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Manuel Rocha Medal Recipient Rock Fracture And Collapse Under Low ...

Rock Mech. Rock Engng. (2003) 36 (5), 339–381
DOI 10.1007/s00603-003-0015-y
Manuel Rocha Medal Recipient
Rock Fracture and Collapse Under Low
Conﬁnement Conditions
By
M. S. Diederichs
Department of Geological Sciences and Geological Engineering,
Queen’s University, Kingston, Ontario, Canada
Received February 1, 2003; accepted September 1, 2003
Published online October 31, 2003 # Springer-Verlag 2003
This paper summarizes some of the work presented by Dr. M. S.
Diederichs during the ISRM Rocha Medal ceremony at the ISRM
International Symposium, EUROCK 2002, in Madeira during
November 2002. Dr. Diederichs was born and educated in Canada
and received his PhD from the University of Waterloo in 2000. He is
currently an Assistant Professor in the Dept. Geological Sciences
and Geological Engineering at Queen’s University and is a mem-
ber of the Geo-Engineering Research Centre at Queen’s and RMC
(www.geoeng.ca).
Summary
The primary objective of this work was an examination of the complimentary roles of tensile
damage and conﬁnement reduction (or stress relaxation) on excavation response of ‘‘hard’’ rock-
masses. Tensile damage and relaxation are examined with respect to structurally controlled or
gravity driven failure modes as well as to strength controlled or stress driven rockmass damage
and yield. In conventional analysis of both structurally controlled and stress driven failure, the
effects of tensile damage and tensile resistance as well as the elevated sensitivity to low conﬁne-
ment are typically neglected, leading to erroneous predictions of groundfall potential or rock yield.
The important role of these two elements in underground excavation stability in hard rock
environments is examined in detail through a review of testing data, case study examination
and a number of analytical and numerical analogues including discrete element simulation,
statistical theory and fracture mechanics. This rigorous theoretical treatment updates, validates
and constrains the current use of semi-empirical design guidelines based on these mechanisms.
Keywords: Rock fracture, strength, discrete elements, underground stability.

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M. S. Diederichs
1. Introduction
Rockmass instability in underground excavations, from an engineering point of view,
can be classiﬁed as structurally controlled gravity driven fallout or as strength controlled
stress driven rockmass yield. The dominant behaviour is a function of the relative in situ
stress and degree of jointing and fracturing in the rockmass. This work deals with both
behavioural extremes in massive to moderately jointed rockmasses with ubiquitous
structure. Instability caused by the presence of continuous faults and discrete shear
features and rock failure in squeezing conditions are not considered here.
While structural analyses normally consider full persistence of bounding disconti-
nuities, non-persistent jointing is more common at depth in hard rocks, away from
major fault or folding zones, where the mechanics of tectonic joint development are
essentially strain controlled, leading to stable fracture conditions such as those
described theoretically by Ingraffea (1987). As will be demonstrated, intact rock
bridges, in hard rock formations, need only occupy a very small percentage of the
joint-coplanar area in order to provide internal or self-supporting load carrying capac-
ity equivalent to conventional underground support systems. Consideration of this
internal support mechanism, at least for short-term or ‘‘ﬁrst-pass’’ applications, could
lead to reduced primary support requirements and more efﬁcient tunnel development.
In Fig. 1a, the joint-normal tensile strength allows load transfer normal to a wedge-
bounding surface or to laminations. This affords direct gravity support in the ﬁrst case
and effectively thickens the active beam in the second, laminated, case – increasing
stability in both situations.
Fig. 1. Issues for structural instability examined in this work: a) Residual tensile strength due to rock
bridges; b) Excavation-parallel conﬁnement (top) and abutment relaxation (bottom)

Rock Fracture and Collapse
341
Fig. 2. a) Increased conﬁnement around well-designed civil excavation; b) relaxation or conﬁnement loss
(shaded areas) due to complex mining geometries
Delayed failure in mining environments is often the result of induced abutment
relaxation. This is considered in the cases of wedge instability and fallout of blocky
rockmasses as in Fig. 1b. Changes in rock quality, excavation geometry, mining
induced stress changes or surface deﬂection can lead to relaxation induced collapse
of otherwise stable rockmasses. It is necessary to recognize the potential for this
mechanism in order to improve mine sequencing and support design to minimize
these types of failures. This is an issue of particular importance to mining. In civil
engineering applications at low or moderate depth (tunnels, caverns, etc), roof geo-
metries are typically arched to attract compression or clamping in the roof, thereby
increasing stability (Fig. 2a). In contrast, complex mining geometries, driven by
operational constraints and orebody geometries actually reduce conﬁnement and
induce structural instability (Fig. 2b).
In the case of stress driven failure, commonly applied shear-based geomechanical
constitutive models have proven to be limited in their ability to accurately represent in
situ failure of massive or moderately jointed rockmasses at depth around excavations.
Past research has suggested that the origins of compressive damage and yield in hard
rocks such as granite are tensile in nature (Brown and Trollope, 1967; Hoek, 1968;
Tapponier and Brace, 1976; Stacey, 1981; etc.), induced by extension strain normal to
the direction of maximum compression, '1. Microcracks, once initiated, tend to prop-
agate parallel to '1 or, more correctly, normal to '3. An understanding of this damage
process is essential in order to explain the observed in situ strength of hard rock-
masses. The role of internal tensile fracture and extension cracking (Fig. 3a) on rock
damage and yield under high, compressive stress is explored in this work. Of partic-
ular interest is the process of spalling around deep excavations in hard rock (Fig. 3b).
The dominant role, under low conﬁning stresses resulting from excavation, of this
form of damage initiation and propagation (Fig. 3c) in hard rock yield processes is
investigated and veriﬁed. The initiation of crack damage is relatively insensitive
to conﬁnement (I1 ¼ '1 þ '2 þ '3 or simply '1 þ '3 in two dimensions). A ratio

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M. S. Diederichs
Fig. 3. Issues for stress driven instability examined in this thesis: a) Compression-induced tension cracking;
b) Field-scale boundary-parallel spalling; c) Crack accumulation vs. propagation: yield strength and
conﬁnement sensitivity
approaching 1:1 is shown to exist between '1 (at ﬁrst damage) and applied '3 in
laboratory experimentation, model simulations and ﬁeld observations of rockmass
yield around openings (Martin, 1994, 1997). Experience (e.g. Pelli et al., 1991; Martin
et al., 1999; Castro et al., 1996) has shown that the in situ strength of the rockmass
near excavations in massive to moderately jointed granitoid rock consistently falls to a
lower bound ('1 À '3 ¼ 0.35 to 0.45 UCSlab), coincident with the damage initiation
threshold for intact rock samples (Fig. 4). Other rock types have similar ratios between
0.3 and 0.6 (as in Brace et al., 1966; Martin 1994; Eberhardt et al., 1998 and Fonseka
et al., 1985, for example). A primary goal of this work is to explain this observation
through a detailed examination of the damage process.
Damage initiation, accumulation and interaction are shown to be predictable pro-
cesses controlled by material properties. These are primarily tensile processes; with
shear mechanisms becoming important only after sufﬁcient tensile damage accumula-
tion and interaction has occurred. While crack initiation is only marginally sensitive to
conﬁning stress, crack propagation, a requisite process for macroscopic spalling, is
highly sensitive to the low conﬁnement conditions (as in Hoek, 1968) near an excava-
tion boundary (Fig. 3c). In such environments, this results in a reduction in in situ
yield strength, ultimately to a lower bound deﬁned by damage initiation.
In addition to observational and empirical evidence, numerical experimentation
based on simple behavioural analogues was employed in this work to illuminate key
aspects of tensile strength and relaxation. A semi-analytical voussoir model (Fig. 5a)
for a jointed beam, corroborated using a discrete element simulation, was modiﬁed to
account for interlaminate tensile strength and for abutment relaxation. It was then
applied to the study of structurally controlled instability around tunnels and under-
ground mining stopes. A bonded disc contact model (Fig. 5b) was used to explore

Rock Fracture and Collapse
343
Fig. 4. Empirical threshold for in situ damage in moderately jointed hard rockmasses
Fig. 5. Behavioural analogues: a) voussoir beam; b) bonded contact discrete elements

