl

e (de

gree

)

Number of cycles

c = 5.0

c = 4.5c = 4.0

c = 3.5

c = 3.0 c = 2.5

c = 2.0

c = 1.5c = 1.0

1 2 3 4 5 6 7 8 9 10 11

Figure 11: Variation of initial dilatancy angle with cycles at different c values.

Table 2: Error estimation table of friction coefficient estimated value and test value.

K� 1.0 (%) K� 1.2 (%) K� 1.4 (%) K� 1.6 (%) K� 1.8 (%) K� 2.0 (%) K� 2.2 (%) K� 2.4 (%) K� 2.6 (%) K� 2.8 (%)c� 1.0 12.44 10.51 8.59 6.67 4.96 4.59 4.33 4.31 4.63 5.82c� 1.1 11.69 9.71 7.73 5.75 4.55 4.17 4.13 4.32 5.15 6.87c� 1.2 10.94 8.90 6.86 4.83 4.14 3.95 4.11 4.56 6.11 7.95c� 1.3 10.18 8.09 5.99 4.11 3.77 3.87 4.11 5.26 7.15 9.04c� 1.4 9.43 7.27 5.12 3.70 3.61 3.86 4.50 6.24 8.19 10.14c� 1.5 8.67 6.45 4.24 3.36 3.59 3.90 5.24 7.24 9.24 11.27c� 1.6 7.90 5.63 3.44 3.30 3.58 4.42 6.18 8.24 10.31 12.58c� 1.7 7.14 4.80 2.98 3.28 3.72 5.05 7.14 9.25 11.57 13.90c� 1.8 6.37 3.97 2.95 3.25 4.20 5.92 8.10 10.43 12.83 15.23c� 1.9 5.59 3.27 2.91 3.50 4.80 6.83 9.18 11.65 14.11 16.57c� 2.0 4.82 2.90 2.89 3.84 5.46 7.82 10.34 12.87 15.39 17.92

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fric

tion

coeffi

cien

t

Number of cycles

TestEstimated

1 2 3 4 5 6 7 8 9 10 11

Figure 12: Variation of friction coefficient with cycles.


+e value of c influences the trend of friction coefficientwith cycles. When c � 2.0, the estimated value of thefriction coefficient is close to that of the test value. Whenc � 2.0 and K � 1.4, the error between the estimated valueof the friction coefficient and the test value was 2.89%, andthe estimated value agreed well with the test value. +echange of the estimated value and the test value withcycles is shown in Figure 12. From Figure 12, it can be seenthat the larger error between the predicted value and thetest value in the fourth sliding was due to the unevendistribution of the asperities on the surface of theSandstone Joint, which led to a larger fluctuation of thetest value of the friction coefficient and increasing themargin of error.

5. Conclusions

For the prediction of the friction coefficient of a SandstoneJoint during the sliding process, a method of combiningtheoretical analysis with a sandstone sliding friction test wasused, and the following conclusions were obtained:

(1) Based on the assumption that the sliding of sand-stone is a cyclical process where the asperities on thesurface of the joint were sheared off and worn flatand new asperities were sheared off, a predictionmodel of the friction coefficient of Sandstone Jointwas proposed, which was a function of the surfaceroughness, wear area, and wear mass. +e compo-nents of the friction coefficient of Sandstone Joint aredifferent at different stages: there is only the meshingfriction coefficient during the initial stage, whichtransforms into a combination of meshing frictioncoefficient and residual friction coefficient with anincrease in cycles.

(2) Analyzing the change trend of friction coefficientpredicted value and test value with the number ofcycles, it is found that the error caused by the heightand cone angle of the asperity on the surface has agradually reduced influence on the overall trend offriction coefficient with the cycles increasing. +eproposed model can give a good prediction of thefriction coefficient. When the wear difference coef-ficient c� 2.0 and the meshing friction amplificationcoefficient K� 1.4, the minimum error was 2.89%,and the predicted value is in good agreement withthe experimental value.

(3) Analyzing the influence of the wear area and wearmass on the friction coefficient during multiplesliding, it was established that, in the single sliding,the larger the wear mass, the larger the frictioncoefficient, and the larger the wear area, the smallerthe friction coefficient. With the cycles increasing ofsandstone, the friction coefficient gradually de-creased and finally reached a stable value.

+e above research results are significant to the controlof slope sliding.

Data Availability

+e data used to support the findings of this study are in-cluded within the article.

Conflicts of Interest

+e authors declare no conflicts of interest.

Acknowledgments

+is research was financially supported by the NationalNatural Science Foundation of China (Grants nos. 51474121and 2017YFC1503102) and Liaoning Natural ScienceFoundation (Grant no. 20180550869). +e authors wouldlike to acknowledge Editage (http://www.editage.cn) forEnglish language editing.

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analysis of the variation of friction coefficient of sandstone ...2020/06/29 ·...

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