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The inf luence of non l inear ity and anisotropy on stress measurement results
EinfluB von Nicht-LineariUit und Anisotropie auf die Ergebnisse von SpannungsmessungenL'effet de la non-Iinearite et de I'anisotropie sur les resultats des mesures de contrainte
R.CORTHESY &D.E.GILL, Department of Mineral Engineering, Ecole Polytechnique de Montreal, Canada
ABSTRACT: The non linear elastic and anisotropic behaviour found in many rocks has a direct effect on the stress intensity and orientation
calculated from stress measurement data. The development of a stress calculation model that allows to take into account simultaneously non
linear elastic and anisotropic behaviour gives the opportunity to quantify the error introduced in the evaluation of stresses if these behaviours
are not accounted for. Stress measurement simulations on rocksalt and Barre granite allowed a validation of the proposed calculation model., . I
REsUME: Les comportements 61astique non lin6aire et anisotrope que presentent de nombreuses roches ont un effet direct sur l'intensit6 et
I'orientation des contraintes calcul6es partir de donn6es de mesure de contrainte. Le d6veloppement d'un modele de calcul permettant de
tenir compte simultan6ment de la non lin6arit6 et de I'anisotropie des roches rend possible 1'6valuation des erreurs que I'on introduit si I'on
n6glige cos caract6ristiques de leur comportement m6canique. Des simulations de mesure de contrainte sur du sel gemme et du granite Barre
ont permis de valider Ie modele de calcul propose.
ZUSAMMENFASSUNG: Die nicht-linear elastischen und anisotropoachen Eigenschaften von vielen Gesteinen haben eine direkten Einfluss
auf die Intensitit und Richtung der errechneten Werte von Felsspannungsmessungen. Ein mathematisches Model1 wurde entwickelt welches
es ermoglicbt, beide diese Eigenschaften in die Berechnungen einzubeziehen, und auch die inhiirente Ungenauigkeit durch die nlcht-
Beriicksichtigung derjenigen in den Berechnungen zu qaantiflzieren. Simulierte Spannungsmessungen mit Steinsalz und Barre Granit haben
es ermoglicht, die Giiltigkeit des vorgeschlagenen mathematischen Modelles zu beweisen.
1- IntroductionThe design methodology in rock mechanics requires that the
mechanical properties of the rock and rock mass and the in situ
stress field be known. The confidence one has in the determina-tion or measurement of mechanical properties is general1y not
questioned. Stress measurement results on the other hand, are
often suspected of being erroneous because of a number of factors
which can be divided in two categories: 1) technical factors; 2)
theoretical factors.If we consider stress measurement techniques requiring stress
relief drilling, technical factors include all the possible sources of
error related to the experimental measurement of strains or
displacements. Means of detecting and correcting these errors
have been dealt by many authors (Blackwood, 1978; Gill et al.,
1987; Corth6sy and Gill, 199(1).Theoretical factors could be identified as the difference
between the hypotheses on which the stress calculation model is
based and reality, the usual hypotheses being that the rock is linear
elastic, isotropic and homogeneous and reality being that the rock
has non linear stress-strain relationships,. is anisotropic and.
heterogeneous.The purpose of this paper Is to show the Influence of
anisotropy and non linearity on the In situ stress tensor characteris-
tics obtained from In situ measurements. This Is done by compar-
ing the calculated stress tensor using the usual hypotheses and
introducing alternatively anisotropy and non linearity in the stress
calculation model. The stress measurement technique used to
lIIustrate this Is the doorstopper technique. It was chosen because
the methodology used to obtain the deformablllty parameters allows
to isolate the anisotropic and non linear behaviours and the
Influence of local heterogeneities can be eliminated (Corth6sy and
Gill, 199Oc).
