Compaction Characteristics and Minimum Void Ratio Prediction Model for Gap-Graded Soil-Rock Mixture

applied sciences

Article Compaction Characteristics and Minimum Void Ratio Prediction Model for Gap-Graded Soil-Rock Mixture

Tao Wang *, Sihong Liu *, Yan Feng and Jidu Yu

College of Water Conservancy and Hydropower, Hohai University, No.1, Xikang Road, Nanjing 210098, China; [email protected] (Y.F.); [email protected] (J.Y.) * Correspondence: [email protected] (T.W.); [email protected] (S.L.)

Received: 6 November 2018; Accepted: 9 December 2018; Published: 12 December 2018

Featured Application: Prediction of the minimum void ratio of gap-graded soil-rock mixture.

Abstract: Gap-graded soil-rock mixtures (SRMs), composed of coarse-grained rocks and ﬁne-grained soils particles, are very inhomogeneous materials and widely encountered in geoengineering. In geoengineering applications, it is necessary to know the compaction characteristics in order to estimate the minimum void ratio of gap-graded SRMs. In this paper, the void ratios of compacted SRMs as well as the particle breakage during vibrating compaction were investigated through a series of vibrating compaction tests. The test results show that gap-graded SRMs may reach a smaller void ratio than the SRM with a continuous gradation under some circumstances. When the particles in a gap interval play the role of ﬁlling components, the absence of them will increase the void ratio of the SRM. The particle breakage of gap-graded SRMs is more prominent than the SRM with continuous gradation on the whole, especially at the gap interval of 5–20 mm. Based on the test results, a minimum void ratio prediction model incorporating particle breakage during compaction is proposed. The developed model is evaluated by the compaction test results and its validation is discussed.

Keywords: soil-rock mixtures; compaction; gap gradation; void ratio; particle breakage

1. Introduction Soil-rock mixtures (SRMs), which are composed of coarse-grained rocks and fine-grained soils particles, are very inhomogeneous materials and widely encountered in geoengineering [1,2]. Due to the inhomogeneous characteristics of SRMs, gap-graded SRMs are commonly found in geoengineering applications [3–5], such as waste rock and tailings from mining [6], clay-aggregate mixtures in rockfill dams [7–9] and glacial tills [10]. Gap-graded SRMs have a range of missing particles [11,12] in which fine particles may more easily be washed out of the matrix of the coarse particles by seepage forces. Therefore, great attention has been paid to the internal erosion of gap-graded SRMs [13–16]. However, little research has been done on the compaction characteristics of gap-graded SRMs. The compaction characteristics significantly influence the density that SRMs can reach under a certain compaction effort, and in turn influence the mechanical behaviour of SRMs [17]. Therefore, it is important to investigate the compaction characteristics of gap-graded SRMs, especially when used in projects where settlement has to be strictly controlled, such as highway and airport foundations [18,19]. On the other hand, a suitable model for estimating the compacted void ratio of gap-graded SRMs is needed for their compaction quality control. Studies show that the void ratio is closely related to the engineering properties of gap-graded materials, such as compressibility [20], shear strength [21] and permeability [22]. The existing void ratio prediction models are built mainly for sand-silt mixture

Appl. Sci. 2018, 8, 2584; doi:10.3390/app8122584 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, x FOR PEER REVIEW 2 of 16

Appl. Sci. 2018, 8, 2584 2 of 16 related to the engineering properties of gap-graded materials, such as compressibility [20], shear strength [21] and permeability [22]. The existing void ratio prediction models are built mainly for orsand-silt sand-clay mixture mixtures or sand-clay [7,23]. However, mixtures SRMs [7,23]. have However, coarser particleSRMs have sizes coarser than sand-silt particle mixture sizes than and sand-claysand-silt mixture mixtures, and resulting sand-clay in moremixtures, particle resulting breakage in more in gap-graded particle breakage SRMs during in gap-graded compaction SRMs [24]. Therefore,during compaction building a [24]. void Therefore, ratio prediction building model a incorporatingvoid ratio prediction particle breakagemodel incorporating is meaningful particle for the predictionbreakage is of meaningful compacted for void the ratio prediction of gap-graded of compacted SRM. void ratio of gap-graded SRM. InIn thisthis paper, aa seriesseries ofof vibratingvibrating compactioncompaction teststests are ﬁrstlyfirstly conductedconducted on SRMs with different continuous and gap gradations.gradations. The void ratios of compacted SRMs as well as the particle breakagebreakage during vibratingvibrating compactioncompaction are investigated.investigated. Then, a minimum void ratio prediction model for gap-graded SRMsSRMs incorporatingincorporating particleparticle breakagebreakage isis proposed.proposed.

2. Vibrating Compaction TestTest

2.1. Test ProgramsPrograms InIn thisthis study,study, thethe gapgap gradationgradation isis generatedgenerated byby removingremoving particlesparticles withinwithin aa gapgap intervalinterval ofof aa continuous gradationgradation (called (called basic basic gradation), gradation), as shownas shown in Figure in Figure1. The mass1. The of mass the missing of the particlesmissing withinparticles a gapwithin interval a gap is interval transferred is transferred into the remaining into the remaining particles in particles proportion in proportion to their mass to ratios.their mass In a gapratios. gradation, In a gapparticles gradation, are particles divided are into divided two groups into attwo two groups sides at of two the gapsides interval, of the gap fine-grained interval, groupfine-grained and coarse-grained group and coarse-grained group. The mass group. content The of mass particles content in the of fine-grainedparticles in groupthe fine-grained is defined as fc. The basic gradation is characterized using the fractal dimension D, defined as the gradient in a group is defined as fc. The basic gradation is characterized using the fractal dimension D, defined as doublethe gradient logarithmic in a double plot oflogarithmic mass of particles plot of againstmass of particleparticles size against [25]. particle size [25]. < 3−D MMd(d()< R) RR R 3−D == (()) (1)(1) MMRR T TLL < inin whichwhich MM(d(d< RR)) is is the the mass mass of of particles particles with with diameters diameters finer finer than thanR ,RM, TMisT the is the total total mass, mass,RL isRL the is = = maximumthe maximum diameter. diameter. In this In this test, test,RL R60L mm.60mm .

Figure 1. DefinitionDeﬁnition of a gap gradation generated from aa continuouscontinuous gradation.gradation.

The lithology of the test SRM is slightly-weatheredslightly-weathered dolomite,dolomite, with a saturatedsaturated uniaxialuniaxial compression strength of of 30~60 30~60 MPa MPa (medium (medium hard hard ro rock),ck), which which was was taken taken from from a rockfill a rockfill dam dam site sitein China. in China. The Thespecific specific gravity gravity of the of SRM the SRM is measured is measured at 2.65 at and 2.65 its and shape its shape is subangular. is subangular. Five basic Five basicgradations gradations with withdifferent different fractal fractal dimension dimension D wereD were tested. tested. For For each each basic basic gradation, gradation, four four gap gradations werewere generated, generated, with with gap gap intervals intervals of 5–20of 5–20 mm, mm, 2–5 mm,2–5 mm, 0.5–2 0.5–2 mm andmm 0.25–0.5 and 0.25–0.5 mm, which mm, correspondwhich correspond to medium to medium gravel, fine grav gravel,el, fine coarse gravel, sand coarse and mediumsand and sand, medium respectively, sand, accordingrespectively, to Chineseaccording standards to Chinese on soilstandards classification. on soil Furthermore,classification. forFurthermore, the gap gradation for the withgap gradation the gap interval with the of 5–20 mm, seven different values of fc were considered. The detailed test program is given in Table1. Appl. Sci. 2018, 8, 2584 3 of 16 Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 16

From these tests, we attempt to explore the effects of different basic gradations, gap intervals and mass gap interval of 5–20 mm, seven different values of fc were considered. The detailed test program is contentsgiven in Table of the 1. ﬁne-grained From these group tests, we (fc) attempt on the compaction to explore the characteristics effects of different of SRMs. basic gradations, gap intervals and mass contents of the fine-grained group (fc) on the compaction characteristics of SRMs. Table 1. Program of vibrating compaction tests.

