Fracture initiation and propagation in elastic brittle materials subjected to compressive stress fields — An experimental study

MECH. RES. COMM.

FRACTURE

Vol. 1, 225-231, 1974.

INITIATION

SUBJECTED

TO

AND

COMPRESSIVE

PROPAGATION STRESS

Pergamon Press.

IN E L A S T I C

FIELDS

N.J. A l t i e r o a n d D.L. S i k a r s k i e D e p a r t m e n t of A e r o s p a c e E n g i n e e r i n g , Ann Arbor, Michigan, U.S.A.

-

The

AN

Printed in USA.

BRITTLE

MATERIALS

EXPERIMENTAL

University

STUDY

of M i c h i g a n

(Received 2 July 1974; accepted as ready for print 10 July 1974)

Intr oduction

I n a s e r i e s of p r e v i o u s p a p e r s [ i , 2, 3] a t h e o r y h a s b e e n o u t l i n e d f o r t h e initiation and propagation of macrofracture in e l a s t i c b r i t t l e m a t e r i a l s subj e c t e d to n o n h o m o g e n e o u s compressive stress fields. In t h e p r e s e n t p a p e r this theory is examined experimentally. Results show excellent agreement with the predicted fracture initiation location and force levels and reasonable agreement with the predicted fracture propagation. The linear

theory,

as developed

in [1, 3] , d e f i n e s

F (o-~ , o-z ;V.) = ~" (o-1 + o-z) +

vrTv

a fracture

(o-1

-

functio'n:

o-z) (1)

u I = W iL/P

Uz = Wz L / P

w h e r e 0-1 , o-z a r e d i m e n s i o n a l p r i n c i p a l s t r e s s e s w i t h ~-1 > Wz a n d ~ i s t h e s l o p e o f t h e e n v e l o p e o f a s e t o f M o h r c i r c l e s d e f i n i n g s t r e n g t h f a i l u r e of t h e material [4]. Stresses are nondimensionalized w i t h r e s p e c t to P , L w h e r e P represents the total force and L the characteristic length shown in Figure (1). F r a c t u r e i n i t i a t e s in t h e f i e l d w h e r e F (ul , uz ; ~ ) i s m a x i m i z e d a n d w h e n the stresses are such that F(0- t ,0- z ; ~ )

= ~ = CL/P I

(2)

w h e r e C i s t h e T- ( s h e a r s t r e s s ) i n t e r c e p t of t h e s t r e n g t h f a i l u r e e n v e l o p e (cohesion) and PI is the fracture initiation load. Fracture propagation is c o n s i d e r e d to b e a s u c c e s s i o n o f f r a c t u r e i n i t i a t i o n s [1, 3 ] , i . e . , the microfractures emanating from critically oriented Griffith flaws coallesce to form the final fracture path; the macrofracture. Scientific Communication 225

226

N.J.

ALTIERO

and D.L.

SIKARSKIE

V,

.1, Nc.~

Analysis

The boundary

value p r o b l e m

to be a n a l y z e d (and subsequently studied experi-

mentally)is that s h o w n in F i g u r e

~

~

(1).

P

LOAD

CELL STEEL PLATE . . . . . STEEL BAR

DISPLACEMENT

-

MEASUREMENT I

©

STEEL PLATEN

~J

I I I

I L //

r~

~

\

L--I"

/

-

N

/ /

N

\ N

/ /

\N

/ /

\\

/

45

\N

/ / /

\\

4,,

/

\

/

:

ij

L

\

D

i~'

,,i I

,'~

L~


! lo

\

Plaster M o d e l

This is a m i x e d

i

:

.J r~

(-~ L1

2

i 1

4,5~

\ PIASTER

MODEL ~

FIG. l in L o a d i n g Fixture

p r o b l e m ; i.e., zero traction on D B

cal d i s p l a c e m e n t a n d zero shear stress on B A procedure

,iLi

and AC,

and CD.

specified verti-

A n integral equation

particularly convenient for fracture analysis a n d d e s c r i b e d in de-

tail in [3, 5, 6] is u s e d to obtain the stress field crI ,0-z . T h e m a t e r i a l studied experimentally was a modeling

plaster having the following m a t e r i a l constants:

E = 4.Z x 105 psi v = .34 C = 590 psi ~=

.93

I T h e s e constants w e r e d e t e r m i n e d using s t a n d a r d tests a n d test s p e c i m e n s m a d e a n d c u r e d u n d e r the s a m e conditions as the m o d e l s p e c i m e n s .

Vol.l, No.4

Results

FRACTURE UNDER COMPRESSIVE STRESS

of the analysis

are

the corners

A,B

components

at the corners

BA.

of the specimen.

For Pincreasing

ing contour

shown in Figure

Fracture

here initiates

T h i s i s d u e to t h e l a r g e

caused

by the uniform

and greater

than PI'

within which microcracking

ted that the direction

(2).