2- Non Ideal mechanical behaviour
Depending on the scale at which it is looked at, the same rock
Can be considered isotropic or anisotropic, homogeneous or
heterogeneous and linear elastic or non linear 'elastic, since the
phenomena responsible for these different behaviours are found at
different scales. It is therefore necessary to define the scales
which are of Importance with respect to stress measurements. Inrelation with anisotropic and non linear behaviour of rocks, three
scales can be defined. The first is the strain measurement scale
which for the doorstopper technique, is the strain gauge active
length. The second is the stress measurement scale which for the
same technique, is the volume of rock upon which boundary
conditions are modified, like stress relief caused by prolonging the
borehole, and stress application required to determine the deforma-
bility parameters as proposed by Corth6sy and Gill (199Oc). These
boundarie~ delimit the core on which the doorstopper cell is glued.
Finally, the third scale, is defined as the volume of rock to which
the stress measurement scale can be extrapolated. It is independent
of the measurement technique and depends on the uniformity of the
stress field surrounding the measurement point. Now that these
scales have been defined, the phenomena responsible for non ideal
mechanical behaviour will be described.
2.1- Deformational anisotropy
Many factors contribute to making rocks anisotropic. First of
all, at the microscopic scale, most minerals are intrinsically
anisotropic and show different types of anisotropy, depending on
the crystallographic lattice they present (Lekhnitskii, 1963). But
the random orientation of the crystallographic axes eliminates the
effect of this anisotropy at a greater scale, unless a preferential
orientation of these axes is present.
, Another cause of anisotropy found at the microscopic scale is
related to the presence of microcracks oriented in a' preferential
direction. This has been observed, amongst others, by Douglass
and Voight (1969) in Barre granite, by Ribacchi (1988) in gneiss
and schists and by Lajtai and Scott Duncan (1988) in rocksalt,
At a greater scale, anisotropy related to primary geological
structures or structures set in place as the rocks formed is encoun-
tered. Sedimentary bedding is a good example of this type of
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structure. Secondary structures or structures induced after the
rocks have formed are also responsible for anisotropy. These
structures are visible in metamorphic rocks of sedimentary origin
like slate or metamorphic rocks of plutonic origin like gneiss.
From this brief review of the potential sources of deformational
anisotropy it is clear that most rocks are bound to present it at
different scales and with different intensity and in most cases,
transverse anisotropy will be found. If anisotropy is ignored in the
stress calculation model, it will affect both the intensity and
orientation of the calculated stresses.
2.2- Non linear elasticity
Non linear stress-strain relationships in the elastic domain or
relationships in which the strains are completely recovered after
unloading, have been associated to the presence of microcracks that
close upon loading as the mean or hydrostatic stress increases and
re-open as this stress decreases, amongst others by Walsh (1965)
and Ribacchi (1988). As mentioned in the previous section
microcracks are also responsible for anisotropy, so the simulta,
neous presence of both phenomena is often encountered and their
intensity will obviously increase simultaneously.
The major effect of non linear elasticity is that the generalized
Hook law cannot be used for multiaxial loadings since this law is
based on the superposition principle and this principle is only
applicable in the case of linear stress-strain relationships. In thecase of a uniaxial loading, no superposition is required and the
secant Young modulus can be used to predict stresses or strains for
given strains or stresses. On the other hand, the solution to
multiaxial loadings requires the use of the fundamentals compo-
nents of material behaviour which are volume change or mean
strain associated to the hydrostatic component of the stress tensor
or mean stress through a parameter known as the bulk modulus K
and the change in shape or deviatoric strains associated to the
deviatoric stresses through a parameter known as the shear
modulus G. This approach was used by Leeman and Denkhaus
(1969) for stress measurement calculations in isotropic non linearelastic rock.
2.3- Heterogeneity
A set of hypotheses which are usually accepted for stress
measurement interpretation, relate to the homogeneity or consist-
ency of the stress and strain fields in a given volume of the rock
mass, as the pointwise stress measurements are extrapolated to the
scale of the excavation being designed. In reality, the consistency
of these fields is related to the heterogeneity of the rock mass at a
scale equal to or greater than the stress measurement scale. As
this subject is rather complex and has been partly covered by
CortMsy and Gill (199Ob, 1991), it will not be dealt with in this
paper.