Fractal dimensionTable 1. ProgramD of basic of gradation vibrating compaction 1.9, 2.1, tests. 2.3, 2.5, 2.7 Fractal dimensionGap interval D of basic (mm) gradation 5–20, 2–5, 1.9, 0.5–2, 2.1, 2.3, 0.25–0.5 2.5, 2.7

fc forGap the interval gap interval (mm) of 5–20 mm 0, 0.2, 5–20, 0.4, 2–5, 0.5, 0.6, 0.5–2, 0.8, 0.25–0.5 1.0 fc for the gap interval of 5–20 mm 0, 0.2, 0.4, 0.5, 0.6, 0.8, 1.0 The vibrating compaction tests were carried out on a steel-made compaction cylinder with an innerThe diameter vibrating of 300compaction mm and tests a height were of carried 500 mm. out Each on a SRMsteel-made sample compaction has 75 kg incylinder mass and with was an dividedinner diameter into three of equal300 mm parts. and Then, a height each of part 500 was mm. dropped Each SRM carefully sample into has the 75 cylinder. kg in mass As shownand was in Figuredivided2, into a vibrating three equal compactor parts. (Frequency:Then, each part 45 Hz; wa Excitings dropped force: carefully 5.4 kN) into was the placed cylinder. on the As top shown of the in sampleFigure 2, and a vibrating compacted compactor the sample (Frequency: for eight minutes. 45 Hz; Exciting It was found force: that 5.4 thekN) height was placed of the sample on the wouldtop of hardlythe sample change and after compacted a ﬁve-minute the sample vibrating for compaction.eight minutes. Therefore, It was found the sample that the was height considered of the tosample be in thewould densest hardly state change after the after eight-minute a five-minute vibrating vibrating compaction. compaction. After each Therefore, test, the totalthe heightsample of was the sampleconsidered was to measured, be in the anddensest then state its void after ratio the waseigh calculated.t-minute vibrating Also, grain-size compaction. analysis After was each performed test, the andtotal the height particle of the breakage sample during was measured, vibrating compactionand then its was void analyzed. ratio was calculated. Also, grain-size analysis was performed and the particle breakage during vibrating compaction was analyzed.

Figure 2. Diagram of the vibrating compaction on soil-rock mixtures. 2.2. Results and Discussion 2.2. Results and Discussion 2.2.1. Influence of gap intervals 2.2.1. Influence of gap intervals Figure3 shows the change of the void ratios of the SRMs with different gap gradations, generated fromFigure the basic 3 gradationshows theD change= 2.3, with of the gap void intervals. ratios The of testthe results SRMsdemonstrate with different that gap the influencegradations, of thegenerated absence from of particles the basic in gradation different gapD = 2.3, intervals with gap on voidintervals. ratios The of SRMs test results is different. demonstrate If the particles that the withininfluence 5–20 of mmthe absence are absent, of particles the void inratio different of the gap SRM intervals is smaller on void than ratios that of SRMs the basic is different. gradation, If contrarythe particles to the within conventional 5–20 mm belief are absent, that gap-graded the void ratio SRMs of the are SRM less compactible. is smaller than This that is becauseof the basic the particlesgradation, within contrary 5–20 to mm the are conventional relatively coarse belief and that difficult gap-graded to fill into SRMs the voidsare less of thecompactible. skeleton particles. This is Thebecause absence the particles of the particles within 5–20 within mm 5–20 are mmrelatively avoids co thearse probability and difficult of enlargingto fill into thethe voidsvoids ofof the skeleton particles. The The absence particles of smaller the particles than 5 wi mmthin usually 5–20 mm act asavoids the fillingthe probability components of enlarging in SRMs. Thethe absencevoids of of the these skeleton particles particles. means thatThe voids particles of the smaller skeleton than particles 5 mm cannot usually be fullyact as filled the duringfilling vibratingcomponents compaction, in SRMs. The so theabsence void of ratio these of theparticles SRMs means increases. that voids As the of mass the skeleton content particles of the particles cannot withinbe fully 0.25–0.5 filled during mm is relativelyvibrating fine,compaction, the absence so the of these void particlesratio of affectsthe SRMs slightly increases. on the As void the ratio mass of thecontent SRMs. of the particles within 0.25–0.5 mm is relatively fine, the absence of these particles affects slightly on the void ratio of the SRMs. Appl.Appl. Sci. Sci.2018 2018, ,8 8,, 2584 x FOR PEER REVIEW 4 of4 of16 16 Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 16

Figure 3. Void ratios of the soil-rock mixture (SRM) with basic gradation D = 2.3 at different gap FigureFigure 3. 3. Void ratios of of the the soil-rock soil-rock mixture mixture (SRM) (SRM) with with basic basic gradation gradation D = D2.3= at 2.3 different at different gap intervals. gapintervals. intervals. = Figure 4 presents the changes of non-uniform coefficients Cu (C d / d ) and coefficients of Figure4 presents the changes of non-uniform coefﬁcients Cu (Cu = d6060/10d10) and coefﬁcients of Figure 4 presents the changes of non-uniform coefficients Cu (C d / d ) and coefficients of = 2 2 u 60 10 curvaturecurvatureC Ccc ((CCc =dd 2/(/d(d )d) of)) the of theSRMs SRMs before before and andafter afterthe test thes at tests different at different gap intervals. gap intervals. It is c = 3030 10 1060 60 curvature Cc (Cc d30 /(d10d60 ) ) of the SRMs before and after the tests at different gap intervals. It is Itseen is seen that that after after vibrating vibrating compaction, compaction, Cu increases,Cu increases, while while Cc decreases,Cc decreases, especially especially in the in absence the absence of seen that after vibrating compaction, Cu increases, while Cc decreases, especially in the absence of ofparticles particles within within 5–20 5–20 mm. mm. This This means means the the SRM SRM gradations gradations have have changed changed during during the the vibrating vibrating particles within 5–20 mm. This means the SRM gradations have changed during the vibrating compaction,compaction, resulting resulting fromfrom thethe particleparticle breakagebreakage that will be discussed discussed later. later. compaction, resulting from the particle breakage that will be discussed later.

