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G A - A 1 5 2 2 8

uc-77 I

A MODIFIED W E I B U L L THEORY

F O R THE STRENGTHOF GRANULAR BRITTLE MATERIAL

bY

F. HO

Prepared under

Contract DE - A T 0 3 - 7 6 E T 3 5 3 0 0

for the San Francisco Operat ions O ff ice

Department o f Energy

D A T E P U B L IS H ED : M A Y 1979

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DISCLAIMER

This report was prepared as an account of work sponsored by anagency of the United States Government. Neither the United StatesGovernment nor any agency Thereof, nor any of their employees,makes any warranty, express or implied, or assumes any legalliability or responsibility for the accuracy, completeness, orusefulness of any information, apparatus, product, or processdisclosed, or represents that its use would not infringe privatelyowned rights. Reference herein to any specific commercial product,process, or service by trade name, trademark, manufacturer, orotherwise does not necessarily constitute or imply its endorsement,recommendation, or favoring by the United States Government or anyagency thereof. The views and opinions of authors expressed hereindo not necessarily state or reflect those of the United StatesGovernment or any agency thereof.

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DISCLAIMER

Portions of this document may be illegible inelectronic image products. Images are producedfrom the best available original document.

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-

ABSTRACT

A modi f i ed Weibul l theory i s d ev e lo p ed t o d e s c r i b e t h e s t r e n g t h o f

a g ra nu la r b r i t t l e mate r i a l . The p r e se n t t h e or y r e c o n c i l e s t h e e f f e c t

o f g r a i n s i z e , i n a d d i t i o n t o t h e e f f e c t s of v olum e a nd s t ress d i s t r i -

b u t i o n . T h i s i s done by in t ro duc ing a f o u r t h mate r i a l parameter h,

c a l le d t h e c h a r a c t e r i s t i c g r a i n s i z e . I t i s a pp l i e d t o over 2000 da t a

p o i n t s of t h e g r a d e H 4 5 1 g r a p h i t e . E x c e l l e nt c o r r e l a t i o n i s o b t a i n e d

by choos ing h t o be the maximum gr a i n s i z e . The d i f f e r en ce be tween

t he t he o r y a nd t he mean s t r e n g t h i s l e s s t h a n 3% i n a l o g . The d i f f e r e n c e

i s about 6% f r o m l o g t o l o g . Th e t h e o r y i s a l s o a pp l i ed t o EGCR-type

AGOT g r a p h i t e w i t h v e r y good r e s u l t s .

0

iii

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CONTENTS

TABLES

1. FLEXURAL STRENGTH TEST DATA OF H451 GRAPHITE - - - - - - - - 16

2. STRENGTH OF EGCR-TYPE AGOT GRPAHITE - - - - - - - - - - - - 17

3. STRENGTH RATIOS OF EGCR-TYPE AGOT GRAPHITE - - - - - - - - - 1 8

FIGURES

10. INTERNAL PRESSURE TESTS ON TUBULAR GRAPHITE SPECIMENS - - -

V

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1 . INTRODUCTION

G r a p hi t e, i n g e n e r a l , i s a n i nhom ogeneous , a n i s o t r o p i c , po r ous

and g r an u la r b r i t t l e s o l i d . I t s mechanica l behavior i s ve ry complex i n

na tu re . S t r eng th measurements show a h ig h v a r i a b i l i t y and a dependence

o n t h e p h y s i c a l s i z e , t h e s h a p e, t h e t y p e o f t e s t , t h e o r i e n t a t i o n and

t h e l o c a t i o n i n a l og , and t h e m a nuf a c tu r i ng p r o c e s s , e t c . Some of

t h e t y p i c a l f a c t s a r e t h a t t h e s t r e n g t h de c re as e s a s t h e vo lume i nc r e a s e s .

The f l e x u r a l s t r e n g t h i s known t o be40

t o 100% h i g h er t h an t h e t e n s i l es t r e ng t h . F u r t he rm or e , when t h e l e a s t dimension of a specimen i s n e a r

t h e g r a i n s i z e , i t s s t r e n g t h goe s down r a p i d l y . T he e f f e c t O f t h e l e a s t

d i m ens i on t he n ou t w e ighs t he volume e f f e c t .

A t tempts have been made by many au th or s ( s ee re f e r en ce s c i t e d i n

Refs . 1 and 2 ) t o d e v e l o p a s t r e n g t h t h e o r y t h a t c a n a d e q u a t e l y d e s c r i b e

t h e m a t e r i a l . Among them a r e P r ic e and Cobb (Ref . 2 ) , who app l i e d th e

Weibul l s t a t i s t i c a l t h eo r y t o g r a p h i t e mate r i a l .

p r e d i c t s t h e e f f e c t o f n on -u ni fo rm s t ress d i s t r i b u t i o n , b u t t h e ex pe ct ed

dependence on volume i s n o t o b s er v ed i n t h e t e s t . A p p l i c a t i o n of t h e

t h e o ry t o o v e r 2000 d a t a p o i n t s on H451 g r a p h i t e i n R ef. 3 reveals t h e

same c o n c l u s i o n .

The t h e o r y s u c c e s s f u l l y

A g e n e r a l s t r e n g t h t h e o ry w a s developed by T s a i a nd Wu ( Ref . 4 ) .

They assume th e ex i s t e nc e of a s c a l a r q u a d r a t i c f u n c t i o n o f t h e s t r e s s

t e n s o r i n s t r e s s sp a c e. Chang and Weng (R ef. 5) a p p l i e d i t t o s e v e r a l

g r a p h i t e mate r i a l s . Good c o r r e l a t i o n b e tw ee n t h e g e n e r a l t h e o r y a nd t h e

a v a i l a b l et e s t

d a t a w a s obt a ine d . However , th e r ea r e s e v e r a l

drawbacks in

p r a c t i c a l a p p l i c a t i o n , a l t ho u gh i t i s m a t h e ma t i ca l l y c o n s i s t e n t .

F i r s t , t h e c o n s t a n ts a r e d i f f i c u l t t o c h a r a c te r i z e e x pe ri me nt al ly .

