Brittle Fracture of Steel -Hasofer

8/13/2019 Brittle Fracture of Steel -Hasofer

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T h e I n t e r n a t i o n a l J o u r n a l o f F r a c t u r e M e c h a n i c s , V o l. 4 , N r . 4 , D e c e m b e r 1 9 6 8 .

W o l t e r s - N o o r d h o f f P u b l i s h i n g - G r o n i n g e n .

P r i n t e d i n t h e N e t h e r l a n d s .

A S T A T I S T I C A L T H E O R Y O F T H E B R I T T L E F R A C T U R E O F S T E E L

A M Hasofer

U n i v e r s i t y o f M e l b o u r n e )

A B S T R A C T

A s t a t i s ti c a l m o d e l f o r t h e b r i tt l e f r a c t u r e o f s t ee l i s f o r m u l a t e d a n d a n a l y z e d . F r a c t u r e i s a s s u m e d t o o c c u r

b y c o a l e s c e n c e o f a n u m b e r o f a r r e s t e d c r a c k s i n i t ia t e d a t d i f f e r e n t p o i n t s . T h e u p p e r t a i l o f t h e d i s t r i b u t i o n

o f m a x i m u m s t re s s i s s h o w ~ t o b e g iv e n b y t h e f o r m u l a 1 - X / O o ) 0 , w h e r e 7o a n d 0 ar e f u n c t i o n s o f t h e

m i c r o s t r u c t u r a l c o n s t a n t s o f t h e m a t e r i a l . It i s f u r t h e r s h o w n t h a t t h e s i z e e f fe c t i n t h i s m o d e l i s m u c h l e ss

m a r k e d t h a n i n th e c a se o f w e a k e s t - - l i n k m o d e l s . F i n a ll y , s o m e e x p e r i m e n t a l r e s u l t s f r o m a p i l o t p r o g r a m

a re p r e s e n t e d a n d s u g g e s t i o n s fo r f u r t h e r e x p e r i m e n t a l w o r k g i ve n .

I N T R O D U C T I O N

M o s t o f t h e w o r k d o n e s o fa r o n t h e s t a ti s ti c a l t h e o r y o f t h e b r i tt l e f r a c t u r e o f s t ee l h a s u s e d

w e a k e s t l i n k m o d e l s . S e l e c t e d r e f e r e n c e s a re , i n c h r o n o l o g i c a l o r d e r , P i e r c e ( I ) , W e i b u l l (2 , 3 ) , F r e n -

k e l a n d K o n t o r o v a ( 4 ) , a n d F i s h e r a n d H o l l o m o n ( s ) . T h e e ss e n ti a l p r i n c ip l e o f s u c h m o d e l s is t h a t

onc e a f r a c tu r e i s i n i t i a t e d :in t he m a te r i a l , i t s p r opa g a t ion c a n no t be a r r e s t e d a n d c o m p le t e c o l l a pse

o c c u r s . I t i s a s s u m e d t h a t t h e r e a r e f l a w s o r c r a c k s i n t h e m a t e r i a l , a n d t h a t t h e s e a r e c o m p l e t e l y

i n d e p e n d e n t o f e a c h o t h e r . T h e s t re n g t h o f a s p e c im e n is t h e n d e t e r m i n e d b y th e m a x i m u m s tr es s

tha t t he w e a ke s t f l a w c a n su s t a in be f o r e f r a c tu r e i s i n i t i a t e d .

S u c h m o d e l s a r e n o t w e ll a d a p t e d t o t h e d e s c r i p t io n o f t h e b r it t l e b e h a v i o r o f s te e l a n d

o t h e r d u c t i l e m a t e r ia l s . F o r a r e c e n t d i s c u s si o n o f t h e p h y s i c a l m e c h a n i s m s i n v o l v e d , s ee M c M a h o n

a n d C o h e n ( 6 ) a n d t h e r ef e .r e nc e s l i st e d t h e r e i n . T h e f e a t u r e s o f t h e f r a c t u r e m e c h a n i s m d e s c r i b e d

w h i c h a re o f i m p o r t a n c e a s f a r a s t h e s t a t i s ti c a l m o d e l p r e s e n t e d i n t h is p a p e r i s c o n c e r n e d , a r e :

( a ) C r a c ks , onc e i n i t i a t e d , a r e ge ne r a l l y a r r e s t e d a t som e ba r r i e r w h ic h a bso r bs t he e ne r gy

r e l e a se d by c r a c k ing .

( b ) F r a c t u r e g e n e r a ll y o c c u r s b y c o a l e s c e n c e o f a n u m b e r o f c ra c k s in i t i a te d a t d i f f e r e n t p o i n t s .

B e c a u s e o f t h e a b o v e - - m e n t i o n e d f e a t u r e s , t h e b r i t t le f r a c t u r e o f st e el is b e t t e r d e s c r i b e d b y

s t a ti s ti c a l m o d e l s o f th e b u n d l e o f f i b e r s t y p e i n t r o d u c e d b y D a n i e ls ( 7 ) .

I n t h a t t y p e o f m o d e l t h e s p e c i m e n is c o n s i d e r e d a s a b u n d l e o f p a r a ll e l fi b er s . W h e n i t is

s u b m i t t e d t o s t r e ss , t h e w e a k e s t f i b e r b r e a k s f i rs t , w e a k n i n g t h e e n ti r e c r o s s - s e c t i o n . H o w e v e r , a s

t h e r e m a i n i n g f i b e r s a r e c o m p a r a t i v e l y s t r o n g e r t h a n t h e b r o k e n o n e , c o ll a p s e d o e s n o t n e c e s s ar i ly

o c c u r i m m e d i a t e l y . W h e n t h e l o a d is i n c r e a s e d , m o r e f i b e rs b r e a k u n t i l t h e u l t i m a t e l o a d is re a c h e d ,

a t w h i c h p o i n t a ll fi b e r s ar e r u p t u r e d . T h i s t y p e o f m o d e l s h a s b e e n u s e d b y W i ll ia m s a n d K l o o t ( 8 ) ,

C o i e m a n ( 9 ) a n d G u c e r a n d G u r l a n d ( 1 ° ) .

Th e m a in f e a tu r e t ha t ha s m a d e the D a n ie l s m o de l una t t r a c t i v e t o e ng ine e r s is t ha t i t i s m a the -

44 A M Hasofer

m a t i c a l l y i n t r a c ta b l e a n d t h a t s im p l e f o r m u l a e h o l d o n l y a s y m p t o t i c a l l y f o r a la rg e n u m b e r o f f ib e rs .