(a) Cu (b) Cc (a) Cu (b) Cc FigureFigure 4.4. ChangesChanges ofof non-uniformnon-uniform coefficientscoefficients Cuu andand coefficients coefficients of of curvature curvature CCc cafterafter vibrating vibrating Figure 4. Changes of non-uniform coefficients Cu and coefficients of curvature Cc after vibrating compaction.compaction. compaction. 2.2.2.2.2.2. Influence Influence ofof basicbasic gradationsgradations 2.2.2. Influence of basic gradations FigureFigure5 5gives gives the the changes changes ofof thethe voidvoid ratiosratios ofof thethe SRMsSRMs with gap intervals under under the the basic basic Figure 5 gives the changes of the void ratios of the SRMs with gap intervals under the basic gradationsgradations withwith differentdifferent fractal dimension dimension DD. .It It can can be be seen seen that that the the changing changing trend trend of the of thecurves curves is gradations with different fractal dimension D. It can be seen that the changing trend of the curves is issimilar similar when when D D= 2.3,= 2.3, 2.5 and 2.5 and2.7 (as 2.7 shown (as shown in Figure in Figure 3), but3 ),it is but different it is different from that from of D that = 1.9 of andD 2.1.= 1.9 similar when D = 2.3, 2.5 and 2.7 (as shown in Figure 3), but it is different from that of D = 1.9 and 2.1. andFrom 2.1. the From definition the definition of D in of EquationD in Equation (1), the (1), particle the particle size sizein an in anassembly assembly increases increases with with the the From the definition of D in Equation (1), the particle size in an assembly increases with the decreasingdecreasingD D.. WhenWhen D = 1.9 1.9 and and 2.1, 2.1, the the particles particles within within 5–20 5–20 mm mm may may act actas filling as filling components components in decreasing D. When D = 1.9 and 2.1, the particles within 5–20 mm may act as filling components in inSRMs. SRMs. The The absence absence of of these these particles particles leads leads to to insufficient insufficient filling filling of ofthe the voids voids among among the the skeleton skeleton SRMs. The absence of these particles leads to insufficient filling of the voids among the skeleton particles,particles, resultingresulting inin thethe bigger void void ratios ratios of of the the SRMs SRMs than than those those of ofthe the basic basic gradations gradations when when D = D particles, resulting in the bigger void ratios of the SRMs than those of the basic gradations when D = =1.9 1.9 and and 2.1. 2.1. Figure Figure 55 alsoalso demonstrates demonstrates that that there there are are the the vital vital grain grain size size groups groups acting acting as as filling filling 1.9 and 2.1. Figure 5 also demonstrates that there are the vital grain size groups acting as filling componentscomponents inin SRMs.SRMs. TheThe vitalvital graingrain size group is is 0.5–2 0.5–2 mm mm when when DD == 2.3, 2.3, 2.5, 2.5, 2.7 2.7 and and it itis is2–5 2–5 mm mm components in SRMs. The vital grain size group is 0.5–2 mm when D = 2.3, 2.5, 2.7 and it is 2–5 mm whenwhenD D= =1.9, 1.9, 2.1.2.1. The absence of the vital vital grain grain size size group group will will significantly significantly increase increase the the void void ratio ratio of of when D = 1.9, 2.1. The absence of the vital grain size group will significantly increase the void ratio of thethe SRM, SRM, which which should should bebe takentaken seriouslyseriously in engineering practice. practice. the SRM, which should be taken seriously in engineering practice. Appl. Sci. 2018, 8, 2584 5 of 16 Appl. Sci. 2018, 8, x FOR PEER REVIEW 5 of 16 Appl. Sci. 2018, 8, x FOR PEER REVIEW 5 of 16

FigureFigure 5. 5.Void Void ratios ratios of of the the SRMs SRMs withwith gapgap intervalsintervals under under different different basic basic gradations. gradations. Figure 5. Void ratios of the SRMs with gap intervals under different basic gradations. AtAt the the same same gap gap interval, interval, the the void void ratioratio ee ofof the SRM decreases decreases with with the the increasing increasing D Dofof the the basic basic gradation.gradation.At the For Forsame example, example, gap interval, at theat the gap the gap intervalvoid interval ratio of e 5–20 ofof the 5–20 mm, SRM mm,e decreasesdecreases e decreases fromwith the 0.36from increasing to 0.36 0.11 to when D0.11 ofD thewhenincreases basic D fromincreasesgradation. 1.9 to 2.7. from For Mcgeary 1.9example, to 2.7. [26 Mcgeary]at found the gap that [26] interval the found mixture thatof 5–20 ofthe two mixturemm, grain e decreasesof size two groups grain from wouldsize 0.36 groups haveto 0.11 would less when void have ratioD lessincreases void ratiofrom when1.9 to the2.7. meanMcgeary particle [26] sizefound difference that the betweenmixture ofthe two two grain groups size is groups coarse. would We use have the when the mean particle size difference between the two groups is coarse. We use the ratio of d’50/d50 less void ratio when the mean particle size difference between the two groups is coarse. We use the to representratio of d’50 the/d50 degreeto represent of particle the degree size differenceof particle betweensize difference the fine-grained between the group fine-grained and coarse-grained group and coarse-grainedratio of d’50/d50 to group, represent where the ddegree50 and ofd’ 50particle are the size mean difference diameters between of the the fine-grained fine-grained groupgroup and group, where d50 and d’50 are the mean diameters of the fine-grained group and coarse-grained group, coarse-grained group,group, respectively.where d50 and The d values’50 are ofthe d ’50mean/d50 at diameters different gapof theintervals fine-grained are plotted group against and respectively. The values of d’50/d50 at different gap intervals are plotted against D in Figure6. It can be Dcoarse-grained in Figure 6. Itgroup, can be respectively. seen that the The ratios values of ofd’ 50d/d’5050/d increase50 at different with gapD irrespective intervals are of plotted gap intervals. against seen that the ratios of d’50/d50 increase with D irrespective of gap intervals. According to Mcgear’s DAccording in Figure to 6. Mcgear’s It can be finding, seen that it isthe understandable ratios of d’50/d 50that increase the void with ratio D irrespective e of the SRM of decreases gap intervals. with finding, it is understandable that the void ratio e of the SRM decreases with the increasing D of the theAccording increasing to Mcgear’s D of the basicfinding, gradation. it is understandable that the void ratio e of the SRM decreases with basicthe gradation.increasing D of the basic gradation.

Figure 6. Changes of d’50/d50 with fractal dimension D at different gap intervals. Figure 6. Changes of d’50/d50 with fractal dimension D at different gap intervals. Figure 6. Changes of d’50/d50 with fractal dimension D at different gap intervals. As aforementioned, particle breakage will inevitably happen during the vibrating compaction onAs SRMs.As aforementioned, aforementioned, Commonly, the particle particle degree breakage ofbreakage particle will willbreakage inevitablyinevitably is quantitatively happen happen during during evalua the theted vibrating vibrating using three compaction compaction indices: onBon SRMs.15 SRMs.[27], Commonly, B Commonly,g [28] and the Bther degree[29], degree where of of particle particle B15 is breakage breakagethe ratio is ofquantitatively quantitatively the diameter evalua evaluated beforeted usingtest using and three three after indices: indices: test B15 [27], Bg [28] and Br [29], where B15 is the ratio of the diameter before test and after test B15corresponding[27], Bg [28] and to B15%r [29 finer], where passingB15 isin thethe ratiogradation of the curve; diameter Bg is before the maximum test and afterdistance test correspondingbetween the corresponding to 15% finer passing in the gradation curve; Bg is the maximum distance between the to 15%original ﬁner gradation passing incurve the gradationand gradation curve; curveBg is after the maximumtest; Br is the distance ratio of between Bt to Bp, thewhere original Bt is the gradation area original gradation curve and gradation curve after test; Br is the ratio of Bt to Bp, where Bt is the area curvebetween and gradationoriginal gradation curve after curves test; Bandr is thegradation ratio of curveBt to Bafterp, where test, BBtp isis the area area between between original original gradationbetween original curve and gradation the line curvesd = 0.074 and mm. gradation Figure 7 curveshows afterthe part test,icle B breakagep is area ofbetween SRMs withoriginal gap gradation curves and gradation curve after test, Bp is area between original gradation curve and the gradation curve and the line d = 0.074 mm. Figure 7 shows the particle breakage of SRMs with gap lineintervalsd = 0.074 under mm. the Figure basic7 gradations shows the with particle different breakage fractal of dimension SRMs with D. gapThe intervalstest results under demonstrate the basic intervals under the basic gradations with different fractal dimension D. The test results demonstrate gradationsthat the particle with different breakage fractal of gap-graded dimension DSRMs. The is test more results prominent demonstrate than the that SRM the particle with continuous breakage of that the particle breakage of gap-graded SRMs is more prominent than the SRM with continuous Appl. Sci. 2018, 8, 2584 6 of 16 Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 16 gap-gradedgradation SRMs on the is whole, more prominent especially thanat the the gap SRM inte withrval continuousof 5–20 mm. gradation The degree on of the particle whole, breakage especially at thedecreases gap interval gradually of 5–20 with mm. the Theincreasing degree fractal of particle dimension breakage D decreasesof the basic gradually gradation. with This the is increasingbecause fractalthe dimensionparticle sizesD ofin theSRM basic decrease gradation. with the This increasing is because D the, and particle consequently sizes in less SRM particle decrease breakage with the will happen during the vibrating compaction. increasing D, and consequently less particle breakage will happen during the vibrating compaction.

(a) B15~D

(b) Bg~D

FigureFigure 7. Particle7. Particle breakage breakage of of SRMs SRMs with withfractal fractal dimensiondimension D at different different gap gap intervals. intervals.

