Abstract
Although intact rock is frequently represented by linear Mohr–Coulomb shear strength envelopes, the actual true behaviour, if taken over a wide range of confining stress, is extremely non-linear. Why is this important? Probably because the real stress across points of contact in both rockfill and rock joints is approaching (or trying to exceed) the crushing strength, and if local confined stresses are equally high, strong non-linearity will be experienced. If we utilize the unconfined compression strength of the rockfill, or of the rock joint surfaces, suitably scaled-down due to size effects, we are part way towards a useful strength criterion. In fact, as we will see, we are “one-third” of the way, as the roughness (of particles and asperities) and a measure of the non-dilatant (residual) frictional strengths are also needed for what we will see are two very closely related strength criteria.
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Quotation
Terzaghi (1920) quoted by Marsal [18]
The fundamental error was introduced by Coulomb, who purposely ignored the fact that sand consists of individual grains, and who dealt with the sand as if it were a homogeneous mass with certain mechanical properties. Coulomb’s idea … developed into an obstacle against further progress as soon as its hypothetical character came to be forgotten by Coulomb’s successors.
The response in 2016
The ‘Coulomb error’ is still present in rock mechanics. In fact most rock-related strength envelopes are non-linear, the exception being planar rock joints. Even rough-surfaced rock joints have no cohesion unless with actual steps preventing shearing. If a rock mass has a cohesive component due to intact ‘bridges’ (or such steps) it is not correct to add the cohesive and frictional components: they are mobilized at successively larger shear strain or displacement. Rock masses are also ‘particulate’ media, with blocks instead of sand grains or gravel or stones. Use of linear Mohr–Coulomb is a tragic error pervading structural geology, petroleum engineering and a lot of rock mechanics. It gives many misleading results in relation to reality, including an exaggeration of petroleum reservoir production potential in the weaker rock types.
Dedication
Thirty-five years ago, while preparing a paper summarizing some work on the shear strength of rockfill [13], the writer came across some of the work by the person we are now honouring. With degrees from MIT and Harvard and lectures from Terzaghi and Casagrande, and more importantly a distinguished career in famous institutes in Mexico City, it is very clear that Raul Marsal was a person whose work had a high status in the world of rockfill and embankment dam engineering.
One of his landmark papers was the following: Marsal, R.J. ‘Mechanical Properties of Rockfill’, Embankment-Dam Engineering, Casagrande Volume, R.C. Hirshfeld and S.J. Poulos, eds., John Wiley and Sons, Inc., New York, N.Y., 1973, pp. 109–200. Note the large number of pages: this was a major chapter by Dr. Marsal, honouring Casagrande. I quote from part of his abstract:
The analysis and measurement are described of particle breakage, which is a function of the contact forces and the crushing strength of the particles. The devices developed to investigate the shear strength and stress–strain characteristics of rockfills and the techniques for preparation of specimens are described, as well as the measurement of shear strength of rockfill in the triaxial and plane-strain apparatus (Fig. 1).
Introduction
Although intact rock is frequently represented by linear Mohr–Coulomb shear strength envelopes, the actual true behaviour, if taken over a wide range of confining stress, is extremely non-linear. Why is this important? Probably because the real stress across points of contact in both rockfill and rock joints is approaching (or trying to exceed) the crushing strength, and if locally confined stresses are equally high, strong non-linearity will be experienced. If we utilize the unconfined compression strength of the rockfill, or of the rock joint surfaces, suitably scaled-down due to size effects, we are part way towards a useful strength criterion. In fact, as we will see, we are ‘one-third’ of the way, as the roughness (of particles and asperities) and a measure of the non-dilatant (residual) frictional strengths are also needed for what we will see are two very closely related strength criteria.
