Orthotropic Lame formulation of the constitutive equations for brittle rock

Mechanics of Materials 13 (1992) 193-205 Elsevier

193

Orthotropic Lam6 formulation of the constitutive equations for brittle rock F.A, Vreede Division of Earth, Marine and Atmospheric Science and Technology, CSIR, Pretoria 0001, South Africa Received 11 April 1991; revised version received 15 November 1991

Analyzing the results of standard uniaxial compressive tests on rock, the two terms of the isotropic Lam6 equation are viewed as the description of two models. By letting the cohesion model become orthotropic through continuous fracturing, as described by Dougill, and the bulk model become orthotropic through overcoming Coulomb friction, as described by Vreede, constituent equations for brittle materials are formulated. Assuming only tensile fracturing, the fit with the test data is excellent. The equation for brittle failure is derived from the incremental form of the orthotropic equations and is a function of tangential stiffnesses and boundary conditions. Cleavage failure depends strongly on tensile strain. The effect of confining pressure and the orientation of the plane of shear failure depend on the friction angle, but also on cohesion. The application of the theory to a number of geomechanical topics is discussed shortly. Rock with high friction, such as granite, exhibits Class II post-failure behaviour as described by Wawersik. The theory explains all the features of brittle behaviour. Formulae for Young's moduli E i and Poisson's ratios viy are given. The equations are therefore quite suitable for engineering applications.

1. Introduction

The behaviour of engineering materials is usually described in terms of the mathematical notions of continuity and homogeneity over macroscopic distances. Although it is known that these notions are invalid over microscopic distances, the results are adequate for practical use when the mass is large compared with the intervals between discontinuities and inhomogeneities. Stress and strain can then be described by means of 2nd order tensors in a phenomenological model. Classical engineering theory assumes that material behaviour is isotropic and linear up to the point of failure, but real brittle materials, such as rock and concrete, exhibit important anisotropic and non-linear features. Correspondence to: F.A. Vreede, Division of Earth, Marine and Atmospheric Science and Technology, CSIR, P.O. Box 395, Pretoria 0001, South Africa.

Crack formation has been identified as the main cause, starting before failure and continuing after failure (Bieniawski, 1969). Dougill (1976) introduced the progressively fracturing model, which is almost purely phenomenological. His theory is based on the generalised Hooke's law and establishes a flow rate for the degradation of the generalized modulus in 4th order tensor mechanics. Dougill's model was an important milestone in the development of a relatively new branch of continuum mechanics called "damage mechanics" (Janson and Hult, 1977). In a comprehensive study under this title (Krajcinovic, 1989), the section on phenomenological models begins with the sentence: "It is often argued that the ultimate task of engineering research is to provide not so much a better insight into the examined phenomenon, but to supply a rational predictive tool applicable in design." The present study was undertaken in order to

0167-6636/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

F.A. Vreede / Orthotropic equations for brittle rock

194

cq {MPo) t rol~ve r,~

)00

50 I i I

0

300

J 2

i

I

3

4

[ 5

Jl

tions in 2nd order tensor mechanics and then allows the stiffness of the cohesive structure to deteriorate according to Dougill's model. The ultimate results are orthotropic constituent equations, able "to supply a rational predictive tool applicable in design", but also "to provide better insight in the examined phenomenon". Microscopic phenomena are not analyzed, but are accepted as fundamental.

o-~ (MPa]

2. The Lam~ formulae in tensor notation

2SO I

1501

I00

5O

0

/

///

(b) GI~ANfI'E

/ I I

I 2

I 3

((10 -3 ) I • 4

Fig. 1. R e s u l t s of s t a n d a r d uniaxial c o m p r e s s i v e tests. (~r =

stress, e = strain).

find suitable engineering parameters for describing the response of rock samples to standard uniaxial compression tests. Figure 1 shows the characteristic strain-softening behaviour, just visible in the axial strain and pronounced in the transverse strain. It was assumed that the transverse tensile strain softening is the cause of the axial compressive strain softening in the following manner: (i) loose material is compressed in the axial direction and expands in transverse directions; (ii) a cohesive structure, reacting to the axial and transverse strains, keeps the material confined; (iii) the confinement decreases under increasing tensile strain; (iv) the axial strain increases proportionally. The paper starts with isotropic elastic equa-

The following conventions are used for the notation of 2nd order tensors: (a) multiplications in ordinary brackets are summated over all identical indices. Two identical indices on one symbol imply equality of the indices, but no summation; (b) two different indices may be equal, but a comma between them establishes that they are unequal; (c) a single index refers to a principal axis of the tensor. The system consisting of loose material (bulk model), working in parallel with a cohesive structure (cohesion model) is embodied in the classical formulation of an isotropic elastic material by G. Lam6 (1795-1870). The internal energy U is the sum of two functions of the strain tensor Eii (Saada, 1974) u =

