Mixed-Mode Fracture of Brittle Cellular Materials

8/8/2019 Mixed-Mode Fracture of Brittle Cellular Materials

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JOUR NAL OF MATERI ALS SCI ENCE 31 (1996) 2647 2652

M i x e d - m o d e f r a c t u r e o f b r i tt le c e llu l a r m a t e r ia l s

J . S . H U A N G * , J . Y . L I NDe partm ent of C iv i l Engineering, Na tional Ch eng Kun g U niversity, Tainan, 70101 Taiwan

Di m ens i ona l a rg um en t ana l ys i s and nea r - t i p s i ngu l a r i n -p l ane shea r s tress o f a con t i nuu mmo de l have been em p l oy ed t o de r i ve t he exp ress i on f o r m ode II f r ac t u re t oughne ss o f b r it t le

ce l l u l a r ma t e r i a ls . I t was f ou nd t ha t bo t h m ode I and I I f r ac t u re t oughne sses have t he samedependence on ce l l s i ze , r e l a t i ve dens i t y and modu l us o f r up t u re o f so l i d ce l l wa l l s , excep t

a m i c ros t ruc t u re coe f f i c i en t i nc l uded i n t he i r exp ress i ons . I n add i t i on , t he l i nea r

sup e rpos i t i on p r i nc i p l e was app l i ed t o ca lcu l a te the ben d i ng m om en t exe r t ed a t t he f i rs t

unb rok en ce l l wa l l f o r b r i t t le ce l l u la r ma t e r i a l s unde r a com b i ned l oad i ng o f un i f o rm t ens i l e

and i n -p l ane shea r s t resses . T he resu l t i ng m i xed -mode f rac t u re c r i t e r i on was compared t o

e x i s t in g e x p e r i m e n t a l d a t a i n P VC f o a m s ; a g r e e m e n t w a s f o u n d t o b e g o o d .

1 . I n t r o d u c t i o n

There is a growing interest in the use of sandwich

panels with ceramic cellular cores as load-bearing

components in lightweight structures. For example,

sandwich panels with a cementitious foam core and

gypsum faces are typically used in building. Ceramic

cellular materials have excellent thermal insulation

and fire resistance but are brittle. Pre-existing cracks

in brittle cellular materials, resulting from manufac-

turing or machining, might cause catastrophic failure

at a tensile stress much less than the yielding strength.Under some circumstances, the crack surface may not

be perpendicular to the imposed in-plane shear stress

which is the primary loading of core materials in

sandwich panels, producing a mode II or mixed-mode

fracture. The mixed-mode fracture criteria for solid

materials are invalid for cellular materials because

failure mechanisms are different. Therefore, mixed-

mode fracture of brittle cellular materials needs to be

fully exploited to understand crack propagation in

cellular materials and to ensure structural integrity of

sandwich panels.

Gibson and Ashby [1] proposed a bending model

of cell walls to analyse mechanical properties of cellu-

lar materials. They found that mechanical properties

of cellular materials are related to cell geometry and

material properties of solid cell-wall materials.

Fowlkes [2] measured mode I fracture toughness of

a rigid polyurethane foam from various types of speci-

mens to verify the applicability of linear elastic frac-

ture mechanics to the fracture of foams. McIntyre and

Anderton E3] confirmed the dependence of relative

density of rigid polyurethane foams on their mode

I fracture toughness: K*c increases with increasing

relative density. Bulk and microscopic models for the

mechanical behaviour of cellular glass were attempedby Zwissler and Adams E4]. It was found that fracture

strength, tensile elastic modulus and fracture tough-

ness increase linearly with density. The influence of

anisot ropy on the fracture toughness of woods, which

have a similar microstructure to honeycombs, was

studied by Ashby e t a l . [5]. The fracture toughness of

cellular materials is directional ly dependent i f they are

not isotropic. The mode I, II and mixed-mode fracture

of PVC foams were investigated by Zenkert and

Backlund [6, 7]. They found that K*c is slightly larger

than K*c.

Maiti e t a l . [8] utilized the bending model of cellwalls in cooperation with the near-tip singular tensile

stress of a con tinuum model to derive the expression

for mode I fracture toughness of cellular materials.

Results indicated that K*c of cellular materials de-

pends on cell size, relative density and modulus of

rupture of solid cell walls. In their modelling, the

modulus of rupture o f solid cell walls was assumed to

be constant. In practice, the modulus of rupture of

a brittle cell wall is mainly controlled by its volume,

giving a cell-size effect. Cell-size effects on fracture

strength of foamed glass [9] and on mode I fracture

toughness of reticulated vitreous carbon foam [10],

were observed. Huang and Gibson [11, 12] studiedthe variable strength of solid cell walls using Weibull

statistic analysis. It was noted that the Weibull

modulus of solid cell walls plays an important role in

determining optimum cell size to obta in a higher value

of K~c.

