Size effect in rock testing

GEOPHYSICALRESEARCHLETTERS, VOL. 8, NO. 7, PAGES 671-674,

Size

Effect

in Rock Testing

Gregory Herbert

Abstract. and strength

effect

not

so the

effect

mechanism.

on

B. H.

Baecher* Einstein*

size

although this assumption is not necessary for the present conclusions.[Bieniawski, 1967; Brace, Paulding and Scholz, 1966] Test results indicate that specimen size affects these stress levels

and

differently.

Empirical relations between size have found ample treatment in the

literature,

underlying

fracture

of

size

This paper

mechanism

in

on the

examines

unconfined

JULY 1981

explained

triaxial tests between intact specimens of various sizes and on jointed specimens with various spacings. Statistical size effect and volume dependent strain energy effect can be distinguished and are shown to differently affect crack propagation and failure mechanism.

The stress ob appears to be by a statistical

model of size effect. seems unaffected

(i.e.,

tors

by specimen

in addition

A strain

Introduction

size.

The peak

to the presumably

statistical

Ob)(Figure lb).

energy

effect

due to specimen size

is believed to influence od.

Under this theory

the increased strain energy available in larger specimens provides an increased source for the propagation of cracks.[Glucklich and Cohen, 1967] The presumption is difficult to prove theoretically, but can be demonstrated empirically. By loading a "spring" in series with uniaxial compression specimens, stored strain energy can be increased without increasing specimen size. The

The influence of specimen size on measured material properties has been an issue of discussion for years, reflected in a generous literature. This note briefly summarizes the results of size effect studies conducted by the authors and their colleagues over the past ten years. Material

value)

stress od seemsto decrease with size more steeply than ob (i.e., may be influenced by facones affecting

Test

extreme

The stress range Oc-Ob

and Equipment

stresses ob and oc are unaffected by this spring, but od is sharply

The results presented were obtained from uniaxial and triaxial compression tests (o2=o 3) on gypsum modeling material which is brittle at low confining pressure.[Hirschfeld and Einstein] Axial

and

by strain axial

circumferential

gauges fixed

loads

are

strains

are

specimens have a height

with

load

cells.

to diameter

and

of 2.

statistics. Unconfined

Early cimens

tests to

on the should

Strength

were conducted

examine

on unconfined

statistical

size

effects

lc).

If

ob corresponds approximate-

ly to the onset of crack propagation or some other mechanism reflecting the Largest or least favorably oriented flaws in the specimen, then one would expect it to reflect extreme value

All

ratio

reduced (Figure

reported here suggest a descripfracturing not inconsistent with

current thought.

measured

to specimen surfaces,

measured

The results tion of brittle

population.

spe-

The range Oc-O b is then conditioned

extreme (or nearly extreme) crack and be unaffected by the size of the crack

If the range Od-Oc reflects

ble propagation and crack coalescence, depend more on averages than extremes

in

compression. While theoretical models based on extreme value statistics seem from empirical evidence to describe tensile strengths of brittle materials fairly well, their extension to compressive strengths has been less obvious. These

fore

be affected

than

the

pliance

statistics

(if

at of

all)

by factors

extremes.

of the loading

unsta-

it should and thereother

Increased

system appears

com-

to reduce

od by about the sameamountas an increase in

of the largest flaw in a homogeneous population of flaws within a specimen. As specimen size

specimen volume corresponding to the same increased strain-energy, and therefore strainenergy effects are inferred to be causing this

increases

decrease

models

are

based

so does

on the

the

statistical

number

distribution

of

flaws,

and

the

largest of that number becomes larger in a statistical sense. However, compressive strength does not depend uniquely on the largest flaw in a specimen. Therefore, an understanding of the effect of specimen size on fracture mechanism was sought in order to better understand the effect of size on peak strength. Measured strength behavior has been summarized by points of inflection of the axial and volumetric stress-strain curves, as shown in Figure la.

ble

workers

with

the

crack propagation

onset

of

stable

in uniaxial

and

Strength

Although unconfined strengths (Od) are influenced by specimen volume, the Mohr envelopes for D=i, 1.5, and 2.0" specimens are fairly similar. An interesting result of the triaxial tests is

that oc for each specimen size appears unaffected by confining

pressure

(Figure

2).

The range of

crack propagation and coalescence (Od-Oc) increa-

unsta-

compression,

ses with constant

specimen size peak strength.

from

unconfined

the

to give an approximately This is quite different

behavior.

The interpretation of the triaxial tests with respect to failure mechanisms is not yet satisfactory. Extrapolating the inferences from uni-

* Associate Professor of Civil Engineering Massachusetts Institute of Technology

axial

tests,

pressure

the constancy of oc over confining

but not

over

an extreme value effect

Copyright 1981 by the American Geophysical Union. Paper number 1L0677. 0094-8276/81/001L-0677501.00

strength.

Triaxial

The stresses ob and oc have been associated by some

in peak

671

specimen

size

would

imply

uninfluenced by o3.

It

672

Baecher and Einstein:

Size Effect

in Rock Testing

(A) TYPICAL STRES-STRAIN CURVES OF BRITTLE MATERIAL

(B) SIZEEFFECT ON STRESS LEVELS'b','c' AND 'd' FOR GYPSUMCYLINDœRS {C) STRESSLEVELS'b','c' AND'd' FORGYPSUM CYLINDER 0 AND FOR GYPUM CYLINDER IN SERIES WITH SPRING 1:3

(c) 2.6

(a) AXIAL STRESS

2.4

2.2

22

I.E 1.6

1.2•rb

•

I.C 0.8 0.6 0

-------

0.6 I

2

STRAIN Figure

1.