Some s op h i s t i c a t e d t ype s of t e s t s a r e r e q u i r e d . S e co n dl y , t h e g e n e r a l

t h e o r y i s o n l y a s t ress f u n c t i o n e v a l u a t e d a t a p o i n t a n d i t does n o t

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recognize the non-uniform s t r e s s d i s t r i b u t i o n i n i t s neighborhood.

On th e o t he r hand, th er e i s i n c r e a s i n g e v id e nc e s u p p o r t i n g t h e g r a d i e n t

e f f e c t on t h e s t r e n g t h . T h i rd l y , g r a p h i t e i s porous and has f laws

d i s t r i b u t e d t h r o ug h o u t t h e b od y. The l a r g e r t h e v olume, t h e l o wer o ne

s t r e ng th . The volume e f f e c t cannot be accommodated by th e gene ra l the ory

w i t ho u t m o d i f i c a ti o n , F i n a l l y , t h e g e n e r a l s t r e n g t h t h eo r y l a c k s a

s t a t i s t i c a l i n g r e d i e n t f o r p r a c t i c a l c o n s i d e r a t i o n . The two a fo re me nt io ne d

t h e o r i e s a r e n o t m u t u a l l y e x c l u s i v e . I n f a c t , t h e y c a n b e b le nd ed i n t o a

more g en e r a l s t r e n g th t h eo r y .

I n t h i s p ap er a new theory i s presen ted by modi fy ing th e Weib u l l

d i s t r i b u t i o n . The m o di f ie d W e i bu l l t h e o r y r e c o n c i l e s t h e e f f e c t of t h e l e a s t

d im en sio n ( r e l a t i v e t o t h e g r a i n s i z e ) i n a d d i t i o n t o t h e e f f e c t s of

volume and s t r e s s d i s t r i b u t i o n . I t h a s t h e s t a t i s t i c a l a s p e c t of t h e

s t r e ng t h f o r p r a c t i c a l a p p l ic a t i o n. T h e Weib u l l d i s t r i b u t i o n i s b r i e f l y

r ev ie we d i n t h e n e x t s e c t i o n , f ol lo we d by d i s c u ss i o n of t h e t e n s i l e

ver sus bend s t r e ng th . The modi f ied Weibu l l the ory i s t h e n i n t r o d u c e d .

I t i s a p p l i e d t o H451 g r a p h i t e m a t e r i a l . Some resul ts a r e summarized,

and recommendation i s made f o r f u tu r e t e s t work.

I t s h o u ld b e w o r th t o men tio n t h a t t h e Weibu l l t h eo r y i s p r e s e n t l y

a p p l i e d i n t h i s paper t o t h e s t r e n g t h of g r a p h i t e w i t h in t h e s co pe of t h e

c l a s s i c a l t h eo ry of e l a s t i c i t y . T h is i s b e c au s e t h a t t h e We i bu l l t h e o r y

c a n n o t b e a p p l i e d u n l e s s t h e s t re s s d i s t r i b u t i o n i s known. But t h e th eo ry

d o e s n o t l i m i t i t s e l f t o t h e above c o n s t i t u t i v e r e l a t i o n . I f a d i f f i c u l t

c o n s t i t u t i v e l a w were ad o p ted , t h en t h e mo d i f i ca t i o n p r e s en t e d h e r e would

b e d i f f e r e n t a nd p o s s i b l y e ve n un n ec e ss a ry .

We s h a l l r e s t r i c t t h e a p p l i c a t i o n o f t h e W e i bu l l t h e o r y t o t h e u n i -

a x i a l s t ress s t a t e o n ly . I t i s premature a t t h i s moment t o ex tend i t t o

t h e m u l t i -a x i a l s t re s s s t a t e s . Fu r th e r r e s ea r ch w o r k i n c lu d in g ex p e r imen -

t a t i o n n e ed s t o be d on e i n t h i s d i r e c t i o n b e f o r e a n y c o n cl u s io n c a n be

drawn.

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2. THE WEIBULL DISTRIBUTION

This distribution is gaining rapidly in popularity for use in analyz-

ing data from strength tests of brittle solids. The three parameter Weibull

distribution in the uniaxial stress state can be written as follows:

where Sv = the cumulative survival probability for the given

specimen,

Go = scale parameter or characteristic strength,

oU = location parameter or the lower bound of strength,

m = shape parameter or Weibull slope,

v = volume o f the specimen,

0 = stress in the specimen, the random variable of interest.

The lower bound of strength, oU, for graphite material is possibly

In Ref. 6 the experimental strengthThe results obtained in the fit

very low and is assumed to be zero.data of POCO graphite are analyzed.

of the data are insensitive to the value of CI,, whether taking 0, = 0

or

in analyzing fatigue life data of mechanical components and electronic

devices in the Reliability Engineering field.

throughout the present discussion.

t a k in g t h e b e s t - f i t v a l u e f o r oU. The same assumption is a l so made

Hence, it is assumed oU = 0

The Weibull theory is then applied to the three commonly used

strength test configurations.

stress distribution. Th e non-uniform stress distribution includes the four

point bending test and a hollow cylinder subjected to internal pressure.

The uniaxial tensile test has a uniform

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where

The s u r v i v a l p r o b a b i l i t y d e r i v e d f ro m e q u a t i o n ( 1 ) c a n b e e x p r e ss e d i n

th e fo l l owi ng form (Ref . 2 )

sV = exp

[-v(kT]V = V t , t e n s i l e t e s t

= Vb, bending t e s t

= V , i n t e r n a l p r e ss u r e t e s tP

V t = t h e l oa d - c a r r y i ng vol um e of a t e n s i l e s pe ci me n

1vb -a v,/2 + vos/(m+l)) , r e c t a n g u l a r s pe ci me n

g , ( m) , c i r c u l a r s pec im e n'b = ( cs + - +3 '0s)

3

g2(R ,m ), ho l low c y l i nd r i c a l s pec im e nL 1

v = v -R2-1 (R? l ) m

o = ot, t e n s i l e s t r e s s , t e n s i l e t e s t

= G maximum bending s t r e s s , bending t e s tb '

= cl s t r e s s a t t h e i n n e r r a d i u s , i n t e r n a l p r e s s u r e t e s tP '

V = t h e vol um e o f t h e c e n t e r s pa ncs

Vo s = the volume of the outer span

R = t h e r a t i o of O.D. t o I.D.

