On the micromechanics of crushable aggregates · tive stress (Bolton, 1986), since at high...

On the micromechanics of crushable aggregates

G. R. McDOWELL� and M. D. BOLTON�

This paper presents a study of the micromecha-nical behaviour of crushable soils. For a singlegrain loaded diametrically between ¯at platens,data are presented for the tensile strengths ofparticles of different size and mineralogy. Thesedata are shown to be consistent with Weibullstatistics of brittle fracture. Triaxial tests ondifferent soils of equal relative density show thatthe dilatational component of internal angle offriction reduces logarithmically with mean effec-tive stress normalized by grain tensile strength.The tensile strength of grains is also shown togovern normal compression. For a sample ofuniform grains under uniaxial compression, theyield stress is related to the average grain tensilestrength. If particles fracture such that thesmallest particles are in geometrically self-simi-lar con®gurations under increasing macroscopicstress, with a constant probability of fracture, afractal geometry evolves with the successivefracture of the smallest grains, in agreementwith the available data. A new work equationpredicts that the evolution of a fractal geometrygives rise to a linear compression line whenvoids ratio is plotted against the logarithm ofmacroscopic stress, in agreement with publisheddata.

KEYWORDS: compressibility; constitutive relations;plasticity; sands; statistical analysis.

Cet exposeÂ preÂsente une eÂtude du comportementmicro meÂcanique des sols concassables. Pour unseul grain chargeÂ de manieÁre diameÂtrale entredes platines plates, nous preÂsentons les donneÂesde reÂsistance aÁ la rupture des particules dediffeÂrentes dimensions et mineÂralogies. Nousmontrons que ces donneÂes correspondent auxstatistiques de Weibull sur la rupture de fragi-liteÂ. Les essais triaxiaux sur divers sols de meÃmedensiteÂ relative montrent que le composant dedilatation de l'angle interne de friction baisse demanieÁre logarithmique en meÃme temps que lacontrainte effective moyenne normaliseÂe par lareÂsistance aÁ la rupture du grain. Nous montronsaussi que la reÂsistance aÁ la rupture des grainsgouverne la compression normale. Pour uneÂchantillon de grains uniformes sous compres-sion uniaxiale, la limite eÂlastique est lieÂe aÁ larupture moyenne des grains. Si la rupture desparticules est telle que les plus petites particulessont dans des con®gurations aÁ similitude geÂomeÂ-trique intrinseÁque sous un effort macroscopiquede plus en plus grand, avec probabiliteÂ constantede rupture, une geÂomeÂtrie fractale apparaõÃt avecla rupture successive des grains les plus ®ns, enaccord avec les donneÂes disponibles. Une nou-velle eÂquation de travail preÂdit que l'eÂvolutionde la geÂometrie fractale provoque l'apparitiond'une ligne de compression lineÂaire quand letaux de pores est repreÂsenteÂ sous forme decourbe par rapport au logarithme de l'effortmacroscopique, en accord avec les donneÂes pub-lieÂes.

INTRODUCTION

It is well known that particle fracture plays a majorrole in the behaviour of crushable soils. For anaggregate of particles under normal compression,the yield stress of a calcareous sand is lower thanthat of a siliceous sand, simply because the calcar-eous particles are more friable (Golightly, 1990).The dilatational component of the internal angle offriction of a sand measured in a triaxial test is

known to reduce logarithmically with mean effec-tive stress (Bolton, 1986), since at high stresses,crushing eliminates the dilatancy. The normal com-pression of a sand is known to give rise to theevolution of a distribution of particle sizes(Fukumoto, 1992). In particular, it has been foundthat the particle size distributions of broken andcrushed granular materials tend to be self-similaror fractal (Turcotte, 1986). The particle gradingcan be characterized by de®ning a fractal dimen-sion which, remarkably, often tends to be about 2´5for aggregates subjected to pure crushing (Turcotte,1986; Palmer & Sanderson, 1991).

The detailed micromechanics of the phenomenamentioned above are not well understood. The

McDowell, G. R. & Bolton, M. D. (1998). GeÂotechnique 48, No. 5, 667±679

667

Manuscript received 17 July 1997; revised manuscriptreceived 4 February 1998.Discussion on this paper closes 1 January 1999.� University of Cambridge.

tensile strength of particles must govern the beha-viour of an aggregate, but how, for example, doesnormal compression give rise to a distribution ofparticle sizes? The dif®culty begins with de®ninggrain fracture, in order to obtain a consistentde®nition of particle strength.

This paper aims to relate the micromechanics ofgrain fracture to the macroscopic deformation ofcrushable aggregates. To this end, it is necessary®rst of all to obtain a reliable de®nition of grainfracture. Data from tests on grains crushed between¯at platens are explained in terms of the Weibullstatistics of fracture of brittle ceramics (Weibull,1951), and this permits a consistent de®nition ofgrain tensile strength. The paper uses this de®ni-tion to quantify the behaviour of various crushablesoils. A study is made of the in¯uence of grainstrength on the internal angle of friction for dila-tant, crushable aggregates. In addition, the micro-mechanics of normal compression are examined,with the speci®c purpose of explaining the exis-tence of approximately linear compression lines ineÿlog ó 9 space. This leads to an expression for thecompressibility index of the aggregate, in terms ofmore fundamental particle parameters.

