Russell, A. R. (2011) Géotechnique Letters 1, 5 9, htt://dx.doi.org/10.1680/geolett.10.00003 A comression line for soils with evolving article and ore size distributions due to article crushing A. R. RUSSELL* In this study it is suosed that energy is dissiated in two ways when article crushing occurs. The first is the release of strain energy stored in the crushed article and is analogous to the creation of surface. The second is due to the load redistribution and change of stored strain energy of the surrounding soil. Terms for these two mechanisms are included in a Cam-clay tye energy equation. By defining article and ore size distributions during the crushing rocess using fractals, and by equating article and ore surface areas, a closed-form exression is obtained for a limiting comression line in the double logarithmic voids ratio stress lane. The limiting comression line and the evolving article size distribution are matched well by the theory for two silica sands loaded in oedometric comression to high stresses. Evidently, energy dissiation due to load redistribution is significantly larger than energy dissiation due to the creation of surface. KEYWORDS: comressibility; fractals; article crushing ICE Publishing: all rights reserved INTRODUCTION Many exressions have been roosed for comression lines for granular soils exhibiting article crushing. Most are henomenological in origin and rely on emirical fitting arameters. However, a few theoretical studies have derived closed-form exressions for comression lines. McDowell et al. (1996) and McDowell & Bolton (1998) used a fractal descrition for the article size distribution and sizedeendent article strength to define a linear comression line in the e{lns lane, where e reresents the voids ratio and s the alied stress. No reference was made to how ore sizes evolve; rather, volumetric deformation was indirectly included through an assumed energy balance. McDowell (2005) later roosed that e is roortional to the volume of the smallest articles and suggested the comression line may be more aroriately defined in the ln e{ln s lane. Definition in this lane seems sensible as a straight line may continue indefinitely with increasing s and reducing e while maintaining a ositive value. In this study, a new aroach is resented to derive a theoretical exression for a limiting comression line. For the first time, fractal characteristics of the ore sace are considered along with those of the articles. A more comlete energy balance equation is also used, which accounts for energy dissiation through the creation of surface as well as load redistribution in the surrounding soil. The derivation is made ossible by assuming many material arameters relevant to comression behaviour are constant (e.g. surface energy, fractal dimensions and shae factors for articles and ores) and is reasonable once a significant amount of crushing has occurred. Manuscrit received 17 December 2010; first decision 11 January 2011; acceted 19 January 2011. Published online 8 February 2011. * Centre for Infrastructure Engineering and Safety, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, Australia ENERGY CONSIDERATIONS When a article in a granular soil is crushed, the energy that is dissiated may be divided into two arts (Nguyen & Einav, 2009). The first is due to release of strain energy stored in the article and is analogous to energy associated with the creation of surface area of the fragments, denoted DW surface. The second is due to load redistribution (and associated dislacement) of the surrounding soil. This energy dissiation mechanism, denoted DW redist, is rarely considered, although Nguyen & Einav (2009) showed it to be significant. Assuming that only lastic work inut is dissiated, a simle link between lastic work inut er unit volume and the main dissiation rocesses is given by s ij de ij ~DW frictionzdw surface zdw redist (1) This can be written in a form that is an extension of the Cam-clay equation de zqde q ~Mde q z CdS s V s ð1zeþ ð1zrþ (2) where DW friction ~Mde q, DW surface~cds s =V s ð1zeþ and R~DW redist =DW surface. The mean stress and deviatoric stress q cause lastic increments of volumetric strain e and deviatoric strain e q. DW friction reresents dissiation through friction (rimarily due to sliding between articles other than those that crush) and M is a function of the internal frictional strength of the soil. DW surface zdw redist accounts for dissiation due to article crushing. DS s is the increase in surface area of articles within a soil having volume of solids V s and C reresents the surface energy (with units of N/m). Cð1zRÞ is an assumed material constant for a given load ath and boundary condition (Nguyen & Einav, 2009) as discussed later. An additional frictional dissiation belonging to newly formed fragments of a crushed article may also exist, even for isotroic loading. Rather than using a searate dissiation term for this, it is assumed to be contained in DW surface. When R 5 0, DW redist ~0 and the work equation reduces to that given by McDowell & Bolton (1998). 5 Downloaded by [] on [24/01/18]. Coyright ICE Publishing, all rights reserved.