2.2.3.2.2.3. Influence Influence of of mass mass content content in in fine-grained fine-grained group group ((fcfc)) In engineeringIn engineering practice, practice, the the gap gap intervals intervals and and thethe gradationsgradations of the the fine-grained fine-grained group group and and the the coarse-grainedcoarse-grained group group of of SRMs SRMs are are known, known, and and they they areare dependentdependent upon the the available available materials. materials. It Itis is Appl. Sci. 2018, 8, x FOR PEER REVIEW 7 of 16

Appl.necessary Sci. 2018 to, 8 study, 2584 the mass content fc of the fine-grained group in an SRM to achieve the minimum7 of 16 Appl.void Sci. ratio 2018 of, 8 ,the x FOR SRM. PEER Figure REVIEW 8 presents the relationship between the void ratios e of the SRMs7 with of 16 different basic gradations and the mass contents fc at the gap interval of 5–20 mm. necessary to study the mass content fc of the ﬁne-grained group in an SRM to achieve the minimum necessary to study the mass content fc of the fine-grained group in an SRM to achieve the minimum void ratio of the SRM. Figure 88 presentspresents thethe relationshiprelationship betweenbetween thethe voidvoid ratiosratios e of the SRMs with different basic gradations and the mass contents fc at the gap interval of 5–20 mm. different basic gradations and the mass contents fc at the gap interval of 5–20 mm.

Figure 8. Changes of void ratios of the SRMs with fc under different basic gradations.

The relationship is approximately bilinear, and the turning point corresponds to the minimum void ratio ofFigure the 8. SRM, ChangesChanges which of void can ratios be explained of the SRMs with with the fc underunder help different differentof Figure basic 9. gradations. Figure 9a illustrates coarse-grained particles enclosing a void space in the case of fc = 0. When fc increases gradually, the fine-grainedThe relationship particles is isenter approximately approximately into the voids bilinear, bilinear, as fillin and ang thed components, the turning turning point pointleading corresponds corresponds to the todecrease the to minimumthe of minimum the void voidratio ofratioof thethe of SRM, SRM the which (FigureSRM, canwhich 9b). be explainedFigure can be 9c explained withshows the th help ewith state of the Figure when help9 .all Figureof voids Figure9a are illustrates 9. fully Figure occupied coarse-grained 9a illustrates by the coarse-grainedparticlesfine-grained enclosing particles particles a voidand enclosing spacethe minimum in a thevoid case voidspace of raf cintio= the 0.of Whencasethe SRMoff cfc increases =is 0. achieved. When gradually, fc increasesThe corresponding the gradually, fine-grained fthec is fine-grainedparticlescalled the enter optimal particles into mass the enter voids contents into as filling the of thevoids components, fine-grained as filling leading components,group. to If the fc continues decrease leading ofto thetheincrease, voiddecrease ratio coarse-grained of of the SRMvoid ratio(Figureparticles of9 theb). will FigureSRM be separated(Figure9c shows 9b). theby Figurethe state fine-grained when 9c shows all voids thparticlese arestate fully whenand occupied “float” all voids in by the theare matrix fine-grainedfully occupied of fine-grained particles by the fine-grainedparticles,and the minimum as particlesshown void in and ratioFigure the of minimum9d,e, the SRM resulting isvoid achieved. raintio the of The increasethe corresponding SRM of is theachieved. voidfc is ratio calledThe correspondingof the the optimal SRM. massAs fc isfc calledincreasescontents the of upoptimal the to fine-grained1.0, mass the coarse-grained contents group. of If thefc continuesparticles fine-grained disappear to increase, group. and If coarse-grained f conly continues fine-grained to particles increase, particles will coarse-grained be exist separated in the particlesmixtureby the fine-grained (Figurewill be 9f), separated particles where the andby mathe “float”ximum fine-grained in void the matrix ratio particles is of reached. fine-grained and “float” particles, in the asmatrix shown of in fine-grained Figure9d,e, particles,resultingThe optimal inas theshown increase mass in Figurecontent of the 9d,e, voidfc and resulting ratio the ofminimum the in SRM.the voidincrease As ratiofc increases of of the gap-graded void up to ratio 1.0, SRMs theof the coarse-grained are SRM. important As fc increasesparticlesparametersdisappear up in to engineering 1.0, andthe coarse-grained only practice. fine-grained In particlesthe particles next disappearpa existrt, we in attempt the and mixture only to fine-grainedbuild (Figure a model9f), where particles for predicting the exist maximum in the mixturevoidoptimal ratio mass(Figure is reached. content 9f), where fc and the minimum maximum void void ratio ratio of gap-gradedis reached. SRMs. The optimal mass content fc and the minimum void ratio of gap-graded SRMs are important parameters in engineering practice. In the next part, we attempt to build a model for predicting the optimal mass content fc and minimum void ratio of gap-graded SRMs.

Figure 9. IllustrationIllustration of of the the mixture mixture of coarse coarse-grained-grained and fine-grained ﬁne-grained particles.

3. A NewThe optimalModel for mass Predicting content fMinimumc and the minimumVoid Ratio void of Gap-Graded ratio of gap-graded SRMs Incorporating SRMs are important parametersParticle Breakage inFigure engineering 9. Illustration practice. of the In mixture the next ofpart, coarse we-grained attempt and to fine-grained build a model particles. for predicting the optimal mass content fc and minimum void ratio of gap-graded SRMs. In this part, the existing models for predicting the minimum void ratio of gap-graded SRMs by 3. A New Model for Predicting Minimum Void Ratio of Gap-Graded SRMs Incorporating Vallejo [5] and Chang et al. [18] are evaluated by comparing the measured and predicted minimum Particle Breakage In this part, the existing models for predicting the minimum void ratio of gap-graded SRMs by Vallejo [5] and Chang et al. [18] are evaluated by comparing the measured and predicted minimum Appl. Sci. 2018, 8, 2584 8 of 16

3. A New Model for Predicting Minimum Void Ratio of Gap-Graded SRMs Incorporating Particle Breakage In this part, the existing models for predicting the minimum void ratio of gap-graded SRMs by Vallejo [5] and Chang et al. [18] are evaluated by comparing the measured and predicted minimum void ratios from compaction tests. Deﬁciencies of the existing models are identiﬁed and a new model for predicting minimum void ratio of gap-graded SRM is proposed.

3.1. Existing Model Vallejo [7] measured porosities of sand-clay mixtures. He found that the minimum porosity of the mixture occurs when all the void space in the sand is completely ﬁlled by the bulk volume of clay. He also proposed an equation for estimating the minimum porosity of the binary mixtures, as Equation (2).

nmix−min = n f nc (2) where nmix−min is the minimum porosity of the sand-clay mixture, nc is the porosity of pure sand, n f is the porosity of pure clay. However, the ideal condition in which all the void space of coarse-grained particles is completely filled by the bulk volume of fine-grained particles can hardly be achieved. During the process of achieving minimum void ratio of the mixture, mutual interference between the coarse-grained particles and fine-grained particles will happen, that is, the so-called “wall effect” and “loosening effect” [30], as shown in Figure 10a. Wall effect means that when some isolated coarse-grained particles are immersed in a sea of fine-grained particles (which are dominant), there is a further amount of voids in the packing of fine-grained particles located in the interface vicinity. Loosening effect refers to when fine-grained particles are inserted in the porosity of a coarse-grain packing (coarse particles dominant), and if it is no longer able to fit in a void, there is locally a decrease of volume of the coarse-grained particles. Chang et al. [23] considered the mutual inferences between fine-grained and coarse-grained particles when studying the minimum void ratio of sand-silt mixtures. They found that the added fine-grained particles would inevitably distort the structure of coarse-grained particles and cause a change of total void volume. They assumed that the change of void volume is proportional to the amount of fine-grained particles added to the mixture. Similarly, for a fine-grained particles dominant structure, the added coarse-grained particles will distort the structure of fine-grained particles and cause a change of total void volume, which is proportional to the amount of coarse particles added to the mixture. They gave the optimal fc corresponding to the minimum void ratio as Equation (3).