344
M. S. Diederichs
aspects of grain-scale tensile damage accumulation under both macroscopically ten-
sile and compressive conditions. Other investigative tools were used in this work to
complement these analogues including case study and laboratory test evidence, ana-
lytical relationships based on fracture mechanics, and a new statistical model devel-
oped herein for damage accumulation. In both structurally controlled and stress driven
failure environments, the inﬂuence of tensile damage and relaxation have been quan-
tiﬁed and used to explain observed behaviour and validate empirically based design
guidelines.
2. Structurally Controlled Gravity Driven Modes
2.1 Rockmass Residual Tensile Capacity
A primary function of artiﬁcial rock reinforcement is to retain the rock’s self-support-
ing capacity. If shear strain is dominating the rock mass behaviour, reinforcement can
improve the self-supporting capacity by maintaining interlock and by suppressing
dilation. In highly stressed rock, fractures form from nucleation cracks and ﬂaws,
creating surface-parallel fractures. While crack initiation is a small strain phenom-
enon, relatively insensitive to conﬁnement, signiﬁcant rock mass degradation as a
result of fracture propagation can be effectively arrested by stiff reinforcement com-
ponents that prevent the opening of fractures.
Stiff reinforcement can also preserve rock bridges formed by incomplete joint
plane formation. Joints are often assumed to be fully persistent for stability analyses
even though this is normally not the case in moderately jointed rock masses. Joints are
often ﬁnite in dimension or are punctuated by bridges of incomplete separation. Where
these rock bridges exist and where they can be preserved by careful blasting and the
use of stiff tendon reinforcement, the self-supporting capacity of the rock mass,
through these rock bridges, can be quite signiﬁcant under tensile loading as shown
in Fig. 6. These values for rock mass tensile strength were calculated using a fracture
mechanics approach modiﬁed here after Kemeny and Cook (1987) as illustrated in
Fig. 7 and detailed in Appendix I.
A detailed comparison of residual tensile capacity to conventional support systems
is given in Table 1 based on a gravity loaded mass such as the non-sliding wedge
shown in Fig. 1a. Of particular interest here is the very small relative area of rock
bridging required to provide capacity equivalent to practical support systems. Note
that a 1% intact rock bridge area corresponds to a 10 cm by 10 cm intact bridge within
a 1 m by 1 m joint plane. For the 2D (prismatic) wedge in Fig. 1 above a 10 m span, a
standard pattern of rebar reinforcement will not support the wedge (factor of safety,
fos < 1) while rock bridges accounting for less than 0.5% of the joint plane (99.5%
fracture development) will be sufﬁcient to support the wedge (factor of safety ¼ 1.1).
For the simplest of three-dimensional joint distribution assumptions (rectangular
islands on rectangular joint surfaces), an observed linear joint trace persistence of 90%
corresponds to a relative intact area of 1%. That is, if careful joint trace mapping in the
roof of an excavation reveals joint traces 90 cm long with collinear 10 cm intact
intervals, the residual tensile capacity of the rockmass normal to this joint set can
be estimated to be just slightly less than that of a standard double strand cablebolt

Rock Fracture and Collapse
345
Fig. 6. a) Reduction in rock tensile strength due to isolated cracks and residual tensile capacity due to rock
bridges. Compare residual strength for 90% cracked area (10% rock bridges) with average distributed
capacity of conventional support systems. Left and right extremes are calculated using the respective models
in Fig. 7 using fracture properties for granite. See Appendix I for details
Fig. 7. Schematic of crack (left) and rock bridge (right) models used to generate Fig. 6 and Table 1
Table 1. Support patterns (capacities from Stillborg, 1994) and equivalent rock bridge area (effective residual
tensile capacity)
Support type
Support
Equivalent
Maximum
Capacity equivalent
pattern
pressure
supported
rock bridge area
mxm
thickness
(% cross section)
Rockbolts
2 Â 2
20 kPa
0.7 m
0.1%
Rebar
1.3 Â 1.3
60 kPa
2.0 m
0.4%
Double Strand Cablebolts
2 Â 2
130 kPa
4.3 m
1.2%
Double Strand Cablebolts
1.3 Â 1.3
300 kPa
10 m
4.0%

346
M. S. Diederichs
Fig. 8. Independent voussoir beams (left); beam ‘‘stacking’’ due to rock bridge tensile strength (right)
pattern (2 Â 2 m). If the degree of joint development (fracture persistence) could be
accurately evaluated, this would lead to signiﬁcant reduction in short term sup-
port requirements for mine development and tunnelling, facilitating optimized
excavation=support cycling and more rapid advance. In a dynamic environment, of
course, these rock bridges cannot be considered for long term natural support as
humidity and load cycling effects will lead to further propagation and rock bridge
reduction. For primary, at-the-face support, however, these ﬁndings have real economic
value.
Another demonstration of the impact of rock bridges and residual joint-tensile
strength involves the voussoir analogue for a jointed beam or more generally a blocky,
regularly jointed rockmass adjacent to an excavation as shown in Fig. 8. The original
voussoir formulations of Evans (1941), Beer and Meek (1982), Brady and Brown
(1993) have been updated with further enhancements and modiﬁcations developed
in this work. The quantitative results in Fig. 9 are obtained by incorporating the
inﬂuence of interlaminate strength into an iterative voussoir beam model.
In this analogue the rock beam cannot carry lateral tensile stresses due to beam-
normal jointing. As a result the standard elastic beam formulation (as in Obert and
Duvall, 1966) is not valid. The voussoir solution iteratively solves for moment balance
based on a compression arch of initially unknown thickness, stress magnitude and
deﬂection. The general voussoir solution is described in detail in Diederichs and
Kaiser, 1999a (and summarized in Appendix II).
Rock bridge internal support is accounted for by relating the minimum support
pressure or distributed capacity required to couple adjacent voussoir roof laminations
together into a composite voussoir beam. Increased rock bridge capacity results in a
thicker composite beam which enhances stability as a higher order relationship (com-
pared to support pressure alone). Figure 9 is generalized to be independent of initial
lamination thickness.

Rock Fracture and Collapse
347
Fig. 9. Impact of residual rock bridges (as in Fig. 8) on critical span for blocky rockmasses based on
modiﬁed voussoir analogue (fractured granite, rockmass modulus ¼ 10 GPa). Lamination thickness here is
assumed to be minimal (actual thickness would limit lower bound)
These rock bridges, while unstable under sustained tensile load beyond the critical
levels, can yield in a stable fashion if the tensile strains are controlled (Ingraffea,
1987). A stiff reinforcement normal to these bridges acts to limit the propagation of
the fractures and the rupture of the rock bridges. It is prudent to include stiff, fully
coupled reinforcement components such as resin-grouted rebar in a composite support
system. Even if a rebar breaks in discrete locations along the shaft, the remaining
segments continue to act to suppress shear localization and to preserve rock bridges,
lessening the demand on the holding components of the support system.
The preservation of rock bridge capacity in hard rock masses is particularly
important at the excavation face. If careful blasting is employed, the installation of
full support can be delayed. Perhaps only a spray-on lining or a thin shotcrete layer is
required for worker safety at the heading. Without the installation of stiff tendon
reinforcement, the rock bridges may eventually rupture, freeing unstable wedges
and laminations. In many cases, however, the installation of major reinforcement
can be delayed several rounds. In a mining or tunneling environment, this allows
for the simultaneous installation of permanent support along with heading develop-
ment and drilling for the next round (the two crews now occupy different space in the
tunnel). In a typical mine, this represents a time saving equivalent to a full crew-shift
and has a signiﬁcant economic impact for mine development.
2.2 Abutment Relaxation
Stress paths are often highly complex around underground mine openings, involving
both elevated as well as highly reduced or even tensile stresses, often in the same
location at different mining stages. High compressive stress is normally associated
with a higher potential for failure of underground openings as evidenced by the
consideration of high stress in conventional empirical design tools. Local stress reduc-
tion or relaxation, however, can occur both normal and tangential to excavation

348
M. S. Diederichs
Fig. 10. Relaxation of surface parallel conﬁnement, resulting in wedge failure. Sample wedge calculation
illustrating support equivalency of minimal conﬁnement
boundaries and can signiﬁcantly reduce the inherent stability of the rockmass causing
failure of the excavation wall. Typical limit equilibrium calculations (e.g. for wedge
stability) or empirical design techniques for the assessment of stope wall stability
often neglect the impact of abutment conﬁnement or relaxation.
Relaxation, as discussed here, refers to the removal of compressive stress in the
vicinity of and in a direction parallel to the surface of an excavation wall or roof as
illustrated in Fig. 10. The wedge illustrated, bounded by continuous joint surfaces, is
typically assumed to be free falling. At depth in conditions such those found in the
Canadian Shield, horizontal compressive stresses ﬂowing around this isolated excava-
tion generate frictional strength on the joint surfaces. The relaxation or removal of this
clamping stress, due to subsequent mining of nearby stopes, for example, will cause
the delayed failure of this wedge, if deadload support is not provided.
In the right-hand example in Fig. 10, the increase in support pressure (distributed
support capacity) required to replace the loss of relatively minor conﬁnement for an
example wedge (20 m span) is quantiﬁed:

