I am pleased to learn that the authors' one-dimensional compression tests on crushed slate having a maximum particle size of 40 mm as well as their theoretical investigations support the Cα/Cc law of compressibility (Mesri & Godlewski, 1977; Mesri, 1987, 2001; Mesri & Castro, 1987; Mesri et al., 1990, 1997; Mesri & Ajlouni, 2007). According to this law of compressibility, Cα/Cc is a constant at all instances during secondary compression, where at any instant (e, σ′v, t) during secondary compression Cc = Δe/Δlog σ′v is the slope of e against log σ′v, and Cα = Δe/Δlog t is the slope of e against log t. The concept is illustrated in Fig. 14 using one-dimensional compression test results reported by Leung et al. (1996) and Yet (1998) on reconstituted specimens of a uniform subangular fine (D50 = 0.2 mm) silica sand with Cu = 2.4, emax = 0.9820 and emin = 0.5904. It is important to note that a constant Cα/Cc = 0.018 corresponds both to very low pairs of values of Cα/(1 + e0) and Cc/(1 + e0) observed at low effective vertical stress range where particle damage was either absent or minimal, and to high pairs of values of Cα/(1 + e0) and Cc/(1 + e0) that were observed at high effective vertical stress range where fracturing and splitting of particles occurred. The empirically established constant Cα/Cc for any soil was interpreted by Mesri & Godlewski (1977) to mean that the same mechanisms of compression (including through particle deformation by compression and bending, and through particle rearrangement by interparticle slip and rotation and by particle damage) that operate during primary compression continue into secondary compression.
Cα/(1 + e0) and Cc/(1 + e0) data for silica sand (Leung et al., 1996; Yet, 1998)
In all soils, particle rearrangement into a more compact configuration is achieved by overcoming interparticle friction through interparticle slip and rotation. In some soils and stress conditions, particle rearrangement is also facilitated by overcoming particle strength through particle damage, as in granular soils. However, particle damage may not be limited to growth of a central crack. It may consist of abrasion or grinding of particle surface asperities, breaking or crushing of particle surface protrusions and sharp particle corners and edges, and fracturing, splitting, or shattering of particles (e.g. Nakata et al, 2001a, 2001b; Chuhan et al, 2002, 2003). A phenomenological interpretation of constant Cα/Cc behaviour, that the same mechanisms that operate during primary compression continue into secondary compression, based on the observed direct linear relationship between Cα and Cc for a wide range of geological materials from granular soils including rockfill to fibrous peats (e.g. table 16.1 in Terzaghi et al., 1996), is, I believe, more powerful and general than a theoretical interpretation assuming one mechanism of compression. As an example, clearly secondary compression in clays or fibrous peats does not result from growth of cracks in clay particles or peat fibres.
A theoretical interpretation of the magnitude of Cα/Cc for granular soils in terms of 1/n, where n is the ‘slow crack growth exponent’, has also been previously reported by McDowell (2003). However, the values of n in the range of 10 to 100 used by McDowell (2003), or 20 to 200 employed by the authors, imply magnitudes of Cα/Cc that are outside the range 0.02 ± 0.01 that has been generally observed for granular soils including rockfill.
The authors' data on the influence of suction on compressibility of the rockfill, as in their Fig. 5, are quite interesting, and are of considerable practical importance. In this connection, the true nature of the rockfill material may be significant. In the section on experimental observations, the rockfill is described as ‘slate’ (which is a fine-grained metamorphic rock formed by low-grade regional metamorphism of shale). Later, however, in the text the rockfill is referred to as ‘shale’ (which is a fine-grained sedimentary rock that could be susceptible to shrinkage and swelling). A more precise description of the rockfill used in the tests could be helpful for interpretation of the effects of suction on compressibility of rockfills. Finally, it would be useful to know whether the reported magnitudes of suction were directly measured, or whether they were computed from the imposed relative humidity environment condition.
Authors' reply
We are grateful to the discusser for his interest in the paper, and for his comments on some significant aspects of the reported work.
The discusser suggests an explanation for the experimentally observed relationship between Cα/Cc (numerically identical to the ratio λt/λ used in the paper), in the sense that the same mechanisms of compression that operate during primary compression continue into secondary compression. This is a general idea that helps in understanding, in a qualitative manner, why the Cα/Cc ratio remains rather constant for a wide range of soils. However, the real problem is to show, by some kind of consistent physical or physico-chemical interpretation, that the general idea leads formally to a constant Cα/Cc ratio, irrespective of the type of soil. In support of the discusser's idea, however, the concept of equivalent deformation mechanisms for primary and secondary behaviour is implicit in most of the paper, and it is explicit in the final section, on model development. Some steps towards finding a general explanation for the secondary behaviour of geomaterials have been given. One should recall the work by Kuhn & Mitchell (1993), explaining creep behaviour in soils by assuming interparticle sliding as a thermally activated rate process. It is interesting to note that the stress corrosion phenomenon, evoked in the paper as the basic mechanism underlying the time-dependent behaviour of rockfill, was also described by means of the rate theory (Wiederhorn et al., 1980, 1982; Atkinson, 1984), which leads to a formulation (in this case for crack propagation rate) that is formally identical to the one applied by Kuhn & Mitchell (1993) for explaining time-dependent interparticle sliding. The formulation of a general theory was beyond the aims of the paper.