ec − eb fc−optimum = (3) 1 + ec + e f − ea − eb where ec is the void ratio of pure coarse-grained particles, ef is the void ratio of pure ﬁne-grained particles, ea and eb are material constants. Chang et al. [23] also proposed an equation for estimating the minimum porosity of the binary mixtures, as Equation (4).

nmix−min = ec fc−optimum + e f (1 − fc−optimum) − b12ec fc−optimum (4) where b12 is material constant. Appl.Appl. Sci. Sci. 20182018, 8,, 8x, FOR 2584 PEER REVIEW 9 of9 of16 16 Appl. Sci. 2018, 8, x FOR PEER REVIEW 9 of 16

Figure 10. The two parts constituting the change of void volume ΔV: (a) Mutual interference betweenFigureFigure 10.coarse-grained10.The The two two parts parts and constituting fine-grainedconstituting the particles changethe change ofΔV voidA , of(b volume)Particlevoid volume∆ breakageV:(a )Δ MutualV :Δ V(aB). interferenceMutual interference between

coarse-grainedbetween coarse-grained and fine-grained and fine-grained particles ∆particlesVA,(b)Particle ΔVA, (b breakage)Particle breakage∆VB. ΔVB. The minimum void ratio of gap-graded SRMs computed using Vallejo and Chang et al. models are shownTheThe minimumminimum in Figure void void 11 ratio ratioand of of gap-gradedcompared gap-graded with SRMs SRMs measured computed computed usingvalues using Vallejo Vallejofrom and compaction and Chang Chang et al. ettests. modelsal. models The are comparisonshownare shown in Figure shows in 11Figure thatand comparedthe11 predictabilityand withcompared measured of thewith valuesmodel measured fromfor sand-clay compaction values orfrom tests. sand-silt compaction The comparison mixtures tests. is shows notThe suitablethatcomparison the for predictability gap-graded shows that of SRMs. thethe modelpredictability It is forbecause, sand-clay of unthelike or model sand-silt sand-clay for mixturessand-clay or sand-silt is or not sand-silt suitablemixtures, mixtures for gap-graded gap-graded is not SRMSRMs.suitable will It undergofor is because,gap-graded more unlike particle SRMs. sand-clay breakageIt is because, or during sand-silt un vibratinglike mixtures, sand-clay compaction, gap-graded or sand-silt as SRMthe mixtures, test will results undergo gap-graded show more in SectionparticleSRM will2.2.1. breakage undergo The particle during more breakage vibratingparticle breakageduring compaction, the during process as thevibrating of test compaction results compaction, show will in generate as Section the test some2.2.1 results. much The show particle finer in particles,breakageSection 2.2.1.and during these The the particle finer process particles breakage of compaction will during further the will fill process generate the voids of compaction some of the much mixture will finer generate particles,(shown somein and Figure much these 10b). finerfiner Therefore,particlesparticles, will theand furthereffect these of fillfiner particle the particles voids breakage of will the mixtureshouldfurther (shownbefill takenthe voids in into Figure considerationof the10b). mixture Therefore, when (shown the building effect in Figure of models particle 10b). forbreakageTherefore, predicting should the minimum effect be taken of voidparticle into ratio consideration breakage of gap-graded should when SRMs. be building taken modelsinto consideration for predicting when minimum building void models ratio offor gap-graded predicting minimum SRMs. void ratio of gap-graded SRMs.

(a) D = 1.9 (b) D = 2.1 (a) D = 1.9 (b) D = 2.1 Figure 11. Cont. Appl.Appl. Sci. Sci. 20182018, 8, ,8 x, 2584FOR PEER REVIEW 1010 of of 16 16

(e) D = 2.7

FigureFigure 11. 11. ComparisonComparison of of measured measured minimum minimum void void rati ratiosos of of gap-graded gap-graded SRM SRM and and predicted predicted void void ratiosratios using using models models from from Vallejo Vallejo and and Chang Chang et et al. al.

3.2. A Model for Predicting Minimum Void Ratio of Gap-Graded SRMs Incorporating Particle Breakage 3.2. A Model for Predicting Minimum Void Ratio of Gap-Graded SRMs Incorporating Particle Breakage AA gap-graded gap-graded SRM SRM consists consists of oftwo two grain grain size size particles: particles: Fine-grained Fine-grained and coarse-grained. and coarse-grained. The The volume of the two grain size particles is denoted as V and V . The void ratio of the pure two grain volume of the two grain size particles is denoted as Vf andf Vc. Thec void ratio of the pure two grain size groups is e and e . Our objective is estimating the minimum void ratio of the gap-graded SRM. size groups is ef andf ec. cOur objective is estimating the minimum void ratio of the gap-graded SRM. Firstly,Firstly, we we consider consider the the coarse-grained particlesparticles asas thethe dominantdominant materials. materials. The The phase phase diagram diagram of V ofpure pure coarse-grained coarse-grained particles particles is shown is shown in Figure in Figure 12a. 12a. The solidThe solid volume volume of coarse of coarse particles particles is c, and is V thec, V andvoid the among void coarse-grainedamong coarse-grained particles particles is vc. Then, is V asvc. shownThen, as in Figureshown 12inb, Figure fine-grained 12b, fine-grained particles are particlesadded into are the added pure coarse-grainedinto the pure particles. coarse-grained In a limiting particles. case, allIn thea limiting added fine-grained case, all the particles added fill fine-grainedinto the voids particles among coarse-grainedfill into the particlesvoids among without coarse-grained altering the structure particles of coarse-grainedwithout altering particles. the V V structureTherefore, of thecoarse-grained solid volume particles. of fine particles Therefore, ( f )the occupies solid volume a space inof thefine void particles volume (Vf) ( occupiesvc) and thea spacetotal in volume the void remains volume constant. (Vvc) and However, the total asvolume stated remains above, theconstant. limiting However, case can as hardly stated be above, achieved, the limitingand the case total can void hardly volume be willachieved, change and due the to thetotal particle void volume breakage will and change altering due of theto the structure particle of breakagecoarse-grained and altering particles. of the The structure change of of total coarse-g volumerained is ∆ Vparticles., and it is The caused change by twoof total parts, volume one is is caused ΔV, andby theit is altering caused ofby thetwo structure parts, one of coarse-grainedis caused by the particles altering∆ ofV Athe, the structure other is of caused coarse-grained by particle particles breakage V V = V + V Δ∆VA,B the, ∆ other∆ isA caused∆ B. by particle breakage ΔVB, ΔV = ΔVA + ΔVB. Appl. Sci. 2018, 8, 2584 11 of 16 Appl. Sci. 2018, 8, x FOR PEER REVIEW 11 of 16

Appl. Sci. 2018, 8, x FOR PEER REVIEW 11 of 16

FigureFigure 12.12. Phase diagramsdiagrams forfor coarse-grainedcoarse-grained particlesparticles dominant:dominant: (a) purepure coarsecoarse particlesparticles (before(before

ﬁne-grainedfine-grained particlesparticles added),added), ((bb)) mixturemixture (limiting(limiting case),case), ((cc)) mixturemixture (general(general case).case). Figure 12. Phase diagrams for coarse-grained particles dominant: (a) pure coarse particles (before

(1)(1)fine-grained CalculationCalculation particles ofof ∆ΔVV added),AA: According (b) mixture toto ChangChang (limiting etet case), al.al. (2015),(2015), (c) mixture thethe changechange(general ofcase).of voidvoid volume volume∆ ΔVVAA causedcaused byby thethe addedadded ﬁne-grainedfine-grained particles particles is is proportional proportional to to the the amount amount of ﬁne-grainedof fine-grained particles particles added added to the to (1) Calculation of ΔVA: According to Chang et al. (2015), the change of void volume ΔVA caused mixture.the mixture. Therefore, Therefore,∆VA ΔcanVA can be calculatedbe calculated using using Equation Equation (5). (5). by the added fine-grained particles is proportional to the amount of fine-grained particles added to