FRACTURE OF A SINGLE PARTICLE

It is widely accepted that the failure of aspherical particle under compression is in fact atensile failure. There is a tensile stress distributioninduced in the particle and failure occurs when thetensile stress ó at a critical ¯aw of size a is suchthat the stress intensity factor K � Yó

pða (where

Y is a dimensionless number relating to the pro-blem geometry) for the ¯aw reaches the mode Ifracture toughness of the material KIc, according toGrif®th's criterion (Grif®th, 1920). Consequently,the `tensile strength' of rock grains can be indir-ectly measured by diametral compression between¯at platens (Jaeger, 1967). Lee (1992) compressedindividual grains of Leighton Buzzard sand, ooliticlimestone and carboniferous limestone, in such amanner shown in Fig. 1. For a grain of diameter dunder a diametral force F, a characteristic tensilestress induced within it may be de®ned as

ó � F

d2(1)

following Jaeger (1967) and Shipway & Hutchings(1993). This is also consistent with the de®nitionof tensile strength of concrete in the Brazilian test.Fig. 2 shows a typical plot of platen load F as afunction of the platen displacement ä. It can beseen that there are some initial peaks in the load±displacement curve, which correspond to the frac-turing of asperities, and the rounding of the parti-cle as small corners break off. For a particle ofsize d under load F, an asperity of size a may be

considered to have an induced tensile stress ofF=a2, and the asperity would break off when thisstress attained the critical value for the asperity.However, this cannot be described as failure of theparticle: the asperity under the platen which hasjust broken off would have fractured irrespective ofthe size of the parent particle. Hence, a morereliable de®nition of particle breakage is required.It can be clearly seen in Fig. 2 that the initialsmall peaks are followed by a large peak corre-

Ff 2 fracture force

Polishedsurfaces

Fig. 1. Particle tensile strength test set-up (Lee, 1992)

8

6

4

2

0

Platen displacement

Bearing failuresat contact points

For

ce: k

N

Ff

Fig 2. Typical load-de¯ection plot (Lee, 1992)

668 McDOWELL AND BOLTON

sponding to the maximum load, and a catastrophicfailure as the particle splits, and the load dropsdramatically. Particle fracture will henceforth beinterpreted as particle `splitting'. With this de®ni-tion of fracture, Lee calculated the tensile strengthof grains as

ó f � Ff

d2(2)

where the subscript f denotes failure. He foundthat for particles of a given size and mineralogy,the tensile strength is not a constant but has astandard deviation about some mean value. Further-more, he found the average tensile strength to be afunction of particle size d. Fig. 3 shows the meantensile strength ó f as a function of the averageparticle size d. The data are described by therelation

ó f / db (3)

where typical values of b are given by ÿ0:357,ÿ0:343 and ÿ0:420 for Leighton Buzzard sand,oolitic limestone and carboniferous limestone, re-spectively. This size effect on particle strength wasalso evident in particle crushing tests performed byBillam (1972), and it is a direct consequence ofthe statistical variation in the strength of brittleceramics. Because ceramic materials contain a dis-tribution of ¯aw sizes, small samples are strongerthan large samples, since there are fewer and smal-ler ¯aws. This will now be quanti®ed more rigor-ously using Weibull statistics of fracture, which iswidely accepted to describe the tensile strength ofbrittle ceramics.

Weibull (1951) recognizes that the survival of a

block of a material under tension requires that allits constituent parts remain intact (i.e. a chain is asstrong as its weakest link). Weibull stated that fora volume V , under an applied tensile stress ó, the`survival probability' Ps(V ) of the block is givenby

Ps(V ) � exp ÿ V

V0

ó

ó0

� �m" #

(4)

where V0 is a reference volume of material suchthat

Ps(V0) � exp ÿ ó

ó0

� �m" #

(5)

which is plotted in Fig. 4. The parameter ó0 is thevalue of tensile stress ó such that 37% (i.e.100 exp(ÿ1)%) of the total number of tested blockssurvive. The exponent m is the Weibull modulus,and decreases with increasing variability in tensilestrength. For chalk, stone, pottery and cement m isabout 5. A similar value would be expected forcarbonate sands, which have similar intraparticleporosity values. The engineering ceramics, such asAl2O3 have values for m of about 10, and thevariation in strength is much less. For soils, weexpect 5 , m , 10. It can readily be shown that themean tensile strength for samples of volume V0 isgiven by

ómean � ó0Ã(1� 1=m) (6)

where Ã is the gamma function, and returns avalue of about 1 for the values of m being consid-ered. It is also readily seen in Fig. 4 that the

1 2 3 5 10 20 30 10050

Average particle size: mm

0.2

0.5

1

2

5

10

20

50

σ f: M

Pa

b 5 20.357

b 5 20.343

b 5 20.420

Mean andmean 62s

Oolitic limestone

Carboniferous limestone

Leighton Buzzard sand

Fig. 3. Mean tensile strength as a function of particle size (Lee, 1992)

MICROMECHANICS OF CRUSHABLE AGGREGATES 669

maximum rate at which samples of volume V0

fracture as the applied tensile stress increases oc-curs at around ó � ó0. A more rigorous analysisshows that dPs=dó is a minimum (i.e. dPf=dó is amaximum, where Pf is the fracture probability) ata value of stress given by

ó � ó0

mÿ 1

m

� �1=m

(7)

It is evident, then, that ó0 is a signi®cant anduseful parameter in describing the strength of cera-mic materials. For a block of material of volumeV1 under tension, equation (4) gives the 37%strength ó01 as

ó01 � ó0

V0

V1

� �1=m

(8)

and the mean strengths scale in the same way.It is now necessary to examine the applicability

of Weibull to the data presented by Lee in Fig. 3.If particles are approximately geometrically similar,and have the same number and distribution ofcontacts, the size of the zones of tensile stressinduced within them must scale with their volume.In this case, Weibull applies. The survival prob-ability of a particle size d under diametral com-pression is therefore given by

Ps(d) � exp ÿ d

d0

� �3 ó

ó0

� �m" #

(9)

where ó is the characteristic tensile stress inducedin the particle given by equation (1), and ó0 isnow the value of F=d2 at which 37% of the testedparticles survive, and is approximately equal to(and is proportional to) the mean tensile strengthof particles of size d0. It is evident from equations(8) and (9) that the average tensile strength ofgrains scales with particle size according to therelation

ó0 / dÿ3=m (10)

which is equivalent to equation (3). Apparently,values of m in the range 5±10 cover Lee's data forrock grains in Fig. 3.

