1-s2.0-016766369090050p-main

Mechanics of Materials 8 (1990) 309-331 309

Elsevier

S T R A I N S O F T E N I N G A R O U N D C A V I T I E S IN ROCK-LIKE MATERIALS

D.J. GATES

CSIRO Division of Mathernatics and Statistics, P.O. Box 1965, Canberra A C T 2601, Australia

Received 18 May 1988; revised version received 19 September 1989

Recent experimental work casts doubt upon the adequacy of classical and related mathematical models for explaining the stress and fracture within rocklike bodies near cavities, For example, the models predict tangential stresses which are too small very near the cavity walls and zones of fracture which extend too far from the cavities. Here we analyse first a spherical cavity then a circular bore in an infinite rock body with isotropic stress at infinity, using a model introduced recently. The new model is based upon a stochastic network of breakable bonds, in which broken bonds contribute to microscopic fracture surfaces. The new model stresses appear to be more in line with data and the zones of softened rock have smaller extent than those of the classical models. Dilatancy arises naturally in the new model. Softening criteria and ground-response curves are obtained, along with a fairly complete discription of the stress and strain components within the material. A family of surfaces governing potential fractures within the softened zones are described. In the spherical case, simple exact results, of practical relevance, are obtained in the limit of infinite isotropic stress.

1. Introduction

Recently Bandis et al. (1987) reported labora- tory experiments on rocklike material containing a circular bore. By implanting miniature pressure cells in the material they were able to measure both radial and tangential (hoop) stress at various distances from the bore wall and to observe the variation of these stresses as the isotropic stress (pressure), at a relatively large distance from the bore wall, is increased. The experimental data so obtained, although not as extensive as one would really desire, provide now a means of testing the large number of mathematical models for bores that have appeared in the literature (see Brown et al. (1983) for a review). Obviously, such work is important for the understanding of well-bores mine-shaft and tunnels.

Bandis et al. also compare their results with classical theory that involves the Coulomb criterion in the fracture zone and elasticity theory in the intact zone. One notable feature of this comparison (their Figs. 3 and 4) is that the model predicts too large a fracture zone relative to the data. Since the model ignores departure from axial symmetry or any form of local destruction of the

0167-6636/90/$3.50 © 1990, Elsevier Science Publishers B.V.

cavity wall, one would expect it, if anything, to underestimate rather than overestimate the size of the fracture zone. So the discrepancy is definitely in the wrong direction, and casts doubt upon the adequacy of the classical model.

Another notable discrepancy is that the model seriously underestimates the hoop-stress near the cavity wall. Indeed the whole variation of hoop- stress with distance shows little accord between model and data (Fig. 1).

To see these problems more quantitatively for the circular bore, we may look at the variation of the radius P0 of the fractured zone with the hydro- static pressure p at infinite radius. According to the Coulomb failure criterion we find (e.g., Duncan Fama, 1984, eqn, (9); Brady and Brown, 1985, eqn. (7.15)):

P o / a ~ cp 1/~K-1) as p ~ ~ , (1.1)

where a is the bore radius, c a positive constant and

= (1 + s in ~ ) / ( 1 - sin ~) ,

where ~ is the angle of internal friction of the rock. Bandis et al. measured 18 ° < ~ < 20 °, which gives x = 2. Thus (1.1) increases approximately

310 D.J. Gates / Cavities in rocks

B o r e No.1

2 -

1.5

0.5

0 0

V , ' , I a I , ~ 1

1 2 3 4 5

,o/a

1.5

0.5

0 0

B o r e No.2

(b)

+ : ; / ÷

1 2 3 4 5

p/a

1.5

0.5

1.5

: .. , o ¢ 1

• 0.5

O ' p

0 1 2 3 4 5 0

(d)

1 2 3 4 5

p/a p/a Fig. 1. Example of the data of Bandis et al. (1987) from their Figs. 3 and 4 for a miniaturized rock body with a circular bore of radius a = 17.5 mm, and pressures (a) 200 kPa, (b) 350 kPa, (c) 400 kPa and (d) 450 kPa remote from the bore. Squares show measured hoop stresses % and circles show radial stresses % at distances p from the bore axis. The lines show the predictions of the classical model

based on the Coulomb criterion.

linearly with p for their material as p --* oo. Al- though this asymptotic analysis is not necessarily a good mathematical approximation for the pressures of interest, we shall find it a useful basis for comparison of models and discussion of basic principles. The indications then are that, for materials like those used by Bandis et al., (1.1)

gives too rapid an increase of P0 with p, for physically realistic values of r. There are other well known limitations of the Coulomb criterion (e.g., Brady and Brown, 1985, para 4.5.2).

Brown et al. (1983) summarize a number of variants of this model and present an analysis of a model based on the Hoek -Brown failure criterion.

D.J. Gates / Cavities in rocks 311

This criterion is essentially an empirical summary of a large number of triaxial tests and contains several parameters which can be adjusted to fit most triaxial data quite well. Applying this criterion to the circular bore problem they obtain an expression [their eqn. (12); or Brady and Brown (1985, Appendix A3.2, eqn. (6)] for P0 which gives

Po/a - exp(cp 1/2) as p ~ oo. (1.2)

This is a rapidly increasing function of p, so that the model may also tend to overestimate the size of the fracture zone. Brown et al. (their p. 32) note that very large fracture zones are sometimes predicted by their model.

These various models are based upon criteria where failure is characterized by peak stresses, so it is not surprising that the fracture zone increases fairly fast as p increases. While such criteria seem tenable in triaxial, homogeneous problems, they may not be adequate in more general problems. In the present paper, we apply a model (Gates, 1988a, henceforth referred to as GI) based on failure of microscopic bonds under critical strain, rather than stress. The model was shown (Gates 1988b, henceforth referred to as GII ) to explain many features of triaxial tests. We note that, within the framework of classical plasticity theory, strain- space formulations have been studied for some time: e.g., Naghdi and Trapp (1975) and Iwan and Yoder (1983).

To see why strain-based failure might tend to produce the physically more realistic, smaller fracture zones, we consider the circular bore in a perfectly elastic material. The radial component of strain is then tensile for

o /a < [(1 + v ) / ( 1 - 2 v ) ] 1/2, (1.3)

and compressive for larger p, where 1, is the Pois- son ratio. The notable feature is that the critical p does not increase with p. The failure criterion in GI allows internal bonds to break only under extension, so one can see heuristically the tendency for fractures to be confined to the region (1.3). This argument is only an indication of the outcome, and the fracture zone can be somewhat larger in our model. We shall find however that the model

fracture zone radius P0 tends to a finite limit as p ~ oo, in sharp contrast with (1.1) and (1.2).

Incidentally, (1.3) highlights the importance of the Poisson ratio effect in producing fractures around cavities: for if l, -- 0, there is no region of tensile strain.

To explain data on brittle material, one re- quires a model based upon fracture and major, permanent loss of strength (rather than a version of classical plasticity theory, which would allow deformation with relatively small loss of strength, more relevant to metals and soils). The new model, discussed in Section 2, is designed specifically for such brittle behaviour.

For our model, the analysis of the spherical cavity is mathematically very much simpler than that of the circular bore. For the spherical cavity, every radius is an axis of symmetry and hence admits an axially symmetric distribution of small fractures. This feature is lacking in the cylindrical problem. As a consequence, the analysis can be carried further and greater insights gained in the spherical case. Unfortunately we have no spherical data comparable to that of Bandis et al. (1987), but we can at least make tentative, qualitative comparisons with their circular bore data. Strictly speaking, however, our spherical results constitute merely a prediction which awaits experimental test.

The spherical cavity is a simple model for a cave or excavated chamber. On a much smaller scale it is also a model for pores produced by gas bubbles during solidification. Hence, spherical cavities are of interest in their own right.

We therefore develop the analysis first for the spherical cavity. The more difficult circular bore is studied in the later sections (12 to 16) where comparisons with the Bandis et al. data are made. Where the bore theory is similar to the spherical theory, it will be presented only in outline.

We emphasise that the solutions given here, like those in GII , do not describe rock behaviour after large cracks (comparable in size to the cavity) have formed. Such cracks imply departure from spherical or axial symmetry, while this paper considers only symmetric solutions. Our solutions imply the creation of small cracks (much smaller than the cavity) and we refer to the resulting states of rock as softened.