The tested material reported in the paper is a crushed slate (the term ‘shale’ is an erratum in the paper) obtained from the Pancrudo River outcrop (Aragón, Spain), which belongs to the Almunia formation from the Cambric period. On the basis of microscopy on thin sections and X-ray diffraction analysis, the parent rock was described as a fine-grained phyllite with low-grade metamorphism, containing quartz, muscovite, altered biotite, pyrite, plagioclase, clinochlore, calcite and dolomite. Some index rock properties, the water retention curve of the rock, the grain size distribution of the tested rockfill and a photograph of an oedometer specimen can be found in Oldecop & Alonso (2001). Moreover, rock cores with various moisture contents were subjected to the Brazilian splitting test (Oldecop, 2001). Fig. 15 displays some results of such tests. The energy of the water was characterised by means of the total suction.
Brazilian splitting test on slate cores under various total suction values. Legends next to data points indicate saturated solution used for controlling relative humidity in environment where specimens were stored
Brazilian splitting test on slate cores under various total suction values. Legends next to data points indicate saturated solution used for controlling relative humidity in environment where specimens were stored
It is acknowledged that an equivalent relationship to equation (32) in the paper has already been proposed by McDowell (2003), although the authors were not aware of this contribution until the paper was already published.
However, it is believed that the developed approach involves a more accurate analysis of the problem, resulting in wider implications than McDowell's approach. In fact, McDowell derived the λt/λ = 1/n relationship directly from the relationship proposed by Davidge (1979) for ceramic specimens undergoing time delayed brittle failure,
where σs0 is a nominal stress value causing failure at time t0, and σs is the stress causing failure at time t. The basic hypothesis from which this relationship was derived is that both specimens, the one failing at t0 and the second failing at t, are identical. This means that, in order to apply equation (33), they must have the same shape, same size, and same initial crack length and position. In general, this is not the case with rockfill or sand particles, or, at least, showing that equation (33) holds for granular particles undergoing breakage is not a trivial matter. And this is, indeed, a significant contribution of the paper. On the basis of the recognition that the crack propagation process in rocks is a two-staged process, it was possible to show that the shape and size of the rock particle and the position of the initial crack or flaw within the particle do not influence the relationship of stress against time to failure (expressed as a stress intensity factor against time to breakage in the paper). As a result, equation (32) in the discussed paper becomes applicable to any particle within a rockfill element. Moreover, this demonstration is not only a theoretical exercise; it has some important implications, as discussed in the next paragraph.
According to the fractal approach to the compressibility of granular materials (McDowell & Bolton, 1998), in the paper by McDowell (2003), the λt/λ = 1/n relationship is derived on the hypothesis that particles undergoing breakage during normal compression are only those belonging to the smallest fraction among the grain sizes contained in the granular medium. This would occur only in the so called clastic yielding regime (McDowell & Bolton, 1998), which implies a linear void ratio–log(stress) normal compression line. So McDowell's equation would apply only to the clastic yielding stage. On the other hand, the conceptual model developed in the paper leads to the conclusion that equation (32) in the paper holds for almost all the normal compression process, eventually excluding a short initial stage of particle rearrangement (Oldecop & Alonso, 2001). This means that equation (32) holds on both curved and linear stages of the normal compression line (see Figs 4 and 5 and accompanying text in the paper). In fact, this result is in better agreement with the experimental observations shown in the paper, and also with those previously published by Mesri et al. (1990).
As far as the conclusion that the position of the crack within the particle does not influence the relationship of stress with time to breakage, it could be thought that the derivations of the proposed theoretical model are applicable to any particle breakage mechanism involving crack propagation, being either the splitting mechanism explicitly considered in the paper or local crushing in the vicinity of interparticle contacts. However, it was not possible to demonstrate the validity of this statement in the paper.
As noted by the discusser, the range of values of 1/n obtained from fracture propagation tests for most rocks (0·005–0·05) is somewhat wider than the range of experimentally observed values of Cα/Cc (0·01–0·03) for soils. Especially notable is the disagreement in the low boundary. The authors gave a possible explanation for such a discrepancy based on the fact that the material exhibiting low values of Cα/Cc (0·005) was in a very dry state (total suction = 255 MPa). In such a low-moisture environment, crack propagation would occur by so-called ‘region 3 stress corrosion’ (Wiederhorn et al., 1980). In region 3, the crack propagates faster than the flow of water vapour (by molecular diffusion) from the macro-voids towards the crack tip. So vapour is no longer able to reach the crack tip, and hence the crack continues to grow as a thermally activated process, but with no influence of water. The values of n measured in region 3 (usually in experiments under vacuum) are systematically larger than the values obtained in region 1 (Wiederhorn et al., 1980; Atkinson, 1984; Fig. 9 in the discussed paper), a fact that fits quite well with the low Cα/Cc values measured in the very dry environment.
In the experiments reported in the paper, changes in the specimen moisture were induced by means of an air flow with controlled relative humidity (Oldecop & Alonso, 2004). Total suction values indicated in the plots of experimental data are used as a convenient measure of water presence in the specimen. Total suction values were derived from RH measurements by means of the psychrometric relationship. However, this should not be interpreted as an absence of liquid water in the specimen. In the so called vapour equilibrium technique, water is transported into or out from the specimen in vapour state, but water will condense within the rock pores. This is a necessary conclusion when the amount of water introduced in the specimen during a single test is considered: some 100 g in a 14 l specimen. Such a mass of water could never be stored as water vapour only, since saturation of the specimen air volume is reached with less than 1 g of water vapour. Moreover, the amounts of water introduced in (or extracted from) the specimens during tests were measured, and it was found that suction–water content experimental points were nearly in agreement with the retention curve of the rock (Oldecop, 2001).