A Δ = the mixture. Therefore, ΔV can be calculated usingVVA = EquationaV f (5). (5) ∆ A f (5) Δ = VA aV f (5) wherewhere aais material is material constant. constant. (2) Calculation of ΔVB: g where(2) Calculationa is material of constant.∆VB:The The data data of of particle particle breakage breakage index index BBg, ,which which is is measured measured in thethe compaction tests, is plotted versus fc in Figure 13. The particle breakage index Bg is found to be compaction(2) Calculation tests, is plottedof ΔVB: The versus datafc inof Figureparticle 13 breakage. The particle index breakageBg, which index is measuredBg is found in the to be c linearlylinearlycompaction relatedrelated tests, toto f fcis. Here, plotted we versus only showfc in Figure the cases 13. ofThe D particle= 2.3 and breakage 2.5 for theindex limitationlimitation Bg is found of thethe to paperpaperbe length.length.linearly Therefore,Therefore, related to BB fggc. canHere, be we represented only show with the cases Equation of D (6):= 2.3 and 2.5 for the limitation of the paper length. Therefore, Bg can be represented with Equation= + (6): Bg k1 fc b1 (6) Bg == k1 fc + b1 (6) Bg k1 fc b1 (6) where k1 and b1 are material constants. where k1 and b1 are material constants. where k1 and b1 are material constants.

Figure 13. Changes of particle breakage index Bg with fc under fractal dimension D = 2.3 and 2.5. FigureFigure 13.13. ChangesChanges ofof particleparticle breakagebreakage indexindexB Bgg withwith ffcc under fractal dimension D = 2.3 and 2.5. Appl. Sci. 2018, 8, 2584 12 of 16

Bg reﬂects the degree of particle breakage of coarse-grained and ﬁne-grained particles. The more particle breakage occurs, the larger Bg is, and thus the void volume change compared with the pure coarse-grained particles (Figure 10a) due to particle breakage ∆V is larger. We assume ∆V is B B V V B − B proportional to the product of solid volume ( c + f) and ( g g fc=0 ), thus:

V = k (V + V )(B − B ) ∆ B 2 f C g g fc=0 (7) where k is material constant. 2 B = k f + b B = b g 1 c 1, g fc=0 1 and thus:

∆VB = k2(Vf + VC)k1 fc (8)

Combining Equations (5) and (8):

∆V = ∆VA + ∆VB = aVf + k1k2(Vf + VC) fc (9)

According to the phase diagrams of coarse-grained particles dominant structure, the minimum void ratio of the mixture can be expressed as Equation (10)

VVC − Vf + ∆V eM = (10) VC + Vf

According to the deﬁnition of fc:

m f γ f Vf fc = = (11) mc + m f γcVc + γ f Vf where mc is the mass of coarse-grained particles, mf is the mass of fine-grained particles, γc is the unit weight of coarse-grained particles, γ f is the unit weight of fine-grained particles. Because the coarse-grained particles and fine-grained particles used in this paper have the same lithology, their unit weights are almost the same, γc = γ f , and thus:

Vf fc = (12) VC + Vf

Substituting Equations (9) and (12) for Equation (10), and together with VVC = ecVC, the minimum void ratio of the mixture can be expressed as:

M e = (a + k1k2 − 1 − ec) fc + ec (13)

Let k = a + k1k2 − 1, thus Equation (13) can be rewritten as:

M e = (k − ec) fc + ec (14) where k is a material constant. It can be seen from Equation (14) that the minimum void ratio of gap-graded SRM is linearly related to the fine particles content, which is consistent with the compaction test results shown in Figure8. In the case of fine-grained particle dominant, the phase diagram is shown in Figure 14. The solid volume of fine-grained particles is Vf, and the void among coarse-grained particles is Vvf. Then, as shown in Figure 14b, coarse-grained particles are added to the pure fine-grained particles. In a limiting case, all the added coarse-grained particles are embedded in the sea of fine-grained particles without altering the structure of fine-grained particles. The total void volume Vvf remains constant. Appl. Sci. 2018, 8, 2584 13 of 16

Appl.However, Sci. 2018 in, 8, ax FOR general PEER sense, REVIEW the total void volume will change due to the particle breakage13 andof 16 altering the structure of fine-grained particles. The change of total volume is ∆V, and it is caused by breakage and altering the structure of fine-grained particles. The change of total volume is ΔV, and two parts: One is caused by the altering the structure of fine-grained particles ∆VA, the other is caused it is caused by two parts: One is caused by the altering the structure of fine-grained particles ΔVA, by particle breakage ∆VB, ∆V = ∆VA + ∆VB. ∆VA is proportional to the amount of coarse-grained the other is caused by particle breakage ΔVB, ΔV = ΔVA + ΔVB. ΔVA is proportional to the amount of particles added to the mixture. ∆VB is assumed to be the product of solid volume (Vc + Vf) and coarse-grained particles added to the mixture. ΔVB is assumed to be the product of solid volume (Vc B − B ( g g fc=1 ). Similar to the derivation of Equation (14), the minimum void ratio of gap-graded SRM + Vf) and ( B − B ). Similar to the derivation of Equation (14), the minimum void ratio of with fine-graing dominantg fc=1 can be expressed as: gap-graded SRM with fine-grain dominant can be expressed as: eM = (e + k0) f − k0 (15) M = f + c − e (e f k')f c k' (15) where k’ is material constant. where k’ is material constant.

Figure 14.14.Phase Phase diagrams diagrams for for ﬁne-grained fine-grained particles particles dominant: dominant: (a) pure (a) ﬁne-grainedpure fine-grained particles particles (before (beforecoarse-grained coarse-grained particles particles added), added), (b) mixture (b) mixture (limiting (limiting case), (c )case), mixture (c) mixture (general (general case). case).

M For a given mass content in fine-grained group fc, two values of minimum void ratio e can be For a given mass content in fine-grained group fc, two values of minimum void ratio eM can be calculated: One is from Equation (14), the other is from Equation (15). The greater of the two void ratio calculated: One is from Equation (14), the other is from Equation (15). The greater of the two void values is likely to be achieved, because it requires less energy to reach the state. Therefore, the greater ratio values is likely to be achieved, because it requires less energy to reach the state. Therefore, the of the two values is considered to be the solution. greater of the two values is considered to be the solution. The linear relationship between the minimum void ratio of gap-graded SRM and fc is shown in The linear relationship between the minimum void ratio of gap-graded SRM and fc is shown in Figure 15. The line represented by Equation (14) is shown in Figure 15 as the line AP. When fc = 0, Figure 15. The line represented by Equation (14) is shown in Figure 15 as the line AP. When fc = 0, the minimum void ratio is ec. The line represented by Equation (15) is shown in Figure 15 as the line the minimum void ratio is ec. The line represented by Equation (15) is shown in Figure 15 as the line PB. When fc = 1, the minimum void ratio is ef. The value of fc corresponding to point P can be solved PB. When fc = 1, the minimum void ratio is ef. The value of fc corresponding to point P can be solved from Equations (14) and (15), and this value is the optimal mass content of fine-grained particles. from Equations (14) and (15), and this value is the optimal mass content of fine-grained particles. 0 e+c + k = ec k' fc−foptimum= 0 (16) c-optimum e+c + e+f +−k − k (16) ec ef k k' The model for estimating minimum void ratio of gap-graded SRM employs 4 parameters, that is, The model for estimating minimum void ratio of gap-graded SRM employs 4 parameters, that ec, ef, k and k’. ec and ef are the void ratio of pure coarse-grain particles and pure fine-grained is, ec, ef, k and k’. ec and ef are the void ratio of pure coarse-grain particles and pure fine-grained particles, which can be easily obtained by conducting the compaction tests on pure coarse-grained particles, which can be easily obtained by conducting the compaction tests on pure coarse-grained and fine-grained particles. k and k’ are material constants. Data of k and k’ are plotted versus fractal and fine-grained particles. k and k’ are material constants. Data of k and k’ are plotted versus fractal dimension D of basic gradation in Figure 16. It can be found that the relationship between k, k’ and D dimension D of basic gradation in Figure 16. It can be found that the relationship between k, k’ and can be well fitted by a quadratic polynomial function, which is given by k or k0 = AD2 + BD + C. A, B D can be well fitted by a quadratic polynomial function, which is given by k or k' = AD2 + BD + C . and C are material constants irrespective of gap intervals. Therefore, if the parameters of D, ec, ef of a A, B and C are material constants irrespective of gap intervals. Therefore, if the parameters of D, ec, gap-graded SRM are given, we only need to conduct compaction tests of one gap interval to obtain the ef of a gap-graded SRM are given, we only need to conduct compaction tests of one gap interval to parameters of ec, ef, A, B, C, and we can predict the minimum void ratio of gap-graded SRM with other obtain the parameters of ec, ef, A, B, C, and we can predict the minimum void ratio of gap-graded gap-intervals. It is noted that we can obtain the parameters k and k’ by conducting some compaction SRM with other gap-intervals. It is noted that we can obtain the parameters k and k’ by tests if the gradation information is not given. conducting some compaction tests if the gradation information is not given. Appl. Sci. 2018, 8, 2584 14 of 16 Appl. Sci. 2018, 8, x FOR PEER REVIEW 14 of 16 Appl. Sci. 2018, 8, x FOR PEER REVIEW 14 of 16