The present work was aimed at deriving the expres-

sion for mode II fracture toughness and a mixed-mode

fracture criterion because they are essential for the

analysis of crack propagat ion in brittle cellular mater-

ials. The bending model of cell walls, dimensional

argument, as well as the near-tip singular stress field of

linear elastic fracture mechanics, were employed toanalyse the mode II and mixed-mode fracture for

*Authorto whom allcorrespondenceshouldbe addressed.

0022-2461 9 1 9 9 6 C h a p m a n & H a H 2647



Y

T > x

(y%

t t t t tt

a ~

(5-%

Figure ! An infinite honeyco mb plate w ith a central crack undera uniform tensile stress.

h o n e y c o m b s a n d f o am s . T h e m i x e d - m o d e f r a c t u re c ri -

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

p a r e d w i t h e x is t in g e x p e r i m e n t a l d a t a i n P V C f o a m s

[ 7 ] .

2 . H o n e y c o m b sA t y p i c a l h o n e y c o m b w i t h a c e n t r a l c r a c k , a * , c e l l

l eng th , l , and ce l l wa l l th icknes s , t , i s unde r a remote

u n i f o r m t e n s i l e s t r e s s , ~ * , a s s h o w n i n F i g . 1 . T h e

e x p r e s s i o n f o r m o d e I f r a c t u r e t o u g h n e s s i s f o u n d t o

be E 8] ( p , ~ 2K F c = c ~ = ( ~ l ) ~ / ~ \ T s l ( 1 )

w h e r e C 1 is a m i c r o s t r u c t u r e c o e f f i c ie n t a n d w a s n u -

m e r i c a l l y f o u n d t o b e 0 .1 8 b y H u a n g a n d C h i a n g [ 1 3 ] ,

a n d cyf~ s t h e m o d u l u s o f r u p t u r e o f so l i d c e l l w a l ls , p *

a n d p , ar e t h e d e n s it i es o f h o n e y c o m b s a n d t h e s o l id

m a t e r i a l f r o m w h i c h t h e y a r e m a d e , r e s p e c ti v e ly .

2 .1 . M o d e II f r a c t u r e o f b r i t t l e h o n e y c o m b s

F i g . 2 i l l u s t r a t e s a h o n e y c o m b p l a t e w i t h a c e n t r a lc r a c k , a * . I t i s a s s u m e d t h a t t h e c r a c k l e n g t h i s m u c h

l a r g e r t h a n t h e c e ll si ze o f h o n e y c o m b s . A r e m o t e

u n i f o r m i n - p l a n e s h e a r s t r e s s , ~ * , i s i m p o s e d o n t h e

o u t e r m o s t l a y e r o f t h e h o n e y c o m b p l at e , g e n er a t in g

a s i n g u l a r n e a r - t i p s t r es s f i e ld in l i n e a r e l a s t i c f r a c t u r e

m e c h a n i c s [ 1 4 ] . T h e r e s u l t i n g i n - p l a n e s h e a r s t r e s s ,

"cxy, a lo ng the c ra ck s u r face i s

K * T*(rca *) 1/2

"Cxy - (2n r)a /2 - - (2for)l~ 2 (2 )

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

a t t h e c e n t r e o f t h e f i r s t u n b r o k e n c ei l, as s h o w n i nF i g . 2 . K * i s t h e m o d e I I s t r e s s i n t e n s i t y f a c t o r a n d i s

p r o p o r t i o n a l t o z * ( n a * ) ~ /z. T h e e x p r e s s i o n f o r m o d e I I

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

u s i n g d i m e n s i o n a l a r g u m e n t a n a l y s i s i n c o n j u n c t i o n

2 6 4 8

> > ) ) > > ~*

a * 1 7

z* < < < < , ( <

Figure 2 An infinite honeyco mb plate with a central crack undera uniform in-plane shear stress.

w i t h E q u a t i o n 2 . T h a t i s, t h e i n t e g r a t i o n o f t h e r e s u l t -

i n g i n - p l a n e s h e a r s t r e s s o v e r a d i s t a n c e o f o n e c e l l s iz e

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

ce l l wa l l

f 2 1 c o s O

V o c % y b d r0

o c z * b ( a * l ) 1 /2 (3 )

