Rock Mechanics and Rock Engineering
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Overview
Rock mechanics is the theoretical and applied
science of the mechanical behaviour of rock and
rock masses. Rock mechanics deals with the
mechanical properties of rock and the related
methodologies required for engineering design.
The subject of rock mechanics has evolved from
different disciplines of applied mechanics. It is a
truly interdisciplinary subject, with applications in
geology and geophysics, mining, petroleum and
geotechnical engineering.
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Rock Mechanics and
Rock Engineering
Rock mechanics involves
characterizing the intact strength
and the geometry and mechanical
properties of the natural fractures
of the rock mass.
Rock engineering is concerned with
specific engineering circumstances,
for example, how much load will the
rock support and whether
reinforcement is necessary.
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Nature of Rock
A common assumption when dealing with the
mechanical behaviour of solids is that they
are:
· homogeneous
· continuous
· isotropic
However, rocks are much more complex
than this and their physical and mechanical
properties vary according to scale. As a
solid material, rock is often:
· heterogeneous
· discontinuous
· anisotropic
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Nature of Rock
Homogeneous
Continuous
strength
equal in
all directions
sandstone
Heterogeneous
Discontinuous
fault
shale
sandstone
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Isotropic
joints
Anisotropic
strength
varies with
direction
high
low
Rock as an Engineering Material
One of the most important, and frequently neglected, aspects of rock
mechanics and rock engineering is that we are utilizing an existing
material which is usually highly variable.
intact
‘layered’ intact
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highly fractured
Rock as an Engineering Material
Rock as an engineering material will be used either:
… as a building material so the structure will be made
of rock
… or a structure will be built on the rock
… or a structure will be built in the rock
In the context of the mechanics, we must establish:
… the properties of the material
… the pre-existing stress state in the ground (which will be
disturbed by the structure)
… and how these factors relate to the engineering objective
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Influence of Geological Factors
Five primary geological factors can be viewed as influencing a rock
mass. In the context of the mechanics problem, we should consider the
material and the forces applied to it.
We have the intact rock which is itself divided by discontinuities
to form the rock structure.
We find then the rock is already subjected to an in situ stress.
Superimposed on this fundamental mechanics circumstance are
the influence of pore fluid/water flow and time.
In all of these subjects, the geological history has played its part,
altering the rock and the applied forces.
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Influence of Geological Factors – Intact
Rock
The most useful description of the mechanical
behaviour of intact rock is the complete
stress-strain curve in uniaxial compression.
From this curve, several features of interest
are derived:
· the deformation modulus
· the peak compressive strength
· the post-peak behaviour
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Influence of Geological Factors – Intact
Rock
high stiffness
high strength
very brittle
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medium stiffness
medium strength
medium brittleness
low stiffness
low stiffness
low strength
low strength
brittle
ductile
Influence of Geological Factors –
Discontinuities and Rock Structure
The result in terms of rock fracturing is to produce a geometrical
structure (often very complex) of fractures forming rock blocks. The
overall geometrical configuration of the discontinuities in the rock mass
is termed rock structure. It is often helpful to understand the way in
which discontinuities form. There are three ways in which a fracture
can be formed:
Mode 1
(tensile)
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Mode 2
(in-plane shear)
Mode 3
(out-of-plane shear)
Influence of Geological Factors –
Discontinuities and Rock Structure
In practice, failure is most often associated with discontinuities which
act as pre-existing planes of weakness. Some examples of the way in
which the discontinuity genesis leads to differing mechanical properties
are:
… open joint which will
allow free flow of
water.
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… stylolitic discontinuity
with high shear
resistance.
… slickensided fault
surface with low shear
resistance.
Influence of Geological Factors –
Pre-Existing In Situ Rock Stress
When considering the loading conditions imposed on the rock structure, it
must be recognized that an in situ pre-existing state of stress already
exists in the rock.
In some cases, such as a dam or
nuclear power station foundation, the
load is applied to this.
In other cases, such as the
excavation of a mine or tunnel, no
new loads are applied but the preexisting stresses are redistributed.
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Influence of
Structure &
In Situ Rock
Stress Together
… types of failure which occur in
different rock masses under low and
high in situ stress levels.