Figure 2 is a representation of the complete shear strength envelope for intact rock, as suggested by Barton [5] following a wide review of other researchers’ high-pressure triaxial data for intact rock, in particular the well-known and numerous studies of Byerlee and Mogi from the 1960s. Both these researchers were concerned with the brittle–ductile transition. The present author suggested a simple ‘definition’ of the top (horizontal) part of the strength envelopes—in rock mechanics terminology called the ‘critical state’. The complete shear strength envelopes of rock, and how much they deviate from linear Mohr–Coulomb have recently been quantified in a new criterion which may become known as the Singh–Singh criterion [21].
The horizontal part of the shear strength envelopes for a large group of silicate and carbonaceous rocks, suggested the following simple relation:
It will be noted that the uniaxial (unconfined) circle (#2) and the critical confining pressure circle (#4) are drawn as nearly tangent to one another. This potential simplicity has recently been confirmed by Singh et al. [21], who found that the majority of rocks exhibited this tendency, i.e. \( \sigma_{{3{\text{ critical}}}} \approx \, \sigma_{\text{c}} \). This actually implies that if we reach a confining pressure (over the small area/volume in contact) approximately equal to UCS or \( \sigma_{\text{c}} \), a local critical state may potentially be reached if (less confined) crushing has not already occurred. The maximum local rock strength will likely have been exceeded, both for the case of rock joint asperities and the contacting areas/volumes of rockfill particles/stones.
In Fig. 3, we see the near ‘super-position’ of two envelopes (rock joints and rockfill). It should be noted that the top three envelopes, together with the fifth (lowest) envelope, symbolize the four components of the shear strength of rock masses. For the last 50 years, since the beginning of rock mechanics, this important in situ strength has been assumed to consist of the Mohr–Coulomb \( \tau \; = \; c + \sigma_{n}^{\prime } \tan \, \varphi^{\prime } \) (or more recently as the non-linear complex-algebra GSI-based Hoek–Brown criterion). In reality, contrary to the M–C or H–B assumptions, the shear strength of rock masses consists of the degradation of the cohesion (the intact ‘bridges’) followed at larger strain by the mobilization of friction and roughness causing dilation. At still greater shear ‘strain’, any clay-filled discontinuities present may also be mobilized. The failure is, therefore, a progression of components mobilized at different levels of shear ‘strain’ (or displacement).
From tension fractures to rock joints
A promising start to an understanding of the shear strength of rough rock surfaces with opposed asperities in contact was experienced by the writer in about 1968, while developing a method to create ‘2D’ models of variously jointed (fractured) rock masses. Figure 4 (top-left) shows shear-displaced roughness profiles of brittle tension fractures which have been sheared and dilated exactly as measured in direct shear tests of the same (2D profiled) samples. Very small areas of contact (and, therefore, high contact stresses) are seen as a result of the pre-peak and post-peak shearing. The direct shear tests produced strongly curved peak strength envelopes, especially at very low normal stresses, as shown in the top-right diagram. (Such results were the forerunner of tilt tests developed 10 years later, illustrated in Fig. 5). The two sets of experimental data with results expressed as a function of peak dilation angles (bottom-left) required a small rotation to the dotted lines, to produce the forerunner of the Barton [4] non-linear peak strength criterion. In 1968, ‘JRC’ was 20 (very rough tension fractures), ‘JCS’ was UCS (due to no weathering), and \( \varphi_{\text{b}} \) was 30° (also due to no weathering).
The terms JRC (joint roughness coefficient) and JCS (joint-wall compression strength) were first used in Barton [4]. In subsequent experimental work with fresh and partly weathered rock joints (130 samples), acquired from road cuttings in the Oslo area, Barton and Choubey [12] showed that \( \varphi_{\text{b}} \) should be replaced by \( \varphi_{\text{r}} \) which could be several degrees lower in cases with weathered joint surfaces. The writer developed the tilt test at this time, as a formal way to back-calculate the variable joint roughness JRC. Examples of this simple technique are shown in Fig. 5 (Do three tilt tests per sample plus roughness profiling).
