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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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