The above analysis concludes that the meanvalue of F=d2 at fracture for grains compresseddiametrically between ¯at platens is a proper statis-tical measure of the tensile strength of ceramicparticles.

INFLUENCE OF GRAIN STRENGTH ON ANGLE OF

INTERNAL FRICTION

Bolton (1986) proposed an empirical relation forthe extra component of internal angle of frictionÄö due to dilatancy above the critical statestrength ö9crit, as a function of the mean effectivestress p9 and the initial relative density ID:

Äö � ö9max ÿ ö9crit � AI R8 (11)

0 0.5 1 1.5 20

0.2

1

0.4

0.6

0.8

1.2

Increasing variabilityin strength

m 5 10

m 5 5

Ps(

V0)

σ /σ0

Fig. 4. Weibull distribution of strengths


where A � 3 for triaxial strain and A � 5 in planestrain conditions, and

IR � ID(Qÿ ln p9)ÿ 1 (12)

where p9 is in kilopascals. The parameter Q relatesto the mean effective stress required to suppressdilatancy, and takes a value of 10 for the quartzand feldspar sands discussed by Bolton. Bolton(1986) noted that triaxial tests by Billam (1972)had shown that reducing the crushing strength ofthe grains reduced the critical mean effective stressrequired to suppress dilatancy, implying that thevalue of Q in equation (12) should be reduced forsoils of weaker grains. Bolton suggested values forQ of 8 for limestone, 7 for anthracite and 5´5 forchalk. However, since the average grain strength asmeasured by Lee (1992) has now been shown tobe a proper statistical measure of tensile strength,it seems likely that the dilatational component ofthe internal angle of friction Äö should be a func-tion of the mean effective stress normalised by theaverage grain tensile strength ó0 for soils of equalrelative density.

Figure 5 shows results of triaxial tests performedby Lee (1992) on samples of Leighton Buzzardsand and oolitic limestone at the same initialrelative density. The results include two differentparticle size gradings for each soil: samples con-tained approximately uniformly sized particles, butthe size of the grains, varied by a factor of about10 between the two gradings. The ®gure shows thedilatational component of angle of friction Äöplotted against p9=ó0 on a logarithmic scale. It isclear that Äö reduces linearly with log( p9=ó0),thus con®rming the hypothesis that the tensile

strength of grains should govern the dilatant beha-viour of crushable soils. More micromechanicalinsight can therefore be achieved if equation (12)is rewritten as

IR � ID lnBó0

p9

� �ÿ 1 (13)

where B is a scalar multiplier. As Bolton (1986)remarked, this expression is applicable for IR > 0,and should not be taken to apply as p9 approachesBó0. The onset of contraction is the subject of thenext section.

ONE-DIMENSIONAL COMPRESSION

There now follows a study of the micromecha-nics of one-dimensional compression. In particular,the aim is to determine the role of grain fracture inthe uniaxial compression of crushable soils. Fig. 6(Golightly, 1990) shows typical plots of voids ratioagainst the logarithm of vertical effective stress forsamples of carbonate and silica sands which havebeen compressed in an oedometer. Plots for looseand dense silica sand are shown for comparison.Consider the compression of dense silica sand. Ageotechnical engineer might describe the form ofthe curve by the relationship

e � f (ó 9v) (14)

which is dimensionally inconsistent: the verticaleffective stress ought to be normalized by somematerial parameter with the dimensions of stress.Bolton & McDowell (1996) suggested that for thesmall deformations in region 1, the normalizing

0.001 0.01 0.1 0.5

20

15

10

5

0

p′/σ0

∆φ

5 φ

′ max

2 φ

′ crit

7/14LBS

52/100 LBS

52/100 OLS

7/14OLS

σ0 5 26 MPa 54 MPa 3.5 MPa 7 MPa

Fig. 5. Dilatational component of angle of internal friction plotted againstmean effective stress normalized by grain tensile strength (Lee, 1992). LBS,Leighton Buzzard sand; OLS, oolitic limestone


parameter should be the elastic modulus of aparticle, so that one might write

e � f (ó 9v=Gp, ö) (15)

where ö is the angle of internal friction. Even witha dense sand, small irrecoverable deformations mayoccur by particle rearrangement, since grains not intheir closest possible packing may still rearrange.This explains why the relative density of a vibro-compacted sand increases with increasing timespent on vibro-compaction.

It is evident in Fig. 6 that a well-de®ned yield-ing region 2 exists. This cannot be due to particlerearrangement alone: for this dense sand, whichhas exhausted all possible particle rearrangement atregion 2, particle breakage is a prerequisite forfurther compaction. For an array of approximatelyuniform particles under compression, the averagecharacteristic tensile stress ó induced in a grainmust be proportional to the applied macroscopicstress ó (where the overbar will now be used toindicate macroscopic, compressive). For particlesloaded in this manner, there will exist a value ofcharacteristic induced tensile stress ó0 such that37% of particles survive. As previously noted, ó0

also represents (approximately) the characteristictensile stress at which the rate of particle fracturewith increasing stress dPf=dó is a maximum,which we may consider to correspond to the yield-ing region 2. The yield stress, then, must be pro-portional to the average tensile strength of grains,

as measured by crushing between ¯at platens, sothat for irrecoverable strains in region 2 we maywrite

e � f (ó=Gp, ö, ó=ó0) (16)

Bolton & McDowell (1996) called the behaviour inregion 2 `clastic' yielding, whereby major irrecov-erable deformations are permitted by the onset ofparticle fracture. A clastic yield stress ó0 may bede®ned as the value of macroscopic stress whichcauses the maximum rate of grain fracture underincreasing stress, for particles loaded in this con-®guration. For an aggregate of brittle particles ofvery high Weibull modulus (i.e. a unique particletensile strength), it is evident that catastrophiccompression of the aggregate would occur whenthe applied macroscopic stress attained a valueequal to the clastic yield stress. McDowell (1997)showed, using a simple numerical model, that thisis indeed the case.