Figure 6 shows some of the joint samples tested (with increasing roughness from #1 to #10), and two sets of shear strength results. The three shear strength criteria (M–C, Patton, and Barton/Choubey) are compared. By chance the writer’s career in rock mechanics started in 1966, just after the first ISRM congress in Lisbon, where Patton [20] presented his bi-linear equation, which was developed from tests on interlocked ‘saw-teeth’. It was an immediate goal to try to improve upon this ‘\( \varphi + i \)’ criterion, since ‘i’ was not sufficiently defined. In due course of time, it was found that ‘i’ was both stress and scale dependent. (The term \( {\text{JRC}}_{n} \log_{10} {\text{JCS}}_{n} /\sigma_{n} \) is required).
Scale effects for rock joints
A ‘final’ stage concerning the writer’s non-linear shear strength criterion was the correction of the input parameters JRC and JCS for scale. As we shall see later, a similar scale correction was also necessary for rockfill. The impetus for scale corrections was the result of a milestone series of direct shear tests on joint replicas of different size. By happy coincidence, Bandis started his Ph.D. studies just after the work of Barton and Choubey [12], and made active use of JRC and JCS to record strong scale effects. Figures 7b and 8 summarize what Bandis found. As may be noted from Fig. 7a, performing tilt tests on jointed blocks is also a scale-dependent process.
The various stages of development of the shear strength criterion for rock joints, starting with artificial tension fractures, and ending with weathering and scale corrections are listed below. As we shall see, the proposed empirical equation for the shear strength of rock fill and for rockfill/rock-foundation interfaces have their origin in the highly stressed asperities shown in Fig. 4a.
The collection of simple index tests for obtaining input data for fractures or joints, using tests on core or on samples sawn or drilled from outcrops is shown in Fig. 10. In the left-hand column, direct shear tests are also illustrated, which can be used to verify the results from the empirically developed index tests. Note that the Schmidt hammer should be used on clamped pieces of core to estimate the uniaxial compression strength UCS or \( \sigma_{\text{c}} \). This is done on dry pieces of core (rebound R, use top 50 % of results). For the joint-wall strength JCS, this is done on saturated samples (rebound r, use top 50 % of results). These samples also need to be clamped (e.g. to a heavy metal base). Three methods of joint-wall or fracture-wall roughness (JRC) can be used: tilt tests to measure tilt angle α (i.e. Fig. 5), or a/L (amplitude/length) measurement, or roughness profile matching; the latter obviously more subjective.
From rock joints to rockfill
As indicated in the Dedication to this article, the non-linear shear strength envelopes for rockfills (see Fig. 1) presented by Marsal [18] made a strong impression when these were first seen by the writer in about 1979. Non-linear strength envelopes were also the dominant theme for rock joints, in the author’s work since about 1967: first with artificial tension fractures (in brittle model materials) then with rock joints, with the latter contributions starting in Barton [4]. In fact, it is interesting to reproduce one of the early diagrams when JRC and JCS were first suggested, since six of the curved envelopes (Fig. 12) contain within their range, something similar to the shear strength of rockfill. It seems that non-linearity was a common but independent pre-occupation in Mexico (for rockfill) and in Norway (for rock joints) in the early 1970s (Fig. 11).
Figure 12 shows that there are strong similarities between the shear strength of rockfill and that of rock joints. This is because they both have ‘points in contact’, i.e. highly stressed contacting asperities or opposing stones. In fact, these contacting points may be close to their crushing strength, such that similar shear strength equations to those used for rock joints can apply to rockfill, and indeed to rockfill/rock interfaces. This can be seen in the progression of equations.
The three equations to be utilized in the remainder of this paper have the shear strength of rock joints as a starting point. The first equation listed was Eq. 4:
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1.
\( \tau /\sigma_{n} = \, \tan [{\text{JRC}}.\log \, ({\text{JCS}}/\sigma_{n} ) \, + \varphi_{\text{r}} ] \ldots {\text{ for rock joints}} \)
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2.