V = the vo lume of t h e h o ll ow c y l i n d e rC

2.4.6 . (m-1) 2- m o dd ' i n t e g e r1.3.5 * . . ( m + Z ) ' i T '

gl(m > and g2(R,m) a re d e fi n ed he r e f o r , i n t e g e r m o n ly .

non- in teger m c a n b e s t b e o b t a i n e d by i n t e r p o l a t i o n me th od s.

T h e i r v a l u e s f o r

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3 . D I S C U S S I O N ON T E N S I L E AND FLEXURAL STRENGTH

I t i s a well-known f a c t t h a t t h e v a l u e of t h e f l e x u r a l s t r e n g t h

b as ed on t h e c l a s s i c a l b eam th eo r y i s c o n s i d e r a b ly h i g h e r t h a n t h e u n i-

a x i a l t e n s i l e s t r en g t h i n b r i t t l e s o l i d s .

ma te ly 1 . 4 t o 2 . 0 f o r g r a p h i t e d e pe nd in g on t h e s pe ci me n s i z e , t h e

specimen shape, t h e a r r ang emen t o f l o ad in g , t h e t y p e of mate r ia l , and

th e s pecimen lo ca t i o n . Bes id es , a h ig h v a r i a b i l i t y e x i s t s i n s t re n g t h

m ea su re me nt s o b t a i n e d i n e i t h e r t y p e of t h e t e s t s .

w i l l g iv e a c r i t i c a l review o n th e main d i f f e r en ce s o f t h e two t e s t s .

H o p e f u l l y , t h i s w i l l l ea d t o some exp la nat ion s t ha t accommodate t he

a f o r emen t io n ed d i s c r ep an c i e s .

T h e r a t i o i s f rom approxi-

I n t h e f o l lo w i ng w e

G r ap h i t e i s , i n gen era l , inhomogeneous , a n i so t r op ic and porous .

S p a t i a l v a r i a t i o n o f t h e m a t e r i a l p r o p e r t i e s i n a g e n e r i c l o g i s observed .

I t c a n b e s e p a r a t e d i n t o t wo p a r t s , a s y s t e m a ti c g l o b a l v a r i a t i o n and a

random lo ca l v a r i a t i o n . The v a r i a t i o n can b e e s t ima ted fr om th e t e s t d a t a .

The t ru e va r i a t io n can never be known.

L e t u s examine t h e e f f e c t of t h i s v a r i a t i o n o n t h e u n i a x i a l t e n s i o n\

t e s t . I n t h e c l a s s i c a l beam theory there w i l l b e no e x t e r n a l b e nd i ng

moment a c t i n g on a g i v en s e c t i o n of t h e beam i f t h e a p p l i e d a x i a l l o a d

i s p as s in g t h r o u g h th e e l a s t i c a x i s . The e l a s t i c a x i s i s d ef in ed b y

EXdA = 0 where E i s Young's modulus and x i s t h e d i s t a n c e fro m t h e

g e o m e t r i c a x i s . S i n c e E i s a random v a r i a b l e , t h e l o c a t i o n of t h e e l a s t i c

a x i s c ha ng es f ro m s e c t i o n t o s e c t i o n a l o n g t h e b eam a x i s .

f o r a c i r c u l a r c r o s s s e c t i o n w i t h l i n e a r v a r i a t i o n o f E a lo n g o n e d i ame te r ,

A s an example

i . e . , E(x) = El ) ( + E o, t h e e l a s t i c a x i s i s s h i f t e d by a d i s t a n c e o f

E , r / 4 E 0 t o wa r ds t h e h i g h s i d e of E , where r i s t h e r a d i us of t h e c i r c u l a r

c r o s s s e c t i o n .

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C o ns id e r t h e u n i a x i a l t e n s i o n t e s t w i t h s pe c i m e ns ha v i ng d i a m e t e r

of 0.25" and 0.505".

E o , t h e s h i f t o f t h e e l a s t i c a x i s i s 0.00156" and 0.00316" f o r 2 r = 0.25"

and 0.505", r e s p e c t i v e l y . The e x tr em e f i b e r s t resses f o r a l o a d P a c t i n g

Assuming a 4. 5% v a r i a t i o n a b o u t t h e mean v a l u e

normal ly a t t h e e l a s t i c c e n t e r of t h e s e c t i o n a r e

P

A -y = - ( I + E l r h o )

With a -5% v a r i a t i o n i n E

0 = 1 . 0 5 P /Amax

augmented by an amount equa l t o one ha l f o f th ehe maximum f i b e r s t r e s s i s

p e r c e n t a g e o f t h e r a n g e of v a r i a t i o n i n E . T h e r ef o r e , t h e c a l c u l a t e d

P / A s t r e s s i s always smal le r t ha n what a pp e a r s i n t h e s pec im e n, a nd t h e

a m oun t o f d i f f e r e nc e i s random.

I n a d d i t io n t o t h e mate r i a l e c c e n t r i c i t y , t h e r e may b e l o a d e c c e n t r i c i t y

due t o i m pr ope r a li gnm e n t o f t he l oa d t r a i n . A d i r e c t a p p l i c a t io n of a

p u r e t e n s i o n f o r c e a l o n g t h e g e om e tr ic a x i s of t h e specimen i s v e r y d i f -

f i c u l t , and i s f u r t he r c om pli c a t ed by s e c onda r y s t resses induced by the

e nd g r i p s . S uppose t h a t a t e n s i l e t e s t machine i s c h a r a c t e r i z e d t o ha ve

a maximum load ec ce nt r i c i ty of s a y , a'' w i t h r e s p e c t t o t h e g e o m et ri c a x i s

o f t h e specimen. The c o r r e s po nd i ng pe r c e n t a ge i nc r e a s e i n t he maximum

stress i s ( 4 a/r) (6 a/h for rectangular cross section of thickness h).