The compression plot in Fig. 6 for loose silicasand shows that at higher voids ratios more particlerearrangement can occur before the onset of crush-ing, in agreement with data published by Hagertyet al. (1993). Furthermore, the clastic yield stressis lower, since at higher voids ratios the averagecoordination number (number of contacts per parti-cle) in the aggregate reduces (Oda, 1977). Jaeger(1967) showed that for circular particles undercombinations of surface forces, the maximum ten-sile stress in a particle reduces as the coordinationnumber increases.

The compression plot for the dense carbonatesand also shows a yielding region, although it isless pronounced than that for the dense silica sand.This should be expected from a material with alower Weibull modulus, which is what one mightanticipate for a calcareous sand which is composedof very porous grains.

`NORMAL' COMPRESSION

Figure 6 shows that for a crushable aggregatewhich has been one-dimensionally compressed,clastic yielding is followed by a region of plastichardening, in which the macroscopic stress mustbe increased in order to produce any further com-paction. Geotechnical engineers call this `normal'compression. Furthermore, there is a linear rela-tionship between voids ratio and the logarithm ofeffective stress. From early publications in soilmechanics, the one-dimensional plastic compres-sion of soils has been shown to satisfy this rela-tionship. The resulting plots are known as `normalcompression lines', and the slope of such a plot,the compressibility index, is taken to be an empiri-cal soil constant, given the symbol ë when usingnatural logarithms, in Cam Clay (Scho®eld &Wroth, 1968), and subsequent models based on

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

00.1 1.0 10 100

σ ′ν: MPa

e

Dense carbonate sand

Loose silica sand

Dense silica sand

12

Fig. 6. One-dimensional compression plots for carbo-nate and silica sands (Golightly, 1990)


plasticity theory (Roscoe & Burland, 1968; Miuraet al., 1984):

e � e0 ÿ ë ln ó (17)

Terzaghi (1948) used a similar equation with com-mon logarithms, and the slope was given thesymbol Cc. Equation (17), like equation (14), isdimensionally inconsistent and physically incom-pleteÐthe macroscopic stress should be normalizedby a material parameter with dimensions of stress.The popular use of atmospheric pressure, or astandard stress (1 kPa, 1 kgf=cm2, etc.) is, ofcourse, no answer to this dilemma. The principlesof physics demand that behaviour be described interms which are dimensionless with respect toparameters involved in the physical process.Furthermore, the physical origins of ë have re-mained a mystery to geotechnical engineers: it hasseemed impossible to formulate a non-dimensionalgroup for ë in terms of particle parameters. Thelinear±log relationship described by equation (17)applies to a wide range of geotechnical materials(Novello & Johnston, 1989), and the compressibil-ity index ë is remarkably in the range 0´1±0´4 fora wide range of granular materials.

It is well known that the normal compression ofcrushable soils leads to particle size disparity(Fukumoto, 1992; Hagerty et al., 1993). The aimof this paper is now to relate the evolution of adistribution of particle sizes to the existence of thenormal compression lines evident in Fig. 6.

THE EXISTENCE OF FRACTALS

It has long been recognized that a wide varietyof scale-invariant processes (magni®cation does notalter the picture) occur in nature. The concept offractals provides a means of quantifying theseprocesses (Mandelbrot, 1982). Turcotte (1986) ex-amined the fragment sizes for a wide range ofcrushed materials and found the size distributionsto have a fractal character. A fractal de®nes asimple power law relation between the number ofparticles and their size, so that the number offragments which have a diameter (or other charac-teristic linear dimension) size L greater than size dis given by

N (L . d) � AdÿD (18)

where A is a constant of proportionality and D isthe fractal dimension. The fractal dimension D isoften about 2´5 for materials subjected to purecrushingÐfor example Turcotte (1986) gives bro-ken coal a value of D � 2:50; granite fragmentsfrom an underground nuclear explosion have D �2:50; and basalt fragments from projectile impacthave D � 2:56. The materials which exhibit fractaldimensions signi®cantly different from 2´5 areusually not the result of pure crushing, and have

often been subject to sortingÐfor example, glacialtill has an observed fractal dimension of 2´88(Turcotte, 1986). It seems likely that there is amechanical explanation for the existence of afractal dimension of about 2´5ÐPalmer & Sander-son (1991) produced a simple fractal crushingmodel for ice, and showed that if the crushingforce on a block of ice is determined equally bythe fracture of fragments of all sizes (i.e. allfragment sizes make an equal contribution to thecrushing force), then the fractal dimension must be2´5, in agreement with observed values of D.Steacy & Sammis (1991) used computer simula-tions to examine the evolution of a fractal geome-try under different conditions, in order to explainthe observed fractal dimensions in fault gouge of2:6� 0:1 (Sammis et al., 1987). Their model con-sists of a cube of material which splits into self-similar cubes, such that no neighbours are of thesame size at any scale, since this would maximizethe stress concentrations on a particle surface.Steacy & Sammis showed that if neighbours arede®ned as blocks sharing faces or edges, then it ispossible to generate arrays of particles with afractal dimension of about 2´5.

It will therefore be assumed here for simplicitythat normal compression gives rise to a fractaldistribution of particles with a value of D � 2:5.McDowell et al. (1996) have given a more generalanalysis of aggregate compaction, in which thefractal dimension need not be 2´5; this will bediscussed brie¯y later. It is essential at the outsetto recognize that although a strictly fractal distribu-tion of particle sizes obeying equation (18) wouldextend over an in®nite range of scales, the fractaldistribution must, in reality, be limited (Fig. 7).