I n f a c t t h e o n l y c o n c l u s i o n o f a n y p r a c t ic a l v a lu e r e a c h e d b y D a n i e l s a f t e r a b o u t t h i r t y p a g e s o f

h e a v y m a t h e m a t i c s , is t h e r a t h e r u n i l l u m i n a t i n g o n e t h a t t h e f a il u r e st re s s o f t h e b u n d l e f o l l o w s a

n o r m a l d i s t r ib u t i o n , a s y m p t o t i c a l l y f o r a v e r y l a rg e n u m b e r o f fi b e rs . E v e n t h i s c o n c l u s i o n d e p e n d s

o n t h e f o l l o w i n g a s s u m p t i o n s , w h i c h o b v i o u s l y d o n o t h o l d i n th e c a se o f s te e l b r i t t le f r a c t u re .

( i) T h e f i b e rs b e h a v e i n d e p e n d e n t l y .

( i i) A l l f i be r s a r e s t a t i s t i c a ll y i d e n t i c a l . I n o th e r w or ds , a ll c r a c ks a r e o f t he s a m e s iz e ( t he

d i a m e t e r o f th e f ib e r ).

A c tua l ly , i n a s t e e l spe c im e n the r e a r e no inde pe nde n t pa r a l l e l f i be r s . O n the c on t r a r y , a l l pa r t s

o f t h e m a t e r i al a re i n t e r c o n n e c t e d .

I n t h i s pa p e r , a s t oc ha s t i c f o r b r i t t l e t e n s i l e f r a c tu r e o f s te e l i s p r e se n t e d . F u nd a m e n ta l l y , the

m o d e l is o f th e D a n i e l s t y p e . H o w e v e r , t h e s t o c h a s t i c a s s u m p t i o n s m a d e a re v e r y d i f f e r e n t f r o m

t h o s e o f D a n i e ls , a n d l e a d t o a s i m p l e f o r m u l a e f o r t h e u l t i m a t e s t r e n g t h o f t h e s p e c i m e n , t h r o u g h

t h e u s e o f t h e t h e o r y o f s t o c h a s t i c p r o c es s e s . I t is t o b e n o t e d t h a t t h e r e s u l t in g f o r m u l a e i s t h a t o f

t h e P a r e t o d i s t r i b u t i o n , w h i c h i s v e r y d i f f e r e n t f r o m e i th e r t h e n o r m a l d i s t r i b u t i o n y i e l d e d b y t h eD a n i e l s m o d e l , o r t h e e x t r e m e v a l u e d i s t r i b u t i o n s y i e l d e d b y t h e w e a k e s t l i n k m o d e l s .

T h e f o r m u l a f o r t h e d i s t r i b u t i o n o f t h e m a x i m u m s t re s s , Crmax is

P (O m a x ~ X ) = l

w h e r e r o i s th e m i n i m u m s t re s s a t w h i c h a c r ac k s t a r ts , a n d 0 d e p e n d s o n t h e d i s t r i b u t i o n o f c r a c k

s i z e. Thu s t he f o r m u la r e l a te s t he s t a ti s t i ca l p r ope r t i e s o f t he f a i l u re s t r e s s w i th t he m ic r o s t r u c tu r a l

p r o p e r t i e s o f t h e m a t e r i a l.

S i n ce 1 9 4 5 , t h e y e a r o f p u b l i c a t i o n o f D a n i e l s p a p e r , t h e r e h a s n o t b e e n , t o t h e a u t h o r s

k n o w l e d g e , a n y s u c c e s s f u l a t t e m p t t o a n a l y z e f r o m f i r s t p r i n c i p l e s t h e p a r a l l e l f a i l u r e m o d e l ,

b e c a u s e o f t h e g r e a t m a t h e m a t i c a l d i f fi c u l ti e s . T h e m o d e l p r e s e n t e d i n t h i s p a p e r is t h e f ir s t s t e p

f o r w a r d in t h i s f i e ld s inc e D a n ie l s w or k .

THE MO EL

W e a s s u m e t h a t t h e s p e c i m e n s c o n s i d e r e d i n t h i s p a p e r h a v e a g a u g e l e n g t h o f t h e o r d e r o f t h e

s iz e o f t h e c r o s s - s e c t i o n , s o t h a t a fi n i te e x t e n s i o n o c c u r s b e f o r e f r a c tu r e . T h e a n a l y t i c a l t r e at -

m e n t o f a p r o c e s s o f c r a c k s in s u c h s p e c i m e n s p r o d u c e s a n u m b e r o f d i f f i c u l t p r o b l e m s in t h e

t h e o r y o f m a t e r i a ls . T h i s p a p e r i s n o t , h o w e v e r , c o n c e r n e d w i t h t h e m e c h a n i c a l p r o b l e m s i n v o l v e d

i n t h e a d d i t i o n o f c r a c k s e f f e c t s , b u t w i t h a s c e r ta i n i n g t h e e f f e c t o f t h e a c c u m u l a t i o n o f c r a c k s

o n t h e s t a ti s ti c al d i s t ri b u t io n o f t h e u l ti m a t e l o a d . W e t h e r e fo r e b y - p a s s t h e a b o v e - m e n t i o n e d

p r o b l e m s b y m e a n s o f t h e f o l l o w i n g de v ic e .

W e a s s u m e t h a t f o r t h e t y p e o f b r i t tl e f r a c t u r e c o n s i d e r e d , t h e e x t e n s i o n o f t h e s p e c i m e n

b e t w e e n t h e p o i n t a t w h i c h a c r ac k i s a r r es t e d a n d t h e p o i n t a t w h i c h a n e w c r a c k is n u c l e a t e d is

e s s e n t ia l l y e l a s ti c . I t f o l l o w s t h a t t h e s t if f n es s o f t h e s p e c i m e n , d e f i n e d a s t h e r a t io o f l o a d t o e x -

t e n s i o n , c a n b e t a k e n a s c o n s t a n t d u r i n g s u c h p e r i o d s . H o w e v e r , w h e n a c r a c k is n u c l e a t e d a n d

a r r e s t e d , t h e s t if f n e ss o f t h e s p e c i m e n is r e d u c e d , b e c a u s e o f t h e w e a k e n i n g o f t h e s p e c i m e n b y t h e

a d d i t i o n a l c r a c k . T h e e l a s t i c i ty a s s u m p t i o n c o u l d b e d o n e a w a y w i t h a n d t h e m o d e l g e n e r a l iz e d

t o i n c l u d e t h e e f f e c t o f p l a s t i c i t y a s w e l l . H o w e v e r , w e w o u l d n o t b e d e a l in g w i t h p u r e b r i t t le

f r a c t u r e a n y l o n g e r , a n d i t s e e m s b e t t e r , o n a f i rs t a t t a c k , t o c o n f i n e o u r a t t e n t i o n t o t h e p u r e

b r i t t l e f r a c t u r e m o d e l .