%'hS2
Ks cos 2þKn sin 2
sinð0 À Þ þ F
4 tan
Ks cos cos 0þKn sin sin 0
F:S: ¼
;
ð1Þ
%S3
24 tan
where S is the span, is the rock unit weight, Kn, Ks are the joint normal, shear
stiffnesses (normally assume Kn ) Ks or simplicity), is the half cone angle, 'h is
the average horizontal stress across the wedge, 0 is the joint friction angle and F is the
total bolt load. It can be seen in this example that for the wedge in question, a
minimum lateral conﬁning stress of 2 MPa, acting across the back of the drift, is
required for stability. This minimal excavation stress is present even at depths of less
than 100 m. If relaxation (conﬁnement loss) occurs, however, as a result of geometry

Rock Fracture and Collapse
349
Fig. 11. Horizontal stresses above 6 Â 4 m drift before (top) and after (bottom) mining of adjacent stope.
Mining of the stope causes a drop in tangential stress, above the drift roof, from 80 MPa to À5 MPa. This
calculated tension (elastic) would manifest as open joints and block=wedge unravelling. (In situ vertical
stress 25 MPa; horizontal stress 50 MPa)
changes due to nearby mining, (as in Fig. 11), this wedge requires a heavy support
system approximately equivalent to a standard pattern of double strand cablebolts.
From a practical perspective, it is necessary to be aware of this relaxation=
conﬁnement effect so that support systems can be economized where clamping is
available and enhanced where relaxation is expected. This selective approach leads
immediately to signiﬁcant cost reductions (less standard support) and greater safety
(more support where needed).
Data collected from mine sites in Sudbury (Hutchinson, 1998) indicates that sup-
port requirements in isolated and stable mine drifts (i.e. removed from mining blocks
or complex geometries) are signiﬁcantly less than that predicted by a typical ubiqui-
tous joint and wedge or beam analysis. On the other hand these steeper wedges can
still be present and are likely to be released if clamping stresses are reduced. Likewise,
blocky rockmasses are more likely to unravel in these areas. Stope access drifts and
cross-cuts are particularly susceptible.
Extreme relaxation occurs when the absolute boundary-parallel strain (compres-
sion positive) drops below zero (datum at zero stress). This manifests itself as the
opening of joints and is equivalent to an outward absolute displacement of the abut-
ments or as an equivalent boundary parallel tension in an elastic stress model. In the
example illustrated in Fig. 11, a previously isolated drift experiences the effects of

350
M. S. Diederichs
Fig. 12. Typical laminated hangingwall rockmass (voussoir) schematically simulated with discrete elements
(left); cablebolt array in hangingwall (Winston Lake Mine, Ontario, Canada) before and after stoping
advance (right). Abutment relaxation due to increased deformation into cross-cuts (‘‘x-cut’’) lead directly
to wall failure (modiﬁed after Diederichs and Kaiser, 1999a)
nearby mining. One effect of the stope is to reduce the horizontal stresses in the back
of the drift (to tensile values in this elastic example).
The complexity of typical ore zones and the operational constraints inherent in
mining often result in groupings of multiple openings favouring the creation of relaxa-
tion zones. Blocky rockmasses within these zones are subject to unraveling or struc-
tural collapse. In addition to safety concerns, this failure leads to production delays
(due to oversized muck) and to costly dilution. Stope geometry and operational deci-
sions can also lead to abutment relaxation and hangingwall failure as shown in Fig. 12.
In this example the advancing stope is oriented, with respect to in situ stress, such that
hangingwall conﬁnement is reduced. Coupled to this is the creation of a cross-cut to
the bottomsill which locally increases the downward displacement of the lower hang-
ingwall abutment. The combined effect results in the failure illustrated. Discrete
element simulations showed that the hangingwall was indeed stable until the lower
abutment was softened to simulate the creation of the cross-cut. Aspects of this case
study are described by Diederichs and Kaiser (1999a, b) and also by Maloney and
Kaiser (1991) and Kaiser et al. (2001).
In order to quantify this effect and incorporate relaxation into existing design
tools, the voussoir beam analogue presented by numerous authors in the past was
updated and modiﬁed to account for abutment relaxation. Extreme relaxation, equiva-
lent to boundary-parallel tensile stresses in an elastic model, can be represented as the
outward displacement of the abutments, a, as shown in Fig. 13. This results in
increased beam deﬂection at equilibrium and smaller critical spans for a given beam
thickness, T, and rockmass modulus, Erm. Using the modiﬁed voussoir model devel-
oped here, this effect is demonstrated in Fig. 14.
In order to apply this new analogue to design, the Modiﬁed Stability Graph, now in
common use in Canadian mining for stope dimensioning, is further adapted for relaxa-
tion. The original technique has been well documented in the literature (Bawden,
1993; Diederichs and Kaiser, 1999b; Potvin and Milne, 1992; Hutchinson and
Diederichs, 1996; etc.). The method compares the ratio of stope face area=perimeter

Rock Fracture and Collapse
351
Fig. 13. Elastic tension (continuum model) equivalent to 1 mm of outward abutment displacement due to
relaxation (modiﬁed after Diederichs and Kaiser, 1999a)
Fig. 14. Reduction in critical unsupported span due to relaxation for rockmass modulus Erm ¼ 10 GPa. (left)
and lamination thickness, T ¼ 1 m
(‘‘hydraulic radius’’) to a rockmass stability number N, a direct product of the factors:
‘RQD=Jn’ (block size); ‘Jr=Ja’ (joint surface condition); factor ‘A’ accounting for
induced compressive stress=strength; factor ‘B’ accounting for joint-face interaction
angle and ‘C’, a factor accounting for face and structure inclinations with respect to
gravity. The ﬁrst two terms are directly from the Q system (Barton et al., 1974) while
the others are summarized in the literature previously listed (e.g. Bawden, 1993).
Factor A is of particular importance here and is illustrated in Fig. 15. This factor
considers only the increase in compression tangent to the stope wall and does not

352
M. S. Diederichs
Fig. 15. The relative stress factor included in the Modiﬁed Stability Number, N0 (after Hutchinson and
Diederichs, 1996)
Fig. 16. a) Unsupported stope database and resultant no-support limit for Modiﬁed Stability Chart (data
from Potvin, 1988 and Nickson, 1992); b) Calibration of voussoir model (relating key model parameters to
N0). HR is directly related to nominal span
include the effects of relaxation. The ﬁnal no-support stability limit (upper bound in
this case) in Fig. 16a is the result of calibration based on the original database of
unsupported stopes by Potvin (1988).
An interpretation of the voussoir analogue with respect to the no-support limit is
shown in Fig. 16b. Here the model is assumed to have near-rigid abutments. The
voussoir model developed here is calibrated with respect to rockmass factor N0 (result
of rockmass classiﬁcation, stress and joint factors) and stope geometry factor, HR,
using a multi-parametric procedure described in Diederichs and Kaiser, 1999a. Above
the no-support limit the voussoir beam model is stable. Below the limit, the beam
model is adjusted so as to fail (Fig. 16b).
Now the effect of abutment relaxation can be assessed and the impact on the
empirical no-support limit can be estimated (recall that the factor A for stress effect

Rock Fracture and Collapse
353
Fig. 17. Adjusted no-support limits for Stability Graph based on relaxed stope walls. Data points show
independent veriﬁcation case histories. Backs are in compression while hangingwalls are relaxed
does not consider loss of conﬁnement or elastically calculated tensile stress). Using
this calibrated model and the relationship between abutment displacement and calcu-
lated elastic tensile stress in Fig. 13, the revised no-support limits shown in Fig. 17 are
obtained.
The data points in Fig. 17 represent an independent veriﬁcation set based on
documented case histories from Bawden (1993) and from (Greer, 1989). The hanging-
walls in question were relaxed (showing nominal tensile stress in a 3-dimensional
elastic model) while the stope backs were under compression. The original no-support
limit does not accommodate this relaxation. The revised no-support limits (calculated
according to the equivalent relaxation stress) show a considerable predictive improve-
ment. The relaxation adjustment is integrated into the N0 factor (through a modiﬁed
stress factor, A) for future design:
À
Á
'T À'0
A ¼ 0:9e11 UCS
for 'T < 0;
ð2Þ
where 'T is the minimum tangential stress and '0 is an offset term normally set to
zero. In this relationship, the deﬁnition of tension may require adjustment due to the
accuracy of elastic models in the near-zero stress region. Experience with three-
dimensional direct boundary element formulations (linear elements) has shown that
modelled stresses up to 5 MPa (positive compression) can be indicative of zero stress