1025 1024 1023 1022 1021 1

102

1

104

106

108

1010

N(L

>d

)

d /d0

D

d0

ds

Fig. 7. A limited fractal of dimension D. The smallestparticle size reduces under increasing stress


There will be a `largest' particle of size d0 corre-sponding to the original population, and a smallestparticle of size ds which will generally reduce asstress increases. This is of uttermost importance inunderstanding the micromechanics of normal com-pression. If the size distribution in Fig. 7 werestrictly a fractal without limit, the aggregate wouldhave a voids ratio uniquely de®ned by the fractaldimension D: scale invariance would demand itand make the process of normal compression im-possible. The model for `plastic' or `clastic' irre-coverable compression which will now bepresented is wholly dependent on the reduction ofthe smallest particle size ds by successive splitting,so that voids ratio reduces as the broken fragments®ll pre-existing voids.

FRACTAL CRUSHING

The probability of fracture of a particle isdetermined (McDowell et al., 1996) by the appliedmacroscopic stress, the size of the particle and thecoordination number (number of contacts withneighbouring particles). The fracture probabilityPf (d) must increase with any increase in macro-scopic stress ó , but reduce with a decrease inparticle size, or an increase in the coordinationnumber (since an increase in the number of con-tacts must reduce the induced tensile stresses(Jaeger, 1967)). For an aggregate with particle sizedisparity, the largest grains will have very high-coordination numbers, since they will be sur-rounded by many smaller particles. The smallestparticles in the aggregate must have the minimumcoordination.

It might be anticipated, at ®rst sight, that be-cause small particles are stronger than large parti-

cles the largest particles are always the most likelyto fracture. This would lead to the evolution of auniform matrix of ®ne particles from the compres-sion of an aggregate of coarse grains, behaviourwhich is not evident in the geotechnical literature.However, although the smallest particles are thestrongest, they also have the fewest contacts. Thereare two opposing effects on particle survival, sizeand coordination number. If coordination numberdominates over particle size in the evolution of theaggregate, then the smallest particles always havethe highest probability of fracture. In this case, thecompression of an aggregate of uniform grainswould lead to a disparity in particle sizes, in whicha proportion of the original grains is retained underthe protection of a uniform compressive boundarystress created by its many neighbours. This type ofbehaviour is evident in Fig. 8, which shows theparticle size distribution curve which evolves forOttawa sand. McDowell et al. (1996) have shown,using a simple numerical model, that when coordi-nation number dominates over particle size indetermining the fracture probability of a particle, afractal distribution of particle sizes evolves.

Suppose that it is the smallest particles whichcontinue to fracture under increasing macroscopicstress, and that the probability of fracture of thesmallest particles is a constant p, and that thenumber of fragments produced when a particlesplits is a constant n. The scale-invariant para-meters p and n will be shown to de®ne a fragmentsize distribution which is a fractal. Consider ahierarchy of splitting grains such that the largestparticles are size d0, and that subsequent `orders'of particle size are d1, d2, . . ., ds, where ds is thesmallest particle size which decreases under in-creasing macroscopic stress. If there were origin-

100

80

60

40

20

00.01 0.1 1

Grain size: mm

Per

cen

t fin

er b

y w

eigh

t 100 MPa

55 MPa

35 MPa

7 MPa

Fig. 8. Evolving particle size distribution curves for one-dimensionally com-pressed Ottawa sand (Fukumoto, 1992)


ally ù particles of order 0, then the number ofparticles of order i must be ù(n p)i(1ÿ p). Thenumber of particles of size di or greater is givenby

N(L . di) � ù(n p)i(1ÿ p)[1� 1=n p

� 1=(n p)2 � . . .] (19)

which reduces to

N (L . di) � ù(n p)i(1ÿ p)[1ÿ 1=n p]ÿ1 (20)

for n p . 1. The number of fragments of size di�1

or greater is given by replacing i by i� 1 inequation (20), so that the ratio of the number offragments of order i or greater, to the number offragments of order i� 1 or greater is easily calcu-lated as

N(L . di)

N (L . di�1)� 1

n p(21)

This de®nes a fractal distribution. Equation (18)gives

N(L . di)

N (L . di�1)� di

di�1

� �ÿD

� (n1=3)ÿD (22)

so that by comparing equations (21) and (22), thefractal dimension can be related to p and n:

D � 3 1� ln p

ln n

� �(23)

The idea of having a probability of fracture whichis constant for the smallest particles is consistentwith the idea that the smallest particles remain ingeometrically similar con®gurations, with a mini-mum coordination. In this case, it is also evidentthat the smallest grains will, on average, have acharacteristic induced tensile stress which is pro-portional to the current macroscopic stress. Micro-scopically, strong columns of force may form atintervals of a few particle diameters to transmit themajor principal stress (Cundall & Strack, 1979).However, the organization of particles must changewhen any particle fractures, so that the snapshot oforder at any instant should be irrelevant. Thetensile stress induced in the smallest particles canbe considered to take some characteristic valuewhen integrated over many fractures.

We now proceed to relate the smallest particlesize ds to the current macroscopic stress ó . Con-sider a material containing Grif®th ¯aws (Grif®th,1920), and for which the maximum ¯aw size in aparticle is proportional to the size of the particle(i.e. there is statistical self-similarity in the ¯awsize distribution), which corresponds to a disor-dered granular material. Following Carpinteri(1994), such a material would exhibit Grif®th's lawof fracture, with a size index in equation (3) of

b � ÿ0:5. Since the smallest particles of size ds

may be considered to have a characteristic tensilestress which is proportional to the macroscopicstress ó , and if the splitting of grains is governedby linear elastic fracture mechanics so that the onlymaterial parameter involved is the fracture tough-ness KIc, then the only dimensionless group thatcan be formed from these parameters isó��ds

p=KIcÐwhich must remain constant for geo-

metrically similar situations, that is,

ó��ds

p/ KIc (24)

Alternatively, we may write

ó��ds

p/ ó0

��d0

p(25)

where ó0 is the clastic yield stress for the aggre-gate, and the constant of proportionality in equa-tion (25) may be derived by equating the fractalprobability of fracture p to the Weibull fractureprobability Pf (McDowell, 1997), and may betaken to be about 1. Equation (25) relates thecurrent smallest particle size to the applied uniax-ial compressive stress for a material which obeysGrif®th's laws of linear elastic fracture mechanics,and provides a mechanism for plastic hardening ofthe aggregate. Bolton & McDowell (1996) havetermed this behaviour `clastic hardening'. It shouldbe noted that granular materials having ¯aw sizedistributions other than that which is describedabove have been treated by McDowell et al. (1996)using Weibull statistics, and have been omittedhere for clarity.

