Darley Dale and Pennant sandstones were tested under conditions of both axisymmetric shortening and extension normal to bedding. These are the two extremes of loading under polyaxial stress conditions. Failure under generalized stress conditions can be predicted from the Mohr–Coulomb failure criterion under axisymmetric shortening conditions, provided the best form of polyaxial failure criterion is known. The sandstone data are best reconciled using the Mogi (1967) empirical criterion. Fault plane orientations produced vary greatly with respect to the maximum compressive stress direction in the two loading configurations. The normals to the Mohr–Coulomb failure envelopes do not predict the orientations of the fault planes eventually produced. Frictional sliding on variously inclined saw cuts and failure surfaces produced in intact rock samples was also investigated. Friction coefficient is not affected by fault plane orientation in a given loading configuration, but friction coefficients in extension were systematically lower than in compression for both rock types. Friction data for these and other porous sandstones accord well with the Byerlee (1978) generalization about rock friction being largely independent of rock type. For engineering and geodynamic modelling purposes, the stress-state-dependent friction coefficient should be used for sandstones, but it is not known to what extent this might apply to other rock types.
The Mohr–Coulomb failure criterion is widely applied to the description of intact rock failure and to the description of rock-on-rock frictional sliding (e.g. Byerlee, 1978; Al-Ajmi and Zimmerman, 2006; Rutter and Glover, 2012). It assumes that failure occurs at particular combinations of the greatest and least principal stresses, that the intermediate principal stress has no effect on failure and that failure criteria can be set out in terms only of the stress state, without any consideration of the state of strain or the deformation mechanisms at work that lead to failure. It is easy to adapt to many geomechanical modelling problems and it is widely applied to problems that involve polyaxial loading (e.g. Vernik and Zoback, 1992; Castillo et al., 2000), often because nothing otherwise is known about the behaviour of particular rocks under polyaxial stress conditions. There are many applications that demand knowledge of failure or frictional sliding under generalized stress conditions. These include modelling reservoir or cap rock behaviour, or the estimation of far-field stresses from borehole breakout geometry, but the uncritical application a Mohr–Coulomb failure criterion based on only uniaxially symmetric shortening experiments can result in significant errors (Song and Haimson, 1997).
Compilation of friction data for (mainly crystalline) rocks up to a normal stress of 800 MPa by Byerlee (1978). In this pressure range there is no basis for recognizing two pressure regimes represented by different friction coefficients. Data are also shown for various other sandstones, from Rutter (for Berea sandstone, unpublished), Menéndez et al. (1996), Mair and Marone (1999), Numelin et al. (2007) and Scott and Nielsen (1991a, b), plus Pennant and Darley Dale sandstones (this study). Collectively, the sandstones display a slightly lower friction coefficient (0.718) than the crystalline rocks (0.779).
Back-scattered electron images showing microstructure of
Experiments on the strength of two porous sandstones and on the orientation
of the fault plane produced are described here to evaluate the generality of
the Mohr–Coulomb criterion under the extreme conditions of axially symmetric
shortening and axially symmetric extension. In the former, the intermediate
principal stress
Rock-on-rock sliding friction is defined in terms of the effective stress normal to the sliding surface and to the shear stress resolved in the direction in which sliding occurs (Byerlee, 1968, 1978). It is therefore a 2-D criterion and friction is generally assumed not to depend on the intermediate principal stress. Rock-on-rock sliding friction is important to geomechanical modelling because it limits the differential stresses that can be obtained at any given depth in the upper crust of the Earth. From a compilation of friction data for a wide range of rock types, Byerlee (1978) suggested that it is a property that is, to a useful approximation, independent of rock type (Fig. 1). These characteristics of friction as a rock property have been widely applied to developing understanding of crustal stresses and rock rheology (e.g. Goetze and Evans, 1979; Kohlstedt et al., 1995; Zoback, 2007). To test these generalizations, frictional measurements were therefore made on these same two sandstones under both axially extensional and compressional loading configurations and with variously oriented saw cuts made in the rock cylinder.
Two quartz sandstones of different porosities were used.