354
M. S. Diederichs
or tension. For these analyses set '0 to a positive offset between 0 to 5 MPa. Simple
trial experimentation is recommended for any numerical scheme prior to design
application.
In addition to the effects of relaxation on rockmass integrity, the effects on support
performance can also be compromised as in the case of plain strand cablebolts and
other frictional support systems. The reader is referred to Hutchinson and Diederichs
(1996), and Kaiser et al. (1992) for detailed discussions of this problem.
The examples presented here highlight the importance of abutment relaxation as a
destabilizing mechanism. An understanding of this mechanism enables the engineer to
optimize mine sequencing and stope geometries to preserve clamping as well as to use
more discretion in support speciﬁcations for isolated drifts and for near-stope accesses,
thereby increasing development economy and safety. Using a calibrated voussoir
beam approach, the empirical stability graph for stope design has been updated to
account for relaxation across stope backs.
The importance of both tensile damage mechanics and conﬁnement reduction have
been clearly demonstrated for the regime of structural instability. A survey of report-
able incidents in hard rock mines in Sudbury (Hutchinson, 1998), indicated that
structural failure accounted for approximately 70% of reported incidents (in access
drifts) in hard rock mines in the Sudbury area. Many of these occur in near-mining
zones where relaxation is dominant. These reportable incidents refer to falls some
time after the initial development (groundfalls at the development face were not
counted unless extremely severe). It can be assumed then that stable or subcritical
fracture propagation (as in Atkinson and Meredith, 1987) or dynamic fatigue leading
to rock bridge degradation may have played a part in some of these failures. The
prudent response to these incidents is to increase standard support for all development.
Clearly this is a needless expense if the mechanisms of failure (relaxation and rock
bridge destruction) are not prevalent, as in major isolated development away from
active mining. An appreciation of these mechanisms leads directly to support optimi-
zation in such cases.
3. Stress Driven Damage and Failure Modes
In combination, tensile damage and conﬁnement reduction also play important roles in
the stress-induced yield of hard rock at depth. Tensile micro-cracking, exacerbated by
low conﬁnement conditions near excavations, leads to a unique failure process (slab-
bing in the extreme) that is inconsistent with conventional shear based failure criteria.
Rock mass strength near underground excavations is controlled by damage initiation
mechanisms that are relatively insensitive to conﬁnement and by fracture propagation
(extension) mechanisms that dominate at low conﬁnement. For brittle rock, the
strength envelope can be represented by a multiphase linear failure envelope illus-
trated in Fig. 18. This envelope will be shown to be the result of the mechanics of
tensile fracture accumulation and propagation and the reduction of the yield surface to
the damage initiation threshold as conﬁning stresses are relaxed.
Below the ‘‘damage initiation threshold’’ the rock is not damaged and remains
undisturbed. When this threshold is exceeded, seismicity (acoustic emissions) is

Rock Fracture and Collapse
355
Fig. 18. Schematic of failure envelope for brittle failure, showing four zones of distinct rock mass failure
mechanisms: no damage, shear failure, spalling, and unraveling. 'c is the unconﬁned compressive strength
(UCS) of laboratory samples
observed and micro-crack damage accumulates, leading to a critical crack intensity for
crack interaction and coalescence resulting in macro-scale shear failure if the con-
ﬁnement level is sufﬁciently high (e.g., in conﬁned cylindrical test samples). When a
stress path moves to the left of the low conﬁnement zone, (into the zone marked
‘‘spalling failure’’ in Fig. 18), and exceeds the damage initiation threshold, individual
cracks can propagate beyond grain boundaries leading to macroscopic axial splitting
or spalling normal to the minor principal stress. As a result, the observed in situ rock
mass failure stress is signiﬁcantly lower than predicted from laboratory tests as fewer
cracks are required for yield. In cylindrical lab samples under axisymetric compres-
sion, spalling failure is retarded due to the geometric constraints and the theoretical
yield strength (‘‘long term strength’’) is achieved with some additional system depen-
dant strain-hardening up to peak (not shown in Fig. 18). This will be discussed further
in the following sections.
3.1 Bounding Limits for In Situ Strength
The lower bound in situ threshold for damage, discussed in Section 3.2, is deﬁned by
the limit for damage initiation in laboratory samples. For example, Hommand-Etienne
et al. (1995) showed that the criterion for damage initiation in Lac du Bonnet granite
was approximately:
'1initiation % A Á UCS þ B'3;
ð3Þ
where A ¼ 0.33 and B ¼ 1.5. Brace et al., 1966, showed a similar relationship for
Westerley Granite with a A ¼ 0.3 and B ¼ 1.4 while Pestman and Munster (1996)

356
M. S. Diederichs
Fig. 19. Increase in crack propagation with reduced conﬁning stress. Tensile stresses lead immediately to
unstable propagation and spalling. Low conﬁnement encountered in near-excavation domains leads to
spalling and strength reduction near the excavation boundary with a lower limit deﬁned by crack initiation.
(a is the new extending crack length while c is the initiating ﬂaw or limiting grain dimension)
demonstrated a slope, B, of 2 for the initiation threshold ('1 vs '3) for sandstone.
Interestingly, this is consistent with the so-called ‘‘serviceability limit’’ for concrete
(Illston et al., 1979) which suggests a design limit according to Eq. (3) with A ¼ 0.3
and B ¼ 2. Below this limit no cracks can form and no yield can occur.
It will be shown that the upper bound for in situ strength, examined in Section 3.3,
is deﬁned as the stress threshold at which non-propagating cracks (cracks which
initiate but do not propagate beyond one grain dimension) accumulate to the point
that crack-crack interaction is inevitable and localization ensues. This threshold cor-
responds to the long-term laboratory strength (as tested in Martin, 1994).
The transition between the upper bound (crack interaction) and the lower bound in
situ strength (damage initiation) in Fig. 19 is controlled by the ‘‘spalling limit’’. This
is a conﬁnement ratio limit ('3='1) below which uncontrolled crack propagation can
occur. In other words, intragranular cracks propagate beyond the grain or crystal
boundaries and become macro-cracks or spall fractures (Fig. 19). Hoek (1968) had
suggested that this occurs below a ratio of approximately 0.1 to 0.05.
3.2 Field Evidence for a Lower Bound Damage Initiation Threshold
Pelli et al. (1991) showed that in order to ﬁt the Hoek-Brown criterion to observed
tunnel failures, the value of m had to be reduced to unconventionally low values.
Martin et al. (1999) found that m should be close to zero for hard rock. Similar
ﬁndings were reported by Castro et al. (1995) who showed, using back-analyses of
brittle failure, that stress-induced fracturing around tunnels initiates at approximately
0.3 to 0.5 'c (s ¼ 0.10 to 0.25) and that it is essentially independent of conﬁning stress.
In Diederichs (2000) additional cases were developed to illustrate this effect including

Rock Fracture and Collapse
357
Fig. 20. Perspective view of Creighton Mine Deep Levels. The upper horizon shown is 6400 level. The
upper most VRM stope block as labeled is the study area
observations at Creighton Mine in Sudbury, Canada. Creighton Mine is the largest and
deepest operating nickel mine in North America. It began as an open pit operation in
1901, progressing underground using a variety of mining methods over the next 90
years. In the deep levels of Creighton Mine, illustrated in Fig. 20, the primary mining
method up until the mid 1980’s was Mechanical Cut and Fill (MCF).
It was decided in the mid 1980’s to experiment with the use of the vertical retreat
method, VRM, which is now routinely used throughout the Canadian mining com-
munity. This method had the advantage of removing miners from the open stope for
most of the mining cycle and did not create horizontal sills. The stopes were to be
mined incrementally in 13 m square panels. Ultimately these panels would be mined
full height (65 m). During the trial documented here, however, the panels would be
mined vertically between the topsill and the top of the uppermost MCF horizon as
shown in Fig. 21. Due to the geometric complexities, stresses are calculated with a
3-Dimensional elastic boundary element model (Map3D – Wiles 1996) of the full
Creighton Deep Zone.
In situ stresses (linearly varying with depth) at the 6700 level are '1 ¼ 82 MPa,
'2 ¼ 62 MPa, and '3 ¼ 53 MPa. '1 is horizontal and approximately east west, while '3