A . M . H a s o f e r 4 4 1

W e c a n ta k e t h e a b o v e - m e n t i o n e d r e d u c t i o n i n s ti f f n e ss a s a m e a s u r e o f t h e o v e r a l l e f f e c t o f

t h e n e w c r a c k in w e a k e n i n g t h e s p e c i m e n , w h e t h e r i n c o n j u n c t i o n w i t h o t h e r c r a c k s in t h e s am e

c r o s s - s e c t i o n , o r i n c o n j u n c t i o n w i t h c r a c ks a t o t h e r c r o s s - s e c t i o n s . I n o r d e r t o g i ve a m o r e

p i c t u r e s q u e d e s c r i p t i o n o f t h e s t a t is t i c a l a sp e c t s o f t h e m o d e l , w e d e f i n e a n ef f e c t iv e a r e a A a n d a

m e a n e f f e c t iv e s t re s s o f o r ev e r y s ta g e o f t h e lo a d i n g . T h e s e a r e p u r e l y c o n c e p t u a l m a g n i t u d e s , b u t

t h e y a re d e fi n e d in t e rm s o f l o a d a n d e x t e n s i o n , t h e y d o n o t p r e s e n t t h e p r o b l e m s w h i c h a ri se w h e n

w e d e a l w i t h r e a l a r e a s a n d s t r e s s e s .

T h e e f f e c t iv e a r e a A i s d e f i n e d b y

A = K ( P / e ) ( 1 )

w h e r e K i s t h e m o d u l u s o f e x t e n s i o n o f t h e s p e c i m e n , P i s t h e l o a d a n d e t h e e x t e n s i o n . T h u s A i s

p r o p o r t i o n a l t o t h e s t if f n e s s o f t_ h e s p e c i m e n P / e . T h e c o n s t a n t K c a n b e e v a l u a t e d a t t h e b e g i n n i n g

o f t h e l o a d i n g , i .e . f o r v e r y s m a l l P , a s A i s t h e n j u s t t h e g e o m e t r i c c r o s s - s e c t i o n A o .

O n c e t h e e f f e c t i v e a r e a A h a s b e e n c a l c u l a t e d , w e c a n c a l c u l a t e th e m e a n e f f e c t iv e s tr e s s o b y

m e a n s o f t h e f o r m u l a

o = P / A . ( 2 )

W e s ee t h a t o is a c t u a l l y e q u a l t o e / K , i . e . i s p r o p o r t i o n a l t o t h e e x t e n s i o n e .

L e t u s n o w c o n s i d e r t h e b e h a v i o r o f A w h e n o i n c r ea s e s . T h e e f f e c ti v e a r e a r e m a i n s c o n s t a n t

e x c e p t w h e n a c r a c k is n u c l e a t e d a n d a r r e s t e d . T h e r e i s t h e n a n i n s t a n t a n e o u s d e c r e a s e o f A t o a

n e w v a l u e , w h i c h r e m a i n s c o n s t a n t u n t i l a f u r t h e r c r a c k is n u c l e a t e d . T h u s t h e g r a p h o f A a s a

f u n c t i o n o f o i s a s s h o w n i n F i g u r e ( 1 ) .

F i g u r e ( 1 ) .

F r o m t h e g r a p h o f A , i t is e a s y t o d e d u c e t h e g r a p h o f P , u s i ng f o r m u l a ( 2 ). W e s i m p l y h a v e :

P = erA(or), (3 )

a n d t h e g r a p h o f P i s o b t a i n e d b y m u l t i p l y i n g t h e a b s c is s a b y t h e o r d i n a t e in F i g u r e ( 1 ) , th u s o b -

t a i n in g F i g u r e ( 2 ) . I t i s o b v i o u s t h a t t h e r e a r e t w o o p p o s i n g i n f l u e n c e s a c t i n g o n P : o n t h e o n e

h a n d P i n c r e a se s w i t h i n c r e a si n g o , a n d o n t h e o t h e r h a n d P d e c r e a se s w i t h t h e d e c r e a s e i n e f f e c ti v e

a r e a A ( c r) . T h u s P w i l l h a v e i n g e n e r a l a m a x i m u m v a l u e P m a x w h i c h w i l l r e p r e s e n t t h e u l t i m a t e

l o a d th a t t h e s p e c i m e n c a n m s t a i n .

4 4 2 A . M . H a s o f e r

//~ / / / /- / /

/ / ' / / /

<4 > J

F i g u r e 2 ) .

no r d e r t o m a k e t h e m o d e l a m e n a b l e t o s t o c h a s t i c an a l y s i s, w e m a k e t h e t r a n s f o r m a t i o n s

Q = l o g P , B = l o g A / A o ) , t = l o g O / O o ) , l o g a o = l o g Q o , l o g A o = B o . T h e n

Q t ) = Q o + B o + t + B t ) .

T h e g r a p h o f B t ) F i g u r e 3 ) ) is s i m i la r t o t h a t o f F ig u r e 1 ) , e x c e p t t h a t t h e v a l u e t = 0 c o r r e s-

p o n d s t o a = o o . H o w e v e r , t h e g r a p h o f Q t ) d i f f e rs f r o m t h a t o f P o ) i n t h a t t h e s l o p e i s c o n s t a n t

a n d e q u a l t o u n i t y F i g u r e 4 ) ) .

it l t 2 t3 t t s t

F i g u r e 3 ) .