T e s t s performed a t GA u s i n g s t r a i n ga u ge s o n some 0.5"$ ( d i a m e t e r m e t a l l i c

s p e c i m e n s i n d i c a t e a maximum.varia t ion of ?6%7% i n s t r a i n . T hi s i s conver-

t e d t o a maximum eccent r i c i ty of 0 . 0 0 4 " . T h is e c c e n t r i c i t y i s a l s o random.

T h e t w o e c c e n t r i c i t i e s a r e a d d i t i v e . .The e c c e n t r i c i t y s ho u ld y i e l d

a h ig he r v a r i a b i l i t y i n t h e t e n s i l e s t r e n g t h th an t h e f l e x u r a l s t re n g th .

Thi s indeed i s observed (Ref . 3 ) . I t a l s o l e a ds t o a l o w e r t e n s i l e

s t r e n g t h , b y seve ra l p e r c e n t .

i n d i c a t e s a n a p p r o x i m a t e r a n g e o f 3 t o 6 pe r c e n t de pe nd i ng on t he s pe c i m e n

s i z e .

O ur be s t e s t i m a t i on f r om t he t e s t r e s u l t s

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The maximum t e n s i l e s t r e s s i n a bending specimen i s c a l c u l a t e d b y

u s in g t h e beam fo r mu la fro m th e c l a s s i ca l beam th eo r y . The q u a l i f i ca -

t i o n ' ' c l a ss i c al ' l r e f e r s t o t h e a ss um pt io n i n t h e c a l c u l a t i o n t h a t t h e

s t ress i s p r o p o rt i o n a l t o t h e d i s t a n c e from t h e n e u t r a l a x i s of t h e

beam. While i n r e a l i t y t h e s ha pe of t h e s t r e s s - s t r a i n c u rv e u nd er l o a d s

n e a ri n g f a i l u r e i s known t o b e n o n l i n e a r , n o t t r i a n g u l a r .

The f l e x u r a l s t r e n g t h t h u s o v e r e s ti m a t e s t h e u n i a x i a l t e n s i l e

s t r e n g t h a nd g i v e s a h ig h e r v a lu e t h an wo uld b e o b t a in ed i n a d i r e c t

t e n s i o n t e s t .

Ref e r en ce 3 p r e s e n t s a m e t ho d of c o r r e c t i o n f o r n o n l i n e a r stress-

s t r a i n cu r v e . The mean v a l u e o f t h e r a t i o of t h e u n c or r e ct e d f l e x u r a l

s t r e n g t h t o t h e c o r r e ct e d f l e x u r a l s t r e n g t h i s ab o u t 1.20 for 0 .25"$1 x

1.5" H451 specimen sub je ct ed t o a f o u r p o in t b en d in g t e s t . A d d i t i o n a l

unpubl i shed t e s t d a t a a t GA on H451 g r a p h i t e show a mean r a t i o o f 1 .25

and 1.1 3 f o r 0.25"@ x 1.5" and 0.35" x 0.35" x 1.5" specimens,

r e s p e c t i v e l y .

One po ss ib le re as on t h a t may accommodate the note d dis cre pan cy

be tw ee n t h e f l e x u r a l s t r e n g t h and t h e u n i a x i a l t e n s i l e s t r e n g t h i s

a s c r i b e d t o t h e W e ib u ll t h e o r y . The a p p l i c a t i o n o f t h i s t h e or y t o t h e

s t r e n g t h of g r a p h i t e i s th e main concern of t h i s p a p e r .

An a l t e r n a t i v e t o W ei bu ll t h e o r y c a n b e d e s c ri b e d as f o l l o w s :

I n t h e b e n d t e s t t h e maximum f i b e r s t ress r each ed i s h i g h e r t h a n

i n d i r e c t t e n s i o n b e c au se t h e p r o p ag a t io n of a c r a c k i s blocked by le s s

s t r e s s e d mater i a l n e a r e r t o t h e n e u t r a l a x i s . Thus t h e en er gy a v a i l a b l e

i s b elow th a t n eces s a r y f o r t h e f o r ma t io n o f new c r ac k s u r f ace s . T h i s

i s e s s e n t i a l l y t h e s t r a i n g r a di e n t t h e or y ( Re f. 7 ) . The theory employs a

d i f f e r e n t c o n s t i t u t i v e l a w and can ex p l a i n t h e a f o remen t io n ed d i s c r ep an cy

i n s t r e n g th .

e l a s t i c i t y w i th c ou pl e s t resses (Ref. 8 ) .

Advances have been made considering a l i n e a r t h e o r y of

7

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Some r e s u l t s de r i ve d f rom t h e a pp l i c a t i on of t h e We i bu l l t he o r y t o

g r a p h i t e i n R ef . 3 a r e reviewed.

s t re ng th of smal l - s i ze d specimens and l a rg e -s i zed specimens f rom th e

same l o c a t i o n and o r i e n t a t i o n a re 1.03 an d 1 .0 8 f o r t h e a x i a l and r a d i a l

o r i e n t a t i o n , r e s p e c t i v e l y . The W e i bu l l t h e o r y p r e d i c t s a s t r e n g t h

r a t i o of 1 . 2 8 f o r t h e a x i a l o r i e n t a t i o n a nd 1.38 f o r t h e r a d i a l o r ie n ta -

t i o n . I t i s t h e r e f o r e c o nc lu de d t h a t t h e W e i bu l l t h e o r y g r o s s l y o v er -

est imates th e volume dependence of th e s t re ng th of H451 g r a p h i t e .

The r a t i o s be tw ee n t he mean t e n s i l e

The obs e r ved r a t i o bet we en t h e f ou r - po i n t mean f l e x u r a l s t r e ng t h

w i t h c o r r e c t i on f o r t h e n o n l i n ea r s t ress s t r a i n r e l a t i o n s h i p and t h e

mean t e n s i l e s t r e n g t h a r e 1 .5 2 f o r a x i a l a nd 1.55 f o r r a d i a l . The

p r e d i c t e d v a l u e s o f 1 .5 1 f o r a x i a l and 1 . 6 4 f o r r a d i a l a re i n good

a g r e e m e n t w i t h t he obs e r ve d r a t i o s .