1

2

+

1

(I)

where A a n d / z are the Lain6 constants and 6it is Kronecker's delta. In the second function ~iz is written instead of the more usual /~ in order to avoid a factor 2 in the stress formulae. The first part of the energy 1

U 1 = -~A(~ij~.ij )

2 (2)

belongs to the isotropic bulk model. The bulk stress tensor sir is derived from U 1 by differentiation 8U1 Sir

OEij

X~ij( (~ir(ij) = X(~ire ,

(3)

F.A. Vreede / Orthotropic equations for brittle rock

195

where e = (~se/s) is the volumetric strain. The components of the tensor sis are hydrostatic:

sii = Ae, si, ~ = 0,

normal stress shear stress

(3a) (3b)

The strain tensor satisfying (~iseis)= 0 is known as a deviator and produces no internal energy or stress in the bulk model. The second part of the energy U 2 -~- II.¢(EijEij )

(4)

/7//111 i//

/I //// i I I

iI

I

//

//

I

I

Io) Plosiic A~ pl

T

r AT f

belongs to the isotropic cohesion model. The cohesion stress tensor zij is derived from U2 by differentiation

ou2

i

"rij -- OEij - ],,tEij

(5)

and conforms to Hooke's law: each stress component affects only the corresponding strain component. Substituting Eq. (5) in Eq. (4) yields U2 = -~( 1 Tij'rij) / / ~ .

(4a)

It follows from Eqs. (5) and (4a) that

(6)

El/ = "rij// Id, = O,l.ij

The total stress tr~s is the sum of the bulk stress sis and the cohesion stress ~'ij and is derived from U by differentiation 0U trij = ~eis = sis + ~'is = A6ise + I~eis.

(7)

The components of the tensor o"o. are as follows

shear stress

ATf

¢ i l il I. "1 I . i / i ii il i.i i i " 7 1 1 1 t p-

~i

(C) Plostic- frocturing

av2

normal stress

(b) Fracturing

tr, = Ae + tzeii , 1

tri,s =/xei,s = ~/zyi,s,

(7a) (7b)

The principal directions of the stress a n d strain tensors coincide, since ei, y = 0 makes tri,s = 0.

3. The n o n - l i n e a r c o h e s i o n m o d e l

The stress-strain relation of the cohesion model can become non-linear through strain soft-

Fig. 2. Stress-strain increments decomposed in elastic and inelastic parts (after Bazant and Kim, 1979).

ening due to incrementally inelastic behaviour, i.e., plasticity and fracturing (Bazant and Kim, 1979). Elasto-plastic behaviour is shown in Fig. 2a, where the elastic deformation has constant stiffness and the remaining deformation is plastic. The external work in the cohesion model is then equal to A W = ( r , ae,.D + (zis aer; ).

(8) I The first term is converted to internal energy and the second term is lost to heat. Elasto-fracturing behaviour is shown in Fig. 2b. The external work (~'~sAeis) is converted to internal energy, but fracturing releases internal energy to the amount of

F.A. Vreede / Orthotropic equations for brittle rock

196

causing a decrease in stiffness and an acoustic emission. Figure 2c shows how a stress-strain increment is decomposed into its elastic, plastic and fracturing parts. In the case of brittle materials, fracture behaviour is dominant and plastic deformation will be ignored for the present. Equation (5) for the cohesion stress is therefore replaced by

according to the progressively fracturing model of Dougill (1976). Because of d/~/dE < 0, the tangential stiffness is smaller then the secant stiffness and may well be negative. The energy balance in the cohesion model is illustrated in Fig. 3a:

rij =/.Lij ( n ) eq.

(9)

The variable (n) indicates that the linearly elastic material passes through a finite sequence of stiffness values. At each value there is a maximum state of strain gq at which an acoustic emission takes place and the stiffness changes to the next lower value. Since the acoustic emissions occur at very short strain intervals, the stress-strain data are conveniently linked by a continuous curve r = M(e) yielding the rate equation r = me, where the tangential stiffness is equal to rn=

dM d/x de -/z+g--d-~E'

de>0,

(10)

( Tij Aeij )

external energy

AW=

internal energy

A U = l('TijA,ij

energy release

~% - % A % ) . AQ = 5(% 1

--

+ %

ATij)= (11)

Two incremental paths are of particular importance. Strain-control as in Fig. 3b (Aeij = 0) produces pure fracture

AU = ½(eij A'riy) = - AQ < o.