W e s h a l l a s s u m e t h a t n o c r a c k c a n p r o p a g a t e a t a n e f f e c t i v e s t r e ss l e ss t h a n 0 o , a n d t h a t f o r

o > g o , i . e . f o r t > 0 , B t ) is a h o m o g e n e o u s C o m p o u n d P o i s s o n P ro c e s s . I n o t h e r w o r d s , t h e

p o i n t s a t w h i c h c r a c k s o c c u r , n a m e l y t a , t 2 . . . . , f o r m a P o i s so n p r o c e s s w i t h p a r a m e t e r k , s a y , a n d

t h e r e d u c t i o n s i n B t ) a t t h e p o i n t s t x , t 2 , - . . , s a y X I , X 2 , . . ., a re i n d e p e n d e n t , i d e n t i c a l l y d i s t r i b u t e d

r a n d o m v a r i a b l e s w i t h c o m m o n d i s t r i b u t i o n f u n c t i o n G x ) . In t e r m s o f th e o r i g i n a l p a r a m e t e r o ,

t h i s i s e q u i v a l e n t t o a s s u m i n g t h a t t h e d i s t r i b u t i o n o f t h e n u m b e r o f c r a c k p r o p a g a t i o n s in eq u a l

p r o p o r t i o n a l i n c r e a se s o f a is t h e sa m e . I n r e l a t i o n t o A o ) , t h e a s s u m p t i o n i s t h a t e q u a l p r o p o r t i o n a l

d e c r e a s e s i n e f f e c t i v e a r e a a r e e q u a l l y p r o b a b l e .

A M H a s o f e r 4 4 3

Cl mo~.

Figure 4).

W e sh a ll p o s t p o n e f o r t h e m o m e n t a d i s c u ss i o n o f th e v a l i d i t y o f t h e s e a s s u m p t i o n s a n d c o n -

c e n t r a t e o n t h e i r m a t h e m a t i c a l c o n s e q u e n c e s . W e a re p ri m a r i l y i n t e re s t e d i n t h e p r o b a b i l i t y d i st ri -

b u t i o n o f t h e m a x i m u m o f Q t ) . L e t us w r i te

P m a x Q t ) ~ x ) = K x ) .

o < t < =

The determination of K X is mathematically equivalent to the solution of a w el l -k no w n problem

in the theory o f dams. See Appen dix A) .

Le t t be the time and consider a dam of infin ite capacity w hose inp ut is the Com pound PoissonP r o c e s s B t) . L e t t h e o u t p u t o f t h e d a m b e o n e u n i t o f w a t e r p e r u n i t t i m e . F in a l l y l e t t h e i n it ia l

c o n t e n t o f t h e d a m b e z . T h e g r a p h o f t h e c o n t e n t o f t h e d a m Z t) w i ll b e a s i n F ig u r e 5 ) . W e

I . T / z l . t

Figure 5).

s h al l d e n o t e b y T z ) t h e t i m e w h e n t h e d a m f ir s t b e c o m e s e m p t y . T h i s is a r a n d o m v a r ia b l e w h i c h

m a y t a ke t he va lue +oo w i th f i n i te p r ob a b i l i t y . O b v ious ly

o ~ t < ~

A.M. Hasofer

N o w t h e r e l a t io n b e t w e e n Z ( t ) a n d Q ( t ) i s o b v i o u s ly

Z ( t ) = z + B o + Q o - Q ( t ) .

T h e r e f o r e

P ( T ( z ) = + o o) = P ( m a x Q ( t ) ~< z + B o + Q o ) ,

a n d t h i s c a n b e r e w r i t t e n

P ~ m a x

0 ~ < t < ®

Q ( t ) ~< x ) = P ( T ( x - B o - Q o ) = + o ~ ) .

T h e s o l u t i o n o f t h e p r o b l e m i n t h e t h e o r y o f d a m d e p e n d s o n t h e m e a n v a l ue o f t h e d ec r ea s e

o f B ( t ) pe r un i t t . W e ha ve t - 1E [ B ( t ) ] = ) t in , w h e r e m = ~ x dG ( x ) . I f ~ tm <~ 1 , K ( x ) = 0 f o r a ll x .

°O n t h e o t h e r h a n d , i f X r n > 1 , t h e d i s t r i b u t i o n K ( x ) e x i s t s . N o w w r i t e ~ ( s ) -- e - s x d G ( x ) . T h e n

t h e e q u a t i o n

S - - X [ 1 - - ~ ( S ) ] = 0

h a s e x a c t l y o n e p o s i t i v e s o l u t i o n 0 , a n d

k ( x )= f l 0 - e - ( 0 - B ° - Q ° ) o t h e r w i s e ,o r x ~ > B o + Q o ,

R e t u r n i n g t o P ( o ) , w e n o t e t h a t

P ( m a x P ( a ) ~ < x ) = P ( m a x Q ( t ) -< < l og x ~ ,

O o ~ < O < ~ o ~ t < ®

= 1 - e xp I - -0 log x - B o - Q o ) ] ,

= 1 - - (X/C0 -0 , X >1 '~ ,

w h e r e cz = e x p ( B o + Q o ) = A o ° o '

T h i s i s k n o w n a s t h e P a r e t o d i s t r i b u t i o n , a n d h a s b e e n u s e d e x t e n s i v e l y in v a r io u s fi e l d s ,

p a r t i c u l a r l y i n e c o n o m i c s . ( S e e S t e i n d l ( 1 1 ) ) . I t s g e n e r a l s h a p e a n d t h a t o f i t s d e n s i t y f u n c t i o n s a r e

g ive n in F igu r e ( 6 ) .

L e t u s d e f i n e t h e u l t i m a t e s t r e s s a m a x f o r t h e s p e c i m e n a s o m a× = P m a x / A o T h e n

\ A o a o /

W e sha l l r e p r e se n t P { o n n , ~ x ~ by F o o , x ) a n d t h e c o r r e s p o n d i n g d e n s i t y f u n c t i o n b y f r o o , x ).

A . M . H a s o f e r 5

F x)[ / x -e

~ - ~ k f x ) = eocSx (s*~)

F i g u r e 6 ) .

S O M E PR O P E R T I E S O F T H E P A R E T O D I S T R I B U T I O N

A c h a r a c t e r i s ti c p r o p e r t y o f t h e P a r e t o d i s t r i b u t i o n i s t h e s h a p e o f i t s t a il . L e t u s w r i t e

S ( x ) = P { O m a x > x t = ( X / ° o ) - 0

W e s e e t h a t S ( x ) i s p r o p o r t i o n a l t o s o m e i n v e rs e p o w e r o f x . I f w e p l o t l o g S a g a i n st lo g x , w e

ob ta in a s t r a igh t l i ne w hose s lope i s ~ 9 . Th i s i s a s im p le t e s t f o r t he P a r e to d i s t r i bu t ion .