Figure 15. Changes of minimum void ratio of gap-graded SRM with fc. FigureFigure 15. 15.Changes Changes of of minimum minimum void void ratio ratio of of gap-graded gap-graded SRM SRM with withf cf.c.

(a) k D (b) k’ D (a) k D (b) k’ D FigureFigure 16.16. FittingFitting thethe materialmaterial constantsconstants kk andand kk’’ withwith aa quadraticquadratic polynomialpolynomial functionfunction ofof fractalfractal Figure 16. Fitting the material constants k and k’ with a quadratic polynomial function of fractal dimensiondimensionD D.. dimension D. 3.3.3.3. ValidationValidation ofof thethe ModelModel 3.3. Validation of the Model InIn order order to to evaluate evaluate the the model, model, we we use use the the vibrating vibrating compaction compaction testtest datadata ofof gap-graded gap-graded SRMsSRMs In order to evaluate the model, we use the vibrating compaction test data of gap-graded SRMs withwith gapgap intervalinterval ofof 5–205–20 mm to obtain the the model model parameters parameters like like k kandand k’k by’ by fitting ﬁtting k andk and k’ kwith’ with D. with gap interval of 5–20 mm to obtain the model parameters like k and k’ by fitting k and k’ with D. DTogether. Together with with the the data data of ofec andec and ef, weef, wecan can predict predict the theminimum minimum void void ratio ratio of gap-graded of gap-graded SRM SRM with Together with the data of ec and ef, we can predict the minimum void ratio of gap-graded SRM with withgap gapintervals intervals of 2–5 of 2–5mm, mm, 0.5–2 0.5–2 mm, mm, and and 0.25–0.5 0.25–0.5 mm mm using using Equations Equations (14) (14) and (15).(15). TheThe modelmodel gap intervals of 2–5 mm, 0.5–2 mm, and 0.25–0.5 mm using Equations (14) and (15). The model parametersparameters areare listedlisted inin TableTable2 .2. The The comparison comparison of of predicted predicted minimum minimum void void ratio ratio and and measured measured parameters are listed in Table 2. The comparison of predicted minimum void ratio and measured minimumminimum void void ratio ratio by by vibrating vibrating compaction compaction tests tests is is shown shown in in Figure Figure 17 .17. It canIt can be seenbe seen from from Figure Figure 16 minimum void ratio by vibrating compaction tests is shown in Figure 17. It can be seen from Figure that16 that the predictedthe predicted minimum minimum void ratio void is closeratio tois theclose measured to the values,measured which values, shows which the validation shows the of 16 that the predicted minimum void ratio is close to the measured values, which shows the thevalidation model. of the model. validation of the model. Table 2. Model parameters for the prediction of void ratio of gap-graded SRM with gap intervals of Table 2. Model parameters for the prediction of void ratio of gap-graded SRM with gap intervals of 2–5Table mm, 2. 0.5–2Model mm parameters and 0.25–0.5 for mm.the prediction of void ratio of gap-graded SRM with gap intervals of 2–5 mm, 0.5–2 mm and 0.25–0.5 mm. 2–5 mm, 0.5–2 mm and 0.25–0.5 mm. Gap Interval ec ef k k’ Gap Interval ec ef k k’ Gap Interval ec ef k k’ 2–52–5 mm 0.433 0.433 0.442 0.442 2–5 mm 0.433 0.442 k = 0.094D2 − 0.44D + 0.49 k’ = −0.24D2 + 1.11D − 1.13 0.5–20.5–2 mm 0.413 0.413 0.467 0.467 k = 0.094D2 − 0.44D + 0.49 k’ = −0.24D2 + 1.11D − 1.13 0.5–2 mm 0.413 0.467 k = 0.094D2 − 0.44D + 0.49 k’ = −0.24D2 + 1.11D − 1.13 0.25–0.50.25–0.5 mm 0.403 0.403 0.472 0.472 0.25–0.5 mm 0.403 0.472 Appl. Sci. 2018, 8, 2584 15 of 16 Appl. Sci. 2018, 8, x FOR PEER REVIEW 15 of 16

FigureFigure 17.17. ComparisonComparison betweenbetween measured minimum void ratio andand predictedpredicted minimumminimum voidvoid ratiosratios usingusing EquationsEquations (14)(14) andand (15).(15).

4.4. Conclusions AA series of vibrating compaction tests were conductedconducted on SRMs with different basic gradations, gapgap intervalsintervals andand massmass contentscontents of of fine-grained fine-grained group. group. It It was was found found that that gap-graded gap-graded SRM SRM may may reach reach a smallera smaller void void ratio ratio than than the SRMthe SRM with awith continuous a continuous gradation gradation under someunder circumstances, some circumstances, for example, for theexample, gap-graded the gap-graded SRMs without SRMs 5–20 without mm particles5–20 mm have particles a smaller have void a smaller ratio thanvoid theratio SRMs than with the SRMs basic gradation,with basic ingradation, the case in of theD = case 2.3, of 2.5 D and = 2.3, 2.7. 2.5 If theand particles2.7. If the in particles a gap interval in a gap play interval the role play of the filling role components,of filling components, the absence the of absence them will of increase them will the increase void ratio the of void the SRM. ratioAmong of the SRM. gap intervals Amongfor gap a certainintervals basic for a gradation, certain basic there gradation, is a vital there grain is size a vital group grain that size dominates group that the dominates compaction the ofcompaction the SRM. Theof the particle SRM. breakage The particle of gap-graded breakage SRMs of gap-grad is moreed prominent SRMs is than more the prominent SRM with continuousthan the SRM gradation with oncontinuous the whole, gradation especially on at the the whole, gap interval especially of 5–20 at mmthe gap among interval the selected of 5–20 gapmm intervals. among the selected gap intervals.A new model is proposed for better predicting the minimum void ratio of gap-graded SRMs. The modelA new requires model is three proposed parameters, for betterec, e fpredictingand D, for the prediction minimum of gap-gradedvoid ratio of SRMs gap-graded with different SRMs. mass contents of fine-grained particles. Comparison between the predicted minimum void ratios The model requires three parameters, ec, ef and D, for prediction of gap-graded SRMs with different measuredmass contents void of ratios fine-grained proves the particles. validation Compar of thisison model. between It should the predicted be noted thatminimum the new void model ratios is onlymeasured valid forvoid the ratios lithology proves and the the validation shape of theof this particles model. studied It should in this be noted work. that the new model is Authoronly valid Contributions: for the lithologyConceptualization, and the shape T.W. of and the S.L.; particles Methodology, studied Y.F.; in this Software, work. J.Y.

