An extended Cam Clay model for soft anisotropic rocks

Computers and Geotechnics 2 (1986) 69-88

AN EXTENDED CAM CLAY ~K)DEL FOR SOFT ANISOTROPIC ROCKS

Roberto Nova

Department of Structural Engineering Milan University of Technology Piazza Leonardo da Vinci 32, 20133 Hilano, Italy

ABSTRACT

The mechanical behaviour of soft sedimentary rocks in triaxial compression is modelled by means of a constitutive law which has been formulated by extending the original Cam Clay model. The geological material is assumed to be linearly elastic-plastic, strain-hardening, with fixed transverse isotropy. Theoretical predictions are compared to observed behaviour at different confining pressures of specimens of diatomite with various bedding inclinations. Although quantitative agreement is not always satisfactory, the trend of the tests is qualitatively well matched, what proves the paradigmatic value of this type of model not only for soils but even for soft rocks.

INTRODUCTION Rock formations of sedimentary origin like calcarenites, sandstones are

frequently met in Italy as foundation subsoil.

diatomites,

From the Civil

Engineering viewpoint [1], strength and deformability of the intact the controlling

factors

in

structural discontinuities characteristics of From an

are

widely

design, spaced,

since

fissures

whilst

the

rock and

are other

mechanical

the rocks are rather weak, because of their high porosity.

extensive

calcarenites and

foundation

mr

turfs,

study

performed

Pellegrino

at

the

[1],[2],[3],

University it

of

Naples

on

appears that for small

confining pressures these types of rocks exhibit a rock-like

behaviour,

i.e.

quasilinear elastic behaviour and brittle failure, whilst after a well defined threshold, the

stress

strain relationship becomes non linear, accompanied by 69

C o m p u ~ and Geo~chnics 0266-352X/86/$03.50

© Elsevier Applied Science Publishers Ltd, England, 1986. Printed in Great Britain

7O large plastic strains and ductile in nature, more similar to the behavio1~r

of

a soft soil than to that of a hard rock. Similar results have been obtained diatomite from

the

by

f4ontagne d'Andance,

Allirot

and

Boehler

[4]

on

a

Ardeche, France, whose structure is

made of a regular alternance of diatom valves and fossilized debris. The experimental description of

the

evidence behaviour

therefore of

such

suggests materials

that can

the

mathematical

be based on the same

concepts on which the constitutive models for virgin soils are founded.

]here

appear to be two main intuitive differences.

First, the initial yield surface

does not coincide with the origin of axes in

the

assumed for

virgin soils.

space,

as

usually

In this respect, lithification acts as an apparent

preconsolidation pressure. markedly anisotropic.

stress

Second, the

From

behaviour

of

sedimentary

rocks

is

an engineering standpoint no sensible result can

be arrived at if anistotropy is not taken into account. The aim of this paper is to confirm these intuitive considerations on the basis of a comparison between theoretical predictions and actual behaviour.

First

anisotropy.

the

original

experimental

Cam Clay model will be extended to cope with

Secondly, on the basis of some experimental results, a

to determine the constitutive parameters will be shown. between calculated

results

and

experimental

evidence

in

several triaxial

compression tests on specimens at different inclination with major principal

stress

direction

will

be

procedure

Finally, a comparison

presented.

respect

to

the

A discussion on the

results obtained, on the potential and limitations of the model proposed

will

end the paper. lhe main effort will not be directed, however, to the exact experimental curves would be

necessary,

for

which

but

phenomenological behaviour

rather of

matching

of

an 'ad hoe', extensive experimental programme

soft

towards

the

anisotropic

understanding rocks

as a whole.

of

the

Further

refinement will be necessary to achieve the goal of formulating a constitl~tive model for such materials that enjoys full validity even

from

a

quantitative

point of view. In the following, the term stress will be always used

in

the

sense

of

effective stress.

THEORETICAL CONSIDERATIONS

Sedimentary rocks may often be idealized as homogeneous but isotropic materials,

where

the

axis

of

symmetry

transversely

is normal to the bedding

71 plane.

We shall define as intrinsic reference frame, a cartesian frame

one axis, Xv,

where

say, coincides with the symmetry axis, and, consequently, the

plane of isotropy is the plane (XH1,XH2) , see Figure I. Xv

i

XH2D"

f

Figure 1. Intrinsic reference frame.

