McDowell, G. R. (2005). Geotechnique 55, No. 9, 697–698

697

TECHNICAL NOTE

A physical justification for log e–log � based on fractal crushing andparticle kinematics

G. R. McDOWELL*

KEYWORDS: compressibility; constitutive relations; particlecrushing/crushability; sands

INTRODUCTIONFor granular materials subjected to one-dimensional com-pression, it is usual to plot voids ratio e against thelogarithm of vertical effective stress �. Beyond yield in theregion of maximum curvature, a linear normal compressionline develops over some range of macroscopic stress. Thisis shown for a silica sand in Fig. 1 (McDowell, 2002).McDowell & Bolton (1998) justified the existence of a linearnormal compression line with a theory of fractal crushingand an energy equation: the smallest particles continue tocrush as stress increases, and they protect the large particlesthat remain intact. Approximately one decade of stressbeyond the onset of yield, the curvature changes again atvery low voids ratios. Clearly the linear normal compressionline cannot continue to very high stresses. McDowell &Bolton (1998) proposed that this was due to the comminu-tion limit for particles: that is, the smallest particles becomeso small that they can no longer fracture but yield instead(i.e. plastic flow occurs at particle contacts). However, it hasrecently been found (Lim & McDowell, 2005) that thechange of curvature at low voids ratios also occurs forrailway ballast subjected to oedometric compression: in thiscase the smallest particles are not small enough to be nearthe comminution limit. Butterfield (1979) suggested an alter-native compression law for soils in which the normal com-pression line is linear in log v–log � space (where v isspecific volume). However, this normal compression linecannot remain linear either at high stress levels; in the limitthe specific volume v must tend to unity at high stresses.Pestana & Whittle (1995) suggested a linear limiting com-pression curve in log e–log � space. Typical values for theslope of this curve are around 0.4–0.5 for sands (Pestana &Whittle, 1995). Fig. 2 shows Fig. 1 re-plotted in log e–log�space. The voids ratio at a stress of 30 MPa is 0.4, and at100 MPa is 0.23. This gives a slope for the limiting com-pression curve of log(0.4/0.23)/log(100/30) ¼ 0.46, which iswithin the range given by Pestana & Whittle (1995). To afirst-order approximation, this implies the following relation-ship:

e / � �1=2 (1)

This paper examines whether this compression law can bejustified physically.

FRACTAL CRUSHINGConsider a hierarchical particle-splitting model in which

the smallest particle size is ds and the next larger size (i.e.the next order up) is ds�1. Particles split into self-similarparticles, so the ratio of particle sizes between consecutiveorders is constant. As the smallest particles become smallerand fill the available voids, the void space becomes domi-nated by these smallest particles, and it is proposed that thevoid space should be proportional to the volume of thesmallest grains, so that as the grains become smaller andsmaller, the voids ratio tends to zero. Usually, for granularmaterials subjected to one-dimensional normal compression,the particle size distribution tends to a fractal distributionwith a fractal dimension of 2.5. The number of particlesN(L > ds) greater than or equal to a size ds is given by

N (L > ds) / d�2:5s (2)

The number of particles N(L > ds�1) greater than or equalto a size ds�1 is given by

N (L > ds�1) / d�2:5s�1 (3)

It follows that the number of the smallest particles N(L ¼ds) satisfies

Manuscript received 8 July 2005; revised manuscript accepted 19August 2005.Discussion on this paper closes on 2 May 2006, for further detailssee p. ii.* Nottingham Centre for Geomechanics, School of Civil Engineer-ing, University of Nottingham, UK.