The first, Pennant sandstone, is an Upper Carboniferous quartz sandstone from South Wales
(Kelling, 1974). This grey, durable rock is available from stone merchants in
large homogeneous blocks and is used as a kerbstone and paving stone. Modal
composition (by chemical mapping on the scanning electron microscope)
is 70 % sutured quartz grains and 15 % feldspar; interstices between
these grains are filled with clusters of muscovite, oxides and clay minerals,
with a small amount of remaining porosity, 4.57 %
The second, Darley Dale sandstone, is an Upper Carboniferous quartz sandstone from
Derbyshire, England. This yellow decorative stone has previously been widely
used in rock mechanics investigations (e.g. Zhu and Wong, 1997; Heap et al.,
2009 amongst others). It is available from stone merchants as large,
homogeneous blocks. It consists of 67 % quartz, 16 % feldspar and
3 % detrital muscovite and clay minerals. Porosity is
13.5 %
The permeability
The permeability of Pennant sandstone to argon gas was measured normal to
bedding during the course of the present study using the oscillating pore
pressure method (Bernabé et al., 2006) over the effective pressure range
1 to 70 MPa and is much lower than that of Darley Dale sandstone. It is best
described by
As part of specimen characterization, acoustic velocity measurements were
made under unconfined conditions normal (
For mechanical tests, cores of either 20 or 15 mm nominal diameter were
taken normal to bedding from both rocks and ground to a length: diameter
ratio of
Samples were jacketed in an inner sleeve, 3 mm thick, of soft silicone
rubber and an outer sealing jacket of heat-shrink tubing. Tests were carried
out that showed the jacketing arrangements did not support any significant
differential stress (0.1 MPa or less). To permit testing in extension, a
bayonet connector was used on the lower loading piston to allow the axial
load to be reduced as the upper loading piston was withdrawn (Fig. 3). This
arrangement was first described and used by Heard (1960, 1963, 1972). Thus
specimens are tested in axisymmetric extension (not tension), in which the
radial hydrostatic confining pressure provides the maximum (and intermediate)
principal stresses, and the axial stress becomes
A jacketed sample with a 45
Summary of results of experiments on intact rock cylinders in axisymmetric extension and shortening, for ultimate strength and residual strength (frictional sliding on fault plane formed). Data are shown as differential stress versus mean stress and as resolved normal and shear stress on the fault plane orientation for each rock type. Shear stress and differential stress are shown as negative for extension tests. Errors of measurement are generally smaller than the size of the points plotted.
The apparatus used employs a synthetic hydraulic oil (Reolube DOS synthetic ester) whose viscosity is relatively insensitive to hydrostatic pressure over a range of more than 400 MPa. Axial load was measured using an internal load cell that permitted stress measurements to an accuracy of better than 0.5 MPa. Axial loading and confining pressure regulation was achieved by computer-controlled electromechanical servo-systems.
As a means of testing rock frictional sliding, the use of an inclined saw cut is imperfect. If confining pressure is kept constant and axial load is increased, as the resolved shear stress increases so too does the resolved normal stress. This means that any tendency towards displacement hardening during sliding will be exacerbated, and the shear and normal stresses will migrate along the frictional sliding line as the test progresses. The problem can be overcome by servo-controlling the confining pressure so that the resolved normal stress on the slip surface is kept constant. This was done in the most of the present experiments.
A second problem is that the displacement of the two halves of the specimen
changes the surface area of contact across the slip plane. Opinions vary
about how to deal with this effect, but if friction is measured at 0 or small
displacement of the forcing blocks relative to each other, the effect is minimized. This is feasible particularly in the case
of sliding tests on 45
A third problem is that sideways displacement of the specimen halves induces
a bending moment in the axial column, which increases both the normal and
shear stress resolved across and along the slip surface. The problem was
analysed by Mackwell and Paterson (2002). The additional lateral force across
the specimen,
Some of the results for ultimate strength of Pennant and Darley Dale sandstones in extension and shortening expressed as Mohr circles at failure. Also shown are resolved normal and shear stresses at the point of failure on the fault planes that formed, the orientations of which are half of the angle subtended by the dashed lines with the abscissa. The Mohr envelopes are not shown, but they would lie at higher stresses than the best fits to the resolved stresses on the incipient fault planes. Fault angles are systematically larger in shortening than in extension, and the angles subtended by the normals to the Mohr envelopes are approximately half-way between these extremes.
Photographs of fresh faults in shortened cylinders of
Optical photomicrographs (crossed polars with gypsum plate) of
faulted, initially intact samples of Darley Dale sandstone
The results of all mechanical tests are summarized in Tables 1 and 2. Intact cylinders of both sandstones were tested until shear failure both in axisymmetric extension and in shortening, and displacement was continued so that the resistance to frictional sliding on the fault plane produced could be established. A second suite of tests was performed on saw cut samples to determine friction coefficients in both extension and shortening.