Since experimental evidence suggests that for low the material

hydrostatic

pressures

behaviour is approximately linear and quasi reversible, we shall

assume that linear elasticity is suitahle in this range, lhe constitutive law written in the intrinsic reference frame is then given by

eV

SH1

1 EV

vVH E~4

~VH

EH

~l

, CH2~

E~HH

EH symm.

YVHI I

YVH2 I YHHJ

(1)

1 0

GVH

0 1

GVH

0 1 GHH

where E

is the longitudinal elastic modulus in the direction of the axis or v symmetry, while E H is the corresponding modulus in any direction in the isotropic plane, ~VH and ~HH are Poisson's ratios and GVH and

GHH

are

shear

72 Since GHH is Liae shear modulus in tile plane of isotropy

moduli.

EH

g!i H :

(2) 2(I+~HH) Equafiinn 1 i s v a l i d

threshold value.

when

the

hydrostatic

then

given

On

lower

than

a

the

lhe limit of validity of

Equation

I

by a yield function whose expression is a priori unknown

and that could be determined programme.

is

It is straightforward to assume that this value will depend

on the particular stress path Followed. will be

pressure

basis

only of

by

means

of

an

extensive

experimental

some results published in [I], we shall assume

that the yield function is given by an expression similar to that of Cam Clay, which in triaxial compression reads

f : q + Mp In(p/p c) = 0

(5)

where

q ~ rSl -

'~3

(4)

1 P ~ -- (~1 + 203) 3

(5)

and Pc is the threshold pressure in taken positive

hydrostatic

compression.

in compression, as well as strains.

and minor principal

confining pressure.

stresses

respectively.

The

~I

latter

Stresses

~3

and

is

are

are the major equal

to

the

Figure 2 shows a comparison between the calculated y i e l d

surface and f a i l u r e locus and the observed y i e l d

and

failure

points

for

a

calcarenite (C2). To take account of anisotropy a generalization of Equation 3 to the stress space is necessary.

The simplest way to do that ls to assume that for

an i s o t r o p i e materlal the y i e l d function can be obtained by r o t a t i n g 3 around

full

Equation

the hydrostatic axis, following a procedure similar to that employed

in [5] to smooth the Mohr-Coulomb yield

condition.

If

s.. zj

is

the

stress

deviator and S.

n.. ~ '1.]

,

iJ

(6)

P

it is easy to show that the yield function can be written as 3 ! f = (-- qijnij) 2 + M in (p/pc) = 0 2

(7)

73

q (MPa) 10

f

/, p {MPa) i=,,,=

1'0

5

Figure 2.

Comparison between experimental yield

calculated f a i l u r e locus (heavy line) and yield

and f a i l u r e

curve ( l i g h t

points and

line)

on the

basis of Cam Clay model (M:I.5~, pc= 8.0 Mpa) - data after [ I ] . Io take account of the structural procedure similar

to

condition to orthotropic materials tenser A..

ijrs

= A

of

sedimentary

will

be

followed.

Define

a

..

rs13

= A..

zjsr

= A..

be

a

quadruple

(8)

j1rs

and is invariant with respect to s rotation of axes in the plane of It can

rocks,

which enjoys the following symmetries

1jrs

A..

anisotropy

that proposed by Hill [6] for extending Von Mises yield

shown

that

the

isotropy.

most general tensor enjoying these properties is

given in the intrinsic reference frame by

74 b

A,

.

b

b

d

e

b

e

d

=

(9)

ijrs c

c

0

0

0

c

c

0

0

0

0 0

0

0

c

c

0

0

0

0

c

c

0 0

0 0

0 0

0 0

0 0 d-e d-e d~e ~

T

@

where a,b,e,d and e are five material parameters.

Equation

7

may

then

be

transformed into Equation 10, that reads 3 ± f = (--qijAijrsqrs)2 + M In (p/pc) = 0 2

(io

Equation (10) reduces to Equation (7) when a=d=l,b=e=O,c=l/2. Indeed the tensor I . .

ijrs

obtained in t h i s way, i . e .

I

(11

lijrs ~ ~(~irajs + 8isajr) acts as a unit tensor in the sense that

(12)

qijlijrsqr s s qijqij The special case of isotropy is then encompassed by putting

A..

13rs

= I..