3- Stress calculation model
In order to evaluate the effects of anisotropy and non linear
elastic behaviour on calculated stresses from doorstopper measure-
ments, it is necessary to have a calculation model that can accountfor the simultaneous presence of both these behaviours. Such a
model has been proposed and described in detail by Corthesy and
Gill for measurements in rocksalt (199Ob) and granite (199Oc). It
deals with transverse anisotropy which introduces second order
phenomena, these being volumetric strain associated with
deviatoric stress and distortion associated with mean stress. To
visualize this, let us suppose that a transversally anisotropic sphere
is compressed hydrostatically. The original spherical shape
changes to an ellipsoid even in the absence of deviatoric stresses
whic~ ~ranslates into second order phenomena. If relationshi~
descnbmg first and second order phenomena are available, mean
~ devi~toric strains measured on a non linear transversally
amsoeropic body can be transformed into stresses using the follow-
ing equations:
(1)
s rr
( 2 )
s .!.[C + .ltkflll2(1 + a)rr 3 ~
Bk2f~(1 + a)2 (3)
+ ~2 J
(1 ~ [C + . I tke rr2 (1 +j!) 3 1-~
B k2 e : a (1 + j!)2 (4)
+ (1 _ ~)2 J
where (1.is the mean stress and sa is the deviatoric stress compo-
nent perpendicular to the intersection of the plane of isotropy and
the bottom of the borehole. fand flD2.arerespectively first and
second order volumetric strains and ell! and eZ12are first and
second order strain invariants in the direction perpendicular to the
intersection of the plane of measurement and the plane of isotropy.
The other parameters are obtained by a biaxial isotropic reloading
(BIR) of the core recovered after stress relief drilling, and on
which the doorstopper is glued. It leads to the following equation,
(5)
where P is the applied biaxial isotropic stress and fu the measured
intermediate principal strain. A, B, and C, are the factors of the
second degree polynomial obtained by regression. k is the ratio
between the intermediate and principal strains recorded during the
BIR and is a function of the hydrostatic stress component.
Parameter ~ is a function of " and k and the parameters a and B
are respectively the ratio between first and second order mean
strain components e/flD2 and first and second order strain in-
variants, ea./eZl2' The Poisson ratio" is obtained from a diametralcompression (brazilian test) on the recovered core. A sensitivity
analysis has shown that putting the Poisson ratios for a
transversally anisotropic material, " I and "2 equal, " I = "2= " ,has very little effect on the calculated stresses and greatly sim-
plifies the calculations (Corthesy and Gill, 199Oc). Equations 1to
4 are solved to find (1 , the mean stress and sa, the deviatoric
stress perpendicular to the intersection of the plane of isotropy and
the plane of measurement, and the ratios a and B . Since theborehole bottom is In plane stress state, the deviatoric stress
perpendicular to this plane, s" is equal to -(1, and since Esu = 0 (first invariant of stress), it is simple to solve for the other stress
invariant in the plane of measurement, as Su = (1 sa' Finally,the shear stress Suis related to the shear strain measured through
the shear modulus G2. This procedure gives the complete stress
state at the bottom of the borehole. In order to derive the far fieldstresses, stress concentration factors as the ones derived by Rahn
(1984) for anisotropic materials can be used.
4- Sensitivity analyses
Sensitivity analyses showing the influence of the degrees of
anisotropy and non linearity on the calculated stress tensor have
been performed. The methodology used consisted in interpreting
stress measurement data obtained from laboratory stress measure-
ment. simulations on rocksalt and Barre granite, both rock types
showmg simultaneous anisotropy and non linear elastic behaviour.