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Influence of Geological Factors –
Pore Fluids and Water Flow
Many rocks in their intact state have a very low
permeability compared to the duration of the engineering
construction, but the main water flow is usually via
secondary permeability, (i.e. pre-existing fractures).
Thus the study of flow in rock masses will generally be a
function of the discontinuities, their connectivity and the
hydrogeological environment.
A primary concern is when the water is under
pressure, which in turn acts to reduce the
effective stress and/or induce instabilities.
Other aspects, such as groundwater chemistry
and the alteration of rock and fracture surfaces
by fluid movement may also be of concern.
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Influence of Geological Factors – Time
Rock as an engineering material may be
millions of years old, however our engineering
construction and subsequent activities are
generally only designed for a century or less.
Thus we have two types of behaviour: the
geological processes in which equilibrium will
have been established, with current geological
activity superimposed; and the rapid
engineering process.
The influence of time is also important given
such factors as the decrease in rock strength
through time, and the effects of creep and
relaxation
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Scalars, Vectors and Tensors
There is a fundamental difference, both conceptually and mathematically,
between a tensor and the more familiar quantities of scalars and
vectors:
Scalar: a quantity with magnitude only (e.g. temperature, time,
mass).
Vector: a quantity with magnitude and direction (e.g. force,
velocity, acceleration).
Tensor: a quantity with magnitude and direction, and with reference
to a plane it is acting across (e.g. stress, strain, permeability).
Both mathematical and engineering mistakes are easily made if this
crucial difference is not recognized and understood.
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Normal and Shear Stress Components
On a real or imaginary plane through a material, there can be normal
forces and shear forces. These forces create the stress tensor. The
normal and shear stress components are the normal and shear forces per
unit area.
It should be remembered that a solid can sustain a shear force, whereas
a liquid or gas cannot. A liquid or gas contains a pressure, which acts
equally in all directions and hence is a scalar quantity.
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Force and Stress
We are now in a position to obtain an initial idea of the crucial
difference between forces and stresses.
When the normal force component, Fn, is
found in a direction θ from F, the value is
F cos θ (i.e. Fn = F cos θ ).
However, when the normal stress
component, σn, is found in the same
direction, the value is σ cos2 θ (i.e. σn =
σ cos2 θ ).
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Force and Stress
The reason for this is that it is only the force that is resolved in the
first case (i.e. vector), whereas, it is both the force and the area that
are resolved in the case of stress (i.e. tensor).
In fact, the strict definition of a second-order tensor is a quantity that
obeys certain transformation laws as the planes in question are rotated.
This is why the conceptualization of the stress tensor utilizes the idea
of magnitude, direction and “the plane in question”.
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Stress as a Point Property
We can now consider the stress components
on a surface at an arbitrary orientation
through a body loaded by external forces
(e.g. F1, F2, …, Fn).
Consider now the forces that are required
to act in order to maintain equilibrium on a
small area of a surface created by cutting
through the rock. On any small area ∆A,
equilibrium can be maintained by the normal
force ∆N and the shear force ∆S.
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Stress as a Point Property
Because these forces will vary according to the
orientation of ∆A within the slice, it is most
useful to consider the normal stress (∆N/∆A)
and the shear stress (∆S/∆A) as the area ∆A
becomes very small, eventually approaching
zero.
Although there are practical limitations in reducing the size of the
area to zero, it is important to realize that the stress components
are defined in this way as mathematical quantities, with the result
that stress is a point property.
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Intact Rock
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Uniaxial Compression Test
… typical record from a uniaxial compression test. Note that the force
and displacement have been scaled respectively to stress (by dividing by
the original cross-sectional area of the specimen) and to strain (by
dividing by the original length).
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Stages of Stress-Strain Behaviour
As the rock is gradually loaded, it passes through several stages:
σaxial
σ ucs
peak
strength
σ cd
crack damage t hreshold
σ ci
crack initiation threshold
crack closure t hreshold
ε later a l
Stage I - Existing cracks
preferentially aligned to the
applied stress will close (σ cc).
Stage II - Near linear elastic
stress-strain behaviour occurs.
σ cc
Contraction
ε a x i al
Dilation
∆ V/V
T o t al
Measured
∆ V/V
Calculated
Crack Volumetric
Strain
Crack
Crack
Closure
Growth
a x i al
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Stage III - Initiating cracks
propagate in a stable fashion (σ ci).