227

displacement

F(0-],~z;~)

(damage)

of macrofracture

normal

traction

of the surface

= B represents

is occurring.

growth is coincident

in

a grow-

It i s p o s t u l a -

with the maximum

P

3.

,

8

9

lo

2~

I

"

"

I/2

3~

I

/

\0

\

\\

/

."

"\ -

F°

tO

080

/.

*2 13

~

//

FracturePath'

~

t4

FIG. 3 Contour Map for Plaster; Uniform Traction Along BA

FIG. 2 Contour M a p for Plaster; U n i f o r m Displacement of B A growth

direction

presents

a discretizecl

macrofracture stress

of this contour.

approximation

path as shown.

state in the undamaged

for fracture progresses distribution)

initiation

load,

map

material. location

shown in Figure

of this contour

It i s a p p r o x i m a t e

beyond the initiation changes

The contour

point,

growth with a predicted

because

it is based

Note that there

and initial

and the approximation

field

becomes

on t h e

is no approximation

direction.

the stress

(2) r e -

As macrofracture

(and s u r f a c e

poorer.

traction

Z A second

z The microfracturing o c c u r r i n g w i t h i n t h e i n i t i a t i o n c o n t o u r F ( ~ I , 0-z ; ~ ) = i n t r o d u c e s d i l a t a n c y w h i c h h a s a m a r k e d e f f e c t on t h e s u r r o u n d i n g s t r e s s field. Work is in progress on determining t h e s t r e s s f i e l d in t h e p r e s e n c e of such damage. When complete, no stress field approximation will be necessary.

228

N.J. A L T I E R O and D.L. S I K A R S K I E

boundary sidered

value problem, and the results

ofmacrofracture represents erical

, uniform

are presented

propagation,

a better

results

i.e.

surface

in F i g u r e

normal

approximation

normal

Vol.l, No.4

(3).

traction,

is a l s o c o n -

D u e to t h e i n i t i a l d i r e c t i o n

to t h e l o a d e d s u r f a c e ,

it is felt t h a t this

to f i n a l c h i p d e p t h a n d f o r m a t i o n

for both the uniform

displacement

and uniform

load.

traction

Num-

cases

a r e g i v e n in T a b l e I.

Experimental

Kesults

The experimental

apparatus

i s s h o w n in F i g u r e

b e t w e e n t w o s t e e l p l a t e s to i n s u r e • 005" thick Teflon sheets were

(1).

plane strain

conditions.

d u e to f r i c t i o n .

s i s t e d of a u n i f o r m

The boundary

vertical

was applied quasi-statically BA w a s m e a s u r e d

was measured

placement

plots were

example,

Figure

with a Shaevitz model

w h i c h i s o f t e n u s e d to s i m u l a t e

made

the plaster-water

of a medium

The plaster-water

moved

and the mixture

set modeling

mixture

was stirred

h a d b e g u n to t h i c k e n

(10 m i n u t e s ) .

in 15-Z0 minutes

were

carefully

dis-

Load dissee,

for

To insure

to insure

flat,

of 6 0 ° F w a s

The models removal.

Prior

rewere The

from the mold.

(70°F) for six months.

position during this time.

brittle

by w e i g h t of p l a s t e r

to f a c i l i t a t e

and was then removed

cured at room temperature

machined

an elastic

until all air bubbles were

mixture

in an u p r i g h t

plaster,

Tap water at a temperature

molds which had been greased

stored

The load

X-Y plotter;

rock [7,8].

c a s t in a l u m i n u m

were

con-

T o t a l l o a d on the s u r f a c e

r a t i o c h o s e n w a s l o w (70 p a r t s

by weight of water).

used.

models

shear

(4).

brittle material

hardened

traction.

500 H R L . V . D . T .

on a H e w l e t t - P a c k a r d

were

to 100 p a r t s

of

m o d e l U3G1 l o a d c e l l w h i l e t h e v e r t i c a l

Test specimens

behavior,

Double layers

B A to e l i m i n a t e

and zero shear

(50 p o u n d s p e r s e c o n d ) .

recorded

confined

c o n d i t i o n s on f a c e B A , t h e r e f o r e ,

displacement

w i t h a B. L . H .

placement

were

placed between the steel plates and the model

faces and between the steel platen and loaded surface forces

The models

to t e s t i n g ,

They were the models

smooth faces and a uniform

thickness

(.450").

Experimental

results

are presented

in T a b l e I a n d F i g u r e s

(4,5).

The

Fracture

VOl.l, No.4

FRACTURE UNDER COMPRESSIVE STRESS

229

TABLE I Re sults

T e s t No.

Fracture Initiation Load Pi (pounds)

chip Formation Load Pf (pounds)

1 2

1030 1125

1400 1500

•~

3 4 5

1250 1050 ll60

1525 1400 1575

6

1050

1600

. 65 .65 .65 .60 .60 .60

x

7

1200

8

1180

1500

.55

1130

1500

. 61

1070 --

3200 2400

Average Uniform displ. Uniform traction

Depth A (inches )

....

