PAGEOPH, Vol. 137, No. 4 (1991) 0033-4553/91/040337-3051.50 +0.20/0 1991 Birkh~iuser Verlag, Basel Deformation of Rock: A Pressure-Sensitive, Dilatant Material A. ORD 1 Abstract--Permanent (plastic) deformation of rock materials in the brittle regime (cataclastic flow) is modelled here in terms of Mohr-Coulomb behaviour in which all three of the parameters cohesion, friction angle and dilation angle follow hardening (or softening) evolution laws with both plastic straining and increases in confining pressure. The physical basis for such behaviour is provided by a sequence of uniaxial shortening experiments performed by EDMOND and PATERSON (1972) at confining pressures up to 800 MPa on a variety of materials including Gosford sandstone and Carrara marble. These triaxial compression experiments are important for the large range of confining pressures covered, and for the careful recording of data during deformation, particularly volume change of the specimens. Both materials are pressure-sensitive and dilatant. It is therefore possible to derive from these experiments a set of material parameters which allow a preliminary description of the deformation behaviour in terms of a non-associated, Mohr-Coulomb constitutive model, thus providing the first constitutive modelling of geological materials in the brittle-ductile regime. These parameters are used as input to a finite difference, numerical code (FLAC) with the aim of investigating how closely this numerical model simulates real material behaviour upon breakdown of homogeneous deformation. The mechanical and macrostructural behaviour exhibited by the numerical model is in close agreement with the physical results in that the stress-strain curves are duplicated together with localization behaviour. The results of the modelling illustrate how the strength of the upper-crust may be described by two different but still pressure-depen- dent models: the linear shear stress/normal stress relationship of Amontons (that is, Byerlee's Law), and a non-linear, Mohr-Coulomb constitutive model. Both include parameters of friction and both describe brittle deformation behaviour. Consideration of the non-linear bulk material model allows investigation of the geologic regimes which favour localization of the deformation over a continuing homogeneous deformation. More complex models are required to describe the deformation of rock with increasing depth and temperature as the behaviour becomes increasingly temperature-sensitive and rate-dependent. Key words: Localization, constitutive modelling, Mohr-Coulomb behaviour, brittle, plastic, deforma- tion, numerical modelling, crustal strength. I. Introduction The predictive capability of any mathematical constitutive relationship formu- lated to describe rock behaviour during deformation depends critically on the data on which it is based. However, designing and performing a sequence of experiments which will provide such useful constitutive data is not a trivial task. Also, we need to determine those material parameters which are necessary and sufficient for a ~CSIRO Division of Geomechanics, PO Box 54, Mt Waverley, Victoria 3149, Australia.
D e f o r m a t i o n o f Rock: A Pressure-Sensi t ive , D i l a t a n t M a t e r i a l
A. ORD 1
Abstract--Permanent (plastic) deformation of rock materials in the brittle regime (cataclastic flow) is modelled here in terms of Mohr-Coulomb behaviour in which all three of the parameters cohesion, friction angle and dilation angle follow hardening (or softening) evolution laws with both plastic straining and increases in confining pressure. The physical basis for such behaviour is provided by a sequence of uniaxial shortening experiments performed by EDMOND and PATERSON (1972) at confining pressures up to 800 MPa on a variety of materials including Gosford sandstone and Carrara marble. These triaxial compression experiments are important for the large range of confining pressures covered, and for the careful recording of data during deformation, particularly volume change of the specimens. Both materials are pressure-sensitive and dilatant. It is therefore possible to derive from these experiments a set of material parameters which allow a preliminary description of the deformation behaviour in terms of a non-associated, Mohr-Coulomb constitutive model, thus providing the first constitutive modelling of geological materials in the brittle-ductile regime. These parameters are used as input to a finite difference, numerical code (FLAC) with the aim of investigating how closely this numerical model simulates real material behaviour upon breakdown of homogeneous deformation. The mechanical and macrostructural behaviour exhibited by the numerical model is in close agreement with the physical results in that the stress-strain curves are duplicated together with localization behaviour. The results of the modelling illustrate how the strength of the upper-crust may be described by two different but still pressure-depen- dent models: the linear shear stress/normal stress relationship of Amontons (that is, Byerlee's Law), and a non-linear, Mohr-Coulomb constitutive model. Both include parameters of friction and both describe brittle deformation behaviour. Consideration of the non-linear bulk material model allows investigation of the geologic regimes which favour localization of the deformation over a continuing homogeneous deformation. More complex models are required to describe the deformation of rock with increasing depth and temperature as the behaviour becomes increasingly temperature-sensitive and rate-dependent.
The predictive capability of any mathematical constitutive relationship formu- lated to describe rock behaviour during deformation depends critically on the data on which it is based. However, designing and performing a sequence of experiments which will provide such useful constitutive data is not a trivial task. Also, we need to determine those material parameters which are necessary and sufficient for a
~ CSIRO Division of Geomechanics, PO Box 54, Mt Waverley, Victoria 3149, Australia.
338 A. Ord PAGEOPH,
given constitutive model, and which may then be used in numerical modelling of the rock behaviour under various loading conditions, and therefore in testing of the formulation.
EDMOND and PATERSON (1972) performed a sequence of triaxial compression experiments at room temperature and at constant displacement rate on Gosford sandstone and Carrara marble, among other materials. These experiments are important for covering such a large range of confining pressures, up to 800 MPa, and for continuously monitoring the volume change throughout the experiments, as well as the axial displacement and load.
A constitutive model is required on which to base any derivation of a set of material parameters from physical experiments. The EDMOND and PATERSON (1972) results are pressure-sensitive and this is confirmed for Carrara marble by FREDRICH et al. (1989). FISCHER and PATERSON (1989) show that Gosford sandstone and Carrara marble are both temperature and rate sensitive, the sand- stone rather less so than the marble. Further, FREDRICH et al. (1989) demonstrate for Carrara marble that both twinning and dislocation glide are active, together with microcracking, at higher confining pressures. A constitutive model is required to describe such complex behaviour. As a first step in this direction, we assume temperature-insensitivity for simplicity. We also assume that loading of the rock is sufficiently slow that its behaviour is rate independent.