358
M. S. Diederichs
Fig. 21. Left: Map3D model geometry for completed MCF and VRM stopes for Creighton deep (to 7200 ft
level). Arrow shows the 6600–6700 VRM block. Middle: Major principal stress around the completed
66–6700 VRM (West). Right: Stope extraction sequence for study block (plan view)
is vertical (in the absence of mining). The stopes in ﬁgure were monitored (as in
Landriault and Oliver, 1992) using an extensive cluster of extensometers, stress cells
and borehole camera devices. The borehole camera information was used in combina-
tion with drilling logs (ﬁnes, ease of drilling, borehole breakout or visible closure,
stuck rods, etc.) from the blastholes in neighbouring panels to delineate the extent of
the damaged or yielded zone. Zones of actual observed yielding are shown for three
mining stages in Fig. 22.
In this study, for each simulation of 3-D elastic stresses around the completed
mining geometry, a grid of points was overlain in the unmined zone and was sampled,
recording the calculated stress state and noting whether the point fell inside or outside
the observed yielded zone. The results are shown in Fig. 23. The ﬁlled squares
(‘‘yielded’’) indicate points inside the observed yielded zone while the circles
(‘‘intact’’) represent stress points outside this zone.
The lower bound yield stress for low and moderate conﬁnement ('3) follows the
relationship:
'1 ¼ ð90 to 100ÞMPa þ ð1:0 to 1:2Þ'3;
ð4Þ
where 95 MPa is approximately 0.4 to 0.5 times the UCS of the intact Creighton
granite and norite respectively (250 MPa and 190 MPa as speciﬁed by Wiles, 1989).
This lower-bound in situ threshold is consistent with the laboratory threshold for
damage initiation. This relationship scales up in this way due to the moderately jointed
character and high integrity of the rockmass (joints are tightly closed and do not
impact on the rockmass strength at this scale of observation).
3.3 A Bonded Solid Analogue for Crack Accumulation and Interaction
Returning to a laboratory scale, in order to better understand the relationship between
crack initiation (lower bound strength) and crack interaction (upper bound in situ
strength), a discrete element simulation based on elastic bonded discs and breakable

Rock Fracture and Collapse
359
d
groun
yielded
of
ations
observ
ents
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Hatched
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1992
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of
ations
Observ
22.
.
Fig

360
M. S. Diederichs
Fig. 23. Correlation between observed damage and elastic stress calculations at randomly sampled locations
in the model of Fig. 21. Damage threshold (minimum major stress) corresponds to a Hoek-Brown envelope
with m ¼ 0
contact bonds is used to investigate damage propagation in heterogeneous solids. The
model, based on the user-modiﬁable PFC code (Itasca, 1995) considers rock as a
heterogeneous assembly of discs bonded together at contacts with elastic springs
resisting interparticle shear and normal translation. As a result of heterogeneity,
stresses are carried through the sample as a tortuous network of contact forces. Under
conﬁned compression, this tortuousity results in numerous contact bonds with tensile
forces as well as those with compression (Fig. 24). The sample is formed by creating a
random assembly of particles with varying radii and inﬂating the particles until max-
imum contact density is achieved. At this point bonds are formed and normally
distributed contact stiffnesses and strengths are assigned. In a biaxial sample, wall
conﬁnement is achieved during testing with rigid walls or ﬂexible ‘‘membranes’’.
Simulated rigid top and bottom platens converge, resulting in increasing deviatoric
stresses. When either a tensile normal-force or a shear-force limit is reached, the
bonds break and cannot carry tension thereafter (frictional sliding is still resisted).
Cracks accumulate as irreversibly broken bonds until sample failure occurs.
At the scale of a single crystal grain, the nominal tensile bond strength used was
one quarter of the shear strength (based on results from experimental fracture
mechanics by Okubo and Fukui 1996 and Laqueche et al., 1986), and therefore
dominated the local damage process in unconﬁned and conﬁned test simulations, even
though the eventual failure mode resembled macroscopic shear (Fig. 25). A typical
axial stress versus axial strain curve from these simulations is shown in Fig. 26.

Rock Fracture and Collapse
361
Fig. 24. Disc arrangement (simple contact bond model) and contact forces for a random assembly
Fig. 25. Conﬁned compression test on a bonded disc sample. Cracks show as segments normal to ruptured
bond. Right-hand image is a manually generated schematic of major rupture coalescence based on individual
cracks in middle ﬁgure

362
M. S. Diederichs
Fig. 26. Typical conﬁned ('3 ¼ 20 MPa) compression response (same sample as Fig. 25). Shear=normal
strength ratio ¼ 4; 7200 discs and 16000 contacts. Mean normal bond strength ¼ 0.3 MN
The stress-strain curve shows the characteristic damage initiation at about 0.3 to
0.4 of the peak strength and rapid strain softening immediately after peak. Note that
even though the sample is conﬁned with 20 MPa, the total amount of tensile cracking
dominates shear cracking by a ratio of approximately 50:1. In the extreme, if shear
bond failure is prevented completely (only tensile bond failure allowed), coalescence
of exclusively tensile cracks still results both in realistic spalling (Fig. 27b) and in
shear=rotation zones reminiscent of macro-shear failure as in Figs. 27c and 27d.
Heterogeneity (both in grain size and material properties) is key in generating
tensile stresses in a compressive stress ﬁeld. These tensile stresses (and strains) occur,
of course, within individual crystal grains, giving rise to crack initiation. Composite
heterogeneity, however, also results in meso-scale tensile zones spanning numerous
crystals. In these areas, crack propagation beyond the limiting grain scale is possible.
A numerical simulation (bonded disc model) is used in Fig. 28 to illustrate the gen-
eration of both grain scale tension and more regional tensile stress through hetero-
geneity. This simulation is under applied lateral conﬁnement (2.5 MPa) and a vertical
compression of 100 MPa.
While the program PFC permits the use of more sophisticated bonding models, the
bonding model used here is a simple tensile bond with no moment effects. In addition,
the discretization does not permit a singular stress concentration effect as in real
microcracks. As a result, grain-scale cracks do not readily propagate in isolation using
this simple contact bond model. As the simulations are performed with the element
size equated to the mean grain size, crack blunting caused by crystal=grain boundaries

Rock Fracture and Collapse
363
Fig. 27. Macro failure zones in discrete element model: a) and b) discs and cracks for unconﬁned sample;
c) '3 ¼ 20 MPa; d) '3 ¼ 60 MPa. All micro-damage is tensile – shear bond failure is prevented. Shear zones
result from tensile crack coalescence
is reﬂected in the model. These cracks do, however, accumulate and interact with other
cracks. After the ﬁrst crack initiation (bond rupture) a period of uniform accumulation
of isolated cracks ensues as illustrated in Fig. 29. Once two cracks interact (i.e. initiate
within proximity of each other) a meso-ﬂaw is created, spanning two to three grain
dimensions. This ﬂaw is now large enough to spontaneously drive failure of adjacent
particle bonds. Crack interaction, therefore, marks the onset of true yield in these
simulated specimens.
The model is used to convincingly demonstrate that true sample yield (deviation
from a non linear response) is coincident with the ﬁrst crack interaction (Fig. 30). This
interaction is a probabilistic phenomenon and occurs when a critical crack density is
reached. After initiation cracks begin to form oblique to the maximum compression
and the crack anisotropy declines as shown in Fig. 30. Furthermore, this critical crack
density, the onset of stress-strain non-linearity was shown to correspond to a consistent
level of lateral extension strain (Fig. 31). This is, of course, consistent with the
criterion developed by Stacey (1981) for yield deﬁned by a critical extension critical
extension strain threshold (e.g. for granite: critical extension strain "cr ¼ 0.03%,
E ¼ 60 GPa, v ¼ 0.2):
E
1 À v
'1 ¼
"
'
v cr þ
v
3 ¼ 72 þ 4'3:
ð5Þ
In the simple contact bond model used here, this density critical extension strain
was indeed independent of conﬁning stress and was even consistent in the tensile
regime (Fig. 32). The slope of the threshold deﬁned by Eq. (5) is approximately
equivalent to a Mohr-Coulomb envelope with a friction angle of 37 degrees. The
important point here is that this conﬁnement dependency for crack interaction, and