(12)

In stress-control as in Fig. 3c (Azq = 0) the released energy is over-compensated by external work, yielding U = ½(rij Ae,i ) = + A Q > O.

(13)

This behaviour is not plastic at all.

4. The consequences of flaw formation

~'°a2a2~ ~2"2

The acoustic emission associated with fracture is caused by breaking of cohesion, i.e., flaw formation. A flaw has a certain orientation and can be modelled after Kachanow (1980) as a small slot-like discontinuity. The average value of scalar H f, ascribed to each flaw f over a volume V containing many flaws, can then be described by a "fabric tensor" hit as follows:

AU,U2-u ~ = ~(TIA,I÷( i At)

~"

B,I z~

1"2

~

(oi

'T

///'•

Ar
A,,o

(b)

[ %

Ar •0

A*

/4 (c)

Fig. 3. Energy balance in the cohesion model: (a) general elasto-fractioning; (b) fracturing under strain control; (c) fracturing under stress control.

1 y(v) Y'~ (Hnini)/,

hit = g f = l

(14)

where f ( V ) is the total number of flaws in volume V, and n i, nj are the components in directions i, j of the unit vector perpendicular to the plane of each flaw. The formation of a flaw in the cohesion stress field [~'] causes an energy release A Q / and it is

F.A. Vreede / Orthotropic equations for brittle rock reasonable to assume that AQ f is a function of the stress vector {r r} in the plane of the flaw:

197

The principal axes of the compliance tensor

C~j are also axes of orthotropy, since for Cz.j = 0 Eq. (19) reduces to

AQ f = } ( H{ r} 2) f.

(15)

Under stress control, the increase of the internal energy U2 according to Eq. (13) will be equal to

AUz= + h Q = p

1 % = -~(C i + Ci)rii

and its inverse is simply

rij = 2(C i + Cj)-1%.

1 f(v)

E ½(g{r}2) f.

(16)

f=l

(20)

(21)

The extensional moduli on the orthotropic axes are

According to stress theory (Saada, 1974) {r} 2= ((nirn)(nffjt)), yielding with Eq. (14): 1 f(V) AU2 = -V f~=l {((Hninj)f(~'il'i'jl))

= l(hij~'il'r'l)

The fabric tensor h~j provides a measure of fracture damage. Equation (4a) for the internal energy U2 of intact isotropic material may be written in the form v2 =

The internal energy of the fracturing cohesion model is therefore

Uc= U2.~_ A U2_~_ 2(CijTilTjl), 1

1

Eij = O,Tij = ~(CilTjl @ 'TilCjl),

(23)

There are only three independent moduli. The orthotropic cohesion model still conforms to Hooke's law on the orthotropic axes.

5. Analysis of uniaxial compression test results In triaxial compression tests on rock cores, the directions of the principal stresses and strains are constant. Fracturing cahses the cohesion model to become orthotropic and Eq. (7a) takes the form:

o.i = A(~pEp) "1-].LicEi, (i = 1, 2, 3).

(19)

or written out for normal strain and shear strain = CiiTii "1- Ci,jTi, j -t- Ci,kTi,k,

~i'J= ~ i i = ( Cii + qJ)'l'i'J "l- Ci,j( 'Tii "l- Tjj)

ri,j/Yi,; = ( Ci + Cj)-' = lzilzj/(l~ * + tzj).

(18)

where Cij = 6ij/~ + hij. According to Eq. (6), the stress-strain relations are therefore

~}rii

(22)

and the shear moduli on the orthotropic axes are

(17)

l~ii

"rii//E ii = C Z 1 = I.t,i

(19a)

+ Ci,kTj, k + Cj,kTi, k. Essentially, the same result was obtained by Oda (1983).

(24)

In standard confined tests transverse stresses are equal (are =°'3 =o't) and transverse strains are tensile (% = e 3 = - ~ t ) Equation (24) can be written in the explicit form o'1 = (a -~ ~.L1)eI - 2act,

(24a)

o't = Aq - (2A + ~t)at.

(24b)

Since the cohesion stresses are proportional to strain, r~ will be compressive and rt will be tensile. Experience shows that the fracture effect of tensile stress is high compared with the effect of compressive stress. In that case, only /.t t will diminish strongly, while the parameters A and ~ will be constant. The values of (A +/z 1) and ( - 2 A ) in Eq. (24a) can then be calculated as correlation factors between the three variables

198

F.A. Vreede / Orthotropic equations for brittle rock

Table 1 Stiffness values and standard deviations for the standard tests of Fig. 1 Material