Size

Effect

Hirschfeld,

5

I0

15

SPECIMEN VOLUME, in. 3

Results

for

Oniaxial

Compression

[from H.H.

Einstein,

SPECIMEN VOLUME, in. 3 G.B.

Baecher

1970].

5

4-

3-

I•1 0':5=1500 psi (•) 0':5 =1000psi

/• 0':5 =500psi

O'

I

!

1.0

1.5

I

2.0

SPECIMEN DIAMETER, inches Figure

2.

Size

Effect

Results

for

Triaxial

Compression

[from D.D.

Hunt,

1973].

and R.C.

Baecher and Einstein'

I.•

i

i

Size Effect

i

in Rock Testing

673

i

500 psi (3.45x106

o'$=0

O !• MN/m2)

ID

0.8

bJ

•

0.6

0.4

0.2

o-$. Confining Pressure (O-l-O-$) t = DeviatorStress at Failure

o0

i 5 NUMBER

Fñgure

3.

Number of Joñnts

,

OF

JOINTS

Tntersected

.

I0

INTERSECTED by •a•[u•e

BY

Su•ace

[•om

I

FAILURE R.C.

.

15

20

SURFACE •sch•eld

a•d •.•.

g•ste•,

1973].

is not immediately

apparent

what this

Effect

might be;

modified Griffith theory would predict a 03 dependence. ob could not be measured in these tests. The disappearance of size effect on od

of

Joints

In early test series the strength of specimens decreased as the number of joints intersected by the failure surface increased (Figure 3). In a

is also difficult to explain. The simple strainenergy hypothesis no longer seems to apply. It is possible that surface effects play a larger role in triaxial strengths than in uniaxial strengths, and that a compensation of effects results, but this is mere speculation.

subsequent study, cylindrical specimens 1.35" in diameter and 3" high with joints perpendicular to the major principal stress were tested in triaxial compression. Nine (9) specimens per spacing and confining pressure were tested and the

(B)

4

_•

•

0

yo'$=600psi 400psi

_

I/2"

0

0

•'

I

2

I

$

I

4

I

NUMBER

Figure 4.

Peak Compressive Strength

I

I

!

I

[

5 6 7 u 9 I0 OF

!

15

I

20

30

JOINTS

of Jointed

Specimen [from K.A. Seeler,

1978].

674

Baecherand Einstein' Size Effect in RockTesting

•

---G_• •..•

ß 0

"'

•UNLOADING

o'B= 200psi

ß G o'• = 400psi

2

ß

o'• = 600psi

--

o

o

•'

2

I/2"

I

% I

I

I

I

I I

4

5

6

7

8

9

NUMBER

Figure

5.

Deformation

Moduli of Jointed

mean results are shown in Figure 4. Peak strength decreased logarithmically with the number of joints per specimen, and this trend did not appear influenced by confining pressure. Perhaps the most obvious explanation is that asperities on the joint surfaces serve as stress concentration loci, whose number increases in proportion to the number of joints. If initial

crack propagation starts at the most critical of these (i.e., at the largest of their number), extreme value theory would apply and would predict a logarithmic relation to the number of joints. While stress concentrations mostly affect stable crack propagation, more joints also lead to an increase in stored strain energy and may affect unstable propagation as well. In addition to this effect on strength, an interesting effect on deformability was observed

(Figure 5). Modulus, defined as the slope of the longest straight line segment of the prefailure (or post failure, respectively) portion of the load-displacement curve, decreases less quickly than in joints.

inverse

proportion

I/8"

to the number of

I

I

15

20

OF JOINTS

Specimen [from K.A.

Seeler,

1978].

lescence of many propagating microcracks). third is that loading compliance and strain energy should be considered in similitude.

The

References

Bieniawski, fracture

Z.T. (1967). "Mechanisms of brittle in rock", International Journal of

Rock Mechanics and Mining Science, 4(4):

395-

406.

Brace,

W.F.,

"Dilatancy

B.W. Paulding

and C. Scholz

in the fracture

(1966).

of crystalline

rocks", Journal of Geophysical Research, 77 (16): Einstein,

3939-3953. H.H., G.B.

(1970).

Baecher

"The effect

of brittle

and R.C.

rock", Proceedings, 2nd Interna-

tional Congress on Rock Mechanics, Glucklich, J. and L.G. Cohen (1967). tor

in

the

Hirschfeld

of size on the strength

brittle

to

ductile

Belgrade. "Size fac-

transition

and

the strength of some materials", International Journal of Fracture Mechanics, 278-289. Hirschfeld, R.C. and H.H. Einstein (1973). "Model Studies on Mechanics of Jointed Rock",

Journal of the Geotechnical Engineering Division,

Implications Aside

from

served intact logarighmically

strengths

the

for

Large

obvious

conclusions

that

ob-

strengths decrease more or less with specimen size, and jointed

decrease logarithmically

is

that

mechanisms

dependent

ASCE, 99 (SM3).

Hunt, D.D. (1973). pressure on size

Scale Testing

with numbers

of joints, the present results carry implicit suggestions about scale effects in rock testing. The first is that changing scale may change the relative importance of failure mechanisms. The second

I0

on extreme

elements (e.g., the first microcrack to propagate) of a physical system must be distinguished from those dependent on averages (e.g., the coa-

"The influence of confining effect", Thesis submitted to the Massachusetts Institute of Technology in partial fulfillment of the requirements for the degree of Master of Science, 254 pp. Seeler, K.A. (1978). "The Influence of Joint Intensity on the Strength of a Rock Model", Thesis

submitted

to

the

Massachusetts

Institute

of Technology in partial fulfillment of the requirements for the degree of Master of cience, 253 pp.

(Received

March I8,

accepted April

1981;

14, 1981.)