T h e s i m p l e P a r e t o d i s t r i b u t i o n c a n n o t b e v e r y s a t i s f a c t o r y f o r s m a l l v a l u e s o f x . I n p a r t i c u l a r,

t he a s su m p t ion tha t no c r a c ks o c c u r a t a s t re s s l e s s t ha n som e f ixe d va lue o o is un r e a l i s t i c . W e c a n

h o w e v e r o v e r c o m e t h i s d i f f i c u l t y b y a s s u m i n g t h a t o o i t s e l f i s a r a n d o m v a r ia b l e w h i c h h a s s o m e

d i s t r i b u t i o n f u n c t i o n H ( x ) , a n d d e n s i t y f u n c t i o n h ( x ) . T h e n t h e d i s t r i b u t i o n o f { ) m a x b e c o m e s ,

u s in g t h e t h e o r e m o f t o t a l p r o b a b i l i t y ,

x [ y l 0 1( x ) = P { O m a x ~ < x } = J [ 1 - h ( y ) d y .

T h i s c a n b e w r i t t e n

F ( x ) = H ( x ) - x - 0 ; y O h ( y ) d y .

I f w e a s s u m e t h a t t h e d i s t r i b u t i o n H ( x ) h a s m o m e n t s o f s u f f i c i e n tl y h i g h o r d e r s , th e i n t eg r a l

t 0 = ~ y O h y ) d y w i l l be f i n i t e . W e sha l l t he n c onc lude tha t

Jim x 0 [ 1 - F ( x ) ] = l im x 0 [ 1 - H ( x ) ] + t O ,

X - - ~ X - - ~

44 6 A.M. Hasofer

s in c e th e e x is t en c e o f / l 0 im p l ie s t h a t l im x 0 [ 1 - H ( x ) ] = O . T h u s 1 - F ( x ) ~ / l o x - O f o r l ar ge x ,

a n d t h e c h a r a c t e r is t i c p r o p e r t y o f t h e t a il o f t h e P a r e t o d i s t r i b u t i o n i s p r e s e rv e d .

W e n o w c a l c u la t e t h e m o m e n t s M n ( a ) o f t h e d i s t r i b u t i o n F ( a , x ) . W e h a v e

n (~ ) = f f x n f ( ~ , x ) d x = Oot0 x - ( O - n + l ) d x ,

= O~n n<O

T h u s a ll m o m e n t s e x i s t u p t o t h e o r d e r o f t h e l ar g e s t i n t e g e r s m a l le r t h a n O . I n p a r ti c u l a r t h e m e a n

i s a O / ( O - 1 ) , a nd the va r i a nc e i s

0 t~2 _ _ t~2

v = 0 - 1 ) 2 o - 2 ) o - 1 ) 2

I f w e n o w c o n s i d e r t he m o m e n t s M n o f t he g e n e ra l iz e d d i s t ri b u t io n F ( x ) , w e f i n d

M n = M n (Y ) h ( y ) d y = n ;

w h e r e n = J y n h ( y ) ° d y i s t h e m o m e n t o f o r d er n o f t h e d i s tr ib u t io n H ( x ) o f o o •

O n e f o r m o f t h e d i s t r i b u t i o n H ( x ) w h i c h c o u l d b e u s e f u l i n p r a c t i c e , a s i t i s e a s y t o f i t a n d is

a n a l y t i c a l l y s i m p l e , i s t h e G a m m a d i s t r i b u t i o n d e f i n e d b y

I t s m o m e n t s a r e g i v e n b y

h ( x ) = ~ e a x p - 1a P P ( p )

~ n = a n r p + n ) / r p ) ,

a n d t h e m o m e n t s M n a r e t h e r e fo r e g i v en b y

M n = O a n P ( p + n ) / ( O - n ) r ( p ) .

T h e d e n s i t y f u n c t i o n f ( x ) i s g i v e n b y

x )aO P O+p ,f x ) = x O + , r p )

w h e r e I ' ( x , p ) is t h e i n c o m p l e t e G a m m a f u n c t i o n , d e f i n ed b y

F ( x , p) = .J e - Y y P - l d y .

A.M. Hasofer 447

T h i s f u n c t i o n h a s b e e n e x t e n s i v e l y t a b u l a t e d ( S e e e .g . P e a r s o n ( 1 2 ) ) .

T h e d i s t r i b u t i o n f u n c t i o n F ( x ) i s s i m i l a r ly g i ve n b y

W e n o t e t h a t r ( x , p ) x P / p a s x t e n d s t o z e r o , a n d t h e r e f o r e l im f ( x ) = 0 a s l o n g a s p > 0 + 1 .

x - + O

T h e s i m p l e s t w a y t o f i t t h e d i s t r i b u t i o n t o o b s e r v a t i o n s i s t o e q u a t e t h e f i r st th r e e o b s e r v e d

m o m e n t s t o t h e c a l c u l a t e d o n e s . T h u s w e o b t a i n t h e t h r e e e q u a t i o n s

m 1 = a p ,

m2 p p+l ) ,

m 3 = a 3 p ( p + l ) ( p + 2 ) ,

w h e r e r n ~ , m 2 , m 3 a r e t h e t h r ee o b s e r v e d m o m e n t s . E l i m i n a t i n g a a n d p , w e o b t a i n a c u b i c e q u a -

t i o n f o r 0 , a n d w e c a n t h e n c a l c u l at e a a n d p f r o m t h e f i rs t t w o e q u a t i o n s .

A l t e r n a t i v e l y , w e c a n o b t a i n 0 b y f i t t in g t h e t a i l o f a P a r e t o d i s t r i b u t i o n t o t h e o b s e r v e d ta i l .