\( \tau /\sigma_{n} = \, \tan [R.\log \, (S/\sigma_{n} ) \, + \varphi_{\text{b}} ] \ldots {\text{for rockfill}} \quad \quad (6)\)
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3.
\( \tau /\sigma_{n} = \, \tan [{\text{JRC}}.\log \, (S/\sigma_{n} ) \, + \varphi_{\text{r}} ] \, \ldots \;{\text{for interfaces}}\; ( {\text{see later)}}. \quad \quad (7)\)
The equation to be used for a rockfill/rock-foundation interface will depend on whether there is ‘JRC’ control, or ‘R’ control. This distinction: preferential sliding on the interface or shearing within the rockfill is described and illustrated later.
Concerning the potentially very high friction angles at low stress levels, or close to the surface of a rockfill dam embankment (e.g. Fig. 13), Marsal [18] makes mention of a basalt rockfill having \( \varphi^{\prime } \) of up to 70° under a normal effective stress of only 0.1 MPa. Although the ‘asperity strength’ component (Fig. 8) is uncertain for rockfill, if it were of similar magnitude to the peak dilation angle (d n peak), as in the case of rock joints, then the latter could be in the region of 20° at similarly low stress, compatible with the low stress predictions of peak dilation given in Fig. 12 (bottom-left) and in Table 1. Marsal [18] considered that within a rockfill dam, the friction angle of the materials within the range of confining stresses of interest was usually less than the 40°–45° previously assumed, due to the effect of the confining pressure and the non-linearity. This will be explored further.
The non-linearity implied by the rockfill data assembled by Leps [16] shown in the left diagram of Fig. 15 is provided by the linear-phi and log-stress format, which is exactly consistent with the experiences of the shear strength of rock joints shown in the diagram on the right side of Fig. 15. Note the direct ‘substitution’ of R for JRC and S for JCS, to demonstrate similarity to the data for rockfill. A similar gradient to that drawn by Leps (the middle-of-the-data line) seems to be provided by R = 7.5 and S = 45 MPa.
Large-scale vacuum triaxial cylinders as seen in Fig. 16 have a limit of confinement of <100 kPa. Somewhat smaller but higher pressure triaxial cells, as also used by Marsal [18], and by many others during several decades, were also used in these studies, to reach the higher confining pressures. Of particular note, besides the non-linear stress-dependent friction angles, is the steepness of their decline with increasing stress.
Simple inspection of one order of magnitude stress change, from 0.16 to 1.6 MPa, bringing \( \varphi^{\prime } \) from 51° down to almost 39° suggests an equivalent R value as much as 10.5, even steeper (rougher) than the highlighted data shown in Fig. 15. [If smaller particle gradings had been used at the higher stresses, the decline would be much reduced due to higher S values (see Fig. 17)]. Note from Fig. 14 that if quarried rock could be so well graded (thereby filling its voids) and so well compacted that a porosity of 20 % was reached, then an R value as high as 10 can be predicted (close to the ‘Q’ of quarried rock in Fig. 14). Beneath 500 m of cover all is possible.
The necessity to have a scale-effect adjustment for the uniaxial compressive strength of rockfill, following approximately the trend obtained in rock mechanics studies is clear. For simplicity and consistent reference, the strength of the d 50 size is recommended. In the case of the UCS of rock, the following equation suggests itself, based on an averaging of Hoek and Brown and Wagner versions with their slightly diverging exponents of 0.18 and 0.22: \( \sigma_{\text{c}} = \sigma_{{{\text{c}}50}} \left( {50/d_{50} } \right)^{0.2} \). The term \( \sigma_{{{\text{c}}50}} \) is the standard (50 mm diameter) size of UCS specimens. The diameter d 50 is the stone size. The way that R (or JRC) causes a change of R° (or JRC°) (i.e. change of angular degrees) per decade change of effective stress or equivalent overburden depth is demonstrated in Table 2. It seems logical to assume that 20, and JRC (and JRC n ) seen earlier in Eqs. 1 through 6 actually have units of degrees, as in the case of Patton’s ‘i’ value shown in Fig. 6 (bottom-left diagram). The log10 (strength/stress) adjustment factor determines how many decades (or fractions of a decade) the change of stress ‘multiplier’ of R (and JRC) will be.