Tests on intact rock cylinders. Fault angle is measured from
cylinder axis. Axial displacement rate is 0.05 mm min
Frictional sliding data and saw cut specimens, with an axial displacement rate
Figure 4 shows results for intact cylinders in extension and shortening for
both rock types, as differential stress at failure versus mean stress (to
permit meaningful comparison between shortening and extension tests) and also
as shear stress resolved along the plane of the fault produced versus
resolved normal stress. Figure 5 shows experimental results for intact rock
failure expressed as Mohr circles and also compared to shear stress at peak
strength resolved along the fault plane produced versus resolved normal
stress. The angles of the fault planes produced, with respect to
Preparatory to the formation of the fault plane, intact rock samples tend to
develop arrays of intragranular cracks that form parallel to maximum
compressive stress. To determine whether some systematic refraction of
Where possible, saw cut samples were tested both in extension and shortening.
There was no difference in the friction coefficients calculated as resolved
Shear stress versus shear displacement curves for Pennant sandstone
sample Pa1a1 saw cut at 45
For some combinations of saw cut angle and rock type, the formation of fresh fault surfaces was observed rather than frictional sliding on an unfavourably oriented saw cut plane. This happens when the cohesive strength of the intact rock is too low. The cohesive strength of Darley Dale sandstone is about 55 % of that of Pennant sandstone; hence Pennant sandstone is better for minimizing this effect. These observations were entirely in accord with expectations from the Mohr–Coulomb description of rock failure and frictional sliding.
Figure 9 shows a compilation of frictional behaviour for both rock types
tested both in extension and in shortening, as well as over the range of all three
saw cut angles used. Frictional sliding behaviour on faults produced from
failure of intact rocks is shown in Figs. 4 and 5 and reproduced for
comparison with saw cut data in Fig. 9. For both rock types the coefficient of
sliding friction is not influenced by the orientation of the saw cut and is
within 2 SD the same as for a freshly faulted surface. It is striking,
however, that in all cases the coefficient of sliding friction is smaller in
extension than in shortening by about 25 %. In shortening, the
coefficients of sliding friction for Darley Dale and Pennant sandstones are
respectively 0.653 and 0.685. Combined they are 0.661
Frictional sliding data on saw cuts (including saw cut angles 35, 45
and 55
Axisymmetric shortening and extension represent the two end-member stress
configurations for polyaxial loading, respectively
The compilation and statistical analysis of polyaxial failure data by
Colmenares and Zoback (2002) show (a) that for most polyaxial strength data
the intermediate principal stress lies closer to
This is an example of a failure criterion that postulates that failure occurs
when some function of the stress invariants reaches a critical value (Lade,
1977; Ewy, 1999):
Test of how well the modified Lade polyaxial failure criterion
predicts the results of extension tests from the Mohr–Coulomb description of
the shortening tests for Pennant and Darley Dale sandstones. Dashed lines
link experimental results for specific extension tests to the expected
Mogi (1967) proposed a modified form of the Mohr–Coulomb criterion to
describe his data in terms of maximum shear stress and a modified expression
for the normal stress across the hypothetical failure plane, increased by a
fraction (
Fits to failure data for Pennant and Darley Dale sandstones for
Test of how well the Mogi (1967) polyaxial failure criterion
predicts the results of extension tests from the Mohr–Coulomb description of
the shortening tests for Pennant and Darley Dale sandstones. Dashed lines
link experimental results for specific extension tests (inverted triangles)
to the expected
Data for Shirahama sandstone (Takahashiu and Koide, 1989;
porosity
Polyaxial stress data at failure for Shirahama sandstone (Takahashiu
and Koide, 1989) fitted by Colmenares and Zoback (2002) to the modified Lade
criterion
Mogi (1971) also proposed a failure criterion that is a generalization of the
von Mises yield criterion:
There has generally been a lack of attention to the possible role of strength anisotropy in the determination of failure criteria under polyaxial stress states. Dehler and Labuz (2007) reported axisymmetric extension and shortening test data for Berea sandstone (22 % porosity, acoustic anisotropy at atmospheric pressure 1 %) at confining pressures up to 5 MPa, on samples cut with a cylinder axis both normal and parallel to bedding. Their data did not show any influence of bedding orientation on strength beyond the effects of experimental variability. Nevertheless, for rock types that are significantly mechanically anisotropic it may prove impossible to separate the influence of anisotropy from obtaining a failure criterion. This is because the orthorhombic symmetry of the stress tensor combined with a different symmetry for strength variations arising from anisotropy means that, for example, equivalent tests cannot be carried out, say, in extension and shortening. This is illustrated in Fig. 14 for the case of different forms of transition from axisymmetric shortening normal to bedding to axisymmetric extension, whilst keeping the bedding orientation fixed. This may lead to different forms of failure criteria being required for different rock types or even for the same rock type.