These assumptions are covered by adopting a framework of non-associated plasticity provided by a Mohr-Coulomb yield condition and flow rule (VERMEER and DE BORST, 1984; HOBBS et aL, 1990a). Note here that the term "plastic" is used to denote any permanent deformation, independent of the micro- or macro-scale mechanisms of deformation.
The material parameters derived from such a model may then be incorporated as input to a numerical model in order to test the constitutive model and its assumptions. Such procedures prove the usefulness of the resultant numerical model for predicting the deformation behaviour of the materials under imposed boundary conditions different from those of the physical experiments.
Such modelling demonstrates the importance of separating material from system behaviour (see discussion by HOBBS et al., 1990a). It provides results on the mechanical behaviour of the system and on the macrostructural behaviour of the material, but provides no information on the microstructural deformation mecha- nisms other than that they lead to pressure-sensitive, dilatant behaviour. Hence crystal-plastic deformation mechanisms are not allowed as the dominant deforma- tion mechanism in this instance and one assumes mechanisms involving intra- and inter-granular cracking and sliding, grain boundary sliding and the like.
In conjunction with testing the theory and the code, using material parameters derived from physical experiments, we examine the range of physical conditions which promote localization in these materials, and consider the implications the results have for the strength of the crust.
Vol. I37, 1991 Deformation of Rock 339
2. Constitutive Behaviour
We use as the basis for our considerations a Mohr-Coulomb constitutive model for which yield in Mohr 's space is defined by
] z , l=C--aN tan dp
where e is the cohesion, q~ is the angle of friction, and ou and ~, are the normal and shear stress respectively acting on a plane oriented at (re/4 _+ 4~/2) to the maximum
principal stress. Written in terms of the principal stresses, oN and a2, the yield function, f , becomes
f = r * + o* sin q~ - e cos 4)
where z* is the maximum shear stress,
~* = IOl - a21/2;
and a* is the mean stress,
a * = ( g t + a 2 ) / 2 ;
for compressive stresses assumed negative. The shear and normal components of stress across some arbitrary surface dement in the material satisfy f = 0 at yield. Negative 'values for f imply elastic deformation, inside the yield surface, and
positive values of f a r e not allowed since plastic or permanent deformation occurs to keep the stress on the yield surface described by the yield function.
The manner in which the material deforms plastically is described by the flow rule which provides a relationship between the orientation of the plastic strain rates and the orientation of the stress in six-dimensional stress space. The flow rule may
be represented as ~P = 2(Sg/Oa), where it is a non-negative multiplier which vanishes upon neutral loading and unloading. The function g describes the plastic potential surface, and it is defined in a manner similar to the (Mohr-Coulomb) yield function, f(VERMEER and DE BORST, 1984; after RADENKOVIC, t961)as
g = r* + a* sin O - constant
so that the dilation angle 0 satisfies the same requirement in the plastic potential function as the friction angle does in the yield function.
The flow rule says that the plastic strain rate vector is normal to the potential surface described by g. The relative magnitudes of the friction and dilation angles control whether or not the yield and potential surfaces are parallel. Equivalence of the friction and dilation angles results in normality of the plastic strain rate vector to the yield surface leading to an associated flow rule (DRUCKER, 1951, 1956, 1959; see HOBBS et at., 1990a, for a discussion). Non-equivalence of these parameters, so that the plastic strain rate vector is normal only to the plastic potential surface, not
340 A. Ord PAGEOPH,
to the yield surface, leads to a non-associated flow rule. In pressure-sensitive, dilating materials which lack cohesion, associated behaviour appears to be physi- cally unreasonable since ~b = ~ leads to zero plastic work being done during plastic deformation. Thus non-associated behaviour is to be expected in cohesion- less, frictional materials. Examples of such relationships have been discussed by RUDNICKI and RICE (1975), CHEN (1982) and VERMEER and DE BORST (1984). Under these conditions, the dilation angle must be less than the frictional angle, otherwise zero or negative plastic work is done during plastic deformation. However, in materials with cohesion, that is in most geological materials, particularly brittle ones, which show pressure-sensitive and dilational responses to stress, situations in which the dilation angle may be greater than the friction angle are allowed (ORD, 1990), a situation not normally considered in soil or rock mechanics.
This model then defines the material parameters which need to be derived from the results of physical experiments in order to describe the constitutive behaviour; the elastic moduli, for example Young's modulus and Poisson's ratio, for descrip- tion of the elastic deformation, and cohesion, friction angle and dilation angle for description of the plastic deformation.
In the elastic linear range (provided such a range can be identified) we define the Young's modulus, E = al/e~, where e~ is the maximum principal elastic strain (equivalent to the axial strain for a state of uniaxial stress, and coaxial constitu- tive behaviour). E may be obtained from Figures such as la and 2a. Poisson's ratio, v is obtained from Figure 3 using the tangent to the initial values for volume strain and axial strain, that is, v = ( 1 - e , / e l ) / 2 . Then for a state of uniaxial stress, the elastic volume change is (1 -2v)a~/E (JAEGER, 1969).
The pressure sensitivity of the plastic flow stress of the material is reflected in the magnitude of the friction angle which can be obtained by plotting the flow stress at a particular plastic strain against the pressure, as described by PATERSON (1967). Then
1 ~ tan ~ } ~b = t a n - ~ 2n / 1 + tan ~ (1)
is obtained from such a plot for experiments performed at a number of different confining pressures where ~ is the angle which the curve of flow stress at constant strain makes with the pressure axis (PATERSON, 1967, Figure 4). The cohesion may then be obtained for the same plastic strain and confining pressure for which ~b has just been derived using
( 1 + sin qS) 2e cos ~b (2) a l = a 2 ( l _ s i n q ~ ) (1-s in~b)
which follows from the geometry of a Mohr diagram. This approach has been used to analyse the data of EDMOND and PATERSON (1972), for Carrara marble
Vol. 137, ]991 Deformation of Rock 341
1000
~. 8O0
Boo
400
200
- a
800 600
400
100-
50
I I I 1
5 10 15 20
AxialStrain %
1000 -
b
800 a.