312 D.J. Gates / Caoities in rocks

All the results presented here are mathematical deduct ions f rom the basic s t ress-strain relation (2.1) and the familiar equilibrium equations (4.1) and (4.2), and there are no uncontrol led approximations. Addi t ional physical reasoning or plausible a rgument is invoked only to provide mot iva t ions for and in terpre ta t ions of the mathematical results. We are thus able to show that some quite complex material behaviour can be unders tood solely in terms of (2.1).

2. Stress-strain relations

The model employed here leads to a relation between the strain tensor (dyadic) e and stress tensor a taking the form

o = f d n f ( e , ) n n + aAI (2.1)

where

ke if e~<e 0 (2.2) f ( e ) = 0 if e>e0 ,

e , , = n . e . n (2.3)

A = trace e, I is the identity tensor and k and c~ are constants characteristic of the material, with a << k. The integral in (2.1) is with respect to unit vectors n, so it consti tutes an integral over the surface of a sphere of unit radius. Here o and are functions of location in the material.

We give an outline of the origin of this relation and refer to GI for a detailed derivation. The model begins with a network of spring-like bonds connect ing nodes. The nodes might be the centres of grains within the material, or of some convenient set of elements, and the bonds represent the elastic forces between the grains. If bonds do not break, such a model is known to generate the classical theory of elasticity on a macroscopic scale (see GI , and Love (1944, No te B)). In the new model, bonds break at some critical tensile strain: broken bonds contr ibute to microscopic cracks between the grains or elements. In the special case considered here (and in GI I ) this critical strain has the same value e 0 for all breaka-

ble bonds. One takes the locations of the nodes and the stiffnesses of the bonds to be r andom variables, with a randomness that is homogeneous and isotropic throughout the material (see GI for a more precise statement). Then the mean stress o at some point is related to the mean strain e at the point by (2.1). The first term in (2.1), involving the integral, comes from the breakable bonds and describes the progressive softening of the material as the strain e increases (it is eqn. (1) of GII) . The second term aA1 in (2.1) comes f rom the unbreakable bonds and describes the residual strength of the material at very large strains. If e is so large (tensile) that e, > e 0 for all n, then only the second term remains. We call it a stabilizing term.

The strain componen t e,, is the con t inuum ana- logue of the strain in a bond with direction n at the location in question. The constant k in (2.2) is a measure of the mean stiffness of the breakable bonds. The constant c~ is related to the tensile and bending stiffnesses of the unbreakable bonds (see Section 7 of GI).

For the triaxial s tudy in GI I , no such stabilizing term was included. Its effect is of minor importance there. For problems involving cavities however, strains at cavity walls diverge without such a term.

Our assumption a << k has the consequence that the a term has negligible effect on stresses and strains except very near the cavity wall. One can use forms other than a A l to remove the divergences, so this postulated form is not crucial to most of what follows. We warn against drawing false analogies between a and measures of residual strength or cohesiveness in other models.

At the other extreme (small strains) where e,, < e 0 for all n, (2.1) reduces to

= k f d n nnnn: e + aAl . (I

Using the equat ion following (7.8) in GI for the integral we recover the familiar relation for an isotropic elastic material

o = 2~e + hA1, (2.4)

where

= 4,~k/15 and ~ = 4"rrk/15 + c~, (2.5)


which are the Lam6 constants for the material in its unfractured elastic state. This gives a Poisson ratio

r = ¼(1 + a / ~ t ) / ( 1 + ½a/t~). (2.6)

For a = 0 we obtain the case p = ¼ studied in GII . M a n y brittle rocks have p close to 0.25, so we might confine a/l~ to the range (0, 0.1).

It is impor tant to note that, as soon as e, > e 0 for some set of n values at a location in the material, the relation (2.1) implies that the material is no longer isotropic at the point in question. The broken bonds (and hence small cracks) will have preferred directions so the material will have different degrees of stiffness or softness in different directions. This proper ty of rock-like materials is well known, and it would seem to be a virtue of the model that it incorporates this proper ty in a plausible way (see GII) . Our model differs in this respect f rom other models that preserve isotropy and allow stiffness parameters to vary with location and local stress. Other virtues of our model include its ability to explain, in a natural way, (and give quahtatively correct predictions for) various phenomena occurring in triaxial compressive tests (GII) ; such as peak stress and failure, residual stress, di latancy and the brittle-ductile transition.

3. Stress-strain relation in spherically symmetric geometry

We take spherical polar coordinates (r, 0, q,) for the position x of a point relative to the centre of the spherical cavity. As shown in GI, o and e share principal axes (they commute) and, by the usual symmetry arguments, these may be taken as the basis vectors of the polar coordinates. The principal components are denoted o r, a o, % and er, eO, t ~ , Symmetry implies that o o = % and % = % and that all components depend only on r (despite the unfor tunate but universal use of variables as suffices). The axial symmetry about any radius vector implies that the equations derived in

G I I (6 and 7) carry over, so that (2.1) reduces to

or = 4"rr d v v 2 f [ e ( v ) ] +O(er+2ee)

= 2 " ~ 1 do(1 - v 2 ) f [ e ( v ) ] + ol(~ r Jl- 2e0) , o0

( 3 . 1 )

where v = c o s O and 0 < O < - ~ / 2 is the angle between n and the radius vector x, and

2 (3 .2)

The reader may verify that the spherical cavity solution for a perfectly elastic, isotropic material, with excess pressure at r = ~ , has everywhere the properties

e r > % and e 0 < 0 . (3.3)

We seek solutions with these properties in our inelastic problem.

If t r < t o , t h e n e ( v ) - e r - ( e r - % ) ( 1 - o 2 ) < e r < e o for all v, so that f i e ( v ) ] = k e ( v ) for all v. Then (3.1) reduces to

o r =/~(3e r + 2%) + a(e r + 2e0),

0"0 = ]'~(Er + 4%) + a(e r + 2e0), (3.4)

with our previous /x, and these are the s tandard elastic equations corresponding to (2.4). To under- stand this, we note that for a purely elastic medium, tensile strains are greatest in the radial direction. If the radial strain is insufficient to break bonds (e r < %) then no bonds at all are broken and the rock remains intact and hence elastic.

If e r > e 0 then e(v) < e 0 when v < v o, where

v0 = [(Co - e0) / (Er -- e0)] 1/2 (3.5)

Using (3.3) we find 0 < v 0 < 1. Then (3.1) reduces to

fo °° Or=4~rk d v v Z e ( v ) + a ( e r + 2%)

=/~v3(3eo + 2%) + a ( e r + 2%) (3.6)


and

= 2"~kfo ~° d v ( l - o2)e (v ) + a(e~ + 2eo) oo

= + 7 o- 3 o14]

+ a ( e r + 2e01. (3.7)

If we put v c = [eo/(e o - er)] 1/2 and define O 0 and O c by v 0 = cos O 0 and v¢ = cos O¢, then, at a fixed point in the rock, bonds making an angle O (0 < O < ~r/2) relative to the radical direction are

under compression if O < O c

intact but under tension if O 0 < O < O¢ (3.8)

broken if O < O 0.

(This differs from the triaxial compression case of G I I in that now there is tensile strain along the axis of symmetry and compressive strain in perpendicular directions). In Section 10, these broken bonds are related to fracture surfaces.

while o r is continuous at a spherical interface between softened and elastic regions.

Spherical symmetry also implies that the displacement vector has only a radial component, denoted u(r) , and that

e~ = d u / d r and e o = u / r . (4.6)

Now equations (4.3) to (4.6) and the stress-strain relations (3.4), (3.6), and (3.7) yield non-linear, second-order differential equations for u(r ) with appropriate boundary and interface conditions.

We note that the problem in a medium which is elastic throughout has solutions satisfying

e~(r)><eo for r><r 0

and some r 0, if p is large enough. We look for solutions of this form in general, where r 0 is the radius of the softened zone.

5. Formal solutions--spherical cavity

4. Equilibrium equations--spherical cavity

In GI it was shown that the equilibrium equations of the random network reduce, after taking statistical means, to the familiar equations

V - o = 0 (4.1)

at interior points in the absence of body forces, and

o . U = G (4.2)

at surfaces and interfaces, where N is the unit normal to the surface and G the surface force.