T h e b e n d i n g m o m e n t e x e r t e d a t th e fi r st u n b r o k e n c e ll

w a l l i s p r o p o r t i o n a l t o g l

M o c z * b l ( a * l ) 1 /2 (4 )

T h e c r i t i c a l sk i n s t r e s s o f t h e f i r s t u n b r o k e n c e l l w a l lc a n b e c a l c u l a t ed f r o m t h e e l e m e n t a r y m e c h a n i c s o f

m a t e r i a l s

M~ c o c b t

T h e c r a c k a d v a n c e s w h e n t h e c r i t i c a l s k i n s t r e s s

r e a c h e s t h e m o d u l u s o f r u p t u r e o f so l i d c el l w a l ls . A t

t h e m o m e n t , t h e i m p o s e d u n i f o r m s h e a r s t r e s s h a s

a m a x i m u m v a l u e c a ll e d t h e f r a c tu r e s t r e n g th , z ~

z'~' oc ~ f = \ a * ) ( 6 )

O n c e f r a c t u r e s t r e n g t h a n d c r a c k l e n g t h a r e k n o w n ,

t h e m o d e I I f r a c t u r e t o u gh r i es s o f t h e h o n e y c o m b c a n

b e c a l c u l a t e d

KFc = ~( ~a *) ~i2

o c ~ = ( n l ) * /2 ( 7 )

T h e r e l a ti v e d e n s it y o f t h e h o n e y c o m b , p * / p , , i s p r o -

p o r t i o n a l t o t i t . A s a r e s u lt , t h e e x p r e s s i o n f o r m o d e I If r a c t u r e t o u g h n e s s o f h o n e y c o m b s c a n b e w r i t te n a s

K i l o = C 2 ( Y f s ( g l ) l / 2 ( 8 )



> > - > > >t t t t t

r

< _ < < I . . . . <

o.-x-

Figure 3 An infinite honeyco mb plate with a central crack undera co mbined oading of uniform tensile and in-plane shear stresses.

w h e r e C 2 is a m i c r o s t r u c t u r e c o e f f i c ie n t a n d m u s t b e

d e t e r m i n e d e m p i r i c a ll y o r n u m e r i c a l l y .

2 . 2. M i x e d - m o d e f r a c t u r e o f b r i tt le

h o n e y c o m b s

A n i n f i n i t e h o n e y c o m b p l a t e u n d e r a c o m b i n e d l o a d -

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

s h o w n i n F i g . 3~ I t i s a s s u m e d t h a t b r i t t l e h o n e y c o m b s

a r e l i n e a r e l a s t i c u p t o f r a c t u r e , r e s u l t i n g i n t h e a p -

p l i c a b i l i ty o f l i n e a r s u p e r p o s i t i o n p r i n c i p l e i n e l a s ti c -

i ty . T h e b e n d i n g m o m e n t a c t in g a t t h e f i rs t u n b r o k e n

c e ll w a l l f o r t h e h o n e y c o m b s u b j e c t t o a s i n g l e te n s i l e

s t r e s s a n d a s i n g l e i n - p l a n e s h e a r s t r e s s c a n b e c a l -

c u l a t e d , r e s p e c ti v e l y . L i n e a r s u p e r p o s i t i o n o f th e t w o

r e s u l t i n g b e n d i n g m o m e n t s g i v e s

M = d l ~ * b I ( a * l ) 1 /2 + d 2 z * b I ( a * l ) 1 /2 (9 )

w h e r e d ~ a n d d 2 a r e c o n s t a n t s d e p e n d i n g o n c e l l

g e o m e t r y o f h o n e y c o m b s . B e c au s e b e n d in g m o m e n td o m i n a t e s c e l l , w a l l d e f o r m a t i o n i n c e l l u l a r m a t e r i a l s ,

t h e c r i t i c a l s k i n s t r e s s o f t h e f i r s t u n b r o k e n c e il w a l l is

h e n c e f o u n d t o b e

MO " o C ~ 2

(d lc l* + d2r*) l (a* I ) 1 /2oc t2 (10)

W h e n t h e c r i ti c a l s k i n s t re s s e x c e e d s t h e m o d u l u s o f

r u p t u r e o f s o l id c e l l w a l ls , m i x e d - m o d e f r a c t u r e w i l l

o c c u r . T h e r e f o r e , t h e m a x i m u m c o m b i n e d l o a d i n g

o f u n i f o r m t e n s i l e a n d i n - p l a n e s h e a r s t r e s s es

( d t cr * + d 2 z * ) f, a t w h i c h c r a c k p r o p a g a t e s , i s:

(d~c~* + d2"c*)f(rca* ) 1/2 oc cyf, - (~/)1/2 (11)

a ~

o.-x-

Figure 4 An idealized mo del of foam pla te with a central crackunder a uniform ensile stress.