CLASTIC HARDENING

It is now proposed to relate the evolution of afractal geometry to the evolution of a `normalcompression line' which is linear in eÿlog ó space.For this purpose, a new work equation will beintroduced. It is ®rst of all useful, however, toconsider the original Cam Clay work equation(Roscoe et al., 1963; Scho®eld & Wroth, 1968),which has been used very widely to model thedeformation of a porous aggregate:

qäåpq � p9äåp

v � M p9äåpq (26)

The left-hand side represents the plastic work doneper unit volume by deviatoric stress q and meaneffective stress p9 (with corresponding irrecover-able strain increments äåp

q and äåpv). The right-hand

side was identi®ed as internal frictional dissipation.If elastic strains are ignored, then the equationreduces to the Granta Gravel work equation(Scho®eld & Wroth, 1968), which will be usedhere for simplicity: elastic strains may easily besuperimposed (McDowell et al., 1996). We nowadd a term on the right-hand side of equation (26)for the energy dissipated in the successive fracture


of brittle particles and get (McDowell et al.,1996):

qäåq � p9äåv � M p9äåq � Ã dS

Vs(1� e)(27)

where dS is the increase in surface area of avolume Vs of solids distributed in a gross volumeVs(1� e), and Ã is the `surface energy', related tothe critical strain energy release rate Gc byÃ � Gc=2 (not to be confused with the gammafunction in equation (6)). For the special case ofone-dimensional normal compression with effectiveaxial stress ó and uniaxial strain å, we obtain

ó då � 29M(1� 2K0)ó då� Ã dS

Vs(1� e)(28)

where K0 is the lateral/axial effective stress ratio,and may readily be estimated using JaÃky's formula(JaÃky, 1944) K0 � 1ÿ sinö. Substituting

då � ÿ de

(1� e)(29)

into equation (28) relates the reduction in voidsratio to the increase in surface area in the soilsample:

de � ÿ Ã dS

(1ÿ ì)óVs

(30)

where ì is a weak function solely of the angle ofinternal friction (McDowell, 1997), varying from0´4 for ö � 208 to 0´6 for ö � 408, and so mayconveniently be assumed to take a value of 0´5. Itis worth noting that equation (27) removes alimitation of the original Cam Clay work equation(Scho®eld & Wroth, 1968) by permitting plasticvolume changes for simple isotropic loading. Thisnew work equation predicts that under isotropicloading (i.e. q � 0, dåq � 0), an increase in thesurface area of particles produces a ®nite volu-metric strain, so that dåv=dåq is in®nite. In thiscase, if the normality rule were to be applied inorder to derive a yield surface, the new workequation (27) would tend to round off the cornerof the Cam Clay yield locus.

The total surface area of particles in the samplescan be found by consideration of equation (18). Bydifferentiating equation (18), it is evident that thenumber of particles with a size in the range of dto d � äd is

äN � ADdÿDÿ1 äd (31)

The surface area of a particle of size d may begiven by

S(d) � âsd2 (32)

where âs is the surface shape factor, so that thetotal surface area of particles of size in the ranged to d � äd is

äS � âs ADd1ÿD äd (33)

The total surface area of particles in the sampleS(L . ds) is given by integrating equation (33):

S(L . ds) � 5âs Adÿ1=2s (34)

using D � 2:5. Incidentally, the mass distributionmay be calculated in the same way, so that thevolume of particles less than or equal to size d isgiven by

M(L , d) � 5âv Ad1=2 (35)

where âv is the volume shape factor for thematerial. This is the ®nal mass distribution whichevolves during the successive fracture of the smal-lest grains. The total volume of solids in thesample Vs is given by equation (35) with d � d0.Combining equations (34) and (25) relates the totalsurface area in the sample to the current appliedmacroscopic stress:

S � 5âs Aó

ó0

� �1��d0

p (36)

Combining equation (36) with equation (30) pro-vides a clastic hardening law:

de � ÿ 1

1ÿ ì

âs

âv

Ã

ó0d0

� �dó

ó(37)

which is a normal compression line, of slope

ë � 1

1ÿ ì

âs

âv

Ã

ó0d0

(38)

It is evident, then that the evolution of a fractalgeometry provides a micromechanical commentaryfor the evolution of normal compression lines. It isinteresting to substitute some typical values for theparameters in equation (38). For a quartz sand,with ö � 308, ì � 0:5, Harr (1977) quotes valuesof âs=âv for crushed quartz in the range 14±18(where the shape factors are de®ned using theprojected diameter of a particle), so we mightassume a value 16; Ashby & Jones (1986) givevalues of surface energy Ã for rocks of 25 J=m2;and the clastic yield stress for a silica sand com-prising 1 mm diameter particles might be 10 MPa.These values give a compressibility index ë � 0:1in equation (38), which is of the correct order ofmagnitude, comparing with available data for gran-ular materials (Novello & Johnston, 1989).