For spherical symmetry (4.1) reduces to (Love, 1944, eqn. (iv), p.91)

dor 2 d--7 + 7 ( o r - o01 = 0, (4.31

for functions Or(r ) and oo(r). For pressures p at r = oo and Pw at the cavity wall, r = a say, (4.2) gives

Or(m ) = - p , (4.4)

° r (a ) = -Pw, (4.5)

Here we find solutions in a form convenient for later explicit calculation and find a condition for the creation of a softened zone. The equilibrium equations can be reduced to a first-order equation as follows. Equation (3.6) gives the explicit function or(e~, eo), so, using (4.6), we can write

dor ,Or dr ~e7 i ) + ~ e o - ~ , '

With (4.3) this gives an expression for ti in terms of known functions. Combining this with the relation

der de r ~ d e o r5 - U r / T r -

gives

d e r

de0 Orl/ o

which formally gives e r as a function of e 0, say

e r = ~(e0). (5.2)

The function (5.2) takes rather different forms in

D . J . G a t e s / C a v i t i e s in r o c k s 3 1 5

the softened and elastic zones, but is cont inuous at an interface. F r o m (4.6),

G = ee + r d e e / d r , (5.3)

which combines with (5.2) to give a differential equat ion for eo(r ) :

r d e e ~ d r = l~(eo) - e o. (5.4)

This integrates to

[ /'~0(a) dx ] r / a = e x p [ j~ o x ---~( x ) ' (5.5)

giving e e implicitly as a funct ion of r, with eo(a )

yet to be determined. Knowing or(G, eo) and getting e r f rom (5.3), we can determine eo(a ) f rom (4.5). Then er, e o, o r, o o are known.

The value eo. o - e o ( r o ) at the softened-elastic interface r = r 0 is ob ta ined by put t ing G = eo in the elastic equat ions ( r > r 0). These have the solu- t ion

u = A r + B / r 2. (5.6)

At r = ~ , w e h a v e G = e e = A and o r = - p . W i t h (3.4) these relations give

eo, o = - 3 p / [ 2 ( 5 # + 3 a ) ] - ½e o. (5.7)

Finally, we can determine r 0 f rom (5.5) and (5.7) to give explicitly

[ f r0(a) d x ] r ° / a = e x p [ j~o, o x - - ~ ( x ) ' (5.8)

provided there is such a softened zone. N o w the elastic solution for the whole med ium has its m a x i m u m radial strain G at the cavity wall r = a. If er (a ) < e0, then all strain componen t s for all r are less than e 0 and there is no softened zone. Thus the condi t ion f o r creat ion o f a so f t ened z o n e is that, for the elastic solution, er (a ) > e 0. This reduces to

(5# + 3a ) P > 3 ( # + a ) ( 2 # e ° + P w ) = P c ' s a y " (5.9)

6. The non-stabilized case--spherical cavity

One of the at t ract ive features of the present model for the spherical cavity is the simple exact solutions that are possible for the non-stabi l ized case a = O. We have already ment ioned (Section 2) the physical inadequacies of such solutions, but they are useful as a limiting case of small a, while their inherent simplicity provides a n u m b e r of insights. It is convenient to change to d imension- less variables

e r = _ e r / e 0

e e = _ e o / e o

s, = - o , / #e 0 (6.1)

so = - Oo/#eo .

These are positive for compress ive strain and stress, in line with the rock mechanics convention. Putt ing

~b = e o - e r

and put t ing the eqns. (3.6) and (3.7), for the (partially) softened state, into (5.1) gives, after some remarkable cancellations,

d~p 5(2e o - 1)~p - 2, (6.2)

dee (2ee- 3)(ee + 1)

a l inear equat ion in ~p. It has solution

(e o + 1)3(2e0-- 3) 2 ~P(eo, e0 ,o )=

( % 0 + 1)2(2ee,o - 3) 2

- 2 ( e e + 1)3(2ee - 3 ) 2

feeO d x (6.3) X o,o ( x + 1)3(2X - - 3) 2

where eo, o - ee(ro) is the value of e e when e r =

- l ( e r = eo). Equat ion (5.7) gives this as

ee.o = 3 p / l O e o # + ½ (6.4)

The condit ion (5.9) for the existence of a sof tened zone, and hence a solution (6.3), reduces to

ee. o > 3 / 2 + p w / ( 2 # e o). (6.5)

We note that the integral in (6.3) can be evaluated explicitly by part ial fractions, giving a relat ion


0,5

0 ~r

-0.5

-1

-1.5

E

I , I , I

1.2 1.4 1.6 r/a

CO

, I , I , I

1.2 1.4 1.6 r/a

1

1

- A o 20

1 1.2 1.4 r/a

0"/.

0.5

, 0 1.6

m

(d)

, I , I , I

1.2 1.4 1.6 r/a

1.5 {,

cr o

1.25 -

/ 1

1

z ' o o

Pw/P 0 . 4

0 . 2

I , I , I 0

1.2 1.4 1.6 0 \ 0.5 I 1.5 r/a 1 A

Fig. 2. (a) to (e) give no rma l i zed s t ra ins and stresses at a d i s tance r f rom the centre of a spher ical cavi ty of rad ius a , as p red ic ted by the new model wi th zero wall pressure (Pw = 0) and no s tab i l iz ing te rm ( a = 0). The different curves co r respond to P/Pc = 1, 1.2, 1.5, 2 and 20, where p is the pressure at r = oo and Pc the cr i t ical pressure g iven by (5.9). (a) Radia l s t ra in G, (b) t angen t i a l s tress ca, (c) vo lumet r i c con t rac t ion - ZL (d) rad ia l s tress o r and (e) t angen t i a l s tress o e. Also shown is (f) no rmal i zed rad ia l convergence A versus normal ized cavi ty wal l pressure p. , , /p for P/Pc values 1 ( the axes of the graph), 1.2, 2 and 20. The d i scon t inu i t i es in g rad ien t s in (a), (c) and (e) m a r k the in ter faces of sof tened wi th elast ic rock.


between e r, e 0 and p in terms of s imple functions. N o w (5.5) becomes

F o r Pw = 0 , we see that the wall condi t ion or(a ) = 0 reduces (3.6) to v 0 = 0 , which implies wi th (3.4) that e r = - - ~ . To see if this is consistent, we note from (6.3) that

- b e ~ a s e 0 ~ o o , (6.7)

if eo, o > 3 / 2 , where b is a posi t ive constant . Thus

e r - -be~ a s e 0--*oo, (6.8)

which is cons is ten t with v 0 = 0. Thus there is inf ini te compress ive hoop strain and inf ini te tensile rad ia l s t ra in at the cavi ty wall ( the divergence of er be ing much s t ronger than that of e0) and all bonds are broken. These divergences are the physical inadequacies , men t ioned in Section 2, of the model with a = 0.

Figures 2a,b show the var ia t ion of e r and e 0 (normal ized by d iv id ing by their value - p / 5 # at = ~ ) with r /a for a few values of the pressure p at infinity. These were ob ta ined by numerica l

in tegra t ion of (6.6) to give r vs. e 0. Then (6.3) gives e r vs. r. We note that bo th e r and e 0 are con t inuous funct ions of r, while e 0 also has a con t inuous gradient . These proper t ies follow from (3.6) and the cont inu i ty of o r. W h e n e r and e 0 are normal ized in this way, we note that they tend to l imi t ing curves as p ~ o¢: in Fig. 2b, the % curve

for P/Pc = 4 would be bare ly d is t inguishable f rom p = ~ a l though its r 0 is s ignif icant ly smaller. This l imit ing behav iour is ana lysed exact ly in Section 9, and shown to provide some s imple and useful formulae. We can find no da ta to compare with these results.

Stresses in the two zones are easily compu ted now from (3.4) to (3.7). F igures 2d,e show the var ia t ion of the normal ized stresses or/p and Oo/p with r / a for a few values of p.

We can get some s imple forms for stresses and s t ra ins near the cavi ty wall as follows. Since v 0 = 0 at the cavi ty wall, Sr(a ) = 0 by (3.6). By (6.7), we have to next o rder

V 0 -- b - l / Z e o 2 a s e0-~ ~ , (6.9)

so that

s o _ c e 81 a s e 0 ~ o o , (6.10)

by (3.7), for a cons tan t c > 0. Thus so(a ) = 0 also, as born out by Fig. 2. Heur i s t ica l ly one would see this vanishing hoop stress at the cavi ty wall as

being a result of the d o m i n a n t d ivergence of e r relat ive e 0 (it is ha rd to give a more sensible physical in terpre ta t ion) . A no teab le fea ture of Fig.