T }~ e l e f t - h a n d s i d e o f t h e a b o v e e x p r e s s i o n r e p r e s e n t s

a l in e a r c o m b i n a t i o n o f m o d e I a n d I I s t re s s i n t e n s it y

f a c t o r s . T h u s , t h e m i x e d - m o d e f r a c t u r e c r i t e r i o n f o r

h o n e y c o m b s c a n b e e x p r e s s e d a s

d , K ~ + d 2 K * = d 3 ( Y f s ( ~ l ) 1 /2 \ ~ s J (12)

H e r e d 3 is a n o t h e r m i c r o s t r u c t u r e c o e f f ic i e nt o f h o n e y -

c o m b s . F o r a s p ec i fi c h o n e y c o m b , t h e r i g h t - h a n d

s i d e i n E q u a t i o n 1 2 i s c o n s t a n t , r e g a r d l e s s o f t h e

m a g n i t u d e o f i m p o s e d t e n si l e a n d i n - p l a n e s h e a r

s t re s se s . D i v i d i n g b o t h s i d es o f E q u a t i o n 1 2 b y

( Y f s ( T C l ) l / e ( p * / p s ) 2 a n d t h e n e m p l o y i n g t h e re l a t i o n -

s h ips be tw een K * , K~ c and c~fs( rc /) /2 (9* /ps ) 2 (E qu a -

t i o n s 1 a n d 8 ) t o r e a r r a n g e t h e m i x e d - m o d e f r a c t u r e

c r i t e r i o n a s

d t K ~ ' d 2 K ~

K * c / C , + K * c /C ~ 2 - d 3 (13)

T h e a b o v e m i x e d - m o d e f r a c t u r e c r i t e r i o n m u s t b e

a p p l i c a b l e f o r t h e t w o s p e c i a l c a s es , K ~ = K ~c f o r

m o d e I f r a c t u r e a n d K ~ = K i ]c f o r m o d e I I f r a ct u r e .

T w o r e l a t i o n s h i p s a r e o b t a i n e d , d l C ~ = d 3 a n d

d 2 G 2 = d 3 . T h e m i x e d , m o d e f r a c t u r e c r i t e r i o n f o r

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

K?K *c + K * ~ = 1 (14)

3 . F o a m s

M e c h a n i c a l p r o p e r t i e s o f f o a m s a r e d e s c r i b e d w e l l b y

t h e b e n d i n g m o d e l o f c el l w a l ls p r o p o s e d b y G i b s o n

a n d A s h b y [ 1 ] . A n i d e a l i z ed m o d e l o f a f o a m p l a t ew i t h a c e n t r a l c r a c k u n d e r a u n i f o r m t e n s i l e s t r e s s i s

s h o w n i n F i g . 4 . U s i n g d i m e n s i o n a l a r g u m e n t a n a l y s i s ,

M a i t i e t a l . [ 8 ] w e r e a b l e t o d e r i v e t h e e x p r e s s i o n f o r

m o d e I f r a c t u r e t o u g h n e s s o f f o a m s a s a f u n c t i o n o f

2 6 4 9



> > > > > ~*

(

a ~

r

,' k

) k

~k

) k

T* < < < < <

Figure 5 A n i d e a l i z e d m o d e l o f f o a m p l a t e w i t h a c e n t r a l c r a c k

under a u niform n-plane shear stress.

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

s o l id ce l l wa l l s

= ( 1 5 )\Psl

w h e r e C 3 is a m i c r o s t r u c t u r e c o e f f i c i e n t a n d w a s e x -

p e r i m e n t a l l y f o u n d t o b e 0 . 6 5 [ 8 ].

3 .1 . M o d e II f r a c t u r e o f b r i t t l e f o a m s

W h e n a n i n f i n it e f o a m p l a t e w i t h a c e n t r a l c r a c k u n d e r

a u n i f o r m i n - p l a n e s h e a r s t r e ss , a s s h o w n i n F i g . 5 ; i so f c o n c e r n , t h e n e a r - t i p s i n g u l a r i n - p l a n e s h e a r s t r e ss ,

"cxy, c a n b e u t i l i z e d a g a i n t o c a l c u l a t e t h e t o t a l s h e a r

f o r c e c a rr i e d b y t h e f i rs t u n b r o k e n c e ll w a ll

oc f l ~xy /d r

oc "c* ( a* l) 1/2 (16)

T h e c r i t i c a l b e n d i n g m o m e n t e x e r t e d a t t h e f i r s t u n -