(13)

zJrs

From Equation

10

we

see

that

only

Four

of

tile

five

parameters

characterizing

A.. are independent parameters, so that for the sake of ijrs convenience we shall assume d=1. Further, For the sake of simplicity and to keep the

number

of constitutive parameters to the minimum, we also take e=O.

The terms that relate to the isotropic plane are then identical to those of an isotropic material. constants that

The

parameters

characterize

the

a,b

and

c

are

additional

material

anisotropy of a specific rock and should be

determined experimentally. The plastic Flow rule will be plastic potential

assumed

to

and yield Function coincide.

be

associated

so

that

the

Such a hypothesis is made for

75 the sake of convenience. parameters and

avoids

In the

fact

it

limits

choice

of

a

the

number

of

constitutive

different function for the plastic

potential, which would be rather awkward for lack of

consistent

experimental

data. Prager's consistency rule may be therefore written as: @f

~f

~Ohk~hk + ~

0c = 0

(14)

~Pc where a dot d e n o t e s i n c r e m e n t or r a t e .

I t w i l l be

v a r i e s with p l a s t i c

vP .

volumetric strain,

Again on t h e b a s i s o f t h e e x p e r i m e n t a l r e s u l t s

further

assumed

shown i n [ 1 ] ,

[4],

that

Pc

it will

be assumed t h a t

Pc = Peo exp (vP/X p)

(15)

where Pco is the threshold pressure for which the rock hehaviour ceases to linear and

reversible

in

material constant,

whose

compressibility.

For

a

hydrostatic

physical example,

meaning for

compression is

another

test,

that

of

calcarenite

hydrostatic compression in [I] the appropriate value of Xp shown in

Figure

3

while Xp a

be is a

logarithmic

(CI) tested is Xp

in

=.025.

As

the agreement between calculated and experiments1 data is

satisfactory, if the bulk modulus K of the considered calcarenite is taken

to

be K=128 Mpa and Pco =2 Mpa. From the very definition of plastic potential and

by

the

hypothesis

of

an

associated flow rule, plastic strain increments are given by Bf ~j

= C~jhk~hk ~ A

(16) ij

where A

is a

scalar

multiplier and C~jhk is the plastic compliance tensor.

Thus by substituting Equation 16 in Equation 14 and solving for A

, we get

Bf ~hk

A = - ~f

Bpc

8vp ~f (-- --) @Pc Bvp B¢~k@~hk

Finally, taking

account

(17)

of

Equations

10,15,16

and

17,

tile plastic

compliance tensor is given by

C~jhk =

qijqhk H

(IB)

76

P (MPa) I0

,'0

o Figure 3. Calculated

and

experimental

,'s v'/. data

for

calcarenite in hydrostatic

compressinn (C1)-data after [I].

where

3 ~ -

qij =

' lm~s

'lm

1

[~

2

A 3 ~in

zr

~

js

Aqq = [ 7 Almrs qlm qrs ]~

- -

1 (~rs + Rrs)Si j ] + - tl,S.. 3 3 ]3

(19) (20)

and Mp

H : Z p (X

Aq~)

(21)

is tile hardening modulus. E1asEic and plastic strains could be Ehen calculated from Equations I and 18 for any given stress path once appropriate constitutive parameters had been

77 chosen. When H is positive the yield surface expands for outward the ductile.

current

stress

rates

For H=O, hardening is no more possible and unlimited

may take place under very small stress increments. however, we

directed

yield surface and the rock behaviour is irreversible but shear

strains

At variance with Cam Clay,

shall assume fully brittle behaviour for negative H.

ductility of rocks after peak is generally very limited.

Indeed, the

Moreover, the strain

pattern within the sample is non uniform but concentrated in a thin shear band where fracturing takes place.

Any description af such a

Continuum Mechanics

would

misleading.

concepts

be

probably

situation

illegitimate

Io avoid this shortcoming, we shall assume that a

based

on

and possibly specimen

will

abruptly fail when the yield surface will be attained in the region where H i8 negative.

This

shown e.g.

assumption

very

often agrees with experimental evidence, as

in [4].

Figure 4 illustrates the difference of behaviour in ductile ranges.