T h i s i s e q u i v a l e n t t o f i t ti n g a s t r a i g h t li n e t o t h e g r a p h o f l o g [ 1 - F ( x ) ] a g a i n s t l o g x . W e c a n t h e n

c a l c u l a te a a n d p f r o m t h e f i r st t w o o b s e r v e d m o m e n t s , u s i n g t h e f o r m u l a e

( 0 - 1 ) 2

P k 2 0 ( 0 - 2 ) - ( 0 - 1 ) 2

m l O - 1 )

w h e r e

D I S C U S S IO N O F T H E S S U M P T I O N S O F T H E M O D E L

A s m e n t i o n e d i n th e p r e c e d i n g s e c t i o n s , w e h a v e a s s u m e d t h a t t h e d i s t r i b u t i o n o f o c c u r r e n c e

o f c r a c k s i n e q u a l p r o p o r t i o n a l i n c r e a s e s in s t r es s is t h e s a m e . T h e r e i s n o t , t o t h e w r i t e r s k n o w -

l e d g e , a n y c o n t r o l l e d e x p e r i m e n t a l e v i d e n c e f o r t h i s a s s u m p t i o n , b u t s o m e r e c e n t e x p e r i m e n t s i n

b r i t t l e f r a c t u r e o f s t e e l d o t e n d t o i n d i c a t e t h a t t h e r e l a t iv e f r e q u e n c y o f a r r e s t e d c r a c k s is c o m -

p a r a t i v e l y h i g h a f t e r a th r e s h o l d o f s t re s s h a s b e e n r e a c h e d a n d d e c r e a s e s ro u g h l y i n in v e r s e p r o -

p o r t i o n t o t h e s t r e s s .

T h e s e c o n d a s s u m p t i o n , n a m e l y t h a t e q u a l p r o p o r t i o n a l i n c r e a s e s i n e f f e c t iv e a r e a a r e e q u a l l y

p r o b a b l e , c a n b e m a d e p l a u s i b le b y t h i n k i n g o f t h e a r r e s te d f r a c t u r e s as o r i g i n a t in g at s o m e d e f e c t

i n th e m a t e r i a l , b e i t a l a t t i c e d e f e c t o r a r e g i o n o f r e s i d u a l s t r e s s , a n d b e i n g a r r e s t e d a s h o r t d i s t a n c e

o u t s i d e t h e r e g i o n o f t h e d e f e c t . I t is o b v i o u s t h a t t h e l a r g e r d e f e c t s w i l l o r i g i n a t e c r a c k s a t t h e

8 A . M . H a s o f e r

l ow e r s t r e s ses , a nd the sm a l l e r de f e c t s a t h ighe r s t re s se s. Th us , w e c a n e xp e c t sm a l l e r c r a c ks a t

h ighe r s t r e s se s , w he n the e f f e c t i ve a r e a ha s a l r e a dy be e n r e duc e d .

F i n a l l y , i t is i n te r e s t i n g t o d i s cu s s t h e r e l e v a n c e o f t h e G a m m a d i s t r i b u t i o n t o t h e d e s c r i p t i o n

t h e v a r i a b i li t y o f t h e t h r e s h o l d o f c r ac k i n g . I t is w e l l k n o w n t h a t t h e G a m m a d i s t r i b u t i o n o f o r d e r

p is t h e s u m o f p i n d e p e n d e n t r a n d o m v a r i a b le s , e a c h h a v in g th e n e g a t iv e e x p o n e n t i a l d i s t r i b u t i o n .

I f w e t h i n k o f th e t h r e s h o l d a s b e i n g re a c h e d t h r o u g h a n u m b e r o f i n d e p e n d e n t s t ag e s , f o r i n s ta n c e

t h e s u c ce s si v e a c c u m u l a t i o n o f d i s l o c a t i o n s a t a g r a i n b o u n d a r y , a n d i f w e a ss u m e t h a t e a c h s t ag e

h a s th e e x p o n e n t i a l d i s t r i b u t i o n , t h e n t h e n a t u r a l d i s t r i b u t i o n t o a s s u m e f o r t h e t h r e s h o l d w i ll i n

f a c t b e t h e G a m m a d i s t r i b u t i o n .

T h e w r i t e r d o e s n o t c l a im t h e a b o v e a r g u m e n t s t o b e m o r e t h a n a n a t t e m p t t o m a k e t h e

a s s u m p t i o n s o f t h e m o d e l p l a u s i b le . T h e r e i s a v a s t fi e ld h e r e f o r e x p e r i m e n t a l w o r k t o b e d o n e .

M o r e o v e r , i f e x p e r i m e n t a l e v i d e n c e s h o w s t h a t t h e a b o v e a s s u m p t i o n s a r e n o t c o r r e c t , t h i s w i l l n o t

a f f e c t t h e f u n d a m e n t a l s t r u c t u r e o f t h e m o d e l , b u t o n l y t h e p a r t i c u l a r s o l u t i o n g iv e n i n t h e

p r e c e d i n g s e c t i o n . A t t h e s a m e t i m e , i t m u s t b e p o i n t e d o u t t h a t t h e s o l u t i o n o f t h e d a m p r o b l e m

w i t h a s s u m p t i o n s o t h e r t h a n t h e o n e s m a d e is n o t k n o w n e x p l ic i t ly , so th a t f u r t h e r m a t h e m a t i c a l

i nve s t i ga t i on i s r e qu i r e d a s w e l l a s f u r th e r e xp e r im e n ta l w o r k .

T H E S I Z E E F F E T

O n e o f t h e f e a t u r e s o f b r i tt l e f r a c t u r e o f st e el w h i c h i s th e m o s t d i f f i cu l t t o e x p l a i n o n a

w e a k e s t l i n k t h e o r y is t h e v e r y s m a ll s iz e e f fe c t i n c o m p a r i s o n w i t h , s a y , c o n c r e t e , f o r w h i c h t h e

w e a k e s t l i n k p r e d i c t i o n s a re i n g o o d a g r e e m e n t w i t h e x p e r i m e n t a l r e su l ts . I n f ac t , f o r st e el , t h e

s iz e e f f e c t i s m o s t m a r k e d f o r v e r y s m a l l s p e c i m e n s , a n d b e c o m e s l es s a n d l e s s m a r k e d a s s iz e

i n c re a s e s. T h i s b e h a v i o r is re a d i l y e x p l a i n e d b y o u r m o d e l . I t s e e m s r e a s o n a b l e t o a s s u m e t h a t

t h e v a l u e o f t h e t h r e s h o l d o f c r a c k i n g w i ll n o t b e a f f e c t e d b y s i ze , s i nc e a c c o r d i n g t o o u r m o d e l , i t

is r e a c h e d t h r o u g h a n u m b e r o f s t a g es w h i c h h a v e t o b e r e a li z ed s i m u l t a n e o u s l y a t t h e v a r i o u s

r e g i o n s o f i n c i p i e n t c r a c k i n g .