600
400
a 200
_ ~ 20~176 c
�9 5Yos
0 ~ I f I I I I I I I
200 400 600 800 1000
Confining Pressure MPa
Figure 1 Carrara marble. (a) Experimental results o f EDMOND and PATERSON (1972). The differential stress versus axial strain curves are plotted for different values of the confining pressure, (b) Values of
differential stress versus confining pressure are obtained for different axial strains from (a).
(Figure 1) and for Gosford sandstone (Figure 2). (a) in each case presents the results of the physical experiments described by EDMOND and PATERSON (1972), while (b) presents the derived data of flow stress at constant shortening strain versus confining pressure from which the angle ~ and hence the friction angle are obtained for that strain from Equation (1). The accuracy of this derivation varies with the pressure sensitivity of the material and therefore with the number of experiments conducted at different confining pressures. The equivalent cohesion for the same strain and confining pressure is calculated from Equation (2) as
a2( 1 + sin ~) -- a~ ( 1 - sin ~) C
2 cos
1000 -
- a f 8001
6oo
400
200
342 A. Ord PAGEOPH,
600
400
200
100
i i
5 10 Axial Strain %
i
15 i
20
1000 - 20% ~:
.e = 600 ~ 5'/0 E
400
200
0 I i i i i i i 0 200 400 600
Confining Pressure MPa
Figure 2
I I I
8 0 0 10oo
Gosford sandstone. (a) Experimental results of EDMOND and PATERSON (1972). The differential stress versus axial strain curves are plotted for different values of the confining pressure. (b) Values of
differential stress versus confining pressure are obtained for different axial strains from (a).
The d i la t ion angle is descr ibed for t r iaxial compress ion by
~C sin ~k = -_ 2g~ + ~C
(VERMEER and DE BORST, 1984) where ~C is the plas t ic vo lume s t ra in rate and ~i
(i = 1, 3) are the pr incip le s t ra in rates. ~O is pe rhaps more intui t ively c o m p r e h e n d e d
Vol. 137, 1991 Deformation of Rock 343
as the uplift angle for a dilating simple shearing deformation (VERMEER and DE BORST, 1984, Figure 4.2); that is
tan ~ = _ u
where ~ is the vertical velocity and u is the horizontal velocity of a deforming material point.
The magnitude of ~ for Gosford sandstone and for Carrara marble may therefore be derived from plots of volume strain versus axial strain as provided by EDMOND and PATERSON (1972) (Figure 3). At this stage, since the dilation angle is associated only with the plastic potential of the material, the elastic volume strain
lO
8 o~
g ~ 4 r -
E 2 _= 0
o
-2 0
i!
10
800
I 1 ] J 5 10 15 20
Axial Strain %
4
g o
E -2
o >
-4
-6 0
b
100
400
I I I I 5 10 15 20
Axial Strain %
Figure 3 Experimental results of EDMUND and PATERSON (1972). The volume change versus axial strain curves are plotted for different values of the confining pressure. (a) Carrara marble. (b) Gosford sandstone.
344 A. Ord PAGEOPH,
must be subtracted f rom the values of e, given by Edmond and Paterson since their values of e~ are total volume changes. However, as shown in Figures 5c and 4c of
EDMOND and PATERSON (1972), the elastic volume strain is approximately con- stant after only a few percent plastic strain, so that the dilation angle may be
derived directly f rom the tangent to the plastic volume strain versus axial strain
curves,
tan(e~/51) sin r -- 2 + tan(e~/el ) '
In a non-hardening, Mohr-Coulomb material, the friction and dilation angles
and the cohesion remain constant, However, some materials may strain-soften or
strain-harden, that is, the differential stress may decrease or increase with increasing
deformation, irrespective of the yield stress. In this case the friction and dilation
angles and the cohesion may increase or decrease with continuing deformation. The deformation history of a material may then be described in terms of the histories of
these parameters. Little is known of the form of these histories for rocks, or of how
changes in these parameters may arise f rom the changing microstructure and
dominant deformation mechanisms within the material.
i i
==
a I ~ P2 b
g
g* e P2 P
Figure 4 (a) Differential stress versus axial strain for different values of the confining pressure P (see inset). A and B are the differential stresses at strain e* for confining pressures P~ and P2, respectively. (b) Differential stress at e* versus confining pressure for A and B. ~ is the angle used in Equation (1) for derivation of
the friction angle. (After HOBBS et aL, 1990b.)
Vol. 137, 1991 Deformation of Rock 345
VERMEER and DE BORST (1984) propose simple models using results from the deformation behaviour of concrete for friction and dilation hardening and for cohesion softening (see, for example, VERMEER and DE BORST, 1984, Figure 7.2). These models are based on various physical results interpreted microstructurally as representing the evolution of inter- and intra-granular microcracking and sliding. VERMEER and DE BORST (1984) remark that their models describe generalised material behaviour, they are not derived by fitting any existing data accurately. Here we examine the consequences of deriving histories for these parameters from the data of EDMOND and PATERSON (1972). The evolution laws defined here do not resemble those proposed by VERMEER and DE BORST (1984).
3. Experimental Results
The results of such analyses on the EDMOND and PATERSON (1972) data are shown in Table 1 and Figures 5a, c and e for Carrara marble, and in Table 2 and Figures 5b, d and f for Gosford sandstone. The magnitudes of the parameters have associated with them both the experimental errors (EDMOND and PATER- SON, 1972) and the errors incorporated by reading the data off the published curves. The cumulative errors are greatest up to about 5% strain, and are considered to be 10% or less. Values of q5 were derived using the values of the differential stress at values of the total axial strain corresponding to first yield, total axial plastic strains of approximately 0.5 and 1.0 for Gosford sandstone (Figure 5b) and of 0.62 and 1.64 for Carrara marble (Figure 5a), and total strains of 5, 10, 15 and 20 percent.