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

o f t o t a l s h e a r f o r c e a n d c e l l s i ze

M o c V 1

oc z*12 (a*l ) 1 /2 (17)

W h e n t h e c r i t i c a l b e n d i n g m o m e n t r e a c h e s t h e m a x -

i m u m r e s i s t a n c e m o m e n t M r = ~fs t3 /6 o f s o l id ce l l

w a l l s w i t h a c r o s s - s e c t i o n a l a r e a o f t 2 , t h e c r a c k p r o p a -

g a t es a n d m o d e I I f r a c t u r e o c c u r s . C o n s e q u e n t l y , t h e

f r a c t u r e i n - p l an e s h e a r s t r e n g t h is

T h e r e l a ti v e d e n si t y o f f o a m s i s p r o p o r t i o n a l t o t h e

s q u a r e o f t f l . T h e e x p r e s s i o n f o r m o d e I I f r a c t u r e

t o u g h n e s s o f f o a m s c a n b e o b t a i n e d o n c e t h e f r a c t u r ei n - p l a n e s h e a r s t r e n g t h a n d c r a c k l e n g t h a r e k n o w n

( p * ~ 3 / 2

K~I C = C 4(~ fs(TCI)1 / 2 \ ~ s / I ( 1 9 )

2 6 5 0

), > > > f

a *

*~ < < ( < <

(3"*

Figure 6 An idealized mod el of foam plat e with a cen tral crackunder a comb ined loading of uniform tensile and in-plane shearstresses.

w h e r e C A i s a m i c r o s t r u c t u r e c o e f f i c i e n t o f f o a m s a n d

m u s t b e d e t e r m i n e d e m p i r ic a l ly .

3 .2 . M i x e d - m o d e f r a c t u r e o f b r i t t l e f o a m sW h e n a f o a m p l a t e i s u n d e r a c o m b i n e d l o a d i n g o f

u n i f o r m t e n s i l e a n d i n - p l a n e s h e a r s t r e ss e s a s s h o w n i n

F i g . 6, th e b e n d i n g m o m e n t a c t i n g a t t h e f ir s t u n b r o -

k e n c e l l w a l l is a l i n e a r c o m b i n a t i o n o f t h e i n d u c e d

b e n d i n g m o m e n t s b y t h e u n i f o r m t e n s il e s t re s s a n d b y

t h e u n i f o r m i n - p l a n e s h e a r s t r e s s . T h e c r i t i c a l s k i n

s t r es s o f th e f i r s t u n b r o k e n c e ll w a l l i s f o u n d t o b e

(d4~* - t - ds'c* )12 (a* l) 1/2~c oc t3 (20)

w h e r e d ~ a n d d s a r e c o n s t a n t s . T h e c r i t i c a l s k i n s t r e ss

i n c r ea s e s u n ti l i t r e a c h e s t h e m o d u l u s o f r u p t u r e o fs o l i d c e l l w a l l s . T h e m a x i m u m c o m b i n e d l o a d i n g o f

u n i f o r m t e n s i l e a n d i n - p l a n e s h e a r s t r e s s e s

(d4c~* + ds ~*) f , a t wh ic h c ra ck p ro pag a te s , i s

(&,(5* + d5"c*)f(~a*) 1/2 oc O'fs ( % 1 ) 1 / 2 (21)

T h e l e f t - h a n d si d e o f t h e a b o v e e x p r e s s i o n is a g a i n

a l i n e ar c o m b i n a t i o n o f m o d e s I a n d I I s t re s s i n te n s i t y

f a c t o r s . T h u s , t h e m i x e d - m o d e f r a c t u r e c r i t e r i o n f o r

b r i t t l e f o a m s c a n b e e x p r e s s e d a s

/ ~ , , \ 3 / 2

d 4 K ~ + d s K ~ = d 6 ~ fs ( rr 1 /2 (~- -} (22 )\ p s /

w h e r e d 6 i s a m i c r o s t r u c t u r e c o e f f i c i e n t o f f o a m s . T h e

r i g h t - h a n d s id e o f E q u a t i o n 2 2 , w h i c h is d e p e n d e n t o n



j v ~

, , S J

I < > 1

, J J

I < / = > 1

Figure 7 Two solid cell walls for brittle honey combswith differentcell size but the same relative density.

c e ll si z e, re l a t i v e d e n s i t y a n d m a t e r i a l p r o p e r t i e s , c a n

b e c o n v e r t e d to m o d e I f r a c t u r e t o u g h n e s s a n d m o d eI I f r a c t u r e t o u g h n e s s i n d i v i d u a l l y

d 4 g ~ d s K ~

K~c/C~ 4- K~lc/C4 - - d6 (23)