6 Russell For oedometric comression, and suosing elastic strains are very small comared with lastic strains so can be neglected for simlicity (elastic comression of the solid articles and its influence on e is not considered), equation (2) becomes S st ds 2{Ds max {d2{ds d 3{D s s max {d 3{D s ~ 3{D s b ss V st 2{D s b sv 0 1 S T ~ 3{D 2{D b S d max {d 2{D @ min A V T 2{D b V d 3{D max {d 3{D min (11) s v de ~ 2 9 M ð 1z2K 0Þs v de z CdS s V s ð1zeþ ð1zrþ (3) where s v is the vertical stress and K 0 is the ratio between horizontal and vertical stresses. FRACTAL DEFINITIONS Fractal roerties of a granular material undergoing article crushing may now be combined with equation (3) to derive a comression line linking e to s v. It is suosed that the number of ores being of any size L larger than d and the number of articles being of any size L larger than d s obey {D N Lwd d and N s ðlwd s Þd s {Ds (4) where D and D s are the fractal dimensions characterising the distributions of ore and article sizes, resectively. The combined surface areas of ores S and articles S s larger than size d and d s, resectively, then obey 2{D S Lwd d and S s ðlwd s Þds 2{Ds (5) The total surface areas of ores and articles obey: S T d 2{D min and S std 2{Ds (6) where subscrit min indicateimum size and T indicates total. Russell (2010) assumed the total article and ore surface areas are equal (suosing the combined contact area of touching articles is very small comared with the total surface area) to derive d 2{D min ~ d 2{Ds (7) d max d s max where subscrit max indicates maximum size. Equation (7) imlies that a geometrical relationshi exists between cumulative surface areas of ores and articles that scales roortionally to their size. The volumes of ores V and articles V s larger than d and d s, resectively, obey 3{D V Lwd d and V s ðlwd s Þds 3{Ds (8) For ores, this can be written more comletely as 0 1 d 3{D max {d 3{D V Lwd ~VT @ A (9) d 3{D max {d 3{D min Furthermore, the article size distribution curve, which exresses the ercentage by mass of articles of size L smaller than d s as a function of d s, becomes %M s ðlvd s Þ~100 d3{d s s {d 3{Ds d 3{D s s max {d 3{D s (10) The total surface areas of articles or ores er total volumes of articles or ores are then given by in which b ss and b S are surface area shae factors for articles or ores and b sv and b V are volumetric shae factors for articles or ores. Harr (1977) reorted that the ratio b ss =b sv lies between 14 and 18 for articles belonging to real granular soils. b ss =b sv and b S bv are assumed constant. Modelling article and ore size distributions using fractals rovides a good fit to exerimental data (e.g. McDowell et al., 1996; Coo et al., 2004; Einav, 2007a, 2007b; Yu et al. 2009; Russell, 2010). For article size distributions, however, a constant D s is reasonable only if a significant amount of article crushing has occurred. Using equation (10) with a constant D s to define the article size distribution for a soil before loading (uncrushed) and its evolution to when a significant amount of crushing has occurred may not be ossible. A more elaborate model that can handle this may introduce into the article size distribution a state variable in addition to d, as in Einav (2007a, 2007b). The alicability of the constant D s assumtion, and what roves to be a significant amount of crushing, is demonstrated later. For most soils formed by fragmentation or crushing rocesses, D s takes on a value between 2?2 and 2?8 (e.g. Perfect, 1997; Coo et al., 2004). DERIVATION OF A COMPRESSION LINE The following derivations aly to when D s and D may be taken as constants (within the range 2 3) once a significant amount of crushing has occurred. It is assumed that, as soil articles crush, the maximum article size (d s max ) remains unaltered as the minimum article size (d ) becomes smaller; this is suorted by exerimental data (e.g. Lade et al., 1996; Nakata et al., 2001; Luzzani & Coo, 2002). The first art of equation (11) can then be written in rate form ds st ~ d ð S st=v st Þ dd V st dd (12) Recalling that the surface areas of ores and articles are assumed equal, e may be written as e~ V T ~ S st=v st (13) V st S T VT This will be a function of