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Fig. 1. Normal compression of silica sand (McDowell, 2002)

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Fig. 2. Fig. 1 re-plotted in log e–log � space

N (L ¼ ds) / d�2:5s (4)

Given that the volume of a smallest particle is proportionalto ds

3, the total volume of all of the smallest particlesV(L ¼ ds) is given by

V (L ¼ ds) ¼ N L ¼ dsð Þ � d3s / d1=2

s (5)

The smallest particle size is a function of stress level: thisprovides the hardening law for the soil. For the sand in Fig.1, McDowell (2002) found the Weibull modulus m to beabout 3, and the size effect on average strength �o to be afunction of size according to the approximate relationship

�o / d�3=m / d�1 (6)

This was shown to be reasonably consistent with the varia-tion of the yield stress in Fig. 1 as a function of the initialparticle size for three uniformly graded silica sands withdifferent initial particle sizes (McDowell, 2002). For thesand in Fig. 1, the initial particle size distribution comprisedapproximately 80% by mass in the 1.18–2 mm size rangeand 20% in the 0.6–1.18 mm size range. Given that thecurrent macroscopic stress is proportional to the averagestrength of the smallest grains (McDowell, 2003), substitut-ing equation (6) into equation (5) implies

V (L ¼ ds) / � �1=2 (7)

If, as is proposed, the void space is proportional to the totalvolume of all the smallest particles once a fractal distribu-tion emerges, then the voids ratio e satisfies the relationship

e / � �1=2 (1)

so that the slope of the limiting compression curve in log e–log� space is 0.5. Clearly, in log e–log � space other slopesof the limiting compression curve are possible; the analysisabove is based simply on typical data for the fractal dimen-sion and size effect on strength. Given that the volume ofthe smallest particles must be related to the particle size viaa power law, and given that the smallest particle size mustbe related to stress level according to the size effect onstrength power law, the void space must be a power functionof stress level, so that the normal compression line is linearin log e–log � space, at least until the comminution limit isreached.

KINEMATICSNow consider the implications for the kinematics when

the smallest particles break. Given that the current voidspace es is given by

es ¼ kd1=2s (8)

where k is a constant, the previous void space es�1 beforethe current smallest particles were formed must be given by

es�1 ¼ kd1=2s�1 (9)

so that the reduction in void space ˜e due to the fracture ofthe particles of size ds�1 must satisfy

˜e / d1=2s�1 (10)

Given that the number of particles of size ds�1 satisfyequation (4), the reduction in void space ˜e (and thereforethe reduction in sample volume) due to the fracture of asingle particle must satisfy

˜e / d1=2s�1

d�2:5s

/ d3s (11)

This means that the reduction in void volume when aparticle breaks is proportional to the volume of that particle.This seems physically reasonable. This holds for any fractaldistribution of particles, independent of fractal dimension.

CONCLUSIONSIt has been shown that a linear limiting compression curve

in log e–log � space is consistent with a theory of fractalcrushing, in which the void space is proportional to the totalvolume of the smallest particles. The theory predicts that thereduction in sample volume when a particle breaks isproportional to the volume of that particle. In addition, thepredicted slope of the limiting compression curve usingmicromechanical parameters has been shown to be consistentwith the value measured for tests on silica sand.

ACKNOWLEDGEMENTSThe author would like to thank Professor Hai-Sui Yu for

his helpful comments on this paper.

REFERENCESButterfield, R. (1979). A natural compression law for soils (an

advance on e–log p9). Geotechnique 26, No. 4, 469–480.Lim, W. L. & McDowell, G. R. (2005). Discrete element modelling

of railway ballast. Granular Matter 7, No. 1, 19–29.McDowell, G. R. (2002) On the yielding and plastic compression of

sand. Soils Found. 42, No. 1, 139–145.McDowell, G. R. (2003) Micro mechanics of creep of granular

materials. Geotechnique 53, No. 10, 915–916.McDowell, G. R. and Bolton, M. D. (1998) On the micro mech-

anics of crushable aggregates. Geotechnique 48, No. 5,667–679.

Pestana, J. M. & Whittle, A. J. (1995) Compression law forcohesionless soils. Geotechnique 45, No. 4, 611–631.

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