9
Scale Effects in Macroscopic Behavior Due to Grain Breakage
This chapter and Chapter 10 deal with the incidence of particle breakage in the macroscopic behavior of granular materials used in civil engineering.
First, an introduction to mineral particle breakage related to fracture mechanics is presented, with its statistical features related to Weibull statistics, concluding on the central trend typical parameters in the statistics of mineral particle failures.
The next section presents well-known characteristics of rockfill shear strength, evidencing scale effects from experimental data. The micromechanical relation of this scale effect in shear strength with particle breakage is then analyzed with both theoretical and experimental proofs. This scale effect approach results in an explicit “scale effect rule” operating on the shear strength envelope.
9.1. Introduction to grain breakage phenomenon: a framework of the analysis
Grain breakage in granular materials while loading was emphasized earlier by studies of grain size distributions before and after testing [MAR 72]. However, the analysis of the influence of this phenomenon on the granular material’s mechanical behavior is more recent, as the results presented by Bolton and McDowell [BOL 98] show the impact on the non-reversible compressibility. The original approach presented here consists in analyzing the effect of local breakage on the maximum strength of the granular assembly and the induced scale effect. The latter can be explicitly expressed by a scale effect relation acting on the expression of the intrinsic failure criterion, or shear strength envelope. This result leads to practical applications for the stability analysis and the design of engineering works [FRO 09], as shown in Chapter 10.
9.1.1. Elementary grain breakage
Figure 9.1. Main features of grain breakage – (a) and (b) basic failure patterns and (c) typical experimental results from [MAR 72]. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip
A simplified description of elementary grain failures when internal forces evolve during motion in granular media may be summarized as follows (Figure 9.1):
- – As contact forces increase, maximum strength is reached in some of the grains that fail (Figure 9.1(a)).
- – These local failures result from the sudden propagation of pre-existing micro-cracks in the grains, controlled by the theoretical laws developed in fracture mechanics.
- – These failures are mainly indirect tensile failures, similar to the ones developing in the “Brazilian test.” The failure mode is mainly fracture mechanics Mode I due to tensile stresses induced in the grains by the loading of the granular assembly.
- – In a grain with cracks of width c, the failure stress in Mode I is given by the following expression:
where KIc is the propagation threshold in Mode I of fracture mechanics.
This basic description of grain breakage raises the following several questions:
- – What is the relationship between micro-crack size and grain size?
- – What is the relationship between local stresses within the grains and the macroscopic stresses applied to the granular assembly?
- – What is the effect of the grain size distribution?
The answers to these questions, often through simplified assumptions, lead to the description of the macroscopic effects of grain breakage.
9.1.2. Statistical representations
The influence of grain sizes was investigated in [MAR 72], the results showing that the average crushing force, Fcr, of gravels or rock fragments is a power function of the average grain diameter (Figure 9.1(c)).
This expression can be connected to Weibull’s theory, which gives the probability of survival within a population of brittle objects subjected to stress conditions near failure
As the volume, V, of a grain is proportional to the cube of its diameter, Weibull’s approach for a given value of the probability of survival Ps also leads to an average crushing force proportional to a power function of the grain diameter:
The average crushing force being proportional to the crushing stress multiplied by the average grain section, the comparison of the two approaches gives a simple mean to fit a Weibull distribution for a given material from a set of crushing tests on grains of different sizes
9.1.3. Central trend in the statistics of mineral particle failures
In the wide range of materials investigated by Marsal et al. [MAR 72] at the University of Mexico (UNAM), the values of the exponent, λ, between 1.2 and 1.8 were found, which correspond to the values of Weibull’s parameter, m, between 4 and 15, with a mean value of λ equal to 1.5 and a corresponding mean value of m equal to 6. The subject of the suitability of Weibull’s theory to natural grains has been recently updated by Lobo-Guerrero and Vallejo [LOB 06].
This central value of λ = 1.5 has a physical meaning. It corresponds to the configuration of a microcrack distribution within grains, which verifies the geometrical similarity with their diameter. For this specific distribution, the size of the significant cracks is statistically proportional to the grain diameter. In this case, the failure stress given by fracture mechanics in the above-mentioned equation [9.1] becomes inversely proportional to the square root of the grain diameter. If we also assume that the significant stress in a grain is proportional to the average macroscopic stress within the granular medium, then the maximum macroscopic stress endurable by the granular medium before significant grain crushing develops is inversely proportional to the square root of a characteristic grain diameter.
These central assumptions are the core of the “clastic process” theory developed by Bolton and McDowell [BOL 98], which provides a clear relationship between the compressibility of granular materials and grain breakage, depending on the material parameters.