Illustration of different polyaxial loading test sequences for a transversely isotropic rock with constant bedding/foliation orientation. Relative lengths of arrows indicate relative values of principal stresses. Potential fault plane orientations and slip senses are indicated. Upper pathway corresponds to that used in the present experiments. Axial stress changes from maximum to minimum, and the relative values of the principal stresses change with respect to the transversely isotropic plane. Lower pathway corresponds to the sequence that might be employed in `true' triaxial loading. Here, maximum stress can always be normal to the plane of transverse isotropy, but contractional loading is applied across the foliation whereas in the upper path constriction is parallel to the isotropic plane.
A further well-known complication arises when rocks are strongly anisotropic and the bedding/foliation plane (weak plane) is variously inclined to the principal stress directions (Jaeger, 1960; Donath, 1961; Smith and Cheatham, 1980; Ambrose 2014). There is a range of bedding/foliation plane orientations with respect to the principal stress orientations such that shear failure with frictional sliding can occur preferentially on the weak plane, even under simple loading conditions such as axisymmetric shortening.
The friction sliding coefficient from shortening tests on Darley Dale and Pennant sandstones can be usefully compared with the Byerlee (1978) generalization (often erroneously termed “Byerlee's law”) that rock friction is to a useful extent independent of rock type. Figure 1 shows Byerlee's compilation of rock friction data from a wide range of rock types up to a normal stress of 800 MPa, leaving out those materials such as swelling clays and talc that are known to have substantially lower friction coefficients. Despite the generalization offered, it is apparent that even Byerlee's compilation shows significant variations in frictional behaviour, especially at low stresses (see also Handin, 1969). At low stresses there are also more data, which increases the likelihood that a greater spread will be observed.
Our friction data for Darley Dale and Pennant sandstones under axisymmetric shortening, together with friction data for other porous sandstones, are also plotted in Fig. 1. These clearly show a tight clustering about a friction coefficient of 0.718, slightly lower than the 0.779 mean of the Byerlee compiled data fitted to one single friction line. The sandstone data therefore support the Byerlee generalization to a useful extent, which might be applied to the modelling of the behaviour of sandstone reservoirs. Rutter and Glover (2012) also examined sandstone friction in relation to the Byerlee generalization and argued that the critical state line, separating dilatant from compactive cataclastic deformation at high pressures, is equivalent to the friction line.
The experimental frictional sliding data in axisymmetric shortening and
extensional tests on Darley Dale and Pennant sandstones showed that the
friction coefficient is higher in axisymmetric shortening than in extension.
This is counter-intuitive because friction coefficient is expected to be a
2-D concept – the ratio of resolved shear stress in the slip
direction to the normal stress acting across the slip plane and hence
should not depend upon
To show how
Jaeger (1964) describes the extension of the Mohr circle construction into
3-D. Figure 15b shows the relations that exist between the stress state and
the frictional sliding line
The slip vector is expected to be parallel to the maximum resolved shear
stress, thus in general oblique slip is expected in a 3-D stress field.
Bott (1955) and Jaeger (1964) derived equations for the resolved dip- and
strike-parallel shear stress components,
The comparative experimental results in axisymmetric extension and in shortening impact upon the ways in which experimental data are used in modelling brittle rock behaviour for geophysical and rock engineering purposes. The results of axisymmetric shortening tests on rocks to failure, and fitted to a Mohr–Coulomb failure criterion, are widely used in geoengineering under shallow conditions (excavations, tunnelling) and increasingly in hydrocarbon reservoir geomechanics (e.g. Zoback, 2007; Castillo et al., 2000). It is commonly assumed that the 2-D Mohr–Coulomb failure parameters can be applied directly to the prediction of conditions for failure under polyaxial stress states. This should give at least a conservative prediction of initial failure because it ignores the strengthening that arises with increasing mean stress as one moves away from axisymmetric compression conditions, but a more realistic result should be obtained using a polyaxial failure criterion, even if not constrained by true polyaxial or axisymmetric extension test data. Unfortunately, different polyaxial criteria can give widely different predictions, and it seems clear that a given criterion does not apply equally to all rock types, perhaps especially when rocks are anisotropic. However, as we show in Fig. 12, the few porous sandstones for which data are available seem to give consistent results using the Mogi (1967) criterion.