As already described, values for the cohesion were then derived using these ranges of ~b (Figures 5c and 5d), and values for ~k were derived from the volume change plots (Figures 3, 5e and 5f). This latter information is required for subsequent back calculation to the corresponding incremental plastic tensor shear strain. We follow here the terminology of VERMEER and DE BORST (1984) which describes an increase of the friction angle (or cohesion or dilation angle) with increasing strain as friction (or cohesion or dilation) hardening and a decrease in the same parameters with increasing strain as parameter softening.
Carrara Marble
At low confining pressures Carrara marble demonstrates strong friction hard- ening of 12 ~ to 16 ~ over about 10% total axial strain, an effect which decreases with increasing confining pressure. At these former confining pressures, there is only minor friction hardening at higher total axial strains, a maximum of 6 ~ from 10% to 20% axial strain. At higher confining pressures, friction hardening is low, less than 5 ~ over 10% axial strain, and only 2 to 3 ~ from 10 to 20% axial strain.
346 A. Ord PAGEOPH,
Table 1
Carrara Marble
EDMOND and PATERSON (1972) o = 0.125 =~ G = K
Pc (MPa) % • % 7 p't~ (~ Cohesion (MPa) ~ E (GPa) G (GPa)
Cohesion behaves similarly with increasing total axial strain, but inversely with confining pressure so that now, cohesion hardening is pronounced at low strains and high confining pressures with a maximum increase of 130 MPa over 10% axial strain at 800 MPa, and decreases with increasing strain and with decreasing confining pressure. At low confining pressures of 50 MPa and 100 MPa, cohesion is continually decreasing, by 32 to 36 MPa.
Dilation angle decreases at low confining pressures from a high value of 16 ~ to 8 to 10 ~ over a total axial strain of 10%. At moderate and high confining pressures, the dilation angle is very low (1 to 2 ~ or negative and is either constant or increases slightly with increasing total axial strain.
Vol. 137, 1991 Deformation of Rock 349
The dilation angle is less than the friction angle in most cases. However, the situation is reversed at first yield at 100 MPa confining pressure (~b = 8 ~ 0 = 15~ and at first yield (qS=0 ~ ~ = 1 ~ and 1% axial strain (qS=0 ~ q)=0.6 ~ ) at 800 MPa.
Gosford Sandstone
The l~riction angle for Gosford sandstone is strongly strain hardening for all confining pressures at low total axial strains, l0 to 17 ~ over 10% axial strain, and is approximately constant above about 10% total axial strain.
Cohesion is strongly strain softening at 100 MPa (38 MPa) and at 600 MPa (23 MPa), although in the latter case, cohesion increases again at higher total axial strains. At 200 MPa, cohesion strain hardens by 21 MPa up to 5% axial strain and then strain softens, while at 400 MPa, cohesion strain softens by 35 MPa to 10% axial strain, and strain hardens at higher total axial strains.
At 100 MPa, the dilation angle increases initially from only 1 ~ to 2.9 ~ then decreases to a constant value after 5% axial strain but only by 1 ~ At all other confining pressures, the dilation angle decreases by up to 9 ~ for the first 5 to 10% axial strain, and then increases by 9 ~ to 18 ~ over the next 10% axial strain. The dilation angle for Gosford sandstone is always less than the friction angle, by 2 ~ or more.
Summary
In summary, the derived material parameters for Carrara marble and Gosford sandstone are strongly contrasting. In particular, the parameters for Carrara marble tend to vary strongly with confining pressure but little, if at all, with axial strain while conversely, though less consistently, the parameters for Gosford sandstone vary strongly with axial strain and less so with confining pressure. This behaviour is most obvious for the friction angle. It is not well displayed by the cohesion for Gosford sandstone, nor by the dilation angle, at low axial strains, which is strongly pressure dependent.
The dilation angles for Carrara marble of 12 to 15 ~ at low axial strains and 50 and 100 MPa are remarkably high in contrast to Gosford sandstone which is only very slightly dilatant, otherwise contractant, under the same conditions. The situation is reversed at high axial strains and high confining pressures under which conditions Carrara marble is contractant whereas Gosford sandstone is dilatant (0 is 8 to 9~
The contrasting material behaviour reflects the differences in the active deforma- tion mechanisms and evolving microstructures for the sandstone and the marble, as described by EDMOND and PATERSON (1972), FISCHER and PATERSON (1989) and
350 A. Ord PAGEOPH,
FREDRICH et al. (1989). This analysis shows that the stress-strain behaviour observed by EDMOND and PATERSON (1972) may be given a quantitative constitu- tive basis. Only after such an analysis may the interpretations of EDMOND and PATERSON (1972) regarding the association between experimental response and the brittle-ductile transition, and therefore regarding stability criteria (see HOBBS et al.,
1990a, for a discussion) be rigorously investigated. The following section investigates this problem by use of a numerical
model.
4. Numerical Modelling
Numerical modelling of the deformation behaviour of Carrara marble and Gosford sandstone provides us with improved insight as to the realistic nature of our constitutive models.
In what follows, the following assumptions are made, as already described, that both rock types may be represented by strict non-associated Mohr-Coulomb constitutive behaviour. Elastic behaviour is strictly linear and is not pressure dependent. The elastic-plastic transition is described by a yield surface, and purely plastic deformation is non-linear, described by a plastic potential. The plastic potential and the yield surface do not necessarily coincide under any conditions so that both materials are subject to a non-associated flow rule, and both have cohesion. The possibility therefore exists (see HOBBS et al., 1990a, for a review) that instability may occur during strain hardening, that is, the stability measure, here the rate of energy over the volume, V, of the material Sv 6 �9 ~ d V may be negative, where a is the stress and e is the total (elastic plus plastic) strain. A dot indicates differentiation with respect to time.
The computer code FLAC (Fast Lagrangian Analysis of Continua, CUNDALL and BOARD, 1988) is used for numerical modelling of the deformation behaviour of Carrara marble and Gosford sandstone. It has already been used for the study of deformation localization for frictional-dilatant materials undergoing plane strain extension (CUNDALL, 1990), plane strain shortening (HoBos and ORD, 1989) and simple shearing (ORD, 1990) deformation histories, and some results are shown also in HOBBS et al. (1990a).