variables d max, d min and d. d min can be removed using equation (7). d max can be removed using the finding of Dodds & Weitz (2002) (further verified by Delaney et al. (2008)) that it is roortional to the total number of articles raised to the ower { ð3{d s Þ and therefore, from equation (4), must be roortional to d raised to the ower ð3{d s ÞD s. This can be exressed mathematically as ð d 3{Ds ÞD s d max ~C 1 d s max (14) d s max in which C 1 is a dimensionless material constant. e then becomes a function of d only, and the rate form is de~ d ð S st=v st Þ S T VT dd (15) dd Downloaded by [] on [24/01/18]. Coyright ICE Publishing, all rights reserved.
Using the volumetric strain definition it can be shown that de ~{ de 1ze ~{ 1 1ze d ð S st=v st Þ S T VT dd (16) dd An exression for s v then becomes 9Cð1zRÞ dsst 1 s v ~{ 9{2Mð1z2K 0 Þ V st de ~ 9Cð1zRÞ d ð SsT =V st Þ { 9{2Mð1z2K 0 Þ dd A comression line for soils with evolving article and ore size distributions due to article crushing 7 " # 1{l ð{3zd s Þ {2zD bv N~ C 1 {3zD ð {2zDs " Þ b S b # ss 1 91zR ð Þ l (22) b sv D s 9{2Mð1z2K 0 Þ dd d ðs st =V st Þ S T VT which is also a function of d, so has the rate form: (17) ds v ~ ds v dd (18) dd The comressive resonse is then defined by de~ d(s st=v st ) S T VT dd ds v (19) dd ds v It so haens that when d max &d min and d s max &d (which is the case once a significant amount of article crushing has occurred), the comressive resonse aroaches, as an asymtote, a straight line in the ln e{ln s v lane. This asymtote, referred to as a limiting comression line, is defined in non-dimensional form as s v ln e~ln N{l ln (20) C=d s max where l and N are dimensionless material constants. C and d s max are also material constants during the crushing rocess. After mathematical maniulations, the comlete exressions for l and N are ð l~{ 3{D sþd s 2{4D s zd 2 (21) s and Significant simlifications would aear in the definition for N when l 5 1, and this would be the case when D s 5 2. N would then deend on only b ss =b sv, R, M and K 0,which would be more or less constant for all granular soil tyes in oedometric comression and indeendent of ore roerties. It is noted that C and R may actually deend on load ath and stress state as well as article size and article contact geometry, although there have been very few studies in this area. One examle (Nguyen & Einav, 2009) used a 1D model of linear srings and breakable connectors to motivate an R value deending on the average number of articles along a single force chain. The length of the chain is deendent on stress and article size. In any case, it is worth attemting to validate eqs (20), (21) and (22) for constant C and R and oedometric comression. VALIDATION Oedometric comression data and evolving article size distributions for two sands comrising different sizes of silica sand articles were considered (Nakata et al., 2001). Before crushing, sand 1 contained article sizes that ranged between 0?25 and 2 mm; sand 2 articles were in the range 1?4 1?7 mm. Fitting only comression data is unsatisfactory as there would be no check on the evolution of d. Ideally, the evolving ore size distributions would also feature in the validations, but data with which to do this are not available. The comression data and theoretical limiting comression lines are resented in the ln e{ln ½s v = ðc=d s max ÞŠ lane in Figs 1(a) and 2(a). Measured article size distribution curves for the later stages of loading along with those redicted by equation (10) are shown in Figs 1(b) and 2(b). Table 1 summarises the material arameters used. The same values of D s, D, b ss =b sv, b S bv, M and K 0 were assumed to aly to both sands. These were selected based on tyical values, excet for D s which was found by fitting equation (10) to the article size distribution curves and equation (21) to the asymtotic sloes of the comression Fig. 1. Sand 1: (a) oedometric comression data; (b) article size distribution. The symbols reresent the exerimentally measured data and the curves reresent theoretical simulations. In (b), the diamond and square symbols indicate loading to s v 5 62 and 92 MPa, resectively Downloaded by [] on [24/01/18]. Coyright ICE Publishing, all rights reserved.