2e is that - o o ( r ) rises rap id ly f rom its zero value, so that there are large hoop stresses at small distances from the cavity wall. This features dis- tinguishes our model markedly from the earlier models. The da t a of Fig. 1 seems in line wi th this feature and we shall make a more de ta i led compar i son when we come to the c i rcular bore. W e note in our model that bo th o r, and o 0 are con- t inuous funct ions of r, while o r also has a con t inuous derivative.

Since e 0 ~ ~ as r $ a, (6.6) and (6.7) give

] r /a - exp d x / ( ax 5 ) [ eo

=exp[1 / (4be4 ) ] ,

so that

e o - - D ( r - a ) -1/4 a s r S a (6.11)

for cons tan t D > 0. This with (6.10) shows that o o - - c ( r - a ) 1/4 as r $ a , which c lear ly d e m o n - strates the large hoop stress near the wall, men- t ioned above and i l lus t ra ted in Fig. 2e.

Typica l ly the process of rock sof tening and fracture a round cavit ies takes m a n y weeks to develop fully, af ter the excavat ion has been made. In the labora tory , Bandis et al. (1987) a l lowed hours for the process to reach equi l ib r ium in their small scale specimens. The inward radia l d i sp l acemen t of the wall, which occurs dur ing this process , is cal led the radial convergence (RC) and its var ia- t ion with the cavi ty wall ( suppor t ) pressure Pw is the ground response curve. Such curves are im- po r t an t for the design of suppor t s t ructures wi th in excavat ions. Brown et al. (1983) take R C = - u(a) where u(r) is their mode l d i sp lacement for their sof tened rock equat ions . Since the elastic response of the rock should occur ins tan teous ly af ter the


excavation, a better representation of the measured radial convergence might be

F = [ f i ( a ) - u ( a ) ] / a = ~o(a) - eo(a ), (6.12)

where

ko(a) = - [ p / ( S ~ + 3a) + ( p - p c ) / 4 / t ]

and the bar indicates solutions for perfectly elastic rock. Figure 2f shows the variation of the normalized radial convergence A = 1"/(p/5t~), with Pw/P for a = 0 and for a range of values of P/Pc- Again we have no data for comparison, but the results are qualitatively similar to the results of other models (cf. Brown et al., 1983). As usual, RC increases with depth of the cavity below ground (with p ) and decreases as support pressure Pw is increased, just as one would expect.

The ground response curves, normalized in this manner, tend to a limiting curve as p ~ oo, this being indistinguishable f rom the P/Pc = 20 curve in Fig. 2f; it would be very close to a P/Pc = 4 curve. The asymptot ic curve is given analytically in Section 8: it has a simple form and is a good approximat ion for moderately large pressure p.

7. Dilatancy--spherical cavity

Dilatancy is one of the most interesting and impor tant properties of brittle solids. In problems with spherical symmetry, for perfectly elastic isotropic solids, the volumetric expansion (dilatation)

A ~ ~r -[- 2e0 (7.1)

has the same (possibly negative) value throughout the solid. For an infinite solid with pressure p > 0 at r = ~ , this value is

- 3p/(5/~ + 3a ) . (7.2)

representing a contraction. When (5.9) holds there is a fracture zone, and (7.2) does not apply in this zone. For a = 0, it follows from (6.8) that

A l e - b e s-~ oo a s r ~ a , (7.3)

so there is infinite dilatation at the cavity wall. For r > a, dilatation decreases cont inuously to the value (7.2) at r = r 0. There is a radius r d, where

a < r d < r 0, such that dilatation occurs in the zone 0 < r < r d and contract ion occurs for r > r d. This is shown in Fig. 2c. where A / ( -- 3p/5l~) is plotted against r /a for a few values of p.

We have found no experimental data on dilatation for cavities or bores, but the ample evidence of triaxial tests suggests its occurrence is general. In the Cou lomb and related models, it does not arise automatically, but can be added in by suita- ble modificat ions (e.g., Duncan Fama, 1984). In the Brown et al. model (1983, Figs. 4 and 5) dilatancy is included as a specific postulate. An advantage of our model is that dilatancy arises purely from the stress-strain relation (2.2), as it did in the triaxial case (GII) .

8. The stabilized case--spherical cavity

Here we briefly examine the effects of including the stabilizing term in (2.1). For c~ ~ 0, the simplifications leading to (6.2) do not apply. In- stead one writes eqn. (5.1) in terms of the variables (6.1) and integrates it numerical ly using Huen ' s (second-order R u n g e - K u t t a ) method. The procedure then follows Section 5. The main result is that divergences at r = a are removed, showing that the second term in (2.1) provides some stabil- ity at the cavity wall, even when Pw = 0. The effect of quite small a/t~ (0.01 in Fig. 3) is very marked on the behaviour of e 0 and o 0 near r = a . The variation of o 0 in Fig. 3a with r looks qualitatively quite similar to the Fig. 1 data. This por tends well for the circular bore where such a compar ison is more meaningful.

The dilation is now finite at r = a. The ground response curves now show finite radial convergence at p,,, = 0, as shown in Fig. 3b. The zone radius r 0 increases slightly with a. The radial stress o r is not markedly affected for a/tx < 0.1.

9. Limiting results for large p--spherical cavity

In Section 1 we pointed out that earlier models, where failure is governed by a peak stress condition, gave softened zones larger than the Bandis et al. data. We used the asymptot ic results (1.1) and


1.5

O" 0

1.25 20

i ~ i , i

1.2 1.4 1.6 r/a

0.6

pw/p f (b) 0.4

0"20

5 1.2 A

Fig. 3. Spherical cavity. Some new model results in the stabilized case with a /~ = 0.01, for various values of P/Pc: (a) normalized tangential stress o 0, showing non-zero o 0 at the cavity wall (cf. Fig. 2e) and (b) normalized radial convergence A versus normalized cavity wall pressure Pw/P, showing finite radial convergence at Pw = 0 (cf. Fig. 2f). The values of P/Pc are (a) 20 and (b) 1.2, 2 and 20.

(1.2) to illustrate this tendency. We have already seen f rom Figs. 2 and 3 that our new model gives relatively small softened zones in the cases considered. Here we examine the asymptotics of r 0 for our new model, showing it to be bounded as p ---> oo. Stresses and strains in this limit are also found, and the results for P/Pc as small as 4, or even 2, are quite close to these limits, showing the asymptot ic formulae, below, to be useful approximations.

We are not asserting that our spherically symmetric solutions are valid as p ~ oo: one would expect large cracks and internal collapse to occur for large enough p. We are merely using the limit p---, oo as a mathematical device for obtaining good approximations to our solutions when p is large but not so large as to cause departure f rom spherical symmetry (about which we make no predictions here; but see the end of Section 17).

First we take a = 0 and Pw = 0, noting that r 0 $ as Pw T. For ~k(e 0, e0.0) given by (6.3), we put

co(y, eo,o) = ~b( yeo.o, eo,o)/eo, o. (9.1)

Substituting this in (6.2) and letting e0, 0 ---, oo (i.e. p ~ oo) gives

dl2 /d y = 5 0 / y - 2, (9.2)

where

12(y) = lim co(y, e0.0). (9.3) eo. o ~

Now ~(eo,o, ea,o)= eo, o - 1 because e r = - 1 when e o = eo. o. Thus $2(1)= 1, and so (9.2) has unique solution

I2= ½(y + y5). (9.4)

Then from (5.8), we have as eo, o ~ oo

] ro/a ~ exp 2 d y / ( y + y ' ) = v~-. (9.5)

Further, ro/a increases monotonica l ly to its limit ~ , so this value is never exceeded. This result contrasts sharply with (1.1) and (1.2). It seems likely that larger zones would occur if spherical symmetry were relaxed, but this is a much harder problem (see Section 17).