T h e a b o v e m i x e d - m o d e f r a c t u r e c r i t e r i o n m u s t b e

v a l i d f o r t h e c a s e s K {~ = K * c f o r p u r e m o d e I a n d

K ~ = K I *C f o r p u r e m o d e I I . T h e r e f o re , t h e m i x e d -

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

r e d u c e d t o a s i m p l e f o r m

KF KIK*c 4 - K I I C * - 1 ( 2 4 )

4 . D i s c u s s i o n

K * c a n d K * c o f b r i t t l e c e l lu l a r m a t e r i a l s i n c r e a s e w i t h

r e l a t i v e d e n s i t y a n d c e l l s iz e if t h e m o d u l u s o f r u p t u r e

o f s o li d c e l l w a l l s i s r e g a r d e d a s a c o n s t a n t . A l s o , e i t h e r

K * c o r K * c i s c e l l g e o m e t r y d e p e n d e n t . A m i c r o s t r u c -

t u r e c o e f f i c i e n t i n c l u d e d i n e a c h e x p r e s s i o n o f K ~'c a n d

K i ] c s h o u l d b e d e t e r m i n e d n u m e r i c a l l y o r e x p e r i -

m e n t a l l y .

T h e b r i t t l e n e s s o f s o l i d c e l l w a l l s , h o w e v e r , w i l l

a f f e c t t h e i r m o d u l u s o f r u p t u r e ; b r i t t l e s o l id s w i t h

a la r g e r v o l u m e h a v e a h i g h e r v a l u e o f m o d u l u s o fr u p t u r e . A s a r e s u l t o f th a t , t h e m o d e I I f r a c t u r e

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

t h e p r e - e x i s t i n g c r a c k - s i z e d i s t r i b u t i o n i n s o l i d c e l l

w a l ls . O n e w a y t o d e s c r i b e t h e b r i t t l e n e s s o f s o li d c e l l

w a l l s is a p p l y i n g W e i b u l l s t a t i s t i c a n a l y s is f o r v a r i a b l e

m o d u l u s o f r u p t u r e . F i g . 7 il l u s t r a t e s t w o s o l i d c e ll

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

s am e re l a t ive den s i ty . T h a t i s, V1 > 1 /2 , t l > t2 and

I1 > I2 but t l / l l = t2/ l> T h e r a t io o f m o d u l u s o f r u p -

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

W e i b u l l s t a t i s t i c a n a l y s i s [ 1 1 ]

O'fs, 1 ( V 2 ~ M

= \ b t ~ l l } (25)

w h e r e m , l a r g e r t h a n z e r o , is t h e W e i b u l l m o d u l u s o f

s o l i d ce l l w a l ls . S o l i d s w i t h a l o w e r v a l u e o f m a r e m o r e

b r i t t l e . B e c a u s e t h e t w o s o l i d c e l l w a l l s h a v e s a m e

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

r u p t u r e i s f u r t h e r r e d u c e d t o

- ( 1 2 ) 2 : = ( 2 6 )~fs, 1

I t i s c l e a r t h a t t h e m o d u l u s o f r u p t u r e o f s o l i d c e l lw a l l s i n c r e a s e s w i t h d e c r e a s i n g c e l l s iz e f o r s a m e d e n -

s i ty h o n e y c o m b s a n d t h e m a g n i t u d e o f in c r e a s e de -

p e n d s o n t h e W e i b u l l m o d u l u s .

T h e r a t i o o f m o d e I I f r a c t u r e t o u g h n e s s f o r t w o

s a m e - d e n s i t y b u t d i f fe r e n t c e ll s iz e h o n e y c o m b s c a n b e

c a l c u l a t e d b y s u b s t i t u ti n g E q u a t i o n s 2 6 in t o E q u a t i o n

8, g ives

K I I C , 1 ( ~ fs , 1 ( 1 1 ) 1 / 2 "

K i ~ c , 2 O'fs,2(12) /2

\ ; 2 2 ( 2 7 )

T h e a b o v e r e s u l t i n d i c a t e s t h a t t h e r e i s a c e ll - si z e e f f e ct

o n t h e m o d e I I f r a c t u r e t o u g h n e s s o f b r i t tl e h o n e y -

c o m b s . W h e n m > 4 , h o n e y c o m b s w i t h a l a r g e r c e ll

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

w h e n m < 4 , *u c i n c r e a s e s w i t h d e c r e a s i n g c e l l s i z e ;

wh en m = 4 , t he r e i s no ce l l - s iz e e f fec t. A s imi la r r e s u l t

i s o b t a i n e d f o r b r i tt l e f o a m s u s i n g t h e s a m e p r o c e d u r e :

K ~ c o f b r i tt l e f o a m s i n c r e a s e s w i t h i n c r e a s i n g c e l l si z e

i f m > 6 ; wh en m < 6 , KI* dec r ea s e s w i th in c rea s in g

c e ll s iz e . T h e W e i b u l l m o d u l u s e f f e c t o n K i* c i s t h e

s a m e a s th a t o n K ~ c d e r i v e d b y H u a n g a n d G i b s o n

E l l , 12 ] .