The c o e f f i c i e n t of v a r i a t i o n f o r sm a l l -s i z e d t e n s i l e sp ec im en s f ro m

t h e same zo ne i n t h e p a r e n t l o g a v e r ag e d 1 2 . 1 % an d 15.3% f o r a x i a l and

r a d i a l o r i e n t a t i o n , r e s p e c t i v e l y . The l a r g e - s i z e d s pe ci me ns show a

s l i g h t l y sma l l e r c o e f f i c i e n t of v a r i a t i o n o f 1 0. 2% a nd 14.3% f o r t h e

a x i a l a n d r a d i a l or i e n t a t i o n . The bend t e s t specimens have the sma l l e s t

c o e f f i c i e n t o f v a r i a t i o n of a l l , a ve r a g i ng 7 . 7 % f o r t h e a x i a l o r i e n t a -

t i o n and 9.0 % f o r t h e r a d i a l o r i e n t a t i o n . The t r e n d of t h e v a r i a t i o n

c on fo rm s t o w ha t ha s be en d i s c u s s e d p r e v i ous l y .

I n a p p l i c a t i o n s t h e m o d if i ed W e ib u ll t h e o r y l a t e r , c o r r e c t i o n s a r e

made on l y f o r t he non - l i ne a r s t ress s t r a i n r e l a t i o n . Av erage c o r r e c t i o n i s

a bou t 1 7 p e r c e n t f o r H451 g r a p h i t e ( R e f . 3 ) . The e f f e c t of e c c e n t r i c i t i e s

i n m ost cases c an n o t be e a s i l y a s s e s s e d . I t i s b e li ev e d t h a t t h e e f f e c t

i s n o t s i g n i f i c a n t . U sin g t h e H451 g r a p h i t e d a t a f ro m R ef . 3 , t h e c o r r e c t i o n

i s e s t i m a t e d t o b e no more t h a n o ne h a l f o f t h e s t a n d a r d d e v i a t i o n . B e s i d es ,i t l a c k s o f a s y s t e m a t i c way t o a cc o u n t f o r t h e e f f e c t . T h e r e f o r e n o c o r-

r e c t i o n f o r t h e e c c e n t r i c i t i e s w i l l be c ons i de r e d . The s t ress d i s t r i b u t i o n

t e r m i n t h e W e ib u ll t h e o r y c an b e m o d if i ed t o i n c l u d e t h e s t r a i n g r a d i e n t

e f f e c t . I t i s n o t w i t h i n t h e s co p e of t h i s r e p o r t a nd no e f f o r t s a re made.

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4 . THE MODIFIED WEIBULL THEORY

The f u n d amen ta l b a s i s f o r t h e Weib u ll t h eo r y i s t h e s o - c a l l e d

" weak es t l i n k h y p o th es i s " . T h i s h y p o th es i s acco u n ts o n ly f o r p a r t of

observed phys ica l phenomena when app l ied t o t he s t r e ng th o f a b r i t t l e

g r a n u l a r s o l i d , s uc h as g r a p h it e . I n t h i s t h eo ry , m a t e r i a l i s charac-

t e r i z e d a s an ag g r eg a t e o f m ic r o - s t r u c tu r a l " l i n k s " . Thus i t l e a d s t o

a c o n c l u s i o n t h a t t h e c o m bi n at i on o f t h e h i g h e s t l o c a l s t r ess concen-

t r a t i o n w i t h t h e most c r i t i c a l f l a w c o n t r o l t h e s t r e n g t h of t h e s pe cim en.

The m ot i v a t i o n t o m od if y t h e W e ib u ll t h e o r y t o a c c o un t f o r t h e s i z e

e f f e c t stems f rom observat ions of many t e s t r e s u l t s . T h e t e s t r e s u l t s

r ep o rt e d i n t h e l i t e r a t u r e (R efs . 1 , 2 and 9 ) i n d i c a t e a r a p i d l y d e c re a s -

i n g s t r e n g t h a s t h e l e a s t dimens ions of th e specimens approach th e so-

c a l l e d " c h a r a c t e r i s t i c g r a i n si z e ". Near t h i s s i z e t he e f f e c t of t h e

l e a s t d imen sio n s ou twe ig h s t h e volume e f f e c t p r e d i c t ed b y t h e Weib u l l

t h eo r y . A s i t t u r n s o u t l a t e r i n a p p l i c a t i o n o f t h e m od i fi e d W e i bu l l

t h e o r y , t h e c h a r a c t e r i s t i c g r a i n s i z e c o nc u r s w i t h t h e maximum g r a i n

s i z e .

T e s t d a t a o f t u b u l a r s p ec im en s u nd er i n t e r n a l p r e s s u r e r e p o r t e d i n

Refs . 2 and 1 2 a re p r e se n t e d i n F i g. 1 . The m a x i m u m t e n s i l e f a i l u r e s t r e s s

i s c a l c u l a t e d u s i n g t h e c l a s s i f a l f or mu la f o r t h i ck -w a ll c y l i n d e r un de r

i n t e r n a l p r e s s u r e . No c o r r e c t i o n , s u c h as volume e f f e c t has been made.

The maximum t e n s i l e s t re s s i s ex p r es s ed as a f u n c t i o n of w a l l t h i c k n e s s

i n t h e d iag ram. The s t ress d e c r e a s e s r a p i d l y a s t h e w a l l t h i c k n e s s g e t t i n g

t h i n n e r and c l o s e t o t h e c h a r a c t e r i s t i c g r a i n s i z e . The t r en d man i f e s t s

th e combined ef fe c t of gr a i n s i z e and volume. The same t r e n d i s a l s o

o bs er ve d i n t h e u n i - a x ia l t e n s i l e t e s t s , e x c e p t t h a t i t d i f f e r s by a s c a l i n g

f a c t o r .