N o w we look at stresses and strain in this limit. Putting

y ( r ) = lira eo(r)/eo.o, p--* ~

we have further, for r < ~/2a,

r /a = exp 2 d z / ( z + z 5 )

= (1 +y-1/4)~/2,


so that

y ( r ) = [ ( r / a ) ? - - l ] - ' /4

Combining this with the elastic region solution gives, as p ---, o¢, the very simple formula

eo( r ) /eo( OO )

~ { 3 { ( r / a ) 2 _ l } '/4

1 + v ~ ( a / r ) 3

for 1 < r /a <~ V~

for v~- ~< r /a

(9.6)

Fig. 2d. Even P/Pc as small as 1.5 now gives a curve which is barely distinguishable from (9.8).

Simple formulae for o 0 and other quantities can be obtained easily in this fashion. A wall pressure Pw > 0 can also be included. For example the limiting normalized ground response (c.f. (6.12))

a = lim[eo(a ) - ~o(a)]/~o(oc). (9.9

as p, p,, ~ zc, Pw/P ~ % , is given by

~r w 3 2 3 / 2 y / ( 1 4X 3/2 = . + y ] , (9.10)

where

This almost exactly describes the upper curve in Fig. 2b. We note too that the curves with P/Pc >1 2, in this figure are barely distinguishable f rom the upper curve, so we conclude that (9.6) is quite an accurate solution for P/Pc > 2, a l though ro/a is still somewhat smaller than v~- for P/Pc = 2.

We further have, as p ~ oo, and r < x/2a

e,/eo, o --* y - fa = ½(y _ y S ) ,

which, combined with the elastic region solution, gives another simple formula

er(r)/er(O0)

t - ( 3 / 2 ) { 1 - ½ ( r / a ) 2 } ( ( r / a ) 2 - 1}-5 /4

' - ' ~ for l < r / a ~ v / 2

( l - ( v ~ a / r ) 2 for v/2 <~r/a.

(9.7)

This almost exactly describes the lower curve in Fig. 2a. Curves with P/Pc > 4 are very close to this curve. By a similar argument we obtain from (3.4) and (3.6) as p ~ oo

Or(r)/Or(o0)

(6v~ /5 ) ( a / r )3 { ( r /a ): -- 1} , /4

for 1 < r /a <~ v~

1 - ( 4 v ~ / 5 ) ( a / r ) 3 for v/2 ~< r /a ,

(9.8)

which almost exactly describes the lower curve in

-2-A

We note that % = 3 / 5 when A = 0 . The curve with P/Pc = 20 in Fig. 2f is indistinguishable from (9.10), while a curve with P/Pc = 4 would be very close.

For the physically more realistic model with a > 0, (9.2) is replficed by

d~2 _ 5(~2/y) - 2 + 6 ( n / y ) 5/: (9.11)

dy 1 + a(n/y) '/2

where 8 = a/l~. This has a "cri t ical" solution ~2 = cy for a specific constant c, but this applies only for suitably chosen p, a and /~. More generally (9.11) admits an asymptot ic solution

~2 - ay as y --, oo, (9.12)

but no asymptot ic solution with higher powers of y. This contrasts sharply with (9.4). Thus e , / e o tends to a limit as e 0 becomes large, in contrast with (6.8). The cavity wall condi t ion s t ( a ) = 0 implies that eo(a)/eo, o ~ l i m i t , y (a ) say, as e0, o

oo. Then (6.6) gives after a change of integration variable

ro/a = exp[ feo(.)/ ..... d y/oo( y, e0.o) ] [ r-~(") ]

e x p [ j , d y / f ~ ( y ) (9.13)

as e0, o ~ oo. Thus r o again remains bounded as p ---, oe (in contrast with (9.5). 1/~2 is not integrable on (1, ~ ) but the finiteness of y (a ) now gives a finite integral).


10. Potential fracture surfaces--spherical cavity

We pointed out in Section 3 that broken bonds are confined to angles O < 8 0 relative to any radius vector, where O 0 depends on r. The broken bonds contribute to microscopic fracture surfaces whose normals are these bonds. Hence, any surface whose normal at every point is a broken bond is a potential fracture surface (PFS): for example, all spherical surfaces concentric with the cavity and inside the softened zone. In the present analysis, where spherical symmetry of solutions is imposed, these surfaces cannot be realized as macroscopic fractures. They may indicate possible macroscopic fractures when symmetry is relaxed (remembering that the present non-linear model may have multiple solutions).

One can define a family F of surfaces which form envelopes of these PFSs. Then a surface f rom F has a normal, at each point, that makes an angle 8 0 with the radial direction, while other PFSs have normals that make angles O < ~9 0 with the radial direction. Surfaces in F are called enveloping fracture surfaces (EFS).

Taking a fixed radial direction, say the vertical, and referring spherical polar coordinates (r, 0, ~) to it, we can write an EFS in the form 0 = O(r) (it being a surface of revolution about the vertical, and its equation thus being independent of ~). By simple geometry, the function O(r) is given by

dO r~-~r = cot tgo(r ), (10.1)

which reduces to

_ - - - 7 - < - - s - ' ds , (10.2)

where eo(r ) and er(r ) are our previous solutions within the softened zone (remembering e 0 > 0, e r < - 1). Equat ion (10.2) is most easily computed as an integral w.r.t, e 0 using (5.5), and taking an appropr ia te sequence of steps converging to the integrable singularity at e r = - 1 .

Figure 4 illustrates such EFSs for P/Pc = 16 and ( a ) a = 0, (b) a//~ = 0.5. EFSs are tangent to the elastic-softened interface ( r - - r o ) in view of (10.2). For ct = 0 (Fig. 4a) they meet the cavity

Fig. 4. Spherical cavity. Enveloping fracture surfaces in the model when Pw = 0 and P/Pc = 16. In case (a) a = 0, giving ro/a = 1.37; in case (b) a/tt = 0.5, giving ro/a = 1.51. The surface is obtained by rotating the curve about a central vertical axis.

wall at right angles, in view of (6.8). For a > 0 this is not so: further the fracture zone is larger and the EFS is more extensive. We chose a large a to accentuate this effect in Fig. 4b. The whole family F of EFSs, for given p, a, etc., is obta ined by taking all orientations of the axis of revolution of one such EFS (in the manner of Fig. l l a ) .

We have no data for comparison, but Figs. 8 and 9 of Bandis et al. (1987) are qualitatively similar to the PFSs implied by our Fig. 4 (remembering that our model allows any PFS whose normal, at every point on its surface, is closer to the radial direction than the EFS normal at each point).

As the pressure p ~ oo, we can use (9.5), (9.6)


and (9.7) to obtain another simple formula

O(r)--~ f//-2a[ s2._._/a221 ]'/2s _ , 1 - 1 s 2 / / a 2 ] ds

=f2arcos[ ( r2 /a2-1) '/2]

- a r cos [7~- (1 - a2/r2)l/2]. (10.3)

In particular, at the cavity wall

O( a ) --* (v/2 - 1)~r/2,

so that the limiting surface subtends, at the centre of the sphere, and angle 2 0 ( a ) = ( ¢ 2 - 1 ) v = 74.6 °. The surface with P/Pc = 16 in Fig. 4a is very close to the limiting surface given by (10.3), so the latter is a very convenient explicit formula, appropriate for larger pressures.

11. Uniqueness of the spherically symmetric solution

In the non-stabilized case (a = 0) with no wall support (Pw = 0) our solution has infinite compressive hoop strain at the wall (eo(a) = - ~ ) . We note from (3.6) that the wall condition or(a) = 0 can be satisfied also with e0(a) = - 3 e 0 (rather than v 0 = 0) representing a finite compressive hoop strain. Can we find a solution of this sort?

One finds that a solution of the form (6.3) implies q; = 0 at r = a in this case, so that G ( a ) = - 3 e 0 also. Thus all components of strain are compressive near the cavity wall, implying perfectly elastic behaviour in our model, and that (6.3) is inapplicable. Hence our model admits no solutions of this sort by reductio ad absurdum.

12. Stress-strain relations for axially symmetric geometry

We now set out to find solutions of the model for the circular bore, and make comparisons with the Bandis et al. data. The mathematics here is a good deal more complex than before, but the methods are mainly elaborations of the preceding ones: so the latter provide a helpful basic frame-

work for the present endeavour. Some readers may prefer to go straight to Section 14.

We consider only axially symmetry solutions, (although the model, being nonlinear, has other solutions). Consequently the solutions admit small cracks and describe strain-softening, but do not admit cracks comparable in size to the bore diam- eter.