364
M. S. Diederichs
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Rock Fracture and Collapse
365
Fig. 29. Stages of damage within bonded disc model – representative of actual laboratory test sample
Fig. 30. Crack interaction indicated by rapid accumulation of neighbouring crack pairs (separation S is
equal to or less than grain diameter d), axial non-linearity (drop in tangent modulus and peak anisotropy,
(normalized second invariant of crack tensor: ¼ 0 for isotropy, ¼ 1 for parallel cracks)

366
M. S. Diederichs
Fig. 31. Constant extension strain and crack intensity (A ¼ sample area; d ¼ grain size) at the point of
critical crack interaction for conﬁned compression tests. Crack intensity at peak deviatoric stress is shown
for comparison
Fig. 32. Crack damage initiation and crack interaction (yield) thresholds for a) bonded contact simulations
(no unstable crack propagation), b) actual rock samples (unstable crack propagation in tension)
therefore for upper bound rockmass strength, is not the result of conventional sliding
friction, but rather of the elastic generation of extension strain and tensile crack
accumulation in which friction plays no part.

Rock Fracture and Collapse
367
Fig. 33. a) Generation of feedback conﬁnement during microcrack dilation; b) No such restraint in crack
near excavation wall; c) Resultant drop in in situ strength due to crack propagation near boundary
Interestingly, this critical crack intensity, critical extension strain and the linearity
crack interaction threshold in stress space are consistent for discrete element simula-
tions (using simple bond model) of Brazilian tests, conﬁned tension tests and uniaxial
tension tests (Fig. 32a). In the absence of unstable isolated crack propagation, as is the
case in the PFC models, crack accumulation and interaction must still take place as in
conﬁned compression. On the other hand, if crack propagation is permitted, a more
realistic behaviour results as in Fig. 32b. In real rock, isolated cracks propagate under
tensile conditions resulting in instantaneous failure after initiation in direct tension
tests as shown.
The mechanics of crack propagation are the key to understanding the reduction in
in situ strength below the true yield threshold for lab samples in Fig. 32b. In standard
tests on cylindrical laboratory samples, initiating cracks within the sample must dilate
in order to extend. This dilation creates a hoop tension in the rock radially beyond the
crack which in turn creates increased conﬁning stress arresting the crack as shown in
Fig. 33a. Adjacent to an excavation, however, this feedback conﬁnement is not pres-
ent, and the initiating crack is free to propagate. In the extreme, in combination with
other factors discussed in the next section, the in situ strength (observed failure
strength) drops to the crack initiation threshold under near-boundary low conﬁnement
conditions favourable to fracture propagation and spalling as in Fig. 33b.
3.4 In Situ Strength Reduction
Other mechanisms compound to exacerbate crack propagation near excavation bound-
aries even if the apparent (average) stress ﬁeld is fully compressive. These include
scale effects related to the inclusion of weak links in a larger volume as well as scale
effects related to the exchange of strain energy between a volume and a propagating
surface. These effects also include unloading damage and oblique damage due to
stress rotation, conﬁnement loss due to open cracks, crack-surface interaction (and

368
M. S. Diederichs
Fig. 34. a) Simple series-parallel reliability model based on Jardine (1973) – failure of system occurs when
both units within an individual cell fail (two interacting cracks). b) a modiﬁed model representing potential
interactions with pre-existing cracks
enhanced crack propagation), and heterogeneity and induced local tension. These
factors combine adjacent to an excavation to reduce the upper bound yield strength
towards the lower bound deﬁned by initiation as shown in Figs. 18 and 33b.
To help explain the impact of some of these factors on interaction (yield) potential,
a new statistical model for interaction was developed based on a series-parallel com-
bination of interacting elements with Weibull strength distribution as illustrated in
Fig. 34. The functions deﬁning the probability of yield at a given stress and alternately
the yield stress at a given probability are:

À Á
V
' À '
m!kx2 V
i
0
PV ð'Þ ¼ 1 À 1 À 1 À exp À
for ' > 'i
ð6Þ
'0

À ÁÀ111
x
2
m
' ¼ '
Á V
2 V
0
À ln 1 À 1 À ð1 À PV ð'ÞÞ
0
þ 'i;
ð7Þ
where PV(') is a speciﬁed yield probability; m, 'i and '0 are Weibull probability
parameters, V0 is the grain volume (or area), V is the sample volume (or area), and x is
the number of possible of interactions per crack (approximately 4 for PFC samples). In
the series-parallel combination in Fig. 34a, the system fails (yield threshold) when two
adjacent elements have yielded. The model is calibrated for crack initiation with real
samples or in this case, with the discrete element model. Scale effects related to weak
link inclusion are incorporated through increasing V while heterogeneity is deﬁned by
the parameter m.
Both the discrete element simulation and the statistical model can be used to
demonstrate the effects of preexisting damage. The PFC model can incorporate dis-
tributions of pre-loading cracks in both isotropic and preferentially oriented fashion.
Such cracks are a part of the unloading process and are discussed extensively by

Rock Fracture and Collapse
369
Martin and Stimpson (1994). Pre-existing cracks do not necessarily contribute to a
lower initiation threshold for new damage but rather create more possibilities for crack
interaction as new damage develops. The total number of new cracks required for
interaction, and therefore the total deviatoric stress level required for yield decreases.
The revised model is schematically illustrated in Fig. 34b where a ¼ 41V=V0 repre-
sents the number of possible interactions with pre-existing cracks and b ¼ (2 À 41)V=V0
represents the number of potential ‘‘virgin’’ interactions. The probability of yield for
damaged samples with a pre-existing crack intensity of 1, is given by:

' À '
m
i
41V
PV ð'Þ ¼ 1 À exp À
Á
'0
V0

' À '
mkð2À41ÞV
V0
Â
i
1 À
1 À exp À
:
ð8Þ
'0
Figure 35 compares the yield predictions for samples with pre-existing isotropic
damage based on a modiﬁed statistical model to PFC simulations of samples with
both isotropic and preferentially oriented cracks prior to this loading stage.
Crack propagation length is incorporated by allowing the number of potential
interactions, x, to become a function of the square of crack propagation length. The
probability of yield at a given stress, ', for a sample with pre-existing damage and the
Fig. 35. Comparison of statistical yield model (including effects of pre-existing cracks as in Eq. (8) with
results from PFC modelling. Model samples had varying degrees of pre-loading damage (isotropic and
preferentially oriented as shown). Annotation above horizontal axis shows crack intensities at stages in a
UCS test without pre-existing damage

370
M. S. Diederichs
freedom for crack propagation, after initiation, to a relative length of LÃ (normalized
to grain size) is given by:

' À '
m
i
4f ðLÃÞ10V
PV ð'Þ ¼ 1 À exp À
Á
'0
V0

Ã
Ã
' À '
mk V ð2ðL Þ2À4f ðL Þ1
V
0 Þ
0
Â
i
1 À
1 À exp À
:
ð9Þ
'0
In this model it is assumed that cracks grow to relative length LÃ immediately upon
initiation (stable crack propagation length is a function of increasing stress after
initiation). It can also be assumed that pre-existing damage does or does not extend
upon loading. In the ﬁrst case pre-existing damage spontaneously extends to length LÃ
upon loading such that f(LÃ) ¼ (LÃ)2 in Eq. (9). In the second extreme, pre-existing
cracks do not extend (but can still interact with extending new cracks) and have a
constant length equal to the elemental dimension such that f(LÃ) ¼ ((1 þ LÃ)=2)2 in
Eq. (9). In Fig. 36 the composite result is shown for both assumptions regarding the
propagation of pre-existing cracks. The model parameters are based on the P ¼ 50%
calibration (probability for systematic damage initiation).
The impact of enhanced propagation on yield strength is clearly demonstrated
using the PFC simulations and the statistical model. While the PFC model does not
adequately reﬂect unstable propagation of cracks, fracture mechanics can be employed
to investigate the impact of conﬁnement reduction near excavation boundaries on
crack extension and spalling. Combining the critical Mode I stress intensity factor
Fig. 36. Results of statistical yield model reﬂecting the effects of both crack extension and pre-existing
damage (Eq. (9))