8 Russell Fig. 2. Sand 2: (a) oedometric comression data; (b) article size distribution. The symbols reresent the exerimentally measured data and the continuous lines reresent theoretical simulations. In (b), the diamond and square symbols indicate loading to s v 5 46 and 92 MPa, resectively lines. The different initial article size distributions for the two sands meant that different values of d s max were used. A tyical value of C 5 25 N/m was assumed to aly to both sands. Finally, arameters R and C 1 were found through their influence on N and by fitting equation (20) to the asymtotic comression lines. It was necessary to adot different R and C 1 values for each sand to ensure a close match between theoretical and exerimental s v and e values. It was also necessary to select aroriate values of d that rovide agreement between theory and exeriment. The adoted values and the corresonding values of s v and e, both exerimental and theoretical, are summarised in Table 2. As l was close to unity, C 1, D and b S bv have only a minor influence on N; b ss =b sv, R, M and K 0 have a major influence. Of these arameters, R cannot be measured directly in indeendent exeriments. However, this study indicates that for oedometric comression, tyical values of C(1 + R) are around 400 N/m, and that energy dissiation due to load redistribution has a much greater influence than that due to the creation of surface. The N values turned out to be almost identical for the two sands considered, as the differences in C 1 were offset by the differences in R. A unique limiting comression line would then emerge if C 1 and R were included in the nondimensionalising stress term along with C and d s max. Consideration of a wider range of exerimental data is necessary to conclude whether this is coincidental or an inherent feature of comression behaviour. Table 1. Material roerties Sand 1 Sand 2 D s 2?3 2?3 D 2?5 2?5 b ss =b sv 16 16 b S bv 16 16 d s max : m 1?3 6 10 23 1?7 6 10 23 M 1?2 1?2 K 0 0?5 0?5 C: N/m 25 25 R 15?4 13?4 C 1 62 110 l (from equation 21) 0?84 0?84 N (from equation 22) 225 221 A good match between theoretical and exerimental article size distribution curves was only ossible for data oints having e values less than 0?26; this imlies that e 5 0?26 reresents an (aroximate) limit below which the amount of crushing becomes significant enough for the fractal dimensions and other arameters to be held constant. The exerimentally observed soil resonse aroached a state of limiting comression for e, 0?26. Based on the data considered here, attemting to detect a limiting comression line for e. 0?26, as was done by Pestana & Whittle (1995) and McDowell (2005), may lead to smaller (and non-unique) values of l. CONCLUSION Searate energy dissiation mechanisms for the creation of surface and for load redistribution have been included in a Cam-clay tye energy equation. Combining this equation with fractal definitions for evolving article and ore size distributions led to a closed-form exression for a linear limiting comression line in the ln e{ln ½s v = ðc=d smax ÞŠ lane. A oint on the line corresonds to a characteristic smallest article size, thus roviding a direct link to the article size distribution. The sloe of the comression line is a function of the fractal dimension of the article size distribution. The osition of the line is a function of article and ore fractal dimensions, along with the shaes of articles and ores, maximum article size, frictional strength, a surface energy constant, a dimensionless constant linking maximum ore Table 2. Exerimental and theoretical arameters relevant to article size distributions d :mm s v : MPa Exerimental e s v : MPa (equation 17) Theoretical e (equation 20) Sand 1 0?025 62 0?259 61?6 0?250 0?020 92 0?171 94?3 0?174 Sand 2 0?023 46 0?260 45?6 0?252 0?0155 92 0?132 96?9 0?133 Downloaded by [] on [24/01/18]. Coyright ICE Publishing, all rights reserved.