DISCUSSION

The value of compressibility index ë � 0:1 gi-ven by equation (38) must be partly fortunate: theleast certain parameter is the surface energy Ã,which is dif®cult to estimate. Furthermore, thedependence of ë on d0 is surprising, although thereis a lack of experimental data to suggest that the


compressibility index should be independent of theinitial particle size. One possible source of expla-nation is the surface energy Ã. The surface energyhas traditionally been taken to be a material con-stant (Ashby & Jones, 1986), related to the mode Ifracture toughness KIc by KIc �

��2EÃp

(where Eis Young's modulus) when the principles of linearelastic fracture mechanics apply. However, recenttensile testing experiments on concrete (Carpinteri,1994), and tensile splitting experiments on shortrods of sandstone and marble (Scavia, 1996) haveshown that over a range of scales for these regulargeometries,

Ã / d Dsÿ2 (39)

where d is some characteristic length of the speci-men and Ds is the surface fractal dimension, with2 , Ds , 3. In fact, the kinematics of the fractureprocess puts an upper limit on Ds of 2´5, whichcorresponds to an extremely disordered material,with a wide distribution of ¯aws. For a materialcontaining Grif®th ¯aws, with the maximum ¯awsize proportional to the size of the specimen, thenÃ / d1=2. If we now rewrite equation (38) as

ë � 1

1ÿ ì

âs

âv

Ã=��d0

p

ó0

��d0

p (40)

then ó0

��d0

p / KIc, which is a material constant(Xie, 1993), so that in the application of linearelastic fracture mechanics it might seem plausibleto scale the surface energy in proportion to d

1=20 ,

so that ë is seen to be a material constant. Never-theless, what is important is that equation (38)predicts a unique value of ë for a particular aggre-gate, so that if a normally compressed soil isunloaded and reloaded, the normal compressionline will be rejoined at the preconsolidation stress.The current preconsolidation or yield stress isdetermined by the tensile strength of the smallestgrains. If the smallest particle size is known, thenthe current voids ratio may be uniquely de®ned bythe relationship

e � f (ó=G p, ö, ó=ó0s) (41)

where ó0s is the tensile strength of the smallestparticles.

The derivation of the eÿlogó relationship inthis paper has been for disordered materials inwhich the average size of a critical ¯aw is propor-tional to the size of a particle, and which formparticle size distributions with a fractal dimensionof 2´5. A more general analysis has been given byMcDowell (1997), which shows that the voidsratio±effective stress relationship is given by

de � ÿë(ó )(m=3)(Dÿ2)ÿ1 dó

ó(42)

so that for the majority of granular materials withD � 2:5, and 5 , m , 10, the power on stress in

equation (42) is negligible, which explains whymost normal compression curves are approximatelylinear in eÿlog ó space. However, the purpose ofthis paper has been to elucidate the fundamentalmicromechanics of normal compression and pro-vide a mechanism of plastic (clastic) hardening.

There is a lack of experimental data to showthat the existence of linear normal compressionlines for crushable soils is consistent with theevolution of a fractal geometry. However, the theo-ry is supported by the data in Fig. 9, which showsa normal compression curve for petroleum coke,together with the particle size distribution which

Fig. 9. (a) One-dimensional normal compression curvefor petroleum coke. (b) Evolving particle size distribu-tions for one-dimensionally compressed petroleumcoke

10 1 0.1

100

80

60

40

20

0

d: mm

(b)

Initialgrading

% b

y m

ass

finer

than

σ 5 5.0 MPaσ 5 7.5 MPaσ 5 10 MPaσ 5 20 MPaσ 5 40 MPaσ 5 57 MPaσ 5 100 MPa

2.5

2.0

1.5

1.0

0.5

0.01023 1021 10 103

Voi

ds r

atio

, e

σ′ν: MPa

Brittle grains

Petroleum coke

5 MPa 10 MPa 20 MPa 40 MPa 57 MPa100 MPa

d60/d10 5 8

(d60/d10)i 5 1.4

5 12

5 14

5 12

5 10

(a)


evolves (Bard, 1993; Biarez & Hicher, 1994). It isclear that the evolution of the linear portion of thecurve in Fig. 9(a) is consistent with the evolutionof a fractal distribution of particle sizes in Fig.9(b) as the uniformity coef®cient tends to someconstant value. The change in curvature of the plotat low voids ratios must be due to the comminu-tion limit (Kendall, 1978) for petroleum cokeÐFig. 9(b) (Bard, 1993) shows that at high stressesthere is signi®cant breakage of the larger grains,since the smallest particles are so small that theyyield before fracture. However, the data clearlysupport the hypothesis that the micromechanicalorigin of normal compression lies in the evolutionof a fractal geometry. It seems that `normal com-pression' would be better termed `fractal compres-sion'.

CONCLUSIONS

The tensile strengths of soil grains compressedbetween ¯at platens satisfy the Weibull statistics offracture of brittle ceramics, which permits a usefuland consistent de®nition of grain strength. Thisgrain strength governs the strength and dilatancy ofcrushable soils, so that at a given relative densitythe dilatational component of the angle of internalfriction is proportional to the logarithm of meaneffective stress normalized by grain tensilestrength. For an aggregate under one-dimensionalnormal compression, the yield stress is proportionalto the grain tensile strength. The successive frac-ture of the smallest particles under increasingmacroscopic stress to form a limited fractal geome-try provides a micromechanical insight into theexistence of linear `normal compression lines'. Inthis case, the current yield stress of the aggregateis determined by the tensile strength of the smallestparticles.

NOTATIONb Lee index on tensile strengthd particle size

d0 size of largest particle, order 0di size of a particle of order ids size of smallest particle, order sD fractal dimensione voids ratioF diametral force for a particle compressed

between ¯at platensFf crushing force for a particle compressed

between ¯at platensGp shear modulus of particle materialID relative densityIR relative dilatancy index

KIc mode I fracture toughnessK0 lateral/axial effective stress ratio in one-

dimensional compressionm Weibull modulus

M(L , d) volume of particles ®ner than size dn number of fragments produced when a

particle splitsN (L . d) number of particles larger than size d

p fractal probability of fracturep9 mean effective stress

Pf (d) fracture probability for a particle of size dPs(d) survival probability for a particle of size dPs(V ) survival probability for a volume V

q deviatoric stressQ Bolton constantS surface area

S(L . d) total surface area of particles larger thansize d

V volumeV0 reference volumeVs volume of solidsâs surface shape factorâv volume shape factoråq triaxial shear strainåv volumetric strainå uniaxial macroscopic compressive strainö angle of frictionÃ surface energyë compressibility indexì frictional constant for one-dimensional

normal compressionM critical state frictional dissipation constantó characteristic tensile stress induced in a

particleó f Lee tensile strength of a particleó0 Weibull 37% tensile strengthó 9 effective stressó 9v vertical effective stressó uniaxial macroscopic compressive stressó0 clastic yield stress

REFERENCESAshby, M. F. & Jones, D. R. H. (1986). Engineering

materials 2. Oxford: Pergamon Press.Bard, E. (1993). Comportement des materiaux granu-

laires secs et a liant hydrocarbone. TheÁse de doctorat,Ecole Centrale de Paris.