Stage IV - Cracks begin to
coalesce and propagate in an
unstable fashion (σ cd)
Elastic Constants
Focussing on the interval of near linear behaviour, we can draw analogies
to the ideal elastic rock represented by our elastic compliance matrix.
Remembering that the Young’s modulus, E, is defined as the ratio of
stress to strain (i.e. 1/S11), it can be determined in two ways:
Tangent Young’s modulus, ET – taken
as the slope of the axial σ-ε curve at
some fixed percentage, generally
50%, of the peak strength.
Secant Young’s modulus, ES – taken as
the slope of the line joining the origin
of the axial σ-ε curve to a point on
the curve at some fixed percentage
of the peak strength.
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Elastic Constants
… differentiation between elastic and plastic strains, with
definition of the Young’s modulus, E, and Poisson’s ratio, ν.
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Elastic Constants
… typical values of
Young’s modulus and
Poisson’s ratio for
various rock types
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Compressive Strength
Another important parameter in the uniaxial compression test is the
maximum stress that the test sample can sustain. Under uniaxial loading
conditions, the peak stress is referred to as the uniaxial compressive
strength, σc.
It is important to realize
that the compressive
strength is not an intrinsic
property. Intrinsic material
properties do not depend on
the specimen geometry or
the loading conditions used in
the test: the uniaxial
compressive strength does.
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Compressive Strength
The compressive strength is probably the most widely used and
quoted rock engineering parameter and therefore it is crucial to
understand its nature. In other forms of engineering, if the
applied stress reaches σc, there can be catastrophic consequences.
This is not always the case in rock engineering as rock often
retains some load bearing capacity in the post peak region of the
σ-ε curve.
Whether failure beyond σc is to be avoided at all costs, or to be
encouraged, is a function of the engineering objective, the form of
the complete stress-strain curve for the rock (or rock mass), and
the characteristics of the loading conditions. These features are
crucial in the design and analysis of underground excavations.
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Effects of Specimen Size
Having described how the complete σ-ε curve can be obtained
experimentally, we can now consider other factors that affect the
complete σ-ε curves of laboratory tested rock.
If the ratio of sample length to
diameter is kept constant, both
compressive strength and brittleness
are reduced for larger samples. Rock
specimens contain microcracks: the
larger the specimen, the greater the
number of microcracks and hence the
greater the likelihood of a critical
flaw and effects associated with
crack initiation and propagation.
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Effects of Loading Conditions
Intact rock strength is dependent on the types of stresses applied to it.
In other words, rock has strength in tension, compression and shear.
… these different
strengths may be tested
either directly (e.g.
uniaxial tension test,
direct shear test, etc.)
or indirectly (e.g.
Brazilian tensile test,
triaxial compression test,
etc.).
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Effects of Loading Conditions
With the application of a confining load an additional energy input is
needed to overcome frictional resistance to sliding over a jagged rupture
path. Most rocks are therefore strengthened by the addition of a
confining stress.
As the confining pressure is
increased, the rapid decline in
load carrying capacity after
the peak load is reached
becomes less striking until,
after a mean pressure known
as the brittle-to-ductile
transition pressure, the rock
behaves in a near plastic
manner.
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Pore Pressure Effects
Some rocks are weakened by the addition of water, the effect being a chemical
deterioration of the cement or binding material. In most cases, however, it is
the effect of pore water pressure that exerts the greatest influence on rock
strength. If drainage is impeded during loading, the pores or fissures will
compress the contained water, raising its pressure. The resulting effect is
described by Terzaghi’s effective stress law:
… as pore pressure “P” increases the effective normal stresses are reduced and
the Mohr circles are displaced towards failure.
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Time-Dependent Effects
We have indicated that during the complete σ-ε curve, microcracking
occurs from the very early stages of loading. Through these processes,
four primary time-dependent effects can be resolved:
Strain-rate - the σ-ε curve is a
function of the applied strain rate.
Creep – strain continues when the
applied stress is held constant.
Relaxation – a decrease in strain occurs
when the applied stress is held constant.