I t i s t h e n u m b e r o f c r a c k s , a n d t h e r e f o r e t h e p r o b a b i l i t y o f a c r a c k o c c u r r i n g i n a sm a l l s t re s s

i n c r e as e s , w h i c h w i l l b e p r o p o r t i o n a l t o t h e v o l u m e V o f t h e s p e c i m e n . In o t h e r w o r d s , th e

p a r a m e t e r X c a n b e t a k e n a s p r o p o r t i o n a l t o s i ze . F i n a ll y , th e d i s t r i b u t i o n G x ) o f r e la t iv e c r a c k

s i z e c a n a l so be t a ke n a s be ing r e l a t i ve ly i n se ns i t i ve t o spe c im e n s i z e .

I t t h u s a p p e a r s t h a t t h e s iz e e f f e ct c a n b e a c c o u n t e d f o r b y a s s u m i n g t h a t X i s p r o p o r t i o n a l t o

s p e c i m e n s iz e a n d t h a t a ll o t h e r p a r a m e t e r s a re i n d e p e n d e n t o f s iz e . N o w X o b v i o u s l y a f fe c t s o n l y

t h e v a l u e o f 0 ; t h u s a s t h e m e a n a n d v a r i a n c e o f u l t i m a t e s t re s s a re g i v en b y

M = a p

5 2 _ 0 a 2 p 2 a 2 p ,

0 - 2 ) 0 - 1 ) z

the e f f e c t o f 0 on M a nd S w i l l be r e l a t i ve ly sm a l l , a nd w i l l be c om e l e ss a nd l es s a s 0 i nc r e a se s .

B y de f in i t i on 0 s a t i s f i e s t he e qua t ion

0 - x [ 1 ¢ 0 ) ] = o .

A M H a s o f e r 449

Now, as can be seen from the Appendix, when X increases, 0 increases without bounds. As 4(0)

tends to zero when 0 ~ o~, we see that for large values of X, 0 is approximately equal to X.

S O M E E X P E R I M E N T L E V I D E N C E

In 1965-66, tests were made at the University of Melbourne on a number of buttonhead

cylindrical sharp-notched steel specimens. There were 89 specimens of diameter 0.75 and 90

specimens of diameter 0.5 . The mean failure stress for the large specimens was 103.805 kips per

sq. in. and for the small ones 110.166. Assuming that 0 is proportional to size, and using the

above formula for the mean, we find that 0 = 11.6 for the small specimens and 26.1 for the large

specimens. The size of the samples was too small to estimate the variance accurately enough to

calculate the other parameters, but the shape of the tail of the distribution was in rough agree-

ment with the above-quoted values of 0.

S U G G E S T IO N S F O R F U R T H E R E X P E R IM E N T L W O R K

A recent survey of current literature has highlighted the fact that adequate statistical data on

brittle fracture of steel are not available, apart from the pilot tests described in the preceding

section. For example, the data given in the above-quoted paper of McMahon and Cohen on the

stress dependence of microcrack format ion, though in qualitative agreement with the assumptions

of the model discussed in this paper, are useless for quantitative analysis because the sample is not

large enough, and because the data refer to surf ce microcracks in i n t e rrup ted loading tests, while

the model of this paper requires data on v o l u m e cracks in un in t e rrup ted loading.

Because of this lack of data, it is not possible to correlate the model with the physical facts.

What is required is a large-scale experimental program involving testing of several thousand

specimens, with particular emphasis on the following aspects:

(a) separation of the variability due to the testing equipment from that due to the intrinsic

properties of the material.

(b) close study of the shape of the probabili ty distribution of the failure stress,

(c) accurate evaluation of the size effect on bo th the mean failure stress and its variance.

The model presented in this paper is a working hypothesis on which such an experimental pro-

gram could be based.

A direct evaluation of the microstructural constants involved, namely the volume distribution

of cracks and their size as a function of applied stress is also required, but experimental techniques

for such a study do not seem to be available up to now. It is hoped that publication of this paper

will spur on interest in this direction.

C K N O W L E D G E M E N T

The research embodied in this paper was carried out by the Department of Civil Engineering

of the University of Melbourne as part of a project on the application of Statistical Methods in

Civil Engineering, suppor ted by a grant from the Australian Research Grants Committee.

Received May 22, 1967; revised May 25, 1968.

1. F.T. Pierce

2. W. Weibull

3. W. Weibull

4. J.I. Frenkel and T.A. Kontorova

5. J.C. Fisher and J.H. Hollomon

6. C.J. McMahon and M. Cohen

7. H.E. Daniels

8. E.J. Williams and N.H. Klo ot

9. B.D. Coleman

10. D.E. Gucer and J. Gurland

1 1. 1. Stei ndl

12. K. Pearson

A.M. Hasofer

REFERENCES

J. Text. Inst., 17 355 (1926).

'A Statisti cal Theory of the Stren gth of Materials, ' Ing. Vetenskaps.

Akad. Handl. No. 151 (1939).

'The P henom enon of Ruptu re in Solids, ' Ing. Vetenskaps. Akad. Handl.

No. 153(1939).

J. Phys. USSR, 1 108 (1943).

Trans. AIME, 171 546 (1947).

Acta Met., 13 591-604 (1965) .

Proc. Roy. Soc. (Lon.), A183 405 (1945).

Aust. J. Appl. Sci., 3, 1 (1952).

J. Mech. Phys. Solids , 7 60 (1958).

J. Mech. Phys. Solids , 10 365 (1962).

Rand om Processes and Growth of Firms: A S tudy of the Pareto Law

Charles Griffin Co., Ltd., London (1965).

Tables of the Incomple te Gamma Function London, H.M.S. Office

(1922).

A P P E N D I X

S i n c e a s h o r t p r o o f o f t h e r e s u l t in D a m T h e o r y w h i c h w a s u s e d i n t h e p a p e r i s n o t r e a d i l y

a va i l a b l e i n t he l i t e r a tu r e , w e g ive one he r e . Le t u s w r i t e

P T z ) ~< t } = G t , z ) ,

P p , z ) = E [ e - p T z ) ] = f e - p t d t G t , z ) .

W e n o w n o t i c e t h a t t h e f i r s t p a s s a g e t i m e a t z e r o c o n t e n t , s t a r t i n g f r o m z , i s t he s a m e a s t he

f ir s t p a s s a g e t i m e a t c o n t e n t y , s t a r t i n g f r o m z + y . T h u s T y + z ) h a s t h e s a m e d i s t r i b u t i o n a s t h e

s u m o f t w o i n d e p e n d e n t v a r i a b le s T y ) , T z ) . I t f o l l o w s t h a t F p , z ) s a t is f ie s th e f u n c t i o n a l e q u a -

t i o n

F p , y + z ) = r p , y) P p , z ) ,

a n d m u s t t h e r e fo r e b e o f t h e f o r m P p , z ) = e x p { - 0 z } .