A + P,l

2A

unit

granite sandstone

79.5 +_0.8 23.4 5:0.3

16.9 _+2.6 4.67 + 0.30

GPa Gpa

o"1, e I and e t. The data of uniaxial tests (o"t = 0) on granite and sandstone specimens, as shown in Fig. 1, were analyzed in this way, and the results are given in Table 1. Figure 4 shows plots of the calculated cohesion stresses 3"1 = /"Llel = (3"1 --

A'(el -- 2et),

T t = [£tet = ~ ( E 1

-- 2 e t )

GRANITE

-5/~-

\

•

1

(a)

,

f 3 ~

U] = - ~ ( b q % )

, .t(,O-..~ _3~

~NDS~

~ X"~34GPO (b)

30<:- T=(MPo)

200

The bulk tensor bij is variable with a constant second invariant (bijbi)= B. The stress tensor is proportional to the bulk tensor:

sij = bij ( bpq epq ). In the ing

(26)

isotropic mode we have bq = eij/u, yield-

~

~

(27)

GPo

b3 1 - sin q~ 1>~-~1~> l + s i n q =tan2(¼"rr-½q~).

(28)

/

/it 0

(25)

.

The Mohr stress circles, as shown in Fig. 5, define a tangent line with slope q~, the Coulomb angle of friction. The ratio of principal values of the bulk tensor is limited to

~/E

I O0

2

Sij = b i j ( b p q b p q ) U = ( b p q b p q ) e i j = B e i j .

250

150

6. The orthotropic bulk model The appropriate amendment to the theory is to change the isotropic bulk model into the orthotropic bulk model (Vreede, 1992), which is summarised here. The first term of Eq. (1) is replaced by

,

~'vt(MPo) IOf ~

which have the following characteristics: (a) the axial plots are predominantly straight, confirming that there is hardly any compressive fracture effect; (b) the lateral plots have different shapes, exhibiting the different fracture behaviour of the two materials; (c) the lateral plots continue into the region of negative stress, implying negative stiffness. Since negative stiffness is not possible, the theory needs to be amended.

I

2

3

4

5

Fig. 4. Plots of cohesion s t r e s s v e r s u s strain using data from Fig. 1 and Table 1: (a) transverse % = A(e 1-2et)=/.6tCt; (b) axial ~'1 = o'l - Tt =//'1~1 •

s3

SI

Fig. 5. Definition of Coulomb friction angle 4~ from Mohr circles for orthotropic bulk stress.

F.A. Vreede / Orthotropic equations for brittle rock

The value of E3//E 1 has the same limits in the isotropic mode. When the strain tensor satisfies ( b p q % q ) < O, instead of tensile stress no stress is produced at all. When the strain tensor satisfies (bpqepq)> 0 the strain can be decomposed in the Coulomb strain Eb with e~/E1b = (1 - sin ~0)/(1 + sin q~), producing the stress si = Be~ and the orthotropic ! deviator e' with (bpqepq) = 0, producing no stress at all. This behaviour is called the deviatoric /'node.

199

7. The constituent equations of rock The behaviour of rock is described by combining the stress tensors of the orthotropic bulk and cohesion models (30)

O'ij = "l'ij( E ) "t- Sij( C.) .

Substituting Eqs. (21) and (26) in Eq. (30) yields for the conforming mode

trij= [ B + 2(Ci + Cy) -1] •ij,

(31)

The deviatoric mode comprises three states, distinguished by the value of the intermediate principal stress s2: (i) the squeezing state with s 2 = Sl; (ii) the confined state with s z = s3; (iii) the shear state with s 2 = Be 2. The intermediate component of the deviator e~ is positive in the squeezing state and negative in the confined state. In the shear state, the deviator is of the plane type

when stress and strain are measured along the principal axes of the compliance tensor [C]. These axes are also orthotropic axes of the rock. In the deviatoric mode, the rock is only orthotropic when the axes of orthotropy of both models coincide with the axes of principal stress, as in test specimens. In that case, the models produce the homotropic equation

E', = (1 - sin ~o)u,

In matrix notion, this equation has the form

e~=0,

o"i = bi(bpep) + Ixie i.

(32)

(29)

E~ = - ( 1 + sinq~)u.

Details of the deviatoric states are given in Table 2. Deviators produce positive dilatancy, which creates an opportunity for plastic deformation through recompaction. Important permanent deformation usually takes place in the post-failure region, but in this paper, plasticity will not be considered.

= [D,j]C,3, Di / = bibj + 6i/~/.