Rock Fracture and Collapse
371
for an isolated sliding crack of length 2c and " ¼ tan 0 and with propagating wing
cracks each of length a is given by Ashby and Hallam (1986):
ﬃﬃﬃﬃ
p ﬃ
(
)
À'1 %c
1
KI ¼
f1 À ! À "ð1 þ !Þ À 4:3!Lg 0:23L þ
;
ð10Þ
3
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð1 þ LÞ2
3ð1 þ LÞ
with a further relationship incorporating the effect of conﬁnement and crack length on
crack interaction with a free surface (Ashby and Hallam, 1986):
2

3
2
1
1 À
4
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c
p ﬃ
L þ 1ﬃﬃ
p
!
1
c
2
pﬃﬃﬃﬃﬃ6
ð3 2Þ t
2
7
Ksurf ¼ p
'
%c6
7
I
ﬃﬃﬃﬃﬃﬃ
1
4

2 2%
t
2
5;
ð11Þ
1 þ 12 c2 L þ 1ﬃﬃ
p
'1
%2 t2
2
E
where ! ¼ '3 and L ¼ a. The orientation, (angle of crack normal with respect to
'1
c
horizontal), of the critical ﬂaw is incorporated into this derivation and is equal to
À Á
1 tan À1 1 . E is the Young’s Modulus of the intact material. For simplicity, a 45
2
"
degree crack is used in the derivation of Ksurf . Near a free boundary, a beam of
I
thickness, t, is formed by the propagating crack. As the beam becomes more narrow
(as the crack grows) it bends and allows further freedom of movement on the initial
sliding ﬂaw. This additional displacement, in turn, increases the stress intensity factor
at the crack tip leading to further propagation. A similar result is obtained using an
alternative formulation by Dyskin and Germanovich (1993). Combining the effects of
the propagating ﬂaw and the surface interaction and presuming that extension occurs
when ðKI þ Ksurf Þ exceeds K
I
IC (critical intensity factor), the following relationship
can be derived for critical (minimum) '3 required for stability.
Combining Eq. (10) and (11), setting ðKI þ Ksurf Þ to K
I
IC , and then solving for
critical '3:

'1 C1ð1 À "Þ þ
C2
À KIC
ﬃﬃﬃ
p%ﬃ
1þC4'1
c
'3 ¼
;
ð12Þ
C1ð1 þ " þ 4:3LÞ þ C2C3
ð1þC4'1Þ
where:

1
1
1
1
c
2
C1 ¼
0:23L þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
;
C2 ¼ pﬃﬃﬃﬃﬃﬃ
;
3ð1 þ LÞ
ð1 þ LÞ3=2
2 2%
t

2
4 1 þ 1ﬃﬃ
p

2 2
2
c
12
1
c
C3 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
p ﬃ
ﬃﬃﬃ
and
C4 ¼
1 þ pﬃﬃﬃ
:
t
%2E
t
ð3 2Þ
2
In the case of tensile stresses large enough to separate the surfaces of the initial
(sliding) ﬂaw, the critical '3 is given by:
ÀKIC
'3 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :
ð13Þ
%cðL þ sin Þ
Using the standard Kirsch solution for elastic stresses around a circular opening in
an anisotropic stress ﬁeld, and iteratively solving for the critical crack extension (L in

372
M. S. Diederichs
Fig. 37. Impact of near surface conﬁnement and surface interaction on propagation of individual cracks
after initiation as in Eqs. (10), (11) and (12). Stresses based on Kirsch solution
Eq. (10)) with and without the added impact of surface feedback (Eq. (11)), Fig. 37
can be generated to illustrate the effect of stress gradient and surface effects on the
propagation of newly initiated cracks around a tunnel or borehole in hard rock (granite
used for parametric purposes). If the impact of conﬁnement on crack length as in
Fig. 37 is considered along with the impact of crack propagation on the reduction of
yield stress shown in Fig. 36, it is clear that the real near-excavation in situ strength
must be considerably lower than the upper bound interaction threshold in laboratory
testing.
Another strength reduction mechanism examined in Diederichs (2000) is the effect
of internal heterogeneity and internal local tension. It is often argued that tensile
failure cannot occur in a conﬁned state. However, most rocks and rock masses are
heterogeneous at the grain or rock block level and this introduces internal stress
variations as illustrated by Fig. 28. When continuum models are adopted to determine
the stability of an excavation, uniform stresses are predicted (implicit in homogeneous
continuum models) with mostly conﬁned conditions near excavations, unless irregular
geometries or high in situ stress ratios cause tension zones. Figure 38 illustrates that
this is not the case in heterogeneous rock masses. Here, the average stresses sampled
within smaller regions of the overall conﬁned specimen (20 MPa) are shown for
applied axial stress levels of 80 and 250 MPa, respectively. The relative dominance
of tensile zones (leading to the deﬁnition of a spalling limit deﬁned by the slope
'1='3) will depend on the degree of heterogeneity within the rock, the level of in situ
damage, the degree of stress rotation and disturbance during excavation.
Simulations at different conﬁning stresses conﬁrm that a constant conﬁning stress
ratio corresponds to equivalent spatial coverage of tensile zones (Fig. 39). The spatial

Rock Fracture and Collapse
373
Fig. 38. Variation in local average stress (each dot represents the average stress within a sampling circle –
Fig. 30a.) within a heterogeneous sample tested in compression under an applied conﬁning stress of 20 MPa.
Applied vertical stress states at 80 MPa and at 250 MPa are shown. Heterogeneity leads to internal tension
which in turn leads to crack propagation and strength reduction in real solids
coverage required for spalling failure and the corresponding critical '3='1 ratio is
governed by heterogeneity, excavation damage and variability in surface geometry.
This ratio, in practice, is normally between 10 and 20. The spalling limit is then
combined with the damage initiation (lower bound) threshold and the crack interaction
threshold (upper bound) to give the composite stress path limit in Fig. 18.

374
M. S. Diederichs
Fig. 39. Spalling limits (after Diederichs, 2002) deﬁned as zones of equal spatial coverage for tensile zones
(from discrete element simulations)
The combination of the factors summarized here leads, in the extreme, to a
complete collapse of the in situ yield limit from an upper bound. The end result is
the multiphase in situ strength envelope introduced in Fig. 17, incorporating the
effects of tensile damage initiation and accumulation with strength reduction due to
uncontrolled propagation in near-excavation conditions.
4. Conclusions
A brief summary of key ﬁndings of Diederichs (2000) is presented here, separated into
two sections relating to the inﬂuence of tensile damage processes an conﬁnement loss
or relaxation.
4.1 Structurally Controlled Instability
The signiﬁcant impact of remnant rock bridges is demonstrated in this thesis. As
shown using a fracture mechanics approach, even a relatively small percentage of
intact area in the plane of a joint (e.g. less than 1%) can provide a distributed load
capacity equal to heavy support systems used in hard rock mining today. This impact
is quantiﬁed using wedge stability calculations and using the modiﬁed voussoir beam
approach.
This internal or natural gravity support may be sensitive to long term weakening
effects (stress corrosion, dynamic disturbance) but can be relied upon to provide
adequate short-term support for efﬁcient excavation cycling, development and stope
extraction.
While civil excavations (tunnels, caverns) are designed to maintain compressive
stress ﬂow parallel to the surfaces of the openings (thereby increasing structural