9

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0

a

0

Q

0 \e

\

0 06

0 0 0

0 0 0

M

0 0 0

c

ISSUlS3SIu

UfV'J

0

0

0 z

z % 0 c0

c0

.-0 0

10

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To r e c o n c i l e t h i s , i t i s assumed t h a t a f t e r a l l ow i n g c r e d i t f o r t h e

n o n l i n e a r s t re s s s t r a i n r e l a t i o n s h i p , t h e ob se rv ed s t r e n g t h ( c a l c u l a te d

from the t e s t d a t a ) i s a f u n c t i o n of t h e r a t i o s o f t h e t h i c k n e ss and

t h e w i dt h o f t h e sp ec im en t o t h e c h a r a c t e r i s t i c g r a i n s i z e . The s t r e n g t h

CJ i n t h e W e ib u ll t h e o r y w i l l be rep laced by Go f ( h o 9 h 1 9 h 2)0

h = t h e c h a r a c t e r i s t i c g r a i n s i z e

h, = t h i c k n e s s of specimen

h2 = wid th of specimen

f i = o i f h i - ho

0

f = 1 .0 i f hi >> ho i = 1,2i

And i n ad d i t i on , f and f a re m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n s of1 2

h. when h > h .1 i - 0

The mo d i fi ed Weib u l l d i s t r i b u t i o n t h en h as t h e f o l l o w in g f or m (*)

All relevant e q u at i on s f o r t h e s u r v i v a l p r o b a d . y are va

t h e m o di f ie d t h e o r y i f t h e p r o pe r e x p r e s si o n f o r 0 i s used. For

ex amp le , t h e r a t i o of t h e t e n s i l e s t r e n g t h of two d i f f e r e n t s i z e d

c i r c u l a r s p e c i m e n s i s

d i n

( *) An ot he r way t o i n t e r p r e t t h e m o d if i e d W e i bu l l d i s t r i b u t i o n i s t h a t

i t i s i d e n t i c a l t o t h e o r i g i n a l W ei bu ll t h e or y e xc e p t t h e s t re s s CT

th ickness -an d -wid th dependent .

i s

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A p p a r e n t l y , f o r a g i v e n vo lu me r a t i o , t h e r e a r e i n f i n i t e l y many m

a nd ho va l ue s w hi ch g i ve t h e same s t r e n gt h r a t i o .

o n l y on e p a i r o r small r a nge of t he s e pa r a m e t e r s w h i c h can accommodate

a l l t e s t e d c a s es .

g r a i n size of t h e mater ia l .

However, there i s

The par a m e t er ho ob t a i ne d c o r r e s ponds t o t he maximum

A c onve n i e n t e xp r e s s i on f o r t h e f unc t i on f w hi ch c l o s e l y a pp r ox im a t e si

t h e t r e n d of s t r e s s w i t h r e s p e c t t o t h e t h i c k n e s s i n F ig . 1 i s

We s h a l l u t i l i z e t h i s i n a p p ly i ng t h e mo di fi ed W ei bu ll d i s t r i b u t i o n

t o t he t e s t r e s u l t s r e p or t e d i n Re fs . 3 and 10 . A f t e r c hoos ing h t o be

th e maximum gra in s i z e , some grap hi ca l so lu t i on s may be used t o ob ta i n

the Weibul l modulus m.

0

3 2

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5. AP P LI C ATI ONS TO GRAPHITE

A. H451 g r a p h i t e

Re sul t s o f more than 2000 t e n s i l e an d f o u r p o i n t b en d t e s t s

f rom one log of e x tr u d ed , n e a r i s o t r o p i c , g r a de H451 g r a p h i t e a r e r e p o r t e d

i n R e f. 3 . T h e m o d i f i e d W e i b u l l d i s t r i b u t i o n w i l l b e ap p l i ed t o t h e d a t a .

Ef fe c t o f Specimen Volume i n t h e Tens ion Tests

The s m a l l t e n s i l e s pe ci me n i s 0.25" i n d i a m e t e r a nd 0 . 9"

l ong .

c o r r e c t i o n f a c t o r f i s

The l a r g e t e n s i l e s pe ci me n i s of 0 . 5 0 5 " $ x 3.0" i n s i z e . The

2 -1f ( ho , h ) =yOS (ho /h)

= 0.8391 h = 0.25''

= 0.9210 h = 0.505"

where h, = 0 .0625 " , t h e m a x i m u m g r a i n s i z e .

The p r ed i ct e d s t r e n g t h r a t i o s are

a, (0.25)/a2(0.505)

= 1.053

= 1.110

A x i a l , m = 1 1

R a d i a l , m = 9

These va lues a re i n good a g re em e nt w i t h t h e o b se r v ed r a t i o s o f 1.03

f o r t h e a x i a l o r i e n t a t i o n and 1.08 f o r t h e r a d i a l o r i e n t a t i o n . The

d i f f e r e n c e is a b o u t 2 . 8 % .

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Re l a t i ons h i p be tw ee n t h e F l e xu r a l S t r e n g t h and t he U n i a x i a l

T e n s i l e S t r e n g t h

The obse r ve d mean r a t i o s of 1 .52 a nd 1 .55 f o r t h e a x i a l and

r a d i a l o r i e n t a t i o n s r e p o r t e d i n R ef. 3 a re used f o r compar i son . The r a t i o s a r e bet

t h e f o u r p o i n t f l e x u r a l s t r e n g t h w i t h c o r r e c ti o n f o r t h e n o n li n e ar s t re s s

s t r a i n r e l a t i o n s h i p and t h e t e n s i l e s t r e n g t h of s m a l l specimen. The

s m a l l t e n s i l e s p e c i m e n i s of 0 .25"$ x 0.9" a nd t h e f l e xu r a l s pe c im e n i s

of 0.25"(1 x 1.5 '' w i th supp or t s a t two ends and th e two inid- th i rd po in ts .

Using the m and h

f o ll o wi ng r a t i o s

va l ue s recommended p r e v i ous l y , t h i s y i e l d s t h e0

a- = 1.508

1.587d T

Ax ia l (m = 11)

Rad i a l (m = 9)

The d i f f e r e n c e bet we en t he p r e d i c t e d a nd t he obs e r ve d i s l e ss t h a n

2.5%.