Biarez, J. & Hicher, P. (1994). Elementary mechanics ofsoil behaviour. Rotterdam: Balkema.

Billam, J. (1972). Some aspects of the behaviour ofgranular materials at high pressures. Stress±strainbehaviour of soils. Proceedings of the Roscoe Mem-orial Symposium (ed. R. H. G. Parry), pp. 69±80.

Bolton, M. D. (1986). The strength and dilatancy ofsands. GeÂotechnique 36, No. 1, 65±78.

Bolton, M. D. & McDowell, G. R. (1996). Clastic mech-anics. IUTAM symposium on mechanics of granularand porous materials (ed. N. A. Fleck and A. C. F.Cocks). Dordrecht: Kluwer.

Carpinteri, A. (1994). Scaling laws and renormalisationgroups for strength and toughness of disorderedmaterials. Int. J. Solids Struct. 31, 291±302.

Cundall, P. A. & Strack, O. D. L. (1979). A discretenumerical model for granular assemblies. GeÂotechni-que 29, No. 1, 47±65.

Fukumoto, T. (1992). Particle breakage characteristics ingranular soils. Soils Found. 32, No. 1, 26±40.


Golightly, C. R. (1990). Engineering properties of carbo-nate sands. Ph.D. dissertation, Bradford University.

Grif®th, A. A. (1920). The phenomena of rupture and¯ow in solids. Phil. Trans. R. Soc. Lond. A221, 163±198.

Hagerty, M. M., Hite, D. R., Ullrich, C. R. & Hagerty, D.J. (1993). One-dimensional compression of granularmedia. J. Geotech. Engng, ASCE 119, No. 1, 1±18.

Harr, M. E. (1977). Mechanics of particulate media.New York: McGraw-Hill.

Jaeger, J. C. (1967). Failure of rocks under tensileconditions. Int. J. Rock. Min. Sci. 4, 219±227.

JaÃky, J. (1944). A nyugalmi nyomaÃs teÂnyezoÈje [The co-ef®cient of earth pressure at rest]. Magyar MeÂrnoÈk eÂsEpiteÂsz-Egylet KoÈzloÈnye [J. Union Hungarian EngrsArchitects], 355±358.

Kendall, K. (1978). The impossibility of comminutingsmall particles by compression. Nature 272, 710±711.

Lee, D. M. (1992). The angles of friction of granular®lls. Ph.D. dissertation, University of Cambridge.

McDowell, G. R. (1997). Clastic soil mechanics. Ph.D.dissertation, University of Cambridge.

McDowell, G. R., Bolton, M. D. & Robertson, D. (1996).The fractal crushing of granular materials. J. Mech.Phys. Solids 44, No. 12, 2079±2102.

Mandelbrot, B. B. (1982). The fractal geometry of nature.New York: Freeman.

Miura, N., Murata, H. & Yasufuku, N. (1984). Stress±strain characteristics of sand in a particle crushingregion. Soils Found. 24, No. 1, 77-89.

Novello, E. A. & Johnston, I. W. (1989). Normally con-solidated behaviour of geotechnical materials. Proc.12th Int. Conf. Soil Mech., Rio de Janiero 3, 2095±2100.

Oda, M. (1977). Co-ordination number and its relation to

shear strength of granular material. Soils Found. 17,No. 2, 29±42.

Palmer, A. C. & Sanderson, T. J. O. (1991). Fractalcrushing of ice and brittle solids. Proc. R. Soc. Lond.A 433, 469±477.

Roscoe, K. H. & Burland, J. B. (1968). On the generalisedstress±strain behaviour of `wet' clay. In Engineeringplasticity (ed. J. Heyman and F. A. Leckie), pp. 535±609. Cambridge: Cambridge University Press.

Roscoe, K. H., Scho®eld, A. N. & Thurairajah, A. (1963).Yield of clays in states wetter than critical. GeÂotech-nique 13, 211±240.

Sammis, C., King, G. & Biegel, R. (1987). The kine-matics of gouge deformation. Pure Appl. Geophys.125, No. 5, 777±812.

Scavia, C. (1996). The effect of scale on rock fracturetoughness: a fractal approach. GeÂotechnique 46, No.4, 683±693.

Scho®eld, A. N. & Wroth, C. P. (1968). Critical statesoil mechanics. London: McGraw-Hill.

Shipway, P. H. & Hutchings, I. M. (1993). Fracture ofbrittle spheres under compression and impact loading.I. Elastic stress distributions. Phil. Mag. A 67, No. 6,1389±1404.

Steacy, S. J. & Sammis, C. G. (1991). An automaton forfractal patterns of fragmentation. Nature 353, No.6341, 250±252.

Terzaghi, K. & Peck, R. B. (1948). Soil mechanics inengineering practice. Chichester: Wiley.

Turcotte, D. L. (1986). Fractals and fragmentation. J.Geophys. Res. 91, No. B2, 1921±1926.

Weibull, W. (1951). A statistical distribution function ofwide applicability. J. Appl. Mech. 18, 293±297.

Xie, H. (1993). Fractals in rock mechanics. Rotterdam:Balkema.


On the micromechanics of crushable aggregates · tive stress (Bolton, 1986), since at high...

Documents

Transcript of On the micromechanics of crushable aggregates · tive stress (Bolton, 1986), since at high...