(33) I

The parameters b i a n d / z i can be calculated from the six independent matrix elements by the formulae

b2 = Di,jDi,k/DLk, Izi = Dii _ bi2"

(34)

Table 2 Principal values of the bulk tensor for the deviatoric states s i = b i (bpep) State a

bl2

Squeezing

B(1 + sin ~o)2

$2 = S1

3 + 2 s i n ~ +3sin2~

confined s2 = s3 shear S2 = 8~2 a

with b 2 + b 2 + b 2 = B

b~

b2

b~

3 + 2 s i n ~ +3sin2~

b2

3 - 2 sin ~o+ 3 sin2~

B(1 - sin ~)2

B ( I + sin ~)2 3 - 2sin q, + 3 sin2q~

B ( 1 - s i n ~o)2

B(1 + sin ~o)2 2(1 + sin2~o)

B ( 1 - sin ~o)2 0

2(1 + sin2~)

200

F.A. Vreede / Orthotropic equations for brittle rock

When test conditions are axisymmetric, Eq. (33) can be written explicitly in the same form as Eqs. (24a) and (24b):

The specimen becomes unstable when a failure increment ~i > 0 exists that satisfies the incremental boundary conditions:

0-1 = (b 2 + tzl)e 1 - 2 b l b t e t ,

d-i = - k i W i ,

(35)

0-t = b , b t e , - ( 2b2 +/zt)et.

These equations solve the problem of negative/z t encountered before, since b 2 is smaller than b i b r The four parameters can, however, not be calculated from the three independent matrix elements. Equation (33) can be inverted to the matrix equation {ei} = [Ei~:'] {% } ,

(36)

which can also be written in the Young-Poisson formulation E.i = E i i

l/i'J ~

(36a)

- vi, , Ek k "

In the shear state, these equations are two-dimensional, with E~.21= 0 and E2z =/z 2. The calculations in the Appendix yield the results: Eii=l~i+b

Eij=

'

- bi--~j

1 + b2 + b~ /.t/

1 + --

/dq

+ --

/2, 2

(37a)

•

+

1

]J'3 ]

=E~,

'

]-1 * V/,i.

(37C)

The matrix [Eij] is conveniently symmetric, but not the coefficients rid and v/, i. In the shear state, Eqs. (37) have b k = b 2 = O.

8. Brittle failure mechanisms of rock

The compressive stability of a rock specimen in the deviatoric mode can be studied by means of the incremental form of Eq. (32) 6"i=bi(bp~p)+mi~i.

bi(bp?:p) + m i ~ i = - k i E i,

(38)

(39)

where m i is a function of Ei. The failure increment ~i=bift/(mi+ki),

ti = - ( b p ~ p ) > 0

(40)

exists, when the tangential stiffnesses m i satisfy ( b 2 ( m i + k/) -1) = -- 1.

(41)

This is the general equation of brittle failure, where at least one tangential stiffness m i must be negative. Substituting ( m i + k i ) / b 2 = a 4= 0, Eqs. (40) and (41) can be written in the forms /i ~i = blot i ,

/~ > 0,

a21 + a f 1= - ( 1 + a l l ) .

(37b) bj P'i Izi b2 1.)i,j= ~ 1 + b-7, + iZkb 2

with k i/> 0,since increasing strain in the specimen causes decreasing boundary stress. The failure increment must therefore satisfy the homogeneous equation

(40a) (41a)

The signs of ~i are the same as the signs of a v For two limiting values of al, viz., a I = l a2[ and al >>l a21, the hyperbolic relation (41a) between a 2 and a 3 is plotted in Fig. 6. For the three deviatoric states mentioned in Section 6, failure may take different forms. In the squeezing state, only e 3 is extensional and the failure values a 2 and a 3 are found between the two curves at the right-hand side of Fig. 6 (a 2 > 0). The extensional deformation will be manifested (trapped) in cracks perpendicular to direction 3 and form a set of parallel cleavages. Behind an unsupported rock face (k 3 = 0), cleavage parallel to the face will occur when the tangent stiffness m 3 equals m3(E3) = -b32(1 + a l I + a21) -1.

(42)

The right-hand side of this equation is practically constant for a given material.

F.A. Vreede / Orthotropic equations for brittle rock -2

-I

II

,/

colum

2

I

I[

0.~,~ 1

~

0

cleavage - ~

I

where/3 is the angle between directions 1 and n. Incremental failure strains are therefore zero when

3

°'>'°21

tan/30 = ~-- ~1/e3 •

In a plane passing through the intermediate principal axis at an angle/3o with the major principal axis, all incremental strains are zero. Such a plane becomes a band of shear failure as shown in Fig. 7. Substituting Eqs. (44) and (45) in Eq. (46) yields

. mi+ki

~

l

"

oi >>(o21

-- 3

[~

a i

b i

Fig. 6. Plots of failure equations for a t = 1~21 and a t >~ a2; for cleavage and column failure: a~ I + a~-l = - ( 1 + ai-t); and for shear failure: a~-t = - ( 1 + a i- t).