The e r r o r i n t h e p r e d i c t i o n c a n b e mi ni mi ze d t o w i t h i n 1 . 8%

by choos ing m = 11.3 and m = 9 .3 f o r t h e a x i a l and r a d i a l o r i e n t a t i o n ,

r e s p e c t i v e l y .

t o t h e m v a l u e u s e d . For t h e pur pos e of t h i s p a p e r , t h e n e a r e s t i n t e g r a l

I t can b e se e n t h a t t h e r e s u l t s a r e n o t ve r y s e n s i t i v e

m v a l u e w i l l be u s e d .

F l e x u r a l S t r e n g t h T e s t s Using Two D i f f e r e n t S i z e d / S ha pe d

SDec mens

F l e x u r a l t e s t s ( T a b l e 1) have been pe r formed u s ing two typ es

of specimens . Both ty pe s of specimens a r e

t o f o u r p o i n t b e n d i n g t e s t w i t h s u p p o r t s a t t h e e nds and t he m i d - t h i r d

p o i n t s .

specimen is 0.35" square .

( r e c t an g u l a r v e r s u s c i r c u l a r c r o s s s e c t i o n s ) a r e 0 . 9 1 9 and 0.873 f o r

a x i a l and r a d i a l o r i e n t a t i o n s , r e s p e ct i v e l y. (*I The modi f i ed theory

g i v e s r a t i o s o f 0 .950 a nd 0. 926.

1 .5" l ong and subj ec t ed

T he c y l i nd r i c a l s pe c i m e n is 0 .25' ' i n d i a m e t e r a nd t h e r e c t a n gu l a r

The mean r a t i o s of t h e f l e x u r a l s t r e n g t h

T he p r e d i c t i on i s of f by on l y 6%.

3 4

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Table 1

I

~ Orientat ion

FLEXURAL TEST RESULTS O F H 4 5 1 GRAPHITE

Location

in Log

Specimen

Cross Section

Midlength-edge 0.25" diameter

0. 5" x 0.35" square~~

End-Center 0.25" diameter

0. 5" x 0.35" square

End-edge 0.25" diameter

0.35" x 0.35" square

Midlength-center 0.25" diameter

0.35" x 0.35" square

*Corrected f o r non-linear stress-strain curve.

Mean Bend*

Strength

( m a )

26.2

23.9

23.3

19.7

22.7

18.8

38.6

16.7

Standard

Deviation

( m a )

1 .3

1.5

1 . 8

1.7

1 2

2 .8

2.5

1.9

No. of

Specimens

3 5

2 3

3 5

2 5

3 6

2 6

3 5

36

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Type of1 T e s t Specimen Size

U n i a x i a l

T e n s i l e

* F r a c t u r e

S t r e s s , p s i

I F l e x u r a l

Table 2

S t r e n g t h of EGCR-Type AGOT G r a ph i t e ( Re f . 10 )

smal l (5/ 16"Q) 1 6 1 0

l a r ge ( 5 / 8" $ ) 1540

smal l (5/ 16" square) 2120

l a r g e (5,' 8" s q u a r e ) 1960

C o e f f i c i e n t of

V a r i a t i o n

20% I1 8%I

5%

* C o rr e ct e d f o r n o n l i n e a r s t ress s t r a i n c u rv es

1 7

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Strength Predicted

Ratio Rat io*

Table 3

STRENGTH RATIOS OF EGCR-TYPE AGOT GRP AHI TE

% Difference

Between Measured

and Predicted

Large Tensile/small tensile

Large Flexural/small FlexuralSmall Flexural/small tensile

Large Flexural/large tensile

0 . 9 5 7 0 . 9 2 2

0 . 9 2 5 0 . 9 2 21 . 3 1 7 1 . 2 6 7

1 . 2 7 1 1. 6 7

3 . 7 %

0.3%

3.8%

0 . 3 %

*Using m = 14 , h, = 1 / 3 2 "

38

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6. SUMMARY AND CONCLUS IONS

The Weibul l theory i s m o d if i ed h e r e by i n c l u d i n g t h e e f f e c t o f t h e

g r a i n s i z e . The smaller t h e r a t i o of t h e bo dy d im en si on t o t h e c h a r ac -

t e r i s t i c g r a i n s i z e , t h e lo we r t h e s t r e n g t h . The t h eo r y i s used to

d e s c r i b e t h e mater i a l b e h a v i o r o f g r a p h i t e .

g r a p h i t e a re u t i l i z e d i n t h e c o r re l a t i on .

t e s t s , u n i a x i a l t e n s i l e an d f o u r p o i n t be n di ng t e s t s , a re d e a l t w i t h

h er e . I n t h e t e n s i l e t e s t s , two d i f f e r e n t s i z e d s p e ci me ns a r e a na l yz e d

t o d e t er m i ne t h e e f f e c t o f p h y s i c a l s i z e .

t h r e e d i f f e r e n t s i z e d a n d / o r s ha pe d sp ec im en s a r e t e s t e d t o e x a m i n e

t h e e f f e c t of s t re s s d i s t r i b u t i o n an d p h y s i c a l s iz e. , I n a p p l y in g t h e

t h eo r y , t h e f l e x u r a l s t resses ob t a i ne d f r om t he c l a s s i c a l beam theory

h av e be en c o r r e c t e d f o r t h e n o n l i n e a r s t re s s s t r a i n b eh av io r .

Over 2000 d a t a p o i n t s of H 4 5 1

Two t y p e s of s t r e n g t hI

I n t h e b end t e s t s ,

De pen dence o f t h e s t r e n g t h o n o r i e n t a t i o n i s r e t a i n ed i n t h e

a n a l y s i s . But t h e s p a t i a l v a r i a t i o n of t h e s t r e n g t h ac c or d in g t o t h e

l o c a t i o n i s e xc lu d ed . I n s t e a d , t h e a v e r ag e v a l u e s of f o u r l o c a t i o n s ,

i . e . , end-edge ,end-cente r , mid length-edge and mid leng th- cen te r , a re

c o r r e l a t e d w i t h t h e t h e o r y . The t h e o r y s h o u l d a p p l y e q u a l l y w e l l

w i t h o ut e x c lu d i ng t h e e f f e c t of t h e s p a t i a l v a r i a t i o n.