The

behaviour

of

the

yield locus (paths AB and CD). H
At

D

rock

and

brittle

is considered to be elastic within the

At B the first

sample

fails

abruptly

since

the second sample yields since H>O and loading may be carried on

until point E associated with H = 0 and theoretically infinite strains. From Equation 20 we see that the failure locus

for

the

ductile

locus

region,

for

is

which

given

by

H=O,

which

failure condition.

it

reduces

to

a

purely

the

a cone, which is not

centered on the hydrostatic axis because of the anisotropy of the isotropic materials

is

rock.

For

frictional Drucker-Prager [5]

The parameter M is then linked to the

angle

of

internal

friction of the rock.

DETERMINATION OF CONSTITUTIVE PARAMETERS

The constitiitive characterize a

parameters

transversely

to

be

following tile model presented, are then Yo,Jng's moduli,

two

determined

volumetric compressibility

as

(logarithmic),

constants

compression should

he

and

specified.

constitutive parameters

many

as

are

one

finally

to

fully

eleven.

In

fact

two

For

necessary.

friction

Not

parameter,

three

the yield pressure in hydrostatic isotropic

materials

only

five

Anisotropy is then responsible for a

formidable complication of both the theoretical and the modelling.

order

Poisson's ratios, one elastic shear modulus, the plastic

plastic anisotropy

of rock

in

isotropic elastic-plastic strain hardening rock,

experimental

aspects

only the stress strain law is much more complicated

78

q

H<0

0

A

C

p

Pco

A,C

Em

Figure 4. Brittle and ductile ranges according to the model.

than the corresponding one for an isotropic material, but also the

number

of

parameters to be experimentally determined is much larger. In the following,

we shall attempt to characterize

the

behavio~r

of

a

diatomite on the basis of a very extensive experimental programme performed in [4], [7],

which,

to

the author knowledge, constitutes the widest study ever

conducted on rock anisotropy. In [4], samples of diatomite with different inclinations to plane have

been

tested

confining pressures. specimens undergo

in

standard

triaxial

After an initally linear stress strain

either

the

with

bedding different

behaviour,

rock

brittle fracture or ductile hardening, depending on

sample inclination and confining pressure. rock tends

compression

As could be easily

expected,

the

to become more and more ductile for increasing confining pressure.

However, the stress strain relationship and the

failure

modes

are

markedly

influenced by sample inclination. ]o start with, we shall consider the initially linear part of the strain law,

that according to the model will Be assumed to !~e elastic.

stress Since

79 the initial slope of pressure, at constant. diagram

the

least

curve

is

roughly

for ~) < 4 MPa ,

the

independent

elastic

of

moduli

tile confining

will

be taken as

Since Ao (Cv,q)

inclination.

is the only non-zero stress increment, the slope of tile v gives the longitudinal elastic modulus for each sample

Dots in Figure 5 give the estimated values of the

the basis of the experimental data. longitudinal modulus

is

stiffness

on

When the bedding is horizontal (~9=0°) the

Ev=18z$ MPa,

while

Equation 1 and the tensor transformation rules, e compliance C1111 along the direction x I of (vertical), which is inclined of an angle $

EH=640 MPa.

for ,%=90° , we

get

the

that

the

From

elastic

major principal stress

to the axis of symmetry

of

the

specimen, in a triaxial test at constant cell pressure, is given by 1 I e 4 Vvh C1111 = -- cos ~ + 2(- ~ + ~)sin Ev EH. 2GvH

I 4 2 ~ cos 2 ~ + --sin ~ EH

By taking C~111=.00333 MPa -1 for &=45 ° we get the dotted

(23)

line

shown

in

Figure 5, which nicely fits the experimental data. For the type of test we have intention to model in the following, the e C1111(~) is enough. The other parameters could be determined in

knowledge of

principle by measuring the strains in the directions perpendicular to that loading for

~=90 °

of

but these are unfortunately not available and we shall not

make any attempt to guess their values. The values of X p

and Pco could be

hydrostatic compression test.

determined

from

the

results

of

a

In that case in fact the volumetric strain v is

given by

v = p/K

(23)

if p is less than Pco' K being the bulk modulus, and by

(24)

v = p/K + X p in(p/pc o) for larger

values

of

p, as can be inferred from Equation 18, after algebra.