We describe here the results of numerical experiments performed under three types of boundary conditions: plane strain shortening, an isochoric simple shearing deformation, and a non-isochoric simple shearing deformation. We first modelled the axially-symmetric axial shortening deformation imposed in the experiments of EDMOND and PATERSON (1972). The modelling technique is similar to that described in Section 4.1. Both the mechanical and the macrostructural results exhibited in these experiments are similar but not identical to those described in
Vol. 137, 1991 Deformation of Rock 351
Section 4.1. These results were considered sufficient to validate the use of F L A C for
the following experiments.
4.1. Uniaxial Compression in Plane Strain
In all cases in these numerical experiments, the material undergoes a plane strain
compression, imposed by steel platens moving towards each other at a constant
imposed velocity, to represent more closely the design of the physical experiments.
A stress state, equivalent to the imposed confining pressure, is initially imposed on
the ma'~eriat so that, before loading, o~ = o~, and the differential stress is zero. The
initial imposed velocity is applied th roughou t the specimen in an a t tempt to model
a quasi-static rather than a dynamic situation, varying down to zero a long the
central axis o f symmetry (Figure 6). The shortening displacement rate is 0.5 x 10 3
length units per time step, which for a material 50 units long results in an axial
shortening o f 1% every 1000 steps. The material has an initial aspect rat ion o f 2:1,
. . . . . . . . . . . . . . . . . ii ..... III I[]11 ..... ' .... ill IIIII . . . . " ' I I l l l . . . . . ~ I~1 I l l l l i k l L i i ]
I I l ~ J ~ I L I ~ I I I I I I I 1 ~ 1
111 I I ~ l l ~ l J E I J l l l l I k l ~ l . . . . . . . . . . . . . . . . . ~ / I l l l . . . . .
Figure 6 (a) Finite difference grid for the initial model. The material is composed of 50 zones in the x direction and 35 in the y direction to provide the same specimen shape ratio as the specimens of EDMOND and PATERSON (1972). (b) Numerical specimen showing schematically the magnitudes and orientations of the initially imposed x-velocities. The x-velocities are kept constant on the platens throughout each
numerical experiment.
352 A. Ord PAGEOPH,
and a friction angle of 2 ~ is arbitarily imposed at the material/piston interface. The normal and shear stiffnesses at the interface are 100 GPa. The steel pistons are given a shear modulus of 81 GPa, a bulk modulus of 166 GPa, and a density of 7889kg-m -3 (JAEGER, 1969, p. 58, Table 1; Handbook of Chemistry and Physics, 61st edition). Both the Carrara marble and the Gosford sandstone are given a density of 2650 kg- m-3. Gravity is not applied.
The magnitudes and histories of the material parameters, the friction and dilation angles and the cohesion, are applied to each finite difference zone in the form of a table relating each value to its equivalent plastic shear strain. The model is therefore initially homogeneous with respect to all its properties, but if the deformation deviates from homogeneity, for example as a result of interac- tion of the sample with the end platens, then the finite difference zones may attain different plastic shear strains, and the material will become heterogeneous with respect to these material parameters. The values given in Tables 1 and 2 (see also Figure 5) are the values used in the data files for the different experiments.
The mechanical results of these numerical experiments are given in Figure 7 for Carrara marble and in Figure 8 for Gosford sandstone, together for compari- son with the original EDMOND and PATERSON (1972) results. It is of interest to investigate the differences in localization behaviour between deformation under axially-symmetric axial shortening versus plane strain uniaxial compression. Fur- ther, the bulk structural behaviour of the specimens is shown in Figure 9 for Carrara marble and in Figure 10 for Gosford sandstone.
As observed by EDMOND and PATERSON (1972) for axially-symmetric, axial shortening of Carrara marble, the tendency towards a sharply-defined narrow shear is well developed at 50 MPa, and changes to a widely distributed deforma- tion at 100 MPa, with increasing confining pressure. The results of the plane strain FLAC experiments would put this transition at a higher confining pressure, between 100 and 200 MPa.
EDMOND and PATERSON (1972) comment regarding their results for Gosford sandstone that "the absence of a single discrete shear fracture places the be- haviour as ductile." FLAC is unable to open along any surface; however the absence of a sharp gradient in velocity is in agreement with the conclusion of Edmond and Paterson. Otherwise, the following features observed by Edmond and Paterson may be observed also in the FLAC runs:
1. "After 20 percent deformation, however, all specimens showed some degree of non-uniform deformation, in addition to localized end effects."
2. "At 1-kb broad conjugate shear bands appeared, while at higher confining pressures specimens barrelled in a somewhat irregular manner without show- ing distinct shear zones."
3 . . . . "specimens strained only 10 percent at 4 and 6kb appeared to be uniformly deformed." Again, shear zone formation appears to be favoured in the numerical, plane strain, deformation experiments.
Vol. 137, 199l Deformation of Rock 353
~~176176 F a : 800 I--
' ' oo
200
00 5 ~0 ~ ___ ~ i , 2'0
Axial Strain %
,ooo ~ b
t. 800 l-
m F- ~oo
400 ~-- /_./~ . . . .
Or ..... 1 . . . . . . . . I,, t J 0 5 10 15 20
Axial Strain %
Figure 7 Carrara marble, Differential stress is plotted versus axial strain for different values of the confining pressure. (a) After EDMOND and PATERSON (]972). (b) Results of numerical modelling experiments.
In these numerical experiments, the initial finite difference grid is square and 50
zones on edge. The initial imposed y-veloci ty is zero a]ong the left-hand edge,
parallel to the y-direct ion, and increases linearly to 50 • 10 -4 length units per time
354 A. Orct PAGEOPH,
el G. :E el .=
N el
== el _= O
1 0 0 0 -
_ a
8 0 0 -
6 0 0 -
2OO
0 "' I I 0 5 10
Axia l Strain %
600
400
200
100
I I 15 20
1000
8OO
( 0
600
"i
400
200
0 0
b 6 0 0
I L I |
5 10 15 20
Axial Strain %
Figure 8 Gosford sandstone. Differential stress is plotted versus axial strain for different values of the confining pressure. (a) After EDMOND and PATERSON (1972). (b) Results of numerical modelling experiments.