9.2. Scale effects in shear strength
9.2.1. Shear strength of rockfill
Numerous experimental results on the shear strength of rockfill have been gathered and published by different authors, such as Marachi et al. [MAR 69, 72], Leps [LEP 70], Charles and Watts [CHA 80] (Figure 9.2), Duncan [DUN 04] and many others. They show a wide dispersion of the values and a pronounced curvature of the shear strength envelope: the friction angle measured in triaxial compression tests significantly decreases when the confining stress increases. This reduction of the friction angle has been attributed to grain breakage that induces a decrease of the dilative behavior (see Chapter 6, section 6.1.1). The amplitude of grain breakage was measured by comparing the grain size distribution before and after testing.
Figure 9.2. Charles and Watts’ compilation of rockfill shear strength envelopes. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip
In Figure 9.2, the results presented by Charles and Watts – curves A, B1, and C – appear shifted toward the upper right side of the diagram when compared to the range of the results obtained earlier by Marsal or Marachi. This remark also holds for Material B2 which corresponds to weaker grain strength and was chosen by the authors for its peculiar properties; therefore, curve B2 should be located below the range of usual rockfill properties.
9.2.2. Evidence of scale effect
In another study, Barton [BAR 81] displayed a very peculiar set of large triaxial tests results on granular materials used in the construction of large dams, extracted from a wider work from [MAR 69]. For each material, the tests have been performed on granular materials issued from the same mineral stock, with homothetic grain size distributions, and tested at the same density. In Figure 9.3(a), the results obtained on two different groups show a clear size effect: the shear strength is lower when the grain size is coarser.
Figure 9.3. Shear strength envelopes for homothetic groups of granular materials – (a) original compilation by Barton [BAR 81] and (b) schematic of scale effects. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip
These two groups having parallel grain size distribution curves. A comparison of their properties may be examined in light of the grain strength resulting from the probabilistic approach with the central trend distribution (see section 9.1.3):
- – for homothetic grain size distributions, the grain resistance is in principle inversely proportional to the square root of a given characteristic diameter, e.g. equal to DMax;
- – therefore, to obtain in a granular material A with a characteristic grain size DA the same amount of grain breakage, and then the same shear strength as in a material B with characteristic grain size DB tested under a confining stress σB, we have to exert on the material A a confining stress σA equal to:
- – so, the shear strength envelope of the rockfill material (DA = 150 mm) can be obtained by a simple geometrical similarity applied on the shear strength envelope of the gravel material (DB = 12 mm) with a similarity factor equal to
applied on the confining stress.
Such transformed curves are plotted in dotted lines in Figure 9.3(b) by extrapolating the data obtained for DMax = 12 mm in order to predict the properties of a granular material with DMax = 150 mm. The comparison with the experimental data shows a quite satisfactory agreement. It appears, therefore, to be possible to determine the shear strength of very coarse granular materials (here 150 mm maximum) by using the results obtained on finer materials (here 12 mm maximum), provided that they have the same mineral origin, parallel grain size distribution, and the same density.
The original data reported by Barton come from an exceptional experimental study by Marachi et al. [MAR 69], which was performed on three groups of materials at University of California (Berkeley):
- – a rockfill made of fine-grained argillite produced by quarry blasting, having very angular particles of comparatively low strength, used for the construction of the Pyramid Dam in California;
- – a rockfill made of crushed basalt extracted from a quarry, with angular sound particles;
- – a coarse alluvium, predominantly made of sound unweathered rounded gravels and cobbles of fine-grained amphibolite, used for the construction of the outstanding Oroville Dam in California.
These three groups of materials, although strongly different, display the same qualitative trend concerning the grain size influence, suggesting the existence of a fairly general scale effect rule.
An independent validation of this approach can be made by using the data gathered by Charles and Watts (Figure 9.4). Their data were obtained on materials with maximum grain sizes DMax = 38 mm, whereas the results previously obtained by Marsal and Marachi concerned coarser materials (DMax = 150–200 mm). A correction of Charles and Watts curves by the method presented above on the confining stress corresponds here to a simple shift of their results toward the left of the diagram, as confining stresses in abscissae are displayed in logarithmic scale. Figure 9.4 shows the curves A, C, and B2 from Charles and Watts [CHA 80] corrected by a horizontal similarity factor equal to 
. The transformed curves A′ and C′ in the diagram are now in far better agreement with the results obtained by Marsal and Marachi than the original curves A and C. For the slate, B2, with low-strength particles, the transformed curve is now located below the usual range for rockfill materials, which is more in agreement with the poor quality of this material, which was why it was selected.