1.00 .80

i n i t i a t i o n a n d g r o w t h of t he d a m a g e r e g i o n (to f o r m t he f i n a l m a c r o f r a c t u r e ) a r e e v i d e n c e d i n F i g u r e (4).

For increasing

l o a d t h e s p e c i m e n b e h a v e s in a

l i n e a r e l a s t i c m a n n e r up to p o i n t I.

200C

A t p o i n t I f r a c t u r e i n i t i a t e s at t he c o r TT

n e r s A , B in F i g u r e

Frr

(2).

The location

a n d d i r e c t i o n of i n i t i a t i o n w e r e d e t e r mined experimentally

by s t o p p i n g t h e

t e s t j u s t beyond point I and o b s e r v i n g the fracture. ~ooo

R e s u l t s a r e in a g r e e -

m e n t w i t h t he t h e o r e t i c a l p r e d i c t i o n s , i.e.,

Figure

microdamage

(2).

F r o m p o i n t I to II

o c c u r s i n two s e p a r a t e

i

r e g i o n s e m a n a t i n g f r o m p o i n t s A , B on

o!

Vertical

.05

10

Displacement of BA (in.)

FIG. 4 Typical Force-Displacement Plot o f U n i f o r m D i s p l a c e m e n t T e s t on P l a s t e r M o d e l ( T e s t N o. 5)

Figure

(2).

A t p o i n t II t h e s e r e g i o n s

meet.

T h i s w a s d e t e r m i n e d by s t o p p i n g

t h e t e s t at t h i s p o i n t a n d c a r e f u l l y r e m o v i n g the c o m p l e t e d chip, see F i g u r e (5).

T h e p o r t i o n of t he c u r v e f r o m

p o i n t s II to III r e p r e s e n t s

t he a c t i o n of

f o r c i n g the c o m p l e t e d chip into the cavity.

Tensile stresses

are introduced

230

N.J. ALTIERO and D.L. SIKARSKIE

FIG. 5 P h o t o g r a p h of C o m p l e t e

Voi.l , No .4

Chip

at the base of the cavity and, at III, a m o d e

I (tensile) fracture e m a n a t e s

this point, splitting the model.

results are presented in Table I.

Numerical

The average of the initiation loads is within 7 % of the predicted value. chip formation load and chip depth are in poorer a g r e e m e n t

from

The

as expected due

to the use of the prefracture stress field. A s discussed previously, it is felt that the u n i f o r m traction case provides a better approximation for the ultimate chip characteristics.

This is evident in Table I.

Dotted lines are s h o w n on Figures

(2,3) near the surface B A .

T h e s e lines

represent regions in w h i c h the intermediate principal stress is not n o r m a l to the s p e c i m e n plane (under plane strain conditions and for lJ= .24, see [3]). This introduces the possibility of oblique fracture in this region.

Such

fractures w e r e observed experimentally. While m u c h

additional w o r k is yet to be done in refining both the theoretical

predictions and the experimental observation of fracture growth (both m i c r o and m a c r o ) ,

the results presented s h o w significant promise.

Vol. l, No. 4

FRACTURE UNDER COMPRESSIVE STRESS

2 31

Reference s l.

D . L Sikarskie and N.J. Altiero, "The Formation of Chips in the Penetration of Elastic-Brittle Materials (Rock)," J. Applied Mechanics 4___00, 3, Trans. A S M E 95, p. 791 (1973).

2.

N.J. Altiero and D.L. Sikarskie, "Fracture Initiation in Elastic-Brittle Materials Having Nonlinear Fracture Envelopes," to appear in Intl. J. Fracture (1974).

3.

N.J. Altiero, Fracture Initiation and Propagation in Nonuniform C o m pressive Stress Fields, doctoral dissertation, Univ. of Michigan (1974).

4.

W.F. Brace, "Brittle Fracture of R o c k s , " State of Stress in the Earth's Crust, ed. W . Judd, Elsevier, N e w York (1964).

.

C. Massonnet, "Numerical Use of Integral Procedures," Stress AnalysisRecent Developments in N u m e r i c a l and Experimental Methods, ed. O.C. Zienkiewicz and G.S. Holister, J. Wiley, N e w York (1965).

.

R. Benjumea and D.L. Sikarskie, "On the Solution of Plane, Orthotropic Elasticity Problems by an Integral Method, " Journal of Applied Mechanics 399, 3, Trans. ASME 94, p. 801 (1972)

7.

B. Stimpson, "Modelling Materials for Engineering R o c k Mechanics," Intl. J. R o c k Mech. and Mining Sci. 7, p. 77 (1970).

8.

K. B a r r o n and G.E. Larocque, "Development of a M o d e l for a Mine Structure," Proc. R o c k Mech. Syrup. , McGill Univ., p. 145 (1962).