Since, as shown in Figure 3, the first linear, K

can be easily determined.

shown by the curve, see e.g.

part

of

the

strain

law

is

The pressure Pco corresponds to the kink

Figure 3, and finally X p

means of a best fitting procedure.

stress

may be

determined

by

80

E 8 (MPa) 800

600

/

J

/° / 400

°/ / . _ ---&

200

0

Figure 5. Comparison

I

I

30

60

of calculated

@

IO

and experimental

]nngitudinal

stiffness

for

different sample inclinations.

In this case, since complete available tria×ia]

Lhe

va]ue

compression

of

o

~CO

tasts.

information

and

Xp

on experimental

will be inferred

~!e shall assume henceforth

results

is

not

from the results of

that

Pco=6.8 MPa

and

xP=.I. For confining pressures ductile in

nature.

asymptotically

In

large enough,

triaxial

the model predicts that failure

compression

approached which depends on the

a

limiting

sample

is

stress

rstio is

inclination.

Indeed,

since in this ease

"~11

Sli

2q

2

p

3p

3

'~22 = ~33 = From Equation given by

"fi

(25)

s33 q ~ " = - - - -- - -p 3p 3 21

we

get

the

asymptotic

value q L

for which H = O, that is

81

(26)

qL = (J6 M)/(4A1111 + 2A2233 + A2222 + A3333 - 4Al133 - 4Al122 )½ ~hece from Equation 9 and the tensor transformation rules we have = a cos 4 Al111 A2233 = b s i n 2

&+ s i n 4

~+ 2(b + 2 c ) s i n 2

$cos 2

A2222 = 1 A3333 = a s i n 4 & + cos 4 ~ + 2(b + 2 c ) s i n 2 5 cos 2 9 Al122 = h cos ~

(27)

Al133 = b ( s i n 4 6 + cos 4 6) + (a + 1 - 4 c ) s i n 2 & cos 2

I% can be seen by differentiating the denominator of Equation

(26)

that

qL

has three extrema for ~=0 °, ~=90 ° and for .

6c

+

4b

-

2a

-

I

= tan -I [(

(28)

)~] 6c + 2b - a - 2

To these

values

of

the

inclination correspond the following stress ratios,

qO = / 6 M/(~a + 2 - 8b) ½

(29)

respectively

t

~90 = / 6 H / ( 5 + a - 2b) ~

(30)

n~* : M

(31)

If n~. is a local minimum then qO and vice versa.

and q90 should necessarily be local maxima,

By performing triaxial compresssion

differently inclined

samples

at

a

tests

on

a

number

confining pressure such that all samples

fail in the ductile range, it is possible to determine the extreme the stress ratio (generally a minimum) for 00<9<90 ° Equation 30 find c. once M

of

value

and then to find M.

for From

it is possible tc~ determine (a-2b) and then, from Equation 28, to

Unfortunately Equation 29 is useless since ~0 and

(a-2b) are chosen.

is completely

defined

The most convenient values for a and b should

be chosen on the basis of a best fitting procedure. For the considered diatomite the following values have been chosen on the basis of the experimental results for a confining pressure equal to 2 MPa: 1.69,

a = . 4 , b= .038,

together with calculated

the

e=

.5.

The

calculated

tt

values

are

the difference

is slightly

values,

although

more than 5%.

plotted

may be noted t h a t t h e

c u r v e F i t s v e r y w e l l t h e e x p e r i m e n t a l d a t a f o r 45°<~<90 °

agreement i s t e s s good f o r t h e o t h e r ratios,

failure

e x p e r i m e n t a l p o i n t s i n F i g u r e 6.

M:

in

terms

while the of

stress

82

O'tf - ~3

•

0" 3 = & MPa

x

20

=2

I0

x

o

Figure 6

'

Calculated

3b

and

4-

'

9o

experimental

failure

values of deviator stress at

various confining pressures.

In F i g u r e 7 the c o r r e s p o n d i n g y i e l d l o c i plotted, HPa.

having

The f a i l u r e

c a l c u l a f i e d and is brittle When o}

assumed

thafi

for different

t o observed v a l u e s .

in nature for all

1

HPa

are

then

The model p r e d i c t s t h a t f a i l u r e

i n c l i n a f i i o n s when °3

i s l e s s than a b o u t ,

.6 HPa.

= 1.MPa, failure is brittle for $>45 ° while it is ductile for $<45 ° .