Stresses were calculated with the same strains for various degrees
of anisotropy and non linearity. The advantage of these laboratory
simulations over in situ measurements is that the applied stresses
are known and can be compared to the calculated stresses, and theadvantage over numerical examples is that the validation of.a
calculation model on real materials allows the inclusion of the
"technical factors" mentioned in section 1, thus giving information
on the overall accuracy of the measurement technique and calcula-tion model.
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4.1- Effect or anisotropy
The sensitivity analysis of anisotropy was done by considering
the material to be linear elastic and by calculating the stresses for
different degrees of anisotropy expressed as the ratio ElEz, E1being the Young modulus in the plane of isotropy of a transversally
anisotropic body and ~, the YOIlngmodulus perpendicular to the
same plane. A1Jthis ratio Is varied, the average strain measured
during the BIR was kept constant. Figure 1 shows how the
calculated principal stresses CJIand CJDvary as the degree of
anisotropy changes from Isotropy, EI~ = I to extreme valuesthat must respect some strain energy considerations as the ones
proposed by Pickering (1970). The dashed and dotted horizontal
lines give the calculated principal stresses CJIand CJDif the material
Is considered Isotropic. The difference between the full and dashed
lines are an indication of the error induced by ignoring the rock
anisotropy in the stress calculation model. The Barre granite
sample used in the simulation showed a ratio EI~ of 1.62 which
on the graph gives I1I= 9.27 MPa and CJD= 8.43 MPa, while theapplied load was CJI- CJD- 8.69 MPa, a difference of +6.7% and-3.1% respectively. Had the rock been considered isotropic, thedifferences would have been of +28.4% and -19.3% respectively.
Similar analyses were performed to evaluate the influence of
anisotropy on principal stress orientation. It was found that this
orientation was not too affected when the differences between
deviatoric stresses were important. In the case of a uniaxial stress
field, the principal stress orientation varied from 00 to 100 for
variations for ratios EI~ ranging from 1 to 6.64. Greater stress
orientation variations will occur for stress states where the
difference between the stress invariants are smaller, but again, the
closer to a hydrostatic stress field, the less important are principal
stress orientations in relation to the excavation design.
1200 00 0
, . . . . . 1 0o
a ..~. . . . . .8IIIQ)III
" IIIIII
.b 6III
" 0
III. . .~ 4 a l"(iiio tropYF.f):.~" .
~ all (isotropy)--
o 2 g \ i ' ( f a ~ ~ ' i ~ Q o ~ Q : : ; ~ . . . . . . . . . . . . . . . . .. . .0)".0'11= applied stress _
DO 1 2 3 4 5 6Degree of anisotropy E1/E2
Figure 1: Effect of anisotropy on the calculated principal
stress intensity.
4.2- Ef1'ect or non UnearityAs for anisotropy, the analysis performed here Is based on data
obtained from laboratory stress measurement simulations. The
strains used to calculate the stresses were obtained from a uniaxial
stress field applied on a block of Barre granite. Although the
material showed anisotropy, it was considered isotropic and the
four strains measured during the BIR were averaged, leading to
curve b on figure 2. The other curves were calculated by modify-ing the parameter B of equation S and by keeping parameters A
and C constant. When confronted with non linear stress strain
relationships, the usual procedure consists in taking a secant mod-
ulus somewhere on the stress strain curve (Aggson, 1975). This
modulus is comprised between the slope at the origin and the
60 . . . .~ n .secant slope at maximum strain~ 50 ~!~I?~..~~..~.i~!,:, .n...6.6D .
curve b-
]20)(
g,
:0'010.sQ.