Comparison of t h e t h e o r y w i t h t h e t e s t r e s u l t s l e a d s t o t h e

f o l l o w i n g c o n c l u s i o n s :

1. E f f e c t o f vo lu me -- th e mean s t r e n g t h r a t i o of t h e l a r g e - s i z e d

a nd t h e s m a l l - s i z e d t e n s i l e s pe ci me ns i s p r e d i c t e d b y t h e t h e o r y w i t h i n

3% a c c u r a c y .

2 . E f f e c t o f stress d i s t r i b u ti o n - - t h e d i f f e r e n c e i n t h e f l e x u r a l

t o t e n s i l e s t r e n g t h r a t i o b etween t h e t h e o ry and t h e t e s t d a t a i s o n l y

2 . 5 % .

39

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3. Effect of volume and s t re s s di s t r i bu t i on -- a maximum of 6 %

d i f f e r e n c e w a s f ou nd by ex am in in g t h e f l e x u r a l s t r e n g t h r a t i o of r e c t a n g u l a r

t o c i r c u l a r c r o s s s e c t i o ns . The mater ia l p a r a m e t e r s , m and h , used

o b t a i n e d f o r a d i f f e r en t l o g . The d i f f e r e n c e may b e r edu ced i f m an d /o r

h f o r t h a t p a r t i c u l a r l o g a r e used .

0

0

4 . The t h e o r y e x p l a i n s t h e f a c t t h a t t h e l o we r s t r e n g t h i s observed

when th e specimen s i z e i s a p pr o ac h in g t h e g r a i n s i z e . T h i s i s because

t h a t i n t h e n orm al s i z e r a ng e of t h e p r e s e n t t e s t specimens t he volume

e f f e c t i s weak when compared t o t he s t r on g e f f e c t of t h e g r a i n s i z e .

5. For H451 graphite, ho = 0.0625", m = 1 1 ( a x i a l ) a n d 9 ( r a d i a l )

f i t t h e da ta q u i te w e l l .

E x c e l l e n t c o r r e l a t i o n i s a l s o s een when t h e mo d i f i ed t h eo r y i s

a p p l i e d t o EGCR-Type AGOT g r a p h i t e .

I n t h e c o r r e l a t i o n i t i s fo und t h a t t h e r e s u l t s a re n o t s e n s i t i v e

t o m v a l u e . C a l c u l a t i o n by de s k t o p c a l c u l a t o r o r by h and g i v e s

r e l a t i v e l y a c c ur a te r e s u l t s .

The m o di f i ed W e i bu l l d i s t r i b u t i o n c a n c o r r e c t l y p r e d i c t t h e e f f e c t s

of volume, s t re s s d i s t r i b u t i o n and g r a i n s i z e . I n or d e r t o m in im iz e t h e

g r ai n s i z e e f f e c t , t h e sma l l e s t dimension of a t e s t specimen should be a t

l e a s t 10, p r e f e r a b l y 1 5 t i m e s t h e m a x i m u m g r a i n s i z e , u n le s s t h e a c t u a l

component has a t h i c k n e s s l e s s t ha n t h i s o r t h e e f f e c t c a n b e p ro pe r ly

accommodated i n th e ana ly s i s o f th e t e s t d a t a .

F u r t h e r wo rk, i n c l u d i n g c o r r e l a t i o n o f t h e m o d i fi e d W e i bu l l t h e o ry

wit h hi gh ly peaked non-uniform stress f i e l d s , n e e d s t o b e p er fo rm ed b e f o r e

a ny p r a c t i c a l a p p l i c a t i o n c a n b e made.

20

8/8/2019 1064588


REFERENCES

1. Brocklehurst, J. E., "Fracture of Polycrystalline Graphite,"

Chemistry and Physics of Carbon, 13, 1977.-2. Price, R. J. and H. R. W. Cobb, "Application of Weibull Statistical

Theory to the Strength of Reactor Graphite," Proceedings of the

Conference on Continuum Aspects of Graphite Design (CONF-701105),

USAEC Technical Information Center, 1972, p. 547.3. Price, R. J., "Statistical Study of the Strength of Near-Isotropic

Graphite," GA Report GA-A13955, May 24, 1976.

4. Tsai, S. W. and E. M. Wu, "A General Theory of Strength for

Anisotropic Materials," J. Composite Materials, Vol. 5, 1971.

5. Chang, T. Y. and T. Weng, "A Strength Criterion for Graphite

under Combined Stresses ' I pfGyTnted at ASME Pressure Vessel and PipingConference, San Francisco,

6. Stevens, R., and T. D. Clausen, "Strength Distribution and

Fracture Behavior o f Structural Ceramics of Low Neutron Absorption

Cross Section," Atomic Energy of Canada Ltd. Report AECL-3422,

October 1969.

7. Mindlin, R. D., "Second Gradient of Strain and Surface Tension in

Linear Elasticity," Int. J. Solids Structures, 1, 1965, p . 4 1 7 .

8. Kao, B. G., F. K. Tzung, F. H. H o , "Influences of the Couple Stresses

on the Pure Bending of a Circular Cylinder," GA Report to be published.

9. Brocklehurst, J. E., and M. I. Darby, "Concerning the Fracture of

Graphite under Different Test Conditions," Materials Science and

Engineering, 16, 1974, p. 91.

10. Greenstreet, B. L . , etc., "Room Temperature Mechanical Properties

of EGCR-Type AGOT Graphite," ORNL-3728, January 1965.

11 . Bullock, R. E . , and J. L . Kaae, "Mechanical Properties of Glassy

Carbon," 13-C-78P, Presented at American Ceramic Society 31st

Pacific Coast Regional Meeting, San Diego, CA, Oct. 25-27, 1978.

12. Brock1ehursL J - E. and K. E. Gilchrist, "The Fracture of Graphite Ring

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