We again use the stress-strain relations (2.1). Since the stress and strain tensors o and e commute (GI), they share principal axes, giving the usual cylindrical components %, %, ~ and %, %, ez. Through an arbitrary point (0, ~, z) in the rock, draw a radius vector perpendicular to the z axis. Relative to this radius vector take spherical polar angles (O, ~), where O = 0 on the radius vector and ~ = 0 in the z direction, and take associated basis vectors in the standard manner. Then the unit vector n in (2.1) may be written

n = (cos O, sin 0 sin ~, sin @ cos q)), {12.1)

giving

E n ~ Ep

= % - { G - e(q})}( 1 - U2)'

where v = cos O and

e ( ~ ) = e, sin2~ + e Z cos2q ~,

cos20 + e, sin20 sin2q) + e: sinZO COS2(] ~

(12.2)

(12.3)

which is the strain component perpendicular to G" Since dn = sin @ dO d~ , we obtain (c.f. eqn. (2.3) of GII )

£ /0 % = dO sin O cos20 24 dq~f(e ,) + aA

= fo2~ dr~ fol dv o2f(en) q- otA. (12.4)

We turn now to the particular case of the circular bore in rock of infinite extent with constant isotropic compressive pressure at infinite distance from the bore. For the perfectly elastic solution of this problem, in isotropic rock, we note that

e 0 > e z > ~ , for a l l o . (12.5)

(In fact, G is constant while e o $ G and ~, ? G as p ~ oe). We look for solutions of our more gen-


eral equations satisfying (12.5), noting that it implies

G > e. > e ( ~ ) . (12.6)

If e o < e o then, from (12.2) and (12.6), e. < e o for all n. Thus (12.4) reduces to

Op= 2kfo2"~ d~fol dv v2e,,+aA

=/~(3G + % + G) + aA, (12.7 3

where /~ = 4~rk/15 is the rigidity modules. By a similar argument

% = ~t(e o + 3e , + ez) + ,~,a

% = ~(G + % + 3G) + aA. (12.8)

These elastic results simply state that if G is not large enough to produce broken bonds in the radial direction, then no bonds at all are broken.

I f e o > e o then e , < e o if V<Vo, where

OO(f~)= [(F,O--E(I~))}//{F,O--E(~)}] 1/2. (12.9)

In view of (12.6) we have 0 < v o < 1 if G > Co" Now (12.4) becomes

% = 2kfo2~ dq)foV°(~')dv

+ + ° a

__ ].t f02~r 2~r d ~ Vo(dP)3[3eo + 2e(dP)] +aA.

(12.1o)

in terms of the 3 kinds of elliptic integrals but this seems to offer no advantage, so we shall simply evaluate them by Simpson's rule.

We can now give a more detailed interpretation of the equations in terms of broken bonds. This leads to an interpretation in terms of fracture surfaces, given in Section 16. The condition e,, > e 0 or v > v 0, for bonds to be broken, can be written

(co - % S in2~ - G c°s2~) sin20 < % - Co,

(12.12)

which defines the interior of a cone of elliptical cross section, whose axis is the chosen radius vector. We call this the fracture cone. It has principal axes in the z and q, directions (q~ = 0 and ~r/2) and semi-angles 0 z and O, in these directions, where

tan O: = [ ( G - e o ) / ( e o - e z ) ] 1/2= R~, say,

(12.13)

and

tan O , = [ ( G - eo) / (eo _ %)] , /2 = R~, say.

(12.14)

Taking a section perpendicular to the axis of this cone, at unit distance from its vertex, gives the ellipse illustrated in Fig. 5. It has semi-axes R z and R , with R Z > R, . One can think of the cone being defined by the curve formed by the intersec-

Similarly (c.f. eqn. (4) of GII) we find

= 2kf02~ d ~ sin2(/)f0v°(~' d r (1 - v 2) %

× + + ° a

_ i tt fO 2~ 2~r d ~ sin2~vo ( ~ )

× [ 1 5 e ( ~ ) + ( 5 % - 7 e ( ~ ) - 3e o } v o ( ~ ) z]

+aA, (12.113

while o~ is obtained by replacing the factor sin2~ by cos2~ in (12.11). These integrals can be written

compressed

R;

R,

ti t Fig. 5. Circular bore. Projecting the fracture cone (12.12) and the extension cone (12.16) onto the plane perpendicular to a radius vector.


tion of the quadratic surface X . e. X = e 0 with the unit sphere I X I = 1.

To summarize, every point within the softened (fractured) zone has an associated elliptic cone: the broken bonds at that point have directions lying within that cone.

Likewise, bonds which are under tension are contained within the larger elliptical cone

( e o - e , s i n 2 ~ - e: COS2~ 1) sin2@ < eo, (12.15)

with semi-angles O,' and O~ where

tan O~' = I%/e: I ' /2 = R'., say (12.16)

and

tan O~, j eo/eq ,Ix~2 = , ' = R~, say. (12.17)

The normal section (as above) is an ellipse with !

semi-axes R" and R , , and illustrated in Fig. 5. We note that, by contrast, the fracture cone by the spherical cavity ( O < O 0 in (3.8)) was circular, resulting in relatively simply stress-strain relations.

13. F o r m a l s o l u t i o n s - - c i r c u l a r b o r e

The equilibrium equations are (4.1) and (4.2). For axially symmetric, z independent geometry, o and e depend only on the radius coordinate 0, and the equations reduce to

do 1 d~ + 0 ( % - o * ) = 0 , (13.1)

with

%(oo) = - p (13.2)

and

Go(a) = -Pw, (13.3)

where a is the bore radius, p the pressure at infinity and Pw the internal pressure on the bore wall. Further, % is continuous at a cylindrical interface between softened and elastic regions.

In this geometry the displacement vector has no q, component, uniform z component and variable

p component u(p), say. Then

~ = - p / ( 5 1 ~ + 3c~) e o = d u / d o (13.4)

% = u / o .

With the stress-strain relations, we thereby obtain a second order differential equation for u ( p ) with appropriate boundary and interface conditions. We can reduce this to a first-order equation in the manner of eqn. (5.1), giving

/

de° % % + (13.5)

where the partial derivatives of % are taken from the function %(e o, %) given by (12.7) and (12.10). We note that the problem in a medium which is elastic throughout has solutions satisfying

e o(p)><e 0 forp><p0

and some 00, if p is large enough. We look for solutions of this form in general, where P0 is the radius of the softened zone. Writing the solution of (13.5) formally as

eo = ~ (e , ) ` (13.6)

we obtain, following eqns. (5.2) to (5.5),

exp ~,(a~ (13.7) o / a = o x - ~ ( x ) '

giving e, implicity as a function of p, with eq,(a) yet to be determined. Knowing %(eo' e , ) and getting e o from (13.6), we can determine e , ( a ) from (13.3).

The value e,, o =-%(0o) at the softened-elastic interface p = Oo is obtained by putting e o = eo in the elastic equations (p > Oo)- These have the well known solution

u = A p + B / p .

Hence at p = o c we have e o = e , = e Z = A and o o = - p . With (12.7) these relations give

~,.o = - 2 p / ( 5 ~ , + 3~) - ~o. (13 .8)

The condition ~(e,, 0) = e 0 fixes the integration


0 .5

0 Cp

-0 .5

-1

-1 .5

(a)

1.5

f

i I

2 p / a

c~

3

1 1 1.5 2

p/a

1 1

1 /

-3 ,

0

-1

)

1.5

O'p

0 . 5

, I 0

2 p / a

(d)

/2f , I , I

1.5 2 p/a

2 1

1--'-"

~ O" z

1.5 0.5

1 0 1 1 . 5 2

I a I

1.5 2 p/a

Fig. 6. Circular bore. Predictions of the new model, with a = Pw = 0, for variation of certain normalized quantit ies with distance P from the bore axis. The different curves correspond to P/Pc = 1, 1.2, 1.5, 2 and 20, where p is the pressure at O = o0 and Pc is the critical pressure given by (5.9). (a) Radial strain %, (b) hoop strain %, (c) volumetric contraction - A (d) radial stress %, (e) hoop stress o,~ and (f) axial stress o~ for pressures P/Pc = 1, 1.2, 1.5, 2 and 20. Gradient discontinuities mark the interface between the elastic and softened zones in (a), (c), (e) and (f).


constant, so that % is given uniquely by (13.7). Then e o, %, % and o z are determined functions of p, and Po is given as in (5.8) provided P0 > a, i.e. that there is a softened zone. As in Section 5, the

condition for occurrence o f such a zone is that ~ p ( a ) > e 0, where ~p(a) refers to an elastic medium. Using (13.3) this gives (coincidentally) (5.9) again.