Fatigue – an increase in strain occurs due to
cyclic changes in stress.
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Temperature Effects
Only a limited amount information is available indicating the effect of
temperature on the complete σ-ε curve and other mechanical properties
of intact rock.
The limited test data does show
though, that increasing
temperatures reduces the elastic
modulus and compressive
strength, whilst increasing the
ductility in the post-peak region.
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Failure Criterion
Rock fails through an extremely complex process of microcrack initiation
and propagation that is not subject to convenient characterization
through simplified models. Building on the history of material testing, it
was natural to express the strength of a material in terms of the stress
present in the test specimen at failure (i.e. phenomenological approach).
Since uniaxial and triaxial testing of rock are by far the most common
laboratory procedures, the most obvious means of expressing a failure
criterion is:
Strength = ƒ (σ1, σ2, σ3)
Or with the advent of stiff and servo-controlled testing machines:
Strength = ƒ (ε 1, ε 2, ε 3)
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Mohr-Coulomb Criterion
The Mohr-Coulomb failure criterion expresses the relationship
between the shear stress and the normal stress at failure along a
hypothetical failure plane. In two-dimensions, this is expressed as:
τpeak = c + σn tan φ
Where:
φ is called the angle of internal friction (equivalent to
the angle of inclination of a surface sufficient to
cause sliding of a block of similar material);
c is the cohesion (and represents the shear strength of
the rock when no normal stress is applied); and
τpeak is the peak shear strength.
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Mohr-Coulomb Criterion
This can be presented graphically using a Mohr circle diagram:
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Mohr-Coulomb Criterion
The Mohr-Coulomb criterion is most suitable at high confining pressures when
rock generally fails through the development of shear planes. However, some
limitations are :
- it implies that a major shear fracture exists at peak strength, at a
specific angle, which does not always agree with experimental observations;
- it predicts a shear failure in uniaxial tension (at 45-φ/2 with σ 3) whereas
for rock this failure plane is perpendicular to σ 3. A tension cutoff has been
introduced to the Mohr-Coulomb criterion to predict the proper orientation
of the failure plane in tension.
- experimental peak strength envelopes are generally non-linear. They can be
considered linear only over limited ranges of confining pressures.
Despite these difficulties, the Mohr-Coulomb failure criterion remains one of
the most commonly applied failure criterion, and is especially significant and valid
for discontinuities and discontinuous rock masses.
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The Hoek-Brown Empirical Failure Criterion
The Hoek-Brown empirical criterion was developed from a best-fit curve
to experimental failure data plotted in σ1- σ3 space. Since this is one of
the few techniques available for estimating in situ rock mass strength
from geological data, the criterion has become widely used in rock
mechanics analysis.
σ1 = σ3 + (m σc σ3+ sσc2)0.5
where σ c is the intact compressive strength, s
is a rock mass constant (s=1 for intact rock,
s<1 for broken rock), and m is a constant
(characteristic of the rock type where values
range from 25, for coarse grained igneous and
metamorphic rocks to 7 for carbonate rocks).
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Discontinuities
It is the existence of discontinuities in a rock
mass that makes rock mechanics a unique subject.
The word ‘discontinuity’ denotes any separation in
the rock continuum having effectively zero tensile
strength and is used without any generic
connotation (e.g. joints and faults are types of
discontinuities formed in different ways).
Discontinuities have been introduced into the rock
by geological events, at different times and as a
result of different stress states. Very often, the
process by which a discontinuity has been formed
may have implications for its geometrical and
mechanical properties.
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Discontinuities
In fact, all rock masses are fractured, and it is a very rare case
where the spacings between discontinuities are appreciably greater
than the dimensions of the rock engineering project. Very often major
discontinuities delineate blocks within the rock mass, and within these
blocks there is a further suite of discontinuities.
Thus, we might expect that a
relation of the form:
should exist.
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Geometrical Properties of Discontinuity
The main features of rock
mass geometry include
spacing and frequency,
orientation (dip
direction/dip angle),
persistence (size and
shape), roughness,
aperture, clustering and
block size.
There is, however, no standardized method of measuring and
characterizing rock structure geometry, because the emphasis and
accuracy with which the separate parameters are specified will depend
on the engineering objectives.