In the confined state, both E2 and £3 are extensional and the failure values of a 2 and o~3 are found within the boundary at the left-hand side of Fig. 6 ( a 2 < 0). If the confinement is symmetric ( a 2 = 19l3 = O~t) , failure requires - 2 ,%< a t ~< - 1 . The cracks may occur in any direction parallel to direction 1 and the resulting column structure may be shattered through secondary instability. This instability does not occur if a t > - 1, requiring either b 2 > (--mt)ma x o r k t > ( - - m t ) m a x -- b 2 > 0. In the shear state, a plane type of failure (~2 = 0) is most probable. Equations (38) for the failure increments ~1 and ~3 then become b](ba~ a + b3~3) + mad 1 = - k l ~ l ,

(43)

tanflo=~b343 ~ b3/bl blal = i--~--~l l and using Eq. (28) yields

The condition of instability (41a) becomes

b343

~3

bl°tl

9. E l a s t o - f r a c t u r i n g

°rl=[bt

t

1+~- 1

+2bt~-i

+ 2bt - - , bl

"t,

(48a)

(48b)

J major

principal ~oxis

bond of shear failure

~n = ~l C0S2/3 + ~3 sin2/3,

(47)

behaviour of rock

("i)'

(44)

The incremental strain in a plane through the major and minor principal axes is equal to

½~),

For uniaxial compression (trt = 0) Eqs. (35) can be written in the parametric form

~l =

(45)

-

for the orientation of the failure plane. The value of /3o is smaller than the M o h r - C o u l o m b value - ~q~) which is valid in cohesionless material (Taylor, 1948).

For a 1 = a 2 and for a I >> oL2, the limits of shear failure are indicated in Fig. 6. The critical values of a 3 lie between the values causing cleavage and column formation. Equation (40a) takes the form ~1

42)-1tan(¼"rr

tan/30 = (1 +

b 3 ( b l ~ 1 + b3~3) + m3~ 3 = - k 3 ~ 3.

4 3 1 = -- (1 + a l l ) .

(46)

shear foiluce

-2

k

201

3 k minor . ~ principal "~oxis

Fig. 7. Definition of orientation angle /30 for shear failure plane.

F.A. Vreede / Orthotropic equations for brittle rock

202

Table 3 P a r a m e t e r s for granite, isotropic at Et = 0 b 1 = 6.5 G P a 1/2 b t = 1.3 G P a I/2 B = 45.6 G P a = 41.8 ° P,1 = 37.2 G P a /z t = ( 3 7 . 2 - 12.3 ~t)

For triaxial compression (~rt @ 0) the right-hand side of Eq. (48a) has the extra term b 1 + #zl/b l bt

causing a normalised increase in strength equal to

GPa

~0 = 3.02 × 1 0 - 3 / ) n=l

°'t'

Eq. (49)

b 1 + t z l / b I trt bt

where ill,t is a function of Et. A simple formula which appears to fit experimental data adequately is: ~t=/z0[1-(et/e~)],

(eta
where o-0 is the uniaxial compressive strength. Following ISRM (1983) this increase is interpreted in terms of an angle of internal friction

(49)

f m-l)

= sin- 1~

Expressing the volumetric strain in the parameter e t yields e=e 1-2e t=

~

2 1-

e t.

cro '

where, in this case, b 1 + ~i/bl

(50)

m = bt

The parameter values of Table 3 fit the data for the granite specimen of Fig. 1 reasonably well, assuming isotropy at zero strain. In Fig. 8, the axial, lateral and volumetric strain are plotted against the axial stress, extrapolated to ~t E0"

Introducing the Coulomb angle ~o by using 1 -

bt = b i

sin q~

1 + s i n q~ '

=

~- 300 O.

/J @@

,4'

.~ 250 O0

tl

......

200

7S

150

t00

50

o

/e I

(5)

(4)

(31

(2)

i11,

o

I

3

Stra~ (1 0^-3) Fig. 8. Plots of e l a s t o - f r a c t u r i n g b e h a v i o u r of g r a n i t e using d a t a from Fig. 1 and T a b l e 3.

F~4. Vreede / Orthotropicequationsfor brittlerock

internal length scale is recognised in fracture mechanics, since the energy loss of a flaw must be supplied by a finite volume (Bazant, 1982).

according to Eq. (28) yields (2bl2 + / z l ) sin ~o +/"/'1 sin q~i = (2b 2 +/~1) +/xl sin q~ /x 1 cos2~ = sin ~0 + 2b2 +/zl(1 + sin ~0).

203

10.3. Latent stress (51)

The angle of internal friction is larger than the Coulomb angle.