The interpretation of borehole breakouts, analysis of borehole stability and estimation of far-field stresses provide good examples of where the use of a polyaxial failure criterion may be important. Around a vertical borehole wall, according to the Kirsch solution, the circumferential stress varies sinusoidally (e.g. Ewy, 1999; Zoback, 2007). The limits of the width of a borehole breakout can be taken to correspond to the stress state at which failure occurs under stress conditions at the borehole wall. Initial approaches to this problem applied the 2-D Mohr–Coulomb criterion (Barton et al., 1988) assuming the circumferential stress at the onset of failure corresponded to the axisymmetric, unconfined compressive strength. It was quickly realized that the uniaxial compressive strength would underestimate the strength of the borehole wall and that a polyaxial failure criterion was required (Vernik and Zoback, 1992). The vertical stress parallel to the borehole wall is due largely to the depth of burial but is modulated by the circumferential stress through the Poisson ratio effect so that it too varies sinusoidally around the borehole wall. There may also be a radial non-zero stress component arising from mud weight, and its influence will vary according to whether or not the rock permeability allows the fluid to enter the rock pores. Thus the mean stress in the rock adjacent to the borehole wall is greater than it would be if it arose only from the circumferential stress.
Song and Haimson (1997) demonstrated experimentally for two rock types how a polyaxial failure criterion could estimate better the stress state in the borehole wall and hence provide an improved estimate of the far-field maximum in-situ stress. Unfortunately, the most appropriate polyaxial criterion to use varies with rock type, and in the absence of any experimental constraints it may not be self-evident which is the best one to use. However, as we demonstrate above, for porous sandstones the Mogi (1967) criterion may be generally useful and is simple to apply.
The stress conditions for the onset of frictional sliding are commonly taken
to impose a bound on the load bearing capacity of rock masses, whether it be
fractured and jointed rock encountered in geoengineering or modelling the
behaviour of the Earth's upper crust. The results presented here suggest that
the frictional behaviour assumed should be modified according to the nature
of the stress state, at least for porous sandstones. Under constrictional
(extensional) loading (high
Darley Dale and Pennant sandstones were tested under conditions of both
axisymmetric shortening and extension normal to bedding, corresponding to the
two extremes of loading under polyaxial stress conditions. The intact rock
strengths for the sandstones studied are best reconciled using the
Mogi (1967) criterion for failure under general triaxial conditions:
Failure under generalized stress conditions can be predicted from a knowledge of the Mohr–Coulomb failure criterion under axisymmetric compression conditions provided the best form of the polyaxial failure criterion is known. Unfortunately, a single generalized failure criterion does not appear to apply equally to all rock types. Nevertheless, where possible the appropriate polyaxial failure criterion should be used for engineering and other modelling applications.
The orientations of the fault planes produced are radically different with respect to the maximum compression direction in the two loading conditions studied. The normal to the Mohr–Coulomb failure envelope does not in either case predict the orientation of the fault planes that are eventually produced.
Frictional sliding on variously inclined saw cuts and failure surfaces
produced in intact rock samples was also investigated. Friction coefficient
is not affected by fault plane orientation in a given loading configuration,
but friction coefficients in axisymmetric extension were systematically lower
than in axisymmetric shortening. This effect may also apply to general
intermediate stress states (
Friction data for these and other porous sandstones in axisymmetric shortening accord well with the Byerlee (1977) generalization about rock friction being largely independent of rock type. For engineering and geodynamic modelling purposes the stress-state-dependent friction coefficient should be used for sandstones. It is not known to what extent this finding might apply to other rock types.
This work was carried out whilst Abigail Hackston was in receipt of a NERC research studentship. We are grateful to Experimental Officer Steve May for equipment maintenance and help with the development of techniques. Kate Brodie carried out the scanning electron microscope imaging of samples. We appreciate the helpful and constructive reviews provided by Tom Blenkinsop and Albert Griera. Edited by: F. Bastida