These results are in very good agreement with experimental evidence. the quantitative predictions are values.

are

i n h y d r o s t a t i c compression Pco i s equal t o 6.8

v a l u e s £or c o n f i n i n g pressures o f 0.5 and compared

values of $

somewhat

less

than

the

However,

observed

failure

This may be due to several reasons but it is the assumed shape of the

yield function

that

is

probably

the main responsible for this discrepancy.

Nevertheless, on the ~#hole, the qualitative agreement looks

reasonabty

good.

ql

83

(MPa 6

e=°3o

i !

I

p (MPa) Figure 7. Predicted initial yield loci for various sample inclinations.

COMPARISON BETWEEN CALCULATED AND EXPERIMENTAL RESULTS

The model has been checked against the experimental data Figure 8

given

in

[4].

shows calculated and observed results for different values of sample

inclination, in triaxial tests at constant cel]

pressure

on

the

considered

diatomite. Although in some cases quantitative agreement is especially for o]=4 MPa, it match on the

whole

particular the

the

general

transition

brittle for

is

o3=1MPa

experimental curve

has

trend

from

brittle

reproduced by the model for any pressure, failure

brittle when been

concentrate. uniform and

effects

along

for

from

the

to

experimental

being

and

&=90 °.

results.

Indeed

under

zero

In

In

those

confining

Failure is also two

cases

the

chopped off just after peak since after a drop,

the

again. plane

of

within

the

This

has

slip

where

sample

been

is

attributed

strains then

to

tend

to

largely

non

makes no sense a comparison with calculated results which, by

their very nature, assume uniform distribution of stresses and strains the sample.

good,

ductile behaviour is quite well

all sample inclinations.

&=60 °

The state of stress it

of

inclination.

the experimental curves tend to rise interlocking

far

is clear from Figure 8 that the model predictions

Ibis is a common feature when softening occurs.

within

For this reason,

84 ”

0

-3

>

t

‘c-l

0

/ -

u

I

\

I

7% I

L.7

85 whenever the model predicts the occurrence of a peak the ideal test is stopped at peak,

although

the

model itself would predict a strength reduction until

the line for which H=O is reached, theoretically with infinitely

large

shear

strains. Failure for all other tests is calculated and

experimental

ductile.

The

only

difference

between

curves occurs for ~=90 ° and ~3 = 2 MPa, where the

latter shows the existence of a peak, while the former tends to

a

horizontal

asymptote.

However, the large ductility of the actual sample after peak is an

index that

the

state

of

the sample is close to the transition from brittle

(existence of a peak) to ductile (asymptotic) behaviour. Another feature described by the model is the fact that while the elastic stiffness increases passing from ~=0 ° trend with

a

minimum

for ~

to ~=90 °, the strength has

close to 60 ° .

an

the model predicts that the elastic range is abandoned first, i.e. deviatoric stresses,

when

°3

is

the largest.

however, corresponds the largest strength.

clearly

shown

also

by

for

lower

To that confining pressure,

This fact causes a mutual crossing

of the stress strain curves for different confining pressures. behaviour is

inverse

Moreover, for all inclinations,

This

the experimental curves.

type

of

Finally, the

model predicts that the strength in unconfined compression is smaller but very close t~ that for a triaxial for &=O ~

where failure

is

test

with

ductile.

o3=I MPa for This

is

also

all

inclinations

clearly

shown

but

by the

experimental data. An evident difference between calculated and experimental curves lies

in

that, while for the former the crossing over the first yield function causes a sharp change in stiffness, the actual behaviour is much smoother. due to

two

facts.

First,

real specimens are non uniform and non uniformly

stretched because ef various types of imperfections.

This means that yielding

occurs under different loadings in different points within results in

a

Secondly, it inadequate in

stiffer is

response

possible

that

and

a

smoother

the

classical

the

transition elastic

sample,

what

to plasticity.

plastic

model

is

that phenomenological behaviour of all materials is such that s

smoother transition than that predicted by elastoplasticity is overcome this

This may be

difficulty

observed.

In

the concept of bounding surface and elastic nucleus

forwarded by Dafa]ias and Popov [8] for metals and by [9] for soils could be employed.

Dafalias

and

Herrmann

However, this would clearly render even more

complex the constitutive law presented.

86 CONCLUSIONS

It has been shown in this paper thaL an elastic model

similar

to

Cam

Clay

may

describe

the

plasLic

behavio~r

strainhardening in

axisymmetric

compression of soft sedimentary rocks with satisfactory q~,alitative with experimental

evidence.

agreement

Two modifications bare heen necessary.