10. Discussion of results

Constituent equations are relevant to nearly all mechanical phenomena. In an introductory paper there is only scope for a few remarks and literature references. 10.1. Orthotropy The constituent equations contain six parameters, but it is known that a further three independent shear parameters are possible (Saada, 1974). These shear parameters will be present when "shear flaws" occur in the cohesion model, causing the loss of shear stiffness, but not extensional stiffness. In ductile material large plastic shear deformation takes place; whether any shear stiffness is lost or whether shear flaws are significant in brittle material is an open question. 10.2. Characteristic length The two models give rise to two superimposed stress fields in equilibrium with internal loads (gravity) and external stresses at free surfaces (tractions). Cohesive stress does not exist at a free surface, so tractions are borne by the bulk model. This implies that a transition zone exists where cohesion stresses are built up; a similar zone exists around a crack. The extent of the zones is determined by a characteristic length, probably related to the grain size. The existence of an

When fractions are absent, latent stress may still be present. The pressure in the bulk model is then compensated by tension in the cohesion model: z I = - s I. A practical example occurs in swelling rock, when water is absorbed on the grains. The increasing volume of the rock may cause the cohesion model to fail in all directions, i.e., to disintegrate. 10.4. Time dependent behaviour The phenomena of stress relaxation and creep suggest that "haphazard failures" may occur in brittle material: either the cohesion model develops flaws or the bulk model sustains plastic recompaction. The two mechanisms can be distinguished because flaws decrease the cohesive stiffness, while during plastic deformation no damage is caused. 10.5. Forming and closing of cracks The microscopic "penny shaped" crack which often appears in the literature (Jaeger and Cook, 1969) is replaced by Kachanow's slot-like flaw. The tensile stiffness of the flaw is permanently lost, but it seems unlikely that a dosing flaw can again provide compressive stiffness. 10.6. Strain softening Strain softening under compression is not a true property of brittle material. Read and Hegemier (1984) proved that dynamic instability at the critical strain would cause the local density to increase without limit, which is physically impossible. Strain softening under extension, however, will cause the local density to decrease to zero, which is physically realized as a crack.

204

F.A. Vreede / Orthotropic equations for brittle rock

10. 7. Strain as a failure criterion

For rock in the squeezing state, tensile strain can be used as a failure criterion (see Eq. (42)), even though the failure mechanism depends on all the deformation parameters. Critical strain has been investigated since J.V. Poncelet (1788-1867) and tensile strain has emerged as important. Axial cleavage (Gramberg, 1965) and the slabbing of faces in tunnels and side walls in mine haulages (Stacey, 1981) confirm the validity of this particular failure mechanism. 10.8. Elastic post-failure behaviour

The axial stress-strain curve of Fig. 8 passes quickly through the negative slopes from 0 to - ~ , making the failure stress virtually independent of the stiffness of the loading machine. Thereafter the curve shows the characteristics of Class II materials according to Wawersik (1968). The volumetric strain is strongly dilatant.

II. Conclusion

The constituent equations for brittle rock are based on a few simple mechanical assumptions: (i) the orthotropic bulk model is controlled by friction; (ii) this bulk model is stiffened by an orthotropic cohesion model; (iii) cohesion stiffness is diminished by the formation of flaws; (iv) tensile strain is the typical cause of flaw formation. The resulting model produces all the features of a brittle material (Yazdani and Schreyer, 1988): (i) enhancement of strength by lateral pressure; (ii) significant difference in tensile and compressive strength; (iii) post-peak response with strong position dilatancy; (iv) anisotropic stiffness degradation under increasing strain. The theory shows an excellent fit with the results of uniaxial compressive tests on rock and

also provides an adequate description of brittle failure mechanisms: (i) explaining why brittle failure is usually sudden; (ii) predicting both the cleavage and shear failure phenomena; (iii) calculating the shear plane angle. The case of homotropy (co-axial orthotropy) has been investigated in detail. The parameters B, ~b, ~1 and ~t must be established by testing; the last parameter is always a variable, but the others might not be truly constant either. For instance, the strain hardening that is often observed at the start of compression tests is probably due to closure of voids, causing an increase in the value of B. In multi-axial testing, sufficient data are obtained to calculate all four parameters, but not in standard uniaxial or triaxial testing. The general case of allotropy (hetero-axial orthotropy) is more complicated. In the theory, a "fracture surface", as defined by Dougill (1976), is expected to play a central role. Experiments in this field must wait until the homotropic case has been investigated thoroughly. A crucial part of the theory depends on the difference between plasticity and "damage". In plastic deformation external energy is lost (converted to heat) but the internal energy is not affected. Damage points to loss of stiffness (gain in compliance) through the loss of internal energy (acoustic emission). It is proposed to call this fundamental mechanical process "klasticity", from the Greek work "klastos" meaning broken. Greek transliteration is used because "clasticity" looks too much like "elasticity". The formation of sheer cracks mentioned in Section 10.1 is klastic, while ductile shear failure is plastic. Both processes can be going on at the same time.