Funding:Author Contributions:This research was Conceptualization, funded by National T.W. Natural and S.L.; Science Methodology, Foundation Y.F.; of China Software, (U1765205; J.Y. 2017YFC0404805). The APC was funded by Hohai University. Acknowledgments:Funding: This researchThis work was was funded supported by byNational the “National Natural Natural Science Science Foundation Foundation of ofChina China” (U1765205; (Grant No. U1765205)2017YFC0404805). and “National The APC Key was R&D fund Programed by Hohai of China” University. (Grant No. 2017YFC0404805). ConﬂictsAcknowledgments: of Interest: ThisThe authorswork was declare supported no conﬂict by the of “National interest. Natural Science Foundation of China” (Grant No. U1765205) and “National Key R&D Program of China” (Grant No. 2017YFC0404805).

ReferencesConflicts of Interest: The authors declare no conflict of interest. 1. Wang, Y.; Li, X.; Zheng, B.; Mao, T.Q.; Hu, R.L. Investigation of the effect of soil matrix on flow characteristics Referencesfor soil and rock mixture. Géotech. Lett. 2016, 6, 226–233. [CrossRef] 2.1. Xu,Wang, W.J.; Y.; Xu, Li, Q.; X.; Hu, Zheng, R.L. Study B.; Mao, on the T.Q.; shear Hu, strength R.L. Inve of soil–rockstigation mixture of the byeffect coarse of scalesoil directmatrix shear on flow test. Int.characteristics J. Rock Mech. for Min. soil Sci.and 2011rock, mixture.48, 1235–1247. Géotech. Lett. 2016, 6, 226–233. 3.2. Kumara,Xu, W.J.; J.J.;Xu, Hayano, Q.; Hu, K.;R.L. Kikuchi, Study Y.on Deformation the shear strength behaviour of soil–rock of gap-graded mixture fouled by ballastcoarse evaluatedscale direct by shear a 3D discretetest. Int. elementJ. Rock Mech method.. MinKSCE. Sci. 2011 J. Civ., 48 Eng., 1235–1247.2016, 20 , 2345–2354. [CrossRef] 4.3. Zhong,Kumara, Q.M.; J.J.; Hayano, Chen, S.S.; K.; Deng,Kikuchi, Z. Y. A simplifiedDeformation physically-based behaviour of gap-graded breach model fouled for aballast high concrete-facedevaluated by a rockfill3D discrete dam: element A case study.method.Water KSCE Sci. J. Eng.Civ. Eng2018. ,201611, 46–52., 20, 2345–2354. [CrossRef ] 5.4. Li,Zhong, L.K.; Q.M.; Wang, Chen, Z.J.; Liu, S.S.; S.H.; Deng, Bauer, Z. A E. simplified Calibration physically-based and performance breach of two model different for constitutive a high concrete-faced models for rockfillrockfill materials.dam: A caseWater study. Sci. Water Eng. 2016Sci. Eng, 9, 227–239.. 2018, 11 [, CrossRef46–52. ] 5. Li, L.K.; Wang, Z.J.; Liu, S.H.; Bauer, E. Calibration and performance of two different constitutive models for rockfill materials. Water Sci. Eng. 2016, 9, 227–239. Appl. Sci. 2018, 8, 2584 16 of 16

6. Khalili, A.; Wijewickreme, D.; Wilson, G.W. Mechanical response of highly gap-graded mixtures of waste rock and tailings. Part I: Monotonic shear response. Can. Geotech. J. 2010, 47, 552–565. [CrossRef] 7. Vallejo, L.E. Interpretation of the limits in shear strength in binary granular mixture. Can. Geotech. J. 2001, 38, 1097–1104. [CrossRef] 8. Fei, K. Experimental study of the mechanical behavior of clay–aggregate mixtures. Eng. Geol. 2016, 210, 1–9. [CrossRef] 9. Ma, C.; Zhan, H.B.; Yao, W.M.; Li, H.Z. A new shear rheological model for a soft interlayer with varying water content. Water Sci. Eng. 2016, 11, 131–138. [CrossRef] 10. Wan, C.F.; Fell, R. Assessing the Potential of Internal Instability and Suffusion in Embankment Dams and Their Foundations. J. Geotech. Geoenviron. Eng. 2008, 134, 401–407. [CrossRef] 11. Langroudi, M.F.; Soroush, A.; Shourijeh, P.T.; Shafipour, R. Stress transmission in internally unstable gap-graded soils using discrete element modeling. Powder Technol. 2013, 247, 161–171. [CrossRef] 12. Zhang, X.Y.; Baudet, B.A. Particle breakage in gap-graded soil. Géotech. Lett. 2013, 3, 72–77. [CrossRef] 13. Le, M.V.T.; Marot, D.; Rochim, A.; Bendahmane, F.; Nguyen, H.H. Suffusion susceptibility investigation by energy based method and stat. Can. Geotech. J. 2018, 55, 57–68. [CrossRef] 14. Chen, S.K.; He, Q.D.; Cao, J.G. Seepage simulation of high concrete-faced rockfill dams based on generalized equivalent continuum model. Water Sci. Eng. 2018, 11, 250–257. [CrossRef] 15. Slangen, P.; Fannin, R.J. The role of particle type on suffusion and suffosion. Géotech. Lett. 2017, 7, 1–5. [CrossRef] 16. Wang, S.; Chen, J.S.; He, H.Q.; He, W.Z. Experimental study on piping in sandy gravel foundations considering effect of overlying clay. Water Sci. Eng. 2016, 9, 165–171. [CrossRef] 17. Shen, C.M.; Liu, S.H.; Wang, Y.S. Microscopic interpretation of one-dimensional compressibility of granular materials. Comput. Geotech. 2017, 91, 161–168. [CrossRef] 18. Alam, M.F.; Haque, A.; Ranjith, P.G. A Study of the Particle-Level Fabric and Morphology of Granular Soils under One-Dimensional Compression Using Insitu X-ray CT Imaging. Materials 2018, 11, 919. [CrossRef] [PubMed] 19. Sandeep, C.S.; Senetakis, K. Effect of Young’s Modulus and Surface Roughness on the Inter-Particle Friction of Granular Materials. Materials 2018, 11, 217. [CrossRef] 20. Chang, C.S.; Meidani, M.; Deng, Y. A compression model for sand-silt mixtures based on the concept of active and inactive voids. Acta Geotech. 2017, 12, 1301–1317. [CrossRef] 21. Fragaszy, R.J.; Su, J.; Siddiqi, F.H. Modeling Strength of Sandy Gravel. J. Geotech. Eng. 1992, 118, 920–935. [CrossRef] 22. Zhou, Z.; Fu, H.L.; Liu, B.C. Orthogonal tests on permeability of soil-rock-mixture. Chin. J. Geotech. Eng. 2006, 28, 1134–1138. 23. Chang, C.S.; Wang, J.Y.; Ge, L. Modeling of minimum void ratio for sand–silt mixtures. Eng. Geol. 2015, 196, 293–304. [CrossRef] 24. Jiang, Q.; Cui, J.; Chen, J. Time-Dependent Damage Investigation of Rock Mass in an In Situ Experimental Tunnel. Materials 2012, 5, 1389–1403. [CrossRef] 25. Tyler, S.W. Fractal scaling of soil particle size distributions: Analysis and limitations. Soil Sci. Soc. Am. J. 1992, 56, 362–369. [CrossRef] 26. Mcgeary, R.K. Mechanical Packing of Spherical Particles. J. Am. Ceram. Soc. 1961, 44, 513–522. [CrossRef] 27. Lee, K.L.; Farhoomand, I. Compressibility and crushing of granular soil in anisotropic triaxial. Can. Geotech. J. 1967, 4, 68–86. [CrossRef] 28. Marsal, R.J. Coarse-scale testing of rockfills materials. J. Soil Mech. Found. Eng. ASCE 1967, 93, 27–44. 29. Hardin, B.O. Crushing of Soil Particles. J. Geotech. Eng. 1985, 111, 1177–1192. [CrossRef] 30. De Larrard, F. Concrete Mixture Proportioning: A Scientific Approach; Taylor & Francis: London, UK, 1999.

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