For example, a large-scale tilt test (as in Figs. 18, 19) with a normal stress at failure of about 0.01 MPa, using a rockfill material having a scaled compressive strength of 10 MPa for the d 50 particle size, involves log10 (10/0.01) = 3 (×R) for the case of the tilt test, while it will be log10 (10/1) = 1 (×R) inside the dam at, e.g. 45 m depth. If we assumed R = 7, there would, therefore, be a 14° lower friction angle inside the dam than in the tilt test: for instance 48° in the tilt test, reducing to 34° inside the dam where the confining stress is assumed to be 1 MPa, or 100 times higher than in the tilt test.
Figure 18 shows the Barton and Kjærnsli [13] suggestion for large-scale tests on dams under construction. By chance, NGI staff had arranged for such tests in Italy, just before the writer returned to NGI after 4 years in the USA in 1984. Due to pressing new administrative duties, the writer was not able to physically accompany the testing in Italy. The sequence of photographs later shown to the writer (some examples are given in Fig. 19) suggest that the Italian contractor at the dam did not follow the recommendation of ‘standard compaction’ within the ‘next lift’ of the dam, followed by ‘excavation’ of the tilt-shear box. Possibly they found this to be too burdensome. So it appears that other, probably less efficient methods may have been used to compact the rockfill before tilt testing. A bulldozer on a ramp is seen in one of the photographs. It appears that the tilt angles were mostly in the range of 45°–50°.
A particular feature of these large-scale tests is that the gravity loading provides a much more uniform loading than the usual top-platten-and-end-loading in a standard shear box. The relative absence of any tendency to rotate when tilted is easy to see. [In the case of some very rough rock joints, the top half of the tilt test sample needs to be made slender (low H/L ratio) to prevent toppling (rotating) instead of parallel sliding. There seems to be no such problem with the 1/5 ratio seen in Fig. 19].
From rockfill to interfaces
Because some dam sites in glaciated mountainous countries like Norway, Switzerland, and Austria have insufficient foundation roughness to prevent preferential shearing along the rockfill/rock-foundation interface, artificial ‘trenching’ is needed. Various scales of investigation of interface strength have been published, and these were synthesized by Barton [6]. The physical reality of a rockfill/rock interface is illustrated in Fig. 20, and potentially correct input parameters for estimating the shear strength are suggested.
Figure 21 illustrates two rockfill dams with moraine cores, while they were under construction. Both appear to be on relatively smooth rock foundations, in particular the Norwegian example on the left. Here, it is uncertain if the foundation is rough enough to prevent the interface from being ‘the weakest link’.
Figure 22 illustrates examples of these two categories of shearing, in which the ‘weakest link’ determines the mode of sliding: whether the interface is smooth enough and the particles big enough to prevent good interlock (JRC-controlled), or the opposite R-controlled behaviour, with preferential failure within the rockfill, due to good interlock across the interface. Numerous cases reported in the literature were analysed in unpublished research performed by the author, and can be summarized by the data points plotted in Fig. 22 (left). The a/L versus d 50 ratios for the Norwegian dam shown in Fig. 21 are not known. However, JRC-controlled interface behaviour seems likely in places. The (discarded) boulders in the foreground of the Brazilian dam (right) would give JRC-controlled behaviour. They are too big to interlock with the smooth rock surface.