14. Numerical results and the circular bore data

Unlike the spherical case, no great simplifica- tion is achieved in the non-stabil ized case a = O. but we shall focus mainly on this case for comparison with the spherical results and because it is the limiting case of small a. The numerical procedure is then formally the same as that for the spherical case with a 4= O.

B o r e No .1

1.5

0.5

0 0

(a,) 2 -

1.5 -

o',~

0.5 -

I , I , I , I O ,

2 3 4 5 0

p/a B o r e N o . 2

(b)

, I ~ I , I = I

2 3 4 5

p/a

1.5

0.5

|

0

(c) 2

O'p •

/. , I , I I

2 3 4

p/a

1.5

0.5

(d)

o-~

f •

, I , I

2 3

p/a

I 0 i i I

5 0 1 4 5

Fig. 7. Circular bore. Comparison of model predictions and data of Bandis et al. for normalized radial and hoop stresses. The model results are for P/Pc > 4 (for which all curves are virtually concident). The data comes from (a) model tunnel 1 Fig. 3a of Bandis et al., (b) model tunnel 1 in their Fig. 3b, (c) model tunnel in their Fig. 4a, (d) model tunnel 2 in their Fig. 4b. The bar lines represent the edge views of their disc shaped sensors drawn to scale. A sensor presumably measures an average stress over its surface.


Stresses and strains are normalized so that they tend to 1 as p ~ oo. Figure 6 shows the variation of the normalized quantities with p/a for a = 0, Pw = 0 and P/Pc = 1, 1.2, 1.5, 2 and 20, where Pc is given by (5.9) and reduces here to Pc = 10/~e0/3. We note that the softened zones are somewhat larger than those for the spherical model: their radius Po is found numerically to tend to a limit Oo/a ~ 1 .65 . . . a s P/Pc ~ oo, compared to ro/a - -*~- in the spherical case. The continuity properties at the softened-elastic interface are the same as in the spherical case. Computa t ion suggests that stresses and strains normalized in this way, tend to limits as p ---, oo. One finds that the curves with P/Pc >~ 20 are indistinguishable on the scale shown, so the P/Pc = 20 curves are essentially the limits. Further, the curves with P/Pc = 4 are very close to the latter, so that the limiting curves are a useful approximat ion for rather small values of P/Pc. The limiting functions were obtained exactly for the spherical cavity in Section 9. The more difficult cylindrical analysis is not carried through here.

In compar ing these solutions with the data of Bandis et al., we note that they measure the uniaxial compressive strength of their material as --- 100 kPa. Equat ing this to the value (5/2)v~/~e 0 predicted by our model (GII , Section 6) we find /~e 0 = 16.3 kPa. Then f rom (5.9), Pc = (10/3)/~e0 = 54.4 kPa. Bandis et al., in their bore experiments, take p in the range 200 to 450 kPa, so we have P/Pc in the range 4 to 9. For such values of P/Pc, our model predictions for (normalized) radial and hoop stresses are very close to their p ~ 0¢ limits. These latter are compared with the Bandis et al. data in Fig. 7. The fit is considerably better than that of the classical theory based on the Coulomb criterion (Fig. 1). In particular, the hoop stress, which the classical theory predicted very poorly, now seems to be adequately represented.

The normalized radial convergence A is defined by (cf. (6.12))

A = [e¢(a) - } , ( a ) ] / k , ( o e ) , (14.1)

where ~, refers to the solution for a perfectly elastic medium. Figure 8 shows the variation of A with normalized wall pressure Pw/P, for P/Pc =

0.6

0.4

Pw/P

0.2

0 a 2 3 1.2 A

Fig. 8. Circular bore. Ground response curves showing the variation of normalized wall pressure Pw/P with normalized radial convergence A given by (14.1), for pressures P/Pc = 1.2, 2 and 20.

1.2, 2 and 20. Again the curve with P/Pc = 20 is virtually the limiting curve as p --* oe. The curves have a vertical tangent on the A = 0 axis at the point

Pw/P = 3( 1 - P J P ) (14.2)

and they asymptot ic to Pw = 0 as A---, oo. This divergence in A in the absence of wall pressure is eliminated by including the stabilizing term ( a > 0), giving results like Fig. 3b.

For cylindrical problems, with O dependence only, in perfectly elastic isotropic solids, the volumetric expansion (dilatation)

A = e ° + e¢ + e z (14.3)

has the same (possibly negative) value throughout the solid. Figure 6c illustrates the case a = 0 showing the variation of normalized A with p/a, for P/Pc = 1, 1.2, 1.5, 2 and 20. As in the spherical case there is a zone of volumetric expansion or dilatance, extending f rom the cavity wall part way into the softened zone.


15. Asymptotic results for the circular bore

The behaviour of stresses and strains near the cavity wall can be obtained in the manner of Section 6. Here we require an asymptot ic analysis of the more complex stress-strain relations (12.10) and (12.11), and this is quite cumbersome. We merely quote the main results. If a = 0 and Pw = 0, we find that, as p $ a,

% - - A ( p - a ) -1/4

ep -- B ( p -- a ) -5/4

(15.1) a~ -- - - C ( p - a) 1/'

-- ) 5 / 4 % - D ( p - a

where A, B, C and D are positive constants. We recall (Section 6) that the spherical results showed a similar r dependence. Again, these illustrate the large compressive hoop stress % near the wall.

If a > 0 or Pw > 0 (or both) the above divergences in % and ep are removed, as in the spherical case (cf. Fig. 3).

We can a t tempt to s tudy model behaviour in the limit p --) oo, in the manner of Section 9. The analysis here is rather daunt ing and to data I have only a p roof that (like the spherical case) P0 has a limit as p -~ o¢ for all a and Pw. I have no explicit expression for this limit, but as ment ioned in the previous section, numerical analysis gives po/a --) 1 .65 . . . f o r a =pw = O.

16. Fracture surfaces--circular bore

As discussed in Section 12, broken bonds at a point x in the softened zone are confined within an elliptical cone (12.12), the fracture cone C, whose axis is the radius vector ( _1_ z axis) through x. Its principal axes are in the z and ~ directions and are given by (12.13) and (12.14), these de- pending on p through the strain components .

As in Section 10, any surface whose normal at every point is a broken b o n d is a potential fracture surface (PFS): for example, all circular cylindrical surfaces concentr ic with the bore and within the softened zone. The family F of PFSs that have

C

Fig. 9. Circular bore. A typical fracture cone C, the orthogonal Monge cone M and a possible potential fracture surface (PFS). Any PFS must lie outside the cone M, and the PFS normal n must lie within the cone C.

normals on the surfaces of the cones (i.e. they are generators of the cones) envelope the whole set of PFSs. In other words, the tangent planes of all PFSs at a point are bounded by the envelope M of the tangent planes of the surfaces f rom F at that point (Fig. 9). We note that M itself is an elliptic cone, and we call the PFSs in F the enueloping fracture surfaces (EFSs).

A basic theorem (Courant and Hilbert, 1962, Chapter II, para. 3) is that the family F of surfaces can be represented by a first-order (non-linear) partial differential equation whose Monge cones are the Ms. In general, F contains a two-parame- ter family of surfaces. In our case it is clear that they are obtained by taking one surface and making arbitrary translations along and rotat ions a round the bore axis.

We look for an EFS representat ion of the form p = p(q,, z) in terms of our usual cylindrical coordinates (O, ~, z). Such a surface has normal

, ,= ( -1 , o-' ap/aq,, op/Oz). (16.1)

The surface of the cone C at (p, ,/,, z) has, accord-


ing to (12.12), the equation

(% - e, sin2~ - e: cos2~) sin20 = ep - e0.

(16.2)

N o w O and • give the polar angles of n, with O measured relative to the p axis and • relative to the z axis• Thus

t an I

tan 0 (16 •3) , a: j ] Substituting these formulae for O and • in (16•2) gives the required differential equation

(16.4)

where ep(p), e , ( p ) are our previous solutions of the equilibrium equations in the softened zone.