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Discontinuity Spacing and Frequency
Spacing is the distance between adjacent discontinuity intersections
with the measuring scanline. Frequency (i.e. the number per unit
distance) is the reciprocal of spacing (i.e. the mean of these
intersection distances).
… quantifying discontinuity occurrence
along a sampling line, where frequency
λ=N/L m-1 and mean spacing x=L/N m.
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Rock Quality Designation
A natural clustering of
discontinuities occurs through
the genetic process of
superimposed fracture phases,
each of which could have a
different spacing distribution.
An important feature for
engineering is the overall quality
of the rock mass cut by these
superimposed fracture systems.
For this reason, the concept of
the RQD was developed.
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Discontinuity Orientation
If we assume that a discontinuity is a
planar feature, then its orientation can
be uniquely defined by two parameters:
dip direction and dip angle. It is often
useful to present this data in a graphical
form to aid visualization and engineering
analysis.
It must be remembered though,
that it may be difficult to
distinguish which set a particular
discontinuity belongs to or that in
some cases a single discontinuity
may be the controlling factor as
opposed to a set of
discontinuities.
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Discontinuity Persistence
Persistence refers to the lateral extent of a discontinuity plane, either
the overall dimensions of the plane, or in terms of whether it contains
‘rock bridges’. In practice, the persistence is almost always measured by
the one dimensional extent of the trace lengths as exposed on rock
faces. This obviously introduces a degree of sampling bias that must be
accounted for in the interpretation of results.
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Discontinuity Roughness
The word ‘roughness’ is used to denote
deviation of a discontinuity surface from
perfect planarity, which can rapidly become a
complex mathematical procedure utilizing 3-D
surface characterization techniques (e.g.
polynomials, Fourier series, fractals).
From the practical point of view, only one
technique has received some degree of
universality – the Joint Roughness Coefficient
(JRC). This method involves comparing
discontinuity surface profiles to standard
roughness curves assigned numerical values.
The geometrical roughness is naturally related
to various mechanical and hydraulic properties
of discontinuities.
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Discontinuity Aperture
The aperture is the distance between adjacent walls of a
discontinuity. This parameter has mechanical and hydraulic
importance, and a distribution of apertures for any given
discontinuity and for different discontinuities within the same rock
mass is expected.
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Mechanical Properties of Discontinuities
The mechanical behaviour of discontinuities
is generally plotted in the form of stressdisplacement curves, with the result that
we can measure discontinuity stiffness
(typically expressed in units of MPa/m)
and strength.
In compression, the rock surfaces are
gradually pushed together, with an obvious
limit when the two surfaces are closed. In
tension, by definition, discontinuities can
sustain no load. In shear, the stressdisplacement curve looks like that for
compression of intact rock, except of
course failure is localized along the
discontinuity.
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Mechanical Properties - Strength
It is normally assumed that the shear strength of discontinuities is a
function of the friction angle rather than the cohesion. This is done by
using the Mohr-Coulomb failure criterion, τ = c + σtanφ, and setting the
cohesion to zero.
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Mechanical Properties - Strength
The bi-linear failure criterion introduces
the idea that the irregularity of
discontinuity surfaces could be
approximated by an asperity angle i onto
which the basic friction angle is
superimposed.
Thus, at low normal stresses, shear
loading causes the discontinuity surfaces
to dilate giving an effective friction of
(φ+i). As the shear loading continues, the
shear surfaces become damaged as
asperities are sheared and the two
surfaces ride on top of one another,
giving a transition zone before the failure
locus stabilizes at an angle of φ.
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Rock Masses
Building on our examination of first intact rock behaviour and then
discontinuity behaviour, we can now concentrate on extending these ideas
to provide a predictive model for the deformability and strength of rock
masses.
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Rock Mass Deformability
As an initial step in determining the overall deformability of a rock
mass, we can first consider the deformation of a set of parallel
discontinuities under the action of a normal stress, assuming linear
elastic discontinuity stiffnesses.
To calculate the overall
modulus of deformation, the
applied stress is divided by
the total deformation. We will
assume that deformation is
made up of two components:
one related to the intact rock;
the other to the
discontinuities.