- )
Figure 9 Carrara marble, differential stress is plotted versus axial strain for different values of the confining pressure: (a) 50 MPa; (b) 100 MPa; (c) 200 MPa; (d) 400 MPa; (e) 600 MPa; (f) 800 MPa. The insert diagrams show the contours of the instantaneous y-velocity (normal to the shortening direction) for the numerical specimens at different values of the axial strain. The more concentrated the contours, the greater the tendency towards localization. The contour interval is 5.0 • 10 -5 in (a), (b) and (c), and 2.5 • 10 - s in (d), (e) and (f). Also insert is the finite deformed grid for 20% axial strain in each case.
Vol. 137, 1991 Deformation of Rock 355
C-? /,
d
. . . . 2
edN SS~qS Pq~mueNI3 j ~
N
J I I J I i I " ' 4 " - o ~ ~ ~ o
~IW mm~S lel~muetllQ
c
W
edW ~OZlSII~1UOJIS~ a ~ udW s s e , ~ 9 1 e ~ u ~ Q
E
71
I-
5 E I F J
J
l i ~ ~- .....
edw sse~S I~UOJo~O
o
e d n ~ I ~ e J e ~ Q
0
o~
356 A. Ord PAGEOPH,
1000:
800
I
i = 400
a 200
a o
1000
=~ 200
b o
0
i i i i
5 10 15 20
Axial Strain %
i i i i
5 10 15 20
A x i a l S t r a i n %
10 15
A z ~ Strain %
_ _ J
2O
I
J m ~
=== =
!
d o 5 10 15 20
Axbl l Strain %
Figure 10 Gosford sandstone. Differential stress is plotted versus axial strain for different values of the confining pressure: (a) 100 MPa; (b) 200 MPa; (c) 400 MPa; (d) 600 MPa. As in Figure 9, the insert diagrams display the contours of the instantaneous y-velocity at different axial strains. The contour interval is 5.0 x 10 - s in (a) and (b) and 2.5 x 10 s in (c) and (d). Also insert is the finite deformed grid for 20%
axial strain in each case.
Vol. 137, 1991 Deformation of Rock 357
3000
2500
2090
1500
1000
5O0
00.0
800
i 0oo
400
g, 200
5O0
4O0
300
200 o
100
J i i i i = r i i
0.2 0.4 0.6 0,8
Shear S t r a i n
L t i ~ i i i i 0.2 0.4 0.6 0,8 1.0
Shea r St ra in
3000 b
2500
2000
1500
1000
500
/ i
1.0 00.0~
500 I- d
e
f
O0 I i I I T i I I I I .0 0.2 0,4 0.6 0.8 1.0
S h e a r S t ra in
J 0.2 0.4 0.6 0.8 1,0
S h e a r S t ra in
0.0 0.2 0.4 0.6 0.8 1.0
S h e ~ r S t r a i n
~oo ~ f
soo
200
100
0.0 0.2 0.4 0.6 0.8 1.0
S h e a r S t ra in
Figure 1 1 Carrara marble. Isochoric simple shearing deformation. Shear stress is plotted versus shear strain for different values of the confining pressure: (a) 50 MPa; (b) 100 MPa; (c) 200 MPa; (d) 400 MPa; (e) 600 MPa; (f) 800 MPa. As in Figure 9, the insert diagrams display the contours of the instantaneous y-velocity but at different shear strains. The contour interval is 5.0 x 10 -5 throughout. In cases where these diagrams are not present, the instantaneous y-velocity contours are representative of a homoge- neous deformation, being equally spaced and parallel to the shearing direction. Also insert is the finite
deformed grid for a shear strain of 1 in each case.
step along the right-hand edge so that the simple shearing direction is,parallel to the y-direction, as described by ORb (1990).
The effect of the initially high dilation angle for Carrara marble is demonstrated by the extremely high shear stresses (1000 MPa or greater at a shear strain of 1) attained at low confining pressures (Figures l la, b and c). In real physical experiments involving isochoric simple shearing at low confining pressures (see for example WILLIAMS and PRICE, 1990), such high stresses are not observed; the mechanism of deformation presumably changes to involve processes with lower
358 A. Ord PAGEOPH,
dilation angle so that the deformation can proceed at lower stresses. Shear stresses
at the higher confining pressures (Figures 1 ld, e and f) are all less than 1000 MPa
at a shear strain of 1. The localization of the deformation apparent in the pattern
of instantaneous y-velocity contours at low confining pressures (Figures 1 la and b)
are not reflected by the apparently homogeneous grid pattern at a shear strain of 1.
At higher confining pressures, Carrara marble under these conditions undergoes a
completely homogeneous deformation. The strong dependence of shear stress on the dilation angle for an isochoric simple
shearing deformation (ORb, 1990) is reflected in the behaviour of Gosford sandstone
(Figure 12). The shear stress decreases for contractant behaviour and increases for dilatant behaviour, a phenomenon particularly obvious for example in Figure 12d for
which the dilation angle decreases from - 1 . 0 ~ to - 9 . 1 ~ and then increases to 8.8 ~ In all cases, a strong localization of the instantaneous y-velocity contours is
apparent, under both strain-softening and strain-hardening conditions, but, except
for Figure 12a, the finite deformed grid is homogeneous at a shear strain of 1. This
contrasts with the strong localization behaviour of Gosford sandstone under a
Figure 12 Gosford sandstone, Isochoric simple shearing deformation. Shear stress is plotted versus shear strain for different values of the confining pressure: (a) 100 MPa; (b) 200 MPa, (c) 400 MPa; (d) 600 MPa. As in Figure 11, the insert diagrams display the contours of the instantaneous y-velocity at different shear strains. The contour interval is 5.0 • 10 -5 throughout. Also insert is the finite deformed grid for a shear
The design of the numerical model for the non-isochoric simple shearing deformation experiments is similar to that for the isochoric case except that there are no boundary constraints imposed on the two edges which initially lie normal to the shearing direction.