F r o m E q u a t i o n s 1 4 a n d 2 4, i t i s n o t e d t h a t t h e

m i x e d - m o d e f r a c t u r e c r i t e r i o n f o r b r i t t le c e l l u l a r m a -

t e r ia l s is a l i n e a r c o m b i n a t i o n o f K */ K *c a n d K*/K~Ic.

F o r s o l i d m a t e r i a l s , t h e e n e r g y - b a l a n c e c r it e r i o n re -

q u i r es t h a t t h e t o t a l e n e r g y r e le a s e r a t e i n m i x e d - m o d e

f r a c t u r e i s t h e s u m m a t i o n o f m o d e I e n e r g y r e le a s e

r a t e a n d m o d e I I e n e r g y r el e a se r a t e [ 1 4 ] . N o r m a l l y ,

K ~'c i s n o t e q u a l t o K ~ c . T h e r e f o r e , a m o d i f i e d m i x e d -

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

m a t e r i a l s

T h e a b o v e e q u a t i o n i s a e l l i p t i c f u n c t i o n o f K */ K *c

a n d * *n / K n c . T h a t i s , t h e m i x e d - m o d e f r a c t u r e c r i -

t e r i o n f o r b r i t t l e c e l l u l a r m a t e r i a l s i s c o m p l e t e l y d if f e r-

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

T h e m i x e d - m o d e f r a c t u r e c r i t e r i o n f o r b r i t t l e

h o n e y c o m b s a n d f o a m s m u s t b e v e r i f ie d b e f o r e it is

u s e d t o e x a m i n e i f a c r a c k u n d e r c o m b i n e d l o a d i n g o f

u n i f o r m t e n s i l e a n d i n - p l a n e s h e a r s t r e ss e s w il l p r o p a -

g a te . H o w e v e r , t h e e x p e r i m e n t a l r e s u lt s o f m i x e d -

m o d e f r a c t u r e i n b r i t t le c e l l u l a r m a t e r i a l s a r e l i m i t ed .

E x i s ti n g e x p e r i m e n t a l r e s u l t s o f m i x e d - m o d e f r a c t u r ei n P V C f o a m s b y Z e n k e r t [ 7 ] a s s h o w n i n F i g . 8 a r e

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

t h a t t h e e x p e r i m e n t a l d a t a o f m i x e d - m o d e f r a c t u r e in

P V C f o a m s a r e cl o s e t o a s t r a ig h t l in e c o r r e s p o n d i n g

2 6 5 1



0 . 5 0

0 . 4 0

0.30E

D ..

v 0 .20

0 . 1 0

0.00

9 I

9 I Q ~ 1

t

"- I

0 . 0 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0

K ]~ ( M P a m v2 )

I0 . 5 0

Figure 8 ( - - ) T h e m i x e d - m o d e f ra c t u r e c r i t e ri o n f o r b r i t t le c e l lu l a r

m a t e r i a l s ( E q u a t i o n 2 4 ) c o m p a r e d t o ( I ) e x p e r i m e n t a l r e s u l t s i n

P V C f o a m s [ 7 ] .

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

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

t h e m o d u l u s o f r u p t u r e o f s o l i d c e ll - w a l l m a t e r i a l s .

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

in to acc o u n t , th e r e i s a ce l l - size ef fec t: K~c an d

K * c i n c r e a s e s w i t h i n c r e a s i n g c e l l s i ze f o r h o n e y c o m b s

w i t h m > 4 a n d f o r f o a m s w i t h m > 6 . I n a d d i t i o n , t h e

m i x e d - m o d e f r a c t u r e c r i t e r i o n f o r b r i t t l e c e l l u l a r m a -

t e r ia l s , d i ff e r e n t f r o m t h a t f o r s o l i d s, is c o m p a r e d w i t h

e x p e r i m e n t a l r e s u l t s i n P V C f o a m s ; a g r e e m e n t i s

g o o d . B e c a u s e b r i t t le s o l i d c e l l w a l l s a r e l i n e a r e l a s t i c

u p t o f r a c t u r e , t h e m i x e d - m o d e f r a c t u r e c r i t e r i o n i s

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

t e n s il e , i n - p l a n e s h e a r a n d o u t - o f - p l a n e s h e a r s t r e ss e s .