Rock Fracture and Collapse
375
stability), mining openings are constructed in accordance with operational require-
ments and ore geometries. As such, conﬁnement loss and complete relaxation are
common around geometrically complex mining excavations. The associated impact
on stability is demonstrated and quantiﬁed.
Small amounts of conﬁnement are required to mobilize frictional resistance on
joint surfaces such that wedges or blocks become stable under gravity loading. Loss of
this conﬁnement is responsible for delayed failure of structurally deﬁned rockmasses.
Loss of conﬁnement in laminated ground leads to increased gravity induced displace-
ments and failure as demonstrated by an updated voussoir beam analogue.
A calibrated voussoir model is used to compute the impact of relaxation on
empirical stability limits which do not explicitly consider this effect. Comparison to
ﬁeld data in which relaxation is identiﬁed as a major inﬂuence, show that the adjusted
stability limit, determined with consideration of relaxation, accurately quantiﬁes and
predicts relaxation-induced instability in mining openings.
4.2 Stress Driven Instability
Both spalling and macroscopic shear rupture are shown to be the result of tensile
damage initiation and accumulation. Microscopic shear initiation mechanisms only
become dominant at very high conﬁnements or at ultra-slow loading rates. At low to
moderate, shear zone formation is the result of extension crack interaction.
Macroscopic or inter-granular friction is not a factor in the damage process
until well after the peak strength has been exceeded and fully localized failure has
developed.
Crack initiation (crack nucleation at the weakest elements) is dependent on devi-
atoric stress and is relatively insensitive to conﬁnement. Crack accumulation is a
stochastic process in a heterogeneous solid. Yield is related to a critical probability
of crack interaction which in turn is associated with a critical amount of accumulated
lateral extension strain (normal to major compression). Crack interaction marks the
onset of true yield, and determines the upper bound for long-term, sample and ge-
ometry independent strength in laboratory tests. If crack extension length is increased,
crack interaction and yield occur with less crack accumulation (fewer individual crack
nucleations) and therefore at a lower compressive stress level.
A number of mechanisms, all investigated in this thesis, reduce the crack interac-
tion threshold near excavations in situ. These include scale effects, pre-existing and
excavation induced damage, crack – surface interaction and enhanced crack propaga-
tion, and heterogeneity-induced local tension.
The stress threshold for crack initiation is unaffected by these factors. The cumu-
lative impact of these mechanisms, however, is to reduce the in situ yield strength,
near excavation boundaries, to a lower bound deﬁned by the threshold for crack
initiation. The important implications for tunnel design and support optimization, of
this lower bound strength and of the characteristics of spalling failure, are developed
and discussed in more detail in Kaiser et al. (2000).
Elastic stress path analysis can be compared to a multiphase threshold (Fig. 18) to
predict relaxation induced blockfall, boundary parallel spalling or conﬁned shear
accompanied or preceded by micro-seismicity.

376
M. S. Diederichs
Acknowledgements
This work has been supported by the National Science and Engineering Research Council of
Canada (NSERC). Special thanks to my supervisors, Dr. Peter Kaiser and Dr. Maurice Dusseault,
and to a number of others who provided insight and inspiration, including Drs. Derek Martin,
Dr. Leo Rothenburg, Dr. Evert Hoek and Dr. Dave Potyondi.
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Appendix I: Tensile Fracture Strength Models
Stress intensity relationships for internal and external cracks in plates (Irwin, 1957;
Sih, 1973; Kemeny and Cook, 1986) can be extended for isolated cracks and rock
bridges in three dimensions. For a given Mode I stress intensity factor at crack
extension KIC, the tensile strength for a partially cracked solid can be computed for
circular non-interacting cracks of radius c, with crack normals oriented at angle to
the direction of tensile loading:

%
1
'T ¼ KIC
pﬃﬃﬃﬃﬃ
:
ðI:1Þ
2 %c
cos 2

Rock Fracture and Collapse
379
Using Kemeny and Cook’s (1986) external crack solution, the tensile strength of a
cylinder of rock with a total cross sectional area, A, containing a circular rock bridge
of radius a (surrounded by a planar, annular crack) is given by:
pﬃﬃﬃﬃﬃﬃ
2a %a
1
'T ¼ KIC
:
ðI:2Þ
A
cos 2
If N=V is the number of regularly distributed cracks or rock bridges in a unit cubic
volume (V ¼ 1 m3), then the total coplanar cross sectional area (cracked and
uncracked) associated with the crack or rock bridge, A is:
1
A ¼
:
ðI:3Þ
N2=3
If (A Ã
Ã
Ã
c )A is the area of the crack and (Aa )A is the area of the rock bridge (Ac
and
A Ã
a
are the ratios of cracked and intact area, respectively, to the total cross section):
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
A Ã
c
c %
;
ðI:4Þ
%N2=3
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
A Ã
a
a %
;
ðI:5Þ
%N2=3
where A Ã
Ã
a
¼ 1 À Ac .
The tensile strength with respect to the percentage of cracked cross sectional area
for cracks perpendicular to loading ð cos 2 ¼ 1Þ is:
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
%
N1=3
'
Ã
T ¼
KIC
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Ã ﬃ for Ac ( 1;
ðI:6Þ
2
%ðAc Þ
or inversely, with respect to percentage of intact rock bridges:
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ðA Ã
a Þ3=2N 1=3
'
Ã
T ¼ 2KIC
pﬃﬃﬃ
for Aa ( 1:
ðI:7Þ
%
For comparison with measurements of relative linear joint trace persistence, P, a
regular array of circular cracks yields the approximate relationship for average linear
persistence, P:
c
A Ã
C
P ¼
¼
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Ã
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Ã ﬃ ;
ðI:8Þ
c þ a
1 þ 2
AC ð1 À AC Þ
pﬃﬃﬃ
where is a factor which ranges from 1= 2 to 1 for small (isolated) cracks and from
pﬃﬃﬃ
1 to
2 for rock bridges (extensive cracking), depending on the linear persistence
measurement direction in the plane of the circular crack or rock bridge.
Appendix II: Voussoir Beam Model
The voussoir beam analogue of Brady and Brown (1993) was reviewed, updated and
veriﬁed in Diederichs and Kaiser (1999b). The effects of relaxation were incorporated
in Diederichs and Kaiser (1999a). The voussoir beam forms in laminated or blocky

380
M. S. Diederichs
ground when the resistance to tensile stress parallel to the laminations is reduced to
zero by through-going fractures perpendicular to the beam. The symmetrical distribu-
tion of compression and tension through a cross-section of the elastic beam of thick-
ness, T, is replaced by a compressive arch which varies in thickness but is typically
between 0.5 and 0.75 times the beam thickness for a highly stable beam. Failure of
slender beams is by snap-through or by crushing at the upper side of the beam at
midspan where the compressive stress is the highest. Thicker beams can also fail
through slip along the lamination normal joint set. The moment generated at the
abutment by the self weight of the half-span must be balanced with the opposing
moment generated by the offset reaction force, F, at the midspan. Two key indepen-
dent unknowns are the thickness of the compressive arch, NT, and the moment arm
between the reaction resultants at the abutment and at midspan, Z. The problem is
statically indeterminate but can be solved in an iterative fashion.
The reaction distributions at the midspan and at the abutments are assumed to be
identical such that the initial (Z0) and ﬁnal (Z) moment arms are given by:

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2
3S
8
Z0 ¼ T 1 À N ;
Z ¼
Z2 À ÁL :
ðII:1Þ
3
8
3S 0
The deﬂection, D, at midspan is given by (Z À Z0) and a negative value for the term
under the square root sign in Eq. (4.2) indicates that the critical beam deﬂection for the
assumed thickness, NT, has been exceeded. For a square span, elastic shortening of the
effective internal arch ÁL, due to compression is calculated as:

S2
2
N
8
ÁL ¼
þ
S þ
Z2 ð1 À vÞ:
ðII:2Þ
6ENðZ
0
0 þ ZÞ
9
3
3S
An iterative solution as described in Chapter 3 is used to ﬁnd the parametric pair
(N, Z(N)) which minimizes fmax, giving the equilibrium solution for the stable beam.
The limit of stability is determined when no solution is possible in-range value of N
and the beam fails by ‘‘snap-through’’. For the two dimensional case this point
corresponds to a deﬂection at the midspan equivalent to approximately 25% of the
thickness. A more conservative stability threshold is introduced based on the onset of
snap-through instability (deviation from a linear deﬂection-thickness relationship).
This limit corresponds to a deﬂection of 10% of the thickness and is deﬁned by the
parametric set which yields a minimum of 35% invalid values of N in the range 0 to 1.
Maximum compressive stress can also be calculated and compared with limiting
values for crushing failure.
Prior to calculating the shortening of the arch due to deﬂection and compression,
it is possible to introduce an symmetrical displacement A, acting in opposite
directions at each abutment. This displacement yields a reduced initial moment
arm, ZÃ:
0
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

3S
8
ZÃ ¼
Z2 À 2
ðII:3Þ
0
A
8
3S 0
for substitution into Eq. (II.1).

Rock Fracture and Collapse
381
For support pressure p or for surcharge loading, s such that p ¼ Às distributed
evenly over the length of the beam, a solution can be obtained by substituting an
equivalent unit weight, Ã:
p
Ã ¼ À
ðII:4Þ
T
into Eq. (II.2).
Author’s address: Ass. Prof. Mark Stephen Diederichs, Ph.D., Department of Geological
Sciences and Geological Engineering, Queen’s University, Kingston, ON K7L 3N6, Canada;
e-mail: mdiederi@geol.queensu.ca