M o r e o v e r , T z ) h a s a l s o th e s a m e d i s t r i b u t i o n a s z + T [ B z ) ] , f o r i f t h e i n it i al c o n t e n t o f t h e

d a m i s z , a f t e r a p e r i o d o f t im e o f l e n g t h z , t h e in i t ia l c o n t e n t h a s b e e n e x h a u s t e d , a n d t h e n e w

c o n t e n t i s t h e i n p u t i n t h e p e r i o d 0 , z ) , B z ) . U s i n g t h e t h e o r e m o f t o t a l p r o b a b i l i ty , w e n o w

c o n c l u d e t h a t

E [ e - p X z ) l = e - p z E { E [ e - p T I B z ) I [ B z ) ] } .

W e n o w u s t h e tw o r e s u l ts

a n d c o n c l u d e t h a t

A . M . H a s o f e r

E [ e - P T I B ( z ) 1 B ( z ) ] = e - 0 B ( z ) ,

E [e - b u z ) ] = e x l 1 - ~ 0 ) 1 ,

e - t) z = e - p z - X z l 1 - ~ ( 6 ) 1 .

T h u s 0 m u s t s a t i s f y t h e e q u a t i o n

= p + X [ 1 - - ~ ( 0 ) ] .

O b v i o u s l y w e a r e o n l y c o n c e r n e d h e r e w i t h p o s i t iv e r o o t s . L e t u s w r i t e t h e e q u a t i o n in t h e

f o r m ( t) - p ) /X = 1 - ~ ( 6 ) .

T h e g r a p h o f y = 1 - 6 ( z ) is n o n - d e c r e a s i n g , c o n v e x , p a ss e s t h r o u g h t h e o r ig i n an d i s a s y m p -

t o t i c t o y = 1 . I ts s l o p e a t t h e o r i g in i s m . I t i s t h e n o b v i o u s f r o m F i g u r e ( 7 ) t h a t th e r e i s o n l y o n e

y : l - f x ),SLope,m / \ / /

F i g u r e 7 ) .

pos i t i ve s o l u t i on f o r p > 0 . IV l o re over , a s p - + O , w e h a ve t w o pos s i b i l i t i e s :

( a ) I f - < m , t) t e n d s t o a l i m i t i n g v a l u e 0 > 0 .X

( b ) I f - ~ > m , 6 t e n d s t o z e r o .X

T h u s

S e - P t d t G ( t z ) = f d t G ( t z ) = P { T ( z ) < ~ } .i ra

P - + O o o

P { T ( z ) < ~ } = e - O z ,

P ( T ( z ) = ¢ ¢ } = I e - O z , i f X m > O .

4 5 2 A . M . H a s o f e r

R E S U M I ~ - O n f o r m u l e e t o n a n a l y s e u n m o d u l e s t a t i s t i q u e a p p l i c a b l e ~ l a r u p t u r e f r a g il e d e l ' a c i e r. L a r u p t u r e• J •

e s t se n s e e r e s u l t e r d e l a c o a l e s c e n c e d ' u n g r a n d n o m b r e d e f is s u r e s ~ I 'a r re t, q u i o n t e t e• a m o r e e e s e n d e s p o i n t s

d i f f e ~ r e n t s .r

O n d d m o n t r e q u e la p o r t i o n s u p e r l e u r e d e l a d i s t r i b u t i o n d e s c o n t r a i n t e s m a x i m a l e s e s t d on ne ~e p a r la

r e l a t i o n :

1 - x / tT O 0

o u G o e t 0 s o n t d e s f o n c t i o n s d e s c o n s t a n t e s m i c r o s t r u c t u r a l e s d u m a t e r i a u .

O n m o n t r e ~ ( ga le m e nt q u e l ' e f f e t d i m e n s i o n n e l i n t e r v e n a n t d a rt s c e m o d u l e e s t b e a u c o u p m o i n s p r o n o n c ~

q u e d a r ts l e c as d e s m o d u l e s q u i s o n t b a s e s u r la r e s i s t a n c e m i n i m a l e .

E n f i n , u n c e r t a i n h o m b r e d e d o n n g e s e x p ( r i m e n t a l e s r ff su lt an t ~ fl '~ u n p r o g r a m m e p i l o t e s o n t p re s en t~ g es , e t

d e s s u g g e s t i o n s s o n t d m i s e s p o u r l e x e c u t i o n d ' u n p r o g r a m m e e x p e r i m e n t a l .

Z U S A M M E N F A S S U N G - E i n s ta t i s t i sc h e s M o d e l l f u e r d e n s p r o e d e n B r u c h in S t a h l w u r d e f o r m u l i e r t u n d

a n a l y s i e r t . E s w u r d e a n g e n o m m e n , d a s s d e r B r u c h b e i V e r s c h m e l z u n g v o n e i n e r A n z a h l a u f g e h a l t e n e r R i s s e,

w e l c h e a n v e r s c h i e d e n e n P u n k t e n b e g a n n e n , e n t s t eh t .

D a s o b e r e E n d e d e r V e r te i l u n g d e s m a x i m a l e n D r u c k s w u r d e m i t d e r f o l g e n d e n F o r m e l g e z e i g t :

1 - X / O o ) 0

w o b e i Oo u n d 0 d i e F u n k t i o n e n d e r m i k r o s t r u k t u r e l l e n K o n s t a n t e n d e s M a t e r i a ls e r g e b e n .

E s w u r d e w e i t e r h i n g e z e ig t , d a ss d e r G r o e s s e n e f f e k t i n d i e se m M o d e U v i el w e n i g e r g e k e n n z e i c h n e t i s t a l s i m

F a U e d e r s c h w a e c h s t e n V e r b i n d u n g s m o d e l l e .

A b s c h l i e s s e n d s ei e r w a e h n t , d a s s e in i ge e x p e r i m e n t e l l e E r g e b n i s s e v o n v o r h e r g e g a n g e n e n T e s t e n p r a e se n -

t i e r t w u r d e n u n d e s w u r d e n V o r s c h l a e g e f u e r w e i t e re e x p e r i m e n t e l l e A r b e i t g e m a c h t .

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