A c k n o w l e d g e m e n t

T h e f i n a n ci a l s u p p o r t o f th e N a t i o n a l S c ie n c e C o u n c il ,

T a i w a n , u n d e r c o n t r a c t n u m b e r N S C 8 4 - 2 2 1 1 - E 0 0 6 -

0 18 , i s g r a t e f u l l y a c k n o w l e d g e d .

t o E q u a t i o n 2 4 . A g r e e m e n t i n F i g . 8 s u p p o r t s t h e v i e w

w e p r o p o s e d , g i v i n g t h e c o n f i d e n c e o f u t i l i z i n g E q u a -

t i o n 2 4 t o c h e c k i f c r a c k p r o p a g a t i o n i n b r i tt l e f o a m s i s

l i k e l y t o o c c u r .

A t t h e s a m e t i m e , t h e m i x e d - m o d e f r a c tu r e c r i t e ri o n

f o r b r it t l e c e l l u l a r m a t e r i a l s c a n b e e x t e n d e d t o t h e

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

s h e a r a n d o u t - o f - p l a n e s h e a r s t r e s s e s . B e c a u s e s o l i d

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

m o m e n t e x e r t e d a t t h e f i rs t u n b r o k e n c e ll w a ll a h e a d

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

a s in g le ten s i le s tr es s , a s in g le in -p lan e sh e ar s t r es s an da s i n g l e o u t - o f - p l a n e s h e a r s t r e s s . T h e m i x e d - m o d e

f r a c t u re c r i t e r io n t h u s b e c o m e s

K* K I~ , KI* ~ _ 1 (2 9)

K * c + K t i c

w h e r e K I ]I a n d *m c a r e t h e m o d e I I I s t r e s s i n t e n s i t y

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

5 . C o n c l u s i o n

T h e e x p r e s s i o n fo r m o d e I I f r a c tu r e t o u g h n e s s a n d t h e

m i x e d - m o d e f r a c t u r e c r i te r i o n f o r b ri t tl e h o n e y c o m b s

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

R e f e r e n c e s

1 . L . J . G I B S O N a n d M . F . A S H B Y " C e l l u l a r s o l id s : s t r u c t u r e

a n d p r o p e r t i e s , ( P e r g a m o n P r e s s , O x f o r d , 1 9 8 8 ) .

2 . C . w . F O W L K E S , Int. J. Fraet. 10 (1974) 99.

3 . A . M c I N T Y R E a n d G . E . A N D E R T O N , Polymer 20 (1979)

247 .

4 . J . G . Z W I S S L E R a n d M . A . A D A M S , i n " F r a c t u r e M e c h a n i c s

o f C e r a m i c s " , V o l . 6 , e d i t e d b y R . C . B r a d t ( P l e n u m P r e s s , N Y ,

1983) p . 211 .

5 . M . F . A S H B Y , K . E . E A S T E R L I N G , F . H A R R Y S S O N a n d

S . K . M A I T I , Proe. R. Soc, Lond. A398 (1985) 261 .

6 . D . Z E N K E R T a n d J . B A C K L U N D , Compos. Sci. Teehnol. 3 4

(1989) 225.

7 . D . Z E N K E R T , Mater. Sci. En9. A108 (1989) 233 .8 . S . K . M A I T I , M . F . A S H B Y a n d L . J . G I B S O N , Scripta

Metall. 18 (1984) 213.

9 . J . S . M O R G A N , J . L . W O O D a n d R . C . B R A D T , Mater. Sci.

En 9. 47 (1981) 37 .

1 0. R . B R E Z N Y a n d D . J . G R E E N , Acta Metall. Mater. 38 (1990)

2517 .

1 1. J . S . H U A N G a n d L. J . G I B S O N , ibid. 39 (1991) 1617 .

12 . Idem, ibid. 39 (1991) 1627.

1 3. J . S . H U A N G a n d M . S . C H I A N G , Eng. Fract. Mech. (1996)

a c c e p t e d .

1 4. D . B R O E K , " E l e m e n t a r y E n g i n e er i n g F r a c t u r e M e c h a n i c s "

( M a r t i n u s N i j h o f f , H a q u e , 1 9 82 ).

R e c e i ve d 2 0 F e br uar y

and accepted 1 Decem ber 1995

2652

Mixed-Mode Fracture of Brittle Cellular Materials

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