In the case of Carrara marble (Figure 13), these models are unable to sustain the high shear stresses attained by the isochoric models at low confining pressures. Shear stresses of only 50 to 100 MPa are accompanied by localization of the deformation into a single dilating zone and the numerical experiment is terminated as the quadrilateral finite difference zones collapse into triangles. At higher con- fining pressures, such strong localization does not occur hut rather a more homo- geneous dilation of the upper and lower regions of the model. At 600 MPa and at
a
4UU 400
300 300 = ==
b
0.2 0.4 0.6 ~ 0
Shear Strain
�9
4OO
300
2oo
0 0,2 0.4 O.fi 0 8 1 O
Sh~u " Strain
e
ago =
"~ 300
2QO
100
400
30O
~' 200
100
O.O 0+2 04 g6 Smelt S~a+.
500 d
0 ~ I i p I , ___ j 0.0 0.2 0.4 0.6 0.8 1.0
Shear Strain
0,0 0+2 b+4 D5 OB 1.0 Sh~t~m~D
400
200 ~
100
.0 0.2 0.4 D.6 0.8 1.0
She~r~min
Figure t3 Carrara marble. Non-isochoric simple shearing deformation. Shear stress is plotted versus shear strain for differenl values of the confining pressure: (a) 50 MPa; (b) 100 MPa; (c) 200 MPa; (d) 400 MPa; (e) 600 MPa; (f) 800 MPa. The insert diagrams disl~lay the finite deformed grid for the shear s~rain I~o~ed
in each case.
360 A, Ord PAGEOPH,
500
400
m 300
~ 200
g~ IO0
0
a
i z i i i i -
0.2 0.4 0.6 S h e a r S t r a i n
i
0.8 1.0
8~176 F b 400 ~ ~
300
200 , I
100
0 ~ L O.O 0.2 0.4 0.~
S h e a r S t r a i n
I I
0.8 1.0
500
4DO
3O0
2OO
m
lOO
( :
0 0.8 1.0 i I L - - I i I
0,2 0.4 0,6 S h e a r S t r a i n
d
0 Q 0.2 0.4 0,6 0.8 1.0 S h e a r S t r a i n
Figur e 14 Gosford sandstone. Non-isochoric simple shearing deformation. Shear stress is plotted against shear strain for different values of the confining pressure: (a) 100 MPa; (b) 200 MPa; (c) 400 MPa; (d) 600 MPa. As in Figure 13, the insert diagrams display the finite deformed grid for the shear strain noted
in each case.
Vol. 137, 1991 Deformation of Rock 361
800 MPa, the shear stresses at a shear strain of 1 are similar to those attained in the isochoric models.
Similar macrostructural behaviour with increasing confining pressure is ob- served for Gosford sandstone (Figure 14) although the material behaviour never becomes sufficiently stable for the shear strain to reach a value of 0.5. The extreme localization at 100 MPa is associated with a lower shear stress than in the isochoric case. However, at 400 and 600 MPa, between shear strains of about 0.1 and 0.4, the shear stress is slightly higher than for the isochoric experiments, and is approxi- mately constant or slightly increasing in contrast to the decreasing then increasing shear stress in the isochoric experiments.
5. Discussion
5.1. Crustal Behaviour
We now have sufficient information to allow us to modify the classical model for the strength of the crust which allows frictional slip on pre-existing surfaces near the surface and crystal-plastic behaviour at greater depths (see GOETZE and EVANS, 1979). This modification is independent of the breakdown in the BYERLEE (1968) relationship for which shear resistance becomes independent of the normal stress at some confining pressure (ORD and HOBBS, 1989).
The BYERLEE (1968) relationship refers only to the shear stress required for initiation of slip on a pre-existing plane of weakness at some known orientation with respect to the stress tensor; that is, it describes bulk deformation of the crust only in as much as the deformation is accomplished by movements on pre-existing planes, separated by essentially rigid bodies.
However, we have discussed above deformation of continuum materials accord- ing to Mohr-Coulomb constitutive behaviour. As in the case of deformation accommodated by slip on pre-existing planes of weakness, the material is pressure- sensitive, and independent of variations in temperature and strain-rate.
The EDMOND and PATERSON (1972) experiments were performed without any fluids within either the Gosford sandstone or the Carrara marble, so that the strengths of these materials may only be compared with the shear stress for movement along a fault also under such dry conditions, that is 2 = 0 (SIBSON, 1974). The shear stresses, (o-~ + o-3)/2 , are therefore much higher than those consid- ered by ORD and HOBBS (1989).
Then for Carrara marble undergoing axially symmetric shortening (Figure 15a), the shear stresses at yield are just higher than those for slip along pre-existing faults at depths less than about 5 kin. However, as depth increases, the shear stress increases less with confining pressure and eventually becomes independent of confining pressure, and therefore with depth, in a manner similar to breakdown in
362
a 0
10
i 20
Q
30
40
A. Ord
Shear Stress MPa 200 400 600 800 1000
i 0
10
20 a
30
40
PAGEOPH,
Shear Slrea.~ MPa
, ~oo' ,oo ooo, ~, 1ooo
Shear Stress MPa 0 500 1000 1500 2000 2500 3000
4 0
d Ol
10
E
30
40
Shear Stress MPa 200 400 600 800 1000
T o25 o5
e 0
10
i 20
30
Shear Stress MPa , 200 J 400 , 600 , 8?0 , 1000
f 0
10
20
30
40
Shear Stress MPa 200 400 600 800 ~ 000
l i l T i r I l l I
Figure 15 Shear stress versus depth for the three sets of boundary conditions described and for the two materials. The three lines in each case represent the mechanical behaviour for movement along pre-existing thrust, strike-slip and normal faults for a situation with no fluids present and a static coefficient of friction of 0.75 (SmSON, 1974). (a) and (b) plane-strain and axially-symmetric shortening; (c) a n d (d) isochoric simple shearing; (e) and (f) non-isochoric simple shearing; (a), (c) and (e) Carrara marble; (b), (d) and
(f) Gosford sandstone.
the BYERLEE (1968) relationship for friction. The shear stress at these depths also becomes strongly strain dependent. Estimates of the strength of the crust based on the BYERLEE (1968) friction relation for normal and strike-slip faulting would be underestimates down to about 15 km compared to estimates based on material behaviour. Localization of the material behaviour which might result in a simple
Vol. t37, 199t Deformalion of Rock 363
shearing deformation (Figure 15b) is not favoured at depths shallower than about 10 km because of the high shear stresses required. The shear stresses required for axially symmetric shortening and for a simple shearing deformation at depths greater than about 15 km are similar. However, localization is not favoured at these higher pressures.