A comression line for soils with evolving article and ore size distributions due to article crushing 9 size to article size and the ratio of load redistribution energy to surface energy dissiation. The comression line was matched well by two silica sands loaded in oedometric comression. The data considered suggest that energy dissiation due to load redistribution is significantly greater than that due to the creation of surface. REFERENCES Coo, M. R., Sorensen, K. K., Bodas Freitas, K. K. & Georgoutsos, G. (2004). Particle breakage during shearing of a carbonate sand. Géotechnique 54, No. 3, 157 163. Delaney, G. W., Hutzler, S. & Aste, T. (2008). Relation between grain shae and fractal roerties in random Aollonian acking with grain rotation. Phys. Rev. Lett. 101, 120602. Dodds, P. S. & Weitz, J. S. (2002). Packing-limited growth. Phys. Rev. E 65, 056108. Einav, I. (2007a). Breakage mechanics art I: Theory. J. Mech. Phys. Solids 55, No. 6, 1274 1297. Einav, I. (2007b). Breakage mechanics art II: Modelling of granular materials. J. Mech. Phys. Solids 55, No. 6, 1298 1320. Harr, M. E. (1977). Mechanics of Particulate Media. New York: McGraw-Hill. Lade, P. V., Yamamuro, J. A. & Bo, P. A. (1996). Significance of article crushing in granular materials. J. Geotech. Engng ASCE 22, No. 4, 309 316. Luzzani, L. & Coo, M. R. (2002). On the relation between article breakage and the critical state of sands. Soils and Found. 42, No. 2, 71 82. McDowell, G. R. (2005). A hysical justification for loge logs based on fractal crushing and article kinematics. Géotechnique 55, No. 9, 697 698. McDowell, G. R. & Bolton, M. D. (1998). On the micro mechanics of crushable aggregates. Géotechnique 48, No. 5, 667 679. McDowell, G. R., Bolton, M. D. & Robertson, D. (1996). The fractal crushing of granular materials. J. Mech. Phys. Solids 44, No. 12, 2079 2102. Nakata, Y., Hyodo, M., Hyde, A. F. L., Kato, Y. & Murata, H. (2001). Microscoic article crushing of sand subjected to high ressure one-dimensional comression. Soils and Found. 41, No. 1, 69 82. Nguyen, G. & Einav, I. (2009). The energetic of cataclasis based on breakage mechanics. Pure Al. Geohy. 166, No. 10 11, 1693 1724. Perfect, E. (1997). Fractal models for the fragmentation of rocks and soils: a review. Engng Geol. 48, No. 3 4, 185 198. Pestana, J. M. & Whittle, A. J. (1995). Comression model for cohesionless soils. Géotechnique 45, No. 4, 611 631. Russell, A. R. (2010). Water retention characteristics of soils with double orosity. Eur. J. Soil Sci. 61, No. 3, 412 424. Yu, B., Cai, J. & Zou, M. (2009). On the hysical roerties of aarent two-hase fractal orous media. Vadose Zone J. 8, No. 1, 177 186. WHAT DO YOU THINK? To discuss this aer, lease email u to 500 words to the editor at journals@ice.org.uk. Your contribution will be forwarded to the author(s) for a rely and, if considered aroriate by the editorial anel, will be ublished as a discussion. Downloaded by [] on [24/01/18]. Coyright ICE Publishing, all rights reserved.