Figure 9.4. Evidence of scale effects in Charles and Watts’ compilation. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip
Thus, the scale effect in Barton’s data, as well as the apparent distortion in the Charles and Watts diagram, appears to be due to this physical similarity rule resulting from fracture mechanics laws governing grain breakage. The adjusted data compilation shown in Figure 9.4 also suggests a typical shear strength envelope for coarse granular materials: the dotted line plotted in the central part of the experimental data range. Complemented by the data of 226 large triaxial tests published by Duncan [DUN 04], this central trend line corresponding to the shear strength envelope of granular materials with grain size DMax = 150 mm can be fitted by a power law:
9.2.3. Scale effect rule on shear strength envelope (failure criterion)
9.2.3.1. General case
In the above analysis, the reasoning is based on the “central trend distribution” where grain-crushing resistances follow Weibull’s theory with a parameter m = 6. Therefore, the failure stresses vary with the grain diameter according to a power law with an exponent −3/m = −1/2 (section 9.1.3).
All the data reported by [MAR 72] and others on grain breakage show a scattering of the values of parameter m for the different materials. This scattering can be attributed to differences in the defect distribution according to the nature and origin of these materials. A direct use of the parameter m determined on each material appears to improve the representation of particle breakage statistics. Therefore, the analysis presented above can be reconsidered with the use of 3/m instead of 1/2 in the effect of the characteristic grain diameter, which leads to the following result called “scale effect rule on shear strength envelope”.
Let us consider two granular materials A and B from the same homogeneous mineral stock, compacted at the same density, with homothetic grain size distributions GA and GB and characteristic diameters DA and DB (e.g. DMax, or any significant diameter such as D80). The two materials are, then, geometrically similar in a ratio DA/DB:
- – In order to mobilize the same internal friction within the two materials, the maximum dilatancy rate has to be the same during shearing, and the amount of grain breakage or the probability of survival also has to be the same, which means that the stresses applied to the grains must verify the following relation of similitude:
- – The link between macroscopic stresses and stresses applied on the grains being enforced by the geometrical similitude of the two materials, the macroscopic stress states, required for mobilizing the same internal friction, must verify the following second relation of similitude, which is identical to the first one:
Equation [9.8] represents the scale effect rule, which generalizes equation [9.6].
This result found here from a reasoning at the macro-scale has also been proven by a reasoning at the micro-scale between two homothetic granular media, Appendix A.9, using the relations that link the macroscopic stresses to the intergranular forces and the geometry of granular arrangement, and the Love–Weber relation used in section 1.3.2 (see also [FRO 12b] which includes a wider set of experimental proofs of this scale effect rule). This theory on size effects, originally developed on the basis of classical axisymmetric “triaxial” tests data, has been recently validated on more general stress-paths, tested on true three-dimensional compression apparatus [XIA 14]. Another consequence of equation [9.8] is that within a given family of similar materials with the same initial compacity, the correspondence between the amount of grain breakage and internal friction (i.e. the maximum mobilized principal stress ratio during motion) is unique, regardless of their characteristic size. This is precisely what was found experimentally by Marachi et al. [MAR 69], as displayed and commented on in [FRO 12b].
In equation [9.8], the scale effect rule resulting from grain breakage under brittle failure Mode 1 is not fundamentally linked to any particular expression of the shear strength envelope: it establishes a simple and direct geometric correspondence between the shear strength envelopes of granular materials A and B, as illustrated in Figure 9.5.
Figure 9.5. Geometric correspondence between shear strength envelopes of materials A and B, set by scale effect rule
If a stress state {σA} is known on the shear strength envelope of granular material A, then the corresponding stress state {σB} on the shear strength envelope of granular material B can be obtained by applying a homothety with the ratio 
on the vector 
. Therefore, from the simple geometric constructions shown in Figure 9.5, the shear strength envelope of granular material B can be obtained from that of granular material A, regardless of any analytical formulation of the shear strength envelope.
9.2.3.2. Shear strength envelope τ = f (σn, D): De Mello’s criterion
When the expression of material A’s shear strength envelope is given by
then, reversing equation [9.8] results in the following relationships between corresponding shear stresses and normal stresses in granular materials A and B:
Then the above relations together with equation [9.9] lead to
Thus, equation [9.10] gives the expression of the shear strength envelope for granular material B, on the basis of the expression of the shear strength envelope for granular material A.
In the particular case of a power law 
with b < 1 (De Mello’s criterion [DEM 77]), commonly used in stability analysis computations, the scale effect rule leads to the following expression of the shear strength envelope:
Note that in relation [9.11], the coefficient AA is modified by a size effect factor, but the power exponent bA remains unchanged.
9.2.3.3. Shear strength envelope σ1 = h (σ3, D): Hoek–Brown’s criterion
If the expression of material A’s shear strength envelope is given by
then, for the same characteristics of material B as the ones given above, combining equations [9.8] and [9.12] gives the expression of the shear strength envelope of material B:
In the case of the Hoek–Brown criterion [HOE 80], widely used in rock mechanics:
The scale effect rule leads to the following expression of the shear strength envelope:
Note that the influence of the scale effect, in this case, affects only parameter σc, but not the other parameters mHB, SHB, and a.