The behaviour of the less-dilatant, more contractant Gosford sandstone is quite different. Under axially symmetric shortening (Figure 15d), shear stress increases approximately linearly with depth, and the values of shear stress are only slightly higher than those for slip on a strike-slip fault. Bulk material deformation would therefore occur in preference to slip on a thrust fault. Localization of the deforma- tion, which appears likely at shallower levels, would result in zones of well-deve- loped simple shearing (Figure 15e), at least up to shear strains of 0.75, as the shear stresses under these conditions are lower than for axially symmetric shortening at 5 to 20% axial strain, and are lower also than the shear stress required for slip on a normal fault at a shear strain of 0.25 and depths greater than 10km. Such localization and changing boundary conditions would also be associated with a drop in shear stress of about 100 MPa.
However, in the event that the boundary conditions for a strictly isochoric simple shearing deformation history are not imposed in the crust, we should investigate the ramifications of considering a non-isochoric simple shearing defor- mation history.
In this case, Carrara marble does not develop the large shear stresses at shallow depths. In fact, the shear stresses for a non-isochoric simple shearing deformation (Figure 15c) are less than for an axially symmetric shortening so localization and continuing shearing along a shear zone would be preferred, except possibly at 25 to 30 km depth where the stresses for all three sets of boundary conditions considered are about the same.
A shear strain of 0.25 only was obtained for the non-isochoric Gosford sandstone numerical experiments (Figure 15f). They show a lower shear stress at shallower depths than in the isochoric case so that localization and continuing shearing along the shear zone would be preferred for all depths with this material.
FtSCHER and PATERSON (1989) show for Carrara marble that the Terzaghi law of effective pressure is reasonably obeyed at room temperature. In this case, the relative magnitudes of the shear stress for plastic yield of an intact material and for initiation of slip on a pre-existing surface will remain similar so that the above discussion will hold, regardless of whether or not fluids are present. However, it is obvious from geological considerations, and from the results of EDMOND and PATERSON (1972) and FISCHER and PATERSON (1989), that such a dry, tempera- ture-insensitive, rate-independent model becomes increasingly unrealistic at greater depths and higher temperatures, and when rate-dependent deformation mechanisms dominate the deformation behaviour, particularly when fluid is present. Consider- ation must be given to determining and modelling material constitutive behaviour
364 A. Ord PAGEOPH,
in this manner for crustal levels dominated by cataclasis and movement of elastic grains, rather than by crystal-plastic deformation described by an empirically- adjusted, temperature and rate sensitive plastic flow law.
Of course, one cannot assume that the upper crust is comprised of Carrara marble or Gosford sandstone. Nevertheless, the results described in this paper suggest a model for the crust illustrated in Figure 16, where the uppermost crust is dominated by the Byerlee relationship, which is replaced with increasing depth by a constitutive relationship which describes the brittle deformation of intact rock. At greater depths such relationships would be replaced by increasingly temperature dependent and pressure independent relationships until finally crystal-plastic be- haviour dominates.
6. Conclusions
(i) Constitutive parameters for brittle behaviour are derived for the first time for the deformation of bulk material.
Shear Stress MPa
09
E o
(-
r'~
0 100 200 0 ~ '
10
20
30
40
300 400 I I
Byerlee Behaviour
Brittle Deformation of
,, Intact Rock
Transition from Brittle to Crystal-Plastic
Behaviour
Crystal-Plastic Behaviour
Figure 16 Schematic diagram of shear stress (MPa) versus depth (kilometres) for the deformation behaviour of the
crust.
Vol. 137,
(ii)
(iii)
(iv)
(v)
1991 Deformation of Rock 365
The material properties of friction angle, cohesion and dilation angle have been derived for Carrara marble and Gosford sandstone from ax-
ially-symmetric shortening experiments of EDMOND and PATERSON (1~972) according to a hardening, non-associated, Mohr-Coulomb con- stitutive model. A strong pressure dependence is demonstrated of all these parameters for Carrara marble and of dilation angle at low strains
for Gosford sandstone. There is a strong dependence of friction angle on the axial strain, not on confining pressure, for Gosford sandstone. At low axial strains and confining pressures, Carrara marble is highly dilatant and Gosford sandstone only slightly dilatant or contractant, whereas at high axial strains and confining pressures, Gosford sand- stone is dilatant and Carrara marble is contractant. The contrasting material behaviour reflects the differences in the active deforma- tion mechansims and evolving microstructures for the sandstone and the marble.
Consideration of the material properties and physical conditions required for localization of the deformation for a cohesive, hardening, non-associ- ated Mohr-Coulomb material under different boundary or loading condi- tions allows an improvement in the constraints on the geologic regimes under which shear bands may form in the crust. The results for Gosford sandstone and for Carrara marble predict that localization of the deformation into shear bands is favoured at shallow levels for plane strain and axially-symmetric shortening, and is not favoured under conditions of isochoric simple shearing. Relaxing the boundary conditions in the latter case so that bulk volume change is allowed favours localization. However, in this case, localization is favoured at greater depths for Gosford sandstone but not for Carrara marble.
The strength of the crust may be simply modelled by Byerlee's law for slip on pre-existing faults at the highest and surface levels, by a Mohr- Coulomb constitutive relationship for bulk material deformation at shal- low to deeper levels, and by a flow law describing material deformation by crystal-plastic mechanisms beneath that.
Acknowledgements
I should like to thank Bruce Hobbs for his continuing support for this work, Peter Cundall for discussions regarding the power of FLAC, Hans Miihlhaus for discussions on constitutive models, and Marco Cassetta for drafting the figures. I acknowledge ITASCA for use of the code FLAC.
366 A. Ord PAGEOPH,
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(Received February 17, 1991, accepted November 6, 1991)