~00 100 200 300
,Average strain from a.I.R.400
Figure 2:' Biaxial isotropic reloading curves for differentdegrees of non linearity O .
secant slope at a certain strain value, usually at the maximum
strain obtained after a stress measurement. Taking these two slopevalues for each of the curves in figure 2, the slope at the origin
being the same for all curves, the stresses shown in figure 3 were
calculated and plotted against the degree of non linearity, 0,
expressed as the ratio between the secant slope at the maximum
recovered strain and the slope at the origin. From this figure, we
see that the effect of non linearity is reflected directly on the calcu-
lated stresses, since the latter are directly proportionnal to the
value of the Young modulus. The greater the non linearity, the
broader the span of Young moduli available to calculate the
stresses. Contrarily to anisotropy, non linearity does not affect the
principal stress orientation. Finally, on figure 3 are also plotted
the applied stresses and the stresses calculated using the approach
proposed by Leeman and Denkhaus (1969). This approach gives
a unique value, independent of an arbitrary choice of secant mod-
ulus.
4.3- Combined effects or anisotropy and non Unearity
Since it would be too complex to represent on a single graph
the combined effects of anisotropy and non linearity, the bar chart
in figure 4 shows the results of the stress measurement simulations
on rocksalt and Barre granite. On the horizontal axis are reported
a series of simulations for which are plotted, on the vertical axis,
the average relative error on CJIand CJDif the effects of anisotropy
and non linearity are neglected, and the average relative error on
CJIand CJIIcalculated by using the proposed model. For each
material used in the simulations, the degrees of anisotropy and non
40
al secant modulus _all secant modulus
al modulus at origin ....................................................................II modulus at origin--
01 Leemon and Denkhaus -a I Leemonand Denkhaua A
... op plLed ..........................II applied
rtI
[l20.h
rtI
.... ..O 0- ..0 .......E)
Figure 3:
2 :5 4 5 6 7Degree of non linearityn
Effect of the degree of non linearity 0 on the
calculated principal stress intensity.
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35
30
~ 25
I . . . .
o
t20(1)(1)
.~ 15. . . . .o(1)
0:::10
(1) (1)I . . . . I . . . .I . . . . I . . . .
o 0ID ffi
(1)
. _ c ! : ! ooI . . . .(1)
>.0_
5
o
l-inear ~ Non linear Isotropic ~ a,nisotropic
FilUre 4: Average relative error on principal stress intensity
from stress measurement simulations on rocksaltand Barre granite.
....r
linearity are reported. The graph shows that the average error for
the stresses calculated when the real mechanical behaviour of the
rock is considered, is comprised between S " and 10", which isin the range of what is usually considered to be the effect of
"technical factors". .
5- DiscussIon
. Many authors have dealt with the effects of anisotropy on
stress calculation. As an example, Amadei (1984) has presented
a sensitivity analysis for stress measurements performed with the
CSIRO cell using a numerical example. No comparison of his
results with measurements performed on anisotropic rocks were
presented. Fewer have studied the effects of non linear stress-
strain relationships. The only valid approach is the one proposed
by Leeman and Denkhaus (1969) for isotropic materials. No sensi-
tivity analysis was performed and again, no comparison with actual
measurements performed on rock were made. Nevertheless, the
validity of their calculation model was confirmed by the experi-
mental results presented here. The model proposed in this paper
is the first that can account' for the, combined presence of
anisotropy and non linearity which occurs frequently as a result of
microcracks. Although it was developed for the doorstopper
technique, the applicability of this model can be extended to other
stress measurement techniques. For example, it would be advan-tageous to use this approach with the CSIR or CSIRO cells, since
measurements in three dimensions are performed and volumetric
stress-strain relationships are readily obtained.' '
6- Conclusions ,
Through the use of a calculation model developed for stress
measurements using the doorstopper cell in anisotropic and non
linear elastic rocks, sensitivity analyses showing how the calculated
stresses vary as the degrees of anisotropy and non linearity vary
have been presented. From these sensitivity curves, it is possible
to evaluate the errors induced on principal stress intensity and
orientation when anisotropy or non linearity are neglected. A
series of stress measurement simulations' on rocksalt and Barre
granite aiso show how the proposed model improves the quality of
stress calculations when the real mechanical behaviour of the rockis considered. These simulations also showed that the approach
proposed by Leeman and Denkhaus (1969) for non linear isotropic
rocks gives good results.
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