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Rock Mass Deformability
The contribution made by the intact rock to the deformation, δ I, is σL/E
(i.e. strain multiplied by length). The contribution made by a single
discontinuity to the deformation, δD, is σ/ED (remembering that ED relates to
displacement directly). Assuming a discontinuity frequency of λ, there will be
λL discontinuities in the rock mass and the total contribution made by these to
the deformation will be δDt, which is equal to σ λL /ED. Hence, the total
displacement, δ T, is:
Hence, the total displacement, δ T, is:
With the overall strain being given by:
Finally, the overall
modulus, EMASS, is given by:
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Rock Mass Deformability
… variation of in situ rock
deformability as a function of the
discontinuities (for the idealized
case of a single set of
discontinuities).
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Rock Mass Strength
In the same way as we considered the deformability of a rock mass,
expressions can be developed indicating how strength is affected by the
presence of discontinuities, starting with a single discontinuity and then
extending to any number of discontinuities.
… scale dependent strength of a single discontinuity.
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Rock Mass Strength
The initial approach is via the ‘single plane of weakness’ theory,
whereby the strength of a sample of intact rock containing a single
discontinuity can be established. Basically, the stress applied to the
sample is resolved into the normal and shear stresses on the plane of
weakness and the Mohr-Coulomb failure criterion applied to consider the
possibility of slip.
Given the geometry of the applied loading
condition:
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Rock Mass Strength
The strength of the sample thus depends on the orientation of the
discontinuity. If the discontinuity is, for example, parallel or
perpendicular to the applied loading, it will have no effect on the
sample strength. At some angles, however, the discontinuity will
significantly reduce the strength of the sample.
The lowest strength
occurs when the
discontinuity normal is
inclined at an angle of
45° + (φ°/2) to the
major applied principal
stress.
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Rock Mass Strength
The plot of rock strength and the discontinuity angles at which the
sample strength becomes less than that for intact rock can be derived
by substituting the ‘single discontinuity’ normal and shear stress
relationships into the Mohr-Coulomb criterion:
Substituted into
|τ| = cw + σntanφw gives:
Where cw and φw are the cohesion
and friction for the discontinuity.
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Rock Mass Strength
An alternative presentation
of the ‘single plane of
weakness’ rock strength
theory is via the Mohr’s
circle representation. The
Mohr-Coulomb failure loci
for both intact rock and the
discontinuity are given.
Circle A – case where the failure locus for the discontinuity is just reached,
i.e. for a discontinuity at the angle 2β w=90°+φ w.
Circle B – case when failure can occur along the discontinuity for a range of
angles.
Circle C – case where the Mohr circle touches the intact rock failure locus, i.e.
where failure occurs in the intact rock.
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Rock Mass Strength
We can consider, on the basis of this single plane of weakness theory,
what would happen if there were two or more discontinuities at
different orientations present in the rock sample. Each discontinuity
would weaken the sample as shown below, but the angular position of
the strength minima would not coincide.
As a result the rock is weakened in
several different directions
simultaneously.With increasing
fractures, the material tends to
become isotropic in strength, like a
granular soil.
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Rock Mass Strength – Hoek-Brown
A methodology of assessing
rock mass strength that does
not depend on the ‘single
plane of weakness’ theory is
the Hoek-Brown failure
criterion. The criterion is
especially powerful in its
application to rock masses
due to the constants m and s
being able to take on values
which permit prediction of
the strengths of a wide
range of rock masses.
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Rock Mass
Strength
… Hoek-Brown
representation and
summary of rock
mass conditions,
testing methods and
theoretical
considerations.
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Rock Mass Strength – Hoek-Brown
For intact rock, the
Hoek-Brown criterion may
be expressed as:
The more general form,
however, was derived to
take into account the
fractured nature of the
rock mass through the
parameters s and a.
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Rock Mass
Strength –
Hoek-Brown
… Hoek-Brown ‘m’
values for different
rock types.
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Rock Mass
Strength – HoekBrown
… estimation of HoekBrown constants and rock
mass deformation constants
based on rock mass
structure and discontinuity
surface conditions.
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Rock Mass Strength – Hoek-Brown
… the Hoek-Brown empirical
criterion applied to a sandstone
rock mass. The criterion
represents best-fit curves to
experimental failure data
plotted in σ1- σ3 space.
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