No te that on the softened-elastic interface, where ep=eo , we get (since e , < 0 and e z < 0 ) f rom (16.4) 0 p / O ~ = 0 p / 0 z = 0 , so the EFS is tangential to this interface. Since the fracture cone C degenerates to a radius vector here, the result is consistent.

On the bore wall ( p = a ) we note that, for Pw = a = 0, we have O,~ = @: = ~r/2 (e o 1" oo fastest in (12.13) and (12.14)). Thus, at a point where an EFS meets the wall, the fracture cone degenerates to a tangent plane to the wall and the Monge cone degenerates to a normal line to the wall. Thus O p / O ' l ' = O p / O z = o0 at the wall, and this is consistent with (16.4) where ep --, oo, e J e o ~ 0. Thus an EFS meets the bore wall at right angles in this case.

Consider the particular EFS through the point P = P0, ~ = 0, z = 0. An indication of its shape is obtained by looking at curves formed by its inter- sections with the planes z = 0 and ~ = 0. In the former case, we note by symmetry that O p / ~ z = O,

so the intersection gives a curve with equation

l d p _ ( eo--eo ] 1/2 '

p d~ e o - e, ]

(b) Fig. 10. Circular bore. Projections of an enveloping fracture surface given by (16.4) for pressure P / P c = 20 (a) onto a section perpendicular to the bore axis and (b) onto a section containing the bore axis.

having solution

fro dp,1 = P %(P')- eo

One such curve is shown in Fig. 10a and a set of them in Fig. l l a . The PFSs implied by these would seem to be qualitatively consistent with the crack distribution observed by Bandis et al. (1987). When a > 0, Fig. 10a is modif ied in a manner resembling Fig. 4b.

On the section ~ = 0 we get Op /O~,=0 by symmetry, giving a curve with equation

d__p_p= _ ( ep - e____~o/1/2

d z e o - e: ] '


(b) . / h / 1

Fig. 11. Circular bore. Sets of the projected surfaces shown in Fig. 10.

having solution

1/2 z = d p' e o - e~

Figure 10b gives one such curve and Fig. l l b a set of curves. Figures 10a and 10b give a fairly clear impression of the bowl shaped EFSs .generated by (16.4). We have no data to compare with Figs. 10b and l l b ; only the frequent reporting that "flakes" of rock are typically shed by bore walls.

17. Discussion

The work presented here has a very specific and limited purpose; to note the inadequacies of classical solutions to the cavity and bore problems in brittle, rock-like materials; to present the predictions of a new model; and to compare these predictions with recent data. The comparison

shows an improvement over the classical predictions (cf. Figs. 1 and 7) especially with regard to hoop stress.

Having set these limits, it seems undesirable to widen the discussion greatly at this point. Mecha- nisms and models for deformation, rupture and fracture comprise a massive literature and are currently under active study. Relatively few models have been developed to the extent that predictions for bores are available (see Brown et al., 1983). Broadly speaking, models that involve classical plasticity theory, or modifications thereof, are least successful for the brittle rock problem. Models that try to summarize the effects of cracks (with permanent damage and major loss of strength, like the Brown et al. (1983) model) are more successful.

It seems very desirable that other authors should test their models against the new data. Also at- tempts should be made to duplicate the experiments of Bandis et al., and variants thereof, obtaining more information about stresses and strains within the material in a variety of geometries (two parallel bores, for example). Previously, uniaxial and triaxial tests have been the most extensive and accurate sources of data for assessing macroscopic models. They provide an indispensible but never- theless a very limited data-base for inventing and testing models of a field-theoretic type. (Hypothet- ically speaking, observations on direct currents in straight wires provide a similarly limited data-base for the invention of electromagnetic theory). One must experiment with other geometries. Experi- ments of the Bandis et al. type are certainly difficult to perform and replicate consistently, but their practical importance would seem to justify a major effort.

We close by listing some of the appealing features of the new predictions, and mention some unsolved problems. On the practical side, (i) the model shows how isotropy of the material stiffness is destroyed by large enough anisotropic strains. (ii) The model predicts dilatancy in the softened zone in a natural way (Figs. 2c and 6c). (iii) The model predicts fracture surfaces (the PFSs of Sec- tion 10 and 16) which, although constrained (by the EFS), are flexible enough to encompass the variability and irregularity of observed fractures (a


feature not shared by classical models). (iv) The stress and strain components are all cont inuous through the softened-elastic interface. The radial derivatives of both tangential strain (e 8 or e , ) and radial stress (o r or op) are also continuous, a proper ty not shared by classical and related models (Jaeger and Cook, 1979, Section 16.5; Brown et al., 1983; Duncan Fama, 1984). Ceteris paribus,

such continuity seems desirable. On the mathematical side, (v) the simplicity and linearity of eqn. (6.2) for the spherical cavity is quite remarkable, in view of the strong nonlinearities involved in eqns. (3,5), (3.6), (3.7) and (5.1). Adherents to the truth in economy view of science would see this discovered (not introduced) simplicity (along with the simplicity of the triaxial equations in GI I ) as lending some credence to the model. (vi) For large loading pressures p (spherical cavities deep underground), simple algebraic expressions can be obtained for stresses, strains and fracture surfaces (eqns. (9.5) to (9.10), and (10.3)). It would be useful to carry through this large p analysis for the circular bore.

An outs tanding unsolved problem for the present work is to find spherically, or axially, non-

symmetric solutions for the spherical cavity and circular bore with isotropic p, and to make predictions about large, strain-created cracks and about incipient collapse of cavities and bores. One would expect the phenomenon to manifest itself mathematically as a spontaneous breakdown o f s ymme t ry

at higher pressures p. The physical concept of localization is a similar idea (Rudnicki and Rice, 1975). The free energy principle given in G W I and G W I I provides a means of selecting from the multiple solutions that arise when such symme- tries are broken, but it remains a daunt ing problem. One would certainly expect to find larger zones of softened and fractured material than those provided by our symmetric solutions. Obvi- ously, predictions of this sort are potentially of great engineering importance, especially to the

mining industry. However, it seems doubtful that the data now available could provide an acid test of such predictions.

Acknowledgement

The author is indebted to S.K. Arya, E.T. Brown, F. deHoog, M.E. D u n c a n Fama, P. Hornby, M.S. Paterson and M. Westcot t for helpful comments.

References

Bandis, S.C., J. Lindman and N. Barton (1987), Three-dimen- sional stress state and fracturing around cavities in over- stressed weak rock, in: Proc. ISRM, Montreal, p. 769.

Brady, B.H.G. and E.T. Brown (1985), Rock Mechanics (for Underground Mining), Allen and Unwin, London.

Brown, E.T., J.W. Bray, B. Ladanyi and E. Hoek (1983), Ground response curves for rock tunnels, J. Geotech. Eng. ASCE, 109, 15.

Courant, R. and D. Hilbert (1962), Methods of Mathematical Physics, Vol. H, Interscience, New York.

Duncan Fama, M.E. (1984), A new constitutive equation for a Coulomb material, in: Design and Performance of Under- ground Excavations, ISRM/BFS, Cambridge, p. 139.

Gates, D.J. (1988a), A microscopic model for stress-strain relations in rock 1. Equilibrium equations, Int. J. Rock. Mech. Min. Sci. 25, 393.

Gates, D.J. (1988b), A microscopic model for stress-strain relations in rock-like materials II, Triaxial compressive stress, Int. J. Rock. Mech. Min, Sci. 25, 403.

lwan, W.D. and P.J. Yoder (1983), Computational aspects of strain-space plasticity, J. Eng. Mech., ASCE, Vol. 109, 231-243.

Jaeger, J.C. and N.G.W. Cook (1979), Fundamentals of Rock Mechanics, Chapman and Hall, London, 3rd edn.

Love, A.E.H. (1944), A Treatise on the Mathematical Theory of Elasticity, Dover, New York.

Naghdi, P.H. and J.A. Trapp (1975), The significance of for- mulating plasticity theory with reference to loading surfaces in strain space, Int. J. Eng. Sci. 13, 785-797.

Rudnicki, J.W. and J.R. Rice (1975), Conditions for the localization of deformation in pressure-sensitive dilatant materials", J. Mech. Phys. Solids 23, 371.

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