First iL

was assumed that an initial elastic region of finite dimensions exisl-s, due t~ the lithification

process.

ovecconsolidation

in

This

soils.

is

seen

Clearly,

as

the

mechanically

destructuration

accompanies yielding could damage the material and properties.

eq~Jivalent

change

process

even

its

t')

which elastic

This fact has been neglected For the sake of simplicity, since no

clear indication

of

this

fact

exists,

at

least

Fur

the

type

of rocks

considered in this study° Secondly, it was necessary to extend the original Cam Clay model to

cope

with transverse isotropy, which is a common Feature for all sedimentary rocks. How large

is

the

relevance of anisotropy on the behaviour of such materials

can be inferred from the experimental results of Figure 8. convenience the

anisotropy

the deformation process.

of

three

the

sake

of

The extension of the model has been performed in the

simplest possible way, involving three new parameters for and another

For

the samples has been taken to be fixed during

for plastic.

elastic

anisotropy

A total of eleven constitutive parameters is

then necessary to fully characterize the material behaviour.

This number

may

look quite large, but the author does not see how the model presented could he made simpler, if account should be taken of anisotrooy. Indeed the very simple extension performed of Cam Clay is heel of

the model.

the

Achille's

In fact, since for an isatropic material Equation 7 would

give rise to a large

overpredir.tion

of

strength

in

plane

strain

and

in

triaxial extension tesLs, it is very likely that similar erroneous predictions would be

consequence

of

the anisotropic model, as well.

A more apprNpriate

" e x t e n s i o n " o f Cam Cla,, t o 3D c o n d i t i o n s t a k i n q a n i s ~ t r o p y i n t o account produce a

v e r y c o m p l i c a t e d e x p r e s s i o n f o r the y i e l d

w~uld

f u n c t i o n and c o n s e q u e n t l y

would make very cumbersome, although still conceptually sim[~le, the derivation of the stress strain law.

Although the calculated results

quantitative is

not

agreement

always

between

satisfactory,

experimental

picture of the behaviour of sedimentary rocks which seems with experimental

evidence.

strength with

inclination

described and

the

also well matched.

The and

transition

trend

confining

of

data

and

the modet gives an overall

the

in

Fair

agreement

variation of stiffness and

pressure

is

qualitatively

well

level between brittle and ductile behaviour is

However, calculated strength

is

generally

too

h]w

for

87 small confining

pressures

and

becomes dramatically higher than the observed

values for high confining pressures. To fit actual experimental results in a more satisfactory way also from a quantitative point of view, some modifications are

necessary.

For

instance

the yield function should allow for some strength in tension and the lines for wich 11=0 should

be

concave

downward.

Also,

a way to describe a smoother

transition from elastic to elastic plastic range should be sought. more

refined

expression

for

yield

function

should

be

Finally

found

to

a

avoid

overestimation of strength in plane stain and triaxial extension. Although the model implementation for

the

is

therefore

solution

not

yet

ready

for

finite

elements

of boundary value problems, it appears that

the framework of strainhardening plasticity is adequate not only for soils but also for soft sedimentary rocks. theory constitutes

a

geologic materials. evidence, a

new

unifying Should

paradigm

It is then natural

this for

to

argue

that

such

a

framework for the description of all types of opinion

geologic

be media

confirmed

by

experimental

would emerge that joins in a

unique description stiffness and strength and overcomes the existing dichotomy between the classical description of the mechanical characteristics

of

these

kinds of materials.

ACKNOWLEDGMENTS

This research has been partly supported by CNR (National Research Council of Italy) and MPI (Ministry of Education).

REFERENCES

1. Pellegrino A., Mechanical behaviour of

soft

rocks

under

high

stresses.

Proc 2nd ICISRM, Belgrade (1970), 2, paper 3-25. 2. Pellegrino A., Proprieta' Napoletano, Proc.

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dei

terreni

vulcaniei

e resistenzs

a

rottura

del

tufo

giallo

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4. Allirot D., Boehler J.P., Evolutions des proprietes meeaniques d'une stratifiee sous pression de confinement, Proc. 1, 15-22.

del

8th Italian Geotechnical Conference, Cagliari 1967.

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roche

4th ICISRH, Montreux, 1979,

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analysis

or

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