Acknowledgement

The assistance of the Head and Staff of the Geomechanics Programme of the Division of Earth, Marine and Atmospheric Science and Technology, CSIR, is acknowledged.

F.A. Vreede / Orthotropic equations for brittle rock

205

References

yields

Bazant, Z.P. (1982), Crack band theory for fracture of geomaterials, in: Z. Eisenstein, ed., 4th Int. Conf. on Numerical Methods in Geomechanics, Edmonton, Vol. 3, 1137-1152. Bazant, Z.P. and S.-S. Kim (1979), Plastic-fracturing theory for concrete, ASCE J. Eng. Mech. Div. 105 (EM3), 407-428. Bieniawski, Z.T. (1969), Deformational behaviour of fractured rock under multiaxial compression, in: M. Te'eni, ed., Structure, Solid Mechanics and Engineering Design, (Proc. Southampton 1969 Civil Engineering and Materials Conf.), Part I, Wiley, New York, pp. 589-598. Dougill, J.W. (1976), On stable progressively fracturing solids, J. Appl. Math. Phys. (Z. Angew. Mech. Phys.) 27, 423-436. Gramberg, J. (1965), Axial cleavage fracturing, a significant process in mining and geology, Eng. Geol. 1, 31-72. International society for rock mechanics (Committee on Standard Laboratory and Field tests) (1983), Suggested methods for determining the strength of rock material in triaxial compression, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 20, 283-290. Jaeger, J.C. and N.G.W. Cook (1969), Fundamentals of Rock Mechanics, Methuen, London. Kachanow, M.L (1980), Continuum model of medium with cracks, ASCE J. Eng. Mech. Div. 106 (EM5), 1039-1051. Krajcinovic, D. (1989), Damage mechanics, Mech. Mater. 8, 117-197. Oda, M. (1983) Elasticity of rock-like materials with random cracks - Theory and experiment, Geomechanical Conf. Houston, Applied Mechanical Division of ASME, New York, pp. 23-28. Read, H.E. and G.A. Hegemier (1984), Strain softening of rock, soil and concrete - A review article, Mech. Mater. 3, 271-294. Saada, A.S. (1974), Elasticity: Theory and Applications, Pergamon Press, New York. Stacey, T.R. (1981), A simple extension strain criterion for fracture of brittle rock, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 18, 469-474. Taylor, D.W. (1948), Fundamentals of Soil Mechanics, Wiley, New York. Vreede, F.A. (1991), The elasto-plastic bulk model for compact cohesionless materials in continuum mechanics, Mech. Mater. 13 (3), 185-192. Wawersik, W.R. (1968), Detailed analysis of rock failure in laboratory compression tests, Ph.D. Thesis, University of Minneapolis. Yazdani, S. and H.L. Schreyer (1988), An anisotropic damage model with dilation for concrete, Mech. Mater. 7, 231-244.

e i = [o"i - bi(be)]tz71.

Multiply by b i and summate (be) = (btr/x -1) - (b2/z-l)(be)

and solve (be)=(b~rlz-')[1

Z = 1 + (b2p. -1) 2 -1 +bktXk 2 -1 =l+bZ/x/-1+ b /l~i

and substituting Eq. (A2) into Eq. (A1) yields: e i = oril..gZ 1 -- b i t x z l ( b o . p , - I ) z - I

b/2

O"i

Ixi L

~i Z

o) +bk~kk bj7

--

Id,i z

Comparing this with Eq. (36): ei = ~riEi71 + °)Ei71 + ~rkE~ ~ yields Eii=P.iZ[Z-b21j.~-l]-I 2 -i Ixi + b2 + [~i[ bflx} - 1 + bklXk ]

2 -1 1 + b ~ ; - 1 + bk~k =tzi+b2[l

2 k-1 ] +bffl.tTl +bkla,

tzjZ

/*i

(37b)

txilzjZ

]_1

Vi,j= -- Ei--~j = Z_b21a,~-i /

---

(37a)

bib/

E~j

[

1,

/zi/zJ [1 + (b2/z-I)]

txilxJZ bibj

EO

bE

b2

l+--+-/xi /xk

In tensor summations the indices are omitted.

bibj

] -'

Equation (33) ori = bi( b e ) + Id,iei

(A2)

+ ( b 2 / . t - 1)] - l

Putting

bib ~ Appendix

(A1)

= b / ~/2+1+~--~/2]

(37c)