Discussion
This paper has seen an attempt to link the shear strengths of rockfill—with its many possible varieties, materials, porosities, stress levels—to the shear strength of rock and rock joints. Naturally, the key to such a linkage is the highly stressed points of contacts. Of course, in 1967, when the writer started experimenting with the shear strength of rough tension fractures, in an attempt to improve upon Mohr–Coulomb and Patton for describing rock joints, the full significance of the asperities in contact as seen in Fig. 4 (top-left) was not understood. Nevertheless, the first shear strength criterion involving the ratio of material compressive strength and effective normal stress was produced, and 10 years later, even for the case of weathered joints, this ratio continued to be a central part of the strength criterion, in the form \( {\text{JCS}}/\sigma_{n} \). For rockfill, we have found that \( S/\sigma_{n} \), with both JCS and S corrected for block size (L n ) or particle size (d 50), has strong relevance to the understanding of peak shear strength.
In remarkably painstaking work by Marsal [18], pre-occupation with particle contacts and particle contact forces is reflected by the drawing of particle contacts in Fig. 23, and by an example of the statistics Raul Marsal produced, with observations of particle contacts numbering in their thousands.
The method used by Marsal to record the (undisturbed) number of contacts was novel in the extreme. He flooded the rockfill or gravel samples from the bottom upwards with paint, allowed it to dry for 24 h, and then began the careful work of ‘excavating’ up to several thousand grains counting the paint-coated contacts as he progressed. This deserves our strong admiration.
Of course, Marsal was characteristically modest in his conclusions about work remaining to be done. Presumably, due to capillary effects, the paint-covered contacts, while an accurate reflection of numbers, must have presented a remaining question about actual areas in contact and, therefore, actual stress levels. In work on rock joints, the writer has assumed that the small percentage of asperities in contact must raise actual stress levels up to UCS or JCS levels. Whether this could hold for rockfill is less certain, as too much particle adjustment is possible.
Conclusions
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1.
Rock material, which inevitably controls the behaviour of rock joints and the particulate components of rockfill, plays an extra sensitive role in shear strength development, because the true contact areas transferring the shear and normal stresses are very small, and therefore, the stress levels are correspondingly very high. So the crushing strength of the rock becomes important.
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2.
The asperities in contact on either side of a rock joint and the small contacting points around the rockfill particles appear to follow the same basic non-linear relation between shear strength and effective normal stress. Both materials dilate strongly at low stress, causing high apparent total friction angles. At high stress levels there is reduced dilation and asperity strength is mobilized due to crushing.
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3.
It is now known that when the local confining pressure reaches approximately the level of the uniaxial compressive strength, the rock in question will generally have reached its maximum possible strength, or ‘critical state’. This is important at these points in contact.
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4.
The strong non-linearity and lack of actual cohesion for both rock joints and rockfill give a special bonus. One can perform extremely low-stress gravity-loaded tilt tests in both cases, using the non-linear strength criteria to extrapolate results to many orders of magnitude higher stress.
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5.
Raúl Marsal played a strong indirect role in linking rockfill and rock joint behaviour because of his painstaking investigations of the particles-in-contact phenomena, and his presentation of triaxial test results in a format familiar in rock mechanics, namely \( \tau \) versus \( \sigma_{n}^{\prime } \), as a result of analysing the triaxial strengths in relation to the failure planes.
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6.
This paper has mostly addressed only the peak shear strengths of rock joints and rockfill. The mobilization of friction and dilation up to peak strength, followed by degradation of roughness towards ultimate and residual strength is also important if numerical modelling is to be performed. The appendix shows by means of three figures how this can be estimated for the case of rock joints.
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Appendix
Appendix
Peak strengths have been the subject of this paper. In case the shear strength-displacement is required, for instance as input to distinct element modelling such as UDEC or PFC, then the JRCmobilized concept can be used (for the case of rock joints). The matching of Bandis experiments with Barton predictions is shown [8]. There seems no reason why ‘R mobilized’ could not be used for rockfill.
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Barton, N. Non-linear shear strength for rock, rock joints, rockfill and interfaces. Innov. Infrastruct. Solut. 1, 30 (2016). https://doi.org/10.1007/s41062-016-0011-1
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DOI: https://doi.org/10.1007/s41062-016-0011-1